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Forces between dissimilar colloidal plates for various surface conditions
Forces between Dissimilar Colloidal Plates for Various Surface Conditions I. General Method G. M. BELL Mathematies Department, Chelsea College, University of London, Manresa Road, London SW3, England AND
G. C. P E T E R S O N I Unilever Research Port Sunlight Laboratory, Port Sunlight, Wirral, Merseyside L62 4XN, England Received April 15, 1976; accepted August 18, 1976 A method is developed for considering the double-layer interaction between parallel colloidal plates with general surface conditions which may be different for the two plates. The general solution of the Poisson-Boltzmann equation is graphed as a set of isodynamic slope curves where the rate of change with distance of the diffuse-layer potential is plotted against the potential itself for a given value of the double-layer force between the plates. The charge-potential isotherms for the two plates (either experimentally or theoretically derived) are then superimposed on the isodynamic slope curves. The range of values taken by the force, as the plate separation is reduced from infinity to zero, together with the existence of maxima or minima in the force, can be seen from the graph. For a given value of the force the potentials of the plates can be read off and any extreme value of the potentials of the plates can be found. The force-separation curve can then be calculated. This method is applicable for any electrolyte type or any mixture of electrolytes. As illustration, the interaction of plates with constant but unequal charges of various signs is considered, and also the case with one plate at constant potential and the other at constant charge. 1. INTRODUCTION
The electrical (double-layer) forces (1, 2) between colloidal particles in electrolyte solution depend on the conditions determining the density of the electric charge on the surface. Such charges may result from the adsorption of ionic species from the medium, dissociation of the surface groups, a surface imbalance between ions of opposite sign forming the material of the colloidal particle, or a combination of these effects. If At~(o is the change in free energy resulting from the 1 Author to whom correspondence should be addressed.
presence of an additional ion of species i, charge ei, on the surface then there is an equilibrium relation At~(i) = A~i + ei~ = 0,
[1.1]
where xI, is the mean potential at the diffuse layer boundary adjacent to the surface. Besides bulk electrolyte terms, Attl may contain surface terms, due to entropy of distribution on the adsorption sites, the mean potential difference between the adsorption plane and the diffuse-layer boundary (termed the inner and outer Helmholtz planes), and the surface fluctuation (self-atmosphere) potentials. The effect of these various surface terms 376
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977 ISSN 0021-9797
Copyright ~ 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
DISSIMILAR
COLLOIDAL
on the interaction between identical parallel plates at large separation was discussed by Levine and Bell (3, 4). If there is only one equilibrium equation of type [-1.1-1 and A~i is constant for a given bulk electrolyte composition, then ~ is independent of particle separation (5). The variation of Agi with the concentration in bulk electrolyte of the appropriate ionic species is then given by the Nernst equation (6). The constantpotential condition was used in the original DLVO theory of colloidal interaction. It is most likely to be applicable in the "imbalance" case where the surface ionic species also form the bulk of the colloidal particle. An alternative "simple" surface condition is that of constantsurface-charge density. This was justified by Frens (7) on the grounds that changes in particle separation due to Brownian motion are too rapid for equilibrium between adsorbed ionic species and bulk electrolyte to be maintained. It may also happen that the surface concentration of a "primary" ionic species will remain effectively constant, due to large adsorption energy or a high relaxation time, while "secondary" adsorbed ionic species remain in equilibrium with the electrolyte medium (8). In this series of papers the terra "dissimilar particles" denotes colloidal particles for which the values of ,I, at infinite separation are unequal. Derjaguin considered dissimilar plates and spheres at constant potential (9) and a comprehensive survey for dissimilar plates at constant potential, with tables, was given by Devereux and de Bruyn (10). Hogg et al. (11) and Wiese and Healy (12) used the small potential (linear) approximation for dissimilar spheres at constant potential and constantcharge density, respectively. Parsegian and Gingell have used the same approximation for dissimilar plates (13). However, for constantcharge density this approximation is known to be unreliable (14, 15). The present authors (14) discussed the various cases of interaction between plates at constant-charge density as a preliminary to their treatment of spheres, but
PLATE FORCES
377
gave no force-distance curves for plates. A common feature for the constant-potential and constant-charge conditions is that the doublelayer force tends to infinity as the width of the diffuse layer tends to zero. The only exceptions are the cases of equal constant potentials and equal and opposite constantcharge densities. This suggests that it is necessary to consider the interaction between dissimilar plates under equilibrium adsorption conditions. Some years ago this was in fact discussed by Bierman (8), who used Langmuirtype adsorption isotherms. However, Bierman's only quantitative calculation was of the separation at which the force changes sign. It is desirable to find a procedure which will yield the fullest information about the doublelayer interaction for general types of surface isotherm and this is the object of the present paper. The assumptions made about the state of the surface phase yield a relation between the potential ~I, and the surface charge density at each plate. This determines the boundary conditions under which it is necessary to solve the Poisson-Boltzmann equation for the potential in the diffuse charge layer between the plates. According to the width of the layer the solution may correspond to negative force or positive force between the plates and with positive force there may or may not be a potential minimum. When equilibrium conditions of type ['1.1] apply, the boundary conditions involve a nonlinear relation between the potential and its slope and with dissimilar plates there are different forms of such a relation at each boundary of the diffuse layer. The determination of a force-separation or a free-energy-separation curve thus appears at first sight to be complex and difficult. We present here a method which, with given conditions for the surface phase on each plate, enables the general form of the double-layer force/plate-separation curve to be seen immediately. The method gives the plate potentials for various force values graphically and with these improved, if necessary, by computation
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
378
BELL AND P E T E R S O N
it is straightforward to calculate accurate values of the separation, either numerically or by tables. It should be stressed that we are not attempting to supersede the electronic computer by graphical methods; indeed, the graphs themselves could, if desired, be produced by computer. What we hope is that these methods can help research workers to gain an understanding of complex systems without recourse to masses of numbers, thus enabling them to make effective and economic use of the computer. In this paper (which is Part I of a series) we illustrate the general method by considering the formally simple case of constant-charge density on each plate and also the mixed case where one plate is at constant-charge density and the other at constant potential. The more physically realistic case, where the plates are charged with ions obeying equilibrium relations, is mathematically more complex and will be discussed in Part II. 2. MEDIUM AND SURFACE RELATIONS FOR PARALLEL PLATES
may be replaced by ~d'z~b/dx 2 = - - eo ~
zini ~ i
[-2.23
Xexp(--z/e#/kT),
the variation of ~b in the y and z directions being negligible. The first integral of U2.23 may be written = kT Z
n / ( O > { e x p ( - - z ~ e o ~ / k T ) --
1}
i
2 \dx/"
Here the constant of integration ff has the physical meaning of force per unit area between the plates, the first term on the righthand side of [2.3] being the difference between the osmotic pressure and its bulk value and the second term the electrical tension. If the first term is greater in magnitude than the second the force is repulsive (~ > 0), while if the second term is greater the force is attractive (~ < 0). The bulk Debye-Huckel constant K is defined by
We consider a pair of parallel plates of area K2 = eo2 Y~ zi2n/<°>/(ekT), [2.4-I i A immersed in an infinite electrolyte medium. Away from the influence of the plate charges K-1 being the characteristic Debye-Huckel the number of ions of type i per unit volume is distance. We define a dimensionless potential denoted by n~<°~ and the electrical potential ~b q~, a dimensionless distance ~, and a dimensionis taken as zero. As in the classical DLVO theory (1, 2), ~bis assumed to obey the PoissonBoltzmann equation
¢
V2~b = --(e0/~) Z z/n/(") i
Xexp(--zieo~b/kT).
