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t l
Electrochemical Processes Within the Slimes Layer
of Lead Anodes During Betts Electrorefining
by
Jose Alberto Gonzalez Dominguez
B.Sc. National Autonomous University of Mexico
(U.N.AM), 1984
M.Sc. National Autonomous University of Mexico
(U.N.AM), 1985
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
in
THE FACULTY OF GRADUATE STUDIES
Department of Metals and Materials Engineering
We accept this thesis as conforming to the required standard
The University of British Columbia
March 1991
© Jose" Alberto Gonzalez Dominguez, 1991
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In presenting this thesis in partial fulfilment of the requirements for an advanced
degree at the University of British Columbia, I agree that the Library shall make it
freely available for reference and study. I further agree that permission for extensive
copying of this thesis for scholarly purposes may be granted by the head of my
department or by his or her representatives. It is understood that copying or
publication of this thesis for financial gain shall not be allowed without my written
permission.
Department
The University of British Columbia
Vancouver, Canada
Date
DE-6 (2/88)
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Abstract
In the Betts process for lead electrorefining the noble impurities originally
prese nt in the bu ll io n for m a strong an d adherent layer of sli mes . W i t h i n this
layer the established ionic concentration gradients can lead to secondary
reactions. The following processes were analyzed from a thermodynamic
perspective: (A) hyd rol ysi s of the a cid (B) precipita tion of seconda ry produc ts (C)
reacti on of noble c omp oun ds.
The na tur e of the con cent rat ion gradients wi th in the sli mes layer an d related
seconda ry processes was s tudied by usin g transi ent electrochemical techniques
which inc lude: (A) current interruptio n, (B) AC impedance, and (C) a vari at ion of
S A C V (Smal l Ampli tude Cycl ic Voltammetry) . These studies were complemented
by: (A) phys ico-chem ical da ta on electrolyte properties, (B) "insitu" and "industriallyrecovered" sli mes electrolyte comp osi tio ns, (C) S EM a nd X-ray diffraction analysis
of the slimes layer. Fo r com par is on purp ose s the electroc hemic al beha viou r of
"pure" Pb electrodes was also studied.
Upon current interruption the anodic overpotential decays, first abruptly,
(as the uncompensated ohmic drop disappears) and then slowly (due to the
presence of a back E .M.F . created by ionic concentration gradients that decay
slowly). Cu rr en t int erru ptio n measu remen ts showed that: (A) concentration
gradients exist across the slimes layer, (B) inne r sol utio n potentials wit hi n theslimes layer can be larger than those measured from reference electrodes located
i n the bu l k electrolyte, (C) seco ndar y pro duc ts ca n shift the in ne r sol uti on potenti al
to negative val ues wh i ch reverse up on re-dissolution an d (D) ioni c diffusi on is seen
u po n cu rre nt i nt err upt ion bu t it is complex an d difficu lt to mod el due to the
presence of processes that can supp or t the passage of in ter nal cur ren ts.
The ano dic polariza tion components were obtained by an aly zin g the pot ential
an d current dependance u po n applic ation of a sma ll ampli tude s inu soi dal
wavef orm. Th is depend ance was f oun d to be line ar in the low overpotential region
(< 250 mV) . Th us , up on sub tra cti on of the unc om pen sa te d ohmi c drop, the
remain ing polarizat ion is due to the "apparent" ohmic drop of the sl imes electrolyte
and to l iquid ju nc ti on and concentrati on overpotentials. These comp onen ts are
directly linked to the electrolysis conditions and to the slimes layer structure.
Furthermore, the ratio of these components can be used to obtain the point at
wh ic h the prec ipi tat ion of secon dary produc ts starts. Cha nge s in thi s ratio can
also be related to the anodic effects caused by the presence of addition agents.
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AC imp eda nce mea sureme nts performed i n the presence of a net Fara dai c
current showed that the impedance increases uniformly as the slimes layer
th ic kens up to the poi nt at wh i ch noble impur it ies start to react. Three electri cal
analogue models were used to describe the impedance spectra.
A steady-state math em at ic al mod el that predicts conc ent rat ion an d potenti algrad ient s acro ss the slimes layer was developed. On ly wh en a pos it ion dependent
eddy diffus ion ter m was incorporate d in the nume ri cal solut ion, were reasonable
local ionic concentrations and overpotentials obtained.
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Table of Cont ents
Abstract ii
List of Tables iz
List of Figures xi
Nomenclature xz
Acknowledgments zzv
Chapter 1 Literature Review 3
I Introduction 3
II Meta l lurgy of Lead 3
III Plant Practice i n Lead E lectroref ining 7
IV The Anodi c Process 12
I. Introduction 12II. The physi cal metal lurgy of the lead anodes 12III. Industrial practice 14IV. Slimes electrochemical behavior 15
V The Cat hodi c Process 22I. Additives control and electrochemistry 22II. Starting sheet technology 23III. Cel l electrolysis parameter optimi zation 23IV. B ipolar ref ining of lead 25
Chapter 2 Fundamenta ls of the E lectrochemical Measurem entProcedure 26
I. Components of the Anod ic Overpotential 26II. Transient Ele ctrochemical Techniques 28A. Cur ren t interruptio n techniques 28B. A C impedance techniques 31
1. Ai ms and l imitations of the A C impedance studies 32
Chapter 3: Exper imental Procedure 34I. E lectrochemical Experiments 34
A. E lectrochemical cel ls 341. Beake r Cel l 34
2. Rectangular cell 35
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B. Ele ctr odes 371. Wo rk in g electrodes 37' (a) Ma te ri al s 37
(b) Be ak er cel l 38(I) Pure lead wo rk in g electrodes 38(II) Lead bu ll io n wo rk in g electrodes 38
(c) Re ct an gu la r cel l 38(I) Pure lead wo rk in g electrodes 39(II) Lea d bu ll io n wo rk in g electrodes 39
2. Reference electrodes 393. Co un te r electrode 40
C. Ele ctr oly te 41D. Tem per atu re con tro l 43E. Inst rume ntat ion 43
1. We nk in g potent iostat 432. Solartron electrochemical interface and frequency responseanalyzer 47
(a) Gen era lit ies : 47
(b) De sc ri pt io n of the exper iment al set-up 48II. Elect rolyt e Phy sic o-C hem ica l Properties 50
A. p H mea sure ment s 50B. Elec tri cal conduc tivit y 50C. Kin ema tic viscosi ty 50D. De ns it y 51
C h ap t e r 4 Elect rorefi ning of Lead In A Small-Scale Eleclx orefin ing Cell:
A Cas e St ud y 52
I. Int rod uct ion 52IL Pre sen tat ion of Res ult s 54
A. An od ic overpotential meas urem ents 561. Stage I 562. Stage H 653. Stage m 67
B. Ana ly ti ca l chem istr y 691. B u l k electrolyte con cent rati ons 692. Inner sli mes electrolyte conc entr ati ons 71
C. Ch ar ac te ri za ti on of the sli mes layer 741. Met al logra phy of the sta rti ng lead anode 752. An al ys is of the sli mes layer phases an d co mp ou nd s 80
(a) S E M ana lys is 80(b) X-r ay dif fracti on 85
C h ap t e r 5 Anodic and Rest Potential Behavior of Pure Lead inH 2 S i F 6 - P b S i F 6 Elect rolyt es 86
I. Overview of Pure Lead Dis sol ut i on in H 2 S i F 6 -PbS i F 6 ElectrolytesUn de r Galv anost atic Cond iti ons 86
A. Ano di c overpot ential i n the absence of large co nce ntr ati ongrad ient s in the anode bo un da ry layer 86
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B. Cor re lat io n between the anod ic overpotential an d the presenceof add it io n agents 88C. Cor re lat io n between the anodi c overpotential an d the presenceof seco nda ry produ ct s that precipi tate on the anode surface 89
II. Establishment of Ionic Concentration Gradients in the AnodeBoundary Layer an d their Relat ionshi p to the Ano di c Ov erpotentia l
90A. In the absence of addi ti on agents i n the bu l k electrolyte 92B. In the presence of addi ti on agents i n the bu l k electrolyte 98
III. AC Impedance 102A. Int rod ucti on 102B. Impedance spectra obtained in an electrolyte without additionagents 103C. Impedance spectra obtained in electrolytes containing additionagents 108
Ch a p te r 6 Electrochemical Behavior of Lead bull io n Electrodes i n thePresence of Sl im es 115
I. In tr oduc ti on 115II. A C Impedance Char acte riza tion of the Storting Wor ki ngElectrodes 117
A. AC behaviour in the absence of addition agents in the bulkelectrolyte 118B. AC behaviour in the presence of addition agents in the bulkelectrolyte 120
III. DC and AC Stu die s i n Cor rod ed Elect rodes 124A. Studi es und er galvanostatic, potentiostatic, and curr ent
in terrupt ion cond iti ons 1241. Ex pe ri ment al resu lts 124
(a) Variation of the anodic overpotential as a function of theelectrol ysis time an d the cur ren t in te rrupt io n time 124(b) Cha ng es i n the impe dan ce as a fu nc ti on of the s limeslayer th ic kne ss an d of the cur ren t in te rrupt io n time 133
2. An al ys is of the expe rime ntal da ta 151(a) Relationship between the DC anodi c overpotent ial an dthe DC cur ren t dens ity 151
(I) Da ta anal ysi s 153(b) Proposed analogue representation of a lead bullionelectrode covered wi th a laye r of sli mes 161
(I) Da ta ana ly si s 169(i) Case I: impedance spectra obtained in the
fwesence of a net Fa ra da ic cu rr en t 170ii) Case II: impedance spectra obtained in the
abse nce of a net Fa ra da ic cu rr en t 183
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C h a p t e r 7 Physico-Chemlcal Properties of H 2 S i F 6 -Pb S I F 6 Electrolytesan d thei r Rel ati onsh ip to the Tr ans por t Processes Acr os s the Sli mesLayer 191
I. In tr odu ct io n 191II. Average Sli mes Electrol yte Co mpo si ti on 191III. Eh -p H Di ag ra ms 194
A. (F)-Si-H2
0 sys tem 195B. (Pb-F)-Si-H aO sys tem 196C. (Sb-F)-Si-H 2O f (As-F)-Si-H20, and (Bi-F)-Si-H 20 sys tems 199
IV. Physico-Chemical Properties of H 2 S i F 6 - P b S i F 6 Elect rolyte s 201A. pH, density, viscosity, and activity of H 2 S i F 6 sol uti ons 201B. Density, viscosity, and electrical conductivity of H 2 S i F 6 - P b S i F 6
electrolytes 203V. Math ema ti cal Model : Nume ri cal Solu tio n of the Nems t-Pl anckF l u x Equ ati ons 214
A. Case A: const ant y, val ues 21 5B. Case B: constant yt val ues in the presence of eddy di ffusi on .... 21 7C. An od ic overpotential valu es derived from the mat hem ati cal
model 222
S u m m a r y 226C o n c l u s i o n s 234R e c o m m e n d a t i o n s f o r F u r t h e r W o r k
B i b l i o g r a p h y 238
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Appendices
Append ix 1 Mathematical Model : Numerical Solution of theNernst-P lanck Fl ux Equati ons and Its App li cat ion to the Betts
Process 265
Append ix 2 Ana lyt ica l Sol uti on of the Nernst-Planck Fl ux
Equations 276
Append ix 3 Time Do ma in To Frequency Doma in Transfor mation:
The Fourier Transform In Current Step Electrochemical Techniques291
Append ix 4 Ana lyt ica l Che mis tr y of Electrolyte Solut ions
Conta in ing PbS iF 6 -H 2 S iF 6 312
Append ix 5 Comp ut er Interfacing of the We nk in g Potentiostat :Cal ibrat ion of the Rou ti nes use d to Interr upt the Cu rr en t 322
Append ix 6 Current Interruption and AC Impedance Meas urem ent s
us ing the Sol art ron Devices 330
Append ix 7 Solubil i ty of PbSiF 6 4 H 2 0 334
Append ix 8 Sol uti on of Fick' s Second Law Equ at io n Un de r Cu rr en tInter ruptio n Cond iti ons 337
Append ix 9 Ext end ed Ve rs io n of Tabl es Presented i n Ch ap te r 6 34 8
Append ix 10 Krame rs-Kr onig Transf ormat ions of NUD Elem ents 360
Append ix 11 Programs Used to Generate the Eh-pH DiagramsPrese nted i n Cha pt er 7 36 3
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L ist of Tables
Chapter 1
Table 1.1 Bet ts lead eleclxorefining in the wor ld 10
Table 1.2 Chemi ca l composit ion and X-Ray di ffraction analys is of thelead anode slimes 19
Chapter 4
Table 4.1 Character is t ics of experiment 1X2 55
Table 4.2 X-ray d iffraction analysi s of outer and in ner sli mes powde rsamples 85
Chapter 5
Table 5.1 Resu lts of the a nalysis of the conc ent rat ion o verpotentialincreases during the first second after application of the cur rent stepsFick's Second Law approximation 96
Table 5.2 Var ia t ion of the B x and parameters wi th curr ent density .. 107
Chapter 6
Table 6.1 Character is t ics of the experiments presented i n chapter 6 116
Table 6.2 Average t imes requ ire d to measure the AC impedance
spec t rum 117
Table 6.3 S u m m a ry of the va lues of the electrical analog ue para mete rsobtained under rest potential conditions 117
Table 6.4 Ana lys is of the spikes produced dur in g the appl icat ion of theAC waveform, i n the presence of a net DC curre nt (Exp. CA 2, Figs. 6.32to 35) 156
Table 6.5.a Analysis of the spike s prod uced dur in g the appl icat ion of
the AC waveform, i n the presence of a net DC current (Exp. CA6) 158
Table 6.5.b Analysis of the spikes produc ed duri ng the application of
the AC waveform, und er curren t inte rrup tio n condit ions (Exp. CA6) 159
Table 6.6 Ana lys is of the spikes produced dur ing the appl icat ion of the
AC waveform, i n the presence of a net DC current (Exp. CC1) 159
Table 6.7 Pa rameters derived fro m the f itting of the imped anc e dat aobtained i n Exp. C A2 to the ZZARC-ZZARC analogue (Circuit A . 2 ,Figs. 6.38,39) 175
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Tabl e 6.8 Param ete rs derived fr om the fitt ing of the imp eda nce spectr aobt ain ed i n Exp . CA 2 (Ci rcu it B.2, Figs. 6.40, 41) 179
Tab le 6.9 Parame te rs deri ved fr om the fit tin g of the Impedance dataobtained in Exp CA5 to the Z^c-Z^^e analogue (Ci rcui t A.2) 181
Tabl e 6.10 El ec tr ic al analog ue param eter s derived fr om the fitt ing ofthe imp eda nce dat a obta ined i n Ex p. CA 5 (Cir cuit B.2) 181
Tabl e 6.11 Para mete rs derived fro m the fitti ng of the imp eda nce dat aobtained in Exp. CA2 to the Randies Analogue Circuit (Figs. 6.43, 44)
184
table 6.12 Electrical analogue parameters derived from the Fitting ofthe impedance data obtained in Exp. CA5 to the Randies AnalogueCi rcu i t (Fig. 6.45) 188
Tab le 6.13 El ect ri cal Anal ogue parameter s derived fro m the fitt ing ofthe A C impedance data obtained in Exp. CA4 to the T^SC'^ZJ^ analogue(C ir cui t A. 2, Fi g. 6.47) 190
Chapter 7
Tabl e 7.1 Co mp os it io n of the electrolyte samp les extracted fr om ano deslimes obtained under industrial operation o f the B E P (Corninco Ltd.) .. 192
Table 7.2 Va lue s of the coefficients A an d Q i n Eq ua ti on 7.17 20 2
Table 7.3 Ch an ges i n the osmo ti c an d acti vit y coefficients as a
funct ion of the Ionic stre ngth 20 3
Tabl e 7.4 Physi co-c hemical properties of H 2 S i F 6 - PbS i F 6 electrol ytes 20 4
Tabl e 7.5 Coeffici ents i n the emp ir ica l electri cal conduc tivi ty, vis cosi tyan d dens ity corre lati ons 20 6
Tabl e 7.6 Ch ang es in the indi vi dua l ioni c mobil itie s as a fun ct io n of
the electrolyte com pos it io n 21 2
SummaryTable S. 1 Su mm ar y of info rmat ion that can be derived fro m us in gtrans ient electro chemical technique s 233
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L ist of Figures
Chapter 1
Fig. 1.1
F lowsheet for lead extracti on from sulph idi c concentr ates 4
Fig. 1.2 A generalized flowsheet for the pyrometal lurgic al ref ining oflead 5
Fig. 1.3 A generalized flowsheet for the pyro metal lurgical/electrometal lurgical ref ining of lead 6
Fig. 1.4 Changes i n the anodi c overpotential value, r\ A ,during leadelectxorefining 9
Fig. 1.5 Lead anode microstructure 13
Fig. 1.6
S tra ight type horiz ont al lead anode casti ng sys tem 15
Fig. 1.7
B i sm ut h content of the ca thod ic dep osit as a funct ion of theanodic polarizat ion a nd the electroly sis time
17
Fig. 1.8
D imens ions of the anod es u se d i n Wenzei 's experi mentsshowing the pos i t ion and size of the electrolyte sa mp li ng wells
18
Fig. 1.9
Cha nge s wi th electrolysis t ime of the relat ive conc entra tions ofP t r 2 and H
+ w i t h i n the sli mes layer wi th respect to their bul k values .... 19
Fig. 1.10 Lead specific weight loss (corrosion) as a funct ion of theimmer s ion t ime i n a 2 M Pb(BF4)2-1 M H B F 4 so lut ion i n the presence of0, 2, 5, and 10 m M of B i + 3
21
Fig. 1.11
Cont i nuou s pur i f i ca t ion of electrolyte v ia puri f ica t ion c ol um n22
Fig. 1.12 Influence of the c ycle l engt h and cur ren t reversal r at io on theanodic overpotential va lue dur ing PCR 24
Chapter 2
Fig. 2.1 E lect roch emical cel l arrangement 27
Chapter 3
Fig. 3.1 Asse mbly used i n the experiments per formed wit h the beakercell 35
Fig. 3.2 Assembly used i n the experimen ts performed usi ng therectangular cel l
36
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F ig . 3 .3 Sections of the lead bu l l i on anodes used to prepare working
electrodes 3 7
F ig . 3 .4 Detail of the Luggin-Haber reference electrode arrangement 4 0
F ig . 3 .5 Exp eri men tal set-up us in g the Wen kin g potentiostat 4 4
F ig . 3 .6 Det ai l of the connect ions requ ired to int err upt the curr ent an d
to follow the cell response 4 5
F ig . 3 .7 Cur re nt step resu lt in g fro m hal ti ng the flow of cur ren t to theelectrochemi cal cell us in g the Wen kin g potentiostat 4 7
F ig . 3 .8 Con nec ti ons fro m the electrochemical cell to the Sol art ron
Electrochemica l Interface 4 9
Chapter 4
F ig . 4 .1
Lead anode top view 5 4
F ig . 4 .2 An od ic overpotent ial (uncorrected for changes as a fun ct io nof the slime s layer thick ness 5 7
F ig . 4 . 3 Ou te r an d in ne r A, B, an d C reference electrodes anod ic
overpotential respo nse to cur rent int err upt ion s (duri ng an otherwisegalvanostatic experiment) 5 8
F ig . 4 . 4 Det ai l of the TIa response of the i nne r A reference electrode tocurrent interruptions (during the whole electrorefining cycle)
5 9
Fig. 4 . 5 Detail of the rj A response of the i nne r B (Fig. A) an d i nne r C
(Fig. B) reference electrodes to current interruptions (during the wholeelectrorefining cycle)
5 9
Fig. 4 . 6 Det ai l of the r\ A response to current interruptions measured by
the outer reference electrode (at different slimes layer thickness) 6 0
F ig . 4 . 7 Changes In the va lue of the uncom pensa ted ohm ic drop, T| Q, as
a function of the slimes thickness 6 1
F ig . 4 . 8 Det ai l of the T\ A response to current interruptions measured by
the i nne r A reference electrode (at different slimes layer thickness) 6 2
F ig . 4 . 9 Detai l of the i i A response to cur ren t inter rup ti ons mea sur ed by
the i nne r B reference electrode (at different slimes layer thickness) 6 3
F ig . 4 . 1 0 Det ai l of the T\a response to current i nterr upti ons meas ured
by the i nne r C reference electrode (at different slimes layer thickness) .. 6 4
F ig . 4 . 1 1 An od ic overpotent ial (corrected for T
J
Q) changes as a function
of the slimes layer thic kness 6 5
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F ig . 4.12 An od i c overpotent ial (uncorrected for i i d changes as a
fun ct io n of the cur ren t int err upt ion time 66
F ig . 4.13 Changes In the va lue of the ano dic overpotenti al ( uncorre ctedfor TI J as a fu nc ti on of the electrolysis time 67
F ig . 4.14 Anodic overpotential response upon current interruption
(Stage m Table 1 68
Fig. 4. 15 Chang es i n com pos iti on of the bu lk electrolyte as a f unc ti on
of the sl imes layer th ic kn es s 70
Fig. 4. 16 Cha nge s i n the bu l k electrolyte compos iti on as a fun ct ion of
the cur ren t in te rru pti on time 70
Fig. 4. 17 Chan ges in the loca l compo si ti on of the slim es electrolyte as afu nc ti on of the movement (from the samp li ng point) of theano de/ sli mes interfac 71
F ig . 4.1 8 Change s in the loca l con cen tra tio n of the tota l Si a nd Fpre sen t i n the sl ime s electrolyte as a fu nc ti on of the mov eme nt (fromthe sa mp li ng point) of the anode /s li mes interf ac 72
F ig . 4.19 Change s in the loca l compo si ti on of the slimes electrolyte as afu nc ti on of the cur ren t int err upt ion time 73
F ig . 4.20 Chan ges in the local conce ntr ati on of the total Si and Fpre sen t i n the sl ime s electrolyte as a func ti on of the cu rr en tint erru pti on time 74
Fig. 4.21 Sect ion of the lead anode an d of the slim es layer st ud ie dmetallographically 75
Fig. 4.22 Lead anode microstuctures. Anode "A", Air cooled face. Allmic rog rap hs co rre spon d to the same observat ion point( point # 1F ig . 4.21) 76
F ig . 4.23 Lead anode micr ostuc tures . Anode "A" . Al l mic rogr aphscorrespon d to the sam e obse rvat ion point( poi nt #2 Fi g. 4.21) 78
F ig . 4.24 Lead anode microstuctures. Micrographs correspond to
different observati on poi nts 79
F ig . 4. 25 Mi cr os tr uc tu re of the slimes layer @2mm away from the
sli mes/ elec tro lyt e interface (position #2 Fig. 4.21) 81
F ig . 4.26 Deta il of the micr ost ruc tur e of the slimes layer @2mm awayfr om the slimes/elec trol yte interface (position #2 Fi g. 4.21) 82
F ig . 4.27 Mi cr os tr uc tur e of the slim es layer @12m m away from theslimes/ elec tro lyt e interface (position #3 Fi g. 4.21) 83
F ig . 4 .28 Detai l of the micr ost ruc tur e of the slimes layer @12 mm awayfr om the sli mes/elec tro lyt e interface (position 3 Fi g. 4.21) 84
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Chapter 5
F ig . 5.1 Pot ent ial difference between a fixed reference electrode an d acorroding anode in the absence of addition agents 87
F ig . 5.2 Overpote ntia l changes dur in g the galvanosta tic di sso lu ti on ofpure lead (in the presence of excess quantities of addition agents) 89
F ig . 5.3 An od ic overpotenti al response (uncorrected for TJQ) of pu re lead,to the application of successive current step 90
F ig . 5.4 Cu rr en t step fun cti on use d to study the est abl ishm ent ofconcentration gradients in the anode boundary layer. The rise time ofthe current steps was smaller than 10 usee 91
F ig . 5.5 An od ic overpotenti al response (uncorrected for TIJ of a pu relead electrode to the current steps described in Fig. 4, i n the presence
of addition agents (compare with Fig. 5) 92
F ig . 5.6 Chang es in the uncomp ensat ed ohmi c drop [r\o) as a function
of the applied current density 93
Fig. 5.7 An od ic overpotential respons e (corrected for in it ia l TIJ of a pu relead electrode to the current steps described in Fig. 4 94
F ig . 5.8 Ch an ge s i n the dimens io nl es s ove rpo ten tial, <!>!, as a fu nc ti on
of the squar e root of time, 96
F ig . 5.9 Decay i n the ano dic overpoten tial (corrected for TJQ) as afunct ion of the in te rrupt io n time, t 97
F ig . 5.10 An od ic overpotent ial response (uncorrected for TIJ of a purelead electrode to the current steps described in Fig. 4 98
F ig . 5.11 An od ic overpotentia l response (corrected for in it ia l of a
pure lead electrode to the current steps described in Fig. 4 99
F ig . 5.12 Co mp ar is on between the anodic overpotential va lue obtainedr ight after app li cat ion of cur ren t (*) an d the unc omp ens at ed ohm icdrop obt ained from the hi gh frequency inte rcep t of the impedance
spec t rum 100
F ig . 5.13 Cha nge s i n the anodi c overpotenti al up on app li cat io n of acurrent step 101
F ig . 5.14 Changes i n the ano dic overpotent ial as a fu nc ti on of the
current i nterru ption t ime 102
F ig . 5.15 Impedance diagram of pure lead under rest potential
condit ions (in the absence of addition agents) 104
F ig . 5.16 Det ai l of the impe danc e curve sho wn i n Fig . 15 (after
subtract ing the Rg value) 105
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F ig . 5.17 Impedance d iagram of pur e le ad i n the presence of an anodiccur re nt 1=150 A m p m* 2 (after subtract ing the Rg value) 107
F ig . 5.18 Impedance d iagram of pur e lead under rest potentia l
condit ions (in the presence of addi tio n agents) 108
F ig . 5.19 Impedance diagrams of pur e lead un de r rest potentia lcondit ions obtained at two different ampli tude s of the applied AC
waveform (in the presence of addi ti on agents) 109
F ig . 5.20 Analogue circuits used to model the hi gh frequency respon se
of the impe dance curve sho wn i n F ig. 15 110
F ig . 5.21 Hi gh frequency section of the impedance d iag ram show n i n
F ig . 18 112
F ig . 5.22 Impedance diagrams of pur e lead obta ined i n the presenceand i n the absence of a net Faradaic current (in the presence ofaddit ion agents) 113
F ig . 5.23 Deta i l of the impeda nce dia grams obtained i n the presence of
a net Faradai c current (in the presence of addi ti on agents) 114
Chapter 6
F ig . 6.1 Impedance diagram (Argand plot) of a typical lead bullionelectrode (Exp. CC1-6) und er rest potent ial cond iti ons an d i n theabsence of addit ion agents 119
F ig . 6.2 Impedance diagram (Argand plot) of a typical lead bullionelectrode (Exp. CA2-J) under rest potential conditions an d in thepresence of addit ion agents 120
F ig . 6.3 Deta i l of the hi gh frequency r egion of the imped ance dia gramshown i n F ig . 2 121
F ig . 6.4 Deta i l of the hi gh frequency regions of the imp eda nce dia gra msobtaine d und er rest potential conditi ons 122
F ig . 6.5 Impedance spectra obtained under potentiostatic (solid line)
and galvanostatic control (dashed line) 123F ig . 6.6 Overpote ntial response of a typic al lead bul li on anode as afunct ion of the sli mes thickn ess (Exp. CA2) 125
F ig . 6.7 An od ic overpotenti al (corrected for in i t ia l r y measured by thecounter and reference electrodes as a funct ion of the slime s thi ckne ss(Exp. CA5 ) 127
F ig . 6.8 Cu rr en t density changes as a funct ion of the electrolysis timeand of the amo unt of lead d issol ved (Exp. CA 4, potentiostaticcondit ions £^^,,=220 mV) 128
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Fig. 6.9 Changes in the anodic overpotential (corrected for TJQ) as afu nc ti on of the am ou nt of lead dis sol ved (Exp. CA4) 129
F ig . 6.10 Anodic overpotential changes upon current interruption
(Exps. C A 2 . C A 5 , an d CA4) 130
F ig . 6.11 Anodic overpotential (corrected for initial TJQ) as a function ofthe sli mes th ic kn ess (Exps. CA 6 an d CC1 ). .: 132
F ig . 6.12 Anodic overpotential changes upon current interruption
(From Ex p. CA6 ) 133
F ig . 6.13 Impedance spectr a obtai ned du ri ng Ex p. CA 2 at sli mes layerth ic kn es s betwe en 0.8 an d 7.8 m m 134
F ig . 6.14 Impedance spectr a obtai ned du ri ng Ex p. CA 2 at a s lime s
laye r th ic kn es s of 8.4 m m 135
F ig . 6.15 Impedance spec tr um obtained du ri ng Ex p. CA 2 at a sl imes
layer th ic kn es s of 8.65 m m 136
F ig . 6.16 Changes in the va lue of the unc omp ens ate d ohm ic re sistance,
Rg, as a fu nc ti on of the sli mes th ic knes s (Exp. CA2) 137
F ig . 6.17 Impedance spectr a obtai ned du ri ng Ex p. CA 5 at slim es layerth ic kn es s betwe en 0.64 an d 1.89 m m 138
F ig . 6.18 Impedance spectra obtained during Exp. CA5 at a slimeslayer thickness of 2.2 mm (Rg was subt ract ed from the com pon ent ofthe impedance ) 139
F ig . 6.19 Changes in the va lue of the unc omp ens at ed ohm ic resist ance,
Rg, as a fu nc ti on of the sl imes th ic kn es s (Exp. CA5) 140
F ig . 6.20 Impedance spectra obtained dur in g Ex p. CA 4 at sli mes layerth ic kn ess between 0.87 an d 2.87 m m 141
F ig . 6.21 Changes in the va lue of the unc omp ens at ed ohm ic r esist ance,
Rg, as a fu nc ti on of the sli mes th ic kn es s (Exp. CA4) 142
F ig . 6.22 Ar ga nd plot sho win g the changes i n the imp edan ce spec tra
obtained in the presence of a layer of slimes and in the absence of a netFa rad ai c cur ren t (Exp. CA2) 143
F ig . 6.23 Changes in the va lue of the unc omp ens at ed oh mic resistance,
Rg, as a fun ct io n of the cur ren t int err upt ion time (Exp. CA2) 143
Fig. 6.24 Argand plot showing the changes in the impedance spectraobtained in the presence of a layer of slimes and in the absence of a netFa rad ai c cur ren t (Exp. CA5) 144
Fig. 6.25 Changes in the va lue of the unc omp ens at ed oh mic resist ance,
Rg, as a fu nc ti on of the cu rr en t in te rr up ti on time (Exp. CA5) 145
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F ig . 6.26 Ar ga nd plot sho win g the changes in the impe danc e spec traobtained in the presence of a layer of slimes and in the absence of a netFa radai c cur ren t (Exp. CA4) 146
F ig . 6.27 Impedance spec tra obtai ned du ri ng Ex p. C A6 at sli mes layerth ic kn es s between 0.43 an d 7.7 m m 147
F ig . 6.28 Impedance spec tra obtai ned du ri ng Ex p. CC 1 at sli mes layerth ic kn es s between 0.14 an d 5.3 m m 148
F ig . 6.29 Changes in the va lue of the unc omp ens at ed oh mic resis tance,
Rs, as a fun ct io n of the slim es thi ckn ess (Exps. CA 6 an d CC1 ). 149
F ig . 6.30 Ar ga nd plot sho win g the change s in the impe danc e spec traobt ained in the presenc e of a layer of sl ime s an d in the absence of a netFar ada ic cur ren t (Exp. CA6) 150
Fig. 6.31 Changes in the va lue of the unc omp ens at ed oh mic resis tance,
Rg, as a fu nc ti on of the cu rr ent in te rr up ti on time (Exp. CA6) 150
F ig . 6.32 Detail of the "spikes" observed at -0.8 mm slimes (Exp. CA2,Ta bl e 6.4) 154
F ig . 6.33 Variations in the anodic overpotential as a function of theanodic current density at various slimes thickness (Exp. CA2,Tabl e 6.4) 155
F ig . 6.34 Changes in the va lue of the res ist ance of the sl imeselectrolyte as a fu nc ti on of the slim es th ic knes s (Exp. CA 2 , Tabl e 6.4)
157F ig . 6.35 Changes in the parameters b and IRn, with relationship to theexpe rime nta l vari ati ons of the anodi c overpotential (Exp. CA 2,Tabl e 6.4) 157
F ig . 6.36 Propos ed analogue m odel repres entat ion of a lead bu ll io nelectrode covered wi th a layer of sli mes 164
F ig . 6.37 Ele ctr ic al cir cui ts us ed to analyze the impe danc e spec tra 171
F ig . 6.38 Cor rel ati on between the exper iment al an d theoret ical
imp edan ce spec tra (Exp. CA 2, cir cui t A. 2, Table 6.7). 174
F ig . 6.3 9 Va ri at io n of the derived electrical analogue param eters as afun ct io n of the sli mes thi ckn ess (Exp. CA 2, Ci rc ui t A. 2, Table 6.7) 177
F ig . 6.40 Det ai l of the imped ance s pec tr um obtai ned i n Ex p. C A2 at
0.80 m m of sl imes (Exp. CA2 , ci rcui t B.l) 178
F ig . 6.41 Va ri at io n of the derived elec tric al anal ogue par ameter s as afu nc ti on of the sli mes thi ckn ess (Exp. CA 2, Ci rc ui t B.2, Table 6.8) 179
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Fig. 6. 42 Va ri at io n of the ma x i m u m valu es of the real, Z*. and theimaginary parts, -Zg , of the impedance as a function of the amount oflead dis sol ved (Exp. CA4) 182
Fig. 6.43 Mod ifi ed Ran die s analog ue cir cui t 183
F ig . 6.44 Cor rel at ion between the exper imen tal an d th eoreti calimped ance spe ctra (Exp. CA 2, curre nt inte rrup tio n condi tion s,Tabl e 6.11) 184
F ig . 6.45 Cor rel at ion between the exper imen tal an d the oreti calimped ance s pectra (Exp. CA 5, curre nt inte rrup tio n condi tion s,Table 6.12) 186
F ig . 6 .46 Ar ga nd plot of a typica l lead bu ll io n electrode (Exp. CC 2) i n
the presen ce of a 2.2 m m layer of sl imes 187
Fig. 6.47 Cor rel ati on between the exper imen tal an d the oreti cal
impedan ce spect ra (Exp. CA4, curren t interru ptio n condit ions,Table 6.13) 189
Chapter 7
Fig. 7.1 Sys t em (F)-Si-H 20 at 25 °C 196
F ig . 7.2 Sys t em (Pb-F)-Si-H 20 at 25 °C 197
F ig . 7.3 Cha ng es in p H as a fun ct ion of loga SiF -2 and loga p b + 2 198
Fig. 7.4A Sys tem (Sb-F)-Si-H aO at 25 °C 199
Fig. 7.4B Sys tem (Bi-F)-Si-H 20 at 25 'C 200
Fig. 7.4C Sy st em (As-F)-Si-HaO at 25 °C 20 0
Fig. 7.5 Cha ng es i n p H as a fu nc ti on of the electrolyte compos it io n 201
F ig . 7.6 Ele ct ric al cond ucti vity , density, an d visc osit y of H 2 S i F 6 - P b S i F 6
electrolytes 207
F ig . 7.8 Changes in Walden's product as a function of the square rootof the ion ic stre ngth 21 3
Fig. 7.8 Va ri at io n i n the slime s electrolyte com pos iti on as a fun ct io n ofthe distance from the slimes/bulk electrolyte interface, assuming nochanges in activity coefficients (Case A) 21 6
F ig . 7.9 Chang es i n the poten tia l of the slim es electrolyte as a fun ct io n
of the dist ance fro m the anode/ sli mes interface 218
Fig. 7.10 Changes in the potential difference between the outer andin ne r reference el ectrodes as a fu nc ti on of the distance from the
s l imes/bulk electrolyte interfa ce 21 9
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F ig . 7.11 Var ia t ion of the eddy diffu sion constant, D E
, as a funct ion of
the distance from the sl im es /b ul k electrolyte interface, (oc=lxlO 2 m m
a n d p ^ x l O " 4 c m 2 sec"1) 220
F ig . 7.12 Variation i n the slimes electrolyte composition as a fu nctionof the distance from the sli me s/b ul k electrolyte interface, when
changes i n activity coefficients and in eddy diffusi on are accounted for(CaseB ) 221
F ig . 7.13 Variation i n the slimes electrolyte composition as a fu nctionof the distance from the sli me s/b ul k electrolyte interface, whenchanges i n eddy diffusi on are accounted for (Case C) 222
F ig . 7.14 Variation i n physico-chemical properties of the sli meselectrolyte as a funct ion of the di stance fr om the s l imes /bulkelectrolyte interface 223
Fig. 7.15 Compar ison between the experimental (unsteady-state) andpredicted (steady-state) anodic overpotentials 225
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Nomenclature
a, molar activity coefficient of species i: a, = C,
a pbdXi) Activity of Pb+2
as a function of the distance from the slimes/electrolyteinterface.
a +2(bulk) Activity of Pb +2 in the bulk electrolyte (i.e. outside the slimes/bulk electrolyte
interface).
b slope, mV (Eq. 5, Chapter 6).
Bj Frequency independent parameter, [Q cm2
sec V zc
], (Eq. 6, Chapter 5)
b,b2b0 Frequency independent parameters, [£2cm2sec"1^], (Eq. 10. Chapter 6)
bD Dimensionless parameter defined in Eq. 4, Chapter 5.
Capacity of the electrical double layer, [uF cm"2]
Cg Geometrical capacitance, [|xF cm"2]
C° t 2 Bulk Pb+2 concentration, [mol cm"3]rb
C,o Concentration of species i at the electrode surface (at x=0), [mol cm"3]
C, Concentration of species i, [mol cm"3]:Species a: Pb +2 Species b: SiF6"
2 Species c: H +
Clo o Concentration of species i in the bulk electrolyte (at x=°°), [mol cm"3]
CPE! Analogue parameter that represents the distributed nature of the
anode/slimes and the slimes/slimes electrolyte interfaceCPE2 Analogue parameter that represents the presence of a distributed
capacitance generated by the concentration gradients present in the slimeselectrolyte.
D Diffusion coefficient, [cm2 sec"1
]
D t Diffusion coefficient of species i, [cm2 sec"
1
]
D E Eddy diffusion constant, [cm2 sec"
1
]
D D^2 Mean diffusion coefficient for Pb +2 ions, [cm2
sec"1
]
D™ Molecular diffusion coefficient of species i, [cm2 sec"1]
<Di Overall diffusion coefficient of species i, [cm2 sec"1] : T>i = DT+D E
e(t) Potential as a function of time
Econtroi Difference in potential between the reference and working electrodes underpotentiostatic control.
F Faraday's constant. 96487 C eq"1
iron. Steady-state corrosion current density, [Amp m"2]
i\, Exchange current density, [Amp m"2]
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I& 0 Current density at the electrode surface, [A cm"2].
I (t) Superficial current density at the anode/slimes interfaceas a function of the electrolysis time, [A cm"
2
].
It Total molar ionic strength = /, = 0.5{Af u(v,Z2+v 2)+M m{y& + VjZ2
)} = 4x[PbSiF(5]
+ 3x[H2SiFe]IiHzSifj Molar ionic strength of H2SiF6 in a H2SiF 6-PbSiF6 mixture = 3x[H2SiF6]
hbsiFj Molar ionic strength of PbSiF6 in a H2SiF 6-PbSiF6 rnixture= 4x[PbSiF6]
j Imaginary number , V --T
L Characteristic length of the electrode, [cm]
mD Slope defined in Eq. 4, Chapter 5, [sec"0 ^
m Molality, [mol of solute/Kg of solvent]
M Molarity, [mol/1]
N, Water mol fraction with respect to species i.
n Number of electrons involved in electrode reaction.
I r Ia, I yr 12 Statistical parameters defined in Appendix 9.
R, Analogue parameter that represents charge transfer resistances associatedwith the lead dissolution processes, [Qcm ].
Analogue parameter related to the DC conductivity of the slimes electrolyte,[Qcm2]
ra Analogue parameter related to the DC resistance of the slimes electrolyte,[Qcm2]
rb Analogue parameter related to the charge transfer resistance associated withthe lead dissolution process, [Qcm2]
Rb Resistance of the electrolyte present between the reference electrode and theanode boundary layer, [Clem ]
R,t Charge transfer resistance, [Qcm2]
RD Diffusional (DC) resistance, [Qcm2.]
Rata Resistance of the film created by the addition agents, [Qcm2]
Rm "Apparent" average resistivity of the electrolyte present across the slimeslayer, [Qcm2].
Rp Polarization resistance, [Qcm
2
]Rs "Uncompensated" ohmic resistance, [Qcm2].
R Universal gas constant, [8.3114 J mol"1 deg"1]
s, Stoichiometric coefficient in electrode reaction
T Absolute temperature, [K]
v+ and v_ Number of cations and anions into which a mole of electrolyte dissociates.
w Solution composition, [wt% H2S1F<J
x, Distance from the slimes/bulk electrolyte interface at which point D e is to
be computed, [ x, < x^], [mm]
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Xjotai Total slimes thickness, [mm]
x Mixing fraction (Eq, 22, Chapter 7).
Z, Charge number of species i, [eq mol"1]
Z(jco) Impedance as a function of frequency,. [Qcm2]
Zj Imaginary component of the impedance, [Qcm2]Z* Real component of the impedance, [Qcm2]
IZI Absolute value of the impedance, [Qcm2], (| Z |= -N/ZI+Z2)
Z a / s e Faradaic impedance at the lead anode/slimes electrolyte interface.
Z a / s l Electronic impedance at the lead anode/slimes interface.
ZCP E Impedance of the CPE analogue element.
Z 2/lBC Impedance of the ZARC analogue circuit.
ZnJt) Changes in D C impedance as a function of time.
Z s l / b e Faradaic impedance at the slimes/bulk electrolyte interface.
Z s I / s e Faradaic impedance at the slimes/slimes electrolyte interface.
Warburg ionic diffusional impedance in the slimes electrolyte/bulkelectrolyte interface.
Zw Warburg ionic diffusional impedance throughout the slimes electrolyte.
[M™] Ionic concentration of ions M, [Ml.
[M fn]b Ionic concentration of ions M, at the slimes/bulk electrolyte Interface, [Mj.
[M+n]e Ionic concentration of ions M, at the anode/slimes electrolyte interface, [Mj.
{M+n}r Concentration of ions M at the anode/slimes interface with respect to their[Pfr+
^V
concentration at the slimes/bulk electrolyte interface (e.g. {Pb+
%=—^)
It Total molar ionic strength = 4x[PbSiF6l + 3x[H2SiF6]
[HaSiFglu H2SiF6 concentration at the total molar ionic strength of H2SiF 6-PbSiF6
mixtures, [M],
[PbSiFg] + [HaSiFJ PbSiF6 and H2SiF6 concentrations in the electrolyte rnixtures, [M],
[PbSiFelK PbSiF6 concentration at the total molar ionic strength of H2SiF6-PbSiF 6
mixture, [M].
a and p Arbitrary positive constants whose value depends on the electrolysisconditions: a [mm] and (3 [cm2 sec"1] (Eq. 31, Chapter 7).
pa Anodic Tafel slope
Pc Cathodic Tafel slope.
8 Thickness of the hypothetical Nernst boundary layer, [cm]
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y, Individual molar activity coefficient.
Y± Mean activity coefficient
AO Migration or "liquid junction" potential, [mV]
Oe Maximum value of the migration potential for a fixed slimes thickness, [mVj
<J>1( 02 Dimensionless parameters in Eqs. 4 and 5, Chapter 5.
^zc Fractional element in CPE analogue element
A ^ Equivalent conductivity of the H 2SiF 6-PbSiF6 mixtures, [cm2 eq"1 ft"1]
A ^ and ApK^ Equivalent conductivity of the pure H2SiF6 and PbSiF6 solutions at
the total ionic strength of H2SiF 6-PbSiF6 mixtures, [cm2 eq"1 CI' 1 ]
X, Individual equivalent conductivity of ions i, [cm2 eq'1 Ci' 1 ]
T) a c Activation overpotential, [mV]
r)n "Uncompensated" ohmic drop, [mV]•nA Anodic overpotential, [mV]
r|c Concentration overpotential. [mV]
rj^, Total ohmic drop across the slimes layer, [mV] (Eq. 32, Chapter 7)
r\c Concentration overpotential due to Pb+2, [mV] (Eq. 33, Chapter 7).
ru Steady-state anodic overpotential from the solution of the Nemst-Planck flux
equations, [mV] (Eq. 34, Chapter 7).
TI Dynamic viscosity [coefficient q/), [cP]
K Electrical conductivity, [mmho cm"1]K [x^ Electrical conductivity changes as a function of the distance from the
anode/slimes interface, A* [mmho cm"1]
K fc Electrical conductivity of H2SiF 6-PbSiF6 mixtures, [mmho cm"1]
KiPbsiFj,, Electrical conductivity of PbSiF6 at the total ionic strength of H2SiF 6-PbSiF6
mixtures, [mmho cm"1.]
K_ Bulk electrolyte electrical conductivity, [mmho cm"1]
H, Absolute ionic mobility of ion i, [cm2 sec"1 volt"1]
v Kinematic viscosity [coefficient oJ), [cSt].
pm Specific electrical resistivity of the electrolyte entrapped within the slimeslayer, [ftcm].
p Solution density, [g cm"3]
ov Warburg Coefficient, [CI cm2
sec"05]
x Relaxation time, [sec]
tD Dielectric relaxation time of the bulk electrolyte, [sec], TD=Cg.Rb
xf Relaxation time of the Faradaic reaction taking place upon discharge of thedouble layer, [sec]
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0)
©min
©max
AAS
ACBEP
cd
C.P.V.
CPE
DC
E.M.F.
EPMA
FFT
FRA
PCR
RM.S.
SACV
SEI
SSM
SEM
SSR
Relaxation time of the diffusional processes across the hypothetical Nemstboundary layer, [sec].
Relaxation time required to equilibrate the charge of the electrical doublelayer, [sec]
Frequency, [rad sec"1
]
Minimum frequency at which the AC impedance was acquired (or analyzed),[rad sec"
1
]
Maximum frequency at which the AC impedance was acquired (or analyzed),[rad sec"
1
]
Abbreviations
Atomic Absorption Spectroscopy
Alternating currentBetts electrorefining process
Current density
Cathode polarization voltage
Constant phase angle element
Direct current
Electromotive force
Electron probe microanalysis.
Fast Fourier transform.
Frequency Response Analyzer
Periodic current reversal
Root mean square
Small amplitude cyclic voltammetry
Solartron Electrochemical Interface
Secondary solidified material
Scanning Electron Microscopy
Solid State Relay
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Acknowledgments
I want to express my sincere appreciation to my research advisor, Dr. Ernest
Peters who guided me constantly throughout my stay at U.B.C. I want to thank
him for all his teachings, and most of all for having the persistence of bearing
with me.
My stay at U.B.C. was made possible due to scholarships from; Universidad
National Autonoma de Mexico (U.NJLM.) and The University of British Columbia,
and through supplementary funding from the Natural Sciences and Engineering
Research Council of Canada. I want to thank these institutions for their support.
I want to express my gratitude to Dr. R.C. Kerby (Corninco Ltd.) and Dr.
C.C.H. Ma (Dept. of Electrical Engineering, U.B.C.) for providing me with a variety
of new ideas. My sincere thanks to Dr. Charles Cooper for taking the task ofproofreading the thesis and contributing with constructive comments.
I want to thank my beloved wife, Diana, whose positive and encouraging
attitude gave me the strength to accomplish this work.
I also would like to thank my parents and brothers for giving me moral
support throughout this work.
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Introduction
The Bet ts proces s for lead electrorefining treats lead bu ll io n cont airi ing 1-4%
impur i t i es . Antimony, arsenic, bismuth, an d a variety of min or metals i ncl udi ng
silver and gold are amongst these impuri ties . The bu l l ion i s cast into anodes of
about 1 m 2 , wei ghi ng between 20 0 and 30 0 kg. These anodes are electrolytically
corroded with the s imult aneous pl at ing of relatively pur e lead ( >99.99% Pb) on
cathodes (pure lead starting sheets) of about the same area as the anodes. The
electrolyte is typical ly an aqueous lead f luosil icate sol utio n (0.2 to 0.5 M PbSiFg)
containing excess f luosi l icic acid (0.5 to 0.8 M HaSiFg). The impur i t ies are largely
retained on the anode scr ap as an adheri ng slime. Betts ref ining is the preferred
lead refini ng process when b i smut h must be separate d fr om lead. S ince b ismuth
forms a soli d solu tion wit h lead at concentrations nor mal ly foun d i n lead bull ion,
it is necessary for the anode s lime s to adhere so tha t they can cement out the
bismuth f rom the electrolyte after it has corroded with l e a d 1 .
Th is stu dy was designed to obt ain a fund amen tal und ers ta ndi ng of the anodic
proces ses that take place up on elect roche mical dis so lut ion of lead anodes as us ed
i n the Bet ts proc ess. Du ri ng the refi ning of lead by the Bet ts process , ideally, only
lead would dissolve an d the noble impur itie s wou ld rem ai n unreacte d and
attached to the anode formi ng a strong , adherent, an d hi ghl y por ous slime s layer.
The extent to wh ic h this ideal operation ca n be achieved i n practice is a complexfu nc ti on of the lead anode phys ic al meta llu rgy and of the electrol ysis con diti ons.
This dissertat ion has focused on s tudying how the electrolysis cond iti ons affect
the behavior of typ ica l lead anodes . The ma i n objectives of thi s resear ch were:
1) To analyze f rom a therm odyn amic perspective, the condit ions und er whi ch
hydrolysis of the ac id, precipitat ion of secondary produ cts, and d issolut ion of
noble compounds can take place .
2) To obt ai n the component s of the anodic polar iza tio n an d relate them w it h:
(a) transport processes across the slimes layer (b) hydrolysis and secondary
products precipitat ion (c) noble compounds dissolut ion (d) bu lk electrolyte
composit ion (including addition agents) (e) cur ren t densi ty.
3)
To formulate a mathematical model that ca n be used to predict
concentrat ion and poten tial gradients across the sl ime s layer.
1 If bismuth does not corrode with lead it would enrich at the anode surface until lead corrosion is
stopped.
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The single, most important parameter to study was found to be the ionic
concentration gradients in the electrolyte entrapped within the slimes layer. One
of the direct effects of the presence of these concentration gradients is the
precipitation of secondary products within the slimes layer. The characterization
of these secondary products and their effect on the dissolution of noble compoundswas studied by several methods. Among these, transient DC (direct current) and
AC (alternating current) electrochemical techniques were used to find the extent
to which precipitation of secondary products and entrapped electrolyte
concentration gradients affect the refining cycle. The electrolyte composition
within the slimes and the accompanying potential gradient were obtained by
mcorporating tap holes in typical lead anodes. The study and characterization of
this entrapped electrolyte required the use of several analytical procedures,
including titTimetic analysis and atomic absorption spectroscopy. Thecharacterization of the physico-chemical properties of this entrapped electrolyte
was achieved by measuring the electrical conductivity, viscosity, and density of
synthetically prepared solutions whose compositions approximated those of inner
electrolytes as found from anode tap holes or calculated from theory. In addition
to this, sampling of the slimes layer and characterization of the contained
secondary products was an important part of this work. Scanning electron
microscopy and X-ray diffraction techniques were employed to detect the phases
and elements in the slimes layer.
A mathematical model based on the Nernst-Planck flux equations was
developed to describe the establishment of concentration gradients within the
slimes layer. This model predicts these gradients when combined with
relationships between concentrations and fundamental solution properties (i.e.
activities, mobilities, diffusion coefficients). The full application of this model will
occur only when more experimental data on these fundamental properties become
available.
To explain the presence of secondary products and their stability range from
a thermodynamic perspective, computer generated Eh-pH diagrams were drawn.
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Chapter 1 Literature Review
I Introduction
This chapter deals mainly with the Betts electrorefining process for lead, as
described in the literature. In this process lead bullion is purified by transferringmost of the lead from a soluble anode to a cathode through a lead-containing
electrolyte while leaving behind impurities in an adherent anode slime. The
electrochemistry of this process involves the properties of anode slimes and of
entrained electrolyte. Thus, the processes of anodic corrosion of lead and transport
of lead ions through the entrained electrolyte are essential to the understanding
of the Betts process. In this chapter, the physical metallurgy of lead anodes, which
affects the slimes adherence is also discussed.
n
Metal lurgy of Le ad
Lead is an ancient metal. It was used by the Romans for components of their
water distribution systems, and for that purpose it had to be malleable and ductile
IH. The ancients made lead of acceptable purity probably by smelting lead ores
under conditions that prevented arsenic, antimony, and other hardening elements
from reducing to the metallic phase. This could be accomplished in most cases
by producing high lead slags such as those still produced in fire assaying. In those
days it was necessary to avoid excessive copper in the ore, but arsenic, antimony,
bismuth, nickel, iron, etc. were reliably held in the slag by maintaining the high
oxidizing conditions of lead silicate based slags. Lead recoveries were low - not
better than 85% from the highest grade hand picked galena ore. When high lead
recoveries were found to be obtainable by coke-based blast furnace reduction,
lead so produced was too hard, usually because of its copper, antimony, and
arsenic content. The function of lead refining became both a softening process
and a method of recovering silver and gold [2].
Nowadays the extraction process can be conveniently portrayed in the
two-step flowsheet shown in Fig. 1. The first step involves bullion production from
the sulphide concentrate and the second step the refining of bullion to the final
product.
The conventional route for bullion production requires sintering of the
concentrate to produce a lead oxide containing product, which is then reduced
in a blast furnace with metallurgical coke to produce lead bullion. The KTVCET
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Metallurgy of Lead
and QSL processes represent two relatively recent commercial developments 131
that replace both the sintering - blast furnace combination with a single furnace
that treats concentrates, and reduce both the costs of lead smelting and the
environmental impact.
The KIVCET [4] process replaces the sintering/blast furnace operations witha flash smelting step. In this process, lead sulphide is oxidized to lead bullion and
sulphur dioxide in a stream of oxygen. In a second step, the bullion and slag flow
under a weir to an electrically heated settling hearth where coke breeze or coal is
added to reduce the residual lead oxide in the slag and produce a final bullion
(for refining) as well as a low-lead slag.
Lead Sulphide
Concentrate
Bullion
Lead Bullion
Production
•
Sulphur
Dioxide
Refining of
Bull ion
Residues
Fig. 1 Flowsheet for Lead
Extraction fromSulphidic
Concentrates.
99.99+ %Pb
Byproduct
Metals
The QSL [5] process consists of a long, horizontal, tubular, brick-lined
converter in which lead concentrates are pelletized and injected near one end into
a bath containing lead bullion, lead oxide - containing slag, and lead sulphide
matte. Oxygen is blown into the bath in the feed injection (and lead - bullion
discharge) zone where it ultimately oxidizes sulphide sulphur to sulphur dioxide
gas. Slag is tapped from the far end of the reactor after passing through a zone
where reducing coal-air mixtures are injected through tuyeres to lower its lead
oxide content.
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Metallurgy of Lead
The refining of bullion is carried out by either a pyrometallurgical route or
a combined pyrometallurgical/electrometallurgical route. Comparisons between
these two routes show that the pyrometallurgical route is usually used when ores
with low Bi content are treated 17,81. The generalized pyrometallurgical flowsheet
is shown in Fig. 2 \ This process consists of a series of steps which capitalize ona complex series of phase relationships to extract all the impurities contained in
the lead bullion down to very low levels. Reviews on the chemistry and technology
of these refining steps are available in the literature [4,6,9-11].
LEAD BULLION
<Co,Sb.At.SnAg,Au.Bf)[F»^n3/*.01
T
DROSSING
COPPER DROSS
OR MATTE1T
COPPER DROSS
OR MATTE
SOFTENING
ANTIMONIAL
SLAG
l
[Sb]
T
Fig. 2 A Genera lized Flowsheet for
the Pyrometallurgical Refining of
DESILVERISING
ZINC-SILVER
CRUST
L e a d [6].
1
(ZnS)
IS>I
T
ZINC-SILVER
CRUST
0 brackets indicate a major impurity
component
DEZINCING METALLIC ZINC
0 brackets indicate a major impurity
component
i
(Bi)
T
[] brackets indica te a mi nor impurity
component
DEBISMUTHISING
•
BISMUTH
DROSS
[] brackets indica te a mi nor impurity
component
1
<C.Mg)
IZnJb)
T
FINAL REFINING
CAUSTIC DROSS
T
MARKET LEAD
> 99.99% Pb
The combined pyrometallurgical/electrometallurgical route is shown in
Fig. 3.
Here, copper dressing is performed to remove the bulk of the copper as a
combination of matte and arsenide - antimonide for further treatment, thus
allowing for the removal of some arsenic and antimony. Arsenic and antimony
are sometimes reduced further as sodium arsenate - antimonate dross by oxidizing
in the presence of caustic soda, because their levels in bullion must be controlled
to produce suitable anodes for successful electrorefining practice.
1 Fig.
2 was
taken
as
is
from
the
literature
and
does
not
contain inputs required
for
material
balances.
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Metallurgy
of Lead
Molten .crude lead
f
Slag and
matte
Cu
recovery
Continuous drossina
tumace
Decopperized lead
I Casting
kettle I
Anode
casting machine
Electrolyte
Anode
7 3 _ J c
I Electrolytic cell
Scrap
anode
Mechanical scraper I
Starling sheet kettle
i
I .Starling sheel machine
^ m
Starling sheet
Scrap
I
Filtrate-
Deposit cathodes
T
Wa
Fig. 3 A Generalized Flowsheet
for the Pyrometallurgical
/Electrometallurgical Refining of
Lead [22].
Cenlrituge
Slime
T
Precious metals recovery
1 Pig castin
kettle 1
1 Casting machine 1
Pig lead
Electrorefining is carried out in either a fluosilicic, fluoboric or sulphamic
acid electrolyte and produces a commercial lead cathode product and an anode
with an adhering slime [12] \ The purity of the produced lead is usually higher
than 99.99% [131. The slimes, representing only 2 to 4% of the anode weight, are
treated by a variety of processes to recover silver, copper, antimony, gold, bismuth,
and sometimes tin and indium [ 14-21].
Processes that entirely avoid smelting (and its attendant gas and dust
treatment systems and associated environmental risks), utilizing
hydrometallurgical/ electrometallurgical flowsheets, have also been proposed to
replace the current technology [23]. These include (a) the U.S. Bureau of Mines
ferric chloride leach process [24] which recovers lead via the molten salt electrolysis
of PbCl2, (b) the Minemet Recherche ferric chloride leach process [25] which
recovers lead from chloride leach solutions using aqueous electrolysis and (c) the
U.S. Bureau of Mines process [26] for leaching lead concentrates in waste fluosilicic
1 Nitric acid media are not suitable because,
in
the presence of free
acid,
nitrate is reduced (to nitric oxide
gas) at the cathode, preferentially to plating of lead. At higher pH, where the nitrate ion is much more inert
to
reduction,
the electrical conductivity of the electrolyte
is
much too low for an economic practice.
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Plant Practice
in
Lead
Electrorefining
acid with an oxidant (hydrogen peroxide or lead peroxide) followed by aqueous
electrolysis from the fluosilicic acid leach solution. All of these processes have
been piloted, but none has been commercialized \
HI Plant Pract ice i n Lead E lectrore f ining
A detailed description of the Betts Electrorefining Process (BEP) can be found
in Betts' book [271 and numerous patents [28-32]. The fundamentals of the process
described there remain applicable to all the plants which electrorefine lead
nowadays.
The BEP process is used in Canada [33-36], China [22,37,38], East Germany
[39], Italy [40-42], Japan [43-51], Peru [52-541, Rumania [55.56], Russia [57-591, U.S.A.
[60] and West Germany [61-63]. The average annual production of lead by BEP is
approximately 1,000,000 tons. Since the production of refined lead in the
non-socialist countries (for which good figures are available) is close to 4,700,000
tons per year [64], it can be assumed that up to about 20% of the world lead
production is refined by the BEP. Two variations of the BEP, the sulphamic acid
and the fluoboric acid processes are operated in Italy 141] and in West Germany
[61] respectively.
The Betts Electrorefining Process normally utilizes a fluosilicic acid (HaSiFe)
electrolyte containing lead fluosilicate to electrorefine impure lead anodes into
pure lead cathodes. The fluoborate and the sulphamate processes are identical
to the fluosilicate process except for the substitution of the electrolyte. Fluoboric
acid has not been used extensively due to its relatively high cost. HBF 4is a stable
acid with a good electrolytic conductivity and a high solubility of the lead salt. It
is also used in lead plating baths [65-67], lead fluoborate-fluoboric acid
rechargeable batteries and in the recovery of lead from spent batteries [68-72].
The sulphamic acid process has two main limitations when compared to the
fluosilicic acid process [40,58,73]: firstly, sulphamic acid decomposes rapidly at
current densities larger than 100 Amp/m 2
resulting in high reagent replacement
cost; secondly, the free acid is a crystalline solid of limited solubility, and limited
ionization in aqueous solution, leading to a low conductivity of sulphamic acid
solutions (at most, half of the conductivity of equivalent fluosilicic acid solutions).
This results in larger power costs in refining. The significant advantage of the
1 Commercialization of any new lead processes
faces
a lack of need for plant expansion in
this
industry
and so must be justified on the basis of conversion or replacement of existing capacity.
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Plant Practice
in
Lead Electrorefining
sulphamic acid process is that it is the most efficient process for the removal of
tin from impure lead (Sn remains in the slimes) [74-77]. Plants that use the
fluosilicic process remove Sn prior to electrolysis by using the Harris process 1781.
When anodes containing significant Sn concentrations are refined by the Betts
process, tin dissolves with lead at the anode and co-deposits at the cathode. Thelead - tin alloy may be sold, or some post treatment of the cathodes is necessary
to remove Sn [79].
The wide use of H 2SiF 6in lead electrorefining is due to its low cost. Fluosilicic
acid is produced as a by-product of the treatment of phosphate rock in fertilizer
manufacture [801. Fluorides and silica contained in phosphate rock form
fluosilicate and are separated from the fertilizer product. Fluosilicic acid is also
produced as a by-product during the dissolution of apatite with sulphuric acid
[54,80]. During the electrorefining operation, fluosilicic acid is consumed by
entrapment in the anode slimes and by volatilization from the surface of the
electrolyte. The acid develops significant vapour pressures through volatile
decomposition products according to the reaction,
H 2 SiF 6 2HF(g) + SiF 4(g)
SiF 4and HF are both corrosive and toxic and are removed from the tankhouse
atmosphere by adequate ventilation.
Table 1 shows some of the operating parameters of various lead
electrorefining plants. The wide variation in electrolysis conditions seen in this
table does not seem to have a strong influence on the final quality of the refined
lead. For example, lead concentrations in the electrolyte can be varied between
30 and 270 g/1 without affecting seriously the refined lead quality. The electrolyte
recirculation rates are also varied widely, with no apparent correlation to other
operating parameters. Extremely high electrolyte velocities might improve mass
transfer across electrode boundary layers, but can also nullify the additive effects,
worsening the deposit quality and causing short circuits [81].
Table 1 also shows that the current densities employed in the Betts process
fall in the range of 120 to 230 Amp/m 2 . Higher current densities have been
achieved through the use of galvanodynamic techniques such as current
modulation and periodic current reversal (PCR). The current modulation
technique consists of decreasing the current density (e.g. from 220 to 160 Amp/m 2)
in small steps [33,87]. Each constant current density step is determined on the
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Plant Practice in Lead Electrorefining
(a)
ft
a
G
Q
C0)uJ -l
3O
Va
200
<uo
120
ft
IH
>
o80
o
•3o
40
JH
<0
60 90 120 ISO 180
Electrorefining Time, hrs
a
in
Q
C9)IHIH
U
(b)
l
c
,
extrapolated
curve
l
c
, current modulation program
>
aa)
C4
oftIH
>
oT lO
c<
60
90 120 150 180
Electrorefining Time, hrs
Fig. 4 Changes in the anodic overpotential value, riA,during lead electrorefining [87]
(a) Conventional galvanostatic process (b) Galvanodynamic process in which the cell current is continuously
decreased during the refining cycle to fix the value of T | A .
basis of an anode overpotential value which increases as the slimes layer thickens
and decreases when the current density is reduced. The upper limit for the anode
overpotential is usually deterrnined by Bi dissolution from the anode. Fig. 4 shows
the current density program that can be applied to the elechrorefiriing circuit
without reaching the critical overpotential value for Bi dissolution. The higher
average current density possible with current modulation reflects in a shorter
electrorefining cycle and higher refinery production.
Periodic current reversal (PCR) in lead electrorefining is widely employed in
China {381. PCR involves frequent short reversals of the electrolysis current
direction 188]. This reduces the concentration polarization in the slimes layer andlevels the cathodic deposit by selectively dissolving projections. High current
efficiencies, good cathode quality, low electrolyte losses, low energy consumption
and a decrease in the number of short circuits have been reported through the
use of PCR [22]. A 16% increase in free acid was found in the slimes layer due to
PCR. [22]
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Plant Practice in Lead Electrorefining
T A B L E 1 . B e t t s l e a d e l e c t r o r e f i n i n g i n t h e w o r l d
Takehara. Japan U ]
Chiglrlshlma Japan 1471 Harima,Japan 1*6]
Shenyang Smelter,China [sal
Electrolyte:
Pb. g/1Total HjSiF,. g/1Free HjSlFj, g/l
Others: rag/1
Additives Consumption: Aloes, g/tonLigniri Sulphonate, g/tonGlue, g/tonOthers, g/ton
Temperature, CCirculation Apparatus
236-270
52-61
2-5 Bi. 0.5-2 Cu, 1-2 As,130-200 Sb
600-1100
28-43
75135
35-38
90-100
110-115
1000
40-43
1.8 m'/min x 20 m Head
74104-12438-58
400 Sb, 800 Fe,110 As, 28 Zn.290 Sn. 0.2 Ag
300-450200-300
32-45Pressure Tank withcentrifugal copper
pump18-251.5-1.7
Recirculation Rate, 1/min Add loss. Kg/ton
301.6-2.7
402.3
304
74104-12438-58
400 Sb, 800 Fe,110 As, 28 Zn.290 Sn. 0.2 Ag
300-450200-300
32-45Pressure Tank withcentrifugal copper
pump18-251.5-1.7
Current:
Amp/m 1, CathodeCdf Voltage, VCurrent, Kw per Generator
Current Efficiency. %Energy Consumption Kwh/ton Pb:Electrolysis Mechanical
120-1400.5-0.6
15000 Amp S70V
95.6-98.7154-15717.8-20.7
1470.47
10000 Amp 9200V
9314330
1850.55
5000 amp ® 50v 13000amp 8
50v
93175-180
154-1720.46
2800-3500
96.30120-13030-40
Anodes:
Casting Technique
Composition
Length,Width,Thickness, mm 3
Mode of SuspensionLife, DaysScrap. %
Anode Spacing, mm Weight, Kg
Casting Wheel, 18 Moulds
.98% Sb. 0.5% Bi.0.02% Cu. 0.02% Sn.
0.01% As, 67 oz/ton Ag.0.4 oz/ton Au
1150x1000x25-39Cast Lugs
8 (half cycle scrubbing)
380-440
Vertical Casting With Water Cooling On Top Of
The Mould
1.25% Sb. 0.12% Bi,0.06% Cu, 65 oz/ton Ag.
0.07 oz/ton Au
1200x800x24cast lugs
730100250
Casting Wheel With WaterCooing On Top And
Bottom Of The Mold. 15 Mould
.5%Sb. 0.13% Bl. <. 1% Cu. 0.05% Sn.
0.05% As
970x740x35cast lugs
826.5110280
Casting Wheel
.68% Sb.0.073% Cu.0.044% Sn
920x620x23Suspended Lugs
20-2595
140±3
Cathodes:
Starting SheetProduction TechniqueThickness, mm Weight, KgLife, days
0.6-1.010-204-5
.8107
1104
.8-1.8-112.5
Anode Slimes:
Composition
Removed After ? daysPercentage of AnodesScrubbing Technique
36.9% Sb. 17.4% Bl,12.1%Pb. 3.1% Cu.0.1% Sn. 0.4% As.8.9% Ag. 0.06% Au
2.4-3.6Rotating Brush
45% Sb. 4% Bi. 12% Pb.3%Cu. 7-10% Ag,
3.22 oz/ton Au. 30% H O
72.9 1.3
12-15% Pb,8-12% Bl.0.2-0.4% Te
2.51.1-1.4
Tanks:
Length, Width. Depth, cm'
Number of Anodes, cathodesConstruction Materials
500x130x155
42, 43 prefabricated concrete,
PVC lining
300x100x150
28. 29vinyl chloride with steel
frame
1150x920x1450(inner size)
asphalt lined reinforcedconcrete tank
320x75x120
32, 33Reinforced Concrete
with asphalt or PVClining
Pb Annual Production, tonPb Average Content %
4500099.999
6800099.999
2700099.999
5445099.99+
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Plant Practice in Lead Electrorefining
T A B L E 1 . Betts lead electrorefining in the wor ld(continuation)
Cominco,Canadaiss]
"Albert Funk", EastGermanytM]
Cerro dd Pasco, Peru[881
Kamloka.Japan [so]
Electrolvte:
Pb.g/lTotal H,S1F„ g/1FreeH,SiF,.g7l
Others: mg/1
Additives Consumption: Aloes, g/tonLignin Sulphonate, g/tonGlue, g/tonOthers, g/ton
Temperature, T!Circulation Apparatus
Recirculation Rate, 1/min Add loss. Kg/ton
75 (60-80)141
90 (90-100)
8 Bi. 65 Sb, 4000 SiO,.1.6 Cu, 12 Sn. 5 As,17 In, 70 Tl
170 (Aloin)250 (Calcium)
40 (38-43)Centrifugal Pumps
27 (27-45)2
30-50120100
600
35Storage tank with epoxy
lined^gump
15 (32% pure)
7512060
600 HF. 5. Sb. 0.5 BL0 SlOj. solids
550 (Calcium)550
40Centrifugal Pumps
123
7513075
1 Bi. 150 Sb
180600
35-45 Volte Pump 18.5Kw x 2
402
Current: Amp/m 1, CathodeCellVoltaBe. VCurrent, Kw per Generator
Current Efficiency. %Energy Consumption Kwh/ton Pb:
Electrolysis Mechanical
230 (max).3-.5
6300 Amp(5 day), 5400 Amp(7 day)
1000 kw90-95
16850
1850.45150
92
19590
1560.5-0.6
Mercury Rectifier 1520 Motor generator 360
90
14335
1350.55
20000 Amp O60V
96
165130
Anodes:Casting Technique
Composition
Open Mould Casting Wheel
1.2-1.4% Sb, 0.4 %As,0.15%BL0.05%Cu,
100 oz/ton Ag
Casting Wheel, 12 Mould
0.5% Sb. 0.3% Ag.0.3% Bi, 0.1% Cu,
0.002% Sn
Casting Wheel
1.8% Sb. 0.15% As,1.5% Bi. 0.05% Cu.
0.01% Sn. 0.07 oz/ton Au.
140 oz/ton Ag940x690x25
Horizontal Casting WithHorizontal Mould0.4% Ag. 0.3% BL0.1% Cu. 0.6% As,
0.6% Sb
Length.Width.Thickness, mm 3 864x660x30 730x710x25 (immersedsurface)
Suspended Lugs
Casting Wheel
1.8% Sb. 0.15% As,1.5% Bi. 0.05% Cu.
0.01% Sn. 0.07 oz/ton Au.
140 oz/ton Ag940x690x25 1140x990x20
Mode of Suspension Lugs Designed intoCasting
525100206
730x710x25 (immersedsurface)
Suspended Lugs Suspended Lugs Shoulder Type
Life. DaysScrap, %
Anode Spacing, mm Weight, Kg
Lugs Designed intoCasting
525100206
640130200
449100150
645110
Cathodes:Starting, SheetProduction Technique
Continuous DrumCasting
Mechanized production with on line casting of
ribbon
Continuous DrumCasting
Direct Method Machine
Thickness, mm Weight, Kg
16.3
160-70 (Final Wdght)
.680 (Final Wdght)
.713
Life, days 5 or 7 3 4 6
Anode Slimes:
Composition
Removed After ? daysPercentage of AnodesScrubbing Technique
40% Sb, 16% As, 13% Pb.2.5% Cu, 2500 oz/ton Ag
5 or 73
Conveyed By MonorailBetween Rubber
Scrapers
15% Pb. 25% Sb.15% Ag. 10% Bi, 5%Cu
61% (solids)
pneumatic stripping
28% Sb. 10% As.24% Bi. 10% Ag.
1.2% Cu. 0.07%^e,0.52 %Te, 18% Pb,
38.4% HjO. 0.4% SIO,44
Water Sprays
10% Pb, 15% Ag.15% Bi, l%Cu. 20% As,
20% Sb
61.4
Rotating Brush
40% Sb, 16% As, 13% Pb.2.5% Cu, 2500 oz/ton Ag
5 or 73
Conveyed By MonorailBetween Rubber
Scrapers
Tanks:Length, Width, Depth, cm 3
Number of Anodes, cathodesConstruction Materials
268x82x11224. 25
Asphalt Lined Concrete(old) Polymer Concrete
(new)
230x80x120 (inner size)16, 17
Rubberized steel plates
455x95x13040, 41
Glue Lined Concrete
500x130x16043, 44
Vinyl Chloride Resinand Concrete
Pb Annual Production, tonPb Average Content %
14400099.99
1500099.99
7200099.99
3000099.99
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Introduction
TV The Anodic Process
A. Introduction
The Betts process, as practiced by all refineries, depends on the formation
of an adherent, porous anode slimes layer during electrolysis. The slimes layerconsists of undissolved impurities which are removed mechanically from anode
scrap after the anodes are withdrawn \ If the slimes do not adhere to the anode
during electrolysis they will settle through the solution and to some extent be
mechanically entrained in the cathode deposit. There is no indication in the
literature of a lead refining process in which (as in copper refining) slimes fall is
encouraged. Further, the recovery of slimes from the bottom of the cell (as in
copper refining) is perceived to be more costly. To form an adherent slimes layer,
certain elements (mainly As, Sb, and Bi) must be controlled within a narrow
composition range and/or ratio in the anode, and the anode casting process must
be designed to control the rate of solidification to optimize the microstructure of
the cast bullion.
B. The physical metal lurgy of the lead anodes
The lead anode microstructure that is desired for optimum slimes structure
during electrorefining is known as the honeycomb structure, because it consists
of uniform size grains of lead surrounded by impurities on the grain boundaries
(Fig. 5). As the lead grains dissolve they leave behind a skeleton of slimes which
resembles a honeycomb. Any non-uniformity of the matrix will lead to
non-adherence of the slimes layer, and any precipitates present in this matrix
material will contribute to slimes detachment because of their extra weight.
There are four elements present in the lead anodes that seem to exert a strong
influence on the anodic process: As, Sb, Bi and Ag. The interaction of these
impurities with each other and with lead can be deduced from the available binary
and ternary phase diagrams [89-93], which show that both intermetallic
compounds and eutectic structures maybe present. Three different slime forming
systems have been identified and classified from studies on synthetic anodes [94].
1 During
normal
lead
electrorefining
practice
some slimes
do
fall, and are cleaned out
of the cells
at very
infrequent
intervals (months).
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The Physical Metallurgy of the Lead Anodes
Fig. 5 Lead anode microstructure. From an anode currently being used by Cominco Ltd. Chemical composition
as described in Table 1. Top view, air cooled side.
1) Impurity phases of the solid solution type (SST): the Pb-Bi system.
2) Impurity phases of the precipitation type (PT): the Pb-Sb system.
3) Impurity phases of the eutectic type (ET): the Pb-As and the Pb-Agsystems.
The strength of slimes adhesion to the anode was quantified by Tanaka [94]
through observations of slimes fall and slimes morphology. Tanaka summarized
the results of these studies with the following relationships:
1) The greatest slimes adherence is obtained when impurity phases of the
SST type are present in the anode in concentrations greater than 0.23% wt.
A SST concentration lower than this critical value produced a slime that
easily slides off the anode.
2) The addition of a third element to the eutectic systems increases
significantly the slimes adherence.
3) Water quenching of the anodes increase slimes adhesion particularly in
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Industrial practice
the FT and ET systems. However, the very small slimes particles formed
during quenching may also promote slimes detachment and mechanical
entrainment in the cathodes.
The increase of slimes adhesion by water quenching relates directly to thelead anode solidification rate, which influences the growth and distribution of
impurity-containing phases. Especially important are the eutectic forming
systems where it is known that the solidification parameters (growth velocity,
temperature gradient in the liquid and growth mechanism) and system parameters
(volume fraction and impurities content) can lead to anomalous eutectic
structures 1951. Such structures have been reported to occur in the following
systems (95]1 :
Ag-Pb and Ag-Bi: broken lamellar structure type
Pb-Bi: complex structure type
Pb-Sb: complex regular structure and irregular structure type.
It must be emphasized that even though the physical metallurgy of the lead
anodes is of great importance to the Betts process, there remains a lack of
knowledge in this area.
