The interaction of two charged spheres in the Poisson-Boltzmann equation
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1 The interaction of two charged spheres in the Poisson-Boltzmann equation JOSEPH E. LEDBETTER, THOMAS L. CROXTON, AND DONALD A. MCQUARRE' Department of Chemistry, Unioersiry of California, Davis, CA 95616, U.S.A. Received May 22, 1980 This paper is dedicated to Dr. Sam Levine on the occasion of his 70th birthday JOSEPH E. LEDBETTER, THOMAS L. CROXTON, and DONALD A. MCQUARRE. Can. J. Chem. 59, 1860 (1981). The Poisson-Boltzmann equation for two large charged spheres immersed in an ionic solution with either constant surface charge density or constant surface potential is solved numerically. The repulsion between the spheres is calculated from the electrostatic potential in the double layer surrounding the spheres. Good agreement between this numerically calculated force and the force computed using the De jaguin formula for spheres with constant surface charge density is found at small separations of the spheres. JOSEPH E. LEDBETTER, THOMAS L. CROXTON et DONALD A. MCQUARRE. Can. J. Chem. 59, 1860 (1981). On a resolu numeriquement 'equation de Poisson-Boltzmann relative a deux grosses sphtres chargees immergees dans une solution ionique ayant soit une densite constante de charge superlicielle soit un potentiel de surface constant. On a calcule la repulsion entre les spheres a partir du potentiel tlectrostatique de la double couche entourant les spheres. Lorsque les distances stparant les spheres sont courtes, on a trouve un bon accord entre la force calculce numeriquement et celle calculee en utilisant la formule de Derjaguin pour les sphtres ayant une densite constante de charge superlicielle. [Traduit par le journal] ntroduction from the numerical solution of the potential in the The interaction between large charged particles double layer surrounding the particles. in an aqueous ionic solution is a problem of central A numerical solution to the Poisson-Boltzmann concern in colloid science (1). n particular, it is equation for this problem was first presented by important to be able to calculate the forces acting Hoskin and Levine (3) for a surface between such colloid particles since these forces potential boundary condition on the spheres. he determine the stability of colloidal suspensions. A accuracy of these calculations was later improved knowledge of the electrostatic potential in the McCartne~ and Levine (4). We present here a double layer surrounding charged colloid particles somewhat different the allows one to calculate the electrostatic component solution of the than has been of the force between such particles. previously considered. nstead of solving for the n this paper we consider a model system total electrostatic potential we solve a differential posed of two charged spheres immersed in a solution equation for the correction to an of point ions in a uniform dielectric medium. with solution to the Poisson-Boltzmann equation. This the usual assumption of a Boltzmann distribution of approach not only results in a simpler numerical ions (21, Poisson's equation becomes the problem since the correction to the potential is Poisson-Boltzmann equation for electrostatic Po- often a slowly varying function7 but it also shows tential in this model. Application of the Poisson- directly the inadequacy of the approximate solu- Boltzmann equation to the geometry considered tion used in the method. Whereas previous numerhere then becomes a boundary value problem of a ical solutions have only considered the boundary second order non-linear partial differential equa- condition of constant surface potential, our method tion. For the geometry of two charged spheres, allows US to also solve the Poisson-Boltzmann however, an analytical solution to the Poisson- equation for spheres of constant surface charge Boltzmann equation is not possible. We thus present density. These two different boundary conditions in this paper a numerical solution of the poisson- yield quite different results, and it is important to Boltzmann equation as applied to two identical have definitive numerical calculation^ for both large charged spheres in an ionic solution. Surface cases so that a comparison with boundary conditions of constant charge density solutions to the Poisson-Boltzmann equation can and constant potential are treated. The electro- be made- The choice of a particular boundary static repulsion between the spheres is calculated condition is dependent on the physical Wstem which is to be modeled. 'To whom all correspondence should be addressed. Due to its importance in colloid science, the /81/ $01.00/ National Research Council of CanadalConseil national de recherches du Canada
2 LEDBETTER ET AL Poisson-Boltzmann equation has been well studied and many approximate solutions have been presented (5, 6). t is interesting to note that the linearized Poisson-Boltzmann equation applied to the problem of two interacting charged spheres can be solved exactly (7), but for large surface potentials, this linearized form is no longer valid. For small separations between the interacting spheres, a well-known approximation to the electrostatic force is the Derjaguin formula (8). At larger separations, a common approximation to the potential is the so-called linear superposition approximation (5). Bell et al. (5, 6) have combined these two approximations and have shown that the electrostatic repulsion calculated with these approximate methods agrees quite well with the numerical calculations of McCartney and Levine (4) for spheres with constant surface potential. We make a similar comparison at small particle separations of the Derjaguin formula appropriate to spheres of constant surface charge density