Electrical Force Between Two Permeable Planar Charged Surfaces in an Electrolyte Solution BY ANTONY J. DUNNING-/-, JAMES MINGINS,* BRIAN A. PETHICA~ AND PETER RICHMOND Unilever Research, Port Sunlight, Wirral, Merseyside L62 4XN Received 8th November, 1977 Numerical calculations of the pressure and electrical contribution to the Helmholtz interaction free energy for two uniformly charged sheets interacting across 1 : 1 electrolyte solution are presented. The model differs from that considered by most other workers in that the electrolyte is allowed to penetrate the charged surfaces. Further, in addition to the cases of constant charge and constant potential, the case where the surfaces contain reversibly ionised groups is considered. In order that the results may have some relevance to cell-membrane interactions, all the numerical calculations are done using parameters which represent approximately the conditions in physiological saline solution.The results for constant potential are identical to those obtained for impenetrable interacting charged sheets. The results for constant charge differ in an essential way. Specifically, the pressure in the limit of zero separation tends to a constant value whereas for impermeable surfaces it is, for small separations, inversely proportional to the square of the separation and thus diverges as the separation approaches zero. Consequently, the difference between constant potential and constant charge for this model is not as marked as differences obtained by other workers who studied impermeable surfaces.As one might anticipate, the results for reversibly ionised surfaces lie between the two extremes of constant charge and constant potential. Present calculations differ from those of Gingell in that the complete Poisson-Boltzmann equation is solved numerically. The electrical double layer force between impenetrable charged surfaces, of interest to colloid scientists, has been thoroughly studied in recent years. Constant wall charge and constant potential as well as the case where the surfaces reversibly ionise as the surfaces approach have all been considered using smeared potential and charge m~dels.l-~ Extension of the constant charge case to systems with discrete surface charges has been made by Richmond and a general thermodynamic treatment of interacting semi-infinite surfaces in equilibrium with any number of adsorbing species has been given by Hall.' The case where electrolyte may penetrate the surfaces, which is of considerable relevance for membrane biologists, has not been studied in such detail.To the best of our knowledge, the " two-sided " double layer was first treated by Verwey and Niessen in the context ofthe polar oillwater interface and later applied to semi-conductor systems such as the silver iodidelwater interface. ' 9 lo Using the linearised Debye-Hiickel approximation the electrical potential for partially penetrable surfaces or " fuzz " layers at an oil/electrolyte solution interface has been calculated by Gingell l1 who, in collaboration with Parsegian,12 later derived an analytic expression for the pressure between two completely penetrable charged sheets, again using the linearised Debye-Hiickel approximation.Other work on penetrable wall charges includes that of Bell, Levine and Pethica l3 who applied the Haydon and Taylor l4 model in the calculation of the electrical potential across a penetrable -t Present address : Directorate General Scientific and Technical Information and Information Management, Biltiment " Jean Monnet ", Plateau de Kirchberg, Luxembourg (Grand Duchy). $ Present address : Clarkson College, Potsdam, New York, 13676, USA. 261 7261 8 INTERACTION OF PERMEABLE CHARGED SURFACES surfactant layer at an oillwater interface. From the above account it can be seen that analysis of one important case is missing, namely the calculation of the pressure petween two penetrable charged plates containing reversibly ionised groups using the full Poisson-Boltzmann equation.In this paper we treat this case and compare the results with those from constant charge and constant potential assumptions. THEORETICAL We consider the model system illustrated in fig. 1 which consists of two similar plane parallel thin uniformly charged sheets (surface charge density a) surrounded by an electrolyte, taken here as 1 : 1 valent but which could be any symmetrical electrolyte. The electrostatic potential, $, is related to the charge density p(x) via Poisson’s equation d2* 4n dx2 & - - P W -= where E is the dielectric constant of the electrolyte solution. We make the well tried assumption that self consistency can be made using the Boltzmann distribution P ( 4 = qn b P (- Pq$) - exp (Pq$)l (2) where n is the charge number density in bulk solution (where $ = 0), q is the electronic charge