Electrosorption on random and patchwise heterogeneous surfaces: electrical double-layer effects

Electrosorption on random and patchwise heterogeneous surfaces: electrical double-layer effects

Electrosorption on Random and Patchwise Heterogeneous Surfaces: Electrical Double-Layer Effects LUUK K. KOOPAL 1 AND WILLEM H. VAN RIEMSDIJK Departmen...

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Electrosorption on Random and Patchwise Heterogeneous Surfaces: Electrical Double-Layer Effects LUUK K. KOOPAL 1 AND WILLEM H. VAN RIEMSDIJK Department of Physical and Colloid Chemistry, Wageningen Agricultural University, P.O. Box 8038, 6700 EK Wageningen, The Netherlands; and Department of Soil Science and Plant Nutrition, Wageningen Agricultural University, P.O. Box 8005, 6700 EC Wageningen, The Netherlands Received December 10, 1987; accepted April 19, 1988 The electrical double layer of a heterogeneous interface is considered for three types of heterogeneity: random, isolated patches, and interacting patches. For a random heterogeneous surface the electrical double-layer characteristics are equivalent to those of a homogeneous interface. For a patchwise heterogeneous interface the properties of the electrical double layer resemble those of the different patches. Examples of surface charge and differential capacitance curves are given for amphotefic surfaces composed of two types of surface sites. The overall point of zero charge (pzc) is a function of both the surface composition and background electrolyte concentration and the minimum in the capacitance-potential curve is not coinciding with the overall pzc. For a patchwise surface with interacting patches an intermediate situation exists. For high electrolyte concentrations the patchwise situation is approached, whereas for low and intermediate electrolyte concentrations the situation resembles that of a random heterogeneous (or a homogeneous) surface. © 1989AcademicPress,Inc.

INTRODUCTION

analytical isotherm equations have been derived (2). Alternatively the adsorption (free) energy distribution can be calculated on the basis of gas adsorption data using advanced numerical methods (1, 3, 4). In principle these methods can also be applied to (single-solute) adsorption from solution data, although it should be realized that the adsorption free energy is in this case an exchange quantity (5). For adsorption from solution of ionic adsorbates, that is to say for charged heterogeneous surfaces, the situation is considerably more complex because of the electrostatic interactions between adsorbate and adsorbent and between the adsorbate molecules mutually, and only recently have the first steps been made in this field. Investigations of the effect of the heterogeneity on the electrical double layer started with studies of polycrystalline silver (6) and gold (7) electrodes. Models of the electrical double

Adsorption of ions on a surface with a variety of reactive sites is largely governed by the distribution of affinity constants of that ion for the different sites. The same applies if an adsorbent is composed of a mixture of different types of particles, each type with its characteristic sites for the adsorbate. In practice such heterogeneous sorbents or adsorbent mixtures are far more common than homogeneous systems. Understanding of the adsorption behavior of these systems is therefore of great practical importance. Adsorption of uncharged adsorbates on heterogeneous uncharged adsorbents has been given due attention (see, e.g., (1, 2)) with emphasis on adsorption from the gas phase. For certain (free) energy distribution functions, i To whom correspondence should be sent.

188 0021-9797/89 $3.00 Copyright© 1989by AcademicPress,Inc. All rightsof reproductionin any formreserved.

Journalof ColloidandInterfaceScience,Vol. 128,No. 1, March 1, 1989

ELECTROSORPTION

ON HETEROGENEOUS

layer of a polycrystalline metal-electrolyte interrace have been given by Grigoryev (8) and Bagotskaya et aL (9). Isotherm equations for ion adsorption on homogeneous and heterogeneous metal oxide-electrolyte interfaces, based on site-binding models taking into account the electrostatic effects, have been presented and discussed by Van Riemsdijk et al. (10-13). In this work a distinction has been made between "random" and "patchwise" heterogeneity, the first indicating that the equal-energy sites are distributed randomly over the entire surface, the latter that the equal-energy sites are grouped together in macroscopic patches each of which can be treated as homogeneous. Analytical expressions for the adsorption isotherm equations could be derived if the assumption was made that the electric double layer is uniform over the entire surface (11, 12). This simplification is only allowed for random heterogeneous surfaces. For patchwise heterogeneous surfaces, with a discrete double layer per patch, some preliminary work has been done (13). In the present paper the emphasis is on the electrical double-layer models which can be used for the various types of heterogeneity in combination with adsorption models. A distinction will be made between random, patchwise, and patchwise-interacting surfaces. It will be assumed that the surface potential, ~0, is due to adsorption of potential determining (p.d.) ions, and that the p.d. ions for the different site types are the same. Such sites could be present on, for instance, the different crystal faces of a solid. Moreover, for the patchwise heterogeneous surfaces it will be assumed that the patches are amphoteric and that the relation between the surface potential of a patch, $0,p, and the activity of the p.d. ions, X, is following the Nernst equation. Such behavior is characteristic for solids like AgI and well approximated for probably most common oxides (14). On the basis of site-binding models, Nernstian behavior is predicted for oxides if the site density is high (10, 15). The advantage

189

SURFACES

of the assumption of Nemstian behavior is that for the moment adsorption models can be omitted. However, for surfaces for which the site density is not extremely high the Nernst equation is not applicable and an ion adsorption model is required to provide the $o(pX) relation. In a future contribution we will return to this problem. Before considering the electrical double layer for the various types of heterogeneity, the double-layer properties of a homogeneous fiat surface will be briefly reviewed, as these form the basis for the further analysis. HOMOGENEOUS

SURFACES

In modelling the double layer of a homogeneous flat surface, the Stern-Gouy-Chapman (SGC) theory (16-18) can be used. In this theory, the surface charge density a0 and surface potential if0 are both seen as mean field (smeared-out) quantities (19). The relation between ¢o and if0 is defined by the integral capacitance K of the double layer K = a0/~ko.

