A simple algorithm for the calculation of the electrostatic repulsion between identical charged surfaces in electrolyte

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A simple algorithm for the calculation of the electrostatic repulsion between identical charged surfaces in electrolyte
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   Simple lgorithm for the Calculation of the Electrostatic Repulsion between Identical Charged Surfaces in Electrolyte I. INTRODUCTION The colloid scientist is often required to calculate the electrical double-layer interaction between charged surfaces in order to compare theory and experiment. For example, recent experimental measurements of forces between charged surfaces in electrolyte (1) are of sufficiently high precision to require the most precise calculation of the electrostatic repulsion in a variety of electrolyte solutions and boundary conditions (e.g., constant surface charge or constant surface potential). The traditional calculation of the double-layer interaction is involved. It requires the analytic solution of the nonlinear Pois son- Boltzmann equation between parallel half spaces in terms of elliptic functions and the subsequent numerical solution of a complex transcendental equation. Not surprisingly, this has led to the extensive use of tables (2, 3) which are available for only selected values of the surface charge or potential and electrolyte concentrations and valences. This restriction means that in most practical cases, one must interpolate from the tables. In spite of the complexities and inconveniences, the tables remain in common use. This note attempts to present a fast numerical procedure which is capable of computing the electro- static interaction across symmetric electrolytes with high precision for any of the commonly encountered boundary conditions. II. THE METHOD For simplicity, we will discuss the numerical pro- cedure for a 1:1 electrolyte between identical, charged, plane-parallel interfaces. The planar Poisson-Boltz- mann equation for the scaled potential Y (e*/kT) in a 1:1 electrolyte solution of number concentration n is day sinh Y, [2.1] dX 2 where X (Kx) is the scaled distance measured from the midplane (Fig. 1) and 87me ~ K ~ = [2.2] ekT A first integration yields dY Q Sgn (Ym), [2.3] dX where we define the variable Q by Q = (2(cosh Y - cosh Ym)) 1j2, [2.4] where Ym is the scaled midplane potential. Equation [2.3] satisfies the zero-derivative boundary condition at X = 0 (Fig. 1). We note that dQ _ sinh Y dY Q Fle From Eqs. [2.3] and [2.5] we derive the differential equation dX_ Ym) -11-1'2. The midplane in terms of these variables is the point (Q = 0, X --- 0). If we know what value of Q (=Q~) corresponded to the surface of charge, then for a given value of the reduced midplane potential Ym, we can solve Eq. [2.6] from Q = 0 to Q = Qs by a suitable numerical technique (e.g., a fourth-order Runge- Kutta method (4)). Thus we determine Xs = r,L/2, the scaled distance from the midplane to the surface of charge corresponding to the given value of Ym. Repeating this procedure for a set of suitably chosen values of Ym (see below) will generate a set of corresponding KL/2 values. The electrostatic pressure at each value of L is simply calculated from the corresponding Ym value by P(L) = 2nkT(cosh Ym - 1). [2.7] The interaction free energy per unit surface area Ep(L) can be computed from Ep(L) = I~ P(L )dL [2.8] using a numerical quadrature formula (see below). The interaction free energy for spherical surfaces Esp(L) with radii al and a2 at separation L can be computed by a similar numerical quadrature directly from the pressure, Eq. [2.7], if the Derjaguin approximation is invoked: E~p(L) ~ a12~rata2+ 2 IL dL Ep(L ) - ax2~ra~a2+ 2 I] dL (L - L)P(L ). [2.9] 283 Journal of Colloid and Interface Science Vol. 77 No. 1 September 1980 0021-9797/80/090283 -03 02.00/0 Copyright © 1980 by Academic Press, Inc. All rights of reproduction n any form reserved.   84 NOTES 0 -Ys Q=Qs) -ym O= O) X' 2 FIG. 1. The potential profile between identical planar double layers. This result should be a good approximation for Kay, Ka~ ~> 10. There are some finer points to the calcula- tion of these integrals and the choice of Ym values to which we return in Section III. It remains to indicate how the surface Q value, viz., Q~, is determined. This is dependent on the type of boundary conditions invoked. i) Constant Surface Potential Y~ In this case, Q~ is calculated directly from definition [2.4] with Y = Ys, Q~ = [2(cosh Y~ - cosh Ym)] 1/~. [2.10] ii) Constant Surface Charge o- t the right-hand surface we have the boundary condition dY 4~re .