and P = (kT)-l.Tis the absolute temperature and k is Boltzmann’s constant. Thus we obtain the Poisson-Boltzmann equation It is straightforward to show that for our system, the solution is given in terms of the transcendental equations du = lcz s” uo J ~ ( c o s ~ u -cash u,) (4) and A = $[(cash U I - cosh u0)* + (cosh U Z - l)+] = 2[(sinh2 uJ2 - sinh2 u0/2)3 + sinh u,/2] (5) It can be shown by a number of different methods ‘ 9 exerted on the charged plates is that the electrical pressure, p , p(1) = 2n kT[cosh Pq$(O) - 13. (7) The electric free energy of interaction A , per unit area may then be obtained by integrat- ing eqn (7) in the usual manner.Thus A,(Z) = 2 Jl p(x) dx = 1’ p ( x ) dx. - IA . J . DUNNING, J . MINGINS, B . A. PETHICA AND P . RICHMOND 2619 The expression (7) for the pressure between the charged sheets is a consequence of our use of the Poisson-Boltzmann equation. However, depending upon whether we assume constant potential, constant charge or some criterion which takes into account ionisable groups on the surfaces, different values of the mid-plane potential at a given separation, 21, arise for the same number density of ionisable groups on the surface and the same electrolyte concentration. These lead to different pressure against distance curves in the three situations. FIG. 1 .-Diagram representing two negatively charged plane parallel membranes, and the electrical potential surrounding them.#(I) is the membrane potential, u is the surface charge density on the membrane and #(O) is the mid-plane potential. NUMERICAL PROCEDURE CONSTANT POTENTIAL [$(2)] If the surface potential is kept fixed, eqn (4) can be used to calculate the plate separation, 21, for a chosen set of values of the mid plane potential. The charge on the surfaces is then given by eqn (5). CONSTANT CHARGE (a) If the surface charge is fixed then the surface potential will vary as the plate separation varies. Now for a chosen value of cr and a chosen set of values of $(O), the value of $(I) is given by eqn (5). Using eqn (4) with the appropriate values of $(O) and $(I) will yield 1 and enable the [$(O), I ] curve to be plotted.IONISABLE SURFACES [BOTH $ ( I ) AND VARYING] In the case where the surfaces or model membranes are composed of ionisable groups, an extra condition must be introduced to account for the equilibrium between an ionisable molecule and its ionised state. We follow previous workers and assume for simplicity the reaction at the surface takes the form HA + H++A- (9)2620 INTER4CTION OF PERMEABLE CHARGED SURFACES where HA and A- represent neutral and negative surface sites respectively. Their relative concentrations are determined by the concentration of the H+ ion at the surface [H:], i.e., where K is the effective acid ionisation constant for equilibrium. If a is the fraction of dissociated molecules with charge q, and S is the total number of molecules per unit area then 101 = aqs.25 20 5 0 0.1 0.6 1.1 f/nm FIG. 2.-Mid-plane potential #(O) plotted against I for constant charge (C) and constant potential (P). Curves A, B and D are for a at equilibrium varying with separation of the membranes for am = 0.1 (A), am = 0.5 (B) and am = 0.9 (D). Introducing the familiar notation p . . . = - log,, . . . and using Pe . . . for the loge . . . equivalent we may write eqn (10) in the form Since the ions in solution obey the Boltzmann law, clearly we also have PeHs = PeHbulk-PqlCI(O*A . J . DUNNING, J . MINGINS, B . A . PETHICA AND P . RICHMOND 2621 From eqn (12) and (1 3) we have a = [exp - ~ A - - P ~ ~ ( O l ~ + 13-' (14) A = Peffbulk- PeK* (15) where When the sheets are infinitely far apart @(O) = 0 and from eqn ( 5 ) we have A, = 4 sinh (u,/2).I 1 1 0 K-1 1.5 3 Ilnm FIG. 3.-Comparison of normalised degrees of ionisation (cC/a,) for constant charge (C) and constant potential (P) for a varying at equilibrium with a , = 0.1 (A), a , = 0.5(B) and a, = 0.9 (D). Now if a, is the corresponding degree of dissociation we have from eqn (14) and (16) that A = l n (1 - :;J+um and 2n12, 2nA Ica, ica s = la,l/a,q = - = -. From eqn (9, (14), (16) and (18) it follows that We may now proceed as follows : for a set of values a, and u, compute S, A and A, using eqn (16)-(18). Set ul and obtain a from eqn (14). The values of a and S may be used to obtain a from eqa (18). Using the same value of ul, uo may be obtained from eqn (19). Finally uo and uz may be used to obtain I from eqn (4).2622 INTERACTION OF PERMEABLE CHARGED SURFACES For each of the three situations we can, therefore, obtain curves of p(Z) and A,(Z) as a function of A.A comparison of the results obtained is given in the next section. RESULTS AND DISCUSSION In comparing the three cases outlined above we have chosen parameters which represent approximately the conditions in physiological saline solutions. So, in an idealised sense, we are answering questions about the electrical force and energy 1.25 1.0 7 0.75 E 2 c, 1 X \ v) !i 0.5 0.25 0 0.5 K-1 1.0 1.5 Zlnm FIG. 4.