- Email: [email protected]
The interpretation of electrokinetic measurements using a dynamic model of the stern layer
The Interpretation of Electrokinetic Measurements Using a Dynamic Model of the Stern Layer I. The Dynamic Model C. F. Z U K O S K I
I V AND D. A. S A V I L L E 1
Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544 Received September 2, 1985; accepted December 15, 1985 Results from recent electrophoresis, electrical conductivity, and electro-osmosis experiments on polystyrene latices have been ambiguous. ~"potentials derived from complementary measurements of mobility and electrical conductivity, for example, fail to agree with each other. One mechanism which can account for the discrepancy is the lateral movement of ions within the Stern layer in response to concentration gradients and the tangential electric field. A mathematical model of electrokinetics is presented here in which a dynamic model of the Stern layer is coupled to the standard Gouy-Chapman model of the diffuse layer. Transport within the Stern layer is modeled with a two-dimensional analog of the NernstPlanck equation used to describe electromigration and diffusion in the bulk solution. Adsorption--desorption processes are employed to depict the equilibrium charge using a model which relates concentrations in the Stern layer to those in the adjacent part of the diffuse layer. The model equations are solved numerically and the solutions used to show how various parameters affect the electrophoretic mobility of individual particles and the conductivity of a suspension of particles. It is shown that the presence of mobile counterions in the Stern layer lowers the ~"potentials inferred from mobility measurements and raises those from conductivity measurements, compared to the ~'potentials that correspond to the intrinsic particle charge. In Part II (C. F. Zukoski IV and D. A. Saville, J. Colloid Interface Sci. 114, 45-53 (1986)) the model will be used in the interpretation of experimental measurements. © 1986 Academic Press, Inc.
INTRODUCTION
b y t r a n s p o r t processes w h e n the diffuse p a r t o f the d o u b l e layer is polarized. D y n a m i c p r o cesses i n v o l v i n g the t r a n s p o r t o f ions w i t h i n the Stern layer are outside the scope o f these theories. W e recently tested one o f the a s s u m p t i o n s i n h e r e n t in the s t a n d a r d theory, n a m e l y , t h e n o t i o n t h a t the electrokinetic b e h a v i o r o f a colloidal particle c a n be c h a r a c t e r i z e d b y a single p r o p e r t y , the ~"potential. I n a series o f e x p e r i m e n t s with p o l y m e r latices (5), discrepancies were f o u n d u p o n c o m p a r i n g the ~"p o tentials inferred f r o m m o b i l i t y m e a s u r e m e n t s with those o b t a i n e d f r o m m e a s u r e m e n t s o f the suspension c o n d u c t i v i t y . T h e ~"p o t e n t i a l s inferred f r o m the s u s p e n s i o n c o n d u c t i v i t y were systematically larger t h a n those d e r i v e d f r o m electrophoresis. These results, a n d other recent
T h e existence o f a layer o f a d s o r b e d i o n s a n d solvent m o l e c u l e s at solid/electrolyte interfaces is well established in e l e c t r o c h e m i s t r y a n d colloid science. S o m e characteristics o f this layer, the Stern layer, can be r e p r e s e n t e d with e q u i l i b r i u m a d s o r p t i o n m o d e l s (1-4). O t h e r features, especially the net electric charge, are reflected in the d y n a m i c p r o p e r t i e s o f colloidal particles, which can be investigated t h r o u g h m e a s u r e m e n t s o f the e l e c t r o p h o r e t i c m o b i l i t y or suspension conductivity. However, theories relating particle m o b i l i t y or suspension c o n d u c t i v i t y to the characteristics o f t h e particle are usually b a s e d o n the p r o p e r t i e s o f a static Stern layer, which r e m a i n s u n d i s t u r b e d 1Author to whom correspondence should be addressed. 32
0021-9797/86 $3.00 Copyright © 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
DYNAMIC STERN LAYER MODEL work (6-10), have led us to investigate the role of transport processes operating within the Stern layer. Attention is directed to the Stern layer for several reasons. For example, experimental work with mica (11, 12) demonstrates that solutions of the nonlinear Poisson-Boltzmann equation furnish a very good representation of the electrostatic repulsion between two charged surfaces, even when the separation is of the order of a few nanometers. These findings, coupled with the results of a variety of calculations (13-15), suggest that despite some inherent approximations the Gouy-Chapman model provides a satisfactory basis for theoretical calculations relating to the diffuse part of the double layer. Furthermore, our earlier experimental work indicates that the responsible mechanism must be one that affects the particle mobility and suspension conductivity in different ways, which also suggests that simply altering the fluid viscosity or dielectric constant near the interface is not likely to provide a consistent explanation for the behavior observed. Thus, we are led to consider the presence of transport processes operative in the Stern layer. The approach selected involves introducing lateral ion transport in the Stern layer and represents, therefore, a picture considerably more complicated than that described by the standard theories. Moreover, the approach taken differs from that used in describing the influence of permeable, gel-like layers on the particle surface (16, 17). Although these models treat the flow processes in the surface layer in detail they omit consideration of ion transport and polarization of the layer. The electrokinetic charge of a colloidal particle is related to the net charge behind the hypothetical surface where shear appears in the fluid adjacent to the particle. The location of this envelope--the shear surface--is determined in part by chemical and physical interactions between the electrolyte and the particle. Ions and solvent molecules behind the shear surface, i.e., in the Stern layer, are said to be specifically absorbed. Although bulk flow
33
is absent in the Stern layer, specifically adsorbed ions can still respond to gradients in electrical potential or ionic concentration. If such transport processes occur, the electrokinetic behavior of the particle will depend not only on the net charge but also on the quantity and mobility of the adsorbed ionic species. Surface transport, in one guise or another, has been a part of colloid science for many years. Henry (18) and Booth (19), for example, each developed expressions describing the effect of "surface conductivity" on the electrophoretic mobility. Their theories, designed to account for the fact that the conductivity of the diffuse region differs from that of the bulk fluid in an approximate fashion, were superseded by rigorous treatments. Later, in their monograph on electrokinetic phenomena, Dukhin and Derjaguin (20) described "anomalous surface conduction," charge transport parallel to the surface but within the Stern layer. More recently, Lyklema, Dukhin, and Shilov (21) investigated the effects of surface transport on the low frequency dielectric properties of colloidal suspensions. In their work, transport in the Stern layer was characterized by a single mobility for all the surface species and no attempt made to depict the chemical composition of the layer in detail. In a similar vein, Van der Put and Bijsterbosch (7) employed a model of anomalous surface conduction to analyze their extensive electrokinetic data on porous plugs, concluding that ion transport in a layer of entangled polymer chains adjacent to the particle surface can explain the electrokinetic behavior observed. However the model they employed does not relate the details of the transport process in the diffuse layer to those in the adjacent gellike layer. None of the extant models describe transport within the Stern layer in the detail currently available for transport in the diffuse layer. To do so requires that the individual contributions of the several ionic species be taken into account. Thus, if an expression analogous to the Nernst-Planck equation is used to describe transport by electromigration Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
34
ZUKOSKI AND SAVILLE
and diffusion, then concentrations and mobilities are needed for each ionic species. The utility of such a model is self-evident and given a rigorous mathematical treatment, one can investigate whether or not transport within the Stern layer could play a significant role in electrokinetic phenomena. The purpose of our presentation is to set out such a model wherein transport phenomena within the Stern layer are coupled to the standard theory as applied to electrophoresis (22) and electrical conductivity (23). In the model there is an explicit accounting for contributions of individual species to both the electric charge and transport within the Stern layer. The local concentrations of each species are related to concentrations in the adjacent diffuse layer through mass action relations which describe the adsorption of ions in the Stern region. By balancing the flux of ions from the diffuse layer against electromigration and diffusion within the surface layer, simultaneous polarization of both the diffuse layer and the Stern layer are taken into account. Although our picture of the adsorption phenomena is particularly simple, the mass action model gives a satisfactory picture of the relation between the composition of the Stern layer and concentrations in the bulk of the solution and provides part of the requisite information needed to model the transport of ions. In this context it appears prudent to avoid taking the adsorption terminology too literally. Much more needs to be known about the structure of the Stern layer before the strict validity of different models can be assessed. For the present we view this sort of description simply as one that allows introduction of certain functional relationships which reproduce the empirical results. The details of any adsorption processes, including the role of the surface itself and interactions between different constituents of the Stern layer, remain subjects for further investigation. In the following sections a model of electrokinetics is developed to describe a dynamic Stern layer. Compared to the standard theory the essential modification lies in an alteration Journal of Colh)idand InterfaceScience, Vol. 114, No. 1, November 1986
to the boundary condition on ion fluxes. To establish a context the standard formulation for bulk electrokinetic processes is reviewed first, followed by a description of the Stern layer model. The discussion then turns to ways that Stern layer processes influence two electrokinetic measurements: electrophoresis and electrical conductivity. TRANSPORT IN THE BULK
The system analyzed consists of a suspension of charged spheres of radius a suspended in an electrolyte solution containing N distinct ionic species; the spatially varying number concentration of the kth ion is ~k with valence z k and mobility ~k. Each sphere has an equilibrium surface charge density denoted by 30. The ions in solution have a volume average concentration denoted by ~ . As noted previously (23), ion concentrations beyond the double layer depend on the volume fraction of particles, C, and the charge 30. It is assumed that the particles are placed in the solution as neutral entities whereupon surface groups ionize to produce "added" counterions of valence Z N+I, the (N + 1)st species. In our analysis the particles are presumed to be present at sufficiently low concentration to minimize particle-particle interactions, both hydrodynamic and electrostatic. The balance equations which govern the conservation of mass, momentum, and charge are: 7.0=0, N+ 1
~, ezk~q~76+ #720,
0 =-v~-
k=l
v.j
~ =
~72~ -
[1]
0, e
N+ 1
E z ~ k. k=l
Here 0 is the local fluid velocity,/3 is the pressure, e is the charge on a proton, # is the fluid viscosity, ~ is the electrical potential, and e is the dielectric constant of the suspending medium. It is assumed that the dielectric constant and viscosity of the fluid and the ion mobilities
DYNAMIC
STERN
are independent of electric field strength and position. The divergence of the flux of the kth species, jg, is set equal to zero since there are no sources or sinks for ionic material within the volume of interest. The flux of material in the bulk arises from contributions due to convection, electromigration, and diffusion, viz.,
LAYER
MODEL
~- = %(1 + A~rC).
