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Advances in Colloid and Interface Science 61(1995) 131-160

ELSEVIER

Electrical double layer interactions for spherical charge regulating colloidal particles* Rammile Ettelaie*, Richard Buscall Colloid and Rheology

Unit, ICI Wilton, P.O. Box 90, Middlesbrough,

Cleveland TS90 SJE, UK

Abstract A new way of linearising the Poisson-Boltzmann equation is presented which is capable of producing accurate answers for moderately high surface charge and surface potentials. Unlike the Debye-Htickel approximation, the linearisation is based on the fact that electric potential variation, rather than the magnitude of the potential itself, remains small between two charged plates at close separation. The method can be applied quite generally to any type of surface. The cases of constant surface charge, the constant surface potential and charge regulating systems are considered as examples. The results for constant charge and potentials as high as 5 @Xm2 and 200 mV are within 7% of the exact answers for plate separations of one Debye length or less. In contrast, the Debye-Htickel approximation produces answers that are at least an order of magnitude too large, for the same problems. The application of the method to the problem of calculating the electrostatic forces between charge regulated spherical particles is also addressed in some detail.

1. Introduction For over forty years the DLVO theory of colloidal stability [1,21 has provided a framework upon which many phenomena relating to the behaviour of mesoscopically sized particles can be understood. The two Q Paper presented at the PSC-CISG Conference on Concentrated Dispersions, Bristol, 29-31 March 1995. * Corresponding

author.

0 1995 OOOl-8686/95/$29.00 SSDZ OOOl-8686(95)00263-4

Elsevier Science B.V. All rights reserved.

R. Ettelaie, R. BuscalllAdu.

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main independent components of the theory remain the attractive van der Waals and repulsive electrostatic interactions between an approaching pair of surfaces. The subject of the attractive van der Waals forces has been examined thoroughly and much improvement has been made since the original DLVO theory. As a result there now exist closed form expressions [3], for a variety of particle shapes, which adequately describe these interactions [4,51. Note however that these expressions require a knowledge of the Hamaker constants for various materials involved. Methods for calculating such constants have been discussed by Hough and White 161 amongst others. Practical examples of their application to colloidal systems of everyday use are to be found in a number of recent papers 171. In contrast to the van der Waals forces, the theory of electrostatic repulsion is still mainly based on the original mean-field model, adopted by Debye-Huckel and Gouy-Chapman 181. In such an approach the charges on the surface of the colloidal particles are taken to be smeared out and the ions in the solution are represented by point charge objects. Statistical mechanics considerations then lead to the PoissonBoltzmann equation

V2y = -IE-C &O&r

i

nizi

E

exp

i

(1) B

1

for the electric potential w in the electrolyte solution. The constants E~E~, e, k, and T denote the permittivity of the electrolyte solution, the electronic charge, the Boltzmann constant and the absolute temperature. The valency and the bulk concentration of each species of ions present is denoted by zi and ni, respectively. The continuum description provided by the Poisson-Boltzmann equation is of course expected to break down as the separation between the particles becomes comparable to the size of the ions. However in some instances, particularly those involving symmetric single-valent electrolytes [9], various corrections to the mean-field theory largely cancel each other out. For these, the Poisson-Boltzmann equation then provides a reasonable description of the electrostatic interactions down to remarkably small surface separations. Even for systems where this is no longer true, or where other more complex short range interactions are at work, an accurate determination of the solution to the PoissonBoltzmann equation remains an essential step in understanding the

R. Ettelaie, R. Buscall IAdv. Colloid Interface Sci. 61 (1995)131-160

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properties of the system. Thus for example, if one is interested in determining the magnitude of the steric or hydration forces, say by means of the atomic force microscopy, one should be able to account for the electrostatic and van der Waals forces which are no doubt also in operation at the same time. The methods used to obtain the solutions to the Poisson-Boltzmann equation mainly fall into two classes. The first of these attempt to integrate the equation directly using numerical techniques such as the finite difference or finite element methods. Due to the non-linearity of the equation this is still a moderately complex task. Thus in the past, whenever the nature of the problem allowed, approximations to the Poisson-Boltzmann equation were made [lo, 111. These normally involved the replacement of Eq. (1) with a linearised form

v2y = K2y

(2)

where K is the inverse Debye length given by

K2 =

e2

kgTs,,s, c

ni ”

(3)

i

Equation (2) is a valid approximation to the Poisson-Boltzmann equation as long as the electric potential remains small everywhere. With the recent availability of powerful computers, such approximations are increasingly becoming less necessary. As a result, accurate solutions to problems involving simple boundary conditions (i.e. constant surface potential or constant surface charge) can now be obtained within reasonable computer run times [12,13]. As yet the same cannot be said of problems involving mixed boundary conditions. For these neither the surface potential nor the surface charge are predetermined. Instead there exists a relation between the two quantities, specified by the surface chemistry. The task of obtaining numerical solutions to the full Poisson-Boltzmann equation with such a set of boundary conditions, particularly when the relation between the surface potential and the surface charge itself is non-linear [14,15], is far from trivial. It is only very recently that solutions to this type of problem have been attempted

Ml. Apart from the numerical solutions, much effort has been, and is still, focused on obtaining analytical approximate solutions to the Poisson-

