Adsorption of hard spheres: structure and effective density according to the potential distribution theorem
We propose a new type of effective densities via the potential distribution theorem. These densities are for the sake of enabling the mapping of the free energy of a uniform fluid onto that of a nonuniform fluid. The potential distribution theorem gives the work required to insert a test particle in...
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| Zitieren: | Adsorption of hard spheres: structure and effective density according to the potential distribution theorem / L.L. Lee, G. Pellicane // Condensed Matter Physics. — 2011. — Т. 14, № 3. — С. 33601: 1-15. — Бібліогр.: 31 назв. — англ. |
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irk-123456789-1200082017-06-11T03:04:51Z Adsorption of hard spheres: structure and effective density according to the potential distribution theorem Lee, L.L. Pellicane, G. We propose a new type of effective densities via the potential distribution theorem. These densities are for the sake of enabling the mapping of the free energy of a uniform fluid onto that of a nonuniform fluid. The potential distribution theorem gives the work required to insert a test particle into the bath molecules under the action of the external (wall) potential. This insertion work Wins can be obtained from Monte Carlo (MC) simulation (e.g. from Widom's test particle technique) or from an analytical theory. The pseudo-densities are constructed thusly so that when their values are substituted into a uniform-fluid equation of state (e.g. the Carnahan-Starling equation for the hard-sphere chemical potentials), the MC nonuniform insertion work is reproduced. We characterize the pseudo-density behavior for the hard spheres/hard wall system at moderate to high densities (from ρ*= 0.5745 to 0.9135). We adopt the MC data of Groot et al. for this purpose. The pseudo-densities show oscillatory behavior out of phase (opposite) to that of the singlet densities. We also construct a new closure-based density functional theory (the star-function based density functional theory) that can give accurate description of the MC density profiles and insertion works. A viable theory is established for several cases in hard sphere adsorption. Ми пропонуємо новий тип ефективних густин, отриманих з теореми розподiлу потенцiалу. Цi густини необхiднi для вiдображення вiльної енергiї однорiдної рiдини у вiльну енергiю неоднорiдної рiдини. Теорема розподiлу потенцiалу дає роботу, необхiдну для устромляння пробної частинки у систему молекул, на яку дiє зовнiшнiй потенцiал. Ця робота Wins може бути отримана з симуляцiї Монте-Карло (MК) (наприклад, пiдходом тестової частинки Вiдома) або з аналiтичної теорiї. Псевдогустини є побудованi так, що коли їх значення пiдставляються в рiвняння стану однорiдної рiдини (наприклад, рiвняння для хiмiчних потенцiалiв Карнагана-Старлiнгa системи твердих сфер), то вiдтворюється робота устромляння частинки в неоднорiдну рiдину, отримана з симуляцiї МК. Ми дослiджуємо поведiнку псевдогустини для системи “твердi сфери”-“тверда стiнка” при середнiх та високих густинах (вiд ρ∗ = 0.5745 до 0.9135). Для цього використовуються результати Монте-Карло Гроота та спiвавторiв. Псевдогустини демонструють осцилюючу поведiнку з протилежною фазою до одночастинкових густин. Ми також пропонуємо нову теорiю функцiоналу густини на основi за-микання (теорiю функцiоналу густини зiркової функцiї), що може точно описувати профiлi густини i роботу устромляння. Точнiсть теорiї перевiряється для декiлькох випадкiв адсорбцiї твердих сфер. 2011 Article Adsorption of hard spheres: structure and effective density according to the potential distribution theorem / L.L. Lee, G. Pellicane // Condensed Matter Physics. — 2011. — Т. 14, № 3. — С. 33601: 1-15. — Бібліогр.: 31 назв. — англ. 1607-324X PACS: 68.43.De DOI:10.5488/CMP.14.33601 arXiv:1202.4276 http://dspace.nbuv.gov.ua/handle/123456789/120008 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We propose a new type of effective densities via the potential distribution theorem. These densities are for the sake of enabling the mapping of the free energy of a uniform fluid onto that of a nonuniform fluid. The potential distribution theorem gives the work required to insert a test particle into the bath molecules under the action of the external (wall) potential. This insertion work Wins can be obtained from Monte Carlo (MC) simulation (e.g. from Widom's test particle technique) or from an analytical theory. The pseudo-densities are constructed thusly so that when their values are substituted into a uniform-fluid equation of state (e.g. the Carnahan-Starling equation for the hard-sphere chemical potentials), the MC nonuniform insertion work is reproduced. We characterize the pseudo-density behavior for the hard spheres/hard wall system at moderate to high densities (from ρ*= 0.5745 to 0.9135). We adopt the MC data of Groot et al. for this purpose. The pseudo-densities show oscillatory behavior out of phase (opposite) to that of the singlet densities. We also construct a new closure-based density functional theory (the star-function based density functional theory) that can give accurate description of the MC density profiles and insertion works. A viable theory is established for several cases in hard sphere adsorption. |
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Article |
| author |
Lee, L.L. Pellicane, G. |
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Lee, L.L. Pellicane, G. Adsorption of hard spheres: structure and effective density according to the potential distribution theorem Condensed Matter Physics |
| author_facet |
Lee, L.L. Pellicane, G. |
| author_sort |
Lee, L.L. |
| title |
Adsorption of hard spheres: structure and effective density according to the potential distribution theorem |
| title_short |
Adsorption of hard spheres: structure and effective density according to the potential distribution theorem |
| title_full |
Adsorption of hard spheres: structure and effective density according to the potential distribution theorem |
| title_fullStr |
Adsorption of hard spheres: structure and effective density according to the potential distribution theorem |
