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Nucleation Theory and

Applications

J¨

urnW.P.Schmelzer,GerdR

¨

opke, and

Vyatcheslav B. Priezzhev (Editors)

Dubna JINR 2006

iv

vch 8 Jul 2005 16:33

5 Statistical Theory of Electrolytic Skin

Eﬀects

(1) Rainer Feistel and (2) Werner Ebeling

(1) Leibniz-Institut f¨ur Ostseeforschung, Seestr. 15,

D-18119 Rostock-Warnem¨unde, Germany

(2) Institut f¨ur Physik, Humboldt-Universit¨at,

Newtonstr. 15, 12489 Berlin, Germany

Zwei Dinge sind zu unserer Arbeit n¨otig:

Unerm¨udliche Ausdauer und die Bereitschaft,

etwas, in das man viel Zeit und Arbeit gesteckt hat,

wieder wegzuwerfen.

Albert Einstein

Abstract

Inspired by Ebeling’s Electrolyte Phase Transition, which hypothetically may possess a

two-phase region in the (pV )-diagram with diﬀerent degrees of dissociation of the dis-

solved salt on both sides of the interface, we derive the statistical expression for the

relaxation force at the boundary of a dilute electrolyte. The analogies and diﬀerences to

the classical theories of Onsager-Samaras and Buﬀ-Stillinger for the surface tension of

aqueous electrolytes are discussed. Suitable integral hierarchy equations for reduced dis-

tribution functions are brieﬂy derived, and the solution for the pair distribution functions

is calculated analytically by means of Fourier, Hilbert, and Hankel integral transforms.

5.1 Introduction 89

5.1 Introduction

The current study on the statistical properties of a dilute electrolyte in the neigh-

bourhood of an interface, which is preventing the ions from diﬀusion into the pure

solvent behind it, was inspired by the Electrolyte Phase Transition.Even

though both phenomena are not immediately related, we will brieﬂy introduce

the theory of this phase transition in the following, and explain their potential

connection.

The Electrolyte Phase Transition was discovered theoretically by Ebeling

(1971). The equation of state of dilute, associating electrolytes exhibits a critical

point, and a region of thermodynamic instability in the (pV )-diagram. Since its

theoretical-statistical description is available in analytical form, it may provide

new insights into critical behaviour in general. This aspect was emphasised later

by Fisher and Levin (1993), and Aqua and Fisher (2004). It may contribute to the

ongoing discussion about the eﬀect of gravity on the critical behaviour of ﬂuids

(Anisimov (1991), Wagner et al. (1992), Kurzeja et al. (1999), Ivanov (2003),

Skripov and Ivanov (2004)), since ionic criticality is unlikely to be inﬂuenced

by the symmetry-breaking gravity force, which is suspected to be responsible

for the apparent contradictions between theory and experimental ﬁndings. The

remarkable contrast between the universality claim of the renormalisation group

theory on the one hand (Sengers and Levelt Sengers (1986), Anisimov (1991)),

and the opposite universality claim of the catastrophe theory on the other hand

(Thom (1975), Poston and Stuart (1980)), is still a scientiﬁc challenge. Since the

statistical theory of the electrolytic surface tension is easier treatable than that of

the liquid-gas interface, the current study is mainly devoted to further progress

into this speciﬁc direction.

We consider the dissociation equilibrium of a symmetrical electrolyte, between

the neutral molecule AC and its anion A−and cation C+, i.e.,

AC ←→

K(T)A−+C

+.(5.1)

The equilibrium constant is deﬁned by the cut-oﬀ condition (Falkenhagen et al.