C. Industrial practice
As Table 1 shows, the range of anode impurities used in lead refining varies
from plant to plant. In every refinery the anode composition is kept vrittiin narrow
limits to obtain an adherent slimes layer. The need for production of anodes with
homogeneous properties has led to control methods for cooling rates and casting
techniques. In addition to the use of a casting wheel, straight horizontal 144-46]
and vertical [47] anode casting systems are employed.
Fig. 6 shows the straight type horizontal lead anode casting system currently
used in Japan. Although the vertical and horizontal casting processes were
originally developed to save space (over that occupied by a casting wheel), they
were carefully designed to achieve uniform cooling rates during the casting of the
1
A broken lamellar structure consists of a near regular array of "broken" plates and occurs in systems
that
contain less than 10% of the faceting phase [95]. The
complex structure
consists of an array of plates
that
are regular over small areas around a
well
defined spine [95].[14]
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Slimes
electrochemical behavior
Anode
Conveying
System
Anod*
01 ID
i i I I T I J C
Casting
Mould
Lifter
Cooling
Lifter
1
Fig. 6 Straight Type Horizontal
Lead Anode Casting System [44]
Electrode
Assembling
Machine
anodes. Such new techniques for anode casting as well as for optimizing heat
treatment have been patented by Japanese companies 1961, but no information
was found that would indicate the nature of changes in microstructure resulting
from these newer techniques.
D .
Sl imes electrochemical behavior
The slimes layer formed in the BEP undergoes structural and chemical
changes during its growth. As the slimes layer thickens, the different phases and
compounds present react with the entrained electrolyte creating secondary
products. The Ej, - pH diagram for the PbSiF6-H 2SiF 6 system (ion indicates that
both lead fluoride and silica can precipitate at higher pH values. This has been
related to the effect of electrolysis parameters on secondary processes (such as
these precipitations) that take place witxiin the slimes layer during electrolysis
[97-1001. The transport processes within the slimes layer in the context of these
secondary reactions have also been the subject of several studies [102-108].
A simple physical model to study the role of the slimes layer during anodic
dissolution rate in electrorefining systems was developed by Reznichenko et al.
[1091. In this model, the slimes layer was represented by a silver gauze diaphragm
(representing the noble impurity) electrically connected to the anode (a highly
pure base metal) and placed at a (variable) distance from the anode corresponding
to the slimes - electrolyte interface. The distance between the anode and the gauze
was varied to simulate the effect of an increasing ohmic drop between the anode
and the slimes layer. During electrolysis, concentration gradients normally
[15]
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Slimes electrochemical behavior
present in a slimes layer were replaced by a simple concentration difference
between the solutions on the two sides of the gauze, and this was related to a
modified Nernst equation:
where:
[A/eJ, /^ concentration and valence of noble metal
[Me,],nx = concentration and valence of base metal
A<t> = ohmic drop between the base metal and the noble metal gauze
= rest potential of the metal to be refined
E 2° = rest potential of the noble metal
Equation 1 shows that the larger the ohmic drop the larger the equilibrium
concentration of ions of the noble metal and the larger its dissolution rate.Although this model oversimplifies the different phenomena taking place within
a slimes layer, it provides an insight as to the effect of ohmic drop buildup in the
slimes layer and the dissolution rate of noble impurities normally left in the slimes.
Reznichenko et al. 11091 applied their model to the study of the Cu-Ag system in
which they found a logarithmic relationship between the dissolution rate of noble
impurities and the ohmic drop potential.
Early research on the behavior of BEP slimes has focused on relating the
anodic overpotential to the cathode purity [l io.ni]. Among the impurities mostclosely followed in the cathodic deposit is Bi [112.1131. Even though the BEP has
a large selectivity for the removal of Bi, it has been found that towards the end of
the electrorefining cycle such selectivity can be lost. It seems that the buildup of
concentration gradients and the precipitation of secondary products within the
slimes layer can initiate steep increases or discontinuities in the anodic
overpotential at a certain slimes thickness. This would produce a large increase
in the rate of dissolution of such impurities, and subsequently in the deposition
of impurities in the cathodic deposit. Fig. 7 shows how Bi contamination in the
cathode increases with anodic overpotentials above a critical value (about 200
mV). This has been recognized as a general behaviour for bismuth in lead refining
by the BEP [33,87,110, ill].
On the other hand, the use of periodic current reversal
(PCR) [39,98] has been found to increase the minimum anodic overpotential at
which noble impurities start to dissolve.
...1
[16]
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Slimes electrochemical behavior
500
Electrorefining Time, hrs
Table 1 shows that the slimes layer weight is only 1 to 4% of the original
anode weight; yet, the slimes occupy the whole of the original anode volume. This
indicates that the porosity of the slimes layer exceeds 92% and may be as highas 98% (taking into account a density for slimes phases of about half that of the
bullion). Concentration gradients in the electrolyte confined within this highly
porous layer and the slimes electrochemistry are closely linked.
Wenzel et al. [97,114,1151 measured the change in composition of the
electrolyte contained in the slimes layer as a function of time and of slimes
thickness. Fig. 8 shows the sampling method used by Wenzel, utilizing sampling
wells at different distances from the anode surface. Small amounts of electrolyte
were withdrawn through these wells at carefully selected times (to avoid perturbing
appreciably the system).
According to their results, using a range of anode compositions (Bi from 0
to 1.74% and Sb from 0.45 to 3.01%), the more the amount of secondary solidified
material (SSM) present in the anode, the steeper the pH and the Pb + 2
concentration
gradient throughout the slimes layer. Also they found that the thicker the slimes
layer the steeper the concentration gradients. Fig. 9 shows these changes in
concentration for two different anode compositions. Wenzel et al.proposed that
[17]
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Slimes electrochemical behavior
62
Bath level
oCO
o
CM
Fig. 8 Dimensions of the anodes
used in Wenzel's experiments
showing the position and size of the
electrolyte sampling wells [97].
All measurements in mm.
498
i
®::i : : e - -
tj *i
-e- -Q-
" T T
T * 8
due to the pH increase several reactions may occur: SiF6"2
decomposition to SiF 4
and F" (pH > 3); precipitation of Sb 2 0 3 (pH > 4.9) and precipitation of PbO (pH >
7).Table 2 shows the results of chemical and diffraction analysis, which seems
to indicate the presence of these secondary compounds within the slimes layerl
.
From an analysis of the concentration of noble impurities in the entrapped
electrolyte, Wenzel et al.were able to conclude that when the SSM exceeded 5%,
as determined from the Pb-Bi-Sb ternary diagram, permissible impurities in the
anode were too high to obtain an acceptable impurity level in the refined lead.
1 Note that since slimes oxidize rapidly in air, the analysis may indicate oxide phases
where
metallic
phases
were
present
in
the in-situ slimes.
[18]
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Slimes electrochemical behavior
Fig. 9 Changes with electrolysis time of the relative concentrations of Pb*2 and H* within the slimes layer with
respect to their bulk values [97,115].
Bulk electrolyte composition: [PbSiFJ = 0.4 M , [HjSiFJ^ = 0.6 M, T= 35 *C, I = 200 Amp/m2
, Stationary
Electrodes. Vertical Axis: Pb +2
and H* concentration ratio between the electrolyte sampled within the slimes layer
and the bulk electrolyte.
(a) Anode with 0.46% Sb and 0.24% Bi (= 1.73% SSM) (b) Anode with 0.92% Sb and 0.24% Bi (= 3.80% SSM)
Note: No extrapolation of the inner electrolyte concentrations at zero slimes thickness done here.
Table 2 Chemical Composition and X-ray Diffraction Analysis of the
Lead Anode Slimes [97]
Sample
No.
Anode
Composition
%wt,(rest Pb)
Slimes Chemical Analysis %wt X-ray
Diffraction
Analysis
Sample
No.
Anode
Composition
%wt,(rest Pb)
Pb Sb Bi F Si
X-ray
Diffraction
Analysis
1 0.46% Sb,
0.24% Bi
56.98 7.62 3.95 11.97 6.66 PbO, SbaOg,
PbF2, Bi 20 3
2 0.46% Sb,
0.24% Bi
58.02 9.40 4.28 14.09 3.33 PbO, Sb 203,
PbF2, Pb4Si0 4.
BtjOg
3 0.92% Sb,
0.24% Bi
56.57 6.81 5.55 11.27 6.65 PbO, Sb203, PbF2,
BiaOg
4 0.92% Sb,0.24% Bi
53.77 9.37 9.20 12.22 4.19 PbO. Sb203. PbF2,BisOa
[19]
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Slimes electrochemical behavior
Wenzel's X-ray diffraction analysis on the slimes products agree with the
findings of Isawa et al. 1116,117], who studied the slimes composition of synthetic
and industrial lead anodes and found the presence of the following compounds1
Metallic Bi: In the Pb-Bi, Pb-Bi-As, Pb-Bi-Sb, and Pb-Bi-As-Ag systemsMetallic As: In the Pb-As, Pb-Bi-As, and Pb-Bi-As-Ag systems
Metallic Ag: In the Pb-Ag and Pb-Bi-As-Ag systems
Also the presence of the Bi-Sb solid solution, of the e and e' phases of the
Ag-Sb system, and of water soluble As and Sb (identified as As2
0
3
and of Sb2
0
3
)
was found in some of the above mentioned systems and in the slimes obtained
from industrial anodes [Ii6,ii7].
One compound that has been difficult to characterize is AgaSb 194], whose
presence has been detected in both synthetic and industrial lead anodes. The
morphology of this compound has been found to be a function of the anode cooling
rate and of the electrolysis conditions.
The electrolyte used in lead electrorefining contains impurities whose
concentration ranges from a few ppm and several g/1, and have their origin in the
anode from which they are dissolved (i.e. Cu+ 2
, Sn+2) or from the acid
manufacturing process (i.e. phosphorus species). Their steady state
concentrations in the electrolyte depend on the electrolysis parameters and slimes
thickness. They can affect both the anodic and cathodic processes through
changes in fundamental electrochemical parameters such as the exchange current
density [Q, the symmetry factor (a), and the electrical double layer capacity (CdJ \
Measurement of these parameters usually involve transient electrochemical
techniques, such as polarization scans, current interruptions, and AC impedance
studies.
Miyashita et al. [118-1231 have studied extensively the influence of minor
impurities and addition agents in the electrolyte used in lead refining. These
studies focused on the determination of the fundamental electrochemical
1 The term electrical double layer is used to describe the arrangment of charges and oriented dipoles
constituting the interphase region at the boundary of an electrolyte [144, p.630]
The exchange current density, i
0
, is a measure of the
rate
of
equilibrium
potential and sensitivity to
interference
[145, P.IO]
The symmetry factor, a, determines
what
fraction of the electrical energy resulting
from
the displacement
of
the potential
from
the
equilibrium value
affects the rate of electrochemical transformation
[144, P.923]
[20]
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Slimes electrochemical behavior
parameters [i^, e.d.l., and a) as a function of Sn + 2 , Fe + 3, Zn+ 2 and glue, using a
single current step transient method [118]. He found that under an oxygen
atmosphere the presence of the above ionic species increased the anodic
overpotential value. Thus, they may inhibit the dissolution of some noble species
including lead.
Fig . 10 Lead specific weight loss
(corrosion) as a function of the
immersion time in a2M
Pb(BF4)2-l M HB F 4 solution in the
presence of 0,2,5, and 10 mM of
Bi*3
[66].
50
100 150 200
Immersion Time, hrs
When noble impurities present in the lead anodes transfer to the electrolyte
they can be redeposited by cementing on less noble elements. While such
redeposition is obscured during anodic dissolution, it can be inferred from
corrosion measurements in the absence of a current. For example, the weight loss
rate for a lead sample in the presence of dissolved bismuth has been observed in
the HBF4-Pb(BF4)2 system [661. Fig. 10 shows how Bi concentrations as low as2 mM enhance the corrosion rate of lead.
During the dissolution of the lead anodes in the BEP, bismuth and other
noble impurities accumulate in the proximity of the slimes/bulk electrolyte
interface as well as in the bulk electrolyte [39,124]. Since these noble impurities
will deposit on the cathode at their limiting mass transfer rate, the permitted
[21]
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Additives control and electrochemistry
electrolyte concentration is related to cathode purity specifications (0.12 mM of
bismuth will typically lead to 10 ppm Bi in the cathode! 124]). They can be removed
by one of two methods:
a) Continuous cementation [125.126] through a column filled with lead
particles (Fig. 11).b) Electrodeposition in a separate electrolysis circuit [124]. The purified
electrolyte is then sent to the main electrolyte stream. This method is being used
in Japan [43].
The majority of plants that electrorefine lead do not purify their electrolyte,
but rather operate under conditions where the dissolution of noble impurities
from the anode is limited to tolerable levels.
Fig. 11 Continuous purification o f
electrolyte via purification co lumn
[125,126].
Continuous Purification
of Electrolyte via
Purification
Column
V The Cathodic Process
A. Additives control and electrochemistry
In the absence of additives, lead deposits with a very small overpotential,
and tends to form rough, porous deposits or dendrites that result in short circuits.
To obtain deposits that are flat, smooth, and free from projections, special reagents
are added to the electrolyte. These "addition agents" increase the cathodic
overpotential (actually called "inhibition"), and change the kinetic parameters (to,
a, and e.d.l.) under comparable electrolysis conditions [120-123].
[22]
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Starting sheet technology
Polarization measurements have been used for controlling and monitoring
the concentration of additives in lead electrorefining circuits [81.127-129], as well
as for screening of additives in the electrolytic refining of Cu and for the testing
of impurity levels in the electrowinning of Zn [130-135]. A polarization technique
for determining lignin sulphonate in lead plating baths has also been described[136]. Long term studies and years of industrial practice, especially by Cominco
researchers [81,129] have led to a sufficient understanding of the levelling
mechanisms of lignin sulphonate and aloes, to permit the use of polarization
measurements for controlling the concentration of these species in the industrial
electrolyte for lead refining. This additives control is achieved by keeping the
cathode polarization voltage (C.P.V.) within preset limits, and adjustments are
made by either new additions or by regeneration (in the case of aloes) with a
thiosulphate salt [128]. The level of additives present at a given time is directly
related to the C.P.V.
B. St ar ti ng sheet tech nolog y
The cathode starting sheets used in the BEP are usually cast on a rotary
drum casting machine. The stiffness of these sheets is increased by adding
approximately 15 ppm Sb to the melt and by impressing wrinkles on the cathodes
right after they are produced [791. Procedures to avoid dross incorporation in the
lead cathodes have also been developed [137]. In this section of the electrorefining
process high levels of automation have been achieved [138].
C. Ce ll electrol ysis parameter op tim iza tio n
The overall electrorefining process has been studied using statistical
correlations of some of the variables measurable in an operating plant [39.54.98].
To optimize the electrolysis parameters when the BEP is run at high current
densities (> 200 Amp/m 2), factorial design of experiments at three levels for four
variables has been used by Lange et al.[98]. They varied PbSiF6 and H 2SiF 6
concentrations, temperature, and current density, at constant values of addition
agent (glue), anode composition, cell geometry, and electrolyte recirculation. The
influence of these parameters on the average anodic and cathodic polarization,
on the cell voltage, and on the specific energy consumption was obtained by
employing regression analysis correlations. In addition, changes in these
parameters were correlated with the cathode quality and appearance as well as
[23]
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CeH
electrolysis
parameter
optimization
with the consumption of glue. The optimum electrolysis parameters found for a
current density increase (from 200 to 300 Amp/m 2) were a Pb+ 2
concentration of
60 g/1 (down from 80 g/1), a glue addition maximum of 1500 g/ton Pb (up from
1250 g/1), a free H 2SiF 6 content of 120 g/1 (no change) and an electrolyte
temperature of 35 to 40°C. Under these electrolysis conditions, Lange et al. foundthat Sb and Bi slimes layers could be subjected to anodic polarizations as high
as 280 mV without serious impairment of the cathode quality.
140
120
cO
100
CO
ti80
c u
>
O 4 0
o•rH
T3 20
O
ti
- : ; :
(*)
i . . . i . . . i . . . i . . . i .. .
Fig. 12 Influence of the cycle length
and current reversal ratio on the
anodic overpotential value during
PCR [39].
Bulk composition and temperature:
[PbSiFJ = 0.29 M, [HjSiFj] = 0.84 M,
T= 37.5 ' C .
W f — 1 = = 10 sec
< * ) ^ = £ r ^ = 20.ec
1
backward 1
20 40 60 80 100 120 140
Electroref ining Time, hrs
Optimization of the BEP through half cycle anode slimes scrubbing and
cathode exchange was also investigated by Lange et al. [98]. Not only did the specific
energy consumption decreased by the implementation of this procedure, but
highly pure cathodes were assured by avoiding anodic overpotentials in excess of
the preset limits for impurities dissolution. On the other hand, the exchange of
cathodes and the half cycle scrubbing of the anodes incorporates labour increases
making implementation difficult. The use of this technique is necessary when
anodes containing high impurity levels (>5%) are to be treated. For example, in
Russia [57], lead anodes containing as much as 15% Bi are scrubbed every 48
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Bipolar
refining of lead
hours during the 6-day long anode cycle. In addition to this, cathodes are
exchanged daily to avoid short circuits and to obtain a deposit of acceptable purity
[57].
Lange et al. also studied the use of PCR in the BEP. Fig. 12 shows the anodic
overpotential dependance on the PCR parameters. Thiet [39] found that the useof PCR decreased the anodic overpotential values. The reduction in the anodic
overpotential values did not reflect in lower PCR energy consumption probably
because it was offset by unproductive energy consumption during the reverse
current phase of the cycle. Thiet also showed that the quality of the PCR cathode
deposits was very high [39].
D. Bipolar ref ining of lea d
Among other alternatives to optimize the lead electrorefining process, theuse of a bipolar configuration looks most promising. In this configuration only
the terminal electrodes are connected to the source of current. Between these
electrodes a large number of bipolar electrodes can be incorporated. One side of
these electrodes will corrode anodically, while pure lead will deposit on the other
side. The Impurities will be left behind as a slimes layer on the anodic side. The
advantages of operating the BEP in a bipolar mode include PCR, C.P.V., optimum
lead and free acid contents, and the use of jumbo electrodes (~4m2, possible
because there are no bus-bar connections)\ The process has been operated in a
pilot plant where it proved to be superior to the parallel process [1421. The rationale
for the current use of the parallel system is that a high economic investment is
required for the substitution and implementation of the bipolar process. If new
plants to electrorefine Pb are to be constructed in the future, very likely they will
be assembled on the bipolar configuration.
1 In the bipolar
mode
by-pass currents are reduced by increasing the area of the electrodes so that the
ratio of bypass to electrode area
in
the cell decreases.
[25]
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Chapter 2 Fundamentals of the Ele ctr och emi cal Measurement
Procedure
Early in the development of this project it was found that the form of ionic
concentration gradients within the lead anode slimes layer needed to be studied.
To study these concentration gradients without causing major disruptions to the
system, "in-situ" experimental techniques were considered. Among these,
transient electrochemical techniques appeared to be particularly advantageous
and consequently were used extensively in this work. The study of the
electrochemical response of typical lead anodes was complemented by
measurements on the physico-chemical properties of H2SiF6-PbSiF6 electrolytes.
What follows in this chapter is a description of the electrochemical parameters
germane to this work, and of the transient techniques that were used.
I. Components of the An od ic Over poten tial
The anodic overpotential was measured by following the difference in
potential between the lead anode ("working electrode") and a suitable reference
electrode. A high purity lead wire located within a Luggin capillary and contacted
with cell electrolyte was chosen as the reference electrode. Its use was possible
because of the high reversibility of lead in the H2SiF 6-PbSiF6 electrolyte system.
Furthermore, the amount of current that passes through the reference electrodeis limited to the current drain of the meter (a few nanoamperes). The "anodic
overpotential" \ T\ A , measured by such an electrode would also be free of any
junction potential between its tip and the slimes/electrolyte interface provided
the bulk electrolyte is well mixed. In the present work (unless specified) all
potentials are given with reference to such an electrode. The third electrode (a
lead foil counter electrode) is required to complete the electrical circuit. The
electrochemistry of this counter electrode is of no importance to the anodic
processes. Fig. 1 shows a simplified view of this three electrode arrangement.
The anodic overpotential measured under constant current conditions
increases continuously as the slimes layer thickens during the refining cycle. The
build up of concentration gradients in the proximity of the anode/slimes interface
1 This "anodic overpotential", r\ Al or potential difference between the anode and the reference electrode,
includes an ohmic
component, x\a , and long range potential gradients as well as interfacial overpotential
components.
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changes the value of T | A mainly through changes in its migration and concentration
overpotential components. Additionally, an ohmic contribution is included in this
TJ a measurement.
Fig. 1 Electrochemical Cell ArrangementThe importance of the ohmic component in electrochemical measurements
is evident by the number of studies that have been conducted in order to assess
its effect [1-91. According to Vetter [ioi the ohmic component is not a real
overpotential because its presence does not have any influence on the rate or type
of electrochemical processes under investigation. Ohmic drop or resistance
polarization can be separated from migration and concentration overpotentials
only in regions where the electrolyte composition is uniform as between the
reference electrode tip and the slimes/electrolyte interface.
Concentration overpotential, ric, is the Nernst potential generated due to
activity differences of the electroactive ion. Its presence is primarily due to the
limitation in the ionic transport of species moving from or towards the working
electrode surface. When this concentration overpotential occurs in the presence
of concentration gradients of non-electroactive ionic species, a migration potential
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gradient \ AO, is generated. The migration potential arises concurrently to TJC.
Thus, by subtracting the value of r|Q from the r\ A measurement, an overpotential
value consisting of T]C and AO is obtained.
An additional contribution to the anodic overpotential, the activation
overpotential, t|ac, is a function of the transport rate of charge carriers across theelectrical double layer. The larger the hindrance for this transport process, the
larger T|A C will be. By measuring the difference in potential between the working
electrode and a point located very near its surface, the value of T]AC is obtained.
The presence of other contributions to T| A (crystallization, reaction) will not
be described here, as they are unimportant for lead. Vetter [io] establishes the
conditions upon which these components are to be taken into consideration. We
assume that the contributions of these overvoltages to T|A are either negligible or
can be incorporated within T]AC or r\c .
n. Transient E lec tro chem ica l Techniques
The amount of information that non-transient electrochemical techniques
are able to provide is rather limited [ 11-14]. Significant enhancements in electronics
in the last 25 years have supplied the electrochemist with a wide range of transient
techniques. Reviews on the use, applications, and limitations of these techniques
are available [11.15-21]. Among these transient techniques, current interruption
and AC impedance were used extensively in this work. These techniques were
chosen due to their potential for resolving the T\ A components.
A . Current interruption techniques
Current interruption techniques go back as far as 1937 [221. The original
driving force for their implementation was to obtain the value of the
uncompensated ohmic drop, T| N . Further research has demonstrated that by
studying the polarization decay curves and their dependance with time, kinetic
and mass transport information can be obtained [23-30]. When several phenomena
are superimposed, interpretation of the current interruption decay curves is not
straightforward. For example, Newman [7,31] and other researchers [3,27,32] have
stressed the fact that upon current interruption, internal currents may not be
1 The migration potential,
<t>
can be calculated by solving the Nernst-Planck flux equations. Appendixes 1
and 2 describe how
these
equations can be solved.
A4>
can be neglected in the presence of a substantial
excess of supporting electrolyte.
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halted instantaneously. This will be the case if contained electrical double layers
are not uniformly charged \ The relaxation time required for the double layer to
equilibrate can be estimated from the following relationship 2 :
TD = . . . 1* K
The discharge of the double layer can also take place through Faradaic
reaction. The time constant for this reaction can be estimated using the following
equation:
T/
Ft. -
Z
The third major process that can take place upon current interruption is the
relaxation of concentration gradients. The time constant for this unsteady state
process can be estimated from the following relationship:
X " = D - 3
Upon disappearance of the external current, r\a will vanish almost
immediately (in < IO"2 msec) [3413. On the other hand, residual charge
arrangements in a supposedly uniform charged electrolyte may affect the rate at
which i] a vanishes.
Typical values for x f and x
R are close to 0.5 msec [7]. Thus, the electrical double
layer will be equilibrated and discharged in approximately 1 msec. After this time,
concentration gradients will relax. The larger the time constant the longer it will
take for the system to reach its open-circuit or equilibrium potential. Another
process that can take place upon current interruption is the dissolution and/or
re-precipitation of secondary products within the slimes layer. This process may
originate one or several potential arrests [351 *.
1 Double layers may not be uniformly charged if the difference in potential at the electrode/solution
interface is not uniform.
2 L
is the characteristic length
that
controls the current
distribution [20,33].
3 In
electrolytes of
uniform
composition, potential
drops upon
current interruption
have
time constants of
the order of 7x10* msec
.
4 These potential arrests are sometimes referred as Flade potentials.
[29]
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An estimation of the time required to relax the concentration gradients
present within the slimes layer can be done by using Eq. 3. Assuming that the
entire slimes layer can be considered as a "nearly" stagnant environment and that
D is of the order of 10"5 cm 2/sec, we get for a 0.1 cm slimes thickness a value of
V of 1000 sec. This shows that long relaxation times are required for the levellingof the concentration gradients present within the slimes layer. Thus, current
interruption for short periods of time (~ 0.3 sec) during the galvanostatic
dissolution of lead anodes was not expected to change the system appreciably.
The information provided by analyzing short interruptions was supplemented by
following, for selected cases, the decay in potential over extended periods of time
(several days).
The experimental set-up (using a Wenking potentiostat) allowed the decay
potential to be followed from about 0.5 msec after current was interrupted.
Oscilloscope traces at shorter times after current interruption showed that the
potential at time zero could be obtained by linear or exponential approximation
of the decay curves in the vicinity of the current interruption. These studies
resulted in a semi-quantitative picture of the transport processes taking place
within the lead anode slimes layer.
It has been shown [1,36-39] that analysis of the potential decay curves obtained
upon current interruption is more easily accomplished when the time-domain
data1 are transferred into the frequency-domain through the use of Fourier
transformations. Both analytical and numerical transformation of the
time-domain generated data can be performed. Matching transformations on
equivalent electrical circuits selected by trial and error must be found to interpret
the data in terms of electrical components such as resistors and capacitors. The
input and output signals are subjected to Fourier transformation (i.e. one-sided
Laplace transformation) and a transfer function is obtained. In the case of a
current interruption experiment, both potential, e(t) and current, i(t) are Fourier
transformed to obtain the system impedance from their ratio. Appendix 3 shows
how the response to current pulses of a simple RC circuit was analyzed in the
frequency domain. Numerical Fourier transformation of some of the current
interruption data generated in this work was carried out by using the Fast Fourier
1
The
time
domain data
refers
to the potential and current transients dependance
with time. When these
data are Fourier transformed, the frequency spectrum of the system is obtained (time-domain o
frequency-domain). For
a
more complete description of
this
transformation see appendix 3.
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Transform (FFT) algorithm [40-43]. The software packages used in this work
incorporate this algorithm1
. Knowledge of the FFT algorithm, its application range
and limitations, are required to understand the data generated. Appendix 3
reviews some of the relevant steps required for implementing the Fourier
transformation.
B. AC impedance techniques
The use of AC impedance to study electrochemical systems spans over a
period of more than 100 years. Kohlrausch, as early as 1854 [44] proposed its use
for the determination of the electrical conductivity of electrolytes. Later, with an
AC bridge, he obtained the conductivities of a large number of electrolytes [45],
Contributions by Warburg [46] and Randies [47] set the theoretical and
experimental foundations from which AC impedance measurements have evolved.Nowadays, the technique is routinely used for the analysis of a wide range of
electrochemical systems [48-54]. Electrical engineering theory has been heavily
used by electrochemists to interpret the results, and reviews on its characteristics,
advantages, limitations, and implementation can be found in the literature
[14.19,48.55-57].
In a typical AC impedance experiment, a small sinusoidal signal (either voltage
or current) is applied to an otherwise DC system. The AC output response is
followed as the frequency of the input signal is changed. The ratio of the input
and output signals is known as the transfer function [58,59]. Thus, the impedance,
Z(jco), is a transfer function. The system response can be assumed to be linear by
limiting the amplitude of the input signal to a few mV (or a few mA). The possibility
of obtaining an electrical analogue circuit by using AC impedance is a strong
driving force for its application. Knowledge of this analogue allows the modelling
and prediction of the output signal when the input is known.
The linkage between current interruption and AC impedance techniques
arises through the use of the transfer function. Theoretically, upon Fourier
transformation of the current interruption data, the transfer function obtained
should match the one obtained by AC impedance. That this connection exists is
shown in appendix 3. On the other hand, experimental artifacts make the analysis
of data generated in the time-domain valid only for a very limited frequency range
1
Asyst* version 2.10 and Asystant* version 1.02.
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[40,60-62]. In addition to this, the time span required for an AC measurement is
several orders of magnitude larger than the one required for a current Interruption
measurement.
AC impedance is the best technique available for the determination of i\a [64].
The resistance polarization value obtained using this technique should coincidewith the value obtained from current interruption experiments [65]. Kinetic and
mass transfer information can also be readily obtained using this technique
[53,66-68].On the other hand, most of the impedance theory developed so far has
been focused on the study of very dilute solutions [191 and solid electrolytes [16].
/ . Aims and limitations of the AC impedance studies
AC impedance studies are aimed at finding an electrical analogue that can
be used to study how different physico-chemical (and in some cases mechanical)parameters affect the response of a system given a certain input function. A
comprehensive analogue model ought to be able to incorporate as many
parameters as variables are in the system. It also should predict the output of
the system given the characteristics of the input function. Like any other
mathematical model, an analogue model is established using assumptions which
are based on prior knowledge of some of the fundamental properties of the system.
A wide variety of electrical analogue circuits can match the response of the system
but only a few could represent the physico-chemical process involved (withoutaccounting for different interpretations for the same circuit). An indication that
the model is appropriate is that the electrical parameters ought to change as the
physical variables are altered. Thus, the main limitationoi the impedance studies
is a function of how ambiguous are the parameters involved in the circuit. Other
limitations arise in unstable systems in which the parameters change faster than
the AC measurement, decreasing the frequency in which the AC spectra can be
accurately measured. The search of an electrical analogue that matches the
response of the system has also been extensively pursued using DC transient
techniques such as current interruption [8,69,64] and small amplitude cyclic
voltammetry (SACV) [70,71]. DC and AC studies have been used concurrently to
better characterize the system under study [15,72].
In the Betts process, AC impedance measurements were made across the
slimes layer at preset slimes thickness. Changes in the impedance values were
related to concentration gradients, precipitation of secondary products, and
dissolution of noble compounds. From these studies, kinetic parameters
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(exchange current densities, double layer capacities) were determined. Linearity
in the system response was assumed by limiting the amplitude of the input
sinewaves to very small values (less than 5 mV R.M.S. for potential controlled
experiments and less than 35 Amp/m 2
R.M.S. for current controlled
experiments). r\a values were obtained from both AC impedance measurements
and a current interruption routine built into the Solartron electrochemical
interface.
The search of electrical analogues that match the response of the system
and have physical meaning is strongly pursued in this thesis.
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Chapter 3: Exp eri men tal Procedure
I. E lec t rochemica l Experiments
A. Electrochemical cells
1. Beaker Cell
A I L wide mouth (0=11 cm) polyethylene 1 beaker was adapted to function
as electrochemical cell (Fig. 1). This cell was used to study the anodic behavior
of small working electrodes (i.e. working electrodes whose geometric area 2 was
between 1.4 and 3 cm2). A Lucite tight cover was used to hold the electrodes in
a fixed position and to avoid loss of water due to electrolyte evaporation. The
cover had holes through which the electrical connections to the electrodes and
the tubes used for electrolyte sampling were taken out of the cell. Previous toeach experiment the cell was thoroughly washed several times using deionized
water. After the cell was dried, the electrodes 3 were firmly positioned In the cell.
Then the electrolyte was introduced to the cell through a plastic tube. During
the experiment, this tube was used to obtain bulk electrolyte samples. Electrolyte
was added only at the beginning of the experiment in volumes that ranged
between 320 and 400 ml. Mixing of the bulk electrolyte was by either magnetic
stirring4
or recirculation5,6. Electrolyte recirculation provided the best
experimental reproducibility and was preferred to magnetic stirring. After theelectrodes were positioned and the electrolyte introduced, the cell was covered
and sealed using generous amounts of silicone rubber 7 . Only after the silicone
rubber had "dried to touch" (i.e. after approximately 2 Hrs.) was the cell immersed
in the constant temperature water bath.
1 H
2
SiF
6
containing solutions
can
etch
glass [1]. Thus,
glass laboratory
ware was
avoided
as
much
as
possible.
2 Geometric areas do not consider
surface
rugosities.
3 Electrodes were
immersed in a 10% Vol HN0
3
solution and washed with deionized water prior to
their
introduction to the cell.
4 Magnetic
stirrer bar
coated
with Teflon* (1=2.54 cm, 0=0.95 cm)
spinning
at low speed by using a
Tek-stir magnetic
stirrer
model S8250-1.
5 As
illustrated
in Fig. 1 electrolyte was recirculated through plastic
tubes
located at opposite sides of the
cell
and at different
electrolyte
depths.
6 Electrolyte
recirculation
rates
were
between 5 and 6 ml/min. A
Masterflex pump
catalogue No.
7553-20
and Masterflex Tygon tubing
catalogue
No. 6409-14
were
used.
7 100%
silicone rubber,
RTV
Silastic
732.
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Rectangular cell
F i g . 1 Assembly used in the
experiments performed with
the beaker cell
(A ) W o r k i n g electrode
(B) Coun ter electrode
(C ) Refere nce electrode
(D) C u wire
(E) Plastic tube used to
wi th dr aw bu lk
electrolyte samples
(F) Electrolyte
recirculation inlet
(G) Electrolyte
recirculation outlet.
2. Rectangular cell
Rectangular Lucite cells were used to study the electrochemical behavior
of working electrodes whose exposed geometrical area ranged between 30 and
40 cm2. Fig. 2 describes the dimensions and design features of this kind of cell \
A Lucite tight lid was used to cover the cell. In this lid, holes of appropriate size
were drilled. Through these holes, the electrical connections to the electrodes
and the tubes used for electrolyte sampling were taken out of the cell. Bulk
electrolyte was recirculated2 using lateral cell inlet and outlet facilities.
1 Cell dimensions were modified according to the characteristics of the experiment. Fig. 2
shows
the
actual
dimensions of the electrochemical cell employed in experiment LC2 to be discussed extensively in
Chapter 4.
2 Electrolyte
recirculation
rates
were
between
5 and 6 ml/min.
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Rectangular cell
Electrolyte from the top of the cell was brought to the bottom continuously.
During the electrolysis of lead electrodes, electrolyte samples taken at different
cell locations showed no significant concentration differences.
Fig. 2 Assembly used in the
experiments performed
using the rectangular cell
(A) Working electrode
(B) Counter electrode
(C) Reference electrode
(D) Cu wire
(E) Plastic tube used to
withdraw bulk
electrolyte samples
(F) Electrolyte
recirculation inlet
(G) Electrolyte
recirculation outlet
(H) Lucite side walls
(I) Lucite strips.
Drawings are not scaled.
All measurements in mm.
(A) Cell tridimensional view
(B) Lateral view(C) Frontal view.
Fig. 2 shows the position of the two small Lucite strips used to support the
anode. The presence of these strips allowed the electrolyte entrapped within the
slimes layer to flow downwards. Additionally, wide Lucite strips were used to
center the anode in the cell and to improve current distribution. Electrolyte
volumes were fixed at the begirining of the experiment and ranged between 300
and 400 ml. When bulk electrolyte samples were withdrawn from the cell,electrolyte of the initial composition was added to the cell to maintain constant
the electrolyte volume. As in the beaker cell case, after the electrodes were placed
and electrolyte added, the lid was fitted to the cell and sealed afterwards using
silicone rubber. Again, only after the silicone rubber had dried to touch, was the
cell immersed in the constant temperature bath.
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Materials
B. Electrodes
1. Working electrodes
(a) Materials
Working electrodes were produced from pure lead1
and from typical leadbullion anodes provided by Cominco Ltd. Typical anodes used in the Betts
electrorefining process were taken from the anode refining wheel and some of
its sections were sent to UBC. No particular details regarding the anodes cooling
rates or about the pyrometallurgical steps previous to their casting were
provided. According to Cominco, the anodes were cast under normal operating
conditions and that's where the term "typical" comes from. The electrochemical
studies here pursued required the anodes to form strong and adherent slimes.
As was later found, this set of anodes did produce slimes which did not fall fromthe electrodes and remained attached to them during the dissolution stages.
Anode "A"
A.1
Anode "B"
B.1 B.2
Fig. 3 Sections of the lead
bullion anodes used to prepare
working electrodes.
Fig. 3 shows the anode sections that were cut from the anodes and sentto UBC. The anode A strip was sent in July 1986. This strip was cut in smaller
pieces and from the area indicated in Fig. 3A, electrodes were produced. Sections
of anode B were sent in March 1987. From anode B, electrodes were produced
mainly from its center part (section B - l shown in Fig. 3B). The working
electrodes were prepared so as to study the electrochemical behavior of the
1 Pure
lead working electrodes
were
made out of Tadanac ingots (Pb >99.99%)
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Pure lead
working
electrodes
"mould" cooled anode face and the "air" cooled anode face. Thus, electrodes
were fabricated by cutting perpendicular sections of the anodes. Depending on
the type of electrochemical cell employed, different electrode sizes were used.
(b) Beaker cell
Small working electrodes whose exposed geometric area ranged between
1.4 and 3 cm2
were prepared as follows:
(I) Pure lead working electrodes
Electrodes were made by cutting small pieces of the pure lead ingot. These
pieces were machined to form rectangular shaped electrodes. The machining
marks on the electrode sides were removed by pohshing the electrode using
the 600 grit. The electrical contact was incorporated by soldering a Cu wire to
the back of the electrode. Subsequently, the electrode was encapsulated with
epoxy resin1
exposing the electrode surface by polishing away the unwanted
resin. The exposed surface was polished again using the 600 grit.
(II) Lead bullion working electrodes
Rectangular shaped electrodes were cut and machined out of the anode
sections previously described. The electrode face to be exposed directly to the
electrolyte (either the air cooled or the steel cooled face), was not machined.
The deformed layer produced by the cutting and machining operations was
removed by polishing2
. Electrical contact was made by either soldering a Cu
wire to the back of the electrode or by pressure contact. The later technique
was preferred due to the fact that the soldering process through heating, may
affect the phases originally present in the anode. The pressure contact electrical
connection consisted of manually pressing a bundle of the Cu wire strands to
the back of the electrode and using acrylic tape to sustain the contact. The
electrode was then mounted in epoxy resin and the unwanted resin was
polished away. The exposed electrode surface was polished with the 600 grit.
(c) Rectangular cell
Medium size electrodes whose exposed geometric area ranged between 30 and
40 cm 2 were prepared as follows:
1 Acrylic plastic
resin:
Quick
mount®
self-setting
resin
2
The deformed
layer
was measured metallographically and in some cases was as thick as
0.1
mm.
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Relgrence electrodes
(I) Pure lead working electrodes
Electrodes were cut and machined out of pure lead ingots as in the beaker
cell case. The machining marks were removed by polishing using the 600 grit.
Electrical contact was made by using two Cu rods. These were screwed-in to
the top of the electrode. One of these rods was used to carry current and the
other to measure the electrode potential. The lateral electrode surfaces were
covered with silicone rubber. The bottom, top, and one of the electrode faces
were not coated with silicone rubber. Thus, only the bottom of the electrode
and one of its sides were directly exposed to the electrolyte.
(//) Lead bullion working electrodes
Electrodes were prepared out of the anode sections in the same way pure
lead anodes were produced. To sample the electrolyte present in the slimeslayer, and to follow the inner slimes electrode potentials, holes were drilled in
top of the anode \ Though these holes, plastic tubes were inserted. These tubes
were used either to extract small amounts of electrolyte 2 or to insulate the
pure lead wires. Additionally, one of these holes was used to insert a Pt wire.
This wire was used to study the electrical conductivity of the slimes layer3.
The Pt wire did not have any insulation and during some experiments it was
moved to other locations where slimes were present.
2. Reference electrodes
A pure lead wire 4
was used to measure the difference in potential between
the bulk electrolyte and the working electrode. This wire was mounted in a plastic
tube (0=2 mm). The tube was bent at one end, and a plastic t ip 5
was inserted
there. Fig. 4 illustrates this Luggin-Haber reference electrode arrangement. The
reference electrode tip was placed between 2 and 5 mm away from the original
position of the working electrode. This distance remained constant during the
1
Size and location of the holes is provided
when
specific experiments in which electrolyte samples were
taken and inner potentials obtained are analyzed.
2 From the inner slimes layer 100 u l of electrolyte were
slowly
extracted (over a period of 3-6 hours). A
100
u.L Unimetrics
removable needle
syringe was used to
withdraw
the inner electrolyte samples.
3 The difference in potential between this wire and the anode was used as an indication of the slimes
layer conductivity.
4 Johnson Matthey,
99.95% Pb, 0=1.0 mm.
5 Eppendorf pipette tips 5-100
|xL
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Counter electrode
experiment. The tip of the reference electrode was located facing the geometrical
center of the working electrode. After the cell was assembled and the bulk
electrolyte had penetrated the reference electrode compartment, this was sealed
by using silicone rubber. Thus, the electrolyte surrounding the lead wire had the
same composition, temperature, and atmosphere as the bulk electrolyte.
A
/
Fig. 4 Detail of the Luggin-Haber
reference electrode arrangement
B
/ (A) Pure lead wire
/ (B) Plastic Tube
/ (C) Eppendorf plastic tip.
0!=1.3±O.2 mm, 02=0.5+0.2 mm
c
1 ..« ,*«H J
In the experiments1 in which the difference in potential between the
electrolyte solution present in the slimes layer 2
and the working electrode was
followed, pure lead wires were also used. Plastic tubes (0=1.3 mm) were inserted
in the previously drilled holes up to @2 mm away from the bottom of the holes.
Then, the lead wires were inserted up to @2mm away from the lower end of the
plastic tubes. This particular set-up was chosen to avoid prohibitive corrosion
of the lead wires during the dissolution of the working electrode.
3. Counter electrode
Pure lead foils 3
were used as counter electrodes. In the beaker cell case the
lead foil surrounded the working electrode 4
whereas in the experiments which
1
This sort of experiments were only carried out
using
the rectangular electrochemical cell
in
which the
size
of the working electrode was large
enough
to incorporate these potential measuring probes.
2 The electrolyte entrapped within the slimes layer will be called slimes electrolyte .
3 Lead foils whose lead content was greater than
99.95%
4 The working electrode was concentrically placed
with
respect to the counter electrode.
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Electrolyte
used the rectangular cell, the lead foil was located facing the working electrode
and @42 mm away from it. In both cases, the area of the counter electrode was
larger than the area of the working electrode. This was done to improve the
current distribution and also to avoid formation of dendrites. When the beaker
cell set-up was used, the geometric area of the counter electrode was between10 and 15 times larger than the exposed geometric area of the working electrode.
In the experiments performed using the rectangular cell, counter electrode areas
were between 1.2 and 1.4 times larger than their working electrodes counterpart.
In all the experiments, electrical contact was incorporated by soldering a Cu wire
to the counter electrode. The place where the electrical contact was made was
isolated from the electrolyte by a silicone rubber coating.
C. Electrolyte
Technical H 2 S iF 6
1 obtained as a by-product of the treatment of phosphate
rock [2] 2 was neutralized with either PbC0 3 or PbO to prepare mother electrolyte
solutions. The reactions that take place upon neutralization of the acid are:
H 2 SiF
6 +PbC0
3 => PbSiF
6 + H 20+C02 . . .1
H 2 SiF 6 + PbO => PbSiF
6 + H
20 ...2
Upon neutralization, several insoluble compounds precipitate3. These
precipitates were removed by filtering using Whatman paper #40. After this
operation, a nearly transparent electrolyte solution is obtained. Depending on the
acid strength of this solution, Si0 2 nH 2 0 colloidal particles may be observed 13-7] *.
1
Acid
composition: 2.03 M H2
SiF
6
and 0.40 M Si0
2
. This acid was provided by Cominco Ltd.
2 Phosphate rock is treated
with H
2
S0
4
to produce HF which is later contacted
with
Si0
2
to
produce
H
2
SiF
6
.
The sequence of reactions that take place is [2]:
Ca]0(PO4)sF2+10//2SO4+20//2O => \0CaSOt -2H 2O+6H^POi +2HF ...a
6HF + Si02 => H 2 SiF 6 + 2H 20 ...b
H£iF 6 => SiF A T +2HF ...c
3SiF t + 2H 20 => H 2 SiF 6 + Si02 ...d
3 Depending on the extent of the neutralization of the acid, a mixture of lead oxides and fluorides in
addition to silica compounds can precipitate.
4 The presence of colloidal Si0
2
nH
2
0
in
this system has been reported
in
the literature
[3-7].
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Electrolyte
A titrimetic analysis routine was set up to analyze the electrolyte for H 2SiF 6,
Si0 2, HF, and PbSiF6. In this routine, Pb is analyzed via complexometric titration
with EDTA [8,9]. H 2SiF 6 , Si0 2, and HF (if present) are determined by titration with
LiOH. Details of this procedure are provided in Appendix 4.
To obtain electrolytes of the set compositions, the mother electrolyte solutionswere diluted with H 2SiF 6 and deionized water \
Additives were added to the electrolyte in selected experiments. The additives
used were aloes 2 and calcium lignin sulphonate 3 . These additives were added
from a mother solution at the beginning of the experiment. Precipitates formed
due to the additives addition were removed by filtering the electrolyte prior to its
introduction to the electrochemical cell.
The electrolyte samples withdrawn from the slimes layer were analyzed by
using Atomic Absorption Spectroscopy (AAS)4 and specific ion electroanalytical
techniques. In these samples, Pb was determined via AAS using the absorption
line at 283.3 nm.
In the Sb determination via AAS, matrix effects can affect the analysis [ioi.
Thus, Sb standard solutions were prepared by adding known amounts of PbSiF6
and H 2 SiF 6 so as to match the lead and acid content of the samples. The absorption
line at 231.1 nm was used to determine Sb via AAS. The total Si content5 of the
entrapped electrolyte was also determined via AAS. Si was determined using the
absorption line at 251.6 nm and a nitrous oxide - acetylene flame. Three different
standard solutions were prepared as follows:
a) From a 1000 ppm Si solution prepared by dissolving 5.056 g of Na
metasilicate (Na2Si0 3.9 HaO) in @300 ml of deionized water, adding sufficient HCl
to bring the pH to about 5 and diluting up to 500 ml using deionized water.
b) From H 2 SiF 6 technical solutions diluted so as to obtain standard solutions
with less than 1000 ppm Si.
c) From PbSiF 6-H 2SiF 6 solutions diluted so as to obtain standard solutions
with less than 1000 ppm Si.
1 Deionized water with electrical conductivity lower than 12 u.mhos/cm.
2 Resin from the leaves of certain species of aloes plant native to South Africa.
3 Organic additive obtained as a
by-product from
wood pulping operations
4 Perkin Elmer Atomic absorption Spectrophotometer model 303.
5 The total Si content corresponds to the total amount of Si present in the electrolyte as H
2
SiF
6
, PbSiF
6
,
and Si0
2
.
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Wenking potentiostat
The total amount of fluorine ions present In the entrapped electrolyte was
estimated by using an ion sensitive electrode \ Standards were prepared by using
NaF solutions. Additionally, calibration curves were obtained using H 2SiF 6 and
PbSiF6-H 2SiF6 standards. Gran's plot and standard addition techniques were
incorporated in these measurements [11,12].
D. Temperature control
The electrochemical cells previously described were immersed in a constant
temperature water bath. This bath had a volume of @9 L and was covered with
styroform. The bath was stirred using magnetic bars and/or air sparging. The
bath temperature was controlled by using a YSI model 71 temperature controller.
A thermistor probe 2 was used to monitor the bath temperature and immersion
heaters
3
were used to maintain it. This experimental set-up allowed temperaturecontrol within ±1.5 °C of the set point.
After the bath had reached the set temperature, the assembled
electrochemical cell was introduced and at least 3 hours were allowed for the cell
to reach thermal equilibrium before the electrochemical experiment began.
E. Instrumentation
The electrochemical instrumentation involved the use of a variety of electronic
equipment. A Wenking potentiostat was used in the first half of this work and a
Solartron Electrochemical Interface together with a Solartron Frequency response
analyzer were used in the second half. The improvements in the electronics of the
Solartron devices enable complex experiments to be performed. What follows is
a description of the different electrochemical arrangements used. Also, computer
control of the electrochemical experiments and data acquisition will be explained.
/. Wenking potentiostat
A Wenking potentiostat model 70 HV1/90 was connected to an IBM XT
personal computer. A Data Translation board DT 2805 was installed in one of
the computer slots. This card allowed the computer to interact with the
potentiostat and with the electrochemical cell. A Data Translation DT707 screw
1
An Orion Fluoride electrode model
94-09
together with a double junction reference electrode Orion
model 90-02 filled
with
1M NaN0
3
in the outer chamber was used in
these
measurements.
2 Thermistor probe YSI model 402.
3 Vycor* immersion
heaters
with
100
to
500W
of
power.
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Wenking potentiostat
terminal panel was used to address the DT2805 board. This panel acts as an
extension of the DT2805 card and simplifies the process of computer interfacing.
All the electrical connections are done in the DT707 terminal panel which sends
and receives information from and to the DT2805 board. Besides from this
terminal panel, a control panel was also used to link the potentiostat, computer,
and electrochemical cell connections.
OSCILLOSCOPE
DT 707
SCREW TERMINAL
PANEL
ELECTROCHEMICAL
CELL
CONTROL
PANEL
IBM XT COMPUTER
A S Y S T / A S Y S T A N T S O F T W A R E
DT 2805 BOARD
Fig. 5 Experimental set-up
using the Wenking
potentiostat
WENKING
POTENTIOSTAT
Fig. 5 shows a simplified view as to how the computer interfacing process
was carried out. The flow of digital and analogue data from the computer to the
system under study and vice versa was controlled by using specialized software \
A number of programs were written to control the digital and analogue operations
performed by the DT2805 board . The complexity of these programs varied
depending on the characteristics of the experiment to be conducted. A storage
oscilloscope 2 was used to test the performance of these programs and to follow
the response of the system when required. A digital voltmeter 3
was also used to
check the computer measurements.
1 Asystant® menu-driven software version 1.02, and
Asyst®
command-driven software version 2.10.
2 Tektronix analogue oscilloscope model 5115
3 Beckman
31/2 digit multimeter model TECH 300.
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Wenking potentiostat
Fig. 6 Detail of the connections required to Interrupt the current and to follow the cell response
(A) Three terminal cell
(B) Four terminal cell
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Wenking potentiostat
The Wenking potentiostat was used either as a potentiostat or as a
galvanostat. Potentiostatic operation was converted to galvanostatic by
connecting a resistor of suitable power rating between the working and reference
electrode terminals \ It was in this configuration that most of the experiments
were conducted. During galvanostatic operation, current was interrupted byshort-circuiting the internal battery that controls the flow of current to the cell.
This was done by using a mercury wetted relay2 activated by the computer at
preset times. The connections required to interrupt the current and to follow
the cell response are shown in Fig. 6. As Fig. 6 shows, the current going through
the cell and the difference in potential between the working and the reference
electrodes were continuously logged. Figs. 6A and 6B differ only in the number
of connections made to the working electrode. When two connections are used
(Fig 6B), one of them is used for conveying current and the other is used for
measuring potential. Experiments in which large amounts of current are involved
or when contact resistances are to be avoided call for this particular electrode
arrangement.
Calibration of the routines used for interrupting the current was done by
using "dummy" cell electrical circuits. Appendix 5 provides a description of these
measurements. The algorithm used in the computer programs to perform the
current interruptions is also described in Appendix 5. By doing these calibration
runs, it was found that the Wenking potentiostat halts the flow of current to the
cell almost immediately (within 10 (isec3). On the other hand, after the short
circuit was opened it was found that the current did not immediately recover its
previous value. The rise time of this process was of the order of milliseconds and
was dependent of the current going through the cell (e.g. see Fig. 7). In addition,
due to hardware and software limitations, the data acquisition system was able
to follow the decay in potential only within 1 msec after current interruption4.
Subsequent points were sampled at various acquisition rates. This resulted in
the current interruption routine being able to resolve decays whose time
1 As shown in Fig. 6, the reference terminal of the Wenking potentiostat is no longer used for connecting
the reference electrode of the electrochemical cell.
2 Mercury wetted relay Elect-trol model 31511051.
3 Value obtained
from
oscilloscope readings.
4 Depending on the complexity of the data acquisition program, the interval of time
between
current
interruption and the first set of data points sampled was
between
0.14 msec and 1.0 msec. Oscilloscope
readings
were
used to verify these measurements.
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Generalities:
constants were larger than 10 msec. As the time constant of the diffusion
processes under study were several orders of magnitude larger than 10 msec,
this instrumental constraint was not critical.
158 i —
Fig. 7 Current stepresulting from halting theflow of current to the
electrochemical cell usingthe Wenking potentiostat
Current was halted within
-10 usee.
.88 68. 128 188
Tine , usee
248 380
2. Solartron electrochemical interface and frequency response analyzer
(a) Generalities:
A 1286 Solartron Electrochemical Interface (SEI) and a 1250 Frequency
Response Analyzer (FRA) were connected to an IBM XT personal computer by
using the IEEE-488 interface built into these instruments. To link theseinstruments to the computer, an IEEE compatible board 1
was installed in one
of its slots. This board enables the computer to act as a "controller" of the flow
of information. All the electrical connections of the electrochemical cell were
attached directly to the SEI front panel.
1 Scientific Solutions
IEEE
488 LM
board.
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Description ol the experimental set-up
Prograinining of the IEEE-488 interface is simpler than programming of
the Data translation interface previously described. All analogue to digital
conversions are done directly in the IEEE apparatus. This results in all the data
communications being digital. IEEE instruments are controlled by a series of
pre-established commands. As these commands are executed they sendinformation to the control device (in this case the computer). This information
instructs the control device with respect to the status of the instrument.
The SEI is a complex and powerful instrument that can be used as a stand
alone electrochemical device. In its simplest configuration, it can function either
as a potentiostat or as a galvanostat. By pressing a button In its control panel
(or by sending a command from the computer) switching between these two
operating modes can easily be incorporated. Among other features, the SEI
offers several ways of obtaining the value of the uncompensated ohmic dropIRg \ Among these, the SEI has built in a procedure ("sampled'TR,, compensation)
which allows the potential to be sampled a few microseconds before and after
current is interrupted. However, this IR, drop compensation routine operates
only under potentiostatic conditions. When galvanostatic experiments are
performed, this limitation can be circumvented by switching from galvanostatic
to potentiostatic operation, mteirupting the current, and switching back to
galvanostatic control. Under computer control the whole process takes @5 sec.
The FRA is also a complex instrument. One of its fundamental functions
is to generate waveforms. Sinusoidal, triangular, or square waves in the
frequency range between 10 |iHz and 65.5 Khz can be generated by this device.
The frequency and amplitude of these waveforms can easily be modified. Another
of the FRA main functions is to analyze the gain and phase characteristics of
sinusoidal waveforms. The system transfer function can be obtained by
analyzing concurrently two sinusoidal signals. This is obtained by using two
channels for simultaneous measurements at any two points in the system.
(b) Description of the experimental set-up
Both DC and AC experiments were carried out using the SEI together with
the FRA. Fig. 8 describes the electrical connections that were used in these
experiments. Furthermore, Appendix 6 illustrates the implementation of the
1 This uncompensated ohmic drop
IR
S
is
equivalent
to i i
n
described in
Chapter
2.
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Description of the experimental set-up
technique using durnmy circuits to simulate the response of electrochemical
cells. The experimental data generated using these dummy circuits was used
to calibrate the electrochemical set-up.
WE
RE1
RE2
CE
SolartronElectrochemical
Interface
Working
Fig. 8 Connections from the
electrochemical cell to the Solartron
Electrochemical Interface.
The dissolution of pure lead and of typical lead anodes was studied using
the beaker electrochemical cell previously described. In these studies, all leads
were shielded to avoid pick-up noise. During the galvanostatic and potentiostatic
experiments current was interrupted using the "sampled" IRg compensation
routine built into the SEI. In these measurements, only the value of the
uncompensated resistance was obtained .
The AC impedance of pure lead and of typical lead anodes was measured
under a variety of experimental conditions. Impedance measurements were
made under rest potential conditions, in the absence and in the presence of a
slimes layer, and in the absence and presence of DC current1
. All the AC
impedance spectra were obtained using sampling rates and integration times
set to 200 cycles per frequency followed by a 5 sec break 2
. The SEI bandwidth
1 The difference in potential between the working and reference electrodes and the current flowing to the
cell are continuously recorded by the SEI. During the AC experiments, the SEI removes the DC component
(if present) from the potential and current waveforms before
sending them to
the FRA. The fundamental
frequency of these waveforms was
used
to obtain the impedance of the system.
2 Integration times were reduced by
using
the auto-integration routine built-in the FRA.
A long
integration
cycle
was
chosen
(1%
error
with
90% confidence)
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pH measurements
type chosen for experiments under potentiostatic control was the type E
(maximum bandwidth 24kHz) and for experiments under galvanostatic control
the type B (bandwidth >100KHz).
n. E lect ro lyte Phys ico -Chemica l Properties
A . pH measurements
The pH of H 2SiF 6 containing solutions was measured using a liquid
membrane pH electrode1 and a double junction reference electrode2.
Measurements were made at room temperature. In these measurements, no
attempts were made to correct for liquid junction potentials. pH measurements
were also performed using pH sensitive paper.
B. Electrical conductivity
The electrical conductivity of H2SiF 6-PbSiF 6 electrolytes was measured using
a Radiometer conductivity meter model CDM2 and a YSI conductivity cell 3 model
3417. A Digital voltmeter was connected to the conductivity meter to assist in the
reading of the conductivity measurements. The electrical conductivity of individual
solutions4
was measured by immersing the samples in a water bath whose
temperature was controlled to within 0.2°C. Two thermocouples 5 were used to
monitor the bath temperature at different positions in the water bath. The
thermocouples were connected to a computer which was used as a temperature
control device. Temperature was maintained within the set limits by using heating
elements controlled by the computer via solid state relays (SSR)6.
C. Kinematic viscosity
Kinematic viscosity of H2SiF 6-PbSiF 6 solutions was measured using
Cannon-Fenske routine viscometers for transparent liquids. The viscometers were
1
Orion pH electrode model
93-01.
2
Orion double junction reference electrode model
90-02.
The inner filling solution used
in this
electrode
(Orion
90-00-02)
matches the characteristics of the standard KCI calomel electrode. In the outer chamber
a
1
M NaN0
3
solution was used.
3 Cell constant =
1.05.
Cell constant was obtained by measuring the electrical conductivity of a NaCl
saturated solution and of a
0.100
M KCI solution at different temperatures.
4 Sample
volumes vary
between 25
and
30
ml.
5 Thermocouples type E Chromel(+)-Constantan(-). A 0 "C reference junction was used during the
temperature measurements.
6 Omega
DC
controlled solid
state relay
(SSR)
model
SSR
240 D10.
Control
voltage
2-32
Voc.
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Density
immersed in a constant temperature water bath 1 and calibrated using deionized
water. Temperature was controlled within 0.2°C. From 5 to 12 viscosity
measurements2 were made until average flow times3 readings were within 0.20
sec.
D. Density
The density of H2SiF 6-PbSiF 6 solutions was measured using 25 mL
Binghmam-Type pycnometers. Pycnometers were calibrated by measuring the
density of deionized water. Measurements were made at room temperature. From
3 to 5 density measurements were made until the obtained mean density values
were within 0.2%.
1 Cylindrical
water
bath
with
@2L volume.
2 Viscosity measurements were made using 10 ml sample volumes.
3 Flow
times measured by using a digital stopwatch.
Parallax
errors
were
diminished by using a
magnifying glass.
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Introduction
Chapter 4 E lectroref ining of Lead in a Small -Scale E lectrorefining
Cel l : A Case Study
I. Introduct ion
During the electrorefining of lead by the Betts electrorefining process, ionic
concentration gradients become established within the slimes layer due to the
restriction that the slimes layer presents to the movement of ions. Additionally,
the presence of the electric field created by the passage of current contributes to
the formation of concentration gradients within this layer.
The Betts electrorefining process is carried out under nearly galvanostatic
conditions. Under these conditions, a constant flux of lead ions is generated at
the anode/slimes interface. These ions are driven towards the cathode by
diffusion, migration, and convection. In the absence of a slimes layer, mixing of
electrolytes removes the migration and diffusion restrictions. Thus, if pure lead
were to be dissolved, the concentration gradients generated would be present only
within 100 to 1000 |im of the anode/bulk electrolyte interface where mixing
disappears. On the other hand, during the refining of lead bullion, the slimes
layer generates an environment in which ionic concentration gradients can be
present over distances larger than 10 mm. These concentration gradients
contribute to gradients in electrode potential applied to slimes filaments, and
thereby affect the dissolution of noble impurities. Primarily, this gradient arises
due to the larger concentration of lead ions in the vicinity of the anode interface
with respect to their concentration in the bulk electrolyte. Lead ions move out of
the slimes layer mainly by diffusion and migration. As lead ions are released, their
positive charge must be neutralized to achieve local electroneutrality. This leads
to reverse movement of SiF 6
2
and repulsion of H+ from the anode/slimes interface.
Electrical neutrality has to be observed throughout the slimes layer and this
dictates a strong relationship between the SiF6"
2
, H \ and Pb
+ 2
concentrationgradients.
Given the previous relationships, the slimes layer will have a large average
concentration of Pb + 2
and SiF6"2
ions and a small average concentration of H+
ions
with respect to their bulk electrolyte values. Moreover, there will be gradients of
their concentrations throughout the slimes depending on the slimes thickness
and the current density. The interrelationship between the diffusion coefficients,
ionic mobilities, and activity coefficients of the above ions plays a significant role
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Introduction
in the shape of these concentration gradients. In addition, migration will promote
the movement of anions from the bulk electrolyte towards the anode interface and
would act in the opposite direction for cations. Diffusion will carry Pb + 2
out of the
slimes layer and SiF6"2 and FT towards it. Convection will affect the movement of
all the ions and depending on the electrode size, it can affect significantly theshape of the concentration gradients.
The establishment of ionic concentration gradients within the slimes layer
can shift secondary equilibria that can lead to both new solutes and precipitates
such as fluoride and oxyfluoride ions, hydrated silica, and lead fluorides. The
precipitates can create further hindrances (in addition of those caused by anode
impurity filaments) to the convection of electrolyte and to the diffusion and
migration of solutes.
The stability of the slimes layer and its adherence to the anode surface is a
complex function of the lead anode physical metallurgy and of the electrolysis
conditions. Upon passage of current, lead is selectively removed from the bullion
and a metallic structure of noble impurities is left behind. This structure is
established as a "slimes layer". The success of the Betts process relies on this
structure remaining unreacted during the electrorefining cycle. By limiting the
potential difference across the slimes layer to less than 200 mV, this condition is
practically fulfilled. On the other hand, the complex chemistry of this layer and
the presence of ionic concentration gradients within it, can affect the extent to
which this potential gradient can be observed. If, due to the presence of large
concentration gradients, secondary products precipitate, the slimes layer can
detach. Under these conditions, the noble compounds harbored in the slimes
layer may no longer remain unreacted, and depending on shifts in electrode
potentials, noble impurities can be transferred to the electrolyte, then to the
cathode.
In this chapter, a study of the dissolution of a typical lead anode under
galvanostatic conditions is presented. The establishment of concentration
gradients within the slimes layer, the precipitation of noble compounds, and the
physical metallurgy of the lead anode and the produced slimes layer are reviewed
in this case study. Current interruption techniques were used to find the link
between the above mentioned phenomena.
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n. Presentat ion of Res ults
Presentation
ol Results
The study of the dissolution of a typical lead anode under galvanostatic
conditions was done by using the rectangular electrochemical cell described In
Fig. 3.2.
42
^
Inner
Electrolyte Sam pling Well
Potential Sensing Probe »// maaauremontm In mm
Fig. 1 Lead anode top view.
The location of the holes in which the inner reference electrodes A, B. and C
were inserted, isindicated In this diagram. The position of the well used to extract electrolyte samples from
the slimes electrolyte is also shown.
Fig. 1 is a top view of the lead anode showing the locations where the potentia
sensing probes were incorporated as well as the point from which electrolyte
samples from the inner slimes electrolyte were withdrawn. Particular
characteristics of this experiment are provided in Table 1. Current interruption
at preset slimes thickness was implemented by using the circuit described in
Figs. 3.5 and 3.6.
Experiment LC2 was carried out in three stages which are described in the
following paragraphs:
Stage I: Galvanostatic dissolution during 300 Hrs. During this stage, current
flow to the cell was halted for 138 msec every 3 Hrs.
Stage II : Long current interruption for 190 Hrs
Stage HI: Further galvanostatic dissolution for 30 min., during which current
was halted every 10 min for 1.8 sec.
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Presentation of Results
During these stages, electrolyte samples were taken from the bulk electrolyte
and from the inner slimes solution at preset times. After the completion of stage
m, current was halted and the corroded anode was left standing in the cell for
48 Hrs. After this time, the electrolyte was slowly drained and a dilute H 2SiF 6
solution (pH =1.5) was used to displace the concentrated solution in the slimeslayer. Afterwards, the corroded anode was removed and dried in a vacuum oven
at low temperature (t < 40°C). Slimes samples for observation in the SEM were
prepared by using a low viscosity resin and a vacuum imbibition technique 11,21
2 . Unsupported samples of the same slimes were analyzed by X-ray diffraction.
Table 1 Characteristics of Experiment LC2
Anode Composition
(Anode A Fig. 3.6)
0.01% Sn, 0.02% Cu, 0.14% Bi , 0.25% As, 1.12% Sb, 81 oz/ton Ag.
(Air cooled face exposed to the electrolyte)
Anode Dimensions 69 mm1,42 mm", 29.5 mm'
Cathode Pure lead foil >99.95% Pb
75 mm1,62 mm*, 2 mm'
Reference Electrodes: Pure lead rods, >99.95% Pb:
Outer: Located in the bulk electrolyte, =5mm away from the
slimes/electrolyte interface
Inner A, B, and C: Located within the lead anode at =3,6, and 8.5 mm
away from the slimes/electrolyte interface.
Initial Bulk Electrolyte Composition [H2SiFs] = 0.74 M, [PbSiF6] = 0.28 M, [SiOJ = 0.12 M
Additives Concentration =2 g/1 aloes, =4 g/1 of lignin sulphonate (added only at the beginning of the
experiment)
Electrolyte Volume 320 ml
Electrolyte Temperature 40±1.5' C
Electrolyte Recirculation Rate 6 m 1/min
Current Density during stages I and
III (from geometric surface area)
139 Amp/m2
Stage I Current Interruption Length
and Frequency
138 msec every 3 Hrs
Stage m Current Interruption
Length and Frequency
1800 msec every 10 min
Instrumentation Wenking potentiostat-DT 2805 Data acquisition Board-IBM XT computer
1 Spurr low-viscosity embedding
media, 11 = 60 cP
2 In the imbibition technique, penetration of the epoxy resin is encouraged by extracting the air
within
the
sample using
a
pressure difference
(i.e. a
vacuum).
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Stage
I
A. Anodic overpotential measurements
1. Stage I
The anodic overpotential response of electrode LC2 under galvanostatic
conditions is shown in Fig. 2. As can be seen from all four reference electrodes,
T)A increases quasi linearly as the slimes layer thickens. At the same time, r)A
measured by the inner reference electrode A shows that steep excursions in the
potential of the solution inside the slimes layer also can be present, although,
they do not seem to affect the r\ A value measured by the outer reference electrode.
The relatively uniform slope of these measurements suggests that the same
reaction and the same ions are being seen by all the reference electrodes \
The anodic overpotential response of the outer and inner reference
electrodes to current interruptions are shown In Fig. 3. Details of the r|A behavior
of the inner reference electrodes A, B, and C are provided in Figs. 4 and 5. In
these figures current interruptions 43 to 47 are not shown as oscilloscope
readings taken during that time interval required detachment of the current
interruption triggering device on the data acquisition board 2
.
1
Measurements on the slimes electrical conductivity were made by following the difference in potential
between a bare Pt wire inserted
in
the slimes layer
and
the lead anode. In these measurements,
it
was
found that the difference in potential between the Pt wire and the lead anode was negligible. The Pt wire
and the anode appeared to be short-circuited indicating the high electrical conductivity of the slimes
filaments.
2 Oscilloscope readings
were
consistent
with
computer measurements.
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Stage/
3X
180
160
140
* 120
n
+»
9 100
+»
o
ft
5 80.
0
0 60.
0
c« 40.
20.
00
* Outer
— Inner A
X Inner B
+ Inner C
tttttt
I— *. tt»tt
I I I I I I I I I I I I I I
^ 1 + + + + 1
.0 2. 4. 6. 8. 10
Slimes Thickness, MM
12 14 16
Fig. 2 Anodic overpotential (uncorrected for TJQ) changes as a function of the slimes layer
thickness.
Stage I in Table 1.
Position of the reference electrodes:
Outer: Located in the bulk electrolyte, =5mm away from the slimes/electrolyte Interface
Inner A, B, and C: Located within the lead anode at =3, 6, and 8.5 mm away from the
slimes/electrolyte Interface
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Stage I
2 0 0
>
a
o
&
>O
o
150 -
1 0 0
40 60
Current Interruption Number1 0 0
Fig. 3 Outer and inner A, B, and C reference electrodes anodic overpotential response to current i nterr uptions (during an
otherwise galvanostatic experiment).
Abscissa values reflect current int erruption number. Current interrupti on measurements where made every 3 Hrs. (i.e. every0.138 mm slimes). The first curr ent in terruption was made in the absence of slimes and the 100th curr ent in terru ption was madeat a 13.8 mm slimes thickness.
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Stags/
(A) (B)1.1 | , "0 I 1 1
2 0 4 0 8 0 8 0 100 20 «1 80 80 100
Current
Interruption
Number Current
Interruption
Number
Fig. 4 Detail of the response of the inner A reference electrode to current interruptions(during the whole electrorefining cycle).
Fig. (B) is the same data presented in (A) with an expanded vertical scale.
(A) (B)
M 60 ?0 SO 80 100 S3 " 80 90 100
Current Interruption Number Current Interruption Number
Fig. 5 Detail of the rfo response of the inner B (Fig. A) and inner C (Fig. B) reference
electrodes to current Interruptions (during the whole electrorefining cycle).
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Stage I
6 1 . 1 2 8 1 8 8 2 4 0 3 8 8 . 8 8 6 8 , 1 2 8 1 8 8 2 4 8 3 8 8
1 2 8 1 8 8 2 4 8 8 6 8 . 1 2 8 1 8 8 2 4 8 3 8 8
Fig. 6 Detail of the response to current Interruptions measured by the outer reference
electrode (at different slimes layer thickness).
X axis: Time, msec Y axis: Anodic overpotential, mV
The response of the outer reference electrode to current interruption is
shown in Fig. 6 for different slimes thickness. Upon current interruption, the r\ A
value first drops abruptly, then decays slowly. The abrupt overpotential decay
is nearly equal to the so-called uncompensated ohmic drop, r\a . Furthermore,
upon application of current back to the cell, it can be seen that, the thicker the
slimes layer, the longer it takes to attain the rjA value observed prior to current
interruption (Fig. 6).
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Stage/
30. i-
23.
20.
IS.
10.
S.0
.00
^ ^ ^ ^
Fig. 7 Changes in the value of theuncompensated ohmic drop, r\a, as afunction of the slimes thickness.
From the T | A response of the outerreference electrode to currentinterruptions.
i i i 1.00 2.0 4.0 6.0 8.0 10.
Slines Thickness, nn
12. 14. 16.
As shown in Fig. 7, r\a remains between 14 and 16 mV during the whole
experiment. Only during the first interruption of current (at zero slimes thickness)does a larger potential drop appear. The decrease in this value results from
changes in the electrolyte concentration in the near proximity of the
slimes/electrolyte interface. Within a few milliseconds, steady state is attained
and r|n no longer changes. Upon interruption of current, the concentration
gradients present within the slimes layer begin to relax towards equilibrium. H +
moves from the bulk electrolyte towards the slimes layer and Pb + 2
and SiF 6"2 move
in the opposite direction. This process is currentless and will cause interaction
between the diffusion and migration fluxes so that the potential gradient decaysin the same way as the concentration gradients.
The response of the inner reference electrode A to current interruptions is
different than that observed by the outer reference electrode. Depending on the
slimes thickness, TJ a measured by this electrode can show a random behavior
(see Fig. 4A). Thus, for example Figs. 8A and 8F show that upon current
interruption T)A jumps towards higher values rather than decreasing. There is no
unambiguous explanation for such jumps. The other curves in Fig. 8 show that
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Stage/
upon current interruption r|A decays linearly. The amplitude of this decay is
smaller than that shown by the outer reference electrode at similar slimes
thickness (see Fig. 3). Additionally, there is no initial steep decay in rjA upon
current interruption. This is an indication that between this electrode and the
lead anode the inner slimes electrolyte does not have a uniform composition.
1 2 0 1 8 0 2 4 0 3 0 O 0 0 6 0 . 1 2 0 1 8 0 2 4 0
Fig. 8 Detail of the response to current Interruptions measured by the inner A referenceelectrode (at different slimes layer thickness).
X axis: Time, msec Y axis: Anodic overpotential, mV
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The inner reference electrode B response to current interruptions is shown
in Fig. 9. Upon interruption of current, a very small decrease in r|A takes place
(see also Fig. 5A). This electrode is very close to the lead anode/slimes interface
and the concentration of Pb + 2
in its vicinity is expected to be very high \ As this
interface moves away from this reference electrode, the region over whichconcentration gradients span grows and so decays in riA of larger amplitude can
take place upon current interruption (see Fig. 5A).
300
42 .
38 . * * • * •
34 . - • «
38.
26 .
22 . , .
1 . . 1
10.2 M M
.80 60 .
74 .
70 .
66 .
62 ,
58 ,
54 .
- »
128 188 248 388
13.7 M M
1
300 .00 60 . 120 180 240 300
Fig. 9 Detail of the response to current interrupuons measured by the inner B reference
electrode (at different slimes layer thickness).
X axis: Time, msec Y axis: Anodic overpotential, mV
The response of the inner reference electrode C to current interruption is
depicted in Fig. 10. This electrode does not show any significant decrease in its
r\A value upon current interruption (see Fig. 5B). Furthermore, the amplitude of
the potential decay shown by this electrode is the smallest among all the other
1
The reference electrode potential is related
to
the lead ion concentration. Knowledge
of
the activity
coefficients of the various
species
that
are
in the vicinity of
these
electrodes is
required to
estimate
these
ionic concentrations.
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Stage /
reference electrodes (see Figs. 3 and 5B). The presence of a highly concentrated
Pb + 2
region between this electrode and the anode/slimes layer interface can
account for this behavior.
13
9.8
5.0
1.0
-3.0
j rfWuuum1VBSBuuuin1WSMi
-7.8
8.1 nn
.88 60 ,
40 .
36 .
32 .
28 .
24 .
20 .
120 180 240 300
12.3 nn
i l i i I i i I
26 .
22 .
18 .
14 .
10 .
6.0
49 .
45 .
41 .
37 .
33 .
29 .
<B>
10.2 nn
.00 60 , 120 180 240 300
13.7 nn
.00 60 , 120 180 240 300 .00 60 . 120 180 240 300
Fig. 10 Detail of the T | a response to current Interruptions measured by the inner Creference electrode (at different slimes layer thickness).
X axis: Time, msec Y axis: Anodic overpotential, mV
A plot of the anodic overpotential values obtained right after current
interruptions for the four reference electrodes is shown in Fig. 11. By comparison
with Fig. 2, the correlation between the n.A curves becomes more evident. Fig. 11
shows that the activities of the lead ions at fixed positions change as the
anode/slimes interface moves, reflecting variations in the inner slimes electrolyte
composition. Furthermore, the lines for the various reference electrodes are
almost parallel, indicating a near steady-state in the solution gradients between
any two reference electrodes.
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Stage II
180. I—
2.08 4.00 6.00 8.00 10.0
SI i lies Thickness, nn
12.0 16.0
Fig. 11 Anodic overpotential (corrected for riJ changes as a function of the slimes layer
thickness.
Stage I in Table 1.
Position of the reference electrodes:Outer: Located in the bulk electrolyte, =5mm away from the slimes/electrolyte interfaceInner A, B, and C: Located within the lead anode at =3, 6, and 8.5 mm away from theslimes/electrolyte interface
2. Stage II
After forming a 13.8 mm thick slimes layer, current was interrupted for 190
Hrs and the lead anode polarization was followed as a function of time. By
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mterTupting the current for an extended period of time, the concentration
gradients present within the slimes layer are expected to disappear. Changes in
these concentration gradients are reflected in the polarization values which
decrease as a function of time.
x — x outer 0
D — a inner A
inner B
inner C
Outer
Inner B
-40
i- i.i.i.
crrri vrw, m0 20 40 60 80 100 120 140 160 180 200
Time Since Current Interruption, Hrs
Fig. 12 Difference in potential (uncorrected for r\^) between the reference electrodes and the lead anode as afunction of the current interruption time.