35 [5]
The Aa term represents the O(C) contribution of the suspended particles to the suspension conductivity and will be called the conductivity increment. The conductivity increment and the electrophoretic mobility are electrokinetic properties of the suspended particles which can jk = n ~ -- e z k ~ k ~ 7 ~ -- ~ k k T V ~ k . [2] be measured experimentally. The electrophoretic mobility and the conThe mobility of the kth ion in the electrolyte ductivity increment are related to the electrosolution is ~k and k T is the product of Boltzkinetic properties of the suspended particles mann's constant and the abolute temperature. in different ways. The mobility is a property A frame of reference is chosen centered on of individual particles while the conductivity a test sphere so that the boundary conditions increment is a result of adding the contribuapplied far from the test sphere are: tions of many particles. Measurements of these xT~ = - E o J cos Ok, two properties provide, therefore, an excellent means for testing the consistency of electrov = -~Eo~r cos 0k, r ---, ~ , [3] kinetic theory. In what has been called the N+ 1 standard form of the theory the electrokinetic Z e zk~k = O. model consists of Eqs. [1] through [3] along k=l with requirements that each particle move as These equations prescribe a uniform electric a rigid impermeable body whose surface has field of magnitude E~ in the z-direction with a uniform charge. All the electrical properties neutral fluid streaming past the test particle at of the particle are subsumed in a single paa free stream velocity equal to the product of rameter, the ~"potential. Thus, measurements the electrophoretic mobility of the particle, of the particle mobility and conductivity of a /~e, and the electric field strength, E~. The suspension at a fixed salt concentration ought boundary conditions applied at the surface of to yield the same ~'potential, unless the theory each particle will be discussed in the next sec- is incomplete. In the work reported earlier (5), tion. For the present we simply note that in substantial differences between the two ~ pothe standard theory the particles have a uni- tentials were found. Given experiments free form surface charge characterized by their ~" of gross errors, the explanation must involve another sort of transport process. We turn, potential. By solving the differential equations around therefore, to investigate transport within the a single test particle we can obtain a theoretical Stern layer. relation between the electrophoretic mobility and the ~ potential (22). Solutions valid for a T R A N S P O R T A T T H E P A R T I C L E S U R F A C E - - A DYNAMIC MODEL OF THE STERN LAYER single particle can also be used to formulate a relation between the local current and the avInside the Stern layer the transport mecherage current within the suspension (23), viz., anisms envisioned are electromigration and 1 fV J = -~
UZ+ l
ezkj kdV.
[4]
k=l
From this relation one can relate the conductivity of a dilute suspension, b, to the volume fraction of particles as
diffusion. To represent the flux using the twodimensional analog of the Nernst-Planck equation, ionic mobilities and compositions are required for each species in the Stern layer. A simple adsorption model is used to relate concentrations in the Stern layer to those in the bulk. JournalofColloidandInterfaceScience,Vol. 114, No. 1, November 1986
36
ZUKOSKI
AND
In the conceptual model, particles of uniform radius are added to an electrolyte containing N distinct ionic species to obtain a suspension of volume fraction C. The colloid acquires a uniform surface charge density through the adsorption of ions from the surrounding electrolyte or by the dissociation of covalently bound moieties. It is assumed that these reactions are rapid, yielding a Stern layer which is in dynamic equilibrium with ions in the diffuse double layer. Each ionic species can undergo adsorption-desorption reactions of the form t/k(a) + gk ~ t/~ [61 where h~(a) is the concentration of the kth species evaluated at the particle surface r = a, ~ is the number density of empty sites in the Stern layer for the/cth species, and ff~ represents the number density of sites at which the kth species is absorbed. Competition between the different ionic species is absent and the total capacity of the surface layer for a particular species is N~. The equilibrium constant for each reaction pair is /(Tk- t~k(a)~k r/k
[7]
To relate the adsorption of ions to the charge each adsorption site is assigned a valence z~. Upon defining the fraction of empty sites, ~ k / ~ , as a k, the charge density within the Stern layer can be written as N+ 1
q = e[ Z k=l
N+ 1
+
-
E
k]
[81
k=l
which shows how changes in the concentration of an ionic species in the Stern layer affect the charge. Furthermore, application of an external electric field polarizes the diffuse double layer causing ionic concentrations to vary around the periphery of the particle. Then, because local equilibrium between species adsorbed in the Stern layer and ions in the diffuse layer is maintained, cf. Eq. [7], polarization of the double layer polarizes the surface charge density. It can be shown, however, that the polarization which lends the angular depenJournal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
SAVILLE
dence to the ion distributions does not alter the total charge (24). Ionic transport within the Stern layer arises from the lateral electric field and concentration gradients. This transport is modeled in a fashion analogous to transport in the bulk. The flux of the kth species, ]~, comes from the combined effects of electromigration and diffusion, i.e., ]~ = --ezkCo~g~Ts~
-- kT&~kX7s~ k.
[9]
Here ~sk is the mobility of the kth species, ~7sfi~is the tangential gradient of the surface concentration, and Vsq~is the tangential gradiem of the electric field. It is stressed that Eq. [9] is applied to perturbations from a state wherein the concentration and potential on the surface of the particle are uniform. More complicated relations involving concentration dependent mobilities and surface activities are conceivable but are unnecessary here since we deal with situations which have been linearized about the uniform state. Accordingly, more complicated flux relations reduce to the form shown above. It is possible that the mobility for surface diffusion would differ from that for electromigration but such differences are ignored here. A brief discussion of the linearization procedure is given in the Appendix, along with references to more mathematical treatments. The transport of ions along the surface is coupled with transport in the bulk through a boundary condition applied at the particle surface, viz., jk.n=-V~.j~
at
r=a.
[10]
In this expression jk represents the flux of the kth species in the bulk and n is the outeffacing normal. This expression is a modification to the standard theory in that the normal component of the flux of the kth species is balanced by an ion flux tangential to the surface. Whenever surface transport of ions is absent, jk = 0, and ]k.n = 0.
[11]
DYNAMIC STERN LAYER MODEL Accordingly, when all the &~ are set equal to zero, surface transport disappears and the model reduces to the standard theory for particles with impermeable surfaces. The remaining conditions enforced at the outer surface of the Stern layer are the same as those used in the standard theory, viz., continuity of the tangential components of the electric field, continuity of the velocity, and a relation between charge and potential, written - E ~rr = q.
[12]
We can summarize the main features of the model as follows. The diffuse part of the double layer is pictured in familiar terms: the Gouy--Chapman model with ion transport by the combined effects of flow, electromigration, and diffusion; fluid motion involves a balance between pressure gradients, viscous stresses, and the electrical body force. However, ions cross the shear surface separating the diffuse layer from the compact Stern layer under the influence of concentration gradients and electrical forces normal to the surface. Each such ionic flux is balanced by lateral transport within the Stern layer due to the same forces. In the Stem layer the ion mobilities differ from those in the bulk due to various factors related to the presence of the adsorbed layer. Because the layer is thin, the balance equations reduce to two-dimensional analogs of their counterparts in the bulk and, finally, are coupled to the equations that describe the bulk as boundary conditions. Due to rapid adsorption-desorption steps, the inner part of the diffuse layer and the Stern layer are in local equilibrium, as described by mass action equilibrium relations. The next task is to work out solutions to the model equations numerically. The solution of the differential equations employs methods developed to analyze situations where the applied field strength produces small changes in the equilibrium double layer configuration around each particle. A brief discussion of the mathematical methodology is given in the Appendix, and further details are given by Zu-
37
koski (24). We note here that the computer codes were carefully tested to eliminate errors and, under conditions where transport in the Stern layer is absent, reproduce the results of O'Brien and White (22) and Saville (23). With the numerical scheme we can explore how the electrokinetic behavior of the colloidal particles responds to changes in the various parameters. The results of a series of such calculations will be discussed shortly. In this connection it should be recognized that the ~ potential is no longer a primary parameter as it is in the standard theories where it can be set independent of other considerations. Here the equilibrium particle charge (and hence the ~" potential) is defined by the adsorption equilibria and involves valences, capacities, and adsorption constants for the various species in the Stern layer. The purpose of the material presented next is to explore the effects of the various model parameters on measurable electrokinetic properties, e.g., electrophoretic mobility and conductivity increment. In the second paper we will interpret experimental results obtained earlier (5) using the dynamic model of the Stern layer. MODEL PREDICTIONS To establish how transport in the Stern layer can influence the electrophoretic mobility of individual particles and the bulk conductivity of the suspension, the model equations were solved with different sets of values for the ion mobilities and other parameters. The model parameters are the mobility, adsorption constant, maximum density, and valence of the adsorption site for each ionic species in the Stem layer. The electrophoretic mobility and conductivity increment will be shown as functions of salt concentration as one of the parameters is varied while the others are held constant. The importance of salt concentration is emphasized because most of the parameters are related to the chemistry of the surface and thus depend on bulk ionic concentration. Results were calculated for 0.2-t~mdiameter spheres suspended in KC1 solutions Journal of ColloM and Interface Science, Vol. 114, No. 1, November 1986
38
ZUKOSKI AND SAVILLE
with a viscosity of 0.893 cP and relative dielectric constant of 78.9. It was assumed that cations adsorb at negatively charged sites, Zs~ = - 1 , and that anions adsorb at neutral sites, z 2 = 0. As a consequence the maximum density of adsorbed cationic species (the counterions) was set by the titratable charge and the adsorption of anionic species represents interactions with the hydrophobic portions of the surface. The adsorption model has been set up in a rather general form so as to be able to accomodate a variety of schemes for localizing ions and charge groups in the Stern region. The scheme we have chosen to use to illustrate the predictive capabilities of the model involves adsorption or binding of both co-ions and counterions in the surface layer. Counterion binding is a firmly grounded mechanism (14) but co-ion binding is less well established (25), although it has been used to explain the maximum in the mobility-salt concentration relation (26, 27). This matter will be discussed in some detail in Part II; for the m o m e n t we simply note that this issue is separate from the role of surface transport. Since the influence of the Stern layer depends on several factors, each will be taken
up separately; effects due to the degree of binding of ions within the Stern layer are investigated first. Figures 1 and 2 display how changes in the adsorption constants affect the electrophoretic mobility and conductivity. The trends in the electrophoretic mobility seen in Fig. 1 can be explained as follows. At a low salt concentration (10 - 6 M) the particle charge is determined by the maximum density of (negatively charged) sites at which the potassium ions adsorb, ~rx~, and the binding constant /~1. As the KC1 concentration is increased, more potassium ions adsorb, neutralizing the negative adsorption sites, decreasing the charge density and diminishing the particle mobility. At the same time, the chloride ion concentration in the bulk increases, and more chloride ions adsorb at neutral sites. Therefore, the particle charge density decreases at a slower rate than would be expected without chloride ion adsorption. At higher KC1 concentrations (10 -4 M), the Stern layer is almost saturated with potassium ions that neutralize the covalently bound negative charge, however adsorbed chloride ions maintain a negative charge in the layer. At still higher KCI concentrations (10 -2 M), chloride ions and potassium ions saturate the layer and
22
-6
LClcm z K+
14
_
-O. [
10-3 I0-~~ ~ . I ~
E ~k V_1
MIL VAmAeLE
~ I- 2.0 IXlO-4 0.2
u x.
