561 ANNALS OF GEOPHYSICS, VOL. 48, N. 4/5, August/October 2005 Key words silicate melts – structure – entropy – un- mixing 1. Introduction In Bernal’s words (Bernal, 1959), a liquid is an «homogeneous, coherent and essentially irreg- ular assemblage of molecules containing no crys- talline regions or holes large enough to admit an- other molecule». This acceptance of a liquid structure is particularly in line with the Random Network Model (RNM) of Zachariasen (1932, 1933) and Warren (1933) for silica melts and glasses: silicate tetrahedrals linked through cor- ners without any long-range periodicity. Already in the 1960s, Evans and King (1966) had physi- cally built up an RNM composed of Corning Glassworks tetrapods linked by bent springs, the angles of the bends having a Montecarlo-generat- ed Gaussian distribution centered at 163°. The measured radial distribution of atoms around the silicon and oxygen reference atoms of the physi- cal model appeared to be in good agreement with existing neutron and X-ray scattering data (War- ren et al., 1936; Warren, 1937; Carraro, 1964). More than half a century of experimental re- search essentially confirmed the first illuminating views of Zachariasen, Warren and Bernal, and nowadays we may basically state the following: – The structure of silicate melts and glasses has elements of long-range randomness coex- isting with short- and medium-range order (Gaskell et al., 1991). – The short-range order is dominated by the structure of the SiO4 rigid unit (Greaves et al., 1981), although the distance beyond which the radial distribution function becomes indistin- guishable from the 4π r 2 ρ 0 function, represent- ing homogeneous distribution of electron den- sity, shrinks progressively with the introduction Chemical interactions and configurational disorder in silicate melts Giulio Ottonello Laboratorio di Geochimica, Dipartimento per lo Studio del Territorio e delle sue Risorse (DipTeRis), Università degli Studi di Genova, Italy Abstract The Thermodynamics of quasi-chemical and polymeric models are briefly reviewed. It is shown that the two classes are mutually consistent, and that opportune conversion of the existing quasi-chemical parameterization of binary interactions in MO-SiO2 joins to polymeric models may be afforded without substantial loss of preci- sion. It is then shown that polymeric models are extremely useful in deciphering the structural and reactive prop- erties of silicate melts and glasses. They not only allow the Lux-Flood character of the dissolved oxides to be established, but also discriminate subordinate strain energy contributions to the Gibbs free energy of mixing from the dominant chemical interaction terms. This discrimination means that important information on the short-, medium- and long-range periodicity of this class of substances can be retrieved from thermodynamic analysis. Lastly, it is suggested that an important step forward in deciphering the complex topology of the inho- mogeneity ranges observed at high SiO2 content can be performed by applying SCMF theory and, particularly, Matsen-Schick spectral analysis, hitherto applied only to rubberlike materials. Mailing address: Dr. Giulio Ottonello, Laboratorio di Geochimica, Dipartimento per lo Studio del Territorio e delle sue Risorse (DipTeRis), Università degli Studi di Ge- nova, Corso Europa 26, 16132 Genova, Italy; e-mail: giot- [email protected]
Transcript
561
ANNALS OF GEOPHYSICS, VOL. 48, N. 4/5, August/October 2005
Key words silicate melts – structure – entropy – un-mixing
1. Introduction
In Bernal’s words (Bernal, 1959), a liquid isan «homogeneous, coherent and essentially irreg-ular assemblage of molecules containing no crys-talline regions or holes large enough to admit an-other molecule». This acceptance of a liquidstructure is particularly in line with the RandomNetwork Model (RNM) of Zachariasen (1932,1933) and Warren (1933) for silica melts andglasses: silicate tetrahedrals linked through cor-ners without any long-range periodicity. Alreadyin the 1960s, Evans and King (1966) had physi-
cally built up an RNM composed of CorningGlassworks tetrapods linked by bent springs, theangles of the bends having a Montecarlo-generat-ed Gaussian distribution centered at 163°. Themeasured radial distribution of atoms around thesilicon and oxygen reference atoms of the physi-cal model appeared to be in good agreement withexisting neutron and X-ray scattering data (War-ren et al., 1936; Warren, 1937; Carraro, 1964).More than half a century of experimental re-search essentially confirmed the first illuminatingviews of Zachariasen, Warren and Bernal, andnowadays we may basically state the following:
– The structure of silicate melts and glasseshas elements of long-range randomness coex-isting with short- and medium-range order(Gaskell et al., 1991).
– The short-range order is dominated by thestructure of the SiO4 rigid unit (Greaves et al.,1981), although the distance beyond which theradial distribution function becomes indistin-guishable from the 4πr 2ρ 0 function, represent-ing homogeneous distribution of electron den-sity, shrinks progressively with the introduction
Chemical interactions and configurationaldisorder in silicate melts
Giulio OttonelloLaboratorio di Geochimica, Dipartimento per lo Studio del Territorio e delle sue Risorse (DipTeRis),
Università degli Studi di Genova, Italy
AbstractThe Thermodynamics of quasi-chemical and polymeric models are briefly reviewed. It is shown that the twoclasses are mutually consistent, and that opportune conversion of the existing quasi-chemical parameterizationof binary interactions in MO-SiO2 joins to polymeric models may be afforded without substantial loss of preci-sion. It is then shown that polymeric models are extremely useful in deciphering the structural and reactive prop-erties of silicate melts and glasses. They not only allow the Lux-Flood character of the dissolved oxides to beestablished, but also discriminate subordinate strain energy contributions to the Gibbs free energy of mixingfrom the dominant chemical interaction terms. This discrimination means that important information on theshort-, medium- and long-range periodicity of this class of substances can be retrieved from thermodynamicanalysis. Lastly, it is suggested that an important step forward in deciphering the complex topology of the inho-mogeneity ranges observed at high SiO2 content can be performed by applying SCMF theory and, particularly,Matsen-Schick spectral analysis, hitherto applied only to rubberlike materials.
