Spatially confined system interacting with Yukawa potential
Hard wall confined fluid with Yukawa potential of interaction is considered. A solution of the inhomogeneous Ornstein-Zernike equation for the pair correlation function is obtained. The expression for the particle density profile is found by the method of functional differentiation of Helmholtz fr...
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| Zitieren: | Spatially confined system interacting with Yukawa potential / M.F. Holovko, I.Y. Kravtsiv, E.M. Soviak // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 137-150. — Бібліогр.: 18 назв. — англ. |
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irk-123456789-1199712017-06-11T03:04:01Z Spatially confined system interacting with Yukawa potential Holovko, M.F. Kravtsiv, I.Y. Soviak, E.M. Hard wall confined fluid with Yukawa potential of interaction is considered. A solution of the inhomogeneous Ornstein-Zernike equation for the pair correlation function is obtained. The expression for the particle density profile is found by the method of functional differentiation of Helmholtz free energy with respect to the external field. The contribution to the behavior of particle density near the surface comes from the initial potential as well as from the collective screening interaction effects. It is shown that the contact value of the profile satisfies the condition of the contact theorem. Dependence of the adsorption coefficient on the particle density is calculated. It is also shown that in the case of repulsive Yukawa interaction the sign of the adsorption coefficient changes with the increase of the particle density. Розглянуто обмежений твердою стiнкою плин з потенцiалом взаємодiї Юкави. Отримано розв’язок просторово неоднорiдного рiвняння Орнштейна-Цернiке для парної кореляцiйної функцiї. Методом функцiонального диференцiювання вiльної енергiї за зовнiшнiм полем знайдено вираз для профiлю густини частинок. Показано, що вклад у поведiнку густини частинок поблизу поверхнi вносить як вихiдний потенцiал, так i колективнi ефекти екранування взаємодiї. Розраховано залежнiсть коефiцiєнта адсорбцiї вiд густини числа частинок. Показано, що у випадку притягувальної взаємодiї iз збiльшенням густини числа частинок системи змiнюється знак коефiцiєнта адсорбцiї. 2009 Article Spatially confined system interacting with Yukawa potential / M.F. Holovko, I.Y. Kravtsiv, E.M. Soviak // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 137-150. — Бібліогр.: 18 назв. — англ. 1607-324X PACS: 05.20.Jj, 05.70.Np, 61.20.-p, 68.03.-g DOI:10.5488/CMP.12.2.137 http://dspace.nbuv.gov.ua/handle/123456789/119971 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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| description |
Hard wall confined fluid with Yukawa potential of interaction is considered. A solution of the inhomogeneous
Ornstein-Zernike equation for the pair correlation function is obtained. The expression for the particle density
profile is found by the method of functional differentiation of Helmholtz free energy with respect to the external
field. The contribution to the behavior of particle density near the surface comes from the initial potential
as well as from the collective screening interaction effects. It is shown that the contact value of the profile
satisfies the condition of the contact theorem. Dependence of the adsorption coefficient on the particle density
is calculated. It is also shown that in the case of repulsive Yukawa interaction the sign of the adsorption
coefficient changes with the increase of the particle density. |
| format |
Article |
| author |
Holovko, M.F. Kravtsiv, I.Y. Soviak, E.M. |
| spellingShingle |
Holovko, M.F. Kravtsiv, I.Y. Soviak, E.M. Spatially confined system interacting with Yukawa potential Condensed Matter Physics |
| author_facet |
Holovko, M.F. Kravtsiv, I.Y. Soviak, E.M. |
| author_sort |
Holovko, M.F. |
| title |
Spatially confined system interacting with Yukawa potential |
| title_short |
Spatially confined system interacting with Yukawa potential |
| title_full |
Spatially confined system interacting with Yukawa potential |
| title_fullStr |
Spatially confined system interacting with Yukawa potential |
| title_full_unstemmed |
Spatially confined system interacting with Yukawa potential |
| title_sort |
spatially confined system interacting with yukawa potential |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119971 |
| citation_txt |
Spatially confined system interacting with Yukawa potential / M.F. Holovko, I.Y. Kravtsiv, E.M. Soviak // Condensed Matter Physics. — 2009. — Т. 12, № 2. — С. 137-150. — Бібліогр.: 18 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT holovkomf spatiallyconfinedsysteminteractingwithyukawapotential AT kravtsiviy spatiallyconfinedsysteminteractingwithyukawapotential AT soviakem spatiallyconfinedsysteminteractingwithyukawapotential |
| first_indexed |
2025-07-08T17:00:06Z |
| last_indexed |
2025-07-08T17:00:06Z |
| _version_ |
1837098871403053056 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 2, pp. 137–150
Spatially confined system interacting with Yukawa
potential
M.F.Holovko1,2, I.Y.Kravtsiv2, E.M.Soviak1
1 Institute for Condensed Matter Physics NASU, 1 Svientsitskii Street, 79011 Lviv, Ukraine
2 I. Franko National University of Lviv, Faculty of physics, Dragomanov Str.,12, 79005 Lviv, Ukraine
Received May 12, 2009, in final form May 19, 2009
Hard wall confined fluid with Yukawa potential of interaction is considered. A solution of the inhomogeneous
Ornstein-Zernike equation for the pair correlation function is obtained. The expression for the particle density
profile is found by the method of functional differentiation of Helmholtz free energy with respect to the external
field. The contribution to the behavior of particle density near the surface comes from the initial potential
as well as from the collective screening interaction effects. It is shown that the contact value of the profile
satisfies the condition of the contact theorem. Dependence of the adsorption coefficient on the particle density
is calculated. It is also shown that in the case of repulsive Yukawa interaction the sign of the adsorption
coefficient changes with the increase of the particle density.