Stage II in Table 1. After anodic dissolution up to a 13.8 mm thick layer of slimes (Fig. 2) current was haltedfor 190 Hrs and the polarization was followed as a function of time.
The arrows indicate the polarization values prior to current interruption.
Details of the polarization decay in the first milliseconds after current interruption can be seen in Figs. 6H, 8H,
9D, and 10D.
Fig. 12 shows how the difference in potential between the reference
electrodes and the lead anode decays during current interruption. The outer and
inner A reference electrode polarizations decay to values close to zero within a
few hours. On the other hand, the potential difference measured by inner
reference electrodes B and C is negative. Furthermore, after a certain time has
elapsed, the polarization displayed by these electrodes jumps to near zero values.
The closer the reference electrode is to the anode/slimes interface, the longer it
takes for this polarization jump to occur. This rise in potential difference is
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Stage III
attributed to dissolution of precipitated products that result from changes in
electrolyte composition in the inner slimes layer. Mixed electrochemical processes
that can support internal currents in the absence of an external current, such
as the reduction of an oxidized ion, also can account for the negative polarization
values displayed by these electrodes \
3. Stage III
After allowing the concentration gradients present within the slimes layer
to relax during the long current interruption, current was applied back to the
cell during 30 min (Fig. 13).
>
<
.03•+->
C
CD
-t->O
Q.
i_
0)
>
O
o
T3
O
C
<
100
90
80
70
60
50
40
30
20
10
0
-10
: }: i •
'-
1
'-
/ \ ••**•'
; .^.......\~*-r
f
|
\ - f - Outer;
* •••!'• Inner fAt : j
- I I I
! • F Inner iC
i, i ,
Fig. 13 Changes in the value ofthe anodic overpotential (
uncorrected for as a functionof the electrolysis time.
Stage m in Table 1.After interrupting the current for190 Hrs (Fig. 13), current wasapplied back to the cell and theT | A values shown in this plot wereobtained.
10
15 20
Time, min
25 30
1 Among
these mixed electrochemical processes are:
a)
local concentration cells (b) cementation
reactions
c)
Re-dissolution of PbF
2
:
PbF 2+2e~ = Pb*2
+2F~
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Stage III
Fig. 14 Anodic overpotential response upon current interruption (Stage m Table 1)
Current interruptions were applied at different electrolysis times as indicated by the arrows
shown in each plot:
(1) 0.01 min (2) 10 min (3) 20 min (4) 30 min
Each plot indicates the overpotential response measured by a different reference electrode:
(A) Outer (B) Inner A (C) Inner B (D) Inner C
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Bulk
electrofytB
concentrations
Fig. 13 shows how the anodic overpotential values increased during the
application of current back to the cell. Concentration gradients within the slimes
layer become established very rapidly: In 30 min only =0.02 mm of slimes were
formed yet the T)a values increased by more than 40 m V1
. This shows that the
slimes layer does hinder appreciably the flow of ions. Current interruptionsduring 1.8 sec were applied during this stage to study the characteristics of these
concentration gradients (Fig. 14). At time zero no significant decay in potential
is observed upon current interruption in any of the reference electrodes. As
concentration gradients become established, larger overpotential drops can be
observed upon current interruption.
B. Analytical chemistry
1. Bulk electrolyte concentrations
The changes in composition in the bulk electrolyte during Stage I are shown
in Fig. 15. A continuous depletion of Pb+ 2 in the bulk electrolyte is seen to take
place during the formation of the slimes layer. As the current efficiency for the
refining process was close to 100%, this depletion can only be associated with
Pb+ 2 concentration enrichment in the electrolyte within the slimes layer.
Concurrently to the Pb+ 2
depletion, there is a continuous, yet small increase in
acid concentration which is related to H + depletion within the slimes layer2. No
major changes in the bulk electrolyte concentration of hydrolyzed Si0 2 were
detected during this stage 3 .
The changes in the bulk electrolyte composition during Stage n are shown
in Fig. 16. Upon current interruption, there is an increase in the Pb+ 2 bulk
electrolyte composition, and a decrease in the acid concentration. Once current
is interrupted, Pb+ 2 and SiF6"
2 diffuse out of the slimes layer while H+ diffuses in.
1 In the same period of time, the outer anodic overpotential rose by less than 2 mV in the absence of the
slimes layer (Fig. 2), as
compared
to
a 65
mV rise
in
the presence of
a
13.8
mm
thick layer
of
slimes
(Fig. 11).
2
Material balances indicated that the slimes electrolyte can have average
[PbSiF
6
]
higher than M while
[H
2
SiFe]
can be lower than 0.5
M.
3 Si0
2
as determined by titration via the LiF-LiOH technique described in Appendix 4 is merely
a
composite of all dissolved species containing at least
oxygen atom. The general formula Si
x
(OH)
y
F
z
q
"
with
1<y<4,
and
'- < 6 accounts
for the
existence of these species.
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Bulk electrolyte concentrations
o
E
0.9
0.8
!
0.7
- .a...
2 0.6
cCDO
c
o
O03
2TS
LU
m
0.5
0.4
\ ° " 7[H2SiF6]
[Pb+2]
[Si02]
A-
/S/02/ "~~"~-©-0
_1 I I I I I I I I I I I I I I I I I I I I I I I I I 1 I 1 I L.
Fig. 15 Changes in composition of thebulk electrolyte as a function of theslimes layer thickness.
Stage I in Table 1.At preset electrolysis times bulkelectrolyte samples were withdrawnfrom the bulk electrolyte and analyzedfor the species shown in this plot
From chemical analysis data
2 4 6 8 10 12 14 1<
Slimes Thickness,
mm
0.9
_ 0.8
o
E
o
08
cCDO
eo
OCD
9
TJaLU
CO.
0.7
0.6
0.5
0.4
O GL.
[H2SiF6]
[PbSiF6]
-X-
[Si02]
i 11 i i i i i i i i i i i 11 11 i i i i i i i i i i i i i 11
Fig. 16 Changes in the bulkelectrolyte composition as a functionof the current interruption time
Stage II in Table 1.
From chemical analysis data
20 40 60 80 100 120 140 160 180 200
Time Since Current Interruption, Hrs
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Inner slimes electrolyte concentrations.
2. Inner slimes electrolyte concentrations.
Some information about local ionic concentrations within the slimes layer
was obtained by withdrawing small amounts of electrolyte (=100 |il) from a fixed
point located =3mm away from the slimes/electrolyte interface (as shown in
Fig. 1).
Fig. 17 Changes in the localcomposition of the slimes electrolyteas a function of the movement (fromthe sampling point) of theanode/slimes interface
Stage I in Table 1.From a fixed point located ~3 mmaway from the slimes/bulkelectrolyte interface (Fig. 1)electrolyte samples were withdrawnat preset times and analyzed for thespecies shown in the plot.
From chemical analysis data
0
2 4 6 8 10
Distance
From the
Anode/slimes
interface, mm
Fig. 17 shows the changes in the composition of the inner slimes electrolyte
during stage I as the anode/slimes interface moves away from the sampling
point. As can be seen, Pb + 2
concentrations at this point are not as high as the
average values predicted from mass balance computations. Furthermore, the
acid balance is also somewhat larger than expected . However, Pb+ 2
concentrations were found to be between 3 and 6 times larger than thecorresponding bulk electrolyte values. Differences between mass balances and
local compositions, may be due to presence of precipitates and also to the fact
that sampling was made only at one point within the slimes layer. Fig. 17 also
shows that negative changes in the acid concentration accompany positive PbSiF6
variations. This indicates that SiF6"2 exerts an influence in the transport processes
within the slimes layer. The Si0 2 concentrations shown in Fig. 17 indicate that
large amounts of these species are contained in the inner slimes electrolyte.
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Inner slimes
electrolyte
concentrations
Analysis of ionic species of noble impurities in this electrolyte showed that AsO+
was present in concentrations of -0.17 m M whereas [BiO+] was lower than 0.01
mM . Additionally, [SbO+] in the entrapped electrolyte was of the order of 0.2 mM .
Furthermore, no major changes in the concentration of these noble species were
detected during the refining cycle. These small amounts of noble impurities inthe inner electrolyte indicate that they do not react significantly at the
corresponding overpotential levels shown in Fig. 12 \
( A ) (B)1.9
1.8
1.6
^ 1 5
E
5f 1 4
1.2
^ from Chemical Analysis
from AAS- H2SIF6
Standards
- - from
AAS-PbSiF6-H2SiF6 Standards
i i l I—' ' i I i I I I i i i
O
E
-
<
—A\ \
-
\ \\ v
-
Chemical Analysis
" -o — JSE and Gran's Plot technique
_
..
0
.
ISE and Standard Addition technique
, , I ,
3
2 4 6 8 10
Distance from the Anode/slimes Interface, mm
0 2 4 6 8 10
Distance
from the anode/slimes
interface,
mm
Fig. 1 Changes in the local concentration of the total Si and F present in the slimeselectrolyte as a function of the movement (from the sampling point) of the anode/slimes
interface
Stage I in Table 1.
Electrolyte samples taken from a fixed point located -3 mm away from the slimes/bulk
electrolyte interface were analyzed for total Si and F using three different analytical
techniques.
(A) Changes in the total concentration of Si-bearing species
(B) Changes in the total concentration of F-bearing species
From chemical analysis data
1 Analysis of the cathode at the end of stage III
confirmed
that impurities
dissolution
was not significant.
Cathode
impurities concentrations were: 0.0003%
Cu ,
0.0010%
Sb,
0.0016%
Bi,
<0.0003%
Sn,
<0.0001%
Ag,
and
<0.0001%
Tl.
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Inner slimes
electrolyte
concentrations.
1.2
1 -
S 0.8
o
E
c
o
«3 0.6
[H2SiF6J
- 0
1 3 - •fZT"
/S/027
[PbSiF6]
Fig. 19 Changes in the localcomposition of the slimes electrolyteas a function of the currentinterruption time.
Stage II in Table 1.From a fixed point located ~3 mmaway from the slimes/bulk electrolyteinterface electrolyte samples werewithdrawn at preset times andanalyzed for the species shown in theplot.
From chemical analysis data
40 80 120 160
Time
Since Current
Interruption,
Hrs
200
Chemical analysis of total Si and F in the inner slimes electrolyte during
stage I are provided in Fig. 18. Si and F total concentrations decrease as the
slimes layer thickens. Changes in the local composition of the slimes electrolyte
are unexpected under steady state anode dissolution conditions. Time dependent
processes such as changes in convection due to movement of the anode/slimes
interface and/or gradual precipitation of secondary compounds can account for
the Si and F decrease at a fixed point such as observed in Fig. 18.
Changes in the composition of the inner slimes electrolyte during stage n
are depicted in Fig. 19. Although no major changes in Pb + 2
concentration are
observed during the long current interruption, a significant enhancement in the
acid concentration is observed. Also,
a sudden decrease in [SbO+
] concentration
seems to occur just a few hours after current interruption. As expected, the total
Si and F concentrations increase during the current interruption stage (see
Fig. 20). Thus, as concentration gradients disappear, the driving force for
convection decreases continuously and redissolution of precipitates can take
place.
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Characterization of the slimes layer
(A) (B)
1.2
O
[Si] from Chemica l Analysis
- o - - [Si] from AA S- H2SiF6 Standards
-£}•
[SI] from AAS-PbSIF6-H2SiF6 Standards
I
7.5 |- O
O 6.5
5.5
p o
7 / 13
X Chemical Analysis
—£ ISE and Gran's Plot technique
ISE and Standard Addition technique _L _J_
20 40 60 80 100 120 140 160 180 200
Time Since Current Interruption, Hrs
0 20 40 60 80 100 120 140 160 180 200
Time Since Current Interruption, Hrs
Fig. 20 Changes in the local concentration of the total Si and F present in the slimes
electrolyte as a function of the current interruption time.
Stage n in Table 1.Electrolyte samples taken from a fixed point located -3 mm away from the slimes/bulkelectrolyte interface were analyzed for total Si and F using three different analyticaltechniques.Total Silica and total fluorine were calculated by adding the concentrations of all the Si and
F-bearing species.
(A) Changes in the total concentration of Si-bearing species
(B) Changes in the total concentration of F-bearing species
From chemical analysis data
C. Characterization of the slimes layer
As described in the previous section, the total concentration of noble
impurities present in the cathodic deposit was lower than 30 ppm. Thus, no
significant dissolution of noble phases and compounds present in the original
lead anode should take place. The relationship between the phases and
compounds present in the uncorroded lead anode and those found in the slimes
layer was studied by using metallographic techniques. Electron probe
microanalysis (EPMA) of these samples was done by using energy dispersive
spectrometry. Additionally, X-ray diffraction was used to study the distribution
and presence of these phases and compounds within the slimes layer.
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Metallography ol the staring lead anode
1. Metallography of the starting lead anode
Fig. 21 shows the section of the lead anode used in the metallographic
analysis, and the observation points chosen to correspond to locations where
the slimes layer structure was later studied.
Uncorroded electrode Corroded electrode
de/slimes
Interface F i g . 21 Section of the lead anode andof the slimes layer studiedmetallographically.
The observation points in theuncorroded specimen were chosen tocorrespond to matching locations inthe slimes layer.
Air cooled face
Fig. 22 shows the microstructure of the lead anode ("air" face, location #1
in Fig. 21) \ A variation of the so-called "honeycomb" structure can be observed
in Fig. 22. As found from EPMA2
, the inner grains in this microstructure have
a large concentration of lead-rich phases, whereas the grain boundaries are
somewhat depleted in lead and can contain large concentrations of noble
elements (Sb, As, Bi, Ag). EPMA performed in this sample (Fig. 22C) shows
significant variations in elemental concentrations along the grains and the grainboundaries.
1 This sample was prepared by
polishing
up to 600 grit followed by 5 urn alumina. Afterwards, the sample
was chemically etched
with
a polishing-etching solution of the following
composition: 20 ml CH3COOH(concentrated), 42 ml H2Oj, (30%), 40 ml
HN03 (concentrated), and 70 ml of glycerine [6-8].
2 EPMA in etched samples is not recommended as irregular absorption of
x-rays
resulting from
topography affects the analysis. Thus, the electron probe microanalysis
shown
in Fig. 22C only indicates
qualitative
changes in concentrations.
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(C)Fig. 22 Lead anode microstuctures.
Anode "A", Air cooled face. All
micrographs correspond to the same
observation polnt( point #1 Fig. 21).
Secondary Electron Images
Specimen was polished up to grit
600 and subsequently chemically
polished/etched
EPMA Fig. C. %wt
Point Cu As Ag Sb Pb Bi
1 0 6 1 2 87 3
2 0 13 5 31 51 0
3 0 14 2 23 62 0
4 0 5 0 2 80 14
5 0 11 4 3 82 0
6 0 9 0 2 81 8
7 0 9 2 3 83 2
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Metallography ol the starting lead anode
The changes in the anode microstructure at different parts of the lead anode
can be seen in Figs. 23 and 24 (locations #2 and #3 respectively of Fig. 21) \
EPMA 3 performed on this sample shows that there is less Pb in the grain
boundaries than inside the grains. Furthermore, Bi seems to be present with
lead throughout the whole microstructure. This analysis also shows a eutecticphase that is rich in As and Sb (points 6 and 7) whereas Sb and Ag-rich phases
can be observed In the proximities of the grain boundaries (points 8 and 9) and
in some cases within the grains. The precipitates within the grains have a random
composition. Most of these precipitates are Pb-rich compounds. Additionally,
Cu-As compounds (not identified in Fig. 23) can also be seen inside the grains.
The microstructures compared in Fig. 24, show the continuity of the honeycomb
structure throughout the sample.
1 These samples were not etched. They were prepared by polishing
up
to 600 grit followed by using 0.5
urn
alumina.
Backscattered electrons were used to
reveal
the anode microstructure.
2 As these samples were not etched,
relative
changes in the probe microanalysis are significant and
represent semi-quantitative changes
in
the elemental
composition
of the samples.
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Fig. 23 Lead anode microstuctures.Anode "A". All micrographs correspondto the same observation point( point #2Fig. 21).
Backscattered Electron ImagesSpecimen was polished up to grit 600followed by using 0.5 um alumina(sample was not etched or chemicallypolished).
EPMA Fig. C . %wt
Point Cu As Ag Sb Pb Bi
1 0 8 0 1 87 5
2 0 7 0 1 87 5
3 0 7 0 1 88 4
4 0 7 0 1 89 4
5 0 7 0 0 89 4
6 0 12 2 6 77 4
7 0 11 1 13 73 38 0 4 28 10 54 3
9 0 6 23 7 62 2
10 0 8 0 1 87 5
Metallography ol tie Starting Lead Anode
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Metallography ol the Starting Lead Anode
Fig. 24 Lead anode microstuctures. Micrographs correspond to different observation points
(A) and (B) Observation point #2 Fig. 21 (C) and (D) Observation point #3 Fig. 21
Backscattered Electron Images
Specimens were polished up to grit 600 followed by using 0.5 urn alumina (samples were not
etched or chemically polished)
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SEM analysis
2. Analysis of the slimes layer phases and compounds
(a) SEM analysis
The slimes microstructure1
=2 mm away from the slimes/bulk electrolyte
interface is shown in Fig. 25a
. The Pb-rich phases inside the grains have
dissolved and the noble phases are left behind. The dissolution of the Pb-rich
phases near the grain boundaries can be seen in Fig. 26. EPMA showed that
large concentrations of noble elements are present along the former grain
boundaries and that there are gradients in their concentrations (e.g. compare
points 1 and 2). Additionally, noble compounds and various segregates were
detected in these inicrostructures (see points 3 and 4). Fig. 26C shows the form
of the precipitates of noble phases originally present inside the lead anode grains
which report to the slimes layer.
The slimes layer microstructure1
=12 mm away from the slimes/bulk
electrolyte interface is shown in Fig. 27. Evidence for the presence of Si-rich
compounds is the major difference between this microstructure and the
microstructures shown in Figs. 25 and 27. As can be seen in Fig. 28, Si
surrounds the former grain boundaries and appears randomly throughout the
structure. Si is expected to result from the hydrolysis of SiF 6"2
which should
be more severe near the anode/slimes interface.
1 Total slimes thickness
=13.8
mm.
2 Samples
were mounted using a vacuum imbibition technique and a low viscosity resin. Careful polishing
using
the
0.5u.m
cloth
grit
was
used
to
remove
the excess of resin. Samples were coated with graphite
previous to their observation in the SEM.
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SEM ana/ysis
Fig. 25 Microstructure of the slimes layer @2mm away from the slimes/electrolyte interface
(position #2 Fig. 21)
Backscattered Electron Image
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016929 30KV X 4 0 0 7 5 u m
Fig. 26 Detail of the microstructure of
the slimes layer © 2 m m away from the
slimes/electrolyte Interface (position #2
Fig. 21)
Backscattered Electron Images
EPMA Fig. C , %wt
Point As Ag Sb Pb Si
1 12 8 56 25
2 6 - 79 14 -
3 3 56 40 0 -
4 20 8 35 36 -
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Fig. 27 Microstructure of the slimes layer @12mm away from the slimes/electrolyte
interface (position #3 Fig. 21)
Backscattered Electron Image
SEM analysis
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SEM analysis
( B )
016924 30KV X400"'' '75um
Fig. 28 Detail of the microstructureof the slimes layer @ 12mm away from
the slimes/electrolyte Interface(position 3 Fig. 21)Backscattered Electron Images
EPMA Fig. C, %wt
Point
1
2
Si Cu As Ag Sb
44 1 8 - 47
23 - 9 6 62
- - 15 3 63
Pb
19
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X-ray diffraction
(b) X-ray diffraction
Table 1 shows the results of the X-ray diffraction analysis of unsupported
slimes samples
Table 1 X-ray Diffraction Analysis of Outer and Inner Slimes Powder Samples
Outer Slimes
(=2 mm from the slimes-electrolyte
interface)
Inner Slimes
(=12 mm from the slimes-electrolyte
interface)
PbF 2 X XXX
Si 0 2 X XX
Sb 20 3X XXX
SbA X XX
Bi X XX
Sb X XX
PbO (yellow) -
Ag 3Sb XX
Cu3Sb XX
A s A - X
PbSiOj
Bi 2 0 3X XX
. Phase presence is dubious
- Phase detected in very small concentrations
X Phase detected in low concentrations
XX Phase detected in medium concentrations
XX X Phase detected in large concentrations
The presence of PbF2 in these samples and its relative larger concentration
in the inner slimes is a supplementary indication to the presence of Si detected
by EPMA that hydrolysis of SiF6"2 takes place.
The presence of metallic Bi, and Sb, together with some of their oxides was
as expected from electron probe microanalysis 2
. The presence of mtermetallic
compounds (i.e. A^Sb and Cu 3Sb) was also expected from these analysis.
1 With multiple phases
present
there were overlapping
peaks
that caused problems with positive,
unambiguous identification.
2 The
presence
of
oxides
can be the result of
oxidation
of the
slimes
after they were
dried.
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Chapter 5 Anodic and Rest Pot ent ial Behavior of Pure Lead i n
HaSiFe-PbSiFe Electrolytes
I. Overview of Pure Lead Dis sol uti on in H 2 S i F 6 - P b S i F 6 Electrolytes
Under Galva nostat ic Condit ionsIn this chapter the dissolution of a pure lead electrode is discussed in terms
of DC and AC electrochemical measurements. A qualitative analysis of the
controlling mechanisms for lead dissolution is presented. The different
components of the anodic overpotential are related to phenomena taking place in
the electrode boundary layer. In the case of pure lead, there are no complications
due to the presence of slimes.
A. Anodic overpotential in the absence of large concentration gradients inthe anode boundary layer
During the galvanostatic dissolution of the lead bullion anode described in
Chapter 4, the value of the uncompensated ohmic resistance, r| n, remained nearly
constant during the whole electrorefining cycle (Fig. 4.7). In this case, the presence
of large ionic concentration gradients within the slimes layer results in a counter
E.M.F. that is responsible for the failure of the anodic overpotential to decay to
zero upon current interruption. When pure lead is dissolved, T|
Q
increases
continuously with time as seen in Fig. 1, as a result of the progressive ohmic
resistance created by the movement of the anode/electrolyte interface. Increases
in rj n reflect changes in the distance between the reference electrode and the
anode/electrolyte interface which directly affect the Rg1
value. Under these
conditions, rjQ appears to be the only source of potential between the lead anode
and the reference electrode. By subtracting the calculated ohmic resistance from
the r)A measurements, the extent to which concentration gradients become
established in the anode boundary layer can be studied. As Fig. IA shows, the
corrected anodic overpotential value remains constant during the dissolution of
lead, indicating a constant thickness of the boundary layer across which
concentration gradients persist. Concentration overpotential can be considered
to be the only source of potential under these circumstances, as lead dissolution
1 As explained in Chapters 2 and 4, R,
[Qcm
2
]
is the specific resistance of the electrolyte and is
related
to
T\a
by the
following
relationship:
Ti
n
=IR
s
.
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Anodic overpotential in the absence of large concentration gradients in the anode boundar
occurs nearly reversibly (n,ac=0 mV) \ The small magnitude of the generated
concentration overpotential indicates that large concentration gradients are not
established a .
(A) (B)
80 i I I I | I i I | I I i | i i i | i i i | i i i | i i i | i i i | i I I | i i i
>
Fig. 1 Potential difference between a fixed reference electrode and a corroding anode In the
absence of addition agents.
Experimental conditions: Galvanostatic Experiment, Current Density= 200 Amp m"2
,
Electrode Area 1.50 cm2
, [PbSiF6l=1.31 M , [H2SiF6]=0.30 M . T=4Q±1.5"C, bulk electrolyteelectrical conductivity K=220 mmhos cm"1. Beaker electrochemical cell, electrolyte volume
=300ml, stationary electrolyte. Wenking potentiostat-Data Translation Board-IBM XTcomputer.
(A) f\
A
variation during the dissolution of pure lead
(B) T\a changes obtained by interrupting the current at preset times.
1
iiae
determination was done by using AC impedance techniques
described
in section III.
2 In the
case
shown in Fig.
concentration gradients
in
the Nernst boundary layer span over a region
between 100 and 1000 urn thick.
By
comparison, during the refining of impure lead, concentration
gradients are present throughout the whole slimes layer (i.e. the Nernst boundary layer spans over
several
mm).
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Correlation between the anodic overpotential and the presence ol addition age
B. Correlation between the anodic overpotential and the presence of
addition agents.
During the refining of lead, addition agents (i.e. aloes and lignin sulphonate)
are normally used to modify cathodic reactions. The presence of addition agents
can also affect anodic reactions through complex adsorption mechanisms [lj. In
general, the r|A values observed during the galvanostatic dissolution of pure lead
were found to increase in the presence of addition agents in the bulk electrolyte1.
Moreover, when an excess of these additives is added to the electrolyte, to the
extent that suspended material is visible, they collect on the anode and promote
very large overpotential values. This behavior is shown in Fig. 2A where it can be
seen that in a few hours T|A rises from 0 to =1000 mV. In this case, current
interruption measurements indicate thatTjQ accounts for most of the overpotential
values (Fig. 2B). In the presence of purely resistive films in the anode surface, T) 0
can be described by the following equation:
T]
a
= IRs=I(Rb+Rfi
J
...1
Rt was found to be equal to 1.4 Qcm2 from the ohmic drop value obtained a
few seconds after the electrolysis began. Any extra increases in T I
q are due to the
presence of Rflim. The T j n produced by this film along with its resistance are plotted
in Fig. 2B. It seems as if the addition agents form or generate a highly resistive
film that may be counter productive to the refining process a . Furthermore, the
presence of this film favours the development of concentration gradients in the
anode/electrolyte interface. Thus, the rjA values obtained upon current
interruption increase continuously with time (see inset Fig. 2A) indicating that
the stagnant zone at the anode surface is thickening.
1
See section ll.b.
2 Visual observation of the anode after the refining process showed
that a yellowish
film adhered to the
anode surface. Such a film was visible only when the amount of undissolved addition agents was large.
In
electrolyte solutions where the electrolyte was filtered prior to its introduction to the cell, the addition agent
film was not visible at naked eye.
The amount of
suspended
solids
produced
by excess additives (aloes and lignin sulphonate) seem to
increase
with
increasing lead concentrations
in
the electrolyte. Techniques for increasing the dissolution
of
these
and other additives are reported
in
the
literature
[2].
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Correlation between the anodic overpotential and the presence of secondary products that precipitate on the ano
(A) (B)
Fig. 2 Overpotential changes during the galvanostatic dissolution of pure lead (in the
presence of excess quantities of addition agents)
Experimental conditions: Current Density = 180 Amp m"2
. Electrode Area 31.0 cm2
,[PbSiF6]=1.31 M , [H2SiF6]=0.30 M , =2 g 1
1 aloes and =4 g 11
lignin sulphonate (suspendedmaterial was visible), T=40±1.5°C. Rectangular electrochemical cell, electrolyte volume=320ml, bulk electrolyte recirculation rate =6 ml min"1. Wenking potentiostat-DataTranslation Board-IBM XT computer.
(A) T |A
increases as a function of the electrolysis time(B) Changes in the and R^, as a function of the electrolysis time. Left, axis: T |
Q
due to thepresence of a colloidal film of "undissolved" addition agent [r\a , fflm ). Right axis: Ram obtained
from the following relationship: R fi!m =
C.
Correlation between the anodic overpotential and the presence of
secondary products that precipitate on the anode surface
Increases in the anodic overpotential values as a result of changes in its
ohmic component also can be due to the precipitation of nearly insoluble salts
such as PbF2 and Si0 2. The precipitation of these salts can be observed if
sufficiently large concentration gradients in the anode boundary layer become
established. The presence of these compounds in the anode surface would hinder
the movement of ions and induce even larger concentration gradients as well as
increase T J q . If the concentration gradients are large enough, even the highly
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Establishment of Ionic Concentration Gradients in (he
Anode
Boundary Layer and their Relationship to the Anodic
O
soluble PbSiF 6.4H 20 salt can precipitate \ Fig. 3 shows how the formation of
secondary products can be promoted by dissolution of the anode using high
current densities. rjQ is the main component of the anodic overpotential values
shown in Fig. 3. Moreover, TJ q is related to the porosity and tortuosity factors
resulting from the presence of precipitated products in the anode vicinity.
Concentration overpotential also contributes to the rjA increases shown in Fig. 3.
6 0 0 0
5 0 0 0 -
4 0 0 0
CM
E\CLE
<
' ( / ) 3 0 0 0
c
Q
cOJ1_
13
O
2 0 0 0
1 0 0 0
Anodic
Overpotential
• Current Density
0)
o>
<5
cOJ
-1—'
oCL
i_OJ
>
O
oT5OC
<1 4 0 0 2 8 0 0
i . _ i o
4 2 0 0
Time,
sec
Fig. 3 Anodic overpotentialresponse (uncorrected for TJQ) of purelead to the application of successivecurrent steps
Experimental conditions: Electrodearea 2.34 cm
2
, [PbSiFJ^.35 M,[HjSiFd^.SO M, [SiOJ=0.13 M, =2g 1 aloes and =4 g l'
1 ligninsulphonate, T=40±1.5'C, bulkelectrolyte electrical conductivityK=330mmhos cm"
1
. Beakerelectrochemical cell, electrolytevolume =300ml, bulk electrolyterecirculation rate =6 ml min"
1
.Solartron ElectrochemicalInterface-IEEE Card-IBM XTcomputer
Left axis: Current steps applied as afunction of timeRight axis: Anodic overpotentialresponse (uncorrected for r\^)
EL. Establ ishment of Ionic Concentration Gradients i n the Anode
Boundary Layer and their Relat ionship to the Anodic
Overpotential
In the previous section it was shown that during the dissolution of pure lead,
T |
A
increases almost exclusively due to changes in its ohmic component. By
subtracting rj Q from n,A, the activation and concentration overpotentials can be
1 The
maximum solubility
of
pure
PbSiF
6.4H 20 at
40 °C is 5
M (see
Appendix
7).[90]
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Establishment ol Ionic Concentration Gradients in the Anode Boundary Layer and their Relationship to the Anodic
obtained \
The division of the anodic overpotential in its ohmic, activation, and
concentration components was attempted by studying the anodic response of
pure lead to current steps. Prior to the experiment -0.5 mm of the exposed surface
of the working electrode was removed by anodic dissolution at low current density.This aided in obtaining reproducible results. Variation in rjA as a result of the
presence of addition agents was studied using the same cell and electrodes 2 . The
thickness of the hydrodynamic boundary layer was not controlled, yet, by fixing
the recirculation rate of the bulk electrolyte, reproducible results were obtained 3
.
cCDQ
cCDv_i_
(J
Fig. 4 Current step function used to study theestablishment of concentration gradients in the anodeboundary layer.
The transient period for both current rise and fall wassmaller than 10 usee.
The time tj marks the onset of the current interruption.
Time
Fig. 4 shows the characteristics of the current steps used to study the anodic
behavior of pure lead. After the application of each current step, current was
1 As described in Chapter 2, concentration overpotential ,T\C , develops due to the establishment of
concentration gradients
in
the anode boundary layer and is
a
function of the current density and the
hydrodynamic
conditions.
Activation overpotential, T^ ,
develops due to the transport hindrance that the
charge carriers find during their movement across the Helmholtz electrical double layer, can
be
determined accurately
in
the absence of concentration gradients and only when
it
is
controlling the
reaction rate.
2 The experiment started by analyzing the T]
a
changes using an additives-free electrolyte. Subsequently,
this electrolyte was slowly substituted with an electrolyte of matching
composition
but containing addition
agents.
3 The hydrodynamic
conditions in
the
vicinity
of the anode/electrolyte
interface
are more nearly
a
product
of convection than of electrolyte recirculation. However, electrolyte recirculation assured uniform
electrolyte
composition
between
the reference electrode and the anode/electrolyte interface.
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In the
absence
ol
addition agents
in the bulk
electrol
The Rg value obtained from the slope of the plot shown in Fig. 6, was nearly
equal to the Rg value obtained prior to the application of the current steps. The
activation overpotential, if present, would be observable by a non-zero intercept
or as a curvature near the origin. Evidently, activation overpotential is not
controlling the dissolution of the lead anode.The
T|
A
values compensated for the initial r|n
(Fig. 6) are presented in Fig. 7.
The overpotential obtained immediately after application of current is very close
to zero (which implies that rj
ac
=0 mV) 1
. Subsequent increases in the anodic
overpotential are due to changes in the concentration of ionic species within the
anode boundary layer2
. Thus, increases in r|
A
are nearly equal to changes in its
rj
c
component.
300
240
>E
CL
o
o
E
O
• «—IL ohmic drop right after applying the current step
-
; I/"/7 i i i i i i i i i i I i i i i i i 1 i i i : i i i 1 i i i i i i i 1 i i i
Fig. 6
Changes in T)Q as a function ofthe applied current density.
Ti n values derived from the datashown in Fig. 5.
T|n values obtained from highfrequency AC measurement (under rest
potential conditions) were consistentwith these readings
400 800 1200 1600
Current Density,
Amp.rrf
2
200
1
As
described
by Eq.
2, at time
t=0, T | C= 0. Thus,
the overvoltage at
t=0 becomes equal to the initial
value
of the activation overpotential [3,p. 356].
2 As the anode
dissolves,
an extra increment to t\a can also be present in these measurements. n,
n
changes can be neglected
during
the first seconds
after the
application of current.
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In the
absence
of addition agen ts in the bulk electrolyte
40
r i i i i i i
i
i i | | -
U 1 1 1 1 <
^ j_ Each Curve corresponds to the TJ A
<r
35 j- response to a different current density step
f from lower
to
higher current densities
S
3 0
~~~~
fr t
o c : / "—~~ . ^
Fig. 7 Anodic overpotential response(corrected for initial T|Q) of a pure leadelectrode to the current steps describedin Fig. 4.
. — : / — i increasing
~£ U / T Current
I
OJ j- f /
' Density
From the rjA data shown in Fig. 5 theT ) N values shown in Fig. 6 weresubtracted.
Each curve corresponds to the T | A
values obtained at different currentdensities, as follows (from bottom totop): 10, 50,100, 150,206,250, 300,500, 800, 1000, 1400, and 1600
Amp m'
2
.
u
• • • • •
0 5 10 15 20 25 30
T i m e , sec
The anodic overpotential response during the first seconds after the
application of the current step was modelled by using the analytical solution of
Fick's diffusion equation under unsteady state conditions. To solve this equation
the following assumptions were used:
1) Absence of Migration
2) Absence of Convection
3) Unit Activity coefficients
4) Dissolution of lead is not controlled by kinetics (r|ac=0).
5) Linear semi-infinite conditions
Description of the boundary conditions required to solve this equation are
provided in Appendix 8 along with its analytical solution. From this solution, the
following relationship between the concentration overpotential and the square
root of time should be observed if the dissolution process is dorninated by diffusion:
eXP [ RT J nFC b p.+2 y
By defining:
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In the absence ol addition agents in the bulk electrolyte
O, =exd1 *|_ RT
21 1
Eq. 3 can be expressed as a linear equation:
<b x=m D-jr x+b D ...4
Thus, by plotting d>j vs.'V^a straight line with slope, mo, and intercept,
should be obtained.
By using the data shown in Fig. 7, and assuming that rjJO.tJ^A, the plots
shown in Fig. 8 were obtained. In this figure, a linear relationship between <J>j and
the square root of time is limited to times smaller than 1 sec \ From curve fitting
these data to Eq. 4, the data shown in Table 1 were obtained. Correlation
coefficients close to 1 were obtained in almost the whole range of current densities
studied. The average diffusion coefficient, D pb+2 , agrees with the values reported
in the literature a . This diffusion coefficient, while dominated by the movement
of Pb + 2, also includes the effects of other ions (SiF6
2
and H+ among others). Changes
in the transference number of the lead ions will take place when concentration
gradients are fully developed and this will affect the absolute value of this
coefficient3
. The presence of these concentration gradients is observable in the
analyzed data, as departures from linearity and as changes in the value of the
intercept from unity4
.
The agreement between the solution of Fick's equation and the results of the
current step experiments confirms that in the absence of addition agents, the
lead dissolution process is controlled by diffusion.
1 As seen in Fig. 8, the larger the applied current density the smaller the linear region
between
the square
root of time
and
O,.
2 Diffusion coefficients for lead in HCI0
4
-Pb(CI0
4
)
2
electrolytes at
25
"C reported
in
the literature are:
D
r o + 1
= 9.4x10"* cm
2
/sec [1]; Dp 4 + 1
= 4.8X10"
6
cm
2
/sec [4j.
3 The solution of Fick's second law presented in Appendix 8 implies that the transference number
of
lead
is zero. Any departures from this transference number will affect the obtained diffusion coefficient.
4 Notice also
that
b
D
decreases as the current density increases.
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In the absence of addition agents in the bulk electrolyte
Fig. 8 Changes in <&, as a function of
the square root of time, -v/f
From the data presented in Fig. 7.
Each curve corresponds to the <t>!values obtained at different currentdensities, as follows (from bottom totop): 10,50,100,150,206, 250, 300,500, 800, 1000, 1400, and 1600Amp m"
2
.
Table 1 Results of the analysis of the concentration overpotential increases during the first second
after application of the current steps. Fick's Second Law approximation.
Current Fitting Parameters of Eq. 4
Density, TUry Regression D +2
Amp-m"2
Coefficient, r 2 cm2-sec 1
50 1.00 0.0371 0.927 5.1E-06
100 0.99 0.0720 0.979 5.4E-06
150 0.983 5.0E-06
250 ^ ^ ^ ^ ^ ^ 0.1614 """"""""""" 6.7E-06
300 0.97 0.1946 ii/iiiiiiiiiiii 6.6E-06
400 0.96 0.2775 0.957 5.8E-06
500 0.95 0.3624 0.963 5.3E-06
800 0.91 0.5672 0.951 5.6E-06
1002 0.89 0.8205 0.964 4.2E-06
1398 0.83 1.1494 0.959 4.1E-06
1598 0.80 1.1921 0.940 5.0E-06
5 . 3 ± 0 . 8 x l 06
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In the absence ol addition agents in the bulk electrolyte
If concentration overpotential is the only source of potential (after subtracting
riJ its presence will be observed after interrupting the current. This can be seen
in Fig. 9 which shows the decay in the anodic overpotential corrected for ohmic
drop obtained after interrupting the current \
3 0
>E
<
V—
c
cuc>CL
i_<D>
O
oTJ
O
C
<
Fig. 9 Decay in the anodicoverpotential (corrected for TIQ ) as afunction of the interruption time, t2.
t1-t2=460 sec (see Fig. 4 for a
description of the relationship betweentt and t .
Other experimental conditions as
described in Fig. 5.
Each curve corresponds to the T | A
value obtained after the interruption ofthe applied current density. Appliedcurrent densities were as follows (frombottom to top): 10,50,100,150,206,250, 300, 500, 800, 1000, 1400, and1600 Amp m"
2
.
i i i i i i i i i i i i i i i i i i i i i i i i i i i i4 6 8 10
Time, sec
By solving Fick's equation under current interruption conditions (see
Appendix 8), the following linear relationship should be observed:
0>2 = mD[V^-VS + ...5
where:
0 2 = exp RT
with rrto and b D as defined by Eq. 4.
Eq. 5 is applicable only when ti is very small (less than 1 sec) because
convection will stop the thickening of the anode boundary layer. Fick's second
law could not be used to analyze the T |
A
response upon current interruption
1 T|n was measured prior
to
the interruption of
current
by measuring the impedance at high frequencies
and by using the current interruption routine built into the SEI. Upon current interruption, the observed
ohmic drop coincided
with that
obtained in
these
measurements.
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In the presence ol addition agents in the bulk electrolyte
because the boundary conditions are not known precisely. Nevertheless, what the
data in Fig. 9 show is that, upon current interruption, concentration gradients
relax and this relaxation can be followed by monitoring the anodic overpotential
dependance with time.
B. In the presence of addition agents in the bulk electrolyte
If addition agents are added to the bulk electrolyte, the rj
A
response changes
(Fig. 10). Upon subtracting the initial T |
Q
value from the r\ A readings, the remaining
overpotential was positive (Fig. 11). This rj
A
value decreased during the first
milliseconds after the application of current. After this initial decrease, changes
in TJ a were a function of the applied current density. In any case, TJ a increased only
up to the point at which convection stops the thickening of the boundary layer.
>E
<
.K5- 4— '
COJ
- 4— '
oCL
OJ>
O
u
TJOC
<
200
175
150
125
100
75
50
25
Each Curve corresponds to the 77A
response to a different current density step
from lower to higher current densities
Increasing
Current
Density
Fig. 10 Anodic overpotentialresponse (uncorrected for T|Q) of apure lead electrode to the currentsteps described in Fig. 4.
conditions:2.34 cm2,
Experimentalelectrode area[PbSiFs]=0.37 M,[H2SiF6]=0.82 M, [SiOJ=0.13 M,T=40±1.5"C, =2gl"1
aloes and=A g l'1 lignin sulphonate, beaker
electrochemical cell, electrolytevolume =310ml, bulk electrolyterecirculation rate =6 ml min"1.Solartron ElectrochemicalInterface-IEEE Card-IBM XTcomputer.
i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 i i 11 1 1 i
1 1 11 1 1 i i 1 1 1 11 i 11 1 1 11 i i
5 10 15 20 25 30
Time, sec
Current steps were applied onlyafter the electrode had reached restpotential conditions (r|x=0mV).Overpotential readings were taken0.10 sec after the application ofeach current step. Each curvecorresponds to the T | A response to
a different current density, asfollows (from bottom to top): 10,50, 100, 150, 200, 250, 300, 400,500,600 Amp m"2.
The differences in the r\ A and T |
N
values at the beginning of the application of
the current steps where consistent throughout the range of current densities
studied (see Figs. 12 and 13). Such differences are the result of the presence of
an adsorbed film of addition agents in the electrode interface. Such a film extends
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In the presence of addition agents in the bulk electroly
(A) (B)
18.8 r- 18.8-
688.
Mr 11 i l i 1111 11 l 1111 i i 11 i 1111 i i 111 i I 11111 111
.888 28.6 48.8 (1.8 88.8 188.
• B88t"' ' ' I ' • • I ' ' ' I ' ' • I ' • ' I ' ' • I • • ' I • ' ' I ' ' ' I ' • ' I.888 6.88 12.8 18.8 24.8 38.8
Time, secTime, sec
Fig. 11 Anodic overpotential response (corrected for initial T|J of a pure lead electrode to thecurrent steps described in Fig. 4
From the data shown in Fig. 10 the T|Q values shown in Fig. 12 were subtracted. Eachcurve corresponds to the values obtained at different current densities (in Amp m"
2
) as
shown to the left of every curve. For the different current densities applied, TJA values at t=0were as follows:
Current density. Amp m2
50 150 250 406 600
T|A corrected for T|Q (at t=0), 10 8.2 9.6 11.5 9.1mV
(A) Long range variation of the TIA response (corrected for initial TIJ. T|a increases up to the
point at which convection stops the thickening of the boundary layer.
(B) Detail of the rjA increases.
the region where concentration gradients can be found. Upon passage of current,
concentration gradients become established, changing the resistance of the filmand the kinetics for lead dissolution. The non-zero intercept of the anodic
overpotential curves indicates that inhibition is present as a result of this adsorbed
film. On account of the synergistic effect between addition agents, activation
overpotential, and concentration overpotential, no simple analysis of the T] A
transients can be performed. Yet, the riA response indicates clearly that the
addition agents affect the anodic reaction for lead in such a way that r| a c can no
longer be considered to be zero.
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In the
presence of
addition agents in the
bulk
electrolyte
288.
- * An o d i c O ve r p ot e n t i a l ( r i g h t a f t e r ap p l y i n g current s tep)
—0 l ineonpensated Ohnic Drop ( f r on AC nea sur ane nt)
168
128 ,
88.8
48.8
.888 i ' i I I I
_ I _ I _L
1 I I I
.888 488 688,128. 248 . 36 8.
Current Density, Amp/m2
Fig. 12 Comparison between the anodic overpotential value obtained right after applicationof current (*) and the uncompensated ohmic drop obtained from the high frequency interceptof the impedance spectrum (O)
TJA values derived from the data presented in Fig. 10. T | F T values obtained from the highfrequency AC measurements under rest potential conditions (i\n = IRJ.
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In the presence of addition agents in the bulk electrolyte
1 8 8 .
8 8 . 8
1
T3
C-Mo
&4 8 . 8
c5a
-ao
<2 2 8 . 8
MM M M H M M M M - * M M » » « M » »
cd = 4 8 8 . AMP/M1
1 2 . 8 -
9 . 6 8
7 . 2 8
4 . 8 8 -
2 . 4 8
,0004 -L.. 8 8 8
O v e r p o t e n t i a l decay caused by
the presence of a dd i t io n agents
t mu»»ii*i
1 • • • 1 • ' ' 1 ' ' ' 1 ' ' ' 1 ' ' • 1 ' •' 1 ' • • 1 • • • 1 ' ' 1 1
. 6 8 8 1 . 2 8 1 . 8 8 2 . 4 8 3 . 8 8
• BOO-j- 1
1 1
I
1 1 1
I
1 1 1
I
1 1 1
I
1 1 1
I
1 1 1
I
1
'
1
I
1 1 1
I
1 1 1
I
1 1 1
I
. 8 8 8 . 6 8 8 1 . 2 8 1 . 8 8 2 . 4 8 3 . 8 8
Time, sec
Fig. 13 Changes in the anodic overpotential upon application of a current step.
From the data presented in Fig. 10. Applied current density: 400 Amp m~ 2
.
Upon application of current TJA increases from 0 to 96 mV and decreases slowly afterwards.From A C measurements under rest potential conditions an Rg value of 2.09 £2cm
2 was found.
Thus, Tl n=2.09£2cm2
* 40 mAmp cm 2 =83.6 mV
The inset plot shows the changes in r\ A (mV) as a function of time (sec) after correcting theanodic overpotential for r\
a
.
Finally, the ohmic drop measured by either fast current interruption or highfrequency AC impedance measurements predicted T
)a values which were slightly
larger than measured (Fig. 14). The difference between these values is partially
due to the presence of a film resistance, (so-called "film inhibition').
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Introduction
s
c
II«•0a.
>
a
»
o
i •
20 .0 -
*
i 6 . 0 Y•Unaccounted
• the p r e s e n c e
o h n i c d r o p c a u s e d 1>«J
at a d d i t i o n a g e n t s
88. e 12 .0
8 . 8 8 -
i
66 .0 4 .aa -
.000+' 1
.eee1 1 1 » 1 1 1 1 1 1 1 1 1 1 1 1
.800 1.60
i l i i * 1 i i i 1 i i i 1 i i i 1 i i t 1
2.40 3.20 4 . 8 8
44 . 0
cd= 400.
•8
| 22 .8
Fig. 14 Changes In the anodic overpotential as a function of the current Interruption time.
Experimental conditions as described in Fig. 10.
After applying current (1=400 Amp m"2
) for 320 sec (trt2=320 sec) current was halted and the
overpotential decay was followed as a function of time.
Upon current interruption % decreases abruptly (from 105 to 12 mV). From ACmeasurements prior to the interruption of current an R, value of 2.17 Qcm
2
was found.
Thus. Tin=2.17flcm2
* 40 mAmp cm2 =86.8 mV.
The inset plot shows the changes in T | A (mV) as a function of time (sec) after subtracting T | Q
from the first r^ reading.
HI. A C Impedance
A . Introduction
In the previous section, DC (direct current) transient techniques were used
to study the dissolution of pure lead and its relationship to the components of
the anodic overpotential. In this section, a complementary study using AC
techniques is presented1
.
1
For a description of the
implementation
of the AC techniques
see Appendix 6.
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Impedance spectra obtained in an electrolyte without addition age
In the absence of a net D C current, application of an A C voltage at low
frequencies (axlO3
rad/sec) generates sinusoidal concentration gradients whose
amplitudes decrease exponentially from the electrode surface towards the bulk
electrolyte [5,61. These concentration gradients cause a characteristic AC energy
absorption. As the A C
frequency is increased (up to 10
5
rad/sec), ionic diffusioncannot keep up with the change in direction and so energy absorption disappears
to be replaced by a phase angle characteristic of a capacitor which represents
processes in the vicinity of the electrical double layer. At extremely high
frequencies (larger than 105 rad/sec) only the movements of ions and dipoles in
solution, representing dielectric properties, can keep up with changing potentials
{7,811. This phenomenon is rarely seen because such high frequency
measurements in liquid electrolytes are experimentally difficult or inaccessible.
A C impedance studies in the presence of a net Faradaic current are analogous
to the impedance studies in the absence of a net D C bias provided the
electrochemical processes under investigation behave linearly. If conditions at the
electrode/electrolyte interface change due to the passage of current, they will be
reflected in the A C impedance spectrum. For example, nonlinearities in the
response of the system can produce a net rectification current and a rectified
voltage. The presence of these nonlinearities and their effect on the related
electrochemical processes have been studied by using Faradaic Rectification
Techniques 16,9]. In highly reversible systems where diffusion controls the
dissolution process (such as lead dissolution in H 2SiF 6-PbSiF6 electrolytes), AC
studies in the presence of a D C current can provide a better insight as to the
extent to which concentration gradients become established in the anode
boundary layer. Also, information on the effects produced by the presence of
addition agents can be derived from A C studies.
B. Impedance spectra obtained in an electrolyte without addition agents
The A C
behaviour of pure lead under rest potential conditions (i.e. in the
absence of a net Faradaic D C current) is shown in Fig. 15 2
. The straight line
1 The bulk electrolyte has
a uniform
composition and its properties are a function of its geometrical
capacitance C
g
and bulk resistance R
b
. From
these
values, the dielectric relaxation
time
of the bulk
electrolyte, Td
can be obtained (x
D
=R
b
C
8
)
[7].
2
A
sinusoidal
current waveform with an amplitude of 21
.3 Amp/m
2
R.M.S.
was swept from lower
to
higher
frequencies
while
the impedance was measured at every frequency.
Application
of DC currents of the
same order of magnitude of the amplitude of the AC waveform (see Fig. 7) resulted in overpotentials lower
than
5 mV.
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Impedance spectra obtained in an electrolyte without addition ag
obtained indicates that diffusion in the electrode boundary layer is the only
mechanism controlling the dissolution/deposition of Pb+ 2
. At high frequencies,
the interception of the impedance curve with the Z% axis is equal to the value of
the uncompensated ohmic resistance, Rg1 . When this value was subtracted from
the impedance curve, a 45° straight line was observed in the impedance diagram(Fig. 16)
2
.
U
4£
NI
120 —
. B9B
. 060
. 030
1 —xE 0 ra<l/sco
- 1 .00
-•
2 . 50 —
•
6 . 28z D IS . 78
z_ • 39 . 63
- -4 99 . 58— 6 298 . — O 628 .
- • 1378 .
-*
1771 .
. 000
. O0O . 320 . 640 .968Z r a * l ft-CM*
1 .28 1 .60
Fig. 15 Impedance diagram of pure lead under rest potential conditions (in the absence of
addition agents)
Experimental conditions: electrode area 2.34 cm2
, [PbSiFel=0.35 M, [H2SiF6]=0.84 M,[SiO2]=0.13 M, T=40±1.5"C, beaker electrochemical cell, electrolyte volume =300ml, bulkelectrolyte recirculation rate =6 ml min"1. Solartron Electrochemical Interface-IEEE Card-IBM
XT computer.
Impedance curve obtained under galvanostatic control, A C waveform amplitude 21.3 Amp m'2
RM.S.
A total of 65 experimental points are plotted. Some of the frequencies (In rad sec"1
) at whichthese points were sampled are indicated in the diagramFrom the high frequency intercept of this plot with the Zg, axis, the Rg value can be obtained(Rg= 1.40 Qcm2).
1 For
a
fixed distance between the reference and the working electrodes, this value was very reproducible
(variations
were
less than 0.5%). These
R
s
values were
used to obtain
r\n prior to the application of the
current steps (see section ll.b).
2 Impedance diagrams in which the real part of the impedance is the abscissa, Z„, and the negative of
the imaginary part of the impedance is the ordinate, -Z,, are also known as Argand plots.
In
an Argand
plot the AC frequencies at which the impedance was measured are also shown.
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Impedance
spectra obtained
in an
electrolyte
without
addition ag
iaer-xE a
.896 -
.872
N .848
.824
.eee
Experimental
— Analogue node I
rad/sec
A i.ee* 2.58
• 6.28
• 15.78
• 39.65
« 99.58
6 258.0 628.
• 1578.
1771.
Fig. 16 Detail of the impedance curveshown in Fig. 15 (after subtracting theR, value).
Some frequencies (in rad sec'1) areshown for both the experimental (solidline) and the regressed data (dottedline).
The impedance curve was fitted to a
CPE element (Z„,E = (/cof'") from
which the following values were
obtained: B , ^ . 139 Qcm2sec** and
Ta^O.50.
Quality of Fit Parameters (65experimental points were fitted to the
CPE analogue):r
2
„=0.951, Iyj2=2.19xl0'3 Q2
cm4
r2
3=0.962 Iy3?=1.70xl0"3
£rcm*^=0.958 ly J
2
=3.75xl03
Q W
i i i
.eee .824 .848 .872
Zreal fi-CM"
.896
xE 8
i i i I
.128
To model the diffusion processes that take place in the electrode boundary-
layer, distributed elements have been used [81112. Among these distributed
elements, the Constant Phase Angle Element (CPE) has been used extensively [71.
The impedance of the CPE element can be described by the foUowing equation:
When the fractional exponent, ^zc , approaches a value of 0.5, the CPE
element describes a serin-infinite diffusional process. Under these conditions the
1 All real electrical analogue elements are distributed in space, i.e. their absolute value changes with
position
due to their finite size.
Diffusion
processes are distributed over the electrode boundary layer, and
constitute a classical example of a distributed element [8].
2 A
distributed
element
is
a
component in an analogue
model that
represents properties of the system
distributed over macro distances, such as ionic concentration gradients across the Nernst
diffusion
layer.
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Impedance spectra
obtained
in
an electrolyte
without
addition ag
impedance of the CPE element is equal to the so-called Warburg impedance
[7.10,11] \ The Warburg impedance for the serru-infinite diffusion case is defined
as [7.8.12]:
Zw,~ = (/"w)^5
...7
When *¥7jC =0.5 the Warburg coefficient can be obtained from the Bj value
(Bj = ov). Thus, ^ is a subset of the generalized CPE response. For the Pb/Pb + 2
equilibrium reaction ov is given by the following equation:
°V,- =RT (n ^°-5
(nF)2 C pb«D pb«
RT 1..8
When is different than 0.5 a generalized representation of the Warburg
coefficient 113-15]
2
can be obtained from the following equation:
The impedance curve shown in Fig. 16 was curve fitted to Eq. 6 from which
the following values were obtained: Bx= 0.139 Q cm2 sec'Vzc and 20=0.5. When these
values were incorporated in Eq. 8, D pb+2 was found to be equal to 2. lxlO"6 cm 2/sec.
For different experiments this number varied by as much as one order of
magnitude 3 . Nevertheless, it was always observed that diffusion was the only
controlling mechanism for lead dissolution/precipitation. Activation polarization
if present would have been observed in the Argand diagram as an arc from which
1 Warburg studied the establishment of concentration gradients in the electrode boundary layer upon
application of an AC voltage.
By
solving Fick's second law under AC conditions, Warburg found
that a
square root frequency dependance of the impedance should be observed if the process
was
controlled by
diffusion.
This square root dependance
is
equivalent to observing a 45" relationship
between
the
real
and
the imaginary components of the impedance
(or
a value of 0.5). The assumptions under which
Warburg
solved Fick's second
law
are similar
to
those described
in
the previous section to solve the DCtransient case: presence of
a
supporting electrolyte, unit
activity
coefficients, absence of convection. Any
departures
from these
conditions
will
be reflected in departures
from
the predicted theory.
2 Notice
that
Warburg impedance
is
strictly valid only
when
4*^=0.5. Unfortunately, in the study of
diffusion processes by AC techniques,
values of 0.5 are the exception rather than the rule [13]. By
introducing
a
generalized form of the Warburg impedance the physical meaning of each of the involved
parameters may change. For example, variations in the *F
ZC
value have been related to the presence of
irregularities in the electrode surface at the micrometer level. Also,
D
M + 2
no longer represents
an
absolute diffusional coefficient but rather an integral
value
related to all the ionic species present across
the
diffusion
layer.
3 Lack of well-defined hydrodynamic conditions could be attributed to this large variation in the observed
diffusion coefficients. As can be seen in Eq. 9
when 4 ^
departs from 0.5
it is
not possible to obtain
the
diffusion coefficient unless the thickness of the Nernst boundary
layer
is known.
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Impedance spectra obtained in an electrolyte without addition age
6.eS|-xE-2
i.ee
• 2.SB
• 6.28
• 13.78
D 39.65
4 99.58
a 258.0 628.
• 888.
4> 1487.
xE-2
Fig. 17 Impedance diagram of purelead in the presence of an anodiccurrent 1=150 Amp m'2 (aftersubtracting the R, value).
Experimental conditions: as describedin Fig. 15.
Some frequencies (in rad sec"1) areshown for both the experimental (solidline) and the regressed data (dottedline).The impedance curve was fitted to aCPE element from which thefollowing values were obtained:
B,=0.066 Q cm1 sec"y2C and Vv-OAL
Quality of Fit
Parameters (63
experimental points were fitted to theCPE analogue):rV=0.973, ly.l^.OSxlO^ Q2cm 4
r 23=0.988 Iy3r=8.95xl0"5 Q2cm 4
^=0.988 ly J 2=2.33xl0^ ftW
4.88 6.88
the exchange current density, i^, and the double layer capacitance, C^, could have
been obtained (to obtain the so-called Randies circuit [161). Evidently, for lead
dissolution in the absence of addition agents rjac->0 (i.e. *«-*»).Table 2 Variation of the B, and ^Vjc fitting parameters with current density
CurrentDensity, Amp/m 2
0 0 100 150 150 200
Bi,
Qcm 2secyzc
0.139 0.139 0.074 0.066 0.070 0.056
0.50 0.50 0.43 0.43 0.43 0.43
AC impedance curves obtained ~2000 sec after the application of the set current density.The amplitude of the applied AC waveform was set to 21.3 Amp/m 2 RM.S.The experimental impedance curves were fitted to Eq. 6 using between 60 and 66 pointsRegression coefficients and residuals for the regression curves were ( l<co<2000 rad/sec ]:r 29t>0.96 with 1 y„ 1 a < IO"3 Q2cm 4
r 23>0.95 with 1 y 312 < 10 4 Q2cm 4
A C impedance curves obtained while a net anodic DC current was applied
were similar to those observed under rest potential conditions (Fig. 17). Upon
application of current the AC impedance decreases with respect to the impedance
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Impedance spectra obtained in electrolytes containing addition ag
observed In the absence of a net DC current. Straight lines with slopes close to
39° were obtained (i.e. *Fzc~0.43) while the B x values were between 0.07 and
0.05 Q cm2 sec'** and appear to change with current density (see Table 2).
C.
Impedance spectra obtained in electrolytes containing addition agents
Addition agents are known to affect the electrochemistry of lead
fundamentally through changes in its kinetic parameters [1.17.18]. The 'levelling"
phenomenon found in electrodeposits and attributed to additives is partially due
to a re-distribution of current from changes in the inhibition intensity [19,20] \
3 . 0 0 I —
2 . 40
V
I
«81<*C...NI
X . 80
1 . 20
. 6 8 8
. 0 0 0
r a d / s e c. 8 6 3
« . 4 8
•
2 . s e
•
I S . 7 8
•
9 9 . 3 8
•4 6 2 8 .
3 9 6 S
.
O
2 5 8 1 5 .
1 3 7 8 3 4 .
*
1 9 8 7 8 6 .
. 8 8 8
Fig. 18 Impedance diagram of pure lead under rest potential conditions (in the presence ofaddition agents)
Experimental conditions: electrode area 2.34 cm 2, [PbSiF6]=0.37 M, [H2SiFe]=0.82 M,
[SiO2]=0.13 M, T=40±1.5°C, =2 g l" 1 aloes and =4 g l" 1 lignin sulphonate, beakerelectrochemical cell, electrolyte volume =310ml, bulk electrolyte recirculation rate =6
ml min'1. Solartron Electrochemical Interface-IEEE Card-IBM XT computer. Impedance curveobtained under galvanostatic control, AC waveform amplitude 21.3 Amp m"2 RM .S.
A total of 130 experimental points were obtained. Some of the frequencies (in rad sec"1) at which these points were sampled are indicated in the diagram ( =1.74 Qcm 2).
1 "Inhibition intensity"
in
Winand's terms [19,20] includes
activation
overvoltage as
well
as other
polarizations caused by addition agents.
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Impedance spectra obtained in electrolytes containing addition age
The impedance spectrum obtained when addition agents are present in the
bulk electrolyte is shown in Fig. 18. There are big differences between this
spectrum and that obtained in the absence of addition agents (compare Figs. 15
and 18). The presence of two weU-defined arcs can be seen in Fig. 18. The arc
observed at large frequencies (a»4000 rad/sec) is related to the charge transferfor the Pb/Pb + 2
reaction [i^, C ) whereas the arc which spans over frequencies
smaller than 4000 rad/sec is associated with the presence of an adsorbed layer
of addition agents that affect the movement of ions from/towards the electrode
surface (and the region where concentration gradients can be present) \ The Rg
value was not affected by the presence of the addition agents.
u
ICi
91t
£... N
I
8 . 0 8 i—
6 . 4 0 —
4 . 8 8 —
3 . 2 8
1 . 6 8
. 0 0 0
r a d / s e c
A . 8 6 3
• . 4 0
• 2 . 5 0
• 1 5 . 7 8
• 9 9 . 3 8
-4 6 2 8 .
6 3 9 6 5 .
O 2 5 0 1 5 .
• 1 5 7 8 3 4 .
*
2 2 2 9 5 3 .
2 1 . 3 Anp/n2
4 . 3 A M P / M1
J I L
. 8 8 8 4 . 8 8 7 . 2 0
Z r e a l ft-CM*
9 . 6 0 1 2 . 0
Fig. 19 Impedance diagrams of pure lead under rest potential conditions obtained at two
different amplitudes of the applied AC waveform (in the presence of addition agents)
Experimental conditions: as described in Fig. 18.
The AC amplitude values shown in the plot refer to their RM.S. value
A total of 130 experimental points were obtained. Some of the frequencies (in rad sec ]) at
which these points were sampled are indicated in the diagram.Ra. was subtracted from both impedance curves.
1 This low frequency arc was only observed
after
a small Faradaic current was applied
(less than
50
Amp/m
2
for 400 sec), after which, it was
always
present in the impedance curves obtained under
rest
potential conditions.
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Impedance spectra obtained in electrolytes containing addition age
Under rest potential conditions, a decrease in the amplitude of the applied
A C waveform produced a decrease in the size of the low frequency arc while the
high frequency arc remained virtually unchanged (Fig. 19) . It appears as if by
decreasing the amplitude of the A C waveform, the size of the region where
concentration gradients are present expands resulting in apparent increases inthe impedance.
(A) (B)
Fig.
2 0 Analogue circuits used to model the high frequency response of the Impedancecurve shown In Fig. 15.
The analogue shown in (B) is obtained when Z C P E - ^ . This will happen at high frequencies orsmall values of the Bj parameter.
(A) Analogue used to represent the high frequency region of the impedance spectrum:
Z(j(6) = R,+ —
l+/?aC^O'(0) + C^iO'<fl)
(B) Analogue used to represent processes taking place in the electrical
double layer:
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Impedance spectra obtained in electrolytes containing addition ag
Analysis of the impedance spectrum obtained at high frequencies (co>500
rad/sec) was done by using the electrical analogue shown in Fig. 20. In this circuit,
Ret represents the charge transfer resistance which is related to i„ through the
following equation [21]
The impedance of the analogue circuit shown in Fig. 20A can be described
as follows 2 :
Z(/co)=/?,+ lVJ — ...11
The CPE included in the analogue shown in Fig. 20A is used only to subtract
the high frequency part of the arc observed at low frequencies from the charge
transfer arc. This CPE is not intended to represent any particular process but
only to subtract the higher frequency data.
The values of the analogue elements obtained from the curve fitting process
are as follows (Fig. 21): R ct= 1.665 Qcm2
(±5%), Cd ,=20.6 fiF/cm2
(±3%),
B!=149.25 Qcm 2 sec>l'zc
(±5%), and =0.74(12%). From the Revalue ^ was found
to be equal to 82 Amp/m 2 (±5%). For the same electrode these values were
reproduced within 15%. On the other hand, small changes in the bulk electrolyte
composition, electrolyte temperature, bulk electrolyte recirculation rate, and
electrode roughness appreciably affected the and C values. In any case, for a
fixed electrolyte composition, i„ values were not smaller than 70 Amp/m2
(70<(o<500 Amp/m 2) whereas C values were between 18 and 30 |±F/cm2
. Since
in the absence of addition agents a charge transfer arc was not found (io-x»), it
is concluded that the presence of the addition agents affects significantly the
kinetics for the Pb/Pb + 2 equilibrium reaction. The decrease in the *<, values as a
result of the presence of addition agents can be attributed to: (A) changes in the
1 The Stern-Geary equation has also been used to
relate
the steady-state corrosion current density, 4and the polarization resistance, R
p
[25]:
\ P
'P
< I ' 1_N
^corr 2.303(8, + BC)
2 Analytical representation of the impedance was obtained by using the Laplace plane techniques
described in Appendix 3.
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Impedance spectra
obtained in
electrolytes containing
addition ag
Fig. 21 High frequency section of the impedance diagram shown In Fig. 18.
Impedance curve was fitted to the electrical analogue shown In Fig. 20A (R. was computedfrom the high frequency Intercept of the Impedance curve with the Z, axis).From the curve fitting process, the following values were obtained: B,= 149.250 Q cm
2 sec"¥zc,^=0.741, F^l.SSSQcm 2, 0 =20.6 uF cm 2.
Some frequencies (in rad sec"1) are shown for both the experimental (solid line) and theregressed data (dotted line).
Quality of Fit Parameters (51 experimental points were fitted to obtain the values of 4 parameters):
rVO.973, l y a l ^ ^ x l O1 Q2cm 4
rVO.941 I y312=6.16xl0"2 Q2cm 4
^=0.985 ly^l^l . iexlO^Q'cm 4
electrochemically active surface area and (B) variations in the current distributionin the anode vicinity. An effective decrease in the value (considering that the
electxochemlcally active surface area is the same in the cases presented in Figs.
16 and 21) would mean that the kinetics for lead dissolution and deposition have
become "less" reversible.
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Impedance spectra
obtained in
electrolytes containing
addition
ag
The AC arc observed at co<4000 rad/sec is more difficult to model by an
electrical analogue. Its presence is associated with abounded concentration region
created by addition agents, some of which adsorb on the electrode surface
changing the and values. The "spikes" observed during the application of
the current steps (see Section Il.b) are undoubtedly associated with thephenomena displayed by the A C impedance curves.
2 .ae
1 .68 —
Y 1 .28
a
91
£
N .888
, 488
.888
r a d / s e c A .863
•
.48• 2 .58• 15 .78
• 99 .384 628 .
3965 .
0 25815.
157834.
*
198786.
R e s t P o t e n t i a l C o n d i t i o n s c d = 8
In the p r e s e n c e of a c d = 188 flny/h'
.A
J L_l_ - i I L_ I I I I I.888 . 788 .48 2.18
Z r e a l ft-CM*
2 .88 3 .58
Fig. 22 Impedance diagrams of pure lead obtained in the presence and in the absence of anet Faradaic current (in the presence of addition agents)
Experimental conditions: As described in Fig. 18.
The AC amplitude was the same in both cases (21.3 Amp m 2 RM.S)
Rs was subtracted from both impedance curves.
In the presence of a net anodic current, the impedance decreases significantly
(Fig. 22). As the current density increases, the size of the arc observed at high
frequencies decreases while the low frequency arc does not change to the same
extent (Fig. 23). Eventually, at high current densities (cd >200 Amp/m 2) the high
frequency arc vanishes and only one arc related to diffusion in the anode boundary
layer is observed. Thus, it appears as if during the anodic dissolution of lead the
polarization created by the addition agents decreases as the current density
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Impedance spectra obtained in electrolytes containing addition age
increases *' a. The decrease in the size of the low frequency arc may be the result
of a defined boundary layer whose size has compacted due to the presence of a
fixed electric field created by the passage of a net Faradaic current3
.
I
a91
£
•p.
N
.see A r i d / t i c.863
• .31
_•
•
1 .587.91
. 168
-
•4
39 .63
199 .a 996 .
— O 4991 .• 9959 .
. 128 -*
15783.
. 888
.848
.888
In the p r e s e n c e of a c d = 58 Anp/n>
In the p r e s e n c e of a c d = IBB fthp/h'
I I l
.888 . 868 TJ 128 .188
Z r e a l ft-CM*
, 248 . 388
Fig. 23 Detail of the Impedance diagrams obtained In the presence of a net Faradaic current(in the presence of addition agents)
Experimental conditions: As described in Fig. 18.
The AC amplitude was the same in both cases (21.3 Amp m"2 R.M.S)
R, was subtracted from both impedance curves.
1
The exchange current density is also a function of the concentration of the electroactive ion in the
electrode surface [22-24].
2 Notice how the radius of the high frequency arc
obtained in
the presence of a net Faradaic current
(^-.08 Qcm2
, Fig. 23) is at
least
20 times smaller than under
rest
potential conditions (Ret~1 -6 Qcm
2
,
Fig.
21). The "apparent" charge transfer resistance decreases until at high current densities R ct ~0. The
term "apparent" charge transfer resistance refers to the R* values that
may be observable
upon
passage
of
a net Faradaic current
when
studying irreversible systems
using AC techniques.
3 Notice
that
addition agents are also dispersed by the anodic process because: (A) the interface is
retreating, and (B)
there
is a flux of Pb
+2
in
the opposite
direction.
Thus, decreases
in
the size of the low
frequency arc as the current density increases can be due to a depletion of addition agents in the anode
boundary
layer.
[114]
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Chapter 6 Ele ct roc hem ic al Behavior of Lead bul l i on Electrodes in
the Presence of Sl i mes
I. Introduct ion
In the previous Chapter it was shown that in the absence of addition agentsin the bulk electrolyte, the Pb/Pb + 2 system behaves nearly reversibly (i.e.
PboPb+2
+2e~ with i,,-**0
). The presence of addition agents in the bulk electrolyte
decreases the reversibility of the system through small increases in the activation
overpotential. Moreover, the addition agents were found to enhance the
concentration gradients at the electrode/solution interface through the formation
of specifically and/or electrostatically absorbed films. This Chapter deals with the
establishment of concentration gradients within the slimes layer and their
relationship to the anodic overpotential observed in the presence and in the
absence of a net Faradaic current.
All experiments shown in this Chapter were performed at T= 40±l .5°C . using
the beaker electrochemical cell (Fig. 3.1) and the Solartron equipment (Fig. 3.8).
Data acquisition and control of the experiment were performed using an IBM XT
personal computer via the IEEE interface (Fig. 3.7). Electrodes were prepared from
the same anode1 (anode A, Fig. 3.3) as described in section 3.1 .B. Electrolyte was
recirculated continuously at 6 rnl/min (cell volumes varied between 300 and 320
ml). The geometrical area of the electrodes2
was used to compute the current
density and the impedance per unit area. In experiments in which addition agents
were added to the electrolyte, any insoluble precipitates were removed prior to
the introduction of the electrolyte to the cell. The assumed concentration of
addition agents prior to the filtering operation was 2 g/1 aloes and 4 g/1 lignin
sulphonate. The compositions of the electrolytes used in the various experiments
presented in this Chapter are shown in Table 1.
1
Anode composition: 0.01% Sn, 0.02% Cu, 0.14% Bi, 0.25% As, 1.12% Sb, 81 oz/ton Ag.
2
The
geometrical area
of the
electrodes
was 1.44±0.02 cm
2
.
[115]
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Introduction
T a b l e 1 Characteristics of the experiments presented in chapter 6
ExperimentNumber
Bulk Electrolyte Composition Bulk
electrolyte
Conductivity,K ,atT=40
#
C,
mmhos/cm
Characteristics ofthe experiment
ExperimentNumber
ElectrodeFace
AdditionAgents
[PbSiF6]mol/1
[HaSiFe]mol/1
[SiOJmol/1
Bulk
electrolyte
Conductivity,K ,atT=40
#
C,
mmhos/cm
Characteristics ofthe experiment
CA2 mould yes 0.27 0.69 0.14 300 Galvanostaticdissolution at1=200 Amp/m
2
CA4 mould yes 0.45 0.77 0.14 315 Potentiostaticdissolution at
Econtn>l=220 mV
CA5 mould yes 0.45 0.76 0.14 315 Galvanostatic
dissolution at1=800 Amp/m2
CA6 mould yes 0.45 0.73 0.14 320 Galvanostaticdissolution at1=200 Amp/m
2
CC1 air no 0.35 0.84 0.09 345 Galvanostaticdissolution at1=200 Amp/m2
CC2 air no 0.36 0.83 0.09 345 Galvanostaticexperiment.
In the presence of a net Faradaic current the impedance spectra were obtained under galvanostatic control
except in Exp. CA4 in which the curves were obtained under potentiostatic control. Under current
interruption conditions all the impedance spectra were obtained under Galvanostatic control.
The A C impedance was obtained in a wide frequency range (0.063<co<4xl05
.. rad/sec) 1
under either potentiostatic or galvanostatic control. The amplitude of
the applied A C waveform was set to 5 mV R.M.S. and 35 Amp/m2
R.M.S.
respectively. In either case, 20 data points were obtained per decade of frequency
swept2. All impedance spectra are reported with respect to the time at which the
A C measurement started. Typical A C measurement times are shown in Table 2.
1
Under galvanostatic conditions application
of AC
frequencies in excess of
3x10
4
rad/sec often resulted
in
phase shifts
produced
by the Solartron Electrochemical Interface due to bandwidth limitations. On the
other hand, under potentiostatic control, frequencies as high as 4x10
s
rad/sec could be applied without
observing phase shifts.
A phase shift is a displacement of the capacitative component of the impedance
curve towards negative values [1-3].
2 For example if 0.063<ox6300 there are l og^j = 6 decades of frequency swept and the impedance is
measured at
6x20=120
discrete points.
[116]
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AC Impedance Characterization of the Starting Working Electrode
T a b l e 2 Average times required to measure the AC impedance spectrum
Frequency Range
Time, HrsO W rad/sec o w . rad/sec Time, Hrs
0.063 408407 l . l
0.63 408407 .30
n. A C Impedance Charac ter izat ion of the Sta rti ng Work ing
Electrodes.
Prior to the anodic dissolution of lead bullion electrodes their impedance was
obtained under rest potential conditions. Table 3 summarizes the characteristicsof the obtained spectra 1 . The differences between the various spectra are
explained in the next paragraphs.
Table 3 Summary of the values of the electrical analogue parameters obtained under rest potential conditions
Frequency Range Derived Analogue Parameters
Experiment Addition Pot/Gal R., Ra,
On, ^zc
and Sweep Agents Control rad/sec rad/sec Qcm2 Qcm2 u F c m 2
Qcm
2
sec'
Amp/m2
Number
CA2-7 yes Gal 560 99588 1.42 0.501 64 23.71 0.64 269
CA4-7 yesllltllll
315 250159 1.02llliill iilslii! 1X45 0.39 72
CA5-7 yes Pot 560 111740 1.19 0.725 65.30 23.90 0.64 186
CA5-4 yes Gal 560 111740 48.9liiiilll
0.56 157
CA6-2 yes Gal 177 157834 1.20 1.916 32.6 35.5 0.55 70
CC1-5 no Gal 628 15783 0.90 llliill 0.57
CC1-6 no Gal 628 44482 0.89 - :
2.88 0.45
CC2-7 no Gal 560 17709 0.85 1.88 0.59
CC2-2 no Gal 560 28067 0.85 1.88 0.59
Different electrodes were used in every experiment.Impedance curves were obtained either under Potentiostatic (Pot) control or Galvanostatic (Gal) Control.
Changes in the R, values reflect differences in the distance between the reference electrode and the working
electrode.
The spectra obtained in the absence of addition agents were curve fitted to the impedance function described by
Eq. 1 while the spectra obtained in the presence of addition agents was fitted to the function described by Eq. 2.
1 The statistical parameters related to the quality of the
curve
fitting
procedure
are
presented
in
Appendix 9.[117]
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AC behaviour
in the
absence of addition agents
in the bulk
electroly
A. AC behaviour in the absence of addition agents in the bulk electrolyte
The AC impedance spectrum of a typical lead anode (in the absence of slimes)
under rest potential conditions and in the absence of addition agents is presented
in Fig. 1 a * 3 . The uncompensated ohmic resistance, Rg, has been removed in this
impedance diagram by subtracting it from the component of the impedance 4 .
This impedance spectrum is similar to the obtained using a pure lead electrode
(see Fig. 5.16) and can be described by a Warburg serni-infinite diffusion element
which follows the response of a CPE element 5 :
The impedance curve shown in Fig. 1 was curve-fitted to Eq. 1 to obtain the
Bi and values. B x and *¥zc were found to be equal to 2.88 Q. cm2 sec*Vzc
and 0.45
respectively. By comparing these values with those obtained when pure lead was
studied (Bi-0.14 Qcm2sec"M'ZC7 and ^^-0.5) it is seen that remains practically
unchanged, while Bj shows a marked increase in its absolute value 5 . The increase
1 Due to software limitations in the graphics program the axes of the Argand plots are not marked and
-Zj as
in
the main
text
but as
Zreal
and -Zimag respectively.