Mailing address: Dr. Giulio Ottonello, Laboratorio diGeochimica, Dipartimento per lo Studio del Territorio edelle sue Risorse (DipTeRis), Università degli Studi di Ge-nova, Corso Europa 26, 16132 Genova, Italy; e-mail: [email protected]
562
Giulio Ottonello
of alkalis («network modifiers»; see later) in thesystem (Waseda and Suito, 1977), which is in-dicative of significant break-up of the randomnetwork.
– There are no systematic structural differ-ences between silicate melts and glasses ofidentical composition, and the effects of ther-mal expansion are virtually negligible on both
Table I. Structural data for binary silicate melts and glasses (from Brown et al., 1995). Interionic distance rij(A)approximated to ± 0.01 A; coordination number Nij approximated to ± 0.3 atoms; root mean square atomic dis-placement ∆rij
(1) Waseda and Toguri (1990); (2) Waseda and Suito (1977); (3) Waseda and Toguri (1977); (4) Waseda andToguri (1978); * pair correlations for K-O and O-O overlap, so that coordination numbers and peak widths couldnot be determined independently; # O-O pair correlation overlaps with Fe-Fe correlations, thus O-O distancecould not be determined independently.
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Chemical interactions and configurational disorder in silicate melts
mean structure and single interionic distances(see, for this purpose, the review article ofBrown et al., 1995, and references therein).
Table I lists some important X-ray scatteringdata concerning the structure of binary silicatemelts collected by Brown et al. (1995) and es-sentially in line with the above three points.Table II, based on the extensive compilation byBrown et al. (1995), lists concisely the meancoordination numbers experimentally observedfor the various cations in melts and glasses. Al-though a generalized increase in the coordina-tion number of the central cation is observedwith the increase in the Lux-Flood basicity ofthe corresponding oxide, the scatter in CN ob-served for basic components is a clear indica-tion that network modifiers «adapt» themselvesto the geometry established by the muchstronger (and essentially covalent) bond of net-work formers, and not vice versa. In this light,it is difficult to agree with the hypothesis ofGaskell et al. (1991) that compositionally simi-lar pyroxene or pyroxenoid glasses may be rep-resented as parallel random-fractal sheets of in-terconnected CaO6 octahedra interleaved by Siatoms (occupying intersitial tetrahedral siteslinked in infinite chains). More probably, in thepresent author’s opinion, the structure of sili-cate melts or glasses locally resemble that ofall-Si zeolites, in which «cages» of variable co-ordination number are determined by the rela-tive symmetry-determining arrangements ofSiO4 monomeric units.
Despite this topological controversy, it isquite evident that conformational disorder doesexist in silicate melts and glasses. Exactly howthermodynamics can account for this confor-mational disorder may be purely phenomeno-logical (as in subregular models and polynomi-al expansions) or may be based on more sophis-ticated physical models (as in the GaussianRandom Walk method; see later). In any case,to be satisfactory, the thermodynamic modelmust not only reproduce the expected reactiveproperties of the substance, but also the inho-mogeneity ranges often observed at high SiO2
content. We discuss below the informational content
arising from sound thermodynamic treatmentof silicate melt energetics, and propose some
conceptual guidelines to be followed in the nearfuture, in order to decipher the complex fea-tures of the inhomogeneity ranges observed athigh SiO2 content along simple MO-SiO2 bina-ries, and also in chemically complex systems.
Guggenheim’s quasi-chemical model (Gug-genheim, 1935; Fowler and Guggenheim, 1939)assumes that, in a binary system with compo-nents 1 and 2, «particles» 1 and 2 mix substitu-tionally in a quasi-lattice obeying short-range or-dering dictated by the equilibrium
Table II. P=1 bar mean M-O coordination numberof cations in silicate melts and glasses. Listed figuresare largely based on extensive compilation of Brownet al. (1995). Allocation among acidic, amphothericand basic oxides after Ottonello and Moretti (2004)and this work (#).
The molar enthalpy change associated withequilibrium (2.1) is expressed in terms of pairbond energies ε ij
(2.2)
where N is the total number of atoms and Z isthe coordination number.
The non-configurational part of the molarentropy is also given in terms of bond contribu-tions σ ij (non-configurational entropy of the ijpair bond)
(2.3)
If Xij denotes the molar fraction of the ij particlewith respect to the total of the particles formed inthe system, since each atom i is bonded to Zj
( ) .NZ2
2 12 11 22= - -~ h h h
( )NZ2
2 12 11 22= - -~ f f f
[ ] [ ] [ ] .1 1 2 2 2 1 2,- + - - neighbors, it follows from mass balance that
(2.4)
The enthalpy of mixing then becomes
(2.5)
the non-configurational (excess) entropy ofmixing is
(2.6)
and the configurational (ideal + excess) entropyof mixing is
(2.7)
( )ln ln
ln ln ln
S R X X X XRZ
XXX
XXX
XX XX
2
2
, conf 1 1 2 2
111211
2222
2212
1 2
12
mixing $
$
=- + -
+ +c m
SX2, non conf
12mixing = h-
HX2
12mixing = ~
( ) ( ) .X
X X X X2 2
121 11 22= - = -
Fig. 1a-d. Integral enthalpy of mixing and integral entropy of mixing of quasi-chemical model for differentvalues of interaction parameter ω along MO-Si0.5O2 join ((a) and (b), respectively). Below: if usual molar nota-tion (i.e., component SiO2 instead of Si0.5O) is adopted and parameter ω is kept unvaried, symmetricity about in-termediate composition is lost ((c) and (d) respectively).
a b
c d
565
Chemical interactions and configurational disorder in silicate melts
because when the mixture is completely ran-dom, X11=X1
2, X22=X22 and X12=X1X2.
The equilibrium concentrations of the vari-ous pairs obey the Gibbs free energy minimiza-tion principle
(2.8)
This, through opportune expansions in terms ofeqs. ((2.5) to (2.7)), gives
(2.9)
The name «quasi-chemical» assigned to themodel stems from the fact that the term on theright in eq. (2.9) reduces to a constant, for con-stant values of ω and η, thus resembling the no-tation of a chemical reaction among ideal [1–1],[2–2] reactants and [1–2] product.