[2.13
Here k is Boltzmann's constant, T is the absolute temperature, e0 is the proton charge, zieo is the charge on an ion of type i, and e is the dielectric permittivity, the latter being assumed constant. If the plate separation H is small compared with the linear dimensions of the plates and the x axis is taken normal to the plates then, between the plates, [2.1-]
i
!
OI
FIG. 1. Example of potential profile between two plates: cIh and cI,~ are the reduced potentials at the diffuse-layer boundaries next to the plates.
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
DISSIMILAR COLLOIDAL PLATE FORCES d¢ ~,aW .--
q__
379
/
3+<0 32-
~p
J t
"..~
I]]1
° Jl
¸ ¸¸"
'~
,¢
0
o
"-
/
-3 /
-L, R --
3-~0
. \
[-
/
j
/
_. -5 .
I -2
-I
:
2
3
FIa. 2. Isodynamic slope curves : The negative of the slope d¢/d~ of the potential-distance profile is plotted against potential. Each curve is labeled with the corresponding value of the reduced doublelayer force if+. less force per unit area if+, respectively, by
For a 1-I electrolyte [2.10-1 becomes
e0ff ¢
g(q~, ~+) = (2 c o s h ¢ -- 2 - 25:+)L
kT '
~ = ~x,
~Y ~:+
-
.
k T ~ gi2Ui(0)
[2.5-]
i
Then Eqs. [-2.2-] and E2.,3~ become
d2¢/d~ 2 = ~ zini (°) exp(--zi$)/Y~ zflni (°) [2.6-] i
i
and
~+ = E ni(°){exp(--z@)
-- 1}/)-', zflm (°) --½(ddo/d() 2.
~5+ = coshq~ -- 1 -- ½(dCp/d() 2.
[2.8-]
It is convenient to express [-2.7.] in the form
d ¢ / d ( = -C-g(,, ~+),
We now introduce the convention that the left-hand plate is labeled 1 and the right-hand plate 2, (I)l and a52 denoting the potentials at the diffuse-layer boundaries (See Fig. 1, where a possible potential profile is shown). When the plate separation H = ~ , q~l and q~2 are denoted by ¢1. and ¢2~, respectively. For symmetrical electrolytes the signs of all potentials in any solution of the equations can be reversed to yield an alternative solution and, for definiteness, we assume that (I)l:o >
[-2.7~
For a 1 1 electrolyte, [2.7~ becomes
[-2.9-]
where
[2.113
[-2.123
0.
The charge densities on the plates are denoted by g~ and g2 and are regarded as functions of ~1 and ~2, respectively. At the diffuse-layer boundaries we have
(d~/dx)l = --crl/e,
(d~/dx)2 = c~2/e.
[2.13-]
If we write
g(cb, ~+) = [2 ~_. ni(°){exp(-zidp)
-
-
1 } / ~ zi2ni (°)
i
i
--2~+.] ~ > O.
i-2.10-]
eKk T or1 = - fl(d#l), e0
eKk T c% -
/2@2), [2.143 go
Journal of Colloid and Interface Science, V o l . 60, N o . 2, J u n e 15, 1977
380
BELL AND PETERSON
--d$/d~ (the negative sign being chosen for later convenience) against q~ for given 5:+, using Eq. [2.10-] or Eq. [-2.113. We term such plots "isodynamic slope curves" and in Fig. 2 we show some of them for the 1 1 electrolyte case. Of particular importance are the curves POR and QOS, which divide the plane into the four regions labeled I, II, I I I , ¢ and IV. The isodynamic slope curves lying in I[ regions I and I I I intersect the q~ axis and correspond to positive 5:+, while those lying ,g 0 in regions I I and IV intersect the --dCp/d( axis and correspond to negative 5:+ Sketches of the types of (q~, () curve associated with regions I to IV are shown in Fig. 3. The graphical method consists of superimposing plots of fl and --.f2 against q~ onto FIG. 3. Isodynamic potential/distance curves corre- the isodynamic slope curves. By [-2.15], sponding to the four regions of Fig. 2. values of ¢ and d4/d( at the diffuse-layer boundaries of the plates are given by interthen the boundary conditions [2.13] can be sections of the f1(4) and -- f~(4~) curves with an isodynamic slope curve for the relevant value expressed in terms of the reduced quantities of 5:+. These points of intersection will be defined by [-2.5] to yield termed "plate points." The method is illus(d¢/d~h = - £ ( ~ 0 , trated by Fig. 4, where hypothetical fl(q~) (d~/d~)2 = .f~(®2). F2.15~ and --f2(4,) curves are plotted with the relevant parts of the isodynamic slope curves As the separation H ~ 0, 5: -~ -- oc if qs1 ¢ q52, for a 1 1 electrolyte. The point C, ~here as occurs with the constant-potential assumpfl(q~) = -- f2 (¢), gives a solution for Eq. [-2.16-] tion and 5:--~-[-*¢ if ~1 ~ - - ~ 2 , as occurs with the constant-charge assumption (14). Hence the limit of 5= as H ~ 0 can be finite /// only if there is a value of q5 satisfying the equation
°/
f~(qs) + f2(qs) = 0.
[2.16.]
Whether this equation can be solved depends on the form of the charge potential isotherms fl (qs) and f2 (qs). 3. A GENERAL GRAPHICAL METHOD We now introduce a graphical method which enables the general form of the force-distance curve to be seen once the functions f1(¢) and f2(¢) are specified. We start by expressing the isodynamic curves of Derjaguin (9) in a rather different way. Instead of plotting ¢ against ~ we plot the reduced electric field
O-
-1 FIG. 4. Illustration of plate interaction diagram: f1(4~) and f2(¢) are plate curves (reduced chargedensity/plate-potential isotherms) and the faint lines are isodynamic slope curves.
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
DISSIMILAR COLLOIDAL PLATE FORCES
381
and is the limiting position of both the plate 1 point and the plate 2 point as the plate
plate potentials can also be easily seen in Fig. 4. The potential cp., increases steadily from
separation tt
O, From the coordinates of
1 at H = m to the common value 1,7 at
C, the limiting values of the plate potentials and charges as H --+ 0 are
H = 0. The potential qs~ increases from 1.9 at H = m to a maximum value of 1.96 at the point Aa, which also corresponds to the maximum value of 5 + , and then decreases to 1.7 a t H = 0 . If values of qs, and (1)2 have been obtained for a given value of 5 + from a diagram such as Fig. 4 (and possibly improved by computation) then the separation is given by either
q51 = q52 = 1.7,
ax = --or2 = 3.4eKkT/eo.