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Boltzmann equation. In some cases such solutions are given as closed form expressions. In the others they are obtained as series expansions, the terms of which can individually be evaluated and summed to provide the required solution. It must be remembered however, that all such solutions are only valid when certain criteria are satisfied by the system under study. For example if the electric potential is expected to be small everywhere, then the linearised equation (1) (also referred to as the Debye-Htickel equation) can be used instead of (2). Glendinning and Russel have derived an exact solution to the linearised PoissonBoltzmann equation for the case of two interacting charge spheres [17]. The particle can either be of the constant potential or of the constant charge type. The solution to the equation is given in terms of a multipole series expansion. More recently, Krozel and Saville have extended this method to cases involving linearised charge regulation 1181. When the potential in the system is no longer small, rendering the use of the Debye-Htickel equation unfeasible, or for models with more complicated non-linear charge regulation, it is no longer possible to obtain direct solutions to the problem of two interacting spheres. To make progress the simpler problem of two interacting infinite plates has to be considered instead. In this case the important quantity of interest is the mid-point potential VM between the plates. Once this is determined, the force per unit area fpl acting on the plates is given viz. relation

(4) In writing (4), and throughout the rest of the paper, we assume that the electrolyte with which we are dealing is of symmetric univalent type. Furthermore that the interacting plates are identical. Fortunately, the results of the calculations for the double plate problem can still give useful information regarding the forces between the two spheres. This is provided that Ka >> 1 and that a >> h. In here a and h denote the radius of the spherical particles and their separation, respectively. The above conditions are not too restrictive and are easily met in most cases encountered in practice. Under such circumstances the force f,,(h) between the spheres and the free energy per unit area V(h) for the plates can be related through the use of the Deryaguin approximation [19]. This gives the relation f,,(h) = naV(h)

(5)

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The free energy of interaction V(h) for the plates has to be calculated by integrating the force fpl(x> from an infinite plate separation down to a value x = h. Although there are some doubts regarding the validity of Deryaguin approximation at very small values of h [17,18], these are essentially associated with the constant charge cases. For these the electric potential between the plates diverges (actually logarithmically) as h + 0. It is the existence of such high potential, and hence large electric fields, that are thought to cause the breakdown of the Deryaguin approximation. It must be stressed however, that these conclusions are made on the basis of studies that involved the use of the Debye-Huckel equation. The numerical work of Carnie et al. [16], with full PoissonBoltzmann equation, give a somewhat contrasting result. They find that at high surface potentials, the non-linearity of the equation causes the potential to drop more rapidly away from the surface, improving the accuracy of the Deryaguin approximation. Thus, while the accuracy of the approximation is somewhat uncertain for the constant charge problems, no such ambiguity exists for cases where the potentials remain finite upon contact. For these, Eq. (5) remains valid at all separations, so long as Ka >> 1 and a >> h. The use of Eq. (5) still leaves us with the problem of determining f,,(h) and subsequently V(h). The case of constant potential plates has been extensively studied in the past, Exact solutions to this problem have been available for a long time [21. However, these involve the use of the elliptical integrals, which has made their use tedious in practice. Therefore, many approximate solutions to the problem have been derived over the years [2,20,21]. When used in conjunction with each other, these expressions allow accurate values of V(h), for low and moderate surface potentials, to be calculated at any plate separation h. Less attention has been devoted to the constant charge problem. Nevertheless, even in this case a number of approximate expressions for V(h) are available [22-251. In particular the relation derived by Gregory [25] gives excellent results for any value of h, provided the charge on the surfaces is small (although surface potential can become quite large as h 4 0). One common feature of these expressions however is the fact that, to a large extent, their use is limited to the cases for which they were derived. Thus for example, it is rather difficult to modify a method used to derive V(h) for a constant potential case to one involving mixed boundary conditions. In this paper we are mainly concerned with electrostatic interactions between surfaces following the charge regulation process discussed by Chan et al. [14,15,261. This type of process is believed to provide a much

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more accurate representation of the amphoteric surfaces than the constant charge or the constant potential models. Typical examples of such amphoteric surfaces are various metal oxides like alumina. We will also limit our discussion to close plate or particle separations. Our interest in such small separations arises from the fact that it is the magnitude of the forces at or near the point of contact which are the essential components in determining the low shear rheological behaviour of a dispersion. Furthermore, in a large number of practical systems, the position of the maxima in the free energy potential happens to occur at distances of around one or less Debye lengths. It is of course the magnitude of this maxima that ultimately is responsible for the stability of the dispersion and other properties such as the coagulation rate [3]. A formal set of simultaneous transcendental equations relating the surface charge to the surface potential, for the charge regulating double plate system, was derived and numerically solved by Chan et al. [14]. Although this provides exact solutions to the problem, once again the use of the elliptical integrals and various Jacobi elliptical functions makes its implementation rather complicated. Not many attempts have been made to provide simpler approximate ways of solving the PoissonBoltzmann equation, for plates satisfying the charge regulation conditions of the amphoteric surfaces. One major step in this direction however was taken by Carnie and Chan, who used the Debye-Htickel equation, together with an appropriate linearisation of the charge regulation condition, to provide analytical results for the problem [27]. In the same work solutions to the Debye-Hiickel equation with the full non-linear surface charge-surface potential relation, are also reported [27]. For these latter cases some numerical work is required. However, this only involves the solution of a simple transcendental equation, which is far easier to implement than the numerical integration of a differential equation or the use of Jacobi elliptical functions. Our aim is to extend the work of Chan to problems where the potential or surface charges can no longer be considered small (although the distance between the plates is assumed to be small (-K-l or less)). Since for charge regulated systems a prior knowledge of the surface charge or potential is not available, we feel that this is a rather necessary step. The approximation that we shall introduce also involves replacing the Poisson-Boltzmann equation (1) with a linearised version. However, as will be seen, this is not the same as using the Debye-Hiickel equation (2). In particular, our approximation makes no assumption regarding the

