The effect of an electrolyte on phase separation in colloids

Physica A 202 (1994) 175-195 North-Holland

B

L

~

SDI: 0378-4371(93)E0300-4

The effect of an electrolyte on phase separation in colloids S.V. B u s h m a n o v a a, A . O .

Ivanov a and Yu.A.

B u y e v i c h a'b

aDepartment of Mathematical Physics, Urals State University, 620 083 Ekaterinburg, Russian Federation bMod~lisation en Mdcanique, Universit~ Pierre et Marie Curie, 75252 Paris, France

Received 16 April 1993 Revised manuscript received 8 June 1993 We consider an influence of the solvent ionic strength on thermodynamic properties and conditions at the onset of instability of a monodisperse colloid, as well as on the coefficient of mutual Brownian diffusion of colloidal particles. Special attention is given to collective effects due to competition of the particles for ions to be adsorbed, which are proved to lead to a noticeable drop in the particle electric charge and potential as well as in the ionic strength outside the double electric layers. An increase in the ionic strength above a region of relatively small electrolyte concentrations favours the phase separation of the colloid under otherwise identical conditions and causes the mutual diffusivity to markedly decrease, the more so the larger is the concentration of the particles.

1. Introduction Molecular dispersion forces as well as m a g n e t i c interactions b e t w e e n colloidal particles are k n o w n to be inevitably attractive o n the whole, thus providing a driving force t o w a r d either phase separation of a colloid o r flocculation of the particles. A key practical p r o b l e m o f preventing those p h e n o m e n a , and in s o m e cases o f taking a d v a n t a g e o f t h e m , consists in imparting the colloid stability by introducing small p o l y m e r or electrolyte additives into the a m b i e n t liquid, which p r o d u c e an effective interparticle repulsion [1,2]. T h e n o w classical t h e o r y o f D e r j a g u i n and L a n d a u [3] and V e r w e y and O v e r b e e k [4] allows for the basic interactions in colloids involving van d e r Waals attraction and s c r e e n e d electrostatic repulsion to be u n d e r s t o o d at the level of initial f o r m a t i o n of particle doublets. This t h e o r y has b e e n f u r t h e r d e v e l o p e d in an e n o r m o u s n u m b e r o f publications, a n d its recent versions include also h y d r o d y n a m i c interactions o f pairs o f spherical particles which c o m e closer t o g e t h e r in o r d e r to f o r m a doublet. H o w e v e r , the t h e o r y does n o t 0378-4371/94/$07.00 t~) 1994- Elsevier Science Publishers B.V. All rights reserved

176

S.V. Bushrnanova et al. / Effect of electrolyte on phase separation

address thermodynamic functions in equilibrium states of colloids and so cannot be used in principle to describe the colloid separation into two phases differing in the particle concentration. Such a reversible separation caused by weak attractions resembles gasliquid, liquid-liquid or liquid-crystal phase transitions in molecular systems. It has been repeatedly observed by many researchers (for a review, see ref. [1,2], also ref. [5-9]). A number of representative examples is supplied by ferrofluids or ferrocolloids, in which the attractive dipole magnetic interaction is opposed by a repulsion owing either to steric reasons due to surfactant or polymer layers on the particle surfaces or to screened electrostatic interaction of charged particles in ionic ferrofluids [10-16]. Attempts to describe theoretically the observed phenomena of the reversible phase separation and to build up corresponding phase diagrams are surprisingly scarce, in spite of the fact that general principles to be employed to this purpose are well worked out and widely accepted [1,2,17]. Among those, an endeavour of Victor and Hansen [18] to predict a phase separation by introducing entropy terms alongside with energy ones specific to the theory of [3,4] and numerical calculation of phase diagrams for ferrocolloids with surfactant-coated particles undertaken by Sano and Doi [19] have to be mentioned. Recently a statistical thermodynamic approach to get phase diagrams for ferrocolloids with the purely magnetic attraction and steric repulsion has been successfully brought into existence in ref. [20]. A serious stumbling block of taking into account electrostatic interactions while developing a statistical thermodynamic theory of colloidal systems emerges from the necessity to carefully study properties of the double electric layers occurring at the particle surfaces. Those properties govern the interparticle electrostatic interaction, and the point is that they are greatly influenced by many secondary effects, such as particulars of the surface charge formation, finite size of all the ions and solvent molecules, the interionic interaction in both solution and adsorbed layers, etc. There are a lot of investigations of these properties, beginning with those founded on relatively simple classical methods of solving the Poisson-Boltzmann equation and ending with rather elaborated treatments based on the solving of the BBGKI equations with the help of various approximate schemes of statistical physics (hypernetted and mean-spherical models, etc.), examples of which can be found in [21,22]. However different as regards the complexity those methods may be, they lead to essentially the same final results. A common feature of all of them consists in that the double electric layers are looked upon as thermodynamically equilibrium systems. Just this finally permits to pass from the description of the layers to the evaluation of the electrostatic part of the whole interaction potential of the particles by using firmly established quantities, such as the

