Interaction theory for double electric layers of dissimilar particles for equilibrium regime of surface ionization Low surface potentials

COLLOIDS AND ELSEVIER

Colloids and Surfaces A: Physicochemical and Engineering Aspects 131 ( 1998 ) 51 62

A

SURFACES

Interaction theory for double electric layers of dissimilar particles for equilibrium regime of surface ionization Low surface potentials 1 Yu.V. Shulepov a,,, L.K. Koopal b, j. Lyklema b S.S. Dukhin a bzstitute gf ColloM Chemistry and Chemistry of Water, Ukrainian National Academy of Sciences. 42 Vernadsky Avenue, Kiev, 252680, Ukraine b Department of Physical and Colloid Chemistry, Agricultural University, Dzeijenplein 6, 6703 HB. Wageningen. The Netherlands Received 17 March 1996; accepted 23 February 1997

Abstract

The solution to the problem of the interaction of double electric layers formed by dissimilar particles is obtained on the basis of the linearized Poisson-Boltzmann equation assuming the equilibrium regime of particle surface charging. Three types of particle surface ionization by the ions of the electrolyte whose concentration corresponds to the Henry law region are considered: (a) cations (anions) adsorption on an initially uncharged surface which does not possess any non-compensated charge before the adsorption of ions takes place; (b) ions adsorption (ion exchange) on an initially charged particle surface that possesses this charge; (c) ions adsorption (ion exchange) on amphoteric surfaces. The three cases can be reduced to the case (c) with unequal Nernst potentials of the surfaces. The dependencies of potentials, densities of plate charges, and electrostatic component of the Gibbs free energy of the particle interactions on the nearest distance between the particles are studied qualitatively, and the corresponding analytical expressions are derived. © 1997 Elsevier Science B.V.

Keywords." Electrical hetero-interaction; Stability of colloids; Poisson Boltzmann equation; Nernst potentials

1. Introduction

The theoretical calculation of the electrostatic component for the Gibbs free energy of interaction between a pair of dissimilar charged colloid particles immersed in an electrolyte is based essentially on the development of the DerjaguinLandau- Verwey-Overbeek (DLVO) theory for the

* Corresponding author. 1 The results of this study were, in part, reported at the IXth European Colloid and Interface Society Conference, Barcelona. 1995. 0927-7757/98/$19.00 c© 1998 Elsevier Science B.V. All rights reserved. P H S0927-7757197 )00076-9

double layer (DL) of like particles [1-5]. Three methods were proposed to perform these calculations, namely the "explicit force m e t h o d " formulated by Derjaguin [3], the Langmuir approach involving the calculation of the disjoining pressure as the sum of the osmotic pressure of the ionic atmosphere and the Coulombic tension [5], and the explicit calculation of the variation of the Gibbs free energy related to the interaction of double layers for the pair of particles [4]. From the computational point of view, all these approaches are equivalent and are based on the solution of the non-linear Poisson-Boltzmann

52

Yu. V. Shulepov et al. / Colloids Surfaces A." Physicochem. Eng. Aspects 131 (1998) 51-62

(PB) equation for the potential in the space between interacting particles with the boundary condition requiring the potentials on the particles' surfaces to be constant [5-10]. It was shown that the behaviour of the particles depends essentially on the relation between their potentials. Even in the case when the surface potentials have equal sign, the electrostatic component of the interaction energy has its maximum (positive) value for some distance between the surfaces, and decreases to negative values when the surfaces are brought closer to each other, that is the attraction between the particles arises. For opposite surface potential signs an attraction between the particles exists at any interparticle distance. Even in this simplest case the electrostatic component of the Gibbs free energy and the interaction forces for arbitrary values of the potential can be calculated only numerically. An important step towards the analytical examination of the problem was made in Ref. [ 11 ], where, for the case of low potentials, the linearized PB equation was used to derive the analytical expression for the electrostatic component of the DL energy density of plane surfaces; this result was generalized within the framework of Derjaguin's approximation onto spherical colloid particles. This expression has been widely used, even in recent studies, owing to its relative simplicity [3, 5,12]. The case when the surface potential is constant, while being relatively simple, is, however, deficient, and can be used merely for the simplification of the problem to the extent when it can be modelled mathematically. In actual systems the behaviour of the potential is interrelated with that of the surface charge, and both these values depend on the processes of surface charging due to the adsorption of ions, the dissociation of surface functional groups, and to the interaction of the DLs of the particles [ 13,14]. In this case, one can also consider the limiting regimes for the processes taking place on the particle surface. One of these regimes is, rather obviously, the regime when the surface charge is constant. If the charging of the surface takes place due to the preferential adsorption of the ions of one species only, it is the adsorption potential of

