Extended Electrokinetic Characterization of Flat Solid Surfaces
Carsten Werner,*,1 Heinz Korber,* Ralf Zimmermann,* Stanislav Dukhin,† and Hans-Jo¨rg Jacobasch*,2
* Institute for Polymer Research Dresden, Hohe Strasse 6, D-01069 Dresden, Germany; and†Ukrainian Academy of Science,Institute of Colloid and Water Chemistry, Kiev, Ukraine
An experimental setup has been developed and applied forthe combined determination of the electrokinetic potential andthe surface conductivity of flat surfaces. The key feature of thenew device (designated as microslit electrokinetic setup) is thevariability of the distance between two parallel flat samplesurfaces (10 mm 3 20 mm) forming a slit channel. The setupallows us to decrease this distance down to about 1 mm keepingthe surfaces parallel. In consequence, streaming potential mea-surements can be performed at a given solid/liquid interfaceboth at conditions where surface conductivity is negligible andat conditions where surface conductivity significantly contrib-utes to the total channel conductivity. The zeta potential iscalculated at different channel geometries based on streamingpotential and channel conductivity data and, alternatively,based on streaming current measurements and the dimensionsof the cross section of the slit channel. The results obtainedwere found to agree well if correct conductivity values for thecalculation of the zeta potential based on the streaming poten-tial data are used. The surface conductivity is determined fromthe extrapolation of the channel conductance values gained ata number of sufficiently small distances between the parallelsample surfaces to the distance zero. An additional feature ofthe developed microslit electrokinetic setup is the assessabilityof the hydrodynamic thickness of adsorbed layers of macromol-ecules or particles at the investigated flat surface. In a series ofmeasurements a plasma-deposited fluoropolymer (PDFP) layeron top of a glass carrier and an adsorption layer of the bloodprotein fibrinogen on top of the PDFP layer were characterizedby zeta potential and surface conductivity measurements indifferent aqueous electrolyte solutions (KCl, KOH, HCl). Forthe PDFP/solution interfaces zeta potential up to –100 mV wereobtained in solutions of neutral pH exclusively due to prefer-ential ion adsorption. After adsorption of fibrinogen the zetapotential is considerably reduced. For the PDFP/solution inter-faces surface conductivities were determined in the range of(1–2) 3 1029 S. The contribution of the diffuse layer to thesurface conductivity has been calculated from the zeta potentialaccording to the approach of Bikerman (Kolloid Z. 72, 100(1935)) and compared with the experimentally determined sur-face conductivity. Based on this comparison ions in hydrody-namically immobile interfacial layers were concluded to con-
Electrokinetic surface characterization as performed bystreaming potential or streaming current measurements hasbeen shown to be applicable to sample surfaces of differentgeometries (1, 2). Van Wagenenet al. (3) suggested a mea-suring chamber to be used for streaming potential measure-ments with flat surface sample geometries which has later beenused by several authors (4–8).
Probably the most important systems for electrokinetic in-vestigations with flat surfaces are polymeric materials. Electri-cal surface charge has been found to be relevant for thebiocompatibility of polymers applied in medical devices aswell as for polymer adhesion in technical systems and numer-ous other areas. An automated streaming potential device forthe electrokinetic surface characterization of materials of dif-ferent shapes—including flat surfaces—has been developed afew years ago (9). By means of this device valuable informa-tion about surface properties of many polymer materials and
1 To whom correspondence should be addressed.2 Deceased.
JOURNAL OF COLLOID AND INTERFACE SCIENCE208,329–346 (1998)ARTICLE NO. CS985787
adsorbed layers on polymer materials has been collected (10).A key result concerns the phenomenon of the preferentialadsorption of electrolyte cations or anions onto indifferentsurface sites of polymer surfaces (11–13). Preferential ionadsorption was found to be an important surface charge for-mation process for the vast majority of polymer materials. (Inthe literature preferential ion adsorption has been frequentlydesignated as “specific ion adsorption.” We are not using thisterm in order to distinguish the process from the specificacid–base interactions.) Another important branch of electro-kinetic studies based on streaming potential measurementswith flat surfaces is the characterization of surfactant or mac-romolecule adsorptionin situ. Examples include the investiga-tion of protein adsorption kinetics at the well-defined transportconditions given for the streaming potential experiment asreported by Nordeet al. (14) and others (15, 16). New insightsabout the interrelation of transport conditions and adsorbedamounts have been revealed. However, the zeta potential asdetermined for adsorbed proteins evades the conventional in-terpretation according to the Gouy–Chapman–Stern–Grahamemodel (17) as successfully applied for the so-called “standardelectrokinetic substrates,” i.e. smooth, impermeable, noncon-ducting solid surfaces. Protein layers are frequently to beregarded as porous and penetrable for small electrolyte ionsand the assumed simple layer structure of the double layermodel is very questionable. Additional information on thecharge carrier distribution in the solid–liquid interface is there-fore desirable in order to support the conclusions which may bedrawn from the zeta potential data. Similar reasoning is validfor numerous polymer interfaces in general, namely, if they areporous or swollen.
Several recent studies with dispersed solids have demon-strated that, along with electrokinetic measurements and titra-tions, the measurement of the surface conductivity (Ks) is veryuseful for the comprehensive electrochemical characterizationof solid surfaces (18, 19).Ks is by definition an excessquantity to occur in addition to the liquid bulk conductivity insolid–liquid interfaces due to the charge carrier accumulationin the electrical double layer. In many cases the zeta potentialrepresents only a part of the mobile charge of the double layer,i.e., the charge beyond the hydrodynamic phase boundary, butKs is related to the total mobile charge. The main concept ofthe surface conductivity on solids was recently elaborated indetail by Lyklema (19) including a broad review of the pre-ceding investigations. We shall follow this fundamental workon the current state of electrokinetics using similar terms inequations and similar nomenclature.
The experimentally determined surface conductivity is usu-ally projected on an interfacial plane, i.e., the excess conduc-tivity is assumed to occur in two dimensions. This simplifica-tion has been found to be advantageous since the precisevolume extension of the interfacial layer relevant for surfaceconductivity is hardly known. A general expression ofKs
compatible with that view consists of the sum of integrals over
infinitesimal thin layers parallel to the solid surface each char-acterized by ion concentrations and mobilities:
Ks 5 F Oi
uziu E0
`
@ci~ x! 2 ci~`!#ui~ x!dx, [1]
where Ks is the total surface conductivity,F the Faradayconstant, i the index of the different ionic species,z thevalency of the ions,c the ion concentrations at the interfaciallayer and in the bulk solution, andu the mobility of the ions.It has to be noted that the mobility of the ions may be causedby both conduction and convection (electroosmotic contribu-tion to the surface conductivity).
The theoretical basis for the interpretation of surface con-ductivity phenomena has been prepared only for simple geom-etries of the investigated solids, i.e., for straight, cylindricalcapillaries and for spherical particles (17, 19). In general, thesurface conductivity caused by the hydrodynamically mobilepart of the double layer (Ksd, diffuse layer conductivity) andthe conductivity by the hydrodynamically immobile part of thedouble layer (Ksi, inner layer conductivity or additional sur-face conductivity) are distinguished.
