THE JOURNAL OF CHEMICAL PHYSICS 135, 204706 (2011)
Constructing a new closure theory based on the third-order
Ornstein-Zernike equation and a study of the adsorption of simple fluids
Lloyd L. Leea)
Department of Chemical and Materials Engineering, California State University, Pomona,
California 91768, USA
(Received 23 August 2011; accepted 2 November 2011; published online 30 November 2011)
The third-order Ornstein-Zernike equation (OZ3) is used in the construction of a bridge functional
that improves over conventional liquid-theory closures (for example, the hypernetted chain or the
Percus-Yevick equations). The OZ3 connects the triplet direct correlation C(3) to the triplet total
correlation h(3) . By invoking the convolution approximation of Jackson and Feenberg, we are able to
express the third-order bridge function B3 as a functional of the indirect correlation γ . The resulting
expression is generalized to higher-order bridge terms. This new closure is tested on the adsorption
of Lennard-Jones fluid on planar hard surfaces by calculating the density profiles and comparing with
Monte Carlo simulations. Particular attention is paid to the cases where molecular depletion on the
substrate is evident. The results prove to be highly accurate and improve over conventional closures.
© 2011 American Institute of Physics. [doi:10.1063/1.3663221]
I. INTRODUCTION
For fluids adsorbed on a solid surface, either of two situations may occur: (a) the value of the contact density is higher
than the bulk density. There is a surplus of molecules at the
wall. We call this adhesion. Or (b) the contact density is lower
than the bulk value. Thus, there is a deficit of molecules at the
wall. We call this depletion. There may also be the third possibility of an even distribution. In statistical mechanics, the density profile near and not far from the wall is described by the
singlet probability density ρ w (z) (z being the distance normal
to the wall). Far from the wall, ρ w approaches the constant
bulk density value ρ 0 . Adhesion or depletion is a general behavior, present at either supercritical or subcritical states, with
different substrate affinities, molecular species, and geometries of the inhomogeneities. Accumulation or detachment of
molecules at the interface is the result of competition between
the interfacial wall forces and the coherent fluid-fluid forces
on the fluid molecules vis-à-vis the state conditions (phase diagrams, pressures, and temperatures) (see reviews of Refs. 1
and 2).
On a planar hard wall this can be clearly shown by the
well-known sum rule for the contact density ρ w (zc ):
P0
,
(1.1)
kT
where subscript “w” denotes properties under the influence
of a wall potential w, and “0” denotes the bulk fluid property
(at zero wall potential, i.e., w = 0). P0 is the bulk pressure, k
is the Boltzmann constant, T is the absolute temperature, and
zc is the contact distance of an adsorbate molecule. Therefore, if we know the fluid equation of state (EOS) ahead of
time we can determine if we have fluid adhesion/depletion at
the given temperature T and density ρ 0 . As an example, for
the Lennard-Jones (LJ) fluid we know an EOS due to Nicolas
ρw (zc ) =
a) Electronic mail: [email protected].
0021-9606/2011/135(20)/204706/11/$30.00
et al.3 For LJ fluid adsorbed on a hard planar wall (LJ/HW),
at temperature, say, T* = kT/ε = 1.35 (ε is the LJ energy parameter), we can calculate ρ w (zc ) values at different bulk pressures P0 according to the sum rule equation (1.1). While from
the EOS, we can also calculate the actual bulk densities ρ 0
that give rise to these pressures P0 . If we plot ρ w (1) (zc ) vs. ρ 0
(at the same P0 ) as in Fig. 1, ρ w (zc ) can be either higher than
ρ 0 (higher contact value means adhesion) or lower than ρ 0
(lower contact value means depletion). A diagonal line marks
off the two regions. Thus, the ρ 0 -axis is divided into two halfplanes: one half for depletion (in this case the left panel with
lower densities) and the other for adhesion (the right panel
at higher densities). The line of demarcation of the two at
this temperature is at about ρ 0 * = ρ 0 σ 3 ∼ 0.66 (σ is the
LJ size parameter). The precise value depends on the accuracy of the Nocolas EOS, which was first obtained from an
empirical fit to the simulation data. In the LJ case, adhesion
happens at high densities (ρ 0 * > 0.66), and depletion happens
at low densities (ρ 0 * < 0.66). Of course, at a different temperature, the line might shift and the regions might multiply; and
we shall have a different demarcation. Note that the isotherm
T* = 1.35 is slightly supercritical for the LJ fluid (T* c,LJ
∼ 1.317).