Key words: nonhomogeneous Ornstein-Zernike equation, hard wall, contact theorem, adsorption coefficient,
Yukawa potential
PACS: 05.20.Jj, 05.70.Np, 61.20.-p, 68.03.-g
1. Introduction
The study of systems with the long-range component of a Yukawa-like potential of interaction
is of significant theoretical interest. Despite the simplicity of the actual potential, it allows one
to describe structural and thermodynamic properties. In addition, there exist analytical solutions
in the mean spherical approximation for a system of hard spheres with the Yukawa potential
of interaction [1]. Recently, the density field theory was applied to the description of the fluid
interacting with the Yukawa potential [2,3]. The Yukawa potential can be used in describing the
systems of charged particles as well as neutral-particle systems. Sets of Yukawa potentials are
used to approximate real interaction potentials in simple liquids [4], colloid fluids [5,6] and other
systems [7].
Despite considerable progress in the study of spatially uniform systems of particles with the
Yukawa potential, the research of non-uniform systems remains a relevant task. A lot more results
have been obtained for spatially non-uniform systems of charged particles. Notably, expressions
for the pair and singlet distribution functions for a system of point ions confined by a hard wall
have been found [8–10]. The results of the pioneering work [11] have made it possible to obtain
an analytical form of the contribution of short-range interactions to the structural properties of
spatially confined systems. These and other works have significantly enhanced our understanding
of surface effects in systems with electrostatic interaction.
When calculating structural properties of spatially non-uniform systems, it is important to
verify that the results found satisfy certain exact relationships. Of special importance are exact
relationships for the contact values of density profiles [12] and those of charge profiles [13], the
so-called contact theorems. Specifically, in accordance with the contact theorem for the density
profile, the contact value of particle density near a hard wall is determined by the pressure of a
fluid without a wall. For a system of point ions it is shown that in the random phase approximation
the expressions found for the density profile meet the requirements of the contact theorem [14].
c© M.F.Holovko, I.Y.Kravtsiv, E.M.Soviak 137
M.F.Holovko, I.Y.Kravtsiv, E.M.Soviak
Spatially non-uniform systems with the Yukawa potential have been studied primarily by com-
puter simulations and numerical calculations of integral equations [15].
The present work focuses on the analytical research of thermodynamic and structural proper-
ties of a spatially non-uniform one-component system of particles with the Yukawa potential of
interaction. We shall obtain expressions for the Helmholtz free energy, pair correlation function
and density distribution for such a system. It will be shown that for the latter, the contact theorem
is valid.
2. The model
We shall consider a two-phase system of particles within volume V , in which the phases are
separated by the plain z = 0. The upper part of space ( z > 0 ) is filled with the upper phase having
the particle density ρ+ , whereas the lower part of space ( z < 0 ) is filled with the lower phase
having the particle density ρ− . The interaction potential between any two particles, the locations
of which are given by radius vectors ~R1 and ~R2 in the Descartes coordinate system, is presented
as the sum of short-range u(R12) potential and long-range Yukawa potential
Φ(R12) = A
e−αR12
R12
, (1)
where A is the constant of interaction, R12 is the distance between two particles.
The potential energy of the system consists of the energy of inter-particle interaction and the
energy of particles under an external field
UN =
∑
j<i
u(Rij) +
∑
j<i
Φ(Rij) +
N
∑
a,i
wa(zi) , (2)
where wa(zi) is the external field forming the phase interface, zi is the distance between a wall and
a particle. The index “a” indicates which phase a particle belongs to and takes the value “+” for
the upper phase and “−” for the lower phase,
w+(~Ri) =
{
0, zi > 0 ,
∞, zi < 0 ,
w−(~Ri) =
{
∞, zi > 0 ,
0, zi < 0 .
(3)
From here on we shall consider only the long-range component of the potential energy similar
to the way it had been done within the collective variables method when short-range interactions
were taken into account using the method of functional differentiation [8]. Let us introduce the
microscopic particle density
ρ̂(~R) =
N
∑
i=1
δ(~R − ~Ri). (4)
In (1) let us do the Fourier transformation of the pair potential of interaction Φ(Rij)
Φ̃(k) =
∫
V
Φ(R) ei~k ~R d~R (5)
and separate the self-energy part of the potential energy (i = j), then for the potential energy of
the system we shall get the equality:
U l
N =
1
2
1
V
∑
~k
Φ̃(k)ρ̂~k ρ̂−~k − 1
2
N
V
∑
~k
Φ̃(k) +
N
∑
a, i
wa(zi) , (6)
138
Spatially confined system interacting with Yukawa potential
where the index l indicates that the potential energy takes into account only the long-range com-
ponents of inter-particle interaction, and
ρ̂~k =
N
∑
a,i
exp(i~k ~Ri) (7)
is the Fourier transform of the microscopic particle density.
