INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

Transcription

1 7" IC/93/57 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON THE DEBYE-HUCKEL'S THEORY Ivailo M. Mladenov INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE

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3 IC/93/57 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS 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 [i 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) ON THE DEBYE-HUCKEL'S THEORY in a solution. This electrostatic component can be expressed as an integral: = J W dq (1) Ivailo M. Mladenov * International Centre for Theoretical Physics, Trieste, Italy. ABSTRACT A one-paraineter family of perturbed Debye-Huckel's models, all sharing the inherent spherical symmetry of the classical theory, is introduced and solved in a closed form. When the deformation parameter is set equal to zero one regains the Debye-Huckel's model and results. taken over all charged parts from q = 0 to its final value. It is evident that the problem of finding W, is e I simply 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 such situations the ambient space is modeled as a homogeneous medium in which the potential at a distance from a point charge Q is given by Coulomb's law: = QAT (2) MIRAMARE - TRIESTE March 1993 where 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 the space. In the regions of uniform dielectric constant without free charges the electrostatic potential Laplace's equation at every point r : satisfies * Permanent address: Central Laboratory of Biophysics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., B1.21, 1113 Sofia, Bulgaria. = 0, A = (3) r

4 If there are charges present this equation is replaced by Poisson's equation: A V( r^ = - (4 n/e)p(~r > ) (4) where p (r ) is the charge density at the point r. Solving either Laplace's or Poisgon's equation where appropriate one gets local solutions depending on integration constants. The global solution is obtained by matching different solutions at common boundaries. Using continuity of ip, i.e. V, = V 2 (5) and that of its (normal) gradient: ( d y I (6) arranges the problem with the integration constants and in this way the globalization of y is achieved. When the boundary is a charged surface with surface charge density c, (6) is replaced by: (d v / (d y / d n - 4 n a (7) It should be mentioned that analytical solutions of these equations exist only for some simple geometries and 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 of which free electrostatic energy we are interested in is a low dielectric sphere medium of radius R (region I) surrounded by a solvent with an external dielectric constant c and mobile counterions. In the low dielectric region I and in the ion exclusion region II (sphere of radius R ) we can apply the Laplace's equation, while in the high dielectric region III (which supposed to be infinite) we should apply the Poisson's equation (Fig.I). Spherical geometry means spherical symmetry which can be expressed saying that y satisfies: and consequently: is )y = 0, i, j, k = 1,2,3 (8) * = = 0 (9) We recollect that as the Laplacian operator written in spherical polar coordinates (r, 6, f) looks as follows: where 0 = - L = (1/r 2 ) d/dr(r 2 dy/dr) + (10) is its angular part. Under spherical symmetry this means that in the free of movable charges regions I and II we should solve the equation: 2 2 A y = d y/dr +(2/r)(dv/dr) =0 (11) It is easily seen that the latter is equivalent to:

5 whose solution r 2 = 0 (12) y/ = C + C /r (13) Inserting (14),(15) and (18) into (5-7) and solving the resulting linear algebraic equations specifies all integration constants A, B, B, C and in this way y. Explicitly, we have: depends on two arbitrary real constants C and C. Now, as the first region contains the singular point r = 0 (of the solution) and as we are looking for y bounded, we are forced to make the restriction c = 0, i.e. V = c in region I (U) All points in region II are regular for ^ of the farm (13) and that is why y can be written there as: if - B + S../r in region II (15) 9t ( r R ) V - (q/ 2 )e 2 /(1+*R 2 )r in region III (19) V = -*r/(1 + *R 2 )]/r in region II (20) = [1 -»R 1 /(1+*R 2 )]/R 1 in region I (21) From the mathematical point of view the equations which we are dealing here are of the so-called Bessel This name refers to the second order linear equation of the form: differential with type. The situation with the third region is quite as now we must solve the equation: At/< = d v'/dr + (2/r)(dy/dr) = * V different (16) 2 ' ' ' 2 2 x y + xy + (x -v )y = 0 (22) Here v can be an arbitrary complex number and if it ' s just an integer the function: where x is the so called Debye-Huckel parameter. This equation has solutions of the form: >/' = A e /r + A e /r (1T) Again, we should pose the second constant A = 0 in order to have only bounded solutions (this time a- infinity i >co), i.e. = Z (23) is a solution of this equation. When v is not ;m integer (u+k)! in formula (23) should be exchanged for T(u+k+1) where T is the gamma function (see e.g. [2]). J (x) and J_ (x) (defined by the same formula) form in this case a basis of solutions of (22). Actually, more close to our situation is the equation: -Mr V = A e /r in region III (18) 2 ' ' ' 2 2 xy +xy -(x + y )y = 0 (24)

6 which is known as the modified Bessel equation and whose space of solutions is spanned by: and (25) For real values of i-> and positive x these functions are real.finally, the equation: 2 ' ' ' xy +xy -(Ax + y )y = 0 (26) In fact the new angular part modifies the radial component of the Laplacian operator too. The Poisson equation turns out to be: r 2 d 2 y / d r 2 + 2r / dr) - r 2 + = 0 (30) and after some analysis the bounded solution in the third region can be written as: 1/2 2,11/2 /2 (31) has a basis of solutions given by: I (Xx) and I. (Xx), V K (27) The respective bounded solutions in the regions I and II are represented by: in terms of which we shall express our results. Now, we go on to explain the modification of the Debye-Huckel's theory mentioned before. We will furnish 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 (9)). On the other hand these operators are connected with the Poisson brackets among canonical coordinates in the phase space. If we deform them in a smooth and explicit way as described below: (32) and (33) Placing as before these solutions into (5-7) and solving the so obtained algebraic equations we fix the integration constants A(v), B(IJ), BAV) and C(i->).In order to keep their explicit expressions transparent it seems that the following notations are useful: (x\x j } = 0, {p., x j }= 6\, (p., p.} = -(Ai/r 3 ). jk x k (28) (34) we should change also the momentum operators (see T31 for more details at this point). Fortunately, they are simply related to the old ones. -2v (35) (36) M = L + as well M 2 = L 2 + (29) (2U+1)/2 I (37)

7 With the help of the preceding the integration constants as follows: notation we can write References: B(v) = = 2q(V(v) (v) = 2q/WU') i (38) (39) (40) (41) 1. P.Debye, E.Huckel, Phys.Z. 24, 185 (1923). 2. E.Whittaker, G.Watson, Modern Analysis, Cambridge Univ. Press, I.Mladenov, Int.J.Theor.Phys. 28, 1255 (1989). 4. C.Tanford, J.Kirkwood, JACS. 79, 5333 (1957). A few remarks are in order here. First of all when we let /J -> O,i.e. v -> \/2 any of these functions goes smoothly to the value prescribed by the classical Debye-Huckel's theory. Next, besides p, a new free parameter is at our disposal in the solutions (26-28). This is the dielectric constant c which can certainly be equated to e, but our opinion is that it must be saved different in order to make the transition from low to high dielectric regions more realistic from the physical point of view. Finally,the possibility of parallel to the Tanford-Kirkwood theory 4] development of our results is obvious. We hope to report on this subject elsewhere. \ ACKNOWLEDGMENTS The author would like to thank Professor Abdus Salani, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. This viork was partially supported by the Bulgarian National Science Foundation Project No.K - Z02/92. Pig.l Geometry and Regions I, II, III of the Debye-Huckel Model. 10 1

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