5.0
-2
O
I
-6
-5
I
I
I
I
-4
-3
-2
-I
LOG [KCI]
-2
-6
I
-5
I
-4 LOG
I
-3 [KCI]
I
I
-2
~I
FIG. 1. Electrophoreticmobility and conductivityincrement. Variations with changes in ionic strength and the potassium ion adsorptionconstant,/~. Curvesare calculatedfrom the dynamic Stern layer model using parameters indicated. Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
39
DYNAMIC STERN LAYER MODEL -6
22--
lO-s
i~k
^k.^k ¢us,'w
=C/cm2
M/L
K+
5.0
IxlO -s
]1-
2.0
18
^ u
^k eN T
-0. t
14 ~t
!VARIABU 0 . 2
E ::L
_.l
\\V.=,o-,
-2
0
e
2i
i ,
-6
-5
, -4
, -3
J -2
I -I
-2' -6
I
-5
I -4
I -3
i
-2
--
J -I
• OG [KC'] FIG. 2. Electrophoretic mobility and conductivity increment. Variations with changes in ionic strength and the chloride ion adsorption constant,/~2. Curves are calculated from the dynamic Stern layer model using parameters indicated.
the charge density becomes independent of ionic strength. Here the electrophoretic mobility decreases due to the compression of the diffuse double layer arising from the increase in bulk salt concentration. The importance of transport in the Stern layer relative to that in the bulk can be characterized using a dimensionless "surface conductivity," i.e., N
as =
k=l
N
[13]
a Z k=l
Here ~0~ and a0k are (respectively) the concentration outside the diffuse layer and the fraction of empty sites for the kth species. The maximum density to which ions can adsorb, ]V~, is independent of ionic strength and so at high salt concentrations (large ~ the dimensionless surface conductivity is small. Stern layer transport processes are unimportant here. In Fig. 1, as is approximately 10 -3 at a KC1 concentration of 10-l M. Lyklema et al. (21) came to similar conclusions regarding the role of surface conductivity in their analysis of the frequency dependence of the conductivity of a colloidal suspension
at high salt concentrations. Although simultaneous polarization of the diffuse and adsorbed parts of the double layer were not taken into account, they were able to conclude that the influence of the mobility of bound ions on the complex conductivity of a colloidal suspension is small when the double layer is thin. This is consistent with the analysis presented here where the electrokinetic properties are largely unaffected by transport in the Stern layer for aK >> 1.2 The effect of altering the binding constant of the potassium ions on the conductivity increment is also shown in Fig. 1. Note, however, that there is a large difference in the scale between the mobility and conductivity increment panels. The differences depicted in the conductivity increment diagram would be experimentally significant at salt concentrations below 10-4 M. At a KC1 concentration of 10-4 M, A~ changes by one unit as the adsorption constant varies from 10-s to 10 -3 and this change is accentuated at lower KC1 concentrations. 3 2 In our example ar = 56 with [KC1] = 10-2 M. 3 The experimental uncertainty in the earlier data (5) was +0.2 Aa unit. Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
40
ZUKOSKI AND SAVILLE
At high salt concentration (> 10-3 M), the particles have characteristics independent of the positive ion adsorption constant, K 1, because the Stern layer is saturated. Accordingly, differences between the conductivity increments disappear. At lower salt concentrations, however, a smaller binding constant yields a larger amount of adsorbed potassium ions and, since the adsorption of a potassium ion neutralizes a negatively charged site, a smaller value for K ~ leads to a lower charge density and a smaller Aa. This behavior is reflected in Fig. 1 for salt concentrations below 10 -4 M and again in Fig. 2 where the parameter varied is the binding constant for the chloride ions. The effects of different adsorption capacities for anions and cations are displayed in Figs. 3 and 4. In these figures the maximum density to which the ions can be adsorbed is varied by changing NTl and NT2 , The situations where either anion or cation adsorption is absent illustrate why binding of both ions is important. The mere presence of a mobile surface species will not explain the structure of the mobilitysalt concentration relation reported in the literature (1, 5, 25, 28). At a given coverage of adsorbed ions, the charge density of the particle is fixed but
changing the mobility of the adsorbed ions influences both the particle mobility and conductivity increment strongly. Ion transport in the Stern layer decreases the tangential component of the electrical field which lowers the electrophoretic mobility. At the same time, this transport provides another conduction path which increases the conductivity increment. The effects of increasing the mobilifies of the cation and anion in the Stern layer are shown in Figs. 5 and 6, respectively. In these figures the mobilities of the adsorbed ions have been varied systematically while holding /~k and NTk fixed. This makes the total particle charge depend on the salt concentration alone and in these figures differences in either the electrophoretic mobility or in conductivity increment at a given ionic strength are due solely to dynamic processes in the Stern layer. The curves marked 0~] = 0 in Fig. 5 and 4 2 = 0 in Fig. 6 represent situations where surface transport is absent. At very high salt concentrations effects due to transport in the Stern layer are swamped by transport in the bulk, as expected (~s ~ 1). However, the insensitivity shown at bulk salt concentrations below 10-5 M is unexpected
,o-
-6-
\
""
,~ ~k ~p~ ,c,0m, M/L --
\
A~
"5 -
i
I •
i
l
]
[Kc,]
-5
-4
-3
LOG[KC, ]
-2
-I
FIG. 3. Electrophoreticmobilityand conductivityincrement.Variationswith changesin ionic strength and the maximumnumber of potassium ion bindingsites,N}. Curvescalculatedfrom the dynamicStern layermodel usingparametersindicated. Journal of Colloid and Interface Science, VoL 114, No. 1, November 1986
DYNAMIC STERN LAYER MODEL -6~
41 e~ k
I0--
~
\
8
3
eNT u
6
-4
2 E :L
4
"5 - 2
2
K+I
i~ k
gsk/~ k
~Clcm z
M/L
--
5.0
i×lO-5
O.i
ixIO -4
0.2
VARIABLE
o 0
0
P
0 -6
-5
-4 -3 LOG [KCl]
-2
-2 -6
-I
I -5
I
i
-4 -3 LOG [KCI]
I
-2
-I
FIG. 4. Electrophoretic mobility and conductivity increment. Variations with changes in ionic strength and the maximum number of chloride ion binding sites, ?~T. Curves calculated from the dynamic Stern layer model using parameters indicated.
(cf. Fig. 5). At concentrations where as >> 1 it might be thought that the surface transport processes would dominate the electrokinetic behavior of the colloid. Instead, both the electrophoretic mobility and conductivity increment appear to be insensitive to changes in the mobility of the potassium ion in the Stern layer. The explanation for this behavior lies
in the coupling between transport of ions in the Stern layer and perturbation potential gradient. At low salt concentrations the potential gradient scales as 1/~]. Thus, since the flux involves the product of the ion mobility and the potential gradient, the flux is relatively insensitive to changes in mobility at low ionic strength. At intermediate salt concentrations
2E
-6
et~k p.C/cm 2
~k
~sk/,.;k
M/L
--
-2
-I
IE
.-£
14 =n
E ::L
A~.