| title_full_unstemmed |
Adsorption of hard spheres: structure and effective density according to the potential distribution theorem |
| title_sort |
adsorption of hard spheres: structure and effective density according to the potential distribution theorem |
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Інститут фізики конденсованих систем НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/120008 |
| citation_txt |
Adsorption of hard spheres: structure and effective density according to the potential distribution theorem / L.L. Lee, G. Pellicane // Condensed Matter Physics. — 2011. — Т. 14, № 3. — С. 33601: 1-15. — Бібліогр.: 31 назв. — англ. |
| series |
Condensed Matter Physics |
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AT leell adsorptionofhardspheresstructureandeffectivedensityaccordingtothepotentialdistributiontheorem AT pellicaneg adsorptionofhardspheresstructureandeffectivedensityaccordingtothepotentialdistributiontheorem |
| first_indexed |
2025-07-08T17:04:10Z |
| last_indexed |
2025-07-08T17:04:10Z |
| _version_ |
1837099125231845376 |
| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 3, 33601: 1–15
DOI: 10.5488/CMP.14.33601
http://www.icmp.lviv.ua/journal
Adsorption of hard spheres: structure and effective
density according to the potential distribution theorem
L.L. Lee1, G. Pellicane2
1 Department of Chemical & Materials Engineering, California State University,
Pomona, California, USA
2 School of Physics, University of Kwazulu-Natal, Private Bag X01 Scottsville,
3209 Pietermaritzburg, South Africa
Received April 11, 2011
We propose a new type of effective densities via the potential distribution theorem. These densities are for
the sake of enabling the mapping of the free energy of a uniform fluid onto that of a nonuniform fluid. The
potential distribution theorem gives the work required to insert a test particle into the bath molecules under
the action of the external (wall) potential. This insertion work Wins can be obtained from Monte Carlo (MC)
simulation (e.g. from Widom’s test particle technique) or from an analytical theory. The pseudo-densities are
constructed thusly so that when their values are substituted into a uniform-fluid equation of state (e.g. the
Carnahan-Starling equation for the hard-sphere chemical potentials), the MC nonuniform insertion work is
reproduced. We characterize the pseudo-density behavior for the hard spheres/hard wall system at moderate
to high densities (from ρ∗ = 0.5745 to 0.9135). We adopt the MC data of Groot et al. for this purpose. The
pseudo-densities show oscillatory behavior out of phase (opposite) to that of the singlet densities. We also
construct a new closure-based density functional theory (the star-function based density functional theory)
that can give accurate description of the MC density profiles and insertion works. A viable theory is established
for several cases in hard sphere adsorption.
Key words: potential distribution theorem, hard spheres, adsorption, effective density, structure, integral
equation, closure
PACS: 68.43.De
1. Introduction
Inhomogeneous fluids pose, due to their vast varieties, challenges to the liquid-state theories,
and are of capital importance in many industrial processes. The nonuniformity in the fluid can be
produced by a one-body potential: such as the surface potential from the interfaces separating two
or three coexisting phases (solid/liquid/vapor phase transitions), electrical fields acting on ionic
species (the electric double layer), or magnetic fields on ferromagnetic liquids. The conventional
classical density functional theory (DFT) employed to deal with such systems in the last half
century has been based on the Hohenberg-Kohn [1] formulation of a grand potential, Ω, that is
expressed in terms of an intrinsic Helmholtz free energy functional F [ρ
(1)
w ] (IHFEF):
Ω = F [ρ(1)w ] +
∫
d~rρ(1)w (~r; [w]){w(~r)− µ0} (1.1)
where ρ
(1)
w is the singlet (nonuniform) density, w(r) the external (1-body) potential, and µ0 the
chemical potential of the bulk fluid. At equilibrium, the grand potential is minimized with respect
to the singlet ρ
(1)
w , and it yields the Euler-Lagrange equation
δβΩ
δρ
(1)
w (~r)
=
δβF [ρ
(1)
w ]
δρ
(1)
w (~r)
− βµ0 + βw(~r) = 0 (1.2)
where β = 1/(kT ): k is the Boltzmann constant, T is the absolute temperature. The IHFEF can be
separated into two parts: the ideal part Fid (which is known: F id[ρ] =
∫
d~rρ
(1)
w (~r)[ln ρ
(1)
w (~r)Λ3− 1])
c© L.L. Lee, G. Pellicane, 2011 33601-1
http://dx.doi.org/10.5488/CMP.14.33601
http://www.icmp.lviv.ua/journal
L.L. Lee, G. Pellicane
and the excess part F ex. Thus F [ρ
(1)
w ] = F id + F ex. Recently, we have proposed a star-function
based density functional theory [2, 3] (s-DFT) which expresses F ex succinctly as
− βF ex
[ρ] = −βF ex
0 +
∫
d~rC
(1)
0 (~r)δρw(~r) +
1
2
∫
d~rd~r ′C
(2)
0 (~r, ~r ′)δρw(~r)δρw(~r
′) + S∗
w (1.3)
where S∗
w is the star function [4] defined as the “primitive” of the bridge function Bw
S∗
w =
∫
d~r
δρw(~r)
γw(~r)
γ1
∫
γ0
dxBw(x) (1.4)
and C
(1)
0 (~r) is the singlet direction correlation function (1 −DCF ), C
(2)
0 (~r, ~r ′) is the pair direct
correlation function (2 − DCF ). F ex
0 is the bulk fluid free energy. Subscripts 0 and w indicate
uniform and nonuniform fluid properties, respectively. δρw = (ρ
(1)
w − ρ0) is the difference between
the nonuniform density ρ
(1)
w and the uniform density ρ0 and γw is the indirect correlation. We note
that if the bridge function Bw can be obtained exactly, the star function S∗
w will consequently be
exact from equation (1.4), thus the excess free energy F ex in equation (1.3) will also be exact. The
Euler-Lagrange (EL) equation (1.2) can be expressed as
ρw(~r) = ρ exp
[
−w(~r) + C(1)
w (~r)− C
(1)
0 (~r)
]
(1.5)
where C
(1)
w is the nonuniform singlet direct correlation function (1 − DCF ). Alternatively, it is
possible to show [2] via functional expansions that equation (1.5) can be written as
ρw(~r) = ρ exp [−w(~r) + γw(~r) +Bw(~r)] . (1.6)
Thus, we have the equality [3]
C(1)
w (~r)− C
(1)
0 (~r) = γw(~r) +Bw(~r). (1.7)
Equations (1.3) and (1.6) constitute the basis of the s-DFT theory.
