J . Phys. Chem. 1989, 93, 3761-3768
3761
The Spherical Double Layer: A Hypernetted Chain Mean Spherical Approximation Calculation for a Model Spherical Colloid Particle Enrique GonzClez-Tovar and Marcel0 Lozada-Cassou* Departamento de Fhica, Universidad Autdnoma Metropolitana- Iztapalapa, Apartado Postal 55-534, 09340 D.F., MCxico (Received: April 26, 1988; I n Final Form: September 6. 1988)
The structure of an electrolyte around a spherical, charged colloid particle is studied, using a simple model in which the diameter of the ions in the soution is considered. The hypernetted chain mean spherical approximation (HNC/MSA) equation is derived and solved numerically. As a result the ionic distribution around the colloid particle is obtained. Calculations are made for 1:l and 2:2 electrolytes for various values of the concentration, colloid radius, and electrical charge. By use of these ionic distributions, excess charge adsorption isotherms and zeta potentials are calculated. These quantities are compared with the nonlinear Poisson-Boltzmann (PB) results. Important quantitative and qualitative differences between the PB and HNC/MSA are found. The zeta potential is found to be a nonmonotonic function of various parameters, for example, the charge density. Qualitative agreement with experimental results for the zeta potential is found.
I. Introduction In the study of systems such as protein solutions1 and colloidal dispersions2 the understanding of the interface formed by the colloid or protein particle with the surrounding liquid is very important. When proteins or lyophobic colloid particles come in contact with some polar solvent, they become charged because of the removal of the counterions by the solvent. Often a salt is also present in this type of system. Because the protein molecules or colloid particles become charged when they are in contact with an electrolyte, the ions in the solution will be attracted to (counterions) or repelled from (co-ions) them. The so-called electrical double layer (EDL) is defined as the structure of the electrolyte around the macroparticles in solution. If the EDL is known, the electrostatic free energy and the zeta potential of the macroparticle in solution can be calculated. The electrostatic free energy can be used to study the titration of proteins,] and the zeta potential is the main quantity to be obtained in electrophoresis experiment^.^-^ On the other hand, the interaction between two macroparticles is determined by the interaction of their EDLs. Different methods have been used to calculate this interaction f ~ r c e . ~ -Although ’~
( I ) See, for example, Tanford, C. Physical Chemistry of Macromolecules; Wiley: New York, 1961. (2) See, for example, Vold, R. D.; Vold, M. J . Colloid and Interface Chemistry; Addison-Wesley: Menlo Park, CA, 1983. (3) Haydon, D. A. Proc. R . SOC.London, A 1960, A258,319. Carroll, B. J.; Haydon, D. A . J. Chem. SOC.,Faraday Trans. I1975, 71, 361. (4) Gonzilez-Tovar, E.; Lozada-Cassou, M.; Henderson, D. J . Chem. Phys. 1985, 83, 361. (5) Verwey, E. J.; Overbeek, J. Th. G.Theory of Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (6) Patey, G. N . J . Chem. Phys. 1980, 72, 5763. (7) Medina-Noyola, M.; McQuarrie, D. A. J . Chem. Phys. 1980, 73,6279. Medina-Noyola, M. J . Chem. Phys. 1982, 77, 1428. (8) Lozada-Cassou, M. J. Chem. Phys. 1984,80, 3344. Lozada-Cassou, M.; Henderson, D. Chem. Phys. Lett. 1986, 127, 392. Lozada-Cassou, M. KINAM 1987, 8, Series A, 107. (9) Hayter, J . B.; Penfold, J. Mol. Phys. 1981, 42, 109. ( I O ) Alexander, S.; Chaiken, P. M.; Grant, P.; Morales, G. J.; Pincus, P.; Hong, D. J. Chem. Phys. 1984.80, 5116. ( 1 1 ) Belloni, L. J . Chem. Phys. 1986, 85, 519. (12) Grimson, M. J.; Rickayzen, G. Mol. Phys. 1981,44, 817; 1982,45, 221. Gonzalez, D. J.; Grimson, M. J.; Silbert, M. Mol. Phys. 1985, 54, 1047. (13) Tarazona, P.; Vicente, L. Mol. Phys. 1985,56,557. Tarazona, P.T.; Bettolo-Marconi, U. M.; Evans, R. Mol. Phys. 1987, 60, 573. (14) Khan, S.; Morton, T. L.; Ronis, D. Phys. Rev. A 1987, 35, 4295. Khan, S.;Ronis, D. Mol. Phys. 1987, 60, 637. Ronis, D. J. Chem. Phys. 1984, 81, 2749. ( 1 5 ) Sogami, I. Proceedings of the 19th Yamada Conference; Ise, N.; Sogami, I., Eds.; World Scientific: Kyoto, Japan, 1987; p 624.