Recently accumulated experience concerning the surfacecharacterization of suspended spherical particles demonstratesthe high efficiency of the comprehensive electrokinetic inves-tigations in comparison with the measurement of a singleelectrokinetic phenomenon: measurement of the electrical con-ductance of diluted latex dispersions as a function of volumefraction (20), measurements of low-frequency dielectric dis-persion on latex particles (21) and biological cells (22), com-bined electroacoustic and dielectric measurements (23), andcombined electrophoresis and conductance measurements (24)made it possible to calculateKs and, based on that,sm (mobilecharge density).sm was found to be close tost (titratablecharge density) in the mentioned cases. Such agreement indi-cates the validity of the theory actually used in interpretingexperimental data from particle electrophoresis. Moreover, thesmall values ofsz (electrokinetic charge density) as comparedto sm established in these works confirms thatKs is a mostimportant characteristic of these surfaces due to the existenceof additional surface conductivity. Recently, new experimentalproofs of the importance of additional surface conductivitywere obtained in studies of Razilov (25) and Peula-Garcia (26).The neglection ofKs in the interpretation of electrophoreticmobilities was found to cause considerable errors in the cal-culated zeta potential.
The first correct measurements ofKs for a nondispersedsolid have been carried out with a series of cylindrical glasscapillaries of different diameters by streaming potential mea-surements (27). The dependence of the measured streamingpotential on the capillary diameter has been used in this studyto derive Ks. In turn, the consistency ofz obtained from
330 WERNER ET AL.
electroosmosis and streaming potential measurements has beenachieved by means of the corrected conductivity used in thestreaming potential evaluation (28). However, a small but notnegligible scatter in the zeta potential determined for capillar-ies of different diameters was observed to persist after theconductivity correction. The principal difficulty of the ap-proach concerns the unavoidable use of different capillarieswhich may slightly differ in their surface composition. Thereare furthermore restrictions in the preparation of well-definedcylindrical capillaries from the majority of materials with someexceptions like glass. The preparation of polymer films on thesurface of quartz capillaries has already been used with regardto the calculation ofKs (29). However, the success of thecoating procedure is rather difficult to control in these systems.
In contrary, there is a large variety of materials suitablefor the preparation of flat surfaces. Flat polymer samples canfor example be prepared as thin, smooth, and well-definedlayers through different techniques like spin-coating, solu-tion-casting, adsorption, or plasma-deposition on top ofplane carriers. These flat surfaces are further easily assess-ible for complementary surface analytics to be comparedwith the results of electrokinetic studies, e.g., surface spec-troscopies, microscopical investigations, contact angle mea-surements and ellipsometry. Thus, there is an enormousmotivation for the determination ofKs based on electroki-netic measurements with flat sample geometries. However,the condition that a considerable portion of the total con-ductivity of the slit channel is caused by the surface con-ductivity could hardly be fulfilled using the flat samplemeasuring chambers known so far for streaming potentialmeasurements even with the most diluted electrolyte solu-tions. Therefore, a new design of the experimental setup hasbeen developed and applied in this investigation.
The formation of microslit channels between two parallelflat sample surfaces not only provides a means for the com-prehensive electrochemical characterization of the materialsbut permits us to further determine the hydrodynamic thicknessof adsorbed layers of macromolecules or particles. Until nowthe hydrodynamic thickness of protein layers has been esti-mated in several studies for capillary systems, e.g., made offritted glass (30–32). The determination of the hydrodynamicthickness of adsorbed layers on flat surfaces is advantageoussince it provides the possibility to compare the hydrodynamicand the optical layer thickness (as determined by ellipsometry)for identical substrates.
In the subsequent sections the fundamental principles usedas the basis of the novel microslit electrokinetic setup, thedesign and the operation of the device will be reported. A firstseries of data obtained in a comprehensive electrokinetic in-vestigation of a plasma deposited fluoropolymer surface and anadsorption layer of fibrinogen on top of this substrate will bepresented.
FUNDAMENTAL PRINCIPLES APPLIED IN THEMICROSLIT ELECTROKINETIC SETUP
The phenomena of streaming current and streaming potentialoccur in general due to the charge displacement in the electricaldouble layer caused by an external force shifting the liquidphase tangentially against the solid. The convective transportof hydrodynamically mobile ions in the direction of the liquidflow can be either detected directly by measuring the electricalcurrent between two positions up- and downstream via nonpo-larizable electrodes by an electrometer of sufficiently lowinternal resistance to justify the neglection of the back currentthrough the streaming channel (streaming current measure-ment). Alternatively, the electrical potential can be measuredbetween the two electrodes up- and downstream in the liquidflow if the internal resistance of the used electrometer issufficiently high (streaming potential measurement). The mea-sured streaming potential results from a steady state of thecharge separation due to the streaming current and the backcurrent in the streaming channel due to the specific electricalconductivity of the volume embedded.
To determine the electrokinetic potentialz of the hydrody-namic phase boundary from streaming potentials or streamingcurrents relations were established that involve a quantitativeexpression of the force that creates the observed electrokineticphenomenon (3, 19). For that aim the liquid flow in the slitchannel is determined only by the pressure difference acrossthe channel assuming the fluid flow to be of the Hagen–Poiseuille type. Since in the cases considered here the aspectratio h/b of the cross section of the rectangular channel (seeFig. 1) will not exceed 1/100, the channel can be assumed toconsist of two parallel surfaces. The velocity profile of theliquid between two parallel plates for this case is parabolic andthe wall shear rate is given according to
FIG. 1. Schematic representation of the rectangular slit channel. Forexplanation see text.
331EXTENDED ELECTROKINETIC CHARACTERIZATION OF FLAT SOLID SURFACES
g 5 SdvdxD
x5~h/ 2!
5~h/ 2!Dp
hL, [2]
where g is the wall shear rate,v the liquid velocity, x thedistance from the center axis of the channel to a plane in theliquid parallel to the solid surfaces,h the distance between theparallel flat solid surfaces,h the dynamic viscosity of the fluid,andDp the pressure drop across the distanceL that describesthe extension of the channel in the direction of the liquid flow(Fig. 1). Since the excess charge creating the net charge con-vection in the fluid flow is located in a thin layer close to thesolid surface (as compared to the total distance of the flatsample surfaces), the wall shear rate given above can be usedto give the velocityV of the liquid at any distancey from thewall relevant in terms of the thickness of the diffuse layer (theDebye radius):
V 5 E0
y SdvdxD
x5h/ 2
d y 5~h/ 2!Dpy
hL. [3]
The electrical charge entrained with the liquid flow to createthe streaming current is given according to
I str 5 2b Ey50
y5h/ 2
r~ y!V~ y!d y, [4]
with r( y) is the volume charge density of the solution. As thechannel can be assumed to be formed of two parallel flatsurfaces the one-dimensional variant of the Poisson equation(33) can be applied to expressr( y):
r~ y! 5 2ee0
d2C
d y2 , [5]
whereee0 is the dielectricity of the liquid solution andC theelectrical potential at any positiony.