(1971)),

K(T)=4π

d

a

r2dr exp (−l/r)≈4πa3exp (b)

bfor b1.(5.2)

vch 8 Jul 2005 16:33

90 5 Statistical Theory of Electrolytic Skin Eﬀects

Here, the Landau length is l=q2/(4πεε0kBT), ais the ion contact distance,

qthe ion charge, b=l/a the Bjerrum parameter, and d=l/2 the association

distance. Of course, in Eq. (5.2), d>ais supposed, or b>2. We note that in

the strict theory [10] the mass action constant is given by a modiﬁed expression

which, however, is asymptotically identical with Eq. (5.2). The mass action law

for the neutral (nAC)andion(n±=n+=n−) particle densities

K(T)= nACfAC

n+f+n−f−

=n−n±

n2

±f2

±

(5.3)

can be solved for the conserved total particle density, n,

n=nAC +n+=nAC +n−=nAC +1

8πa3

(κa)2

b,(5.4)

using the Debye parameter, κ2=8πl ·n±, and the activity coeﬃcients (Falken-

hagen and Ebeling (1971)),

ln f±=−κl

2(1+κa),ln fAC =0,(5.5)

resulting in the total particle density expressed by the Debye and the Bjerrum

parameter as

n=1

v=1

8πa3

(κa)2

b1+(κa)2

2b2exp b

1+κa .(5.6)

This formula is to be used in the expression for the osmotic pressure (Falkenhagen

and Ebeling (1971))

p

kBT=nAC +n++n−−1

8πa32ln(1+κa)−κa

1+κa −κa(5.7)

=n+1

8πa3(κa)2

b+2ln(1+κa)−κa

1+κa −κa

=n+κ2

24πa 3

b−κa +3

2(κa)2+...

.

With κa as running dummy parameter, the partial volume, ν(κa) (Eq. (5.6)),

can be plotted versus the osmotic pressure, p(κa) (Eq. (5.7)), as shown in Fig. 5.1

vch 8 Jul 2005 16:33

5.1 Introduction 91

p

b=16.2

b=16

b=15.8

Fig. 5.1 Critical behaviour at the Electrolyte Phase Transition. Shown are osmotic pressure, p,

versus partial volumes, v, for diﬀerent Bjerrum parameters, b

for the critical region and selected values of b. The critical Bjerrum parameter of

Eq. (5.7) is bc= 16, found at the Debye radius (1/κc)=a, and the corresponding

critical temperature is

Tc=q2

64πεε0kB

.(5.8)

For T<T

c, we observe van der Waals’s wiggles in the osmotic pressure curves.

In this region, the electrolyte divides into two regions with diﬀerent ionic concen-

trations. In the limit TTc, one of the phases is a neutral salt solution and the

other one is a fully ionized electrolyte. This situation is the motivation for the

studies in the subsequent sections, since only little is known about the behavior

of the electrolyte in the instability region, (∂v/∂p)T>0. Although the behaviour

resembles very much the van der Waals equation, note that the instability region

appears at supercritical pressures, however. If there is a stable spatial separation

of phases, their thermodynamic and statistical properties at the interface are of

substantial theoretical interest.

In Section 5.2 we brieﬂy introduce the usual approach to the electrolytic surface

tension based on the image force method, and why our problem - possessing

vch 8 Jul 2005 16:33

92 5 Statistical Theory of Electrolytic Skin Eﬀects

a homogeneous dielectric solvent background - is distinct. In Section 5.3, we

introduce the required integral hierarchy equations, specialized for the case of

ions conﬁned to a certain spatial region, as the starting point of the statistical

theory. In Section 5.4, the analytic solution for the pair distribution function

is derived, using the integral transforms of Fourier, Hilbert and Hankel, as an

alternative to the ”classical” mirror image method. That solution is the very aim

of this paper.

5.2 Electrolytic Skin Eﬀects

The physical problem we are going to treat in this article is sketched in Fig. 5.2.

We imagine a membrane separating a half-space, containing a dilute electrolyte,

Debye Cloud

Electrolyte

Relaxation

Force

Pure Solvent

Fig. 5.2 Forces onto an ion near the interface between electrolyte and pure solvent. Due to the

cut-oﬀ part of the ion cloud, the cloud charge centre is displaced from the central ion

position, causing a relaxation force repelling the ion from the surface

from the other half-space, ﬁlled with the pure solvent. The ions are conﬁned to the

half-space of the electrolyte by, say, an idealised thin non-conducting membrane,

not inﬂuencing the electrostatic ﬁelds of the ions.