~1 -2
I0
6
0:E 2 f 0"0~
I
-6
-5
I
I
i
-4 -3 LOG[KC,]
-2
I -I
-z -6
-5
,
~
-4 -3 "OG [KC,]
FIG. 5. Electrophoretic mobility and conductivity increment. Variations with changes in ionic strength and the surface mobility of potassium ion, g~]. Curves calculated from the dynamic Stern layer model using parameters indicated. Journal of Colloid and Interface Science, Vol. ! 14, No. 1, November1986
42
ZUKOSKI AND SAVILLE 22-
-6
J.Clcm;'
18
u
14
E ~L
~
K+
5
]1-
2
MIL
--
x l o -s
0.0
[XlO -4 VARIABLE
•",o- I0
>_1 -2
6
~
~s2/~2-o.9
0 :E 2
-6
i
i
L
i
I
-5
-4
-3
-2
-I
]
-2
-6
I
-5
I -4
L -3
I
I
-2
-I
FIG. 6. Electrophoreticmobilityand conductivityincrement. Variationswith changesin ionic strength and the surfacemobilityof chloride ion, ~2. Curves calculatedfrom the dynamicStern layermodel using parameters indicated. (10 -5 M < [KC1] < 10-2 M), the potential gradient has a much weaker dependence on ~1 and changes in the ion mobility change the electrophoretic mobility and conductivity substantially. Both electrophoretic mobility and conductivity increment appear insensitive to changes in the anion mobility, g~2(Fig. 6). Most of this is due to the type of equilibrium adsorption chemistry chosen here. Since there are far fewer adsorbed chloride ions than there are potassium ions, a larger change in the adsorbed chloride ion mobility is required to produce an equivalent change in transport rate. In addition, the layer does not become saturated with chloride ions until a r >> 1, where the effect of surface transport is small. Finally, to illustrate the influence of ion transport within the Stern layer, we present results calculated in terms of an "apparent" ~" potential (Fig. 7). The results were obtained as follows. For the model system the binding constants, adsorption capacities, and other parameters were set so as to give a particle surface with a modest ~"potential, about - 6 0 mV. Then with the equilibrium properties of the surface held fixed, calculations of particle mobility and suspension conductivity were carded out for several different adsorbed Journal of Colloid and Interface Science, V o l . 114, N o . 1, N o v e m b e r
1986
counterion mobilities using the dynamic model of the Stern layer. Particle mobilities and suspension conductivities so calculated reflect the full influence of transport within the layer, including polarization of the concentrations of adsorbed ions. Next, these mobilities and conductivity increments were reinterpreted in terms of"apparent" ~"potentials using standard theories (22, 23). Figure 7 shows how these apparent ~"potentials depend on the magnitude of the ion mobility in the Stern layer. It is clear that a modest sized ion mobility can have a large influence on the apparent ~"potential. For example, if the counteflon has a mobility, g~, one-tenth of its mobility in free solution, ~1, the ~"potential inferred from an electrophoresis experiment using the standard theory will be about 60% of the true figure. Similarly, from a suspension conductivity measurement the apparent ~"potential will be twice as large. Such differences all arise from the way transport in the Stern layer alters the dipole term that arises in the solution of the transport equations (22, 23). We conclude that transport processes in the Stern layer can produce differences between the ~" potentials derived from mobility and conductivity similar to those reported in earlier experimental studies (5).
43
DYNAMIC STERN LAYER MODEL
~C/cm z
--
K+
5
I xlO -5 V~IASI..E
CI-
2
I x l O -4
¢..--& 0.5 t;-r
°0
M/L
0.0
~v
Ol
02
0 3
oI 0
i
I
I
0.1
0.2
0.3
~/~
FIG. 7. Relationships between the apparent ~"potentials and the counterion mobility in the Stern layer. Left: Apparent ~"potentials derived from particle mobility. Right: Apparent ~"potentials derived from suspension conductivity. ~'Astands for the apparent ~"potential and ~T for the true value.
SUMMARY
In this paper we have presented an electrokinetic model incorporating effects due to transport in the Stern layer. In the model, ionic concentrations in the Stern layer are in local equilibrium with the diffuse double layer so polarization of the diffuse layer affects the Stern layer and vice versa. Transport within the Stern layer arises due to diffusion down concentration gradients and conduction in response to a tangential electric field. The set of model equations was solved using numerical techniques to avoid mathematical approximations. Systematic variation of the magnitudes of the binding constants, the density of binding sites, and mobilities in the Stern layer leads to several conclusions. First, ionic transport has substantial effects on both the electrophoretic mobility, t~e, and electrical conductivity increment, Aa, for intermediate values of aK (i.e., 2 ~< aK ~< 50). At extremes of salt concentration where a~ >> 1 or aK ~< 1, the mobility and conductivity increments each become insensitive to transport in the Stern layer. Furthermore, although the magnitude of Ao- is sensitive to changes in the model parameters, the qualitative features of the relationship are not as sensitive as those of the electrophoretic mobility-salt concentration
relationship. In a subsequent paper this model will be used to interpret the experimental results reported earlier (5). APPENDIX: A DESCRIPTION OF THE MATHEMATICAL TECHNIQUES
The techniques used to carry out the numerical calculation of the fields around a test particle and to construct relations for the electrophoretic mobility and conductivity increment were patterned after the work of O'Brien and White (22), Saville (23), and O'Brien (29). First the model equations are recast in dimensionless form and simplified using a perturbation scheme employing the expansion parameter aeE~/kT, a dimensionless ratio involving the strength of the applied field compared to a characteristic field strength. This leads to a set of equations describing the equilibrium state around a test particle and another set describing the way this situation is perturbed by the applied field. The solutions to the second set are axisymmetric with respect to the azimuthal angle and so the dependent variables can be represented as the product of two functions, one which describes the r dependence and the other the 0 variation. The latter is simply cos 0. Thus the problem comes down to integrating a set of ordinary differJournal of Colloid and InterfaceScience, Vol. 114,No. 1, November1986
44
ZUKOSKI AND SAVILLE
ential equations. A l t h o u g h it is useful to int r o d u c e auxiliary functions o f t h e sort e m p l o y e d in earlier w o r k (22, 23, 29), s o m e o f the a d v a n t a g e o u s features are d i l u t e d b e c a u s e we m u s t calculate t h e p o t e n t i a l a n d i o n conc e n t r a t i o n s at the surface explicitly. T h e sol u t i o n for the fields a r o u n d the test particle p r o v i d e s the e l e c t r o p h o r e t i c m o b i l i t y directly (22). T o o b t a i n the c o n d u c t i v i t y i n c r e m e n t these fields m u s t be averaged a c c o r d i n g to the techniques o u t l i n e d b y O ' B r i e n (29) a n d Saville (23). T h e conductivity i n c r e m e n t involves c o n t r i b u t i o n s f r o m the a d d e d c o u n t e r i o n s , nonspecific a d s o r p t i o n , a n d w h a t is called the d i p o l e term, m o d i f i e d to t a k e a c c o u n t o f t r a n s p o r t in the Stern layer. As n o t e d earlier the c o m p u t e r p r o g r a m s were tested to insure t h a t t h e y r e p r o d u c e d the correct results for the mobility and conductivity increment when t r a n s p o r t in the Stern layer was suppressed. F u r t h e r details are given b y Z u k o s k i (24). ACKNOWLEDGMENTS This work was supported by the NASA Microgravity Sciences and Applications Program and a grant from the Xerox Corporation. We are also indebted to J. C. Baygents for his help with the mathematical modeling. REFERENCES 1. Hunter, R. J., "Zeta Potential." Academic Press, New York, 1981. 2. Healy, T. W., and White, L. R., Adv. Colloidlnterface Sci. 9, 303 (1978). 3. James, R. O., Stiglich, P. J., and Healy, T. W., in "Adsorption from Aqueous Solutions" (P. H. Tewaft, Ed.). Plenum, New York, 1980. 4. James, R. O., and Parks, G. A., Surface Colloid Sci. 12, 119 (1982). 5. Zukoski, C. F., IV, and Saville, D. A., J. Colloid Interface Sci. 107, 322 (1985).
Journal of Colloid and Interface Science. Vol. 114, No. 1, November 1986
6. Van der Put, A. G., and Bijsterbosch, B. H., J. Colloid Interface Sci. 75, 512 (1980). 7. Van der Put, A. G., and Bijsterbosch, B. H., J. Colloid Interface Sci. 92, 499 (1983). 8. Springer, M. M., Korteweg, A., and Lyklema, J., J. Electroanal. Chem. 153, 55 (1983). 9. McDonogh, R. W., and Hunter, R. J., J. RheoL 27, 189 (1983). 10. O'Brien, R. W., and Perrins, W. T., J. Colloidlnterface Sci. 99, 20 (1984). 11. Pashley, R. M., J. ColloidlnterfaceSci. 80, 153 (1981). 12. Pashley, R. M., J. Colloid Interface Sci. 83, 531 (198 I). 13. Levine, S., and Outhwaite, J., J. Chem. Soc. Faraday Trans. 2 74, 1670 (1978). 14. Torrie, G. M,, and Valleau, J., J. Chem. Phys. 73, 5807 (1980). 15. Carnie, S. L., Chan, D. Y. C., Mitchell, D. J., and Ninham, B. W., J. Chem. Phys. 74, 1472 (1981). 16. Levine, S., Levine, M., Sharp, K. A., and Brooks, D. E., Biophys. J. 42, 127 (1983). 17. Wunderlich, R. W., J. Colloid Interface Sci. 88, 385 (1982). 18. Henry, D. C., Trans. FaradaySoc. 44, 1021 (1948). 19. Booth, F., Trans. Faraday Soc. 44, 955 (1948). 20. Dukhin, S. S., and Derjaguin, B. V., in "Surface and Colloid Science" (E. Matijevi6, Ed.), Vol. 7. Wiley, New York, 1974. 21. Lyklema, J., Dukhin, S. S., and Shilov, V. N., J. Electroanal. Chem. 143, 1 (1983). 22. O'Brien, R. W., and White, U R., J. Chem. Soc. Faraday Trans. 2 77, 1607 (1978). 23. Saville, D. A., J. Colloid Interface Sei. 91, 34 (1983). 24. Zukoski, C. F., IV, "Studies of Electrokinetic Phenomena in Suspensions," Ph.D. thesis, Princeton University, Princeton, N.J., 1984. 25. Goff, J. R., and Luner, P., J. Colloid Interface Sci. 99, 468 (1984). 26. Meijer, A. E. J., van Megan, W. J., and Lyklema, J., J. Colloid Interface Sci. 66, 99 (1978). 27. Ma, C. M., Micale, F. J., E1-Asser,M. S., and Vanderhoff, J. W., in "Emulsion Polymers and Emulsion Polymerization" (D. R. Bassett and A. E. Hamielec, Eds.), ACS Symposium Series Vol. 165, pp. 251-262. Amer. Chem. Soc., Washington, D.C., 1981. 28. Ottewill, R. H., and Shaw, J. N., J. Electroanal. Chem. 37, 133 (1972). 29. O'Brien, R. W., J. ColloidlnterfaceSci. 81, 234 (1981).