As C lies on the 5:+ = --4 isodynamic slope curve, the limiting value of 5 + as H--+ 0 is - 4 . We next consider the opposite limiting case H = m when qs, and (d4/d})l must be of opposite sign while ~2 and (d~/d}).e have the same sign, since 1 and 2 label the left- and right-hand plates, respectively. Hence the plate 1 point at H = oo is the intersection of the fl(q~) curve with the 5 + = 0 curve PO while the plate 2 point is the intersection of the -f2(q~) curve with the 5 + = 0 curve QO. In Fig. 4 the plate 1 point at H = m is ALs, giving ~ , = 1.9, and the plate 2 point is Bb giving q)2~ = 1.0. As H decreases from 0¢ to 0 the plate 1 point in Fig. 4 shifts from Aa.5 to C, while the plate 2 point shifts from B1 to C. At any intermediate separation the plate 1 and 2 points are, respectively, the intersection of .f1(4~) and --f2(¢) with the same branch of an isodynamic slope curve, since each branch corresponds to a distinct (q~, () isodynamic curve (see Fig. 3). For instance, at 5 + = --2 the plate 1 and 2 points in Fig. 4 are, respectively, A7 and BT. The general behavior of the force in the Fig. 4 example can now be easily seen. As H decreases from oo the plate points move into the positive force region, the plate 1 point passing through the sequence A,,5, A2,4, Aa, A2.4, and A1,5 and the plate 2 point through the corresponding sequence B1, B2, Ba, B4, and B~. Since the 5 + = 1 isodynamic slope curve is tangent to the --f2(4)) curve at Ba, 5 + has a turning point where it takes the maximum value 1. At A~.5 and Bs, respectively, the plate points cross into the negative force region and pass through the corresponding sequences A1.5, AG, and Av and Bs, B6, and By before they meet at C, where H = 0 and 5 + = --4. The behavior of the
=
+
-
D
or
KH =
2
g
,
[-.3.21
where g is defined by Eq. E2.10-] above. Equation [-3.1~ applies when 5 + > 0 and there is a turning point in the (¢, () curve between the plates, q~,, being the value of 4, at the turning point. Equation [-3.2-] applies when d4,//d( has the same sign everywhere between the plates. It must always be used when 5 + < 0 and also when 5 + > 0 but there is no turning point in the potential curve between the plates. The diagram will always show which fornmla to use. For example, in Fig. 4, Eq. [-3.11 applies when the plate points are B2 and A2.4, respectively, the value of qL,, = 0.96 being given by the intercept of the 5 + = ½ isodynamic slope curve with the q5 axis. On the other hand, when 5 + = ½ and the plate points are B4 and A2,4, respectively, there is clearly no potential turning point between the plates and Eq. E3.2~ applies. For the 1 1 electrolyte at large separations the "superposition" formula 5 + = 32 tanh (~¢P1~) tanh ( ~ , ) X exp(--KH)
[-3.3]
is a good approximation. When 5 + passes through the value zero the separation is
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
382
BELL AND PETERSON
1
- -2
2
""-,\ \
n
~a
FIG. 5. Plate interaction diagram for constant-chargedensity cases: the horizontal lines are "plate curves" and the others are isodynamic slope curves. given by KH = lln{tanh (}Ih)/tanh (}@2)}1. [-3.4] For other values of 5:+ expressions [-3.1] and [-3.2-] can, for the 1-1 electrolyte case, be transferred to elliptic integrals. Formulas for the various cases are given in the Appendix. A possible graphical technical for estimating plate separation would be to plot d~/d@, fl -I, and f2-1 against @ instead of d@/d}, fl, and f2, and the area under the segment of the d}/d@ curve between the plate points would give the plate separation. However, we do not regard this as a desirable alternative to numerical integration or the use of elliptic integral formulas, especially as d(/d@ becomes infinite at a potential minimum. Plate separations could also be obtained from potential-distance isodynamic curves if these were available in the required force range or from similar data in tabular form. It is unfortunate that the tables of de Bruyn and Devereux (10) are not in isodynamic form and thus the method described in the Appendix is, at the moment, the most easily utilized.
method; we shall choose that of constant charge. With the graphical method it is also easy to consider the "mixed" case where one plate is at constant potential and the other at constant charge and we shall give examples of this. With ~1 and a2 constant, the plate curves become lines parallel to the ¢ axis and can never meet except where al = - a 2 , when they are coincident. Except in the latter case, no solution of [-2.14] is thus possible and the force 5:+ and plate potentials @x and @2 all become infinite at H = 0. Figure 5 shows one example of constant charges of the same sign and one of constant charges of opposite sign: (a) @2~ = 2, @1~ = 1, f l =
2.3504, f 2 = 1.0422.
The plate points start at Ax and B1 on the appropriate ~+ = 0 curves when H = oo. At any value of H less than infinity they must lie on the same branch of an isodynamic slope curve. Hence as H decreases from infinity the plate points move into the positive force region and travel to the right along the L
I
I
I
(~2 ~2) 2
to
4. THE CONSTANT-CHARGE AND "MIXED" CASES The theory of dissimilar plates at constant potential (9, 10) or constant charge (14) has already been fully discussed. However, these cases afford simple illustrations of the graphical
J
-I
~
0~-~
1
1
I
2
I
3
FIC. 6. R e d u c e d f o r c e / p l a t e - s e p a r a t i o n c u r v e s f o r constant surface charge: The curves are labeled with t h e p l a t e p o t e n t i a l s a t infinite s e p a r a t i o n .
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
DISSIMILAR COLLOIDAL PLATE FORCES parallel f , and f2 lines, their distance from the 4~ = 0 axis becoming infinite as H ~ 0. The force-distance curve is shown in Fig. 6. (b)
q%~ = 2,
i
i
f.., = -- 1.0422
The plate points s t a r t at A1 and B~' when H = m. As H decreases from infinity the plate 2 point moves to the right and the plate 1 point to the left into the negative force region. The force has a m i n i m u m value when the plate 2 point is at B,/, where the plate 2 line is tangent to the 5 + = - - 0 . 5 4 3 1 isodynamic slope curve. The corresponding plate 1 point A2 gives a m i n i m u m value 1.8372 of ~1, which thereafter increases again as the plate 1 point moves back to A~. At A~ and B3', respectively, the plate points cross into the positive force region and the behavior is then similar to t h a t when the charges are of the same sign, q'l, ~2, and 5 + all increasing to infinity as H ~ 0. The force-distance curve is shown in Fig. 6. F o r c e - d i s t a n c e curves for cases where ~ = = - - ~ 2 ~ and ~,~ = ~2~, respectively, are also shown in Fig. 6. In the former case the plate points on the isodynamic slope diagram are s y m m e t r i c a l l y situated with respect to the dq)/tt~ axis and meet on the axis when H = 0. -
V
i
~ , , = --1,
f~ = 2.3504,
-
2
383
,d_~
,/P
3
~2
" /
2
r<
p2
, •
,,
',B 2
R
~
\
/ / /' /
t
/
-2
1
-3
KN
D
..........
1
I
2
I
3
FIG. 8. Reduced force/plate-separation curves for the mixed case with plate 1 at constant potential and plate 2 at constant charge: The curves are labeled with the plate potentials at infinite separation. I n the latter case the plate points are symmetrically situated with respect to the ¢ axis and move steadily toward infinite potentials as H--~ 0. We now consider the " m i x e d " case where one plate i s , a t constant potential and the other at constant charge. Since the plate curves are,~respectively, a vertical line and a horizontal line they intersect and hence the limiting value of 5 + as H ~ 0 is always finite in the mixed case. I n Fig. 7 plate 1 is at constant potential with ~ , = 2, while plate 2 is at constant charge. Three examples with different values of the constant charge density are shown :
0) %
qh > ~2~ > 0;
;2
~1 = 2,
~
= 1,
f,~ = 1.0422. FIG. 7. Plate interaction diagram for the "mixed" case: The vertical line is the plate curve for the constant-potential plate 1, the horizontal lines are plate curves for the constant-charge plate 2, and the others are isodynamic slope curves.