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137

magnitude of the electric potential w in the electrolyte region, between the plates. Instead it only requires that the variation of such potential remains relatively small, a condition that is increasingly satisfied as plates approach each other more closely. Like the Debye-Htickel equation our approximation has general applicability, in the sense that it can be applied to any type of boundary condition. Nevertheless, with good approximations already available for the constant charge and constant potential problems, we feel that it is most useful when used in conjunction with the charge regulating boundary conditions. The use of approximation does not entirely eliminate the need for the numerical work. But instead, it replaces the process of integrating a differential equation with mixed boundary conditions, with the much easier task of solving a transcendental equation. Nowadays this can be done almost trivially using no more than a programmable calculator, if the corresponding equation is not too complex. Alternatively of course, one can make further approximations to the equation in order to arrive at an analytic result. In this paper however, we shall not pursue such an approach, leaving the discussion of this to future publications. The paper is organised as follows. In the next section we shall present the details of the approximation. The use of the method is exemplified through its application to the constant surface charge and the constant surface potential problems. In each case the results are compared with the exact numerical solutions to the Poisson-Boltzmann equation. Next we discuss the charge regulated cases, exploring the variation of charge and surface potential, for a variety of surfaces, as the plate separation is varied. We examine some of the previous conclusions in the literature regarding such surfaces, in the light of our current results. Finally in Section 4, the calculation of the free energy potential per unit area V(h) for two plates satisfying the charge regulation is addressed. Using expressions derived by Chan and Mitchell [28], identities relating V(h) for the charged regulating system to those for the constant potential (or constant charge) models are derived, once the surface charge and potential for the former are known.

2. Approximate solutions to the Poisson-Boltzmann at close plate separations

equation

Before we proceed further a note on the units used throughout the paper is appropriate. As is customary, we shall use a set of normalised

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Colloid Interface Sci. 61 (1995) 131-160

units in which the unit of electric potential is taken as (knT/e> and the unit of length as K-l. With these units, Eq. (1) for a pair of interacting plates, involving symmetric univalent electrolyte, reads

d2W

~ = sinh(v) dx2

(6)

Similarly, we shall normalise the surface charge density on the plates surface by ~2n0kBTs0s,. The boundary condition relating the surface charge density and the electric field then becomes

at the interfaces. For a typical solution of 0.01 mol/l at room temperature these units correspond to about 3.1 nm for length, 0.59 yC/cm2 for the charge density and 25.7 mV for the electric potential. The popularity of the Debye-Huckel equation (2) is to a large extent due to the fact that it provides simple expressions for v in the entire range between the two plates. This allows one to quickly relate the surface charge (T, and the surface potential ws, on the plates, to the important quantity of the mid-point potential v,. Thus, one has ty, = yrn cosh(h/2)

(8a)

0, = v,

(8b)

sinh(h/2)

where as before h is the distance between the plates. For the fmed surface charge or fixed surface potential boundary conditions, the appropriate equations (8) can easily be inverted to yield the value of vm. Furthermore, for charge regulating surfaces, expressions (8a) and (8b) can be substituted in the equation relating the surface charge and the surface potential. This results in an equation involving the single unknown v,,, which can quickly be solved using one of many numerical schemes for this purpose. Our aim is then to replace Eqs. (8a) and (8b) with another suitable set of approximations that remain valid when vm, v, and os are no longer small. As we shall see this is possible if h is sufficiently small. To introduce the approximation, consider a specific example involving two plates with a fixed surface charge of 6,. As the schematic

R. Ettelaie, R. BuscalllAdv.

Fig. 1. Schematic

Colloid Interface Sci. 61 (1995) 131-160

139

diagram showing the variation of the electric potential between two

similar plates. The maximum O&/2.

variation

in the potential

Iv, - vrnl is always less than

diagram in Fig. 1 shows, the maximum variation in the electric potential in the region between the plates is 1~~- v,.,J. This is always less than (ho$2). Therefore if (ho$2) CC 1, one can safely assume that the quantity y = w - v, is small everywhere in the electrolyte between the two plates. Let us then expand Eq. (6) about \vm,only retaining the linear terms in y. We obtain

Aelf=sinh(yr,) dx2

+ y cash (v,) + o(y2)

At first sight this does not seem to be a particularly useful step. After all one does not have a prior knowledge of the value of vrn. Nevertheless let us persist with the exercise and solve Eq. (9). Taking the mid-point

R. Ettelaie, R. BuscalllAdv.

140

Colloid Interface Sci. 61 (1995) 131-160

between the plates as our origin x = 0, the solution has to satisfy the boundary conditions KO dx

and

y=o

With boundary becomes y = tanh(v,)

conditions

atx=O

(10)

(lo), the solution to the linear equation

[cash (dcosh(v,,J

x) -

11

(9)

(11)

Now consider the plate-electrolyte interface. Putting x = -h/2 in Eq. (11) and its first derivative with respect to x, and using (7), we arrive at the following relationships

ys = ym + tad

(w,> [ cash (-\icosh(~,)hD)

- 11

(124

0S = dcosh (w,)

tanh (w,) sinh (dcosh (I&) h/2)

(1%)

where we have used cosh(-z) = cash(z) and sinh(-z) = -sinh(z). Although somewhat more complex, Eqs. (12a) and (12b) are quite similar to (8) in that they allow us to express v,, and 0, in terms of the mid-point potential v,. Thus, if either the surface charge or the surface potential are specified, v, can be calculated by inverting (12). This is best achieved numerically, though with further approximations reasonable analytical results can be derived. Here we shall limit our discussions of such approximations to a few very simple cases, in order to demonstrate the point. For the constant charge problem, at very small values of h, it is known that the potential everywhere in the region between the plates is high. Hence it is reasonable to put tanh(v,) - 1 in Eq. (12b). This gives %

dcosh Wm>

= sinh (dcosh (w,) h/2)