[ 1]

In order to characterize the shape of the a0($o) curves the differential capacitance C is more convenient: C = (aao/O~o)Cs,

[21

i.e., the slope of the a0($o) curve at constant background electrolyte concentration, cs. In the SGC model the total capacitance is subdivided in a Stern layer capacitance, Cs, and a diffuse layer capacitance (Kd or Cd) connected in series, hence 1

1

1

--Cd

for the integral capacitances or 1

1

1 (--O~d~

[4]

for the differential capacitances if it is assumed Journal of Colloid and lnterface Science, Vol. 128, No. 1, March 1, 1989

190

KOOPAL AND VAN RIEMSDIJK

that Cs is independent of ao. The diffuse layer charge density is denoted by trd. In the absence of specific adsorption of counterions from the background electrolyte, --trd/aO = 1 and -(0ad/0ao) = 1 so that Eqs. [3] and [4] can be further simplified. In the present treatment only this situation will be considered. Equations [3] and [4] show that the total capacitance is composed of a specific (the Stem layer capacitance) and a generic contribution (the diffuse layer capacitance). The Stem layer capacitance is defined as O"o

Cs=f0_fd

~O~S

d '

[51

where fd is the potential at the Stem plane, eo is the permittivity of vacuum, ~s is the relative permittivity of the Stern layer, and d is its thickness. Theoretically the Stern capacitance accounts for specific effects. In the first place Cs allows one to differentiate between different (indifferent) counterions through d. Second and more important with respect to surface heterogeneity, different surfaces have a different effect on the solvent layer adjacent to the surface and thereby on es. In general, in aqueous solutions ~s is smaller than er, the relative permittivity of the bulk solution. For a hydrophobic surface such as AgI (20), rather low values of Cs and hence of es are observed (21), whereas for hydrophilic surfaces such as oxides, with a water-like interface, Cs is high and Es and Er are comparable in magnitude (22). The tendency that Cs increases with increasing hydrophilicity is also found for metal surfaces (9, 23). In principle the surface charge may also affect ~s. At high values of ~o the water dipoles may change their orientation in the electric field which can result in a lowering of Es. In the present treatment this effect will be neglected, Cs is seen as a constant independent of ~ro (and cs), and the integral and differential Stern layer capacitances are the same. The diffuse layer capacitances Kd and Cd follow from the GC theory (19)

Kd

----

ereoKq[ln{q + (1 + q2)O.5}]-1

[6]

Journal of Colloid and InterfaceScience, Vol. 128, No. 1, March 1, 1989

and Cd = Er%K(1 + q2)0.5,

[7]

where

q = (-~rd)(2kT~r%r/ze) -l,

[8]

r is the reciprocal Debye length, which is proportional to the square root of the ionic strength, z is the valency of the background electrolyte (symmetrical), and e, k, and T have their usual meaning. Equations [6] and [7] are independent of the type of surface, and they incorporate the generic effects related to the surface charge density and ionic strength in the double-layer models. For low values of crd, Kd and Co are approximately independent of ~d and equal to eOerr so that Eqs. [3] and [4] can be written as (~0 = - ~ d ) I

1

K-

C

~0¢S

~0¢r

d + r -~ "

[9]

Equation [9] emphasizes that the electrical double layer can be considered as two plate capacitors connected in series. For given values of ao, cs, and Cs, Cd and C can be calculated using Eqs. [7] and [4], respectively, whereafter fro, and hence go(fro), can be obtained by integration: fo =

fo°°l~ da.

[101

The ao(f0) curve can also be obtained by using the integral capacitance of the double layer. RANDOM

HETEROGENEOUS

SURFACES

On a heterogeneous surface the distribution of charges due to adsorption of p.d. ions is determined by the positions of the different sites on the surface, by the lateral electrostatic repulsion, and by the site occupancy. Only the first factor is specific for a heterogeneous surface. For a surface with a random distribution of the different sites, an approximately even distribution of the adsorbed p.d. ions over the entire surface will result. The average surface charge density, ao, can therefore be calculated