I V AND D. A. S A V I L L E 1
Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544 Received September 2, 1985; accepted December 15, 1985 Results from recent electrophoresis, electrical conductivity, and electro-osmosis experiments on polystyrene latices have been ambiguous. ~"potentials derived from complementary measurements of mobility and electrical conductivity, for example, fail to agree with each other. One mechanism which can account for the discrepancy is the lateral movement of ions within the Stern layer in response to concentration gradients and the tangential electric field. A mathematical model of electrokinetics is presented here in which a dynamic model of the Stern layer is coupled to the standard Gouy-Chapman model of the diffuse layer. Transport within the Stern layer is modeled with a two-dimensional analog of the NernstPlanck equation used to describe electromigration and diffusion in the bulk solution. Adsorption--desorption processes are employed to depict the equilibrium charge using a model which relates concentrations in the Stern layer to those in the adjacent part of the diffuse layer. The model equations are solved numerically and the solutions used to show how various parameters affect the electrophoretic mobility of individual particles and the conductivity of a suspension of particles. It is shown that the presence of mobile counterions in the Stern layer lowers the ~"potentials inferred from mobility measurements and raises those from conductivity measurements, compared to the ~'potentials that correspond to the intrinsic particle charge. In Part II (C. F. Zukoski IV and D. A. Saville, J. Colloid Interface Sci. 114, 45-53 (1986)) the model will be used in the interpretation of experimental measurements. © 1986 Academic Press, Inc.
INTRODUCTION
b y t r a n s p o r t processes w h e n the diffuse p a r t o f the d o u b l e layer is polarized. D y n a m i c p r o cesses i n v o l v i n g the t r a n s p o r t o f ions w i t h i n the Stern layer are outside the scope o f these theories. W e recently tested one o f the a s s u m p t i o n s i n h e r e n t in the s t a n d a r d theory, n a m e l y , t h e n o t i o n t h a t the electrokinetic b e h a v i o r o f a colloidal particle c a n be c h a r a c t e r i z e d b y a single p r o p e r t y , the ~"potential. I n a series o f e x p e r i m e n t s with p o l y m e r latices (5), discrepancies were f o u n d u p o n c o m p a r i n g the ~"p o tentials inferred f r o m m o b i l i t y m e a s u r e m e n t s with those o b t a i n e d f r o m m e a s u r e m e n t s o f the suspension c o n d u c t i v i t y . T h e ~"p o t e n t i a l s inferred f r o m the s u s p e n s i o n c o n d u c t i v i t y were systematically larger t h a n those d e r i v e d f r o m electrophoresis. These results, a n d other recent
T h e existence o f a layer o f a d s o r b e d i o n s a n d solvent m o l e c u l e s at solid/electrolyte interfaces is well established in e l e c t r o c h e m i s t r y a n d colloid science. S o m e characteristics o f this layer, the Stern layer, can be r e p r e s e n t e d with e q u i l i b r i u m a d s o r p t i o n m o d e l s (1-4). O t h e r features, especially the net electric charge, are reflected in the d y n a m i c p r o p e r t i e s o f colloidal particles, which can be investigated t h r o u g h m e a s u r e m e n t s o f the e l e c t r o p h o r e t i c m o b i l i t y or suspension conductivity. However, theories relating particle m o b i l i t y or suspension c o n d u c t i v i t y to the characteristics o f t h e particle are usually b a s e d o n the p r o p e r t i e s o f a static Stern layer, which r e m a i n s u n d i s t u r b e d 1Author to whom correspondence should be addressed. 32
0021-9797/86 $3.00 Copyright © 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
DYNAMIC STERN LAYER MODEL work (6-10), have led us to investigate the role of transport processes operating within the Stern layer. Attention is directed to the Stern layer for several reasons. For example, experimental work with mica (11, 12) demonstrates that solutions of the nonlinear Poisson-Boltzmann equation furnish a very good representation of the electrostatic repulsion between two charged surfaces, even when the separation is of the order of a few nanometers. These findings, coupled with the results of a variety of calculations (13-15), suggest that despite some inherent approximations the Gouy-Chapman model provides a satisfactory basis for theoretical calculations relating to the diffuse part of the double layer. Furthermore, our earlier experimental work indicates that the responsible mechanism must be one that affects the particle mobility and suspension conductivity in different ways, which also suggests that simply altering the fluid viscosity or dielectric constant near the interface is not likely to provide a consistent explanation for the behavior observed. Thus, we are led to consider the presence of transport processes operative in the Stern layer. The approach selected involves introducing lateral ion transport in the Stern layer and represents, therefore, a picture considerably more complicated than that described by the standard theories. Moreover, the approach taken differs from that used in describing the influence of permeable, gel-like layers on the particle surface (16, 17). Although these models treat the flow processes in the surface layer in detail they omit consideration of ion transport and polarization of the layer. The electrokinetic charge of a colloidal particle is related to the net charge behind the hypothetical surface where shear appears in the fluid adjacent to the particle. The location of this envelope--the shear surface--is determined in part by chemical and physical interactions between the electrolyte and the particle. Ions and solvent molecules behind the shear surface, i.e., in the Stern layer, are said to be specifically absorbed. Although bulk flow
33
is absent in the Stern layer, specifically adsorbed ions can still respond to gradients in electrical potential or ionic concentration. If such transport processes occur, the electrokinetic behavior of the particle will depend not only on the net charge but also on the quantity and mobility of the adsorbed ionic species. Surface transport, in one guise or another, has been a part of colloid science for many years. Henry (18) and Booth (19), for example, each developed expressions describing the effect of "surface conductivity" on the electrophoretic mobility. Their theories, designed to account for the fact that the conductivity of the diffuse region differs from that of the bulk fluid in an approximate fashion, were superseded by rigorous treatments. Later, in their monograph on electrokinetic phenomena, Dukhin and Derjaguin (20) described "anomalous surface conduction," charge transport parallel to the surface but within the Stern layer. More recently, Lyklema, Dukhin, and Shilov (21) investigated the effects of surface transport on the low frequency dielectric properties of colloidal suspensions. In their work, transport in the Stern layer was characterized by a single mobility for all the surface species and no attempt made to depict the chemical composition of the layer in detail. In a similar vein, Van der Put and Bijsterbosch (7) employed a model of anomalous surface conduction to analyze their extensive electrokinetic data on porous plugs, concluding that ion transport in a layer of entangled polymer chains adjacent to the particle surface can explain the electrokinetic behavior observed. However the model they employed does not relate the details of the transport process in the diffuse layer to those in the adjacent gellike layer. None of the extant models describe transport within the Stern layer in the detail currently available for transport in the diffuse layer. To do so requires that the individual contributions of the several ionic species be taken into account. Thus, if an expression analogous to the Nernst-Planck equation is used to describe transport by electromigration Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
34
ZUKOSKI AND SAVILLE
and diffusion, then concentrations and mobilities are needed for each ionic species. The utility of such a model is self-evident and given a rigorous mathematical treatment, one can investigate whether or not transport within the Stern layer could play a significant role in electrokinetic phenomena. The purpose of our presentation is to set out such a model wherein transport phenomena within the Stern layer are coupled to the standard theory as applied to electrophoresis (22) and electrical conductivity (23). In the model there is an explicit accounting for contributions of individual species to both the electric charge and transport within the Stern layer. The local concentrations of each species are related to concentrations in the adjacent diffuse layer through mass action relations which describe the adsorption of ions in the Stern region. By balancing the flux of ions from the diffuse layer against electromigration and diffusion within the surface layer, simultaneous polarization of both the diffuse layer and the Stern layer are taken into account. Although our picture of the adsorption phenomena is particularly simple, the mass action model gives a satisfactory picture of the relation between the composition of the Stern layer and concentrations in the bulk of the solution and provides part of the requisite information needed to model the transport of ions. In this context it appears prudent to avoid taking the adsorption terminology too literally. Much more needs to be known about the structure of the Stern layer before the strict validity of different models can be assessed. For the present we view this sort of description simply as one that allows introduction of certain functional relationships which reproduce the empirical results. The details of any adsorption processes, including the role of the surface itself and interactions between different constituents of the Stern layer, remain subjects for further investigation. In the following sections a model of electrokinetics is developed to describe a dynamic Stern layer. Compared to the standard theory the essential modification lies in an alteration Journal of Colh)idand InterfaceScience, Vol. 114, No. 1, November 1986
to the boundary condition on ion fluxes. To establish a context the standard formulation for bulk electrokinetic processes is reviewed first, followed by a description of the Stern layer model. The discussion then turns to ways that Stern layer processes influence two electrokinetic measurements: electrophoresis and electrical conductivity. TRANSPORT IN THE BULK
The system analyzed consists of a suspension of charged spheres of radius a suspended in an electrolyte solution containing N distinct ionic species; the spatially varying number concentration of the kth ion is ~k with valence z k and mobility ~k. Each sphere has an equilibrium surface charge density denoted by 30. The ions in solution have a volume average concentration denoted by ~ . As noted previously (23), ion concentrations beyond the double layer depend on the volume fraction of particles, C, and the charge 30. It is assumed that the particles are placed in the solution as neutral entities whereupon surface groups ionize to produce "added" counterions of valence Z N+I, the (N + 1)st species. In our analysis the particles are presumed to be present at sufficiently low concentration to minimize particle-particle interactions, both hydrodynamic and electrostatic. The balance equations which govern the conservation of mass, momentum, and charge are: 7.0=0, N+ 1
~, ezk~q~76+ #720,
0 =-v~-
k=l
v.j
~ =
~72~ -
[1]
0, e
N+ 1
E z ~ k. k=l
Here 0 is the local fluid velocity,/3 is the pressure, e is the charge on a proton, # is the fluid viscosity, ~ is the electrical potential, and e is the dielectric constant of the suspending medium. It is assumed that the dielectric constant and viscosity of the fluid and the ion mobilities
DYNAMIC
STERN
are independent of electric field strength and position. The divergence of the flux of the kth species, jg, is set equal to zero since there are no sources or sinks for ionic material within the volume of interest. The flux of material in the bulk arises from contributions due to convection, electromigration, and diffusion, viz.,
LAYER
MODEL
~- = %(1 + A~rC).