With Eqs. [3–5], the streaming currentIstr may be written as
I str 5 2Dpee0hb
hL Ey50
y5h/ 2
yd yd2C
d y2 . [6]
Through integration by parts of Eq. [6] one obtains
I str 5 2Dpee0hb
hL HFydC
d yGy50
y5h/ 2
2 @C#y50y5h/ 2J [7]
For the solution of Eq. [7] we can assume thatC anddC/d y
at y 5 0 become zero and designateC at y 5 h/ 2 as theelectrokinetic potentialz. Thus, Eq. [7] can be rewritten:
I str 5 2Dpee0hbz
hL. [8]
Solving Eq. [8] forz gives
z 5 2h
ee0
I str
Dp
L
hb. [9]
If the streaming current does not run in a circuit a streamingpotentialUstr is created across the channel due to the chargecarrier transport. The stationary state is characterized by thebalance of the streaming current and a back current caused bybulk conductivity (KB) and surface conductivity in the stream-ing channel:
I back5hbUstrKB
L1
2bUstrKs
L, [10]
and with Istr 1 Iback 5 0 one obtains
hbUstrKB
L1
2bUstrKs
L5
Dpee0hbz
hL. [11]
Solving Eq. [11] forz gives
z 5h
ee0
Ustr
Dp SKB 12Ks
h D . [12]
It is obvious from Eq. [9] and [12] that the streaming currentdepends on the streaming channel dimensions for a givensolution and pressure difference. The streaming potential inthis case is a function of the total specific electrical conduc-tivity of the channel. Equation [12] is frequently used in asimplified form neglecting 2Ks/h, which is justified for largevalues ofh. The ratio of streaming potential and streamingcurrent results in the electrical resistanceR (or its inverse theconductanceC) of the channel according to Ohm’s law:
Ustr
I str5 Rchannel5
1
Cchannel. [13]
The derivation given above assumes the charged flat solidsurfaces to be sufficiently isolated not to affect each other.However, for very small channels the interfacial charge carrierdistribution might be influenced by the interaction between thecharge on the facing surfaces (34, 35). The generalization ofthe streaming current theory for cylindrical capillaries is char-acterized in Refs. (36) and (37) by the multiplierA1(kd). Oneobtains for the streaming current
332 WERNER ET AL.
I str 5 2Dpe0ezS
hLA1~kd!, [14]
where S is the cross-sectional area of the conduit,d is acharacteristic length (the radius of a capillary, orh/ 2 of a slitchannel), andk21 is the Debye radius. The functionA1(kd)expresses the influence of the electrokinetic thicknesskd onthe double layer structure that causes deviation from the gen-eral equation. For broad capillaries, i.e., at large electrokineticradiusA1 approaches 1. For slit channels withkd . 30 thedifference 12 A1 is less then 0.03. The electrokinetic mea-surements in aqueous solutions are mainly related to the con-centration range exceeding 1025 mol/L. As the resultkd in thediscussed situations generally exceeds 30, the correction can beneglected.
Surface Conductivity Measurements with Parallel FlatSurfaces
The combination of the Eqs. [8] and [12] further yields therelationship betweenIstr, Ustr, andKs for the discussed case ofa streaming channel formed of two flat parallel sample sur-faces:
I str 5 ~KBh 1 2Ks!Ustr
b
L. [15]
Using Eq. [15]Ks can be determined from streaming currentand streaming potential data obtained for a known channelgeometry provided that the surface conductivity produces asufficiently large contribution to the total channel conductivity:
Ks 51
2 S I strL
Ustrb2 KBhD . [16]
If the width and the length of the channel are fixed and theheight of the channel (i.e., the distance between the parallel flatsample surfaces) is variable (Fig. 1), the condition for thechannel height to be fulfilled for the proper evaluation ofKs isgiven by
2Ks
KBh$ 1. [17]
The exactness of the determination ofKs is diminished as thechannel height is increased and the condition (Eq. [17]) isviolated.
As an alternative to the use of the streaming current and thestreaming potential the Ohm resistance measured across chan-nels of small, known heights, e.g., by means of a Wheatstonebridge can be used for an analogous evaluation.
Electrokinetic Potential and Surface Conductivity in View ofDouble Layer Models
The parameters ofz and Ks as introduced so far do notinvolve detailed model assumptions regarding the structure ofthe electrochemical double layer. However, to usez andKs forthe evaluation of surface properties of the solid they have to beassigned to model pictures of the interfacial charge carrierdistribution. As pointed out in (19) static double layer modelshave been successfully applied for the interpretation of dy-namic double layer properties like electrokinetics and surfaceconductivity. The widely accepted model for the electrochem-ical double layer according to Gouy, Chapman, Stern, andGrahame is schematically shown in Fig. 2. It is characterizedby the localization of ions both in a fixed layer (Stern layer)and in a diffuse layer of the solid–liquid interface. The fixedlayer is further assumed to be composed by two planes whereimmobile counterions (outer Helmholtz plane) and immobilecoions (inner Helmholtz plane) are adsorbed. The boundarybetween the diffuse layer and the Stern layer (characterized bythe potentialCd) is for the evaluation of electrokinetic mea-surements frequently assumed to coincide with the hydrody-namic solid/liquid boundary, the shear plane characterized bythe potentialz. The existence of a hydrodynamic slip plane hastaken a prominent role among the discussed items concerningthe interpretation ofz. A comprehensive consideration of thedeviations ofe and h at the interface from the bulk solutionvalues has been given with this concern in (38). It is concluded,that the assumption of a discrete shear plane is wrong on amolecular scale but may nevertheless be applied as a usefulsimplification for data evaluation. However, considerable de-viations betweenz and Cd occur for nonideal solid surfacestructures, e.g., due to roughness, porosity, presence of hairylayers (39).
To what extentz approaches the surface potentialC0 de-
FIG. 2. Schematic representation of the electrochemical double layeraccording to the model of Gouy, Chapman, Stern, and Grahame.
333EXTENDED ELECTROKINETIC CHARACTERIZATION OF FLAT SOLID SURFACES
pends further on the degree of surface charge compensation byions in the immobile part of the double layer (i.e., whether thedouble layer structure resembles the Gouy–Chapman–type orthe Helmholtz-type, respectively (40)). This, in turn, dependson the surface charge density and on the kind of ionic compo-nents and concentrations of the liquid phase.
Another important aspect of the interpretation ofz concernsthe molecular discreteness of the surface charge on noncon-ducting solids. Any conclusion drawn on the basis of electro-kinetic measurements on charge formation processes has toconsider the potential heterogenicity of the surface and thedistribution of surface sites.