(13)

Now for any finite crS,at a sufficiently small value of h, the left-hand side (and hence also the right-hand side) of Eq. (13) becomes small. At this point one can justifiably put sinh(x) - x in the right-hand side, reducing (13) to

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141

(14) The logarithmic divergence of vrn as h + 0 has been known for some time. In particular the expression given by Gregory for the constant charge problem gives the value of v, as [25] VI, = sinh-’

(15)

which for small h gives uy, - ln(4vJh). The symbol v,__in (15) represents the value of the surface potential for an isolated plate with the same surface charge as the double plate case. Gregory’s expression is known to be valid only for small values of surface charge density. When o, is small, w_, is also low. The use of the Debye-Huckel approximation then gives OS- v.__+Therefore the results of Eq. (15), for small values of h, are the same as our result (14) for the regime where Gregory’s approximation is valid. For larger values of oS however, more accurate results for small h can be obtained by replacing w_, with cr,. Equation (14) becomes exact in the limit h +O. By substituting (14) into (4), and for small values of h, one finds that the electrostatic repulsive force between the plates varies as h-l. Therefore any attractive forces with a plate separation dependence of the form h-“, where m > 1, will dominate over the electrostatic repulsive force at small distances. The Debye-Huckel approximation on the other hand produces a completely different and incorrect qualitative result. For now one finds vrn - 20, h-l, which gives a repulsive force of the form exp(2oJh) at small plate separations. If this was to be true, then no attractive force varying as an inverse power of h can overcome such a strong repulsive force. In other words any colloidal dispersion with particles having constant surface charges would remain well dispersed, indefinitely. Experience clearly shows this not to be the case. Similar approximations to Eq. (14) can also be derived for the case of plates with a constant surface potential. Considering Eq. (12a) in the limit of very small plate separations, it is easy to show that to the leading order in h h2 ,+,, =: v, - s sinh (v,) Once again Eq. (16) is valid for any value of the surface potential vvs, provided that the distance between the plates is small enough so that

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R. Ettelaie, R. Buscall IAdv. Colloid Interface Sci. 61 (19951 131-160

cosh(v,) << 4/h2. Finally as a test case, it is also interesting to consider Eq. (11) for systems where the electric potential is expected to be small (e.g. low constant surface potentials). Expanding all the hyperbolic functions involving vrn in Eq. (ll), and only retaining terms up to the linear order, one finds y = v’m cash(x) - v,. Hence as expected, w = v, + y = vrn cash(x), the same as the solution given by the Debye-Huckel approximation. Before we compare the results of our approximation to the exact solutions of the Poisson-Boltzmann equation, it is worth mentioning that many perturbative schemes in the past have been devised in order to improve the accuracy of the Debye-Htickel approximation [21]. In these schemes, the solution provided by the linearised Poisson-Boltzmann equation forms the initial term in a series expansion to the exact solution. The approximation introduced here also involves replacing the Poisson-Boltzmann equation with a linearised equation (11). Therefore one might expect that such methods can equally well be used to improve the accuracy of our approximation, when this is required. Figures 2 and 3 show the variation of the electric potential between two plates obtained from Eqs. (11) and (12). In each case the exact solution of the Poisson-Boltzmann equation, numerically obtained using the fourth-order Runge-Kutta method, are also shown for comparison. Both examples concern constant surface charge systems. In the case of Fig. 2 the distance between the plates was chosen as 0.65 and the charge density as 5.0, in the normalised units. This corresponds to a separation of 2 nm and a surface charge density of 3 uC/cm2, where we have assumed the electrolyte concentration to be 0.01 mol/l. Using (12b), the mid-point potential is found to be 3.22 as opposed to the exact result 3.18. By comparison, the Debye-Huckel approximation gives a value 15.12. Similarly for Fig. 3, h = 0.2 (-0.62 nm) and (3, = 8.5 (5 uC/cm2). The Debye-Huckel equation gives vrn = 84.9, which is more than an order of magnitude greater than the exact result 5.00. Using our approximation one obtains vrn = 5.0 1. The failure of Debye-Htickel approximation in this limit, where surface charge densities are high and the plate separations are small, is of course to be expected. Note however that in both cases o&/2 - 1,which at first sight may seem to be at the limit of the applicability of our approximation. Yet as Figs. 2 and 3 show, even in this limit, Eqs. (11) and (12b) produce excellent results for the entire potential profile between the plates. We believe this is due to the fact that in general the value of ]v, - v,] for problems involving small h, can be considerably smaller than (o&/2). This is clearly demonstrated

R. Ettelaie, R. Buscall IAdu. Colloid Interface Sci. 61 (1995) 131-160

4.0

3.9

143

7

\

3.8 F a

3.7

$ 5 3.6 Q) 5 : ‘C ti u w

3.5

3.4

3.3

3.2

3.1 -0.3

-0.2

0.0

0.0

0.1

0.2

0.3

KX

Distance

measured

from the mid-point

between

the plates

Fig. 2. The electric potential between two plates having a constant surface charge of 5.0 (in normalised units of d2n,kgTs,sr = 0.59 @Jcm’ for an electrolyte concentration of 0.1 mol/l). Plates are a distance 0.65 Debye lengths apart. (a) Exact results (solid line), (b) approximate results of Eqs. (11) and (12b) (dashed line).

in Fig. 1. Hence our approximation remains valid even when (O&J/~) 1. In Fig. 4 we have shown the variation of the force with distance, for the double plate system of Fig. 2. A few exact results, represented by open circles, are also displayed. It can be seen that the approximate and exact results are very close over the distance range shown. Using Eq. (12a) instead of (12b) similar calculations can also be performed for constant surface potential problems. The force-distance graph for a double plate system with a constant surface potential of 8.0