~r. [2.11] dX ek T From Eq. [2.3], we see immediately that 47re Q~ = ~ Io-I. [2.12] iii) Surface Charge Regulation When the surface charge is determined by the adsorption of a potential-determining ion onto specific surface sites, neither charge nor potential remains constant during interaction (5-7). Both Eqs. [2.10] and [2.12] still hold but o- and Ys are not determined. To determine Q~ an extra equation is therefore necessary. Applying the mass-action principle to the surface adsorption process leads to another relation- ship between surface charge and surface potential. For example, for a surface weak-acid dissociation, we have (5) 1410 °. eN~ - ~-I = Kae-Y [2.13] Journal of Colloid and Interface Science, Vol. 77 No. 1 September 1980 where Ns is the surface density of acid groups and K~ is the dissociation constant. Equations [2.10], [2.12], and [2.13] are sufficient to determine Q~ (and c~ and Ys) for a given Ym- III. OPTIMIZATION OF NUMERICAL QUADRATURE ACCURACY In a constant potential interaction the scaled mid- plane potential varies smoothly from zero at X = ~ to Y~ at X = 0. Clearly, one must choose the Ym values from between these limits1; the smallest Y,~ value corresponding to the largest distance, Lmax say. The numerical quadrature required to obtain the interac- tion free energy can be performed only over the range L to Lma x. By ensuring that the smallest Ym value is such that KLmax >~ 3, we can perform the tail of the integration from Lmax to infinity analytically since the pressure P(L) can be assumed to decay exponentially for L ~ Lmax. If we decide on a total of N points (L, Ym) between (~,0) and (0,Y0, then the simplest choice of the Ym, viz., equally spaced values of Ym, will produce unequally spaced (L,P(L)) values. With this choice of Ym values, only a very unsophisticated quadrature formula (e.g., trapezoidal rule) can be used to evaluate the integrals in Eq. [2.8] or [2.9]; and, for accuracy, we would need to choose a very large value of N (~100). This can be avoided by choosing the value of Ym SO that the pressure values are equally spaced between zero and P~nax (given by Eq. [2.7] with Ym = Y0- After an integration by parts, we can rewrite Eqs. [2.8] and [2.9] as Po L Ep(L) = (L'(P') - L)dP' [3.1] and _ 7rala2 rD (L' (L'(P') - L)2dP '. [3.2] sp(L) aa -1- a2 )o Because the points have been chosen so that the P values are equally spaced, we can use a much more accurate quadrature rule (e.g., an Adams extrapolation formula (4)). This enables us to use much fewer points to achieve comparable accuracy. The method outlined above is easily generalized to the case of a general electrolyte solution containing mixtures of ions of different valence. Stern layer models can also be incorporated with some straight- forward modification of the surface boundary condition 1 For constant charge and regulated interactions the magnitude of the midplane potential Ym can increase indefinitely. However, by the time Ym has increased to Ys(o~) (the value of the surface potential at infinite separation) we have KL ~ 1. Thus Ys(~) provides a suitable scale on which to choose Ym values for these types of interaction.  NOTES 285 which determines Qs. As a computational algorithm, the method possesses the advantages of speed, accuracy, and simplicity. A Fortran subroutine is available from the authors. REFERENCES 1. Israelachvili, J. N., and Adams, G. E. J. Chem. Soc. Faraday Trans. 1 74, 975 (1978). 2. Honig, E. P., and Mul, P. M.,J. Colloidlnterface Sci. 36, 258 (1971). 3. Devereux, O. F., and de Bruyn, P. L., Interaction of Plane Parallel Double Layers. M.I.T. Pres s, Cambridge, 1963. 4. Abramowitz, M., and Stegun, I. A., Handbook of Mathematical Functions. Dover, New York, 1968. 5. Ninham, B. W., and Parsegian, V. A., J. Theor. Biol. 31,405 (1971). 6. Chan, D. Y. C., Healy, T. W., Perram, J. W., and White, L. R. J. Chem. Soc. Faraday Trans. 1 70, 1046 (1975). 7. Healy, T. W., and White, L. R., Advan. Colloid Interface Sci. 9, 303 (1978). DEREK Y. C. CHAN RICHARD M. PASHLEY LEE R. WHITE Department of Applied Mathematics Research School of Physical Sciences Australian National University Canberra Australian Capital Territory 2600 Australia Received November 26 1979; accepted January 17 1980 Journal of Colloid and Interface Science Vol. 77, No. 1, September 1980
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