s.v. Bushmanova et al. / Effect of electrolyte on phase separation

177

Debye screening length, the particle potential and the surface charge density, and so forth. Having in mind to leave main ideas unencumbered with unnecessary details and to present principal inferences in the simplest possible form, we shall confine ourselves in this paper to familiar methods of the Gouy-Chapman theory without further comments and shall use a number of simplifying assumptions. A serious deficiency of the studies the authors are aware of lies in the fact that the above mentioned quantities are usually regarded as being dependent only on the interaction of a single particle with an electrolyte solution of fixed initial chemical composition and ionic strength. This implies an implicit assumption that all the other particles do not affect the said interaction. Clearly enough, such an assumption cannot hold strictly true, and a question arises merely as to with what accuracy it may be supposed to be approximately correct. As it has been shown in ref. [23], a collective influence of all the particles on the actual state of the ambient electrolyte solution within interstices cannot be ignored in many situations of practical importance. Such an influence is primarily due to the mere fact that adsorption of some part of the ions at the particle surfaces inevitably causes a consequent reduction of the solution ionic strength, which brings about, in its turn, a decrease in the particle charge and potential, a shift of the dissociation-recombination equilibrium, and some other phenomena [23]. These effects can be and below are actually proved to be of crucial significance for colloids with a well-developed specific interphase boundary area and with a sufficiently high specific energy of adsorption of the ions, at relatively low original ionic strengths.

2. Double electric layers In what follows, we consider a colloid with identical spherical particles of diameter d suspended in a dilute electrolyte solution. The electrolyte is supposed to be binary, symmetric and completely dissociated at all temperatures of interest. The energy of Coulomb interactions between ions is thought of as small against the temperature k T expressed in energy units, which permits those interactions to be completely left out of account. Other simplifying assumptions are made in order to provide for the applicability of the classical Gouy-Chapman-Debye theory [24,25] to the description of the double electric layer structure and of the interparticle electrostatic interaction. They can be conveniently put forward in terms of relations between different relevant length scales. We assume the Debye screening length K-1 to essentially exceed the mean distance l i separating an ion from its nearest neighbours in the solution and, second, to be much smaller

178

s.v. Bushmanova et al. / Effect o f electrolyte on phase separation

than the particle diameter d. Then, first, any double layer contains a lot of ions to ensure the use of the classical calculation scheme under common conditions of local thermodynamic equilibrium and, second, the particle surface curvature may be neglected while determining the layer structure. In addition, to simplify the calculation we are not going to take into account the Stern parts of the double electric layers. Then a pertinent problem for the Poisson-Boltzmann equation reads ~2 ffi

2

Ox 2 = K

q~

shqJ,

8~Cq 2

2

¢-kT'

K

-

•k~

'

~01x=0=¢0,

~0lx===0, (2.1)

and has a relevant solution $(x) = 2 In

(~_~ +/32 e x p ( - Kx) '] _/32 ~ / ,

/31 = 1 + exP(½¢o),

q'o = q,(o),

/32 = exP(½¢o) - 1,

(2.2)

the particle potential being related to the surface electric charge density o- by an equation 4"rr

kT

--Or=

•

Oq,

q ~x /=0

~--

exp(%) - 1 q exP(½¢0)

kT

K.

( 2 3~

"-'-"

Here q is the ion electric charge, • is the solution dielectric permeability and C stands for the electrolyte concentration (the ionic strength) outside the double layers. The ion concentrations inside the double layer are Cl(X)

= C

exp(-qJ),

c2(x) = C exp(¢).

(2.4)

As regards relations between characteristic linear scales, we have also to mention that the screening length must be large as compared with the Bjerrum length inherent to the solvent. This condition is not always satisfied for aqueous electrolyte solutions even at moderate concentrations (see, for example, a discussion of the point for ionic ferrofluids in ref. [13]). It suggests account of both pair ion-ion correlations and finite size of the ions, which makes the problem much more difficult to treat. Further we retain the simple technique based on eqs. (2.1)-(2.4) notwithstanding the fact that the above indicated factors cause an appreciable influence on the double layer structure and the interaction between equally charged surfaces [26].