Equation (2.9) yields (Pelton and Blander,1986)
(2.10)
where
(2.11)
The model is resolved first in terms of equilib-rium distribution of particles (eqs. (2.10), (2.11)and (2.4)) and then in terms of enthalpy and en-tropy (eq. (2.5) and eqs. ((2.6) to (2.7)), respec-tively).
The computed enthalpic and entropic termshave a typical V- and M-shaped conformation,the more pronounced as the value assigned to ωbecomes more negative. The quasi-chemical en-thalpy of mixing and entropy of mixing curvescomputed with Z=2, η=0 and various ω areshown for comparison in fig. 1a,b, respectively,for a MO-Si0.5O system. When ω and η are setat zero the mixture is obviously ideal and, whenη=0 and X12 =2X1X2, the mixture is strictly reg-ular. In all cases, mixtures are always symmet-ric in the compositional space of interest.
( ).expX X
ZRTT
1 42
11 2
2
1
= +-
-p~ h
c m; E( 2
X X X2 1
212 1 2=+ p
.expX X
X T
ZRT4
2
2211
122
=- -~ h^ h; E
T( ).
XG
XH S
0mixing mixing mixing
12 1222
22
= =-
2.1. Modified quasi-chemical approach
The symmetricity about intermediate com-position was long regarded as a severe limita-tion to the application of the quasi-chemicalmodel to silicate melts and slags which invari-antly exhibit maximum ordering about the com-position XMO= 2/3 in the MO-SiO2 chemicalspace. In order to shift the maximum orderingcondition depicted by the quasi-chemical mod-el to the condition actually observed in MO-SiO2 melts, Pelton and Blander (1986) pro-posed substituting true molar fractions X1, X2
with «equivalent fractions» Y1, Y2 so that
(2.12)
(2.13)
Coefficients b1 and b2 are chosen in such a wayas to have b1/(b1+b2)=1/3 and b2/(b1+b2)=2/3.Clearly, when X1=2/3 and X2=1/3, we haveY1=Y2=0.5.
Through this substitution it follows that, in-stead of eq. (2.4), the following equation holds(Pelton and Blander, 1986):
(2.14)
The molar enthalpy of mixing is now (Peltonand Blander, 1986)
(2.15)
the non-configurational (excess) entropy of mix-ing is
(2.16)
and the configurational (ideal + excess) entropyof mixing is
(2.17)
( )
( ) .
ln ln
ln ln ln
S R X X X XRZ
XYX
XYX
XY YX
b X b X
2
2
,mixing conf 1 1 2 2
111211
222222
121 2
12
1 1 2 2
$
$ $
$
=- + -
+ +
+
c m
( )SX
b X b X2,mixing non conf
121 1 2 2= + h-
( )HX
b X b X2mixing
121 1 2 2= + ~
( ) ( ) .X
Y YX X2
121 11 2 22= - = -
.Yb X b X
b X2
1 1 2 2
2 2=+
Yb X b X
b X2
1 11
1 1 2
=+
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Giulio Ottonello
Although whatever b1, b2 couple exists, so thatb1/b2=1/2 results in Y1/Y2=1/2, application ofthe additional condition that configurational en-tropy must attain zero when ω=−∞ yieldsb1= 0.6887, b2= 1.3774 for binary MO-SiO2
melts with maximum ordering about XMO=2/3(cf. Pelton and Blander, 1986, for details).
Lastly, in order to assign more computation-al elasticity to the model, parameters ω and ηare expanded as polynomial functions of equiv-alent fractions Y1, Y2, i.e.
(2.18)
(2.19)
Actually, the asymmetricity of MO-SiO2 joins isdue to the difference in the amounts of oxygenin the two end-member components. If we adoptSi0.5O instead of SiO2 as end-member, maxi-mum ordering will be observed at XMO=0.5, andno readjustment in terms of equivalent fractionswill be necessary. Obviously, if we wish to rep-
Y .Y Y nn
0 1 2 2 22
2f= + + +h h h h h
YY Y nn
0 1 2 2 22
2f= + + +~ ~ ~ ~ ~
Table IV. Non-configurational excess entropy parameters of modified-quasi-chemical model for binary silicatemelts (after Pelton et al., 1995).
Chemical interactions and configurational disorder in silicate melts
resent the chemical space of interest in terms ofMO-SiO2 components, then it is sufficient toconsider that two moles of Si0.5O correspond toone mole of SiO2, the resulting molar fractionsXMO, XSiO2 are translated along the abscissa, andan apparent asymmetricity arises, as depicted infig. 1c,d.
Tables III and IV list the extensive modi-fied-quasi-chemical parameterization of binarysilicate melts of Pelton et al. (1995). The listedparameters allow precise conformation ofphase boundaries along the binary joins and arealso consistent with observed unmixing phe-nomena.
3. Configurational disorder in polymermodels
In polymeric models for silicate melts, it ispostulated that, at each composition, for givenvalues of P and T, the melt is characterized by anequilibrium distribution of several ion species ofoxygen, metal cations and ionic polymers ofmonomeric units SiO4
4–.The charge balance of a polymerization re-
action involving SiO44– monomers may be for-
mally described by a homogeneous reaction in-volving three forms of oxygen: singly bondedO-, doubly bonded (or «bridging oxygen») O0,and free oxygen O2− (Fincham and Richardson,1954)
(3.1)
In fact, eq. (3.1) is similar to a reaction betweenmonomers
(3.2)
Polymer chemistry shows that, the larger thevarious polymers become, the more their reac-tivity is independent of the length of the poly-mer chains. This fact, known as «the principleof equal reactivity of co-condensing functionalgroups», has been verified in fused polyphos-phate systems (which, for several properties,may be considered as analogous to silicatemelts; cf. Fraser, 1977) with polymeric chainslonger than 3PO4
4– units (Meadowcroft and
.SiO Si O OSiO 244
2 76
44
, ++ - - --
.O O O2melt meltmelt
0 2, +- -
Richardson, 1965; Cripps-Clark et al., 1974).Assuming this principle to be valid, equilibriumconstant K18
(3.3)
(in which the terms in brackets represent thenumber of moles per unit mole of melt) is al-ways representative of the polymerizationprocess, independent of the effective length ofthe polymer chains.
