Diffusion approach to the linear Poisson

20 February 1998

Chemical Physics Letters 284 Ž1998. 78–86

Diffusion approach to the linear Poisson–Boltzmann equation Veaceslav Zaloj 1, Noam Agmon Department of Physical Chemistry and The Fritz Haber Research Center, The Hebrew UniÕersity, Jerusalem 91904, Israel Received 16 September 1997; in final form 19 November 1997

Abstract The linear Poisson–Boltzmann equation ŽLPBE. is mapped onto a transient diffusion problem in which the charge density becomes an initial distribution, the dielectric permittivity plays the role of either a diffusion coefficient or a potential of interaction and screening becomes a sink term. This analogy can be useful in two ways. From the analytical point of view, solutions of the LPBE with seemingly different functional forms are unified as Laplace transforms of the fundamental Gaussian solution for diffusion. From the numerical point of view, a first off-grid algorithm for solving the LPBE is constructed by running Brownian trajectories in the presence of scavenging. q 1998 Elsevier Science B.V.

1. Introduction Two classical fields of continuum physics are electrostatics w1x on the one hand and heat conduction w2x and diffusion w3,4x on the other. In these fields one studies the solution of elliptic partial differential equations of similar form. Specifically, one considers the Poisson equation ŽPE. or Poisson–Boltzmann equation ŽPBE. on the one hand, and the heat equation, Fick or Debye–Smoluchowski equations ŽDSE. on the other. The analogy between diffusion and electrostatics is quite evident, and forms the basis for the so-called ‘‘probabilistic potential theory’’ w5,6x. Here we extend the analogy to the linear Poisson Boltzmann equation ŽLPBE., which describes the potential of a fixed charge distribution in the presence of screening by ‘‘mobile’’ charges w7x. On the computational side, applications of the theory so far are rather limited. The analogy between the rate constant of diffusing particles hitting a given object and its electric capacitance w8x, has led to Brownian dynamics algorithms for calculating the capacitance w9–11x. The analogy between the dielectric and diffusion constants has been invoked in a random-walk simulation of the effective dielectric constant of composite materials w12x. A path-integral method for calculating the electrostatic potential has been tested for self-energies of one-dimensional systems w13x. We have shown w14,15x how a properly formulated analogy, in terms of a DSE, can serve as the basis for a Brownian dynamics ŽBD. algorithm for solving the PE. Here we continue this work by showing that the addition of a sink term to the DSE allows to extend the BD method to the LPBE. The PBE is a mean field description for the screening effect of mobile ions in solution, developed by Debye w7x. It is a cornerstone for the physical chemistry of electrolyte solutions w16x, as it describes the and Huckel ¨

1

Permanent address: Department of Physics, State University of Moldova, Mateevici Str. 60, MD-2009, Chisinau, Republic of Moldova.

0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 7 . 0 1 3 6 4 - X

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concentration dependence of activity coefficients in ionic equilibria and inert salt effects on ionic kinetics. The PBE has recently undergone a revival within the field of biophysics w17x, as a practical means for obtaining the electrostatic potentials around macromolecules w18–20x. Numerical algorithms developed to solve the PBE are exclusively grid methods. These include finite difference schemes w19–23x, boundary element w24,25x, finite element w26x and even Monte Carlo w27x methods. The last method utilizes the analogy of the LPBE with the steady-state diffusion equation to develop a grid random-walk algorithm for fixed potential boundary conditions. It was stated that ‘‘as yet the method is not suitable for constant surface charge type problems’’ w27x. These are precisely the problems treated below. In the present work we show that the LPBE is equivalent to diffusion with scavenging. Therefore, analytic solutions of the LPBE can be unified in the form of time integrated solutions of the diffusion equation ŽSection 3.. A BD algorithm for solving the LPBE is based on propagating stochastic trajectories with finite lifetimes. We demonstrate the applicability of the algorithm for several test cases ŽSection 4., and conclude that the probabilistic approach to the LPBE can be useful both numerically and as a means of obtaining analytical approximations.

2. Basic definitions Let us begin by defining the basic physical quantities from the theories of electrostatics and diffusion. 2.1. Electrostatics The classical treatment of ionic interactions in solution is based on the Poisson–Boltzmann equation ŽPBE. w16x, as derived by Debye and Huckel w7x. For moderate ionic concentrations this equation can be linearized, ¨ leading to a partial differential equation relating the electrostatic potential, w Ž r ., to the density of fixed charges, r Ž r ., in a dielectric medium of permittivity ´ Ž r ., = ´ Ž r . =w Ž r . y k 2 Ž r . ´ Ž r . w Ž r . s y4pr Ž r . .

Ž 1.

1rk Ž r . is the Debye screening length representing, in a mean field sense, the effect of the mobile ions in solution. In their absence, k s 0, the equation reduces to the PE. In the special case of a point charge at r 0 , r Ž r . s d Ž r y r 0 ., the solution is the Green function w Ž r < r 0 .. The solution of the PBE provides also the ‘dielectric displacement’, Del Ž r . ' ´ Ž r . E Ž r . s y´ Ž r . =w Ž r . ,

Ž 2.

which is proportional to the electrostatic field, E Ž r ., and hence also to the force. The surface integral of its normal component can be written as

HSn Ž r . P D

el

Ž r . d S s 4pH r Ž r . d r y H k 2 Ž r . ´ Ž r . w Ž r . d r , V

Ž 3.