the ions, not the surface electrostatic potential, which remains invariant during the interaction process. The DL model corresponding to this concept was developed in Refs. [14, 15]. There exists no general approach to the problem of how to determine the type of the interaction between the ions and the surface. In each particular case a number of factors are to be considered: the structure and surface properties of the particle, the electrolyte composition, the concentration of the salt, etc. However, some general criteria can be stated. For example, if the relaxation times of adsorbed ions exceed the characteristic coagulation time, then the surface charge remains constant. It is the consideration of relaxational surface phenomena for particle charges which forms the basis for the dynamic concept of slow coagulation of similar lyophobic colloids [16,17]. This approach essentially relies on the assumption that an additional component of the disjoining pressure is to be taken into account, which arises due to the deviation of surface charge from its equilibrium value corresponding to the adsorption potential as the result of the Brownian coagulation. It follows, then, that the colloid particles' interaction potential is defined simultaneously by the DL electrostatic repulsion, the van der Waals attraction, and the relaxation interaction. Therefore, to calculate the total interaction potential for dissimilar particles it is necessary to derive an analytical expression for the electrostatic energy component in the equilibrium regime. The behaviour of the disjoining pressure, the potentials and the charges of dissimilar surfaces possessing amphoteric groups capable of equilibrium dissociation were analysed in Ref. [18]. The dependence of the disjoining pressure on the intersurface distance was expressed via the ordinary and inverse Jacobi elliptic integrals within the potential intervals defined by the relations between the parameters of the problem. The problem of the behaviour of the DL interaction force arising between dissimilar surfaces possessing fixed and zero initial charge density with subsequent adsorption of both counter- and co-ions was solved numerically in Ref. [19]. The Debye-Hiackel energy is extremized for a configuration of two interacting flat plates and

Yu. V. Shulepov et al. / Colloids Surfaces A." Physicochem. Eng. Aspects 131 (1998) 51 62

Derjaguin's method is used to derive an approximate analytic expression for the DL interaction free energy [20] for two dissimilar spherical particles bearing ionogenic surface groups with an equilibrium regime of charge regulation. In the present study, the expression for the electrostatic component of the Gibbs free energy is derived, similar to that obtained using a very cumbersome procedure in Ref. [20].

expressed according to Refs. [21,22] in the form of Langmuir-type isotherms: ('A --JAi CA0i

1+

CA

--JAi

i=1,2

axi(~i) = e2hrlsxi

47r

d: 2

e

/ 5

p, p = - 2 e C o z o s i n h l Z ° e ~ l \ kT /

(1)

where p is the bulk density of ions in the space between the surfaces of particles, Co the bulk number density of the 1: 1 electrolyte ions possessing the charge z0, e the electronic charge, e the dielectric permittivity of the solution, k the Boltzmann constant, T the temperature, and z the coordinate measured from one of the walls. The boundary conditions near the surface of the walls have the form dO ,-:, 47r d6 .-=h 4~ = -0-1 (01)" : --0"2(@2) d: e d= ¢ (2) where h is the distance between the plates and subscripts l and 2 refer to first and second walls respectively. The dependence of the surface density of charges near each plate on the plate potentials ~b~ and 02 depends on the mechanism of surface charging. A simple model of surface charging via the adsorption of ions of both signs will be considered here, i.e. we assume that the particle surface possesses the adsorption centres both for cations and anions. The surface charge densities for the ions of each sign in this ions adsorption model can be

exp(+

e:

(3) b

Oi/kT)

CXOi CA

d2~

exp(--e:b~ki/kT)

CA0i

CA

To determine the disjoining pressure within the space between plane dissimilar surfaces immersed in a symmetric binary electrolyte with the ions charges z o, we begin with the PB equation for the electrostatic potential ~:

e x p ( - eZbOi,/kT)

O-ai(0i) = e~-blls4.i

--Jxi

2. Basic equations

53

1+--

Jxi exp( ('2bOi,/kT) -

CXOi

if the interaction between the ions on the particle surface and in the solution bulk is neglected. Here, aAI(00 and ax/(~q) are the densities of the surface potential-determining charges, i.e. cations (A) and anions (X) respectively; nsAi and nsXi are the densities of the localization sites (centres or surface groups) for the cations (A) and anions ( X ) respectively; CA is the bulk concentration of the electrolyte potential-determining cations (which is equal to the bulk concentration of the potential-determining anions due to the electroneutrality condition), CAOi=CAI/2JAi , CXOi=('AL2jxi, where CA1/2 iS the bulk concentration of the solution cations corresponding to a half-completed monolayer: JAi and./xi are the partition functions of cations and anions respectively, corresponding to internal (electronic, rotational and vibrational) degrees of freedom [21,22]. For the calculation of these partition functions the energy of the ions in the electrolyte solution is taken as the zero energy level. Eq. (3) was derived using the assumption that the chemical potentials of cations (A) (and anions (X)) in the solution and on the surface are equal to one another. The total surface charge density for each particle (i= 1,2) is equal to the sum of surface densities of cations and anions, Eq. (3): O'i(~li)=O'Ai-]-O'Xi,

i=1,2

(4)

If the concentration of potential-determining electrolyte is small, i.e. (CA/CAoi)JAi<<1 and

54

Yu. V. Shulepov et al. / Colloids Surfaces A. Physicochem. Eng. Aspects 131 (1998) 51-62