3.1. Toop-Samis model
Toop and Samis (1962a,b) showed that, in aMO-SiO2 binary melt, in which MO is the ox-ide of a basic cation completely dissociated inthe melt, the total number of bonds per mole ofmelt is given by
(3.4)
where NSiO2 are the moles of SiO2 in the MO-SiO2 melt. The number of bridging oxygens isthus
(3.5)
Mass balance gives the number of moles of freeoxygen per mole of melt
(3.6)
where (1−NSiO2) are the moles of basic oxide.
Equations ((3.3) to (3.6)) yield
(3.7)
(O–) is thus given by the quadratic equation
(3.8)
which may be solved for discrete values of K18
and NSiO2.
( ) ( ) ( )( )
( )
K N
N N
O O4 1 2 2
8 1 0
SiO
SiO SiO
218 2
2 2
- + + +
+ - =
- -
( )[ ( )] [ ( )]
.KN N
4 O4 O 2 2 OSiO SiO
18 2
2 2=- - -
-
- -
( )( )
.NOO
12SiO
22= - --
-
^ h
( )( )
.N
OO
24 SiO0 2=
- -
( )( ) NO O2 4 SiO0 2
2=-
( )( )( )
KO
O O18 2
0 2
= -
-
568
Giulio Ottonello
The Gibbs free energy change involved ineq. (3.1) is
(3.9)
Since two moles of O− produce one mole of O0
and one of O2−, the Gibbs free energy of mixingper mole of silicate melt is given by
(3.10)
3.2. Extended Polymeric Approach (EPA)
Although the original Toop-Samis model is apowerful tool in deciphering the complex rela-tions existing between chemical interaction andconfigurational disorder, there are two minorpoints that must be quantified, i.e. assessment ofthe low polymerization limit, and extension tothe multicomponent field.
The usual expression of the thermodynamicactivity of a component in mixture (Ottonello,2001) yields
(3.11)
As the mean number of monomers in the poly-mer chain is defined as
(3.12)
adopting Temkin model activities of the fusedsalts, it is quite evident that, along any binaryjoin, we may pose
(3.13)
Although the domain of eq. (3.13) spans the en-tire compositional range, it is obvious that can never be lower than one (i.e., a monomer)and, consequently, the limit of maximum de-polymerisation defines a limiting activity rep-resented by
Sior
( )( )
( ).
aO
O
SiO
,MO binary
22
2Si =
-o -
-
r
( )SiO anions2Si =or /
( )( )
.expln
aRT
RT K XX
G2O
1 MOMO
mixing
18
MO22
=+ -
-
> H
( ).lnG RT K
2O
mixing 18=-∆-
.lnG RT K18180 =-∆
(3.14)
Thus, the molar abundances of the various oxy-gen species cannot be expected to be those ofthe original Toop-Samis model, especiallywhen highly basic oxides are present in the liq-uid phase. In solving thermodynamic activityon a Temkin model basis, Toop and Samis(1962a,b) observed that the mean extension ofthe polymer chains is univocally defined by a«polymerzsation path» depicted in terms ofmean numbers of silicon atoms per polymerunit versus the stoichio-metric ratio (O–)/[(O–)+(O–)+SiIV] (cf. figs. 2 and3 in Toop and Samis, 1962a). However, theirfurther assumption – that a single polymeriza-tion path of general validity in the ternary sys-tem CaO-FeO-SiO2 may be proposed on the ba-sis of viscosity data – cannot be shared, becausea different reaction constant, K18, pertains toeach MO-SiO2 system and, as the activity of thebasic dissolved oxide MO is implicitly definedby the partial derivative of the Gibbs free ener-gy of mixing at any point of the compositionalspace of interest, we have (Ottonello, 2001)
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)( )Si SiO2IV =
( )( )
O XO2MO
2 = --
-
( )( )
OX O
24 40 MO=
- - -
( )
( )( )
X X
KK X X
2 4 4 2
8 232 4 1
MO MO
P MO MO
2
18
221
- + - +
-- - -
-( )O =6
@
( )ln
XG
RT KXO
21
MO
mixing
MO182
222
=-
( )
( )
expRT
GG
anionsX
X
O
1mixing MOMO
mixing
2
22
=+ -
-
; E* 4
/
( )N anionsSiOSi 2=or /
( ) ] .SiO#[( )SiO!=( ) ( )
( )a
O SiOO
, minMO binary 22
2
2 2+-
-
569
Chemical interactions and configurational disorder in silicate melts
where the terms in brackets denote the numberof moles and XMO is the molar fraction of ageneric basic oxide MO in a MO-SiO2 binaryjoin.
Distinct «polymerization paths» relating themean number of silicon atoms in the polymericunits ( ) to the relative proportion of singlybonded oxygen in the unit (O–)/ [(O–)+(O0)+SiIV]may be calculated for the two limiting binariesCaO-SiO2 and FeO-SiO2 on the basis of thesystem of eqs. ((3.15) to (3.20)). These pathsdiffer substantially from the general path pro-posed over 40 years ago by Toop and Samis(1962a,b) and, more importantly, they differgreatly from each other.
The apparent complexity introduced by rec-ognizing that each oxide imposes its own im-print on the extension of polymeric units turns
Sior
out to be a powerful tool in deciphering the re-active properties of chemically complex melts.In fact, chemical interactions within the anionmatrix of an n-component system are complete-ly fixed by the interaction properties observedalong the n-1 limiting binaries. As discussedelsewhere (Moretti and Ottonello, 2003), the in-trinsic thermodynamic significance of this evi-dence is apparent in the Flood-Grjotheim ther-modynamic cycle (Flood and Grjotheim, 1952).If KP,A-F, KP,B-F are the polymerization con-stants valid along the limiting A-F, B-F bina-ries, the polymerization constant of the extend-ed Toop-Samis model turns out to be
(3.21)
The above equation states that, in a composi-tionally complex melt, network modifier cationsA, B of charge ν+
A, ν+B interact with free oxygen
O2– (and hence with the polymeric units built upby network former F), with a simple propor-tionality described in terms of equivalent frac-tions (Flood and Grjotheim, 1952; Ottonello,2001; Moretti and Ottonello, 2003)
(3.22)
Application of the Flood-Grjotheim thermody-namic cycle to the CaO-FeO-SiO2 system satis-factorily reproduces the bulge in the thermody-namic activity of FeO experimentally observedlong ago by Elliot (1955) (fig. 2). No high-orderterms are necessary to reproduce the observedcomplex thermodynamic behavior, except for atopologically negligible interaction in the cationsublattice.
