we1 = j+dq + = &/ET, (2)

Transcription

1 EUROPHYSICS LETTERS Europhys. Lett., 24 (S), pp (1993) 10 December 1993 On Debye-Huckel s Theory. I. M. MLADENOV Central Laboratory of Biophysics, Bulgarian Academy of Sciences Acad. G. Bonchev Str., B1. 21, 1113 Sofia, Bulgaria (received 15 March 1993; accepted in final form 22 October 1993) PACS Electrochemistry and electrophoresis. PACS Group theory. Abstract. - A one-parameter deformation family of Debye-Hiickel s model, sharing the appropriately modified rotational symmetry of the classical theory, is introduced and solved in a closed form. It is shown that, when the deformation parameter is set equal to zero, one regains Debye-Hiickel s model and results. When one studies a many-particle system in which the interaction between the constituents is strong enough, the most simple way to account for this interaction is to use the notion of mean or molecular field. This approach has been used by Debye and Huckel [ll in order to find the electrostatic component of the free energy which turns out to be equivalent to the work done for charging a sphere (macroion) in a solution. This electrostatic component can be expressed as an integral we1 = j+dq taken over all charged parts from q = 0 to its final value. It is evident that the problem with finding We, goes through the fundamental problem of classical electrostatics where one tries to determine the (electrostatic) potential (field) at every point in a space for a given distribution of charges. In many of such situations the ambient space is modelled as a homogeneous medium in which the potential at a distance r from a point charge Q is given by Coulomb s law + = &/ET, (2) where E is the dielectric constant, a quantity that indicates the extent to which the electrostatic effect of the charge is screened by the medium. In the most general form the problem covers cases where the dielectric constant can vary in space. In the regions of uniform dielectric constant without free charges the electrostatic potential + satisfies Laplace s equation at every point r: A+(r) = 0, A = a2/ax2 + a2/ay2 + a2/az2. (3)

2 694 EUROPHYSICS LETTERS If there are charges present this equation is replaced by Poisson s equation where p(r) is the charge density at the point r. By solving either Laplace s or Poisson s equation, wherever appropriate, one gets local solutions depending on integration constants. The global solution is obtained by matching different solutions at common boundaries. Using the continuity of $, i.e. and that of its (normal) gradient $1 = $2 (5) arranges the problem with the integration constants and in this way the globalization of $ is achieved. When the boundary is a charged surface with surface charge density 0, (6) is replaced by It should be mentioned that analytical solutions of these equations exist only for some simple geometries, while more complicated cases are treated by numerical methods. Here we shall consider the Debye-Huckel model and its modification. This modification includes the Debye-Huckel model as a special case and, what is also interesting, retains its explicit solvability. The classical Debye-Huckel model is shown in fig. 1. It is assumed that the macroion in whose free electrostatic energy we are interested is a low dielectric spherical medium of radius R1 (region I) surrounded by a solvent with an external dielectric constant E, and mobile counterions. In the low dielectric region I and in the ion exclusion region I1 (sphere of radius R,) we can apply Laplace s equation, while in the high dielectric region I11 (which is supposed to be infinite with the same dielectric constant = E,) we should apply Poisson s equation. It is also assumed that the total charge of the central ion q is distributed uniformly on the boundary between regions I and I1 where its surface charge density is 0 = Q/4?&1. Fig Geometry and regions I, 11, I11 of the Debye-Huckel model.