ELECTROSORPTION

ON

HETEROGENEOUS

as the total charge due to adsorption of p.d. ions divided by the entire surface area. Following the SGC model, this average (mean field) charge density corresponds with an average, smeared-out surface potential, fro, in exactly the same way as for the homogeneous interface. Hence, for given values of the overall surface charge, a0, the overall Stern layer capacitance, Cs, and the ionic strength, cs, the surface potential fo can be calculated with the equations given in the previous section. The corresponding equivalent electrical circuit of such a random heterogeneous surface is shown in Fig. 1. With respect to Cs it should be remarked that for a r a n d o m heterogeneous surface consisting of, say, two different types of sites, A and B, Cs cannot simply be related to Cs,a and Cs,b, the Stern layer capacitances of the homogeneous surfaces A and B. The parameters Cs,a and Cs,b apply to macroscopic interfaces, whereas surface A B is a r a n d o m array of microscopic sites A and B to which these values do not apply. To illustrate the behavior of a r a n d o m heterogeneous surface and to be able to compare the results with those for patchwise heterogeneous surfaces, Figs. 2 and 3 are presented. Figure 2 shows the ao(f0) behavior for two different values of cs and Fig. 3 shows the relation C(fo). The curves are based on the arbitrarily chosen value of Cs = 85 # C / c m 2. All curves are symmetrical around the point of zero charge (pzc) and equivalent to those of a homogeneous surface with the same Cs value. Note that for different surfaces with the same value of Cs, the shape of the ao(fo) and C(fo) curves is the same. In general, it m a y be expected that the Cs value of a random heterogeneous oxide will be large (22) and not

~ bo

191

SURFACES

20

15.

~H 10 S

o.~~o

40o

r~--J

10o

,

~o

~o

~o

%(mV)

-5 U -10 i

i

i

h

i

i

FIG. 2. The dependence of the surface charge, ao, of a random heterogeneous surface on the surface potential, fro, at two concentrations of indifferent electrolyte. For the Stern layer capacitance Cs,,b, a value of 85 #F/cm2 is chosen. The curves also apply to a patchwise interacting surfaceAB with Cs, a = 20/~F/cm2, Cs,b= 150 uF/cm2, a common diffuse layer, andf~ = j~ = 0.5. strongly dependent on the surface composition. It m a y therefore be expected that ao(fo) and C(fo) curves of such heterogeneous oxides will be nearly independent of the composition and the same as those of the comparable homogeneous oxides. In a previous article (11) it has been shown that also a0(pH) curves for r a n d o m heterogeneous oxides, calculated on the basis of sitebinding models in combination with the SGC model, are very similar to those of a homogeneous oxide with the same pzc and Cs value,

80i =

~

pzcAB

~

~

104H

20-

-"

'I l u l CS

lI

0

-

Cd

FIG. 1. Equivalent electrical circuit of the electrical double layer of a random heterogeneous surface with a uniform surface potential.

-~0

-100

1 0

160

z00

~0

~0 ~(mv)

FIG. 3. The dependence of the differential capacitance 6t of a random heterogeneous surface on the surface potential (Cs,ab = 85 ~tF/cm2). The curves also represent a patchwise surfaceAB (fa = J~)with Cs,o= 20 uF/cm2, Cs.b = 150/~F/cm2, and a common diffuse layer. Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989

192

K O O P A L A N D VAN RIEMSDIJK

GS,a Cd,a but not exactly the same. Apparently the ffo(pH) behavior is not exactly the same for "I random heterogeneous and homogeneous surfaces. Comparing this with the present resuits, it may be concluded that if the fr0(ApH) -I curves (ApH = pH - pH(pzc)) for different Cs,b Cd,b homogeneous or random heterogeneous metal oxides with about the same Cs value differ FIG. 4. Equivalent circuit for the electrical double layer considerably, this should be due to differences of a patchwise, noninteracting, heterogeneous surface. in the ~bo(ApH) curves rather than to differences in the a0(~bo)curves. For patchwise het- and Ca.p, the Stern layer and diffuse layer caerogeneous metal oxides, this reasoning does pacitance of patch p, respectively. The overall not apply (see below). capacitance of the total surface, Ct, is the result of connecting up the patch capacitors, Cv, PATCHWISE HETEROGENEITY: parallel. For the surface considered it follows N O N I N T E R A C T I N G PATCHES that

,Ht

General Description With patchwise heterogeneity and large patches it can be assumed that each patch behaves as a homogeneous surface in equilibrium with the bulk solution. In practice this type of heterogeneity may occur in systems made up of a mixture of different homogeneous particles or for crystalline particles with different (large) crystal planes. We will restrict the treatment to amphoteric surfaces for which the different types of patches respond to the same p.d. ions (X). For such surfaces the surface potentials of the (amphoteric) patches and that of the entire surface are all related to the pX. The precise relation is determined by the conditions. In the present situation the Nernst equation will be assumed, which predicts that for each patch P the surface potential ¢/o,p is directly proportional to ApX(P) = pX pX(pzc P). Under these conditions the specific properties of a patch are characterized by its pX(pzc P) and Stern layer capacitance. As all patches behave Nernstian, the average or equivalent surface potential ~0 of the entire surface S is also obeying the Nernst equation so that if0 is proportional to ApX(S) = pX - pX(pzc S). For such a patchwise amphoteric surface with sites A and B, the equivalent electrical circuit is shown in Fig. 4. The differential patch capacitance, Cp, is given by Eq. [4] using Cs,p -

Journal of Colloid and Interface Science, Vol. 128,No. 1, March 1, 1989

I

Ct =faCa +ACb

[lla]

Ct = Z £ C v ,

[llbl

or in general P

where£ is the fraction which each patch contributes to the total surface area. According to Eq. [11] the heterogeneity will show up most clearly when the Cp values are rather different, in that case Ct is strongly dependent on £ . The total surface charge density, fro, is the weighted sum of the patch charge densities, fr0,p,

fro = fafro,a + J;fr0,~,

[12a]

which can be generalized to fro = ~fpfro, p.