(CA/Cxoi)'Jxi<
parable to each other:

and total surface density of the particle, Eq. (4), transforms into

ffi(t~')=eT"bCA IAAdA'exp(

Ai e x p ( - Wbi)= ~/AgjniAxJxi,

ezbt~ikT)

A A/JAi = e x p < 2 Z ~ N i ) ,

- a x a x i exp \ k r J / '

i= 1,2,

axJxi

(5)

(8)

where ZIAi:nsAi/CAo i and Axi :(nsxdAi)/(CAOflXi) a r e the thicknesses of adsorbed cations and anions monolayers respectively. Assuming next that the only difference between the state of each ion at the surface and that in the bulk is the difference between the corresponding electron densities, and imposing an additional requirement that the ions of one sign are adsorbed preferentially, one obtains the expression for charge density which was derived in Ref. [15] and used in Refs. [23,24] as the most simple model of surface charging: O"i(IPi)= -[-CaeZbAai,x i exp

- I/VbAi'Xi ~-

eZblPl "~

kT J'

where ~N~ is the Nernst potential (for ~,~= ~Ni the surface charge of each particle vanishes). It is to be noted that in each case of the ions adsorption, i.e. on uncharged surface, Eq. (6), on the surface possessing uncompensated charge, Eq. (7), or the adsorption of the ions of both charge on the amphoteric surface, the behaviour of surface charge densities as a function of the potentials is qualitatively similar. We next transform the equations to the dimensionless form, introducing the dimensionless potential ~=ezb~/kT. Then, for the case of small potential Iq~l<<1, Eq. ( 1) and boundary conditions in Eqs. (2) and (6)-(8) can be transformed into d2~b --

i= 1,2

(6)

d~ 2

-~b=0

(9)

-~b~ =~a(~bx)= +6,(l~,b~), where WbAi,Xi is the bonding energy of cations or anions adsorbed preferentially on the surface of the particles. Here, the plus and minus signs refer to the adsorption of cations (A) and anions (X) respectively. If the surface of colloid particles is charged, i.e. possesses the non-compensated charge when there are no adsorbed ions, then the resulting charge density is equal to the sum of the non-compensated charge density aoi and the charge density defined by Eq. (6) (see also Ref. [25]) becomes O'i(Ipi) = aOi -1- CA CZb A Ai,X i

×exp

eZblPi"] -- WbAi,XiT- k T J' i= 1,2, (7)

= +52(I T-~2)

(10)

= 6~(~;~ - ~)

(11)

--~)1 = ~ 1 ( ~ 1 ) = ) ' 1 (~bN1- (~1),

q~' = ff2(~2)

~---)~'2(~ N2 - - 4 2 )

(12)

ZbK

a,(~b0 = - -

2ez2co ~ i ( ~ i ) ,

CA

6i=

_2 KHsAi,Xi~b

CA

Zb K ~]Oi -- - -

tlOi 71-6 i

2C0z2 ' ~Ni=

CA(l/2)

7i = CA(l/2) -From Eq. (5) the expressions can also be derived for the amphoteric surfaces case, for which the values of cations and anions adsorption are com-

q~=62(~b2)

6i

KVHsAiHsXiZ2 1 nsA i Coz2 , ~bNi = ~ 1og--,nsxi

2ez~co

aoi, i=1,2,

Z=~:z,

H=~:h,

55

Yu. bi Shulepov et al. / Colloids" Surfaces ,4: Physicochem. Eng. Asl,ects 131 (1998) 5l 62

where ¢i are the derivatives of the potentials calculated near the left (i= l) and right plates (i=2) respectively. In the expressions for the dimensionless charge densities Eq. (10) and the potentials q~ the upper and lower signs correspond respectively to the adsorption of cations A and anions X on each particle surface: ~,~=V8ZrzoeZco/ekT is the inverse Debye screening radius. To derive Eq. (12) the values CN~ (i= 1,2) were assumed to be small. The equalities for (~ and ~,~ show that the condition used, CA/CA~:2<<1, leads to the inequalities 6~<<1 and /~<<1 for i= 1,2. This is the consequence of two inequalities: CA/CAll2<< I, which follows from the applicability of the linear Henry isotherm, and t,n~A~,x~Zh/CoZo<
dz/

13)

where the integration constant/'/is the dimension-

less component of the electrostatic disjoining pressure:

H=kTcol'I,

(14)

and /7 is the disjoining pressure. The value of H can be determined using the boundary conditions of Eq. (12) when the integration Eq. (9) is performed. The result of this integration depends on the relation between the potentials and charge densities of the plates (Fig. I ) [3] (see below). The surface density of the electrostatic component of the Gibbs free energy for the interaction of plates can be found from the integration of disjoining pressure Eq.(14) over the distance between the plates [3]:

H(h)dh=kTc~?c

Us(h)=

(15)

1 ~Ts(Xl,

i

Us(x ) =

i:' .I( )

----/'/(x)dx

x=exp(-H),

(16)

V

where E;s(x) is the dimensionless density of the electrostatic component of the Gibbs free energy for the interaction between the plates. The electrostatic component of the Gibbs free energy of interaction of two colloid particles with radii R~ and R2 can be calculated from the integ-

f a)

b)

c)

4, >o 4'; =-o-,>0, Fig. 1. Qualitative behaviour of the solutions to the PB equation Eq. (9) for three possible types of relation between the potentials and charge densities of the plates: (a) positive potentials and charge densities of the plates; (b) positive plates potentials and opposite signs of plates charge densities: tcl opposite signs of potentials and opposite signs of plates charge densities.