0022-3654/89/2093-3761$01 S O / O
some of these methods have been criticized by Teubner,I9 and one of they seem to be reasonably successful when compared with experimental results.20s2’ In any case, whether the force between two macroparticles is calculated through a Yukawa potential (where a renormalized macroparticle charge density is u~ed),”’9~~ density functional^,'^'^ the solution of the three species Ornstein-Zernike equation (where the macroparticles are or are not at infinite d i l ~ t i o n ) , ~ , ~or, ~three-particle ~ J ~ s ~ ~ distribution functions,8.l6the structure of the single EDL plays a very important role in the calculation of the force itself and/or as the limit of infinite separation of the interacting EDLs. Depending on the shape of the macroparticle, the EDL is called the planar, cylindrical, spherical, ellipsoidal, etc., electrical double layer. The spherical electrical double layer is particularly important since there is a large number of proteins’ and colloidal particles2 that have a nearly spherical shape, and there is a large number of reported experiments for this type of system. This paper is about the EDL of spherical macroparticles. In the past, the spherical double layer has been calculated through the Poisson-Boltzmann5 (PB) equation. In this theory the ions are taken to be point ions, the solvent is a uniform dielectric medium, and the macroparticle is a uniformly charged, hard sphere, composed of a material with the same dielectric constant, t, as that of the solvent (hence image forces need not be considered). In addition, it is assumed that the macroparticle concentration is sufficiently low so that interactions between them can be neglected. Needless to say that this is a very simple model to represent any real system. However, improvements to the model such as a discrete charge or a different dielectric constant for the macroparticle might not be that relevant in the calculation of the EDL.4 On the other hand, solvent structure23and ionic size4~2e2g (16) Lozada-Cassou, M.; Diaz-Herrera, E. Proceedings of the 19th Yamada Conference; Ise, N.; Sogami, I., Eds.; World Scientific: Kyoto, Japan, 1987; p 555. (17) Henderson, D.; Lozada-Cassou, M. J . Colloid Interface Sci. 1986, 114, 180. (18) Sheu, E. Y.; Wu, C.-F.; Chen, S.-H. Phys. Reu. A 1985, 32, 3807. Wu, C.-F.;Sheu, E. Y.; Bendedouch, D.; Chen, S.-H. KINAM 1987,8, Series A, 37. (19) Teubner, M . J. Chem. Phys. 1981, 75, 1907. (20) Schaefer, D. W. J . Chem. Phys. 1977.66, 3980. (21) Chen, S.-H. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M., Ed.; Elsevier: New York, 1985. Chao, Y. S.; Sheu, E. Y.; Chen, S.-H. J . Phys. Chem. 1985, 89, 4862. (22) Beresford-Smith, B.; Chan, D. Y . C. Faraday Discuss. Chem. SOC. 1983, 76, 65; Chem. Phys. Lett. 1982, 92, 474. (23) Carnie, S. L.; Chan, D. Y . C. J . Chem. Phys. 1980, 73,2349. Blum, L.; Henderson, D. J . Chem. Phys. 1981, 74, 1902. Blum, L.; Henderson, D.; Parsons, R. J. Electroanal. Chem. 1984, 161, 389. (24) Henderson, D.; Blum, L.; Smith, W. R. Chem. Phys. Lett. 1979, 63, 381. Henderson, D.; Blum, L. J . Electroanal. Chem. 1980, 111, 217.