Density functional theories4 (DFT) have been employed
to describe the adsorption behavior. There are several approaches in DFT: for example, the weighted-density approach (e.g., the fundamental measure theory5–8 ), the squaregradient approach,9, 10 and the perturbation/closure-based
approaches11–14 (see reviews of Refs. 15–18). Historically,
the closure-based theories (e.g., the wall-particle OrnsteinZernike (WOZ) integral equation approach19, 20 and its variations) were developed in the 1970s and were found to
treat simple systems with adhesive behavior quite well; but
did not always give quantitative answers for those cases
with severe depletion.21–23 The closures used in these works
were from the common uniform fluid theories, such as the
135, 204706-1
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204706-2
Lloyd L. Lee
J. Chem. Phys. 135, 204706 (2011)
1.0
1.4
0.9
0.8
Adhesion
Depletion
HNC
0.7
1.0
0.6
ρw
Contact density, ρ w
1.2
0.8
0.5
0.4
0.6
0.3
Diagonal
PY
0.4
0.2
0.2
0.1
0.0
0.0
0.5
0.0
0.2
0.4
0.6
0.8
MC
z
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
1.0
Bulk density, ρ 0
FIG. 1. Determination of the “depletion” and “adhesion” regions of adsorption for the Lennard Jones fluid on hard wall at T* = 1.35. The hard wall
sum rule Eq. (1.1) is used to calculate the contact density ρ w at a given bulk
pressure P0 . The bulk density ρ 0 at the same pressure P0 is obtained from
an equation of state (Nicolas et al.3 ). A diagonal line (where ρ w = ρ 0 ) is
used to divide the abscissa into two regions where ρ w < ρ 0 (depletion) or ρ w
> ρ 0 (adhesion).
Percus-Yevick24 (PY) closure (see below, Eq. (3.2)) or the
hypernetted chain25, 26 (HNC) closure (Eq. (3.1)). Note that
we have shown earlier27 that the WOZ approach19, 20 and the
closure-based DFT (Refs. 11–14) are theoretically equivalent.
Recently, there have been notable advances in the closurebased DFT.11–14, 28 Applications were made to various fluids
(Lennard-Jones, Yukawa, and soft matters) and geometries
(planar surfaces, parallel slits, spherical cavities). More sophisticated closures have been applied.11–14, 28
One can however pose the question: were the noted inadequacies of the WOZ for depletion adsorption due to the
inherent errors in the approximate closure relations used (as
found in the PY or HNC equations)? If so, can we improve
the performance by correcting the errors and incorporating
more fundamental physical principles that had been neglected
earlier? In this work, we shall examine this question. We
shall formulate a new closure theory based on the third-order
Ornstein-Zernike (OZ3) equation (see below) and as a partial
check of its validity apply it to the LJ/HW system, especially
for cases of depletive adsorption.
For LJ/HW, Balabanic et al.29 and Lutsko30 have provided Monte Carlo (MC) simulation data at T* = 1.35 and
ρ 0 * = 0.50. The singlet density profile (symbol = diamonds)
is shown in Fig. 2. Near the wall (zc = 0.5 σ ) the contact density ρ w (zc )σ 3 ∼ 0.2, while the bulk density is much higher
ρ 0 * = 0.50. Thus, this case belongs to the depletion region.
However, if we solve the Euler-Lagrange equation (see below,
Eq. (1.4)) for the density profile with the PY and/or the HNC
closure, we would obtain the two top curves shown in Fig. 2.
They are overly oscillatory, and yield contact values that are
too high. None of the closures (PY or HNC) shows any depletion (contact densities > 0.50). These two classical closures,
successful for uniform fluids, are clearly not appropriate for
the nonuniform LJ/HW system. The question is what is missing in PY and HNC that caused their poor performance for
these nonuniform cases?
FIG. 2. The singlet density profile ρ w(z) for LJ/HW system at ρ * = 0.5 (T*
= 1.25). Diamonds = Monte Carlo data from Balabanic et al.22 Upper line
= HNC closure. Lower line = PY closure. PY and HNC fail to give quantitative predictions of the MC ρ w (z).
To establish nomenclature, we briefly review the commonly employed density functional theory based on the grand
potential, . It is defined for a nonuniform
fluid with an ex
ternal potential w(r ) as ≡ F [ρw ] + d rρw (r )[w(r ) − μ0 ],
where F is the intrinsic Helmholtz free energy functional, being a functional of ρ w (r ), and ρ w (r ) is the singlet density. μ0
is the chemical potential of the bulk fluid. Subscript w indicates nonuniform properties (in the presence of the wall potential w(r )), while subscript 0 indicates uniform properties
(when the wall potential is zero, w = 0). The grand potential
is minimized at the equilibrium singlet density and the EulerLagrange (EL) relation for this minimization is
−
δβF [ρ]
= βw(r) − βμ0 = Cw(1) (r) − ln[ρw (r)3 ],
δρw (r)
(1.2)
where β is the reciprocal temperature 1/(kT), k is the Boltzmann constant, T is the absolute temperature, and is the de
Broglie wavelength. Rearrangements give
(1.3)
r ) − C (1)
ρw (r ) = ρ0 exp − βw(r ) + C (1)
w (
0 .