The Helmholtz free energy of the system can be determined from the relation
F l
N = F id
N + F l,ex
N = F id
N − T ln Ql
N , (8)
where F id
N is the Helmholtz free energy of the system without particle interaction,
F id
N = −T
N
∑
a,i
Na
{
1 − ln
(
ρaΛ3
)}
, (9)
Ql
N is the configuration integral of a system of particles with the long-range interaction
Ql
N =
1
V N
∫
V
N
∏
i
~Ri exp
(
− 1
T
U l
N
)
, (10)
and F l,ex
N is the contribution of long-range interactions to the Helmholtz free energy, T is the
temperature measured in units of energy, N+ and N− are the numbers of particles in the upper
and the lower half-space accordingly, and Λ is the thermal de Broglie wavelength.
3. Random phase approximation for the Helmholtz free energy of the sys-
tem
Let us evaluate the configuration integral Ql
N, which has the following form in terms of the
collective variables method:
Ql
N =
1
V N
∫
∏
i
d~Ri
∫
∏
~k
dρ~k
∫
∏
~k
dω~k exp
−1
2
∑
~k
ν̃(k)ρ~k ρ−~k
+
1
2
N
∑
~k
ν̃(k) − 1
T
N
∑
a, i
wa(zi) + 2iπ
∑
~k
ω ~−k
(
ρ−~k − ρ̂−~k
)
, (11)
where for convenience we have introduced
ν̃(k) =
1
T V
Φ̃(k) =
A
T V
4π
k2 + α2
. (12)
Integration of Gaussian form with respect to collective variables and making use of the cumula-
tive development technique [8] for the repulsive interaction makes it possible to do the integration
in the space of individual variable particles. Then, by taking into account only the zero, the first,
and the second cumulants, for the configuration integral in the random phase approximation we
shall get
Ql,RPA
N = exp
M0 +
1
2
N
∑
~k
ν̃(k) +
1
2
∑
~k
ln
2π
ν̃(k)
∫
∏
~k
dω~k
× exp
2π2
∑
~k1,~k2
g̃−1(~k1, ~k2)ω~k1
ω~k2
− 2πi
∑
~k
M1(~k)ω~k
, (13)
139
M.F.Holovko, I.Y.Kravtsiv, E.M.Soviak
where M0(~k), M1(~k), M2(~k1, ~k2) denote the zero, the first, and the second cumulants, respectively
M0 = N+ ln
1
V
∫
V
d~R e−
1
T
w+(z)
+ N− ln
1
V
∫
V
d~R e−
1
T
w+(z)
, (14)
M1(~k) = δ~p, 0
ρ+
∫
V
d~R e−
1
T
w+(z)eiqz + ρ−
∫
V
d~R e−
1
T
w−(z)eiqz
, (15)
M2(~k1, ~k2) = δ~p1+~p2,0ρ+
∫
V
d~R e−
1
T
w+(z) ei(q1+q2)z + δ~p1+~p2,0ρ−
∫
V
d~R e−
1
T
w−(z) ei(q1+q2)z
− δ~p1,0δ~p2,0ρ+
1
V+
∫
V
d~R e−
1
T
w+(z) eiq1z
∫
V
d~R e−
1
T
w+(z)eiq2z
− δ~p1,0δ~p2,0ρ−
1
V−
∫
V
d~R e−
1
T
w−(z) eiq1z
∫
V
d~R e−
1
T
w−(z) eiq2z . (16)
V+ =
∫
V
d~R e−
1
T
w+(z) , V− =
∫
V
d~R e−
1
T
w−(z) , (17)
and ~k = ~p +~izq (~iz is a unit basis vector perpendicular to the surface and ~p is a vector parallel to
the surface), δ~p, 0, δ~p1+~p2, 0 is the Kroneker delta.
