Sum rules for inhomogeneous Coulomb fluids, and ideal conductor boundary conditions

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Sum rules for inhomogeneous Coulomb fluids, and ideal conductor boundary conditions

B. Jancovici

To cite this version:

B. Jancovici. Sum rules for inhomogeneous Coulomb fluids, and ideal conductor boundary conditions.

Journal de Physique, 1986, 47 (3), pp.389-392. �10.1051/jphys:01986004703038900�. �jpa-00210218�

Sum rules for inhomogeneous ^Coulomb fluids,

and ideal conductor boundary ^conditions

B. Jancovici

Laboratoire de Physique Théorique ^{et Hautes} Energies (*),

Université de Paris-Sud, ^Bâtiment 211, ⁹¹⁴⁰⁵ Orsay Cedex, ^France

(Reçu le 30 août 1985, accepti le 12 novembre 1985)

Résumé.

On montre que la fonction de corrélation de charge ^{d’un fluide} coulombien inhomogène ^satisfait

une règle ^{de somme} qui ^met en jeu ^{les moments} électriques multipolaires ^d’ordre quelconque; ^cette règle ^{de somme}

est une généralisation ^de plus ^{de la} règle ^{du second moment de} Stillinger ^et Lovett. On donne une application

à un fluide limité par ^une paroi parfaitement conductrice; ^on montre comment les potentiels chimiques ^contrôlent

à la fois les propriétés ^{de volume et de surface.}

Abstract

The charge correlation function of an inhomogeneous Coulomb fluid is shown to obey ^{a sum rule} involving electrical multipole ^{moments of} arbitrary order; ^{this sum rule is a further} generalization ^{of the} Stillinger-

Lovett second-moment condition. An application ^is given ^{for a fluid bounded} by ^an ideal conductor wall; ^{it is}

shown how the chemical potentials control both the bulk and surface properties.

Classification Physics ^Abstracts

05.20

50.00

82.45

1. Introduction.

The present paper is about classical equilibrium

statistical mechanics of Coulomb fluids (plasmas, electrolytes...). ^The fluid is modelled as a system ^of

particles interacting through Coulomb’s law plus

some short-range interaction (a ^{hard core} repulsion

for instance).

Coulomb fluids exhibit the fundamental property of screening, with the consequence that their charge

correlation functions obey ^a variety ^{of sum rules.}

If p(r) ^{is the} microscopic charge density ^{at a} point ^r,

the static charge ^structure function is defined as

where > ^{denotes an} equilibrium average. In the

simplest ^{case of a classical} homogeneous ^fluid (for

a homogeneous ^fluid ( p(r) > ⁼ 0, the truncation T in (1.1) may be omitted, and also S depends only ^on

the distance r’

r 1), S is known to obey ^{the Stil-} linger-Lovett [1] ^{rules which} give its zeroth and second moments :

and

(where fl ^{is the inverse} temperature). ^{In the more} complicated ^{case of a classical} inhomogeneous fluid,

i.e. a fluid which is not invariant by translations and rotations (for instance because of the presence of walls or of external charges), (1.2) ^is unchanged ^and (1. 3) ^{is to be} replaced, ^{as shown} by ^{Carnie and Chan} [2, 3], by

(with ^an arbitrary choice of the origin); ^one readily

shows [4] ^that (1.4) ^{reduces to} (1.3) ^{in the} homoge-

neous case.

The first Stillinger-Lovett ^rule (1.2) ^{is a} simple

consequence of the following statement : any charge q

introduced into the fluid is screened by ^a polariza-

tion cloud of charge - q, and therefore the total

excess charge (i.e. ^the charge q plus ^the charge ^{of its} polarization cloud) vanishes. When applied ^{to the} particles of the fluid themselves, ^{this statement leads}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703038900

390

to (1.2). However, ^{a more} general property ^{of the}

excess charge distribution is that all its electrical

moments vanish [5, 6], provided ^the correlations in the fluid have good decay properties (faster ^than algebraic); ^these decay properties ^{will be assumed to}

hold for the systems ^to be considered here. There-

fore, (1.2) ^{can be} generalized ^into

where t’ stands for the polar angles of r’ and Y,m(i’)

is any spherical ^{harmonic. Of course,} (1.5) ^{is trivial}

for I =1= 0 by rotational invariance in the homogeneous

case, but it becomes a non-trivial statement for ^an

inhomogeneous ^fluid.