We have shown earlier27 that an equivalent form of the EL can
be written as
ρw (r ) = ρ0 exp − βw(r ) + γ w (r ) + Bw (r ) , (1.4)
where Bw is the bridge functional in the presence of the external potential w. Comparing Eqs. (1.3) and (1.4) gives
r ) − C (1)
r ) + Bw (r ).
C (1)
w (
0 = γ w (
(1.5)
The nonuniform indirect correlation function (icf) γ w (r ) is
defined, by analogy with uniform fluids, as
γ w (r ) ≡ d r C0(2) (|r − r |)δρ w (r ),
(1.6)
where δρ w (r ) ≡[ρ w (r ) – ρ 0 ] ≡ ρ 0 hw (r ). hw is the nonuniform total correlation function. γ w (r ) as defined is a wallparticle correlation (not a particle-particle correlation). Note
that the kernel in Eq. (1.6) is the uniform fluid pair direct
correlation, C0 (2) .
In this study, we shall re-examine the general basis of
the closure relation for both nonuniform and uniform fluids.
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204706-3
Closure and the third-order OZ relation
J. Chem. Phys. 135, 204706 (2011)
We start by examining the triplet function C(3) (i.e., the triplet
direct correlation function, DCF-3) based on the third order
Ornstein-Zernike equations (Sec. II). OZ3 is an exact equation
connecting the DCF-3 C(3) to the triplet total correlation function (TCF-3), h(3) . It allows interchange of expressions either
in C(3) or in h(3) . Thus, we are able to express in closed form
the third-order bridge function term B3 (see the Appendix and
Sec. II) in terms of either C(3) or h(3) . By making appropriate approximations to the TCF-3 h(3) , we are able to construct a new and improved closure for the bridge functional
for nonuniform fluids (Sec. III). In numerical implementation
of the theory, we also visit the issue of renormalization of
the base function used in the closure (Sec. IV). Tests on the
LJ/HW systems are made (Sec. IV) and new and more accurate predictions are obtained for the “depletion” cases of
adsorption. Conclusions and anticipated future directions are
given in Sec. V.
II. THE THIRD-ORDER ORNSTEIN-ZERNIKE
EQUATIONS
We have shown earlier31 that there are four members of
the OZ3, based on the methods of Lebowitz and Percus32 on
the functional derivatives. It is a generalization of the secondorder OZ2:
(2)
(
r
,
r
)
−
C
(|
r
,
r
)
≡
d r C0(2) (r , r )ρ 0 h(2)
r , r ).
h(2)
0
0
0 (
(2.1)
As is known,32 Eq. (2.1) can be derived via the inverse matrix
relation between the functional derivative δρ w (1)/δW(2) and
its inverse δW(2)/δρ w (1), where W is the intrinsic chemical
potential defined by W(r )≡βμ0 – βw(r ) (notation: arguments
1 = r1 and 2 = r2 , etc.):
δρw (1) δW (3)
= δ3 (1, 2),
d3
(2.2)
δW (3) δρw (2)
where δ 3 (1,2) is the three-dimensional Dirac delta function
δ 3 (r1 – r2 ). The functional derivatives are given explicitly by32
δρw (1)
= δ3 (1, 2)ρw (1) + F (2) (1, 2),
δW (2)
and
δ3 (1, 2)
δW (1)
=
− C (2) (1, 2),
δρw (2)
ρw (1)
(2.3)
where F(2) is the Ursell distribution. OZ3 follows a similar
pattern based on higher order functional derivatives, for example, δ 2 ρ w (1)/[δW(2)δW(3)]. OZ3 has four alternative forms,
because one can choose different combinations of the partial differentials. We retrieve one of the forms31–33 explicit in
C(3) (we shall drop the superscripts and subscript “w” to save
space):
C123 = h123 − C12 C13 − C21 C23 − C31 C32
3
2
−ρ d4d5d6 C14 C25 C36 h456 +ρ d5d6 C15 C26 h356
−ρ
d5 C15 h235 + ρ 2
d4d5 C15 C34 h245
−ρ
d6 C26 h136 + ρ 2
−ρ
d4d6 C26 C34 h146
d4 C34 h124 + 2ρ
d4 C14 C24 C34
(OZ3).
