Adsorption of hard spheres: structure and effective density according to the potential distribution theorem

Abstract

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 W_ins 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 \rho^*= 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.

Full text

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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
1Department of Chemical & Materials Engineering, California State University,
Pomona, California, USA
2School 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 Winscan 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)
?
where ρ(1)
w
is the singlet (nonuniform) density, w(r) the external (1-body) potential, and µ0the
chemical potential of the bulk fluid. At equilibrium, the grand potential is minimized with respect
to the singlet ρ(1)
w ] (IHFEF):
Ω = F[ρ(1)
w] +d? rρ(1)
w(? r;[w]){w(? r) − µ0}
(1.1)
w , and it yields the Euler-Lagrange equation
δβΩ
δρ(1)
w (? r)
=δβF[ρ(1)
δρ(1)
w ]
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: Fid[ρ] =?d? rρ(1)
c ? L.L. Lee, G. Pellicane, 2011
w (? r)[lnρ(1)
w (? r)Λ3−1])
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L.L. Lee, G. Pellicane
and the excess part Fex. Thus F[ρ(1)
based density functional theory [2, 3] (s-DFT) which expresses Fexsuccinctly as
?
where S∗
w ] = Fid+ Fex. Recently, we have proposed a star-function
− βFex
[ρ]= −βFex
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)
wis the star function [4] defined as the “primitive” of the bridge function Bw
S∗
w=
?
d? rδρw(? r)
γw(? r)
γ1
?
γ0
dxBw(x)
(1.4)
and C(1)
correlation function (2 − DCF). Fex
uniform and nonuniform fluid properties, respectively. δρw= (ρ(1)
the nonuniform density ρ(1)
that if the bridge function Bwcan be obtained exactly, the star function S∗
exact from equation (1.4), thus the excess free energy Fexin equation (1.3) will also be exact. The
Euler-Lagrange (EL) equation (1.2) can be expressed as
?
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
0(? r) is the singlet direction correlation function (1 − DCF), C(2)
0
is the bulk fluid free energy. Subscripts 0 and w indicate
0(? r,? r?) is the pair direct
w − ρ0) is the difference between
w and the uniform density ρ0and γwis the indirect correlation. We note
wwill consequently be
ρw(? r) = ρexp
−w(? r) + C(1)
w(? r) − C(1)
0(? r)
?
(1.5)
ρw(? r) = ρexp[−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)
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
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
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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)
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
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]. Bwas 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 Bwcan
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 (` a la mode of the Percus-Yevick equation [17])
that relates Bwto 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 Bwis in fact a functional [3, 21] Bw= Φ[ρ(1)
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 Bwin [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 . To illustrate the PDT,
0) of the uniform fluid to those (βWins) of the nonuniform fluid. In this
w ,w] of the singlet density ρ(1)
w and/or
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)
The thusly obtained βWinsis 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 ρpseudoand compare with the commonly used weighted densities found in literature. In
w as input.
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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
?? pN?+ UN
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
HN(? pN,? rN) = KN
?? rN?+ WN
?? rN?,
(2.1)
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(? rN−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(? rN−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(? rN−1) + WN(? rN) − WN−1(? rN−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”)
N−1
?
Ψ−w(? rN) =
i=1
u(2)(? ri,? rN).
(2.7)
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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−1exp[−βUN−1− βWN−1]
?N−1
i=1
× exp
−β
?
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−1exp[−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= lnQN−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
since
?
ρ(1)
w(? rN)Λ3?
(2.12)
?exp[−βΨ−w]?N−1;w
=
exp[βw(? rN)]
QN−1
?
?
?
d? rN−1exp[−βUN−1− βWN−1]
?N−1
i=1
× exp
−β
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(ρ0Λ3))
βWins
=
−ln?exp[−βΨ−w]?N−1;w= βµw− βw(? rN) − ln
βµex
0− βw(? rN) − ln
?
ρ(1)
w(? rN)Λ3?
=
?
ρ(1)
w(? rN)/ρ0
?
.
(2.14)
The quantity Winscan 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) = 0,
u(2)(r) = ∞,r ? σ,
r > σ.
(3.1)
33601-5