J . Phys. Chem. 1986, 90, 3248-3250
3248
Ion Dlstributions in a Cylindrical Capillary V. Vlachyt Department of Chemistry, University of California at Berkeley, Berkeley, California 94720
and D. A. McQuarrie* Department of Chemistry, University of California at Davis, Davis, California 9561 6 (Received: November 18, 1985; In Final Form: February 11, 1986)
The Poisson-Boltzmann equation corrected for the ion-ion correlations (HNC/MSA) has been applied to the model of the uniformly charged cylindrical cavity containing electrolyte solution. The results suggest, in agreement with previous studies of the electrical double layer, that the uncorrected Poisson-Boltzmann equation may yield unreliable conclusions in the case of high surface charge and/or for divalent counterions present.
Introduction The model of the cylindrical capillary with a charge uniformly distributed on the inner surface and immersed in an electrolyte solution has been applied to explain electrokinetic phenomena,'+ to describe. ion selectivity in certain types of ion-exchange resins,I0 and to obtain the mean activity coefficient of electrolyte within the capillary." In all these studies, the mean electrostatic potential has been evaluated either by the Debye-Huckel approximation'*l2 or by the solution of the nonlinear Poisson-Boltzmann equation by numerical or by some approximate methods.l3-I5 The Torrie-Valleau simulation studies of the planar electrical double-layer indicateI6J7 that the Poisson-Boltzmann equation is a useful approximation only at low to moderate ionic strengths of added electrolyte and only if the charge density on the planar surface is not too high. The Poisson-Boltzmann approach treats the ions as uncorrelated charged points embedded in a dielectric continuum. However, the influence of ion-ion correlations may be important and deviations from the Poisson-Boltzmann mean-field theory are expected. This may modify the conclusions whenever the Poisson-Boltzmann theory is used to extract the information about the charge (or the potential) on a capillary from the experimental data. It is the aim of this paper to examine the validity of the Poisson-Boltzmann approximation for the model of the cylindrical cavity with a uniformly charged inner surface, immersed in an electrolyte solution. The corrections due to ion-ion correlations are calculated within the MSA approximation.Is This approximation has been successfully applied to studies of an electrolyte solution in contact with a charged planar and recently also to solutions of cylindrical polyelectrolyte^.^^*^^
In this expression, the superscript R denotes the reference solution (a = 0), where all external fields vanish. Further, = l/kT, where k is the Boltzmann constant and T the absolute temperature. A useful result may be obtained from eq 1 only if ~ ( ~ ) ( 7 , , 7 ~ ; a p ) can be well approximated in the region where p ( 7 ) varies. One approach19*21,23*25 is to simply ignore the a dependence in eq 1 and to use the mean spherical approximationla for the direct correlation function of the solution of charged particles
(1) Oldham, I. B.; Young, F. J.; Osterle, J. F. J. Colloid Sei. 1963, 18, 328. (2) Dressner, L. J . Phys. Chem. 1963, 67, 1635. (3) Rice, C. L.; Whitehead, R. R. J . Phys. Chem. 1965,69, 4017. (4) Morrison, F. A.; Osterle, J. F. J . Chem. Phys. 1965, 43, 21 11. ( 5 ) Jacazio, G.; Probstein, R. F.; Sonin, A. A.; Yung, D. J . Phys. Chem. 1972, 76, 4015. (6) Levine, S.; Mariott, J. R.; Neale, G.; Epstein, N. J . Colloid Interface Sei. 1975, 52, 136. (7) Olivares, W.; Croxton, T.; McQuarrie, D. A. J . Phys. Chem. 1980,84, 867. ( 8 ) Olivares, W.; McQuarrie, D. A. J . Phys. Chem. 1985, 89, 2966. (9) Christoforou, C. C.; Westermann-Clark, G. B.; Anderson, J. L. J . Colloid Interface Sei. 1985, 106, 1. The Model and Method (10) Dolar, D.; Vlachy, V. Vestn. Slou. Kem. Drus. 1981, 28, 327. The