In this article, we shall explore two essential aspects of nonuniform fluids: (1) the effective
density that enables the mapping between the uniform-fluid free energies and the nonuniform-fluid
free energies; (2) the closure relation between the bridge function and other correlation functions.
The first task is mediated through the potential distribution theorems [5, 6]; the second is based
on a successful uniform liquid theory [7] developed in recent years. Historically, there have been
interchanges and cross-fertilization of uniform and nonuniform liquid theories [8, 9]. In this paper
we test the new formulations on hard spheres adsorbed on hard planar walls.
In weighted-density density functional approaches (WDA) [10, 11] the “mapping” of the free
energy is done through some weighted density: ρ. The weighted density is obtained from the true
density ρ
(1)
w via a convolution integral with a weighting function, ω, as the kernel: i.e. ρ(~r) =
∫
d~r ′ρ
(1)
w (~r ′)ω(~r, ~r ′). We shall, however, develop an effective density without going through this
weighting procedure. This is made possible by the potential distribution theorem (PDT), since
PDT naturally lends itself to yield the work of insertion (the free energy of an inhomogeneous
system).
The PDT is a fundamental theory of statistical mechanics [5, 6, 11–13]. It is concerned with the
work required to insert test particles in an equilibrium ensemble representing a uniform fluid or a
nonuniform fluid. In an earlier paper [14] we have generalized the PDT from the chemical potential,
to the cavity functions, and to higher-order correlation functions. The PDT has previously been
extended to nonuniform systems [5, 6]. However, here we distinguish two types of test particles.
The first type is for a test particle subject to the wall force as well as to the forces of the bath
molecules. This is the commonly studied type. The second type is for a test particle free of the
wall force. Its interaction with the wall is intentionally removed (similar to the cavity function
33601-2
Adsorption of hard spheres
in homogeneous fluids). The PDT of the second kind is directly related to the Euler-Lagrange
equation [3]. The PDT establishes a connection among (i) the insertion work Wins, (ii) the bulk
fluid chemical potential µ0, and (iii) the singlet distribution function ρ
(1)
w . To illustrate the PDT,
we carry out calculations on the system of hard spheres on a hard wall (HS/HW), for which MC
data are available (e.g. from Groot et al. [15]).
The mapping between the uniform and the nonuniform systems can be constructed as follows:
given an equation of state for the uniform fluid (which is utilized to calculate the uniform chemical
potential βµ0), we solve for the hypothetical densities ρpseudo(z) that will reproduce the insertion
works Wins(z) obtained from the MC, i.e. βµ0(ρpseudo(z)) = βWins(z). This density profile is an
artificial construct and is called the pseudo-density ρpseudo(z). Its sole role is to reproduce the
nonuniform free energy via the bulk equation of state. For instance, in the HS/HW system, the
bulk fluid equation is taken to be the Carnahan-Starling (CS) [16] equation which is known to be
highly accurate. It yields the uniform hard-sphere chemical potential βµ0. If we know the insertion
work βWins(z) (from MC) in HS/HW, a series of values of ρpseudo(z) can be generated so that
the equality βµHS
0 = βWins(z) holds. The ρpseudo(z) obtained thusly will enable the mapping of
the free energies ( βµHS
0 ) of the uniform fluid to those (βWins) of the nonuniform fluid. In this
respect, the pseudo-densities perform precisely the same function as the weighted densities ρ(z)
in the WDA. The difference is that our approach is based on the PDT, an exact theory, not on
approximate theories (such as the Percus-Yevick equation [17]).
Should the bridge function Bw be known, one can obtain the fluid structure at the wall via
the Euler-Lagrange equation (1.6) [2, 3]. Bw as a correlation function is well defined in terms of
infinite series of cluster diagrams [18] or functional expansions [19]. However, direct evaluation from
its definition is numerically intractable. But it can be inverted from machine data (as in reverse
engineering [20]) if we consider equation (1.6) as the defining equation [19]. The inverted Bw can
be given as a set of numerical data [3], say, in the form of tabulated entries; or it can be expressed
as a function Bw = f(r) for some analytical function f (e.g. a polynomial). Another approach, and
a more appealing one is to formulate a liquid theory (à la mode of the Percus-Yevick equation [17])
that relates Bw to other correlation functions Bw = f(γ) , γ being a well-defined quantity in liquid
state theory. This in uniform liquid theory is called the closure relation. This relation may or may
not exist, because Bw is in fact a functional [3, 21] Bw = Φ[ρ
(1)
w , w] of the singlet density ρ
(1)
w and/or
the external potential w (Φ[.] being the functional sign). Thus, it may not be simply related to any
one correlation function. For uniform liquids, the determination of the function B = f(γ), γ being
some correlation, is already a major task (examples such as the Rogers-Young [22] (RY) closure
or the Martynov-Sarkisov [23] (MS) closure). We have proposed a theory upon resummation of
the functional expansions of Bw in [3]. This resummation seems to work well for a number of pair
potentials. For nonuniform fluids, the PY and hypernetted-chain [24] (HNC) closures have been
tried and shown not to work well in the past [25]. And this prompted the proposition that closure-
based approaches might not be suitable for nonuniform systems. In this work, we shall propose
a zero-separation type (ZSEP) closure derived from the study of uniform liquid theory [7]. This
closure will be tested here on selected cases of the HS/HW systems. Once the singlet density is
obtained from equation (1.6) given Bw, one can use the Euler-Lagrange equation (1.5) to obtain
the 1-DCF C
(1)
w . The 1-DCF is simply related to the insertion work as
βWins(~r) = −C(1)
w (~r). (1.8)
Thus, we can access the insertion work through (1.5), using the MC data [15] on ρ
(1)
w as input.
The thusly obtained βWins is considered as the MC-derived insertion work. In addition, we shall
construct a theoretical method for obtaining the insertion work.