The plate points start at A~ and B~, respectively, when H = oo and meet at C~ u hen H = 0, where 5 + = 2.2191. The force 5:+ is always positive and passes through a m a x i m u m value u h e n the plate 1 point is at A2, where
Journal of Colloid and Interface ,~eience, V o l . 6 0 , N o . 2, J u n e
15, 1 9 7 7
384
BELL AND PETERSON
the plate line is tangent to the 5:+ = 2.7622 isodynamic slope curve. The plate 2 point is then at B2 on this curve and @2 has its maximum value of 2.1392. The force distance curve is shown in Fig. 8. (ii) @l > --@2~ > 0;
@1 = 2,
@2~ = --1,
f2 = --1.0422. The plate points start at A1 and BI', respectively, when H = ~ and meet at Cx' when H = 0. The force 5:+ is negative for large / / and decreases to a minimum when the plate 1 point is at As' and the plate 2 point at B2' where @2 = 0 and the 5:+ = - - 0 . 5 4 3 1 isodynamic slope curve is tangent to the plate 2 line. The force increases as H decreases further and the plate points then cross into the positive force region at A1 and B J, respectively. The force finally reaches a maximum value of 2.2191, at C1 when @1 = @2 = 2. The force distance curve is shown in Fig. 8. (iii) @ 2 ~ > @ 1 > 0 ; @1=2, @2~=3, f2=4.2586. The plate points start at A1 and D1, respectively, when H = oo and meet at E when H = 0. The force 5:+ is positive for large H and increases to a maximum value of 2.7622 when the plate 1 point is at As and the plate 2 point at D2 where @2has its maximum value of 3.2434. The force then decreases with H and the plate points cross into the negative force region at A3 and D1, respectively. The force finally reaches a minimum value of --6.3055 at E. The force distance curve is shown in Fig. 8. Figure 8 also includes force distance curves for cases where @1 = @2:¢ > O,
@1 = --@2~ > O.
--@2= > @ 1 > 0.
5. D I S C U S S I O N
Since both the constant-potential and the constant-charge assumptions give forces tend-
ing to infinity as the plate separation H tends to zero, it can be seen that neither assumption can be physically realistic at small plate separations. The mixed case gives a finite force at all separations but is hardly likely to apply to a wide range of phenomena. In the next paper of the series it is hoped to apply the methods developed in this one to situations where either all the surface ions or a secondary adsorbed layer of ions is in equilibrium with the bulk electrolyte. This means that in determining the charge-potential isotherm for each plate some of the factors discussed in the Introduction in connection with Eq. [1.1] must be taken into account. Conditions under which the potentials and force remain finite as the separation tends to zero can be obtained from considering the curves of the reduced charge density functions fl(~b) and f2(4~) defined above. Although the curves shown in this paper are all for the medium with symmetrical 1-1 electrolyte the methods developed can be used for any type of electrolyte. The expression on the right-hand side of Eq. [2.10], from which the isodynamic slope curves are obtained, is easy to evaluate for nonsymmetrical or mixed electrolytes. Equations [3.1] and [3.2] are applicable for any type of electrolyte. Although the integrals cannot, in general, be expressed in terms of standard functions they can always be evaluated by numerical integration. APPENDIX
We give a table for the calculation of the reduced separation KH with a 1-1 electrolyte medium. As may be seen from Fig. 3 (cases I and III), the boundary potentials @1 and @2 must have the same sign where the force 5:+ > 0. Again from Fig. 3 (cases I I and IV), the boundary slopes (d4~/d()l and (ddp/d()2 must have the same sign when 5:+ < 0. It should be noted that from Eq. E2.15] the ratio of the boundary slopes is related to that of the
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
385
DISSIMILAR COLLOIDAL PLATE FORCES sin ai = [-k~ cosh ( ~ . / 2 ) ' ] - ' ,
reduced charge densities fx and f2 b y
sin 13i = [ c o s h (~i/2)-] -1,
fl../ f.2 = - - (d~b/d~)1// ((l~.../(t~)2.
T h e following definitions are necessary for the table, i t a k i n g v a l u e s of 1 and 2 in each case. if+
Plate conditions
~+ > 0
f,/f'z > 0
if+ > 0
/l/f2 < 0
- - 2 < if+ < 0
,1,,/'4,2 > 0
--2 < 3:+ < 0 if+ < --2 5:+ < --2
(I,,/~I,2 < 0 cI,,/~ > 0 q%/q~ < 0
5. 6.
7.
k~2 = (5 :+ + 2)//5 +.
I n the r i g h t - h a n d c o l u m n the s y m b o l F denotes an elliptic integral of the first kind. ~H
1. DERJAGUIN, B. V., AND LANDAU, L. O., Acta Physicochim. 14, 633 (1941). 2. VERWEY, E. J. W., AND OVERBEEK, J. TfI. G.,
4.
kt~2 = i -J- ½~+,
k~[2F(k~, 7r/2) -- F(k, cq) -- F(k, a2)] k~,[F(k~, a2) -- F(k~, a,)] F(ka, [%) -- F(k~, ~ ) 2F(k~, 7r/2) -- F(k~, ~2) -- F(k~, ~,) (I -- kv2)~[F(kv, "yj) -- F(k~, "Y2)] (l -- kv~)~[F(kv, ~1) + F(kv,'r2)]
REFERENCES
3.
sin 3'~ = t a n h (~./'2),
k~ -2 = 1 + ½~+,
"Theory of the Stability of Lyophobic Colloids." Elsevier, New York/Amsterdam, 1948. LEVINE, S., AND BELL, G. M., J. Colloid Sci. 17, 838 (1962). LEVINE, S., AND BELL, G. M., J. Phys. Chem. 67, 1408 (1963). BELL, G. M., AND LEVINt':, S., Trans. Faraday Soe. 53, 143 (1957). CHEN, C. S., AND LEVINE, S., J. Chem. Sot., Faraday I I 68, 1497 (1972). FRENS, G., Thesis, "The Reversibility of Irreversible Colloids." Utrecht, 1968.
8. BIERMAN,A., J. Colloid Sci. 10, 231 (1955). 9. DERJAGUIN, B. V., Discuss. Faraday Soc. 18, 85 (1954). 10. DEVEREUX, O. F., AND DE BRUYN, P. L., "Interaction of Plane Parallel Double Layers." M.I.T. Press, Cambridge, Mass., 1963. 11. HOGG, R., HEALY, T. W., AND FURSTENAU, D. W., Trans. Faraday Soc. 62, 1638 (1966). 12. WIESE, G. R., AND HEALY, D. W., Trans. Faraday Soc. 66, 1490 (1970). 13. PARSEGIAN, V. A., AND GINGELL, D., Biophys. J. 12, 1192 (1972). 14. BELL, G. M., AND PETERSON, G. C., J. Colloid Interface Sci. 41, 542 (1972). 15. GRFGORY, J., J. Chem. Soc. Faraday I I 69, 1723 (1973).