[12b]

p

Per patch the frO,p(~,b0,p)behavior is exactly the same as that of an equivalent homogeneous surface under the same conditions. This is an important difference with a random heterogeneous surface where the charge density per group of equal-energy sites depends on the overall surface potential. It should be noted that the fr0,pvalues of a patch cannot be chosen independently, they depend on the pX or rather on ApX(P). As a consequence of the assumption that all patches behave Nernstian, ApX(P) is directly propor-

ELECTROSORPTION

ON HETEROGENEOUS

tional to ~k0,p= ff - ffp(pZc), ff being a potential axis relative to an arbitrary reference potential. Hence, only for a series of patches with known values of pX(pzc P) or ff(pzc P) and Cs,p is Eq. [ 12] operational. According to Eq. [12] the pzc of an amphoteric heterogeneous surface is reached for the bimodal surface if fago,, + J~gO,b = 0

(pzc AB)

[ 13a]

(pzc).

[13b]

or in general fpg0,p = 0

According to Eq. [13] the pzc of the entire amphoteric surface will be somewhere in between the patch pzc's. Its precise position depends on the conditions, that is to say, on the patch pzc's and capacitance values. In order to illustrate the behavior of patchwise surfaces, some examples will be worked out for surfaces composed of two types of patches, each with its own characteristics. In the present model the specific nature of the patches is in the first place represented by their Cs,p values. To model the heterogeneity we therefore consider three cases: (1) the Stern layer capacitances of both types of patches are very low, (2) one of the Stern layer capacitances is low and the other high, and (3) both Stern layer capacitances are high. To a first approximation a low Stern layer capacitance represents a hydrophobic patch and a high Stern layer capacitance a hydrophilic patch, but this distinction may not be rigorous. A second distinction between different patches is ff(pzc P). In the examples ff(pzc A) and ff(pzc B) have been given fixed values. VERY LOW STERN LAYER CAPACITANCES

to ~k- ff(pzc P), i.e., on the RHS of their ff(pzc) the patches are considered to be positive, and on the LHS negative. The charge density on each patch is found as gO,p =

p

Cpd~ =

dO

Cpd~,

[14]

where Ap indicates that ~khas to be integrated from its value at the patch pzc to the actual value. In terms of surface potentials this corresponds with integrating from 0 to if0,;. The overall value of go is found by combining Eqs. [14] and [12] at a given value of ft. For the example shown in Fig. 5 the hatched areas represent g0,a and Cro,b. In the case that go,a = --gO.b, as shown in Fig. 5, the net surface charge is zero, provided fa = Jb = 0.5. Note that although f~ = J~, pzc AB is not halfway between pzc A and pzc B because Ca and Cb have different values. In this extreme case where Ca and Cb are independent of ff and the indifferent electrolyte concentration, the pzc of AB is located at a fixed value offf independent of cs. Moreover the overall surface charge density, go, as a function of ~kis a straight line with a slope determined by f~, Ca, Cb, and ff(pzc B) - ff(pzc A) but independent of cs. The g0(ff) curves of such a heterogeneous surface with constant patch capacitances cannot be distinguished from that of a homogeneous surface with a constant total capacitance. [p

i

pzcA

t

%;°

The simplest situation exists when the Stem layer capacitances of both patches are so low that Cd,p >>Cs,p. According to Eq. [4], Cp is then determined by Cs,p. The situation is shown in Fig. 5, where Cp values are plotted versus the potential ft. The patch pzc values chosen are indicated as ffo,a = 0 and ~o,b = O. The surface potentials of each patch are equal

193

SURFACES

L

pzcAB pzcB

_ i

i

%°¢° %;°

-y

i~0. 5. Double-layer capacitancesof patchesA and B of a heterogeneoussurfaceAB as a function of the surface potential for the case t h a t the p a t c h capacitances are d o m i n a t e d b y the Cs,p values (Cs, p ~ Cd,p). T h e h a t c h e d areas indicate the patch c o n t r i b u t i o n s to the overall surface charge ~r0 o f a surface for w h i c h f ~ = J~ = 0.5 in the case t h a t or0 = 0. The pzc values of b o t h patches a n d t h a t o f surface A B are indicated. Journal of Colloidand InterfaceScience, Vol. 128, No. 1, March 1, 1989

194

KOOPAL AND VAN RIEMSDIJK

The situation illustrated in Fig. 5 is of course an oversimplification, for in general Cp is not a constant but depends on Csand go. In practice the condition Cs,p ,~. Cd;p cannot be matched at low indifferent electrolyte concentrations and Cp will depend on cs and go,p. Nevertheless it may be concluded that the effects of heterogeneity will be small for surfaces composed of patches with low Cs, p values, especially at higher electrolyte concentrations.