S.V. Bashmanova et al. / Effect of electrolyte on phase separation

179

Now we must identify the origin of surface electric charges. Although different mechanisms of the surface charge formation can be considered without much ado, for the sake of definiteness we have recourse to situations when those charges are entirely due to the adsorption of ions of, say, the first type, whereas ions of the second type are assumed to have no affinity to the surfaces and so to take no part in adsorption. On making use of the adsorption isotherm by Stern [24,25] we are able to write a relation between C and the surface area fraction covered with the first type ions in the following form:

Col = ys(1 - ys) -1 exp(x + qto),

(2.5)

where 7 and s are understood as the surface concentration of the adsorbed ions and the area per ion, respectively, v 1 is the ion volume and k T x < 0 stands for the specific energy of adsorption. It is evident that or = yq. It should be noted right away that interionic interactions in the solution are proved to considerably affect the equilibrium adsorption as against that governed by the isotherm (2.5), up to an order of magnitude for concentrated solutions of polyvalent electrolytes [27]. We entirely overlook this effect, which complies with the above neglect of the energy of the said interactions.

3. Collective effects

Eqs. (2.3) and (2.5) are sufficient to find out ~/'0 and or as functions of C. It means that a complete description of the double electric layers in accordance with eqs. (2.2) and (2.4) will be attained if the ionic strength of the ambient solution is known. However, the original ionic strength C O of the solution without particles is usually given, and a relation between C and C O is needed. Such a relation is supplied by the material balance law for the ions. When bearing in mind the first type ions we can write

"rtd2yn + (1 - &)C + ~rd2n i [Cl(X) - -

C] dx =

(1 - &)C 0 ,

(3.1)

0

where n is the number and & = (,rrd3/6)n is the volume particle concentration. A similar relation for the second type ions gives nothing new since it follows from eq. (3.1) and a condition of the colloid being electrically neutral that is implicitly involved in the formulation of problem (2.1). By using the concentration profile from eqs. (2.4) and (2.2) we get from eq. (3.1), after integration,

S.V. Bushmanova et al. / Effect of electrolyte on phase separation

180

64' ( d

2C~z

'~

7 - K e x P ( ½ ¢ o ) ] = (1 -- 4 ' ) ( C o - C ) .

(3.2)

Eq. (3.2) suffices to find C in t e r m s of C o and d and, thus, closes the p r o b l e m . Influence of the ion a d s o r p t i o n on the particle potential, the surface electric charge density and p r o p e r t i e s of the double layers and the interstitial solution is illustrated by figs. 1 - 3 , respectively. B o t h ¢0 and cr as well as C m o n o t o n o u s l y decrease and the characteristic d o u b l e layer thickness K -1 increases as 4> grows, those changes b e c o m i n g m o r e p e r c e p t i b l e w h e n Co and d diminish, all o t h e r things being equal. It is quite clear that these p h e n o m e n a are caused by the effective r e m o v a l of the a d s o r b e d coions f r o m the solution and by the resulting c o m p e t i t i o n of the particles for ions that they could adsorb. T h e m o r e p r o n o u n c e d they are, the lower is the original c o n c e n t r a t i o n C O and the finer are, at fixed 4', the colloidal

,

!

.99

a

.98

I

i

0

O.|

0.2

i

i

0

0.1

0.2

.98

.96

Fig. 1. Relative particle potential d~o = ~0(~)/¢0(0) at (a) CO= 10-3 mol/l, X = -12 (curve 1), -14 (curve 2) and (b) at X = -12, CO= 1 0 - 3 (curve 1), 10 4 mol/l (curve 2).

S.V. Bushmanova et al. / Effect of electrolyte on phase separation !

181

I

0.9

0.8

I 0.1

0

!

I 0.2

F

!

0.9

0.8 b

oo7

0

i

i

O. I

0.2

_/

,,~

Fig. 2. Relative surface charge density & = tr(~b)/~(0) at the s a m e conditions as fig. 1.

particles. The first effect is self-explanatory, the second one is a consequence of an increase in the specific interphase boundary area that is accessible for the ions to be adsorbed. In the last case, the charge of a single particle decreases even at constant o- due to reduction of the particle surface, but the total electric charge Q = 'rtd2trn = 6trcb/d per unit volume grows, which signifies an increase in the overall amount of adsorbed colons. As pertains to the counterions, their equivalent amount is collected within relatively thin diffusive parts of the double layers to neutralize the surface charges produced by the adsorbed coions. It is worth noting that various more subtle factors influencing the ion adsorption which are not accounted for in eq. (2.5), such as the dependence of the specific adsorption energy on the surface ion concentration and the deviation of real adsorption from that being carried out in a monolayer, are capable of quantitatively affecting the curves in figs. 1-3 but by no means change their qualitative character. Of significant interest is a peculiar limiting case of almost complete adsorp-

182

S.V. Bushmanova et al. / Effect of electrolyte on phase separation C,

~d

~

,

0.9

0.8

0.7 0

C,

~d

I

I

0.1

0.2

i

!