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
G. C. P E T E R S O N I Unilever Research Port Sunlight Laboratory, Port Sunlight, Wirral, Merseyside L62 4XN, England Received April 15, 1976; accepted August 18, 1976 A method is developed for considering the double-layer interaction between parallel colloidal plates with general surface conditions which may be different for the two plates. The general solution of the Poisson-Boltzmann equation is graphed as a set of isodynamic slope curves where the rate of change with distance of the diffuse-layer potential is plotted against the potential itself for a given value of the double-layer force between the plates. The charge-potential isotherms for the two plates (either experimentally or theoretically derived) are then superimposed on the isodynamic slope curves. The range of values taken by the force, as the plate separation is reduced from infinity to zero, together with the existence of maxima or minima in the force, can be seen from the graph. For a given value of the force the potentials of the plates can be read off and any extreme value of the potentials of the plates can be found. The force-separation curve can then be calculated. This method is applicable for any electrolyte type or any mixture of electrolytes. As illustration, the interaction of plates with constant but unequal charges of various signs is considered, and also the case with one plate at constant potential and the other at constant charge. 1. INTRODUCTION
The electrical (double-layer) forces (1, 2) between colloidal particles in electrolyte solution depend on the conditions determining the density of the electric charge on the surface. Such charges may result from the adsorption of ionic species from the medium, dissociation of the surface groups, a surface imbalance between ions of opposite sign forming the material of the colloidal particle, or a combination of these effects. If At~(o is the change in free energy resulting from the 1 Author to whom correspondence should be addressed.
presence of an additional ion of species i, charge ei, on the surface then there is an equilibrium relation At~(i) = A~i + ei~ = 0,
[1.1]
where xI, is the mean potential at the diffuse layer boundary adjacent to the surface. Besides bulk electrolyte terms, Attl may contain surface terms, due to entropy of distribution on the adsorption sites, the mean potential difference between the adsorption plane and the diffuse-layer boundary (termed the inner and outer Helmholtz planes), and the surface fluctuation (self-atmosphere) potentials. The effect of these various surface terms 376
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977 ISSN 0021-9797
Copyright ~ 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
DISSIMILAR
COLLOIDAL
on the interaction between identical parallel plates at large separation was discussed by Levine and Bell (3, 4). If there is only one equilibrium equation of type [-1.1-1 and A~i is constant for a given bulk electrolyte composition, then ~ is independent of particle separation (5). The variation of Agi with the concentration in bulk electrolyte of the appropriate ionic species is then given by the Nernst equation (6). The constantpotential condition was used in the original DLVO theory of colloidal interaction. It is most likely to be applicable in the "imbalance" case where the surface ionic species also form the bulk of the colloidal particle. An alternative "simple" surface condition is that of constantsurface-charge density. This was justified by Frens (7) on the grounds that changes in particle separation due to Brownian motion are too rapid for equilibrium between adsorbed ionic species and bulk electrolyte to be maintained. It may also happen that the surface concentration of a "primary" ionic species will remain effectively constant, due to large adsorption energy or a high relaxation time, while "secondary" adsorbed ionic species remain in equilibrium with the electrolyte medium (8). In this series of papers the terra "dissimilar particles" denotes colloidal particles for which the values of ,I, at infinite separation are unequal. Derjaguin considered dissimilar plates and spheres at constant potential (9) and a comprehensive survey for dissimilar plates at constant potential, with tables, was given by Devereux and de Bruyn (10). Hogg et al. (11) and Wiese and Healy (12) used the small potential (linear) approximation for dissimilar spheres at constant potential and constantcharge density, respectively. Parsegian and Gingell have used the same approximation for dissimilar plates (13). However, for constantcharge density this approximation is known to be unreliable (14, 15). The present authors (14) discussed the various cases of interaction between plates at constant-charge density as a preliminary to their treatment of spheres, but
PLATE FORCES
377
gave no force-distance curves for plates. A common feature for the constant-potential and constant-charge conditions is that the doublelayer force tends to infinity as the width of the diffuse layer tends to zero. The only exceptions are the cases of equal constant potentials and equal and opposite constantcharge densities. This suggests that it is necessary to consider the interaction between dissimilar plates under equilibrium adsorption conditions. Some years ago this was in fact discussed by Bierman (8), who used Langmuirtype adsorption isotherms. However, Bierman's only quantitative calculation was of the separation at which the force changes sign. It is desirable to find a procedure which will yield the fullest information about the doublelayer interaction for general types of surface isotherm and this is the object of the present paper. The assumptions made about the state of the surface phase yield a relation between the potential ~I, and the surface charge density at each plate. This determines the boundary conditions under which it is necessary to solve the Poisson-Boltzmann equation for the potential in the diffuse charge layer between the plates. According to the width of the layer the solution may correspond to negative force or positive force between the plates and with positive force there may or may not be a potential minimum. When equilibrium conditions of type ['1.1] apply, the boundary conditions involve a nonlinear relation between the potential and its slope and with dissimilar plates there are different forms of such a relation at each boundary of the diffuse layer. The determination of a force-separation or a free-energy-separation curve thus appears at first sight to be complex and difficult. We present here a method which, with given conditions for the surface phase on each plate, enables the general form of the double-layer force/plate-separation curve to be seen immediately. The method gives the plate potentials for various force values graphically and with these improved, if necessary, by computation
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
378
BELL AND P E T E R S O N
it is straightforward to calculate accurate values of the separation, either numerically or by tables. It should be stressed that we are not attempting to supersede the electronic computer by graphical methods; indeed, the graphs themselves could, if desired, be produced by computer. What we hope is that these methods can help research workers to gain an understanding of complex systems without recourse to masses of numbers, thus enabling them to make effective and economic use of the computer. In this paper (which is Part I of a series) we illustrate the general method by considering the formally simple case of constant-charge density on each plate and also the mixed case where one plate is at constant-charge density and the other at constant potential. The more physically realistic case, where the plates are charged with ions obeying equilibrium relations, is mathematically more complex and will be discussed in Part II. 2. MEDIUM AND SURFACE RELATIONS FOR PARALLEL PLATES
may be replaced by ~d'z~b/dx 2 = - - eo ~
zini ~ i
[-2.23
Xexp(--z/e#/kT),
the variation of ~b in the y and z directions being negligible. The first integral of U2.23 may be written = kT Z
n / ( O > { e x p ( - - z ~ e o ~ / k T ) --
1}
i
2 \dx/"
Here the constant of integration ff has the physical meaning of force per unit area between the plates, the first term on the righthand side of [2.3] being the difference between the osmotic pressure and its bulk value and the second term the electrical tension. If the first term is greater in magnitude than the second the force is repulsive (~ > 0), while if the second term is greater the force is attractive (~ < 0). The bulk Debye-Huckel constant K is defined by
We consider a pair of parallel plates of area K2 = eo2 Y~ zi2n/<°>/(ekT), [2.4-I i A immersed in an infinite electrolyte medium. Away from the influence of the plate charges K-1 being the characteristic Debye-Huckel the number of ions of type i per unit volume is distance. We define a dimensionless potential denoted by n~<°~ and the electrical potential ~b q~, a dimensionless distance ~, and a dimensionis taken as zero. As in the classical DLVO theory (1, 2), ~bis assumed to obey the PoissonBoltzmann equation
¢
V2~b = --(e0/~) Z z/n/(") i
Xexp(--zieo~b/kT).