3.3. Hybrid Polymeric Model
In the Hybrid Polymeric Model (HPM) (Ot-tonello, 2001), the Gibbs free energy of mixingin binary melts is first subdivided into a (domi-nant) chemical Toop-Samis interaction termplus a (subordinate) elastic Hookian-like strainenergy contribution term, as is observed inblock copolymers (see later)
= .N n nn
AA A
A A
B Bf+o oo
+ +
+
+ol
f+ln K+N+ln K+ln K N , ,P A P A F B P B FO= - -+ +ol l
Fig. 2. FeO activity surface at 1600°C for liquidFeO-SiO2-CaO ternary system. Graphic representa-tion obtained by inverse-distance contouring of 9900activity values generated by extended polymericmodel, adopting, as limiting Kp, 0.21 for FeO-SiO2
join and 0.002 for CaO-SiO2 join, respectively. As-sumed cationic interaction WCa-Fe=−33 kJ/mol.Strain energy contributions included (from Ottonel-lo, 2001, with modifications).
570
Giulio Ottonello
(3.23)
In eq. (3.23), is the number of monomericunits of statistical length a extended to a dis-tance x, and the remaining symbols assume theusual thermodynamic significance. The elasticstrain energy depends on bending term x/a(which basically represents the effect of cova-lent bonding on the relative arrangement ofmonomers in the polymer chain). This term isarbitrarily expanded into a polynomial of type
(3.24)
Strain coefficients χ1…χn depict linear or sec-ond-order T dependence. Since the polymeriza-tion path along any join is defined in terms of
versus NSiO2, the strain energy calculatedalong a given binary at various T conditions isintimately related to the extent of polymeriza-tion along any particular compositional path(i.e., the mean number of silicon atoms in thechain, ). The existence of a maximum de-polymerization limit and the approximate na-ture of the entropy terms depicted by the adopt-ed parameterization of bending angles meansthat the problem cannot be solved exactly(more precise definition of entropic intermedi-ate ordering effects on the bulk stability of thephase would require application of the self-con-sistent mean field theory – see later).
To overcome these complexities, Ottonelloand Moretti (2004) modified eq. (3.23) as fol-lows:
(3.25)
In eq. (3.25), S18 is the entropic contributionembodied in the Toop-Samis formulation ofchemical interaction, and η represents the straincontributions arising from the conformationalarrangements of the various polyanions in theanion matrix. This term is again arbitrarily ex-
( ).G H T S T
2O
mixing 18 18= - - h∆ ∆ ∆-
^ h
Sior
Sior
f+N$|+N$|+N$|+N$|=ax
SiO SiO SiO SiO1 2 22
33
44
2 2 2
Sior
( )lnG G G RT K
RTax
2O
23
mixing chemical strain
Si
18
2
= + = +
+o
∆ ∆ ∆-
rb l
panded into a polynomial on NSiO2, the coeffi-cients of which are inversely dependent on T
(3.26)
(3.27)
The new strain energy term −Tη is more akin tothe modified-quasi-chemical formulation of thenon-configurational entropy of mixing (Peltonand Blander, 1986) with respect to the originalformulation (Ottonello, 2001) and allows bettercomparison of the two model energetics. Obvi-ously, eq. (3.27) implies that, here, strain energyis not purely entropic, but embodies enthalpicand entropic sublattice interactions within theanion matrix which are not explained by theoriginal Toop-Samis formulation. This amountof energy is far less than that arising from chem-ical interaction, and important only in quantify-ing the observed unmixing phenomena at highSiO2 content in the mixture (Ottonello, 2001).
Tables V and VI list the results of non-linearminimization calculations carried out to bring themodified quasi-chemical parameterization ofPelton et al. (1995) (tables III and IV) for binaryMO-SiO2 melts to an HPM formulation. Thepolymerization constants were first calculatedfor a single join at various T conditions of inter-est in the specified T range, and then regressed onan Arrhenian function of absolute T. Stemmingfrom the obtained regression coefficients, there isno doubt that the adopted functional has precisethermodynamic significance which results in ex-pansion (3.25) of the previous hybrid model.
The first striking evidence that arises fromHPM treatment of interactions in MO-SiO2
melts is the clear distinction between basic, am-photeric and acidic oxides. Basic oxides, whichare essentially network modifiers in an MO-SiO2
liquid, exhibit purely enthalpic contributions tochemical interactions in mixture, whereas acidicoxides (network formers) give rise to athermalbehavior when mixed among themselves; and,clearly, amphoteric oxides exhibit both entropicand enthalpic contributions to chemical interac-tions in the melt phase.
If we now analyse the standard state en-thalpic contribution to chemical interactions inMO-SiO2 melts we note that they are simply re-
.T, ,1 0 1 1 1= +| | |
N$|+N$|+N$|+|= SiO SiO SiO1 2 32
43
2 2 2h
571
Chemical interactions and configurational disorder in silicate melts
Table V. Chemical interaction of HPM (from Ottonello and Moretti, 2004).
(1) This work; (2) Ottonello and Moretti (2004), based on experiments of Eliezer et al. (1978); (3) Ottonello andMoretti (2004), based on modified quasi-chemical parameterization of Pelton et al. (1995); (4) Ottonello (2001);(5) Ottonello et al. (2001).
Table VI. Strain energy of HPM (from Ottonello and Moretti, 2004).
lated to the atomistic properties of the centralcation, like Pauling’s electronegativity (Paul-ing, 1932, 1960) (fig. 3) or to spectroscopicallyderived magnitudes, such as differences in opti-
cal basicity between metal oxide and silica(Duffy and Ingram, 1974; Gaskell, 1982; Sosin-sky and Sommerville, 1986; Duffy, 1989, 1990;Ottonello et al., 2001) (fig. 4).
