J.C.S. Faraday I, 1980,76,2575-2586Analysis of Electrical Double-layer MeasurementsBY DENVER G. HALL* AND HENRY M. RENDALL?Unilever Research, Port Sunlight Laboratory, Wirral, Merseyside L62 4XNReceived 14th December, 1979We describe methods of presenting experimental electrokinetic and potentiometric titrationmeasurements which are more informative than traditional representations. Simple procedures,based on the application of a theorem recently proved by one of us, make it possible (a) to identifyregions where a simple double-layer model is adequate to fit the results ; (b) to calculate values ofmodel parameters ; (c) to recognise the breakdown of the simple model assumptions ; and ( d ) tosuggest, in many cases, likely causes of the breakdown. A generalisation to more complex systems,with several adsorbed species, is outlined. The application of the theorem to determine the surfacearea of a dispersion is discussed.Electrokinetic measurements and potentiometric titrations are widely used toobtain information about the state of charge at the solid aqueous solution interface.The results are often discussed in terms of the Stern-Grahame model for the doublelayer.Recent publications 1-4 have developed the application of this model toelectrokinetic measurements near the isoelectric point (i.p.). In the simplest case,the following assumptions are made : (1) There is a region (the Stern layer) close tothe surface which is devoid of counterions and coions. (2) The Poisson-Boltzmannequation applies outside the Stern plane.(3) A procedure is available to calculatethe [-potential, at the plane of shear, from electrokinetic measurements. (4) Theplane of shear coincides with the Stern plane.This limiting model will therefore fail if any of the following occur : (a) The planeof shear is displaced from the Stern plane by a distance A, which is not necessarilyconstant. A procedure has been given to identify and in principle to quantify thissituation.2 (b) The basis of the calculations (including the applicability of the PBequation) breaks down. (c) Ions, other than potential determining ions, are (speci-fically) adsorbed within the Stern plane.However,to calculate, from titration results, the charge density at the interface, it is necessaryto know the surface area, A, of the sample and the point of zero charge (P.z.c.),which is less easily obtained than is the i.p.The choice of experimental techniquedepends on the nature of the sample and on the information required.Although measurements in the immediate region of the i.p. are well under~tood,~'~the analysis of results becomes tedious and unsatisfactory at higher charge densitieswhere the simple model assumptions are increasingly likely to break down and wherein particular the specific adsorption of counterions may be expected. A simplecriterion to establish limits, outside which the assumptions are clearly not applicable,was proposed recently.' The purpose of the present paper is to explore the sensitivityof the proposed test and to establish whether, from the form of the deviations, areasonable diagnostic inference about the likely cause of the breakdown may beobtained.For potentiometric titrations, only assumptions (1) and (2) apply.t Present address : Paisley College of Technology, High Street, Paisley, Renfrewshire PA1 2BE.2572576 ANALYSIS OF DOUBLE-LAYER MEASUREMENTSTHEORYConsider a simple double-layer model consistent with assumptions (1)-(4), in thepresence of a z : z supporting electrolyte and with a surface excess rl of potentialdetermining ion.In terms of rationalised units we haveandgo = zleorl (1)2 k T a sinh (-). zleO$d2kT - g d = -z1eoWe definePl = In'l-z,eo$a (3)(4)wherepy = pF( T ) + kT In n'; .The theorem proved in ref. (5) shows that p 1 -@ is the same for all points with thesame rl.Thus, at constant C T ~ , ~where X is a potential determining ion and the constant is a function of do. Underthese conditions, therefore, $d follows a " Nernst " relation with pX. An equivalentstatement is that, over the entire experimental range of pX and ionic strength ( I )where assumptions (1)-(4) hold, plots of go (or ad) against ,ul as defined by eqn (3)should be congruent.When the surface potential tjo follows a Nernst relation with pX, eqn (5) has theformZleOtjd = const -2.303kT pX ( 5 )2.303 kTPX2.303kTwhere pX" is the value of pX at the isoelectric point and K the integral capacity of theStern layer. For a surface with two ionising groups of the type described in ref.