model is the same as that used several times before.6~8~10~11 (1 1) Dressner, L. J. Phys. Chem. 1963, 67, 2333. The ions are modeled as charged hard spheres with a distance (12) Booth, F. J . Chem. Phys. 1951, 19, 821. of closest approach a and are distributed inside a cylinder of radius (13) Sigal, V. L.; Ginsburg, Yu. Ye. J . Phys. Chem. 1981, 85, 3730. R a / 2 . The solvent is treated as a uniform dielectric continuum (14) MacGillivray, A. D.; Swift, J. D. J . Phys. Chem. 1968, 72, 3573. whose dielectric constant ereo is taken to be equal to that of pure (15) Martynov, G. A,; Avdeev, S. M. Kolloidn. Zh. 1982, 44, 702. water. A fixed charge e is distributed uniformly over the inner (16) Torrie, G. M.; Valleau, J. P. J . Chem. Phys. 1980, 73, 5807. surface of the capillary, with a charge density u = e / 2 a ( R (17) Torrie, G. M.; Valleau, J. P. J . Phys. Chem. 1982, 86, 3251. a/2)h,where h is the length of the capillary. The internal solution (18) Waisman, E.; Lebowitz, J. L. J . Chem. Phys. 1972, 56, 3086. is assumed to be in contact with some external reservoir of (19) Carnie, S. L.; Chan, D. Y . C.; Mitchell, D. J.; Ninham, B. W. J . Chem. Phys. 1981, 74, 1472. electrolyte which maintains a constant chemical potential. Finally, (20) Lozada-Cassou, M.; Saavedra-Barrera, R.; Henderson, D. J. Chem. the capillary is assumed to be very long ( h >> R ) so that end Phys. 1982, 77, 5150. effects are unimportant. (21) Grimson, M.; Rickayzen, G. Mol. Phys. 1982, 45, 221. The equation which relates the singlet density p ( 7 ) to the (22) Gonzalez-Tovar, E.; Lozada-Cassou, M.; Henderson, D. J . Chem. cup) and to the external two-particle correlation function ~(~)(7~,7~; Phys. 1985, 83, 361. field v(7) for a one-component system is24 (23) Vlachy, V.; McQuarrie, D. A. J . Chem. Phys. 1985,83, 1927. (24) Saam, W. F.; Ebner, C. Phys. Reu. A 1977, 15, 2566. 'On leave from the Department of Chemistry, E. Kardelj University, 61000 (25) Fixman, M. J . Chem. Phys. 1979, 70, 4995. Ljubljana, Yugoslavia.
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0022-3654/86/2090-3248$01 S O / O
0 1986 American Chemical Society
Ion Distributions in a Cylindrical Capillary
The Journal of Physical Chemistry, Vol. 90, No. 14, 1986 3249
In eq 2 ei and ej denote the charges of the ions, both of diameter a, with their centers a distance r12apart. This approach, which assumes that the correlation between two particles depends only on their separation and not on the position relative to the charged surface, has been successfully applied in several previous studi e ~ . ’ ~ -A~ generalization l of eq 1 together with eq 2 to a multicomponent systemZ3J5is
10.0 C
1,o
10‘
counterions o /
where w = 4?ra3/3 is a domain of integration. Equation 3 reduces to the Poisson-Boltzmann equation for zero ionic diameter. An alternative derivation of eq 3 for a cylindrical polyelectrolyte solution has been presented previously by Lozada-Cassou.26 The mean electrostatic potential, +(q,relative to the reference solution (+R = 0), is given by Poisson’s equation (4) where x denotes the radial distance from the cylinder axis. An alternative path to eq 3 is to make a functional Taylor expansion in powers of [ p ( q - pR] of the difference in single-particle direct correlation functions, c(7) - $ (this is equal to the integral term on the right-hand side of eq 1; see also eq 6 of ref 23), ignore all but the linear term in this expansion, and use the mean spherical ,7~) approximation for ~ ( ~ ) ( 7 ~again. Equations 3 and 4 need to be solved simultaneously subject to the following boundary conditions
In this study the reference solution is the external electrolyte is taken at the solution with concentration po and c(Z)(r12;pR) concentration, pR N p0.”23*2s The numerical solution requires an integration of eq 3 within the spherical domain w over a cylindrical coordinate. Considering that a