2 An impedance diagram is also known as an Argand plot.
3 FL=0.90
Qcm
2
.
R
s
was obtained from the high frequency interception of the impedance
curve
with
the
real axis.
4 For a description of the relationship between the CPE element and the semi-infinite Warburg
impedance see Chapter 5 (section III.B, Eqs. 6 to 9).
5 Different CPE analogues may
have
the same slope but quite different B, values. By
using
de Moivre's
theorem, the impedance of the CPE analogue can be expressed as follows:
thus,
Z a ^ w ^ c o s ^ V z c Z 3 = -B 1 (o"
¥ z c
ysin^ 2 c
and the slope, m, between Zj, and -Zj is given by:
Jt
m = tan-Yzc
from which:
2 i
V z c = -tan m
Thus, it can be seen
that
*P
ZC
does not depend on B, which is only a multiplying factor (i.e. the impedance
curve
shrinks or contracts according to its
value).
[118]
ZaB = Bl V0i)* = Bl <Q
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AC behaviour in the absence of addition agents in the bulk electrolyte
. 1 38 p i t 8
. 128 —
E x p e r i M e n t a l — • f l n a l o o u t nodel
A.
•
•
aa0
•
628.
1117.
1987.
3533 .
£ 2 8 3 .
11173
19869
33333
28866
44482
.888
.888
J L
' • • ' I • • ' I ' ' ' I ' ' ' I
.1 28 . 168 .288
Zred, Qcm 2
Fig. 1 Impedance diagram of a typical lead bullion electrode (Exp. CC1-6) under rest potential conditions and In the absence of addition agents.
R, was subtracted from the Z* component of the impedance (Fv=0.89 Qcm 2).Some frequencies (in rad/sec) are shown for both the experimental (solid line) and the curvefitting data (dotted line).The impedance curve was fitted to a CPE element from which the following values were obtainedB!=2.88 Qcm 2sec"* ^=0.45.
in the B x value is attributed to differences in the roughness and electjochemically
active surface areas between these electrodesl. Also, the distinct electrochemical
characteristics of the impure lead bullion electrodes may account for the observed
increases in the B x values. As indicated in Table 3 (see sets CC1 and CC2), the
Bj values change significantly from one experiment to the next but remain within
0.2 iicm 2 during the same experiment. The relatively constant values of the
exponent (0.40<*FZC<0.60) is a clear indication that diffusional processes are the
only ones being observed in the impedance spectra. While quantitative values of
1 As a result of surface irregularities at the micrometer level, the thickness of the electrode boundary layer
may not be uniform
and
that
results in increases
in
the
B,
value.
1119]
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AC behaviour in the presence of addition agents in the bulk electroly
the diffusion coefficients cannot be obtained 1
the shape of the impedance curves
unequivocally indicates that there are no charge transfer limitations for the
dissolution/precipitation of Pb + 2
from/to the lead bullion electrodes.
B. AC behaviour in the presence of addition agents in the bulk electrolyte
The AC spectrum obtained in the presence of addition agents is shown in
Fig. 2. Three different regions can be identified depending on the frequency range:
a) A high frequency arc (co>4000 rad/sec) assigned to the charge transfer
process taking place across the electrode Helrnholtz double layer. The point where
the arc intercepts the Z% axis is equal to the Rg value.
b) A distorted arc observed at medium range frequencies (100<co<4000
rad/sec) assigned to addition agents effects in the diffuse double layer.
c) A quasi-linear impedance region (at ox 100 rad/sec) assigned to diffusionalprocesses in the hypothetical Nernst boundary layer.
r a d / s e cA * 06 3
* .40
* 2 .50• 15 .78
• 9 9 . 5 8
4 6 2 8 .
a 3 9 6 5 .
O 2 3 8 1 3 .
• 6 2 8 3 2 .
* 9 9 5 8 8 .
I I I I
I I I
I
I I I I
I I
l_l I
I I I I I I I I I ' I I ' ' I ' ' ' I i i ' I1.28 1.88 2. 48 3. 88 3. 68 4.2 8
Fig. 2 Impedance diagram (Argand plot) of a typical lead bullion electrode (Exp. C A2-7) under rest potentialconditions and in the presence of addition agents (R, was not subtracted from Z^)
A . D O
1 .68
S i
. 48 8
ana
1 The
lack
of knowledge of the electrochemically
active
surface area poses a serious hindrance
for
the
computation of the
diffusion
coefficient: the penetration depth of the AC wave and the thickness of the
boundary layer have to be uniform across the electrode in the whole range of frequencies for meaningful
diffusion
coefficient
values
to be obtained.
[120]
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AC behaviour in the presence ol addition agents in the bulk electrolyt
Analysis of the impedance spectra obtained at co>200 rad/sec was carried
out using the electrical analogue shown in Fig. 5.20. The impedance of this
analogue can be described by the following equation l - 2 :
Z(/co) =/?,+-^ e , + fl1(/Q>)
The Rg values were obtained directly from the high frequency intercept of the
impedance curves with the axis. The four remaining parameters in Eq. 2 (R^,
C^, B l t and zc). were obtained by curve fitting the experimental data to Eq. 2.
Su
aSo
1csf
3 6 8 .
11.17 .
•
2 2 2 9 .
a
4 4 4 8 .
• 8 B 7 6 .•* 1 7 7 B 9 .
a 3 3 3 3 3 .
o 71364.>• 6 2 B 3 2 .
* 9 9 3 3 9 .
a a ' ' ' ' _L_i_
J I I I I I I I I L_ '
. . . ' . . . I
. 42
,2
Z^,, Qcm2
Fig. 3 Detail of the high frequency region of the Impedance diagram shown in Fig. 2
Rg was subtracted from the Z* component of the impedance (Rs=1.42 Qcm 2).
Some frequencies (in rad/sec) are shown for both the experimental (solid line) and thecurve fitting data (dotted line).
The impedance curve was fitted to the analogue circuit shown in Fig. 5.20 from whichthe following values were obtained B,=23.71 Q cm
2
sec'*^. =0.64, Fc^O.501 Qcm 2, and 0^=64
uF cm'2.
As can be seen in Figs. 3 and 4, the analogue model describes accurately
the experimental data. The values of the analogue parameters varied from one
1
For
a
description of the characteristics
of
this circuit
see
Chapter 5 section
III.c.
2 Again, it
is worth repeating that the presence of
a
CPE in the circuit shown in Fig. 5.20 does not
represent a purely diffusional process and is included only to aid in
the curve
fitting procedure to obtain
the
R
d
and
C
d
,
values.
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AC behaviour in the presence of addition agents in the bulk electroly
G B I — x E B
Ol
8
r A d / f i c
sea.*
1 1 1 7 .
•
3 3 3 9
.
D
4 4 4 8 .
• 8 8 7 6
.
4 1 7 7 B 9 .
a 3 3 3 3 5 .
a 7 8 9 8 4 .
7 0 3 8 4 .
*
1 1 1 7 4 0 .
x E 8
-J— 1—I — l — l — l— 1 —l—l—l I l I 1 I I I I I l l I I l l l I l l l I l l l I l l i _ J. 0 0 . 2 0 . 4 8 . 6 8 . 8 0 l . O
Fig. 4 Detail of the high frequency regions of the impedance diagrams obtained underrest potential conditions
(A) Exp. CA4-I fB) Exp. CA5-J
The values of the derived analogue values are shown In Table 3.
experiment to the next. Nevertheless, the impedance curves reproduced within
10% the indicated parameter values in the same experiment (i.e. for the same
electrode and electrolyte composition).
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AC behaviour in the presence of addition agents in the bulk electrolyt
In experiments CA4, CA5, and CA6 the electrolyte composition was kept
constant, yet, the derived kinetic parameters varied widely \ These variations
can be attributed to differences in the electrode roughness among the different
electrodes a
. In any case, the exchange current densities were not lower than 70
Amp/m
2
(70<io<270 Amp/m
2
) while the double layer capacitances varied between25 and 66 |±F/cm
2 3
. These large values of ^ indicate that dissolution of the lead
present in the lead rich phases takes place under nearly reversible conditions
and that the noble impurities present in the lead bullion are not significantly
affecting the kinetics for lead dissolution (or deposition).
a a rad/se o
. B63
. 4 a•
2 . 30
- IS . 78
e a - • 99 . s ee a- 4
£28 .
z 3965 .
O 238 13.
»- 63833.
-*- 99388.
. 4 o e
— Potvirit i os-t.-axt.ic: Control
- GAIvAnosiatlo Control
Zrea,, Qcm2
Fig. 5 Impedance spectra obtained under potentiostatic (solid line) and galvanostaticcontrol (dashed line).
From Exp. CAS (Sweeps CA5-1 and CA5-4 in Table 3).
Finally, the impedance spectrum obtained under potentiostatic control was
found to be a variation of that obtained under galvanostatic control (Fig. 5). As
in the pure lead case, the observed changes are related to different concentration
1 The same electrolyte was used in Exps. CA4, CA5, and CA6 but since the amount of solids filtered prior
to the introduction of electrolyte to the cell varied in each case, the final addition agent content may vary.
2 Small
changes in the bulk electrolyte recirculation rate, electrode microstructure, and cell temperature
may also produce the observed changes in the parameter values.
3 Notice
that
large capacitance
values
are associated with high exchange current densities indicating
that
electrochemically
active
surface
areas
are different in
every
case.
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Variation of trie anodic overpotential as a function of the electrolysis time and the current interr
waves created by the different amplitude of the perturbating signal \ The
parameters obtained from both curves are close to each other indicating that
similar information was obtained from both techniques (see Table 3, Exp. CA5
sets 1 and 4).
HI. DG and AC Studies i n Corroded E lect rodes
A. Studies under galvanostatic, potentiostatic, and current interruption
conditions.
1. Experimental results
Lead bullion working electrodes were either galvanostatically or
potentiostatically dissolved. The movement of the anode/slimes interface was
computed assurning that 100% of the current flow was at that interface
2
. Duringthe dissolution process, the A C impedance was measured at preset slimes
thicknesses. After dissolving the electrodes up to a certain slimes thickness the
overpotential decay was followed as a function of time. During this decay, the AC
impedance was also measured. The D C current and anodic overpotential recorded
during the A C measurements were analyzed to obtain the components of the
concentration overpotential.
(a) Variation of the anodic overpotential as a function of the electrolysis
time and the current interruption time
Fig. 6A shows the anodic overpotential response observed during the
galvanostatic dissolution of lead at a current density of 194.44 Amp/m 2
(Exp. CA2 in Table 1). The cell potential follows closely the recorded anodic
overpotential values (Fig. 6B). Upon subtraction of the initial T ]Q from both
measurements, it is seen (Fig. 6C) that even though a net current was being
passed through the counter electrode (a pure lead cathode), it can act very well
as a reference electrode due to the high reversibility of lead in this system.
1
Theoretically both curves
should
have
been identical if the processes under study behaves linearly.
Instrumental
artifacts (i.e. Potentiostat bandwidth) may also contribute to the observed discrepancies.
5 mV R.M.S does not exactly produce a sinusoidal current waveform of 35 Amp/m
2
of amplitude and
vice
versa.
2 The fraction of electronic current
going through
the slimes filaments was assumed to be negligible
compared to the amount of current
crossing
the anode/slimes interface. The validity of this assumption
was confirmed by measuring the distance between the anode/slimes electrolyte interface and the
slimes/bulk electrolyte interface at the end of the experiment
which
was in agreement with the computed
value. Furthermore, the slimes
composition
does not seem to change much
with
slimes thickness.
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Variation of the anodic overpotential as a function of the electrolysis time and the current interrupt
(A) (B)
1
acu-Mo
&1>
o
o
1.88 3.SB 3.4B 7.28 9.88 -.BBS 1.88 3.68 3,48 7.28 9.88
mm Slimes(C)
mm Slimes
I73
C-Mo
&Sio
o
o
70B .
360
420
200 . —
140 . —
. 00
Fig. 6 Overpotential response of a typical lead bullion anode as a function of the slimesthickness (Exp. CA2).
The AC impedance was measured at preset slimes thicknesses indicated by the spikesshown in the curves.
CA) Anodic overpotential as a function of the slimes thickness (uncorrected for initial
(B) Cell potential as a function of the slimes thickness (uncorrected for initial T IJ.(C) Anodic overpotential measured by the counter and reference electrodes as a function
of the slimes thickness (corrected for Initial TIJ.
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Variation of the anodic overpotential as a function of the electrolysis time and the current interr
Further analysis of Fig. 6 shows that as the anode dissolves, T|
A
increases
quasi-linearly up to the point at which impurities dissolution occurs at a
significant rate (at ~8 mm slimes and -350 mV) \ At this point, increases in riA
are also the result of the precipitation of secondary products (such as PbF2 and
SiCy which hinder the movement of ions and increase the concentrationgradients across the slimes layer.
In Fig. 6, the small departures of the anodic overpotential (so-called
"spikes") result from the application of the AC waveform used to measure the
impedance of the system at preset slimes thicknesses. Such overpotential
changes were detected only at low AC frequencies (ox6.3 rad/sec). Potential
variations produced by the AC waveform at higher frequencies were either very
small or were undetected by the potentiostat digitized readings a
.
Fig. 7 shows the anodic overpotential changes (corrected from initial rj J
observed during the dissolution of lead at a current density of 800 Amp/m 2
(Exp. CA5 in Table 1) . By comparing the r|A response at low and high current
densities (Figs. 6C and 7 respectively), it can be seen that increasing the current
density resulted in decreasing the time required for impurities to dissolve at
excessive rates 3
. Moreover, increases in current density do not appear to have
a significant effect on the critical point at which the anodic overpotential rises
exponentially. This indicates that if impurities are to dissolve to a large extent,
a minimum overpotential value must be overcome.
The large T|
A
values obtained at high current densities as compared to those
values obtained at lower current densities, arise primarily as the result of
changes in the concentration of the slimes electrolyte. Also, at high current
densities steeper concentration gradients become established. This results in
potential differences that increase monotonically with the concentration
gradients. Steeper concentration gradients develop larger potential differences
1
Abrupt
dissolution
of phases containing noble impurities was not observed in the case study presented
in Chapter 4 because a low
current
density was applied, the electrode was larger, only a 14 mm thick
slimes layer was formed, and the maximum value of T|
a
was <200 mV.
2
The potentiostat follows continuously (among other parameters) the difference in potential
between
the
reference
electrode and the current. At finite sampling
times
(i.e once every 3
min.),
these data are
digitized
and saved as "DC" data.
If
during the digitization process the AC waveform was being applied,
a
net DC overpotential and DC current
may be observed as "spikes".
3 A 400% increase in the
current
density (from 200 to 800 Amp/m
2
) resulted in
a corresponding
reduction
in the amount of lead that
could
be removed before impurities dissolve at an excessive rate (i.e. at similar
T1 A values, the
equivalent
amount of
lead
dissolved is
~4 times smaller).
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Variation of tie anodic overpotential as a function ol the electrolysis time and the current interrupti
.88.aa
.68 1.2 1.8
rnrn Slimes2 . 4 3 . 8
Fig. 7 Anodic overpotential (corrected for initial TIJ measured by the counter andreference electrodes as a function of the slimes thickness (Exp. CAS).
Galvanostatic conditions 1=800 Amp-m 2
Detween the slimes and the electrolyte promoting their earlier dissolution. In
addition, passage of larger currents increases the "ohmic" drop component of
the anodic overpotential \
Fig. 8A shows the changes in current density as a function of the
electrolysis time during a potentiostatic experiment (Exp. CA4 in Table 1). In
this experiment the dissolution of noble impurities present in the slimes layer
was restricted by liiinting the potential difference between the reference electrode
and the lead anode to 220 mV. From the numerical integration of the data
presented in Fig. 8A, the amount of lead dissolved as a function of the electrolysis
time was found (Fig. 8B). At short electrolysis times, large current densities can
flow because ionic transport proceeds relatively unhindered. As the slimes layer
thickens, its presence restrains the flow of current up to the point at which only
very small currents can flow through the cell. These decreases in current are
1 See
section
III.2.a
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Variation ol the anodic overpotential as a function of the electrolysis time and the current interrupt
( A ) ( B )
2.S <r£'3<
. 0 0 . E a 1 .2 JL . 8 2 . 4 3 . amm Dissolved
Fig. 8 Current density changes as a function of the electrolysis time and of the amountof lead dissolved (Exp. CA4. potentiostatic conditions Econtrol=220 mV)
IA) Changes in the anodic current density as a function of the electrolysis time(B) Changes in the amount of lead dissolved as a function of the electrolysis time. This
curve was obtained from numerical integration of the data presented in CA)(C) Current density changes as a function of the amount of lead dissolved
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Variation of the anodic overpotential as a function of the electrolysis time and the current inte
also the result of the presence of secondary products which block the movement
of ions even further until the lead dissolution process is nearly halted (Fig. 8C).
At long electrolysis times, lead ions are still being generated whereas
concentration overpotential no longer changes and the precipitation of
secondary products must take place for rjc
to remain constant.During experiment CA4 the difference in potential between the reference
and working electrodes was kept constant and equal to 220 mV (£ ^^,=22 0
mV). As a result of the presence of r|n between such electrodes, the potential
difference applied between the working electrode and the slimes/bulk electrolyte
interface is not constant. The anodic overpotential changes as a function of Rg
and of the current density can be described by the following equation:
TjA changes as a function of the amount of lead dissolved are shown in
Fig. 9 1
. The anodic overpotential increases continuously as the current density
decreases, up to the point at which current flow is negligible and r|A = Eco , ^.
Fig.
9 Changes in the anodicoverpotential (corrected foriln)
as a function of theamount of lead dissolved(Exp. CA4).
mm Dissolved
1
R„
was
obtained from AC impedance measurements
at
preset electrolysis times.
1129]
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Variation
of the anodic overpotential as a function of the electrolysis time and the current interruptio
(B)
786
548
1f-H
S <uo
&
I
u
o
228
68.1
-188
788
568
428
281
148
.88 .32 .64 .96 1.3 1.6
.u.!...,.»M; f M
24.8 48.8 72.8
Time, Hrs.
( C )
I73
fi<u-MO
&IT3
O
.88 18. 28. 38. 46. 38.
Time, Hrs.
Time, Hrs.
Fig. 10 Anodic overpotential changes upon current interruption (Exps. CA2, CA5, and CA4).
T | n prior to current interruption was not subtracted from these measurements
The AC impedance was measured at preset times indicated by the spikes shown in the curves.
(A) From Exp. CA2 (B) From Exp. CA5 (C)
From Exp . CA4
[130]
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Variation of the anodic overpotential as a function ol tie electrolysis time and the current interru
Changes in anodic overpotential upon c u r r e n t i n t ecases described previously (Exps. CA2. CA5, and CA4) are shown in Fig. 10. A
steep decrease in the overpotential is observed at the beginning of the
interruption cycle. This steep change is followed by a quasi-exponential decrease
in the r|A value. On the other hand, the shape of the r\ A
decay at times smallerthan 1.6 Hrs is different for the three cases shown in Fig. 10. This indicates
that upon current interruption, processes with different time constants can
take place. The presence of these processes appears to be a function of the r\ A
value and the electrode's history.
In the three previous cases AC impedance measurements did not seem to
affect the pseudo-equilibrium present within the slimes layer. On the other
hand, when the impedance measurements were made more frequently, transient
excursions in the anodic overpotential were observed (in Exps. CA6 and CC1,see Fig. 11). As in the case study presented in Chapter 4, these potential
excursions cannot be unambiguously explained. These excursions in potential
were reproducible and indicate that the changes created by the AC waveform
induce a shift of part of the Faradaic current towards the slimes filaments l
.
The fraction of the current going to the slimes filaments may be insignificant;
yet it increases the corrosion potential of the electrode. The transient character
of these excursions and the fact that after a certain time the overpotential
decreases to a value that can be obtained by extrapolation of the r|A curve (see
Fig. 1 IA), indicates that the reaction at the anode/slimes interface was taking
place at its normal rate even during the excursions in potential. Finally, upon
current interruption, the anodic overpotential decay did not indicate any abrupt
changes related to the potential excursions observed during the passage of
current (Fig. 12).
1 The AC wave
may
have
changed the conditions at the anode/slimes electrolyte interface by promoting
precipitation
and/or
hydrolysis reactions
at that
interface. Under
these conditions, Faradaic currents can
divert
to the slimes
filaments
and
cause
the excursions in potential
shown
in Fig. 11.
[131]
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Variation of the anodic overpotential as a function ol the electrolysis time and the current intenvpti
(A)
7 8 0 c -
(B)
7 B B . pr
5 6 8 ,
4 2 0
e*»oa.
01 2 8 0
»
O
9 . 0
. 0 0 0t
— L
. 0 0 0
_J
I
I I I I 1_
i . . 8 0 3 . 6 0 S . 4 0
S l l n e s T h i c k n e s s , nn
Fig. 11 Anodic overpotential (corrected for initial r\^j as a function of the slimesthickness (Exps. CA6 and CC1).
(A) From Exp. CA6 (B) From Exp. CC1
[132]
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Changes in the impedance as a function ol the slimes layer thickness and of the current interruptio
788 r
560< r
Z 428
SI
¥
0
J 288
J
0
1481
668 .
548'
428
388
189;
68..88 .82 .84 .86 .88 .18
Fig. 12 Anodic overpotential
changes upon current
interruption (From Exp. CA6)
T|Q prior to currentinterruption was notsubtracted from thesemeasurements
The AC impedance wasmeasured at preset timesindicated by the spikes shownin the curves.
128
Tine, Hrs
(b) Changes in the impedance as a function of the slimes layer thickness
and of the current interruption time
The AC spectra obtained at different slimes thickness are presented in
Fig. 13 (Exp. CA2). Each spectrum was taken in the p r e s e n c e of a net Far
current and at different stages of the electrolysis cycle (see Fig. 6). Both the
reactive (Z*) and the capacitative (-Zg) parts of the impedance increase as the
slimes layer grows.
The impedance spectra in Fig. 13 show the presence of a low frequency
arc which at high frequencies bends quasi-linearly towards the axis.
AC impedance spectra obtained in the overpotential region above 350 mV
are shown in Figs. 14 and 15 . These spectra are significantly different from
those presented in Fig. 13. The presence of high frequency arcs (i.e. arcs whose
time constant is at least of the order of msec) shown in the respective Bode plots
(Figs. 14B and 15B) indicates that reaction of noble compounds present in the
slimes layer takes place in this region.
1133]
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Changes in the impedance as a function olthe slimes layer thickness and ol the current interrup
I
ft
c
N
I
4 . 0 r* ad/s e c
—
•m- .31
•
1 . 56
• 7.91
3 . 2 a 39 . 65
<
199 .
£* 996 .
O 4991 .
»- 15783.
25015.2 . 4
1 . 6
M M SI i ne -s
A 0 . 8B 1 . 6C 2 . 2
n 3 . 1E 3 . 7F 4 . 7
G 5 . 3H 6 . 0
I 6 . 6
J 7 . 2K 7 . 8
000 5 . 40
ft-CM*
00
F i g . 1 3 Impedance spectra obtained during Exp. CA2 at slimes layer thicknesses between 0.8 and 7.8 mm.
Each impedance curve was obtained at a different slimes thickness as indicated in this Argand plot
[134]
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Changes in the impedance as a function ol the slimes layer thickness and ol the current interruptio
(A)
2 . a a
x . 6 8
Y 1 . 2 B
a
Dl
t
N . 8 0 0
r a d / s e a. 0 6 3
*
. 3 5
• 1 . 3 9
a 1 1 . 1 7
a 6 2 . 8 3-« 3 5 3 .
1 9 8 7 .
O 1 1 1 7 4 .3 9 6 4 7 .
6 2 8 3 2 .
1 . 2 8
. 4 0 0
8 0 0 1 — 1 » '—
I
— I—I—1—1
—I—I—I 1 I I I I I I I I I I I I I I I I
1_
(B)
- . 8 3 0
1 5 8
•0
1
u
91
Fig. 14 Impedance spectra obtained during Exp. CA2 at a slimes layer thickness of 8.4 mmi
(A) Argand plot (B) Bode plot
1 In a Bode plot, the
high
frequency
arcs
are
better
resolved by
analyzing
the
variations in
the phase
angle (dotted
line, right
vertical axis)
as a function of the logarithm of the frequency.
[135]
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Changes in
the impedance
as a
function
ol
the slimes
layer
thickness a nd
of
the current interruptio
(A)
2 . 0 0
X . 6 0
M
U 1 . 2 8
r A d / l • c
A . 0 6 3
» . 4 4
• 3 . 1 3
n 2 2 . 2 9
a 1 5 8 .
4 1 1 1 7 .
ti 7 9 1 1 .
O 3 6 8 8 2 .
¥• 2 2 2 9 3 3 .
* 3 5 3 3 5 4 .
1 bi4E
N . 0 0 0
1
s
~ ~~^
/
. 4 0 0
. 0 0 0
2 .
\• i l i i i l
i i
1
i i i
1
i i i
1
i i i
1
i i i
1
i i i
1
i i •
1
i i i
. 0 0 0
2 . 4 0 4 . 0 8 5 . 6 8 7 . 2 8 8 . 8 8 1 8
Z r e a l ft-CM*
(B)
. 4
1 0 .
"i " — ^ . 0 3 0
8 . 8 _
M
5 T • 21
a
N
a . 6
- » ^ ^ '
*
t1
- 1 - ' '
- . 0 3 0 /
f
~ "~ " '
" - ^ _ \ - - o^fo- X . t
x ** X . t
X /
X .
- . , 1 5 0x
>^ ' 1
1
1
1
1
1
1
1
P h a s e
A n g l e ,
r a d
4 . a
%
— > \ '
S 1
* \ . '
—
*- ^ ' ^ 1
I
I
2 . 4
- 1 . - 1 1 1 1 1 1 1 1
• i i i i i i i i i i i i i i
t
t i i i i i i i i i i i i i i
2 . 4
- 1 . 4 8 - . 8 8 8 1 . 4 8 2 . 8 8 4 . 2 0 5 .
L O G < F r e < i , r a d / s e c )
6 0
Fig. 15 Impedance spectrum obtained during Exp. CA2 at a slimes layer thickness of8.65 mm l.
CA) Argand plot (B) Bode plot
1 In a
Bode
plot, the high frequency
arcs
are better
resolved
by analyzing the
variations
in the
phase
angle (dotted line, right vertical axis) as a function of the logarithm of the frequency. Thus, for example, in
Fig.
15B a hump can be observed at log
oo
=
4.20.
Thus,
x=
6.3x10
s
sec (t is the
time
constant,
x =
co"
1
).
[136]
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Changes in the impedance as a function of the slimes layer thickness and ol the current interruption
The high frequency intercept of the impedance curves with the axis was
used to find the changes in Rg as a function of the slimes thickness. As Fig. 16
indicates, the Rg values measured by AC impedance were in agreement with
those obtained by current interruption l i 2 . Discrepancies between the Rg values
were only observed at slimes thicknesses larger than 8 mm.
s Xo•0
41
*»
<
*stl&seoe
3.90 1 —
3 .38 —
2 .70
2 . 10
1 . SO
* F r a n OC D * t *O F r o n C u r r e n t I n t e r r u p t i o n
•= ft 6 6 6 6
i.90S- .000 1.80 3.60 5.4 0
S I i n e s T h i c k n e s s , n n
7 .20 9 . 00
Fig. 16 Changes In the value of R,, as a function of the slimes thickness (Exp. CA2).
From the AC and DC data obtained in Exp. CA2 .
A C impedance measurements done while the anode was being
galvanostatically dissolved at 800 Amp/m 2
(Exp. CA5) are shown in Figs. 17
and 18. Changes in Rg as a function of the slimes thickness are plotted in Fig. 19.
Rg changes the most at slimes thickness larger than 2 mm where dissolution
of noble impurities takes place. The small changes in the Rg values at small
slimes thickness are partially due to changes in the concentration of the
electrolyte between the reference electrode and the slimes/bulk electrolyte
interface as a result of the large current density applied.
1
Current interruption was done
prior
to obtaining the
AC impedance spectrum. Appendix 6 describes how
the current interruption measurement was implemented.
2 The invariability of the
R,
values at slimes thicknesses smaller than 8 mm
confirms
the results obtained
in Chapter 4 in
which
upon
current interruption
t\a
remained constant (Fig. 3.8).[137]
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Changes in the impedance as a function ol the slimes layer thickness and ol the current interwpti
1 . 2
I
ft
c N
I
. 96
72
—
A. 6 . 3
- *
±7 . 7
—•
49 . 9
• 141 .
•
396 .
1117 .
— & 3149 .
— O 8876 .
•> 14067 .
-*
22295.—
48
n n S I i M G S
ft 0 . 64B 1 . 20C 1 . 76D 1 . 89
- J * 0.63 rad/sec
000 400 800 1.20
Zreal , ft —CM 2
1 . 60 2 . 00
Fig. 17 Impedance spectra obtained during Exp. CA5 at slimes layer thicknesses between 0.64 and 1.89
mm.
Each impedance curve was obtained at a different slimes thicknesses as indicated in this Argand plot. R,
was subtracted from the Z„ component of the impedance.
[138]
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Changes in the impedance as a function of the slimes layer thickness and of the current interruptio
(A) (B)
.Mr-
;
riJAtc
-A 63> J.M
.64
-
• 9.96
.64
D
39,63
•a 158.
_4 628.
•& 2381.
0 9959,
.48 • 39646.
t
S68B2.
3.8 r-
Q
f I I I I
I I I I
I I I I
I I I I
I I 1 I
I I I I I I t I I I I I I I I I
I I I I
.888 .248 .488 .728 .968
,2
1.28
1.8
L
-.588
4.58588 1.58 2.58 3.58
Z^j, flcm2
Log (Freq, rad/sec)
Fig. 18 Impedance spectra obtained during Exp. CA5 at a slimes layer thickness of 2.2 mm (R, was subtracted from the Z* component of the Impedance).
(A) Argand plot fB) Bode plot
Fig. 19 Changes in the valueof R, as a function of the
slimes thickness (Exp. CA5).
From the AC data obtained inExp. CA5.
i.ee.eee .6ee l.ae l.se
Slimes Thickness, mm2.48
3.98
[139]
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Changes in the impedance as a function ol the slimes layer thickness and of the current intanvptio
(A)
I— r a d l / s e c
1 . 68
1 . 28
. 888
. 488
.888
.63
• A .99
• 6 .28
• 19 . 87
a 62 . 834 199 .
628 .
O 1987 .
6284 .
*
7911 .
n n D i s •» o I v e dl
ft 8 . 87B 1 .39C 1 .77n 2 . 86E 2 . ISF 2 . 52
. 888 1 . 28 2 . 48 3 . 68
Z
real. ftcm*(B)
4 . 88 6 . 88
688 ,
- nn D i t t o l va«1
_
G 2 . 76— H 2 . 834 — I 2 . 837
J 2 . 841K 2 . 865
368 .
248 . A. . 863
. 44
• 3 . 15
• 22 . 29
• 158 .
4 1117 .
tk
7911 .
O 56882.
»- 396469.
4> 488487.
I
I I I
_L
208 . 880 ,488. 608.
Fig. 20 Impedance spectra obtained during Exp. CA4 at slimes layer thickness between0.87 and 2.87 mm.
Each impedance curve was obtained at a different slimes thickness as indicated in this Argand plot. Ft, was subtracted from the Z* component of the impedance.
[140]
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Changes in tie impedance as a function of the slimes layer thickness and of the current mtemiptio
While the impedance curves obtained during Exps. CA2 and CA5 increased
uniforrnly with the slimes thickness, those obtained during Exp. CA4 showed
marked increases and variations in their magnitude and shape (Fig. 20). This
different behaviour is believed to be the result of the precipitation of secondary
products across the slimes layer. Thus, impedance arcs whose time constantis large are present throughout the whole electrolysis cycle 1. The absence of
high frequency arcs is a clear indication that Faradaic dissolution of noble
impurities did not occur in this experiment.
1.30 r-
"aU
1.24o
a
8 1.18
cj
O J . . J . 2
T 3
IoC
1.06
1.00
Fig. 21 Changes in the valueof the R, as a function of theslimes thickness (Exp. CA4).
From the AC data obtained inExp. CA4.
j i_ _] L J
l_ I I
.000 .600 1.20 1.80
mm Dissolved2.40 3.00
Changes in Rg as a function of the amount of lead dissolved are shown in
Fig. 21 2 . Again, minor variations in the Rg values result from changes in the
concentration of the electrolyte between the reference electrode and the
slimes/bulk electrolyte interface. As lower currents go through the cell Rg
returns to its original value because such variations in concentration disappear.
1 As indicated by the frequency values at which the capacitative
pan"
(i.e. the imaginary part) of the
impedance
reaches a maximum value,
the
relaxation
processes
nave very
large time
constants
(of the
order of sec).
2 These changes in
R
9
were
used
to compute the anodic
overpotential
as a function of the amount of
lead
dissolved
(Fig. 9).[141]
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Changes in tie impedance as a function of the slimes layer thickness and of the current interruptio
2 .00 r a d / s e c17 .7
• 62 .8•
223 .
•
791 .
1 .£0 • 2807 .
- 4 99S9 .— a 3S33S.
a 125375.— • 222953.
- 353354.1 .28
.880
.488
.000
A A f t e r 8 . 17 Hrs
B A f t e r 86 .9 Hrs
A
6- 4 - - - - ' - Q -
J l_I • • •
I
2 .88 2 .68 3.28 3.88
Z r e a l ft-CH*4.48 5 .88
Fig. 22 Argand plot showing the changes in the impedance spectra obtained in the presence of a layer
of slimes and in the absence of a net Faradaic current (Exp. CA2).
The anode was corroded up to the formation of -8.7 mm of slimes (Fig. 6). Subsequently, current was
halted for -115 hrs (Fig. 10A). During this period, impedance spectra were obtained at preset times asindicated in this Argand plot.
Curve A was obtained -0.17 Hrs after current interruptionCurve B was obtained -86.9 Hrs after current interruption
The AC impedance spectra obtained in the a b s e n c e of a net Faradaic cu
(i.e. under current interruption conditions) are significantly different from those
obtained during the passage of a net D C
current. The impedance curves indicate
the presence of a linear region which bends towards a small arc as high
frequencies are approached (Fig. 22, Exp. CA2). The impedance decreases as a
function of the current interruption time and the anodic overpotential. Also, Rg
decreases up to a liiniting value often higher than observed at the beginning ofthe experiment (compare Figs. 16 and 23). Such a difference arises partially as
a result of the prior reaction of the slimes compounds which changed the
microstructure of the slimes layer.
The impedance spectra obtained at the end of Exp. CA5 (Fig. 24) are siinilar
to those obtained in Exp. CA2 (Fig. 22). The decrease in Rg in this experiment
follows the pattern previously explained because dissolution of the slimes layer
also took place in this experiment (Fig. 25).
[142]
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Changes in the impedance as a function of tie slimes layer thickness and of the current interrup
o , . s .
OJ4-1
CO
s.
6
o
o
3 .28
2.98
O 2.68i
2.38
2.88 S .888
Fig. 23 Changes in thevalue of R, as a functionof the current interruptiontime (Exp. CA2)
From AC data obtained inExp. CA2.
24.8 48.8 72.8
Time, Hrs.96.8 128 .
. s a r a d / s e a. 6 3
*
2 . 5 0
:
• 9 . 9 63 9 . 6 5
. 4 0 •4
1 3 8 .
6 2 8 .
_ 2 5 0 1 .
o9 9 5 9 .
3 9 6 4 6 .
. 3 0*
4 4 4 8 5 .
. 2 8
. 1 0
. 0 0
A A f t a r 0 . 6 3 H r s
BA f t e r
7 . 8 8H r s
C A f t e r 3 8 . 4 H r s
1 . 4 1 . 7 2 . 0 2 . 2 2 . 5 2 . 8
Z^u, Qcm 2
Fig. 24 Argand plot showing the changes in the impedance spectra obtained in the presence of a layer of slimesand in the absence of a net Faradaic current (Exp. C A5).
The anode was corroded up to the formation of -2.2 mm of slimes (Fig. 7). Subsequently, current washalted for -46 hrs (Fig. 10B). During this interval, impedance spectra were obtained at preset times as indicated
in this Argand plot.
Curve A was obtained -0.63 Hrs after current interruption
Curve B was obtained -7.88 Hrs after current interruption
Curve C was obtained -38.4 Hrs after current interruption
[143]
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Changes in the impedance as a function of the stones layer thickness and of the current interrup
In experiment CA4 slimes dissolution was restricted by holding the anodic
overpotential at values lower than 220 mV (Fig. 9). This resulted in nearly
constant Rg values during both the potentiostatic dissolution (Fig. 21) and the
current interruption cycle1
. During current interruption, ionic concentration
gradients within the slimes layer relax. This results in re-dissolution ofprecipitates and in ensuing impedance decreases (Fig. 26 Exp. CA4). The shape
and magnitude of the impedance arcs are different from those observed in Exps.
CA2 and CA5 as the precipitated products can generate an electrical double
layer which affects the dielectric properties of the slimes filaments and of the
lead electrode through changes in their relative permittivity a
. This results in
impedance arcs with a large capacitative component.
Fig. 25 Changes in the valueof R, as a function of thecurrent interruption time(Exp.
CA5)
From AC data obtained inExp. CAS.
18.a 2a.a 3e . e
Time, Hrs.48.a se.e
1 During current interruption conditions, R, remained nearly constant (1.09<R,<1.13 Qcm2 ).
2 The (static) relative permittivity is
defined
as t„=§-C is the capacitance of a parallel
plate
condenser
with
plates
of large area separated by
a
small gap, the whole
being
in a
vacuum
whereas
C 0 is the
capacitance of
a
parallel plate
condenser
when an isotropic material is present between the
plates [5].
[144]
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Changes in the impedance as a function of the slimes layer thickness and ol the current interruptio
Fig. 26 Argand plot showi ng the changes In the Impedance spectra obtained In the presence of a layer of slime s and In the absence of a net F ara dai c
curr ent (Exp. CA4)
T he anode was corrode d up to the for mation of -2. 8 mm of slimes (Fig. 9). Subseque ntly, cur ren t was halt ed for -2 3 hrs (Fig. IOC). D u r i n g this
interval, impedance spectra were obtained at preset times as indicat ed in this Argand plot.
Curve A was obtained -0. 27 Hrs after current interrupt ion
C u r v e B was obtained -2.08 Hrs after current interruption
[145]
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Changes in the impedance as a function of the slimes layer thickness and of the current mtenvption
(A)
a . e
s
a1 . 2
. 8 0
. 4 0
r a d / s e c. 6 3
» 2 . s a
•
9 . 9 6
a 3 9 . 6 5
a
1 5 8 .
4 6 2 8 .2 5 8 1 .
a
9 9 5 9 .
*- 1 V 7 8 9 .
• 2 8 8 6 8 .
r t n S I t n e sA . 4 3B . 9 8
1 . 4 7C. 9 81 . 4 7
D 2 . 3 9E 3 . B 6F S . 3 7G 5 . 7 9H 6 . 3 2I 6 . 6 4
J 7 . 8 6X 7 . 4 9
J i i i L — J i i L
5 . 0
(B)Z^, Ocm
2
(C)
Zreai, Qcm 2
Fig. 27 Impedance spectra obtained during Exp. CA6 at slimes layer thicknesses between 0.43 and 717 mm.
Each impedance curve was obtained at a different slimes thickness as indicated in this Argand plot. R, was subtracted from the component of the impedance.
(A) Impedance spectra acquired in the region where potential excursions where not observed(Fig. 11A).OB) Impedance spectrum obtained at ~3.7 mm of slimes(C) Impedance spectrum obtained at ~7.7 mm of slimes
[146]
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Changes in
the
impedance
as a
function
of the
slimes layer thickness
an d of the
current interw
As previously described, In Exps. CA6 and CC1 excursions in potential
appeared to be triggered by the AC measurements (Fig. 11) 1 . The impedance
spectra obtained in Exp. CC1 are shown in Fig. 27 while those obtained in
Exp. CA6 are shown in Fig. 28. The spectra measured in the region in which
no potential excursions were observed increase uniformly as the slimes layerthickens (Figs. 27A and 28A). On the other hand, the impedance spectra
measured in the region where the potential excursions were observed indicate
the presence of arcs whose time constant is of the order of 10"5 sec 2 . These
spectra were reproducible and did not show any large changes in magnitude at
different electrolysis times. Such high frequency phenomena can only be related
to very fast processes such as found across the Helmholtz electrical double
layer.
In Exps. CA6 and CC1, changes in Rg were observed only in the regionwhere the anodic overpotential excursions occur (Fig. 29). Upon current
interruption, the impedance spectra show the presence of high frequency arcs
whose size decreases as the current interruption time increases (Fig. 30). Also,
Rg decreases as a function of the current interruption time up to a limiting value
which in the case of Exp. CA6 is nearly equal to that observed at the beginning
of the experiment (compare Figs. 29A and 31).
1 Exps. CA6 and
CC1 were
performed using working electrodes from different sides of the lead anode
and
in
the presence and absence of addition
agents
(see Table 1). In both experiments the potential
excursions appeared at about the same slimes thickness (-3.2 mm). Consequently, the outset of the
excursions in potential must be related to changes in the slimes electrolyte rather than in the slimes layer
or
in
the anode.
2 In Figs. 27B, 27C, 28B and 28C, the maximum of the imaginary part occurs at 2x10
4
<co<3x10
3
rad/sec,
thus
5x10"
5
<T<3.3
X10^
sec.
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Changes in the impedance as a function olthe slimes layer thickness and of the current interrupt
(A)
1.08 r—
.880
S .608
a
.480
.280
i
r«d/sec A 6.3
• 17 .7• 49 .9• 141 .
• 39 6 .4 11 17 .3149 .
O 8876 .• 11174 .
*
17789.
nn S i t H I S
A . 14
B .51C 1 .37D 2 .22E 3 .50
«i 8.63 r a d / s e c
.8884 • i I i i i I i i i I i i i I
. 00 0 .330 .640 . 960 1 . 28 1 .60
sCJ
C
8.68
rid/sec
6.3
• t 25.8
•99.6
• 396.
6.48
0
1378.
•
4 6283.
625813.
0 99582.
•396463,
* 488487
4.88
3.28
-
1.68
-
(B)
AC
tueei dene it 4.5 m Slinei
Z ^ . Qcm2
a
a
8,88
rid/sec
6.3
«25.8
•99.6
• 396.
6.48
0
1378.
4 6283.
• 0 2581S.
0 99582.
•396463,
* 488487
4.88
3.28
1.68
(C)
AC tvtep dene
it 5.3 nn Slinet
12.8
I I I I I I
12.8
Z ^ . Qcm2
Fig. 28 Impedance spectra obtained during Exp. CC1 at slimes layer thicknesses between 0.14 and 5.3
,800
1
1 1 1 1
'
1
'
1
' ' '
1
' ' '
1
' ' ' '
1
'
1 1
' ' '
1
' ' '
.888
2.48 4.88 7.28 9.68
mm.Each impedance curve was obtained at a different slimes thickness as indicated in this Argand plot. R,
was subtracted from the component of the impedance.
(A) Impedance spectra acquired in the region where potential excursions where not observed (Fig. 11B) (B)Impedance spectrum obtained at -4.5 mm of slimes (C) Impedance spectrum obtained at -5.3 mm of slimes
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Changes in the impedance as a function of the slimes layer thickness and of the current interruptio
Fig. 29 Changes in the value of the R, as a function of the slimes thickness (Exps. CA6and CCD.
(A) From the AC and DC data obtained in Exp. CA6
(B) From the AC data obtained in Exp. CC1
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Changes in the impedance as a
function
oltte
slimes
layer
thickness
and of the
current interruption
CJ
. 3. 6 3
-*
3 . I S
•
•1 3 . 7 87 9 . 1 1
. e
n4
3 9 6 .1 9 8 7 .
— 9 9 3 9 .
o 4 9 9 1 8 . — 1 3 7 8 3 4 .
. s
3 3 8 1 3 2 .
1 . 8
. 8 8
A A4> t « r- 8 . 6 3 H r .
B A f t e r 2 . 4 4 H r s
C A f t a r 1 1 2 . 8 H r »
Fig. 30 Argand plot showing the changes In the Impedance spectra obtained in the presence of a layer of slimes and in the absence of a net Faradaic current [Exp. CA6).
The anode was corroded up to the formation of -8 mm of slimes (Fig. 11A).Subsequently, current was halted for -113 hrs (Fig. 12). During this interval, impedancespectra were obtained at preset times as indicated in this Argand plot.
Cury.e A was obtained -0.63 Hrs after current interruptionCurve B was obtained -2.44 Hrs after current interruptionCurve C was obtained -112.8 Hrs after current interruption
M
6 9 ee ;
ua
o 4 0 8 !
s
3 ea i
O h r
j
\•a 2 ea r
V
ea r
n s
UP .
S 1
oCJ '
aP
.08 2 4 . 48 . 72.
Time. Hrs.96. 120
Fig. 31 Changes in the valueof R, as a function of thecurrent interruption time(Exp. CA6).
From AC data obtained in
Exp. CA6.
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Relationship between the DC anodic overpotential and the DC current density
2. Analysis of the experimental data
In the study of the establishment.of concentration gradients across the
slimes layer an electrical analogue could be used to describe how these gradients
affect the dissolution of noble impurities present in the slimes layer. This
analogue has to evolve from fundamental D C and A C studies and must include
parameters that provide a physico-chemical insight of the system. In the following
section the main characteristics of such a model are introduced. The D C
data are
analyzed prior to the A C data to provide a framework for the elaboration of the
analogue model. Subsequently, this model is used to relate the experimental D C
and A C behavior of lead bullion electrodes covered with a layer of slimes.
(a) Relationship between the DC anodic overpotential and the DC current
density
As shown in Chapter 4, concentration gradients produce concentration
overpotentials which can be linked to the resistivity of the electrolyte present
within the slimes layer. Concentration overpotential is also the major component
of the r)A curves presented in this Chapter.
According to Newman 123,24]
concentration overpotential can be defined
as follows x > 2 :
Eq. 4 can be expressed as a linear equation:
T ]c=IRm + b
...5
With
1 Despite the different definitions of concentration overpotential available
in
the literature
there
is no
conclusive evidence
that
any of them describes accurately the physical phenomena involved, yet,
there
is
an agreement that because of the presence of concentration gradients, an ohmic drop is included in the
concentration overpotential measurement (Compare refs.
[23-24] and
[8-9]).
2 The ohmic part in Eq. 5
of
TI0
has more physical meaning
when
it is equal to /
I = 0
t h a n
when
it is
equal to i Jj^-^dx
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Relationship between the DC
anodic overpotential and
the DC
current density
Eq. 4 indicates that rjc is composed of at least 2 contributions: (A) an ohmic
drop due to variations of conductivity in the diffusion layer and (B) the potential
difference of a concentration cell.
In Chapter 4 the determination of the electrical conductivity of the slimes
electrolyte was attempted by using current interruption techniques. Suchmeasurement was not really applicable because as Eq. 4 indicates, even though
the ohmic term ought to disappear upon current interruption (I-»0), the
relaxation of concentration gradients creates a counter E.M.F. that avoids the
direct measurement of the conductivity of the slimes electrolyte \
Under steady state conditions a , Eq. 4 indicates that small changes in the
applied current density ought to result in changes in TJc due exclusively to the
ohmic component of the slimes electrolyte 3 . If linearity between applied current
density and the observed concentration overpotential is observed, the averageresistivity of the slimes electrolyte, F^, could be obtained from Eq. 5. In addition,
the parameter b should provide complementary information about the extent
to which concentration gradients vary across the slimes layer4.
The validity of Eq. 5 in the determination of the resistivity of the slimes
electrolyte is studied in this section by analyzing the current and anodic
overpotential changes observed during the application of a small amplitude
sinusoidal waveform 5 . Thus, these changes are analyzed according to Eq. 5
using the following assumptions:
1) Upon subtracting TJq from the anodic overpotential observed upon the
dissolution of lead, the main component of the remaining overpotential is due
to the presence of concentration overpotential.
1 As explained in Chapter 4, even though the external current may had been halted, processes that
support
the passage of internal currents may still be present after current interruption.
2 Steady state conditions
are
such that
the concentration gradients across the slimes layer remain
constant
during
the measurement of the
T|C-I relationship.
3 As Eq.
4 indicates, Ohm's law is not useful in
a region where
concentration gradients are present.
Nevertheless, in such
a
region, an integral value of the changes
in
conductivity can be obtained.
4 Notice
that R
m
and b are a function of the local electrolyte conductivity and concentration gradients
across the slimes layer.
5 AC and DC currents and overpotentials are terms
that
can easily be
confused: An AC wave
varies
sinusoidally
as
a
function of time and can be
described
by the following equation:
/(f,a)) = M<)sin((iM + (|))
where:
M
e
is the amplitude of the waveform, co is its frequency, * is the phase angle, and t is the time. By
knowing M
0
,
©,
and
<B,
the
DC "instantaneous" component of the AC
waveform
can be obtained.
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Data analysis
2) Upon application of a small amplitude sinusoidal current waveform (less
than 35 Amp/m 2) at low frequencies (ox6.3 rad/sec) and for a short period of
time, ionic concentrations and concentration gradients throughout the slimes
electrolyte remain unchanged \
3) The only overpotential increase that results in a linear dependancebetween potential and current is that due to the ohmic drop of the slimes
electrolyte (Rn, term in Eq. 5).
4) The restriction to ionic flow caused by the slimes can be obtained from
the R, , , value and its dependance with the electrolysis conditions.
5) The part of the overpotential that does not depend on the current density
is due to the presence of concentration gradients across the slimes electrolyte
(b term in Eq. 5) and to related increases on the corrosion potential of the lead
anode.
6) The lead dissolution process proceeds urihindered ( R ^ O ) whereas the
slimes layer remains unreacted (Ret-***)2
7) Ionic concentration gradients are present throughout the slimes layer
(i.e. 5 is equal to the thickness of the slimes layer). Furthermore, these gradients
are only observed in the direction normal to the anode.
(I) Data analysis
The steps that were followed to analyze the rjA spikes produced by the ACcurrent waveform are as follows:
A) From the TJ a readings obtained during the application of the AC current
waveform, r)n
3 , was subtracted *.
B) The resulting anodic overpotential and the current density were
curve-fitted to a straight line according to Eq. 5.
1 The exact values
for
the amplitude and frequency of the waveform may change depending
on
whether
a oc current is applied or not.
2 This is equivalent to
implying
that all the current flow is at the anode/slimes interface without any
significant Faradaic current
crossing
the slimes/electrolyte interface.
3 R, is known prior
to
and after the application of the current steps. An average Ft, value can be used to
correct the r\
readings.
4 Notice
that
the anodic overpotentials shown in Figs.
6C, 7, and
11 are all corrected only
for the
initialr\
a
Onn=IR«)-
The approach
in
this section is to analyze the anodic overpotential corrected forr|
n
present
at
the
local
time
the current sweeps
were
applied.
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Data analysis
(A) (B)
268.
236.
2 1 2 ,
C
cu
Qti
U
188,
164.
M.lr
i
i i t I i i i I i i i I i i i I i t i I i i i I i i i I i i i I i i i I
12,1 1 2 . 3 1 2 . 6 1 2 . 9 1 3 . 2
Time, Hrs.(C)
13 .5
3 8 . 8
33,0' i i i i i i i i i i i i i i i
1 2 . 8 1 2 . 3 1 2 . 6 1 2 . 9 1 3 . 2 1 3 . 3
Time, Hrs.
38.8
r
>-> 48.8
•a
C 38.8
<U
o&
I...O
^ 11"
* *
MI r 1111111
11111 ' ' ' ' ' ' 1 1 1 1 1 1 1 1
1
1 1 1 '
.888 52.8 184. 156.
288.
268.
Current Density, Amp/m 2
Fig. 32 Detail of the "spikes" observed at -0.8 mmslimes (Exp. CA2, Table 4).
From Exp. CA2 (see Fig. 6)
The data points shown in (A) and (B) were linkedusing cubic splines interpolation. Had more pointsbeen available these curves would have looked likedistorted sinusoidal waveforms with equalamplitudes.
(A) Variation of the anodic current density as a
function of time(B) T | A variation as a function of time (corrected for
tin)-
(C) Current density vs anodic overpotential curve[obtained from the data shown in (A) and (B)]
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Data analysis
The current density changes observed during the application of one of the
A C sweeps (Exp. CA2) are shown in Fig. 32A. The anodic overpotential spikes
(corrected for TJQ) that result from variations in the cell current are shown in
Fig. 32B. The linear relationship between these two quantities can be seen in
Fig. 32C. This relationship was found to be frequency independent, indicatingnearly steady state conditions. Further analysis of the r\ A spikes produced
during the application of the A C waveform at different slimes thickness showed
that linearity was observed only up to anodic overpotentials smaller than 250
mV (Fig. 33).
2 4 0 . r
2 0 0 . -
1 6 0 . h
1 2 0 .
8 B . e
4 0 . 01 4 0 .
3 6 0 . r
3 2 0 .
2 8 0 .
2 4 0 .
2 0 0 .
1 6 0 .
1 6 4 . 1 8 8 . 2 3 6 . 2 6 1
1 i I • i •
2 3 6 . 2 6 0 . 1 4 0 . 1 6 4 , 1 8 8 . 2 1 2 . 2 3 6 . 3 6 0 .
Fig. 33 Variations in the anodic overpotential as a function of the anodic currentdensity at various slimes thickness (Exp. CA2, Table 4).
X axis: Anodic current density, Amp/m 2
Y axis: Anodic overpotential (corrected for T IJEach curve corresponds to the analysis of the spikes observed at the following slimes
thickness: (A) 1.6 mm (B) 4.7 mm (C) 6.6 mm (D) 8.6 mm
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Data analysis
Table 4 Analysis of the spikes produced during the application of the A C waveform,
in the presence of a net DC current (Exp. CA2, Figs. 32 to 35)
Parameters Derived from RegressionAnalysis Computations
From Eq. 4 Experimental
SlimesThickness
mm
SlopeQcm
2 Intercept,b, (mV)
|y r|
2
,mv
3
b_
IR,
p
m
, QcmIR.. mVmV mV
0.80
2.23
3.10
5 33
5.95
6.56
7.18
7 79
8.41
8 65
0.62
1.45
2.15
3 93
4.57
5 23
5.71
7 22
4.96
9 40
30.8
45.8
54.4
75.3
82.4
93.1
113.9
124.9
361.0
360.6
0.850
0 986
0.988
0.981
0.961
0.920
0.905
0888
0.051
0.558
6.8
6.4
12.6
59.8
163.7
375.1
633.5
1192.5
4386.0
5688.7
2.54
1.63
1.30
098
0.93
092
1.03
089
3.74
1.97
7.78
6.48
6.93
7.38
7.68
7.98
7.95
9.27
5.90
10 86
12.1
28.1
41.8
76 5
88.8
1017
110.9
140 3
96.5
1827
42.9
73.9
96.1
151.8
171.2
194.8
224 8
265.2
457.5
543 3
41.9
73.3
95.5
150.4
170.0
193 3
221.8
2614
436.0
520.0
Between 18 and 20 experimental points were used to obtain the regression coefficient. These points
were collected during -55 min and correspond to digitized samples taken when the frequency of the applied
AC waves was between 0.063 and 6.3 rad/sec.
Abstracted from Table 4, Appendix 9
As seen in Table 4, both R , and b increase as a function of the slimes
thickness (see Eq. 5 and Fig. 34A) indicating that larger concentration
gradients across the slimes layer generate larger ohmic drops. Thus, the ohmic
drop generated by these concentration gradients can promote the dissolution
of the slimes layer only if large concentration gradients are also present.
Furthermore, the ohmic drop generated by the slimes electrolyte (IRJ is smaller
than the "current independent' term b only up to -5.3 mm slimes (i.e. ^- reaches
a value of -1 at 5.3 mm slimes). After this slimes thickness, the IR™ term not
only becomes larger than the b term, but it increases at a faster rate. The large
variations in the values of these terms above -5.3 mm slimes are related to
the precipitation of secondary products. Precipitation of secondary products
changes both local ionic concentrations and local electrolyte conductivities.
Moreover, the changes in the pm value, (Fig. 34B)1 also indicate increases in
its value towards the end of the electrolysis cycle. Dissolution of slimes
compounds at slimes thicknesses larger than 8 mm results in non-linear
1 p
m
is the resistance of the slimes electrolyte per
cm
of slimes:
Rm [Qcm2 ] pm[Qcm] =SlimesThickness[ cm ]
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M e
h
i
i
I i
•
I
•
i I
i
i I
i
i I i i I • i I i i I i • I i
•
I
.
M
i i
i I i i I i i
I
i i I
i i I
i i I i i I i
i I i
i
I
i
i.eaa l.sa s.se
s
.4 a 7.2a 9.ea
.aaa l.sa 3.6a 3.4a 7.2s 9.aa
mm Slimes mm Slimes
Fig. 34 Changes In the value of the resistance of the slimes electrolyte as afunction of the slimes thickness (Exp. CA2, Table 4)
(A) R,,, (Qcm 2) changes (B) p m (Qcm) variations
. e e e I . S B 3 . 6 a s . 4 e 7 . 2 a 9 . 0 0
mm Slimes
Fig. 35 Changes in the parameters b and IR™ with relationship to the experimentalvariations of the anodic overpotential (Exp. CA2, Table 4).
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Data analysis
relationships between overpotential and current \ Under these conditions, D C
current enters the slimes electrolyte from both the anode/slimes electrolyte
interface and the slimes/slimes electrolyte interface and this produces
non-uniformities in the distribution of current across the slimes layer.
The relationship between r|
a
, b, and IRm is graphically presented in Fig. 35.The jump in the value of b at slimes thicknesses larger than 8 mm indicates
that noble compounds have reached the potential at which they can react
Faradaically. By adding the value of IRn, to the b value, the experimental r|A
is
obtained 2
. This is a very important relationship: It links the slimes solution
properties to the changes taking place in the lead anode and in the slimes
layer. Thus, for different electrolysis conditions and anode compositions,
optimum parameters for lead electrorefining can be obtained by studying the
variations in the b and IRm values at different slimes thickness. Moreover, itappears as if by merely applying a sinusoidal galvanodynamic scan (i.e. one
scan whose amplitude is -20 Amp/m 2
at co~0.1 rad/sec) at preset slimes
thickness, the same information obtained using the FRA can be derived 3
.
Table 5 .A Analysis of the spikes produced during the applicat ion of the AC waveform.
in the presence of a net DC current (Exp. CA6)
Parameters Derived from Regression From Eq. 4 Experimental
Analysis ComputationsSlimes Slope Intercept, r2 |y r|
2, b pm, Qcm » » , mV
Thickness Rn.. b, (mV) mV2
mV mV
mm Qcm 2
il? m m m 0.170.14
3.561.96
5.144.75
32.057.8 m
3.065.37
1.52189 m 0.99
0.850.4015.64
1.592.55
4.983.52 m 76.6
130.578.31341
6.647.49
2.703.13
113.6140.9
0 880.97
10 164.09
2162.31
4.064.18
52.560.9
166.1201.9
168.2,205.2
4 experimental points were used to obtain the regression coefficient These points were collected during
-12 min and correspond to digitized samples taken when the frequency of the applied AC waves was between
0.63 and 6.3 rad/sec.Abstracted from Table 5.a, Appendix 9
1
Non-linearities
are
identified when
|y| increases and r
2
decreases.
2 e.g. compare columns
IRm+6) and TIa
in Table 4.
3 Use of linear
polarization techniques such
as SACV (small
amplitude cyclic
voltammetry)
should
provide
the same results
[10-13].
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Data analysis
Table 5.B Analys is of the spikes produced during the application of the AC waveform,
under current interruption conditions (Exp. CA6)
Parameters Derived from RegressionAnalysis Computations
FromEq.4
Experimental
Tune, hrs Slope R m,
Qcm2
Intercept, b,
(mV)
r2
|yrr\mv
2 pm, Qcm TR^+b, mVmV
0.63
2.44
3.83
43.07
112.8
2.23
3.62
3.83
1.64
0.99
60.8
i i i i^ i i i i ; :
44.5
lllliiliillli0.2
0.89
0.91
0.95
88
21.95
41.19
62.43
22.09
21.63
2.82
4.57
4.84
m
60.8
48.4
44.5
5.3
0.2
82.1
WmmMi44.0
9 experimental points were used to obtain the regression coefficient These points were collected
during -27 min and correspond to digitized samples taken when the frequency of the applied AC waves was
between 0.63 and 6.3 rad/sec.
Abstracted from Table 5.b, Appendix 9
The TJ a spikes observed in Exps. CA6 and CC1 were also analyzed according
to Eq. 5. The results obtained from such an analysis are shown in Tables 5A
and 6. A linear relationship between overpotential and current was only found
in the region where the potential excursions were absent. Furthermore, a
quasi-linear relationship between overpotential and current was also found
under current interruption conditions (Table 5B, Exp. CA6) *.
Table 6 Analysis of the spikes produced during the application of the AC waveform,
in the presence of a net DC current (Exp. CC1)
Parameters Derived from Regression
Analysis Computations
From Eq. 4 Experimental
Slimes
Thickness
mm
Slope R „
Qcm2
Intercept,
b, (mV)
r2
|y rl2
,mV
2
b
lRm
pm, Qcm IRm.mV IR .,+6,
mV mV
1.37
1.79
2.22
2.65
3.07
3.50
0.52
0.63
0.74
0.81
0.92
0.96
32.4
38.0
42.8
48.0
52.4
58.4
0.98
0.97
0.98
0.95
0.93
0.93
0.1
0.3
0.4
1.1
1.7
2.3
3.21
3.11
2.97
3.06
2.94
3.12
3.80
3.51
3.34
3.05
2.99
2.74
10.1
12.2
14.4
15.7
17.8
18.7
42.6
50.2
57.2
§Mmm70.3
i l 7
! l l l :
42.8
50.6
57.8
64.4
70.9
78.1
5 experimental points were used to obtain the regression coefficient These points were collected during
-12 min and correspond to digitized samples taken when the frequency of the applied AC waves was between
0.63 and 6.3 rad/sec.
By comparing the data presented in Tables 4, 5A, and 6 it can be seen
that:
1 In Table 5B, r2
and |y|
vary widely
yet the predicted (IR
m
+o) and experimental (T|a
) overpotentials are
nearly
equal.
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Data analysis
i) The anodic overpotential values observed at slimes thicknesses lower
than 3.5 mm (before potential excursions appeared in Exps. CA6 and CC1) are
different in the three analyzed experiments: T |A (Exp CA2) > rjA (Exp CA6) > r|A
(Exp CC1). This results in differences among the computed RQ, and b values
ii) The smallest and b values obtained at a fixed slimes thickness are
observed in Exp. CC1. The fact that in this experiment p m decreases as the
slimes layer thickens is significant.
iii) A relationship that relates the slope b to the average slimes electrolyte
resistance. Rn , , and to the local electrolyte concentrations cannot be inferred
yet \
The observed differences in the values of the r|A components (IRn, and b)
at a fixed slimes thickness are related to the bulk electrolyte composition, to
the presence of addition agents, and to the changes in the slimes
physico-chemical properties (i.e. porosity, tortuosity).
Addition agents appear to play an important role in the anodic
overpotential increases . This was also observed in the sulphamic acid system
1251a
. Thus, in Exp. CC1 the distinct R^, and b values indicate that the addition
agents increase n A mainly by restricting the flow of ions 3
. In the presence of a
Faradaic current, I, this restriction can be related to the ratio: The largerm
this ratio, the smaller the restriction for the movement of electrolyte across the
slimes layer and the lower the observed overpotential. In Exp. CC1 the ^- ratio
is ~3 and remains nearly constant. On the other hand, at similar slimes
thickness, for the other experiments this ratio shows marked decreases (from
2.5 to 1.3 for Exp. CA2 and from 3.4 to 1.6 for Exp. CA6).
1
i.e. b values cannot be derived from
R
m
values and
vice
versa
:
Average electrical conductivities may be
equal yet local concentrations may be different.
2 The amount and nature of addition agents incorporated
during
the refining of Pb
using a
sulphamic acid
electrolyte has also been shown to have a
strong
impact
on
the permeability of the anode slimes
[25].
3 Notice that whereas Exps. CA2 and
CA6
were carried out
using
the lead anode
"mould" cooled
face,
Exp. CC1 was
performed
using the "air" cooled face. The microstructures of
these
electrodes were found
to be similar.
Thus,
the anodic overpotential variations can hardly be
related
to microstructural differences
between
the different electrodes.
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Proposed analogue representation of a lead bullion electrode covered with a layer ol s
Analysis of the TJ a spikes obtained upon current interruption also indicated
a linear relationship between rjA and current (Table 5B). Under current
interruption conditions 1=0 and rjA does not have an ohmic component, yet, by
slowly displacing the dynamic pseudo-equilibrium observed during the
relaxation of the concentration gradients, it was possible to obtain the integralvalue of the resistivity of the slimes electrolyte, Rn , \ As seen in Table 5B this
value decreases as the concentration gradients relax. Furthermore, at the end
of the current interruption cycle, T]A~0, yet, as indicated by the finite Rn , value,
the electrolyte present within the slimes was found to have a different
conductivity than the bulk electrolyte.
(b) Proposed analogue representation of a lead bullion electrode covered
with a layer of slimes.
So far in this thesis it has been shown that the response of electrochemical
systems to AC
waveforms can be used to obtain kinetic and diffusional
parameters. Resistors, capacitors, and CPE's have been used to link changes
in the A C
spectra to associated physico-chemical parameters. A general model
is to be proposed to analyze the observed A C impedance spectra. This model is
based on a set of assumptions. From this model and from the experimental
data, several electrical analogues are developed to examine the A C
impedance
data and to find the link between these analogues and the physical phenomenathey may represent.
Fig. 36 shows a general analogue model of a lead bullion electrode covered
with a layer of slimes. Six interfaces can be identified in this figure. Each one
of these interfaces has an associated impedance 2 :
1 i.e. the pseudo-steady state observed during the relaxation of the concentration gradients in the
entrained
electrolyte
was
slowly
displaced by applying a sinusoidal current waveform.
2 The impedance of the
reference
electrode is neglected
in
this
analogue.
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Proposed analogue representation of a lead bullion electrode covered with a layer of s
za/se = Faradaic impedance at the lead anode/slimes electrolyte interface.
= Faradaic impedance at the sltaes/slimes electrolyte interface.
Zgi/be = Faradaic impedance at the slimes/bulk electrolyte interface.
Z a / 8 , = Electronic impedance at the lead anode/slimes interface.
Zw = Warburg ionic diffusional impedance throughout the slimeselectrolyte.
Zyj„ = Warburg ionic diffusional impedance in the slimes electrolyte/bulk
electrolyte interface.
The value and mathematical expressions that each of these impedances
can adopt is a function of the current density, the slimes electrolyte composition,
the slimes layer microstructure and composition, and the electrode's thermal
history and composition. Thus, a single mathematical representation of the
overall impedance may be too difficult to determine unambiguously. Moreover,some of these impedances are distributed (i.e. their value changes as a function
of the position) and coupled (i.e. they change only if other impedances change).
Despite the complex interrelationships existing among the different impedances,
their individual contribution to the total impedance can be assessed by analyzing
each of them separately. From this analysis and from experimental data, the
relative magnitude of each of these impedances with respect to the total
impedance can be inferred.
The individual analysis of the components of the impedance has to start
from the simplest scenario. This will be the case when only a DC
current is
applied to the electrode. In this case the total impedance of the system has only
a real component [Z DC = Z(j(a)(O=0 ]. Thus, while the impedance is not an explicit
function of time, if measured in a system where changes are taking place, its
value at a fixed frequency will be a time dependent quantity. Thus at (0
=0 the
changes in impedance as a function of time, Z^f), can be defined as follows:
where:
TJ (t) = overpotential (compensated for rid observed upon passage of current
as a function of the electrolysis time. By using a reference electrode reversible
to Pb + 2 no corrections for liquid junction at the reference electrode/electrolyte
interface have to be incorporated in the T\(t) values.
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Proposed analogue representation of a lead bullion electrode covered with a layer ol s
During the galvanostatic dissolution of a lead bullion electrode, rj increases
as the slimes layer forms. Thus, all the information on the impedance comes
from knowledge of the changes in rj as a function of time. By subtracting T|
Q
1
from the r\(t) values one of the most obvious contributions to the impedance has
been identified and subtracted from the overall impedance. However, othercontributions to the overall impedance cannot be as easily identified and/or
quantified.
By analyzing the path of the DC current on its way from the anode to the
bulk electrolyte a deeper insight in the Z^. components can be obtained. Thus,
as shown in Fig. 36 there are two main paths for the DC current: One through
the anode/slimes electrolyte interface and the second through the anode/slimes
interface. If current enters the slimes filaments it can either leave them through
the bulk electrolyte (path A) or through the slimes electrolyte (path C). On theother hand if any current crosses the anode/slimes interface it can either go
across the slimes electrolyte (path B) or return to the anode via a ground loop
(path D).
For current to go through the slimes filaments it will first have to overcome
a resistance associated with Z a / S l . As the slimes filaments were found to be
grounded to the anode 2
such resistance must be negligible. Any current
entering the slimes filaments can only produce Faradaic work at the
slimes/slimes electrolyte interface if and only if the activation energy barrier of
such process is overcome. This requires large overpotentials and ionic gradients
across the slimes layer. If any current diverts towards the slimes filaments it
can either go across the slimes/slimes electrolyte interface (path A) or across
the slimes/bulk electrolyte interface (path C). The experimental evidence is that
Faradaic reactions of the slimes compounds are insignificant at overpotentials
less than 200 mV. For example, the amount of bismuth corroded is less than
30 ppm, too small to account for significant Bi corrosion currents crossing the
slimes filaments. Thus, during most of the electrorefining cycle, any current
going through the anode/slimes interface can only result in charging of the
1 T j a is equal to the current density, I,
times
the
uncompensated
ohmic resistance, R, (nn
=IR
9
).
2
See
Chapter
4
section
II.A. 1
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Proposed analogue representation ola lead bullion electrode covered with a layer of s
A node/ S l i mes Interfacei
i
S l i m e s •: f i l amen t
z
Sl imes/Electro lyte Interface
ii
f
Zsl/be
sl/se
W
5 * a/se
L e a d A n o d e S l i m e s E l e c t r o l y t e
llllllllllillllllllllllllll . >• ;
:
?.;
:
?/
:
S
;
1 irnne"•• ^H&ihneshii
Bulk
Electro ly te
S l i m e s L a y e r
Fig. 36 Proposed analogue model representation of a lead bullion electrode covered with a layer of slimes.
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Proposed analogue representation of a had bullion electrode covered with a layer ol sl
electrical double layer of the slimes filaments *and are insignificant. Thus, from
this analysis it can be assumed that for the D C case, up to r\(t) values equal to
200 mV: Za/ ai->0, Z ^ - ^ , and Z a l / b e -*>° .
From the previous description it can be assumed that almost all of the
applied D C
current flows across the anode/slimes electrolyte interface andcontinues in its way towards the bulk electrolyte by overcoming the diffusional
impedance Z^ r
(path B). All this current is transferred to the electrolyte mainly
as a result of the Faradaic transfer of Pb + 2 to the slimes electrolyte. For such a
transfer to take place, an energy barrier associated with Z a / s e has to be overcome.
Such impedance is small (lead dissolves almost reversibly), yet it can increase
if conditions at the interface change (i.e. as a result of the presence of secondary
products blocking the interface). Once Pb + 2 ions are transferred to the slimes
electrolyte, an ionic current is established. The resistance that the ions find intheir movement across the slimes is a function of Zw which is an intensive and
position dependent quantity. The larger the effective distance the ions have to
travel before reaching the bulk electrolyte the larger this impedance. The
complex migraUonal/diffusional processes taking place across the slimes
electrolyte can be described by Z ^. Furthermore, if as a result of the movement
of ions across the slimes electrolyte/bulk electrolyte interface a diffusion layer
is established, a semi-infinite Warburg impedance has to be included as well,
although for well mixed bulk electrolytes its presence can be neglected. Once
the Pb + 2 ions reach the bulk electrolyte they are transferred to the cathode by
convection, migration and diffusion.
Thus, the analysis of the D C experimental data presented in Section III.2
of this chapter can now be related to the analogue model shown in Fig. 36. In
the DC analysis of the anodic overpotential changes, the lead dissolution
processes were assumed to proceed unhindered whereas the slimes layer was
assumed to remain unreacted. This is equivalent to assuming that Z a / s e - » 0 and
that Zsj/ae- oo , and Z s ] / b e — w h i c h is in agreement with the statements on the
characteristics of the electrical analogue here presented. Thus, the Rm value
obtained from the analysis of the changes in overpotential as a function of
1 Notice that under
DC conditions a
capacitor
will act as an open circuit and will have an infinite
impedance. The impedance of a capacitor, Zc, is given by: z c=^,
thus at co=0, Z,.-*».
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Proposed analogue representation ola lead bullion electrode covered with a layer of s
current can only be related to changes in the value of Z^. A relationship between
the observed b values and Zy, may also exist, yet, impedance values obtained
at frequencies other than zero have to be provided (see Eq, 9).
The previously introduced D C model also must be consistent with the
experimental evidence observed under current interruption conditions.Depending on the extent to which concentration gradients had been established
prior to the current interruption, an overpotential decay will always be observed.
Eventually, the electrolyte compositions inside and outside the slimes electrolyte
equilibrate, ionic movement virtually stops, and the impedance disappears as
the overpotential vanishes (i.e. as n .^ -» 0). On the other hand, as a result of
the ionic concentration differences between the slimes electrolyte/bulk
electrolyte and the anode/slimes electrolyte interfaces, a series of concentration
cells may be established because electronic current can flow across the slimesfilaments (path D). This process may consist of (as an example) continued
corrosion of the lead anode, accompanied by deposition of noble elements in
the electrolyte, as B i + 3 , SbO+, or Ag* on slimes filaments. These internal currents
may be too small or difficult to measure, however, they can affect the
characteristics of the overpotential decay and consequently of the impedance.
As concentration gradients disappear, the rate of these internal processes
declines, up to the point at which they become negligible.
The phenomena associated to the current interruption case can also be
represented by the analogue circuit shown in Fig. 36. Diffusional processes
can be represented by Zy, and Z^^. However, as concentration gradients within
the slimes layer relax, Z^ describes diffusional processes that resemble more
semi-infinite diffusion than diffusion in a bounded region. As described
previously, Z^ is linked to the components of the concentration overpotential,
and for a purely diffusional process, Rn , should be equivalent to the diffusion
resistance Ro (i.e. to the value of Z^ at co=0).
In the analysis of the components of the A C impedance, the same line of
thought followed during the D C analysis will be used: First, the different paths
that the A C wave can follow will be traced. Then the relative contribution of each
of the associated impedances to the overall impedance will be estimated. On
the basis of this and from experimental evidence, simplifications of the general
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Proposed analogue representation ol
a toad
bullion electrode covered
with
a layer o
analogue model will be proposed. The validity of the proposed model will be
tested by analyzing the changes in impedance in the presence and in the absence
of a net Faradaic D C current.
In the previous analysis it was found that all the applied D C current can
be assumed to flow across the anode/slimes electrolyte interface. On the otherhand, a superimposed AC wave can either cross the same interface and/or divert
through the slimes filaments. If it diverts to the slimes filaments, it can cross
the slimes/slimes electrolyte interface without actually perfonriing any Faradaic
work. This will happen if the slimes layer can be considered to have only a
capacitative component. For these processes to take place, the A C waveform will
have to produce alternating potential fields at the slimes/slimes electrolyte
double layer interface. This will result in large capacitative effects being observed
in the impedance spectrum. Thus, if any A C current crosses the slimes/slimeselectrolyte interface, equivalent impedances related to the capacitative part of
the slimes impedance should be observed even in the absence of a net Faradaic
current. As the experimental impedance spectra obtained under current
interruption conditions did not indicate the presence of large interfacial areas,
A C current transfer across the slimes layer in the presence of a net Faradaic
current can be neglected.
The experimental data indicate that while a net D C current is being applied,
formation of Pb + a is the preferred reaction. Such a reaction takes place without
a significant energy expenditure and is the path of least resistance for the flow
of current. If this is the path for least resistance for D C current, it can also be
assumed to be the path of least resistance for the A C
current. Thus, assurning
that all the A C current crosses this interface, the overall impedance will have
only three components: Z a / a e , 7^,, Zw„ (notice the similarity between the D C and
the A C cases)1. Accordingly, in the absence of a net Faradaic current, the polarity
of the double layer can be switched and/or its potential difference changed
simultaneously in the slimes/electrolyte and the anode/slimes interface. Only
then, the capacitative and resistive process associated with these impedances
can be observed as changes in the impedance of the system. Consequently,
1 Notice that in the worst scenario in
which
in the presence of a DC current the AC current actually crosses
the slimes/slimes electrolyte interface and causes
Faradaic
reactions, a
wide non-uniform current
distribution
can result.
If
this had taken place the system would had been changed to the extent that
steady state concentration gradients during the AC measurements would not have been observed, the
system would
have
oscillated to the extent
that
the impedance measurement
could
not have
been taken.
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Proposed analogue representation of a lead bullion electrode covered with a layer ol sl
other processes overlooked by the anodic overpotential measurements under
current interruption conditions may be better analyzed by A C
impedance
techniques.