(1) Based on modified quasi-chemical parameterization of Pelton et al. (1995); (2) based on modified quasi-chemical parameterization of Pelton and Blander (1986) and Pelton (pers. comm.).
N$|+N$|+N$|+|=
=| T, ,
SiO SiO SiO1 2 32
43
1 0 1 1 1
2 2 2
+
h| |
573
Chemical interactions and configurational disorder in silicate melts
The discrimination capability of optical ba-sicity in terms of Lux-Flood basicity is evenclearer when looking at polymerization con-stant KP. As shown in fig. 5, the relationship be-
tween the natural logarithm of KP and the opti-cal basicity difference is so evident as to allowit to be used as a predictive tool for the chemi-cal interaction of basic oxides with silica. Each
Fig. 3. Reaction enthalpy of equilibrium 2 plotted against Pauling’s electronegativity of MO metal cation.
Fig. 4. Reaction enthalpy of equilibrium 2 plotted against difference in optical basicity of MO and SiO2. Adopt-ed basicity values are those of column 6 in table A1 of Ottonello et al. (2001). Selection of other basicity valueswithin uncertainty ranges (columns 1 to 6 in same table) do not greatly alter observed relations.
Fig. 5. Polymerization constant in MO-SiO2 melts at various T (with MO=basic oxide) plotted against differ-ence in optical basicity of MO and SiO2 (adopted basicity values are those of column 6 in table A1 of Ottonel-lo et al., 2001). Selection of other basicity values within uncertainty ranges (columns 1 to 6 in same table) donot greatly alter observed relations. Dashed lines: extrapolations at T beyond T limits investigated here for mostbasic oxides. Solid heavy line: functional form adopted by Ottonello et al. (2001) and Moretti and Ottonello(2003) when role of T was still unclear.
3 4
574
Giulio Ottonello
of the depicted straight-line dependencies infig. 5 has a correlation coefficient R 2=0.9985,and their slopes and intercepts also exhibit asecond-order dependence on absolute T.
4. Configurational disorder in blockcopolymers
Although there is no experimental evidenceabout the possibile formation of local blockcopolymeric structures in silicate melts or glass-es, recent findings in this branch of materials sci-ence are also important for proper understandingof the unmixing phenomena which take place inSiO2-rich melts. The physics of microphase sep-aration in block copolymers is understood interms of an expanded Flory-Huggins model.
The Flory-Huggins model belongs to theclass of quasi-chemical approaches. Flory him-self defined his method as «a restricted variationof the quasi-chemical method used by Guggen-heim» (cf. Flory, 1953, p. 507). Basically, themain merit of the Flory-Huggins approach is toacknowledge that molar fractions are unsuitablechemical parameters whenever molecules differ-ing greatly in size interact among themselves andwith much smaller stoichiometric units in themelt phase. Volume fractions are thus substitutedfor molar fractions, based on the identity
(4.1)
where φi is the volume fraction of the ith compo-nent in mixture, ni its molar amount, υi its partialmolar volume, and the summation at the denom-inator is extended to all k components in mixture.
Block copolymers are composed of two ormore distinct blocks interconnected by linear orbranched sequencing. The relative arrangementsof the various blocks gives rise to an astonishingnumber of distinct configurations, which is re-sponsible for the many useful and desirable prop-erties exploited by the industry of rubberlike ma-terials. The simplest architecture is the linear ABdiblock: a long sequence of type A monomers co-valently bonded to a chain of type B monomers.The relative arrangements of the two blocks is
n
ni
i i
i
ki i
1
=zy
y
=
/
characterized by «a fluidlike disorder on the mo-lecular scale and a high degree of order at longerlength scales» (Bates and Fredrickson, 1999).
In interpreting the structural and reactiveproperties of diblock copolymers, the followingworking assumptions are commonly made:
– Each molecule in the melt is composed ofν segments, f A-type and (1−f) B-type monomers.
– A and B segments represent a sufficientlength of polymers such that they can be treat-ed as Gaussian, where the internal conforma-tional states of the segments produce a Hookianentropic penalty of stretching.
– The statistical length of a segment is itsaverage RMS end-to-end length when no ten-sion is applied, and is related to the effectivespring constant.
– The average segment concentration isforced to be uniform (incompressibility con-straint).
The interaction between A and B is de-scribed in terms of a χAB Flory-Huggins interac-tion parameter which, in units of thermal ener-gy kT, is expressed as follows:
(4.2)
where Z is the number of nearest-neighbormonomers to a copolymer configuration cell(note the analogy with eq. (2.2)). If χAB is large,chemical interaction leads to a macroscopicsegregation of A and B (binodal or spinodal de-composition). If instead χAB is sufficiently low,the thermodynamic forces driving separationare counterbalanced by entropic restoring con-tributions (i.e., «chain elasticity»; intimatelyconnected to the covalent character of the bondbut Hookian in nature) reflecting the require-ment that, to keep the dissimilar A and B por-tions of each molecule apart, copolymers mustadopt extended configurations (Bates andFredrickson, 1999).
4.1. Mean field approximation theories
Helmholtz free energy F (the incompress-ibility constraint) (Fredrickson et al., 1994,Fredrickson and Liu, 1995) is
kTZ
21
AB AB AA BB= - +| f f f] g: D
575
Chemical interactions and configurational disorder in silicate melts
(4.3)
where q is the reciprocal of the distance in thesearch space and SA, SB are form factors of theA and B block copolymers (Fredrickson et al.,1994; Fredrickson and Liu, 1995; Chen et al.,2000, 2002). These «form factors» assume pre-cise values depending on the extent and geom-etry of the block. For a linear chain L ofmonomers of statistical Kuhn length b, the formfactor is simply SL=νLg(x) with νL=degree ofpolymerization and
(4.4)
(4.5)
For branched polymers the form factors aremore complex and are obtained by a combina-
( )( )
.g xx
x2 e 1x
2=+ --
x q b 6L2 2= o
dqlnq= ( ) ( )FkT
S q S q4 A A B B2
2
0
+r z z3
7 A#tion of arm and backbone contributions (seeFredrickson et al., 1994, 1995, for appropriatetreatment).
4.2. Matsen-Schick self-consistent mean fieldmethod
The spectral analysis developed by Matsenand Schick in a series of articles (Matsen andSchick, 1994a,b; Matsen and Bates, 1996; Mat-sen, 1998) is a powerful predictive tool in deci-phering the complex microphase separationprocesses taking place in rubberlike materials.The theory stems from the Self-ConsistentMean Field (SCMF) treatment of Hong andNoolandi (1981). The form factor functionalsare substituted here by space-occupation func-tions rα(s) describing the space curve occupiedby the αth copolymer, and s is the parameteri-zation variable.