Section 2 gives the potential distribution theorems for nonuniform fluids with general interaction
forces. Section 3 shows the calculations of the effective densities based on the MC-derived insertion
works for the hard spheres/hard wall system. The mapping of the nonuniform free energies is
mediated through an accurate equation of state by Carnahan and Starling. We characterize the
behavior of ρpseudo and compare with the commonly used weighted densities found in literature. In
33601-3
L.L. Lee, G. Pellicane
section 4 we test the new closure theory (ZSEP) on the same systems. We compare the theoretical
structures and free energies with the MC data. In section 5 we draw the conclusions.
2. Potential distribution theorems for nonuniform systems
In this section, we shall introduce the potential distribution theorems for two types of test
particles. One type of test particle is the well-known one, i.e. it is subject to interactions with the
bath molecules as well as to forces of the wall. The second type is the one that interacts only with
bath molecules and not with the wall. The latter result will be the PDT that we shall use in this
article. We start with the averages in a canonical ensemble. (This can also be done in the grand
canonical ensemble [5, 6]). The N-body system consists of N fluid molecules and they border on
one side at a solid surface. The Hamiltonian HN is written as
HN (~pN , ~rN ) = KN
(
~pN
)
+ UN
(
~rN
)
+WN
(
~rN
)
, (2.1)
where ~rN ≡ (~r1, ~r2, . . . , ~rN ) is a shorthand for the N -vector of the positions of N particles, and pN
is the N -momenta vector, KN is the kinetic energy, while
UN(~rN ) =
N−1
∑
1=i<j
N
∑
j=2
u(2)(ri, rj) (2.2)
is the total potential energy between fluid particles (assumed to be pairwise additive), and
WN (~rN ) =
N
∑
k=1
w(rk) (2.3)
is the sum of one-body energies arising from the external (wall) potential w. This is the potential
energy that drives the inhomogeneities in the system.
Next, we consider an (N-1)-body system: the sum of pair energies becomes
UN−1(~r
N−1) =
N−2
∑
1=i<j
N−1
∑
j=2
u(2)(ri, rj) (2.4)
and the sum of one-body energies
WN−1(~r
N−1) =
N−1
∑
k=1
w(~rk). (2.5)
A test particle is introduced to the (N-1)-body system as particle with label “N” located at
the distance rN . This test particle can interact with the other N − 1 bath molecules through the
pair potential u(2)(ri, rN ), i < N , as well as with the wall through the one-body potential w(~rN ).
This particle is designated the type-1 test particle. The excess potential energy Ψw(~rN ) due to the
presence of type-1 test particle is
Ψw(~rN ) = UN (~rN )− UN−1(~r
N−1) +WN (~rN )−WN−1(~r
N−1) =
N−1
∑
i=1
u(2)(~ri, ~rN ) + w(~rN ). (2.6)
On the other hand, the type-2 test particle, by choice, does not interact with the wall (w(~rN ) =
0). The excess potential energy Ψ−w(~rN ) is (subscript “–w” stands for “without the wall interac-
tion”)
Ψ−w(~rN ) =
N−1
∑
i=1
u(2)(~ri, ~rN ). (2.7)
33601-4
Adsorption of hard spheres
The potential distribution theorem for the first-type test particle (interacting with w) is obtained
from the (N-1)-body canonical ensemble average of Ψw(~rN )
ln〈exp [−βΨw]〉N−1;w = −βµw + ln
[
ρ(1)w (~rN )Λ3
]
(2.8)
since
〈exp [−βΨw]〉N−1;w =
1
QN−1
∫
d~rN−1 exp [−βUN−1 − βWN−1]
× exp
{
−β
[
N−1
∑
i=1
u(2)(riN ) + w(~rN )
]}
. (2.9)
Furthermore, from the definition of the singlet density ρ
(1)
w (~rN )
ρ(1)w (~rN ) =
N
QN
∫
d~rN−1 exp [−βUN − βWN ] . (2.10)
Equation (2.8) has been obtained earlier [5, 6]. Note that QN is the N-body configurational
integral; Λ is the de Broglie wavelength; and
βµw = ln
QN−1
QN
+ ln(ρΛ3). (2.11)
The potential distribution theorem for the type-2 test particle (without w) is obtained instead
from the (N-1)-body ensemble average of the wall-less excess potential Ψ−w(~rN ) equation (2.7),
i.e.
ln〈exp [−βΨ−w]〉N−1;w = −βµw + βw(~rN ) + ln
[
ρ(1)w (~rN )Λ3
]
(2.12)
since
〈exp [−βΨ−w]〉N−1;w =
exp [βw(~rN )]
QN−1
∫
d~rN−1 exp [−βUN−1 − βWN−1]
× exp
{
−β
[
N−1
∑
i=1
u(2)(riN ) + w(~rN )
]}
. (2.13)
The insertion work Wins required to insert a type-2 test particle with the excess potential
Ψ−w(~rN ) is thus (noting that βµw = βµex
0 + ln(ρ0Λ
3))
βWins = − ln〈exp [−βΨ−w]〉N−1;w = βµw − βw(~rN )− ln
[
ρ(1)w (~rN )Λ3
]
= βµex
0 − βw(~rN )− ln
[
ρ(1)w (~rN )/ρ0
]
. (2.14)
The quantity Wins can be directly simulated via the MC method using the Widom particle-
insertion technique [5, 26], or from the EL equation (1.5). Wins can also be considered as the
intrinsic work of insertion.
3. Effective density and work of insertion
For the hard sphere system, we adopt the MC data by Groot et al. [15] as the basis of calculation.
The number densities chosen for the hard spheres are ρ∗ = 0.5745, 0.715, 0.758, 0.813, and 0.9135.
The hard spheres are adsorbed on a planar hard wall stretching over the x-y plane. Thus, the
fluid-fluid pair potential u(2) is
{
u(2)(r) = ∞, r 6 σ,
u(2)(r) = 0, r > σ.
(3.1)
33601-5
L.L. Lee, G. Pellicane
Figure 1. The effective (pseudo) density ρpseudo(z) (symbol ∗; from PDT) and the insertion
work βWins (symbol △; from MC) at the bulk hard-sphere density ρ∗ = 0.5745 for the HS/HW
system. Symbol (♦) ≡ the singlet density profile ρ
(1)
w (z) from MC data of Groot et al. [15].