56

Yu. V. Shulepov et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 131 (1998) 51-62

ration of the surface energy density Eq. (15) over the surface of both spherical particles using Derjaguin's approximation [3]: 2rcRlR2

U~(h) = R1 +R2 =

L

C•2

2/~

C/312

2

Us(h) dh

2 ~ R 1 R 2 k Tco ~ - 2

t)~ (x),

cpZ

(17)

RI + R2

(7o(x)= fo -(7~(x) -dx,

(18)

a)

X

where (7~(x) is the dimensionless electrostatic component of the Gibbs energy of interaction between spherical particles. Thus, it is possible to determine the electrostatic component of the Gibbs free energy for the interaction of dissimilar colloid particles in the equilibrium charging regime on the basis of the expression for disjoining pressure Eq. (13).

, ~l'Ni-

\

3. The case when Nernst potentials of particle surfaces possess equal sign

TIi(~)i)=~ 2 --~2(~bNi--~bl) 2, i = 1,2.

(19)

(It will be shown below how the results obtained in this way can be transformed for the case of negative Nernst potentials.) As the equilibrium exists within the system, the pressure exerted on plate 1 is to be equal to that exerted on plate 2:

fl~(4~)=fl~(~2)=fl The

behaviour

(20) of

the

functions

S

J

To study the general features of the behaviour of disjoining pressure, potentials and charge densities in more detail, the qualitative approach can be used, which was first introduced in Ref. [18]. Eq. (13) for each plate (1 and 2) in the case when their Nernst potentials are positive, which corresponds to the boundary conditions in Eq. (12), can be represented in the form

~b2 and

),2(~Ni-~bi) 2 for i = 1,2 which enter the expression

for the disjoining pressure Eq. (19) is illustrated qualitatively in Fig. 2 (a). It follows from Eq. (19) that the disjoining pressure is determined by the

7

b) Fig. 2. Qualitative behaviour of the functions ~bl 2 and )'/2(q~Ni-4i)2 (i= 1, 2) for (a) equal and (b) opposite signs of Nernst potentials of the plates. difference between these functions, which becomes zero when the distance between the plates is infinitely large. Therefore, for the potential of infinitely separated plates one obtains the expression (see Fig. 2(a))

~bai--

~i~Ni

i = 1,2.

(21)

1 + ?'i '

With a decrease in the distance between the plates their surface potentials increase, which corresponds to the transition from the points ~,i (i= 1,2) to the right, see Fig. 2(a). In this case the difference between the parabolas

~,~- ~ ( ~ N , - 4 , )

2

(i=1,2) increases monotonously and, therefore, the disjoining pressure also increases monoto-

"t)c [~ Shulepov et aL /Colloids" Surlaces ,4. Physicochcm. Eng. Aspects 131 i 199,~4) 51 62

nously with the decrease of the distance between the plates ( Fig. 3 ). From the condition in Eq. ( 20 ), according to which the pressures exerted on each plate are to be equal to each other, it follows that, if, for example, for any value of ~b the inequality 3'1(4)N1--~1)2 >)'2(4)N2--4)2) 2 holds, then the relation 4~ >q S, exists between corresponding potentials of the plates, see Fig. 2. Therefore, the relations

41: <~4',, ~a, <~4~,<~ qS~"--

",' 1 (/'N 1 -- 3'2 q~N2

;'l -,'e

i= 1,2,

(22)

max(q~al,gba2) ~ 4~1 ~
are valid, where ~b~ is the value of the potential at

57

which the relation between the potentials of both plates inverts. Thus, within the interval ~ba~~b2 is held. If the distance between the plates is further decreased, the potentials, while being within the interval qS~"~
;"1 (~N1 -~-,'2~N2 -

(24)

;:1 +i'2

/-/

% 9," q)at

w, k/

0

('~i = ('}~ = (J-" =

;'1 (q~N 1 -- (/~*) = "'2 ( (/)N2 -- (fl*).

(25)

j-f -6"

The behaviour of the plate potential q~i(H) (/= 1,2 ), is illustrated in Fig. 3. The case of the inversion of the relation between the potentials of both plates corresponding to the condition in Eq. (23) is illustrated in Fig. 3 by a dashed line which shows the potential ~be(H). The behaviour of the surface density of plate charge ai(H) (i= 1,2 ), Eq. (12), corresponds to the potential behaviour. When the distance between the plates decreases, these surI:ace densities also decrease, and for zero distance these surface charge densities are equal to each other in absolute wtlue and opposite in sign:

./

k/

,

Fig. 3. Qualitative behaviour of disjoining pressure, potentials and charge densities of the plates for equal signs of Nernst potentials of the plates. (Dashed line--see text).