In a system of n starblock copolymers com-posed of M AB diblock arms joined together ata central core, s=0 at the core, s= f at the ABjunction, and s=1 at the end of the first arm. Inthe second arm, s is 1, 1+ f, 2, and so on, untilthe end of the last arm, where s=M is reached.The space occupation function is thus piece-wise continuous with discontinuities at integervalues (fig. 6).
The partition function for a system of ppolymers of polymerization ν and density ρ0 is
(4.6)
P is the probability density functional for a giv-en curve (see eq. 2 in Matsen and Schick,1994a, for the approximate form adopted to de-scribe P); , are (dimensionless) Aand B monomer density operators
(4.7)
(4.8)( ) ( ) ( )ds s sr r r1B
M f
p0
0
$= - -z to c d
+
t^ h 6 @#
( ) ( ) ( )ds s sr r rA
M f
p0
0
$= -z to c d
+
t6 @#
( )rBzt( )rAzt
pD ;
( ) ( ) .exp
Q P M f
d
r r
r r r
1p A B
A BAB
0
#
#
= + - -
-
d z z
ot
| oz z
t t
t t
6 8@ B
' 1
#
#
Fig. 6. Positional parametrization of starblock co-polymers. Copolymer melt is composed of starblockcopolymers made up of alternating blocks consist-ing of f A-type and (1−f) B-type monomeric frac-tions. Positional vector defining relative position ofvarious branches of the ith copolymer is defined interms of polymer number and branch number. Spacefunction is piecewise continuous with discontinu-ities at integer values.
576
Giulio Ottonello
γ(s) is a discontinuity function which attains avalue of 1 when s corresponds to an A-monomerregion and a value of 0 for a B-monomer region.The product of density operators in eq. (4.6)represents the various space-dependent interac-tions in the copolymer melt.
To render the problem mathematicallytractable, the following steps are performed:
1) A Delta functional integral
(4.9)
allows the chain density operator to be replacedby segment volume fractions φΑ.
2) Segment volume fractions are expandedas linear combinations of basis functions
(4.10)
3) Basis functions (Fourier transforms) pos-sessing the symmetry of the phase in questionare selected, i.e., for the gyroid phase with sym-metry (Matsen and Schick, 1994b)
(4.11)
with X=2πx/d and d=size of the cubic (gyroid)unit cell.
Several passages yield SCMF equations, re-ducing the problem to that of average densitiesof the A,B monomers at r in the ensemble ofnon-interacting polymers subject to self-consis-tent external fields (cf. eqs. 11-15 in Matsen andSchick, 1994a).
In various configurational states, Helm-holtz free energy F=−ktlnQ is computed, andthe minimum value (saddle point), attainedthrough steepest descent techniques, is selectedas representative of the stable configuration.
( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) / ( ) ( ) ( )
( ) ( ) ( )
( )
( )
/ cos sin sin cos
sin sin cos sin sin
cos cos cos
coscos cos
f r
f r X Y Z Y
Z X Z X Y
f r X Y
Z Z X
f r
f r
Y
1
2
2 2
3 2 2 2
2 2 2
8 3
4
1
4
2
3
5
$
$
$
$
f
f
=
= +
+
= +
+
=
=
6
6
@
@
Q 3Ia d-
( ) ( ) ( ) ( )r r f r f rf, 2 ,A A A1 2 331 f= + + +z z z z
D1 A A A= -d zΦ Φ t8 B#
5. Discussion
For the sake of clarity, the discussion con-cerning information arising from the thermody-namic treatment of binary silicate melt interac-tions is divided into three distinct sections: 1) in-sights into short- and medium-range structuralarrangement; 2) chemical interaction and long-range disorder; 3) and inhomogeneity ranges.
5.1. Short- and medium-range disorder:insights from the EPA
In fig. 7, the polymerization paths of theEPA for various binary joins overlap the dis-crete valuesfor structural units thought to be present in sili-cate melts and glasses (Toop and Samis,1962a,b). For comparative purposes, in thesame plot, the consolute point in the high-SiO2
portion of each binary join also overlaps. Thissort of plot is very informative, since the fol-lowing evidence may be retrieved:
– In most joins, the melt is fully depoly-merized over a large part of the compositionalrange.
( ) ( )/ [( ) ( ) ]O O O SiSiIV0- + +o - -r
Fig. 7. Polymerization paths of EPA model com-pared with compositions of discrete polymeric unitswhich may form in melt at various SiO2 contents.Adopted polymerization constants in constructingvarious curves are those of table V. Location of con-solute points overlap, for comparative purposes.
577
Chemical interactions and configurational disorder in silicate melts
– Ring connectivity at the consolute pointdepends on the Lux-Flood basicity of the dis-solved oxide (i.e., 4-rings-4-membered forFeO, 10-rings-6-membered for Na2O).
Figure 8 compares experimental evidenceconcerning the speciation of the various formsof oxygen present in a CaO-SiO2 melt at T==1600°C, P=1 bar, obtained by Park and Rhee(2001) by XPS (X-ray Photoelectron Spec-troscopy) with EPA predictions. This plot, be-sides experimentally confirming for the firsttime (as far as we know) the actual presence ofoxide ions in the silicate melt, leaves fewdoubts about the soundness of the polymericapproach in depicting oxygen speciation. Thefit between calculated and observed specia-tion is particularly satisfactory at low SiO2
amounts. At high silica content, the polymericmodel appears to underestimate the degree ofcondensation of the polymeric units to someextent. This may be interpreted (in line withthe preceding observations) as due to strainenergy contribution effects (not accounted forin the model), which may stabilize high-mem-bered rings in the vicinity of the two-phase re-gion.