In (13) we have introduced an infinite matrix G
−1 whose elements are
g̃−1(~k1, ~k2) = − 1
ν̃(k)1
δ~k1+~k2, 0 − M2(~k1, ~k2) (18)
and the elements of the inverse matrix G are the Fourier transforms of the so-called screened
potentials and can be determined from the equation
g̃(~k1, ~k2) = − ν̃(k1) − ν̃(k1)
∑
~k
M2(−~k1, ~k2)g̃(~k1, ~k2) . (19)
In (13) we shall convert the square matrix form with respect to variables ω~k into the diagonal
form similar to the way it was done in work [10]. Then, as a result of integration in the random
phase approximation, the configuration integral of a two-phase spatially non-uniform system of
point particles with the Yukawa interaction can be written in the form
Ql,RPA
N =exp
M0 +
1
2
N
∑
~k
ν̃(k) − 1
2
ln det{1 + ν̃M2} +
1
2
∑
~k1,~k2
g̃(~k1, ~k2)M1(~k1)M1(~k2)
. (20)
Let us examine the last term in the equality (20) taking (19) into account. We have
∑
~k1,~k2
g̃( ~k1, ~k2)M1(~k1) M1(~k2) = −
∑
~k1
ν̃(k1)M1(~k1) M1(−~k1)
−
∑
~k1,~k2,~k
ν̃(k1)M2(−~k1, ~k)g̃(~k, ~k2)M1(~k1) M1(~k2)
= −
∑
~k1
ν̃(k1)M1(~k1) M1(−~k1) , (21)
140
Spatially confined system interacting with Yukawa potential
at the thermodynamic limit (N → ∞, V → ∞, ρ = const)
∑
~k1,~k2,~k
ν̃(k1)M2(−~k1, ~k)g̃(~k, ~k2)M1(~k1) M1(~k2) = 0. (22)
Then, taking into account only the long-range inter-particle interactions, for the Helmholtz free
energy of the system in the random phase approximation we shall have
1
T
F l,RPA
N =
1
T
F l,id
N − M0 −
1
2
N
∑
~k
ν̃(k) +
1
2
ln det {1 + ν̃M2} +
1
2
∑
~k
ν̃(k)M1(~k) M1(−~k) , (23)
where 1 + ν̃M2 is a matrix whose elements can be determined from the following expression
δ~k1+~k2, 0 + ν̃(k1) M2(−~k1, ~k2). (24)
4. Particle density profile
We shall be looking for the density of the system using the method of functional differentiation
of Helmholtz free energy F l
N (23) of the system with respect to the external field
ρ(z1) =
1
T
δ
δwa(z1)
F l
N . (25)
The functional derivative applies only to the terms that contain cumulants. Then, in the random
phase approximation
1
T
δF l,RPA
N
δwa(~R1)
= − δM0
δwa(~R1)
+
1
2
δ
δw(~R1)
ln det{1 + ν̃M2} +
1
2
δ
δw(~R1)
∑
~k
ν̃(k)M1(~k) M1(−~k) . (26)
The first term in (26) is equal to
− δM0
δwa(~R1)
= ρa e−
1
T
wa(~R1) . (27)
In the second term, it should be taken into account that the derivative of M2(~k1,−~k1) in the
thermodynamic limit is equal to zero. Then,
δ
δwa(~R1)
ln det{1 + ν̃M2} =
∑
~k1,~k2
δM2(~k1, ~k2)
δwa(~R1)
δ ln det{1 + ν̃M2}
δM2(~k1, ~k2)
= ρa e−
1
T
wa(~R1) ga(~R1), (28)
where ga(~R1) is the regular part of the screened potential
ga(~R1) =
∑
~k1,~k2
[
1 − δ~k1 +~k2, 0
]
g̃(~k1, ~k2) e−i~k1
~R1−i~k2
~R1 . (29)
The third term in (26), considering the expression for the first cumulant (15), can be written in
the form
δ
δwa(~R1)
∑
~k
ν̃(k)M1(~k) M1(−~k) = −ρa e−
1
T
wa(~R1)
∑
~k
ν̃(k) [1 − δ~k, 0] e
−i~k ~R1 M1(−~k)
= − ρa
1
T
e−
1
T
wa(~R1)
∑
b
ρb
∫
V
d~R2 Φab(|~R1 − ~R2|)
[
e−
1
T
wb(~R2) − 1
]
. (30)
141
M.F.Holovko, I.Y.Kravtsiv, E.M.Soviak
Thus, considering (25–30) the density profile of a spatially non-uniform system of point particles
with the Yukawa interaction in the random phase approximation is:
ρa(~R1) = ρa e−
1
T
wb(~R1)
1 +
1
2
ga(~R1) −
1
T
∑
b
ρb
∫
V
d~R2Φab
(
|~R1 − ~R2|
)[
e−
1
T
wb(~R2) − 1
]
. (31)
The last term in (30) represents the potential of interaction between the surface and the parti-
cles. In a similar way one can get the surface “3–9” potential from the spatially uniform Lennard-
Jones “6–12” potential. It should be noted that in ionic systems the last term, which comes from
the contribution of the initial potential to the density profile, is absent due to the condition of
general electroneutrality of the system.
5. The pair correlation function of the system
Equation (19) for Fourier-transforms of screened potentials in the coordinate space has the
form
gab(~R1, ~R2) = − 1
T
Φab(R12) − 1
T
∑
c
ρc
∫
V
d~R3 e−
1
T
wc(~R3) Φac(|~R1 − ~R3|) gcb(~R3, ~R2) . (32)
This equation coincides with the spatially non-uniform Ornstein-Zernike equation
hab(~R1, ~R2) = cab(~R1, ~R2) +
∑
c
∫
V
d~R3 ρc(~R3) cac(~R1, ~R3) hcb(~R3, ~R2) , (33)
if, like it is done in the mean spherical approximation for a system of point particles, we set the
direct correlation function equal to the potential of particle interaction and if for a zero approxi-
mation we use the density of free particles under an external field w( ~R).
ρa(~R1) = ρae−
1
T
wa(~R1) , cab(~R1, ~R2) = − 1
T
Φab(R12). (34)
In this case the screened potential and the pair correlation function coincide hab(~R1, ~R2) =
hl
ab(
~R1, ~R2) = gab(~R1, ~R2).