It is seen from Fig. 2(a) that as the interparticle distance decreases, the potential q~ which corresponds to the plate possessing a lower Nernst potential, increases continuously, approaching the value ~bN~.The surface charge density of this plate changes its sign. At the same time, the potential of the surface possessing a larger Nernst potential never approaches this value, and the charge density for this plate does not change its sign at any interparticle distance, see Fig. 3. The behaviour of the functions fl(H),(o~(H) and ~(H), (i= 1,2), is shown qualitatively in Fig. 2(a)

58

Yu. K Shulepov et at / Colloids Surfaces A: Physicochem. Eng. Aspects 131 (1998) 51 62

and Fig. 3 for the case of positive Nernst potentials of both plates. For the case when these Nernst potentials are both negative, the behaviour of H(H) and, consequently, the Gibbs energy of interaction, remains unchanged. The behaviour of potentials and charges remains qualitatively the same, only the potential and charge axes are turned in opposite directions. The behaviour of the disjoining pressure, potentials and plate charge densities shown in Fig. 3 corresponds to one of the regimes considered in Ref. [18], where the case of like surfaces was studied. The qualitative considerations presented above enable one to make an important conclusion: the behaviour of the charges and potentials at large distances up to the distance where the potential 4~=4N1 corresponds to the type shown in Fig. 1(a), i.e. when the signs of the potentials and charge densities of both planes are equal to each other. At shorter distances, starting from the potential value 41=4N1, the behaviour of the system corresponds to the type illustrated by Fig. 1(b), i.e. the signs of the plate potentials are the same but the signs of the charge densities are opposite to each other. Consider the integration of Eq. (13) within the first interval, from infinite distance and up to the distance at which 41 =4N1. As the minimum of the electric potential exists between the charged plates, see Fig. 1(a), the integration yields x -1 =

The equations for the potentials of the plates, 41 and ¢2, c a n be obtained introducing the expression for the root in Eq. (27) into Eq. (26), and using the relations in Eqs. (19) and (20): 41( 1 -I-0{10)1)--42 ( 1 --0~20.12)X=(.01 -~-U)2X

The equations in Eq. (28) transform into one another when the subscripts 1 and 2 are interchanged. Assuming that the potentials are small, one can solve this equations set to obtain 1 4i-- l _ r 2 x 2 (a]i)-k-a(2ilx+a(3i)x2),

a]l)=a~31)=0)l,

COi~'~i4Ni,

~ i = 4 N i 1,

(27)

i=1,2

In Eq. (27) the expression (in Eq. (13) for the potential derivative near each plate is also used, with the sign before the root chosen according to Fig. l(a). As the estimate 4Ni
a~21~=20)2, 0)1<<1, 0)z<
/F/(X) = [( 1 --g10)1)41 + 0)1][( 1 -- 0~20)2)42 "]- (D2]X

4x

The expressions for the boundary conditions in Eq. (12) can be transformed into

4~ = 0 ) 2 ( 1 - ~242) = zX/Tz2----//,

(29)

where the expressions for the coefficients a~2~ (k= 1,2,3) follow from those for a~1) with the subscripts 1 and 2 interchanged. To derive the expressions in Eq. (23) the potentials were assumed to be small. This condition, which is necessary to linearize the PB equation Eq. (19) and the boundary conditions Eqs. (10)-(12), is equivalent to the inequalities 0)i<< 1 (i= 1,2). The expression for the disjoining pressure f/(x) can be derived from Eqs. (13) and (26) and the relations in Eq. (27) for the potential gradients near each plate:

-

--41 =0)1( 1 --0{141) = ~ ,

i=1,2,

r 2 = 1 - 2~10)l -- 2~20) 2,

(26)

/)

(28)

--41(1 -- 0~10)I)X"[- 42 ( 1 -]- ~20)2 ) = 0)2 -{- 0)1X

( 1 -- r2x 2 )2

[ill(1 +x2)+/~2x],

/~1 =0)1(O2'

/~2 =0)1 -~- O)i"

(30) The expression for the dimensionless density of electrostatic component of the Gibbs free energy for the interaction between the plates can be obtained from Eq. (16):

f2s(x) =

2x(2fla + fi2x) 1 --r2x 2

(31)

and, finally, integrating Eq. (31) according to Eq. (18), one obtains the expression for the dimensionless electrostatic component of the Gibbs free

Yu. I~, Shulepov et al. / Colloids Surfaces A: Phvsicochem. Eng. Aspects 131 (1998) 51 62

energy for the interaction between the particles: ~e(.v) : 2fi: log 1 i"

+rx fl~ l o g ( 1 - r 2 x 2)