0.8

0.6

0ol4

I

0

0.1

I

0.2

,~"

Fig. 3. Relative interstitial solution ionic strength C = C(~)/Co and screening length rd = rd((k)/ xd(0) (solid and dashed curves, respectively); notation is the same as in fig. 1.

tion of the coions, when C ~ C o. On assuming for simplicity that 7s(1 - 7s) -I 1 and exp(qJo) - 1 we get from eq. (2.5) that the last inequality holds true if e x p ( x ) ~ Cool

or

Ixl ~311n(ll/li0)l,

(3.3)

where l 1 is the coion own linear dimension and li0 is the mean distance between neighbouring ions at the original solution concentration C 0. It is easy to see that the inequalities (3.3) can well be satisfied when the original concentration is not very low and the specific ion adsorption energy is sufficiently high. If, in addition, the colloid is dilute ((h ~ 1), we conclude from eqs. (2.5) and (3.2) that 3' (or tr) and C depend on C O and on ~b merely through their ratio, C O/4). This means that not the ionic strength of the original electrolyte solution is relevant in the case of strong adsorption, but rather the total n u m b e r of ions per colloidal particle.


183

4. Interparticle interaction Consider, next, the interaction between colloidal particles. Their repulsion is provided for by the electrostatic contribution, the potential of which can be written out, at Kd >> 1, as [2,25]

Ue

4dE(kT) 2

q2

th2(¼~Oo)exp[K(d - r)],

(4.1)

r being the distance between the centres of interacting particles. An advantage of the expression (4.1) lies in that it is not limited by restrictions that might be imposed on permissible values of the potential and, from this point of view, is consistent with the problem (2.1), whose solution, when supplemented with eqs. (2.5) and (3.2), completely determines qJ0 as a function of the concentrations C O and ~b and of relevant physical parameters. The particle attraction may be due to different reasons. The first and usually most important one stems from action of van der Waals dispersion forces [1,2,24,25], the potential of which can be taken in the following form: d2 {r2-d2\] A [d 2 U r n = - 1--'2 -r5--~ r 2 - d 2 t- 21n~-------~-2~)] ,

(4.2)

A being the Hamaker constant. Another reason of particle attraction is due to dipole-dipole interactions of particles with either permanent or varying magnetic moments. Incorporating effective attraction forces of this origin into analysis meets with serious difficulties, especially so in the presence of an external magnetic field. Those difficulties have been successfully overcome with the help of methods developed in ref. [20], where an expression for the corresponding potential is given. At last, interactions of colloidal particles with a soluble polymer produce nonuniform distributions of the polymer and its osmotic pressure throughout the solution and give rise to an effective force, frequently referred to as that of depletion interaction, whose sign and magnitude depend on the nature of the said interactions. The problem has been thoroughly reviewed and discussed in refs. [2,28] for different modes of polymer-particle surface interaction, a distinction having been drawn between terminally anchored, adsorbing and nonadsorbing polymers present in good and poor solvents. Some representations of the potential of this force, obtained on a basis of the self-consistent field and scaling theories, are also given in the cited textbooks. When keeping in mind that all the mentioned constituents of the interpar-

S.V. Bushmanova et al. / Effect o f electrolyte on phase separation

184

ticle interaction can in principle be included in further considerations, here we focus our attention mostly on a case where only potential contributions (4.1) and (4.2) are relevant, so that the total potential may be presented as U = U e + Urn. This potential is known to exhibit a n o n m o n o t o n o u s d e p e n d e n c e on r with primary and secondary minima positioned at r ~ d and r = d + A and a repulsive potential barrier in between. The solution ionic strength influences the depth of these minima and the height of the barrier, which helps to explain the transition from kinetic stability to rapid irreversible flocculation into the primary minimum and to understand key underlying reasons of colloid stability

[1,2,24,25]. In this paper, we are concerned merely with reversible phase separation on which most of the behaviour of the potential curve in the vicinity of the secondary minimum has a bearing. As follows from figs. 1-3 and from the discussion in the preceding section, this part of the curve is dependent not only on the ionic strength but also on the particle concentration by volume. Since the H a m a k e r constant may be approximately regarded as independent of ~b, the latter quantity affects solely the repulsive part of the potential. The effect of the particle volume concentration on the total potential of interparticle interaction is illustrated in fig. 4. It is easy to see that ¢0, Y and C reach maxima and K 1 comes to a minimum at ~b = 0 under other conditions fixed. This corresponds to a relatively deep secondary minimum and to a high and narrow potential barrier. As th grows, ¢0, Y and C diminish and K-I increases, what leads to a decrease in both secondary minimum depth and

- -

-

I

I

2

1

I • I

r/d

Fig. 4. Dimensionless potential of interparticle interaction in the vicinity of the secondary minimum at C O= 10 -3 mol/l and tb = 0, 0.15 and 0.3 (curves 1-3, respectively); the dashed line shows the supposed behaviour at the particle surface.