Fig. 9. HPM themodynamic parameterization of CaO-SiO2 system (solid lines; Ottonello and Moretti, 2004),compared with estimates of modified quasi-chemical model (dashed lines; Pelton et al., 1995).
Fig. 8. XPS measurements of oxygen speciation inquenched CaO-SiO2 melts (after Park and Rhee, 2001),compared with EPA predictions (Ottonello, 2001).
– Incipient decomposition (intrinsic insta-bility) tendentially takes place when ring con-formations change (e.g., 3-to-4 members, 4-to-5 members, and so on).
578
Giulio Ottonello
5.2. Chemical interaction and long-rangedisorder
As already mentioned, the HPM allows usto discriminate chemical (enthalpic + entropic)from elastic strain energy contributions to thebulk Gibbs free energy of mixing in MO-SiO2
joins, and this discrimination is typical of thebasic-amphotheric and acidic Lux-Flood be-havior of MO oxides. The quasi-chemical toHPM conversion does not affect the bulk valueof the Gibbs free energy of mixing of the join inquestion, although it does modify to some ex-tent the relative magnitudes of enthalpic andentropic terms in the two models. As we see infig. 9, for the CaO-SiO2 join, entropic interac-tions observed for a Lux-Flood basic oxidearise entirely from elastic strain in the anionsublattice. Moreover, enthalpic chemical inter-actions are dominant, and the bulk strain ener-gy contribution to the Gibbs free energy of mix-ing is far less and practically confined to thehigh-SiO2 portion of the system. Instead, whena Lux-Flood acidic oxide is dissolved in the sil-icate melt (Al2O3-SiO2 join, fig. 10), althoughstrain energy contributions are again confinedto high SiO2 content, both enthalpic and entrop-
ic chemical interaction terms are present overthe entire compositional range, leading to anoverall «regular» appearance of the bulk Gibbsfree energy of mixing. Undoubtedly, CaO per-turbs the long-range periodicity of the silicatenetwork with local depolymerization effectswhereas, at least apparently, «clusters» of over-all SiO2 (tectosilicate) stoichiometry coexiststably with clusters of overall Al2O3 stoichiom-etry throughout the compositional range. Obvi-ously, the picture is more complex for oxideswith less definite Lux-Flood reactivity.
5.3. Inhomogeneity ranges
Although we can envisage the role of chem-ical interaction in determining (or modifying)short-, medium- and long-range periodicity insilicate melts and glasses, our appraisal is still in-sufficient to appreciate the complex phenomenawhich take place within the instability region. Inthe present author’s opinion, better appraisalmay be achieved by tentatively applying SCMFprocedures to this topologically complex region.
Application of the Matsen-Schick theory tosilicate polymer melts seems particularly prom-
Fig. 10. HPM thermodynamic parameterization of Al2O3-SiO2 system (solid lines; Ottonello and Moretti,2004), compared with estimates of modified quasi-chemical model (dashed lines; Pelton et al., 1995).
579
Chemical interactions and configurational disorder in silicate melts
ising. As anticipated in the introduction, theshort- and medium-range periodicity of silicatemelts and glasses locally resemble that of all-Sizeolites, with cationic «cages» of variable di-mensions, depending on silicate network con-nectivity (fig. 11). In this sort of arrangement, itis still possible to isolate polymeric subunits (in-set, fig. 11) of quasi-crystalline short- and medi-um-range order, but their relative positions per-turb long-range periodicity. The followingworking assumptions are thus proposed, to ren-der the problem feasible for SCMF treatment:
1) A simple, irreducible, A-A starblock-likerepresentation of the network is selected as re-sponsible for the short- and medium-range or-der observed in silicate melts and glasses (figs.6 and 11).
2) It is assumed that, in the two-phase re-gion, coexisting liquids locally mimic the struc-tures of crystalline phases observed at the mono-tectic point.
3) Basic functions representative of thegeometry of fictive intermixing A-A starblocks(or A-B starblocks, in the case of aluminosili-cate melts) are assumed to be represented bythe Fourier expansions of the spatial groups ofthe two solid phases at the monotectic point.
4) Calculations are conducted in terms ofGibbs free energy on the basis of HPM parame-terization, after subtracting chemical interac-tion terms and adding PV contributions.
6. Conclusions
Polymeric models provide an extremelyuseful tool in deciphering the structural and re-active properties of silicate melts and glasses.They not only establish the Lux-Flood charac-ter of the dissolved oxides through opportuneconversion of existing quasi-chemical parame-terization (Pelton and Blander, 1986; Pelton etal., 1995; Pelton, pers. comm.), but also dis-criminate the subordinate strain energy contri-butions to the Gibbs free energy of mixing fromthe dominant chemical interaction terms. Thisdiscrimination allows us to retrieve importantinformation about the short-, medium- andlong-range periodicity of this class of sub-stances from thermodynamic analysis. Howev-er, the conceptual models developed for silicatemelts and glasses are still insufficient to allowthorough appraisal of the complex phenomenawhich take place within the inhomogeneityrange. It is thus suggested that a further impor-tant step may be made by applying SCMF the-ory and, particularly, Matsen-Schick spectralanalysis, hitherto applied only to rubberlikematerials.
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Fig. 11. Structure of an all-Si zeolite with «cages»of variable dimension and coordination number de-termined by SiO4
4– network connectivity. Inset: irre-ducible representation of «starblock-like» copolymerscomposed of (identical) interacting A-A monomers ofsame Kuhn length. Interaction parameter χAA is thatarising from HPM treatment. Bending between adja-cent polymers depends on value attained by χAA anddetermines medium-range order. Long-range perio-dicity is locally interrupted by depolymerization, andrelative arrangement of starblock-like copolymersmay contribute to structural instability (and conse-quent unmixing) through strain energy contributions.
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