In the present paper, it will be shown, in section 2, that the generalization ^{of the first} Stillinger-Lovett

rule (1.2) ^into (1. 5) ^{has an} analog ^for the second

Stillinger-Lovett rule, ^{the Camie} and Chan form (1. 4)

of which can be generalized ^into

with spherical harmonics of arbitrary ^order.

It should be emphasized ^that (1. 5) ^and (1. ) ^{do not}

hold in general ^{for a Coulomb} fluid bounded by ^an insulating ^wall (1), because the charge correlation functions near the wall do not have the _necessary

good decay properties (in the directions parallel ^to

a plane wall, ^the charge ^{structure function} (1.1) decays only ^as the inverse cube of the distance [7, 8]).

However, good decay properties ^{for the} charge ^corre-

lations ^are expected ^{in a} Coulomb fluid bounded

by ^a conducting ^wall (a ^more general conjecture [9]

is that the charge correlations have a faster-than-

algebraic decay whenever every region ^{of the fluid}

is completely surrounded by conducting media).

In section 3, equation (1.6) ^{will be} used, ^{in its} dipolar

form (I = I’ = 1), ^{for a} Coulomb fluid bounded by ^a plane ^{ideal conductor wall,} and it will lead ^to the

simple ^{sum rule}

where the origin ^{is on the wall and z is} the component of r normal to the wall.

Finally, ^{in *section 4, the} thermodynamics ^{of a}

Coulomb fluid bounded by ^ideal conductor walls will be discussed, ^{in the} grand-canonical ^ensemble.

It will be shown how the chemical potentials ^control

both the bulk and surface properties ^{of the} fluid, ^and

how (1. 7) ^can be retrieved as a consequence of this

more general study.

(’) They ^{however hold for I}

^l’

^0.

2. General sum rule.

We derive the general ^{sum rule} (1.6) by ^a simple

extension of the linear _response argument ^{which can} be used [3] ^for proving (1.4).

At the (arbitrarily chosen) origin, ^we put ^{a small} point ^test multipole ql’m which creates an interaction Hamiltonian

The fluid responds by ^a change ^{of the} charge density

at r

The general screening ^rule [5, 6] ^{discussed in the}

Introduction states that the multipole ^moment (1’, m’)

of bp(r) ^cancels ql,m, while the other multipole ^moments

of bp(r) ^{vanish. Therefore}

Using (2 . 2) ^and (2 .1 ) ⁱⁿ (2. 3), ^{we obtain at once} (1.6).

3. Plane ideal conductor wall

We now apply (1.6) ^{to a} semi-infinite Coulomb fluid,

bounded by ^a plane ^{wall z =} 0; ^{the fluid} occupies ^the region ^z > 0. The wall is assumed to be an ideal con-

ductor. This might ^{be a model for an electrode in an} electrolyte. ^A well-defined model is obtained by ^assum- ing that the interface between the fluid and the wall is impermeable ^{to the} particles of the fluid.

We want to write a sum rule involving ^the charge

structure function of the fluid alone, ^i.e. S(r, r’) ^is

defined by (1.1) ^where p(r) ^{is the} charge density ^of

the fluid particles. ^{There are also} charges ^{induced at}

the surface of the ideal conductor : they will be dealt with by the method of images. ^{To be more} specific,

we put ^{a small test} point dipole ^{in the} fluid, ^{at the} origin, ^{on the wall} (more precisely, infinitely ^close

to the origin, ^{on the fluid} side); ^the dipole is directed

along ^{the z} axis, ^{normal to the wall. We can} carry the argument of section 2, ^{for I’ =} 1, ^{m’ =} 0, ^{with how-}

ever the modification that the dipole ^{now has an} image and the total interaction Hamiltonian with the fluid is easily ^{found to be twice as} large ^as (2. 1).

Bringing ^this modification into (1.6), ^{we find}

Since S depends ^{on the} components ^{of r and r’} parallel

to the wall only through ^their difference, ^{after a sim-} ple manipulation ^{we can rewrite} (3.1) ^{in the form} (1.7).

It should be noted that the integral (1. 7) ^{is not}

absolutely convergent, ^{and the} integrations upon ^z

and z’ cannot be permuted. ^{The first} integration

dr’ S(r, r’) gives ⁱⁿ general ^a non-vanishing result, notwithstanding (1.2), because here there are induced

charges ^{on the} conducting wall, ^{which are not includ-}

ed in the definition of S(r, r’).