0 1989 American Chemical Society
3762 The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 effects seem to be more relevant. Here, the ionic size will be taken into account and the solvent structure will be left for some future publication. In any case, the model proposed above is too simple to represent accurately any real system. Therefore we do not expect to achieve quantitative agreement with experimental data. Nevertheless, we will show that some experimental results can be qualitatively predicted with this simple model. It would be convenient, however, to have Monte Carlo data for this model to test the different theories that can be used to solve this model. Unfortunately, no Monte Carlo data are yet available for this system. Several approximations for the direct correlation function as a closure to the Ornstein-Zernike (OZ) equation have been considered for model electrolyte^,^^^^ model planar interfaces,2e2s and cylindrical polyelectrolytes.4~29~33~34 One of the best of these approximations is the hypernetted chain (HNC). This is the approximation used here. The cylindrical EDL has also been studied successfully through use of modified Poisson-Boltzmann (MPB) t h e o r i e ~ . ~ ~ , ) ~ Two versions of the H N C equations are obtained, the H N C / H N C and the HNC/MSA. In both versions the macroparticle-ion correlations are treated in the H N C approximation, but in the H N C / H N C and the HNC/MSA approximations the ion-ion correlations are treated in the H N C approximation or mean spherical approximation (MSA), respectively. Although the H N C / H N C approximation is the more consistent procedure, the HNC/MSA approximation is usually more successful both for the low concentration e l e c t r ~ l y t and e ~ ~for the planar EDL.2"26 Vertenstein and Ronis3' have given an interesting explanation to this fact for the planar EDL system. However, it is not clear that the same explanation is valid for the pure electrolyte case. Since for whatever reason the fact is that the HNC/MSA theory is better than the H N C / H N C theory, we use the HNC/MSA equations here. This study complements that of Patey6 and Bratko et al.,)* who used the HNC/HNC equation to study the spherical EDL. By use of numerically calculated ionic profiles, various properties, such as the mean electrostatic potential (MEP) and the excess adsorption isotherms, can be evaluated and the effect of various parameters (Le. macroparticle radius and charge, ionic charge and concentration) studied. In particular, the MEP at the distance of closest approach, which can be identified with the zeta potential, 1,of electrophoresis experiment^,^^ is studied. This paper is outlined as follows. In section 11, the HNC/MSA equations for the spherical EDL are derivated and the numerical procedure is outlined. In section 111, numerical results for the (25) Carnie, S. L.; Chan, D. Y. C.; Mitchell, D. J.; Ninham, B. W. J . Chem. Phys. 1981, 74, 1472. (26) Lozada-Cassou, M.; Saavedra-Barrera, R.; Henderson, D. J. Chem. Phys. 1982, 77, 5150. Lozada-Cassou, M.; Henderson, D. J. Phys. Chem. 1983, 87, 2821. Mier y T e r h , L.; Diaz-Herrera, E,; Lozada-Cassou, M.; Henderson, D. J. Phys. Chem., in press. (27) Croxton, T. L.; McQuarrie, D. A. Chem. Phys. Lett. 1979, 68,489. Mol. Phys. 1981, 42, 141. Henderson, D.; Blum, L.; Bhuiyan, L. B. Mol. Phys. 1981, 43, 1185. (28) Bhuiyan, L. B.; Outhwaite, C. W.; Levine, S. Mol. Phys. 1981, 42, 1271. Outhwaite, C. W.; Bhuiyan, L. B. J. Chem. SOC.,Faraday Trans. II 1983,79,707. Outhwaite, C. W. J. Chem. Soc., Faraday Trans. 