35 [5]
The Aa term represents the O(C) contribution of the suspended particles to the suspension conductivity and will be called the conductivity increment. The conductivity increment and the electrophoretic mobility are electrokinetic properties of the suspended particles which can jk = n ~ -- e z k ~ k ~ 7 ~ -- ~ k k T V ~ k . [2] be measured experimentally. The electrophoretic mobility and the conThe mobility of the kth ion in the electrolyte ductivity increment are related to the electrosolution is ~k and k T is the product of Boltzkinetic properties of the suspended particles mann's constant and the abolute temperature. in different ways. The mobility is a property A frame of reference is chosen centered on of individual particles while the conductivity a test sphere so that the boundary conditions increment is a result of adding the contribuapplied far from the test sphere are: tions of many particles. Measurements of these xT~ = - E o J cos Ok, two properties provide, therefore, an excellent means for testing the consistency of electrov = -~Eo~r cos 0k, r ---, ~ , [3] kinetic theory. In what has been called the N+ 1 standard form of the theory the electrokinetic Z e zk~k = O. model consists of Eqs. [1] through [3] along k=l with requirements that each particle move as These equations prescribe a uniform electric a rigid impermeable body whose surface has field of magnitude E~ in the z-direction with a uniform charge. All the electrical properties neutral fluid streaming past the test particle at of the particle are subsumed in a single paa free stream velocity equal to the product of rameter, the ~"potential. Thus, measurements the electrophoretic mobility of the particle, of the particle mobility and conductivity of a /~e, and the electric field strength, E~. The suspension at a fixed salt concentration ought boundary conditions applied at the surface of to yield the same ~'potential, unless the theory each particle will be discussed in the next sec- is incomplete. In the work reported earlier (5), tion. For the present we simply note that in substantial differences between the two ~ pothe standard theory the particles have a uni- tentials were found. Given experiments free form surface charge characterized by their ~" of gross errors, the explanation must involve another sort of transport process. We turn, potential. By solving the differential equations around therefore, to investigate transport within the a single test particle we can obtain a theoretical Stern layer. relation between the electrophoretic mobility and the ~ potential (22). Solutions valid for a T R A N S P O R T A T T H E P A R T I C L E S U R F A C E - - A DYNAMIC MODEL OF THE STERN LAYER single particle can also be used to formulate a relation between the local current and the avInside the Stern layer the transport mecherage current within the suspension (23), viz., anisms envisioned are electromigration and 1 fV J = -~
UZ+ l
ezkj kdV.
[4]
k=l
From this relation one can relate the conductivity of a dilute suspension, b, to the volume fraction of particles as
diffusion. To represent the flux using the twodimensional analog of the Nernst-Planck equation, ionic mobilities and compositions are required for each species in the Stern layer. A simple adsorption model is used to relate concentrations in the Stern layer to those in the bulk. JournalofColloidandInterfaceScience,Vol. 114, No. 1, November 1986
36
ZUKOSKI
AND
In the conceptual model, particles of uniform radius are added to an electrolyte containing N distinct ionic species to obtain a suspension of volume fraction C. The colloid acquires a uniform surface charge density through the adsorption of ions from the surrounding electrolyte or by the dissociation of covalently bound moieties. It is assumed that these reactions are rapid, yielding a Stern layer which is in dynamic equilibrium with ions in the diffuse double layer. Each ionic species can undergo adsorption-desorption reactions of the form t/k(a) + gk ~ t/~ [61 where h~(a) is the concentration of the kth species evaluated at the particle surface r = a, ~ is the number density of empty sites in the Stern layer for the/cth species, and ff~ represents the number density of sites at which the kth species is absorbed. Competition between the different ionic species is absent and the total capacity of the surface layer for a particular species is N~. The equilibrium constant for each reaction pair is /(Tk- t~k(a)~k r/k
[7]
To relate the adsorption of ions to the charge each adsorption site is assigned a valence z~. Upon defining the fraction of empty sites, ~ k / ~ , as a k, the charge density within the Stern layer can be written as N+ 1
q = e[ Z k=l
N+ 1
+
-
E
k]
[81
k=l
which shows how changes in the concentration of an ionic species in the Stern layer affect the charge. Furthermore, application of an external electric field polarizes the diffuse double layer causing ionic concentrations to vary around the periphery of the particle. Then, because local equilibrium between species adsorbed in the Stern layer and ions in the diffuse layer is maintained, cf. Eq. [7], polarization of the double layer polarizes the surface charge density. It can be shown, however, that the polarization which lends the angular depenJournal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
SAVILLE
dence to the ion distributions does not alter the total charge (24). Ionic transport within the Stern layer arises from the lateral electric field and concentration gradients. This transport is modeled in a fashion analogous to transport in the bulk. The flux of the kth species, ]~, comes from the combined effects of electromigration and diffusion, i.e., ]~ = --ezkCo~g~Ts~
-- kT&~kX7s~ k.
[9]
Here ~sk is the mobility of the kth species, ~7sfi~is the tangential gradient of the surface concentration, and Vsq~is the tangential gradiem of the electric field. It is stressed that Eq. [9] is applied to perturbations from a state wherein the concentration and potential on the surface of the particle are uniform. More complicated relations involving concentration dependent mobilities and surface activities are conceivable but are unnecessary here since we deal with situations which have been linearized about the uniform state. Accordingly, more complicated flux relations reduce to the form shown above. It is possible that the mobility for surface diffusion would differ from that for electromigration but such differences are ignored here. A brief discussion of the linearization procedure is given in the Appendix, along with references to more mathematical treatments. The transport of ions along the surface is coupled with transport in the bulk through a boundary condition applied at the particle surface, viz., jk.n=-V~.j~
at
r=a.
[10]
In this expression jk represents the flux of the kth species in the bulk and n is the outeffacing normal. This expression is a modification to the standard theory in that the normal component of the flux of the kth species is balanced by an ion flux tangential to the surface. Whenever surface transport of ions is absent, jk = 0, and ]k.n = 0.
[11]
DYNAMIC STERN LAYER MODEL Accordingly, when all the &~ are set equal to zero, surface transport disappears and the model reduces to the standard theory for particles with impermeable surfaces. The remaining conditions enforced at the outer surface of the Stern layer are the same as those used in the standard theory, viz., continuity of the tangential components of the electric field, continuity of the velocity, and a relation between charge and potential, written - E ~rr = q.
[12]
We can summarize the main features of the model as follows. The diffuse part of the double layer is pictured in familiar terms: the Gouy--Chapman model with ion transport by the combined effects of flow, electromigration, and diffusion; fluid motion involves a balance between pressure gradients, viscous stresses, and the electrical body force. However, ions cross the shear surface separating the diffuse layer from the compact Stern layer under the influence of concentration gradients and electrical forces normal to the surface. Each such ionic flux is balanced by lateral transport within the Stern layer due to the same forces. In the Stem layer the ion mobilities differ from those in the bulk due to various factors related to the presence of the adsorbed layer. Because the layer is thin, the balance equations reduce to two-dimensional analogs of their counterparts in the bulk and, finally, are coupled to the equations that describe the bulk as boundary conditions. Due to rapid adsorption-desorption steps, the inner part of the diffuse layer and the Stern layer are in local equilibrium, as described by mass action equilibrium relations. The next task is to work out solutions to the model equations numerically. The solution of the differential equations employs methods developed to analyze situations where the applied field strength produces small changes in the equilibrium double layer configuration around each particle. A brief discussion of the mathematical methodology is given in the Appendix, and further details are given by Zu-
37
koski (24). We note here that the computer codes were carefully tested to eliminate errors and, under conditions where transport in the Stern layer is absent, reproduce the results of O'Brien and White (22) and Saville (23). With the numerical scheme we can explore how the electrokinetic behavior of the colloidal particles responds to changes in the various parameters. The results of a series of such calculations will be discussed shortly. In this connection it should be recognized that the ~ potential is no longer a primary parameter as it is in the standard theories where it can be set independent of other considerations. Here the equilibrium particle charge (and hence the ~" potential) is defined by the adsorption equilibria and involves valences, capacities, and adsorption constants for the various species in the Stern layer. The purpose of the material presented next is to explore the effects of the various model parameters on measurable electrokinetic properties, e.g., electrophoretic mobility and conductivity increment. In the second paper we will interpret experimental results obtained earlier (5) using the dynamic model of the Stern layer. MODEL PREDICTIONS To establish how transport in the Stern layer can influence the electrophoretic mobility of individual particles and the bulk conductivity of the suspension, the model equations were solved with different sets of values for the ion mobilities and other parameters. The model parameters are the mobility, adsorption constant, maximum density, and valence of the adsorption site for each ionic species in the Stem layer. The electrophoretic mobility and conductivity increment will be shown as functions of salt concentration as one of the parameters is varied while the others are held constant. The importance of salt concentration is emphasized because most of the parameters are related to the chemistry of the surface and thus depend on bulk ionic concentration. Results were calculated for 0.2-t~mdiameter spheres suspended in KC1 solutions Journal of ColloM and Interface Science, Vol. 114, No. 1, November 1986
38
ZUKOSKI AND SAVILLE
with a viscosity of 0.893 cP and relative dielectric constant of 78.9. It was assumed that cations adsorb at negatively charged sites, Zs~ = - 1 , and that anions adsorb at neutral sites, z 2 = 0. As a consequence the maximum density of adsorbed cationic species (the counterions) was set by the titratable charge and the adsorption of anionic species represents interactions with the hydrophobic portions of the surface. The adsorption model has been set up in a rather general form so as to be able to accomodate a variety of schemes for localizing ions and charge groups in the Stern region. The scheme we have chosen to use to illustrate the predictive capabilities of the model involves adsorption or binding of both co-ions and counterions in the surface layer. Counterion binding is a firmly grounded mechanism (14) but co-ion binding is less well established (25), although it has been used to explain the maximum in the mobility-salt concentration relation (26, 27). This matter will be discussed in some detail in Part II; for the m o m e n t we simply note that this issue is separate from the role of surface transport. Since the influence of the Stern layer depends on several factors, each will be taken
up separately; effects due to the degree of binding of ions within the Stern layer are investigated first. Figures 1 and 2 display how changes in the adsorption constants affect the electrophoretic mobility and conductivity. The trends in the electrophoretic mobility seen in Fig. 1 can be explained as follows. At a low salt concentration (10 - 6 M) the particle charge is determined by the maximum density of (negatively charged) sites at which the potassium ions adsorb, ~rx~, and the binding constant /~1. As the KC1 concentration is increased, more potassium ions adsorb, neutralizing the negative adsorption sites, decreasing the charge density and diminishing the particle mobility. At the same time, the chloride ion concentration in the bulk increases, and more chloride ions adsorb at neutral sites. Therefore, the particle charge density decreases at a slower rate than would be expected without chloride ion adsorption. At higher KC1 concentrations (10 -4 M), the Stern layer is almost saturated with potassium ions that neutralize the covalently bound negative charge, however adsorbed chloride ions maintain a negative charge in the layer. At still higher KCI concentrations (10 -2 M), chloride ions and potassium ions saturate the layer and
22
-6
LClcm z K+
14
_
-O. [
10-3 I0-~~ ~ . I ~
E ~k V_1
MIL VAmAeLE
~ I- 2.0 IXlO-4 0.2
u x.