Analogous to the discussion ofz the assumption of a discreteplane separating the double layer into two parts has beeninvolved to further elucidate the sources of the surface con-ductivity. In that sense the total surface conductivity (Ks) hasbeen attributed to ions located in the diffuse part of the doublelayer (Ksd, diffuse layer conductivity) and to ions “behind”that plane towards the surface (Ksi, inner layer conductivity oradditional surface conductivity):
Ks 5 Ksi 1 Ksd. [18]
Ks is caused by ion concentrations and mobilities in planesparallel to the solid surface (Eq. [1]) and the diffuse layercontribution toKs can be estimated by means ofz for theboundary potential, the bulk solution values for the ion mobil-ities and the Gouy–Chapman approach for the calculation ofthe local ion concentrations. Firstly considered by Bikerman(41), an electroosmotic transport contributes toKs in additionto the conduction. (This electroosmotic transport may as wellcause apparent anomalous viscosities of the liquid in fluid flowthrough very fine microchannels (42).) Integrating over thewhole diffuse layer and assuming the solution to contain onlytwo ion sorts of equal valency the following equation is ob-tained for the part ofKs caused by the diffuse layer (Ksd):
Ksd 5 Î8ee0cRTH u1
A 2 12
u2
A 2 11
4ee0cRT
hzF
1
A2 2 1J , [19]
whereA 5 coth(2zFz/4RT).Using the known diffusivities of the ions, Eq. [19] can be
rewritten for a symmetrical electrolyte:
Ksd 52F2cz2
RTk@D1~ez / 2 2 1!~1 1 3m1!
1 D2~ez / 2 2 1!~1 1 3m2!, [20]
where m6 5 (RT/F)2(2ee0/3hD6) describes the relativecontribution of the electroosmosis toKsd (z 5 zFz/RT andk21 is the Debye radius). Equation [19] involves the assump-tion that z andCd coincide. If this prerequisit is fulfilled the
calculation provides precise results since the composition ofthe bulk electrolyte phase is known and the ion concentrationin the diffuse layer is simply attributed to electrostatics. (Thenecessity of considering the contribution of OH2 and H3O
1 tothe electrolyte composition of the solution is discussed inAppendix 1.)
Ksi can be caused by mobile ions in the Stern layer. How-ever, not only countercharge carriers have to be considered: ifpreferential ion adsorption is creating the interfacial charge theStern layer may contain mobile ions of both signs. The mobil-ity of ions in the Stern layer cannot be assumed to be identicalto the bulk solution mobilities although they can attain thesame order of magnitude. The concentration of the differentions present in the Stern layer depends on their solution con-centrations in pattern influenced by nonelectrostatic interactionforces (van der Waals forces) occurring between the solidsurface and the ions. Therefore, a similar analytical estimationof Ksi as described above forKsd is not applicable. Instead, thedifference ofKs andKsd can be determined according to Eq.[18] to obtainKsi. Ksi may also originate from the conductionin a surface volume of the solid itself, most probably in casesof swollen or porous solid surfaces.
Determination of the Hydrodynamic Thickness of AdsorptionLayers on Flat Surfaces
The fluid flow through a microchannel at known pressuredifferences across the channel permits to derive information onthe channel geometry (43). Ifb andL are large compared toh(see Fig. 1) the dependence of the liquid volume velocityQ ina slit channel on its geometry is given with
Q 5Dph3b
12hL, [21]
andh can be calculated with the use of the measuredQ:
h 5 Î3 12QhL
Dpb. [22]
Equation [22] can be used for the precise determination of theslit channel height before and after the adsorption of macro-molecules or particles from the solution onto the sample sur-face. From the difference divided by 2 the hydrodynamicthicknessd of the adsorbed layer is obtained. The cubed powerdependence of the fluid flow onh permits us to detect varia-tions in the fluid flow caused by adsorption layers of thicknessd about three orders of magnitude below the total channelextensionh.
The strict linear dependence of the streaming current onhyields an additional possibility to determine small distancesbetween the parallel flat sample surfaces. A large referencevalue hr is determined based ondU/dp and dI/dp data for
334 WERNER ET AL.
channel geometries were the surface conductivity can be ne-glected. Subsequently, Eq. [23] can be used to determineh inany range based on the streaming current:
h 5I str~h!
I str~hr!hr. [23]
Thus, ifdI/dp is measured before and after the adsorption stepfor a reasonable smallh, the determination ofd can be basedon that.
DESIGN AND FUNCTION OF THE ELECTROKINETICMICROSLIT SETUP
The general idea elaborated in this study concerns the com-prehensive electrokinetic characterization of two parallel flatsurfaces (10 mm3 20 mm) at varied distance. It has beenintended to vary the distance between the flat sample surfacesfrom about 50mm down to about 1mm in perfect parallelposition without opening the cell and dewetting the samples.Based on this feature streaming current and streaming potentialmeasurements were expected not only to permit the calculationof the zeta potential but further to provide precise informationon the surface conductivity on the solid–liquid interface.
In order to fulfill the above-mentioned requirements a newelectrokinetic setup has been developed (44) since no setupavailable in the past provided the necessary potentialities. Inthe subsequent section the design and the function of this setupwill be briefly described.
Design of the Microslit Electrokinetic Setup
The microslit electrokinetic setup (Fig. 3) is a home-builtdevice consisting of the following components:
● Sample carriers and positioning unit/microscope;● Pressure generation and control unit;● Tubing system and liquid reservoirs;
● Electrometer, conductivity, temperature, and pH sensors,flow meter;
● Personal computer for data collection and control of mea-surement series.
The size of the two flat sample surfaces has been chosen to10 mm3 20 mm in order to use the samples conveniently incomplementary analytical experiments (e.g., XPS, AFM, ellip-sometry, wetting measurements).
The demand of laminar, established flow across the channelwas scrutinized for different channel geometries referring tothe fundamental equations of flow mechanics (see Appendix2). As the result channel heights up to 50mm were found to beappropriate for the given sample size and pressure differencerange.
It is further concluded that for solution electrolyte concen-trations of 1025 M KCl or higher the effect of overlappingdiffuse layers on the data evaluation according to equations 9and 12 can be neglected. The salt rejection across the streamingchannel creating a concentration drop of the electrolyte isprevented by changing the flow direction of the electrolyteregularly during a series of measurements.
The sample carrier and positioning unit are shown in Fig. 4.The most difficult task has been to preserve the parallelity ofthe sample surfaces in closest distance at about 1mm. Thefollowing efforts were undertaken to guarantee sample paral-lelity:
● The sample surfaces are thin organic layers (about 1–300nm thickness) produced by spin coating, adsorption, or plasmadeposition on top of polished glass carriers (10 mm3 20mm 3 3 mm).
● The glass carriers are fixed on a glass hist and positionedby adjustment screws as shown in Fig. 4 before being filledwith the liquid solution. The parallelity of the sample surfacesalong L (Fig. 1) is controlled by a light microscope (203
335EXTENDED ELECTROKINETIC CHARACTERIZATION OF FLAT SOLID SURFACES
objective) which can be turned above the slit. A second 203objective is equipped with a prism and used to control theparallelity of the samples alongb.
