Surface Effects in Debye Screening*

A thermodynamic system of equally charged, plus and minus, classical particles constrained to move in a (spherical) ball is studied in a region of parameters in which Debye screening takes place. The activities of the two charge species are not taken as necessarily equal. We must deal with two physically interesting surface effects, the formation of a surface charge layer, and long range forces reaching around the outside of the spherical volume. This is an example in as much as 1) general charge species are not considered, 2) the volume is taken as a ball, 3) a simple choice for the short range forces (necessary for stability) is taken. We feel the present system is general enough to exhibit all the interesting physical phenomena, and that the methods used are capable of extension to much more general systems. The techniques herein involve use of the sine-Gordon transformation to get a continuum field problem which in turn is studied via a multi-phase cluster expansion. This route follows other recent rigorous treatments of Debye screening. 0. Introduction The rigorous study of Debye screening was initiated in I-4] by Brydges, with the treatment of a charge symmetric lattice Coulomb gas. This work was greatly generalized by Brydges and Federbush [7]. Their proof applies to continuum Coulomb systems with essentially arbitrary short range forces, and charge symmetry is not required. Imbrie [12] improved the convergence estimates of [7] and removed a restriction on the relative sizes of the activities. He also proved Debye screening in Jellium. All of these treatments of Debye screening impose two important constraints on the system. First, there is a constraint on the activities z i and charges ei which is usually referred to as a "neutrality" condition. This condition may be viewed as essentially saying that ~ z i e i = O. Second, Dirichlet boundary conditions for the i * This work was supported in part by the National Science Foundation under Grants No. PHY83-01011 and PHY 81-16101 A03 362 P. Federbush and T. Kennedy Coulomb interaction are used. Physically, this means that the walls of the container are conducting. The significance of these constraints is that they minimize surface effects at the boundary of the container. This paper is devoted to the study of a Coulomb system in which these two constraints are not imposed. Our system has two species with charges +_ e and acti,dties z + and z_. The "neutrality" condition would require z + = z_; we do not enforce this. Free boundary conditions are used for the Coulomb interaction. This corresponds to a container with insulating walls. We modify the Coulomb potential at short distances in a way which passes easily through the Sine-Gordon transformation [see Eq. (1.2)]. To facilitate computing covariances we take our finite volumes (containers) to be spherical with radius R. The most interesting results we obtain for this system are for a finite volume. The charge density J(x) is non-zero near the surface of the volume (see Theorems 1.1 and 1.2). This surface charge is a result of the unequal activities z+ and z_. Near the surface of the volume Debye screening breaks down. The correlation function between two charges near the surface decays as 1/r 3 rather than exponentially (see Theorem 1.6). In the infinite volume limit these surface effects disappear, and there is Debye screening. Moreover, the infinite volume correlation functions of this system are equal to those of the system whose activities are both equal to (z+z_) 1/2 (see Theorem 1.5). We will give two explanations of these surface effects. The first explanation uses the mean field treatment of Debye and Hiickel. The second will use the SineGordon transformation and serve as an introduction to our proofs. The DebyeHtickel equation for the mean field potential tp(x) is A~p = ( z + e ~ + z_eP~)X , (0.1) where Z is the characteristic function of the volume (which we call A). In the infinite volume limit ()~= 1), the solution is the constant potential q~o =(2fi)-1 ln(z+/z_) . The solution of this equation for a finite volume, the "instanton," is studied in Appendix A. Contributions of J. Rauch to this study are gratefully acknowledged. It is shown that ~p(x) approaches the constant ~P0 well inside the volume. Since the charge density is A~p, the charge density is essentially zero away from the surface of the volume. Near the surface one finds that there is a charge per unit area of (2Rfi)-1 ln(z+/z_) . It is easy to check that such a charge distribution yields the potential tpo inside the ball. With free boundary conditions the screening breaks down near the surface even if z + = z_. So we will restrict our explanation of the 1/r 3 decay to the simpler case of equal activities. The simplest explanation is that the wall of the container interferes