with our numerical calculation of the force between the particles. Theory Consider two identical rigid spherical colloid particles of radius A, whose centers are separated by a distance R, immersed in a solution of point ions of uniform dielectric constant E. This geometry is shown in Fig. la. The colloid particles are also assumed to be of uniform dielectric constant equal to that of the electrolyte so that image effects can be ignored. With this model, the mean electrostatic potential in the surrounding double layer is given approximately by the Poisson-Boltzmann equation FG. 1. (a) Geometry of two interacting spheres; (b) boundary conditions for the correction potential ye on the half-plane in bispherical coordinates. Equation [4] could be solved numerically for the total potential y, but we choose instead to formally divide this potential into the sum 151 y (r1, r2) = Y lg-1) + Y 2(r2) + yc(r1, r,) = ys + y' [l] V2Q(rl, r2) = (8nnze~) sinh (zeqlkt) The functions yi (ri) are arbitrary, but a reasonable approximation to the Poisson-Boltzmann equation with the appropriate boundary conditions. n this is a linear superposition of the potentials due to equation we have assumed a symmetric electrolyte where n is the number density of one type of ion, T each sphere (5). We are thus led to choose yi(ri) to is the absolute temperature, k is Boltzmann's be the potential due to a single sphere. The 'Onstant, and zr is the magnitude correction term yc(rl, r2) clearly goes to zero as the the charge of particle separation becomes very large, since the One ion. Defining the reduced potential and the superposition solution yp(rl,r2) better approxi- Debye-Hiickel screening constant mates the total ~otential as the particle separation increases (5, 6).*substitution ofeq. [5] into eq. [4] yields a differential equation for yc(r,, r2) we can rewrite eq. [] in reduced form [6] V2y' = sinh (ys + yyc) - [Vr,2~l(rl) + Vr22~z(rz)l [4] V2y = sinh y where n this equation and in all following equations we have reduced all distances by K. Thus A, R, and ri [7] Vr,2yi = rip2 d(ri2 dyi ldri)ldri are now all reduced by K. since y, (ri) is spherically symmetric.
3 ~ spheres ' 1862 CAN. J. CHEM. VOL. 59, 1981 Since the electrostatic potential due to a single colloid particle is not known analytically, we [41 (ayc/ar,),, = YDH [F] choose the excellent approximation to this potential given by Brenner and Roberts (9) ~i (~i) = YDH/(~O - ~ ~YDH~) [(ao - a3~dh3)2 L81 n this approximation, the parameters a, and a, are determined by a variational principle associated with the free energy of the single sphere system. y,, is the familiar Debye-Hiickel potential P ydh(ri) = YOA{~XP [-(rt - A1l)lrt where yo is the (constant) potential on the sphere. Although it is not obvious because we have not written out the forms of a, and a,, if ydh(a) is substituted into eq. [8], we find that y, (A) = yo. Substitution of the yi (ri) given by eq. [8] into eq. [6] gives the differential equation to be solved for yc(rl, r2) once the surface boundary conditions are given. n the geometry of two charged spheres, it is convenient first to transform the problem into bispherical coordinates. n this system, surfaces of constant q are spheres, and surfaces of constant 6 are either spindles or dimples on the z-axis. For of the same size, it is sufficient to consider the half-space including the median plane (q = 0) and excluding one sphere (S,). This space is defined by the rectangle shown in Fig. 16. The particle surface is defined by the spherical surface given by [lo] qo = cosh- (r12a) Also shown in Fig. 16 are the boundary conditions on y'(6,q) which are required by the symmetry of the problem. n order to complete the description of this problem, the boundary condition on the sphere (q = qo) must be given. A fixed surface potential requires [ll yc(6,q = qo) = -y2(r2) A fixed surface charge density requires that the normal derivative of the potential on the surface of the sphere be proportional to the charge density [21 (a~lar,),~ = -4nzeol~ekT = -o* Using eq. [5] this condition becomes X + 2a3~DH3 ] [&+2tl- Ri] where y,, = ydh(r2), and where we have identified the (uniform) reduced charge density A+ 1- a0+2a3yo3] [151 O* = [A j [(a, - a3y03)2 The boundary condition given by eq. [14] may be manipulated to yield a boundary condition for (ayc/aq),,o in bispherical coordinates. After expressing the Laplacian in bispherical coordinates, eq. [6] may then be replaced by a standard set of finite difference equations (3, 10). Such a replacement assumes that third and higher order differences are negligible, which is justifiable since yc(c,q) is a rather smooth surface. The spherical surface boundary condition is incorporated into this scheme by using a finite difference equation which is appropriate along the q = qo line. Alternate line-by-line matrix inversion is performed over lines of constant q and over lines of constant 6, and this procedure is iterated to a convergent s~lution.~ n the calculations presented here, yc(6,q) is represented by a grid of 50 x 50 points. Previous numerical solutions (3, 4) have used fewer points. Once the electrostatic potential is known, the force between the colloid particles may be obtained by an integration of the Maxwell stress over the median plane (3). This reduced force is given by where c is a parameter given by [17] c = A sinh qo 2c2(c0sl1 y - ) + (1 - cos 6)2 ] sin 6 d6 t is also convenient to define another reduced force [18] f ** = f */2nAyo2 A common and apparently successful approximation to the force between two spherical colloid particles which is good only at small separations is valuation of the partial derivatives in this equa- 2~ copy of the computer program used is available from the tion yields authors upon request.