erence point for if, hence ~ko,~ = ~k and ffO,b = ~b -- 200 mV. In general, Cp goes through a minimum located at the pzc of the patch. This is clearly illustrated for patch B. For patch A in 10 -z M electrolyte this minimum is very shallow and Ca is nearly constant, reflecting the situation of Fig. 5. According to Eq. [ 14], for each value of ff the areas under the curves (with ~k(pzc P) as reference point) represent g0,p. The hatched areas indicate the situation that o-0,a --g0,b or that the total surface charge density g0 = 0 providedfa = j~ = 0.5. In this case the pzc of the entire surface, ~b(pzc AB), is located nearest to pzc B, i.e., nearest to the patch with the highest capacitance. Moreover, the position of pzc AB is dependent on the indifferent electrolyte concentration: the higher Csis, the more pronounced is the effect of the relatively high value of Cs,b and the closer pzc AB is to pzc B. For the present example pzc AB shifts about 34 mV along the potential axis if cs is raised from 10 -3 to 10-1M. On the basis of the Cv(~b) curves and Eqs. [14] and [12], go(~b)curves can be obtained for each electrolyte concentration. In Fig. 7 the results are shown for three electrolyte concen=

A LOW AND A HIGH STERN LAYER CAPACITANCE

For a patchwise surface composed of patches with a low and a high Stern layer capacitance both types of patches are rather different. Let us assume that for patch A, Cs,a = 20 #F/cm:, and that for patch B, Cs,b = 150 #F/cm 2. Furthermore let the difference between ff(pzc A) and ~b(pzc B) be 200 inV. For each patch Cp can be calculated using Eqs. [4], [7], and [8] for different values of g0,p, whereafter ~bo,v is obtained by an integration using Eq. [10]. The dependence of Ca and Cb on the surface potential is shown in Fig. 6 for two electrolyte concentrations. The pzc of A is taken as a ref-

E 1/*O

I

---=

i

j

,,

120-

i

lOO-

1

r pzi 8

j \i 60

~ I

1o.

,

,

'

150

200

/

20- ~ 0

-100

-50

0

50

100

250

300

q)(mV) FIG. 6. Double-layer capacitancesof patchesA and B of a heterogeneoussurfaceAB as a function of the surface potential at two indifferent electrolyte concentrations (10 -3 M and 10-I M 1 - 1 electrolyte). The Stern layer capacitance o f patch A equals 20 #F/cm 2 and that of patch B equals 150 #F/cm2; the difference between the patch pzc's is 200 mV. The hatched areas correspond with the patch contributions to the overall charge density for the case that fa = Jb and go = 0. The resulting pzc values of surface A B are indicated at both electrolyte concentrations. Journal of Colloid and lnteoCaceScience, Vol. 128,No. 1, March 1, i989

E L E C T R O S O R P T I O N ON H E T E R O G E N E O U S SURFACES

. . . .

E

/

i0.~M

2 0

-I00

0

I00 ~,.'/~

2

-6 -8 -10 -12

FiG. 7. The dependence of the overall surface charge density on the (average) surface potential for a patchwise surface A B (f~ = J~), with Cs,~ = 20 #F/crn 2 and Cs,b = 150 # F / c m 2 at three (indifferent) electrolyte concentrations. The potential axis is fixed arbitrary at the pzc of patch A.

trations. Note that ff0,~b= ~ -- ff(pzc AB). The dependence of ff(pzc AB) on the electrolyte concentration is clearly observed. The near c o m m o n intersection point (cip) at about 0.5 #C/cruZ should not be confused with the pzc, nor be interpreted as an indication for specific adsorption (24). The absence of a generally occurring cip at the pzc is an important difference with homogeneous and random heterogeneous surfaces. To analyze the shape of the ao(ff) curves in more detail, in Fig. 8 the overall capacitance Ct is plotted versus ff for different values off~ and two electrolyte concentrations. In each curve only one minimum in Ct, indicated with an open arrow, is observed. For the heterogeneous surfaces this minimum is located near pzc B, its exact location depending on fa and cs. The location of pzc AB is also a function off~ and c~ (see the solid arrows). Comparison of the positions of the open and the solid arrows reveals that the m i n i m u m in Ct cannot be identified with the pzc of AB (unless the surface is homogeneous). This is another important difference between patchwise and

195

random heterogeneous or homogeneous surfaces. The cip's of the curves at 10-3 M (Fig. 8a) correspond with the situation that Ca = Cb = Ct. At 10 -~ M cip's do not occur as Cb > Ca for all values of ft. T W O H I G H STERN LAYER CAPACITANCES