with a test charge's attempt to surround itself with a screening cloud. The clouds of two test charges near the wall will have non-zero dipole moments. This dipole-dipole interaction produces the 1/r 3 decay. To see this 1/r 3 decay in the Debye-Hiickel theory, we start with the DebyeHfickel equation A~p(x) + 2z)~(x) sinh/hp(x) = 6(x y ) . We have included a test charge at y. If we linearize this equation by replacing sinhfl~p(x) by t ip(x), then ~p(x)= ( A + l;)2X) l(x, y). With Dirichlet boundary Surface Effects in Debye Screening 363 conditions this covariance has exponential decay even if x and y are near the surface. This covariance with free boundary conditions is studied in Appendix B. It only decays as 1/r 3 along the surface. (For technical reasons we only prove a 1/r 3 -~ decay.) Jancovici [13-16] studied a system with an infinite insulating plane wall using Debye-Hficket theory. In this approximation he found that the correlations along the wall decay as 1/r a. The Sine-Gordon transformation expresses the partition function as Z= ~ d#(O)exp {~ dx[z+e+~g~(X)+z_e-~Uf~(x)]}. ~b(x) is a free (Gaussian) field whose covariance is essentially ( -A) l (x , y). The correlation functions are given by expectations of products of the observables z+_e +-~Ufo(x). One is tempted to argue as follows. Translate ~b(x) by the purely imaginary constant i~-1/2c. Since All3-1/2c = 0, d#(~b) would be unchanged, while z _+ would change into z + e ~ c. So there would be a one-parameter family of activities which yield the same physical system. In particular, taking c = 1⁄2 In (z +/z_), the two activities would both be equal to (z+z_) 1/2. For technical reasons one cannot translate ~b(x) by a constant. However, in the infinite volume limit the above conclusions are correct. Lieb and Lebowitz proved the invariance of thermodynamic quantities like the pressure and densities under the transformation z~---, z ie e~c [17]. For our model we show that the infinite volume correlation functions are invariant as welt. The objection to translating ~b(x) by a constant is not merely a technical point. In a finite volume the system is not invariant under z+_~z+_e -v-c. The correct approach is to translate ~b(x) to the stationary point of the functional integral. An easy computation reveals that this stationary point is i]/~p(x), where ~p(x) is the instanton defined in Eq. (0.1). Since ~p(x)~ ~Po well inside the ball, this translation essentially replaces z+ and z_ by (z+z_) ~/2 well inside the ball. The translation introduces the term exp[i]/ /~ dx(J(x)A~p(x)]. This term shows the presence of a surface charge distribution. (Recall that under the Sine-Gordon transformation an external charge distribution q(x) becomes a factor of e x p [ i l f ~ dxO(x)q(x)].) The explanation in the language of the Sine-Gordon transformation of the breakdown of screening near the boundary is similar to the first explanation. Debye screening occurs because the quadratic part of the cos~/~b(x) terms acts like a mass for the field ~b(x). Thus the inverse covariance A becomes A + I/~ 2)~. With Dirichlet boundary conditions the absence of this mass outside of the volume is irrelevant. With free boundary conditions the covariance ( A + l; 2Z) 1 feels the absence of this mass, and so doesn't have exponential decay everywhere. As ha the previous rigorous studies of Debye screening we analyze the functional integrals by a multi-phase Glimm-Jaffe-Spencer cluster expansion [-10]. This expansion is complicated by the fact that the cosine interaction has infinitely many minima. The standard approach, which we follow, is to introduce a function h(x) which is constant on cubes and only takes on the values 2zc/~~/2n, where n is an integer, h(x) labels which minima ~b(x) lies near, h(x) is only defined inside of A; one should think of h(x) as being zero outside of A. The sum over the h's is controlled by a small factor e -~. For each face between two cubes inside of A, there is a contribution to E of the order of (6h) 2, where 6h is 364 P. Federbush and T. Kennedy the change in h across the face. So e -~ would control the sum over h except for the translations h(x)~h(x)+2rc~-l/2n. With Dirichlet boundary conditions E also contains contributions of order (6h) 2 for the faces on the boundary of A. So these translations present no problem. With free boundary conditions the situation is drastically different. Let h~ be the function which equals 2re/?1/Zn everywhere. Then the energy E ofh~ is only of order / / lnZR instead of/?lnZR2. As a result of this the approach of [7] will not work. We use the notion of a "sector" to handle the problem. To each h we assign an integer n, and say that h belongs to the n th sector. For intuitive purposes one may define the sector of h to be the closest integer to the average of ~1/2h/2~ over the boundary of A. The