V

where nŽ r . is the vector normal to the surface S and V the volume enclosed by it. When k Ž r . s 0 this is just Gauss’ law, stating that the surface integral equals to the total fixed charges enclosed by the surface. When r Ž r . vanishes outside V, the field perpendicular to S decays asymptotically as 1rr 2 . Then the surface integral on the left hand side Žl.h.s. remains finite as V ™ `. When k Ž r . ) 0 at every point r, the electrostatic field decays exponentially and the l.h.s vanishes as V ™ `. The screening term can then be interpreted as an induced charge density,

r Ž i. Ž r . ' k 2 Ž r . ´ Ž r . w Ž r . r4p .

Ž 4.

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It preserves overall neutrality because

Hr

Ž i.

Ž r . d r s Hr Ž r . d r ,

Ž 5.

whereas the induced and fixed charges have opposite signs. 2.2. Diffusion In the theory of diffusion influenced reactions w4x, the diffusion of particles in a field of force and with position-dependent reactivity is described by a DSE with a sink term

E p Ž r ,t .

s =D Ž r . exp yU Ž r . =exp U Ž r . p Ž r ,t . y k Ž r . p Ž r ,t . . Ž 6. Et Here pŽ r,t . is the probability density for a diffusing particle, whose diffusion coefficient is DŽ r . and interaction potential Žin units of thermal energy. is UŽ r ., to be located at point r by time t, given its initial probability density, pŽ r,0.. For a particle initially at r 0 , pŽ r,0. s d Ž r y r 0 ., the solution becomes the Green function for diffusion, pŽ r,t < r 0 .. When the ‘‘sink term’’, k Ž r ., is constant, one talks about a scavenging process w4x. Denote the k s 0 solution by p 0 Ž r,t .. The solution in the presence of scavenging Ž k ) 0. can then be written as p Ž r ,t . s p 0 Ž r ,t . exp Ž ykt . .

Ž 7.

Using the particle flux definition J Ž r ,t . ' yD Ž r . exp yU Ž r . =exp U Ž r . p Ž r ,t . ,

Ž 8.

the integrated DSE reads d

HV p Ž r ,t . d r dt

s y n P J Ž r ,t . d S y

HS

HVk Ž r . p Ž r ,t . d r .

Ž 9.

Thus the total change in the ‘‘survival probability’’ on the l.h.s is the balance of the overall flux of particles entering the domain V, minus those which react through the sink term within the domain.

3. The correspondence principle 3.1. Theory The correspondence between electrostatic and diffusion problems is based on the following mapping of the two partial differential equations, Eq. Ž1. and Eq. Ž6., D Ž r . exp yU Ž r . ' ´ Ž r . ,

Ž 10a.

k Ž r . ' DŽ r . k 2 Ž r . ,

Ž 10b. Ž 10c.

p Ž r ,t s 0 . ' r Ž r . . By integrating both sides of Eq. Ž6. over time one finds that the electrostatic potential is given by `

w Ž r . s 4p exp U Ž r .

H0

p Ž r ,t . d t .

This extends earlier relationships between the two equations w5,13,14x.

Ž 11 .

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We recognize two special cases of the above mapping. 1. The ‘‘Fick representation’’: ´ Ž r . s DŽ r ., UŽ r . ' 0 and k Ž r . s ´ Ž r . k 2 Ž r .. 2. The ‘‘Smoluchowski representation’’: ´ Ž r . s expwyUŽ r .x, DŽ r . ' 1 and k Ž r . s k 2 Ž r .. The classical ‘‘probabilistic potential theory’’ refers mostly to the first representation w5x. We find the latter more useful for the numerical treatment of inhomogeneous dielectrics w14x. Within the correspondence rules, the dielectric displacement, Eq. Ž2., becomes the time integrated flux `

Del Ž r . s 4p

H0 J Ž r ,t . d t ,

Ž 12 .

see Eq. Ž8.. Likewise, the induced charge distribution within the electrolyte solution becomes

r Ž i. Ž r . s k Ž r .

`

H0

p Ž r ,t . d t ,

Ž 13 .

which is just the ultimate trapping probability from point r. 3.2. A simple example We now demonstrate the correspondence principle by recasting the solutions of the LPBE in dielectric homogeneous media as time-integrals of the corresponding solutions for diffusion. In the Fick representation, ´ Ž r . s ´ s const corresponds to free diffusion. Denote the k s 0, free-diffusion ŽUŽ r . s 0. solution in d-dimensions by f d Ž r ,t < r 0 . s

1

Ž 4p´ t .

exp y dr2

Ž ryr0 .

2

4´ t

.

Ž 14 .

From Eqs. Ž7. and Ž11., the Green functions, wd Ž r < r 0 ., for the LPBE in a d-dimensional uniform medium and in the presence of mobile ions Ž k ) 0., can be written as the Laplace transform `

w d Ž r < r 0 . s 4p

H0

f d Ž r ,t < r 0 . exp Ž ykt . d t

Ž 15 .

with a constant k ' k 2´ . Insertion of Eq. Ž14. gives

w 1Ž r < r 0 . s w3 Ž r < r0 . s

2p

k´

exp Ž yk < r y r 0 < . , 1

´ < ryr0