[2.13
Here k is Boltzmann's constant, T is the absolute temperature, e0 is the proton charge, zieo is the charge on an ion of type i, and e is the dielectric permittivity, the latter being assumed constant. If the plate separation H is small compared with the linear dimensions of the plates and the x axis is taken normal to the plates then, between the plates, [2.1-]
i
!
OI
FIG. 1. Example of potential profile between two plates: cIh and cI,~ are the reduced potentials at the diffuse-layer boundaries next to the plates.
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
DISSIMILAR COLLOIDAL PLATE FORCES d¢ ~,aW .--
q__
379
/
3+<0 32-
~p
J t
"..~
I]]1
° Jl
¸ ¸¸"
'~
,¢
0
o
"-
/
-3 /
-L, R --
3-~0
. \
[-
/
j
/
_. -5 .
I -2
-I
:
2
3
FIa. 2. Isodynamic slope curves : The negative of the slope d¢/d~ of the potential-distance profile is plotted against potential. Each curve is labeled with the corresponding value of the reduced doublelayer force if+. less force per unit area if+, respectively, by
For a 1-I electrolyte [2.10-1 becomes
e0ff ¢
g(q~, ~+) = (2 c o s h ¢ -- 2 - 25:+)L
kT '
~ = ~x,
~Y ~:+
-
.
k T ~ gi2Ui(0)
[2.5-]
i
Then Eqs. [-2.2-] and E2.,3~ become
d2¢/d~ 2 = ~ zini (°) exp(--zi$)/Y~ zflni (°) [2.6-] i
i
and
~+ = E ni(°){exp(--z@)
-- 1}/)-', zflm (°) --½(ddo/d() 2.
~5+ = coshq~ -- 1 -- ½(dCp/d() 2.
[2.8-]
It is convenient to express [-2.7.] in the form
d ¢ / d ( = -C-g(,, ~+),
We now introduce the convention that the left-hand plate is labeled 1 and the right-hand plate 2, (I)l and a52 denoting the potentials at the diffuse-layer boundaries (See Fig. 1, where a possible potential profile is shown). When the plate separation H = ~ , q~l and q~2 are denoted by ¢1. and ¢2~, respectively. For symmetrical electrolytes the signs of all potentials in any solution of the equations can be reversed to yield an alternative solution and, for definiteness, we assume that (I)l:o >
[-2.7~
For a 1 1 electrolyte, [2.7~ becomes
[-2.9-]
where
[2.113
[-2.123
0.
The charge densities on the plates are denoted by g~ and g2 and are regarded as functions of ~1 and ~2, respectively. At the diffuse-layer boundaries we have
(d~/dx)l = --crl/e,
(d~/dx)2 = c~2/e.
[2.13-]
If we write
g(cb, ~+) = [2 ~_. ni(°){exp(-zidp)
-
-
1 } / ~ zi2ni (°)
i
i
--2~+.] ~ > O.
i-2.10-]
eKk T or1 = - fl(d#l), e0
eKk T c% -
/2@2), [2.143 go
Journal of Colloid and Interface Science, V o l . 60, N o . 2, J u n e 15, 1977
380
BELL AND PETERSON
--d$/d~ (the negative sign being chosen for later convenience) against q~ for given 5:+, using Eq. [2.10-] or Eq. [-2.113. We term such plots "isodynamic slope curves" and in Fig. 2 we show some of them for the 1 1 electrolyte case. Of particular importance are the curves POR and QOS, which divide the plane into the four regions labeled I, II, I I I , ¢ and IV. The isodynamic slope curves lying in I[ regions I and I I I intersect the q~ axis and correspond to positive 5:+, while those lying ,g 0 in regions I I and IV intersect the --dCp/d( axis and correspond to negative 5:+ Sketches of the types of (q~, () curve associated with regions I to IV are shown in Fig. 3. The graphical method consists of superimposing plots of fl and --.f2 against q~ onto FIG. 3. Isodynamic potential/distance curves corre- the isodynamic slope curves. By [-2.15], sponding to the four regions of Fig. 2. values of ¢ and d4/d( at the diffuse-layer boundaries of the plates are given by interthen the boundary conditions [2.13] can be sections of the f1(4) and -- f~(4~) curves with an isodynamic slope curve for the relevant value expressed in terms of the reduced quantities of 5:+. These points of intersection will be defined by [-2.5] to yield termed "plate points." The method is illus(d¢/d~h = - £ ( ~ 0 , trated by Fig. 4, where hypothetical fl(q~) (d~/d~)2 = .f~(®2). F2.15~ and --f2(4,) curves are plotted with the relevant parts of the isodynamic slope curves As the separation H ~ 0, 5: -~ -- oc if qs1 ¢ q52, for a 1 1 electrolyte. The point C, ~here as occurs with the constant-potential assumpfl(q~) = -- f2 (¢), gives a solution for Eq. [-2.16-] tion and 5:--~-[-*¢ if ~1 ~ - - ~ 2 , as occurs with the constant-charge assumption (14). Hence the limit of 5= as H ~ 0 can be finite /// only if there is a value of q5 satisfying the equation
°/
f~(qs) + f2(qs) = 0.
[2.16.]
Whether this equation can be solved depends on the form of the charge potential isotherms fl (qs) and f2 (qs). 3. A GENERAL GRAPHICAL METHOD We now introduce a graphical method which enables the general form of the force-distance curve to be seen once the functions f1(¢) and f2(¢) are specified. We start by expressing the isodynamic curves of Derjaguin (9) in a rather different way. Instead of plotting ¢ against ~ we plot the reduced electric field
O-
-1 FIG. 4. Illustration of plate interaction diagram: f1(4~) and f2(¢) are plate curves (reduced chargedensity/plate-potential isotherms) and the faint lines are isodynamic slope curves.
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
DISSIMILAR COLLOIDAL PLATE FORCES
381
and is the limiting position of both the plate 1 point and the plate 2 point as the plate
plate potentials can also be easily seen in Fig. 4. The potential cp., increases steadily from
separation tt
O, From the coordinates of
1 at H = m to the common value 1,7 at
C, the limiting values of the plate potentials and charges as H --+ 0 are
H = 0. The potential qs~ increases from 1.9 at H = m to a maximum value of 1.96 at the point Aa, which also corresponds to the maximum value of 5 + , and then decreases to 1.7 a t H = 0 . If values of qs, and (1)2 have been obtained for a given value of 5 + from a diagram such as Fig. 4 (and possibly improved by computation) then the separation is given by either
q51 = q52 = 1.7,
ax = --or2 = 3.4eKkT/eo.