11 1983.79, 1315. (29) Lozada-Cassou, M. J . Phys. Chem. 1983,87, 3279. (30) Rasaiah, J. C.; Friedman, H. L. J. Chem. Phys. 1968.48, 2742; 1969, 50, 3965. Rasaiah, J. C.; Card, D. N.; Valleau, J. P. J. Chem. Phys. 1972, 56, 248. (31) Waisman, E.; Lebowitz, J. L. J. Chem. Phys. 1972, 56, 3093. (32) Henderson, D.; Lozada-Cassou, M. J. Chem. Phys. 1983, 79,3055. (33) Bacquet, R.; Rossky, P. J. J. Phys. Chem. 1984, 88. 2660. (34) Vlachy, V.; McQuarrie, D. A. J. Chem. Phys. 1985, 83, 1927. Vlachy, V.; Haymet, A. D. J. J. Chem. Phys. 1986,84, 5874. (35) Bratko, D.; Vlachy, V. Chem. Phys. Lett. 1982,90,434; 1985, 215, 294. (36) Outhwaite, C. W. J. Chem. Soc., Faraday Trans. I I 1986, 82, 789. (37) Vertenstein, M.; Ronis, D. J. Chem. Phys. 1987, 87, 4132. (38) Bratko, D.; Friedman, H. L.; Zhong, E. C. J. Chem. Phys. 1986,85, 377. (39) Lyklema, J.; Overbeek, J. Th. G . J. Colloid Sci. 1961, 16, 501. Lyklema, J. J. Colloid Interface Sci. 1977, 58, 242.
GonzBlez-Tovar and Lozada-Cassou
r l
/
I
I
I I I I
---;--.. -! I
>
Figure 1. Schematic representation of the model.
ionic density profiles, MEP, zeta potential, and excess adsorption isotherms are presented. Conclusions are given in section IV. 11. Theory and Numerical Procedure The HNC equations for a multicomponent system of M species are given by
iJ = 1 , 2, ...) M where kIJ(rl2)and gI,(rl2) (=hlJ(rl2) + 1) are the total correlation function and the radial distribution function, respectively, for particles 1 and 2 of species i and j , clJ(r32) is the direct correlation function, pr is the bulk number concentration of species 1, U,j(r12) is the direct interaction potential, and /3 = l / k T , where k is Boltzmann's constant and T is the absolute temperature. Let the species M correspond to hard, charged, spheres of radius R at infinite dilution, and let us assume that we have n ionic species of equal diameter, a, then eq 1 becomes gM,(r) = exp(-WM,(r) + I=~ 1P I S ~ M ~ ( ~ )dV31 C & ) (2) j = 1, 2, ..., n
where r rI2,t rI3,and s r23 are the radial distances between particles 1 and 2, 1 and 3, and 2 and 3, respectively, dV3 = t2 sin 0 d p d0 dt is the volume element in spherical coordinates, and M n + 1. The geometry is shown in Figure 1. Assuming the restricted primitive model to represent the electrolyte of n ionic components, the ion-ion direct correlation function cI,(s) is given by the following MSA expression:
cI,(s) = c,(s) for F
+ zrz,cS'(s) - @e2zrz,/cs
(3a)
> 0, with C,(S)
= -C1 - 6 7 ~ -2 f/zT)C3S3 ~
for s 6 a and c,(s) = cdsr(s) = 0 for s > a, where n
P = T)
CPr I= I
= aa3p/6
+ 27)2/(1 - ?7)4 c2 = -( 1 + ~ / 2 ) ~ /1a-( ~ c, = (1
c3 = cl/a3
7 ) ~
(3b)
The Spherical Double Layer
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3763 The function K,(r,t) in A(r) can be integrated to obtain A(r) =
From eq 2 and 3a we obtain
2qc2(a3-
r
r3)
+ -(aS w3
- rS)
10
1
+
T3[
such that TI = j = 1, 2,
(R
+ a/2)'
- (r
- a)'
1
..., n
(17)
Equation 8 is the HNC/MSA equation for the spherical double layer, from which the ionic density profiles around a macroparticle can be calculated. From the ionic density profiles, the MEP as a function of distance to the macroparticle, +(r), and the charge density on the macroparticle's surface, u, can be calculated through the expressions
where we have used the definitions n
and n