The brown line is the insertion work βWins produced by the FMT theory. Z ≡ distance from
the wall. Unit of length is taken to be the hard-sphere diameter σ. The horizontal line is at
ρ∗ = 0.5745 (a guide for the eye). The pseudo-density ρpseudo(z) oscillates weakly around the
bulk value and in opposite periodicity as compared to ρ
(1)
w (z). The insertion work βWins also
oscillates synchronously with ρpseudo(z) but oppositely to ρ
(1)
w (z).
The wall potential w on the fluid particles (in the z-direction perpendicular to the wall) is
{
w(z) = ∞, ∀z 6 σ
2 ,
w(z) = 0, ∀z > σ
2 .
(3.2)
Thus, the closest approach of the center of a hard sphere to the wall is at z = σ/2 (we shall use
the hard sphere diameter σ as unit of length below). For ρ∗ = 0.5745, the pure hard-sphere fluid
excess chemical potential βµ0 is 4.897, as calculated from the CS equation. The singlet density
profile ρ
(1)
w (z) has been obtained by Groot et al. [15] via MC simulation and is displayed in figure 1
(see the diamond symbols ♦). The insertion work βWins(z) is identified as −C
(1)
w (z) (equation (1.8)
i.e. the (negative) 1-DCF, which is obtained from the MC data via the EL equation (equation (1.5).
The insertion work is plotted as triangles (N). We remark that both the singlet density ρ
(1)
w (z) and
the insertion work βWins(z) are “real” quantities, i.e. they are measurable quantities for the system
of interest. We next formulate the pseudo-densities as mentioned earlier in section 1. We ask at
what fictitious densities ρpseudo(z) for a uniform HS fluid (which follows the CS equation), will we
obtain a chemical potential value βµ0 that is equal to the MC nonuniform insertion work βWins(z)
βµ0 =
4η − 3η2
(1− η)2
+ Z ′HS = βWins(z), (3.3)
where η = π
6 ρpseudo(z)σ
3. In figure 2, we show this mapping via a diagram. Note that Z ′HS is
the non-ideal compressibility factor from the CS equation. The pseudo-density ρpseudo(z) thus
calculated is also plotted (as asterisks ∗) in figure 1. We observe that the pseudo-density stays
fairly flat and close to the bulk value of 0.5745, oscillating only weakly up and down with respect
to the horizontal line. The oscillations are out of phase with respect (opposite in period) to the
singlet density ρ
(1)
w (z). The singlet density ρ
(1)
w (z) has a contact value ≈ 2.296 (z = σ/2). It
oscillates vigorously with pronounced peaks and valleys. We note that the insertion work βWins(z)
also oscillates with trends similar to the pseudo-density: when ρpseudo(z) is up, βWins(z) is also up,
and when ρpseudo(z) is down, βWins(z) is also down.
A question naturally arises: how does this pseudo-density ρpseudo(z) compare with the commonly
used weighted densities, say, nα(z) in the fundamental measure theory (FMT) of Rosenfeld [27]. We
33601-6
Adsorption of hard spheres
Figure 2. “Mapping” of properties between the non-uniform system and the uniform systems.
Figure 3. Comparison of the effective (pseudo) density ρpseudo (symbol ∗) with the weighted
densities n0 (green line) and n3/V (blue line) from the FMT theory of Rosenfeld. Bulk hard
sphere density ρ∗ = 0.5745. Symbol (♦) ≡ the singlet density profile ρ
(1)
w (z) from MC data of
Groot et al. [15].
selected two weighted densities n0(z) (green line) and n3(z)/V (blue line) from FMT for comparison
in figure 3 (there are in fact six weighted densities nα(z) in FMT (α = 0, 1, 2, 3, V 1, V 2) based
on the geometries of hard spheres). The basis set of weights are composed of three functions: ω2
is based on the Dirac Delta function, thus generating n0, ω3 is based on the Heaviside function,
generating n3, and ωV 2 is a vector function (see Rosenfeld [27]). The other linearly-dependent
weighting functions are related to this basis set by scaling through spherical geometries. The first
three weighted densities are related by the relations: n0(z) = n1(z)/R = n2(z)/S (R and S are the
radius and surface area, respectively, of a hard sphere), while n3(z) is normalized by the spherical
volume V : n3(z)/V .
In figure 3 we notice that the magnitudes of both n0(z) and n3(z)/V are close to the pseudo-
density ρpseudo(z) and far from the singlet density ρ
(1)
w (z). There are visible differences between
ρpseudo(z) and n0(z) as well as n3(z)/V . This is not surprising, since these coarse-grained densities
are derived from different theories (the FMT vs. the PDT). ρpseudo(z) is close to n0(z) only near the
wall and at long ranges. The cusp in n0(z) (due to the delta weighting) is missing from ρpseudo(z).
33601-7
L.L. Lee, G. Pellicane
Values of n3(z)/V are quite distinct from both n0(z) and ρpseudo(z). The observations also apply
to other two weighted densities owing to scaling: n1(z)/R and n2(z)/S. On the whole, the weighted
densities and the pseudo-density are commensurate with each other (of similar magnitudes) even
though they came from entirely different origins.
To make further comparisons, we also calculate the insertion work arising from the FMT. The
excess free energy F ex in FMT according to literature [27]) is expressed as: βF ex =
∫
d~rΦ(nα(~r))
where Φ is a free energy density. This quantity is a function of the weighted densities nα and itself
arises from a uniform fluid model (such as the compressibility equation from the Percus-Yevick
equation [17] (PYc) or variations thereof [28]). The FMT insertion work can be evaluated as the
functional derivative [28] δF ex/δρ
(1)
w (z), thus in Fourier space
βWins(FMT)(k) =
∑
α
∂Φ
∂nα
ωα(k). (3.4)
This FMT insertion work (Wins(FMT)) at ρ∗ = 0.5745 is plotted in figure 1 (the brown line). It
quite closely follows the simulated Wins(MC) (triangles). This is to be expected, since both the
FMT and the PDT-based theories are designed to match the nonuniform free energy. The FMT free
energy lies somewhat above the PDT values, because the PYc solution for the chemical potential
is higher than the MC value for hard spheres.