As C lies on the 5:+ = --4 isodynamic slope curve, the limiting value of 5 + as H--+ 0 is - 4 . We next consider the opposite limiting case H = m when qs, and (d4/d})l must be of opposite sign while ~2 and (d~/d}).e have the same sign, since 1 and 2 label the left- and right-hand plates, respectively. Hence the plate 1 point at H = oo is the intersection of the fl(q~) curve with the 5 + = 0 curve PO while the plate 2 point is the intersection of the -f2(q~) curve with the 5 + = 0 curve QO. In Fig. 4 the plate 1 point at H = m is ALs, giving ~ , = 1.9, and the plate 2 point is Bb giving q)2~ = 1.0. As H decreases from 0¢ to 0 the plate 1 point in Fig. 4 shifts from Aa.5 to C, while the plate 2 point shifts from B1 to C. At any intermediate separation the plate 1 and 2 points are, respectively, the intersection of .f1(4~) and --f2(¢) with the same branch of an isodynamic slope curve, since each branch corresponds to a distinct (q~, () isodynamic curve (see Fig. 3). For instance, at 5 + = --2 the plate 1 and 2 points in Fig. 4 are, respectively, A7 and BT. The general behavior of the force in the Fig. 4 example can now be easily seen. As H decreases from oo the plate points move into the positive force region, the plate 1 point passing through the sequence A,,5, A2,4, Aa, A2.4, and A1,5 and the plate 2 point through the corresponding sequence B1, B2, Ba, B4, and B~. Since the 5 + = 1 isodynamic slope curve is tangent to the --f2(4)) curve at Ba, 5 + has a turning point where it takes the maximum value 1. At A~.5 and Bs, respectively, the plate points cross into the negative force region and pass through the corresponding sequences A1.5, AG, and Av and Bs, B6, and By before they meet at C, where H = 0 and 5 + = --4. The behavior of the
=
+
-
D
or
KH =
2
g
,
[-.3.21
where g is defined by Eq. E2.10-] above. Equation [-3.1~ applies when 5 + > 0 and there is a turning point in the (¢, () curve between the plates, q~,, being the value of 4, at the turning point. Equation [-3.2-] applies when d4,//d( has the same sign everywhere between the plates. It must always be used when 5 + < 0 and also when 5 + > 0 but there is no turning point in the potential curve between the plates. The diagram will always show which fornmla to use. For example, in Fig. 4, Eq. [-3.11 applies when the plate points are B2 and A2.4, respectively, the value of qL,, = 0.96 being given by the intercept of the 5 + = ½ isodynamic slope curve with the q5 axis. On the other hand, when 5 + = ½ and the plate points are B4 and A2,4, respectively, there is clearly no potential turning point between the plates and Eq. E3.2~ applies. For the 1 1 electrolyte at large separations the "superposition" formula 5 + = 32 tanh (~¢P1~) tanh ( ~ , ) X exp(--KH)
[-3.3]
is a good approximation. When 5 + passes through the value zero the separation is
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
382
BELL AND PETERSON
1
- -2
2
""-,\ \
n
~a
FIG. 5. Plate interaction diagram for constant-chargedensity cases: the horizontal lines are "plate curves" and the others are isodynamic slope curves. given by KH = lln{tanh (}Ih)/tanh (}@2)}1. [-3.4] For other values of 5:+ expressions [-3.1] and [-3.2-] can, for the 1-1 electrolyte case, be transferred to elliptic integrals. Formulas for the various cases are given in the Appendix. A possible graphical technical for estimating plate separation would be to plot d~/d@, fl -I, and f2-1 against @ instead of d@/d}, fl, and f2, and the area under the segment of the d}/d@ curve between the plate points would give the plate separation. However, we do not regard this as a desirable alternative to numerical integration or the use of elliptic integral formulas, especially as d(/d@ becomes infinite at a potential minimum. Plate separations could also be obtained from potential-distance isodynamic curves if these were available in the required force range or from similar data in tabular form. It is unfortunate that the tables of de Bruyn and Devereux (10) are not in isodynamic form and thus the method described in the Appendix is, at the moment, the most easily utilized.
method; we shall choose that of constant charge. With the graphical method it is also easy to consider the "mixed" case where one plate is at constant potential and the other at constant charge and we shall give examples of this. With ~1 and a2 constant, the plate curves become lines parallel to the ¢ axis and can never meet except where al = - a 2 , when they are coincident. Except in the latter case, no solution of [-2.14] is thus possible and the force 5:+ and plate potentials @x and @2 all become infinite at H = 0. Figure 5 shows one example of constant charges of the same sign and one of constant charges of opposite sign: (a) @2~ = 2, @1~ = 1, f l =
2.3504, f 2 = 1.0422.
The plate points start at Ax and B1 on the appropriate ~+ = 0 curves when H = oo. At any value of H less than infinity they must lie on the same branch of an isodynamic slope curve. Hence as H decreases from infinity the plate points move into the positive force region and travel to the right along the L
I
I
I
(~2 ~2) 2
to
4. THE CONSTANT-CHARGE AND "MIXED" CASES The theory of dissimilar plates at constant potential (9, 10) or constant charge (14) has already been fully discussed. However, these cases afford simple illustrations of the graphical
J
-I
~
0~-~
1
1
I
2
I
3
FIC. 6. R e d u c e d f o r c e / p l a t e - s e p a r a t i o n c u r v e s f o r constant surface charge: The curves are labeled with t h e p l a t e p o t e n t i a l s a t infinite s e p a r a t i o n .
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
DISSIMILAR COLLOIDAL PLATE FORCES parallel f , and f2 lines, their distance from the 4~ = 0 axis becoming infinite as H ~ 0. The force-distance curve is shown in Fig. 6. (b)
q%~ = 2,
i
i
f.., = -- 1.0422
The plate points s t a r t at A1 and B~' when H = m. As H decreases from infinity the plate 2 point moves to the right and the plate 1 point to the left into the negative force region. The force has a m i n i m u m value when the plate 2 point is at B,/, where the plate 2 line is tangent to the 5 + = - - 0 . 5 4 3 1 isodynamic slope curve. The corresponding plate 1 point A2 gives a m i n i m u m value 1.8372 of ~1, which thereafter increases again as the plate 1 point moves back to A~. At A~ and B3', respectively, the plate points cross into the positive force region and the behavior is then similar to t h a t when the charges are of the same sign, q'l, ~2, and 5 + all increasing to infinity as H ~ 0. The force-distance curve is shown in Fig. 6. F o r c e - d i s t a n c e curves for cases where ~ = = - - ~ 2 ~ and ~,~ = ~2~, respectively, are also shown in Fig. 6. In the former case the plate points on the isodynamic slope diagram are s y m m e t r i c a l l y situated with respect to the dq)/tt~ axis and meet on the axis when H = 0. -
V
i
~ , , = --1,
f~ = 2.3504,
-
2
383
,d_~
,/P
3
~2
" /
2
r<
p2
, •
,,
',B 2
R
~
\
/ / /' /
t
/
-2
1
-3
KN
D
..........
1
I
2
I
3
FIG. 8. Reduced force/plate-separation curves for the mixed case with plate 1 at constant potential and plate 2 at constant charge: The curves are labeled with the plate potentials at infinite separation. I n the latter case the plate points are symmetrically situated with respect to the ¢ axis and move steadily toward infinite potentials as H--~ 0. We now consider the " m i x e d " case where one plate i s , a t constant potential and the other at constant charge. Since the plate curves are,~respectively, a vertical line and a horizontal line they intersect and hence the limiting value of 5 + as H ~ 0 is always finite in the mixed case. I n Fig. 7 plate 1 is at constant potential with ~ , = 2, while plate 2 is at constant charge. Three examples with different values of the constant charge density are shown :
0) %
qh > ~2~ > 0;
;2
~1 = 2,
~
= 1,
f,~ = 1.0422. FIG. 7. Plate interaction diagram for the "mixed" case: The vertical line is the plate curve for the constant-potential plate 1, the horizontal lines are plate curves for the constant-charge plate 2, and the others are isodynamic slope curves.