185

barrier height and to an increase in the radial coordinate of the minimum. It must ultimately result in making easier conditions for the equilibrium irreversible flocculation into the primary minimum, what is out of the question in this paper. The same reasons have, however, hindered the nonequilibrium reversible phase separation into the secondary minimum. Such an influence of the particle concentration on the phase diagram of the phase separation into two colloid phases, one rich and one poor in colloidal particles, is confirmed by experiments of refs. [13,14]. The shift of the secondary minimum position that accompanies the particle concentration growth leads to an increase in the effective particle diameter which can happen to be much larger than d. That this effect causes a rather drastic influence on the observable colloid behaviour in different circumstances has been repeatedly recognized, a representative example to be supplied by the disorder-order transition of a colloid in a restricted volume [29]. Below it is discussed in connection with thermodynamic properties and mutual Brownian diffusion of the colloidal particles.

5. Thermodynamic model The temperature and pressure of colloids are usually maintained constant. This makes natural the consideration of the Gibbs free energy of the particulate assemblage that can be written down as [30] t~

0 p, T) + Nl~°(p, T) + N1/Zl(p, = Nolzo( T) + N2/z2(p, 0 o t) + ~i + l~)a + ~o~ + N k T ln(~b - kT) In Q , (5.1)

where No, N, N 1 and N 2 are the total numbers of molecules of the solvent, particles and ions of both types, respectively, and a degree marks standard chemical potentials. The quantities ~i, ~a and ~el are understood as the potentials of thermal motion of the ions in the whole solution volume, of the adsorbed ions and of the ions in an electric field caused by the charged particles, respectively, and Q stands for the configurational integral. The first two potentials can be expressed in a familiar way as

~i = kT(CVo ln(C2uao2) oo

+ ,rtdZN f [cl ln(ClVl) + c 2 ln(c2v2) - C ln(C2VlV2)] dx) , 0

186

S.V. Bushmanova et al. / Effect o f electrolyte on phase separation @a = N k T N s [ l n ( y s )

-

(1 - 1 / y s ) ln(1 - ys) + X],

(5.2)

N S being the number of adsorbed ions per particle. The potential q~e~can be expressed through the energy Ee~ of the electrostatic interactions as follows:

--f (kr)-2eeld(kr),

=

(5.3) 0

where q~ and % are the dimensional potentials inside the double layers and of the particles (see eq. (2.1)). The last but one term in eq. (5.1) describes an ideal gas composed of the particles whereas the last term depends on details of both particle arrangement and interaction. If the potential barrier separating the potential minima is sufficiently high, a possibility of rapid flocculation into the primary minimum can be overlooked. Then a state of the particulate assemblage with no allowance for the flocculation is metastable, but may be considered within the scope of equilibrium thermodynamics, the particles being supposed to interact with the potential U = U e + U m at large distances and as "soft" spheres with hard cores of diameter d e at short distances, d e = d being the radial position of the potential maximum [17,31]. The softness of interparticle repulsion can be taken into account by means of a generalisation of the hard sphere perturbation technique [17,32], which amounts to making use of a modified interaction potential

U,(r)

=

{

w,

r < ds ,

Um + Ue ,

r > dh ,

(5.4)

where the effective core dimension is defined as dh

ds-- f [1-exp(-

dr+ e

de

and

dh

is understood as a root of the equation

[Urn(r) + Ue(r)lr=ah = O .

(5.6)

By considering a system of sterically interacting hard spheres in the capacity

187


of a reference approximation for the assemblage of colloidal particles and using eqs. (5.4)-(5.6) within the frames of a standard perturbation method [33], we get -kT

In Q = F~ - N k T ~ G 4~ri(

,

Um+Ue\ 2

G =~

-kT

o = ~ r d 3,

) r gs(r) dr ,

(5.7)

dh

where gs(r) is the pair distribution function for hard spheres and the Helmholtz free energy of the reference system can be put forward on a basis of the model by Carnahan and Starling [34] in the following form:

Fs = NkT

44)s - 3~b2 (1 - ~bs)z '

4)s = ~,rrn d~.

(5.8)

Interaction parameter G is always positive. When other types of interparticle attraction are relevant this parameter can be defined quite similarly to the relation in eq. (5.7). While calculating the chemical potentials of all the colloid components we must regard the total numbers of particles, N, of solvent molecules, N 0, and of ions of both types, N 1 = N2, as independent variables with respect to which differentiation of the Gibbs free energy has to be performed. The calculation can be simplified, however, if one notices that conditions of local thermodynamic equilibrium require the ion chemical potentials to be constant across the double electric layers. It means that the total numbers of ions outside the layers, N~ = N~, may be regarded as new independent variables instead of N i, i --- 1, 2. In view of Kd being much larger than unity by assumption, the layer volume may be neglected against the whole interstitial volume V0, and then N; = CV0. When choosing N~ in the capacity of independent variables, we must regard parameters 00 and y and functions ci(x) and O(x) to be insensitive to changes in C. Taking this fact into account and dropping out terms proportional to small parameters Cv~, v i / v and (Kd) -1 (what we are compelled to do in order to avoid violation of accuracy), we arrive at the following expressions for the chemical potentials of particles, /x, of solvent molecules, /z0, and of ions, /x~:

o

(

8- 54,~

I~ = l~ + k T l n ~ b - ~ b + q ~ (l_~bs)2

44~s - 24)$ @~- ~b + (] - 4,):