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144

5.5

5.4

5.0

4.9 -0.10

-0.05

0.00

0.05

0.10

KX

Distance

measured

from the mid-point

betwqen

the plates

Fig. 3. Same as Fig. 2 but with plates having a surface charge density of 8.5 and a separation of 0.2 Debye lengths.

mV> is displayed in Fig. 5. Once again we have included a few exact results (open circles) for comparison. Despite the high value of the surface potential, for distances less than 1 Debye length, good agreement between the two sets of data is found. As with the constant surface charge cases, the Debye-Huckel approximation grossly overestimates the forces. For a separation of 1 Debye length for example, the linearised Poisson-Boltzmann equation (2) predicts a force of 600 (in units of 2n,kBT). As Fig. 5 shows even at a separation of 0.2 Debye lengths the actual repulsive force does not attain such a high value. The relative errors in the use of our approximation are just under 7% for \cI,, at a (-200

R. Ettelaie, R. BuscalllAdv. Colloid Interface Ski. 61 (1995)131-160

145

160

140

20

0

r 0.0

0.2

0.4

0.6

0.8

1 .o

Plate separation (in units of IC-~) Fig. 4. Variation of the force with distance for two interacting constant surface charge plates. Plates have a normalised charge density of 8.5 (i.e. 5 PC/cm’). (a) The results of the approximate method introduced in the paper (solid curve), (b) exact data (open circles).

distance of one Debye length. As the plate separation is increased much beyond this distance, the discrepancy between the exact forces and the results of our approximation become more profound. The larger the surface potential the shorter the distance at which the errors become significant. Thus, for somewhat smaller values of v,, one might hope that there exists a range of values of h for which both the superposition [2,3] and our approximation become simultaneously valid. It is known that the superposition method becomes more accurate as h increases

146

R. Ettelaie, R. BuscalllAdu.

Co&id

Interface Sci. 61 (1995) 131-160

300

250

F

200

A? co 5. g 5

150

.z =1 8 ::

100

50

0 0.2

0.4

0.6

0.8

1.0

Distance between the plates (in units of

K-’)

Fig. 5. The same as Fig. 4 but showing the force-distance plot for a constant surface potential problem. Plates have a surface potential of 8 (i.e. 200 mV) in this case.

(i.e. when the overlap between the double layers is small). Our approximation on the other hand does the reverse becoming more accurate at smaller h. In this sense our approximation can provide a good complement to the superimposition method, for problems involving moderate surface potentials.

3. Application

to amphoteric

charge regulated

systems

The results of the preceding section indicate that for both the constant surface charge and the constant surface potential cases the ap-

R. Ettelaie, R. Buscall IAdu. Colloid Interface Sci. 61 (1995) 131-160

147

proximations (11) and (12) work well, at least at small plate separations. Charge regulating surfaces are often thought to exhibit properties that fall somewhere between these two limits. Therefore one might expect that for such surfaces the approximations continue to be valid. Many models have been proposed to account for the behaviour of various charge regulating surfaces. Although the method which has been outlined here can easily be modified to apply to any one of these, we shall principally focus our discussion on the model proposed by Chan et al. [14,X] for the amphoteric surfaces. We begin by giving a brief review of the model. Consider an amphoteric surface with ionisable sites of the form AH, in contact with a solution of 1: 1 electrolyte at a given pH. By gaining or losing hydrogen ions the sites can become positive or negatively charged, respectively. The reactions at the surface are given by AH; H AH + H+ AH t) A- + H+

(K,) (K_)

where K, and K_ are the corresponding dissociation equilibrium the following relations are satisfied

constants.

At

[AHI [H+ls = K+[AH;l

(174

[A-l [H+l, = K-[AHI

(17b)

The concentration [H+], of the hydrogen ions next to the surface can further be related to the pH of the bulk electrolyte

WI s = D-U b exp(-w,>

(18)

where once again the surface potential wS is expressed in normalised units. Denoting the total surface density of ionisable sites as N,, we also have N, = [AH] + [AH;] + [A-]

(19)

and s = [AH;] - [A-] e

(20)

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R. Ettelaie, R. Buscall IAdv. Colloid Interface Sci. 61 (1995) 131-160

Equations (17)-(20) can now be combined in order to eliminate the variables [AH], [A-l and [AH;], finally resulting in a relation between the surface charge density and the surface potential. Chan et al. 1151 have shown that this can most conveniently be expressed as

--0s eN,

6 sinh (VN - w,) - 1 + 6 cash (VN - V,>

In Eq. (21), VN is the value of the potential for an equivalent obeying Nernst’s equation, i.e. WN= 2.303[PH,

- PH,l

(21) surface

(22)

pH, is the pH at the iso-electric point, where cr, = 0, and is equal to (pK+ + pK-)/2. pH, is the bulk pH of the electrolyte solution. Finally, 6 = 2(K_/K+>1’2 = 2x10- APw2 . For amphoteric surfaces then, where neither the charge nor the potential is fixed on the surface, relation (21) provides the necessary boundary conditions in conjunction with which the Poisson-Boltzmann equation has to be solved. For the double plate problem at close plate separations, our approximate relations (12a) and (12b) for wSand o, can be substituted into (21). This results in a rather lengthy equation which will not be quoted here. Nevertheless this is an equation in a single unknown v,,.,, expressed in terms of a combination of hyperbolic functions. As such it can easily be solved numerically with a minimum of computing power. Results obtained in this way are plotted in Figs. 6 and 7. Three different systems with ApK = 1, 3 and 6, represented by dashed line, solid line and the dash-dotted line respectively, are displayed. In all cases pH, = 9 and N, = 3.1 sites/nm2. The plates are assumed to be in contact with an electrolyte solution of concentration 0.01 mol/l, at a bulk pH of 7. The individual points on the same graphs are exact numerical results. These were obtained by integrating Eq. (6) combined with a trial and error type procedure. For all three systems both the surface charge density and the surface potential vary as the distance between the plates is altered. However, it is evident from Fig. 6 that the surfaces with a lower value of ApK can regulate more efficiently. As such they maintain their surface potential more or less constant as h is varied. For the system with ApK = 1, the surface potential varies from 4.34, for two isolated plates, to a value of 4.61 at contact. This is to be compared with the case

R. Ettelaie,

R. Buscall

--\

0.0

IAdv.