1 -- rx

(32)

r-

The expressions for the potentials Eq. (29), the disjoining pressure Eq. (30), the density of the electrostatic component of the Gibbs free energy Eq. (31), and the electrostatic component of the Gibbs free energy Eq. (32) are valid within the interplate distance range from infinity to the distance at which the relations q~=~bN~= 1/:q take place. One also has to calculate these characteristics within the second interval, from the distance corresponding to the above relations, to the zero distance. H = 0. In this case, it is seen from Fig. 3 that the interaction between the plates corresponds to the type illustrated by Fig. 1 (b), i.e. the signs of the potentials are equal, whereas the signs of the charge densities are opposite to each other. Integrating Eq. ( 13 ) one obtains A" 1__

respect to the inversion of the signs of the plate potentials. It can be readily verified that sign inversion of ~i and ~bNi (i= 1,2) in Eqs. (26) and (27) and Eqs. (33) and (34), and subsequent transformations of these equations similar to those performed when Eq. (28) was derived, again lead to the equations set of Eq. (28) for the potentials of the plates. Therefore, the behaviour of the plate potentials, the disjoining pressure, the dimensionless density of the electrostatic component of the Gibbs free energy of the interaction of plates, and the dimensionless electrostatic component of the Gibbs tYee energy of particles interaction, are determined by Eqs.(29)-(32) in the general case, i.e. for both positive and negative Nernst potentials. The values of the interaction characteristics corresponding to the limiting case when the distance between plates vanishes can be derived from Eqs. (29) (32). The potentials in this case are equal to each other, and the corresponding value can be obtained from Eq. (29) at H--*0:

(33) ~b1 = ~b2 -

Using the equality in Eq. (12), one can reduce the expressions for the boundary conditions Eq. (12) within the interval considered, to the form (see Fig. l(b)) q~', =,,,1(~1~/,~- 1 ) = V ~ - H : V~-H

~b~=o~2(1-c~z~b 2) (34)

The equations for the potentials ~bl and ~b2 c a n be derived similarly to Eq. (28), by the transformation of the simultaneous equations set in Eqs. ( 19 ), (20), (33) and (34); the resulting expressions are just the equation set of Eq. (28). Therefore, within the second interval, from the distance where the relation ~bl=~bN1 holds to the point H = 0 , the behaviour of the plate potentials, the dimensionless density of the electrostatic component of the Gibbs free energy for the plate interactions and, finally, the dimensionless electrostatic component of the Gibbs free energy for the particles interaction, are defined by Eqs. (29)-(32) respectively. Turning now to the case when Nernst potentials of the plates are negative, one can easily see that the equations set in Eq. (28) is invariant with

59

~o 1 -+-to 2

H<< 1.

:q ~,)~+ c~2¢o:+ H "

( 35 )

The limiting values for the dimensionless disjoining pressure, the dimensionless density of the electrostatic component of the Gibbs free energy of the interaction of plates, and the dimensionless electrostatic component of the Gibbs free energy of particles interaction follow from Eqs. (30) ( 32 at H ~ 0 : =

((o 1 -}-(02) 2

36 (cq ~ol + ~2¢o: ):

((O 1 "4-(02) 2

37

r2stH=0)= 3( 1 O) l -~- 3(2(O 2

/_)e(H = 0 ) = - (~')1 + ~o2)z log(:q % + ~,~o: I -'((O 1 --(O2) 2

log

2

38

4. The case when the Nernst potentials of particle surfaces possess opposite signs This case corresponds to the interaction of colloid particles with different surfaces, ionized by the charges of opposite signs.

60

Yu. V. Shulepov et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 131 (1998) 51-62

We again begin with the qualitative analysis of the behaviour of the potentials, charges and disjoining pressure of interacting plates. The behaviour of the functions ~2 and ),'/2(~Ni __~i)2 for i = 1,2 which determine the disjoining pressure is illustrated qualitatively in Fig. 2(b). When the distance between the plates is infinitely large, then the potentials are equal to the values ~ba~(i= 1,2) at which the disjoining pressure vanishes:

]~i~)Ni

~bai -

i = 1,2

7*

\

H

H

d h"

(39)

1 + )'i When the distance between the plates vanishes, the potentials approach the value ~b* which can be determined from the requirement for the disjoining pressure values exerted on the plates to be equal to one another: ~ =~b2 =~b* - 7'q~m +'~'2~bN2

:7

/7

9ae %.