5.0
-2
O
I
-6
-5
I
I
I
I
-4
-3
-2
-I
LOG [KCI]
-2
-6
I
-5
I
-4 LOG
I
-3 [KCI]
I
I
-2
~I
FIG. 1. Electrophoreticmobility and conductivityincrement. Variations with changes in ionic strength and the potassium ion adsorptionconstant,/~. Curvesare calculatedfrom the dynamic Stern layer model using parameters indicated. Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
39
DYNAMIC STERN LAYER MODEL -6
22--
lO-s
i~k
^k.^k ¢us,'w
=C/cm2
M/L
K+
5.0
IxlO -s
]1-
2.0
18
^ u
^k eN T
-0. t
14 ~t
!VARIABU 0 . 2
E ::L
_.l
\\V.=,o-,
-2
0
e
2i
i ,
-6
-5
, -4
, -3
J -2
I -I
-2' -6
I
-5
I -4
I -3
i
-2
--
J -I
• OG [KC'] FIG. 2. Electrophoretic mobility and conductivity increment. Variations with changes in ionic strength and the chloride ion adsorption constant,/~2. Curves are calculated from the dynamic Stern layer model using parameters indicated.
the charge density becomes independent of ionic strength. Here the electrophoretic mobility decreases due to the compression of the diffuse double layer arising from the increase in bulk salt concentration. The importance of transport in the Stern layer relative to that in the bulk can be characterized using a dimensionless "surface conductivity," i.e., N
as =
k=l
N
[13]
a Z k=l
Here ~0~ and a0k are (respectively) the concentration outside the diffuse layer and the fraction of empty sites for the kth species. The maximum density to which ions can adsorb, ]V~, is independent of ionic strength and so at high salt concentrations (large ~ the dimensionless surface conductivity is small. Stern layer transport processes are unimportant here. In Fig. 1, as is approximately 10 -3 at a KC1 concentration of 10-l M. Lyklema et al. (21) came to similar conclusions regarding the role of surface conductivity in their analysis of the frequency dependence of the conductivity of a colloidal suspension
at high salt concentrations. Although simultaneous polarization of the diffuse and adsorbed parts of the double layer were not taken into account, they were able to conclude that the influence of the mobility of bound ions on the complex conductivity of a colloidal suspension is small when the double layer is thin. This is consistent with the analysis presented here where the electrokinetic properties are largely unaffected by transport in the Stern layer for aK >> 1.2 The effect of altering the binding constant of the potassium ions on the conductivity increment is also shown in Fig. 1. Note, however, that there is a large difference in the scale between the mobility and conductivity increment panels. The differences depicted in the conductivity increment diagram would be experimentally significant at salt concentrations below 10-4 M. At a KC1 concentration of 10-4 M, A~ changes by one unit as the adsorption constant varies from 10-s to 10 -3 and this change is accentuated at lower KC1 concentrations. 3 2 In our example ar = 56 with [KC1] = 10-2 M. 3 The experimental uncertainty in the earlier data (5) was +0.2 Aa unit. Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986
40
ZUKOSKI AND SAVILLE
At high salt concentration (> 10-3 M), the particles have characteristics independent of the positive ion adsorption constant, K 1, because the Stern layer is saturated. Accordingly, differences between the conductivity increments disappear. At lower salt concentrations, however, a smaller binding constant yields a larger amount of adsorbed potassium ions and, since the adsorption of a potassium ion neutralizes a negatively charged site, a smaller value for K ~ leads to a lower charge density and a smaller Aa. This behavior is reflected in Fig. 1 for salt concentrations below 10 -4 M and again in Fig. 2 where the parameter varied is the binding constant for the chloride ions. The effects of different adsorption capacities for anions and cations are displayed in Figs. 3 and 4. In these figures the maximum density to which the ions can be adsorbed is varied by changing NTl and NT2 , The situations where either anion or cation adsorption is absent illustrate why binding of both ions is important. The mere presence of a mobile surface species will not explain the structure of the mobilitysalt concentration relation reported in the literature (1, 5, 25, 28). At a given coverage of adsorbed ions, the charge density of the particle is fixed but
changing the mobility of the adsorbed ions influences both the particle mobility and conductivity increment strongly. Ion transport in the Stern layer decreases the tangential component of the electrical field which lowers the electrophoretic mobility. At the same time, this transport provides another conduction path which increases the conductivity increment. The effects of increasing the mobilifies of the cation and anion in the Stern layer are shown in Figs. 5 and 6, respectively. In these figures the mobilities of the adsorbed ions have been varied systematically while holding /~k and NTk fixed. This makes the total particle charge depend on the salt concentration alone and in these figures differences in either the electrophoretic mobility or in conductivity increment at a given ionic strength are due solely to dynamic processes in the Stern layer. The curves marked 0~] = 0 in Fig. 5 and 4 2 = 0 in Fig. 6 represent situations where surface transport is absent. At very high salt concentrations effects due to transport in the Stern layer are swamped by transport in the bulk, as expected (~s ~ 1). However, the insensitivity shown at bulk salt concentrations below 10-5 M is unexpected
,o-
-6-
\
""
,~ ~k ~p~ ,c,0m, M/L --
\
A~
"5 -
i
I •
i
l
]
[Kc,]
-5
-4
-3
LOG[KC, ]
-2
-I
FIG. 3. Electrophoreticmobilityand conductivityincrement.Variationswith changesin ionic strength and the maximumnumber of potassium ion bindingsites,N}. Curvescalculatedfrom the dynamicStern layermodel usingparametersindicated. Journal of Colloid and Interface Science, VoL 114, No. 1, November 1986
DYNAMIC STERN LAYER MODEL -6~
41 e~ k
I0--
~
\
8
3
eNT u
6
-4
2 E :L
4
"5 - 2
2
K+I
i~ k
gsk/~ k
~Clcm z
M/L
--
5.0
i×lO-5
O.i
ixIO -4
0.2
VARIABLE
o 0
0
P
0 -6
-5
-4 -3 LOG [KCl]
-2
-2 -6
-I
I -5
I
i
-4 -3 LOG [KCI]
I
-2
-I
FIG. 4. Electrophoretic mobility and conductivity increment. Variations with changes in ionic strength and the maximum number of chloride ion binding sites, ?~T. Curves calculated from the dynamic Stern layer model using parameters indicated.
(cf. Fig. 5). At concentrations where as >> 1 it might be thought that the surface transport processes would dominate the electrokinetic behavior of the colloid. Instead, both the electrophoretic mobility and conductivity increment appear to be insensitive to changes in the mobility of the potassium ion in the Stern layer. The explanation for this behavior lies
in the coupling between transport of ions in the Stern layer and perturbation potential gradient. At low salt concentrations the potential gradient scales as 1/~]. Thus, since the flux involves the product of the ion mobility and the potential gradient, the flux is relatively insensitive to changes in mobility at low ionic strength. At intermediate salt concentrations
2E
-6
et~k p.C/cm 2
~k
~sk/,.;k
M/L
--
-2
-I
IE
.-£
14 =n
E ::L
A~.