Basically, the main difference between the D C and the A C experiments is
that while D C currents cannot cross the slimes/slimes electrolyte interface at r\lower than 200 mV, A C waveforms can cross such an interface and by doing so
induce changes in the measured A C
impedance. However, in the presence of a
net Faradaic current such process may be hindered to the extent that such
transfer does not take place at all. If this is the case, then all the D C and A C
current flow is at the anode/slimes electrolyte interface. Nonetheless, if any A C
current flows across the slimes layer it may result in impedance changes only
in the high frequency region of the impedance spectrum at which point charge
transfer phenomena are isolated from diffusional processes in the slimeselectrolyte. The situation changes at overpotentials at which Faradaic reaction
of the slimes filaments can take place 1 . Under these conditions, both D C and
A C
currents will cross the slimes/electrolyte interface and in doing so, at that
point, abrupt changes in impedance will take place. On the other hand, in the
absence of a net Faradaic current (i.e. under current interruption conditions)
the hindrance for the current flow across the slimes layer disappears (as the
whole electrode can change polarity and/or act as a corrosion or concentration
cell) and impedances associated with the slimes filaments will be observed in
the total impedance.
Summary of assumptions used in the development of the analogue model:
1) A one dimensional representation of the lead bullion electrode covered
with a layer of slimes.
2) Electrolysis takes place under isothermal conditions 2 .
3) The Warburg diffusional impedance can be used to describe the ionic
mass transfer processes that take place across the slimes electrolyte.
4)
The A C
impedance measurement is obtained without significantly
affecting the quasi-equilibrium conditions witJiin the slimes layer.
1
These abrupt impedance changes upon
Faradaic
reaction of the slimes
filaments
can be seen by
comparing Figs. 13 and 14.
2 Isothermal temperature of the slimes electrolyte can be assumed on the basis of sufficient thermal
conductivity of H
2
SiF
6
-PbSiF
6
solutions and of the
large
porosity of the slimes layer, contributing to
convection.
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Data analysis
5) In the presence of a net Faradaic current the lead anode and the slimes
layer are equipotential.
6)
Changes in the microstructure of the slimes layer as a result of blockage
of ionic flow (i.e. by the precipitation of secondary products) can be incorporated
within the Warburg diffusional impedance or by additional impedancesconnected in series with it \
7) Blockage of the anode/slimes electrolyte interface (i.e. by the
precipitation of secondary products) inhibits charge transfer processes and
increases the Faradaic impedance, Z a / s e .
8) Dissolution of noble impurities present within the slimes layer takes
place only at overpotentials larger than 200 mV.
9) In the presence of a net Faradaic current and at overpotentials values
smaller than 200 mV, 100% of the current transfer occurs across theanode/slimes electrolyte interface and: Z a / s e -»0 , Z9l/se-**>, and Z 8 l / b e-»oo.
1 0 ) In the presence of a net Faradaic current capacitance effects associated
with the slimes layer are prone to be observed only in the high frequency region
of the impedance spectrum a .
11 ) The impedance changes attributed to the presence of addition agents
can be incorporated within the proposed Faradaic and diffusional impedances.
1 2 ) Impedances associated with the reference electrode can be neglected.
(I) Data analysis
In this section, several electrical circuits derived from the proposed
analogue model are introduced. These electrical circuits were formulated so
as to follow the assumptions used in the development of the general analogue
model. Among all the analyzed circuits, only those that actually matched the
experimental data are presented and discussed. The characteristics of these
circuits are established by comparing their parameter values with
physico-chemical processes taking place across the slimes layer.
1 Non-uniform porosity across the slimes layer can be expected when secondary products precipitate or
re-dissolve.
2 Capacitance effects associated with the noble
compounds
present
in
the slimes layer are more
significant as the TIa
approaches values larger than 200 mV (i.e.
in
the region where their Faradaic
reaction can occur).
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(i) Case I: impedance spectra obtained in the presence of a net Faradaic current
The impedance spectra obtained in Exps. CA2 . CA5 . CA6, and CC 1
were found to be described accurately 1
by two different yet related electrical
circuits (see Fig. 37). The values of the parameters associated with the
proposed electrical circuits were obtained by curve fitting 3 the experimental
data to the theoretical impedance functions.
The Jirst of these circuits is a ZzARd-ZzARea-CPEo analogue (circuit A . 1,
Fig. 37) 3 . The impedance of each of the components of this circuit is given
by the following equations :
Z CPEL
= b,m^' ZCPE , = Wmf*
1 Z CPEO = Kijnf™
From which the total impedance, Z A , can be obtained as follows:
with
Rj W
z
A
=
-ir+
2 —-+ B 0 U^' ...io
Rl /?2
D 1=— and D 2 = —
The overall impedance of the circuit A . 1 was re-arranged as described
in Eq. 10, so that the relaxation times can be obtained from the D; and D 2
parameters. The relaxation time associated with each one of the ZARC circuits
can be obtained as follows:
• i
x, and \ = D!^
1 As presented in the various Tables shown in Appendix 9 described accurately means
that
from a
statistical
regression
analysis
perspective, good correlation existed
between
the experimental and the
curve-fitted data. The number of parameters involved
in
the
curve
fitting routine was
never in
excess to
that
required to obtain significant
fits
according to
statistical
rules
(i.e. as
derived
from ANOVA statistical
tables).
2 This process required the use of complex non-linear square fitting routines . The Zj component of the
total
impedance was
used to
obtain the
values
of individual parameters in the different electrical circuits.
Once an
initial
set of
values
was
obtained, they were
used to compute the
Z„
component of the
impedance and improve the accuracy of the fitting process.
3 For an in-depth
review
of the characteristics of the
ZARC circuits see ref.
[is].
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Data analysis
Circuit A . l
ZARCl
CPE,—,
—Wv—
ZARC2
ZCP E
n
W Ffgwrey tewree I I F r W 7 "PP™ | | Bthaon
CPE,-,
Wr—
-CPE.
Circuit A . 2
•ZARCl
ZARC2
Fig. 37 Electrical circuits used to analyze the impedance spectra
The electrical parameters associated with circuit A . 1 were correlatec
with the individual impedances shown in Fig. 36 1
. Thus, the first ZARC
circuit was used to represent high frequency phenomena
2
associated withthe Faradaic impedances Z a / s e and Z s l / s e . The second
ZARC
circuit was used
1 R
1
was chosen to represent charge transfer resistances associated with the lead
dissolution
processes,
while R
2
was related to the DC conductivity of the slimes electrolyte. CPE, represents the distributed
nature of the anode/slimes and the slimes/slimes electrolyte interface while CPEa represents the
presence of a distributed capacitance generated by the concentration gradients present in the slimes
electrolyte.
2 The high frequency
term refers
to the
part
of the impedance
that
was observed at
co>
100 rad/sec.
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Data analysis
to represent the low frequency response associated with ionic diifusion
across the slimes layer (i.e. with Zy,). The third component of this circuit, a
CPE element was introduced to represent diffusional processes (i.e. Zw,.).
The second analogue circuit was indirectly assembled by finding an
electrical circuit whose impedance matched the impedance of circuit A.Such a circuit is shown in Fig. 37B (Circuit B . l ) . The impedance of each of
the components of this circuit can be described as follows:
Z r =r Z. =r.z
c
> = QO'co)
From these components the total impedance, Za, can be obtained as
follows:
A direct correspondence between the parameters in this circuit and
those in the analogue model shown in Fig. 36 is not immediately evident.
Yet, this circuit was found to reproduce the experimental data and it was
considered worthwhile to try to find some analogies among both circuits.
Thus, it was found that the Zy,,- component could be associated with the
CPEB component and that the RC circuit could be associated with the
Faradaic impedances Z a / s e and Z s l / s e . Diffusional processes can be included
in both R x and CP EA components.
When diffusional processes in the bulk electrolyte are neglected (i.e.
when Zw „.-»()) \ the two proposed electrical analogue circuits can be related
through their impedances at co=0 (by their D C
resistances). Thus, the total
D C
resistance of these circuits can be obtained as follows 2 :
Z A (oo
z*,
»((,
1
Notice
that
by neglecting Z
w
„,
in
circuits
A and
B the only elements
that
disappear are
CPE„
and CPE
B
respectively.
2 Once the parameter values were obtained the impedance at co=0 was
found
to be nearly equal for the
two electrical circuits shown in Fig. 37.
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Data analysis
In the analysis of the experimental impedance spectra, the impedance
values at co=0, were related to the total D C
resistance, obtained from
analysis of the spikes obtained during the application of the A C waveform.
The values of the parameters obtained by curve fitting the impedancespectra obtained in Exp. CA2 to the Z?ARcrZrzARC2 analogue (circuit A.2 in
Fig. 37) are shown in Table 7. The characteristics of this circuit stressed
the fact that the impedance spectra are composed of at least two distributed
components with different time constants. The presence of two humps in
the impedance spectra was clearly observed at slimes thicknesses lower
than 3 mm. At larger slimes thicknesses the separation of these humps was
not hnmediately evident, yet, from the curve fitting process, it was found
that the spectra could be deconvoluted to produce two arcs whose center
lay below the axis (see Fig. 38).
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Data analysis
Table 7 Parameters derived from the fitting of the impedance data obtained
In Exp. CA 2 to the Z ^ - Z ^ analogue (Circuit A.2,
Figs. 38,39)
High Frequency Parameters Low Frequency Parameters R A^I -
R, + R 2
Slimes ^zci b„ \ ,sec ^ZC2- b2, ,sec
RA,IOUI»
Thickness Q c m2 i i c m 1 sec 1
Qcm2 i i
cm secQ c m
2
nun
0.80 0.21 0.73 33.50 0.0010 0.44 0.77 5.55 0.037 0.65
2.23 0.42 0.64 14.09 0.0042 1.17 0.69 3.84 0.176 1.58
3.10 0.64 0.59 9.39 0.0102 1.62 0.69 4.11 0.260 2.265.33 1.06 0.60 10.19 0.0225 3.17 0.71 4.60 0.592 4.23
5.95 1.54 0.51 IllBili 0.0373 iilill!llllll 4.06 0.795 4.99
6.56 4.70 0.56 iiiillliiii 0.3323 iiMil 0.98 0.82 1.583 5.99
7.18 5.13 0.55 0.3593 2.08 0.96 1.992 7.21
7.79 4.30 0.59 i i . i l 0.2030 4.65 0.90 2.05 2.484 8.95
* All measurements refer to the geometrical area of the electrode.
** Low and high frequency terms refer to the ranges of frequencies used during the deconvolution
process.
*** Low and high frequency arcs were fittedto the whole frequency range (0.063<(IK 30000 rad/sec)
Abstracted from Table 7, Appendix 9
Analysis of the parameter values shown in Table 7 indicate the
following relationships (see also Fig. 39):
a) The impedance obtained at co=0, R ^ t o u i . w a s
found to be similar to
Rn, (see Table 4). This correspondence provides an import ant l in k between
the DC and the A C expe riments \
b) Xg
i
values are small (of the order of msec) while x Ri values are large
(of the order of sec). The separation in the magnitudes of these time constants
decreases as the slimes layer thickens .c) The largest changes in the values of the derived analogue parameters
appears at slimes thicknesses larger than 6.5 mm. These changes can be
1 R totai is expected to be larger than
R
m
because in addition to include ionic processes it also includes
resistances associated
with
charge transfer processes.
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Data analysis
associated with the precipitation and hydrolysis of secondary products.
Moreover, changes in these parameters correspond to changes in the
ratio a
.
Among the previously drawn relationships, one of the most significant
appears to be that related to the precipitation/hydrolysis of secondary
products. If the hydrolysis point actually took place at ~6.6 mm of slimes,
that would have resulted in blockage of the reacting interface (the
anode/slimes electrolyte interface) and in ensuing increases in Za / s e
. This
appears to be in agreement with the observed increases in R t at slimes
thicknesses larger than 6.6 mm. On the other hand, upon precipitation of
secondary products, an increase in Zw (i.e. in the impedance associatedwith ZARC2 ) should have been observed had the movement of ions been
blocked by these products. As observed in Table 7, even though R2 and b 2
decrease between 6 and 6.6 mm of slimes, x R increases almost a 100% in
the same interval. Thus, increases in x Ri seem to be inversely proportional
to R 2 and b 2 because the fractional exponent ^zcz is also changing. The large
values of x R observed after 6.6 mm are an indication that movement of ions
across the slimes layer proceeds more slowly because such movement is
being hindered by precipitated products.
1 See Table 4 and focus on the changes in R
m
, and p
m
, between 6.6 and 7.8 mm of slimes.
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Dataanafysa
(A) (B)
J
.M
O
CO
4J
3 . U *XR I
o
TR
2
1.31
Sl .N
.318
o LitCJ
o
w
o
•4-4
8aCO
7.21
4.81
2.48
.881
3.68 3.48 7.28
mm Slimes
(C)
12.8
*P A , total
o
Pi
X P2
' I I I I I I I
I I
,888 1.88 3.61 3.48 7.28
mm Slimes
I B . B r
8 .81
a
u
t »
2.88
* R A . total
0 R i
X Ra
9,88
3.68 3.48 7.28
mm Slimes
9.68
Fig. 39 Variation of the derived electricalanalogue parameters as a function of the slimesthickness (Exp. CA2, Circuit A.2, Table 7)
From the parameters value presented in Table 7
(A) Changes in the relaxation times tR 1 and x^.
(B) Changes in the resistance values R At oai . Ri andRj. [toon*](C) Changes in the specific resistance P A , ^ , PI
and p2. [Qcm] as obtained from:
/?, [Qcm2 ]0,
[Qcm] = — = — ;
; rK
' Slimes Thickness [cm]
Compare RAil0U i with Rm (Fig. 34k)
I i i I i i I i i9.88
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Data analysis
.7(0A .063
• .31• 1.S8• 7.91
.560a
39.63
- 4 199.6 996.0 4991.
c
• 250IS.
c
* 28068.c 420
a
2 8 0
.140
.080'
*t ,89 nn Stines
Experimental
Anilogue noJel
. 8 8 8 700
Fig. 40 Detail of the impedancespectrum obtained in Exp. CA2 at0.80 mm of slimes (Exp. CA2,
circuit B.l).
Circuit B. l was used to fit theexperimental data. The derivedanalogue parameters are shown inTable 8.
Analysis of the experimental impedance spectra obtained in Exp. CA2
using the electrical circuit B.2 resulted in the parameter values shown in
Table 8. Only the impedance spectrum obtained at 0.80 mm slimes
thickness was fitted to circuit B.l (see Fig. 40). This was possible because
at very low frequencies (less than 1 rad/sec) a tail related to diffusion in
the bulk electrolyte (i.e. to Zy, J was observed in this impedance spectrum1, 2
1 This tail was found to be described by the CPE
a
element shown
in
the analogue circuit B.1.
2 Notice
that
B
a
is nearly equal to the B,
value
obtained under rest potential conditions and
in
the
absence of addition agents [B
a
~1.9
Q
cm2 sec y 2 C
while
B,~2-3 Q. cm2 sec'yzc
(see Table 3, Exps.
CC1-5
to
CC2-2)]
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Data analysis
Table 8 Parameters derived from the fitting of the impedance spectraobtained in Exp. CA2 (Circuit B.2, Figs. 40, 41)
Slimes
Thickness
r a
Qcm 2
r b
Qcm
2
cb
F cm"
2
B a
Q cm1
sec'^
^ZCa Ratotal
Qcm2
mm
0.80
2.23
0.43
1.150.230.49
0.0260.102
11.487.46
0.580.53
0.661.64
3.10„ , . , , ^ , , , , , , .
1.64, , , , , , , , , ™ „ , . , . . „ .
0.66
0.90
0.146
0.398
8.59 — § - j f —
0.56
0.602.304.25
5.956.56
i i i i i i i i i !
4.361.131.60
0.5700.749
9.8810.48
0.610.61
5.005.96
7.18
7.79
4.80
5.29
2.403.82
0.9751.35
11.2512.61
0.620.64
7.209.12
For the AC sweep done at 0.8 mm slimes: B b= 1.92 f2 cm 2 sec""*, and 4/
ZC b= 0.78 (this was the only
sweep fitted to circuit B . l , all the other sweeps were fitted to circuit B.2)
Abstracted from Table 8, Appendix 9
(A) (B)
mm Slimes mm Slimes
Fig. 41 Variation of the derived electrical analogue parameters as a function ofthe slimes thickness (Exp. CA2. Circuit B.2, Table 8)
CA) Changes in the resistance values,'[Qcm 2], Ra.totai
(marked with • ), r a (marked with O) and r b (marked with X).
(B) Changes in the specific resistance, [Qcm], pB
, totai (marked with •), pa (marked with O), and Pt (marked with X).
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Data analysis
As seen in Fig. 41, r a and r b increase continuously during the lead
dissolution process. Moreover, it was found that at slimes thicknesses
smaller than 6 mm, r a=R2 and r^R x (see Tables 7 and 8). This correspondence
agrees with the assigned physical meaning of the parameters present in
the electrical circuit. Thus, r a is related to the DC resistance of the slimeselectrolyte while r b is related to the charge transfer resistance associated
with the lead dissolution process. The only parameter that indicates
significant changes in its value as the slimes layer thickens is r b. These
changes are observed at slimes thicknesses larger than 5.4 mm and indicate
a restriction in the charge transfer process for lead dissolution. Such a
restriction agrees with the proposed mechanism of precipitation of
secondary products and ensuing increases in Z a / s e .
The two electrical circuits shown in Fig. 37 were also used to analyze
the impedance spectra from exp. CA5. The parameters obtained from the
curve fitting process are shown in Tables 9 and 10. From the analysis of
the data presented in these tables, precipitation of secondary products
appeared to take place between 1.2 and 1.76 mm of slimes. Furthermore,
the large R x value at 1.89 mm of slimes and the small time constant
associated with it, appear to indicate that charge transfer processes were
strongly hindered as a result of the continuous precipitation of secondary
products (see Table 10).
By comparing the results obtained in Exp. CA2 with those obtained in
Exp. CA5, it can be seen that increases in current density resulted in
corresponding increases in r a and decreases in r b (compare Tables 8 and
10). This indicates that higher current densities produce larger diffusional
Impedances which can induce precipitation of secondary products and
earlier dissolution of impurities present in the slimes layer.
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Data
analysis
Table 9 Parameters derived from the fitting of the impedancedata obtained in Exp CA5 to the Z-J^-ZZJ^ analogue (Circuit A.2)
High Frequency Parameters Low Frequency Parameters RA^DHI -
R , + R2
Slimes Ri. ^ Z C l bi, \ ,sec b2, X Ri ,sec R A ^ O U I
Thickness
mm
Qcm2Q. cm 2 sec 1
Qcm 2 £2 cm sec Qcm 2
0.64
1.20
1.76
1.89
0.101
0.017
0.069
0.817
0.91
0.68
0.83
0.58
75.04
1.47
0.35
13.64
0.0007
0.0015
0.1423
0.0079
0.32
0.83
1.61
0.99
0.68
0.59
0.53
0.81
8.19
11.10
9.26
6.77
0.00880.0121
0.0373
0.0934
0.42
0.85
1.68
1.81
* All measurements refer to the geometrical area of the electrode.
** Low and high frequency terms refer to the ranges of frequencies used during the deconvolution process.*** Low and high frequency arcs were fitted to the whole frequency range (6.3<ox23000 rad/sec)Abstracted from Table 9, Appendix 9
Table 10 Electrical analogue parameters derived from the fitting of the impedancedata obtained in Exp. CA5 (Circuit B.2)
Slimes r b c b
B a ^ZCA Re.total
Thickness
mm Qcm
2
Qcm
2
F cm'
2
Q cm
2
sec' Vzc
Qcm
2
0.64 0.38 0.04 0.017 16.10 0.65 0.42
1.20 0.72 0.08 0.036 16.10 0.64 0.801.76 0.16 0.063 11.39 0.57 1.60
1.89 1.50 0.32 0.062ifiliilllllf
0.62 1.82
The impedance spectra obtained in Exp. CA4 was not analyzed directly
by any of the two proposed electrical analogues because of the complex
characteristics of the observed spectra. On the other hand, enough
information on the precipitation of secondary products was obtained bysolely plotting the maximum and -Zg values as a function of the slimes
thickness, as obtained from the following relationships:
-Z„ =[Z3]•hnax J
at a> = a><)
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Data
analysis
1888. er {
—* Zraal (nax)
—o -Zinag (nax)
Fig. 42 Variation of themaximum values of the real, Z*,and the imaginary parts, -Z«j , ofthe impedance as a function ofthe amount of lead dissolved(Exp. CA4)
From the data presented inFig. 20
1 ' I ' ' I ' ' I ' i I ' ' I ' ' I ' ' I
1.28 1.88 2.48 3.88
mm DissolvedAs can be seen in Fig. 42, both components increase at the same rate
up to @2.1 mm, after which sudden increases in their values are observed.
This point appears to mark the onset of precipitation of secondary products
This precipitation is continuous and produces exponential increases in the
impedance towards the end of the electrolysis cycle. These impedance
increases result from both charge transfer and diffusional restrictions
created by the presence of these products. Secondary products act as a
barrier for ionic movement across the whole layer of slimes and create a
nearly motionless region in which only very small currents can flow. Thus,
this represents the lirrutlng case in which Z^-***. As there is no other path
for current flow (see Fig. 37) other than through Path A \ the dissolution
process has to stop (so called "anode passivation").
1
Exp.
CA4 was
performed under
potentiostatic
conditions and,
as
found
by
analyzing
the
cathodic
deposit, impurity dissolution did not
take
place.
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Data analysis
(ii) Case II: impedance spectra obtained in the absence of a net Faradaic cu
Upon current interruption the impedance spectra were found to be
capable of being described by the modified Randies analogue circuit 1
shown
in Fig. 43. Such a circuit has the following impedance :
Z
( / 'G>)=-
/?c(+51(/'u))".14
c
DL
G-
l — wv-CPE
- 0
Fig. 43 Modified Randies AnalogueCircuit
The CPE circuit shown in Fig. 46 represents seim-irifinite diffusiona
processes. is the charge transfer resistance and is the double layer
capacitance.
From the analysis of the data obtained in Exp. CA2 (see Table 11 and
Fig. 44) it can be seen that:
1
Notice that the fact that the impedance
could
be fitted to Randies
circuits
implies that: Z ^ - * - , and
Zjiybe-^oo while
and
Z w
have
finite
values.
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Data
analysis
(A) (B)
6
6
77.
1
443.
•1117.
•
28(7.
g
Ttn .
4
17781
t
44483.
a 111741
•28MU
* 333334
• -Aml t f i i no4tl
i
• 6M r*4
/stc•
177.
• • 396.
4 888.
• 1987.
.48*
a
4448.
4
9939.
a 22293.
- a 49914.
•78384.
. •111748,
.3tB
1
' '1
>1 1 1 1 1 1 I l i i
. i i .
1.28
l.M
,2
.248
.128
.888'
* Eiy«rtMnt*1• A IM 10«utftfl4tl
Fig. 4 4 Correlation between the experimental and theoretical Impedance spectra(Exp. CA2, current Interruption conditions. Table 11).
From the A C data presented in Fig. 22The Randies circuit shown in Fig. 43 was used to fit the experimental data. The
derived analogue parameters are shown in Table 11(A) From the Impedance spectrum obtained -0.17 Hrs after current interruption(B) From the impedance spectrum obtained ~86.9 Hrs after current interruption
Table 1 1 Parameters derived from the fitting of the impedance dataobtained in Exp. CA2 to the Randies analogue circuit (Figs. 43, 44)
Time, Hrs R,t. ilcm 2
Cdj, U.F cm"2
B L Qcm 2sec" Vzc ^zc
0.17 0.044 23.75 3.83 0.190.64 0.112 167.42 i!§i6l?:
:
aiii;: i 0.3312.03 0.070 206.32 3.i3 0.3186.90 0.048 274.37 2.23 0.30
Abstracted from Table 11, Appendix 9
a) Charge transfer resistances, R^, are very small and do not seem tochange significantly during the current interruption cycle. This indicates
that only highly reversible processes are present and that charge transfer
phenomena associated with the noble elements present in the slimes layer
do not contribute to the measured impedance spectra.
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Data analyse
b) Double layer capacitances, increase only during the first hour
after the current interruption. The small values do not indicate the
presence of large electrochemically active surface areas x . Consequently,
Ca values in excess of those observed in the absence of slimes, are mostly
due to changes in the relative permittivity of the anode/slimes electrolyteinterface as a result of the presence of secondary products at that interface
2
.
c) B x decreases the most at the begmning of the current interruption,
to reach a limiting value after -70 Hrs. These decreases are the result of
the relaxation of concentration gradients across the slimes layer 3
.
d) is nearly constant4
.
From these relationships it can be concluded that both and C^
represent only phenomena associated with Za / s e
(i.e. with the Pb/Pb + 2
equilibrium). This is a very i mpor tant relations hip because it proves thatcharge transfer and capacitative phenomena related to the slimes filaments
are not affecting significantly the impedance of the system. In addition,
processes that support the passage of internal currents do not appear to
result in characteristic contributions to the impedance, except at very short
current interruption times, at which large C^ changes were found. Finally,
diffusional processes across the slimes electrolyte can be successfully
described by a single CPE element because upon current interruption no
more Pb + 2 ions are being generated at the anode/slimes electrolyte interface
(i.e. ionic concentrations are fixed only at the slimes/bulk electrolyte
interface).
The electrical analogue parameters obtained by analyzing the spectra
obtained in Exp. CA5 are shown in Table 12 (see also Fig. 45). From these
parameters the following relationships can be observed:
1 Remember that both the t\ and C
d
, measurements are given with respect to the geometrical area of the
electrodes.
2 Ro, and C
d
| are also a function of ionic concentrations at the anode/slimes electrolyte interface and the
roughness of the anode.
3 B
1
decreases also as a result of the re-dissolution of secondary products and ensuing tortousity and
porosity changes.
4 For ionic
diffusion
across
a
porous electrode O.S^^O.25
[13,16,19-21]
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Zmu. "cm 2
Z^j. flcm2
Fig. 45 Correlation between the experimental and theoretical impedance spectra(Exp. CA5, current interruption conditions. Table 12).
From the A C data presented in Fig. 24The Randies circuit shown in Fig. 43 was used to fit the experimental data. The
derived analogue parameters are shown in Table 12(A) From the Impedance spectrum obtained -0.63 Hrs after current interruption(B) From the Impedance spectrum obtained -34.6 Hrs after current interruption
T a b l e 1 2 Electrical analogue parameters derived from the fitting of theimpedance data obtained in Exp. CA5 to the Randies analogue circuit
(Fig. 45)
Time, Hrs Ret, Qcm2
Cd,, uF cm'2
B „ Qcm2sec'Vzc
^zc
0.6311.6734.59
m
0.023
804.671588.191308.61
J'.710.70
0.210.250.25
Abstracted from Table 12, Appendix 9
a) 1^ values are small and of the same order of magnitude as those
obtained in Exp. CA2.
b) Even though the slimes layers are not as thick as those formed in
Exp. CA2, double layer capacitances are larger. This indicates that in
Exp. CA5 secondary products precipitated to a larger extent than in
Exp. CA2.
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Data
analysis
c) Bj values are smaller than in Exp. CA2 because the region over which
concentration gradients span is ~3 times smaller.
d) values are smaller than in Exp. CA2. Thus, due the larger
presence of secondary products there is a larger blockage for ionic flow and
a reduction in the available space for ionic diffusion.
7.Br -xE-2
5.6
6 4.2
ci
2.8
— Expertnental
— Analogue node I
rad/sec
A
6.3
t 14.1
• 31.5
• 78.5
D 158.
4 353.
0 791.
0 1771.
• 2229.
* 3533.
2.8 3.8
Z^y, Qcm 2
Fig. 46 Argand plot of a typical lead bullion electrode (Exp. CC2) in the presence of a 2.2 mm layer ofslimes
AC impedance spectra obtained under rest potential conditions 20 Hrs after current interruptionR, was subtracted from the Zj, component of the impedance
The impedance curve was fitted to a CPE element from which the following values were obtained Bj=
0.14 cm2 sec'**, ^^.4 2.Compare with Figs. 5.16 and Fig. 6.1
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Data
analysis
Similar experiments in which neither noble impurities dissolution nor
secondary products precipitation took place, provided analogous spectra
that could be fitted to a modified Randies circuit, from which very small R^
and Cdi values were obtained. Thus, for example, the impedance spectrum
obtained with an electrode that had been previously dissolved (up to -2.2mm at 200 Amp/m 2 , riA -55 mV) shows only a straight line which
indicates the presence of a purely diffusional controlled process that
proceeds without kinetic limitations (Fig. 46). Moreover, the derived BY value
is nearly equal to that obtained when pure lead was studied 1. The absence
of a charge transfer arc confirms that the slimes layer is not contributing
to the measured impedance. Furthermore, as no secondary products were
precipitated, the equilibrium Pb/Pb + 2
was not hindered to the extent that
significant capacitances could be observed.
In Exp. CA4 large amounts of secondary products precipitated and
this resulted in impedance spectra that could be fitted to circuits A. 1 and
A . 2 (see Table 13). The physical meaning assigned previously to each of the
impedance components was found to be valid also under these conditions.
This correspondence is due to the enhanced restriction in ionic flow
produced by the large amount of precipitated products. Furthermore, these
products create a nearly stagnant environment in which bounded
diffusional processes can take place 2 . Additionally, these products block
the anode/slimes electrolyte interface and increase the charge transfer
resistance for the Pb/Pb + 2 equilibrium. Thus, as ionic concentrations across
the slimes layer diminish, secondary products re-dissolve and the
impedance in the whole frequency range decreases as a function of time
1 In the absence of slimes, the A C impedance spectra
were
also fitted to a Randies circuit
from
which a B,
value
of 1.88 Ci cm2 sec'** was obtained (see Table 3). Thus, as a result of the uniform corrosion of the
electrode, surface irregularities observed at the slimes/bulk electrolyte interface diminish at the
anode/slimes electrolyte interface and this results in equivalent B,
values
for lead bullion and pure lead
working electrodes (see section
II.A).
2 Upon current interruption concentration gradients relax and local ionic concentrations decrease. This
results in re-dissolution of the precipitated products. Upon re-dissolution ionic species are released
creating an environment in which bounded diffusional processes take place (i.e. the precipitated products
during their re-dissolution maintain a fixed Pb*
2
concentration at the slimes/slimes electrolyte interface).
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Data analysis
(see Fig. 47). These phenomena can be followed by analyzing the variations
in the analogue parameters as a function of the current interruption time
(see Table 13).
(A) (B)
l.M
•
ExpurtiMitil
' f h u l omc na<lel
tl 2.1 Hri
r*4/t*c
.63
a.St•
3.96
D 39.63
0
138.
4 628.
6 2381.
0 9939.
•25813.
»39646.
I
«t 9.3 Hri
r«4/«ic
& .63
l 2.38
» 9.96
a 39.(3
a
138.
4
628.
2381.
a 9939.
•22293.
33333.
, r i. 1 1 1 1 1 1 1 1i ' '
1
1
1
' ' i
1
• •
i
1
'
1
1
1
1 1 1 1 1 1
1 •
11
1
(D)
1.21 I 68 I.I2
1.88 r
r
*4/ttc
.863
• .33
• 1.99
a
11.17
a 63.834 333.
ft 1987.
a 11174.23813.
»
39647.
.(88
i i i I t i i I i t i I i i i I i i i I i i i I i t i I t i t 1 i i i I i i i I
I .488 .888 1.28 1.(8 2.88
*
ExperineiWI
- Amloiut nodtl
r*4/tic
.863
•33
» 1.99
• 11.17
0
62.83
4 333.
6
1987.
0 11174.
•23813.
39647.
DM*
' i i I i i i I i i i I i i i I i i i I i i i I i i i I i i i I i i i I i i i I
.888
.
488
.
888
1.28 1.(8
2.1
,2
Z ^ , i i c n r Z ^ . Qcm
Fig. 47 Correlation between the experimental and theoretical impedance spectra(Exp. CA4, current Interruption conditions. Table 13).
From the AC data presented in Fig. 26. Circuits A, 1 and A.2 (Fig. 37) were used tofit the experimental data. The derived analogue parameters are shown in Table 17
Each spectrum was obtained at the following current interruption times:(A) 2.1 Hrs CB) 9.3 Hrs (C) 11.2 Hrs (D) 19.6 Hrs
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Data analysis
Table 13 Electrical Analogue Parameters derived from the fitting AC impedance data
obtained in Exp. CA4 to the Z^^-T^^ analogue (Circuit A.2, Fig. 47)
High Frequency Parameters Low Frequency Parameters
Ri + R 2
Time, Hrs ^zci bi, \ ,sec R 2, V R * Z C 2 - b2, x Ri ,sec
Qcm2 il cm2
sec
1
Qcm2 O cm sec Qcm
2
2.08 0.87 0.73 l iPii i i 0.00167 1.77 0.43 0.77 2.64
9.33 0.32 0.79 108.92 0.00062 """'""l.26:"""""" 0.47 0.56 1 58
11.18 034 0.73 lllllillll 0.00057 0.99 0.55 111111111 0.34 1.33
19.55 0.32 0.72 75.11 0.00052 0.99 0.59 1.70 0.40 1.31
The A C sweeps obtained at current interruption times longer than 11 hrs were fitted to the
ZZARCI-ZZARC2-CPEO analogue (circuit A .1). Thus, 2 additional parameters were obtained:
Time, Hrs b„,
Q. cm2 sec
^ Z C o
11.18
iiiiiiiiiiiiiii0.050
0.046
0.74
0.79
* All measurements refer to the geometrical area of the electrode.
** The regression coefficients r/9,2 and r 3
2 were larger than 0.99
Abstracted from Table 13, Appendix 9
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Introduction
Chapter 7 Physico-Chemical Propert ies of HaSiFe-PbSiFe
Electro lytes and their Relat ionship to the Tran spor t Processe s
Across the S l imes Layer
I. Introduction
In the previous chapters, the electrochemistry of lead bullion electrodes and
of the phases and compounds present in the slimes was shown to be linked to
the ionic concentration gradients across the slimes layer. The extent to which
these gradients change was found to be a function of the current density and the
bulk electrolyte composition. From an analysis of the ohmic and diffusional
components of the anodic overpotential, increases in riA were linked to changes
in ionic composition across the slimes layer. riA changes were also related to
precipitation of secondary products and to hydrolysis of the acid. Precipitated
products hinder the ionic movement across the slimes layer, increase TJa, and
thereby enhance the dissolution rate of noble impurities.
In this chapter lead and acid concentration gradients within the slimes layer
are related to hydrolysis of the acid and precipitation of secondary products. The
mean ionic compositions present within the slimes layer during the industrial
operation of the BEP are used to indicate the magnitude of such gradients.
Additionally, Eh-pH diagrams are used to describe the conditions under whichsecondary reactions across the slimes layer can take place. The shape of the
concentration gradients and their effect on the anodic processes are assessed by
solving numerically the Nemst-Planck flux equations in their fundamental form.
Some of the experimental data required to solve these equations were derived
from an analysis of the physico-chemical properties of H2SiF6-PbSiF 6 electrolytes
(pH, viscosity, density, and electrical conductivity) and from data available in the
literature.
n.
Average S l imes E lectrolyte Composit ion
Table 1 shows the range of electrolyte compositions found within the slimes
layer at the end of the electrorefining cycle during normal industrial operation of
the BEP x . Pb + 2
concentrations within this electrolyte are between 5 and 10 times
1 In-situ electrolyte concentrations of the slimes electrolyte are likely to be smaller than the reported
compositions shown in Table 1. During the extraction of electrolyte samples
from
the separated slimes,
some re-dissolution of precipitates
(mainly
of
PbF
2
)
may
have
taken place.
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Average Slimes Electrolyte Composition
larger than their corresponding bulk electrolyte values while H 2SiF 6
concentrations are between 2 and 6 times smaller . Thiet et al. U.21 and Wenzel
et al. [3-51 found similar mean ionic concentrations. Furthermore, Thiet Ul also
found that increases in current density result in corresponding increases in ionic
concentrations \
Table 1 Composition of the electrolyte samples extracted from anode slimes obtained under
industrial operation of the BEP (Cominco Ltd.)
Lead electrorefining test conditions:
Refinery cells, cd = 220 ATTI2
, 5 days, T=35-40*C,
Bulk
electrolyte composition: 0.31 M PbSiF
e
,
0.59 M H2
SiF
8
Test
Anode Composition
Anode
Slime
Cake
Anode Slimes
Filtrate
Current
Number
wt%
Face
Assay, wt% M (mol l
1
)
m M
(mmol l
1
)
Efficiency
i l l i l l i l l l I l l l Ag
Bi 111 I l l l Pb Pb
HjSiF,
Sb
Bi As
%
#1 Compo
1.51
0.42 0.07
0.01 0.31
0.14 air
mold
49.8 11.8
121
82
82
1.86
1.98
0.24
0.21
0.046
0.034
0.043
0.048
0.17
0.20
84
#1 Anode 9
1.49 0.47 010 001 0.28 0.11 air
mold
491
50.6
12.2
12.6
7.8
7.3
2.03
2.20
0.17
0.14
0.038
0.028
0.048
0.033
0.15
0.11
#2 Anode 22
1.65
054 o.n 001 031 012
air
mold
50.0
48
5
14.9
14 8
6.9
7.1
1.28
1.83
0.28
0.17
0.038
0.031
0.057
0.029
0.21
0.29
#2 Anode 6 H I 042 III m 031
0.14
air
mold
47
5
482
12.5
111
12.3
101
2.15
2.63
0.21
0.14
0.018
0.032
0.048
0.033
0.16
0.15
#3 Compo
1.57
0
57
0.12
001 0.36 015
air
mold
49.2
49.2
14.3
i l l
7.5
7.6
2.24
2.24
0.21
0.17
0.040
0.037
0.043
0.043
0.43
0.45
79
#3 Anode 3
i.65
0.6 i l l H I 0.36 0.16
air
mold
48
4
47.8
14.7
i l l
7.6
7.5
1.98
2.15
0.21
0.17
0.038
0.036
0.014
0.033
0.27
0.13
#4 Compo
149
047 010
001
028 011 air
mold
517
51.3
14.7
14 6
7.8
7.2
1.52
1.52
0.28
0.28
0.052
0.037
0.043
0.038
0.17
0.20
77
103-1
1.5
0.44
0.09 0.02 031 0.10
air
mold
45.1
515
10.3
11.7
21.0
9.5
2.97
3.04
0.14
0.10
0.016
0.023
0.038
0.038
0.03
0.04
96
103-2
144
0.37 008
0.01
0.29 0.10
air
mold
46 9
54.3
10.3
116
22.6
9.7
2.24
2.44
0.21
0.17
0.031
0.010
0.033
0.038
0.13
0.08
76
104-1
1.5
0 30 0.09 001 0 35 011
air
mold
47 3
53.4
10.0
11.1
20.8
9.1
2.05
2.10
0.24
0.21
0.018
0.014
0.029
0.033
0.08
0.07
78
104-2
144
0.38
008
0.01
0.36 0.11
air
mold
45
2
53 7
97
11.1
21.3
9.0
2.53
2.20
0.17
0.17
0.027
0.014
0.033
0.029
0.12
0.05
94
In the BEP the only cause for lost current efficiency Is cathode-to-anode short circuits and current lost to ground.
Without shorts, a lead bullion electrorefining cell gives 100% current efficiency 152).
1 Thiet [1] found the following mean ionic concentrations within the slimes layer:
at
200 A /m
2
1.75 M Pb*
2
and 0.20 M H2
SiF
6
at 250 A /m
2
1.85 M Pb*
2
and 0.12 M H2
SiF
6
at
300 A /m
2
2.23
M Pb*
2
and 0.02
M H
2
SiF
6
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Average Sbnes Bectrofyte Composition
As shown in Table 1, the concentration of noble impurities (As, Sb, and Bi)
in the entrapped electrolyte is very small. The indicated Bi and As concentrations
are of the same order of magnitude as found from insitu measurements (see
Section 4.2.B Chapter 4) while Sb concentrations are larger \ Within the slimes
layer the concentration of noble impurities is small because the noble phases donot react to a significant extent and cementation reactions maintain their
concentration at small levels (see Eq. 1.1). Furthermore, the presence of
equilibrium quantities of these impurities in the recirculating electrolyte (see
Table 1.1, Electrolyte composition row) indicates that these impurities can be
tolerated up to a certain level without serious impairment of the cathode quality.
This was also indicated by Thiet Ul and others 16-91, who showed the relationship
between the concentration of noble species in the bulk electrolyte and the cathode
composition.
The mean Pb + 2 concentration across the slimes layer for all the data presented
in Table 1 is ~2.15 M while the mean acid concentration is ~0.20 M . Assuiriing
linear concentration gradients across the slimes layer 2 , the maximum amount
of Pb + 2 at the anode/slimes electrolyte interface can be considered to be close to
4 M while the H 2SiF 6 concentrations at such an interface can be considered to be
smaller than 0.10 M . Even at this large Pb+ 2 concentration, PbSiF 6.4H 20 is not
expected to crystallize because of the low acid concentrations 3 . Furthermore, the
presence of small amounts of acid at such an interface aids in the stabilization
of PbSiF6 by suppressing its hydrolysis.
In an acid depleted environment PbSlF6 can undergo hydrolysis and
decomposition according to the following equations:
PbSiF 6 + 2H 20 <=>PbF 2 + Si02 + 4HF ...I
PbSiF 6 <^>PbF 2 + SiF 4 ' ...2
As indicated by Eq. 1 the hydrolysis of PbSiF6 will be buffered by theformation of the weakly dissociated HF. On the other hand, as soon as PbF2 is
1 Larger Sb concentrations shown in Table
as compared to those shown in section 4.2.B can be
attributed to partial re-solution during the filtering process of antimony oxides present
in
the slimes layer.
2 Because of migrational-diffusional effects,
H*
concentrations are expected to decrease exponentially
rather
than
linearly.
3 The maximum solubility of pure PbSiF
6
.4H
2
0 at 40
'C is
~5
M (see Appendix
7).
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Eh-pH Diagrams
formed, Pb local concentrations will decrease to a steady-state value determined
by the combination of Pb + 2
precipitation and its continuous generation at the
anode/slimes interface.
In the context of lead refining by the Betts process, hydrolysis of the S1F6"2
ion in moderately strong acid solutions (pH between 1 and 2 ) can be betterdescribed by the following equation *•a:
Both PbF2 and Si0 2 have been identified and quantified in the slimes layer
(e.g. see Table 1.2) 3 [5.11,12]. As early as 1908 Betts [ i l l already indicated the
presence of Si0 2 in the slimes layer and related it to H2S1F6 decomposition. He
found that -2% of Si0 2 in the slimes could correspond to ~1 kg of H2S1F6 loss per
ton of lead produced. In 1966 Cominco Ltd. reported Si0 2 assays in well washed
slimes 4
that were of the order of 1.6% [131. Thus, according to Eq. 3 and assuming
a production of slimes of ~25 kg per ton of lead, an estimated loss of H2S1F6 of
-0.96 kg/ton of Pb can be obtained. This can account for as much as 50% of the
acid losses. Furthermore, PbF2 in the slimes can be as high as 19.6% and can
account for most of the lead found in the slimes layer. In Table 1.1 only one refinery
(Cerro del Pasco) reports SiOa assays in the slimes layer. From their reported value
(0.4% SiOJ an estimated loss of H 2 SiF 6 of ~0.13 kg/ton of Pb was obtained 5 .
HI. E h - p H Diagrams
The Eh-pH diagrams shown in this section were generated using the c.s.i.RO.
THERMODATA8
computer program [14,151 . A list of the INPUT (*.inp) files used In
the generation of these diagrams is provided in Appendix 11. Also, a list of the
1 Hydrolysis of the SiF
6
"
2
ion takes place in 3
steps
which involve intermediate splitting of F" and formation
of
SiFsIHaO]" (see section lll.a).
2 At 25 "C the maximum
solubility
of
monosilicic acid
H
2
Si0
3
is 10"
3
M while the
solubility
product for
PbF
2
is
7.8x10"
8 [10].
3 Notice that in
Table
1.2 f 7-).,,% is
between
4.1 and 5 while in PbF
2
(
7-L*
is 5.45. Some
fluorine
may
be
present
in
compounds other
than
PbF
2
but most of it can be
accounted
as
PbF
2
.
4 As shown in section lll.b washing of the slimes can lead to
further
hydrolysis of the acid and removal or
formation
of
precipitates.
5 This
number
was
obtained
assuming that 1.4 kg of
slimes
are
produced
per ton of
lead.
6 c.s.i.R.o. means Commonwealth
Scientific and
Industrial
Research Organization
(Mineral Engineering
Division,
Port
Melbourne,
Australia).
3Pb+1 + SiF? + 3H 20 <=> 3PbF 2+H^iOj + 4H +
...3
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(F)^hH
,0 system
species considered and their free energy values is provided \
A. (Fj-Si-H 3 0 system
As can be seen in Fig. 1 the stability region of SiF 6~ 2 extends over a
considerable pH range. At pH values larger than 6.75, SiF 6"2
hydrolysis takes
place according to the following reaction a > 3
:
5tf
6
"
2
+ 3H
2
0 <=> //jS/O, + 6F "
+ 47/
+
.. .4
The equilibrium constant of reaction 4 is a function of the ionic strength U91.
Moreover, it has been shown [19] that in the presence of F" ([F] > 6[SiF6"2
]) the
maximum thermodynamic stability of SiF 6'2
occurs at a pH between 2.6 and 2.7.
In dilute solutions, hydrolysis of the hexafluosilicate ion takes place in
elementary steps which involve splitting of one F" and formation of SiF 5[H20]"(reaction 5). This species losses a F" ion in a second step forming SiF 4 which
subsequently undergoes hydrolysis towards H 2 Si0 3 (reactions 6 and 7) [18,20-231:
SiF~
+2H
2
0
<=>SiF
5
[H
2
OT + F~
...5
SiF
5
[H
2
OT
+H
2
0
<=>SiF
A
+ F~ +2H
2
0
...6
SiF
4
+
3H
2
0
<=>H^iOs
+ AHF
...7
In the acidic range (pH<l) formation of SiF 5[H 20]' and SiF 4[H 20] 2 takes place[24]:
Si
F;
2
+
H
2
0
+
H
+
<=> SiF
5
[H
2
0
]~+HF ... 8
SiF
5
[H
2
0
]~ + H
2
0
+H
+
<* SiF
A
[H
2
0
\
+
HF .. .9
Conductometric and cryoscopic measurements have shown that the
solubility of Si0 2 in H 2SiF 6 is due to the formation of SiF 4[H 20] 2 [25]. This compound
behaves as a strong acid [26]:
1 The diagrams were drawn assuming
unit
activities for all
the
solid species species and
T=25
°C.
2 Either
H
2
Si0
3
or Si(OH)
4
can be formed during
the
hydrolysis of
the
acid [16,17]. The polymerization of
orthosilicic acid, Si(OH)
4
, has
been
shown to be
very
slow [18].
3 The dashed
vertical lines in
Fig.
extend
the
region
in which
HF, HF
2
\ and
F
may be
present.
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P b-Fj-Si-HjO system
Fig . 1 System (FJ-Si-HaO at 2 5 C
A SiF 4(H 20) 2 + HF
B SiF5(H20)- + HF
C HF + SilL, (g)
D HF2- + SiH,(g)
E F j O ^ + HjSiO,
F SiF6"2
G H 2Si0 3 + F
H F + SiH*(g)
Activity of all species =1
J
-U -3 " -2 - 1 0 L 2 3 U B E 7 B 9
PH
25/F
6
-2
+ Si02 + 4H 30+
«• 3SiF 4 [H 20\ ... 10
Thus, depending on the equilibrium position of reaction 10, F to SI ratios,
£ , other than 6 can be obtained [25-29].
F
=
6 - [ i/2
S t F
<
J
+
[//f]
Si [H-SiFJ + iSiO-]
B. (Pb-F)-Si-H 2 0 system
The stability region of SiF6"2 decreases significantly in the presence of Pb + 2
according to Eq. 3 (see Fig. 2):
3Pb+2 + SiF~2 + 3H 20 » 3PbF 2 + H-SiO^ + AH* .. .3
The rninimum pH at which PbF2 precipitation can take place is a function
of the activities of the species indicated in Eq. 3. The thermodynamic equilibrium
constant for this reaction can be described as follows:
A 3 - — — ...IZU
»~0 a
siF?
a
Pb
+2
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(Pb-F )-Si-Hfi system
Assuming that 0^ 0 = 3 / . ^ = % ^ = 1. the pH values above which PbF2
precipitation can take place are given by the following equation (see also Fig. 3):
pH = 1.78 - 0.25 l o g [ ^ • ... 13
Thus, the larger a$iF -2 and a pb+2 the smaller the pH at which PbF2
precipitation can occur.
F ig . 2 Sys tem (Pb-F) -S i -H 2 0 at 25 C
F 2 0 (g) + U-SiO- +Pb02
SiF 4(H 20) 2 + HF + Pb0 2
SiF 5(H 20)-
+ HF + Pb0 2
SiF6-2
+ Pb0 2
F + PbOj + H j S ^SiF 4(H 20) 2 + HF + Pb*4
F ^ G O + Pb*4
SiF 4(H 20) 2 + HF + Pb
SiF 4(H 20) 2 + HF + P b +2
SiF s (H 2 0)+HF + P b +2
SiF 5(H 20)-
+HF+Pb
SiF6-2
+ P b +2
PbF 2+H 2Si0 3
F + PbSiOj
SiF6'2
+ Pb
F + Pb + HjSiOj
HF + Pb + S i ^ (g)HF 2 + Pb + SO^ (g)
F + Pb + SiH, (g)
Activity of all species =1
Once the activities of the individual species are known, Eq. 13 can be usee
to obtain an upper limit for the ionic concentrations across the slimes layer to
avoid obstruction of the ionic flow due to Si0 2 and PbF 2 precipitation
1 The extra weight of these precipitates can also lead to falling slimes.
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(Pb-Fj-Si-HjO system
1.2 1.4 1.6 1.8 2 2.2
PH
Further analysis of Eq. 3 indicates the difficulties encountered during the
washing of the slimes to obtain accurate PbF 2 and Si0 2 assays. Depending on the
pH and ionic strength of the washing solution, PbF 2 and Si0 2 may either precipitate
and/or re-dissolve indicating erroneous assays. Only by slowly displacing the
concentrated electrolyte with solutions of varying PbSiF 6-H 2SiF 6 composition and
matching pH but different strength, can the extent to which these secondary
reactions take place be diminished1. This process involves a trial and error
procedure using "slices" of slimes taken at different slimes thickness. In a first
try the composition of the slimes electrolyte is measured and solutions with similar
pH but diminishing ionic strength are prepared. The slimes are put in contact
with these solutions until all the SiF 6"2 and Pb + 2 are extracted. Once these ions
are removed, the slimes can be washed with dilute HN0 3 solutions at the same
pH as before.
1 Notice that for the washing
process to be
successful, activities of
Pb*
2
, H*,
and
SiF
6
"
2
are
required
to
remain
in
a
fixed
ratio
given
by the following equation
(see Eq.
12):
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(Sb-FySi-Hfi, (As-F)-Si-H,0,
and (Bi-FJ-St-HjO systems
C. (Sb-F)Si-H 3 0, (As-F)Si-H 2 0, and(Bi-F>SiH 2 0 systems
As shown In Fig. 4, Sb, As, and Bi are noble with respect to lead and their
dissolution will not take place unless the potential and/or the pH rises permit
the formation of oxides and/or soluble species \ Sb is the least noble of these
three elements and is expected to react earlier than the others. Upon dissolution,
Sb forms soluble oxides some of which have been detected in lead slimes (e.g. see
Tables 1.2 and 4.2) [301
3. Arsenic also forms highly soluble oxides such as the
amphoretic arsenious anhydride, As 4 0 6 1311
s
. When As is exposed to moist air it
becomes covered with oxidation products [31]. This poses a problem in its
identification in the slimes layer. Bi does not form oxides to the same extent as
Sb and As, but rather, forms polynuclear complexes such as ByOH)^*6
[301. These
ions have been found to be the predoiriinant species at pH=1.5 and total Bi
concentrations of 0.01 M [301.
Sb + SiFV2
Sb + SiF5(H2
0)- + HF
SbO+
+ SiF6-2
SbtT + SiF^j OY + HF
Sb4
0
6
+ SiF6"2
Sb2
0
4
+ SiF6-2
+ HF
S b A + S i F ^ O V + HF
Sb 2Os + SiF,1
+ HF
Sb + HF
Activity of Sb soluble
species = 10"*
1 The fat dashed lines in
Figs.
4A, 4B, and 4C
mark
the stability region of Pb*
2
according to Fig. 2.
2
In the pH range between 0 and
1, Sb
in solution
is
present mainly
as
SbO* [30].
3
The
solubility
of As (as
As
4
0
6
)
at
25*C
is
-0.17
M [31].
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pH, density, viscosity, and activity of HjSiF, solutions
IV. Phys ico- Chemi cal Propert ies of H 2 S i F 6 - P b S i F 6 E lectrolytes
To model the concentration changes across the slimes layer, data on the
physico-chemical properties of H 2SiF 6-PbSiF6 electrolytes are required. In the next
section, the experimental data generated in this work are analyzed alongside data
available in the literature.
A. pH, density, viscosity, and activity qfH 2 SiF e solutions
The pH of H 2SiF 6 solutions is shown in Fig. 5. These pH values agree with
those reported in the literature [20,32].
44
33
- Fig. 5 Changes in pH as a function
of the electrolyte composition
1 1 2CL
Z T=25"C
Initial Electrolyte Composition:
[H2SiF6]=2.040 M, [SiOJ=0.40 M
1 1 2CL
Z
-
T=25"C
Initial Electrolyte Composition:
[H2SiF6]=2.040 M, [SiOJ=0.40 M
1
01C
T=25"C
Initial Electrolyte Composition:
[H2SiF6]=2.040 M, [SiOJ=0.40 M
1
01C
-
: ;
1
01C"5 10"4 10"3 10~2 10~1 10°
[H
2
SiF
6
],
mol/l
H 2SiF 6 and H 2 S 0 4 are very similar in their electrolytic properties [20,33].
H 2SiF 6 is an acid of moderate strength similar to that displayed by H 2 S 0 4
solutions .The first and second dissociation constants of H 2SiF 6 at 25 °C are as
follows [341:
H 2 SiF 6 <=> HSiF; + H + ...14
HSIF; <=> SiF;2
+ H +
pK 2=\.19 ...15
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pH, density,
viscosity,
and activity ol HgiF, solutions
Arkhlpova et al. [27,28] have indicated that the pH of H 2SiF 6 solutions (among
other properties) is a function of j. and varies according to the way the acid was
produced.
Leonte et al. [351 measured the density, viscosity, specific heat, and electrical
conductivity of solutions containing between 0.3 and 2 M H 2SiF 6 . They showedthat due to the technique under which their acid was produced, minor impurities
such as P 2 0 5 and S04~ 2 were present in concentrations that increased with the
acid composition. On the other hand, no j t values were reported in their samples.
Leonte et al. derived empirical equations for the density and viscosity changes
as a function of the acid composition 1 :
p = 0.9982 + (8.53 • 10"3 -1.96 • 10"5r)w - (2.99 • IO"
1
+ 8.06 • lO^w) (t - 20) ... 16
c
x\=AeT ...17
A and Q = empirical coefficients which depend on the solution composition (see Table 2).
T = temperature Kt = temperature "C
Table
2 Values of the coefficients A and Q in Equation 17 [35]
[H 2SiFJ, M Q
0.00 1.219X10"3 1965.93
0.334 1.908x10* 1857.05
0.663 2.027xlO'J
1860.40
0.972 2,434xlOJ
1834.361.298 2.937X10'1 1803.40
1.708 2.485X10-1 1884.70
Folov et al. [361 measured the osmotic and activity coefficients of H 2SiF 6
solutions at 25°C using isopiestic measurements and H 2 S 0 4 as standard. As can
be seen in Table 3, y± do not change significantly with ionic strength.
1
Sohnel
et
al. [37] also
present
a p-[H2
SiF
6
]
correlation
from which
the apparent partial
molar
volume of
HgSiFs at
infinite
dilution,
,
was obtained: <J>v =23.5 cm
3
mol"
1
at
17.5"C.
[202]
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Density, viscosity, and electrical conductivity o/H£iF,-PbSiF,
electrolytes
Table 3 Changes in the osmotic and activity coefficients as a function of the molal ionic strength, [36]
molality, m molal ionic strength Osmotic Coefficient, <p Mean Activity
Coefficient, y±
0.10 0.30 0.931 0.611
0.20 0.60 0.577
0.30 0.90 0.927 0.562
0.40 0.934 0.555
0.50 1.50 0.944 0.553
0.60 1.80 0.954 0.553
0.70 2.10 0.964 0.559
0.80 0.975 0.559
0.90 2.70 0.987 0.564
1.00 3.00 0.999 0.571
1.20 3.60 1.026 0.588
1.40 4.20 1.060 0.612
B. Density, viscosity, and electrical conductivity of H 2 SiF e-PbSiF g
electrolytes
Some of the experimental data obtained from measurements of the
physico-chemical properties of H2SiF 6-PbSiF6 electrolytes are shown in Table 4.
Each sample was analyzed for H 2SIF6, PbSiF6, Si0 2, and HF 3 - 3 \
Two different values of j. are indicated in Table 4, and each one of them can
be related to the presence or absence of Si0 2 or HF in H 2SlF6-PbSlF 6 electrolytes:
1 H
2
SiF
6
-PbSiF
e
electrolytes
were prepared by neutralizing H
2
SiF
6
solutions with PbO.
2 The dynamic viscosity, TJ is the ratio between the applied shear
stress
and rate of shear.The cgs
unit
of
dynamic viscosity is one
gram
per
centimeter
per
second
and its called one poise (symbol P). One
centipoise,
1 cP
=
10'2
P.
3 The kinematic viscosity, v, i s a measure of the resistive flow of a fluid under gravity, the pressure head
being proportional to the density of the fluid [44]. The cgs unit of kinematic viscosity is one
centimeter
squared per
second
and it is called one
Stoke
(symbol
St).
1 centistoke,
cSt
=
10"2
St.
[203]
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Density, viscosity, and electrical conductivity o/H,SiF,-PbSiF, electrolytes
Table 4 Physico-chemical properties of H2SiF6-PbSiF6 electrolytes Experimental Results
Electrolyte Composition, M £ ratio Electrical Conductivity, Dynamic Kinematic Density,
JC, mmhos cm'1
Vfecosfty, Viscosity, p. g cm"3
TJ.CPv
.cSt
p. g cm"3
Solution [PbSiFd
[HaSiFJ
[SiOJ [HF] (a (51 T=40'C To25.5 "C T-40"C T-40'C T» 23 -
C
s1 0 2040 0.4 0 5.0 5.0 525.1 443.7 1.229
From mother solutions pre
pared using acid sample S1 (Technical
HaSiFg
supplied by Cominco
Ltd.):
v17 0 0.215 0.04 l i i bl l i l l i i o l i 5.0 128.1 108.7 0.697 0677 1029v18 0 0.332 0.05 o 5.2 5.2 190.2 160.5 0.724 0.699 1.036
v19 0 ilblMOlI 0.06 l l lbl l l :111211
l l l s i l l l 215.9 183.1 0754 0.714 1056 v20 o 0.847 0
.13 """""" 'S'"""""""" 5.2 ,,,,,g,^,,,,,. 359.6 303.9 0.822 0.753 1.091
V21
01
.017 0.20 0 l l i i i i 5.0 400.0 341.9 0.893 0.7961
122k6 0.27 0.692 0.09 b " " 5.5 5.3"" 293.7' 245.3
v13 032 iioi»ii 0.09lllblll
i 4llll l l i i l l 237.4 1990 0.843 0.726 1.161vii
0.32 0.837 0.13 5.4 """
354.4 297.9 0.962 0.791 1.216
v12 0.34 0934 011 i i lbl l l l l i M i illliill
i l $ p l i 3312 1.016 0.827 1.228zl4 0.35 0.795 0.07 o 5.7 5.5
331.8 282.4 0.961 0.79 1.217
V14
0.36 l l o M i l 0.04 l l l b l l l iiisliiillellll
1391 0773 0 712 1.085K2 0.36 0.830 0.09 o 5.6 5.4 345.6 290.9 1.215
kt I l K l l 0792 lilolli Iilbll l l l i i i i ; l l l i i l l llililf 295.4 v4 0.36 0.067 b""' 6.0 6.0 80.2 63.9 0.778 0.681 1142
v100.59 1.010 0.15 llf&Ill 11P11i i i s i i i i lliwsill 363.5 1.137
0.861 1320 v9 0.64 0.849 0.14
5.5 •""""•'•5:2 " v " 345.1 2844 1.084 0.831 1.304
v16 l l l i i i l 0 363 005 l i l o l i l i i s l l il lSlll
i l p p l 1971 0.989 0757 1.306
v15
0.81 0.165 0.02 o 5.9 5.4 173.8 143.8 0.935 0.73 1.281
v3 096 0.083 l i l o l l i 0 lllbli•loll11150111
123.4 0977 0 737 1.325 v8 0.98 0.716 0.12 o 5.6 5.2 300.7 251.2 1.170 0.831 1.408
v7 Itilili 0.331 0.06
lllbllllltllil l i i f l l 231.1 lll ill
1.109 0807 1.374
v2 1.21 0.097 o o 6.0 6.0 177.2 144.1 1.090 0.771 1.414
v6 1.21 0.156 0.00 o iiiiii lllSiii l i b l l l l 157.2 1.115 0 785 1421
ALIO
1.45 0.446
0.09 5.7 5.0 249.3 205.8 1.209
AL1 1.46Ilb5i4ill
010 l i i b l l i l l 5 l t l : l pllspll 11 :711i i p p l 1 546k4 1.53 0.317 o o 6.0 6.0 256.2 213.5
v5 0.306 0 0 6.0 l l l i i l l :|l236l5ll:
196.1 l l l i i l l 0.887 1570
vi 1.61
0.092 ob
6.0 6.0 201.3 163.9 1.300 0.846 1.537
sol 42 1.62 0.069 l i i i l l ! l l l i i l l ll&ii l l l i i i i l i w i i l 170.6
pbaOl
1.75 0.119
0 6.0 6.0 209.6 171.6
vO 1.75 0.124 l l i b l i f l l l i i l l ll&ii l i i i l l lliilili llliil 1.387llliill
1582sol 6 1.8 0.151
ij,,,,,.
6.0 6.0 211.5 173.0 1.612
w71 0 3.064 0 0.40 6.1 6.1 1.397
From mother solutions pre
pared using acid sample VV71 (Technical Reactive BDH Chemicals):
w65llllbllili i lbl l l l 0.22
llMii
132.0 108.5 0.923 0.714 1 293w62 1
.46 o b 0.07 6.0 209.9 174.9 1.153 0.785 1.469
w61 0 Iilbllll lioloalll WMtim i iMi i lilblieli 1.500 0.895 1.676
w60 2.58 0 o 0.12 6.0 247.7 203.0 1.958 1.066
1.837
w64 2.74 0 0 0.63 iiifci! llaSSIIl 170.0 1.979 1.064 1860w50 2.88 0 0.66 6.2 241.8 196.7 2319 1.199 1.934
w63ll ll
0 0 llolsfllli ll&all 1:4:207.91:::!!!l67t6lll II'PSSII 1.942w51 3.25 0 .................g, .............. 0.15 6.0 203.5 161.3 2.520 1.26 2.000
Dynamic viscosity
values were
obtained by assuming that p13 .c = p v p
[204]
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Density, viscosity, and electrical conductivity
0i,iSSiF,-PbSiF,
electrolytes
(- 6
• {[H-SiFJ + [PbSiFJ} + [HF]
[HzSiFJ
+ iPbSiFtl + lSiOt]
...18
[HJiFJ + lSiO-]
As can be seen in Table 4, for pure H2S1F6 solutions, the amount of dissolved
SiO a increases with acid strength without significant changes in (samples
VI7 to V21). This relationship holds even in the presence of PbSlF6. The amount
of dissolved Si0 2 appears to decrease with the PbSlF6 concentration but by
comparing the j. values it can be seen that (ji)b remains close to 5 even in the
presence of large amounts of PbSiF6 (e.g. compare and in samples V13,
V7, and AL1). This indicates that soluble silica is present as H2S1F6 reaction
products such as those shown in Eq. 10. Similar Si0 2 solubilities for H 2SiF 6-PbSiF6
electrolytes are reported in the literature [io]. In the absence of H2S1F 6, minor
amounts of HF are required to avoid PbSiF6 hydrolysis (Samples W50 to W65).
As seen in Table 4 only when HF is present can j. reach values larger than 6.
By neglecting the presence of Si0 2 and HF \ empirical equations were used
to correlate K, p, and v, as a function of the H 2 SiF 6 and PbSiF6 concentrations.
These correlations are as follows 2 l 3
:
1
The presence of Si0
2
appears to increase p and
v. On
the other
hand,
HF appears to have
a
larger
effect on K than on p or v.
2 From these correlations K , p, and v, values that are within 6% of the experimental measurements can
be obtained.
3 A simple relationship between K
,
[PbSiF
6
]
and
[HjSiFJ,
for the range of
compositions
shown in Fig. 6A
could
not be
found
because of the changes in K at [H2
SiF
6
]>0.4
M .
Eq.
19
although cumbersome
describes accurately the electrical conductivity changes as a function of
[PbSiFj
and
[H
2
SiF
6
J.
[205]
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Density, viscosity, and electrical conductivity ofHSiF.-PbSiF, electrolytes
-T-AKJX^+^11^3+A 1 2x 3 + A13x3 + A1 4XJXJ+A1 5x 2.x 3 ...19
p ^ A ^ A ^ + A j ^ ...20
v^A^AjV.+Ajvf+A^ ...21
Where:
K „ = electrical conductivity at 40 "C, [mmhos cm'1]
v a - Kinematic viscosity ( coefficient of) at 40 °C. |cSt)Pb = density at 23 'C. [g cm"3]
xlf X2, x3, y l f y2, y 3: variables related to the H 2SiF 6 and PbSlF 6 concentrations as follows:x, = [PbSiFg] Xj = [PbSiF6]+ [HaSiFd x 3 = 2 x [H 2SiF 6]y, = [PbSiF6] y 2 = [H2SiF6l y 3 = [PbSiF6]x[H2SiF 6] Aj to A 1 5 are constants whose value is given in Table 5.
Table 5 Coefficients in the empirical electrical conductivity, viscosity and density correlations
Coefficient Electrical Kinematic Viscosity,
conductivity, K, (Eq. 19)
Density, p. v., (Eq.21)
(Eq. 20)
A, -1.212E+08
9.951 E-01
6.423E-01
l i i f i i t i i i l i i i i i i i i i
5.666E+01
3.223E-01
4.340E-02
A,2.244E+07 1.668E-01 4.320E-02
A* 1.473E+07 1.606E-01
As2.981 E+07
Ae 1.212E+08
A7 -3.090E+02
Ae -3.717E+07
Ag-2.713E+01
A101.273E+01
A,,-6.062E+07
A121.677E+02
A13
9.293E+06
Au3.563E+00
A15 1.129E+01
Eqs. 19 to 21 can be used to compute K , p, and v of solutions whose
composition range within 0<[PbSiF6]<2 M and 0<[H2SiF6]<l M.
Fig. 6 shows changes in K , p, and v as a function of the electrolyte composition.
Both p and v increase uniformly with [PbSiFg] and [H 2SiF 6] while K shows a decrease
in value with PbSiF6 additions at [H 2SiF 6] larger than 0.4 M.
[206]
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Density,
viscosity,
and electrical
conductivity
o/H,SiF,-PbSiF,
electrolytes
T= 40°C
Fig. 6 Electrical
conductivity, density, and viscosity of
H
2
SiF
6
-PbSiF
6
electrolytes
Data points were obtained from the empirical
correlations
shown
in Eqs. 19,20, and 21.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.B 2
[PbSiF6
], mol/l
[207]
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Density, viscosity, and electrical conductivity ol H£iF,-PbSiF,
electrolytes
Further analysis of the conductivity changes as a function of the composition
of the individual salts, indicated that the equivalent conductivity of H 2SiF6-PbSiF6
mixtures can be represented by the following relationship:
= xA PbSiFt + (1 -x)A H iFt .. .22
where:
Kg* W _ Ke W /,
^mixt - n r r D i , c , T i , r i / c . T i i A
J W .
_ - — A H_. W P .2{[PMiF6] + [/W 6]} w
« 2[PbSiF 6 ] lt 2-[/72S/F6]/,
x - -
Thus, for example, for an electrolyte mixture containing 0.36 M PbSiF6 and
0.252 M H 2 SiF 6 at 40 °C:
It = 2.196 I {PbSiF j = 1.44 x = 0.66
[PbSlF6]It = 0.549 W ^8 7
-8 A miF=80.0
[H2SlF 6]It= 0.732 % W /=334.9 A WjOT=228.8
A^BO^xSO + il -0.66) x228.8 = 131.2 1 ^ = 2x0.612x131.2=160.6
obtained from Eq. 22 is within ~4% of the experimentally measured
value \ In general, Eq. 22 could be used to obtain A ^ values that are within 6%
of those experimentally measured. This indicates that both PbSiF6 and H 2SiF 6 are
highly dissociated.
The validity of Eq. 22 to describe A ^ changes in strong electrolytes has been
related to a lack of simple dependance of the ionic mobilities on electrolyte
concentration [45] and is a reflection of Walden's rule 2
:
•nX, = constant ...23
X, and m are related through the following relationship:
1 This
iwvalue
compares
well with
the experimental K value shown in Table 4 (K^= 160.6 vs K =167.1
mmho crrr for solution V14).
2 Walden's rule is based on a model for ionic mobility that obeys
Stokes
law in the laminar flow region
[51].
[208]
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Density, viscosity, and electrical conductivity ofH,SiF
(
-PbSiF,
electrolytes
H . = - ...24
The validity of Walden*s rule has been confirmed in a variety of systems [46].
Recently, Peters et al. 1471 have shown that in strong chloride mixtiires, Walden's
rule is not followed by H+ because its movement in electrolytic solutions is not a
function of the viscosity of the medium but of the activity of water. Similarly, in
H 2SiF6-PbSiF6 electrolytes Walden's rule is expected to be followed by Pb+ 2
and
SiF 6~ 2 while T )X H + values should decrease with increasing ionic strength as the water
activity decreases.
To obtain ^ values from electrical conductivity measurements In
H 2SiF6-PbSiF6 electrolytes the following assumptions were made:
1) H 2SiF 6 and PbSiF6 are completely dissociated and the formation of ion-pairs
can be neglected.
2) Only three ions contribute to the total electrical conductivity: Pb+2
, SiF6"2
,
and FT.
3) The electrical conductivity in concentrated H 2SiF 6-PbSiF 6 electrolytes can
be represented by the following relationship:
K=iic1.zixk
-l
•••
25
i
4) The individual equivalent conductivity of SiF6"2 is equal to the individual
equivalent conductivity of Pb+ 2 (i.e. X Sif -2 = X pb+2 ) \
5) Xi values in pure H 2SiF 6 and PbSiF6 solutions are the same as those of
H 2SiF 6-PbSiF 6 iriixtures of equivalent ionic strength:
=
2
•
C pb+2 • X pb+2 + 2
•
C SiF -2 • X SiF _2 --= 4[PbSiF 6 ] lt .X pb+2..26
%lSP ,6l/ l
= = C H+-X H+ + 2-C SiF;2-X SiF -2 = 2
[HiSiF^-X + l-iHJiF^-X^..27
thus, should be equal to:
1 Individual equivalent conductivities at infinite dilution and at 25 °C reported in the literature are:
[209]
X
OT
_
2
=
59[53] and X
pt
,
2
=65 [46]
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Density, viscosity, and electrical conductivity ot H£iF , -PbS iF , electrolytes
C pb+1' Kb+ 2
' '
+ C
H
+
'
V ...28.a
=
2
• X pb+2 +
2 • [HJiFJ • V + 2 • { t * W J + [PbSiF 6 ]} • ...28.6
=
2 V+ (2
•
[Z/^/FJ
+
4 •
[ / W J }
•
X pb+2
...28
.c
Assumptions 1 and 2 can be considered to be fair approximations of the real
behavior of H 2SlF 6-PbSiF6 electrolytes as indicated by the validity of Eq. 22.
Assumptions 3, 4, and 5 are linked through Eqs. 26 to 28 . They are valid only
to the extent to which X, values from binary solutions (i.e. from Eqs. 26 and 27)
can be used to obtain the conductivity of ternary mixtures (i.e. from Eq. 28). Such
correspondence has been found to occur in chloride electrolytes 148] for which
V =
Kr •Thus, to obtain X, values and their dependance with concentration, the
following steps have to be followed:
1) Given [PbSiFg] and [HaSiFg] in the mixture obtain the total ionic strength,
It= 4x[PbSiF6] + 3x[H2SiF6]
2) Obtain the composition of pure H 2SiF 6 and PbSiF6 solutions at the total
ionic strength and the specific conductivities of these solutions (From Eq. 19):
[PbSiFJn = \ with K =
[H2SiF6]It= \ with K = %2stf -6]/i
3) Obtain X pb+2 and X H + from Eqs. 26 and 27 respectively:
KiPbSiFfo KwjpJn ~ 2 ' -H -$ iF 6 -u • Kb*1
r»+1 = 4[PbSiF 6 ]
It " + = 2[H
2 SiF
6 ]
h
4) Introduce X pb+2 and XH+ in Eq. 28.c and compare the K ^with the obtained
from Eq. 19.
For example, for an electrolyte mixture containing 0.36 M PbSiF6 and 0.252
M H 2 SiF 6 at 40 °C ( r ^ = 0.769):
[210]
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Density, viscosity, and electrical conductivity ol HSiF,-PbSiF, electrolytes
It = 2.196 IPbSWJ= 1.44
[PbSIF6)It = 0.549 =87.8 = 40.0
[H2SiF6]
lt= 0.732 K,W /=334.9
= 2 x 0.252 x 188.8 + {2 x 0.252 + 4 x 0.36} x 40.0 = 172.9
= 188.8
TUA* = 0-769 x 188.8 = 145.2 . = 0.769 x 40.0 = 30.8
Thus, it can be seen that obtained from Eqs. 26 to 28 is only -4% different
from the experimentally measured value \ For other electrolyte compositions,
the maximum deviation between the experimental and predicted values is -8%
(see Table 6). Finally, transference numbers, tj, were also obtained as follows:
Changes in Walden's product as a function of/,"2
are shown in Fig. 7. Large
decreases in X H+ as the ionic strength increases are due to decreases in water
activity. At high It less water is available for H+
to move via a proton jump
mechanism and this reflects in lower X H+ . On the other hand, Pb + 2
transport
depends mostly on the viscosity of the medium rather than on the water activity2
. Furthermore, by comparing Figs. 7A and 7B it can also be seen that in binary
solutions nX* values are nearly equal to those obtained in ternary solutions of
equivalent ionic strength (see also Table 6).
1 This twvalue compares well with the experimental
K value
shown in Table 4 (K nM = 172.9 vs K =167.1
mmho
crrr for
solution
V14). The difference between these values is partially due to innaccuracies in the
computation of K values using Eq. 19.
2
Water activities
for this
system are not available. Yet, by analysis of similar phenomena taking place
in
other systems [47] this analogy can be drawn.
[211]
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Density, viscosity, and electrical conductivity ol H£iF,-PbSiF, electrolytes
Table 6 Changes in the individual ionic mobilities as a function of the electrolyte composition
T=40'C X Sir = X m t J Theoretical Results
Solution Solution Properties lllfifl Binary Individual Walden's Transport
Composition, u From Eqs. 20,21, and Properties Properties Equivalent Product numbers
22 Strang* (PbSlF6]=J [H2S1F6
1
=J Conductivities
[PbSiFJ [H,SiFJ rrrtxt Pmtxt It [PbS.F.1 K [HjSiF,] K
WMmV Eq. 29.C w
200 0.00 I i i ! 0.902 1.640 i i b i i i liool?mi llell izlil 954 I269I1 38.7 .