3 I. M. MLADENOV: ON DEBYE-HUCKEL'S THEORY 695 Spherical geometry means rotational symmetry which can be expressed by saying that the generators L~ = Eijkxja/axk, i, j, IC = 1,2,3 of the so@) Lie algebra [ei, Zjl = EijkZk satisfy and consequently Ei+ = (E$da/axk)+ = 0, i,j, IC = 1,2,3, (8) 22Y = (Z1" +e; +L&b = 0. (9) Here we also recall that, written in spherical polar coordinates (r, e, p), the Laplacian operator looks as follows: A+ = (l/r2)a/3r(r2a+/ar) + ( l/r2)sz$, where SZ = Z2 is its angular part. Spherical symmetry means that in the regions I and 11, free of movable charges, the equation that has to be solved is A$ =d2+/dr2 + (2/r)(d$/dr) = 0. It is easy to see that the latter is equivalent to (10) (11) whose solution d2 (r+)/dr2 = 0 (12) + = c + C/r (13) depends on two arbitrary real constants, C and 6. Now, as the frst region contains the singular point r = 0 (of the solution) and as we are looking for + bounded, we are forced to make the restriction = 0, i.e. + = C in region I. (14) All points in region I1 are regular for a + of the form (13) and that is the reason why + can be written there as + = B + B/r in region 11. ( 15) The situation with the third region is quite different as now we must solve the equation A+ = d2+/dr2 + (2/r)(d+/dr) = I C $ ~, ( 16) where IC is the so-called Debye-Huckel parameter. This equation has solutions of the form + = A exp [ - KY]/Y + A exp [ ICT]/V. (17) Again, we should pose the second constant A = 0 in order to have only bounded solutions (this time at infinity r +- CO ), ie. $ = A exp [- ICY]/~ in region I11. (18) Inserting (14), (15) and (18) into (5)-(7) and solving the obtained linear algebraic equations

4 696 EUROPHYSICS LETTERS allows us to specify all integration constants A, B, E, C, and hence $. Explicitly, we have It should be noted that these solutions depend only upon E ~ while, one expects from the very beginning that they will depend on both dielectric constants. As we shall see, the presence of the missing constant c1 will be restored into the solutions of the modified model. From the mathematical point of view, the equations with which we will deal further on are of the so-called Bessel type. This name refers to the second-order linear differential equation of the form x2y + xy + (x2 - v2)y = 0 * (22) Here v can be an arbitrary complex number and, if it is an integer one, then the function m J,(x) = 2 (- l)k/k!(v + k)!(x/b)y+2k (23) k=o is a solution of this equation. When v is not integer, in eq. (23) (v + k)! should be replaced by r(v + k + l), where r is the gamma-function (see, e.g., [2]). In this case a basis of solutions of (22) is given by J,(x) and J-,(x) (defined by the same formula). Actually, closer to our situation is the equation x2y + xy - ($2 + v2)y = 0, (24) which is known as the modified Bessel equation and whose space of solutions is spanned by the modified Bessel functions of the frst kind (when v 2 ) I,($) = exp[- vxi/21jv)ix) and I-,(x). (25) Whether v is an integer or not, this equation has another basis of solutions provided by Here I,(%) and K,(x). K, (x) = x[i-, (x)- I, (x)]/2 sin vx denotes the modified Bessel function of the second kind. It can be shown that K,(x) is a continuous function of its index v which, in particular, means that (26) lim K, (x) = K, (x), when v + n E 2. (27) Moreover, I, (x) and K, (x) are linearly independent. For real and positive values of v and x these functions are real. Finally, the equation