Figure 4. The effective (pseudo) density ρpseudo (symbol ∗; from PDT) and the insertion work
βWins (symbol ∆; from MC) at bulk hard sphere density ρ∗ = 0.715 for the HS/HW system.
The insertion work is scaled down 50% in order to fit in the graph. Symbol (♦) ≡ the singlet
density profile ρ
(1)
w (z) from MC data of Groot et al. [15]. The brown line is the insertion work
βWins produced by the FMT theory. The horizontal line is at ρ∗ = 0.715. The pseudo-density
ρpseudo(z) oscillates weakly around the bulk value and in opposite periodicity as compared to
ρ
(1)
w (z). The insertion work βWins also oscillates synchronously with ρpseudo(z) but oppositely
to ρ
(1)
w (z).
Similar calculations are made for HS/HW at a higher density ρ∗ = 0.715. Figure 4 shows
the singlet density ρ
(1)
w (z) (diamonds). The pseudo-density (asterisks) oscillates mildly. The FMT
insertion work (brown line) is close to the MC insertion work βWins (triangles) but overestimates
it. The PYc values are higher than the MC values as already noted. (N.B. The values of βWins are
multiplied by 0.5 (or 50%), in order to fit into the same graph).
In figure 5 ρpseudo(z) is compared with n0(z) (green line) and n3(z)/V (blue line). Again, we
see that at a few points, n0(z) and ρpseudo(z) match closely, while n3(z)/V has larger differences
from ρpseudo(z) at this higher density. The comparisons show that the PDT approach is clearly
different from the fundamental measure theory. FMT was constructed based on the PYc theory
(or modifications thereof) in order to map the compressibility free energy (see also later improved
versions [29]) to the nonuniform free energy. In our approach, we use the PDT to map the CS
free energy to the nonuniform free energy, and we use MC data to obtain the insertion work. The
objectives are the same; but the theoretical bases are different.
33601-8
Adsorption of hard spheres
Figure 5. Comparison of the effective (pseudo) density ρpseudo(z) (symbol ∗) with the weighted
densities n0 (green line) and n3/V (blue line) from the FMT theory of Rosenfeld. Bulk hard
sphere density ρ∗ = 0.715. Symbol (♦) ≡ the singlet density profile ρ
(1)
w (z) from MC data of
Groot et al. [15].
To further examine high-density hard spheres, we next examine the case ρ∗ = 0.813. The
results are plotted in figure 6. Similar oscillatory behavior is observed for the pseudo-density and
the insertion work (reduced to 20% in order to fit in the graph). The phases of oscillations of the
singlet ρ
(1)
w (z) and ρpseudo(z) (and βWins as well) differ by a half period π.
Figure 6. The effective (pseudo) density ρpseudo (symbol ∗; from PDT) and the insertion work
βWins (symbol ∆; from MC)) at bulk hard sphere density ρ∗ = 0.813 for the HS/HW system.
The insertion work is scaled down to 20% in order to fit in the graph. Symbol (♦) ≡ the singlet
density profile ρ
(1)
w (z) from MC data of Groot et al. The horizontal line is at ρ∗ = 0.813. The
pseudo-density ρpseudo oscillates weakly around the bulk value and in opposite periodicity as
compared to ρ
(1)
w (z). The insertion work βWins also oscillates synchronously with ρpseudo(z) but
oppositely to ρ
(1)
w (z).
4. Construction of a closure theory
It has been recognized that the DFT for the structures of nonuniform fluids should first be
based on valid uniform-fluid properties as inputs. In particular, the structure of nonuniform fluids
depends critically on the accuracy of the pair direct correlation functions 2-DCF of the uniform fluid
that must be supplied beforehand. We have developed in the last decades an accurate theory [7]
for the uniform hard sphere fluid: the zero-separation-theorem-based closure (ZSEP). We can thus
easily generate accurate 2-DCF C
(2)
0HS for use in the present study of nonuniform hard-sphere fluids.
33601-9
L.L. Lee, G. Pellicane
We shall look at the five densities where the MC data are available for the HS/HW systems:
namely ρ∗ = 0.5745, 0.715, 0.758, 0.813, and 0.9135. The ZSEP closure for the uniform bridge
function B0HS is of the form [7] (caret indicates a function)
B̂0HS(γ
∗) = −
ζ
2
γ∗2
[
1− φ+
φ
1 + αγ∗
]
, (4.1)
where α, φ, and ζ are adjustable parameters, and γ∗ = γ + ρ f
2 . f is the Mayer factor of the repul-
sive WCA (Weeks-Chandler-Andersen) potential. For details, see references in [7]. The adjustable
parameters α, φ, and ζ are determined using structural consistencies (the zero-separation theorem
and the contact-value theorem) and thermodynamic consistencies (e.g., the pressure consistency,
and Gibbs-Duhem relation). The uniform 2-DCF’s C
(2)
0HS obtained are depicted in figure 7. They
have all satisfied closely the consistency relations imposed. For example, in the case of the high
density ρ∗ = 0.90, the C
(2)
0HS is compared with the MC-generated data [30] in figure 8. Diamond
Figure 7. The pair direct correlation functions C
(2)
0HS(r) of uniform hard spheres generated by
the ZSEP closure [7] for five densities: ρ∗ = 0.5745, 0.715, 0, 758, 0.813, and 0.9135 (Curves from
top to bottom, respectively). They all show a non-zero tail at r > 1 (a correct behavior for hard
spheres).