141.1 0.50 0.00
1.88 0.06 208 0.887 1.6107
.72 1.93
208 2.57
624 27.0 94.1 218 38.6 134.5 0.47 0.05
1.52 0.849 1.517 i i i i iilfiiWMW
IliSl l;;
564:lWWii
I9I2II IIMI :l37.9lf 1213 llioii 0.20
1.17 0.43
249 0.823 1.429 6.00 1.50 189 2.00 526 31.5 100.0 262 37.0 117.6 0.33 0.33
0.85 0.61 mm 0.808 1345 liill Ural 174 I I M I 332 110.4
WmW
120.0 0.27 0.47
0.35
0.87 350 0.802 1.2184
.00 1.00 ' 143 1.33
456 35.6 135.0 346 34.8 131.9 0.16 0.68
0.10 0.10 0.663 1.040 0.70 0.18 34 0.23 48.3 238.4, , , , ™ , , , ,
33.3
164.4 0.19 0.62
0 25 0.25 iisi! 0.695 1106 1.72 1614311liii WMW
l i i i 41.7 205.1 Ileal: 321 Islil Iiiii 0.620.49 0.49
248 0.753 1.2153.43
0.86 ' 126 ' 1.14 427 36.8 150.2 25533.7 137.4 0.21 0.58
0 76 liiiii lisnll 0.828 1.3465.47 WMmlire!ll&fwww
32.6 106.4 36.4 118.7 0.24 0.520.88 0.86
337
0.855 1.390 6.16 1.54
192 l05 ' 532 31.1 98.6 338 37.0 117.1 0.24 0.51
0.98 0.98 liii Iflff. 1.434 IIPIlltllIH!iiiii iiiii IS!Illl!wWliiPii1111110.24 0.52
000 007 l l a l l 0.653 l i i i 0.21 005 Ilil l i i i WwW56.0 245 8 367 161.2 019 0810.00 0.48 248 0.720 1.056 1
.45
0.36 62 0.48 249 43.0 214.6 249 32.7 163.1 0.17 0.83
000 0.90 0.786 1.109 11691111616711l i 6 i l 16:9611Wmw
38.6 1378:1 33.6 Isolli 0.18 0.820.00 1.10 421 0.820 1.135
3
.31 0.83 123 1.10 421 37.1 153.6 421 34.5 142.8 0.19 0.81
0.00 1.38 l:;
46i!l 0.664 1 17011111 l-iiil
iilii1.38 iiiii 35.3 131 8 35.7 133.2 iiiii: 0.79
0.00 1.66 491 0.908 1.2054
.97 1.24 167 1.66 49133.7
114.6 491 36.9
125.4
0.23 0.77
0.00 1.86 511 0.941 1.2315.59
1.40 181 1.86 511 32.4 104.9 511 37.5 121.6 0.24 0.76
0.00 Iflll Illl! 1.249
11!?! Illlllwww Mil IllllllUti 120.4 Illl! 0.76
0.10 000 i i i 0.647 1.027 0.40 lito-l Illllll i i f l wWii52.1 :|24|il iiiii Wmw162.8 Iiiii 000
0.40 0.00 66 0.667 1.124 1.60 0.40 68 0.53
268 42.3 209.3 68 31.7 156.8 0.50 0.00
0.90 0.00 0.716ImM
.60 illol&il 131 120 l lal l 36.4 1455 l lal l 335 laMil 0.50 0.00
1.00 0.00 143 0.729 1.3174
.00 1.00 142 1.33
455 35.6 135.1 142 34.2 129.8 0.50 0.00
1.25 0.00 169 0.764 1.3985.00 1.25 168 1.67 492
33.6 114.0 168 35.9 121.7 0.50 0.00
1.50 0.00 189 0.805 1.479 6.00 1.50 189 2.00 526 31.5 100.0 189 37.4 119.0 0.50 0.00
In Table 6 it can be observed that as the electrolyte gets concentrated in
PbSiF6 and depleted in H2SiF 6, tH+
—> 0 while / 2 0.5. Thus, at the anode/slimes
interface most of the current should be carried by Pb + 2 and SiF 6
2
while at the
slimes/bulk electrolyte interface most of the current should be carried by H + . This
should create significant concentration gradients as Pb+ 2
and SiF6"2 are ~5 times
less mobile than H+
.
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Density, viscosity, and electrical conductivity ol H£iF,-PbSiF,
electrolytes
(A) (B)1
300
O
0101
• H
250
1
300
o
CD
01•p'
250
1
300
O
0101
• H
250
• i • 1 • i • i 1 • 1 •
A 2.0 <[PbSiF„]< 0.1 , 0.0 <[H2SiF,]< 1.0
.« 0.1 <[PbSiF6]< 1.0 . 0.1 <[H,SiF6]< 1.0.
1
300
o
CD
01•p'
250
i .
* [PbSiFg] = 0.0, 0.1 <[H2SiF6]< 2.0
« 0.1 <[PbSiF6]< 1.5. [H2SiF 6] = 0.0
ZfV
9•p"
200
-y ' 200
G G
£
150
o
£
150
cj
-CM 1 0 0
|
•
o\
50
-
o\
50
"
77^2
< r.
3 0.5 1 1.5 2 2.5 3 ^ (
3 0.5 1 1.5 2 2.5 3
Ii* It"
Fig. 7 Changes in Walden's product as a function of the square root of the ionic strength
(A) Changes in Walden's product for H2SiF
6-PbSiF
6 mixtures of different compositions
(B) Changes in Walden's product for binary H2SiF6 and PbSiF6 solutions
The Nernst-Einstein relationship [49] can be used to obtain diffusion
coefficients from ionic mobilities1 :
D: =&RT _\RT
.30
Thus, for a solution conteining 0.35 M PbSiF6 and 0.84 M H 2SiF 6, at 40 "C,
^ = 35.8 and ^ = 137.1, from which: D H+ = 3.8 x 10"5
and £ ^ = 5.0x10^. This
diffusion coefficient for Pb + 2
is nearly equal to that obtained from
chronopotentiometric measurements (seeTable 1 Chapter 5). The correspondence
between these values and the validity of Walden's rule for species other than H +
support the assumptions and the model used to derive Individual equivalent
conductivities from electrical conductivity measurements.
1 As shown in Appendices and 2 this equation can be modified to include the
effects
of the effective
charge of species i exposed to the electric field, Z*. When
such modification is included: D, = D/".
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Mathematical Model: Numerical Solution of the Nernst-Planck Flux Equati
V. Mathemat ical Model : Numerical Solut ion of the Nernst -Plan ck
F lux Equations.
To obtain local ionic concentrations and potential gradients across the slimes
layer as a function of the electrolysis conditions the Nernst-Planck flux equations
were solved using a finite interval numerical technique (see Appendix 1). The
differential equations that were solved simultaneously are as follows [50]:
1 dlna„ d
<D /(r)
—CD c - + U . c — = — — ...rn N a
a a
dx * a
"dx Z aF 1 1
1 d\nab rfq>
1 dlnac dQ>
Z aC a+Z bC b+Z cC c=0 ...[/v]
Eq. i states that the only species being generated at the anode is Pb + 2
whereas
Eqs. if and iii indicate that H+
and SiF6"2
do not react and their equilibrium
concentrations are only a function of the migration and diffusion gradients 2
-3
.
Eq. iv is the electroneutrality condition.
As presented in Appendix 1, to solve Eqs. [i] to [iv], relationships between the
involved variables fy, 2>it u,) and the electrolyte composition (Ca, C b , and CJ have
to be known in advance. Thus, subroutines to compute the values of those
variables as a function of the electrolyte composition were incorporated in the
computer program. In these subroutines, individual ionic mobilities, |i„ were
1
©
;
=D?+D E : D E is the eddy diffusion constant
which
can be considered to be equal for all the ions.
Furthermore, it can be modified so as to incorporate changes in porosity and tortuosity of the ionic
flow
across the slimes layer for different anode compositions and/or electrolysis conditions.
2 Notice that when the molecular
diffusion
coefficient is smaller than the eddy diffusion constant, the
overall
diffusion coefficient is nearly the same for all the components. In
this
case, eddy diffusion
practically equalizes the
effective
diffusivities for all the components [so,p. 238]
3 Dp and pn can be linked using the Nernst-Einstein relation:
D? = ?h-
Z;F
where Z* is the
effective
charge of species i exposed to the electric field, z* <Z,.
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Cass A: constanty,
values
obtained from \ values (see Eqs. 25 to 27) \ From the computed ^ values D"
values were obtained using the Nernst-Einstein relationship (Eq. 30). To these
values, D E were added to obtain corresponding 2} values 3
. Individual ionic
coefficients were assumed to remain constant 3
in the whole range of
concentrations. Changes in y,as a function of the ionic strength require knowledgeof activity changes (and water activities) in H 2SiF6-PbSiF6 iriixtures unavailable
in the literature.
The solution of the Nernst-Planck flux equations was obtained by increasing
the number of variables analyzed. This was done by first neglecting eddy diffusion,
then incorporating it in a re-run. Potential and ionic concentration terms are
represented by the following symbols:
[M+n] = Ionic concentration of ions M, [M].[NT"!,, = Ionic concentration of ions M, at the slimes/bulk electrolyte interface, [Ml.[M
+n
]e = Ionic concentration of ions M, at the anode/slimes electrolyte interface, [Ml.{M*
n
}r = Concentration of ions M at the anode/slimes interface with respect to theirconcentration at the slimes/bulk electrolyte interface.
<J>e = Maximum value of the migration potential, mV
A . C a s e A ; constant y( v a l u e s
If activity coefficients are assumed to remain constant 4
, the Nemst-Planck
flux equations can easily be solved numerically or analytically. This also allows
the numerical procedure to be compared with the analytical solution 5
. This
comparison indicated that the numerical method provides the same concentration
and potential profiles if the step size is chosen to be between 25 and 100 |im.
Therefore, step sizes of this order of magnitude were used in the numerical
solution.
1 By using
the X, values obtained at
40
°C, concentration and potential profiles are also obtained at that
temperature.
2D E changes also must be known in advance. D
E
is not only a function of the electrolyte composition
but of the slimes thickness as well. In the initial computation of the concentration profiles this number can
be assumed to be
zero.
This value can be changed once local concentrations are known (i.e. as
a
result
of changes in v and
p
as a function of the slimes thickness).
3
As shown in Table 3
y
±
does not change significantly with ionic strength and therefore this assumption
is not unreasonable.
4 This is equivalent to assuming
that
\ = y b = y c = 0
in Eqs. A1.1 to A1.8 (Appendix 1).
5
The analytical solution for the case
when
activity coefficients are constant, and mobilities and diffusion
coefficients are independent of concentration is presented
in
Appendix
2.
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Case A: constant f values
(A)
s . 6a
E O
•a
33
E Socuc
o
13
Distance from the Slimes/Bulk Electrolyte Interface, mm
(B)
o5
fio
e
<uoao
o
7 . 3 6
43 . 0
33 . 0 -
i
i
0 0 0
2 3 . 0 -
1 3 . 0 -
3 . 0 0 -
i i i1.2
ocuc
o
Distance from the Sli me s/Bulk Electrolyte Interface, mm
Fig. 8 Variation in the slimes electrolyte composition as a function of the distance from the slimes/bulk
electrolyte interface, assuming no changes in activity coefficients (Case A).
[PbSiF6]b = 0.2 M [HzSiFJb = 0.8 MRight scale: Changes in <& Left scale: Changes in [Pb+2L [SiF6'
2], and [H+](A) cd = 50 A m"2. Derived ratios: <&e= 22.0 mV, {Pb**J =22.1, {SiF6"
2
l = 4.71, and {H+}=0.38(B) cd = 200 A m
2
. Derived ratios: Oe= 40.8 mV, {Pb+2)r=76.3 , {SiF6
2}r= 15.4, and {H+}=0.19
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Case B: constant values
in the
presence
of eddy diffusion
Fig. 8 shows the concentration and migration potential 1
changes across the
slimes layer for two different current densities. Both [Pb+2] and [SiF6
2] increase
linearly as the slimes layer thickens. These increases are accompanied by a
quasi-exponential decay in [H+
]. As a direct consequence of the presence of
concentration differences across the slimes layer, a migration potential becomesestablished. Variations in d> are given with respect to the bulk electrolyte (<I> = 0
at the slimes/bulk electrolyte interface). The larger the slope of d> with respect to
the distance from the bulk electrolyte, the larger the electric field.
By comparing Figs. 8A and 8B, it can be seen that: (a) increases in current
density result in corresponding changes in {Pb+2}r, {SiF6"2
}r, {H+}r, and <X>e; (b) the
concentration values found at normal slimes thickness are impossibly high.
B. Case B: constant y< v a l u e s in the presence of eddy diffusion
In the current interruption experiments discussed in Chapter 4, ionic
concentration gradients were found to be present throughout the slimes layer.
The largest decays in overpotential obtained upon current interruption were
observed in the vicinity of the slimes/bulk electrolyte interface (see Figs. 4.7 to
4.11). Correspondingly, the closer the potential probes were to the anode/slimes
interface the smaller the potential decay. The smaller decays were associated with
larger amounts of Pb + 2
and SiF6"2
near the anode/slimes interface as compared
to those found near the slimes/bulk electrolyte interface. The difference inmagnitude of the overpotential decays can also be attributed to the presence of
natural convection.
During lead dissolution from lead bullion electrodes, different convection
patterns may be present. These convection patterns change local concentrations
and potentials across the slimes layer. An estimation of these variations can be
obtained by re-plotting the data presented in Fig. 4.12 with respect to the distance
from the anode/slimes interface and as a function of the electrolysis time (Fig. 9).
The anodic overpotential for a fixed position within the slimes layer decreases as
the slimes layer thickens. This indicates that small yet significant changes in
1
The electric
field, X, at
any
point
within the slimes layer is equal
to
the
negative
gradient of the
electrostatic potential at
that
point:
v - d ° X
~~dx
Thus,
the electrical field increases
in
the
opposite
direction to the electrostatic
potential.
Likewise, positive
charges move
in
the direction
of
the
electric field
[51, p.348].
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Cass B: constant values in the presence
of
eddy diffusion
153. ,_
8.8 1 . . . I •2 4
i .
• I • ' I I
J
I I L J _ J
8 6 8 18 12 14 16
Di s t a n c e from the Anod e/s l i mes Inte rf ac e, mn
Fig. 9 Changes in the potential of the slimes electrolyte as a function of the distance from the anode/slimesinterface.From the experimental data presented in Chapter 4, Fig. 4.12.
As the slimes layer thickens the potential for a fixed point away from the anode/slimes interface decreases.
concentrations are continuously taking place. These changes are not only
associated with natural convection but also with precipitation of secondary
products. This changes the physico-chemical properties of the slimes layer and
significantly hinders the ionic movement. Thus, from the data presented in Fig. 10,
it can be inferred that precipitation of secondary products takes place after -10
mm of slimes have been formed: While the difference in potential in the middle
of the slimes remains nearly constant, a steep increase in the potential difference
between the inner B and inner C reference electrodes can be observed after a
slimes thickness of ~ 10 mm. As a result of the precipitation of secondary products,
the inner potential rises because ions can no longer move at the same rate.
The predicted ionic concentration ratios presented in Fig. 8 are several fold
larger than those experimentally measured (e.g. see Fig. 4.21, and 13,41).
Furthermore, their magnitude indicates that large differences in viscosity and
density across the slimes electrolyte can be present. These differences lead to the
development of natural convection. The extent to which natural convection can
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Cass B: constanty, values in the presence of eddy diffusion
8I\3
00
4»
c0
a,
25
28
15
18
8 j i i_
* fiiiter- 1
0 I * n e r A -
l i n e r A
• Inner B
X i n n e r B - • i n n e r C
J I l_
J L J 1__L
J I l_ _1 1_
8 2 4 6 8 18 12 14 16
Distance from the Slimes/Bulk E l e c t r o l y t e interface) mm
Fig. 10 Changes in the potential difference between the outer and inner reference electrodes as a function of the
distance from the
slimes/bulk
electrolyte interface.
From the experimental data presented in Chapter 4, Fig. 4.12.
The difference in potential between the inner reference electrodes was divided by their separation distance and
plotted in this figure.
be established is a function of the current density, electrolyte concentrationgradients, geometry and size of the electrode, porosity and tortuosity of the slimes
layer, thickness of the slimes layer, and viscosity gradients.
Mixing of electrolytes as a result of the presence of natural convection is
incorporated in the solution of the flux equations through the eddy diffusivity
term, D E
. For a fixed amount of lead dissolved (i.e. for a fixed slimes thickness),
changes in D E were assumed to comply with the following relationship l:
x ; + aD
£
= | p l n - 1 ...31
1 Eddy
diffusion
as expressed by Eq.
31,
incorporates
mixing
of slimes electrolytes as a result of both
natural
and forced convection. It becomes extremely small at the anode/slimes
interface
and reaches a
maximum
value (in this model) at the slimes/bulk electrolyte interface. At a
fixed
distance from the latter,
the contribution of eddy diffusion increases with time, because this point becomes more remote from the
anode/slimes interface. Therefore, eddy
diffusion
leads to an unsteady-state electrolyte
composition
and
thus to a time-dependent polarization at points within the slimes layer that are at a fixed distance from the
slimes/electrolyte interface.
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Case B: constant^ values in
the presence of
eddy diffusion
where:
x to ta i= Total slimes thickness, [mm)
x, = distance from the slimes/bulk electrolyte interface at which point D E is to be computed,
[ x, < XfetJ, [mm]
a and P = arbitrary positive constants whose value depends on the electrolysis conditions:
a [mm] and p [cm2
sec"1
]
4 . ee i — xE-3
, ee a. 0 8 0 4 . 0 0 8 . 0 0 1 2 . 0
Distance from the Slimes/Bulk Electrolyte Interface, mmFig. 11 Variation of the Eddy diffusion constant, D E , as a function of the distance from the slimes/bulkelectrolyte interface, (a=lxl0"
2 mm and P=5xl0"* cm2 sec"1)
D E values were computed for three x, values (4,10, and 14 mm) using the following equation:
F ~ *, + aD
£
= IP-ln^_-1
1 6 . e
Eq. 31 can be used to obtain D E
values across the slimes layer for a fixedslimes thickness. As shown in Fig. 11, D
E decreases logarithmically from the
slimes/bulk electrolyte interface towards the anode/slimes interface. For a fixed
point away from the anode/slimes interface, the thicker the slimes layer, the
smaller D E
. This is equivalent to implying that because of the increasing thickness
of the slimes, natural convection will not penetrate deep within the slimes layer,
even though large density differences can be found there (i.e. even in the presence
of large [Pb+2]e and [SiF6"2]e). Wenzel's [3,4] experiments indicate that such patterns
in convection are operative during the dissolution of lead by the BEP.
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Cass
l:
cons
tan
J-J values
in the presence
of eddy
diffusion
z i . e -
H p i**t i i I i i i i i i i i i i i i i i i i i i i I i i i i i i i i i l i i i i I i i i i i i i
I I
e e e 2 .36 4 . 7 2 7 . 0 8 9 . 4 4 1 1
Distance from the Slimes/Bulk Electrolyte Interface, mm
(B)
I
•a
i 6
o
C
o
33
1 .
e e
1 . 4 4
1 . 08
o
B
.720
oU
. 360
=_ CS1F6-2]
r
[SiF6-2]
e e e ^ ' ' 1 1
. 0 0 0
7 2 . 0 -
36 . O
b * a i « B . 0 -
24.0
8 . 00
1 1 1
i
•ai•s
g
2
1 2 . 0
Distance from the Slimes/Bulk Electrolyte Interface, mm
Fig. 12 Variation in the slimes electrolyte composition as a function of the distance from the slimes/bulk
electrolyte interface, when changes in eddy diffusion are accounted for (Case B).
[PbSiF6]b = 0 .2M [H 2SiF6]b = 0.8M
Right scale: Changes in 4> Left scale: Changes in [Pb*2
], [SiF6'2], and [H
+
]
(A) cd = 50 A m"2. a=lxl0~ 2 mm p = lxlO~* cm2
sec"1
Derived ratios: <De= 21.6 mV, [Pb+2)=3.80, {SiF6
2
}r= 1.30, and {H*}r=0.68
(B) cd = 200 A m"2. a=lxl0" 2
mm P = 5xKT* cm2
sec'1
Derived ratios: <D = 71.6mV, {Pb+2
} =4.51, {SiF6'2} = 1.35, and {H*} =0.56
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Anodic overpotential values derived horn the mathematical mo
Fig. 12 shows the concentration and potential gradients obtained by
incorporating eddy diffusion in the overall diffusion term, CD* . The local
concentrations and potential differences shown in this figure are those predicted
to be present at the point at which the slimes layer has reached a thickness of
10 mm. By incorporating eddy diffusion, large decreases in the magnitude of thepotential gradients are obtained (compare Fig. 12 with Fig. 8) . Furthermore, the
calculated mean slimes electrolyte concentrations are closer to those
experimentally found than those predicted without incorporating an eddy
diffusion term \
C. Anodic overpotential values derived from the mathematical model
The experimental d ata used in Exp. 1X2 (see Chapter 4) will be used to show
how the mathematical model can be used to predict anodic overpotentials.
1 . 60
1 . 28
\ . 9 6 0
1g*33
. 640
i
V
Q
. 320
[SiF6-2
]
i
000-k 1 1 1
. 0 0 0
8 . 00
I I I I I I I I I I I I i i i I i i i
4 . 0 0 8 . 0 0 12 .0 16
Distance from the Slimes/Bulk Electrolyte Interface,- mm
Fig. 13
Variation
in the
slimes electrolyte
composition as a
function
of the
distance
rom
he slimes/bulk
electrolyte interface, when changes
in eddy diffusion are accounted for
(Case
C).
1E e
ii
- c
z 1
[PbSiFg]^0.28M [HzSiFJfc = 0.74M a=lx l0 "
2
mm P = 5xlfX
4
cm
2
sec
1
Right scale: Changes in
<> Left scale: Changes in
[Pb+2], [SiF
6
'
2
], and [H+]
cd = 139 A m
2
. Derived
ratios: «D = 72.0 mV, {Pb+2
}=3.48, (SiFy
2
}
= 1.38, and {H+} =0.57
1
The
predicted
mean
ionic concentrations within
the
slimes
layer
can
be
made
to
match
those
experimentally measured by changing the eddy diffusion parameters
a and p.
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Anodic overpotential values derived from he mathematical mo
(A)
400 .
360 .
320 .
IO
3
"SoO1
BCJ
V
3
280 .
240 .
200 . _L _L
.00O 4.00 8.00 12.0
Distance from the Sli mes/Bulk Electrolyte Interface, mm
(B)
16.0
1 . 40 i — > « E 0-
^1.20 —
Q
i .
0 0
CJ
CO
4J
800
600
- Density g/cm3
Kinematic Viscosity, cm2
/sec
i i i _i i i i i i i_
K E :0
• • i
.000 4.00 8.00 12.0 16.0
Distance from the Slimes/Bulk Electrolyte Interface, mmFig. 14 Variation in physico-chemical properties of the slimes electrolyte as a function of the distance from theslimes/bulk electrolyte interface.
From the data presented in Fig. 13
(A) Variation in electrical conductivity K (B) Variation in density p, and in kinematic viscosity, v.
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Anodic overpotenial values derived from the mathematical mod
The migration potential and concentration profiles shown in Fig. 13 were
obtained for the initial bulk electrolyte composition and current density used in
the case study presented in Chapter 4 (Exp. LC2). To simulate the presence of
eddy diffusion, the following values were assumed: a = lxlO'2
mm and P = 5x10 4
cm
2
sec"
1
.From the data presented in Fig. 13, changes in K , p, and v, as a function of
the distance from the slimes/bulk electrolyte interface were obtained (see Fig. 14) \
As shown in Fig. 14A, K decreases continuously as the slimes electrolyte gets
depleted in H+ and enriched in Pb + 2 and S1F6"
2. From these data, the total ohmic
drop across the slimes layer was computed using the following equation:
The results of the numerical integration of the data presented in Fig. 14A,
according to Eq. 32, are shown in Fig. 15. The concentration overpotential shown
in this figure was computed according to the following equation:
RT
a
nM
Tlc = ™—Trin —— ...33 Z PB+1F a^+2(bulk)
Using Eqs. 32 and 33, and the computed d> values, the total anodic
overpotential can be obtained as follows:
'H>ri>rilf e r t + q> ...34
As shown in Fig. 15, the computed anodic overpotential, r^, is smaller than
that experimentally measured, ri A. The difference in potential between ru and rjA
decreases as the slimes layer thickens until it practically disappears towards the
end of the electrorefining cycle (at -14 mm of slimes). The smaller values with
respect to TJ a at smaller slimes thicknesses, are due to the fact that TJ a was measured
as a function of time, (i.e. with respect to a moving interface) whereas ru was
obtained from concentration gradients computed at a fixed time (i.e. with respectto a fixed interface). That is, the model predicts anodic overpotentials only after
a pseudo-steady state has been established. Thus, the lowest experimental rjA
values shown in Fig. 9 (those obtained at the end of the electrorefining cycle) are
nearly equal to the computed shown in Fig. 15. The small differences between
1
Changes in
K ,
p,
and
v
were
computed
from the local ionic concentration changes using Eqs. 19'to 21.
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Anodic overpotenial values derived from the mathematical mod
the computed and experimental overpotentials can be related to variations in the
corrosion potential of the lead anode as the slimes layer becomes exposed to wide
ranges of electrolyte compositions.
. e e e 4 . e e 8 . e e 1 2 . e l e . a
Slimes Thickness, mmFig. 15 Comparison between the experimental (unsteady-state) and predicted (steady-state) anodic
overpotentials.
Experimental anodic overpotential from Exp. LC2 (outer reference electrode measurements corrected foruncompensated ohmic drop).Predicted anodic overpotential from the solution of the Nemst-Planck flux equation and from the data presentedin Fig. 13.
The computed overpotential contains contributions from ohmic resistance, concentration overpotential (of Pb*1),and migration potential (caused by unequal diffusivities of anions and cations). The ohmic and concentrationoverpotentials are shown.
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Summary
The Betts electrorefining process (BEP) for lead is successful because it
retains most noble impurities in adherent anode slimes while depositing rather
pure (>99.99%) lead cathodes. The anode slimes account for only 1 to 4% of the
weight of dissolved lead, and so represent an enrichment of a factor of 25 to
100 in the noble impurities (particularly precious metals) present in lead
bullion.
A review of the Betts Process as described in the literature was presented
in C h a p t e r 1. From this review, important attributes of the Betts process w
identified:
1) The anode slimes formed upon selective removal of lead from the impure
bullion electrodes are adherent and thick (> 1 cm). X-ray diffraction on theseslimes has identified mainly oxides and lead fluoride, but these results may have
been compromised by accidental oxidation of samples. The actual (in-situ) slimes
are more likely electrically conducting filaments of noble metals and intermetallic
compounds, sometimes supplemented by precipitated PbF2 and Si0 2 (both of
which have been detected in anode slimes precipitates).
2) It is well known that anode polarization must be limited to less than 200
mV to avoid bismuth dissolution and transfer to the cathodic deposit. This,
together with the reported X-ray diffraction data provides evidence that in-situslimes contain metallic bismuth.
3) Electrolyte extracted from within slimes layers shows that it is markedly
enriched in lead fluosilicate and somewhat depleted in fluosilicic acid. The change
in the inner slimes electrolyte concentration is more abrupt towards the
slimes/anode interface and accounts for the hydrolysis of SiF6"2
ions:
3Pb* 2
+ SiF;2 + 3H 20 -> 3PbF 2 + H 2 Si03 + AH*
4) Fundamental electrochemical studies have evaluated the effects of
additives such as lignin sulphonates, glue, and aloes extracts as levelling agents
in cathode deposition. The presence of these additives in the electrolyte for lead
refining may affect both the cathodic and the anodic overvoltages.
Among the most important processes in Betts refining are those which take
place within the slimes layer. Thus, across this layer, ionic concentration gradients
become established and depending on the electrolysis conditions (anode
composition, current density, electrolyte properties) secondary reactions
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Summary
(hydrolysis, secondary products precipitation, dissolution of noble impurities) can
take place. This research studied these processes from a fundamental perspective
in order to get a better understanding of the industrial operation of the BEP. Some
of the fundamental questions this study addressed were:
a ) What are the components of the anodic polarization and how do theyrelate to the hydrolysis of the acid and the precipitation of secondary products?.
b)
Under which thermodynamic conditions can noble compounds present
in the slimes layer dissolve?
c)
Are the filaments of noble metals in the slimes layer electrically
conducting?
d ) What effect do the addition agents have on the anodic process?
Furthermore, this study was directed to the formulation of a mathematical
model to describe the changes in concentration and potential across the slimes
layer.
In C h a p t e r s 2 a n d 3 the experiments designed to address so
questions were presented. In the electrochemical experiments, the use of a high
purity lead wire as a reference electrode was rationalized based on the high
reversibility of lead in H 2SiF 6-PbSiF 6 electrolytes and the absence of a liquid
junction . Also the components of the anodic polarization were individually
analyzed and the use of transient electrochemical techniques was proposed to
assess their magnitude and impact on the anodic processes.
The electrochemical processes taking place at open circuit or during anodic
dissolution of lead electrodes were studied using "in-situ" transient techniques
which include:
a ) Single potential and current step (i.e. potentiostatic and galvanostatic
dissolution including chronopotentiometry).
b)
Current interruption.c) A C impedance in the presence and in the absence of a net Faradaic current.
d ) A variation of the Small Amplitude Cyclic Voltammetry (SACV) technique:
during the A C impedance measurements the transient variations of potential and
current were measured simultaneously.
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Summary
The results obtained from electrochemical measurements were
supplemented with physico-chemical data on experimentally measured electrolyte
properties (electrical conductivity, kinematic viscosity, and density) and with data
from "in-situ" and "industrially recovered" slimes electrolyte compositions. Also
scanning electron microscopy and X-ray diffraction were used to analyze thephases and compounds present in the slimes obtained during dissolution of lead
bullion electrodes as used in the Betts refining process.
C h a p t e r 4 is a case study in which a lead bullion electrode
galvanostaticaUy dissolved. Reference electrodes within the lead anode were
incorporated to follow the difference in potential between the slimes electrolyte
and the lead anode. Measurements on the slimes electrical conductivity were
made by following the difference in potential between a bare Pt wire inserted in
the slimes layer and the lead anode. In these measurements, it was found thatthe difference in potential between the Pt wire and the lead anode was negligible.
The Pt wire and the anode appeared to be short-circuited indicating the high
electrical conductivity of the slimes filaments.
Samples of the slimes electrolyte were withdrawn in-situ and their
composition was related to the transport processes across the slimes layer.
Chemical analysis of Si and F at a fixed point in the slimes electrolyte, show that
their total concentration decreases as the slimes layer thickens. These decreases
were related to time dependent processes such as changes in convection due to
movement of the anode/slimes interface and/or gradual precipitation of
secondary products. The noble impurity concentrations was very small
(SbO+~0.2mM, AsO+~0.2mM, and BiO+~0.01mM) and did not change significantly
during the refining cycle.
Upon galvanostatic dissolution, ionic concentration gradients become
established and the anodic overpotential increases as the slimes layer thickens.
On the other hand, upon current Interruption the anodic overpotential decays,
first abruptly, (as the uncompensated ohmic drop, Tj
n
, disappears) and then slowly
(due to the presence of a back E.M.F. created by ionic concentration gradients
that decay slowly). The uncompensated ohmic drop is caused by ohmic resistance
of the mixed electrolyte between the reference electrodes and the solution within
the slimes layer. Across the slimes layer there is not any measurable
uncompensated ohmic drop because of the presence of ionic concentration
gradients.
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Summary
Current interruption measurements showed that (A) concentration gradients
exist across the slimes layer, (B) inner solution potentials within the slimes layer
can be larger than those measured from reference electrodes located in the bulk
electrolyte (C) secondary products can shift the inner solution potential to negative
values which reverse upon re-dissolution (D) ionic diffusion is seen upon currentinterruption but it is complex and difficult to model due to the presence of
processes that can support the passage of internal currents.
X-ray diffraction and scanning electron microscopy analysis on the slimes
layer indicated that secondary reactions took place: near the anode/slimes
interface the "cellular" microstructure was infiltrated with smca-contoining
products.
I n C h a p t e r 5 further studies on the components of the anodic polar
were performed by using "pure" lead (>99.99%) working electrodes. When pure
lead dissolves, the uncompensated ohmic drop increases as the
electrode/interface retreats. Concentration overpotential accounts for the
remairiing polarization (activation overpotential is negligible) and it is a function
of the electrolysis time and the current density. On the other hand, in the presence
of a net anodic current, dissolution of Pb + 2 does not proceed urihindered when
addition agents are incorporated in the bulk electrolyte. Because of the addition
agents, a finite activation overpotential (rjac<10 mV), was observed using
chronopotentiometry. The DC results were confirmed by AC impedance
measurements which indicated unambiguously that addition agents have an
impact on the anodic and rest potential behaviour of lead electrodes.
I n C h a p t e r 6 the components of the anodic polarization measured
dissolution of lead bullion electrodes were resolved. The polarization components
were obtained by analyzing the potential and current dependance upon
application of a small amplitude sinusoidal waveform. This dependance was found
to be linear in the low overpotential region (< 250mV). Thus, upon subtraction of
the uncompensated ohmic drop, the remaining polarization is due to the
"apparent" ohmic drop of the slimes electrolyte and to liquid junction and
concentration overpotentials. These components are directly linked to the
electrolysis conditions and to the slimes layer structure. Furthermore, the ratio
of these terms can be used to obtain the point at which secondary products
precipitation starts. Changes in this ratio can also be related to the anodic effects
caused by the presence of addition agents
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Summary
A C impedance measurements performed in the presence of a net Faradaic
current showed that the impedance increases uniformly as the slimes layer
thickens up to the point at which noble impurities start to react.
The difference in the magnitudes of the impedance arcs measured under
galvanostatic and potentiostatic conditions was attributed to the presence ofsecondary products blocking the anode/slimes interface and the ionic transport.
After a certain electrolysis time under potentiostatic conditions, secondary
products precipitate continuously. This significantly increases the impedance of
the system which can reach values as high as 1000 Qcm 2. Under galvanostatic
conditions concentration gradients are not as steep and this results in smaller
impedances (of the order of 10 Qcm 2).
Under current interruption conditions the impedance decreases as the
concentration gradients relax and secondary products re-dissolve. The change in
shape of the impedance curves as compared to those obtained while in the
presence of current, permits a distinction to be made between the contribution
from the slimes filaments and that from ionic concentration gradients and charge
transfer at the anode/slimes interface. It was found that charge transfer and
capacitative phenomena related to the slimes filaments do not contribute
significantly to the impedance of the system.
Three electrical analogue models were used to describe the A C impedance
measurements. Two of these analogues were used to model the impedance changes
observed while in the presence of a net Faradaic current while the third was found
useful in describing the phenomena observed upon current interruption. These
models were derived from a proposed analogue representation of a lead bullion
electrode covered by a layer of slimes, as developed from a set of assumptions.
Each of the components in the derived electrical circuits has a reasonable physical
meaning. Changes in the analogue parameters present in the electrical circuits
were related to: (A) the D C conductivity of the slimes electrolyte, (B) the charge
transfer resistances associated with the lead dissolution process, ( c ) the
distributed1 nature of the anode/slimes and the slimes/slimes electrolyte
interface, and (D) the distributed capacitance generated by concentration gradients
present In the slimes electrolyte.
1 A "distributed element" in an analogue model represents properties of the system distributed over
macro distances, such as ionic gradients in solution across the slimes layer, or such as slimes
filament/electrolyte interfaces distributed
from
the anode/slimes to the bulk electrolyte/slimes interface.
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Summary
Overall, A C impedance measurements were found useful in describing the
presence of concentration gradients and the onset of the precipitation of secondary
products.
In Chapter 7 a thermodynamic analysis of the processes that can take place
across the slimes layer was presented. Thus, Eh-pH diagrams for the quaternarysystems Pb-Si-F-H20, Sb-Si-F-H20, As-Si-F-H2O t and Bi-Si-F-H20 were found
useful in describing the sequence of reactions that can take place across the
slimes layer. In the recirculating electrolyte for lead refining and In the electrolyte
samples extracted from the slimes layer, the compositions of the noble impurities
agree with the predictions of the Eh-pH diagrams ([Sb]>[As]>[Bi]). The nature of
these elements and related secondary products (such as PbF2 and Si02) can be
assessed experimentally only if the slimes are properly washed and then kept in
a dry atmosphere.
From changes in K as a function of composition for pure H 2SiF 6 and PbSiF6
solutions, individual equivalent conductivities (A,) in H2SiF6-PbSiF 6 mixtures were
obtained by assuming that A, 2 = X pb + 2 . From the ^ values and ionic
concentrations, transference numbers were obtained. Changes in transference
numbers as the slimes layer thickens can then be derived from local ionic
concentrations. Moreover, the diffusion coefficient for Pb+ 2 obtained from X pb+2
values {D ph+2 = 5.0x IO"6 cm2/sec) agrees with that obtained from
chronopotentiometric measurements [D pb+2 = 5.3±0.%x IO"6
cm2
/sec).
Finally, based on the experimental and thermodynamic findings, a
mathematical model based on the Nemst-PIanck flux equations was developed.
The resulting set of simultaneous differential equations was solved using a finite
interval algorithm so that adjustments in A , y, and £>, can be incorporated \
The model predicts steady state ionic concentrations and potential gradients
across the slimes layer (i.e. it does not predict local concentration changes as the
slimes thicken). Experimentally, this steady state is not observed, with the result
that a convective (eddy diffusion) component must be invoked in order to account
for an unsteady state, that is, for decreasing local lead concentrations and
polarization at a fixed point in the slimes electrolyte as the anode/slimes interface
retreats.
1 a), is the overall diffusion coefficient for species i, and includes the contributions of molecular and eddy
diffusion.
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Summary
The inclusion of mixing due to eddy diffusion (equal for all ions) accounts
for the unsteady-state observations if the eddy diffusion contribution increases
at a fixed distance from the slimes/electrolyte interface as the anode/slimes
interface retreats. The form of the proposed eddy-diffusion component
D = | p • In— | contains this provision even though it is a simple empirical
equation. If a and P in this equation are adjusted so that the model fits the
polarization across the whole slimes layer, it also predicts quite accurately the
polarization of intermediate sections, even though these polarizations decline (i.e.
are in unsteady state) with respect to the outer reference electrode measurements.
The mathematical model can be used to predict (a) ionic concentration
gradients (b) total polarization values across anode slimes. The thermodynamics
of the system can be incorporated to predict the onset of hydrolysis and the
dissolution of noble elements.
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Summary
Table 1 Summary of information that can be derived
from using transient electrochemical techniques
Addition
Agents
DC
Faradaic
Current
Slimes Secondary
Products
DC/AC
Analysis
Comments
No No No No A C
Nearly reversible
Pb/Pb
+2
equilibrium
(tiac- 0): Ra and C
d
, not
measurable.
Impedance
can be described by a CPE analogue.
Diffusion coefficients cannot be obtained unless the electrode
surface is smooth and the hydrodynamic conditions are
controlled.
No No Yes No A C
Under
rest potential conditions and in the
absence
of
concentration
gradients
within
the slimes electrolyte: (Tia.,-^0) for the
Pb/Pb
+2
equilibrium.
The analogue parameters derived from analysis of the
impedance spectrum were similar to those obtained
when
pure
lead was being studied. The slimes
layer
does not appear to
contribute significantly to the measured impedance.
Yes No No No A C Addition agents decrease the
/
0
values
for the
Pb/Pb
+2
equilibrium.
Ro and
C
d
can be obtained from analysis
of
the impedance curve.
Yes/No
Yes Yes Yes/No
D C
The components of the anodic overpotential (n
A
=frflRJ were
obtained by
superimposing
a low amplitude
sinusoidal
waveform
at preset slimes thickness. Changes in the resistivity of the
entrained electrolyte and of the liquid junction and corrosion
potentials were used to analyze the conditions under which
hydrolysis can take place. In the absence of addition agents
m
values
remain nearly
constant whereas
in the presence of addition
agents JJ- decreases. This indicates that addition agents can
enhance the concentration gradients
within
the slimes electrolyte.
Yes/No No Yes Yes/No D C
Upon current interruption concentration gradients relax and
secondary products re-dissolve. Changes in T|
a
as a function of
time were complemented
with b and IR
m
data obtained at preset
current times. Upon current interruption the entrained electrolyte
resistivity decreases monotonically with the corrosion (or "rest-
potential).
Yes/No
Yes Yes Yes/No
A C
The impedance in the
whole
frequency range increases as the
slimes
layer thicken.
These increases are uniform only up to the
point at which noble compounds dissolve. R
s
obtained from the
high
frequency end of the impedance spectrum is nearly
equal
to
that
obtained from current interruption measurements. Two
electrical circuits were used to fit the experimental data. These
circuits were derived from a general analogue model
that
was
based on a set of assumptions derived from empirical data. The
derived electrical analogue parameters were linked to the
resistance of the entrained electrolyte, the charge transfer
resistance for the Pb*
2
dissolution
process,
and to the capacitative
phenomena
associated to the slimes filaments and the slimes
electrolyte. The extent to which ionic concentration gradients
become established and their relationship to the hydrolysis point
were analyzed using these electrical analogues.
Yes/no
No Yes Yes/No
A C
Upon current interruption the impedance decreases. These
decreases were modelled by a modified Randies circuit. R
d
and
C
d i
were small indicating
that
capacitative phenomena associated
to the slimes filaments can be neglected.
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Conclusions
1) A tJiennodynamlc analysis of the system indicates that the slimes electrolyte
being depleted in acid and enriched in PbSiF6 can cross a threshold which
marks the onset of precipitation of secondary products, namely PbF2 and
Si0 2.
2) Addition agents s i g n i f i c a n t l y affect the anodic process
layer (In lead electxorefiiiing addition agents are added for cathodic purposes
and ideally should have no anodic effect).
3) The filaments contained wit±iin the slimes layer are highly conductive and
are grounded to the anode.
4) Upon galvanostatic dissolution, ionic concentration gradients become
established and the anodic overpotential increases as the slimes layer
thickens. Abrupt increases in overpotential can be observed when hydrolysis
and precipitation of secondary products (such as PbF2 and SiOJ occur.
Overpotential increases of sufficient magnitude are accompanied by
dissolution of noble impurities present in the proximity of the slimes/bulk
electrolyte interface.
5) Upon current interruption the anodic polarization decays first abruptly (as
the uncompensated ohmic drop disappears) and then slowly ( due to the
presence of a back E.M.F. created by ionic concentration gradients that decay
slowly).
6) The anodic polarization has four contributions that can be obtained
experimentally using transient electrochemical techniques:
1) Uncompensated ohmic drop
2) "Apparent" ohmic drop of the slimes electrolyte.
3) Liquid Junction or "Migration" potential4) Concentration Overpotential
7) A mathematical model based on the Nemst-Planck flux equations was
developed. This model can be used to predict (a) ionic concentration gradients
(b) total polarization values across anode slimes. The thermodynamics of the
system can be incorporated to predict the onset of hydrolysis and the
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Conclusions
dissolution of noble elements. Thus, the mathematical model could be used
industrially as a control criterion for modifying the current density as the
slimes layer thickens.
These studies show that the process could be "improved" (in a performance
sense) if precipitation of secondary products were prevented or limited. To
do this, the current density could be adjusted as the slimes layer thickens
and the H2SiF 6 content of the electrolyte could be increased. Further, methods
of decreasing the final slimes thickness might be devised, such as casting
txiinner or corrugated anodes.
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Recommendations
fo r
Further Work
Recommendations for Fur ther Work
1) The Nernst-Planck model should be tested and modified to make it predict
the concentration and potential gradients observed in the industrial operation of
the Betts electrorefining process.
2) The eddy diffusion coefficient witiiin the slimes layer is a function of the
electrode height, as well as slimes permeability and slimes thickness. Its value
should be measured and related to electrode polarization for various electrode
heights, especially practical heights found in the industry.
3) The experimental procedures generated in Chapters 4 to 6 can be used to
expand the applicability range of the mathematical model here proposed. An
experimental separation of the parameters accounted for in the model calls for
the use of a horizontal anode (below the cathode) configuration. The immediate
effect of this change will be the presence of large ionic concentrations
approximating those computed from the model in the absence of natural
convection. Furthermore, current interruption decays will have time constants
related strictly to molecular diffusion and migration without the confusion
introduced by natural convection. Vertical electrolysis using electrodes of different
height should also provide information on the presence of natural convection and
its relationship to the establishment of concentration gradients. Studies using
rotating disk electrodes at very low rotation speeds (to avoid detachment of slimes)can also be used to improve the predictions of the model. All these studies can
be complemented by measuring, at preset times, the resistance of the slimes
electrolyte and the liquid junction and concentration overpotentials (e.g. by using
a superimposed small amplitude cyclic waveform as described in Chapter 6,
section III). These values can be compared directly with those predicted by the
mathematical model.
4) The transport processes witxiin the slimes layer could be related to the
Navier-Stokes equation as an ultimate goal in modelling.
5) The extent to which concentration gradients become established across
the slimes layer can be further studied by adjusting the flow of current to avoid
precipitation of secondary products and impurities dissolution. Periodic
interruption of current together with periodic current reversal can be used to
accomplish those objectives.
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Recommendations
for Further Work
6) In the industrial operation of the BEP the components of the anodic
overpotential could be deterrnined "in-situ" using superimposed low amplitude
AC currents at preset slimes thickness. This should provide with enough
information for modulating the current density or changing the bulk electrolyte
composition (including concentration and nature of addition agents).
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BIBL IOGRAPHY Chapter 1
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[47] J.E.B. Randies, "Kinetics of Rapid Electrode Reactions", Disc. Farad. Soc, 1, 11-1(1947).
[48] C. Gabrielli, "Identification of Electrochemical Processes by Frequency
Response Analysis", Monograph Reference 004/83, Solartron InstrumentationGroup, Farnsborough, England, 1981.
[49] J. R Macdonald, "Comparison and Discussion of Some Theories of The Equilib Electrical Double Layer In Liquid Electrolytes" ', J. Electroanal. Chem.,
223,
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[51] J .R Macdonald.. "Impedance Spectroscopy and Its Use In Analyzing Th Steady-State AC Response of Solid and Liquid Electrolytes", J. Electroanal. Ch
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Analysis of Painted Metals", Electrochim. Acta, 26 (9). 681-703 (1986).
[53] Z. Stoynov, "Impedance Modelling and Data Processing: Structural and Param Estimation", Electrochim. Acta, 35 (10), 1493-1499 (1990).
[54] F. Mansfield, "Electrochemical Impedance Spectroscopy (EIS) As A New To Investigating Methods of Corrosion Protection", Electrochim. Acta, 35 (1533-1544 (1990).
[55] S. Venkatesh and D.-T. Chin, "The AUernating Current Electrode Process", Isr
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[58] T.H. Glisson, "Introduction to System Analysis", McGraw-Hill Book Company, US
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[59] RD. Strum and J.R Ward, "Electric Circuits and Networks", 2nd ed.,Prentice-HInc., USA 1985.
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[61] W.H. Smyrl, "Digital Impedance For Faradaic Analysis II. Electrodissolution inHCl', J. Electrochem. Soc, 132 (7), 1555-1562 (1985).
[62] W.H. Smyrl and L.L. Stephenson, "Digital Impedance For Faradaic Analysis i Corrosion in Oxygenated .1 N HCl', J . Electrochem. Soc, 132 (7), 1563-156(1985).
[63] "Techniques in Electrochemistry, Corrosion and Metal Finishing, A
Handbook", AT. Kuhn ed., John Wiley & Sons, Great Britain 1987.[64] M. Hayes, "Ohmic Drop", Chapter 4 in ref. [63].
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[66] K. Jiittner, "Electrochemical Impedance Spectroscopy (EIS) of Corrosion Pr On Inhomogeneous Surfaces", , Electrochim. Acta, 35 (10). 1501-1508 (1990).
[67] D.D. Macdonald, "Review of Mechanistic Analysis by Electrochemical Imped Spectroscopy", , Electrochim. Acta, 35 (10), 1509-1525 (1990).
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[69] M. Kuhn, K.-G. Schiitze, G. Kreysa, and E. Heitz, "Principles and Applications ofComputer Controlled Electrochemical Measurements With Elimination of IR
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[72] D.D. Macdonald, "Some Advantages and Pitfalls of Electrochemical Impeda
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B I B L I O G R A P H Y Chapter 3
[I] "Fluosilicic Acid', Dechema Data Sheet, Serial No. 49.235.050.1, October 1959.
[2] "Trail Operations", A basic description of the Main Production Facilities, Pamphlet#2101, Cominco Ltd., Trail, British Columbia.
[3] LG. Ryss and V.N. Plakhotnik, "Solubility of Si02 in Hydrogen Hexafluorosilica(W, Russ. J. Inorg. Chem., 15 (12) 1742-1745 (1970).
[4] K. Kleboth, "Fluorine Complexes of Silicon in Aqueous Solutions", Monatsch. Ch99 (3), 1177-1185 (1968). (Translation Associated Technical Services, Inc.,
55E148G).
[5] K. Kleboth and B. Rode, "Semiempirical MO Calculations on FkiorosilicoCompounds and their Hydrolysis Products", Monatsch. Chem., 105, 815-82
(1974). (Translation National Translations Center 82- 13494-07b)
[6] K. Kleboth, "Fluoro Complexes of Silicon in Aqueous Solution - Part 2 : For and Properties of Tetrafluorosilicic Acid' , Monatsch. Chem. 100 (3), 1057-106(1969). (Translation Associated Technical Services Inc., 41W119G).
[7] V.M. Masalovich, G JV. Moshkareva and P.K. Agasyan, "Study of Complex FormationinSokitions ofHydrofluoric and Silicic Acid', Russ. J. Inorg. Chem., 24 (2), 196-19
(1979).
[8] G. Schwarzenbach and H. Flaschka, "Complexometric Titrations", 2nd ed.,
Methuen and Co.. London, 1969.
[9] A Vogel, "Textbook of Quantitative Inorganic Analysis", 3rd ed., Longman Inc.,UK. 1978.
[10] R A Mostyn and AF. Cunningham, "Determination of Antimony by Atomi Absorption Spectrometry", Anal. Chem., 39(4), 433-435 (1967).
[II] G.J. Davies and M.A Leonard, "Semi-micro Determination of Fluorine in Orgacompounds by Oxygen Silica-Flask Combustion and Gran-type Potentio
Titration of Fluoride with IxaUhonum Nitrate", Analyst, 110, 1205-1207(198
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[12] KJi. Phillips and C.J. Rix, "Microprocessor-Controlled termination OJ Fluoride Environmental And Biological Samples By A Method Of Standard Additions Fluoride Ion Selective Electrode", Anal. Chem.. 53 . 2141-2143 (1981).
BIBLIOGRAPHY Chapter 4
[1] A R Spurr. J . Ultrastructure Research. 26 , 31 (1969).
[2] A R Spurr and W.M. Harris, Am. J . Botany, 55 , (1968).
[3] AS. Gioda. M.C. Giordano, and V.Z. Macagno, "The Pb/Pb+2 Exchange Reaction In Perchlorate Acidic Solutions", J. Electrochem. Soc, 124 (9), 1324-1329 (1977).
[4] S.H. Glarum and J.H. Marshall, "An Admittance Study of The Lead Electrode", JElectrochem. Soc. 131 (4), 691-701 (1984).
[5] C.J. Bushrod and N.A Hampson, "Anodic Behavior ofLead in Perchloric Acid Continuous and Interrupted Polarization1', Br. Corros. J., 6, 87-90 (1971).
[6] R.M. Slepian, "Improved MetaUographic Preparation of Lead and Lead Allo
Metallography 12, 195-214 (1979).
[7] J. Haernaez, A Pardo and M.I. Aja, "Improved MetaUographic Preparation ofLe
by Electropolishing'', Metallography 19, 5-17 (1986).
[8] V. Patzold and H. Langbein, "An Economical Process of Lead Metallography", NHiitte, 22 (1) 40-43 (1977). (Translation British Industrial and ScientificInternational Translation Service. BISI 16109. August 1979.
BIBL IOGRAPHY Chapter 5
[1] S.H. Glarum and J.H. Marshall. "An Admittance Study of the Lead Electrode", Electrochem. Soc. 131(4), 691-701 (1984).
[2] RC. Kerby, "Addition Agents in Lead Electrodeposition", Canadian Patent1,115,658, January 5, 1982.
[3] K.J. Vetter, 'Electrochemical K i n e t i c s , Theoretical and Experimental
Aspects" , Academic Press, New York (1967).
[4] AS. Gioda. M.C. Giordano, and V.Z. Macagno, "The Pb/Pb* 2 Exchange Reaction i Perchlorate Acidic Solutions", J. Electrochem. Soc. 124 (9), 1324-1329 (1977).
[5] Z. Kovac, "The Effect of Superimposed A.C. and D.C. in Electrodeposition of Alloys", J. Electrochem. Soc, 118 (1), 51-57 (1971).
[6] B. Breyer and H.H. Bauer, "Alternating Current Polarograpby and Tensametry",Chemical Analysis V. Xin, Interscience Publishers, USA 1963.
[7] J .R Macdonald ed., "Impedance Spectroscopy", John Wiley and Sons, New York,1987.
[8] J.R Macdonald., "Impedance Spectroscopy and its use in Analyzing th
Steady-State AC Response of Solid and Liquid Electrolytes", J. Electroanal. Che223 , 25-50 (1987).
[9] S. Venkatesh and D.-T. Chin, "The Alternating Current Electrode Process", IsrChem., 18. 56-64 (1979).
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[10] Z. Stoynov, "Impedance Modelling and Data Processing: Structural and Parame Estimation", Electrochim. Acta, 35 (10), 1493-1499 (1990).
[11] E. Warburg," Uber Das Verhalten Sogenannter Unpolarisierbarer Electroden G Wechselstrom", Ann. Phys. Chem., 67 , 473-499 (1899).
[12] G.W. Walter, "AReview of Impedance Plot Methods Used For Corrosion Perfor
Analysis of Painted Metals", Electrochim. Acta, 26 (9), 681-703 (1986).
[13] J.C. Wang, "Realizations of Generalized Warburg Impedance with RC La Networks and Transmission Lines", J . Electrochem. Soc, 134 (8), 1915-1920(1987).
[14] W. Scheider, "Theory of Frequency Dispersion of Electrode Polarization. Topol Networks with Fractional Power Dependence", J . Phys. Chem., 79 (2), 127-1(1975).
[15] R de Levie, "On the Impedance of Electrodes withRough Interfaces", J. ElectroaChem., 261 , 1-9 (1989).
[16] J.E.B. Randies, "Kinetics ofRapid Electrode Reactions", Disc. Farad. Soc, 1, 11-1(1947).
[17] F. Miyashita and G. Miyatani, "On the Anode Phenomena on Lead Electrolyt Refining", Tech. Rep. Kansai Univ., 13, 81-92 (1972).
[18] F. Miyashita and G. Miyatani, "Influence of Electrolytes and Surface Active AgeontheSurfaceReactionofLeadElectrode", Nippon Kogyo Kaishi, 94 (1085), 485-489(1978).
[19] R Winand, "Electrocrystallization", in Applications of Polarization Measurementsin the Control of Metal Deposition, I.H. Warren ed., Elsevier Science Publishers,
Amsterdam, p. 47-83 (1984).[20] R Winand, "Fundamentals of Electrometallurgy- Part B", in Fundamentals a
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[21] C. Gabrielli "Identification of Electrochemical Processes by Frequency Resp Analysis", Monograph Reference 004/83, Solartron Instrumentation Group,Farnsborough, England, 1981.
[22] J . Reid and AP. David, "Impedance Behavior of a Sulphuric Acid-Cupri Sulphate/Copper Cathode Interface", J . Electrochem. Soc, 134 (6), 1389-1394(1987).
[23] Z. Nagy and D A Thomas, "Effect of Mass Transport on the Determination oCorrosion Rates from Polarization Measurements", J. Electrochem. Soc, 133 (12013-2017 (1986).
[24] J. S. Newman, " E l e c t r o c h e m i c a l S y s t[25] D.D. Macdonald, "Some Advantages and Pitfalls of Electrochemical Impeda
Spectroscopy", Corrosion-NACE. 46 (3), 229-242 (1990).
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BIBL IOGRAPHY Chapter 6
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[2] F. Mansfeld, M.W. Kendig, and S. Tsai "Recording and Analysis of AC Impedance Data For Corrosion Studies II. Experimental Approach and Results",Corrosion-NACE, 38 (11). 570-580 (1982).
[3] M. Kendig and F. Mansfeld, "Corrosion Rates From Impedance Measurements: Improved Approach For Rapid Automatic Analysis", Corrosion-NACE, 39 (1466-467 (1983).
[4] A A Zaki and R Hawley, "Dielectric Solids", Solid State Physics, Dover
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At pH 9' J. Electroanal. Chem., 261 . 23-38 (1989).
[7] D.D. Macdonald, M.C.H. Mckubre, and M. Urquidi-Macdonald, "Theoretical Assessment of AC Impedance Spectroscopy for Detecting Corrosion of Reb Reinforced Concrete", Corrosion-NACE, 44 (1), 2-7 (1988).
[8] K.J. Vetter, "Electrochemical Kinetics, Theoretical and ExperimentalAspects", Academic Press. New York (1967).
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[II] "Electrochemical Corrosion Testing with Special Consideration of Practi Applications" [ProcConf.], E. Heitz, J.C. Rowlands, and F. Mansfeld eds., DechemaMonographs V. 101, VCH, Germany 1986.
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[13] D.D. Macdonald, "Some Advantages and Pitfalls of Electrochemical Imped Spectroscopy", Corrosion-NACE, 46 (3), 229-242 (1990).
[14] B. Breyer and H.H. Bauer, "Alternating Current Polarography and Tensametry",Chemical Analysis V. XIII, Interscience Publishers, USA 1963.
[15] J .R MacDonald. "Simplified Impedance/Frequency-Response Results Intrinsically Conducting Solids and Liquids", J. Chem. Phys., 61 (10). 3977-39(1974).
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y-Manganese Dioxide"Electrochim. Acta, 21 , 575-584 (1976).[22] J. R. Macdonald, "Comparison and Discussion of Some Theories of the Equilibr
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BIBLIOGRAPHY Chapter 7
[1] N.K. Thiet, "The Electrolytic Refining of Lead at High Current Densities", PDissertation, Bergakademie Freiberg, East Germany, 1982.
[2] H. J. Lange. K. Hein and N.K. Thiet, "Electrolytic Lead Refining at Increased Curre Densities", Neue Hiitte, 28 (4), 136-41 (1983).
[3] F. Wenzel and K. Hein, "Influence of the Anode Sludge on the Purity of the Cat Metal in the Electrolytic Refining ofLead', Neue Hiitte. 19 (5), 263-67 (1974).
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[5] F. Wenzel, "Contribution to the Clarification of the Anodic Processes in L Electrorefining;', Ph.D. Dissertation, Bergakademie Freiberg, East Germany, 1971.
[6] E. Nomura, A Aramaki, and Y. Nishimura, "Electrolytic Lead Refining and its
Mechanization at Takehara Refinery", TMS paper selection A74-21, TMS-AIME,Warrendale, PA (1974).
[7] S. Hirakawa and R. Oniwa, "Method for Producing High Purity Lead by Remov Bismuth', Canadian Patent 1023691, Jan. 3, 1978.
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[50] AS. Roy, "A Perspective On Electrochemical Transport Phenomena:', In AdvaIn Heat Transfer, T.F. Irvine and J.P. Harnett Eds., V. 12, p. 196-282, AcademicPress, New York, 1976. (also see references Appendixes 1 and 2)
[51] J.O'M. Bockris and AK.N. Reddy, "Modern Electrochemistry", V. 1 and 2, PlenumPress, New York, 1970.
[52] C.J. Krauss, "Electrometallurgical Practice: Zinc and Lead", in Fundamentals and
Practice of Aqueous Electrometallurgy, Short Course Notes, 20th AnnualHydrometallurgical Meeting, CIM, Montreal, Canada (October, 20-21,1990).
[53] N.T.Crosby, "Equilibria of Fluorosilicate Solutions with Special Reference t Fluoridation of Public Water Supplies", J. Appl. Chem., 19. 100-102 (1969).
BIBLIOGRAPHY Appendix 1
[1] I. Rousar, K. Micka, and A Kirnla, "Electrochemical Engineering I (Parts A -Q",
Chemical Engineering Monographs V. 21A Elsevier, Amsterdam 1986.
[2] N. Ibl and O. Dossenbach, "Convective Mass Transport', in Comprehensive treatise
of Electjochemistry, E. Yeager, J. O'M. Bockris, B.E. Conway and S. Sarangapanieds., Vol. 6, Chapter 1, Plenum Press, New York (1983).
[3] E. Peters, Transport in Electrolytes, to be published.
[4] K.S. F0rland, T. Forland, and S.K. Ratkje, "Irreversible Thermodynamics", JohnWiley and Sons, New York, 1988.
[5] B.G. Ateya and H.W. Pickering, "Effects of Mass Transfer In The Aqueous PhasOn Repassivation of Activated Surfaces and The Stability of Protective FilmPassivity of Metals [Proc. Conf.], RP. Frankental and J. Kruger, Eds., TheElectrochemical Soc, Corrosion Monograph Series, U.SA, P.350-368. 1978.
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Bibliography Appendix 2
[6] B.G. Ateya and H.W. Pickering, "Effects of Ion Migration on the Concentrations and Mass Transfer Rate tn the Diffusion Layer of Dissolving Metals", J . Appl.Electrochem., 11 , 453-61 (1981).
[7] AS. Roy, "A Perspective On Electrochemical Transport Phenomena", In AdvaIn Heat Transfer, T.F. Irvine and J.P. Harnett Eds.. V. 12, P. 196-282, AcademicPress, New York, 1976.
[8] J.J.C. Janz, "Estimation of Ionic Activities in Chloride Systems at Ambient Elevated Temperatures", Hydrometallurgy, 11 , 13-31 (1983).
[9] C.L. Kusik and H.P. Meissner, "Electrolyte Activity Coefficients in Inorga Processing", AIChE Symposium Series, 173 (74), 14-20 (1978).
[10] H. Majima and Y. Awakura, "Measurement of the Activity of Electrolytes and the Application of Activity to HydrometaUurgical Systems", Metall. Trans. B , 12141-147 (1981).
[11] C.Y. Chan and K.H. Khoo, "Re-determination of mean Ionic activity coefficient
the system HCl+KCl+H 20 at 298.15 K and correlations between Horned and Pi Equations", 1371-1379.
[12] "Handbook of Aqueous Electrolyte Thermodynamics", J.F. Zemaitis. D.M. Clark, Ratal, and N.C. Scrivner eds., AIChE, USA 1986.
BIBL IOGRAPHY Appendix 2
[1] E. Peters, Transport in Electrolytes, to be published.
[2] B.G. Ateya and H.W. Pickering. "Effects of Mass Transfer In The Aqueous PhaseOn Repassivation of Activated Surfaces and The Stability of Protective Films",Passivity of Metals [Proc. Conf.], R.P. Frankental and J. Kruger, Eds.. The
Electrochemical Soc, Corrosion Monograph Series, U.SA, P.350-368, 1978.
[3] B.G. Ateya and H.W. Pickering, "Effects of Ion Migration on the Concentrationsand Mass Transfer Rate in the Diffusion Layer of Dissolving Metals", J . Appl.Electrochem., 11 . 453-61 (1981).
BIBL IOGRAPHY Appendix 3
[1] W.D. Stanley, "Transform C i r c u i t Analysis for Engineering and Technology" ,
2nd Ed., Prentice-Hall Inc., USA 1989.
[2] RD. Strum and J.R Ward, "Laplace Transform Solution of Differential
Equat ions" . Prentice-Hall Inc., USA 1968.[3] D. Britz, "D ig i t a l Simulation i n Electrochemistry", 2nd Ed. Springer-Verlag,
Germany 1988.
[4] T.F. Bogart Jr., "Laplace Transforms: Theory and Experiments", John Wiley &
Sons, New York, 1983.
[5] T. Young, "Linear Systems and D ig i t a l Signal Processing", Prentice-Hall
Inc.USA 1985.
[6] K.L. Su, "Time-Domain Synthesis o f l i n e a r Networks", Prentice-Hall Inc.USA
1971.
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Bibliography Appendix 3
[7] AK. Walton, "Network An al ysi s and Practic e", Cambridge University Press, GreatBritain, 1987.
[8] T.H. Glisson, " In t roduc t ion to Sys t em Analys is" . McGraw-Hill Book Company,USA 1985.
[9] A A Pilla and G.S. Mar gules, "Dynamic Interfacial Electrochemical Phenomena
Living CellMembranes: Application to the Toad Urinary Bladder Membrane SysJ. Electrochem. Soc. 124 (11), 1697-1706 (1977).
[10] A A Pilla, "Influence of the Faradaic Process on Non-faradaic ResistanCompensation inPotentiostaic Techniques", J . Electrochem. Soc, 118 (5), 702-7(1971).
[ 11] A A Pilla, "Laplace Analysis of Electrode Kinetics", in Electrochemistry, Computein Chemistry and Instrumentation, Vol 2, J.S. Matsson, H.B. Mark, and H.C.Macdonald Eds., P.139-181. Marcell-Dekker, New York. 1972.
[12] J . Ye and K. Doblhofer, "Some Practical Aspects of the Numerical Laplace
Transformationfor Impedance Analysis ofElectrochemicalSystems", J. ElectroChem.. 261, 11-22 (1989).
[13] D.D. Macdonald and M.C.H. Mckubre, "Impedance Measurements in Electrochemical Systems", in Modern Aspects of Electrochemistry V. 14, Chapter2, J.O'M. Bockris, B.E. Conway, and RE. White Eds., Plenum Press, New York,1982.
[ 14] A A Pilla, "Laplace Plane Analysis of the Impedance of Faradaic and Non-Fada Electrode Processes", J. Electrochem. Soc, 1 18 (8), 1295-1297 (1971).
[15] A A Pilla, "A Transient Impedance Technique for the Study of Electrode Kinet J. Electrochem. Soc, 117 (4), 467-467 (1970).
[16] K. Doblhofer and A A Pilla, "Laplace Plane Analysis of the Faradaic and Non-Faradaic Impedance of the Mercury Electrode", J . Electroanal. Chem., 91-102 (1972).
[17] W.H. Smyrl, "Digital Impedancefor Faradaic Analysis I. Introduction to Digital Analysis and Impedance Measurements for Electrochemical and Corro Systems", J. Electrochem. Soc, 132 (7), 1551-1555 (1985).
[18] W.H. Smyrl, "Digital Impedance for Faradaic Analysis II. Electrcdissolution of HCV, J. Electrochem. Soc, 132 (7), 1555-1562 (1985).
[19] W.H. Smyrl and L.L. Stephenson. "Digital Impedance for Faradaic Analysis
II Corrosion in Oxygenated .1 N HCV, J . Electrochem. Soc, 132 (7). 1563-1567(1985).
[20] H. Gerisher, Z. Phys. Chem., 198, 286 (1951).
[21] D.C. Grahame. J. Electrochem. Soc. 107. 452 (1960).
[22] C. Gabrielli, "Identification of Electrochemical Processes by Frequency
Response Analys is" , Monograph Reference 004/83, Solartron InstrumentationGroup, Farnsborough, England, 1981.
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Bibliography Appendix
3
[23] J. R Macdonald, "Comparison and Discussion of Some Theories of the Equilib Electrical Double Layer in Liquid Electrolytes", J. Electroanal. Chem., 223, 1-2(1987).
[24] I. Epelboln, C. Gabrielli, and M. Keddam, "Non Steady State Techniques", iComprehensive Treatise of Electrochemistry, E. Yeager, B.E. Conway, J . O'M.
Bockris, and S. Sarangapani eds., Vol. 9, Chapter 3, Plenum Press, New York(1983).
[25] G.W. Walter," A Review of Impedance Plot Methods usedfor Corrosion Perfor Analysis of Painted Metals", Electrochim. Acta. 26 (9), 681-703 (1986).
[26] R de Levie, "On the Impedance of Electrodes withRough Interfaces", J. ElectroChem., 261, 1-9 (1989).
[27] R De Levie and L. Pospisil, "On the Coupling of Interfacial and Diffusiona Impedances, and on the Equivalent Circuit of an Electrochemical Cell', Electroanal. Chem., 22, 277-290 (1969).
[28] J.R Macdonald ed., "Impedance Spectroscopy", John Wiley and Sons, New York,1987.
[29] J.R Macdonald, "Impedance Spectroscopy: Old Problems and New DevelopmElectrochim. Acta, 35 (10), 1483-1492 (1990).
[30] D.D. Macdonald, "Some Advantages and Pitfalls of Electrochemical Impeda Spectroscopy", Corrosion-NACE, 46 (3), 229-242 (1990).
[31] K.J. Vetter. "Electrochemical Kinetics, Theoretical and ExperimentalAspects", Academic Press. New York (1967).
[32] "Electrode Kinetics: Principles and Methodology,". C H. Bamford and RG.
Compton eds., V. 26 in series Comprehensive Chemical Kinetics, Elsevier, TheNetherlands, 1986.
[33] F.P. Dousek, "Behavior of a Potassium Electrode Pretreated with Water VapGaseous Solution!', Electrochim. Acta, 32 (7), 1079-1086 (1987).
[34] D. Britz and W.A Brocke, "Elimination oflR-Drop in Electrochemical Cells by Use of a Current-Interruption Potentiostat", J . Electroanal. Chem., 58, 301-3(1975).
[35] D. Britz, "iR Elimination in Electrochemical Cells", J . Electroanal. Chem.. 8309-352 (1978).
[36] M. Hayes and AT. Kuhn, "Techniques for the Determination of Ohmic Drop Half-Cells and Full Cells: A Review", J. Power Sources, 2, 121-136 (1977/78).
[37] M. Kuhn, K.-G. Schutze, G. Kreysa, and E. Heitz, "Principles and Applications ofComputer Controlled Electrochemical Measurements with Elimination of IRPaper 24 in ref [49].
[38] "Techniques in Electrochemistry, Corrosion and Metal Finishing, aHandbook", AT. Kuhn ed., John Wiley & Sons. New York, 1987.
[39] E.J. Muth, "Transform Methods", Prentice-Hall Inc., USA 1977.
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Bibliography Appendix 4
[40] S.J. Orfanidis. "Optimum Signal Processing", MacMlIlan Publishing Company.USA 1988.
[41] AJ . Bouwens, "Digital Instrumentation", McGraw-Hill Book Company, USA
1984.
[42] F. Mansfeld, "Recording and Analysis of AC Impedance Data for Corrosion Stud
I.Background and Methods ofAnalysis", Corrosion-NACE, 37 (5), 301-307 (1981).
[43] F. Mansfeld, M.W. Kendig, and S. Tsai "Recording and Analysis of AC Impedance Data for Corrosion Studies II. Experimental Approach and Results",Corrosion-NACE, 38 (11), 570-580 (1982).
[44] J .R MacDonald, "Simplified Impedance/Frequency-Response Results Intrinsically Conducting Solids and Liquids", J. Chem. Phys., 61 (10), 3977-396(1974).
[45] M.W. Kendig, E.M. Meyer, G. Lindberg, and F. Mansfeld, "A Computer Analysis of
Electrochemical Impedance Data", Corros. Sci., 23 (9), 1007-1015 (1983).
[46] S.D. Stearns, "Digital Signal Analysis". Hayden Book Company. Inc., USA 1975.
[47] K.G. Beauchamp and C.K. Yuen, "DigitalMethodsforSignal Analysis", George Allen& Unwin, Great Britain, 1979.
[48] CD. McGillem and G.R Cooper, "Continuous and Discrete Signal and System Analysis", 2nd ed.. Holt Rinehart and Winston, USA 1984.
[49] "Electrochemical Corrosion Testing with Special Consideration of Practi Applications" [Proc.Conf.], E. Heitz, J.C Rowlands, and F. Mansfeld eds., DechemaMonographs V. 101. VCH, Germany 1986.
[50] F. Mansfeld, "Monitoring of Atmospheric Corrosion Phenomena with Electroche
Sensors", J. Electrochem. Soc, 135 (6). 1354-1358 (1988).
[51 ] W.J. Lorenz and F. Mansfeld, "Determination of Corrosion Rates by Electrochemi DC andAC Methods", Corros. Sci., 21 (8), 647-672 (1981).
BIBL IOGRAPHY Appendix 4
[1] V.N. Krylov and E.V. Komarov, "Some Properties of Fluorosilicic Acid', Russ. Inorg. Chem., 16 (6), 827-829 (1971).
[2] P.M. Brinton, L.A Sarver, and AE. Stoppel, "The Titration of Hydrofluoric and Hydrofluosilicic Acids in Mixtures Containing Small Amounts of Hydrofluo
Acid', Ind. Eng. Chem.. 15 (10). 180-1081.[3] C A Jacobson, "Fluosilicic Acid IE: Method of Titrating and Properties", J. Che
Soc. 119. 506-509 (1921).
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Bibliography Appendix 4
[4] V.N. Plakhotnik. "Equilibrium Constant of the First Stage of Hydrolysis of Hexafluorosilicate Anion", Russ. J. Inorg. Chem.. 48 (11). 1651-1653 (1974).
[5J P.M. Borodin and N.K. Zao, "Equilibria in the Li2SiF 6 -HCl04-H 20 by the 19F Nucle Magnetic Resonance Method', Russ. J. Inorg. Chem.. 16 (12). 1720-1722 (1971).
[6] H. K5nig, "Indirect termination of Silicic Acidby Titration", Z. Analyt. Chem.. 19
401-406 (1963).
[7] H. Bombach, K. Hein, J . Korb and H.J. Jange, "Investigation on the Chemica Stability of Lead Electrolytes Containing Hexafluorosilicic Acid' , Neue Hiitte, 3347-351(1986). (Translation MINTEK TR-1265).
[8] S.M. Thomsen, "Acidimetric Titrations in the Fluosilicic Acid System", Anal. 23 (7), 973-975 (1951).
[9] T.N. Sudakova, V.V. Krasnoshchekov and Yu. G. Ftolov, "Determination of the Ionization Constants ofHydrogenHexafluorosilicate inAqueousandOrganicMeRuss. J. Inorg. Chem., 2 3 (8), 1150-1152, (1978).
[10] W. Lange. "The Chemistry of Fluoro Acids", in Fluorine Chemistry, J.H. Simons ed.,Vol. 1, p. 125-188, Academic Press (1950).
[11] L.N. Arkhipova and S.Ya. Shpunt, "SomeProperties ofFLuorosilicicAcid', Tr. NIUIF208 , 88-103, (1965). (Translation National Translations Center, NTC -99D-85-22456).
[12] Y. Folov. T.N. Sudakova, A A Agcev, and V. Kransnoscherv, "Properties and Behavior of Fluosilicic Acids in Aqueous Solutions", Doklady Timinyaze Vsk Selskokhozya. p. 154-158, 1977. CA 89:2050908.
[13] Gmelins Handbuch der Anorganische Chemie. E.H.E. Pretsch ed.. V. 15 Si [B], •
p. 614-653, Verlag Chemie, Germany (1959)[14] L.N. Arkhipova and S.Ya. Shpunt, "Hydrolysis of Calcium Silicofluoride in Water
25'C, Tr. NIUIFa, 2 0 8 . 69-88 (1985). (Translation National Translations Center,NTC - 99D-85-22457).
[15] K. Kleboth. "Fluorine Complexes of Silicon in Aqueous Solutions", Monatsch. C99 (3), 1177-1185 (1968). (Translation Associated Technical Services, Inc.,55E148G).
[16] K. Kleboth, "Fluoro Complexes of Silicon in Aqueous Solution - Part 2 : For and Properties of Tetrafluorosilicic Acid' ', Monatsch. Chem. 100 (3). 1057-106
(1969). (Translation Associated Technical Services Inc.. 41W119G).
[17] S.Ya. Shpunt and O.V. Vasileva, "Physico-ChermcalStudies on the Reaction betwFluorides during the Treatment with Phosphate Acids" , Tr. NIUIFa, 228 , 12(1976). (Translation to Spanish: ICYT 76-4982).
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Bibliography Appendix 7
[18] K. Kleboth, "Concerning the Theory of the Distillation of Fluorine as Fluosilicic AcMonatsch. Chem. 100.1494-1498(1969). (Translation to Spanish ICYT: 69-4762).
[ 19] V.M. Masalovich, G A Moshkareva and P.K. Agasyan. "Study of Complex FormationinSolutions of Hydrofluoric and Silicic Acid', Russ. J. Inorg. Chem., 24 (2), 196-198(1979).
[20] RH. Busey, E. Schwartz, and RE. Mesmer, "Fluosilicate Equilibria in SodiumChloride Solutions from 0 to 60'C, Inorg. Chem!, 19 (3), 758-761 (1980).
[21] V.N. Plakhotnik, "The Concentration of Hexafluorosilicate Ions in Hexafluorosil Acid Solutions Saturated with Silicic Acid', Russ. J . Inorg. Chem.. 48 (10),1550-1551 (1974).
[22] N.T.Crosby, "Equilibria of FUiorosilicale Solutions with Special Reference toFluoridation of Public Water Supplies", J. Appl. Chem.. 19 . 100-102 (1969).
[23] J.R Cooke and M.J. Minski, "Kinetics and Equilibria of Fluorosilicate Solutions w Special Reference to the Fluoridation of Water Supplies", J. Appl. Chem., 123-1
March 1962.[24] W. Lange, "The Chemistry ofFluoroacids", in Fluorine Chemistry, J.H. Simons ed.,
V. 1, Academic Press, 1950.
[25] N.N.Golovnev, "The Influence of pH on the Formation ofSiF 6 ' 2 in Aqueous Solutio
Russ. J. Inorg. Chem., 31 (3), 367-368 (1986).
[26] O.P. Subbotina, L.N. Arkhipova, and M.N. Tsybina, "Composition and Propertiesof Fluosilicic Acid', Tr. NH Po Udobr. i Insektofungitsidam, 228, 56-60 (1976).(Translation to Spanish ICYT: 5933).
[27] C. Leonte, A Talaba, N. Aelenei, and R Smocot, "Real Properties of Hexafluorosilici
Acid Solutions", Revista de Chimie. 36 (12), 1125-1129 (1985).[28] A A Ennan, V.E. Blinder, and T.S. Borisenko, "The Properties of Fluosilicic Acid (A
Review)", Russ. J. Physical Chemistry. 51 (8), 1255, (1977).