(3),we havePX (7)$d=[- 2.303kTz1eowhere Y = [( 1 - O-)/O-][O+/( 1 - O,)] is constant for constant do.It follows that experimental results which have already been shown to be consistentwith the Stern model will inevitably conform to eqn (5). We may therefore use theseresults to demonstrate the possibility of applying eqn (5) for a preliminary assessmentof experimental results and to examine the relative merits of eqn (5) and of the con-gruence test.RESULTS AND DISCUSSIONCOMPARISON WITH EXPERIMENTFig. 1 shows (interpolated) values of 5 against ApCa for CaCO, and Ca,(PO,),at constant values of dd, with theoretical lines as given by eqn (6). Fig. 2 shows asimilar plot for a nylon The proposed test would have been particularly usefulin this case, where it is not immediately obvious that the results should or could beanalysed on the basis of a simple double layer model.Fig. 3 shows the treatmen3020100>E2ii -10-20-304aD . G . HALL AND H. M. RENDALL 2577I 11-2 -1 0 1 2ApCaFIG. l.-Plots of {-potential against pCa, at constant Ud for CaCO,(O) and Ca3(P04)2 (O)." Solidlines given by eqn (6) with K = 40 pF cm-2 and - Ud = 0.5 (a), -0.5 (6) and - 1.0 pC cm-2 (c).PHFIG. 2.-Plots of (-potential against pH, at constant (Td, for nylon 5 0 1 . ~ Solid lines given by eqn (7) withK = 25 pF cm-2 and - Ud = 0.4 (a), 0.3 (b), 0.2 (c) and 0.1 pC cm-2 ( d ) . Appropriate values of Ywere obtained from the data in ref. (3).1-82578 ANALYSIS OF DOUBLE-LAYER MEASUREMENTSapplied to electrokinetic results with AgI (sols were prepared by precipitation fromKI/AgF, dialysed not less than 24 h and used within 2-3 days.Particle sizes weretypically 50 nm. The supporting electrolyte was KN03).7 The best fit overall isobtained with K = 25 pF cm-2, although a slightly better fit at low charge densitiesis obtained with K = 27 pF cm-2, as was calculated by Peterson from the ionicstrength dependence of the quantity Ns-l.l* In principle, therefore, the presentmethod of representing the results may given information on the value of K =f(o,)while assumptions (1)-(4) are valid. Deviations from the proposed plot set in athigh la1 and at high ionic strengths (all experimental points at ionic strength 0.15 ~ m - ~and for ladl > 1 pC cm-2 at 0.1 mol dm-3).These conditions represent a severetest of the theoretical assumptions and, indeed, of the experimental technique. Thedeviations are, however, in a direction qualitatively consistent with the effect ofspecific adsorption of counterions. Fig. 1-3 demonstrate the application of theproposed test to ascertain whether a simple Stern-layer model can be expected tofit the results (see especially fig. 2) and to identify regions where the simplest modelwill clearly not apply (fig. 3). To confirm this usefulness of the proposed plot,however, it is desirable to show more clearly that experimental results to which thesimple double layer model is not applicable will unequivocally fail the test. Electro-kinetic measurements for quartz or silica interfaces are particularly appropriate forthis purpose.It has already been shown that results for silica do not conform90807060SO > s u4030201c2 3 4 5PAgFIG. 3.-Plots of &potential against pAg, at constant ad, for AgI sol.’ Solid lines given by eqn (6) withK = 25 pF Cm-’ and - ad = 3.0 (a), 2.5 (b), 2.0 (c), 1.5 (d), 1.0 (e) and 0.5 pC cm-’ (f). Ionicstrength (0), lo-’ (@), 5 x lo-* (a), 10-1 (a) and 1.5 x 10-1 mol dm-3 (0)D . G. HALL AND H . M . RENDALL 2579to a stability test based on assumptions (1)-(4). The [-potentials for the samesample *, are shown in fig. 4 as a function of pH. Although reasonably linear plotsof [ against pH at constant (Td are obtained for a number of values of od, the slopesare = 33 mV per pH unit, clearly far removed from the " Nernst " value implied byeqn (5).The proposed test therefore correctly identifies the breakdown of the modelrepresented by assumptions (1)-(4) for a system where this failure has been established.The alternative " congruence " test is shown for these samples in fig. 5-7, whereb d is plotted as a function of pX* = pX+zle()$d/2.303kT [eqn (3)]. This procedurehas the advantage of using directly all the experimental results. Fig. S demonstrates,perhaps more convincingly than fig. 2, that the ionic strength-dependence of themeasured nylon mobilities may be attributed to diffuse layer effects. On the otherhand, fig. 3 demonstrates more graphically than fig. 6 the deviations from the simplemodel of the results for AgI at high ladl and ionic strength.The silica results (fig. 7)-100PHFIG. 4.