Ions located farther away from the interface than the radius of the Debye screening

cloud do not recognise the existence of the boundary; their distribution is the

vch 8 Jul 2005 16:33

5.2 Electrolytic Skin Eﬀects 93

same as for a homogeneous electrolyte. Ions near the surface, however, extend

their unscreened Coulomb ﬁeld into the other half-space and miss the oppositely

charged part of the cloud behind the barrier. This causes the charge centre of the

cloud to be displaced oﬀ the interface, and a resulting relaxation force pulling

the ion away from the surface. Thus, the ion concentration near the surface will

be lowered until the diﬀusion force against the density gradient will balance the

electrostatic force.

Debye Cloud

Image

Force

Image Cloud

Water Air

Fig. 5.3 Forces onto an ion near the surface between an aqueous electrolyte and air. Due to the

image charge, resulting from the diﬀerence between the dielectric constants of water

and air, the image force is repelling the ion from the surface

It is helpful to consider here the similar case of electrolytic surface tension at the

water-air interface, which is studied extensively in the literature. The physical

situation is sketched in Fig. 5.3. Considering the electrostatic problem of a point

charge in a discontinuous dielectric, forces onto the charge appear trying to move

it into a position with lowest potential energy of the polarisation ﬁeld. The case of

a planar interface is analytically solvable and exactly corresponds to the existence

of a virtual image charge behind the surface. This image charge is proportional

to the diﬀerence between the dielectric constants of both media,

q(im)=q(εH2O−εair)

(εH2O+εair).(5.9)

vch 8 Jul 2005 16:33

94 5 Statistical Theory of Electrolytic Skin Eﬀects

It becomes zero if both media have the same dielectric properties, and changes

its sign if the external medium has a higher dielectric constant than that of the

internal solvent. Note that similar closed solutions for this problem are not found

for other physically very interesting but non-planar interfaces like bubbles or

droplets (Landau and Lifschitz (1967)). An overview over known such analytical

solutions can be found in Grinberg (1948) or Smythe (1950).

The limiting law of the surface tension of a dilute electrolyte based on the image-

force models was derived by Wagner (1924), later corrected by Onsager and Sama-

ras (1934). Their result was conﬁrmed and generalised by several authors in the

following. Buﬀ and Stillinger (1956) re-derived it starting from a strictly sta-

tistical approach; we will brieﬂy consider their paper below again because it is

methodically closely related to the calculations performed in this paper.

Nakamura et al. (1982) improved the Onsager-Samaras theory by a self-consistent

approach. Bhuiyan et al. (1991) have extended the Buﬀ-Stillinger method to high

concentrations, using a modiﬁed Poisson-Boltzmann approximation. For the same

purpose, Li et al. (1999) propose a one-dimensional box model of the interface

and the application of Pitzer functions for the ion activities. A similar description,

but based on a modiﬁed mean spherical approximation for the osmotic coeﬃcient,

is given by Yu et al. (2000). Levin and Flores-Mena (2001) replace the grand-

canonical formulation for the computation of surface tension of Onsager-Samaras

by a simpler, canonical one. Hu and Lee (2004), in distinction to Li et al. (1999)

or Yu et al. (2000), propose the use of Patwardhan-Kumar expressions for the

activity coeﬃcients at higher concentrations.

Buﬀ and Stillinger (1956) apply a modiﬁed form of the Kirkwood integral equa-

tion, leading to an integral equation for the pair distribution function

Fab (r1,r2)=1+µzazbf(r1,r2),(5.10)

describing the probability density of ﬁnding one ion ”a” with valence number za

at position r1and another ”b” at r2:

f(r1,r2)+ 1

r12

+1

r(im)

12

+κ2

4π

whole

space

f(r2,r3)

r13

dr3= 0 (5.11)

Here, the distance r12 between the particles 1 and 2 enters into the (scaled)