H

%

9,

(40)

E* The case of positive ~b* values will be examined first; this is illustrated by Fig. 2(b). It is seen that within the interval ~bal~ 1 ~ b l , where ~bbl

~21(/)N1 -

-

1-71

(41)

the disjoining pressure is negative and turns into zero on the boundaries of this interval, i.e. an attraction exists between the plates. There exists a point within this interval in which the disjoining pressure attains its minimum value: this value can be determined by the differentiation of Eq. (19) with respect to ~bl: ~lmin --

1-7 2

(42)

The minimum potential of second plate ~b2mincorresponding to this value can be determined using Eq. (42) and the requirement for the disjoining pressures Eq. (20) to be equal to one another. The behaviour of the disjoining pressure, potentials and charge densities of the plates for the case ~*>4)a2, when the value of ~b* corresponds to the repulsion of the particles (/'/>0) is shown in Fig. 4(a). For q~>~bbl the disjoining pressure is positive, and increases monotonously up to the

/-/ 0

0

j.____------y-

/4

i

a)

b)

Fig. 4. The same as in Fig. 3 for opposite signs of Nernst potentials of the plates. Case (a): ~*> ~b,2;case (b): ~b*< ~ba2.(Dashed line--see text).

value /;r(~*)=/'/* when the distance between the plates is reduced to zero. After the minimum value ~bzmin of the second plate potential is attained, the potential q~2 also increases up to its maximum value ~b2 = ~b*. The behaviour of plate charge densities al and ~r2 is determined by Eq. (12 I) and is shown in Fig. 4(a) along with the behaviour of the potentials: the maximum of charge density a2(Oz) corresponds to the ~b2 potential minimum, and the monotonous increase of ~b1, which takes place when the distance between the particles is decreased, corresponds to the monotonous decrease of charge density al(~bl). The behaviour of the functions /~(H), ~,(H), a,(H) (i= 1,2) when ~blmin<~*<~a2~bbl is shown in Fig. 4(b) by solid lines. The dashed line corresponds to the same functions for

Yu. I'~ Shulepov et al. / Colloids Surfaces A. Phvsicochem. Eng. Aspects 131 (1998) 51 62

0~*~(~lmin: in this case the functions ffl(H), {hi(H), adH) (i= 1,2) possess no extreme values. The behaviour of the functions [I(H), q~(H), a~(H) (i= 1,2) corresponds qualitatively to one of the regimes considered in Ref. [18] for the case of unlike surfaces. It is seen that the behaviour of the potentials and charge densities of the plates in the case of opposite Nernst potential signs corresponds to the cases described by Figs. l(b) and l(c), i.e. equal or opposite signs of the potentials and opposite signs of the densities of plate charges. In this case the integration of Eq. (13) again leads to the expression in Eq. (33). The expressions for the boundary conditions in Eq. (12) can be transformed into Eq. (34) if the equality in Eq. (13) is taken into account. We had thus proved that the behaviour of the functions ~bi(.v) (i = 1,2 ),/'/(x), U~(x) and Udx) for ~b*> 0 is determined by Eqs. (29)-(32) for the case when the Nernst potentials of the particles possess opposite signs. The expressions valid for the alternative case q~*< 0 can be obtained by the inversion of the potential signs (i.e. substitution of ~bi with -~b~and qSN~with -~bN~,i = 1,2) in Eqs. (33) and (34). We had already performed this transformation of Eqs.(33) and (34) and also showed that in the considered case ~b*< 0 the behaviour of the functions 4)i(x) (i=1,2), /~/(x), [TAx) and ~Tfe(A'), for 4~*>0 is still determined by Eqs. (29)-(32).

5. Discussion

In the present study the problem concerning the interaction of double electric layers of colloid particles is solved on the basis of a linearized PB equation, assuming the equilibrium charging regime of particle surfaces. Three types of surface ionization by the potential-determining ions of an electrolyte with a concentration corresponding to the Henry region are studied: (a) adsorption of ions on an initially uncharged surface that possesses no non-compensated charge in the absence of adsorbed ions: (b) ions adsorption (ion exchange) on an initially charged surface which possess some non-compensated charge in the absence of adsorbed ions: (c) ions adsorption (ion

61

exchange) on amphoteric surfaces, for which the cations and anions adsorption values are commensurate. The last type of adsorption on amphoteric surfaces is found to be the most general one, as the types (a) and (b) can be reduced to this type (c) by the redefinition of the constants. The dependence of the potentials, charge density of the plates and disjoining pressure on the distance between the plates was qualitatively studied on the basis of the behaviour of the functions ~b2 and 72(4>Ni _q~i)2 (i= 1,2) which determine the disjoining pressure, see Fig. 2(a,b) and Figs. 3 and 4. It was shown that the behaviour of the principal electric characteristics (the potentials and charge densities of the plates), the disjoining pressure and the electrostatic component of the Gibbs fi'ee energy of particle interactions, depends essentially on the relation between the Nernst potentials of particle surfaces. If Nernst potentials of the particle surfaces possess equal signs, then the surtace potentials, disjoining pressure and electrostatic component of the Gibbs free energy of the particle interactions all increase monotonously with the decrease of the distance between the particles, see Fig. 3. In this case, at low distances between the surl:aces a charge inversion takes place on the surface of the particle whose absolute value of Nernst potential is lower, see Fig. 3. If the signs of the Nernst potentials li)r two particles are opposite, then the signs of the charge densities for these particles are also opposite, see Fig. 4(a,b). In this case of opposite Nernst potential signs the behaviour of the electrostatic component of the Gibbs free energy of particle interactions depends on the relation between the parameters. For example, if 14"1>14>N[, then the disjoining pressure becomes positive when the distance between the plates decreases, see Fig. 2(b). This qualitative study of the behaviour of potcntials and charge densities was necessary to establish the principal types of PB solution (Fig. l(a,b,c)) and, consequently, to perform the integration of Eq. ( 131. As a result, in spite of the fact that the final forms of the PB equation integrals for various types of solulion are different, see Eqs. (26) and (27) and Eqs. (33) and (34). it was l\)und possible to prove that the behaviour of the potentials and charge densities of the plates, the disjoining pres-