Colloid

Interface

Sci. 61 (1995) 131-160

-I-

0.3

0.6

Plate separation

149

--

0.9

1.2

1.5

(K-‘)

Fig. 6. The surface potential for three interacting charge regulated double plates, plotted as a function of plate separation. The plates only differ in their value of ApK. (a) ApK = 1 (dashed line, (b) ApK = 3 (the solid line), (c) ApK = 6 (the dash-dotted line).

ApK = 6, where the variation of the surface potential is from 1.56 to 4.61. This behaviour is to be expected. A small value of ApK implies a large K_ and a small K+. From Eqs. (17a) and (17b), it is obvious that for this type of system, the AH sites have a large tendency to acquire a charge. Thus, even at the iso-electric point there are a large number of sites carrying either a positive or a negative charge. Of course in this case the net charge is zero. A non-zero charge on the surface would then only require a small change in the relative number of AH; and A- sites. That is to say that the values of [AHJIN, and [A-IN, change very little as one moves away from the iso-electric point. To a first approximation then,

150

R. Ettelaie, R. Buscall IAdu. Colloid Interface Sci. 61 (1995) 131-160

0.0

0.3

0.6

Plate separation Fig. 7. Showing the corresponding systems of Fig. 6.

0.9

1.2

(K-l)

variation in the surface charge density for the three

the system attempts to maintain the local pH adjacent to the surfaces the same as pH,, for any value of h. This at once leads to

wsc- --In

= 2.303(PH,

- PH,) = VN

(23)

which is just the Nernstian behaviour. Therefore it is not surprising that colloidal particles of salts like AgI, where all the sites on the surface are ionised (Ag+ or I-), satisfy the Nernst equation perfectly 131.In contrast surfaces with large values of ApK are thought to have properties closer to those with a constant surface charge. Although Fig. 7 shows this to

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be the case, we make the additional comment that in general the values of the surface charge remain very small for such systems. Similarly, the surface potential is low for any value of h, except very close to contact where it steeply increases to attain its Nernstian value. One can already see the beginning of this type of behaviour for the ApK = 6 system, in Fig. 6. The fact that for all charge regulating systems of this kind, the surface potential reaches its Nernstian value upon contact independently of ApK value, was first appreciated by Chan et al. [141, Their explanation for this phenomenon is based on the realisation that as the

-3

-2

-1

0

1

2

3

PH,,- PH, Fig. 8. Variation of the surface potential with pH for the three systems of Fig. 6. The plates are kept at a distance of 0.7 Debye lengths (-2 nm) apart. In addition the dotted line shows a surface with a perfect Nernstian behaviour for comparison.

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plates get closer to each other the charge needs to decrease towards zero, if the surface potential is to remain finite. The potential cannot diverge to infinity, unlike the constant charge case. If it does so, then from Eq. (21) one finds that the signs of os and u/, are opposite to each other. It is clearly not conceivable that a negative surface charge should result in an infinitely large positive surface potential. At the point of contact then (3, = 0, which using (21) gives ill, = VN. The deviation from the Nernst equation (23) is more clearly seen by plotting the surface potential against (pH, - pH,). This is done for the three systems of Fig. 6. The value of h is kept at a fured value of 2 nm throughout. The results are displayed in Fig. 8, together with the potential variation for a perfect Nernstian surface (the straight dotted line). For the ApK = 1 case, ws is very close to the value given by (23). For larger values of ApK however clear deviations from the Nernst equation are evident. At pH, - pH, = -2.5, the value of uy, for the ApK = 6 case is less than half the predicted Nernstian value. So far our discussions have been limited to the problem of interactions between two plates. Ultimately of course our aim is to use the results of the double plate system to calculate the forces between a pair of spherical colloidal particles. In the next section we outline possible ways for doing this.

4. Electrostatic forces for a pair of large spherical particles close separations

at

We begin by considering the Deryaguin approximation (5). This relates the force between two spheres, a distance h apart, to the free energy potential for the two parallel plates of the same kind; the free energy being evaluated at the same separation h. The use of the approximation assumes that the radius of the particles “a” is much greater than the Debye length and that Kh << 1. The first condition is more often than not true in practice. Since we are also interested in close particle separations the second condition is satisfied too. In the introduction we mentioned that for certain classes of problems, such as the constant charge cases, doubts have been raised as to the validity of the Deryaguin approximation at very small values of h [17,18]. We also pointed out that the cause of this is thought to be the escalation of the potential in the narrow gap between the particles. No such problems are expected with the charge regulated systems. For these the largest value