5 I. M. MLADENOV: ON DEBYE-HUCKEL S THEORY 697 has a basis of solutions given by I,,(kx) and K,,(kx), (29) in terms of which we shall express our results. Now, we move to explain the modification of Debye-Huckel s theory mentioned before. We will provide this by a revision of the spherical symmetry of the underlying model. In analytical form this symmetry has been expressed as the properties of the momentum operators (see (8) and (9)). On the other hand, these operators are connected with the Poisson brackets among canonical coordinates in phase space. If we change the canonical symplectic structure on the classical phase space P = T * (R3/{ O}), w = d( zpi dxi) = 2 dpi A dxi (i = 1,2,3) by adding to w a <<magnetic term,) 0, = - (p/2r3) 2 Eijkxidxi A dxk, we will have as a result the deformation of the Poisson brackets among all functions on P. Especially, for the canonical coordinates on the phase space we will have The new so@) Lie algebra {Mi, Mj} = E,j.ki%fk is generated by and its Casimir operator i$ point see [3]): Mi = Li + pxi/r, Li EijkX Pk i=l,2,3, (31) is related to that of the old i2 as follows (for more details on this M 2 = + g. (32) From now on we assume that the potential + is with respect to the modified momentum operators Mi. Substitution of M2 -,U for L2 in (10) modifies the radial part of the Laplacian operator and now the Poisson equation reads r2d2+/dr2 + 2r(d+/dr) - ( I C + p2)+ = 0. After some analysis, the bounded solution in the third region can be written as (33) +=M,(~r)/r /~, v = ( l +4p2) /2/2. (34) The respective bounded solutions in the regions I and I1 are represented by and + = Cr 2-1)/2 (35) + = Br(2v - 1)/2 + Er -(2v + 1)/2 Placing as before these solutions into (947) and solving the so-obtained algebraic equations we fur the integration constants A, B, E and C. In order to keep their explicit expressions transparent, the following notations are useful: x = 1 + EK - 1 (KR2) + K + 1 (K&J1& IK (KR2), (37) Y = [(2v - 1) 2 + X,]R; /[(2v + 1) 2 - XE,], v= ((2, - 1)[ 1(R{ + YR; ) - &{I + (2v + l) ZYR; }Ry/2. (36) (38) (39)

6 698 EUROPHYSICS LETTERS With the help of the preceding notations we can write the integration constants as follows: A = q(r, + YR, )/VK, (KR~), (40) B = a/v, (41) E = qy/v, C = q(r{ + YR; )/VR{. Few remarks are in order here. First of all, when we let p + 0, i.e. v any of these functions goes smoothly to the value prescribed by the classical Debye-Hiickel s theory. Here, we should mention that in [4] and [5] a parameter-dependent permittivity (r) has also been used within the framework of Debye-Huckel theory as a starting point for a more rigorous treatment via the Poisson-Boltzmann equation [6-91. On the other hand, in [lo] the authors have shown that beyond some distance the Debye-Huckel solution is the best approximation to the solution of the Poisson-Boltzmann equation. Now, solutions (34)-(36) are parameter dependent and this makes them more flexible for satisfying the corresponding matching conditions. Next, besides,u, a new free parameter is at our disposal in the solutions (34)-(36). This is the dielectric constant c2 which can be equated again to E ~ but, our opinion is that it must be kept different in order to make the transition from low to high dielectric regions (c1 < < s3) more realistic from the physical point of view. Finally, the need of comparing our results to the Tanford-Kirkwood theory [ll] is obvious. We hope to report on this subject elsewhere. *** This work is partially supported by the Bulgarian National Science Foundation Project no. K REFERENCES [l] DEBYE P. and HUCKEL E., Phys. Z., 24 (1923) 185. [2] WHITTAKER E. and WATSON G., Modem Analysis (Cambdrige University Press) [3] MLADENOV I., Int. J. Theor. Phys., 28 (1989) [4] GOLDSTEIN R. and KOZAK J., J. Chem. Phys., 62 (1975) 276. [5] GOLDSTEIN R., HAY P. and KOZAK J., J. Chem. Phys., 62 (1975) 285. [6] BURLEY D., HUSTON V. and OUTHWAITE C., Mol. Phys., 27 (1974) 225. [7] ORTTUNG W., Ann. N.Y. Acad. Sci., 303 (1977) 22. [81 WARWICKER J. and WATSON H., J. Mol. Biol., 157 (1982) 671. [9] GILSON M., SHARP K. and HONIG B., J. Comp. Chem., 9 (1987) 327. [lo] LEBRET M. and ZIMM B., Biopolymers, 23 (1984) 287. [lll TANFORD C. and KIRKWOOD J., J. Am. Chem. Soc., 79 (1957) 5333.