Coulomb potential 1/rbetween the particles. By including their images, denoted

vch 8 Jul 2005 16:33

5.3 BBGKY-Hierarchy Equations 95

by superscript “(im)”, the integral is extended to the whole space and solved by

those authors using Fourier transform, resulting in a superposition of two Debye

distributions,

f(r1,r2)=−exp (−κr12)

r12

−

exp −κr(im)

12

r(im)

12

.(5.12)

The Debye parameter κ(the reciprocal radius of the ion cloud) and the plasma

parameter µused here are deﬁned in Section 5.4 in which our corresponding

solution will be derived, obtained by a statistical approach similar to that of Buﬀ

and Stillinger. We emphasise, however, that the Buﬀ-Stillinger method brieﬂy

recalled here uses strictly equal expressions for the direct and the image force,

which in the case of water-air surfaces has in fact a ratio of

(εH2O−εair)

(εH2O+εair)≈(80 −1)

(80 + 1) ≈0.98 (5.13)

but not exactly 1. If the diﬀerence between the dielectric constants on both sides

of the interface vanishes, however, the image force disappears, and Eqs. (5.11) and

(5.12) reduce to the normal Debye-H¨uckel bulk theory, without any surface eﬀects.

Thus, the relaxation eﬀect we are going to consider in the following chapters is

completely neglected in the Buﬀ-Stillinger theory.

5.3 BBGKY-Hierarchy Equations

The systematic statistical theory of electrolytes is based on the Bogolyubov-

Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of coupled integro-diﬀerential

equations for the molecular unary, binary etc. distribution functions Fa(r1),

Fab (r1,r2), . . . (see Falkenhagen et al. (1971)), whom we are following in this

section. The s-particle function Fsis deﬁned as

Fs(r1...rs)= Vs

QNexp (−βUN)drs+1 ...drN.(5.14)

Here, QN=exp (−βUN)dr1...drNis the so-called conﬁguration integral, β=

1/kBT,andVis the volume. The N-particle mean interaction potential UNis

vch 8 Jul 2005 16:33

96 5 Statistical Theory of Electrolytic Skin Eﬀects

supposed here to consist of pair and single-particle contributions, the latter in

distinction to other derivations found for only homogeneous systems:

βUN=

N

i=1

Ψi(ri)+1

2

N

i,j=1

ψij (ri,rj).(5.15)

Taking the derivative with respect to the ﬁrst coordinate, we have

β∂

∂r1

UN=β∂Us

∂r1

+

N

j=s+1

∂

∂r1

ψij (r1,rj) (5.16)

with the s-particle potential,

βUs=

s

i=1

Ψi(ri)+

1≤i<j≤s

ψij (ri,rj) (5.17)

and Eq. (5.14) yields

∂Fs

∂r1

=−βVs

QN

∂Us

∂r1exp (−βUN)drs+1 ...drN

(5.18)

−Vs

QN⎡

⎣

N

j=s+1

∂ψ1j(r1,rj)

∂r1⎤

⎦exp (−βUN)drs+1 ...drN.

Comparing this equation with the deﬁnition Eq. (5.14), we get the s-particle

hierarchy equation as

∂Fs

∂r1

=−β∂Us

∂r1

Fs−1

V

N

j=s+1 ∂ψ1,j (r1,rj)

∂r1

Fs+1 (r1,...,rs,rj)drj,(5.19)

which reads for the lowest two orders, after introducing particle classes ”a”, ”b”

etc.,

∂Fa

∂r1

+Fa

∂Ψa

∂r1

+

b

nbdr2

∂ψab

∂r1

Fab =0,(5.20)

vch 8 Jul 2005 16:33

5.3 BBGKY-Hierarchy Equations 97

∂Fab

∂r1

+Fab ∂Ψa

∂r1

+∂ψab

∂r1+

c

ncdr3

∂ψac

∂r3

Fabc =0.(5.21)

Here, kBTψ

ab (r1,r2) is the pair interaction potential between ions of kind ”a”

and ”b”, located at positions r1and r2.

We reformulate the hierarchy equations after separating Boltzmann factors as

Fa=faexp (−Ψa),

Fab =fab exp (−Ψa−Ψb),(5.22)

Fabc =fabc exp (−Ψa−Ψb−Ψc).