● After the adjustment of the sample carriers the channel issealed by soft rubber stripes pressed from the four directionssimultaneously by a connected pneumatic system. Thus, posi-tional changes during sealing can be prevented.
The distance of the two parallel plates is adjusted by meansof a differential micrometer screw that acts on the movableglass hist.
The two sealings facing the 10-mm side of the channel (B inFig. 1) contain holes to realize the liquid flow inlet and outlet.Since the electrical resistance across the slit channel can arise upto a few hundred megaohms for the smallest distances everyconnection between inlet and outlet of the channel through thetubing system had to be avoided. Therefore, a strict “one-wayprinciple” has been used consisting of two liquid reservoirs with-out any connection excepting the one across the slit channel.
Silver pipes are used as electrodes inserted in the tubingsystem at the entrance to the sealings of the channel. Thedistance between the electrode tips at the channel inlet andoutlet has been kept fixed at 4 mm. The inner surfaces of thepipes were covered by AgCl layers as described elsewhere(45). The use of AgCl pipe electrodes was found to preventvery efficiently the polarization of the electrodes in streamingcurrent measurements for the low liquid fluxes applied in thedescribed system.
The two liquid reservoirs were chosen in dimension andposition to keep the liquid levels on the height of the slitchannel with only minor variations (,10 mm) throughout aseries of measurements. The pressure gradient across the slitchannel is adjusted by the gas pressure (nitrogen or argon)above the liquid in the reservoirs. The pressure is measured andcontrolled by a pressure transducer (Burster, Gernsbach, Ger-many). The electrical measurements are performed with a
Keithley 6512 electrometer (Keithley Instruments, Germering,Germany). Alternating measurements of the current and poten-tial were performed for given pressure gradients controlled bythe PC. The accuracy of the measurements is specified by themanufacturer as to the following values: pressure, less than0.25%; current, less than 0.25%; voltage, less 0.05%.
Liquid flow, pH, temperature, and specific electrical conduc-tivity of the solution are measured in a bypass circuit at theoutlet of the channel. The liquid flow is measured by a home-built capillary flow meter (for lowest fluxes) or by a commer-cial liquid flow meter (Microflow meter, Phase Separations,UK), alternatively. Temperature, pH, and specific electricalconductivity of the solution are measured by a KM 200 instru-ment (Meinsberg Sensor GmbH, Meinsberg, Germany). Theliquid is forced to pass through the bypass tube after the currentor potential measurement has been performed.
The measurement control and data collection is performedby a personal computer with an A/D board. A program hasbeen developed for easy monitoring and for automated mea-surements.
Data Collection and Evaluation
Current I, potentialU, and volume fluxQ for every sampledistanceh were measured at varied pressure differences be-tween 5 and 500 mbar across the slit channel in both directions.Discrete pressure differences were kept constant for a period oftime necessary to perform the measurements, usually a fewseconds.
The gained potential and current values were plotted versusthe pressure difference and the slope of the obtained linearfunction is used for further evaluation. This permits us toovercome electrode asymmetry effects.
An example for the resulting streaming potential data versuschannel height is given in Fig. 5, the corresponding streamingcurrent data are given in Fig. 6 (PDFP, 1025 M KCl solution).
FIG. 5. Streaming potential (dUstr/dp) versus distance (h) of the PDFPsample surfaces in 1025 M KCl solution.
FIG. 6. Streaming current (dIstr/dp) versus distance (h) of the PDFPsample surfaces in 1025 M KCl solution.
336 WERNER ET AL.
Both data sets are further used for the calculation ofz accord-ing to Eqs. [9] and [12]. In addition Eq. [12] is used with thebulk conductivity as the total channel conductivity for com-parison. The values ofe andh were assumed to be the data forpure water, and a temperature correction was considered.
The ratios of streaming current and streaming potential (C)for given h values were further plotted versush (examplegiven in Fig. 7). The plot is evaluated for a range of data wherethe condition of Eq. [17] is fulfilled to deriveKs from theintercept of the extrapolated linear regression (Eq. [15], seeFig. 7).
Flow measurements were performed to calculate the dis-tanceh between the sample surfaces according to Eq. [22]. Todetermine the hydrodynamic thicknessd of an adsorption layerthe volume flow/pressure difference ratio (Q/Dp) has beendetermined before and after the adsorption step without vari-ation of the sample position.d is obtained as the difference ofh before and after adsorption processes divided by 2.
MATERIALS AND METHODS
The investigated surfaces (Fig. 8) were a plasma-depositedfluoropolymer (PDFP) layer on top of a glass carrier (GB 504,Noble GmbH, Dresden, Germany), and an adsorption layerformed of human fibrinogen (FGN) on top of the PDFP layer(fibrinogen A 3782, Sigma, Germany).
The plasma-deposited fluoropolymer (PDFP) layers on topof the glass carriers were produced at the Institute for EnergyProblems of Chemical Physics, Russian Academy of Sciences,Moscow/Chernogolovka by plasma polymerization of tetraflu-oroethylene in an argon-glow-type reactor equipped with a RFgenerator at a frequency of 13.56 MHz.
The electrokinetic microslit setup has been described above.Additional streaming potential measurements were performedby means of the EKA electrokinetic analyzer (A. Paar GmbH,
Graz, Austria) equipped with a home-built flat sample cellusing flat surfaces of similar size as used in the microslitelectrokinetic setup (46).
PDFP layers have been characterized by X-ray-photoelec-tron spectroscopy (ESCA-Lab 2, Vison Instruments, UK),by water contact angle measurements (G 40, Kru¨ss GmbH,Hamburg, Germany) and by atomic force microscopy(Nanoscope 2 Multimode, Digital Instruments, Santa Bar-bara, CA). The layer thickness of the PDFP layer has beendetermined by variable angle spectroscopic ellipsometry(M44, Woolam Co., Lincoln, NE).
The XPS spectra of the PDFP layer are given in Figs. 9and 10. Figure 9 shows the survey spectrum. It contains onlycarbon and fluorine signals, and oxygen was found in traces.The C1-s signal of the XPS-spectrum (Fig. 10) shows thatthe layer consists of a highly linear, nonbranched fluorocar-bon compound since otherwise the signal would occur lesssharp.
Dynamic water contact angle measurements revealed thePDFP layer to be very hydrophobic (uadv5 118°,urec5 100°).
FIG. 7. Slit channel conductance (C) versus distance (h) of the PDFPsample surfaces in 1025 M KCl solution. For a range of data where Eq. [17]is fulfilled Ks is obtained as the intercept with they-axis of the extrapolatedlinear regression plot.
FIG. 8. Investigated substrates: A, plasma-deposited fluoropolymer layer(PDFP) on top of a glass carrier; B, adsorbed fibrinogen (FGN) layer on top ofthe PDFP layer.
FIG. 9. XPS survey spectrum of the PDFP layer.