The sum rule (1. 7), which holds for an ideal conduc- tor wall, ^{has some} similarity ^{with a} rule which holds for an insulating ^wall [10, 3] :

where also the integrations ^{cannot be} permuted Although (1.7) ^and (3.2) apply ^{to different} cases, both can be rewritten in the same form

Indeed, ^{for an} insulating wall, ^{z - z’ can be} replaced by - z’ because of (1.2), ^{and for an} ideal conductor

wall, ^{z - z’ can be} replaced by ^z because of (1. 5) ^for

l = 1, ^{m = 0.}

4. Thermodynamics ^{of a} Coulomb fluid with ideal conductor boundary conditions.

We now proceed ^to study ^the thermodynamics ^{of a}

Coulomb fluid bounded by conducting ^{walls, from}

a more general point ^of view, ^{in the} grand-canonical

formalism. Inside a box of volume S2, ^with conducting

walls kept ^at potential ^{zero, we consider a fluid made}

of M different species ^of particles. ^A particle ^of species ⁱ

has a charge ei, there are Ni ^such particles, ^{and their}

chemical potential ^is pi. It is convenient to introduce the M-dimensional vectors e ⁼ (el, ^..., em), ^{N =} (N t, ..., N M)’ J1 = (Pt, ^..., gm).

Coulomb fluids in equilibrium ^are neutral in the bulk. This has been shown, ^{for the case of} systems bounded by insulating walls, in the elaborate mathe- matical investigation of Lieb and Lebowitz [11].

Since we do not expect ^{the bulk} properties ^to depend

on the nature of the walls, ^{we shall admit without} proof ^{that bulk} neutrality ^{also holds for} conducting

walls. Thus, whatever the chemical potentials may be,

in the thermodynamic limit the bulk densities ni ⁼ lim Ni >/Q ^{must be} such that n e ⁼ 0.

U-+ 00

There are only ^{M - 1} independent densities, ^and

M - 1 chemical potentials ^{should be} sufficient for

controlling these densities. The relevant part ^{of the} M-component vector J1 ^can be exhibited by splitting J1

as

where M" ^{is the} component of J1 along ^{e :}

If N is split, ^{in the same} way, ^as

where

the factor exp(pp - N) ^{in the} grand-canonical ^for-

malism becomes exp[P(u’ . ^{N’ + M"} N")]. ^{In the} thermodynamic limit N" >/Q always ^vanishes (this

is the bulk neutrality requirement), ^the grand-cano-

nical distribution function has a sharp peak ^{at N" = 0,}

and therefore the chemical potential p" ^{is that} part of p

which is irrelevant for controlling the bulk densities which depend only ^{on the} (M - I)-component ^vec-

tor p’.

However, in the case of conducting walls, p" ^does

have a physical role : it controls the surface structure of the fluid, ^{as it will now be shown.} Again, ^{we assume}

that the interface between the fluid and the ideal conductor walls is impermeable ^{to the} particles ^{of the}

fluid The region ^of the fluid close to the interface is known as the electrical double layer. Although ^the

bulk fluid must be neutral, ^the electrical double

layer ^{may carry some surface} charge density. ^An important ^related quantity ^{is the} potential ^difference

across the electrical double layer : while the conduct-

ing ^{wall is assumed to be at} potential ^{zero, in} general

the interior of the fluid is at some different electro- static potential 0, ^and, given ^{the bulk} composition ^of

the fluid and the detail of the interparticle ^and particle-

wall interactions, ^{the surface structure is determined} by 0, ^or alternatively by p", because there is a very

simple ^relation between 0 ^and Jl". Indeed, if 0 ^is

varied by do, ^at fixed bulk composition (i.e. ^fixed p’),

the reversible work ^for bringing ^a particle ^{of spe-}

cies i from infinity ^{into the fluid varies} by ei do (this

is the variation of the work _necessary for crossing ^the

electrical double layer). ^Thus

and, from the definition (4.2) ^of p,",

This result (4.6) ^{sustains our} claim that p" ^controls

the surface structure.

Let us now derive a sum rule relating ^the density

and a correlation function, following ^{a method of}

Blum et al. [12]. ^Let hi(r) ^{be the} microscopic ^number density ^of species i ^{at r. The} grand-canonical ^for-

malism gives ^{at once}

392

On the other hand,

Using (4. 5) ^and (4. 7) ⁱⁿ (4. 8), ^{we obtain the sum rule}

This sum rule has been previously conjectured by

Forrester [13]. Note that it can be considered as the

analog ^{for a} conducting ^{wall of the} dipole ^{sum rule}

of Blum et al. [10] which holds at a plane charged insulating ^wall creating ^an electrical field E :

Finally, ^{in the} limiting ^{case of a} semi-infinite fluid bounded by ^a plane ideal conductor wall, ^{we can}

retrieve the sum rule (1.7) ^{as a} consequence of (4. 9).