1 - 4,~ + k ~ r q , ,

4)(2 - ~b)G - ~bg,s(1 - 40

4,~ =

4,,

OG)

188

S.V. Bushmanova et al. / Effect o f electrolyte on phase separation

o

I% = IZo - 2 k T C v o

- kT

__v o( v

20G\

1

¢b 2 G - ~ fb s - ~ s ) ,

( 1 - 4,J ~

o tz~ = tzi + k T l n ( C o i ) ,

(5.9)

where v 0 is the solvent molecule volume. The dependence of G on &s involved in eq. (5.9) is due to the presence in the integrand in eq. (5.7) of the pair distribution function gs(r) of hard spheres. If the colloid under study is of small or moderate concentration (~b~~ d~. Then OG/Ofbs = O. The chemical potentials of ions listed in eq. (5.9) correspond to the theory of dilute solutions. The particle chemical potential involves a term due to the presence of the double electric layers at the particle surfaces. This term is to be written out, after a rather lengthy and cumbersome calculation, in the form

aly = _Trd2(

4~2C

[(]31 _ ~2) ln(Cu1)

- (/3~ +/32) ln(Cv2) + 4/31( ~ +q f[

7--25,2----2516/31/32C( f l ~ - 9 )

K(~I -- ]~2)

- y [ l n ( y s ) - (1 -

1/ys) In(1

~°)1 + 3'00] d(kT) kT

- ys) + X ] } .

(5.10)

The above relations enable one also to find out all other thermodynamic functions of colloids by using standard methods.

6. P h a s e s e p a r a t i o n

If the interaction parameter G identified by eq. (5.7) is large enough, chemical potentials/z and l.% defined by eq. (5.9) can be proved to exhibit van der Waals loops, when being considered as functions of ~b, of the same kind as those characteristic of ferrocolloids devoid of ions [20]. This brings about a conclusion that the colloid is able to experience a phase separating resembling the gas-liquid first type phase transitions in molecular systems. As a consequence of the separation, two colloidal phases I and II distinguished by particle concentrations, &~ and 4'u, emerge. If one neglects the curvature of arising interfaces, necessary and sufficient conditions of the phase separation read

S.V. Bushmanova et al. / Effect of electrolyte on phase separation ~(~)I)

: ~'L(~II) ,

~/~O(~I) = ~'Lo(~)II) •

189

(6.1)

As seen from eq. (5.9), changes in th do not affect ion chemical potentials/xi, which remain uniform throughout the whole volume of the separated colloid since the interstitial ionic strength C is the same for both colloidal phases. The total concentrations C0~ and Con in the phases are, however, different because of presence of the adsorbed ions. These quantities are to be found from material balance equations, (~IVI -1- ~IIVII = C V ,

Co~(1 - (~I)VI at- Coii(1

V-~- V I q- VII , -

-

~bii)Vii

:

Co(1 - ¢ ) V ,

(6.2)

VI and Vn being the total volumes occupied by the phases. Solutions of eq. (6.1), if any, determine ~bI and ~b~i as functions of an initial colloid composition and of physical parameters; after that all the other properties of the colloidal phases can easily be found with the help of eq. (6.2) and equations of the preceding sections. Corresponding phase diagrams in different planes can then be build up numerically. Such calculations for the plane (Co, oh) have been carried out in a broad range of other parameters and, in particular, for conditions specific to accessible experimental data. The trouble with comparing theoretical phase diagrams with those resulting from experiments lies in that a number of pertinent quantities is usually not reported in experimental works. Among the latters, the Hamaker constant or the relative specific energy of ion are to be mentioned, which have consequently to be regarded as some adjustable parameters. Nevertheless, we are above to draw a general conclusion that there exists satisfactory agreement of the theory with available experimental evidence. By way of example, the data of ref. [14] are displayed in fig. 5 together with a theoretically predicted phase diagram. It has already been pointed out that situations are possible when the particles compete for ions, with a decisive influence upon the ion distributions in the colloid: the coions are then mostly adsorbed at the particle surfaces and the counterions form relatively thin ionic clouds within the diffusive parts of the double electric layers. Properties of the colloid have been proved to depend on the original ionic strength in those situations only through the ratio Co/dp. Hence it is natural to infer that the phase separation diagrams must also be determined by this ratio. Such an inference finds an excellent support in experiments by Bacri et al. [13] who have shown that the phase diagrams of ionic ferrofluids with quite different C O and ~b can be reduced to a unique universal form by using the variables Co/d~ and tk when correlating their findings.