This substitution into Eqs. (5.20) and (5.21) leads to the simpliﬁed equations

∂fa

∂r1

+

b

nbdr2exp (−Ψb)∂ψab

∂r1

fab =0,(5.23)

∂fab

∂r1

+fab

∂ψab

∂r1

+

c

ncdr3exp (−Ψc)∂ψac

∂r1

fabc =0.(5.24)

To simulate the ion conﬁnement, we introduce a barrier potential, B1, in the

form

Ψa(r1)=B, if r1/∈Ω,

0,otherwise (5.25)

to keep the ions exclusively within the region (e. g. half-space) Ω in the limit

B→∞. We see from Eqs. (5.23) and (5.24) that discontinuous functions Ψa

lead to discontinuous functions Fa, but leave the functions fasmooth. Applying

Eq. (5.25), we ﬁnally get the desired equations

∂fa(r1)

∂r1

+

b

nb

Ω

dr2

∂ψab (r1,r2)

∂r1

fab (r1,r2)=0,(5.26)

∂fab (r1,r2)

∂r1

+fab (r1,r2)∂ψab (r1,r2)

∂r1

(5.27)

+

c

nc

Ω

dr3

∂ψac (r1,r3)

∂r1

fabc (r1,r2,r3)=0.

vch 8 Jul 2005 16:33

98 5 Statistical Theory of Electrolytic Skin Eﬀects

These hierarchy equations look formally almost identical to those obtained for

homogeneous electrolytes (Falkenhagen and Ebeling (1971)), except that the in-

tegrals are extended only over a limited spatial region Ω here instead of the

whole space. This, of course, destroys the former homogeneity and isotropy of

the mathematical solution functions and makes their analytical calculation sig-

niﬁcantly more diﬃcult, since merely relative coordinates will no longer suﬃce

for their formulation.

The 1-particle equation Eq. (5.26) explicitly expresses the balance between the

diﬀusion force (ﬁrst term) and the relaxation force (second term), controlling

the concentration gradient in the skin layer. For its computation, the 2-particle

distribution is required as the solution of Eq. (5.27), which will be derived in the

next section.

5.4 Analytical Solution for the Pair Distribution

Function

We look for an analytical solution for the pair distribution function Fab =

fab exp (−Ψa−Ψb), obeying Eq. (5.27)

∂fab

∂r1

+fab

∂ψab

∂r1

+

c

nc

Ω

dr3

∂ψac

∂r1

fabc =0.(5.28)

We apply an asymmetric superposition approximation for the ternary function,

fabc ≈fabfbc , which is known to permit the derivation of the correct Debye-H¨uckel

limiting law in the homogeneous case (Falkenhagen et al. (1971))

∂ln fab

∂r1

+∂ψab

∂r1

+

c

nc

Ω

dr3

∂ψac

∂r1

fbc =0.(5.29)

We can write the left-hand side as a gradient of a scalar function and integrate,

yielding

ln fab +ψab +

c

nc

Ω

dr3ψacfbc =0.(5.30)

vch 8 Jul 2005 16:33

5.4 Analytical Solution for the Pair Distribution Function 99

The integration constant is zero, as becomes evident from the limiting case of

inﬁnitely distant particles. We consider only pure Coulomb interaction potentials:

ψab (r1,r2)= zazbl

|r1−r2|=zazb

l

r12

.(5.31)