PMdMd(t) = CZ/P/hM/(t) I= I
(6b)
By use of the geometrical relation
s2 = r2
+ t 2 - 2rt cos 6
(7)
and
the integrals over 6 and cp, indicated in eq 5 , can be easily done and we get ghfJ(r) = exp(-puhfJ(r) + L:(ai2fMS(t)K'(I,f) dt + zJSd+c.,2fMdd(t)Kd(r.f) d t
+ J' L:(a/2fhfd(t)fi(r,t)
dt +
respectively. Equation 8 can be solved for two different, although equivalent, given boundary conditions on the macroparticle: constant surface +(R). In charge density, u, or constant surface potential, the former, the unscreened macroparticle-ion potential, UMj(r), is given by
j = 1, ..., n
UMj(r) =
where
":""(:)
--
(20)
Thus, eq 8 becomes (9)
for (r - tl
< a, and K,(r,t)
= Kd(r,t) = 0 for Ir - t( > a, and
On the other hand, if +o is given, using eq 18 and 19 and after some elementary algebraic manipulation, eq 2 1 becomes I
for R
+ a / 2 < r < R + 3a/2. 1 J, = -(a/+2
(I
+ 2)rt
In eq 10 and 11 - Ir - t1'+2)
Note also that to get eq 8 we have used the fact that the bulk electroneutrality condition is given by
f2(r,t) =
n
CPrZ, = 0
I= I
and that for r
\
\ 0
: 5
2 5
,
-E
\
0 5
I
10
\
I
3 0
sphere radii. All the curves are for a sphere surface charge density of 0.3031 C/m2. The solid and broken curves give the HNC/MSA and nonlinear PB results, respectively. The curves are plotted by using a nonlinear sinh-l scale.
\
,2oP\
2 5
Figure 11. Reduced density profiles for a 0.5 M model divalent electrolyte ( a = 4.25 A, z+ = 1z-I = 2, T = 298 K, t = 78.5) for various
A\
,
2 0
r /a
60
-I
15
10
30
1 P
I
4 5
i 6 5
r /a
Figure 10. Potential profiles for a 1 M model monovalent electrolyte ( a
= 4.25 A, I+ = 12-1 = 1, T = 298 K, t = 78.5) for various sphere radii. All the curves are for a sphere surface charge density of 0.220 C/m2. The solid and broken curves give the HNC/MSA and nonlinear PB results, respectively. The curves are plotted by using a nonlinear sinh-' scale.
as R decreases. This is in contradiction with the results shown in Figures 2, 3, 5, and 6. However, this is just a feature of the 0, as it should. definition of qm. The quantity q,,,R2 0 as R The adsorption around the spherical macroparticle is in general lower than that for the cylindrical macroparticle and both are lower than that for the planar EDL. This, as for the case of Figures 5 and 6, can be understood in terms of the intensity of the electrical field produced by these three geometries for a given surface charge density, which is higher for the plate and lower for the sphere. The HNC/MSA and PB density and MEP profiles of the ions around the spherical macroparticle are shown in Figures 9 and 10, respectively, for three values of R. The calculations were made for a 1 M electrolyte concentration and a surface charge density of 0.220 C/m2. The differences between the H N C / M S A and PB theories are more important for the larger values of R (notice that the abcissa uses a sinh-' scale). The valence, zM, of a macroparticle of R = l A and u = 0.220 C/m2 is 0.17, Le., only a fraction of the charge of an electron. Although clearly this does not correspond to a real situation, it does show the behavior of the density and MEP profiles with R. On the other hand, we did calculations, not shown, for R = a / 2 and zM = 1 ( 0 = 0.2823 C/mz) and zy = 2 (u = 0.5646 C/m2). The density profiles and
- -
0
- 20 0 5
15
2 5
3 5
r/a
Figure 12. Potential profiles for a 0.5 M model divalent electrolyte ( a = 4.25 A, z+ = (z-(= 2, T = 298 K, e = 78.5) for various sphere radii.