190

S.V. Bushmanova et al. / Effect of electrolyte on phase separation mol

C,--- i-

1.O

O.6

@

0.2 0

0.2

Fig. 5. Phase diagram at arbitrarily adjusted X and A; the dots represent the experimental results from ref. [14].

Stabilization of colloids by means of addition of an electrolyte in small amounts is not revealed, of course, by the presented theory since not all of the accepted assumptions hold true at small ionic strengths. The theory correctly describes, however, how a further increase in the ionic strength, which results in making both the potential barrier simultaneously higher and narrower and the secondary minimum deeper, promotes conditions favouring separation. In a sense, such an increase is equivalent to a fall of temperature, as it has been earlier concluded in ref. [14]. The phase separation occurs when interaction parameter G from eq. (5.7) reaches a certain critical value dependent of ~b. A temperature decrease always results in an increase of G. The effect of the ionic strength is more complicated. Eqs. (4.2) and (5.7) evidence that G formally goes to infinity as C O comes to zero. In spite of the theory being inadequate in a range of small C 0, a general conclusion of G being a decreasing function of C O in such a range is valid. As C O grows beyond this range, the length of screening becomes shorter and the depth of the secondary minimum increases, which inevitably induces G to be an increasing function of the ionic strength. Since an increase in the particle volume concentration tends to obliterate this effect, the latter is more pronounced for dilute colloids (see fig. 6). It means, in particular, that a

S.V. Bushmanova et al. / Effect of electrolyte on phase separation G

i

!

10 -4

10 -3

191

I

5

/

/

0 10 -5

C, mol/l

Fig. 6. Interaction parameter G from eq. (5.7) at A = 4.10 20j and 4~ = 0 and 0.2 (curves 1 and 2, respectively); the dashed line shows the expected behaviour at small ionic strengths.

difference between the phase concentrations t~i and ~)II in a separated colloid with a fixed value of C o decrease as ~b grows, which corresponds to the data of ref. [14] as well.

7. Mutual Brownian diffusion The presented findings bear upon equilibrium properties of colloids and are sufficient to envisage the onset of phase separation. They do not help, nonetheless, to predict the rate with which the separation proceeds, if ever. The kinetics of all stages of the phase separation, save for the final coalescence stage, have been recently considered in refs. [35,36]. It is not difficult to understand that nearly all methods of the cited papers can be immediately used for electrolyte containing colloids with practically no changes, if the concentrational dependence of the coefficient of mutual Brownian diffusion of colloidal particles is elucidated. This is why we are going to discuss briefly this point in the remainder of this paper. A general thermodynamic analysis leads to the following expression for the mutual diffusivity [1,2,30,35]:

D_K($) ~ (O/z) 3"rr-qd 1 th ~ p,T' where 77 is the dynamic solution viscosity and

(7.1) ~7/K(qb) plays the

role of colloid

192

s.v. Bushmanova et al. / Effect o f electrolyte on phase separation

viscosity that describes the effective mobility of the colloidal particles. It is usually accepted that K(~b) = (1 - ~b)j with ] being a n u m b e r between 4.5 and 6.5 [2]. On calculating the particle chemical potential derivative in the fight-hand side of eq. (7.1) with the aid of eq. (5.9) and on taking j = 6 to c o m p a r e results with those of ref. [35], we arrive at curves of D / D o, D o = k T / 3 ~ r r l d , as a function of ~b that are plotted in fig. 7. At small ionic strengths, D rapidly increases with th, in contrast to a similar coefficient in colloids with no electrolyte. The difference is caused by a sharp intensification of the steric excluded volume effects due to an increase in the effective sphere diameter (and volume) that accompanies the a p p e a r a n c e of double electric layers at the sphere surfaces. For reasons discussed above, the developed theory cannot explain this difference. H o w e v e r , a further increase in C lies well within the scope of the theory. It m a k e s the indicated effects w e a k e r and, at the same time, the retardation of diffusion due to the particle attraction becomes stronger, as it has b e e n explained in detail in ref. [35]. This causes D(4~) to evolve with C being gradually increased in the way illustrated by fig. 7. The evolution is quite analogous, in essence, to that resulting from the interaction p a r a m e t e r G being increased [35]. Similarly, the onset of phase separation leads to appearance of negative values of D(~b) in the range (~I ~ (~ ~ (~II, just as has been the case in the p a p e r cited. A comparison of theoretical and experimental findings at this stage of theoretical development cannot be other than tentative. It helps, nevertheless,

!

Q

/!