Let us now assume a patchwise surface for which the Stern layer capacitances of both patches are large, say, for example, Cs,a = 100 ~tF/cm 2 and Cs,b = 200 #F/cm z. As before ff(pzc A) - ~k(pzc B) is put equal to 200 mV. Owing to the large Cs,p values the patch capacitances are mainly determined by Cd,p. This is illustrated in Fig. 9, where Ca and Cb are shown as a function of ft. Both Cp(~) curves show a well-defined minimum. For a surface for whichf~ =J~ = 0.5 the shaded areas in Fig. 9 indicate ao,a and tr0,b for the case that cro,a = --a0,b and hence a0 = 0. Due to the large influence of Cd,pboth patches behave similarly and the location of pzc AB is only weakly dependent on cs. Clearly for a surface with f~ = J~ the location ofpzc AB would be independent of cs when Cs,a and Cs,b would be equal. On the basis of the Cp(~) curves, ~o(ff) or Ct(ff) curves can be constructed. As the Ct(ff) curves show differences more clearly, these curves are shown in Fig. 10 for different values of f~ at two electrolyte concentrations. The c o m m o n intersection points of the curves correspond with the values of ff where Ca = Cb SO that Ct is independent offa. As before the minima in the curves are indicated by open arrows, and the pzc AB values by solid arrows. Although the effect of Cd,p on Cp is large, none of the curves shows two minima. Moreover, there is a much more gradual transition of the position of the minimum as a function offa than for the surface composed of patches with a low and a high value of Cs,p. Similarly as in Fig. 8, the positions of the minima do not coincide with the positions of pzc AB. The position of pzc AB as a function of the surface composition is plotted in Fig. 11 for 10-3 and 10-~ M 1-1 electrolyte (solid curves). The curves intersect atf~ = 0.67. This is due Journal of Colloidand InterfaceScience, Vol. 128,No. 1, March 1, 1989

196

KOOPAL A N D VAN RIEMSDIJK

:so

E 100-

,..7 80-

6

So

1oo

~So

2~

2~o

o.2

~0-

o

~

08

20=

os o2 1o[AI

o-,-%

10

,

z

t

08

, os

,,o2

,~oo

120- ~ O0 (B) 100- .

,

(~ ~

~

10-~ M

800.5

,o

4.0-

o8

20-

1.o (A)

~ T

08

10

0--'%

-50

I)

50

I00

1.50

200

2'50 N-"(mV)

FIG. 8. The overall capacitance of patchwise surfaces AB as a function of the surface potential for various surface compositions. The Stern layer capacitances chosen are Cs,a = 20 and Cs,b = 150/=F/cm 2. The area fraction f= is indicated in the figure. The open arrows indicate the position of the minimum in each curve and the solid arrows the pzc AB. Part (a) applies to 10 -3 M and (b) to 10-~ M 1-1 electrolyte.

160 I i

!

140

t

I

120]

i ~

I ! pzcB

pzcAB ~IO-'H

pzcA

/

/

8O 60 tO 2(? (?

-I00

- 50

0

50

100

150

200

250

300 q)lmV)

FIG. 9. As in Fig. 6, but for a patchwise surface AB with Cs,a = 100 #F/cm 2 and Cs,b = 200/zF/cm 2.

Journal of Colloid and InterfaceScience, Vol. 128, No. 1, March 1, 1989

ELECTROSORPTION ON HETEROGENEOUS SURFACES

}-=100N~m

197

® 10-3M

0.5 0.67 0.8

0.67

60-

.,.

08

t~0-

05

/

~

10 10

20N

-50

0

50

~0

150

~00

2'50 (..plmV}

FIG. 10. As in Fig. 8, but for a patchwise surfaceAB with Cs,a= 100 t~F/cm2 and Cs,b= 200 ~F/cm:.

to the fact that in the region between pzc A and pzc B the weighted capacitance curves of both patches (0.67Ca and 0.33Cb) are approximately symmetrical around pzc AB. This being true for 10 -3 and 10-1 M electrolyte, it also holds (approximately) for other electrolyte concentrations. Hence, for this value offa the pzc of AB is independent of c~. In general, a weak dependence of pzc AB on c~ is to be expected if the fpCp curves are similar. In Fig. 11 the results obtained for the previous surface (Cs, p values of 20 and 150 ~tF/cm 2, respectively) are also included (dashed curves). For most values off~ the shift of pzc AB as a function ofc~ is large, due to the fact that the shape of the two fpCp curves (Fig. 6) is rather different. For large values offa, f~ Ca approximates

fbCb and here the pzc of AB is only weakly dependent on cs. The variation of ~(pzc AB) with cs is also a function o f / x ~ = ff(pzc B) - ~k(pzc A). For low values of A~ and ~ values in between pzc A and pzc B the patch capacitances are approximately constant ( Ccp ~ ~K), resembling the situation shown in Fig. 5. Hence, in this case ff(pzc AB) is only slightly dependent on cs, irrespective of the Cs,p values. The trends that ~(pzc AB) is a function of cs and that the pzc AB and the m i n i m u m in the capacitance do not coincide have also been found by Bagotskaya et al. (9) for the metalsolution interface. For surfaces with more than two types of patches the situation is essentially similar. The Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989