4 LEDBETTER ET AL the Derjaguin formula. n this method, the interacting spheres are essentially replaced by sections of charged parallel plates, and the force between the spheres is found to be proportional to the interaction free energy of this sum of parallel plates. For plates with constant surface charge density, the, potential of these interacting plates at infinite separation can be related to the charge density by integrating the Poisson-Boltzmann equation for this one-dimensional geometry. The result is [19] coshy, = 1 + (op*)/2 Derjaguin's method has been used by Bell and Peterson (6) to calculate interaction forces at small separations between spheres of different sizes and different surface boundary conditions. We compare our numerical calculations of the force between spheres with constant surface charge density with the force calculated from the expressions given by Bell and Peterson. Results and Discussion Graphs of the reduced force f ** versus the 1 separation K(R - 2A) between the two spheres are given in Figs. 2 and 3. Results for both constant charge and constant potential spheres with reduced potentials of 2 and 4 and reduced radii of 5 and 15 1 are presented. The forces given here for constant potential spheres agree to within 1% with the force calculations of McCartney and Levine (4). For each radius the charge density is related to the constant surface potential through eq. [15]. Although not shown by the data presented here, the surface potential of the spheres with constant surface charge density does approach the corresponding constant potential yo in each case to within a few percent at a separation of approximately K(R - 2A) = 5. As expected, the force between the spheres of constant charge is always greater than the force between particles of the corresponding constant potential, and the two force curves approach one another with increasing separation. n the Derjaguin formula, the reduced force f ** is independent of the particle radius (6), and Bell et al. (5) have noted that the numerical calculations for spheres with constant surface potential (4) support this approximation. A similar comparison of the force curves in Figs. 2 and 3 for different radii reveals that this approximation is also reasonable for spheres of constant surface charge density. n Fig. 4 the force curve calculated using De rjaguin's formula is compared with the numerical results for KA = 5 and a surface charge density on the sphere of o* = This charge density FG. 2. Force-separation curves plotted as f ** versus~(r - 2A) from our numerical solution, for identical spheres with~a = 5. The lines ---,-,..., and --- are for yo = 4,o* = 7.91, yo = 2, o* = 2.78, respectively. Data with yo = constant are for constant surface potential spheres, while data witho* = constant are for spheres with constant surface charge density. FG. 3. Similar curves as in Fig. 2 for KA = 15. The lines ---, - ;..,and---areforyo=4,0*=7.42,yo=2,ando* =2.51, respective1 y. corresponds to a potential at infinite separation of yo = 2. n the Derjaguin approximation one must use the potential of the plates at infinite separation y, which is related to the constant charge density on the plates by eq. [19]. For a sphere of a constant charge density, however, the potential at infinite separation is dependent on the size of the sphere. One must therefore decide whether to use y, = yo, or y, obtained from eq. [19] by setting op* = o* (constant charge density on the sphere). Figure 4 clearly shows that the correct choice for the potential at infinite separation of the plates is yo since good agreement with the numerical calculations of the force is achieved with this identification. Good agreement between De rjaguin's formula
5 1864 CAN. J. CHEM. VOL. 59, 1981 FG. 4. Comparison off ** calculated from De jaguin's formula (ref. 6) with our numerical results ford = 5 ando* = The lines-and --- are for spheres of constant surface charge density as calculated from De jaguin's formula with y, = 2.00 (a,* = 2.35) and y, = 2.26 (o,* = o*), respectively. Circles are the numerical calculations from this work. and the numerical results for higher reduced potentials and larger spheres is obtained as well with proper choice for y,. n summary, we have presented calculations of the electrostatic repulsion between identical spheres with either constant surface charge density or constant surface potential by numerically solving the Poisson-Boltzmann equation. Forces calculated using the well-known Derjaguin formula appropriate to spheres with constant surface charge density are compared with numerically calculated forces. Good agreement is obtained provided the correct input is used in the Derjaguin method. Acknowledgement This work was supported by a grant from the National nstitutes of Health, GM E. J. W. VERWEY and J. TH. G. OVERBEEK. Theory of stability of lyophobic colloids. Elsevier, Amsterdam D. A. MCQUARRE. Statistical mechanics. Harper and Row, New York N. E. HOSKN and S. LEVNE. Trans. R. Soc. Sec. A, 248 (1956); 433 (1956); 449 (1956). 4. L. N. MCCARTNEY and S. LEVNE. J. Colloid nterface Sci. 30, 345 (1969). 5. G. M. BELL, S. LEVNE, and L. N. MCCARTNEY. J. Colloid nterface Sci. 33, 335 (1970). 6. G. M. BELL and G. C. PETERSON. J. Colloid nterface Sci. 41, 542 (1972). 7. B. ENOS, D. MCQUARRE, and P. COLONOMOS. J. Colloid nterface Sci. 52, 289 (1975). 8. B. V. DERJAGUN. KoHoid-Z. 69, 155 (1934). 9. S. L. BRENNER and R. E. ROBERTS. J. Phys. Chem. 77, 2367 (1973). 10. W. F. AMES. Numerical methods for partial differential equations. Academic Press, New York
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