-Plots of S-potential against pH at constant Ud for silica. (a), Ud = 0.2 pC cm-2, slope =31.7mV ; (b), ad = 0.4 p C cm-2, slope = 32.7 mV ; (c), Ud = 0.5 pC cm-2, slope = 31.8 mV ; (a), Ud =1.0 pC cm-2, slope = 34.8 mV2580 ANALYSIS OF DOUBLE-LAYER MEASUREMENTSfail the congruence test under all experimental conditions, confirming the inapplica-bility of the simple model used here. The same general features are observed in theelectrokinetic properties of quartz.” Note that for silica and quartz the variationof od with ionic strength is in the opposite direction from that which would be expectedfor simple specific adsorption of counterions, or from a constant displacement A ofthe plane of shear.A detailed discussion of the nature of the double layer on silicais outside the scope of the present paper. However, considering assumptions (1)-(4),the form of the curves in fig. 7 appears to imply either the presence of coions withinthe plane of shear, or a breakdown of the model for calculating 5 from electrokineticmeasurements. In the absence of titration curves for the same samples as wereused for the electrokinetic measurements, the Naf adsorption results of Li andde Bruyn lo may give some indication of the plausibility of these suggestions. Onthe assumption that the Na+ adsorption was governed purely by the Poisson-Boltzmann equation, these authors lo deduced double-layer potentials for quartz atmol dm-3 agreeing (except at extremely high potentials) with themeasured electrokinetic potentials.This, therefore, provides support for the validityof the [-potential calculation and eliminates one possible source of the “non-congruence” of the results (fig. 7) at low ionic strength. These Na+ adsorptionresults, unlike (hydrogen) potentiometric titrations, do not allow us to eliminate thepossibility that coions, but not counterions, are found within the shear plane. Athigher ionic strength ( 2 mol dm-3), the potentials deduced from Na+ adsorptionwere of substantially higher magnitude than those from electrokinetics. Here,andI I I II I I I4 6 8 10 -1 .c2pH + zleO 512.303 kTFIG, 5.-Plots of Od against pH* for nylon Ionic strength lo-* (o), (O), 5 x (A) andmol dm-3 (0).Solid line given by eqn (8) with KA = 8.6 pF cm-2therefore, at constantkinetic measurementsor specific adsorptionconsistent explanationsamples.D. G . HALL A N D H . M. RENDALL 258 1ionic strength, intercomparison of " titration " and electro-would suggest outward displacement of the plane of shear,1°of counterions. Hence there is at present no obvious self-. of the failure of the congruence condition for silica/quartzDEDUCTION OF DOUBLE-LAYER MODEL PARAMETERSRearranging eqn (6) we haveHence the value of K may be obtained directly. If we represent the slope[da,/d(pX*)] by the symbol S, the relation with the slope d@,/d(pX), at the i.p.,t t 13 G 5pAg+ (z1e0</2.303 kT)FIG. 6.-Plots of Od against pAg* for AgI.' Ionic strength low3 (O), (@), 5 X lod2 (Q), lo-' (a)and 1.5 x 10-1 mol dm-3 (0).Solid line, eqn (8) with K = 25 pF cm-22582 ANALYSIS OF DOUBLE-LAYER MEASUREMENTSpreviously defined as sl, is given byKEN _ - - 1-- NS1 S (9)where N, the “ Nernst ’’ factor, is defined as -2.303kT/z,eo.However, unlike sl, which is a limiting slope where $d 4 0 and which varieswith ionic strength, the valfie of S is a constant for all points where assumptions(1)-(4) hold, for a “ Nernst ” surface with a constant double-layer capacity K. Thismakes the evaluation of the slope and hence of K simpler and more reliable. IfK = f(ao), or if a low density of ionising sites gives rise to deviations from “ Nernst ”beha~iour,l-~ the prescribed plots will be non-linear, but congruent.Hence theseeffects may be distinguished from specific adsorption. Clearly, also, the sameapparent capacity KA for a “non-Nernstian” interface is obtained at the i.p. asfrom the Ns-1 method.In confirmation of this, a linear regression analysis gives KA = 8.52 pF cm-2(r = -0.963) for nylon (fig. 5, 17 points). This compares well with the reportedvalue of KA = 8 6k0.2 pF cm-2. For AgI (fig. 8, including electrokinetic results2.0 -1.5 -NI Eg 1.0- \ Y80 s -14 6 8pH+ (z1e0C/2.303 kT)FIG. 7.-Plots of ad against pH* for ~ i l i c a . ~ Ionicstrength (a), (0) and mol dm-3 (A)D. G . HALL AND H . M. RENDALL 2583from fig. 6 and titration resu1ts)ll K = 26.4 pF cm-2 (Y = -0.977), below the i.p.The titrated charge densities at pAg values above the i.p.deviate from the linerepresented by K = 26.4 pF cm-2, but apparently remain congruent. This effectwould therefore be attributable to changes in the value of the double layer capacityK rather than to specific adsorption of supporting electrolyte ion.21NI0 3 . Yb"I-1- 2I II I- 1 0 1pAg+ (z1e05/2.303 kT)2FIG. 8.