Let us solve the spatially non-uniform Ornstein-Zernike equation for a system of point particles
in the approximation (35). Considering the symmetry of the potential energy, we shall express the
pair correlation function and the Yukawa potential as follows
hl
ab(
~R1, ~R2) = hl(s12, z1, z2) , Φab(R12) = A
exp (−α
√
s2
12 + (z1 − z2)2)
√
s2
12 + (z1 − z2)2
. (35)
where ~s1, ~s2 are the projections of radius vectors ~R1 and ~R2 onto the plane z = 0, s12 = |~s1−~s2| is
the distance between projections and z1, z2 are particle coordinates in the direction perpendicular
to the surface.
Then, considering the step-like approximation for the particle density of the system, the OZ
equation will look as follows
hl(s12, z1, z2) = −A
T
exp (−α
√
s2
12 + (z1 − z2)2)
√
s2
12 + (z1 − z2)2
− Aρ+
T
∫
S
d~s3
∞
∫
0
dz3
exp (−α
√
s2
13 + (z1 − z3)2)
√
s2
13 + (z1 − z3)2
hl(s32, z3, z2)
− Aρ−
T
∫
S
d~s3
0
∫
−∞
dz3
exp (−α
√
s2
13 + (z1 − z3)2)
√
s2
13 + (z1 − z3)2
hl(s32, z3, z2), (36)
142
Spatially confined system interacting with Yukawa potential
where integration with respect to ~s3 is given in an infinite plane S.
Similar to [16], let us introduce one-sided pair correlation functions hl
+(−)(s12, z1, z2).
hl(s12, z1, z2) = hl
+(s12, z1, z2) − hl
−(s12, z1, z2),
hl
+(s12, z1, z2) =
{
hl(s12, z1, z2), z1 > 0,
0, z1 < 0,
hl
−(s12, z1, z2) =
{
0, z1 > 0,
−hl(s12, z1, z2), z1 < 0.
(37)
After the Fourier transformation of (36) and a few additional simple transformations we shall
obtain the equation for the Fourier transforms of one-sided pair correlation functions
P+(p, q1) h̃l
+(p, q1, q2) − P−(p, q1) h̃l
−(p, q1, q2) = −4π
A
T
δ(q1 + q2), (38)
where
h̃l
+(−)(p, q1, q2) =
∫
S
d~s12e
i~p~s12
∞
∫
−∞
dz1e
iq1z1
∞
∫
−∞
dz2e
iq2z2hl
+(−)(s12, z1, z2) (39)
and the coefficients P+(p, q1) and P−(p, q1) are quadratic polynomials of variable q1
P+(p, q1) = p2 + q2
1 + γ2
+, P−(p, q1) = p2 + q2
1 + γ2
−,
γ2
+ = α2 + κ
2
+, γ2
− = α2 + κ
2
−, κ
2
+ = 4πA
ρ+
T
, κ
2
− = 4πA
ρ−
T
. (40)
The indexes “+” and “−” in these coefficients do not correspond to the regions of their analyticity
but indicate where they belong to in the equality (38).
Equation (38) is known as the Riemann problem [18]. We shall use the technique proposed
in [16,17], to factorize the equation. From here on we shall restrict the problem to the case of
A > −αT/(4πρ), since a strongly attractive Yukawa potential A < −αT/(4πρ) requires a separate
study. We shall present the fraction P−(p, q1)/P+(p, q1) in the form
P−(p, q1)
P+(p, q1)
=
Q+(p, q1)
Q−(p, q1)
, (41)
where the functions Q+(p, q1), Q−(p, q1), being the functions of the variable q1, are analytical and
have no zeros in the upper + or the lower − semi-planes of the complex plane. They are easy to
find since the coefficients of equation (38) are second degree polynomials of a variable q1
Q+(p, q1) =
q1 + iα−(p)
q1 + iα+(p)
, Q−(p, q1) =
q1 − iα+(p)
q1 − iα−(p)
, α+(−)(p) =
√
(p2 + γ2
+(−)) . (42)
Equation (38) can be rewritten as follows
h̃l
+(p, q1, q2)
Q+(p, q1)
− h̃l
−(p, q1, q2)
Q−(p, q1)
= −4π
A
T
1
Q+(p,−q2) P+(p,−q2)
δ(q1 + q2). (43)
In equality (38) let us present the Dirac δ-function as the difference of one-sided functions
δ(q1 + q2) = δ+(q1 + q2)− δ−(q1 + q2), which are analytical in the upper and the lower semi-planes
of the complex plane, respectively. Since the index of problem (43) is equal to zero [18] for the
Fourier-transforms of one-sided pair correlation functions we obtain