partition function is a sum over all the h's. We split this sum up Ct3 into a sum over each sector. Thus Z = ~ Z ("). n = 0 9 Rather than doing a cluster expansion for Z, we do an expansion for each Z ("). Then we show that Z(")/Z (°) is bounded by e -~pI,~R So only the zero sector survives in the infinite volume limit. In a finite volume the contribution of the nonzero sectors to the correlation functions will be of order e -~p'"~g. For small/~ this is much smaller than the zero sector contribution. The correspondence h*-~h + h~ gives a one-to-one correspondence between the terms in Z (°) and Z C"). The corresponding terms differ in two important ways. First, the energy E(h) is different from the energy E(h + h"o). This difference is helpful since E(h + h"o) is greater than E(h) by an amount of order / / lnZR. So this provides the small factor of e -~p ~,~R The second difference is that the integrand in the functional integrals for corresponding terms in Z ("1 and Z (°) differ near the boundary. One would expect a contribution to Z(")/Z (°~ of order e ~R~ from this. However, because the surface charge is only of order 1/R, the difference in the functional integrands is only of order 1/R. So the contribution to Z(")/Z (°) can be bounded by e ~R. Z (") (and Z(")/Z ~°~) are studied by means of a polymer-type cluster expansion. The use of free boundary conditions introduces another technical problem that must be handled differently from the treatment in [7]. In the cluster expansion, factors of N! at each cube arise for various reasons. In ET] the exponential decay of the covariance is used to beat these factorials. We cannot use this "exponential pinning" since our covariance is only slightly better than integrable. The work of Battle and Federbush [2] provides an extra factor of 1/N! at each cube (see also [1, 3, 5, 8, 20]). Thus we can tolerate an N! at each cube. This improvement is essential for our expansion. In [7] the convergence estimates actually contained (N!) v at each cube with p fairly large. By doing these estimates more carefully we obtain p = 1 + e with e small. The (N !)~ can be overcome by a "power law pinning" since our covariance is slightly better than integrable. Some familiarity with [7] is assumed. In particular we recommend reading Sects. 1 through 8 of [7], excluding details of the infinite volume limit (in Sect. 1), and the Mayer Series (Sect. 3). References to a few other sections of [7] are made, but these may be treated as isolated references to any other source. The present cluster expansion is rather different from that in [7], so one may well restrict ones attention to the sections mentioned above. Surface Effects in Debye Screening 365 In the future one may want to treat the most general problem in classical Debye screening involving melding the techniques of [-7, 12], and this paper. At present this seems possibly doable, but complicated much beyond its value. There has been interesting work studying screening in an axiomatic setting (see [-11] for example). Recent work in the physics literature studies surface charge and effective potentials near a plane surface [,13-I6, 19]. In addition to organizational details mentioned above, we wish to outline the paper's development as follows. The basic results are stated in Sect. 1. Section 2 displays the Peierls expansion. Section 3, and Appendix E are concerned with organizing regions of space that for a given term in the Peierls expansions are treated as units in the cluster expansion. Section 5 describes the interpolation procedure, and Sect. 6 the polymer expansion. Sections 9 and 10 present energy estimates, interesting geometrical analyses of the division of Coulomb-like energy over units in the cluster expansion. Sections 7, 8, t 1, 12 handle the combinatoric aspects of the cluster expansion, as well as certain estimates of functional integrals. Appendix D studies some theorems of use to us, that fall in the domain of geometric measure theory. 1. Basics We have two charges, + 1 and 1, with (bare) activities z+ and z_. These need not be equal; this will mean we are not imposing a neutrality condition (for our system the imposed neutrality condition (3.8) of [7] would be z+ = z_). We let the charge density J be J= ~ elan, (1.1) i=+_ where ei = + 1 and a~ is a sum of delta functions at the position of particles of species i. (The notation of [-7] is a basic guide for us.) We let 1 1 \ (1.2) 1 ] " U = A A t ,~ l D T5)$ 2 wilt be a fixed small parameter, l D will be specified, and the natural infinite volume Green's functions are always understood. We set 5± =z±e~/2~"(~'~), (i.3) z 2 =z+z_ , (1.4) 5 2 = 5 + 5 _ , (1.5) l~ = (2zfi)-~, (1.6) ~ = ( 2 e f l ) * , (1.7) U =1⁄2i JuJ. (1.8) The particles are constrained to move inside a ball of radius R. We let A denote this volume A = {x e R 3, txt < R } . (1.9)