Do

2

•

/

/

~

0

o

1

&&~ 0.1

&

•

i

i

0.2

0.3

Fig. 7. Relative coefficient of mutual Brownian diffusion for Co = 10-4 (curve 1), 5 × 10-4 (curve 2), 10 3 (curve 3) and 5 × 10-3 mol/1 (curve 4) at A = 4.10 -20 J; the dots represent the experimental data from refs. [1] (O, A), [37] (O), [38] ([]) and [39] (&).


193

to understand the occurrence of nonmonotonous dependences on 4, of the mutual diffusion coefficient which are so frequently observed in numerous experimental works.

8. Discussion

We have succeeded in showing that the usage of information about ion distributions in colloids and, next, about the influence of the distributions on the effective potential of interparticle interaction, which can be attained by comparatively simple means, appears to be sufficient to explain both equilibrium properties of the colloids, including the onset of phase separation, and some peculiarities of the particle diffusion in qualitative agreement with experiments. The point that should be stressed once again consists in that the conventional methods, however successful they may be while dealing with dilute colloids, prove to be inadequate when applied to concentrated colloids, because of ignoring very significant collective effects due to the shortage of ions to completely satiate adsorption layers on the particle surfaces. Especially those effects provide for the understanding of some phenomena of principal importance that are time and again observed in practice. Validity of the developed theory is restricted by a number of assumptions brought into action to simplify its mathematical part. Some of these assumptions can readily be avoided, others demand a substantial generalisation of the theory. Most of them seem, however, to be fully justified in view of existing uncertainties caused by a lack of our knowledge concerning exact values of various relevant parameters and by actual colloid polydispersivity, that can well mask more subtle effects which could be taken into account by means of an appropriate generalisation. An extremely important exception consists in using the limit inequality Kd>>l, which not only makes the calculation easier but also, what is apparently more essential, specifies a range of electrolyte concentrations within which an increase in the ionic strength happens to favour the phase separation in accordance with fig. 5. This inequality is inevitably replaced by the inverse one, Kd ~ 1, in a range of sufficiently small concentrations. The latter range admittedly corresponds to the stabilization of colloids as a consequence of the mere fact that electrostatic repulsion partly equilibrates, when firstly arises, interparticle attraction due to either dispersion forces or dipole magnetic interactions. It means that electrolytes in a very small concentration must hinder the onset of thermodynamic instability of colloids, as it is universally acknowledged [1-4]. Such a hindering action remains beyond the scope of the

194


presented theory, as was already pointed out when discussing the phase separation. On the other hand, the electrostatic interaction of colloidal particles m a y be described with the help of a screened potential which resembles that of Yukawa form (4.1) in its all significant features, no m a t t e r whether Kd is large or small. Moreover, the competition of the particles for ions to be adsorbed also takes place irrespective of Kd. This proves the influence of the ionic strength on p a r a m e t e r s of the total interaction potential at Kd ~< 1 tO be qualitatively the same as at Kd >> 1, So that the f o r m e r reasons of p r o m o t i n g the phase separation still exist even in colloids with a relatively large screening length. Thus quite opposite tendencies as pertains to the effect of electrolytes on t h e r m o d y n a m i c stability of colloids m a y be simultaneously anticipated, and a conclusion about what tendency will prevail at Kd ~ 1 appears to be ambiguous and dependent on other conditions. Carrying out a thorough analysis needed to this end is by no means a simple task since the double electric layers adjacent to neighbouring particles of a nondilute colloid can no longer be regarded as plane and overlap. Allowance for the latter effect presents serious difficulties even in a case of two interacting participles. W h e n m a n y particles are involved, it amounts to the d e v e l o p m e n t of a theory of effective properties of the colloid, such as the effective dielectric permeability. Although the theory can be constructed by making use of special methods of ensemble averaging c o m b i n e d with ideas of the self-consistent field theory, as has been the case for heat and mass transfer in granular systems [40], it can hardly be considered as one of the aims of this paper. It seems obvious that the future theoretical work on the subject must be conducted, in order not to stick in possible speculations, in close connection with experimental work.

References [1] W.B. Russel, The Dynamics of Colloidal Systems (University of Wisconsin Press, Madison, 1987). [2] W.B. Russel, D.A. Saville and W.R. Schowalter, Colloidal Dispersions (Cambridge Univ. Press, Cambridge, 1989). [3] B.V. Derjaguin and L.D. Landau, Acta Physicochim. USSR 14 (1941) 633. [4] E.J.W. Verwey and J.Th.G. Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1952). [5] J.A. Long, S.W.J. Osmond and B. Vincent, J. Colloid Interface Sci. 42 (1973) 545. [6] I.F. Efremov, Colloids Surf. Sci. 8 (1976) 85. [7] C. Cowell and B. Vincent, J. Colloid Interface Sci. 87 (1982) 518. [8] J.W. Jansen, C.G. de Kruif and A. Vrij, J. Colloid Interface Sci. 116 (1986) 681.

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