337EXTENDED ELECTROKINETIC CHARACTERIZATION OF FLAT SOLID SURFACES
The morphology of the PDFP layer has been characterizedby AFM. The determined degree of roughness is in the rangeof few nanometers (Ra 5 4.8 nm, Figs. 11 and 12). Thethickness of the PDFP layer has been determined by spectro-scopic ellipsometry using an optical two-layer model (glass/PDPF) to 806 2 nm.
The solutions used in this study were prepared from Milli-Qdeionized water, which has been vacuum-degassed and filteredthrough an 100-nm syringe filter prior use. The electrolyteconcentrations were obtained by addition of 0.1 M KCl, 0.1 MKOH, or 0.1 M HCl solution.
RESULTS AND DISCUSSION
Electrokinetic measurements with PDFP surfaces were per-formed at varied slit channel cross sections in 1025, 5 3 1025,1024, and 1023 M aqueous KCl solution at pH 6.0, further indiluted KOH (pH 9.8) and diluted HCl (pH 4.5) solutions. Anadsorption layer formed of fibrinogen (FGN) on top of the
PDFP surface has been investigated in a similar procedure in1023 M KCl solution at pH 6.0.
Figure 13 givesz values determined from differentialstreaming potentials (dUstr/dp) and differential streaming cur-rents (dIstr/dp) versus the sample distanceh for the bare PDFPsurfaces in KCl. In a similar way Fig. 14 givesz values versush from measurements of the PDFP surfaces in KOH and HClsolutions.
The z andKs data corresponding to the different solutionsused are summarized in the Tables 1 and 2. The tables containfurthermore the results of the extended evaluation of the resultswith respect toKsd according to Eq. [20] andKsi according toEq. [18].
Figure 15 shows the pH dependence ofz for the PDFPsurface based on streaming potential measurements with anEKA electrokinetic analyzer (slit channel height 50mm). Dataobtained by the microslit electrokinetic setup in different elec-trolyte solutions are shown as single points in this plot forcomparison.
Human fibrinogen (FGN), a globular blood protein, hasbeen adsorbed onto PDFP surfaces in the microslit electro-kinetic setup from diluted KCl solutions (100 ppm FGN in1023 M KCl). The channel was subsequently rinsed withexcess protein-free solution and characterized by measure-ments of dUstr/dp, dIstr/dp, and Q for different Dp anddifferent h as described above (Fig. 16). The liquid flowQacross the channel has been compared before and after FGNadsorption for givenDp values (ath 5 5 mm). According toEq. [22] the difference ofh caused by the FGN layer, and,thus, the hydrodynamic thickness of the adsorption layerhas been determined. Table 3 gives the results of the elec-trokinetic measurements and the hydrodynamic thicknessfor the adsorbed FGN layer as determined in 1023 M KClsolution.
For convenience we have divided the following discussioninto a set of subsections, each of which permits us to focus onthe specific aims of our study.FIG. 11. AFM image of the PDFP surface.
FIG. 12. Roughness profile of the PDFP surface according to AFMdata.
FIG. 10. C1-s signal of the XPS spectrum of the PDFP layer.
338 WERNER ET AL.
Correctness of the Experimental Determination ofz and Ks
by Means of the Microslit Electrokinetic Setup
Almost invariantz values were gained from the stream-ing current measurements at differenth according to Eq. [9]for the investigated range ofh values (Figs. 13 and 14). Thiswas expected (2) since the channel surface conductivitymanifests itself in the streaming potential measurementsand does not influence the streaming current. The combi-nation of streaming current and streaming potential datagives the channel resistance or its reverse, the channelconductanceC, according to Eq. [13]. Ifh is known,C canbe used to the determineKs according to Eq. [16] or, inan other transformation,Ktotal, the total electrical conduc-tivity of the channel, which is the termKB 1 (2Ks/h) inEq. [12]. If Ktotal is used for the calculation ofz fromstreaming potential data, the result coincides with the results
from the streaming current evaluation through the wholerange ofh values and any apparent channel geometry de-pendence ofz vanishes. It is obvious that for largeh the bulkconductivityKB becomes a good approximation ofKtotal. IfC is plotted versush according to a linear regressionKs
results from the interception with they axis whereas12KB
corresponds to the slope. The correlation of the data gives acriterion of the scatter ofKs. In the investigated cases thecorrelation was very good (i.e., better than 0.99) ifC datawere taken from the range ofh in which the condition of Eq.[17] is fulfilled. In some cases a deviation from the lineardependence is observed for the lowesth values. This hasbeen attributed to deviations from the approximated channelgeometry caused by imperfections of the elastomeric seal-ings. As obvious from Eq. [17] the range ofh which can beused for the determination ofKs is affected by the ionicstrength of the solution and by the order of magnitudeKs
attains.Alternatively to the discussed evaluation ofC the determi-
nation ofKs can be based onUstr andh only. For that aim areference value at largeh is needed to providez without
FIG. 13. z values from streaming potential (dUstr/dp) and streamingcurrent (dIstr/dp) versus the distanceh between parallel PDFP surfaces in KClsolutions.
FIG. 14. z values from streaming potential (dUstr/dp) and streamingcurrent (dIstr/dp) versus the distanceh between parallel PDFP surfaces in HCland KOH solution.
339EXTENDED ELECTROKINETIC CHARACTERIZATION OF FLAT SOLID SURFACES
contributions fromKs. Using that boundary value Eq. [13] canbe modified to give
Ks 5KBh
2 SUstr~ref!
Ustr~h!2 1D . [24]
Thus, there are two different approaches applicable for thedetermination ofKs based on measurements with the microslitelectrokinetic setup.
The self consistency achieved for both the determination ofz and Ks demonstrates the correctness of measurements andproves the reliability of the new microslit electrokinetic setup.
Interpretation ofz and Ks: PDFP
z as determined from streaming potential and streamingcurrent for flat PDFP interfaces in diluted KCl solutions at pH6.0 attains rather large negative values (Fig. 13 and Table 1).This confirms that hydrophobic polymers without dissociatingsurface groups become strongly charged in simple electrolytesolutions due to preferential ion adsorption (8, 11). Althoughzreaches high values the determination ofKs indicates thepresence of mobile charge largely exceeding the electrokineticsurface charge. This is obvious from the comparison ofKs andKsd (Table 1; for the use of Eq. 20 see Appendix 1).
The AFM results (Figs. 11 and 12) show that the roughnessof the PDFP surface remains below 10 nm (Ra). Furthermore,there is no porosity of the surface. Thus, although the deter-mined small roughness should be considered as well the largedifferences ofKs and Ksd have to be attributed mainly toinner-layer conductivity effects. Doubtlessly, contributions of
the outer Helmholtz plane may account to the inner-layerconductivity in considerable degrees since the diffusivities ofions adsorbed here can be assumed to be of the same order asfor the bulk solution and the diffuse layer (19). However, veryhigh Stern potentials (CS) result in the investigated case for theassumption that the total charge derived for the hydrodynam-ically immobile layer would be caused by counterions in theouter Helmholtz plane. An additional explanation for the highinner-layer conductivity is therefore required.