-Plots of adagainst pAg* for AgI. Resultsfromtitration '' (open symbo1s)andelectrokinetics '(filled symbols) at ionic strength (0) and lo-' mol dm-3 (0). Solidline, eqn (8) with K = 26.4 pF cm-2. Bars indicate spread of reported titration results at constantpAg or constant cro.Finally, we consider the case where the shear plane is displaced a constant distanceA into an otherwise undisturbed diffuse layer.Plots of the electrokinetic chargeagainst the pX* evaluated at the plane of shear will no longer be congruent. How-ever, near the i.p. the slopes of these plots will follow a relation of the same form aseqn (9) with the limiting slope s[dc/d(pX)], defined previous1y.l- The slope s isgiven by(0), lo-* (A), 5 xs = s1 exp (-.A). (10)In this situation the present method gives no additional insight into the interpreta-However, because the plots are at least partially linearised, tion of the measurements2584 ANALYSIS OF DOUBLE-LAYER MEASUREMENTSevaluation of the initial slopes and hence of A may be more reliable than from adirect [-potential plot.Some reasonable deductions from the form of the '' congruence plot " for electro-kinetic measurements are summarised in table 1.A quantitative analysis of theresults would normally be required to distinguish the alternative explanations of agiven observation. If, however, potentiometric titration results are available for thesame sample, significant qualitative differences may be observed. For example,outward displacement of the plane of shear would not influence the results of potentio-metric titrations, nor indeed would the breakdown of other assumptions, required inthe calculation of ( from electrokinetic measurements, about the response of thesystem to a non-equilibrium situation. On the other hand, specific adsorption ofcounterions would affect the " congruence plot " for titration results, but in theopposite sense from that quoted for electrokinetics. The final case in table 1,observed here with silica but probably characteristic of a range of materials, is notfully understood and clearly merits further consideration.TABLE 1 .-CONGRUENCE PLOT FOR ELECTROKINETIC MEASUREMENTSobservation inference(1) linear, congruent plot results will be fitted adequately with a" Nernstian " surface potential and a constantcapacity K(2) non-linear, congruent plot (4 K = f(odor (b) non-Nernstian Ic/o : pX relation(a) specific adsorption of counterions(b) (outward) displacement of plane of shear or(a) specific acisorplion of coions(b) breakdown of assumptions for calculating j oror (c) ?SURFACE AREA DETERMINATIONFor any surface where assumptions (1)-(2) hold [or (1)-(4) in the case of electro-kinetic measurements] the approach outlined above suggests a number of possibleprocedures for calculating the surface area of the sample.( A ) If both electrokinetic measurements (giving a surface charge density) andpotentiometric titrations (giving a total surface excess charge) are available, calculationof a surface area is obviously trivial.This surface area determination should bereliable provided that it is confined to a region where the electrokinetic measurementssatisfy the congruence condition.( B ) If only potentiometric titration results are available, but the sample has aclearly defined and identifiable P.z.c., the area may be determined as follows. Thecongruence condition requires that, for a given surface excess charge N', per unitweight of sample, the slope [d(ApX'k)/aN",l, is independent of ionic strength.ProvideD . G . HALL AND H . M . RENDALL 2585that the slope is evaluated through the P.z.c., the Poisson-Boltzmann equation maybe linearised to provide the simple relation:lim( aApX -) = const + Z;f?;N:+O aN; K E 2,303 k TKEATo obtain the best value for the limiting initial slope of the titration curve, it isconvenient to plot ApX/N; against ApX or N,". Within the linear region of thePoisson-Boltzmann equation, ApX/N," is a constant for a given ionic strength,provided that K and d$,/d(pX) are constant. The surface area of the sample maybe calculated from the gradient of the linear relation, eqn (1 l), between the initialslopes and I/KE.I t is easy to show that the ratio of the gradient to the intercept ofeqn (1 1) gives the apparent capacity KA of the Stern layer which would be obtainedfrom the Ns-l procedure described previously lm2 (to which this present plot isdirectly analogous).