~1 -2
I0
6
0:E 2 f 0"0~
I
-6
-5
I
I
i
-4 -3 LOG[KC,]
-2
I -I
-z -6
-5
,
~
-4 -3 "OG [KC,]
FIG. 5. Electrophoretic mobility and conductivity increment. Variations with changes in ionic strength and the surface mobility of potassium ion, g~]. Curves calculated from the dynamic Stern layer model using parameters indicated. Journal of Colloid and Interface Science, Vol. ! 14, No. 1, November1986
42
ZUKOSKI AND SAVILLE 22-
-6
J.Clcm;'
18
u
14
E ~L
~
K+
5
]1-
2
MIL
--
x l o -s
0.0
[XlO -4 VARIABLE
•",o- I0
>_1 -2
6
~
~s2/~2-o.9
0 :E 2
-6
i
i
L
i
I
-5
-4
-3
-2
-I
]
-2
-6
I
-5
I -4
L -3
I
I
-2
-I
FIG. 6. Electrophoreticmobilityand conductivityincrement. Variationswith changesin ionic strength and the surfacemobilityof chloride ion, ~2. Curves calculatedfrom the dynamicStern layermodel using parameters indicated. (10 -5 M < [KC1] < 10-2 M), the potential gradient has a much weaker dependence on ~1 and changes in the ion mobility change the electrophoretic mobility and conductivity substantially. Both electrophoretic mobility and conductivity increment appear insensitive to changes in the anion mobility, g~2(Fig. 6). Most of this is due to the type of equilibrium adsorption chemistry chosen here. Since there are far fewer adsorbed chloride ions than there are potassium ions, a larger change in the adsorbed chloride ion mobility is required to produce an equivalent change in transport rate. In addition, the layer does not become saturated with chloride ions until a r >> 1, where the effect of surface transport is small. Finally, to illustrate the influence of ion transport within the Stern layer, we present results calculated in terms of an "apparent" ~" potential (Fig. 7). The results were obtained as follows. For the model system the binding constants, adsorption capacities, and other parameters were set so as to give a particle surface with a modest ~"potential, about - 6 0 mV. Then with the equilibrium properties of the surface held fixed, calculations of particle mobility and suspension conductivity were carded out for several different adsorbed Journal of Colloid and Interface Science, V o l . 114, N o . 1, N o v e m b e r
1986
counterion mobilities using the dynamic model of the Stern layer. Particle mobilities and suspension conductivities so calculated reflect the full influence of transport within the layer, including polarization of the concentrations of adsorbed ions. Next, these mobilities and conductivity increments were reinterpreted in terms of"apparent" ~"potentials using standard theories (22, 23). Figure 7 shows how these apparent ~"potentials depend on the magnitude of the ion mobility in the Stern layer. It is clear that a modest sized ion mobility can have a large influence on the apparent ~"potential. For example, if the counteflon has a mobility, g~, one-tenth of its mobility in free solution, ~1, the ~"potential inferred from an electrophoresis experiment using the standard theory will be about 60% of the true figure. Similarly, from a suspension conductivity measurement the apparent ~"potential will be twice as large. Such differences all arise from the way transport in the Stern layer alters the dipole term that arises in the solution of the transport equations (22, 23). We conclude that transport processes in the Stern layer can produce differences between the ~" potentials derived from mobility and conductivity similar to those reported in earlier experimental studies (5).
43
DYNAMIC STERN LAYER MODEL
~C/cm z
--
K+
5
I xlO -5 V~IASI..E
CI-
2
I x l O -4
¢..--& 0.5 t;-r
°0
M/L
0.0
~v
Ol
02
0 3
oI 0
i
I
I
0.1
0.2
0.3
~/~
FIG. 7. Relationships between the apparent ~"potentials and the counterion mobility in the Stern layer. Left: Apparent ~"potentials derived from particle mobility. Right: Apparent ~"potentials derived from suspension conductivity. ~'Astands for the apparent ~"potential and ~T for the true value.
SUMMARY
In this paper we have presented an electrokinetic model incorporating effects due to transport in the Stern layer. In the model, ionic concentrations in the Stern layer are in local equilibrium with the diffuse double layer so polarization of the diffuse layer affects the Stern layer and vice versa. Transport within the Stern layer arises due to diffusion down concentration gradients and conduction in response to a tangential electric field. The set of model equations was solved using numerical techniques to avoid mathematical approximations. Systematic variation of the magnitudes of the binding constants, the density of binding sites, and mobilities in the Stern layer leads to several conclusions. First, ionic transport has substantial effects on both the electrophoretic mobility, t~e, and electrical conductivity increment, Aa, for intermediate values of aK (i.e., 2 ~< aK ~< 50). At extremes of salt concentration where a~ >> 1 or aK ~< 1, the mobility and conductivity increments each become insensitive to transport in the Stern layer. Furthermore, although the magnitude of Ao- is sensitive to changes in the model parameters, the qualitative features of the relationship are not as sensitive as those of the electrophoretic mobility-salt concentration
relationship. In a subsequent paper this model will be used to interpret the experimental results reported earlier (5). APPENDIX: A DESCRIPTION OF THE MATHEMATICAL TECHNIQUES
The techniques used to carry out the numerical calculation of the fields around a test particle and to construct relations for the electrophoretic mobility and conductivity increment were patterned after the work of O'Brien and White (22), Saville (23), and O'Brien (29). First the model equations are recast in dimensionless form and simplified using a perturbation scheme employing the expansion parameter aeE~/kT, a dimensionless ratio involving the strength of the applied field compared to a characteristic field strength. This leads to a set of equations describing the equilibrium state around a test particle and another set describing the way this situation is perturbed by the applied field. The solutions to the second set are axisymmetric with respect to the azimuthal angle and so the dependent variables can be represented as the product of two functions, one which describes the r dependence and the other the 0 variation. The latter is simply cos 0. Thus the problem comes down to integrating a set of ordinary differJournal of Colloid and InterfaceScience, Vol. 114,No. 1, November1986
44
ZUKOSKI AND SAVILLE
ential equations. A l t h o u g h it is useful to int r o d u c e auxiliary functions o f t h e sort e m p l o y e d in earlier w o r k (22, 23, 29), s o m e o f the a d v a n t a g e o u s features are d i l u t e d b e c a u s e we m u s t calculate t h e p o t e n t i a l a n d i o n conc e n t r a t i o n s at the surface explicitly. T h e sol u t i o n for the fields a r o u n d the test particle p r o v i d e s the e l e c t r o p h o r e t i c m o b i l i t y directly (22). T o o b t a i n the c o n d u c t i v i t y i n c r e m e n t these fields m u s t be averaged a c c o r d i n g to the techniques o u t l i n e d b y O ' B r i e n (29) a n d Saville (23). T h e conductivity i n c r e m e n t involves c o n t r i b u t i o n s f r o m the a d d e d c o u n t e r i o n s , nonspecific a d s o r p t i o n , a n d w h a t is called the d i p o l e term, m o d i f i e d to t a k e a c c o u n t o f t r a n s p o r t in the Stern layer. As n o t e d earlier the c o m p u t e r p r o g r a m s were tested to insure t h a t t h e y r e p r o d u c e d the correct results for the mobility and conductivity increment when t r a n s p o r t in the Stern layer was suppressed. F u r t h e r details are given b y Z u k o s k i (24). ACKNOWLEDGMENTS This work was supported by the NASA Microgravity Sciences and Applications Program and a grant from the Xerox Corporation. We are also indebted to J. C. Baygents for his help with the mathematical modeling. REFERENCES 1. Hunter, R. J., "Zeta Potential." Academic Press, New York, 1981. 2. Healy, T. W., and White, L. R., Adv. Colloidlnterface Sci. 9, 303 (1978). 3. James, R. O., Stiglich, P. J., and Healy, T. W., in "Adsorption from Aqueous Solutions" (P. H. Tewaft, Ed.). Plenum, New York, 1980. 4. James, R. O., and Parks, G. A., Surface Colloid Sci. 12, 119 (1982). 5. Zukoski, C. F., IV, and Saville, D. A., J. Colloid Interface Sci. 107, 322 (1985).
Journal of Colloid and Interface Science. Vol. 114, No. 1, November 1986
6. Van der Put, A. G., and Bijsterbosch, B. H., J. Colloid Interface Sci. 75, 512 (1980). 7. Van der Put, A. G., and Bijsterbosch, B. H., J. Colloid Interface Sci. 92, 499 (1983). 8. Springer, M. M., Korteweg, A., and Lyklema, J., J. Electroanal. Chem. 153, 55 (1983). 9. McDonogh, R. W., and Hunter, R. J., J. RheoL 27, 189 (1983). 10. O'Brien, R. W., and Perrins, W. T., J. Colloidlnterface Sci. 99, 20 (1984). 11. Pashley, R. M., J. ColloidlnterfaceSci. 80, 153 (1981). 12. Pashley, R. M., J. Colloid Interface Sci. 83, 531 (198 I). 13. Levine, S., and Outhwaite, J., J. Chem. Soc. Faraday Trans. 2 74, 1670 (1978). 14. Torrie, G. M,, and Valleau, J., J. Chem. Phys. 73, 5807 (1980). 15. Carnie, S. L., Chan, D. Y. C., Mitchell, D. J., and Ninham, B. W., J. Chem. Phys. 74, 1472 (1981). 16. Levine, S., Levine, M., Sharp, K. A., and Brooks, D. E., Biophys. J. 42, 127 (1983). 17. Wunderlich, R. W., J. Colloid Interface Sci. 88, 385 (1982). 18. Henry, D. C., Trans. FaradaySoc. 44, 1021 (1948). 19. Booth, F., Trans. Faraday Soc. 44, 955 (1948). 20. Dukhin, S. S., and Derjaguin, B. V., in "Surface and Colloid Science" (E. Matijevi6, Ed.), Vol. 7. Wiley, New York, 1974. 21. Lyklema, J., Dukhin, S. S., and Shilov, V. N., J. Electroanal. Chem. 143, 1 (1983). 22. O'Brien, R. W., and White, U R., J. Chem. Soc. Faraday Trans. 2 77, 1607 (1978). 23. Saville, D. A., J. Colloid Interface Sei. 91, 34 (1983). 24. Zukoski, C. F., IV, "Studies of Electrokinetic Phenomena in Suspensions," Ph.D. thesis, Princeton University, Princeton, N.J., 1984. 25. Goff, J. R., and Luner, P., J. Colloid Interface Sci. 99, 468 (1984). 26. Meijer, A. E. J., van Megan, W. J., and Lyklema, J., J. Colloid Interface Sci. 66, 99 (1978). 27. Ma, C. M., Micale, F. J., E1-Asser,M. S., and Vanderhoff, J. W., in "Emulsion Polymers and Emulsion Polymerization" (D. R. Bassett and A. E. Hamielec, Eds.), ACS Symposium Series Vol. 165, pp. 251-262. Amer. Chem. Soc., Washington, D.C., 1981. 28. Ottewill, R. H., and Shaw, J. N., J. Electroanal. Chem. 37, 133 (1972). 29. O'Brien, R. W., J. ColloidlnterfaceSci. 81, 234 (1981).