Figure 8. The pair direct correlation functions C
(2)
0HS(r) of uniform hard spheres generated by
the ZSEP closure [7] for the bulk density ρ∗ = 0.90. The lines are from ZSEP theory, and the
symbols (♦) from MC data of Groot [30]. The inset shows the magnified view of the non-zero
tail and discontinuity in C
(2)
0HS(r). At r = σ+, C
(2)
0HS(σ
+) = 1.20 (from ZSEP), and 1.05 (from
MC). At r = σ−, C
(2)
0HS(σ
−) = −3.965 (from ZSEP), and −4.12 (from MC).
symbols denote the MC data, and the solid line is the 2-DCF from the ZSEP(4.1). We see very
close agreement of the two curves. The zero-value C
(2)
0HS of MC is −44.90 while the ZSEP value is
−47.70 (within 6%). In particular, the values C
(2)
0HS(r) derived from ZSEP for r > 1 are positive and
non-zero. This is corroborated by the MC data. The positive peak C
(2)
0HS(σ
+) = 1.2 from ZSEP,
33601-10
Adsorption of hard spheres
while MC gives 1.05. This is in obvious contrast to the PY-based theories that would produce
C
(2)
0HS(r) = 0 for r > 1 (an erroneous result). The same observation applies to all five present cases
(figure 7).
For the nonuniform HS/HW case, we developed a closure similar to ZSEP [3]:
B̂w(γw) = −sgn(γw) · ζ
′γ2
w
[
1− φ′ +
φ′
1 + α′γw
]
, (4.2)
where α′, φ′, and ζ′ are adjustable parameters for the nonuniform Bw. sgn(.) is the sign function.
γw is the indirect correlation function defined as
γw(~r) ≡
∫
d~r ′C
(2)
0HS(|~r − ~r ′|)δρw(~r
′). (4.3)
We found from practice that “renormalization” of the indirect correlation is not needed for this
nonuniform fluid. We also note that the expression (4.2) is modeled after the ZSEP equation (4.1)
for uniform fluids [7]. In earlier studies [2, 3] we have shown that this closure was effective for
adsorption of high-density Lennard-Jones fluids. The similarity between the uniform closure and
the nonuniform closure used represents an ongoing effort in the transfer of successful theories for
uniform liquids to nonuniform liquids.
Table 1. Parameters for the ZSEP closure (4.2) and wall sum rule for the HS/HW system.
ρ0 0.5745 0.715 0.758 0.813 0.9135
α′ 0.4 0.4 0.4 0.5 0.62
φ′ 0.4 0.4 0.4 0.4 0.4
ζ′ −0.38 −0.28 −0.22 −0.20 −0.13
†ρ
(1)
w (σ/2) 2.288 4.319 5.131 6.598 10.23
§ρ
(1)
w (σ/2) 2.29 4.27 5.15 6.57 10.28
†Contact value (from ZSEP calculations).
§Contact value (from hard-wall sum rule: ρ
(1)
w (σ/2) = P/kT ) from Carnahan-Starling equation.
Equations (1.6), (4.2), and (4.3) are coupled. The convolution was solved by standard numerical
methods [3] in bipolar coordinates. Trapezoidal rule was used in integration. Picard’s iterations with
mixing (relaxation) on input and output iterates were employed. The grid of numerical integration
was ∆r = 0.01σ, and total grid number N = 2048. Thus, the maximum z-distance reached is
20.48σ. Cauchy’s absolute convergence was enforced for the γw-function with convergence criterion
δ = 0.0001. There were three free parameters to be determined in equation (4.2): α′, φ′, and ζ′.
We needed three conditions. The first condition was the hard-wall sum rule: the contact density
ρ
(1)
w (σ/2) = P/(kT ). In practice, we set φ′ to a default value of 0.4. α′ and ζ′ were adjusted in
tandem until the wall rule was satisfied (see table 1). In the future, other sum rules [31] will be
tested and incorporated.
Figure 9 shows the result of application of ZSEP. The density varies from the moderate ρ∗ =
0.5745 to the highest ρ∗ = 0.9135. The symbols are the MC data [15], the lines are from the
ZSEP theory. We observe overall superb agreement for densities ρ∗ from 0.5745, 0.715, 0.758 to
0.813. At ρ∗ = 0.813, our results compare well with previous accurate theoretical attempts (e.g.
the White-Bear version of FMT [29]). The ingredients for success are from (i) the high accuracy
of the input uniform 2-DCF’s; and (ii) the effectiveness of the ZSEP closure (4.2). For the highest
density ρ∗ = 0.9135 there is some deterioration of the theoretical curve at the second peak. The
heights of the peaks (of ZSEP and MC) are similar, however the predicted location is displaced to
a longer distance. A closer reading of the original MC source [15] reveals that at this high density
(packing fraction η = 0.4783), there are clusters of molecules forming hexagonal structures for the
33601-11
L.L. Lee, G. Pellicane
Figure 9. The nonuniform singlet density functions ρ
(1)
w (z) obtained from the new ZSEP-closure
equation ( 4.2) for the HS/HW system at five densities ρ∗ = 0.5745, 0.715, 0.758, 0.813, and
0.9135 (from bottom to top. Each curve shifted up by 0.5 units, respectively). Symbols = MC
data from Groot et al. [15]. Lines ≡ ZSEP results. We discern close agreement between the
ZSEP curves and the MC curves except at the highest density (0.9135) where the second peak
of the theoretical curve is displaced to a larger z-value. (See discussions in the text).
Figure 10. The insertion work generated by the new closure theory at ρ∗ = 0.5745, 0.813, and
0.9135 (Curves from bottom to top, respectively). Lines ≡ ZSEP theory; triangles ≡ MC data.
Close agreement is in evidence for ρ∗ = 0.5745 and 0.813. The deviations at ρ∗ = 0.9135 reflect
the same cause: the two-dimensional solid-fluid transition of the surface layer of hard spheres at
this high density.
first layer of particles (figure 6 of [15]), and reduced normal-direction diffusion coefficient for the
surface molecules: all indications of two-dimensional quasi-solid-fluid transition. Note that the fluid
limit of homogenous hard spheres is at about η ≈ 0.4948. For inhomogeneous hard spheres this
may occur earlier. This puts stringent demand on a closure theory which is not intended to predict
solid formation.
Next, we calculate the insertion work via equation (2.14) using ZSEP-density ρ
(1)
w (z) as input.