h̃l
+(p, q1, q2) = −4π
A
T
Q+(p, q1)
δ+(q1 + q2)
Q+(p,−q2) P+(p,−q2)
,
h̃l
−(p, q1, q2) = −4π
A
T
Q−(p, q1)
δ−(q1 + q2)
Q+(p,−q2) P+(p,−q2)
. (44)
143
M.F.Holovko, I.Y.Kravtsiv, E.M.Soviak
Substituting (40) and (42) into (44), then for h̃l
+(−)(p, q1, q2) we get
h̃l
+(p, q1, q2) = −4π
A
T
q1 + iα−(p)
q1 + iα+(p)
δ+(q1 + q2)
(q2 − iα−(p))(q2 + iα+(p))
,
h̃l
−(p, q1, q2) = −4π
A
T
q1 − iα+(p)
q1 − iα−(p)
δ−(q1 + q2)
(q2 − iα−(p))(q2 + iα+(p))
. (45)
Let us now find the originals of the one-sided pair correlation functions. To this end we shall
do the inverse Fourier transformation
hl(s12, z1, z2) =
∫
d~p
(2π)2
e−i~p~s12
∞
∫
−∞
dq1
2π
e−iq1z1
∞
∫
−∞
dq1
2π
e−iq2z2
{
h̃l
+(p, q1, q2) − h̃l
−(p, q1, q2)
}
. (46)
Let us calculate the integral with respect to the variable q2. One-sided δ-functions can be
presented as follows
δ+(ζ) = lim
ε→+0
i
ζ + iε
, δ−(ζ) = lim
ε→+0
i
ζ − iε
. (47)
Let us consider a case when the first particle is in the upper phase z1 > 0. Since the function
h̃l
−(p, q1, q2) is an analytical function of the variable q1 in the lower half of the complex plane,
integration of this function with respect to q1 yields zero. Subsequently for z2 > 0, closing the
contour of integration with respect to q2 in the lower half of the complex plane, we find
lim
ε→+0
∞
∫
−∞
dq2
2π
i e−iq2z2
(q2 − iα−(p))(q2 + iα+(p))(q1 + q2 + iε)
=
ie−α+(p)z2
(α+(p) + α−(p))(q1 − iα+(p))
+
eiq1z2
(q1 + iα−(p))(q1 − iα+(p))
. (48)
Now, let us integrate with respect to q1. Since z1 > 0, we have
∞
∫
−∞
dq1
2π
{
ie−α+(p)z2−iq1z1
(α+(p) + α−(p))(q1 − iα+(p))
+
eiq1(z2−z1)
(q1 + iα−(p))(q1 − iα+(p))
}
q1 + iα−(p))
q1 + iα+(p))
=
1
2
{
1
α+(p))
e−α+(p))|z1−z2| +
1
α+(p))
α+(p)) − α−(p))
α+(p)) + α−(p))
e−α+(p))(z1+z2)
}
. (49)
By the inverse Fourier transformation with respect to the surface vector ~p, we can obtain an
expression for the pair correlation function for the case when both particles are in the upper half
of space z1 > 0, z2 > 0.
hl
++(s12, z1, z2) = −A
T
e−γ+R12
R12
− A
T
∞
∫
0
J0(ps12)
pdp
α+(p))
α+(p)) − α−(p))
α+(p)) + α−(p))
e−α+(p))(z1+z2) , (50)
where the first term in (50) represents the spatially uniform part of the pair correlation function
and the second term – the regular part of the pair correlation function contributed by spatial
non-uniformity. J0(x) is the first order Bessel function
J0(ps12) =
1
π
π
∫
0
dϕ ei ps12 cos ϕ . (51)
144
Spatially confined system interacting with Yukawa potential
In a similar way, we can calculate the pair correlation function for other regions of space.
Finally, for the pair correlation function of a two-phase system of point particles with the Yukawa
potential of inter-particle interaction we obtain the expressions:
hl
++(s12, z1, z2) = −A
T
e−γ+R12
R12
− A
T
∞
∫
0
J0(ps12)
p dp
α+(p))
α+(p)) − α−(p))
α+(p)) + α−(p))
e−α+(p))(z1+z2), for z1 , z2 > 0 , (52)
hl
+−(s12, z1, z2) = −A
T
∞
∫
0
J0(ps12)pdp
e−α+(p)z1+α−(p)z2
α+(p)) + α−(p))
for z1 > 0 , z2 < 0 , (53)
hl
−+(s12, z1, z2) = −A
T
∞
∫
0
J0(ps12)pdp
eα−(p)z1−α+(p)z2
α+(p)) + α−(p))
for z1 < 0 , z2 > 0 , (54)
hl
−−(s12, z1, z2) = −A
T
e−γ−R12
R12
+
A
T
∞
∫
0
J0(ps12)
pdp
α−(p))
α+(p)) − α−(p))
α+(p)) + α−(p))
eα−(p))(z1+z2) , for z1 < 0 , z2 > 0 . (55)
From (52–55) it is easy to derive the pair correlation functions for Coulomb systems of point
particles [16,17]. This this end one should set α = 0 and A = Q1Q2, where Q1, Q2 are the electric
charges of particles 1 and 2.