At this point the nature of the interfacial charge on thehydrophobic fluoropolymer surface has to be reconsidered.There are no dissociating surface functions on the PDFP andthe surface charge formation occurs exclusively by preferentialion adsorption from solution. In order to elucidate this processthe pH dependence ofz has been determined in 0.001 N KClsolution (pH 3–9, addition of 0.1 N KOH and 0.1 N HCl) bystreaming potential measurements performed with the electro-kinetic analyzer. The ionic strength in the pH range 9–4 can beconsidered as almost constant in this experiment due to the KClsolution concentration. KOH addition leads to an increase ofzwhich proves the important role of the OH2 ions. In contrary,a strong decrease ofz is observed after addition of HCl. Theoccurrence of the isoelectric point caused by HCl addition (atpH 4.06 0.1) proves that H3O
1 ions as well are preferentiallyadsorbed in the competition with Cl2 ions. (The position of theisoelectric point in the acidic pH range is due to the preferentialadsorption of OH2 ions as competing with H3O
1.) Thus, it isjustified to assume OH2 and H3O
1 to be accumulated at theinterface in general which provides in turn an explanation forthe observed highKs values since the mobilities of OH2 andH3O
1 are considerably higher than the mobilities ofK1 and
TABLE 1Experimentally Determined Values of z and Ks and Calculated Values of Ksd and Ksi
for Different KCl Solution Concentrations (pH 6.0)
Cl2. The situation is further different from the charge forma-tion by covalently bound dissociable surface functions sincethe adsorbed ions creating the surface charge cannot be ex-cluded to contribute to the surface conductivity.
Ks of the PDFP interface can be compared now at differentvalues ofz for the different mentioned electrolytes (KCl, KOH,HCl). The decrease ofz with increasing KCl solution concen-tration (Table 1) corresponds to the compression of the doublelayer. However,Ks is not decreased in a similar way butslightly increased. Thus, an increase of the surface chargedensity occurs with increase of the KCl solution concentrationbut remains invisible when solely consideringz. The increaseof the negative surface charge might be caused by the reducedDebye radius at higher ionic strengths in a similar way as thevariation of pK values of dissociable groups with varied ionicstrength.
TheKs data obtained in HCl and KOH solutions indicate thebackground of the variation ofz with the solution pH for thePDFP interface. The inner-layer conductivity does not decreasestrongly with the charge of the hydrodynamically mobile layer.The conclusion to be drawn is that the variation ofz seems toreflect the degree of the internal charge compensation in the
inner layer but not the total charge carrier concentration in theinner layer. In other words, the sum of the products of ionconcentration and specific interfacial mobilities of the interfa-cial charge dominating ions OH2 and H3O
1 in the immobilelayer does apparently differ only slightly at variations of thesolution pH.
Interpretation ofz, Ks, and d: FGN Adsorbed on PDFP
Human fibrinogen (FGN) is an amphiphilic protein (el-lipsoid shape, radii 45 and 9 nm (47)) that adsorbs irrevers-ibly onto PDFP surfaces. The adsorbed layer behaves hy-drophilically. By means of ellipsometry maximum surfaceconcentrations of about 4.5mg/cm2 were determined instagnant solutions containing 100 ppm FGN (0.1 mg/cm3)(48). The protein bears numerous dissociable functions onamino acid side chains which are exposed to the solution(isoelectric point of dissolved FGN5 5.8 (47)).z is deter-mined to about210 mV for the fully established FGN layeron top of the PDFP surface in 1023 N KCl solution at pH 6.0
FIG. 15. pH dependence ofz for the PDFP surface in 0.001 M KClsolution (streaming potential measurements with the electrokinetic analyzer,EKA) as compared withz data determined by the microslit electrokinetic setup(MES) in different electrolyte solutions.
FIG. 16. z values from streaming potential (dUstr/dp) and streamingcurrent (dIstr/dp) versus the distanceh between the parallel PDFP surfacesFGN layers on top of the PDFP surface in 1023 M KCl solution.
341EXTENDED ELECTROKINETIC CHARACTERIZATION OF FLAT SOLID SURFACES
(z of the bare PDFP surface in similar solutions:242 mV).On the other handKs is about an order of magnitude higheras compared to the corresponding values for the bare PDFPsurface! Thus, the contribution of the hydrodynamicallymobile layer toKs becomes very small. To interpret thisresult it has to be considered thatz is related to the netcharge of the protein layer, i.e., it depends on the differenceof the quantities of positively and negatively chargedgroups. In contrast,Ks is related to the sum of the quantitiesof positively and negatively charged groups. The drasticincrease ofKs might indicate the presence of high quantitiesof ionogenic groups on the protein. The decrease ofz shouldcorrespond to a high degree of compensation of anionic andcationic groups. An additional explanation for the observedhigh value ofKs is the porous character of the adsorbedFGN layer at the scale of small electrolyte ions. It is mostinteresting with respect to this explanation to detect theextension of the adsorbed layer in the direction ofh, i.e., thethickness of the adsorption layer. A common approach forthe evaluation of the thickness of macromolecular adsorp-tion layers on flat surfaces consists in ellipsometric mea-surements (48). Ellipsometry is an integral technique andthe determined optical thickness reflects an average valuefor assumed homogenous adsorption layers. Another inte-gral approach to the thickness of adsorbed macromoleculelayers is the determination of the hydrodynamic thickness.No data of corresponding measurements on flat surfaces aregiven in the literature. In our study the hydrodynamic thick-ness of the FGN layer has been estimated by means of themicroslit electrokinetic setup to 486 5 nm. In similarexperiments the thickness of an assumed homogeneouslayer of mean refractive index 1.375 was determined byellipsometry to 23 nm (48). The difference of the two valuesobtained by the different methods is not necessarily a con-tradiction of the results. It should be considered that the shiftof the shear plane is caused by the outermost boundary ofthe adsorbed proteins whereas the optical thickness ratheraccounts for the center of the layer were the refractive indexdifference to the ambient medium is large.
SUMMARY
A new device for the determination ofz and Ks at flatsolid/liquid interfaces (the microslit electrokinetic setup)
has been developed. The new setup further permits thedetermination of the hydrodynamic thicknessd of adsorp-tion layers.
The interfacial charge on a plasma-deposited inert, hy-drophobic fluoropolymer layer has been studied by the mi-croslit electrokinetic setup in different electrolyte solutions.For the investigated systemsz is primarily determined bythe solution pH, i.e., OH2 and H3O
1 are the charge deter-mining ions that become preferentially adsorbed onto indif-ferent surface sites. The hydrodynamically mobile chargecontributes only to about 10% or less toKs in all casesalthough the polymer surface was found to be smooth on ananometer scale. The resulting high inner layer conductivityis attributed both to the high specific mobilities of the ionsaccumulated in the inner layer (OH2 and H3O
1) and to thecontribution of these surface charge creating species to thesurface conductivity.