62

Yu. V. Shulepov et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 131 (1998) 51-62

sure, the density of the electrostatic component of the Gibbs free energy for the interaction between the plates and the electrostatic component of the Gibbs free energy for the interaction between the particles are, in all cases, determined by Eqs. (29)-(32). The surface density of the electrostatic component of the Gibbs free energy for the interaction between the plates, Eqs. (15) and (31), is found to coincide with that derived in Ref. [20] (see eqs. (49) and (50) therein) by extremizing of Debye-Ht~ckel energy expanded over small potentials for charge-regulated interfaces bearing ionogenic surface groups. It is to be noted that for the case when the adsorption and electric characteristics of two interacting colloid particles are identical, then the expression for the electrostatic component of the Gibbs free energy, Eqs. (17) and (32) reduces to the corresponding expression derived in Refs. [17,26] for the adsorption (ion exchange) on an initially charged surface. The analytical expression for the electrostatic component of the Gibbs free energy for the interaction between the particles derived in the present paper can be useful for the statistical study of colloid particle assemblies [27].

Acknowledgment The authors (Yu.V.Sh. and S.S.D.) are grateful to the International Association for the Promotion of Cooperation with Scientists from the Independent States of the Former Soviet Union for the financing of this investigation.

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[3] B.V. Derjaguin, Theory of the Stability of Colloids and Thin Films, Nauka, Moscow, 1986 (in Russian). [4] J.Th.G. Overbeek, Colloids Surf. 51 (1990) 61. [5] H. Kihira, E. Matijevic, Adv. Colloid Interface Sci. 42 (1992) 1. [6] B.V. Derjaguin, Discuss. Faraday Soc. 18 (1954) 85. [7] B.V. Derjaguin, Kolloidn. Zh. 16 (1954) 425 (in Russian) [8] O.F. Devereux, P.L. de Bruyn, Interaction of Plane Parallel Double Layers, MIT Press, Massachusetts, 1963. [9] J.Th.J. Overbeek, J. Chem. Soc. Faraday Trans. I: 84 ( 1988 ) 3079. [10] H. Kihira, N. Ryde, E. Matijevic, Colloids Surf. 64 (1992) 317. [11] R. Hogg, T.W. Healy, D.W. Fuerstenan, J. Chem. Soc. Faraday Trans. I: 62 (1966) 1638. [12] E. Barouch, E. Matijevich, T.A. Ring, J.M. Finlan, J. Colloid Interface Sci. 67 (1978) 1. [13] V.M. Muller, in: Studies in Surface Forces, Nauka, Moscow, 1967 (in Russian). [14] V.M. Muller, in: Surface Forces and the Stability of Colloids, Nauka, Moscow, 1974 (in Russian). [15] G.A. Martynov, Electrochimija 15 (1979) 474 (in Russian) [16] S.Yu. Shulepov, S.S. Dukhin, J. Lyklema, Kolloidn. Zh. 54 (1992) 182 (in Russian) [17] S.Yu. Shulepov, S.S. Dukhin, Kolloidn. Zh. 54 (1992) 98 (in Russian) [18] D.Y.C. Chan, T.W. Healy, L.R. White, J. Chem. Soc. Faraday Trans. 72 (1976) 2844. [19] G. Bell, G.C. Peterson, Can. J. Chem. 59 (1981) 1888. [20] E.S. Reiner, C.J. Radke, Adv. Colloid Interface Sci. 47 (1993) 59. [21] Yu.V. Shulepov, Zh. Fiz. Chim. 63 (1989) 2824 (in Russian) [22] Yu.V. Shulepov, V.V. Kulik, Zh. Fiz. Chim. 63 (1989) 2824 (in Russian) [23] J. Lyklema, J. Kijlstra, S.S. Dukhin, S.Yu. Shulepov, Kolloidn. Zh. 54 (1992) 92 (in Russian) [24] S.S. Dukhin, Yu.V. Shulepov, J. Lyklema, Kolloidn. Zh. 56 (1994) 641 (in Russian) [25] LK. Koopal, S.S. Dukhin, in: Th.F. Todros, J. Gregory (Eds.), Colloids in the Aqueous Environment, vol. 73, Elsevier, New York, 1993, p. 201. [26] S.Yu. Shulepov, S.S. Dukhin, J. Lyklema, J. Colloid Interface Sci. 171 (1995) 340. [27] Yu.V. Shulepov, S.Yu. Shulepov, Prog. Colloid Polym. Sci. 100 (1996) 148.