The results are shown in figure 10 for three densities ρ∗ = 0.5745, 0.813, and 0.9135. The agreement
between the ZSEP βWins (lines) and MC βWins (triangles) is excellent for ρ∗ = 0.5745 and 0.813.
For ρ∗ = 0.9135, similar discrepancies as for the singlet density ρ
(1)
w (z) are in evidence. The
displacement of the first minimum to a larger distance corresponds to the displacement of the
second peak of ρ
(1)
w (z).
33601-12
Adsorption of hard spheres
5. Conclusions
The two major questions that we set out to explore in this study are (i) how does the effective
density based on the potential distribution theorem look like and how does it behave? (ii) Can
we obtain the structure of hard spheres adsorbed on the hard wall through a closure theory for
nonuniform fluids? We formulate two potential distribution theorems: one for the insertion work of
the usual test particle that interacts with the wall, and the other for the test particle that ignores
the wall. It is the latter insertion work that corresponds to the singlet direct correlation and as a
consequence the second potential distribution theorem becomes equivalent to the Euler-Lagrange
equation.
The pseudo-densities ρpseudo , due to their construction, are much smoother quantities than the
singlet densities. They oscillate around the bulk value with periods out-of phase with respect to
the nonuniform ρ
(1)
w .
The insertion work is derived from the MC data on HS/HW (Groot et al., 1987 [15]) via the
Euler-Lagrange equation (with the identification βWins = −C
(1)
w ). Its behavior for several densities
(ρ∗ = 0.5745, 0.715, and 0.813) is determined. βWins oscillates in phase as the pseudo-density, but
out of phase with respect to the singlet density ρ
(1)
w .
The density functional theories need accurate uniform pair direct correlation functions C
(2)
0HS as
input. Earlier we have formulated an accurate closure theory for the 2-DCF (the ZSEP theory) of
uniform hard spheres. It is computationally cheap now to generate these functions: it took less than
a minute of CPU time on a PC computer to produce an accurate DCF. Here we propose a new
closure theory for the nonuniform hard spheres equation (4.2). Its form is inspired by the uniform
fluid ZSEP equation. This bridge function is used in the star-function based density functional
theory equation (1.6) to generate the singlet density profiles. These ZSEP densities are compared
with the MC data (figure 9) at five densities ρ∗ = 0.5745, 0.715, 0, 758, 0.813, and 0.9135. Except
for the highest density, the ZSEP gives accurate nonuniform density profiles. This shows that
the closure-based density functional theory can perform reasonably well for hard spheres when a
suitable closure (bridge function) has been adopted. We shall have further developments on this in
the future.
These theoretical predictions for the singlet densities are used to calculate the insertion works
(figure 10). They also compare favorably with the MC data.
In summary, we have proposed and evaluated a new type of effective densities via the potential
distribution theorem. We characterize their behavior for the hard spheres/hard wall system at
moderate to high densities (up to ρ∗ = 0.9135). The free energies (the insertion works Wins) of the
nonuniform system are also calculated. They show oscillations opposite in phase to those of the
singlet densities. We also construct a new closure-based density functional theory that can give
accurate reproduction of the computer simulated densities and insertion works.
The newly developed star-based density functional theory can be extended to fluids interacting
with soft/attractive potentials. This task is under active investigation.
Acknowledgements
G.P. acknowledges the Competitive Grant support by the University of Kwazulu-Natal. L.L.L.
had stimulating discussions with Walter Chapman and Ken Cox during my sabbatical leave at
Rice University in 2011. He also had useful discussions with Zhengzheng Feng on the White-Bear
calculations of the FMT theory. L.L.L. thanks them for their hospitality.
33601-13
L.L. Lee, G. Pellicane
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Adsorption of hard spheres
Адсорбцiя твердих сфер: структура та ефективна густина
вiдповiдно до теореми розподiлу потенцiалу
Л.Л. Лi1, Г. Пелiкане2
1 Вiддiл хiмiчної iнженерiї та iнженерiї матерiалiв, Унiверситет Калiфорнiї,
Помона, Калiфорнiя, США
2 Школа фiзики, Унiверситет iменi Квазулу-Наталь, Скотсвiль,
3209 Пiтермарiцбург, Пiвденна Африка
Ми пропонуємо новий тип ефективних густин, отриманих з теореми розподiлу потенцiалу. Цi гус-
тини необхiднi для вiдображення вiльної енергiї однорiдної рiдини у вiльну енергiю неоднорiдної
рiдини. Теорема розподiлу потенцiалу дає роботу, необхiдну для устромляння пробної частинки у
систему молекул, на яку дiє зовнiшнiй потенцiал. Ця робота Wins може бути отримана з симуляцiї
Монте-Карло (MК) (наприклад, пiдходом тестової частинки Вiдома) або з аналiтичної теорiї. Псев-
догустини є побудованi так, що коли їх значення пiдставляються в рiвняння стану однорiдної рiди-
ни (наприклад, рiвняння для хiмiчних потенцiалiв Карнагана-Старлiнгa системи твердих сфер), то
вiдтворюється робота устромляння частинки в неоднорiдну рiдину, отримана з симуляцiї МК. Ми
дослiджуємо поведiнку псевдогустини для системи “твердi сфери”-“тверда стiнка” при середнiх та
високих густинах (вiд ρ∗ = 0.5745 до 0.9135). Для цього використовуються результати Монте-Карло
Гроота та спiвавторiв. Псевдогустини демонструють осцилюючу поведiнку з протилежною фазою
до одночастинкових густин. Ми також пропонуємо нову теорiю функцiоналу густини на основi за-
микання (теорiю функцiоналу густини зiркової функцiї), що може точно описувати профiлi густини i
роботу устромляння. Точнiсть теорiї перевiряється для декiлькох випадкiв адсорбцiї твердих сфер.
Ключовi слова: теорема розподiлу потенцiалу, твердi сфери, адсорбцiя, ефективна густина,
структура, iнтегральне рiвняння, замикання
33601-15
Introduction
Potential distribution theorems for nonuniform systems
Effective density and work of insertion
Construction of a closure theory
Conclusions
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