(C) For a surface where the P.Z.C. is ill-defined or not experimentally accessible(e. g., a surface containing only carboxyl- or sulphate-type charged groups), procedure(B) may not be applied. Here we make use of the fact that, at constant uo (i.e.,constant NE), the quantity pX+~~e~$~/2.303kT is a constant. In the linear regionof the Poisson-Boltzmann equation, therefore,pX+ [zfe:N;/2.303kT~&A] = constant. (12)When measurements of N," may be obtained at sufficiently low charge densities, thearea may be calculated directly from a plot of pX against 1 /rc at constant N,".Other-wise an initial value of A may be calculated at the lowest accessible N,". This estimatemay then be refined by converting the titrated charges to charge densities and adjustingthe value of A to produce the best congruence plot. An error of at most 10 % in thesurface area is sufficient to generate a clear systematic deviation from congruence.In the event of significant specific adsorption of counterions, the curves could not bemade congruent by any choice of area A . In this case, the appropriate value of Awould be that giving congruence at low charge densities and ionic strengths. Oncethe area is known, the proposed tests may be applied to identify the onset of specificadsorption of counterions.With titration curves having a common intersectionpoint, indicating no significant specific adsorption near the P.z.c., the tests may bebased on the area obtained directly from procedure ( B ) .For a totally uncharged surface, the surface area could be obtained by method (C),provided that species 1 (the potential determining ion X) is replaced by a singlestrongly adsorbing ion.EXTENSION TO MORE COMPLEX SYSTEMSA similar treatment to the above can be applied when there are several ionic speciespresent inside the Stern plane (SP). To show this we recall that, according to thetheorem on which the treatment is based, the quantity dL given bydL = XTi dpiiis an exact differential where ( I ) the summation is over all independent ionic speciesfound on both sides of the SP; (2) the p i are defined byp i = @(T, p ) + kT In i$' - zieoi,hd 2586 ANALYSIS OF DOUBLE-LAYER MEASUREMENTS(3) the Ti are the amounts adsorbed in the inner regions of the double layer andsatisfy the expressionEzieorl+ad = 0.(15)iLet species 1 be one of the potential determining or specifically adsorbed species andletWe may use eqn (14) to substitute for the dpu, in eqn (1 1) which then becomesIt is apparent from this equation that at constant T and ei, pl is a function of bdonly and vice versa. Thus when ions of the supporting electrolyte are absent fromthe inner regions of the double layer it follows that at constant T and Oi graphs of bdagainst [kT In n1 - zle,t,hd] for different ionic strengths should be superimposableand that at constant T, 81 and b d graphs of kT In nl against z,e,t,hd obtained from dataat different ionic strengths should be linear with unit slope.It is clear from the above that both the “ Nernst plot ” and the congruence plotcan be applied to systems in which several ionic species are believed to occur on bothsides of the SP in exactly the same way as when there is only one such species, providedthat the for all species concerned are held constant.This condition correspondsto the quantities (n:)>z’/(n;)zi being constant for all ij pairs.CONCLUSIONSTwo simple procedures are outlined for analysing electrokinetic or potentiometrictitration measurements. The suggested plots offer advantages over other methodsfor deriving model parameters from experimental results. Furthermore, a directmeasure is given of the range of validity of the model assumptions, together with auseful diagnostic indication of the most likely source of the breakdown. There isno problem, in principle, in generalising the treatment to take account of specificallyadsorbing species.We thank Dr. G. C . Peterson for supplying us with an extensive set of unpublishedmeasurements with AgI.A. L. Smith, in Dispersions ofpowders in Liquids, ed. G. D. Parfitt (Applied Science, London,2nd edn, 1973), p. 93.A. L. Smith, J. Colloid Interface Sci., 1976, 55, 525.H. M. Rendall and A. L. Smith, J.C.S. Faraday I, 1978,74, 1179,H. M. Rendall, A. L. Smith and L. A. Williams, J.C.S. Faraday I, 1979, 75, 669.D. G. Hall, J.C.S. Faraday 11, 1978, 74, 1757.T. Foxall, G. C. Peterson, H. M. Rendall and A. L. Smith, J.C.S. Faraday I, 1979, 75, 1034.T. Foxall and G. C. Peterson, unpublished results.D. G. Hall and M. J. Sculley, J.C.S. Faraday II, 1977, 73, 869.G. R. Wiese, R. 0. James and T. W. Healy, Disc. Faraday SOC., 1971, 52, 302.J. Lyklema, results compiled from (a) E. L. Mackor, Rec. Trav. chim., 1951, 70, 763; (b)J. A. W. van Laar, Thesis (State University of Utrecht, 1952) ; (c) J. Lyklema, Trans. FuradaySOC., 1963, 59, 418 ; ( d ) B. H. Bijsterbosch, Thesis (State University of Utrecht, 1965) ; (e)B. H. Bijsterbosch and J. Lyklema, J. Colloid Sci., 1965, 20, 665.lo H. C. Li and P. L. de Bruyn, Surface Sci., 1966,5,203.(PAPER 9/1984