-Comparison of the electrical pressure, p , between the charged surfaces as a function of the separation distance, I, for constant charge (C) and constant potential Cp) for a varying at equilibrium with am = 0.5 (B).The dotted lines show the variation of the distances at which selected values of the mid-plane potential ($,,) are obtained for the three cases. relationships in biological cells. The study of a more detailed model in which the cell walls have thickness, structure and discrete ionisable molecules on each side of the membrane is a more complex subject worthy of future investigation but for the present purpose of examining the differences between the three types of assumption we are probably justified in our choice of a simple model. Thus the 1 : 1 valent electrolyte concentration is 0.145 mol dm-3, T = 293.16 K, E = 78.54 and the surface potential [$(a)] when I is infinite is taken as - 18 mV.A . J . DUNNING, J . MINGINS, B .A . PETHICA AND P . RICHMOND 2623 The results of calculating the mid-plane distances, the wall potentials and the degree of ionisation as a function of mid-plane potential for the cases of (i) constant charge, (ii) constant potential and (iii) ionisable membranes with am = 0.1, 0.5 and 0.9 are given in fig. 2 and fig. 3, respectively. When the sheets are a distance k--l apart there is a 12 % increase of the mid-plane potential on going from the constant potential to the constant charge case. The case of ionisable membranes lies between these two situations. The differences between the various cases increases quite markedly as the separation, I, decreases. We note here that for our system k--l = 0.8 nm. The corresponding pressures and free energies of interaction are shown in fig. 4 and 5.5 0 0.5 K-1 0.9 FIG. 5.-Electrical contribution to the Helmholtz free energy of interaction A&) as a function of the separation distance, I, for constant charge (C), for a varying at equilibrium with am = 0.1 (A), am = 0.5 (B). The dotted lines show the variation of the distance at which selected values of the mid-plane potential ( #o) are obtained for the three cases. It is of interest to compare the results for our model with those obtained by other workers who considered the interaction between impermeable dielectric half spaces. For the case of constant potential, both models yield identical results. If we fix the membrane potential then from eqn (4) we see that uo(Z) and hence the pressure will take the same values as for the usual model.The surface charge obtained from eqn (5) will, however, differ from the usual situation where it is determined by the relation II = 2[sinh2 ul/2-sinh2 u0/2]*. (20)2624 INTERACTION OF PERMEABLE CHARGED SURFACES The case of constant charge, however, differs in an essential way. As the plate separation, 2, becomes small, clearly uo --* ul and in the usual case it follows by expanding eqn (4) and (20) that u1 -uo N (g sinh uo 2 and 3, 2c 2[-?) sinh 14.1'. From eqn (21) and (22) we obtain sinh u0 2: A/d, i.e. cosli uo N [I + ( A / K ~ ) ~ ] * and, therefore, p 2: 2nkT{[l+(Ay]i-l} + 2nkT -. a 1+0 K l Thus the pressure in the limit of small separations is divergent. For our system we obtain from eqn (9, again by expanding in powers of (ul - u,) and retaining leading terms, that Therefore Thus the pressure in the limit of small separations tends to a limiting value.This is physically very reasonable, and indeed one might imagine for some fluid-like mem- branes that when the separation is small the ions may penetrate the membrane and the electrical pressure will saturate as indicated above. 3, N, 2 sinh u0/2. (24) p 21 nkTA2. (25) B. V. Derjaguin and L. D. Landau, Actu Physicochim., 1941, 14, 633. E. J. W. Verwey and J. Th. G. Overbeek, Theory of the Stability ofLyophobic Colloids (Elsevier, Amsterdam, 1348). B. Ninham and V. A. Parsegian, J. Theor. Biol., 1971, 31, 405. G. M. Bell and G. C. Peterson, J. Colloid Interface Sci., 1972, 41, 542. D. Chan, J. W. Perram, L. White and T. W. Healy, J.C.S. Furuday 11, 1975, 71, 1046. P. Richmond, J.C.S. Furuduy 11, 1974, 70, 1061 ; 1975, 71, 1154. ' D. G. Hall, J.C.S. Furuduy II, 1977, 73, 101. * E. J. W. Verwey and K. F. Niessen, Phil. Mag., 1939, 28,435. T. B. Grimley and N. F. Mott, Disc. Furaduy SOC., 1947,1,3 ; T. B. Grimley, Proc. Roy. SOC. A , 1950, 201,40. lo E. P. Honig, Truns. Furuduy Soc., 1969, 65,2248. D. Gingell, J. Theor. Biol., 1967, 17, 451 ; 1967, 19, 340. l2 V. A. Parsegian and D. Gingell, Biophys. J., 1972,12,1192. l3 S. Levine, G. M. Bell and B. A. Pethica, J. Chem. Phys., 1964, 40, 2304. l4 D. A. Haydon and F. H. Taylor, Phil. Truns. A, 1960, 252, 225. (PAPER 7/1967)