6. Equation of state for a spatially uniform system with a Yukawa-like po-
tential of interaction
If there are no external fields forming the interface, the equality (23) turns into the expression for
the Helmholtz free energy of a spatially uniform system. At the same time, the part of the Helmholtz
free energy responsible for the inter-particle interaction in the random phase approximation can
be found by integrating with respect to the coupling parameter ξ [8]
F l,RPA,ex
N =
1
2
ρ N
∫
V
d~R12 Φ(R12)
1
∫
0
dξ [ 1 + h(ξ, R12)] . (56)
The spatially uniform part of the pair correlation function can be found from the respective
spatially non-uniform expression by setting ρ+ = ρ− = ρ. Inclusion of the interaction parameter
means that the constant of interaction A gets substituted for ξ A. Then, integrating with respect
to ~R12, we have
A
∫ ∞
0
e−α R12
(
R12 − ξ
A
T
e−R12
√
α2 + ξ κ
2
)
=
A
α2
+
A
4πρ
(
√
α2 + ξ κ2 − α
)
. (57)
After integrating with respect to the parameter of interaction inclusion, for F l,RPA,ex
N we get the
following expression
F l,RPA,ex
N = V T
(
α3
12π
−
(
α2 + κ2
)3/2
12π
+
α κ2
8π
+
ρ κ2
2α2
)
. (58)
Equation of state for a spatially uniform system can be found by differentiating the Helmholtz free
energy with respect to the system’s volume at a constant temperature and a constant number of
145
M.F.Holovko, I.Y.Kravtsiv, E.M.Soviak
particles.
P = −
[
∂
∂V
FN
]
T ,N
= −
[
∂
∂V
(
F id
N + F ex
N
)
]
T ,N
. (59)
This differentiation applies only to density ρ and the parameter κ. Consequently, in the random
phase approximation for the contribution of long-range inter-particle interaction to the pressure
we have
1
T
P l,RPA,ex =
ρ κ2
2α2 +
α2
√
α2 + κ2
12π
−
√
α2 + κ2κ2
24π
− α3
12π
. (60)
Finally, since 1
T P id = ρ, the pressure in the system equals
1
T
P l,RPA = ρ + ρ
κ2
2α2
+
α2
√
α2 + κ2
12π
− κ2
√
α2 + κ2
24π
− α3
12π
. (61)
It should be noted that in the case of ionic systems, the term ρ κ
2
2α2 is absent due to the
condition of electric neutrality of the system. Then, setting α = 0, we get the equation of state for
an ionic system
1
T
P l,RPA = ρ − ρ
κ3
24π
, (62)
which coincides with the results of the Debye-Huckel theory.
7. The contact theorem
For systems of particles that are spatially confined by a hard wall, certain exact relationships
hold true. These relationships establish the connection between structural properties of the system
near the surface and its thermodynamic properties. Notably, the contact theorem connects the
value of the particle density with the pressure
ρa(z1 = 0) =
1
T
P. (63)
In [14], in the random phase approximation, the validity of this relation was shown for a spatially
confined system of charged point particles. Let us now show that the particle density profile found
satisfies the condition of the contact theorem.
In the case of a system confined by a hard wall we shall set the particle density in the lower
phase equal to zero and for density profile in the upper phase z > 0 of the system with (31), setting
ρ+ = ρ, ρ− = 0, we get
ρ(z1)
ρ
= 1 − ρ
1
T
0
∫
−∞
dz2
∫
S
d~s12Φ(
√
s2
12 + (z1 − z2)2) +
1
2
g+(z1) . (64)
Taking into account the form of the Yukawa potential (1) and that of the regular part of the pair
correlation function (29),
g+(z1) = − A
T
∞
∫
0
p dp
α+(p))
α+(p)) − α−(p))
α+(p)) + α−(p))
e−2α+(p)z1 . (65)
For ρ(z1) we get the following expression
ρ(z1) = ρ
1 +
2πAρ
α2 T
e−α z1 − A
2T
∞
∫
0
p dp
α+(p))
α+(p)) − α−(p))
α+(p)) + α−(p))
e−2 α+(p) z1
, (66)
146
Spatially confined system interacting with Yukawa potential
where
α+(p)) =
√
p2 + α2 + κ2, α−(p)) =
√
p2 + α2 , κ
2 =
4πAρ
T
. (67)
From equality (66), setting z1 = 0, we obtain the value of particle density at the surface
ρ(0) = ρ + ρ
κ2
2α2
+
α2
√
α2 + κ2
12π
− κ2
√
α2 + κ2
24π
− α3
12π
, (68)
which is identical to the expression (61) for the pressure of a spatially uniform system with the
Yukawa potential of interaction. Therefore, we have proven the validity of the contact theorem for
a system with the Yukawa-like inter-particle interaction in the random phase approximation.
8. Adsorption
Expression (66) established for the density profile can be written in the form
ρ(z1)
ρ
= 1 +
κ2
2α2 T
e−α z1 − κ
2 A
2T
∞
∫
0
p dp
α+(p)
1
(α+(p) + α−(p))2
e−2 α+(p) z1 . (69)
Contribution of the Yukawa potential to the particle distribution function increases with the in-
crease of the system’s density. It should be noted that the change of the sign of interaction does
not reverse the sign of the term responsible for collective effects while the contribution of the
initial potential reverses that sign. This property causes significant distinctions between surface
properties of Coulomb systems and those of neutral particle systems.
Before moving on to computational calculations, we would like to point out that a system with
the Yukawa potential of inter-particle interaction is characterized by two unitless parameters: the
unitless inverse temperature αA/T = 1/T ∗, which characterizes the intensity of interaction, and
the unitless density of the system ρ∗ = ρ/α3.