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that the potential can attain is vN, occurring at the contact point as discussed in the preceding section. This then leaves us with the problem of calculating the free energy of the interaction VYh) for our double plate problem. The superscript r is added here to indicate that we are dealing with a charge regulated system. A direct method for obtaining V’(h) is to evaluate the force at various values of separation down to h. These can then be integrated numerically to give v(h). Since our approximation breaks down at large separations, this means that we have to rely on some other methods (e.g. superposition approximation) to provide the value of the force at these larger distances. In turn this requires that there should be a range of values of h for which both approximations are valid. Even when this is the case, the direct method remains tedious and time consuming. It is dearly of great advantage if one could just determine the charge and the surface potential at the final required separation h, and calculate V’(h) on the bases of this information alone. Fortunately this is possible if one is willing to make use of the already available data [22] or approximate expressions [2,20-251 for P(h) and V’+‘(h). The quantities P(h) and W(h) denote the free energy potentials for the two parallel plates, a distance h apart, under the constant charge and the constant potential conditions respectively. To do this we first need to derive a set of identities relating r(h) to P(h) and W(h). A similar type of relation between VO(h) and W’(h) was first derived by Frens and Overbeek many years ago [29]. The identity (known as Frens’ equation) states that VO(h) = V”‘(h) + 2 (w,(h) - w,(-))o,

-4[cosh[T)-cosh[y]]]

>

(24)

where all the free energies, charges and potentials are expressed in normalised units. For the free energies the normalised unit is 2n,knT/K. Also in (24), (3, is the value of the fixed surface charge for which we wish to evaluate the free energy potential Wh) between the plates. The surface potentials on the plates with this value of the surface charge density are given by v,(h) and v,(-), at a distance of h and at isolation. For the isolated plates we have the well known result v,(-)

= 2 sinh-’ f 2 ‘I

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154

Vu’(h) is corresponding free energy for a constant surface potential problem, evaluated at a distance of h and for a fixed potential v&h). Thus provided we can be determined v,(h), and have access to tabulated results or an expression for V’Yh), we can easily obtain the free energy potential for the constant surface charge systems. We wish to do the same thing for the charge regulated systems. We start with the work of Chan and Mitchell [281, who have derived the following general expression for the free energy potential applicable to any type of system:

V(h) = 2 jr’“)

[email protected]‘,h)

o,ch) Ap,(o’) 7 do’ 0

i -

do’ + j

2

[email protected]‘,+

do’ +

A-) [email protected]‘)

I” 0

~

do’

e

(26)

!

with w&o’, h) representing the surface potential when the distance between plates is h, and their surface charge density 0’. The first two integrals in (26) give the free energy of the double layer formation when plates are a distance h apart. The last two terms give the same quantity when the same plates are infinitely far from each other. As Eq. (26) shows, as well as the electrical contributions to the free energy there are also some chemical terms. The quantity Al&(c~‘) denotes the chemical part of the electrochemical potential difference between the surface and the bulk electrolyte, for the potential determining ions, when the charge density on the two surfaces is CJ’.In the present discussion the potential determining ions are H+. At equilibrium the electrochemical potential for the two phases has to be the same. Thus, the equilibrium value of o’ has to satisfy the following condition ApL,(o’) + evs(o’,h)

=0

(27)

This value of o’ is denoted by o(h). The corresponding equilibrium value for the surface potential is then v&o,(h),h), which in order to abbreviate the notation will be written as v&h) from now on. Of course, Eq. (27) is also the condition that the free energy of the double layer formation be minimum, as can easily be shown by considering the first two terms in (26). For the constant surface charge problems we have o,(h) = o,(m). Hence the integrals involving Ap&o’) in Eq. (26) cancel each other out.

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155

In this case the exact form of A&o’) is of no relevance. For the constant potential systems on the other hand we have v&h) = v&w), which implies that Ap.,(cr’) is a constant equal to -ev,(m) independent of 0’. The free energy of the double layer formation for an isolated plate of this kind is well known from the original calculations of Verwey and Overbeek 121. This is given by o&m) ~,(o’,c~) do’ - v,(-) I

o,(=~) = - 4 cosh(yr,(+/2)

(28)

0

in our normalised units. The integral in the above equation is the same as the one occurring in the third term in Eq. (26). Since the integral represents the electrical work which is independent of the nature of the surface we are dealing with, the value ~,(c=)o,(=J) - 4cosh(v,(m)/2) can be substituted for this in Eq. (26). For exactly the same reason the results of the constant potential problem can also be used for the first integral in (26). Recalling that the free energy of the double layer formation for two parallel constant surface potential plates, a distance h apart, is V’+‘(h) higher than that for two isolated plates, we have

s

o,(h)

v&W

VW(h) do’- o,(Ws(h) = -yy- -

(29)

0

The quantity V’+‘(h) is evaluated at a surface potential of v&h). Using results (28) and (29) in Eq. (26), we now obtain a generalised form of the Frens’ equation

V(h) = VW(h)+ 2 Ws(h)o,(h)- v,(-NJ,(-)> -4[cosh[~)-cosh(~)]~+2[;;;)~dd

(30)

which relates the free energy potential V(h) for any type of surface to VW(h) for the constant surface potential case. The nature of the surface is reflected through the quantity A~.L,((J’).For example, if we are dealing with a constant surface charge system, o,(h) = (~~(00).The last term in (30) involving the integral in Ap,(o’) becomes zero, reducing (30) to the familiar Frens’ equation (24). Having derived Eq. (30), all that remains is to evaluate the integral involving A~.L,(G’)for the specific case of a charge regulated system

R. Ettelaie, R. Buscall IAdv. Colloid Interface Sci. 61 (1995)131-160

156

defined by Eq. (21). Following the argument of Chan and Mitchell [28], at equilibrium the values of surface charge and potential are related through Eq. (21). But at the same time from (27) we have

