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Statistical Theory of Electrolytic Skin Effects

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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 dissolved 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 distribution functions are briefly derived, and the solution for the pair distribution functions is calculated analytically by means of Fourier, Hilbert, and Hankel integral transforms.
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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 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 Aand 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)
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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+nf
=nn±
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++n1
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
bκa +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 (1c)=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
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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
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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(εH2Oεair)
(εH2O+εair).(5.9)
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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
(εH2Oεair)
(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-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)+
1i<js
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
Fs1
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, kB
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 (ΨaΨb),(5.22)
Fabc =fabc exp (ΨaΨbΨc).
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/,
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
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
dr3
∂ψac (r1,r3)
r1
fabc (r1,r2,r3)=0.
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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 (ΨaΨb), obeying Eq. (5.27)
∂fab
r1
+fab
∂ψab
r1
+
c
nc
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
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
dr3ψacfbc =0.(5.30)
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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
|r1r2|=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, 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 =|x1x2|,
gab (r1,r2)=zazbµ
|x1x2|exp (−|x1x2|)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
|x1x2|(5.34)
is of first order in µ,too.
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100 5 Statistical Theory of Electrolytic Skin Effects
Eq. (5.30) becomes now
ln fab +zazbµ1
x12
+1
4π
za
z2
c
νczc
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
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+¯
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=fg,therelation
h(x1,x2)+ 1
4π
dx3
h(x2,x3)
|x1x3|=1
4π
¯
dx3
g(x2,x3)
|x1x3|.(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 kzksgn(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
)
(kzk)(5.45)
=1
2dkδkzk±i
π(kzk)˜
fkx,k
y,k
.
The corresponding operators P±form projection operators into the particular
half-spaces:
P±=1
2dkδkk±i
π(kk).(5.46)
As formal 3D operators, P±take the shape
P±=1
2dkδkxk
xδkyk
yδkzk
z±i
π(kzk
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+¯
dx3
|x1x3|[(1 + sgn (z3)) h(x3) (5.48)
+(1sgn (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+¯
dx1exp (ikx1)(1+sgn(z1)) h(x1) (5.49)
=1
2k2
Ω+¯
dx3exp (ikx3)[(1 + sgn (z3)) h(x3)+(1sgn (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 (−|x1x2|)
|x1x2|=4πexp (ikx2)
1+k2(5.51)
and apply to the result the Hilbert transform, i.e. the operator P, Eq. (5.47):
˜g(k)=2πdkδkxk
xδkyk
y×(5.52)
×δkzk
zi
π(kzk
z)exp (ikx2)
1+k2
=2πexp (ikx2)
1+k2+2iexp (ikxx2+ikyy2)×
×dk
zexp (ik
zz2)
(kzk
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)
(kzk
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(x2x1))
(1 + k2)2
+i
4π3dkexp (ik(x2x1))
(1 + k2)dk
zexp (i(k
zkz)z2)
(kzk
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 (−|x2x1|)
4+1
4
1
[(1 + |z2z1|t)exp(−|z2z1|t)
exp ((z2+z1)t)] J0ρt21dt
t2.(5.55)
Here, J0is the Bessel function, and ρ=(x2x1)2+(y2y1)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
J0ρt21exp ((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 (z2z1)+(z2+z1)Ei(z2z1)] .
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)=1zazblexp (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
5.5 References
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Leipzig, 1971).
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vch 8 Jul 2005 16:33
5.5 References 107
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27. V. P. Skripov and D. Yu. Ivanov: On the Second Crossover Near to the Critical Point.
Lecture on the 8 Research Workshop on Nucleation Theory and Applications, Dubna,
Russia, October 2004.
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New York, Toronto, London, 1950).
29. A. Sommerfeld, Ann. Physik Chemie 28, 682 (1909).
30. R. Thom, Structural Stability and Morphogenesis (W. A. Benjamin, Inc., Reading,
Massachusetts, 1975).
31. C. Wagner, Phys. Z. 25, 474 (1924).
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Press, 1995).
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(Holden-Day, San Francisco, Cambridge, London, Amsterdam, 1968).
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vch 8 Jul 2005 16:33
... Kombiniert man die Gibbs-Funktion des Meerwassers mit den entsprechenden Potenzialen für reines Wasser und Dampf [31,52], sowie für Eis [23,24], so sind auch die thermodynamischen Gleichgewichte in den wichtigsten Mehrphasensystemen , die im Ozean und an seiner Oberfläche vorgefunden werden , quantitativ umfassend beschrieben. Viele offene Fragen bestehen aber noch bezüglich der Eigenschaften an den Phasengrenzen, wie etwa der Oberflächenspannung von Elektrolyten [14, 44], wobei neuere Untersuchungen z. B. auch auf oszillierende Konzentrationsverteilungen der gelösten Teilchen in der Nähe der Grenzfläche hinweisen [34, 43], im Gegensatz zu den bekannten Grenzgesetzen. ...
Chapter
Full-text available
Wir diskutieren die thermodynamischen Bilanzgleichungen der Ostsee, eines relativ geschlossenen Randmeers, unter den Gesichtspunkten (i) der Zustandsgleichung des Meerwassers, d.h. eines gemischten Elektrolyten mittlerer Konzentration im Gleichgewicht, (ii) von Näherungsl¨osungen des Systems der hydrodynamischen Transportgleichungen, wie sie f¨ur die ozeanografische Praxis relevant sind, und (iii) anhand von Abschätzungen der Jahresbilanzen der Ostsee für Wasser, Salz, Strahlung, Energie und Entropie. Wir begründen unsere Feststellung, dass der mittlere klimatische Zustand der Ostsee von Fluktuationen und einem nichtlinearen Antwortverhalten dominiert ist und sich fern vom Gleichgewicht befindet.
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  • M Abramowitz
  • I A Stegun
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, Inc., New York, 1965).