All the curves are for a sphere surface charge density of 0.3031 C/m2. The solid and broken curves give the HNC/MSA and nonlinear PB results, respectively. The curves are plotted by using a nonlinear sinh-' scale. other calculated quantities reduced to the Henderson et HNC/MSA results for bulk electrolytes, which have been shown to be in fairly good agreement with M C data.48*49 The HNC/MSA and PB density and MEP profiles for a 2:2 salt are shown in Figures 11 and 12, respectively, for three values of R. The agreement between the HNC/MSA and PB results is worse than for the 1:l salt. The charge oscillation seen in the HNC/MSA curves for the higher values of R is of interest as is the crossover of the HNC/MSA curves for different values of R.
IV. Conclusions The HNC/MSA equations for the structure of the double layer around a charged sphere were derived and numerically solved. As a corollary, the integral version of the nonlinear PB theory is obtained by taking the limit of point ions in the HNC/MSA equations. (47) Henderson, D.; Lozada-Cassou, M.; Blum,L. J . Chem. Phys. 1983, 79, 3055. (48) Card, D. N.; Valleau, J. P. J . Chem. Phys. 1970,52, 6232. Valleau, J. P.; Cohen, L. K. J . Chem. Phys. 1980, 72, 5935. Valleau, J. P.; Cohen, L. K.; Card, D.J . Chem. Phys. 1980, 72. 5942. (49)Van Megen, W.; Snook, I. K. Mol. Phys. 1980, 39, 1043.
3768
J . Phys. Chem. 1989, 93, 3168-3170
found, for a similar set of parameters, in the cylindrical or planar For R 2 IO3 A, the H N C / M S A planar EDL results are obEDLs. tained.26 The HNC/MSA equation for planar geometry has been shown to be in agreement with MC data for this model s y ~ t e m . ~ ~ ~ In ~ ~spite of the simplicity of the model proposed, satisfactory qualitative agreement is obtained between the calculated poFor R = a / 2 , the HNC/MSA results for a bulk electrolyte are tential and experimental results for this quantity. obtained.47 The HNC/MSA equation for bulk electrolyte has Finally, we wish to point out that if the surface charge density been shown to be in good agreement with MC data for this on the macroparticle is continuously increased, the concentration system.47@Therefore, although no MC data exist for the spherical of counterions around the macroparticle also increases continuously double layer, the fact that our numerical results do agree with and therefore the EDL becomes continuously narrower. Thus, MC data for these two limits, gives us some confidence in the rest we have detected no singularity or discontinuity. We think that of our results. the so-called “condensation” of ions around macroparticles, which For the planar, EDL, the HNC/MSA equation starts to break has been experimentally observed, is the increase in the ionic down at u 2 0.3 C/m2. However, if R = a / 2 , the HNC/MSA equation has been shown to agree with the MC data for u 0.6 concentration next to the macroparticle due to the stronger electric field the ions feel when the surface charge is increased, but no C/m2.47,48Therefore, we expect the HNC/MSA equation for singularity or other dramatic effect is involved. Similar conclusions the spherical EDL, in general, to give correct results for u > 0.3 have been reached by Gueron and Weisbuch for the PB theory C/m2. The smaller the central particle radius, the higher we can in cylindrical g e ~ m e t r y . ~ ’ go in the surface charge density. The agreement between the HNC/MSA and PB theories for A c k n o w l e d g m e n t . We gratefully acknowledge the support of the spherical EDL is found to be good for low ionic concentration CONACYT, MExico. and surface charge density and is in general better than the one