The plate points start at A~ and B~, respectively, when H = oo and meet at C~ u hen H = 0, where 5 + = 2.2191. The force 5:+ is always positive and passes through a m a x i m u m value u h e n the plate 1 point is at A2, where
Journal of Colloid and Interface ,~eience, V o l . 6 0 , N o . 2, J u n e
15, 1 9 7 7
384
BELL AND PETERSON
the plate line is tangent to the 5:+ = 2.7622 isodynamic slope curve. The plate 2 point is then at B2 on this curve and @2 has its maximum value of 2.1392. The force distance curve is shown in Fig. 8. (ii) @l > --@2~ > 0;
@1 = 2,
@2~ = --1,
f2 = --1.0422. The plate points start at A1 and BI', respectively, when H = ~ and meet at Cx' when H = 0. The force 5:+ is negative for large / / and decreases to a minimum when the plate 1 point is at As' and the plate 2 point at B2' where @2 = 0 and the 5:+ = - - 0 . 5 4 3 1 isodynamic slope curve is tangent to the plate 2 line. The force increases as H decreases further and the plate points then cross into the positive force region at A1 and B J, respectively. The force finally reaches a maximum value of 2.2191, at C1 when @1 = @2 = 2. The force distance curve is shown in Fig. 8. (iii) @ 2 ~ > @ 1 > 0 ; @1=2, @2~=3, f2=4.2586. The plate points start at A1 and D1, respectively, when H = oo and meet at E when H = 0. The force 5:+ is positive for large H and increases to a maximum value of 2.7622 when the plate 1 point is at As and the plate 2 point at D2 where @2has its maximum value of 3.2434. The force then decreases with H and the plate points cross into the negative force region at A3 and D1, respectively. The force finally reaches a minimum value of --6.3055 at E. The force distance curve is shown in Fig. 8. Figure 8 also includes force distance curves for cases where @1 = @2:¢ > O,
@1 = --@2~ > O.
--@2= > @ 1 > 0.
5. D I S C U S S I O N
Since both the constant-potential and the constant-charge assumptions give forces tend-
ing to infinity as the plate separation H tends to zero, it can be seen that neither assumption can be physically realistic at small plate separations. The mixed case gives a finite force at all separations but is hardly likely to apply to a wide range of phenomena. In the next paper of the series it is hoped to apply the methods developed in this one to situations where either all the surface ions or a secondary adsorbed layer of ions is in equilibrium with the bulk electrolyte. This means that in determining the charge-potential isotherm for each plate some of the factors discussed in the Introduction in connection with Eq. [1.1] must be taken into account. Conditions under which the potentials and force remain finite as the separation tends to zero can be obtained from considering the curves of the reduced charge density functions fl(~b) and f2(4~) defined above. Although the curves shown in this paper are all for the medium with symmetrical 1-1 electrolyte the methods developed can be used for any type of electrolyte. The expression on the right-hand side of Eq. [2.10], from which the isodynamic slope curves are obtained, is easy to evaluate for nonsymmetrical or mixed electrolytes. Equations [3.1] and [3.2] are applicable for any type of electrolyte. Although the integrals cannot, in general, be expressed in terms of standard functions they can always be evaluated by numerical integration. APPENDIX
We give a table for the calculation of the reduced separation KH with a 1-1 electrolyte medium. As may be seen from Fig. 3 (cases I and III), the boundary potentials @1 and @2 must have the same sign where the force 5:+ > 0. Again from Fig. 3 (cases I I and IV), the boundary slopes (d4~/d()l and (ddp/d()2 must have the same sign when 5:+ < 0. It should be noted that from Eq. E2.15] the ratio of the boundary slopes is related to that of the
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977
385
DISSIMILAR COLLOIDAL PLATE FORCES sin ai = [-k~ cosh ( ~ . / 2 ) ' ] - ' ,
reduced charge densities fx and f2 b y
sin 13i = [ c o s h (~i/2)-] -1,
fl../ f.2 = - - (d~b/d~)1// ((l~.../(t~)2.
T h e following definitions are necessary for the table, i t a k i n g v a l u e s of 1 and 2 in each case. if+
Plate conditions
~+ > 0
f,/f'z > 0
if+ > 0
/l/f2 < 0
- - 2 < if+ < 0
,1,,/'4,2 > 0
--2 < 3:+ < 0 if+ < --2 5:+ < --2
(I,,/~I,2 < 0 cI,,/~ > 0 q%/q~ < 0
5. 6.
7.
k~2 = (5 :+ + 2)//5 +.
I n the r i g h t - h a n d c o l u m n the s y m b o l F denotes an elliptic integral of the first kind. ~H
1. DERJAGUIN, B. V., AND LANDAU, L. O., Acta Physicochim. 14, 633 (1941). 2. VERWEY, E. J. W., AND OVERBEEK, J. TfI. G.,
4.
kt~2 = i -J- ½~+,
k~[2F(k~, 7r/2) -- F(k, cq) -- F(k, a2)] k~,[F(k~, a2) -- F(k~, a,)] F(ka, [%) -- F(k~, ~ ) 2F(k~, 7r/2) -- F(k~, ~2) -- F(k~, ~,) (I -- kv2)~[F(kv, "yj) -- F(k~, "Y2)] (l -- kv~)~[F(kv, ~1) + F(kv,'r2)]
REFERENCES
3.
sin 3'~ = t a n h (~./'2),
k~ -2 = 1 + ½~+,
"Theory of the Stability of Lyophobic Colloids." Elsevier, New York/Amsterdam, 1948. LEVINE, S., AND BELL, G. M., J. Colloid Sci. 17, 838 (1962). LEVINE, S., AND BELL, G. M., J. Phys. Chem. 67, 1408 (1963). BELL, G. M., AND LEVINt':, S., Trans. Faraday Soe. 53, 143 (1957). CHEN, C. S., AND LEVINE, S., J. Chem. Sot., Faraday I I 68, 1497 (1972). FRENS, G., Thesis, "The Reversibility of Irreversible Colloids." Utrecht, 1968.
8. BIERMAN,A., J. Colloid Sci. 10, 231 (1955). 9. DERJAGUIN, B. V., Discuss. Faraday Soc. 18, 85 (1954). 10. DEVEREUX, O. F., AND DE BRUYN, P. L., "Interaction of Plane Parallel Double Layers." M.I.T. Press, Cambridge, Mass., 1963. 11. HOGG, R., HEALY, T. W., AND FURSTENAU, D. W., Trans. Faraday Soc. 62, 1638 (1966). 12. WIESE, G. R., AND HEALY, D. W., Trans. Faraday Soc. 66, 1490 (1970). 13. PARSEGIAN, V. A., AND GINGELL, D., Biophys. J. 12, 1192 (1972). 14. BELL, G. M., AND PETERSON, G. C., J. Colloid Interface Sci. 41, 542 (1972). 15. GRFGORY, J., J. Chem. Soc. Faraday I I 69, 1723 (1973).
Journal of Colloid and Interface Science, Vol. 60, No. 2, June 15, 1977