For a system of point charges, only three length-like quantities appear in these

equations, the spatial distance |r|, the Landau length l=e2/(4πεε0kBT), and the

Debye radius, rD=1/κ, reﬂecting the ion density by κ2=4πl

a

naz2

a.Notethat

these deﬁnitions are slightly diﬀerent from those used in Section 5.1; here we like

to explicitly include mixed and unsymmetrical electrolytes, and for the purpose

of having a unique Landau length we have deﬁned it based on elementary charges

einstead of ion charges qa, writing valence numbers zaseparately. Scaling with

the Debye radius to dimensionless length variables x=κr,µ=κl,weseethat

the classical Debye distribution, depending only on the dimensionless distance

x12 =|x1−x2|,

gab (r1,r2)=−zazbµ

|x1−x2|exp (−|x1−x2|)≡zazbµ·g(x1,x2) (5.32)

becomes a linear function of the plasma parameter µ. Thus we can expect to ob-

tain the limiting laws, corresponding to the Debye-H¨uckel theory, as lowest order

expansion terms with respect to the plasma parameter, if all lengths including

the concentrations

nc=νcn=1

4π

νc

z2

κ2

l(5.33)

are expressed in dimensionless form. Here, n=

a

nais the total concentration,

νa=na/n are the mole fractions, the average valence

a

zaνa= 0 vanishes due

to electro-neutrality, and its mean-square we abbreviate with z, i.e. z2=

a

νaz2

a.

In these terms, the Debye parameter is given by κ2=4πnlz2.Notethat

ψab (r1,r2)= zazbµ

|x1−x2|(5.34)

is of ﬁrst order in µ,too.

vch 8 Jul 2005 16:33

100 5 Statistical Theory of Electrolytic Skin Eﬀects

Eq. (5.30) becomes now

ln fab +zazbµ1

x12

+1

4π

za

z2

c

νczc

Ω

dx3

1

x13

fbc =0.(5.35)

A series expansion with respect to the plasma parameter, µ=κl,isdoneupthe

linear term, fab =1+zazbµf +Oµ2, resulting in the integral equation

f(x1,x2)+ 1

x12

+1

4π

Ω

dx3

1

x13

f(x2,x3)=0.(5.36)

The solution fis a universal mathematical function, independent of any physical

parameters except the shape of the interface, which, by the way, has not been

restricted to be necessarily planar so far.

The Debye function g, Eq. (5.32), satisﬁes the corresponding ”bulk” equation

(Buﬀ and Stillinger (1956), Falkenhagen et al. (1971)), where the integral is ex-

tended over the whole space Ω + ¯

Ω:

g(x1,x2)+ 1

x12

+1

4π

Ω+¯

Ω

dx3

1

x13

g(x2,x3)=0.(5.37)

In our case, the function fmust converge towards gfor particles far from the sur-

face. We subtract Eq. (5.37) from Eq. (5.32), and obtain for the surface anomaly

function, h=f−g,therelation

h(x1,x2)+ 1

4π

Ω

dx3

h(x2,x3)

|x1−x3|=1

4π

¯

Ω

dx3

g(x2,x3)

|x1−x3|.(5.38)

This surface-eﬀect equation does no longer contain explicit long-range Coulomb

forces. The right-hand side, which is extended only over the ion-free space region

and hence possesses no singularities in the integrand, quantiﬁes the cut-oﬀ parts

of the ion clouds (Fig. 5.2), while the integral on the left describes the internal

reaction of the ion distribution on this external force. Note that both these eﬀects

are of the same order and appear already in the limit of inﬁnite dilution. The

deformation of the ion cloud close to the interface, often expressed in terms of

a distance-dependent Debye parameter, κ(z), is usually neglected in approaches

like those of Onsager-Samaras or Buﬀ-Stillinger (compare Section 5.2).

vch 8 Jul 2005 16:33

5.4 Analytical Solution for the Pair Distribution Function 101

In Eq. (5.38), the position x2plays merely the role of an external parameter;

we will in some cases suppress it in the following calculation steps for simpler

writing. We can solve Eq. (5.38) by Fourier transform. However, for the case of

a planar interface at z= 0, it is helpful to consider ﬁrst some special properties

of half-sided Fourier transforms. Let

˜

f(k)=drexp (ikr)f(r) (5.39)

be a complete 3D Fourier transform, i.e. carried out over the whole space. Now,

we transform backward in one dimension only

˜

fz(z,kx,k

y)= 1

2πdkzexp (−izkz)˜

f(k).(5.40)

We make this function half-sided by means of

˜

f+(z)=1

2(1 + sgn (z)) ˜

fz(z) (5.41)

for positive z,or

˜

f−(z)=1

2(1 −sgn (z)) ˜

fz(z) (5.42)

for negative z, and transform it back into the k-space:

˜

f+(k)=dz exp (ikzz)˜

f+(z) (5.43)

=1

2˜

f(k)+ 1

4πdk˜

fkx,k

y,k

dz exp iz kz−ksgn(z).