~4-W) = -

ys

= -

(31)

g(0’)

e

where the function g(o’) can be obtained by inverting the relation (21), so as to express the surface potential in terms of the surface charge density. Although this last step is a trivial one, it is not necessary for the evaluation of the integral in (30). The integral can be expressed in a more convenient form as follows:

The last step in (32) was obtained performing an integration by parts. Now using Eq. (21) we have

= eN, In

1 + 6 cash (VN - ur,(-1) 1+ 6 cash (VN - vs(h))

1

(33)

A similar type of expression involving double layer formation for an isolated, charge regulating single plate has been derived by Chan [30]. Substituting Eq. (33) into (32) and using the result in (30), we finally arrive at the required expression. V’(h) = V’+‘(h) - 8 [ cash [!$)I-

+ 2eN, In

cash [y)]

I+ 6 cash (VN - Y’s(“)> 1+ 6 cash (VN - v&h)) I

(34)

where once again V’+‘(h) is the free energy potential for a pair of constant potential plates distance h apart, with fixed surface potentials of v&h).

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Equation (34) is the central result of this section. Using it we can calculate the free energy potential for the two interacting charge regulated plates, provided we know the function W’(h). To do this we need only determine the surface potentials at the required separation h and when the two plates are isolated. No information regarding the intermediate states are required at all. The calculation of v&h) for small separations h was explained in Section 3. For yr,(-), we can replace CJ, in (21) with 2sinh(vJ2), solving the resulting equation for v&-j. This can be done numerically with a small amount of computing effort. Note that for the special case of plates at contact one only needs to evaluate

40

30

20

10

0 -2

-1

0

1

2

PHb-PH, Fig. 9. Electrostatic free energy potential at the point of contact, plotted against pH, for two charge regulating plates with a ApK = 3, and N, - 3.1.

158

R. Ettelaie, R. Buscall IAdu. Colloid Interface Sci. 61 (1995) 131-160

v,(-), since v&h = 0) = VN as was discussed in the previous section. Figure 9 shows the variation of the electrostatic free energy potential at h = 0 versus pH, calculated using Eq. (34). In this case the surfaces were assumed to have a ApK = 3 and N, = 3.1 sites per nm2. The data for VW(h) were taken from the work of Honig and Mu1 [221 which are generally thought to be exact. The solid line in Fig. 9 is a best fit curve through the data. It is found that a quartic (order four) equation provides a much better fit to the data than a simple quadratic function. Through the use of Eq. (5) it is easy to convert the results of Fig. 9 to forces for spheres with radii a >> ~‘-1 at contact.

5. Summary

and conclusion

To summarise then, in this paper we have attempted to provide an accurate yet easily implementable method for calculating the forces between spherical charge regulating colloidal particles. Examples of such colloids are various amphoteric surfaces such as hydrous metal oxides. We have focused our discussion on small particle separations, arguing that it is the forces at or close to contact that determine the rheological properties of a dispersion to a large extent. Also the position of the potential energy barrier, important for the colloidal stability of the dispersion, often occurs at relatively short separations (about one Debye length or so). The first step in determining the forces for the spherical particles is to solve the Poisson-Boltzmann equation for the double plate problem. Once this is done the force and the free energy potential between the plates can be computed. Through the use of the Deryaguin approximation the results for the double plate problem can be converted to yield the forces between the spherical particles. Although exact solutions for the double plate problem exist, they are quite tedious to use in practice (particularly for charge regulating systems). Also the nature of the problem is such that various commonly used approximations, while giving useful results at small surface potential or large separations, are not suitable in the current case. To this end we have introduced a new way of linearising the Poisson-Boltzmann equation which is capable of handling relatively large surface charges and potentials at close separations. The approximation exploits the fact that although the electric potential can be quite high between the plates its variation in the inner plate region remains small, at least for small plate separations. The

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approximation itself is of general applicability. We have given examples of its use for the constant surface charge, the constant surface potential and the more important charge regulating surfaces. Despite the rather high surface charges (up to 5 $Ycm2) and potentials (up to 200 mV> specified in some cases, good agreement between the results of the approximation and exact data is found for distances less than one Debye length. The exact data was calculated numerically by integrating the full Poisson-Boltzmann equation using the appropriate boundary conditions. For the same problems the usual Debye-Hiickel approximation gives answers that are at least an order of magnitude too large. With the surface charge and surface potentials calculated, we show how these results can efficiently be used to yield the free energy potentials for the system. This in turn gives the required forces between the spherical particles. We have derived identities similar to Frens’ equation 1291, relating the free energy potential for the charge regulating systems to those for the constant potential problems. This allows the large amount of data and many good approximate expressions for the latter to be used in the calculation of the aforementioned free energy potential. Although as yet we have not fully explored our technique, sufficient examples are given here to demonstrate the principles. Work is currently in progress to extend the method to cases involving unlike charge regulating particles.

Acknowledgement Many useful discussions with T.W. Healy of Melbourne University and M.G. Smyth of ICI are acknowledged. The work was carried out under the Department of Trade and Industry Project in Colloid Technology.

References [l] 121 131

B.V. Deryaguin and L.D. Landau, Acta Phys.-Chim. USSR, 14 (1941) 663. E.J. Verwey and Th.G. Overbeek, Theory of Stability of Lyophobic Colloids. Elsevier, New York, 1948. R.J. Hunter, Foundation of Colloid Science Vol. 1. Oxford University Press, Oxford, 1987; W.B. Russel, D.A. Saville and W.R. Schowalter, Colloids Dispersions. Cambridge University Press, Cambridge, 1989.

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I51 [61 [71

WI I91 [lOI Ill1 [121

[131 [141 [151

1161 I171 I181 [191 [201 I211 I221 I231 I241 I251 [261 1271 [%I 1291 I301

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