Figure 1. The profile of particle density. Curves are labeled by: (1) for 1
T∗
= 10.0 and ρ
∗ =
0.0005, (2) for 1
T∗
= −10.0 and ρ
∗ = 0.0005, (3) for 1
T∗
= 10.0 and ρ
∗ = 0.005, (4) for
1
T∗
= −10.0 and ρ
∗ = 0.005.
In figure 1 the unitless profile of particle density ρ(αz1)/ρ versus the unitless distance to the
hard wall z = αz1. is shown. At small values of the unitless interaction coefficient α A/T < 1 the
147
M.F.Holovko, I.Y.Kravtsiv, E.M.Soviak
term associated with the Yukawa potential plays the major role in the behaviour of the density
profile. With the rise of the coefficient, the contribution of collective effects increases. At the same
time, it remains negative regardless of the nature of interaction. For this reason, at small distances
to the surface, where collective interactions are more important than the initial Yukawa potential,
the value of the singlet function is less than the one for both the attractive and the repulsive
interactions.
As the distance to the hard wall increases, the contribution of collective interactions decreases
at a rate faster than κ
2
2α2 e−α z1 . Consequently, with increasing distance of a particle from the
surface, the density of the system becomes higher than the bulk value for repulsive interaction and
remains lower than the bulk value for attractive interaction.
Another important characteristic of the surface properties of a system is the adsorption coeffi-
cient
Γ =
∞
∫
0
dz [ ρ(z) − ρ ] . (70)
Substitution of the expression (66) into the equality for the adsorption coefficient (70) yields
Γ =
ρ κ2
2α3 − κ2
32π
( 2 ln 2 − 1 ) +
κ2
16π
ln
(
1 +
α√
α2 + κ2
)
− α
(√
α2 + κ2 − α
)
16π
. (71)
Figure 2. Density dependence of adsorption. Curves are labeled by: (1) for 1
T∗
α = 10, (2) for
1
T∗
= 1, (3) for 1
T∗
= −10, (4) for 1
T∗
= −1.
In figure 2, the dependences of adsorption on the particle density ρ/α3 are presented. The
adsorption in the system considerably depends on the sign of interaction as well as on the range
of the Yukawa potential. For systems with an repulsive interaction it is distinctive that the sign of
adsorption reverses with transition to higher densities. As the interaction parameter 1/T ∗ increases,
the point of sign reversal for the adsorption coefficient moves to the region of the higher densities.
9. Summary
In this work we have studied a spatially non-uniform system of point particles with the Yukawa
potential of interaction. Such a system allows one to reveal distinctions between structural and
148
Spatially confined system interacting with Yukawa potential
thermodynamic properties of charged-particle systems and those of neutral-particle systems. These
distinctions are also present in spatially uniform systems, in particular due to the inter-particle
interaction contributing to the expression for the Helmholtz free energy. In systems of charged
particles such a term is absent due to the condition of general electric neutrality [8]. For spatially
non-uniform systems, this term leads to the appearance in density profiles of a coordinate depen-
dence related to the effective interaction; in addition, density profiles acquire a dependence caused
by the finiteness of the range of the initial potential. In the pair correlation function of neutral par-
ticles, no additional functional coordinate dependences appear compared to spatially non-uniform
systems of charged particles. The expression for the density profile of a system obtained via func-
tional differentiation of Helmholtz free energy with respect to the external field contains terms
from the initial potential and the pair correlation function. At the same time, the profile of the
particle density satisfies the condition of the contact theorem. Analysis of the expression found for
the adsorption coefficient shows that in the case of an repulsive interaction, the sign of adsorption
reverses with the increase in density. For sufficiently dense systems, adsorption remains negative
regardless of the nature of interaction.
Acknowledgements
The authors thank Dung di Caprio for the fruitful discussions.
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M.F.Holovko, I.Y.Kravtsiv, E.M.Soviak
Просторово обмежений плин iз потенцiалом взаємодiї
Юкави
М.Ф.Головко1,2, I.Я.Кравцiв2, Є.М.Сов’як1
1 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, Львiв, 79011, Україна
2 Львiвський нацiональний унiверситет iм. I.Франка, Фiзичний факультет, вул. Драгоманова,12, Львiв,
79005, Україна
12 травня 2009 р., в остаточному виглядi – 19 травня 2009 р.
Розглянуто обмежений твердою стiнкою плин з потенцiалом взаємодiї Юкави. Отримано розв’язок
просторово неоднорiдного рiвняння Орнштейна-Цернiке для парної кореляцiйної функцiї. Методом
функцiонального диференцiювання вiльної енергiї за зовнiшнiм полем знайдено вираз для профi-
лю густини частинок. Показано, що вклад у поведiнку густини частинок поблизу поверхнi вносить
як вихiдний потенцiал, так i колективнi ефекти екранування взаємодiї. Розраховано залежнiсть кое-
фiцiєнта адсорбцiї вiд густини числа частинок. Показано, що у випадку притягувальної взаємодiї iз
збiльшенням густини числа частинок системи змiнюється знак коефiцiєнта адсорбцiї.
Ключовi слова: просторово неоднорiдне рiвняння Орнштейна-Цернiке, тверда стiнка, контактна
теорема, коефiцiент адсорбцiї, потенцiал Юкави
PACS: 05.20.Jj, 05.70.Np, 61.20.-p, 68.03.-g
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