The inner z-integral has the principal value (Brychkov and Prudnikov (1977),

Buttkus (2000))

dz exp (ikz)sgn(z)= 2i

k.(5.44)

Thus, the multiplication with the signum function in the conﬁguration space

becomes a convolution with 1/k in k-space. This convolution is well-known as

the so-called Hilbert transform (Poularikas (1996), Buttkus (2000)) which is fre-

quently applied in signal processing, especially for the construction of causal (i.e.

vch 8 Jul 2005 16:33

102 5 Statistical Theory of Electrolytic Skin Eﬀects

half-sided) ﬁlters. This way, we can describe here a half-sided function in r-space

by a Hilbert transform in k-space:

˜

f±(k)=1

2˜

f(k)±i

2πdk˜

f(kx,k

y,k

)

(kz−k)(5.45)

=1

2dkδkz−k±i

π(kz−k)˜

fkx,k

y,k

.

The corresponding operators P±form projection operators into the particular

half-spaces:

P±=1

2dkδk−k±i

π(k−k).(5.46)

As formal 3D operators, P±take the shape

P±=1

2dkδkx−k

xδky−k

yδkz−k

z±i

π(kz−k

z).(5.47)

If ˜

f(k) is the Fourier transform of a function f(r), so ˜

f+(k)=P+˜

f(k)is the

Fourier transform of a function f+(r), which for z>0coincideswithf(r), and

vanishes for z<0.

After these preliminary remarks, we rewrite now Eq. (5.38) in the form of an

integral over the whole space,

(1 + sgn (z1)) h(x1)= 1

4π

Ω+¯

Ω

dx3

|x1−x3|[−(1 + sgn (z3)) h(x3) (5.48)

+(1−sgn (z3)) g(x3)] ,

multiply with exp (ikx1), and integrate x1over the whole space, with subscript

”+” denoting the half-space containing the ions:

˜

h+(k)≡1

2

Ω+¯

Ω

dx1exp (ikx1)(1+sgn(z1)) h(x1) (5.49)

=1

2k2

Ω+¯

Ω

dx3exp (ikx3)[−(1 + sgn (z3)) h(x3)+(1−sgn (z3)) g(x3)]

≡1

k2−˜

h+(k)+˜g−(k).

vch 8 Jul 2005 16:33

5.4 Analytical Solution for the Pair Distribution Function 103

This way, we can easily solve this equation for ˜

h+(k):

˜

h+(k)= ˜g−(k)

1+k2.(5.50)

Thus, we have to calculate the right-hand side function ˜g−(k),andthentoper-

form the backward transform of ˜

h+(k). First, we compute ˜g(k) by Fourier trans-

form,

˜g(k)=−dx1exp (ikx1)exp (−|x1−x2|)

|x1−x2|=−4πexp (ikx2)

1+k2(5.51)

and apply to the result the Hilbert transform, i.e. the operator P−, Eq. (5.47):

˜g−(k)=−2πdkδkx−k

xδky−k

y×(5.52)

×δkz−k

z−i

π(kz−k

z)exp (ikx2)

1+k2

=−2πexp (ikx2)

1+k2+2iexp (ikxx2+ikyy2)×

×dk

zexp (ik

zz2)

(kz−k

z)1+k2

x+k2

y+k2

z.

Inserting ˜g−(k) into Eq. (5.50) yields

˜

h+(k)=−2πexp (ikx2)

(1 + k2)2+2iexp (ikxx2+ikyy2)

(1 + k2)×(5.53)

×dk

zexp (ik

zz2)

(kz−k

z)1+k2

x+k2

y+k2

z.