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KOOPAL AND VAN RIEMSDIJK

I

E

2oo ///

PATCHWISE HETEROGENEITY: INTERACTING PATCHES

Electrostatic interactions extend in general over relatively long distances and for a patchwise surface these interactions will be effective / z/ / ~ 100 in the border zones of neighboring patches. As // / / a measure of these interactions KR can be used, i.e., the ratio between the characteristic di50 mension R of a patch and the Debye length Cs~, Cs.b K-~. For KR >> 1 the interaction zone is small 0 relative to the size of a patch and the patches (pF~mz} -5(~ can be considered as independent. For KR ~ 1 o'.8 o'.6 o L, ~: oo patches are strongly interacting and the surface % can be considered as r a n d o m heterogeneous. FIG. 11. The position of the potential of the pzc of a The intermediate region, i.e., KR ~ 1, is typical surfaceAB as a function of the surfacecompositionat two for interacting patches. Obviously the ionic concentrations of indifferent electrolyte.The potential axis strength, which determines the magnitude of is fixed at the pzc of patch A. ff(pzcB) - ff(pzcA) = 200 mV. Patch capacitances for dashed curves are Cs,a = 20 K, is an important parameter. At high ionic /~F/cm2 and Cs.b = 150 ~tF/cm2 and for solid curves Cs.~ strength a surface m a y show patchwise behavior, whereas at low ionic strength r a n d o m = 100/~F/cm2 and Cs,b= 200/~F/cm2. behavior is shown. For r R ~ 1 considerable lateral doubledistribution of patch pzc values enters as an layer overlap occurs. Due to this overlap the important parameter. Another complication lateral differences in the electrostatic potential not treated here is specific adsorption of the at some distance from the surface diminish. In other words, the patchwise nature of the background ions. To end this section a brief remark should surface is most strongly reflected close to the be made on the behavior of ao(pX) curves for surface, i.e., in the Stern layer. As a first apnoninteracting patchwise surfaces. In general, proximation we therefore assume that patchwhen the ao,p(pX) curves per patch are known, wise interacting surfaces can be described as the ao(pX) curves can be constructed by simply if each patch has its " o w n " Stern layer capacadding the weighted a0,p values for various par itance, but that the diffuse layer is the same values. Similarly as with the ao(ff) curves, the for all patches. Hence, the surface plane is hetao(pH) curves at different salt concentrations erogeneous and nonequipotential, but the will not have a c o m m o n pzc value, Only when Stern plane is considered as an equipotential thefpao,p-pH curves for the different patches plane. The equivalent electrical circuit for this are congruent for each salt concentration and situation is shown in Fig. 12. Under this conthe distribution of patch pzc's is symmetrical dition the Stern layer capacitances of patches will a cip corresponding with the overall pzc A and B are independent and Cs,t can be found occur. For congruency ofthefp ao,p-pH curves, as not only the fpCp-ffo,p curves should be conCS, t = f a G s , a "~"f b C s , b. [15] gruent but also the ffo,p - p H relationships. Two examples of a0(pH) curves for patch- The total capacitance of the entire surface folwise heterogeneous surfaces as calculated with lows from a simple site-binding model in combination 1 1 1 with the SGC model have been presented else= Cs,t + Cd" [161 where (13). ///

150

....

20

150

Journal of Colloidand lnterfaceScience, Vol. 128, No. 1, March 1, 1989

ELECTROSORPTION ON HETEROGENEOUS SURFACES

CS,a

--

I

:

I I

I

:

Cs,b FIG. 12. Equivalent circuit for the electrical double layer of a patchwise interacting surface AB in the case that the Stern plane is equipotential. (Roughly speaking rR ~< 1.)

Due to the fact that Co is dependent on the overall surface charge density, ao, instead of on the individual patch charge densities, Ct has to be calculated as a function of ao, using Eqs. [7], [15], and [16]. With thus obtained values of Ct, the relation between ao and the equivalent or average surface potential ~o can be found on the basis of Eq. [10]. As a consequence of the fact that Cd is independent of the patch properties, Ct is determined by Eq. [ 16], which is equivalent to Eq. [4] for a homogeneous surface. In other words, the Ct(~o) and ao(t~o) curves of this type of patchwise interacting surfaces are identical to those of a homogeneous (or a random heterogeneous) surface with the same Cs value. For instance, the ~0(~o) and Ct(~o) curves for a patchwise interacting surface with a common diffuse layer and Cs,t = 85 #F/cm 2 are identical to the curves shown in Figs. 2 and 3, respectively. The difference between the random and the patchwise interacting surface is that for the patchwise surface Cs,t is related to Cs,a and Cs,b (see Eq. [15]), whereas for a truly random surface no simple relation exists between Cs and the individual Stern layer capacitances. A comparison of the patchwise interacting surface with a common diffuse layer with a patchwise noninteracting surface can also be made, For instance, for a surface with f~ = J~ the value of Cs.t = 85 #F/cm 2 (see Figs. 2 and 3) can be seen as the equivalent capacitance for two capacitors connected parallel with Cs,p values of 20 and 150 ~tF/cm2, respectively. The a0(ff) and Ct(~k)(f~ -- 0.5) curves for the comparable patchwise noninteracting surface are