[29] CE. Roberson and T.B. Barnes, "Stability of Fluoride Complex with Silica and I Distribution in Natural Water Systems", Chemical Geology, 21, 239-256 (1978).
[30] E . Hayek and K. Kleboth, "Concerning the Solubility of Silicon Dioxide i Hexafluorosilicic Acid', Monatsch. Chem. 92 (5), 1027-1034 (1961).
[31] Fluorine, "Determination of Fluosilicic Acid, Hydrofluosilicic Acid, Lead and S in Lead Refinery Electrolytes", Cominco Report, Laboratory Section 17, Report No.
18. July 18, 1973.[32] G. Schwarzenbach and H. Flaschka, "Complexometric Titrations", 2nd. ed.,
London, Methuen and Co., 1969.
[33] A Vogel, 'Textbook of Quantitative Inorganic Analysis", 3rd ed.. Longman Inc.,
1978.
BIBLIOGRAPHY Appendix 7
[1] M. Broul, J . Nyvlt, and O. Sdhnel, "Solubility in Inorganic Two-ComponentSystems", V. 6, Physical Sciences Data, Elsevier, Czechoslovakia, 1981.
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Biography
Appendix
8
[2] K.K. Worthington and M. Haring, "Methods for Determining the Solubilities of SoFUiostiicotes", Ind. Eng. Chem. Anal. Ed.. 3 (1). 7-9 (1931).
[3] W.F. Linke, "Solubilities of Inorganic and Metal-Organic Compounds". Van
Nostrand. New York. 1958.
BIBL IOGRAPHY Appendix 8
[1] M. Sluyters-Rehbach and J.H. Sluyters. "A.C. Techniques", in Comprehensive
Treatise of Electrochemistry. E. Yeager, B.E. Conway, J . O'M. Bockris, and S.Sarangapani eds., Vol. 9, Chapter 4, Plenum Press, New York (1983).
[2] C.J. Bushrod and N.A Hampson. "Anodic Behavior ofLead in Perchloric Acid I.Continuous and Interrupted Polarization", Br. Corros. J., 6, 87-90 (1971).
BIBLIOGRAPHY Appendix 1 0
[1] D.D. Macdonald, "Some Advantages and Pitfalls of Electrochemical Imped Spectroscopy", Corrosion-NACE, 46 (3), 229-242 (1990).
[2] M. Urquidi-Macdonald. S. Real, and D.D. Macdonald, "Application of Kramers-Kronig Transforms in the Analysis of Electrochemical Impedance DTransformations in the Complex Plane", J. Electrochem. Soc. 133 (10). 2018-202(1986).
[3] D.D. Macdonald and M. Urquidi-Macdonald, "Kramers-Kronig Transformation ofConstant Phase Impedances", J. Electrochem. Soc, 137 (2), 515-517 (1990).
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Appendix 1 Numerical Solut ion of the Nernst-Planck Flux
Equat ions and Its App l icat ion to The Betts Process
I. Objective:
To introduce an algorithm to solve numerically the Nemst-Planck flux
equations. From the solution of this equation the concentration of ionic species
and the potential profile throughout the slimes layer will be obtained as a function
of the electrolysis conditions.
n. Assumptions:
1) The Nernst-Planck flux equations can be applied in concentrated solutions
when they are solved in their fundamental form \
2) Dissolution of lead is the only Faradaic reaction. This reaction takes place
exclusively at the anode/slimes interface and proceeds without kinetic limitations
(i.e. i o - * ~ for Pb->Pb+2+2e)2.
3) Noble impurities originally present in the lead anode report to the slimes
and remain unreacted during the whole electrorefining cycle (i.e. i„-»0 for the
dissolution of Sb, Bi, As, and other noble impurities)3
.
4) Hydrolysis and secondary products precipitation can be neglected 4
.
5) Mixing of electrolytes within the slimes layer can be accounted for by
mcorporating the eddy diffusion component, D E in the overall diffusion term, 2}
(i.e. w=0) [1.21
5
.
1
This assumption implies
that
the cross coefficients in the Onsager phenomenological equations can be
neglected.
Interrelationships between the involved ions can be incorporated
within
the absolute
value
of
activities and mobilities.
2 Assumptions 2 and
3 are based on the fact that
upon dissolution
of lead bullion electrodes, lead
dissolution
at the anode/slimes interface is the main reaction. Furthermore, under normal operation of the
BEP the slimes remain unreacted and polarized.
3 Dissolution
of noble impurities is a function the potential gradient across the slimes layer. Once the
potential gradient is known, the point at which noble compounds start to react can be predicted.
4 Assumption 4 implies
that
the concentration gradients across the slimes layer do not depend on the
presence of secondary products or hydrolysis. This assumption can be relaxed once changes in activities
as a function of
position
are known. This will allow the hydrolysis point to be predicted.
5
Eddy
diffusion
will be used to incorporate
mixing
due to natural convection within the slimes electrolyte.
Natural convection arises due to concentration differences
that
are created by the electrolysis itself.
Forced convection takes place in regions
in
which the velocity field is not influenced by the concentration
field around the electrode, and the density of the solution is constant [1,2]. Within the slimes electrolyte,
large concentration gradients exist and natural convection is perceived to be much larger than forced
convection.
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Fundamental Equations
6
Steady state, unidimensional model1 .
7) Electrical neutrality is observed within the electrolyte entrapped in the
slimes layer 2 .
8 Only three ionic species are present (i.e. the PbSiF6 and H2 SiF 6 molecules
are completely dissociated)3
:
Species a: Pb+ 2
Species b: SiF6"2
Species c: H +
m. Fun dam enta l Equat ions
Given the previous assumptions, the Nemst-Planck flux equations that need
to be solved simultaneously are as follows 13-61:
1 dlna
a
d<& I(t)
—o c -+u c — = — — ...rn
N a a
a
dx *
a
dx Z
a
F
1 1
1 d\na
b
dQ>
1 d\na
c
d®
Z.C.+Z
t
C,+Z,C,
= 0 ...[iv]
where:
Species a = Ft/2
Species b = S1F6
2
Species c = H +
Q = concentration of species 1, (mol cm"3
].
Z, = charge number of species 1, [eq mol"1!.
<Di = overall diffusion coefficient of species i, [cm2
sec"1
] : ©; = D™ +D E
Df = molecular diffusion coefficient [cm2 sec"1]
D E = eddy diffusion constant [cm
2
sec"1
] 17] *•8 .
1 The electrolyte
composition
for a
fixed
position across the slimes layer may change as a result of
hydrolysis,
precipitation of secondary products and
natural
convection. Depending
on
the height of the
electrode
(among
other electrolysis parameters) local ionic concentrations may show gradients in both the
vertical and
horizontal
coordinates.
2 Electrical neutrality rules out the presence of charge imbalance at any point throughout the slimes layer.
3 H
2
SiF
e
-PbSiF
6
are strong electrolytes and their
dissociation
is expected to be very high.
4 D E can be
considered
to be equal for all the
ions.
Furthermore, it can be
modified
so as to incorporate
changes in porosity and tortuosity of the ionic flow across the slimes layer for different anode
compositions
and/or electrolysis conditions.
5 Notice
that when
the molecular
diffusion
coefficient is smaller than the eddy
diffusion
constant, the
overall
diffusion
coefficient is nearly the same for all the components. In this case, eddy diffusion
practically
equalizes
the
effective
diffusivities for all the components
[7].
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Fundamental Equations
N, = water mol fraction with respect to species i:
A r > l - - £ - a n d ^ = 10 , + ^
^ total
a, = molar activity coefficient of species i: a, = YC
•y,= individual molar activity coefficient.
*= migration potential, [volt]HJ= ionic mobility of species i, [cm 2 sec'1 volt"1! \
I (t) = net current density as a function of the electrolysis time 2, [Amp cm"2].
F = Faraday's constant, [96487 C eq"1]
R = universal gas constant, 8.3114 J mol"1 deg"1
x = distance from the slimes/electrolyte interface (as defined in Fig. 1), [cm].
Eq. i states that the only species being generated at the anode is Pb + 2 whereas
Eqs. ii and iii indicate that H+
and SiF6"2
do not react and their equihbrium
concentration is only a function of the migration and diffusion gradients. Eq. iv
is the electroneutrality condition.
Anode/Slimes
Slimes/Electrolyte
Interface
1
nterface
\
Lead Slimes
Bulk
Anode Layer
Electrolyte
Fig. 1 Coordinate system used toobtain the numerical solution of the Nemst-Planck flux equations.
i
X
=-x
X
=0
x 0
1 D"
and
u,
can be linked using the Nemst-Einstein relation:
• Z;F
where Z* is the effective charge of species i
exposed
to the electric
field,
Z*
<z,-.
2 The Betts electrorefining process for lead
is
normally carried out under nearly galvanostatic conditions
(i.e.
/
(t)
= constant).
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Derived Equations
TV. Der ived Equations
Equation iv can be differentiated using the following relationship:
then,
4
d_ d_(aAj^-a'i^lda, at dyt
dx {Ci) dx^) jf yt dx y}dx
rearranging terms:
d " dlna{ dlnyC
dx K dx dx _
By incorporating this relationship in equation [iv) and assuming that Z, = Z*
the following relationship is obtained:
= 0
Thus, the electrical neutrality equation (Eq. iv) can be equivalently expressed
as follows:
dlnaa d lnya
dx dx
+z bc„dlnab d\nyb
dx dx
+Z.C.dlnac d\nyc
dx dxBy making the following substitutions in equations U to v:
dlnya _-
dx ~ y"
d\naa _
da>
dx= q>
d lnyb _
—= y
>
d\nab
_
2FC a
=
0 ...[v]
—
=
D
d lnyb -
d\nab _
=n
Eqs. i, ii, iii, and v can be represented by the following equations:
D aaa + ua
O = -n ...1
...2
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Formulation
of
the Equations to
be
Solved Numerically
D eae + u e* = 0 . . .3
2Ca(aa - y.) - 2C „(Z b -yb ) + C C (3C -yc ) = 0 ..A
Equations 1 to 4 constitute a set of simultaneous differential equations that
have to be solved subject to the following initial conditions:
at x=0 (i.e. at the slimes/bulk electrolyte interface)
c,=c;
Y.=7f
0 = 0
Eqs. 1 to 4 have to be solved for 7 unknowns (aa, ab , ac , ya , yb , yc , <S>)
using three additional equations of the form:
_ dlna, dlnC, dlny,1 dx dx dx
These equations can be used to obtain either or ctj once the concentrations
of all the species are known.
V. Formulat ion of the Equat ions to be Solved Numerically
Eqs. 1 to 4 constitute a system of differential simultaneous equations that
have to be solved in order to yield the values for the four unknowns as a function
of the distance from the slimes/electrolyte interface : aa(x),ab(x),ac(x),®(x). By
performing the following transformations, the differential equations are combined
in such a way that they can be solved by a finite difference numerical technique.
This transformation is accomplished as follows:
Multiplying Eq. 1 times u.c minus Eq. 3 times u.a:
D
a u . c aA
- D
c u a a c
= -riu,e
rearranging and solving for ac
1
a-Yl . . . 5
1 For a fixed electrolyte concentration, the individual
activity
coefficients, y, and the individual
activities
can be
derived
from
theoretical
and experimental information
[8-12].
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Finite
Difference
Equations
Multiplying Eq. 1 times u.b minus Eq. 2 times u,a:
rearranging and solving for a
1
D.M*
(Db\i aab-n\i b ) ...6
Eqs. 5 and 6 are equivalent, thus:
and
D.M*
(Dbaaab-Tl \i b )
A L = —— a. ...7
IncoiTJorating Eqs. 5 and 7 into eq. 4:
2C,
D„
D c l i 0 _
a - n
-Ya+
Ce(ac-Ye) = 0
rearranging and solving for ac:
a ( r + 2 C ^ - 2 C D ^=
2C
a
y
a
-2QY
i
+
C
(:
Y
c
+
2 ^ n
Thus if initial values for the Y and other variables are provided, a finite interval
numerical approach could be used to solve for ac \ Once ac is known, ab and a a
can be obtained from Eqs. 7 and 5 respectively. The fourth unknown, cp, can be
obtained from either Eq. 2 or from Eq. 3.
VI. Finite Difference Equations
The differential terms presented in the previous equations can be
approximated using a first-order Taylor expansion.
1 By
incorporating
these transformations the differential equations can be
approximated
by
using
a
Taylor
expansion
series and solved as algebraic equations.
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Finite Difference
Equations
Thus, a, can be represented as:
' a,dx ~ a{ Ax
from which:
aj +l =aj + Ax-af-ai
where the superscripts j and j+1 refer to different nodal positions (see Fig. 1
) and Axis the finite difference distance l
.
A similar equation can be written for the change in the activity coefficients:
- _ = I<*Y,_
71 ytdx'Y, Ax
from which the value of the activity coefficient at the next nodal point can be
obtained:
7j+1
=^ + Ax-7j-Y
(
.
Local ionic concentrations can also be put in a finite difference form by using
the following equations:
dlna, dlnCi dlny,a >
dx dx dx
dlnCi _ _ dlnji
dx = a
' dx~
dQ _ _
From the previous equation, the local values of the ionic concentrations can
be obtained as follows:
C / + 1 = Cj + Cf-Ax-la,-^
The migration potential can also be expressed in finite difference form:
qV +1 = qV + Ax <D
1
Ax is
a
negative quantity (see
Fig.
1).
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Input data
Finally, the values of the D t, m, C l f and II parameters in equations 1 to 4 all
refer to their values at position j.
VII. A lgor i thm for the Numeric al Solut ion
Given the equations presented in the previous section the numerical problem
consists in finding the values of the activities and potential at the next nodal point
given their initial values (a/=>a/+1
, d>{=>Of +1). To obtain these values, the
following variables have to be known at the following position:
DJ N{ a{ i Y T 1
The concentrations at point j are known and the values of the above
parameters can be easily computed at that position. On the other hand the
concentrations at the next nodal point are not known in advance and an estimate
has to be used to compute the Y<+1
values. The values for the i* 1 parameters can
be improved by repeating the iteration procedure until the assumed and computed
concentrations in the next nodal point are nearly equal. In the following
paragraphs the algorithm to solve such iteration problem will be outlined.
A. Input data
To solve the Nernst-Planck flux equations in their finite difference form, the
following information has to be known in advance:
at x=0 (i.e. in the bulk electrolyte):
r =C°
cb = cc
c c =c:
and
q> =
q>°
=
0
For a fixed value of the concentration of the ionic species, the following
variables have to be computed according to pre-established relationships1
:
1 i y values
have
to be provided also as a function of the slimes thickness, x J , because of the presence of
eddy diffusion.
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Algorithm
3)
Assume that for the next interval1 : ya = yb = ye = 0
4) Given the initial electrolyte composition obtain the values of the following
parameters:
c;=c; ci=c° b c/=c;
the value of migration potential at this point is also known:
oV = a>°
From these values compute the following parameters:
& c
vi vi vi
yi yl yi
< at < K AY Nj
oa Di D;
Obtain the value of the n parameter:
Z aFCi
5) Compute ac from Eq. 8:
a, =•
- - - c'-ti2C
a y
a-2C
b y
b + C c yc + 2—
a
6) Compute aa and ab from Eqs. 6 and 7 respectively:
- D«Vi_ah-—:—ar
" miaa=-^-(Di^b-n
j at)
7) Obtain the value of O from Eq. 3:
1 Notice that this assumption implies that
for
the first interval the activity coefficients do not change:
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Algorithm
Cl + Cl-Ax'-fc-yJ
C J
c + C' c-to'-&e-yc ]
9) From these concentration values obtain yi* 1
, yl +1
, yi^and from these
values obtain:
1 2 ) Once the convergence criteria have been fulfilled, go back to step 2 and
compute the concentrations and potentials for the next step.
1
The convergence criteria have to be fulfilled by two of the three computed concentrations (the third
is
obtained from electrical neutrality). A successful convergence criterion was
found
by averaging the
assumed and computed concentrations and
using
the new value to improve the
solution:
yi Ax' b
y J
h Ax'
- _iyi +1
-yiYc
yi Ax'
1 0 ) Repeat steps 3 to 8 using the "new" yt values *.
1 1 ) Continue the iteration until:
C{*\new)-Ci +\old)
' C{ +\new)<0.01 (or any other tolerance value)
C,i+
\next iteration) <•Xnew) + C i
i+
\old)
2
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Appendix 2 Ana ly t ica l Solut ion of the Nernst-Planck Flux
Equations
I. Objective:
To solve analytically the Nernst-Planck Flux equations. This solution is to
be obtained for the case in which activities are equal to concentrations and eddy
diffusion is not present. From this solution the rmgration/diffusion ratio is derived.
Also an equation that describes local changes in resistivity of the entrained
electrolyte is obtained.
n. A s s u m p t i o n s 1 :
1) Dissolution of lead is the only Faradaic reaction. This reaction takes place
exclusively at the anode/slimes interface and proceeds without kinetic limitations
(i.e. for Pb->Pb+2+2e").
2) Noble impurities originally present in the lead anode report to the slimes
and remain unreacted during the whole electrorefining cycle (i.e. Lj-^O for the
dissolution of Sb, Bi, As, and other noble impurities).
3) Hydrolysis and secondary products precipitation can be neglected.
4) Mixing of electrolytes within the slimes layer is neglected (i.e. D £= 0 and
v=0).
5) Activities are equal to concentrations a .
6) The water mol fraction is the same for all the ions and is equal to 1.
7) Steady state, unidimensional model.
8) Electrical neutrality is observed within the electrolyte entrapped in the
slimes layer.
9) Only three ionic species are present (i.e. the PbSiF6 and H 2 SiF 6 molecules
are completely dissociated):Species a: Pb + 2
Species b: SiF 6"2
Species c: H +
1 These assumptions are similar
to
those used to obtain the numerical solution (see Appendix 1) except
for assumptions
4
to
6.
2 It can be
shown that
the solution here presented is also
valid
when individual activity coefficients are
constant (see
Appendix
1).
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m. Fundamental Equations
Fundamental Equations
Given the previous assumptions, the Nernst-Planck flux equations that need
to be solved simultaneously are as follows [l-si:
" dx RT dx Z„F
n dC b Z bFd& n
-Dk _ i + c t-J-=r- =0dx
b
RT dx
„ dC c Z C F d<f> n
dx c
RT dx
Z aC a+Z bC b+Z cC c=0
...1
...2
...3
...4
where:Species a = Pb +2 Species b = SiF 6
2 Species c = H +
C, = concentration of species i, [mol cm'3].
Z, = charge number of species i, [eq mol'1].£>, = molecular diffusion coefficient [cm 2 sec'1]
*= migration potential, [volt]
HJ= ionic mobility of species i, [cm 2 sec'1 volt"1] l.
I = current density [Amp cm"2].
F = Faraday's constant. [96487 C eq"1]
R = universal gas constant, 8.3114 J mol"1 deg"1
x = distance from the slimes/electrolyte interface (as defined in Fig. 1), [cm]
Eq . i state that the only species being generated at the anode is Pb + 2 whereas
Eqs. ii and iii indicate that H+
and SiF6"2
do not react and their equilibrium
concentration is only a function of the migration and diffusion gradients. Eq. iv
is the electroneutrality condition.
IV. Bound ary Con dit ions
Eqs. 1 to 4 constitute a set of simultaneous differential equations that haveto be solved subject to the following boundary conditions (see Fig. 1):
1 £>r andm were linked using the Nernst-Einstein relation:
D =—' Z'F
Z* is the
effective
charge of species i exposed to the electric field, z ' was assumed to be equal to Z r
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Derived Equations
Anode/Slimes Slimes/Electrolyte
Interface Interface
1 I
Lead Slimes
Bulk
Anode Layer
Electrolyte
• A
x=x
x=0
Fig. 1 Coordinate system used to obtain the analytical solution of the Nernst-Planck flux equations,
at x =x at x=0
Ca = C a C a = C a ... 5
C b = C b C t = Cf ...6
C c — C c C c — C c ... 7
<D=q> O = 0 ...8
V . Der ived Equations
Eqs. 1 to 4 are to be rearranged so that the concentration and potential
profiles can be obtained as a function of the distance from the slimes/bulk
electrolyte interface, x. This is done as follows:Dividing Eqs. 1, 2, and 3 by DA , DB , and D c respectively and substituting the
corresponding Z, values in these three equations and in Eq. 4, the following
relationships are obtained:
^ + 2 C = ...
9
dx "RT dx 2FD a
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Derived Equations
dC b F dq> — -2C h —— = 0dx b RT dx
...10
dC c F d <b+ C =0
dx Cc RTdx U...11
2C a + C e = 2C b ...12
adding Eqs. 9, 10, and 11:
dC a dC b dC c F d® I
**+*+*+(2C'+ C<-2C»W=-2FD„...13
substituting Eq. 12 in Eq. 13:
dC a dC b dC c _ I...14
dx dx dx 2FD a
...14
separating variables and integrating using b.c. 1 5, 6, and 7:
c m + c i + c e = c ; + c ° + c ? + - ^ - ...15
Eq. 15 can be expressed in dimensionless form by dividing it by C°:
c a + c b + c c = c * a + c 0
b + c 0
c + I x Q
b c a b 2FD aC°...16
where C, is the dimensionless concentration defined as follows:
c ' c°...17
and
C° = total bulk electrolyte concentration , C ° = C x + C 2 , [mol cm'3] ...18
Ci = PbSiF6 concentration in the bulk electrolyte, [mol cm"3].
C 2 = H 2 SiF 6 concentration in the bulk electrolyte, [mol cm'3].
Eq. 16 can be simplified by defining a new constant, K x:
...19
Thus, Eq. 16 can be expressed as follows:
C a + C b + C =K.+—^—z
a b c 1 2FD aC°
...20
1
"boundary conditions" is
abbreviated
as b.c.
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Analytical
Solution
VI. Ana ly t ica l Solution
All the terms to the right of the equal sign in Eq. 20 are known or can be
assumed. Thus, the problem consists in finding algebraic relationships between
the C a , C b , and C c terms. These relationships were obtained as follows:
rearranging Eq. 10 and separating variables:
^ = 2 C ——dx
b
RTdx
dC„ 2Fd®
C b ~RT dx
Eq. 22 can be integrated using b.c. 6 and 8:
rearranging:
C 4 =Cexp( -2F_
RT
.21
.22
...23
...24
Eq. 24 can be expressed in dimensionless form by dividing it by C°:
C b = C^exp [RT )
...25
Eq. 25 provides a relationship between C b and 4>. A sirmlar relationship
between C c and <& can be obtained from Eq. 11 using b.c. 7 and 8:
...26
A relationship between C b and C c can be found by multiplying Eqs. 25 and
26:
cfr (cc)2
=cT(cTj ...27
the C° b(C°f term in Eq. 27 can be equaled to a new constant, Kj , :
K 2 = C°b{c° c J ...28
Thus, C b can be expressed as follows:
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Analytical Solution
- K 2c „ = ^ - 2
(C cf
.29
Eq. 29 can be substituted in Eq. 12 to obtain the following relationship:
2Ca + C c = — 2
{C cf...30
this equation can be rearranged to obtain:
T
* 2 C c
a~(C ef 2...31
* 2 C c
a~(C ef 2...31
Finally, substituting Eqs. 29 and 31 in Eq. 20 and rearranging:
(CJ3
+ (CJ2
1 FDaC°+ 4K 2 = 0 ...32
Eq. 32 is a cubic equation that has to be solved for C c given K l t K , andlx
FD.C
. From the obtained C c values, C a , C b , and <I> values can be obtained from Eqs.
29, 31, and 25 respectively.
Eq. 32 was used to obtain changes in the dimensionless concentrations and
in the migration potential as a function of the bulk electrolyte composition andof the dimensionless parameter 0 (see Figs. 2 to 4):
Ix0 =
FDaC°...33
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Changes in tie Specific Resistivity of the Entrained Electrolyt
VII. Changes i n the Speci f ic R esist iv i ty of the Entrained
Electrolyte
The specific resistivity of dilute electrolytes can be expressed as follows:
1 RT 1P,=
FXZ.n.C, F'lZfD.Q
where:
Pi= specific resistance, [flcm].
The dimensionless ratio between the resistivity of the bulk electrolyte and
that of the entrained electrolyte, p(/), is given by the following equation:
/ c . _ _ lZ?DiC,(Bulk Electrolyte)- ( / ) = p(0-0)_;
p(0 = O) lZ?D,C,(Slimes Layer)
Changes in p(/) as a function of 0 are given in Fig. 5 for the indicated c o n s t
diffusion coefficient values. p(/) decreases as 0 increases indicating that the
entrained electrolyte is more conductive than the bulk electrolyte \
1 Electrical conductivities within the slimes electrolyte are likely to decrease rather than
increase
(see
Appendix
and Chapter 7).
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Changes
in the Speak resistivity of the entrained electrofyte
0, 2 . C, » .2 mol/1 C, = .2 mol/1
Pi f. A ™.'/L 9i°.:?
C, .8 mot/I C, » .2 mol/1
I I I ' ' ' I '
2 4 6 8 10 12 14 16 18 20
C, = t.S mol/1 C , = .1 mol/1
- i i ' | i i i | i i i | i i i | | i i i | | i i i0 2 4 6 8 10 12 14 16 18 20
Fig. 5 Effect of the variable © on the dimensionless resistivity, p(I), for different bulk electrolytecompositions.
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Computation ol the Diffusion Flux
VUI . Computat ion of the Diffusion Flux
As shown by Eq. 1, the ionic flux has two contributions:
dC a-D„—— = Diffusion Flux
dx
Z aF rfq>
-D„C„ , = Migration Flux° " RT dx
Thus, the diffusion flux can be obtained once the changes are known.
dC
The term was obtained using the following algebraic procedure:
The derivative of Eq. 12 with respect to x is given by:
...34
dC a dC r dC h2 — + — = 2 —
dx dx dx
Eqs. 10 and 11 can be rearranged to obtain a relationship between C a and
1 dC c l dC bF d<Z>
RT dx C c dx 2C b dx
d £]L = _ 2C 1dC 1
dx C c dx..35
Thus, Eqs. 34 and 35 can be used to obtain —, and as a function of
dC a dC c C bdC c
dx dx
dC c
dx
C c dx
dC„=
-2-
dx
dC c 2C C dC a...36
dx C c + 4C bdx...36
dC b 4 C
4
dC a...37
4 C
4
dC a...37
dx ~C c + 4C b dx...37
dC.Incorporating Eqs. 36 and 37 in Eq. 14, and solving for -£
dC a
dx1 +
4 C
h
2C,
C c + 4C b C c + 4C b
dC a
dx
C c + 4C b + 4C b -2C c
C. + 4C 2FD„
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Computaton ol the Migration Flux
dC a _ / \C e + 4C„l...38
dx 2FDa . 8 C , - C C
...38
Eq. 38 can be expressed in dimensionless form my multiplying it by ^
dx 2FD„.38.a
The bracketed term in Eq. 38.a can be further simplified by using Eq. 29:
4iC,
C c + 4C b^Cc + (cc fJC c f + 4K 2
SC b-C c j f i _ c SK 2-(C c f(c«f
U c
Thus, Eq. 38.a can be expressed as follows:
dC^
dx 2FD„
(C c f + 4K 2
8tf 2 - ( C J3
From which the diffusion flux can be obtained:
" dx IF
(C C )3 + 4K
2
8 / T 2 - ( C J3
.39
IX . Computat ion of the Migr atio n Flux
The migration flux can easily be derived once the diffusion flux is known.
This can be inferred by rearranging Eq. 1:
_ F d <b I n dC a-ID C = — +D —-" tt RT dx IF a dx
The left hand terms in this equation can be simplified by using Eq. 39:
UD^ IF ' dx
From which the migration flux can be obtained as follows:
/ / "(C C )3
+ 4AV _ I ' (C c f + 4K 2
SK 2-(C C )\
_ I '4K 2-2{C C T
IF IF SK 2-(C c f ~2F
' (C c f + 4K 2
SK 2-(C C )\ ~2F _ %K 2-(C c f
F d<3> I'4K 2-2{C c f...40-2D C = —
a a RT dx 2F
'4K 2-2{C c f...40-2D C = —
a a RT dx 2F _ %K 2-{C c f
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Computation of the Uagmion/Diffusion Ratio
X. Computat ion of the Migrat ion/Diffusio n Ratio
Both the migration and diffusion fluxes can be obtained once the C c values
are known (see Eq. 32). The rnigration/diffusion ratio, R(I), can be obtained by
dividing Eq. 40 by Eq. 39:
Migration Flux n , T ^ 4/sT2-2(CJ3
.~T .——— = /?(/)= ...41Diffusion Flux (C c f + 4K 2
As shown in Fig. 6, the diffusional flux is always larger than the migrational
flux. Migration becomes increasingly more important at large values of the
dimensionless parameter 0 (e.g. at large slimes thicknesses and/or current
densities).
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Computation ol tfie Migration /Diffusion ratio
1-r
(A)
0.2- ,
legend
C t » .2 mol/1 C, » .2 mol/1
Cjtt_ ._2_ moj l C a .8 mot l
C, = .8 mol/1 C, g .2 mol/1
0 +
10
0 =
12
lxFD n X°
T -
16
18
20
(B)6
o3£COin3
5
o•cni
0.8
0.6
0 .4 -
0.2-
legend
C, • .8 mol/l C , « .8 mol/l
C, = 1.5 mol/l C , = .1 mol/l
0 =
10
-t—12
lx16
18 20
Fig. 6 Effect of the variable 0 on the migration /diffusion ratio, R(I), for different bulk electrolytecompositions.
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Appendix 3 T ime Domai n To Frequency Domain
Transformation: The Fourier Trans form In Cu rre nt Step
E lec t rochemica l Techniques
I. Introduct ion
The theoretical foundations of the Laplace transformation are described
extensively in the literature [1-4]. Laplace plane domain techniques are widely
used in the analysis of electrical circuits (5-71. Particularly, in the area of system
analysis Laplace analysis is extremely valuable 181. The availability of personal
computers with high processing power has boosted the use of this technique in
almost every scientific field. Among other advantages, the power of the technique
resides in its ability to decompose any signal into its fundamental components.
Furthermore, the possibility of obteuxiing the transfer function of the system is a
strong driving force for the implementation of this technique. Electrochemists
have applied Laplace techniques to the study of a variety of electrochemical
systems [9-191.
In this appendix, the one-sided Laplace transformation (or Fourier transform)
will be used to analyze a simple electrical circuit. The circuit here chosen follows
the electrical analogue of the electrode/solution interface proposed by Gerischer
[201 and Grahame [21]. The response of this circuit to a pulse of current will beanalyzed both in the time and in the frequency domains. The transfer function of
the system will be obtained by Fourier transformation of the input and the output
signals. The differences between the values obtained by the numerical (using the
FFT algorithmx
) and the analytical Fourier transformations will be stressed.
n. Response of a S imple RC C i rcu i t to Current Pulses
In the analysis of electrochemical systems the use of electrical analogues is
widespread [22-271. The advantages and limitations of this approach are discussedin the literature (28-301. Fig. 1 shows the analogue that has been extensively used
to model the behavior of electrochemical cells. In this circuit Ri represents the
uncompensated ohmic resistance, Rj the Faradaic resistance for charge transfer,
and c the capacity of the electrical double layer. The quantification of each of this
parameters can be done using a wide array of electrochemical techniques [24,31,32].
1
FFT stands by
Fast
Fourier Transformation.
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Analytical solution
For this simple analogue, the study of the decay in potential upon current
interruption provides accurate information on these parameters. However, as the
system becomes more complex, analysis of the time-domain transients is
extremely difficult and the amount of information that can be obtained is limited.
Very often information on rj n is the only value sought [33-381 but sometimes kineticand mass transport data are also required. Usually a "trial and error" procedure
has to be conducted in order to find an electrical analogue that matches the
response of the experimental system. That is, the configuration of the electrical
circuit is not usually known and experimental data have to be provided to create
an analogue circuit. The use of the Fourier transformation is a strong aid in such
an analysis. More complex circuits can be analyzed following the same technique.
(a) (b)
e(t) E(S)
i(t)
-<s>-R,
i(S) Ri
c S
Fig. 1 Electrical analogue circuit
(a) Time-domain representation, (b) Laplace-domain representation
A. Analytical solution
The problem here consists in finding for a known configuration of the
electrical circuit, the response (potential as a function of time) given the input (a
current pulse). This circuit is depicted in Fig. l.a.
The response of the circuit shown in Fig. 1 to two different input functionswill be analyzed both in the time and in the frequency domains.
Figs. 2.a and 2.b describe the input functions used to exemplify the analytical
procedure. Fig. 2.a describes a current step with zero decay and rise times whereas
Fig. 2.b sketches a similar current step but with a non-zero rise time \
The analytical procedure to obtain the output of the system is as follows:
1 In Fig.
2 L, represents the time that has elapsed since the current
step
was first
applied.
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Analytical solution
1) Obtain the Laplace transform of each one of the components of the
electrical circuit (i.e. i(t), R2, c).
The Laplace transform is defined as:
W(r)]= rf (t)e-*'dt=F{S) t>0 Jo ...1
where:
T ->lfW] = F(S) = Laplace transform of the time-dependent function, JW.
S = new algebraic variable which is independent of time.
2) Obtain the transfer function of the system:
In this case, the transfer function is known as the impedance, Z(S). For the
circuit shown in Fig. 1, Z(S) can easily be obtained using nodal analysis. The
general form of Z(S) is given by the following relationshipl
:
where:
I(S) = Laplace transform of the current. i(t)E(S) = Laplace transform of the potential, e(t).
3) Obtain the value of E(S) from Eq. 2
4) Obtain the inverse Laplace transform of E(S), which is:
V l [E(S)] = e(t) ...3
Thus, given i(t) and the configuration of the electrical analogue, the response
of the system, e(t), is obtained by following the 4 steps previously described. In
the following section these steps will be performed one by one for each current
step waveform (cases A and B).
Case A:
1
Notice
the relationship between Eq. 2 and Ohm's Law: R = j
2 Notice
that
this
step can be performed because both
Z(S)
and
l(S)
are
known
precisely.
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Analytical solution
Fig. 2 Current step functions
used
to analyze the circuit
described
in Fig. 1
(a) Current step with
zero
decay and rise times
(b) Current step with zero decay but with a finite rise time.
Step 1:
The input function, i(t), can be described by the following relationship:
i(0 = /: 1[l(r)-l(r-U + ia-O] -4
where 1.(0 is the unit step function:
1 / x {0 for t<0\
for ,>o}
The Laplace transformation of the R l t c components is described by the
following equations:
L [ / ? J = ...6
L [ / ? J == R2...7
L [ c ] =
1
'cS...8
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Analytical solution
while the Laplace transform of the time dependent function, i(t) can be
expressed as follows:
/ ( S ) = | [ l - ^ + ] - .9
Step 2:
The Laplace domain representation of the electrical circuit shown in Fig. 1 .a
is shown in Fig. l.b. Applying nodal analysis the impedance of the system is
obtained:
Z(S)=Z l (S)+Z 2(S) ...10
where:
Zt(S) = ...11
Z2(S) = R2
l+R2cS...12
thus,
Z ( 5 ) =
^ + l + ; 2 c 5
...13
rearranging:
R.+R.
Z(5)= p - ...14
Step 3:
E(S) is obtained by multiplying Eq. 14 times Eq. 9:
E(S) = I(S)Z(S)=j[i-e-' J + e" bS ]
rearranging:
k R- }^ L
£(S) = - ^ [ l - ^ \ r ' 1 + T ^ [ l - ^ S
+ «f''S] -IS
.15
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Analytical solution
Step 4;
The inverse Laplace transformation of Eq. 16 gives
+ t , [ « , + / ! j { 1 - c " ^ _ [ i _ e - ^ . - . . ) ] i ( , _ , > ) + [ i _ e - ^ . - , ) ] , ( r _ I s ) J ...17
Eq. 17 is the pursued analytical solution for the input function shown in
Fig. 2.a.
Case B:
Step 1:
The function i(t) can be described by
i (0 = *x{ 1 (0 " K' - O + rn [r(t - t b ) - r (t - t c )]} ... 18
where r(r) is the unit ramp function:
r W - f °
f r <
n
0} ...19
= [t for t > 0 J
and m is the slope of the rise time curve which is defined as:
k x
m =— — ...20
tc-h
The Laplace transform of the time-invariant components is given by Eqs. 6
to 8 and the Laplace transform of i(t) is given by:L[i(01 =/(5) =| [1 ~e-' S ] +!jL [ e-*-e-V] ...21
Step 2:
The impedance of the system is independent of the input and output
functions and is given by Eq. 14.
Step 3:
E(S) can be described as follows:
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Analysis ol tie data generated in the time domain
R
x
+R
2
'Si'-"1 -;m
V * 2R~c
.22
Step 4:
Taking the inverse Laplace transform of Eq. 22, e(t) is obtained as follows:
e(f) = k l Rl [e v
'-e ''m-/.)] + ...23
+ [1 _ e-^(,
-,
»)
]ia-o-[1 _/^"'-)
]i(t
-o}
+ m{Rl +RJR2cK '-O-
1
B . A n a l y s i s o f t h e d a t a generated in the time
The response of the circuit shown in Fig. l.a to the input signals expressed
by Eqs. 4 and 18 is described by Eqs. 17 and 23 respectively. The time dependance
of the input and output signals is presented in Fig. 3 assuming the indicated
parameters. Analysis of the response of the system in specific windows of time
can provide the unknown values of some of the parameters present in the circuit.
For instance, considering the time interval between t (i.e. after the capacitor is
fully charged but before current is interrupted) and t (i.e. just before current is
applied back), Eqs. 24 to 26 are derived from Eqs. 17 and 23:
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Analysis of the dab generated in tie time domain
0.006
0.005
CL£ 0.004
<
(a1)
0.003 -,
3 0.002
O
0.001 -
t
0
=
0.000 sec
t= 0.583 sec
t„= 1.000 sec
t
c
- 1.600 sec
J?f 25.0 n
/V
10.00
n
c =
3000. / iF
f
\
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Time, sec
(bi)
0.006
0.005
CLE 0.004
<
0.003
•
Z3 0.002
o
0.001
t
(„= 0.000 sec
(,= 0.101 sec
t„= 0.108 sec
f
e
= 0.115
sec
tf 0.130 sec
Rr
25.0 0
Rz= 10.0
n
c = 300. fjF
0.026 0.052 0.078 0.104 0.13
Time, sec
0.2
0.15
(a2)
>
+->c
oQ_
r
0.1
0.05 -
f
'
\
0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Time, sec
(b2)
Time, sec
Fig. 3 Response of the circuit described in Fig. 1 to the input functions described in Fig. 2 assuming the indicatedsystem parameters Figs, (al) and (a.2) correspond to Eqs. 4 and 17 respectively. Figs, (b.l) and (b.2) comply withEqs. 18 and 23 respectively
A) At t = t':
B) at t = U:
C) between and t
e(t) = k 1(R1+R2 )
steep potential decay = k^
e(t) = klR2e
...24
...25
.26
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Analysis of the data generated
in
the
ime
domain
Thus, R t and Ra can be obtained by solving simultaneously Eqs. 24 and 25.
Knowing R 2 , c is computed using Eq. 26. Analysis of other windows of time should
provide with equivalent values for the resistors and the capacitor.
For this particular example, the computation of the value of the electrical
parameters was straightforward. On the other hand, when the number of analogueelements increases and/or time-dependant components are involved, this
analysis is overwhelmingly difficult. Thus, the use of Fourier analysis is
encouraged.
The Fourier transform is defined as [39-41]:
Fr /(f)] = j ~ = F(j<o) .. .27
where co is the frequency (in rad/sec) and j is the imaginary number (V -T). The
functions used in electrochemistry are only defined for times greater than zero,
therefore Eq. 27 can be expressed as
Fr /(0] =
(~f(t)e- J °'dt=F(j(0) ...28
Jo
Eq. 28 resembles closely Eq. 1 when the variable S is substituted by the
variable jco. Eq. 28 can be called the one-sided or imaginary part expression of
the Laplace transformation. Thus, the Fourier and Laplace transformations whichthemselves are two separate mathematical entities, can be linked through changes
in the S algebraic variable. A real part representation of the Laplace transform
can also be obtained. However, for the following analysis this particular
representation is not explicitly required.
The Fourier transform of the impedance of the circuit described in Fig. 1 can
be obtained in the same way its Laplace transform was derived. Thus, by analogy
with Eq. 13, the following equation is obtained:
R2
v 1
l+R2cj(Si
The variation of Z(jco) with frequency is sketched in Fig. 4.a. In this plot, the
real part of the impedance is the abscissa and the negative of the imaginary part
of the impedance is plotted as the ordinate. Analysis of this plot (so-called Argand
diagram) can provide the value of the elements of the analogue circuit. Plotting
of log (co) vs. I Z(jco) I (so-called Bode plot) complements the overall information
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Analysis of tie data generated in the imedomain
required to analyze the system. The Bode plot for this circuit is shown in Fig. 4.b.
Information on the values and characteristics of the electrical parameters likely
to be present, can be derived from these plots (28.42.431.
The value of the real part of the total impedance can be obtained as the low
frequency interception of the impedance curve in the Argand diagram. In addition,at high frequencies the value of Rj is attained:
when (0 = 0 Z (j (n )=Rl +R2 ...30
when CD — > oo Z (ja)=Rl...31
Eq. 29 can also be written as:
Z
( /G>)=K
,+7
72
1
+ CJ(H
...32
Thus, at co = co 0=^ a maximum in the Argand diagram is found. This
maximum is marked in Fig. 4.a and corresponds to the value of Thus, to obtain
the value of c, I Z(Jco) I is computed at this point and the frequency at which it
takes place is obtained from the Bode plot (Fig. 4.b). Complex non-linear
regression analysis [44,45]
and deconvolution techniques [42,43] can also be appliedin the analysis of impedance data.
At this point, it is worth noting the fact already stated in the literature [28,29]
that the impedance obtained by Fourier transformation of data generated in the
time domain is also in character an AC impedance. That is, the results obtained
from the study of a time-invariant system should be equivalent in both cases.
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Numerical
Solution
- zmag (a)
RtCHWv
J
/
\ y W =0
*2
FVR 2 eal
IZ! ( b )
log w Q log w
Fig. 4 Frequency spectrum of the circuit depicted in Fig. 1(a) Argand Plot (b) Bode Plot
ni. Numer ica l Solution
The relationship between the Laplace and Fourier transformations expressed
by Eq. 28, produce the following representation of the impedance in the frequency
domain:
ZC/co) = Em
/(/co)
...33
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Numerical Solution
By knowing the configuration of the electrical circuit the frequency
dependance of the impedance was found (Fig. 4). As Eq. 33 shows, the impedance
can also be obtained by Fourier transforming the input and output functions and
taking their ratio. In the previous section the system response was derived
analytically from the knowledge of the circuit configuration and of the time
dependance of the input function. The approach in this section is to obtain the
transfer function given the input and output functions in discrete form. Details
of the configuration of the electrical analogue and evaluation of its components
are to be derived from the analysis of the frequency spectrum of the transfer
function.
The numerical representation of the transfer function is obtained by finding
the zero-order approximation of Eq. 27. This approximation to the Fourier integral
is known as the discrete Fourier transformation (DFTj. The DFT is defined as [461:
where:
fn(t) = value of the digitized time dependent signal fl.t) taken at the nth time interval.
N = total number of sampled points taken
At = sampling interval, sec
Assuming that fit) is only defined for t>0 and that it has a finite number of
points, the following relationship is derived from Eq. 34:
F n = F(j<x>)= lf n(t)e l N i
m= 0 , l , . . . , A / - l ...35
»=o
where m is an integral value known as the frequency index.
The frequency index is related to the frequency , co, through the following
relationship:
2nm 17 ,co = —— m =0,1,...N-l ...36
NAt
The period, T, of the function f(t) is given by the product NAt. For a single
period, Eq. 36 shows that coAt spans in values comprehended between 0 and 2n
rad. It also shows that the frequency representation of the data generated in the
time-domain is given in multiples of the fundamental frequency ay
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Numerical
Solution
Accurate results can only be obtained by "matching" o)f so that an integral
number of waveform periods is covered. If this condition is not fulfilled, "leakage"
[461
will be present and incorrect spectra will be attained. Another phenomenon
that can significantly alter the frequency spectrum is known as "aliasing"[461. This
can be avoided by limiting the value of the highestfrequency that can be preciselyknown. This is done by applying the sampling theorem [46]
which can be expressed
by the following relationship:
Thus, even though coAt values as high as 2n can be obtained (Eq. 36) the
highest meaningful coAt value is n rad. In practice, this value is reduced even
further to avoid as much as possible the presence of aliasing.By applying the definition of the DFT given in Eq. 35 to Eq. 33, the discrete
Fourier transformation of the impedance can be expressed as follows:
where the DFT of the different functions involved has been incorporated.
Thus, the problem reduces to find the Fourier coefficients F„ for the e(t) and
i(t) functions and taking their ratio. This transformation has to account for the
restrictions imposed by leakage and aliasing and the F
m
coefficients must be
computed from Eq. 35 using the FFT algorithm.
One of the requirements for the implementation of the FFT algorithm is that
the f(t) functions have to be given as sets of 2q (q being a positive integer number)
equally spaced data points. A large amount of easy-to-use programs are available
in the literature to implement this algorithm [1,46-481. In this particular work, use
of a built-in FFT routine present in a commercial software package was employed.
When the numerical time-domain to frequency-domain transformation is to
be performed, important consideration should be given to the values that the
sampling interval, At, and the function period, T, can adopt. To assure that the
maximum spectral content is to be obtained, At has to be chosen as small as
possible (see Eq. 37). In addition to this, the extent to which information in the
lower frequency range can be attained is regulated by the total number of points
sampled, N (see Eq. 36). Thus, the use of very small At values and a large number
...38
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Numerical
Solution
of data points is advantageous to obtain the full frequency spectrum. However,
due to instrumental and experimental constraints, there is a limit as to how small
or large these parameters can be. Moreover, choice of the signal period is not an
easy task and a "cut" and "try" approach has to be undertaken.
In the numerical analysis of the potential and current transients hereconducted, coy values were restricted even further than indicated by Eq. 36
because a large amount of aliasing was found when the frequencies considered
went as high as Eq. 36 indicates \ Thus, by using a "trial and error" procedure
the maximum allowable value of co was found to be:
1 'rod'...40
2Ai
...40
In Table 1, hypothetical values for NAt were introduced to illustrate howchanges in this parameter can affect the characteristics of the frequency spectrum.
In this table, cOf and 0 ^ were calculated according to Eqs. 37 and 40 respectively.
For a fixed NAt value, increases in the number of sampled points are reflected in
a wider frequency region. Reduction in the NAt values for a fixed sampling rate
produces an increase in the frequency range but at the cost of decreased resolution
(by increasing COf). Thus, changes in COf can only be introduced through
modifications in NAt. Small COf values are helpful as the lower they are the better
resolved the spectrum and the lower the possibility of "leakage". If the fundamentalfrequency is to be decreased, the easiest way of doing it is by extending the period
of the waveform.
Table
Changes
in the maximum
and
fundamental frequencies
for
different waveform
periods
and
number
of
sampled points
NAt=l sec, C0(=2TC NAt=2
sec, C0f=n
NAt=4 sec, C0f=n/2
N
At
w
u
At WW At
w
u
128
0.00781
64
0.01563
32
0.03125
16
256
0.00391
128
0.00781 64
0.01563
32
512 0.00195 256
0.00391
128
0.00781
64
1024 0.00098 512 0.00195 256
0.00391
128
2048 0.00049 1024 0.00098 512 0.00195 256
1 Use of filters can help in obtaining frequencies as high as the ones implied by Eq. 36.
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Numerical Solution
The interaction of the previous parameters will be further exemplified by
analyzing the numerical representation of the frequency spectrum of the transfer
function given in Figs. 5 to 8. The effect that At and N have on (% and coy is shown
in the respective captions. The impedance was obtained by applying the FFT
algorithm to the theoretical discrete and uniformly spaced input and outputfunctions. These functions were generated by using the equations developed in
the previous section. The numerical representation of the frequency spectrum of
the impedance was compared with its analytical counterpart. As can be observed,
good agreement exists between the analytical and numerical solutions. The large
number of data points considered together with small At values contributed to
this correspondence.
One of the most critical choices in the FFT transformation is the one
concerning the point at which the waveform starts. The logic behind this is thatthe input and output waveforms have to considered as a trend, that is, as if they
repeat themselves periodically. Thus, in this case, the time t„ was chosen so as
to avoid the influence of the initial charging of the double layer. By doing this,
truncation errors were avoided and a "clean" frequency spectrum was obtained.
If time t is to be considered equal to zero, the waveform period has to be selected
between zero and tb. In addition to this, tb has to be chosen as large as possible
so that the double layer is fully discharged before current is applied back. By
imposing these limits the waveform can be considered to repeat itself afterwards.
Figs. 5 and 6 show the frequency spectrum obtained when different input
functions and system parameters are employed. By comparing the Argand
diagrams in these figures it can be seen that they are similar. This is to be expected
as the only electrical parameter that was changed was the capacitor value. The
decrease in this value is observed in the Bode plots as a displacement of the
frequency towards higher values. As can be observed, an order of magnitude
decrease in the capacitor value reflected in a tenfold frequency increase required
to define the frequency spectrum.
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Numerical Solution
(a) (b)0.006
0.2
0.005
CL
£
0.004
<
0.003
CDi _
•
0.002
O
0.001
t= 0.550
sec
t= 0.583 sec
t b= 1.000 sec
t= 1.600 sec
Rf 25.0
n
/?= 10.0 Q
c= 3000. yuF
0.55 0.76
0.97 1.18
Time, sec
(c)
1.39
c ncd
EN
l
* Analytical Solution
— Numerical Solution
20
Zreal, Q
0.15 -
>
'•+->CCD
oCL
0.1
0.05
1.6
0.55 0.76 0.97 1.18
Time, sec
(d)
1.39 1.6
G
N
35
34
33
32
31
30
29
28
27
26
25
4 0
10
X Analytical Solution
— Numerical Solution
Freq, rad/secFig. 5 Frequency response of the circuit depicted in Fig. 1 obtained assuming the parameters shown in (a).
The input (a) and output (b) functions were calculated according to Eqs. 4 and 17 respectively. These functionswere digitized using N=2048, At= 5.127x10"* sec.Plots
(c)
and (d) describe the analytical (Eq. 33) and numerical (Eq. 37) frequency spectra. For this set of data cop5.98 rad/sec, 0) =975 rad/sec (Eqs. 37 and 40 respectively)
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Numerical
Solution
0.006
0.005
0.004
- 0.003
CCDk_
k_
13
o
0.002
0.001
G
CO
EN
(a)
tcf 0.100 sec
t= 0.101 sec
t„= 0.110 sec
t e= 0.200 sec
Rf 25.0 fl
Rf 10.0 n
c = 300. yuF
0.1
0.12 0.14 0.16 0.18
Time, sec
(c)
20 25 30 35
Zreal. Q
>
-t-<CCD
oCL
0.2
0.1 0.12
* Analytical Solution
— Numerical Solution
A1 1
c • • • •
35
34
33
32
31
G30
N29
28
27
26
25
0.14
0.16 0.18
Time, sec
(d)
40
10
X Analytical Solution
— Numerical Solution
Freq, rad/secFig. 6 Frequency response of the circuit depicted in Fig. 1 obtained assuming the parameters shown in (a).
The input (a) and output (b) functions were calculated according to Eqs. 4 and 17 respectively. These functionswere digitized using N=2048, At= 4.883xl0'3
sec.
Plots (c) and (d) describe the analytical (Eq. 33) and numerical (Eq. 37) frequency spectra. For this set of dataC0(=62.83. rad/sec, co^10240 rad/sec (Eqs. 37 and 40 respectively)
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Numerical Solution
Figs. 7 and 8 show the effect that a different waveform period and sampling
interval have on the definition of the frequency spectrum at lower frequencies.
Increasing the waveform period from 0.03 to 0.10 sec decreased the value of co
but increased % due to the fact that At was increased. Better resolution of the
frequency spectrum at lower frequencies is reflected in a well defined Bode plot
shown in Fig. 8.d.
Figs. 5,7, and 8 were obtained using the same electrical analogue parameters
and only changes in the characteristics of the input function were studied. Plots
c and d in these figures show that regardless of the shape and magnitude of the
input and output functions, the transfer function remains identical provided the
same system is being analyzed.
The advantages of the Fourier transformation in the analysis of time-domain
transients can be visualized by inspecting plots c and d in Figs. 5 to 8. Thenumerical transfer function obtained matches the one derived analytically. If the
electrical circuit analogue had not been not known in advance, it could be inferred
from analysis of this frequency spectrum. Different circuits could be proposed
and their frequency spectrum calculated so as to match the numerical frequency
response. The numerical frequency response could then be curve fitted to the
theoretical circuit leading to the derivation of the values of the electrical
parameters.
Plots c and d of Figs. 6 to 9 also show that the information that AC impedance
and current interruption can provide is equivalent. This equivalence is due to the
fact that "lumped" time-invariant electrical parameters were considered. The
presence of distributed and/or time dependant parameters may give different
information from each technique as different systems maybe analyzed. If systems
where these parameters are present are to be studied, careful experimental
planning is required. Also, the information provided by each technique may be
used jointly to better characterize the system.
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Numerical Solution
(a)0.006
CL
£ 0.004 -
<
cCDv_• 0.002
O
_i_
0.100 sec
t
4
= 0.101 sec
t b= 0.108 sec
t= 0.115 sec
If 0.130 sec
Rf 25.0 n
Rf 10.0 fl
c = 300. fjf
0.1 0.105 0.11 0.115 0.12
Time, sec
(c)
G
d>CO
E
Nl
X Analytical Solution
— Numerical Solution
20
zreal, Q
0.15
>
CCD
-t—
oCL
0.125 0.13
0.05 -
G
N
0.11 0.115 0.12
Time, sec
(d)
0.13
40
Freq, rad/secFig. 7 Frequency response of the circuit depicted in Fig. 1 obtained assuming the parameters shown in (a).
The input (a) and output (b) functions were calculated according to Eqs. 18 and 23 respectively. These functionswere digitized using N=2048, At= 4.883xl0'5
sec.Plots
(c)
and (d)
describe the analytical (Eq. 33) and numerical (Eq. 37) frequency spectra. For this set of data cop209.44 rad/sec, 0 =34133 rad/sec (Eqs. 37 and 40 respectively)
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Numerical Solution
(a)
0.006
0.005
a£ 0.004
<-M " 0.003
cCDi _
• 0.002
o0.001
t 0= 0.100 sec
\ = 0.101 sec
f„=
0.108 sec
t
e
= 0.115 sec
frf= 0.2(90
sec
R,= 25.0 n
/?
2
= 70.0
0
c = 300. /xF
0.1 0.12 0.14 0.16
Time, sec
(c)
0.18 0.2
G
CC
4 -
N
* Analytical Solution
— Numerical Solution
20
G
N
35
34
33
32
31
30
29
28
27
26
25
0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18
Time, sec
(d)
40 10"'
* Analytical Solution
— Numerical Solution
Zreal, Q Freq, rad/secFig. 8 Frequency response of the circuit depicted in Fig. 1 obtained assuming the parameters shown in (a).
The input (a) and output (b) functions were calculated according to Eqs. 18 and 23 respectively. These functionswere digitized using N=2048, At= 1.465xl0'5
sec.Plots (c) and (d) describe the analytical (Eq. 33) and numerical (Eq. 37) frequency spectra. For this set of data cop62.80 rad/sec, 0^=10240 rad/sec (Eqs. 37 and 40 respectively)
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Summary
IV. Summary
Equations were derived to describe the time domain response to a current
step for a simple RC electrical circuit. These equations were used to generate
theoretical time-dependant discrete functions that were subject to DFT by using
the FFT algorithm The complex process of analyzing electrochemical data in the
time domain was simplified by analyzing it in the frequency domain. The practical
implementation of the FFT algorithm was discussed. The limitations and
application range of the time domain to frequency domain transformation were
also explored. It was stressed that provided the same system is analyzed, current
interruption and A C impedance can provide the same information.
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Appendix 4 Analyt ica l Chemist ry of E lec troly te Sol ut ions
Conta in ing P b S i F 6 - H 2 S i F 6
I. Objectives:
A) To describe the different techniques that have been reported in theliterature to analyze solutions containing H 2SiF 6.
B) To explain how a modified technique for analyzing H 2 SiF 6 solutions was
developed.
C) To describe the analytical technique used in the present study to analyze
H2SiF6-PbSiF6 electrolytes.
A. Chemical analysis ofH aSiF e solutions
1. Overview of the acid properties
The different techniques reported in the literature [1-81 for the analysis of
H 2SiF 6 solutions are based on the very special properties of this acid. Upon
dissociation, H 2SiF 6 behaves like a strong acid, its strength being close to that
of H 2 S 0 4 solutions 191. By potentiometric titration of H 2SiF 6 solutions in water
and in various organic solvents, the first and second dissociation constants of
the acid were determined [9]:
H£iF t <=> HSiFl + H + ...1
HSiFl <=> SiFf + H +
pK 2=\.19 ...2
The H 2SiF 6 molecule is known to exist only in aqueous solutions [1,7,10-13].
On evaporation, H 2SiF 6 solutions decompose into SiF 4 (g) and HF (g) [7,10]l . The
maximum concentration of the acid that has been found in liquid solutions is
61% [3,101.
1 Solutions with more than 13.3% H
2
SiF
6
become enriched with HF due to preferential evaporation of
SiF
4
. Reciprocally, SiF
4
solution enrichment takes place upon evaporation of solutions with less than
13.3% H
2
SiF
6 [7,10].
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Techniques for the analysis of HJSiF, solutions
The fluorine to silica ratio 1 in H2SiF 6 solutions is not always equal to six p
[11.14-16]. ^ ratios larger than 6 are observed in solutions in which HF is present.
Solutions with j. ratios lower than 6 are observed when the acid has an excess
of S i0 2 n H 2 0. Alarge variety of compounds containing silica can be present under
these conditions. Among these, conductometric and cryoscopic measurements
have shown that the solubility of Si0 2 in H 2SiF 6 is due to the formation of
SiF 4-2H 20 [151. This compound behaves like a strong acid [16]. The | number
depends on the technique used to produce the acid [11,14,17]. j. ratios can also
be changed by dilution [18].
In dilute solutions, hydrolysis of SiF6"2 takes place in different steps which
involve intermediate splitting of one F" and formation of SiF 5[H 20]' (reaction 3).
This compound losses its F" atom in a second step forming S iF 4 which then
hydrolyses towards H 2 Si0 3 (reactions 4 and 5 ) [5,14,19-21]:
SiF;2 +2H 20 o SiF 5 [H 2OY " + F~ ...3
SiF 5 [H 20]~ +H 20 o SiF 4 + F~ +2H 20 ...4
SiF 4 + 3H 20 <=> H^iOs + HF ...5
In very dilute solutions, SiF6"2 is completely dissociated
[20.22]
2.This has
lead to the use of fluorosilicates (i.e. NajSiFe) for the fluoridation of water supplies
[23,24]. Data on the physico-chemical properties of H2SiF 6 solutions are reported
extensively in the literature [18,22,24-30].
2. Techniques for the analysis ofH 2 SiF 6 solutions
H 2SiF 6 solutions have been analyzed using titrimetric techniques [3,6,8].
The acid is titrated against alkaline solutions such as NaOH or LiOH. During
the titration, two equivalence points can be observed. The first equivalence point
corresponds to the neutralization of the acid and the second equivalence point
corresponds to the hydrolysis of the hexafluorosilicate ion. The two equivalencepoints can be described by the following reactions:
1 In H
2
SiF
6
solutions the fluorine to silica ratio is defined as:
F 6[//
g
SiF
6
]
+
[//f]
Si ~ [HJIFJ + lSiOJ
2 S iF
6
2
is an
octahedral
ion.
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Techniques
or he analysis of H£iF, solutions
H 2 SiF 6 + 2/Va 0/7
-»
Na2 SiF 6 +H 20 ...6
Na2 SiF 6 + 4NaOH -» 6NaF + H^iO^+H 20 .. .7
Reactions 6 and 7 indicate that the amount of alkali required in the second
equivalence point should be equal to twice the amount used for the neutralization
of the acid. Disagreement in these values indicates the presence of other species
in solution. The presence of H F and soluble SiOa compounds in H
2SiF
6 solutions
have accounted for these discrepancies. In addition to this, it has been found,
that the hydrolysis of S1F6"2 is kinetically slow [201. To circumvent these
limitations, various analytical techniques have been proposed [3,81. In these
techniques, a two-step titration routine has been used to analyze the acid for
H2SiF
6, and either Si0 2 or HF. The first step in this routine consists in adding
a suitable salt to precipitate the hexafluorosilicate ions. When this salt is chosen
to be a fluoride salt, SiOa can be determined in the same titration. Thus, if NaF
is added to the H 2SiF 6 solution, the following reactions take place:
H 2 SiF 6 + 2NaF -»Na2 SiF 6 1+2HF ..
.8
6HF + Si02 -»H
2 SiF
6 + 2H 20 .. .9
Reactions 8 and 9 show that by adding an excess of NaF, all the
hexafluorosilicate ions are present as Na 2SiF 6. If any Si0 2 were present, it wouldreact with HF according to reaction 9. The total amount of HF is obtained by
titrating the solution at low temperatures [io]1 using NaOH:
HF +NaOH ->NaF +H 20 ...10
Reaction 10 reaches completion at a pH close to 8.
Once this first step is completed, the sample is diluted with hot water and
titrated near its boiling point2. This dilution and solution warming brings all the
hexafluorosilicate ions into solution allowing reaction 7 to take place. Thisreaction reaches completion near pH 10. From the stoichiometry of reactions 7
to 10, the acid composition can be found. If the amount of alkali spent during
1 This titration has to be carried out at low temperatures (near
0
*C) to avoid dissolution of sodium
hexafluorosilicate. The solubility of Na
2
SiF
6
is: 6.52 g/l at 17°C and 24.54 g/l at 100*C
.
2 Titration of the solution near its boiling point aids
in
the kinetics of the hydrolysis reaction.
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Background
the second titration was larger than twice as much the amount spent in the first
titration, HF was present in the sample. A reciprocal relation will indicate that
Si0 2 was present in the solution.
Experimentally this two-step titration technique is difficult to perform. The
main difficulties in the implementation of the technique are the cooling andheating operations which have to be performed carefully. Dilution of the samples
can be a source of error as some unaccounted fluorosilicate hydrolysis can occur.
Additionally, any alkali added in excess during the first titration will react with
the hexafluorosilicate ions. Jacobson [3] points out that in addition to the
temperature of the solution during the first titration being very low, the solution
has to be concentrated so that the reaction products of the first stage do not
pass into the second stage before the first stage reactions are completed.
Notwithstanding these difficulties, this technique provides accurate acid analysis
and its use is frequently reported in the literature 15,181.
3. Analysis of H 2 SiF 6 -PbSiF 6 electrolytes
Analysis of the electrolyte used in lead refiriing has been carried out by
modifying the two-stage titration previously described [7,311. In the analysis of
H 2SiF 6-PbSiF 6 electrolytes, Pb has to be removed before the acid is titrated. Use
of Na 2S0 4 to separate Pb from the electrolyte was discarded due to potential
co-precipitation of NaaSiFg with PbS0 4 [311. To avoid this loss of SiF 6~
2
, knownamounts of H 2 S 0 4 are added to the sample to remove Pb + 2
as PbS0 4 (see reaction
11). The precipitation of PbS0 4 releases an equivalent amount of H 2SiF 6 that
can be subsequently titrated using the two-stage titration previously described.
PbSiF^+H^O^ H^iF^PbSO^ i ...11
B. Modification of existing experimental procedure
1. Background
The two-step titration technique was modified so as to determine, in a single
step, H 2SiF 6, and either Si0 2 or HF .The approach that was taken was to titrate
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Straight titration
against
LiOH
in the
presence
o f
excess
SK),.
the acid against LiOH adding LiF and/or HF. H 2 SiF 6 solutions were titrated at
room temperature using an automatic titration system1. The same H 2SiF 6
solution (sample SI a) was titrated under the following conditions:
Case A: Straight titration against LiOH in the presence of excess Si02.
Case B: Titration against LiOH in the presence of excess HF.
Case C: Titration against LiOH in the presence of excess HF and LiF.Case D: Titration against LiOH when LIF is added previous to the titration.
Each of these routines is explained in the following paragraphs.
(A) Straight titration against LiOH in the presence of excess Si0 2 .
When H 2 SiF 6 solutions are titrated against LiOH, the following reactions
take place:
H 2 SiF 6 + ILiOH -> Li 2 SiF 6 + H 20 ...
12
Li 2 SiF 6 + 4LiOH ->•6UF +
H£i03 + H 20 ...
13
Li 2SiF 6 is the most soluble of all the alkali fluosilicates HO]3. Thus, upon
neutralization of the acid (reaction 12), all the SiF 6"2 remains in solution. As
these ions are already in solution, heating and dilution preceding hydrolysis
via reaction 13, are not required. On the other hand, the hydrolysis reaction
has to be carried out very slowly to circumvent its kinetic limitations. Fig. 1 .A
shows the titration curve 4 obtained under these conditions 8 . Two equivalence
points are present during the titration of the acid. The first equivalence pointtakes place at a pH between 3.5 and 4 and corresponds to reaction 12. The
second equivalence point takes place at pH values between 7.7 and 8.4 and
corresponds to reaction 13. While [H2SiF6] can be calculated from the total
amount of LiOH spent in the titration, [Si02] cannot be quantified using this
technique 6 > 7 .
1 Radiometer
end-point titration
system model ETS 822.
2 Acid composition
2.03 M H
2
SiF
6
and 0.40 M Si0
2
. All samples were titrated against 0.8657 M LiOH.
3 The solubility of LiSiF
6
-2H
2
0 is
730 g/l,
T=17'C
.
4 Titration
curves
are shown primarily to indicate
where
equivalence points are observed. Particular
characteristics of these curves may
vary
depending on
the experimental conditions.
5 0.5
ml
of the acid sample S1 was diluted to 20
ml
with
deionized
water and titrated using 0.8657 M
LiOH.
6 The presence of Si0
2
in
the sample will be indicated
by
the difference in alkali volumes spent
during
the
first and second equivalence points.
7 [H
2
SiFe], [PbSiF
6
], [HF],
[SiOJ
=
concentrations of the indicated species in
mol/l,
[ M].
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Titration
against
LiOH
in th e
presence ol excess HF.
CB) Titration against LiOH in the presence of excess HF.
The sample analyzed in Case A showed that SiOa was present. To quantify
it, a known amount of HF was added to the sample1.
HF reacts with Si0 2 according to reaction 9 producing H 2SiF 6. Upon
titration of the acid, any HF in solution is neutralized according to:
HF + LiOH -»LiF + H 20 ... 14
Fig. 1 .B shows the titration curve obtained in a sample containing H 2SiF 6
and HF. This curve shows that reactions 12 and 14 take place very close to each
other (at a pH between 3.1 and 3.8). Furthermore, hydrolysis of SiF6"2 takes
place at a pH between 7.7 and 8.4 (reaction 13). From the alkali volumes spent
to reach the first and second equivalence points, [H2SiF6] and [HF] can be
calculated. By deducting the amount of H2SiF 6 released during reaction 9, [Si02]
could be obtained.
(C) Titration against LiOH in the presence of excess HF and LiF.
To differentiate between reactions 12 and 14 during the first neutralization
reaction, LiF was added to the original acid sample. Additionally, HF was added
to this solution 2 .
LiF reacts with H 2SiF 6 according to the following relationship:
H 2 SiF 6 + ILiF Li 2 SiF 6 + 2HF ... 15
As reaction 15 shows, by adding LiF to the sample, only HF and Li 2SiF 6
are present in solution. Thus, only two well-defined equivalence points are
expected upon titration of this sample. As Fig. l .C shows, these points are well
defined, and agree with reactions 13 and 14. Accurate acid analysis can be
obtained by analyzing this titration curve.
1 2 ml of 1.127 M HF were added to 0.5
ml
of the acid sample S1. Afterwards the solution was diluted to
20 ml using deionized
water.
2 2
ml
of 1.127 M HF were added to 0.5
ml
of the acid sample S1. Afterwards the solution was diluted to
20
ml
using deionized
water.
To
this
solution
=
0.5 g of
CP
LiF was added.
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Titration against LiOH in the presence of excess HF and LiF.
( A ) (B)
, i , . . . i . . . i . . . i. . i 11 i i . i 11.. 11 , i i . i
0 1 2 3 4 5 6 7 8 9
10 0 1
2 3 4 5 6 7 ^ 8 9
10
mM LiOH mM LiOH
F i g . 1 Cha n g e s in p H durin g the titration of a H 2 S i F 6 solution against L i O H (Sample SI, Table 1 ) .
( A ) Case
A : Straight titration
(B) Case
B: A d d e d H F
(C)
Case
C : A d d e d H F + L i F
(D) Case D: A d d e d L i F ( L i F - L i O H Technique).
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Titration against LiOH when LiF is
added
previous to
tie
titration.
(D) Titration against LiOH when LiF is added previous to the titration.
To the original acid sample, LiF was added and the solution was titrated
against LiOH1
. The two equivalence points described by reactions 13 and 14
can be observed in Fig. I.D. As in the previous case, accurate analysis of the
solution can be obtained by analyzing the titration curve.
Comparison of the different analytical routines:
The results obtained by analyzing the same acid sample using the routines
previously described are shown in Table 1. Similar concentrations can be found
by either of these techniques. The technique described in C a s e D
(LiF-LiOH
technique) proved to give accurate acid analysis and was used extensively in
this work. This technique was used to analyze H 2 SiF 6 within ± 1.5 g/1, SiOa
within ± 1 g/1, and HF within ± 1 g/1. When PbSlF 6-H 2SiF 6 solutions were
analyzed by this technique, Pb was removed prior to the acid analysis by using
Li 2 S0 4 :
PbSiF 6 +Li£0 A ->
Li 2 SiF 6 + PbS0 4 i ... 16
The amount of L i 2SiF 6 released by Li 2 S0 4 can be calculated by knowing the
Pb content of the sample. Pb is quantified in a separate sample by using EDTA
complexometric analysis 1321.
Table 1 Comparison of the different analytical techniques.
Technique [HaSiFd. M [Si02l. M [HF], M Comments
Two-step
titration:
Cold-Hot titration
2.03 0.40 Quantification of HF or SiO a
present in the sample is
possible.
One Step U OH Titrations :
Case AStraight titration
2.04- -
Only the total H2SiF
6 content of
the sample can be quantified
Case B:
HF added.
2.05 0.44 Quantification of HF or Si02
present in the sample ispossible.
Case C:
HF and LIF
added.
2.02 0.40 Quantification of HF or Si0 2
present in the sample is
possible.
Case D:LiF added.
2.02 0.40 Quantification of HF or Si0 2
present in the sample is
possible.
1 0.5 ml of the acid sample S1 was diluted to 20 ml using deionized water. To this solution = 0.5 g of CP
LiF was added.[319]
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Analytical routine
C. H 2 SiF 6 and HF or Si0 2 analysis in H 3 SiF e-PbSiF 9 electrolytes (IAF-LiOH
Technique)
1. Analytical routine
Pipette a 2-ml aliquot of the sample into a plastic centrifuge tube \ add
approximately 10-ml of « 0.2 M Li 2 S 0 4
2 . Mix well, then centrifuge for 5 minutes.
Decant the supernatant liquor into a 50-ml beaker. Wash the solid residue in
the centrifuge twice with 2-3 ml of distilled water, and add the washings to the
solution In the beaker. Add ~ 0.5 g of powdered CP LIF. Stir well and titrate the
mixture with LiOH 3
up to a pH of 3-4 [first equivalence point). Record This volume
as V\. Continue the titration adding LiOH at a very slow rate4 up to a pH of
6.8-7.3. Record this volume as V 2 [second equivalence point).
To titrate the acid sample, use of an automatic titration system is highlyrecommended. In this set-up LiOH addition rates can be set to very low speeds.
Also, the equivalence points can be easily determined by obtaining the derivative
of the titration curve.
2.
Reactions
As described previously, the reactions that take place during the LiF-LiOH
titration are as follows 5 :
(a) Neutralization of the lead salt:
PbSiF 6 +LJ'2S04 -> Li 2 SiF 6 + PbS04 i . . . 16
(b) Addition of LiF and reaction of HF with Si0 2 (if present):
H 2 SiF 6 + 2L /F -» Li^iFf, + 2HF . . .15
6HF + Si02 -» H 2 SiF 6 + 2H 20 . . .9
(c) First neutralization point pH= 3-4:
HF+LiOH ->LiF+H 20 . . .14
1 Plastic tube volume larger than 10 ml.
2 Add
enough
L i 2 S0 4 so as to neutralize all the lead salt present
in
the electrolyte (reaction 15).
3 LiOH
=
0.9 M, known titer.
4 LiOH addition rates smaller than 0.5 ml/min.
5
[Pb]
=
Lead concentration
in
the sample as obtained in a separate analytical routine, M .v
sanpi« = Volume of the acid sample, ml.
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Calculation
ol results
Record the volume of LiOH spent as Vi and obtain the moles of LiOH required
in this titration: mx = V x * [LiOH].
(d) Second neutralization point pH - 6.8-7.3
Li 2 SiF 6 + ALiOH -> 6LiF +H 2 Si01i +H 20 .. . 13
Record the volume of LiOH spent as V 2 and obtain the moles of LiOH required
in this titration: n\ = V 2 * [LiOH].
3. Calculation of results
HF is present when: 2V,> V 2 . Compute [HF] and [H2SiF6] as follows:
[H 2 SiF^=-/^—-[Pb]"sample ^
[HF]=-^--2*[H 2 SiF 6 \^sample
Si0 2 is present when: 2 V x < V2 . Compute [SiOa] and [HaSiFg] as follows:
2*[//25/F6]-1''h
[SiO j = —
The t o t a l fluorine to silica ratio is derived from the following relationsh
F _ 6{[H 2 SiF 6 ] + [PbSiF 6 ]} + [HF]
Si " [H 2 SiF
6 ] + [PbSiF 6 ] + [5/OJ
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Appendix 5 Comp uter Interfacing of the We nk in g Poten tios tat:
Cal ibrat ion of the R out in es used to Interrupt the Current
I. Objectives:
A) To explain some of the details involved in the computer interfacing of theWenking potentiostat.
B) To describe the algorithm used to interrupt the current.
C) To demonstrate how by using "dummy" cell circuits the current
interruption routines were checked.
D) To analyze the experimental results obtained in the time domain in the
frequency domain by using the FFT algorithm.
A. Current interruption using the Wenking potentiostat
In Chapter 3 the various connections required to interface the computer to
the potentiostat were described (see Fig. 3.6). Current was halted by using a
mercury wetted relay activated by the computer. This relay short circuits the
battery that controls the amount of current that goes through the electrochemical
cell. To activate this relay, a DC voltage of at least 3.5 V is required. Voltages
smaller than 0.5 V deactivate the relay allowing current to flow back to the cell.
The easiest and fastest way of controlling the relay operations was found to be
through the use of the digital output of the DT2805 board 1 1 2 . By setting the output
value of the digital bits to 1, the relay was activated and the current halted3
.
Accordingly, resetting this value to zero opened the short circuit imposed by the
relay, allowing the re-establishment of the current flow.
During the galvanostatic experiments, at least two data acquisition channels
were used 4
. One of these channels was used to record the difference in potential
between the reference and working electrodes. The other was used to monitor the
current flow to the cell. As Fig. 3.6 shows, all the data logger channels were
1
When
a
digital
value
of ("high" bit status), is output
a
voltage of
=2.5
volts is generated, and
when its
value
is zero
("low"
bit statusltne voltage output
value
drops to
=0.2
V. This signal goes back and forth
in
the shape of
a
pulse.
2 To generate the required voltages to activate the relay three digital output bits
were
connected in series.
3 Oscilloscope readings showed
that
current was halted within 10 usee.
4 Additional channels
were
used
when
the difference in potential between the inner slimes electrolyte and
the working electrode was measured.
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Algorilhm used lo interrupt the current
grounded with respect to the working electrode. This aided in preventing common
mode voltages. Additionally, common mode voltages were avoided by not making
direct connections between the counter electrode and the DT2805 board.
B. Algorithm used to interrupt the current.
Concurrent interruption of current at preset times under galvanostatic
conditions was implemented by the hardware connections shown in Fig. 3.6. In
addition, appropriate software programs were developed. The characteristics of
these programs varied according to the number of channels sampled, the length
of the experiment, and the extent and frequency of the current interruptions.
The computer programs were written so as to handle up to a maximum of 8
analogue input channels. Additionally, control of the digital bits used to generate
the voltages required to control the relay's functions was incorporated in theseprograms. Thus, data acquisition and setting and resetting of digital bits were the
tasks considered in the program's algorithm. In this algorithm, the speed at which
these operations took place was the most important parameter accounted for.
The different operations performed by the computer were programmed by
using the "foreground/background" operation mode. In this mode, all the tasks
performed by the DT2805 board are executed in the "background" while the
computer is free to do other operations in the "foreground". The rate at which the
background operations run is set by fixing the task period. Additionally, thefrequency at which each task runs is controlled by fixing the task modulo.