The potential difference across the electrical double

layer is related by simple electrostatics to the dipole

moment of the charge density :

Since

combining (4. 9) ^and (4 .11 ), ^{we obtain} (1.7).

5. Concluding ^remarks.

Although ^{we have} pointed ^out analogies ^between conducting ^and insulating walls, ^{there is an} impor-

tant difference : for a Coulomb fluid bounded by insulating walls, Il" ^{is a} totally irrelevant parameter

which controls neither the bulk nor the surface _pro-

perties. ^{This can be} easily ^{seen in the} simple ^{case of a}

fluid in an insulating uncharged spherical ^{box of}

radius R. If the fluid had a surface charge density ^a,

the potential ^at the surface would be 4 n u R, ^becom- ing infinite in the thermodynamic ^{limit R -} oo, cr fixed. The reversible work _pic for bringing ^a particle

of species i ^from infinity into the fluid would have an

infinite part ei ^{4 naR. In other} words, ^{in the} grand-

canonical formalism, ^{with finite chemical} potentials,

it is not possible ^to control the surface charge density,

which will remain _zero, like the bulk charge density.

It should also be noted, ^{from the} derivation in section 4 of the conducting ^{wall sum rules} (4.9) ^and (1. 7), ^that they remain valid in quantum ^mechanics without _any change ^{and that} they ^{also hold for the} time-displaced correlation functions, ^{i.e. when the}

densities at r and r’ are taken at different times 0 and t.

This is to be contrasted with the insulating ^{wall sum}

rule (3.2), ^{which can} be extended to quantum- mechanical and time-displaced correlations only ^for

a one-component plasma, ^{and in a modified form} [14]

where wp ^{is the} plasma frequency.

6. Acknowledgments.

I am indebted to L. Blum, ^{J. L.} Lebowitz, ^{and M. L.}

Rosinberg ^for stimulating discussions. Part of this work was done at Rutgers University ^{and at the} University ^{of Puerto} Rico; ^{I thank} J. L. Lebowitz and L. Blum respectively ^for their kind hospitality

at these institutions.

References

[1] STILLINGER, ^{F. H. and} LOVETT, R., ^{J. Chem.} Phys. ⁴⁹ (1968) 1991.

[2] CARNIE, ^{S. L.} and CHAN, ^{D. Y.} C., ^Chem. Phys. ^Lett.

77 (1981) ^437.

[3] CARNIE, ^S. L., ^{J. Chem.} Phys. ⁷⁸ (1983) ^2742.

[4] OUTHWAITE, ^C. W., ^Chem. Phys. ^{Lett. 24} (1974) ^73.

[5] GRUBER, Ch., LEBOWITZ, J. L. and MARTIN, ^Ph. A.,

J. Chem. Phys. ⁷⁵ (1981) 944.

[6] BLUM, L., GRUBER, C., LEBOWITZ, ^{J. L. and} MARTIN, P., Phys. Rev. Lett. 48 (1982) ^1769.

[7] ^USENKO, A. S. and YAKIMENKO, ^I. P., ^{Soviet Tech.}

Phys. ^{Lett. 5} (1979) ^549.

[8] JANCOVICI, B., ^{J. Stat.} Phys. ²⁸ (1982) ^43.

[9] JANCOVICI, B., ^{J. Stat.} Phys. ³⁴ (1984) ^803.

[10] BLUM, L., HENDERSON, D., LEBOWITZ, ^J. L., GRUBER,

Ch. and MARTIN, ^Ph. A., ^{J. Chem.} Phys. ⁷⁵ (1981)

5974.

[11] LIEB, ^{E. H. and} LEBOWITZ, J. L., Adv. Math. 9 (1972)

316.

[12] BLUM, L., LEBOWITZ, ^{J. L. and} HENDERSON, D., unpu- blished.

[13] FORRESTER, ^P. J., J. Phys. ^{A 18} (1985) ^1419.

[14] JANCOVICI, B., LEBOWITZ, J. L. and MARTIN, ^Ph. A.,

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