199

shown in Figs, 7 and 8, respectively. Remember that ~k0 = ff - ff(pzc AB). Comparison of these figures with Figs. 2 and 3 reveals considerable differences. For a discussion of these differences we refer to the previous section. In conclusion, patchwise interacting surfaces for which the Stern plane is equipotential can be treated as equivalent homogeneous surfaces. As a rule of the thumb such a treatment is allowed up to about rR ~ 1. For rR > 1 the patchwise nature of the surface will show up. Finally, it should be remarked that in the case of patchwise surfaces it is not possible to calculate patch properties such as aO,p(~O,p)on the basis of the overall ao(ffo) curves alone. This is only possible if other information is also available such as fp and ~k(pzc P). For patchwise interacting surfaces, separation in patch contributions is only feasible if KR > 1. The same will hold for the ao(pX) curves of patchwise surfaces. That is to say, also with the aid of site-binding models it is not possible to separate the overall ao(pX) curve in its patch contributions without further information. CONCLUSIONS

The double-layer properties of heterogeneous interfaces depend on the type of heterogeneity of the surface, Random heterogeneous surfaces with a smeared-out electrical double layer behave similarly to homogeneous surfaces. Noninteracting patchwise heterogeneous surfaces reflect the individual properties of the patches in their electrical double-layer properties. As a consequence the double-layer behavior is rather different from that of homogeneous surfaces. The a0(~0) curves are asymmetrical around the overall pzc and the pzc is a function of the background electrolyte concentration. The electrical double layer of patchwise surfaces with interacting patches is intermediate between that of random and purely patchwise surfaces. Which of the two situations Journal of Colloid and Interface Science, Vol. 12g, No. 1, March 1, 1989

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KOOPAL AND VAN RIEMSDIJK

dominates depends on the size R of the patches and the electrolyte concentration. As a rule of thumb up to about KR ~ 1 the Stern plane can be considered as equipotential and the behavior is similar to that of homogeneous (or random heterogeneous) surfaces, whereas for KR > 1 the surface can be approximated as patchwise without interactions. REFERENCES 1. House, W. A., ColloidSci. 4, 1 (1913). Specialist Periodical Report, Royal Society of Chemistry, London. 2. Jaroniec, M., Adv. Colloid Interface Sci. 18, 149 (1983). 3. Koopal, L. K., and Vos, C. H. W., Colloids and Surfaces 14, 87 (1985). 4. Vos, C. H. W., and Koopal, L. K., J. Colloidlnterface Sci. 105, 183 (1985). 5. Papenhuijzen, J., and Koopal, L. K., in "Adsorption from Solution" (R. H. Ottewill, C. H. Rochester, and A. L. Smith, Eds.), p. 211. Academic Press, London, 1983. 6. Valette, G., and Hamelin, A., J. Electroanal. Chem. 45, 301 (1973). 7. Clavilier, J., and Nguyen Van Huong, C., £ ElectroanaL Chem. 41, 193 (1973); 80, 101 (1977). 8. Grigoryev, N. B., Dokl. Akad. Nauk SSSR 229, 647 (1976).

Journal of Colloid and Interface Science, Vol. 128, No. 1~ March 1, 1989

9. Bagotskaya, I. A., Damasin, B. B., and Levi, M. D., J. Electroanal. Chem. 115, 189 (1980). 10. Koopal, L. K., Van Riemsdijk, W. H., and Roffey, M., J. Colloid Interface Sci. 118, 117 (1987). 11. Van Riemsdijk, W. H., Bolt, G. H., Koopal, L. K., and Blaakmeer, J., aT. Colloid Interface Sci. 109, 219 (1986). 12. Van Riemsdijk, W. H., De Wit, J. C. M., Koopal, L. K., and Bolt, G. H., .1.. Colloid Interface Sci. 116, 511 (1987). 13. Van Riemsdijk, W. H., De Wit, J. C. M., and Koopal, L. K., Neth. J. Agric. Sci. 35, 241 (1987). 14. Fokkink, L. G. J., De Keizer, A., Kleijn, J. M., and Lyklema, J., J. Electroanal Chem. 208, 401 (1986). 15. Bolt, G. H., and Van Riemsdijk, W. H., in "Soil Chemistry. B. Physico-Chemical Models" (G. H. Bolt, Ed.), Chap. 13. Elsevier, Amsterdam, 1982. 16. Stern, 0., Z. Elektrochem. 30, 508 (1924). 17. Gouy, G., J. Physique 9, 457 (1910); Compt Rend. 147, 645 (1910); Ann. Physik 7, 129 (1917). 18. Chapman, D. C., Philos. Mag. 25, 475 (1913). 19. Barlow, C. A., in "Physical Chemistry," Vol. IXA (H. Eyring, D. Henderson, and W. Jost, Eds.). Academic Press, New York, 1965. 20. Billet, D. F., Hough, D. B., and Ottewill, R. H., J. Electroanal. Chem. 74, 107 (1976). 21. Lyklema, J., and Overbeek, J. T. G., J. Colloid Sci. 16, 595 (1961). 22. Kleijn, M. J., and Lyklema, J., J. Colloid Interface Sci. 120, 511 (1987). 23. Trasatti, S., Croat. Chem. Acta 60, 357 (1987). 24. Lyklema, J., J. Colloidlnterface Sci. 99, 109 (1984).