Moreover, tasks may start synchronously but they may not be instantaneously
activated. The task phase controls how many task periods have to take place
before the task is first executed. Furthermore, completion of a specific task may
activate/deactivate other tasks. The state of each task may also be altered from
the "foreground" operating mode. Also, tasks become idle, once the number of
times the task has been executed equals a preset value .This preset value is defined
as the number of iterations.
Table 1 shows the task set-up used to control the DT2805 operations. Typical
values for the task modulo, task phase, and number of times the task is executed
are provided in this table. The order in which each task is defined is important
as tasks are executed sequentially rather than simultaneously. Every task period,
the status of each task is checked. If the task status is idle, the task is skipped
and the next task is executed.
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Algorithm used
to interrupt the current
Table 1 Tasks description and typical parameter values.
Task Fu n c t i o n Descr ipt ion Task
Modulo
Task
Phase
Nu m be r o f
Iterations
T a s k Status
T y p i c a l valu es wh en tas k
peri od =4 msec
1 Set bits Hal ts f low of cur re nt to the
11111111 llll ill ^^^^9 9 1
Tasks s imul taneous ly
activa ted by Ta sk 4
These tasks
con t ro l the cu rre nt
Interrupti on cycle
2 Reset bits Re-es tablishes flow of current
to the cel l
50 50 1 Tas ks s imul tane ously
activa ted by Ta sk 4
These tasks
con t ro l the cu rre nt
Interrupti on cycle
3 Acqui re data T r a n s ie n t R e c o r d e r Mode".
Logs data conti nuousl y
befo re , aft er , a nd d u r i n g
current in terruption.
1 1 100
Tasks s imul taneous ly
activa ted by Ta sk 4
These tasks
con t ro l the cu rre nt
Interrupti on cycle
4 No oper ati on Onc e activa ted it starts the
curre nt Interrupti on routine
b y s ta rt in g Ta sk s 1, 2, an d 3.
1 1 OO "Foreground" activated
once Task 5 buffer is
f i l l ed . "Background"deactivated once
Task 3 is completed
5 Acqui re data "Data Logger Mode". Logs data
d u r in g normal galvanostatic
conditions . Thi s task controls
the curre nt in te rrupt ion
frequency and the experiment
length. Every cert ain nu mbe r
of acquired points Task 4 is
activated.
2500 0 250 00 390 0 Alway s active
As was previously described, all tasks run at the same rate. This rate is fixed
by the value of the task period which for most of the programs was set to 4 msec.
The frequency at which tasks are executed is fixed by the task modulo. Thus, for
example, Task 5 was used to acquire a set of data points every 100 sec (i.e. 25000
times 4 msec). These data were sequentially put in a buffer array. The size of this
array was selected to control the current interruption frequency. Thus, every time
the buffer array was filled, data were saved in the computer's hard drive, andTask 4 was activated. Once Task 4 is activated, Tasks 1, 2, and 3 start
simultaneously. These three tasks are executed sequentially. However, Tasks 1
and 2 start only after a certain number of tasks periods (or clock-ticks) have gone
by, whereas Task 3 is executed immediately. Task 3 acquires data at the highest
acquisition rate set by the task period (in this case at 250 Hz). As Table 1 shows,
100 data points are collected at this acquisition rate (i.e. during 400 msec). After
9 data points have been collected, Task 1 sends a high bit and halts the flow of
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Calibration
of the
routines
used lo interrupt the current
current. As this task is executed first, there is a time lag between the current
interruption and the acquisition of the tenth set of data points. This time lag is
due to the fact that the computer has to check Task 2 status before executing
Task 3 (i.e. there is a software overload). This time lag was measured with an
oscilloscope and was of the order of 1 msec. The current interruption length iscontrolled by Task 2. Task 2 phase is 50, thus, current was interrupted during
41 clock-ticks (i.e. during 164 msec). After this time, Task 2 sends a low bit and
re-establishes the flow of current. One niillisecond after the short circuit is opened,
data are recorded. Subsequently, samples are taken every 4 msec during a period
of 196 msec. Once Task 3 is completed, Task 4 is deactivated and the data are
saved in the computer's hard drive. As Task 5 is active all the time, the previous
process repeats as often as Task 5 buffer is filled. The data acquisition program
ends when task 5 becomes idle (i.e. after 3900 data points or 108.33 Hrs.).
During the data acquisition run, an interactive procedure for changing the
A / D gain was incorporated in the computer program \ Thus, the DT2805 board
measured potentials lower than 100 mV with an accuracy of ±0 .3 mV. Potentials
between 100 and 1000 mV were measured within ±1 mV.
C . C a l i b r a t i o n of the routines used to interrupt the current
In addition to using an oscilloscope to check the input-output operations
performed by the DT2805 board, "dummy" cell circuits were also used. A typicalRC circuit equal to the one shown in Fig. A3.1 .A was assembled out of commercial
components. Capacitors and resistors of known values a were used to test the
current interruption routines. The nominal value of the components was
compared with the obtained empirically.
Fig. 1 shows the empirical potential and current waveforms obtained by
current interruption. The theoretical values of the elements of the circuit are
shown in the figure. Also, the points at which samples were taken are marked in
the plots. In all the cases, current was interrupted after applying current to the
circuit for several minutes. Thus, the initial charging of the capacitor is not shown
1
The
DT2805
board performs analogue to digital (A/D) conversions at
a
12-bit resolution. Four input
voltage ranges are
available in
the
board: -20
to 20 mV,
-100
to 100 mV,
-1000
to
1000
mV, and -10000
to 10000 mV. These
ranges
can be interactively changed by software control improving the resolution of
the acquired data.
2
Resistors
were measured independently by using a Tech-300 digital
voltmeter.
Capacitor values were
obtained by using AC impedance techniques (see Appendix
6).
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Calibration ol the routines used to interrupt the curre
(A) (B)
Time,
msec
Time,
msec
Fig. 1 Transients obtained upon current interruption using a dummy cell
Experimental data: R t = 15.1 £2, R 2 =10.2 Q, c = 3147 uF.
Theoretical data derived from analysis of the potential decay: Rx = 15.1 Q, R 2 =10.1 fi, c = 3140 uF.
(A) Current as a function of time(B) Potential as a function of time
in these transient curves. As Fig. 1.A shows, current is interrupted almostimmediately after sending a "high" bit from the computer. On the other hand,
several milliseconds are required for the current to recover its previous value once
the short circuit is opened.
The response of RC circuit upon current interruption was studied using the
equations developed in Appendix 3. Eqs. A3.24 to A3.26 were used to obtain the
values of the components of the electrical circuit. The potential upon current
interruption was fitted to Eq. A3.26. From this equation the Ra and c values were
obtained. Extrapolation of this curve at time zero provided the R x value.
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Calibration of tie routines used to interrupt the current
T a b l e 2 Comparison between experimental and theoretical RC componen
values obtained by the interruption technique
Theoretical values Expenmental Values Regression No. points
mAmp Ra, Q C, \lF R x, Q R* n C, uF Coeff., r 2
Considered
4.88 10.3 0
- 10.2 0.0
- - 1
4.89 15.1 0
- 14.9 0.0
- - 1
0.97 10.3 2.3 3147
10.0
2.3 2952
0.984
3
4.93 10.2
2.1
2863 0.991
5
0.95 10.3 4.9
3147
10.1 4.9
3062 0.995
4
4.90 10.2 4.8 2879 0.993 4
0.49 15.1 10.5 3147 149 10.5 3289 0.998
10
0.99 15.0 10.3 3207 0.998
10
2.46 15.1 10.2 3178 0.999
10
2.45 15.1 10.2 3140 0.999
10
4.91 15.1 10.2
3146
0.999
10
24.61 15.1 10.1
3140 0.999 10
0.97 10.3 2.3
1117 10.0
2.3
1339 0.969
3
4.91 10.1 2.2 1022 0.993 4
0.96 10.3 4.9 1117 9.8 5.2 938 0.989 3
4.90
1111161114.9 913 0.993 4
0.50 15.1 10.5 1117 iiii;4
:
i9iii 10.4 1115
0.992
6
0.99
15.0
10.3 1053
0.994
6
2.46
15.0
10.2
1026 0.994
6
2.45 15.2 10.1 1017 0.997 6
4.93
15.0
10.3 992
0.996
6
24.65 15.1 10.2 1005 0.995 6
0.50 15.1 10.5 225 14.4
II?233 Ii 0.985
3
0.99
13.7 11.6
203
0.990
4
2.46 13.7 11.6 177 0.996 4
2.45 12.2 12.9 133 1.000 3
4.91 13.2 12.2 151 0.998 4
24.65
13.6 11.6
159 0.999
5
4.89 10.3 2.3 11.7 12.2 0.0
***0.842
3
,0.20
10.3 4.9 11.7
14.8
0.2
•**0.647 3
4.83
14.9 0.1
***0.647 3
0.50 15.1 10.5
11.7 24.7
0.7
***
0.001 3
0.99
24.9
0.4
***0.800
3
2.47 25.1 0.2
***0.707
3
4.93
25.0
0.3
***0.997
4
24.63 25.2
0.1
***0.998
4
0.49 15.1 10.7 7.8 24.7 0.7
***0.006 3
0.98 25.0 0.3
***-0.004 3
2.45 25.3 0.0
***-0.004
3
24.61 25.2 0.0
***-0.500 3
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Tune domain to frequency domain tans formation: experimental resul
Table 2 shows the values of resistors and capacitors obtained for different
current levels. The theoretical and experimental values of the components of the
electrical circuit are shown in this table. The first two rows of data show the circuit
parameters obtained when a single resistor is used as a dummy cell. In all the
other cases, the circuit shown in Fig. A3.1.A was used. The number of pointsupon current interruption used to fit the data to Eq. A3.26 is also presented in
Table 2. The parameters used in the current Interruption routine to obtain these
data sets were the same in all the cases fTable 1).
Table 2 shows that as long as the time constant 1 of the electrical circuit is
larger than IQ msec, accurate values of ths. parameters of the circuit can be
obtained. These large time constants are present in the sets where the 3147 and
1117 |iF capacitors were used. Circuits in which smaller time constants were used
did not provide accurate parameter values. In these circuits, only the total
resistance value (R x + Ra) can be obtained. In both cases, the amount of steady
state current going through the circuit does not seem to have an effect on the
calculation of the parameters of the circuit.
D. Time domain tofrequency domain transformation: experimental results
In Appendix 3 the fundamentals behind the time domain to frequency domain
transformation were presented. Here, FFT techniques will be used to Fourier
transform the time domain transients presented in Fig. 1.The FFT algorithm requires the data to be given as sets of 2q
(q being a positive
integer number) equally spaced data points. Thus, to satisfy these requirements,
the transient functions shown in Fig. 1 were modified. This was done by curve
fitting these functions using high order polynomials over different time intervals.
Once the transients were curve fitted, they were reconstructed using a uniform
sample interval equal to 1 msec. By doing this, the functions shown in Fig. 1 were
represented by 400 points rather than by the original 100 points. To increase the
range of the frequency spectrum and to have the required 2 q data points, the
transient functions were extended. Thus, a total of 4096 points were Fourier
transformed. The impedance of the system was obtained by Fourier transforming
1 The
time
constant
for the RC
circuit
shown
in Fig. A3.1 .A can be
expressed
by:
x
(sec) = RT C
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Time domain b frequency domain transformation: experimental res
the output (potential transient) and the input (current transient) functions and
taking their ratio. The results of this transformation are shown in Fig. 2. The
theoretical impedance spectrum is also shown in this figure.
(A) (B)
Z r , a l , Q Freq, rad/sec
Fig. 2 Frequency domain representation of the time domain data presented in Fig 1.
The analytical solution was computed using the known parameter values: R t - 15.1 f2 ,R 2 = 10.2 i>,c = 3147 uF.The numerical solution was obtained by applying the FFT algorithm to the experimental data presented in Fig. 1,from which the following values were obtained: R, = 15.3 £2, R 2 =9.9 £2, c = 2864 uF.
(A) Argand Plot(B) Bode Plot
Fig. 2 shows that the circuit parameters derived from the experimental data
are within 10% of the theoretical values. These values are not as good as the ones
obtained by time domain analysis. However, if the electrical analogue had not
been known in advance, Fourier transformation would have shown some of the
characteristics of the circuit. Thus, time domain and frequency domain analysis
can be used concurrently to study and verify experimental data. As the informationprovided by each technique is given by only changing the frame of reference (time
<=> frequency), the data can be interpreted from different perspectives.
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Appendix 6 Current Interruption and AC Impedance
Measurements us ing the Sola rtr on Dev ices
I. Objectives:
A) To describe how the Solartron Electrochemical Interface (SEI) performs
the "sampled" current interruption procedure.
B) To describe the implementation of the AC impedance measurements using
the SEI and the Frequency Response Analyzer (FRA).
A . Current interruption using the SEI
The SEI offers three ways of compensating for the value of IRg *. Two of them
(Feedback and Real Part Correction) require the value of the uncompensated ohmic
resistance, Rg, to be known in advance. The tiiird of these procedures, the"sampled" IRg drop compensation, determines Rg by using a current interruption
routine. Use of this routine involved the quantification of the potential and current
values during current interruption conditions. These values are different from the
ones observed under steady state conditions as the SEI interrupts the current
continuously at frequencies as high as 18.5 KHz. Thus, by executing the "sampled"
interruption routine, the following parameters are obtained:
a) Factual This is the value of the potential actually being applied to the cell
under current interruption conditions.
b) C a c t u a l : This is the value of the current actually being applied to the cell
under current interruption conditions.
C) PsamPie+hoid> This is the value of the potential 5 usee before current is applied
back to the cell.
From these parameters the value of the uncompensated resistance is
obtained using the following equation:
_ (Factual Psample + hold)
The current interruption procedure was tested by using the dummy cell
provided by Solartron 2 . The routine was applied under potentiostatic and under
galvanostatic conditions. A 18.5 KHz current interruption frequency was found
1 This IRg value corresponds to r\a described in Chapters 2 and 4.
2 Dummy Cell circuit 12861
ECI
test
module
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Implementaton of the AC measurements
to give the best results. The Rg values obtained from this technique were witiiin
5%. of the theoretical values. Table 1 shows some of the Rg values obtained under
potentiostatic conditions using the Solartron dummy cell.
T a b l e 1: Parameters obtained by using the "sampled" IRg compensation
routine under potentiostatic conditions. Theoretical Rg value = 1800 ohms
Values obtained under
steady-state conditions
Values obtained under current
interruption conditions
Experimental
value
Potential
mV
Current
uAmp
p A actual
mV
p1 sample+hold
mV
p^actual
uAmp
Experimental
value
200 20.86 234.0 192.3 24.32 1714
500 52.04 593 483 61.67 1784
B. Implementation of the AC measurements
In a typical AC experiment, a low amplitude sinusoidal voltage waveform is
generated by the FRA and sent to the SEI. Depending on whether the SEI is set
to operate under potentiostatic or galvanostatic conditions, the waveform is
superimposed either on the DC potential or on the DC current being applied to the
cell \ The variation in current and potential measured by the SEI is fed back to
the FRA which obtains the transfer function of the system. This is obtained by
repeating this process for a range of frequencies. From this information, plots
such as those shown in Figs. 1 and 2 are obtained. In these plots, imaginary
numbers are required to express the transfer function of the system. In a Bode
plot (Figs. 1 .B and 2.B) the frequency of the sinusoidal waveform is plotted on the
abscissa, whereas the absolute value of the impedance is plotted on the ordinate.
In an Argand diagram, the real and the negative parts of the imaginary impedance
are plotted on the ordinate and in the abscissa respectively (Figs. 1.A and 2.A).
From the analysis of these plots, kinetic and mass transfer parameters can bededucted when electrochemical systems are studied.
1 If
the system is controlled under galvanostatic conditions, the voltage waveform input by the FRA
is
converted to
a
current waveform by the
SEI.
This current signal is
superimposed
on the
DC current being
applied to the cell. The amplitude of this superimposed signal is obtained by dividing the sinusoidal
voltage amplitude by the value of the resistor used
in
the
SEI
to measure current.
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Implementation ofthefc measurements
(A) (B)
eooo c,Hh
c2
Hh7000 OWi
6000
-
5000
<g
E
MI
4000
3000
2000
1000
•
' • • •
•
AKMX) 2500 4000 5500 7000 8500 10000
Zrea|, ft
M
4000
2500
Fig. 1 Impedance diagrams generated using the Solartron Dummy cell
R, = 1800 Q, R 2 = 1000 ii , R3 =6800 ft, c, = 0.1 uF, =4.7 uF
(A) Argand Plot (B) Bode Plot
\
10° 10' 10* 103 104
Freq,
rad/sec
10s
To illustrate the use of the AC impedance technique dummy cells were
assembled. The impedance of these circuits was measured using the experimental
set-up shown in Figs. 3.7 and 3.8. The frequency spectrum was obtained under
potentiostatic control using 5 mV R.M.S. sinusoidal waveforms. Frequency was
swept between 1 and 60000 Hz. Figs. 1 and 2 show the Argand and Bode plots
obtained experimentally. These curves were analyzed following the procedure
described in Appendix 3, Fig. A3.4.
The curves shown in Fig. 1 correspond to the frequency spectrum of theSolartron dummy cell. This circuit has two RC components connected in series.
As the time constants of these RC components are far from each other, two well
resolved humps are obtained. By analyzing each hump separately, the values of
the parameters of the circuit were obtained. The magnitude of these parameters
was witliin 0.1% of the expected values.
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Implementation oftheAC measurements
(A) (B)
Z n Freq, rad/sec
Fig. 1 Impedance diagrams generated using a RC Dummy cell
R t = 10.2 Q, R 2 = 10.2 fl, c = 3147 uF
(A) Argand Plot (B) Bode Plot
Fig. 2 shows the frequency spectrum for a dummy cell with only one RC
component. Only one hump is obtained in the Argand diagram. The R x and Rj
values obtained were within 1.0% of the expected values. Capacitor values
measured by this technique were within 5% of the nominal value (i.e. the value
provided by the manufacturer) \
1 The
capacitor measurements
done by
AC impedance
were
used as standards for
calibrating
the current
interruption routines presented in appendix
5.
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Appendix 7 So lub i l i t y of P b S i F 6 4 H 2 0
I.Objective:
To obtain the solubility of PbSiF6-4H 20 in water and in H2S1F 6 as a
function of temperature and acid composition. Solubility changes are computedfrom data available in the literature.
EL Solubil i ty of pure PbS iF 6 4 H 2 0 as a funct ion of temperat ure
The solubility of pure PbSiF6-4H 20 solutions in terms of the anhydrous
mole fraction is given by the following relationship [ l ] :
2464 4418
log;t 2 = -62.56499
+ — ^ — - +
21.54633 log T .. .
1
where:
jt, = anhydrous mole fraction (number of moles of the i-th component divided by thenumber of moles of the mixture).
T= Temperature. K
From the x 2 value at a fixed temperature, the value can be obtained
from the following relationship:
w = 2 x 2 M ank + (l-x 2 )M l
where:
W mh = weight of the anhydrous component contained In 1 Kg of solution [Kg ofcomponent/ Kg solution!.
Manh = molecular weight of the anhydrous component (For PbSiF6 M mh is equal to 349.34g/mol)
M, = molecular weight of the solvent ( for water M, = 18 g/mol)
From the value obtained in Eq. 2, the value can be obtained as
follows:
where:
W)^ = weight of the hydrated component contained in 1 Kg of solution [Kg of component/Kg solutionl.
Afhyj = molecular weight of the hydrated component (For PbSlF6.4H20 is equal to421.34 g/mol)
Af, = molecular weight of the solvent
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Solubility 0lPbSiF
t
.4H
,0 in the
presence
of HjSF, at T
=20
'C
Once W and and W anh are obtained, the solubility of the salt can be obtained
from any of the following relationships:
IOOOWU
^ A W l - W ^ )...4
...5
lOOOW^ p
c, =2
...6
where:
rrx, = molality of the solution [mol/Kg of solvent
c
2
=
molarity of the solution [mol/1]
P,
= weight
percent: Kg of anhydrous i-th component contained in 100 Kg of solution,
p
=
solution density,
[g/cm
3
]
Table 1 shows the solubility changes as a function of the temperature for
pure PbSiF6-4H 20 solutions obtained from Eqs. 1 to 6.
Table
Solubility changes
as a function
of
the
temperature
for pure
PbSiF
6
4H
2
0 solutions.
Temp *C
mj, mol/kg Pi,% c
2
*, mol/1
0 -1.0472 0.0897 0.6567
3.81
5.47 65.67 4.57
20 -1.0015 0.0997
0.6824
4.65 6.15
68.24
4.75
25
-0.9842 0.1037
0.6919
5.04
6.43
69.19 4.81
30
-0.9649
0.1084
0.7024 5.54
6.76
70.24
4.89
40 -0.9208 0.1200 0.7258
7.02
7.58 72.58 5.05
50
-0.8702
0.1348 0.7515 9.68 8.66 75.15 5.23
57
-0.8314 0.1474 0.7704
13.13
9.61
77,04
5.36
* c2 values obtained using a density
value
p = 2.43 g/cm
3 [2].
ni .Solubil ity of P b S i F 6 . 4 H 2 0 i n the presence of H 2 S1F 6 at T=20 °C
Table 2 shows the solubility of PbSiF 6.4H 20 as a function of the H 2SiF 6
concentration as reported in the literature[3l.
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Solubility of PbSiF,.4Hfi in the
presence
of HjSiF, at
T=20 'C
Table 2 Solubility of PbSiF6.4H 20 as a function of
the HjSiFj concentration at T=20 #
C [3].
P, [PbSiFJ ,
0 68.97
0.98 67.96
7.34 56.5
13.93 43.1
25.82 23.95
39.65 10.38
P, = weight percent: Kg of anhydrous i-th
component contained in 100 Kg of solution.
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Appendix 8 Solut ion of F i ck ' s Second Law Equat ion Under Cur rent
Interrupt ion Condit ions
I. Objective:
To solve analytically Fick's Second Law equation in the presence and in the
absence of a net Faradaic current. This solution will be used to predict local
concentrations from which the concentration overpotential across the
hypothetical Nernst diffusion layer will be derived.
n. Assumptions:
1) Dissolution of lead is the only Faradaic reaction and proceeds without
kinetic limitations (i.e. io-**> for Pb->Pb+2+2e").
2) Mixing of electrolytes witfiin the hypothetical Nernst diffusion layer is
neglected (D E
=0 and v =0).
3) Migration is absent.
4) Activities are equal to concentrations.
5) Dissolution of lead takes place under serin-infinite, unidimensional
conditions.
m. Fundamental Equations
Unsteady state diffusion can be described by Fick's second law [1,2] 1:
where:
C = concentration of species
i [in
this case
Pb
+2
],
[mol cm"
3
].
D =
Pb*
2
molecular diffusion coefficient
[cm
2
sec"
1
]
t
= time,
[sec]
x
= distance
from the electrode/solution
interface,
[cm]
For any problem the initial concentration, C \ is known2
:
1
In this appendix
the
equations presented by Bushrod
et
al.
[2] are re-derived and explained in greater
detail. Furthermore, a relationship between
concentration overpotential and local concentrations
(not
included in
Bushrod's derivation)
is
obtained.
2 i.e.
C=C*
for
t
=0 at any
point
x.
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Fundamental Equations
Fig. 1 Coordinate system used toobtain the analytical solution ofFick's second law.
A A
x
=0 x=x
I •'0
x
C* = C(x ,0) ...[ii]
Eqs. [i] and [ii] can be expressed in a general form by substituting the variable
C for the variable ix.
u* = u(x ,0) ...2
The general solution of Eq. 1 subject to the initial condition 2 can be obtained
from their Laplace transform.
The Laplace transform, L, of the left and right terms of Eq. 1, is as follows:
= D—- Li—\=Su-udx 2 I Bt J
where u =u(x,S) is the Laplace transform of u =u{x,t) .Thus, Eq. 1 can be
expressed as follows:
Pure
Lead
Nernst
Bulk
Pure
Lead
Boundary
Electrolyte
Anode
Layer
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Fundamental Equations
dx 2
DU
~D
Eq. 3 can also be expressed by the following relationship:
in which y(x) = u(x,S) and P =
The solution of Eq. 3.a can be obtained from the complementary
(homogeneous) and the particular (nonhomogeneous) solutions:
The complementary solution satisfies the source free equation:
Eq. 4 can be solved by assuming a trial solution:
y(x) = k exp[m x] . . . 5
where m is a dummy variable. Substituting Eq. 5 in Eq. 4 and rearranging:
m 2 + p = 0 ...6
Eq. 6 is the characteristic or auxiliary equation. Thus, solving for m, the
following complex conjugate roots are obtained:
from which the complementary solution of Eq. 4 is obtained:
y(x) = k x
exp[-*^^j + k
2 e x
p [yv/f]
Similarly, the particular solution is obtained by assuming a trial solution:
y(x) = *, . . . 8
Incorporating Eq. 8 in Eq. 3.a and solving for k 3:
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Boundary Conditions andAnalyScal Solution
Finally, the general solution is obtained by adding the complementary and
particular solutions:
y (x) = k x exp^^!~| + *2 e x l j * ^ ! ] + J - 1 0
from which:
u(x,S) = M(S) e x r j - x - ^ +N(S) exp[x^|j +±. ...11
IV. Boundary Condit ions and Analyt ica l Solution
For semi-infinite linear diffusion C(°°,t) = C* and M(°°,S) = y . Consequently,
N(S)=0 and Eq. 10 is given by:
< H I Iu(x,S) = M(S)ex l i-x\l^\ + j ...12
Eq. 12 will be solved for three different cases (see Fig. 2):
A) Application of a galvanostatic step (t[ < t)
B)
Current Interruption in the presence of concentration gradients in the
Nemst-Boundary layer (tl < t < t" 2 )
C Application of constant current after current interruption (t > t" 2 ) .
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Application ol a galvanostatic step
tc
O
c1_L.
CJ
-
Time
1 , .
t 1
Fig. 2 Current step function for
which the concentration andoverpotential changes were solved/
ti starts when the current step is firstapplied
ts starts when the current is
interrupted (at t[).
t3 starts when the current is applied
back (at Q.
+ ti
• t 2
A. Application of a galvanostatic step
The boundary condition that expresses the application of a galvanostatic
anodic current step is a follows:du, . h
where:
U = current density, [Amp-m"2].
Z= Charge number of Pb +2 [eqmol M
The Laplace transform of Eq. 13 is given by:
<Tx{o' S)=-zT5s
...13
...14
Thus, Eq. 14 is to be incorporated in Eq. 12 to obtain M(S), from which the
changes in concentration as a function of time can be obtained. This is done as
follows:
Derivating Eq. 12, rearranging and solving for M(S):
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Application ol a galvanostatic step
.15
Incorporating Eq. 16 in Eq. 12:
u(x,S)=-
ZF^DS' 2
exp
hVI]
+
J
...17
The inverse Laplace transformation of Eq. 17 can be obtained using the
following relationship:
L-!
jA«p[^VS]|
= 2^I^p[.
At
-k erfck '
2>/r
with
Thus,
k =
Vo
"
( J :
' ' '
) =
Z
FW | 2
V ? X { - 4 D J
...18.a
...18.6
From Eqs. 18.a and 18.b, the concentration overpotential can be obtainedfrom the following relationship:
T] c(x,t)= — ln-^T
J - Lb u
...19
from which:
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Current interruption
r x J t
f
l
(
exp
-A
liDh ...20.a
...20.6
B . C u r r e n t interruption
After applying current for t[ sec, a concentration profile that follows Eq. 18.a
is present. Thus, Eq. 12 is to be solved for the current interruption case using
Eq. 18.a as initial condition and Eq. 21 as boundary condition:
d u
dx(0
,0 =
0
| (0 ,5 ) = 0
Thus, Eq. 12 can be expressed as follows:
...21
u(x,S) = M(S)ex - +u\x,tx )
.22
Incorporating Eq. 18.a in Eq. 22 and derivating:
+
ZF J D] ^
"v^ D
e x p[4oJ .
evaluating Eq. 23 at (0,S) and solving for M(S):
du.
dx(0
'
5)=
V?
M(S)- 1 1
ZF^D^DS=
0
..23
...24
M(S) = -
ZF^DS 1..25
Finally, incorporating Eqs. 25 and 18.a in Eq. 22, obteuxiing the inverse
Laplace transform and rearranging:
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Application of current back to the electrode
u(x,tr ) =ZFyfD
x (
erfc -erfc
ZFynD
+ u ..26.a
...26.6
The the concentration overpotential can be obtained from Eq. 26.b:
...27
C . A p p l i c a t i o n o / current back to the electrode
After time tl current is applied back (see Fig. 1). The initial concentration
profile follows Eq. 26.a, and the boundary condition is given by Eq. 28:
dU . . h
du h
dx-{o' S)=-Wm
Thus Eq. 12 can be expressed as follows:
...28
u(x,S)=M(S)e\Hfi- + ...29
Incorporating Eq. 26.a in Eq. 29 and derivating:
'll
5
.30
evaluating Eq. 30 at (0,S) and solving for M(S):
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Application of current back to the electrode
M(S)= — ; ...32
ZF I DS' 2
Finally, incorporating Eqs. 32 and 26.a in Eq. 29, obtaining the inverse
Laplace transform and rearranging:
The concentration overpotential can be expressed as follows:
Figs. 3 and 4 show the variation in concentration and overpotential for
different step functions according to Eqs. 18.b, 20.b, 26.b, 27, 33.b, and 34.
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Application of current back t> the electrode
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- • , . , , , , , , , . , , , , , , , i i i 1 _J
•IBB i a .a ae. a 3 a .
a 4 e . a se. aT i M e , s e c
Fig. 3 Changes In u(0,t) and iic(0,t) upon application of the current step described inFig. 2.
C'=C(x,0)=[Pb+2lBulk=0.8 moll"1, i1=i2=50 Ampm
2, £>=5xlO6 cm 2sec"1. /,'= 16 sec, t\ = 32 sec.
(A) Changes in [Pb+2] (B) Changes in T | c corresponding to the data presented in Fig. A
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Application of current back to the electrode
488. 688. 888
Time, sec
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
488. 688.
Time, sec
Fig. 4 Changes In r
\c{0,t)
upon application of the current step described in Fig. 2.{C*=C(x,0)=[Pb+alBulk=0.8 moll
1}, D=5xlO 6cm 2sec 1
Fig. ij, AmpnV 2 i 2 , AmpnV 2 t[ =, sec t'2=, sec
A 50 100 16 32B 200 100 16 32C 200 200 100 800D 200 200 200 600
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Appendix 9 Extended V ersion of Tables Pres ente d i n Chapter 6.
Objective:
To present the complete set of tables presented in chapter 6 and their
associated statistical regression parameters.
Foreword:
The abstracted tables presented in chapter 6 are shown in this appendix
in their complete form. The names of the tables in this appendix were chosen to
correspond to those presented in chapter 6. Furthermore, the regression
parameters associated with the data shown in these tables are also presented
in this appendix alongside the main tables.
The regression parameters shown in this appendix were obtained from
ANOVA analysis \ The square of the multiple regression coefficient, r2, and the
square of the residual errors, l y r l 2 , were obtained from such analysis. These
quantities are defined by the following relationships:
\yT NY-Y'\2
in which:
y = Experimental data
y = Regressed data
y x = parameter related to the variations vector 2 , y with y = y r + y x .
y r = remairiing residual error
r = multiple regression coefficient.
The regression parameters r 2 and I yj 2 obtained from the curve fitting of
the AC impedance data, are given with respect to the real, Z* , the imaginary,-Zj,and the absolute impedances, IZI, (| Z |= ^Z^ + Z^
1 ANOVA stands for Analysis Qi Variance.
2 y and yx
are
related
through the
following
relationship:
|
y, f=\ y |2
-1Y- Y' |2
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Table 3 Summary of the values of the electrical analogue parameters obtained under rest potential conditions
Frequency Range Derived analogue parameters
Experiment Addition Pot/Gal 4*zcand Sweep Agents Control rad/sec rad/sec Qcm
2
Qcm2
pFcm' 2 Ci cm2
sec *Amp/m
2
Number
pFcm' 2
Amp/m2
CA2-/ yes Gal 560 99588 1.42 0.501 64 23.71 0.64 269
CA4-7 yes Pot 315 250159 1.02 1.855 lllPilf 12.45 IIKl! 72CA5-i yes Pot 560 111740 1.19 '"'0725''''' 65.30 23.90 0.64 186
CA5-4 yes Gal 560 111740 1.21 0.854 liii!!! llliill!!! 0.56 157CA6-2 yes Gal 177 157834 1.20 1.916 "liS"""" 0.55 70
CC1-5 no Gal 628 15783 0.90Illlll
iiiiioiii 0.57CC1-6 no Gal 628 44482 0.89
- - 2.88 0.45
CC2-J no Gal 560 17709 iiliii i i i i i ! lllslli0,01-2
no Gal 560 28067 0.85 1.88 0.59
Values of statistical parameters related to the quality of the fit:
El
Experimentiy
r
i
2
.
r
2
iy
r
i
2
.
r
2
iy
r
«
2
,
r
2
and Sweep iy
r
«
2
,
NumberQ
2
cm
4
Q W
CA2-7 4.99X102
0.972 2.43X103
0.977 4.75X10"2
0.971
CA4-7 2.92xia' 0.995 4.74X103
0.986 2.45X10"1
0.996CA5-7 1.05x10' 0.976 6.53xl0'3
0.967 9.81xia2
0.976
CA5-4 2.09X10'1
0.969 1.23X102
!lll!l§ii!!ll1.96x1a
1
0.970CA6-2 3.10x10' 0.995 5.06X10
2
0.992 5.59X101
0.996
CC1-5 5.09X10"4
0.876 2.19x10* 0.966 3.51x10^ 0.967CC1-6 1.02xia3
0.970 3.11x10* 0.987 1.03xl0-3
0.982
CC2-Z 8.00X1O'5
0.944 6.39X10"5 0.975 1.16X10"4 0.971
CC2-2 1.48X10-4
0.913 4.75x10s
0.984 1.60X10 -4 0.966
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Table 4 Analysis of the spikes produced during the application of the A C
waveform,
in the presence of a net DC current (Exp. CA2, Figs. 6.32 to 35)
Parameters Derived from Regression AnalysisComputations
From Eq. 4 Experimental
Slimes
Thickness
mm
Slope R m,
Qcm2
Intercept,
i,(mV)
r
2 |y r |2
,mv2
b
pm,Qcm K» , mV
mVTI A.
mV
0.80
1.61
3
3.10
3.72
4.72
5.33
5.95
0.62
1.19
L45
2.15
2.50
3.16
3.93
4.57
30.8
38.1
45.8
54.4
i i i i i i i73.7
llill&iiif82.4
0.850
0.983
0.986
0.988
0.992
0.952
0.981
0.961
6.8
5.1
6.4
12.6
8.8
86.0
59.8
163.7
2 54
1.65
1.63
1.30
1.26
1.20
0 98
0.93
7.78
7.37
6.48
6.93
6.72
6.70
7.38
7.68
12.1
23.1
28.1
41.8
48.6
61.5
76.5
88.8
42.9
61.2
73.9
96.1
109.7
135.2
151.8
171.2
41.9
60.4
733
95.5
109.3
133.5
150.4
170.0
6.56
7.18
7.79
5.23
5.71
7.22
llliiili113.9
124.9
0.920
0.905
0.888
375.1
633.5
1192.5
0.92
1.03
0.89
7.98
7.95
9.27
101.7
110.9
140.3
194.8
224.8
265.2
193.3
221.8
262.48.41
8.65
4.96
9 40
361.0
360.6
0.051
0.558
4386.0
5688.7
3.74
1.97
5.90
10.86
96.5
182.7llliiili:
543 3
436.0
520.0
Between 18 and 20 experimental points were used to obtain the regression coefficient. These points were
collected during -55 min and correspond to digitized samples taken when the frequency of the applied C waves
was between 0.063 and 6.3 rad/sec.
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Table 5 .A Analys is of the spikes produced during the application of the AC waveform.
in the presence of a net DC current (Exp. CA6)
Parameters Derived from Regression From Eq. 4 Experimental
Analysis Computations
Slimes Slope Intercept, r2
ly rl2, b pm, Qcm nc mV
Thickness b, (mV) mV
2
TD mV mVmm Qcm 2 'Km
0.70 0.36 24.9 0.95 0.17 3 56 5.14 7.0 32.0 32.6
0.98 0.48 28.4 0.99 0.03 3.05 4.90 9.3 37.7 39.0
1.25 0.67 29.6 0.98 0.11 2.28 5.36 13.0 42.6 44.2
1.47 0.56 35.7 0.98 0.04 3.28 3.83 10.9 46.7 47.9
2.11 1.00 ! ! t l 3 l i i ! l 0.99 0.14 1.96 4.75 19.5 57.8 59.5
2.39 1.21 40.0 0.92 1.80 1.69 5.08 23.6 63.6 65.7
2.66 1.15 i i i ^ i i i i i 0.97 0.71 209 4.31 22.3 68.8 70.5
3.06 1.52 47.0 0.99 0.40 1.59 4.98 29.6 76.6 78.3
5.37 1.89 llllPi!!!! 0.85 15.64 2 55 3.52 36.8 130.5 134.1
5.58 2.87 81.4 0.88 15.37 1.46 5.14 55.8 137.2 138.3
5.79 2.16 98.6 0.97 4.33 2.35 3.74 42.0 140.6 143.76.00 2.08 105.7 0.96 6.85 2.61 3.47 40.5 146.2 149.7
6.22 1.95 113.0 0.92 10.92 2.98 3.13 37.9 150.8 154.7
6.43 2.53 106.1 0.82 42.73 2.16 3.94 49.2 155.3 160.0
6.64 2.70 i l l i l i i l i 0.88 10.16 2.16 4.06 52.5 166.1 168.2
6.85 2.76 120.3 0.92 7.49 2.24 4.02 53.6 173.9 176.3
7.06 3.36 118.6 24.54 182 4.75 65.3 183.8 184.5
7.28 3.98 112.3 0.66 83.22 1.45 5.47 77.4 189.6 193.7
7.49 3.13 140.9 0.97 4.09 2.31 4.18 60.9 201.9 205.2
4 experimental points were used to obtain the regression coefficient. These points were collected during -12
min and correspond to digitized samples taken when the frequency of the applied AC waves was between 0.63 and
6.3 rad/sec.
Table 5.B Analys is of the spikes produced during the application of the AC waveform.
under current interruption conditions (Exp. CA6)
Parameters Derived from Regression From Eq. 4 ExperimentalAnalysis Computations
Time, hrs Sloped, Intercept, r 2
|y r |2
,mv2
pm. Qcm IRm, mV IR^+ft, mV*u,
Qcm2
b, (mV) mV
0.63 ,.,,,,,2.23,,,,,,. 60.8 0.89 21.95 2.82 0.0 60.8 82.1
2.44 3 62 48.4 0.91 41.19 QQ 48.4 49.43.83 3.83 44.5 0.95 62.43 484 0.0 44.5 44.0
9.08 2.93 31.5 0.79 154.01 3.70 0.0 31.5 30.014.33 2.59 24.0 0.64 208.29 3.27 0.0 24.0 21.7
41.96 1.92 6.7 0.88 30.14 2.42 0.0 6.7 6.743.07 1.64 5.3 0 87 22.09 2.07 0.0 5.3 5.1
54.97 0.98 3.3 0.37 22.74 1.24 0.0 3.3 2.965.17 1.00 2.3 066 21.53 1.26 0.0 2.3 1.4
91.44 1.15 -6.2 0.95 7.60 1.45 0.0 -6.2 -6.1102.00 0 90 -0.4 0.70 17.22 1.14 0.0 -0.4 0.3
112.8 0.99 0.2 0.69 21.63 U 5 0.0 i i i i i i i i t t 0.3
9 experimental points were used to obtain the regression coefficient These points were collected during -27
min and correspond to digitized samples taken when the frequency of the applied AC waves was between 0.63 and
6.3 rad/sec.
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Table 6 Analysis of the spikes produced during the application of the AC waveform,
in the presence of a net DC current (Exp. CC 1)
Parameters Derived from Regression
Analysis Computations
From Eq. 4 Experimental
Slimes
Thicknessmm
Slope R„
Qcm
2
Intercept,
b, (mV)
r2
ly r|2,
mV
2
b pm, Qcm IRm. mV IRm+fc,
mV mV
1.37
1.79
2.22
2.65
3.07
3.50
0.52
0.63
0.74
0.81
0.92
0.96
32.4
38.0
42.8
48.0
52.4
58.4
0.98
0.97
0.98
0.95
0.93
0.93
0.1
0.3
0.4
1.1
1.7
2.3
3.21
3.11
2.97
3.06
2.94
3.12
3.80
3.51
3.34
3.05
2.99
2.74
10.1
12.2
14.4
15.7
17.8
18.7
42.6
50.2
57.2
63.7
70.3
77.1
42.8
50.6
57.8
64.4
70.9
78.1
5 experimental points were used to obtain the regression coefficient These points were collected during -1 2
min and correspond to digitized samples taken when the frequency of the applied AC waves was between 0.63 and
6.3 rad/sec.
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Table 7 Parameters derived from the fitting of the impedance data obtainedin Exp. CA2 to the Zz^-Z^c analogue (Circuit A.2, Figs. 6.38.39)
High Frequency Parameters Low Frequency Parameters R -A ,tottl —
R t + R 2
Slimes ^zci bi, \ ,sec ^zcz- D2, b 2 , 1R2 ,sec
Thickness
mm
Qcm2
secCi cm 2 sec Qcm2
secil cm sec
Qcm2
0.80
1.61
2.23
3.10
3.72
4.72
5.33
0.21
0.31
0.42
0.64
0.52
1.10
1.06
0.73
0.67
0.64
0.59
0.59
0.55
0.60
0.0062
0.0165
0.0296
0.0683
0.0503
0.1511
0.1045
33.50
18.76
14.09
9.39
10.34
7.25
10.19
0.0010
0.0023
0.0042
0.0102
0.0064
0.0321
0.0225
0.44
0.88
1.17
1.62
2.22
2.50
3.17
0.77
0.69
0.69
0.69
0.68
0.70
0.71
0.079
0.21
0.30
0.39
0.42
0.59
0.69
5.55
4.17
3.84
4.11
5.30
4.26
4.60
0.037
0.104
0.176
0.260
0.276
0.464
0.592
0.65
1.19
1.58
2.26
2.74
3.59
4.235.95 1.54 0.51 0.1866 llliill 0.0373 3.45 !§§llllliill 4.06 Illlll 4.99
6.56
7.18
7.79
4.70
5.13
4.30
0.56
0.55
0.59
0.5415
0.5671
0.3873
iiiiieiiii9.04
11.11
0.3323
0.3593
0.2030
1.29
2.08
4.65
0.98
0.96
0.90
llliiili1.94
2.27
0.82
1.07
2.05
lislli!1.992
2.484
5.99
7.21
8.95
All measurements refer to the geometrical area of the electrode.1 Low and high frequency terms refer to the ranges of frequencies used during the deconvolution process.
'* Low and high frequency arcs were fitted to the whole frequency range (0.063<co<30000 rad/sec)
Values of statistical parameters related to the quality of the fit:
EZt
Slimes |yr|2,QW r
2 |yr|2,QW r
2 |yr|2,oW r
2
Thickness,
mm
0.80 0.0067 0.999 0.0661 0.777 -0.0594 1.009
1.61 0.0147 0.999 0.0185 0.978 -0.0038 1.000
2.23 0.0242 0.999 0.0136 0.991 0.0106 1.000
3.10 0.0174 1.000 0.0107 0.997 0.0067 1.000
3.72 0.0574 0.999 0.0143 0.997 0.0431 1.000
4.72 0.0328 llllifiiollill! 0.0095 0.999 0.0233 1.0005.33 0.1136 1.000 0.0449 0.997 0.0688 1.000
5.95 0.4281 0.999 0.1344 0.993 0.2936 0.999
6.56 0.0563 1.000 0.0083 1.000 0.0480 1.000
7.18 0.1286 i!i!iH!i!!l 0.0054 lillllHIII 0.1232 1.000
7.79 0.3369 1.000 0.0045 1.000 0.3324 1.000
A total of 6 parameters were curve fitted using between 110 and 113 experimental points.
For the impedance curves obtained at 0.8 and 1.6 mm slimes, r2
3>0.99 when 1<CK 30000 rad/sec
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Table 8 Parameters derived from the fitting of the impedance spectraobtained In Exp. CA2 (Circuit B.2, Figs. 6.40, 41)
Slimes r a r bc b B a ^ Z C a ^B. tota l
Thickness Qcm 2
Qcm 2
F cm"2
Qcm2
sec"¥ z c Qcm 2
mm
0.80 0.43 0.23 0.026 11.48 0.58 0.661.61 0.84 0.38 0.068 8.06 1.22
1.15 0.49 0.102 "•••"••••••"7 "— 0.53
3.10 1.64 0.66 0.146 0.56 2.303.72 2.03 0.73 0.178 8.84 0.57 2.764.72 2.81 0.80 IIIIIIITIIIII 0.59 3.615.33
, , , , , , , , 3 : ^ , , , , , , ,
0.90 0.398 0.60 4.255.95 l l l i l l l l l i l l l l l 9.88 0.61 5.006.56 4.36 0.749 10.48 0.61 5.96
7.18 2.40 0.975 0.62 7207.79 5.29 3.82 12.61 0.64 9.12
For the AC sweep done at 0.8 mm slimes: Bb= 1.92 Clem2sec'¥zc, and 4/
ZCB= 0.78 (this was the only sweep
fitted to circuit B . l , all the other sweeps were fitted to circuit B.2)
Values of statistical parameters related to the quality of the fit:
Frequency Range (ZI
Slimes llliiili |yr|2. r2 | y r | 2 ,nW r2 | yR |
2
,QW r2
Thickness, Q W
mm
0.80 0.063 28068 0.012 0.998 0.059 0.791 -0.047 1.0071.61 0.063 28068 0.031 0.999 0.013 IllilPiil. 0.018 0.9992.23 0.063 28068 0.057 0.998 0.008 0.995 0.049 0.999
3.10 0.063 28068 0.015 1.000 0.007 0.998 0.008 1.0003.72 0.063 28068 0.028 1.000 0.009 0.998 0.019 1.000
4.72 0.063 28068 0.077 1.000 0.019 0.998 0.058 1.0005.33 0.063 28068 0.124 0.999 0.028 0.998 0.096 1.000
5.95 0.063 28068 0.181 illiRil! 0.038 0.998 0.143 0.9996.56 0.063 28068 0.226 0.999 0.054 0.998 0.171 1.000
7.18 0.063 28068 0.241 1.000 0.072 0.999 0.169 1.0007.79 0.063 28068 0.253 1.000 0.086 0.999
0.167 LOOO
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Table 9 Parameters derived from the fitting of the impedancedata obtained in Exp CA5 to the analogue (Circuit A.2)
High Frequency Parameters Low Frequency Parameters
R, + R 2
Slimes
Ri,
^ Z C l
D „
bi,
\
,sec
R
2
,
D
2
,
b
2
,
\
,sec
RA.total
Thickness
mm
Q c m
2
sec
Qcm sec
Q cm
2
sec
O. cm1 sec
Q cm
2
0.64
1.20
1.76
1.89
0.101
0.017
0.069
0.817
0.91
0.68
0.83
0.58
0.0013
0.0115
0.1972
0.0599
75.04
1.47
0.35
13.64
0.0007
0.0015
0.1423
0.0079
0.32
0.83
1.61
0.99
0.68
0.59
0.53
0.81
0.039
0.075
0.174
0.146
8.19
11.10
9.26
6.77
0.0088
0.0121
0.0373
0.0934
0.42
0.85
1.68
1.81
* All measurements refer to the geometrical area of the electrode.
** Low and high frequency terms refer to the ranges of frequencies used during the deconvolution process.
*** Low and high frequency arcs were fittedto the whole frequency range (6.3<ox23000 rad/sec)
Values of statistical parameters related to the quality of the fit:
Slimes |y ri2,Q 2cm
Thickness,
mm
0.64 0.005
1.20 0.013
1.76 0.045
1.89 0.062
ly,l2.QW |yr|2,QW
0.996
0.997
0.995
0 998
0.001
0.002
0.002
0005
0.990
0.994
0.998
0 997
0.005
0.012
0.043
0 057
0.996
0.997
0.995
0 998
A total of 6 parameters were curve fitted.
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Table 10 Electrical analogue parameters derived from the fitting of the impedance dataobtained in Exp. CA5 (Circuit B.2)
Slimes r aC b B a ^ Z C A
Thickness
mm Qcm 2
Qcm 2
F cm"2
Q c m
2
s e c '
V z c Qcm 2
0.64 0.38 0.04 0.017
16.10 0.65
0.42
1.20 0.72 0.08 0.036 16.10 0.64 0.80
1.76 1.44 0.16 0.063 0.57 1.60
1.89 1.50 0.32 0.062 iiiiiiiiiiiiiiii 0.62 1.82
Values of statistical parameters related to the quality of the fit:
Frequency
Range
tzi
Slimes |yr|2, r2 |yr
l
2.nW r2 |yr|2,QW r2
Thickness,
Q2
cm
4
mm
0.64 6.28 22295
0.002
0.999 0.001
0.978 0.001 1.000
1.20 6.28
fiiiii0.002
1.000 0.001
0.997
0.001
1.000
1.76 6.28 "22295'"" 0.019 0.998 0.001 0.999 0.018 0.998
1.89 0.63 28068 0.056 0.998
0.019 0.991
0.037 0.999
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Table 11 Parameters derived from thefitting of the impedance dataobtained in Exp. CA2 to the Randies analogue circuit (Figs. 6.43. 44)
Time, Hrs Rct. Qcm2
Cdj. nF cm'2
B L il cm2 sec fzc
0.17 0.044 23.75 3.83 0.190.64 0.112 167.42 3.63 0.3312.03 0.070 206.32 3.13 0.3121.32 0.063 247.86 2.69 0.3258.58 0.063 267.78 2.62 0.32
lillililllll0.066
illllll9l)ililil;sii i i i i i t i iHii i i i i 0.32
72.08 0.049 284.79 2.25 0.3076.82 0.044 272.15 2.23 0.3078.06 0.046 276.70 - 0.3086.90 0.048 274.37 2.23 0.30
Values of statistical parameters related to the quality of the fit:
iZI
Time, Hrs|y
r
|
2
,Q
2
cm
4 r2
|y
r
|
2
,QW
r2
|y
r
|
2
,"W r2
0.17 0.28 0.98 0.01 Il!;ii:i9iiiiill 0.24 0.980.64 0.11 0.95 0.00 0.99 0.08 0.97
12.03 0.06 0.97 0.00
Illli9l!f;!IIII0.04 0.98
21.32 0.04 0.97 0.00 0.99 0.03 0.98
58.58 0.07 0.95 0.00 0.99 0.05 0.9667.57 0.07 0.94 0.00 0.99 0.05 0.96
72.08 0.07 0.93 0.00 0.99 0.05 0.9576.82 0.05 0.95 0.00 0.99 0.04 0.97
78.06 0.05 0.95 0.00 0.99 0.04 0.9786.90 0.05 0.95 0.00 0.99 0.04
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Table 12 Electrical analogue parameters derived from the Fitting of theimpedance data obtained in Exp. CA5 to the Randies analogue circuit (Fig. 6.45)
Time, Hrs Ret, Qcm2
Cd,, nF cm'2
B i , Q cm2 sec"heZC
063
2.444.26
6.07
7.88
9.70
11.67
17.58
23.45
30.27
34.59
0010
0.0180.030
0.041
0.046
0.046
0.021
0.024
0.009
0.023
0023
804.67
1350.321502.48
1625.51
1707.71
1719.34
1588.19
1579.10
1431.30
1452.57
1308.61
1.11
0.850 83
0.82
082
0.83
0.71
0.71
0.66
0.69
0.70
0.21
0.240 26
0.27
0.28
0.28
0 25
0.26
0250.26
0.25
Values of statistical parameters related to the quality of the fit:
Frequency Range llllllllllllllllll IZI
Time, HrsIIIBII lyrl
2
. r2 |yr|
2,QW r2 |yr|
2,nW r2
tfcm4
0.63 177.1 44485 0.026 0.958 0.002 0.956 0.012 0.979
2.44 177.1 35335 0.023 0.913 0.000 0.983 0.016 0.9374.26 177.1 31492 0.014 0.936 0.000 0.987 0.009 0.955
6.07 177.1 31492 0.014 IIollllll: 0.000 ilfJilll 0.010 0.9517.88 177.1 31492 0.017 0.913 0.000 0.985 0.011 0.939
9.70 177.1 31492 0.015 0.922 0.000 0.986 0.010 0.946
11.67 17.7 31493 0.013 0.979 0.001 0.990 0.010 0.985
17.58 17.7 31493 0.013 0.980 0.000 0.992 0.009 0.985
23.45 11.2 31493 0.019 0.971 0.001 0.989 0.015 0.978
30.27 17.7 31493 0.012 0.978 0.001 0.990 0.010 0.983
34.59 17.7 39647 0.015 0.976 0.001 0.989 0.012 0.981
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Table 13 Electrical analogue parameters derived from the fitting of the AC impedance data
obtained in Exp. CA4 to the Z a ^ - Z a ^ c analogue (Circuit A.2, Fig. 6.47)
High Frequency Parameters Low Frequency Parameters
R, + R 2
TImt.
Hra
Ri.
Qcm2
z c i Di,
V2C,
sec
bi,
CI cm
1
sec
\ ,sec Ra,
Qcm2
TCI- D2,
sec
b2,
i i c m2
sec
T R
,sec R
AJOISI,
Qcm2
208
3.89
5.70
7.52
9.33
11.18
15.37
19.55
0 87
0.43
0 35
0.33
0.32
0.34
0.33
0.32
0 73
0.83
0 82
0.80
079
0.73
071
0.72
00092
0.0031
0.0026
0.0026
0.0029
0.0042
00047
0.0043
94.77
139.84
137.18
126.81
108.92
79.97
71.53
75.11
0.00167
0.00092
0.00071
0.00059
000062
0.00057
0.00054
0.00052
1.77
1.96
165
1.44
1.26
0.99
0.93
0.99
043
0.35
0 39
0.43
047
0.55
0.59
0.59
0 89
0.94
0.86
0.82
0.77
0.55
0.53
0.58
198
2.08
1.91
1.75
165
1.79
1.76
1.70
0 77
0.85
0 69
0.63
0.56
0.34
0.34
0.40
2.64
2.38
2.00
1.77
1.58
1.33
1.27
1.31
The A C sweeps obtained at current interruption times longer than 11 hrs were fitted to the Z Z A R C r Z Z A R C 2 - C P E 0
analogue (circuit A.l). Thus, 2 additional parameters were obtained:
Time, Hrs b0, ^ZCo
Q. cm 2 sec *'
11.18 0.050 0.74
15.37 0.054 0.73
19.55 0.046 0.79
* Al l measurements refer to the geometrical area of the electrode.
** The regression coefficients r*2 and r 3
2
were larger than 0.99
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Appendix 1 0 Krame rs-Kr onig Transformations of NUD E lements
I. Objective:
To obtain the Kramers-Kronig transformation of the non-uniform diffusion
element (NUD) used to fit the experimental data presented in Chapter 6.
n. Fundamental Equations
The Kramers-Kronig transformations (KKT) have been proposed as a way of
validating experimental data obtained from AC impedance measurements [l]. They
imply that causality, linearity, and stability were observed during the experimental
measurement. Basically, KKT indicate that there is a unique relationship between
the real (Z*) and the imaginary (-2y parts of the impedance. Thus, for a sufficiently
wide impedance spectrum, the real part of the impedance can be obtained from
the imaginary part and viceversa.
One of the mathematical forms of the KKT is as follows [2,3]:
Z3(G)) = -(2(0^ f-rZaCO-Z^G))"
^ 7C JJ o [ X 2 -(£> 2dx ...1
where the total impedance as a function of the frequency, Z(co), is given by the
following relationship:
Z(to)=Z9l((o)-yZ3(co) ...2
Eq. 1 indicates that if Zj, is known in the whole frequency range, -Zg can be
obtained. Thus, if the derived -Zg values match those experimentally measured,
the data follows the KKT.
To evaluate Eq. 1 at least two problems (one experimental and another
mathematical) have to be overcome:
1) Impedance data are not usually available in the whole frequency range
(0<uK°o).
2) Eq. 1 has singular points at x=co.
Several experimental procedures and mathematical arrangements have been
used to evaluate the KKT [2,3]. The approach followed here is to evaluate Eq. 1 by
assuming that the NUD element fits the experimental data in the whole frequency
range:
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Fundamental Equations
Z NUD((0) = B.O'co) tanhfl2(yco)y2C
.. .3
The second assumption, is that when Eq. 1 is solved numerically, at the
singularity point (i.e. at x=co) the bracketed term in Eq. 1 is equal to one \
Thus, with these two assumptions the problem consists in evaluating
numerically Eq. 1 for the NUD element represented by Eq. 3. Thus, the KKT to be
solved is as follows:
%{Bjjjxf^ tanhfl2(/x)Vac -fltQco)" tanhfl2(/co)¥2C}
x2
-co2
The steps followed to obtain the KKT of the NUD element were as follows:
1) From the e x p e r i m e n t a l d a t a obtained in Exp. C
^zc values (e.g. for the sweep taken at 1.79 mm, Bx= 1.03 r2c/w2secVzc,
B2=0.91 sec"^, and ¥ ^ = 0 . 3 3 , Table 14, Chapter 6).
2) Assume the impedance was measured in the frequency range between
0.0001 and 10000 rad/sec at 2500 equidistant discrete intervals.
3) Evaluate numerically Eq. 4 for each of the 2500 discrete frequency points
assuming that at x=co, the bracketed term in Eq. 4 is equal to one.
By following these steps the data presented in Fig. 1 were obtained. The
theoretical (obtained directly from the imaginary component of Eq. 3) and the
computed (obtained from Eq. 4) -Zj values are shown to be very similar. This
appears to indicate that the NUD follows the KKT. Furthermore, the fulfillment of
the KKT implies that the data fitted to such a function adhere to experimental
stability, causality, and linearity which are the foundations of impedance
spectroscopy.
1 i.e.
Z^-Z^co)
-co'
0
0
= ll,
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Fundamental
Equations
(A)
.238
xE 8
I
<s
t
£
N
I
.
eee
23a —xE e
. 288
isa
108
,038 —
. 8 8 8 »
.
eee 40 . a
88.8 128.
F r a q , r j i d / s a c
168
288 .
Fig. 1 Kramers-Kronig Transforms of the impedance data fitted to the NUD element.
From the experimental data presented in Table 6.14 (sweep taken at 1.79 rnm.Bp 1.03 Q cm 2 sec'Vzc, B2=0.91 sec'yzc,
and ¥zc=0.33).(A) Comparison of the theoretical imaginary impedance component, -Z3(co), with -Z co) obtained by KKT of thereal component. (B) Detail of Fig. A at low frequencies.
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p p e n d i x 1 1 P r o g r a m s U s e d t o G e n e r a t e t h e E h - p H D i a g r a m s P r e s e n t e d i n C h a p t e r 7The Eh-pH diagrams presented in Chapter 7 were generated using the
THERMODATA computer program. In this appendix, the i n p u t f i l e s required to draw
these diagrams are presented alongside the o u t p u t f i l e s .
D - S i - H ^ O S y s t e mn p u t F i l e :
pH diagram for the (F)-Si-H20 system
of
all
components = 1
DIAGRAM
AXES
<-> (AQ)
H<+>(AQ)
H2 0
LIGANDS
H2SI03
HSI03<1->(AQ)
SI03<2-> (AQ)SI H4 (G)
AREAS
SIF6<2->(AQ)
SI F4 [H20]2 (AQ)
SIF5[H20] <1->(AQ)
F<->(AQ)
HF (G)
HF (AQ)
HF2 <1-> (AQ)
F20(G)
LINES
2(G)
H2(G)
UNITSOU
FILE
cpdaiber
gibalber
cpdjandat
cpdnpldat
cpdnbsdal
cpdsgtdat
ACTIVITY
1.
COMPONENTS
1,1AXES
ABSCISSA PH
H <+> (AQ)
ORDINATE EH
<-> (AQ)
LIMITS
abscissa 4.9.
ORDINATE-1.0 2.5
TEMPERATURE
298.15
PRESSURE
1.
AREASgibbs
check
all
OVERLAY
H2(G) = 2<->(AQ) + 2H<+>(AQ)
OVERLAY
2 H20 = 02 (G) + 4 <-> (AQ) + 4 H <+> (AQ)
1.,1.,1.,1.
PLOT
VISUAL
mark lower abscissa 12
-3.,-2.,-1.,0.,1.,2.,3.,4.,5.,6.,7.,8.
MARK LOWER ORDINATE 6
-.5,0.,.5,1., 1.5,2.0label lower abscissa 14
4.,-4
-3.,-3-2.,-2
0.01.0 1
2.0 2
3.0 3
4.045.0 5
6.0 6
7.0 7
8.0 8
9.0 9LABEL LOWER ORDINATE 8
-1.-1
-.5-0.50. 00. 5 0.5
1. 11.51.5
2. 22.5 2.5
TITLE LOWER ABSCISSA
PH
TITLE LOWER ORDINATE
Eh
TITLE UPPER ABSCISSA
SYSTEM (F)-Si-H20 at 25 C
LEGEND
linetype dashed 0.5 0.5OVERLAY
linetype full
IDENT
linetype dashed .3.7.3.7
LIGAND
linetype full
O u t p u t F i l eINPUT WILL BE TAKEN FROM F:\PB07_0.INP
OUTPUT WILL BE WRITTEN TO F:\PB07_0.OUT
SPECIES 17 ELEMENTS 5 E H O Si F
UNITS CHANGED TO JOUMAIN SPECIES PER AREA 1
LIGAND SPECIES PER AREA 1
ORDINATE EH MULTIPLIER 1.0
SPECIES <-> (AQ)ABSCISSA PH MULTIPLIER 1.0SPECIES H<+>(AQ)
CURRENT PRESSURES IN ATM
1.0000
TEMPERATURE OF SYSTEM 298.15 K
G VS T COEFFICIENTS FOR SPECIES JOU
SPECIES GA GB
<->(AQ) 0.901164E-12 0.000000H<+>(AQ) -16.9655 0.000000H2 0 -237190. 0.000000
H2 SI 03 -.102300E+07 0.000000HSI03<1->(AQ) -955460. 0.000000SI03<2-> (AQ) -887000. 0.000000SI H4 (G) -39310.0 0.000000SI F6 <2-> (AQ) -.213800E+07 0.000000SI F4 [H20]2 (AQ) -.200200E+07 0.000000SIF5[H20] <1->(A -.207300E+07 0.000000F<->(AQ) -278820. 0.000000H F (G) -273220. 0.000000HF (AQ) -296850. 0.000000HF2<1->(AQ) -578150. 0.000000F2 O (G) -4600.00 0.000000
02(G) 0.183206E-11 0.000000
H2(G) 0.180604E-11 0.000000
TOTAL LIGAND AREAS 2TOTAL MAIN AREAS 7A SI F6 <2-> (AQ)
B SI F4 [H20]2 (AQ)
C SIF5[H20] <1->(AQ)D F<->(AQ)
E F2 0(G)
F H F (AQ)G HF2<1-> (AQ)
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(Pb-F)-Si-H 2Q System
P b - F J - S i - H a O S y s t e mn p u t F i l e
TITLEEh vs pH diagram for the (PW>SI-H20 systemActivities of all components = 1DIAGRAM
AXES
<-> (AQ)H <+> (AQ)
H2 0
LIGANDS
H2SI03
HSI03<1->(AQ)
SI 03 <2-> (AQ)
SIH4 (G)
AREAS
SIF6<2->(AQ)
SIF4[H20]2(AQ)
SIF5[H20] <1->(AQ)
F<->(AQ)
HF(G)
HF (AQ)
HF2<1-> (AQ)
F20(G)
Pb
Pb<2+>(AQ)
PbO
Pb2 03
PB3 04
PB 02
PB02H2
PB02H<1->(AQ)
PB<4+>
PB F2 (S)
PBSI03
LINES
02(G)H2(G)
UNITS
JOU
FILE
cpdalbergibalber
cpdjandat
cpdnpldat
cpdnbsdat
cpdsgtdat
ACTIVITY1.
ALL
COMPONENTS
2,1
AXES
ABSCISSA PH
H<+>(AQ)
ORDINATE EH
<-> (AQ)
LIMITS
abscissa -4.9.
ORDINATE-1.0 2.5
TEMPERATURE
298.15
PRESSURE
1.
AREAS
gibbs
check
all
OVERLAY
H2 (G) = 2 <-> (AQ) + 2 H <+> (AQ)
1..1..1.
OVERLAY
2 H20 = 02 (G) + 4 <-> (AQ) + 4 H <+> (AQ)
1.,1.,1.,1.
PLOT
VISUAL
mark lower abscissa 12
-3.,-2.,-1.,0.,1.,2.,3.,4.,5.,6.,7.,8.
MARK LOWER ORDINATE 6
-.5,0.,.5,1.,1.5,2.0
label lower abscissa 14
4.,4
-3.,-3
-2.,-2
0.01.0 12.0 2
3.0 3
4.045.056.067.0 7
8.089.09
LABEL LOWER ORDINATE 8
-1.-1-.5-0.5
0. 00. 5 0.5
1. 1
1.51.5
2. 2
2.5 2.5
TITLE LOWER ABSCISSA
pH
TrTLE LOWER ORDINATE
Eh
TITLE UPPER ABSCISSA
SYSTEM (Pb-F)-SI-H20 at 25 C
LEGEND
linetype dashed .2 .2OVERLAY
linetype DASHED .5.5
IDENTlinetype DASHED .2 .4LIGAND
linetype full
O u t p u t F i l eINPUT WILL BE TAKEN FROM F:\PB013_0.INPOUTPUT WILL BE WRITTEN TO F:\PB013 _0.OUT
SPECIES 28 ELEMENTS 6 E H O Si F Pb
UNITS CHANGED TO JOU
MAIN SPECIES PER AREA 2
LIGAND SPECIES PER AREA 1
ORDINATE EH MULTIPLIER 1.0
SPECIES <-> (AQ)
ABSCISSA PH MULTIPLIER 1.0
SPECIES H<+>(AQ)ABSCISSA LIMITS 4.000 TO 9.000
ORDINATE LIMITS -1.000 TO 2.500
CURRENT PRESSURES IN ATM
1.0000
TEMPERATURE OF SYSTEM 298.15 K
G VS T COEFFICENTS FOR SPECES JOU
SPECIES GA GB
<-> (AQ) 0.901164E-12 0.000000
H<+>(AQ) -16.9655 0.000000
H20 -237190. 0.000000
H2 SI 03 -.102300E+07 0.000000
HSI03<1->(AQ) -955460. 0.000000
SI03<2-> (AQ) -887000. 0.000000
SI H4 (G) -39310.0 0.000000
SI F6 <2-> (AQ) -.213800E+O7 0.000000
SI F4 [H20]2 (AQ) -.200200E+07 0.000000
SIF5[H20] <1->(A -.207300E+07 0.000000
F<->(AQ) -278820. 0.000000
HF (G) -273220. 0.000000
HF (AQ) -296850. 0.000000
HF2<1->(AQ) -578150. 0.000000
F20(G) 4600.00 0.000000
Pb 0.168155E-11 0.000000
Pb<2+>(AQ) -24310.0 0.000000
PbO -188500. 0.000000
Pb2 03 411780. 0.000000
PB3 04 -616200. 0.000000PB 02 -218990. 0.000000
PB 02 H2 420900. 0.000000
PB 02 H <1 -> (AQ) -339000. 0.000000
PB <4+> 302500. 0.000000
PB F2 (S) -619600. 0.000000
PBSI03 -.100000E+O7 0.000000
02(G) 0.183206E-11 0.000000
H2(G) 0.180604E-11 0.000000
TOTAL LIGAND AREAS 2
TOTAL MAIN AREAS 23
A SI F6 <2-> (AQ) + Pb
' B SI F6 <2-> (AQ) + Pb <2+> (AQ)
C SIF6<2->(AQ) + PB 02D SIF6<2->(AQ) + PBF2(S)E SIF4[H20]2(AQ) + Pb
F SI F4 [H2012 (AQ) + Pb <2+> (AQ)
G SIF4[H20]2(AQ) + PB 02H SIF4[H20]2(AQ) + PB<4+>I SIF5[H20] <1->(AQ) + Pb
J SI F5 [H20] <1-> (AQ) + Pb <2+> (AQ)
K SIF5[H20] <1->(AQ) + PB 02
L F<->(AQ) + Pb
M F<->(AQ) + PB 02N F<->(AQ) + PBF2(S)O F<->(AQ) + PBSI03P F20(G) + PB 02
Q F20(G) + PB<4+>R Pb + PBF2(S)S Pb<2+>(AQ) + PBF2(S)T PB 02 + PB F2 (S)
U PB F2 (S) + PB SI 03
V H F (AQ) + Pb
W HF2<1-> (AQ) + Pb
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(Sb-F)-Si-H 20 System
S b - D - S i - H a O S y s t e mn p u t P i l e :ITLEh vs pH diagram for the (Sb-F SWfiO systemctivities of afl Sb soluble components = 1e-6
M
(AQ)
<+> (AQ)
0
ANDS
(AQ)
>
4 06(c)
2 05(c)
(c)
->(aq)
F6
<2->
(aq)
<1->(AQ)
>(AQ)
(AQ)
(AQ)
TS
<+>
0.
>
0.
4
06 (C).
2 04(c)
SbOF(aq)
487400.
Sb02H2F(aq)
724700.
Sb06H6<1->(aq)
514093.
FIE
cpdalber
cpdjandat
cpdnpldat
cpdnbsdat
cpdsgtdat
ACTIVITY
1.1.1.
1.1.1.1.
1.10-610-61.1.1.1.1.1.1a-e 1.1.1.1a-e 1.1.1.1.1.
1.1.1.
1. 1.
ALL
COMPONENTS
2,1
AXES
ABSCISSA PH
H
<+> (AQ)
ORDINATE EH
<1->
(AQ)
LIMITS
abscissa
0.4.
ORDINATE-0.51.
TEMPERATURE
298.15
PRESSURE
1.
AREAS
check
all
OVERLAY
H2(G) = 2<1->(AQ)
+
2H<+>(AQ)
1..1..1.
OVERLAY
2
H20
=
02
(G) + 4
<1->
(AQ) + 4
H <+>
(AQ)
1.,1.,1.,1.
PLOT
VISUAL
16.5.16.5.2..1.
mark
lower
abscissa
7
0.5,1.,1.5,2.,2.5,3.,3.5
MARK LOWER ORDINATE 5
-0.25,0.,0.25,0.5,0.75
label
lower
abscissa
9
0.0
0.5.5
1.0
1
1.5 1.5
Z02
2.52.5
3.03
3.5
3.5
4.04.0
LABEL LOWER ORDINATE 7
-.5-0.5
-0.25 -0.25
0. 0
0.25
0.25
0.5 0.5
0. 75 0.75
1. 1.
TITLE LOWER ABSCISSA
pH
TITLE LOWER
ORDINATE
Eh
TITLE
UPPER
ABSCISSA
SYSTEM (Sb-F)-SI-H20 at
25
C
LEGEND
linetype dashed
.2 .2
OVERLAY
linetype
DASHED .5.5
IDENT
linetype DASHED
.2 .4
LIGAND
linetype
full
O u t p u t P i l e :INPUT
WILL
BE TAKEN FROM F:\SB017_2.INP
OUTPUT
WILL BE
WRITTEN TO
F:\SB017_2.0UT
Eh vs
pH
diagram
for
the
(Sb-F)-Si-H20
system
Activities
of all Sb soluble components = 1e-6
ACTIVITIES OF SPECIES
<1->(AQ) 1.00000
H<+>(AQ) 1.00000
H20
1.00000
H2SI03
1.00000
HSI03<1->(AQ) 1.00000
SI03<2->
(AQ)
1.00000
SIH4(G)
1.00000
Sb 1.00000
SbO<+> 0.100000E-05
Sb02<1-> 0.100000E-05
Sb4 06(c) 1.00000
Sb204(e) 1.00000
Sb2
05(c) 1.00000
SbH3(g)
1.00000
HSb02(aq) 1.00000
Sb03H3(aq) 1.00000
Sb04H4<1-> 0.100000E-05
SbF3(c) 1.00000
SbOF(aq)
1.00000
Sb"02H2F(aq) 1.00000
Sb06H6<1->(aq) 0.100000E-05
SiF6<2->(aq) 1.00000
Si F4 [H20]2
(AQ) 1.00000
SIF5[H20] <1->(A 1.00000
F<1->(AQ) 1.00000
HF(G) 1.00000
HF
(AQ)
1.00000
HF2<1->(AQ) 1.00000
F20(G)
1.00000
02(G) 1.00000
H2(G) 1.00000
MAIN SPECIES
PER AREA
2
LIGAND SPECIES
PER
AREA
1
TOTAL LIGAND AREAS 2
TOTAL
MAIN AREAS
9
A Sb
+
SiF6<2->(aq)
B Sb + SIF5[H20] <1->(AQ)
C
Sb
O <+>
+
Si F6 <2-> (aq)
D
SbO<+> +
SIF5[H20] <1->(AQ)
E Sb406(c)
+
SiF6<2->(aq)
F Sb2 04(c)
+
SiF6<2->(aq)
G Sb2 04 (c) + SI F5 [H20] <1-> (AQ)
H
Sb2 05 (c) + Si F6 <2-> (aq)
I
Sb+HF
(AQ)
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(Bi-F)-Si-H20 System
( B i - D - S i - H a O S y s t e mn p u t F i l e :
TITLEEhvspH diagram for the (BI-F)-SWtiO systemActivities of Bi soluble components = 1e4
RAM
•> (AQ)
(AQ)
NDS
<2-> (AQ)
)
i
<+3>
(aq)
0 H <+2>
012H12<+6>(aq)
O20H20<+7>(aq)
id
021 H21 <+€>(aq)
i9
022 H22
<+5> (aq)
i2 03
0
<+>
H3 (g)
<2->(AQ)
<1->(AQ)
(AQ)
(AQ)
(AQ)
TS
s
i <+3> (aq)
20.
i 0 H <+2>
.
i6
012H12<+€>(aq)
00.
i9 021 H21
<+€> (aq)
H22 <+5>(aq)
144446.
Bi
H3(G)
231431.
FIE
cpdalber
cpdjandat
cpdnpldat
cpdnbsdat
ACTIVITY
1. 1.1.
1.1.1.1.
1.1e41e41e41e41e41e41.1.1.1.1e41.1.1.1.
1.1.1.1.1.
1.1.
COMPONENTS
2,1
AXES
ABSCISSA
PH
H<+>(AQ)
ORDINATE EH
<->
(AQ)
LIMITS
abscissa 0.4.
ORDINATE -0.51.
TEMPERATURE
298.15
PRESSURE
1.
AREAS
check
all
OVERLAY
H2(G) = 2<->(AQ) + 2H<+>(AQ)
1..1..1.
OVERLAY
2 H20 = 02 (G)
+
4
<-> (AQ) +
4 H <+> (AQ)
1.,1.,1.,1.
PLOT
VISUAL 16.5,16.5,2.,1.
mark lower abscissa
7
0.5,1.,1.5,Z,2.5,3.,3.5
MARK
LOWER
ORDINATE 5
-0.25,0.,0.25,0.5,0.75
label lower abscissa 9
0.0
0.5.5
1.0
1
1.5 1.5
2.0
2
2.5 2.5
TRIE LOWER ABSCISSA
pH
TITLE
LOWER ORDINATE
Eh
TITLE UPPER ABSCISSA
SYSTEM
(Bi-F)-SI-H20at25C
LEGEND
linetype
dashed
.2 .2
OVERLAY
linetype DASHED
.5.5
IDENT
linetype DASHED .2 .4
LIGAND
=
1e4
O u t p u t F i l e :Eh
vs pH diagram for
the
(Bi-F)-Si-H20
system
Activities
of all Bi soluble components
=
ACTIVITIES
OF SPECIES
<->(AQ) 1.00000
H<+>(AQ) 1.00000
H20 1.00000
H2SI03 1.00000
HSI03<1->(AQ) 1.00000
SI03<2->
(AO)
1.00000
SIH4(G) 1.00000
Bi 1.00000
Bi<+3>(aq) 0.100000E-03
BiOH<+2> 0.100000E-03
Bi6 012H12<+«>(a 0.100000E-03
Bi9
020
H20 <+7> (a 0.100000E-03
Bi9
021 H21 <+€>
(a 0.100000E-03
Bi9
022
H22 <+5> (a 0.100000E-03
Bi203
Bi205
Bi204
Bi4
07
BiO<+>
BiH3(g)
SI F6 <2 > (AQ)
1.00000
1.00000
1.00000
1.00000
0.100000E-03
1.00000
1.00000
SI F4 [H20]2 (AQ) 1.00000
SIF5[H20]
<1->(A 1.00000
F<->(AQ) 1.00000
HF (G) 1.00000
HF
(AQ)
1.00000
HF2<1-> (AQ) 1.00000
F2 0(G) 1.00000
02(G) 1.00000
H2(G) 1.00000
MAIN SPECIES PER AREA 2
LIGAND SPECIES PER AREA
TEMPERATURE OF
SYSTEM
298.15
K
TOTAL
LIGAND
AREAS
2
TOTAL MAIN AREAS
6
A Bi + SIF6<2->(AQ)
B Bi +
SIF5[H20J
<1->(AQ)