The adsorption of the human plasma protein fibrinogen(from 100 mg cm23 fibrinogen solutions) onto the flu-oropolymer surface was found to considerably decreasez atthis solid–liquid interface. Simultaneously,Ks is increasedabout an order of magnitude due to the formation of theswollen, hydrodynamically immobile adsorption layer. Thehydrodynamic layer thickness of FGN has been determinedto 48 6 5 nm.
The novel microslit electrokinetic setup is in conclu-sion regarded to provide a powerful tool for the compre-hensive electrochemical characterization of flat solid/liquidinterfaces including the investigation of adsorption phe-nomena.
APPENDICES
Appendix 1
The use of Eq. [19] for the calculation ofKsd neglectingany ions other then K1 and Cl2 is fully justified for the dataobtained in 1024 and 1023 n KCl solutions since for thesesolutions at pH of 6.0 the diffuse layer can be assumed to becomposed of K1 and Cl2 almost exclusively. Even forlower solution concentrations the diffuse layer is not ex-pected to accumulate H3O
1 and OH2 ions since the com-position of this layer is caused by pure electrostatics. How-ever, the mobilities of the water ions exceed the mobilities
TABLE 3Experimentally Determined Values of z, Ks, and d and Calculated Values of Ksd and Ksi
after Adsorption of FGN in 0.001 m KCl Solution (pH 6.0)
of K1 and Cl2 considerably and their contribution to thetotal conductivity can therefore not be neglected in general.At pH 6 the contribution of OH2 ions to the total conduc-tivity can be neglected in comparison with the KCl conduc-tivity at concentrations below 1025 M. The solution con-ductivity increases due to the presence of H3O
1 ions andcan be described by
Kb 5F2
RT~2nD 1 n1D1!, [A.1.1]
wheren1 and D1 are concentration and diffusivity of H3O1
ions andn andD the concentration and diffusivity of K1 andCl2 ions, respectively. The deviation of the conductivitycaused by the presence of H3O
1 ions can be expressed as acorrection multiplierB:
B 51 1 ~b/ 2!
1 1 b, [A.1.2]
whereb 5 D1n1/Dn.Due to high diffusivity of the H3O
1 ionsb is not very smallfor n 5 1025 M and even forn 5 5 3 1025 M. For the firstconcentrationB 5 0.84, and for thesecond oneB 5 0.95.
FIG. A.2.1. Schematic representation of the fluid flow in the rectangular slit channel.
343EXTENDED ELECTROKINETIC CHARACTERIZATION OF FLAT SOLID SURFACES
In a similar way for the calculation ofKsd Eq. 20 has to beextended:
Ksd 52F2z2
RTk@~nK1DK1 1 n1D1!~e
z / 2 2 1!~1 1 3m1!
1 ~nCl2DCl2!~ez / 2 2 1!~1 1 3m2!#. [A.1.3]
Appendix 2
The conditions to be fulfilled for the application of Eq. [2]include the fluid to behave Newtonian and the flow to be ofthe Hagen–Poiseuille type. Hagen–Poiseuille type flow re-quires the following conditions to be fulfilled: Incompress-ible fluid (density and viscosity of the fluid are independenton the applied forces), laminar fluid flow (no velocitycomponents normal to the flow axis) and completely estab-lished (or developed) flow profile. The assumption of theNewtonian character of the aqueous solutions, incompress-ible fluid, and steady fluid flow are in general justified forthe investigated case. The assumption of laminar and estab-lished flow will be discussed subsequently. According to(49) the liquid flow between to parallel plates (L andb @ h,see Fig. 1) is characterized for any steady state by the totalpressure difference across the channel. The frictional flowresistance is a function ofh, the liquid flow velocitycm, thedynamic viscosityh and the density of the liquidr. Thesevalues are combined in the Reynolds number Re which iscalculated according to
Rei 5Dglcmir
h, [A.2.1]
whereDgl is an equivalent diameter defined by
Dgl 54bh
2~b 1 h![A.2.2]
(The index i is used since the liquid flow velocity is set to bevariable in the discussed case.)
The Reynolds number instantly gives the type of the liquidflow, and for Re, 2300 the liquid flow can be regarded aslaminar. The decay of the total pressure across the slit channelcan be described by means of the Bernoulli equation. If inletand outlet of the slit channel are located on similar hydrostaticlevels the Bernoulli equation attains the following form:
Dpi 5l iLrcmi
2
2Dgl, [A.2.3]
whereDp is the pressure difference across the slit channel,lthe friction coefficient andr the density of the liquid.
The friction coefficient varies in the so-called entrance re-gion of the conduit before the parabolic profile of the laminarvelocity distribution becomes fully developed because the fric-tional flow resistance is not constant in this region. The en-trance region lengthai is given by
ai 5 0.02h Rei. [A.2.4]
FIG. A.2.2. Total pressure drop versus volume flux across a slit channel of50 mm height (L 5 20 mm, B5 10 mm).
FIG. A.2.3. Reynolds number versus volume flux for a slit channel of 50mm height (L 5 20 mm, B5 10 mm).
344 WERNER ET AL.
In this range the friction coefficient is calculated according to
lai 596
Rei1
1
ReiF 0.774
~ai/ 2h Rei!2
0.00089
~ai/ 2h Rei!2G . [A.2.5]
The pressure decay in this entrance region is given according to
Dpai 5lairai
2Dgl~cmi!
2. [A.2.6]
For the fully developed parabolic flow profile the frictioncoefficient is obtained as the first term of Eq. [A.2.5],
lSi 596
Rei, [A.2.7]
and the pressure decay is given by
DpSi 5l ir~L 2 ai!
2Dgl~cmi!
2. [A.2.8]
At the outlet of the streaming channel the cross section isconsiderably enlarged and the liquid flow becomes turbulent.The dynamic flow energy is at this point transformed into heatwhich is related to a loss of pressure:
DpCi 5r
2~cmi!
2. [A.2.9]
Thus, the total pressure difference across the streaming channelis calculated according to
Dptotali 5 DpCi 1 Dpai 1 DpSi. [A.2.10]
In Figs. A.2.2 and A.2.3 the discussed deviations are given forthe slit channel withL 5 20 mm,b 5 10 mm, andh 5 50mm at varied liquid flow conditions.
The given example shows that Re is far below 2300. Theassumption of laminar liquid flow is therefore justified for theused setup. Deviations of the fluid flow behavior due to inlet andoutlet effects are just recognizable at 50mm for the entranceregion (0.5%) and for the dynamic pressure drop (0.7%). Thesedeviations decrease for values ofh smaller than 50mm.
ACKNOWLEDGMENTS
The preparation of fluoropolymer films by Dr. Victor Vasilets (RussianAcademy of Sciences, Chernogolovka, Russia) and the contributions of Dr.Frank Simon, Andreas Janke, and Dr. Karina Grundke (Institut fu¨r Polymer-forschung, Dresden, Germany) to the surface characterization of the flu-oropolymer films are gratefully acknowledged. The authors also thank Dr.Petra Welzel (Institut fu¨r Polymerforschung Dresden, Germany) for the helpfuldiscussion of the manuscript.
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