The backward Fourier transform leads to the 4-dimensional integral

h(x1,x2)= 1

(2π)3dkexp (−ikx1)˜

h+(k) (5.54)

=−1

4π2dkexp (ik(x2−x1))

(1 + k2)2

+i

4π3dkexp (ik(x2−x1))

(1 + k2)dk

zexp (i(k

z−kz)z2)

(kz−k

z)1+k2

x+k2

y+k2

z.

vch 8 Jul 2005 16:33

104 5 Statistical Theory of Electrolytic Skin Eﬀects

The required calculation using known standard techniques and integral tables

(Abramowitz and Stegun (1965), Brychkov and Prudnikov (1977), Prudnikov et

al. (1981)) is somewhat lengthy and eventually results in

h(x1,x2)=−exp (−|x2−x1|)

4+1

4

∞

1

[(1 + |z2−z1|t)exp(−|z2−z1|t)

−exp (−(z2+z1)t)] J0ρt2−1dt

t2.(5.55)

Here, J0is the Bessel function, and ρ=(x2−x1)2+(y2−y1)2is the par-

ticle distance parallel to the interface, scaled to the Debye radius, in cylinder

coordinates.

Referring to tables of Hankel transforms (Bateman and Erdelyi (1954), Wheelon

(1968), Oberhettinger (1972)), we mention the mathematical identity

∞

01+βy2+α2exp −βy2+α2J0(xy)ydy

(y2+α2)3/2=

=1

αexp −αx2+β2.(5.56)

This very special but physically important Hankel transform was used in similar

form already by Sommerfeld (1909); its proof is found e.g. in Watson (1995).

Its application to Eq. (5.55) shows that the bulk terms, i.e. those independent

of the distance from the surface, cancel each other completely as was expected

for physical reasons. Then, the solution simpliﬁes signiﬁcantly to only a surface

correction of the Debye distribution:

h(x1,x2)=−1

4

∞

1

J0ρt2−1exp (−(z2+z1)t)dt

t2.(5.57)

Making use of Eq. (5.56) again in a modiﬁed form, we can further reformulate

Eq. (5.57) into the simpler integral

h(x1,x2)=−1

4

∞

1

exp −ρ2+(z2+z1)2t2dt

t2.(5.58)

vch 8 Jul 2005 16:33

5.4 Analytical Solution for the Pair Distribution Function 105

An explicitly evaluated analytical form of this integral is not known to the au-

thors. For two ions on the surface itself, z1=z2= 0, the special case solution is

simply

h(x1,x2)|surface =−exp (−ρ)

4.(5.59)

Another integrable, interesting special case is ρ= 0, i.e. two ions located on a

line normal to the interface, at diﬀerent distances.

h(x1,x2)|normal =−1

4

∞

1

exp (−(z2+z1)t)dt

t2(5.60)

=−1

4[exp (−z2−z1)+(z2+z1)Ei(−z2−z1)] .

It has a constant limit of −1/4 at the surface with ﬁrst deviations in the order of

(z1+z2)ln(z1+z2), and an exponential decay towards the bulk.

To summarise, the generalized Debye law for the binary distribution function of

a dilute electrolyte at a planar interface, impenetrable for ions, is, after returning

to usual, unscaled length units

fab (r1,r2)=1−zazblexp (−κr12)

r12

+ (5.61)

+κ

4

∞

1

exp −κρ2

12 +(z2+z1)2t2dt

t2⎫

⎬

⎭

+Oµ2.

This is the ﬁnal result we have been looking for.

Acknowledgements

The authors thank Wolfgang Wagner for providing his experimental results on

the critical behaviour of ﬂuids under gravity, and Michail A. Anisimov for helpful

discussions about this issue. They are grateful to J¨urn W. P. Schmelzer for oﬀering

the opportunity to present this study on the Nucleation Workshop 2005 in Dubna,

and his assistance in preparing a printable version. The authors further thank A.

Schr¨oder and B. Sievert for their permanent help in accessing special literature.

vch 8 Jul 2005 16:33

106 5 Statistical Theory of Electrolytic Skin Eﬀects

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