(2.4)
This equation is exact, since it is based on the following exact
inverse matrix relation:32
δ 2 W (1)
δW (4) δW (6) δW (1)
= − d4d5d6
δρw (2)δρw (3)
δρw (3) δρw (2) δρw (5)
×
δ 2 ρw (5)
.
δW (4)δW (6)
(2.5)
The notation is self-explanatory: i.e., C12 = C(2) (1,2), C123
= C(3) (1,2,3), and h12 = h(2) (1,2), h123 = h(3) (1,2,3), etc. We
observe that C(3) necessarily involves the triplet TCF h(3) in
convolution with pair correlations. If we know h(3) , we can
calculate C(3) (vice versa). The knowledge of C(3) will impact
on the closure relations.
A. Closure theories
A closure expresses the bridge function B(r) in terms of
a function B̂(γ ), or in terms of a functional B[γ (r)], of an argument function γ (r), i.e., equations postulated between the
bridge function B and other known liquid-theoretical correlation functions (such functions can, in different approximations, be the thermal potential ω, the cavity function y, and/or
the indirect correlation γ ). Most existing closure equations
are approximate in one way or another because they make
plausible simplifications to the exact relation (which is a functional). In statistical mechanics, the bridge function B(r) itself is a formally defined quantity: either as an infinite cluster
series34 or as a functional Taylor expansion.35 However, direct
numerical evaluation from these definitions is extremely cumbersome if not impossible (due to the infinite number of terms
and lack of knowledge of high-order correlations). Thus, truncations were made entailing questionable convergence problem. On the other hand, with computer simulations, the bridge
function B(r) can be inverted directly from the simulation data
based on its definition (see Eq. (2.6) below). This type of reverse engineering (i.e., inversion from the machine data) has
been done many times before.36 The definition of Bw is (we
shall switch back to the nonuniform Bw from the uniform B
without loss of validity)
Bw (r ) ≡ ln yw (r ) − γw (r ),
(2.6)
which is in fact a rearrangement of the Euler-Lagrange equation (1.4). The cavity function yw is defined as ln[yw (r )]
≡ ln[ρ w (r )/ρ 0 ] + βw(r ). We, equipped with the simulated
density distributions h(2) (r) or ρ w (r ) (from MC or molecular dynamics (MD)), can obtain γ (r) = h(2) (r) – C(2) (r) from
Eq. (2.1) (in the homogeneous case) or γ w (r ) from Eq. (1.6)
(in the inhomogeneous case). With MC inputs to the righthand side of Eq. (2.6), the bridge function Bw (r ) can be
calculated. This raw data-inverted bridge function has been
determined earlier and reported for the LJ/HW system in
Ref. 27. For the state of ρ * = 0.50 (T* = 1.35), the
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204706-4
Lloyd L. Lee
J. Chem. Phys. 135, 204706 (2011)
2
1.5
1
w
0.5
Z
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Since we have derived, from OZ3, the exact expression for
C(3) , it is a simple matter to substitute Eq. (2.4) into Eq. (2.8)
(the full result is given in the Appendix). We have at hand
an exact expression (A1) for B3 in terms of h(3) . However,
the final equation is not amenable to numerical calculations
because of the lack of knowledge of the triplet TCF h(3) . We
shall thus consider making “approximations” to h(3) .
-0.5
-1
-1.5
III. CONSTRUCTION OF A CLOSURE THEORY BASED
ON HIGHER-ORDER OZ RELATIONS
B
w
-2
-2.5
FIG. 3. The nonuniform bridge function Bw (z) (lower line) at ρ * = 0.50 as
obtained from inversion of the MC data39 according to the definition (2.6).
Also shown is the MC- inverted indirect correlation γ w (z) (Eq. (1.6), upper
line). Both are derived from the MC data, without any approximation.
bridge function Bw (z) is depicted in Fig. 3 (where we have
also shown the MC-inverted icf γ w (z)). Thus, the bridge
function Bw (r ) as a function of r “exists” for nonuniform fluids, contrary to speculations otherwise. The next question is
whether this curve (in Fig. 3) is related (based on distribution
theory) to other well-known correlations (beyond the definition (2.6))? The relation Bw = B̂ w (γ ), if it exists, is called
a closure (the name was originally derived from the complementary relation supplied in order to “close” the OZ2 equation (2.1), to make it solvable). It constitutes a liquid-state
theory that is the crux of the matter in establishing a closurebased DFT. We reiterate: the bridge function Bw (r ) is a welldefined function of rand it exists (as defined by Eq. (2.6))
and is obtainable by numerical inversion of simulation data;
while a closure relation Bw = B̂ w (γ ) is an equation (theory) between this bridge function and other correlation functions. This equation is, generally speaking, a functional Bw
= Bw [γ ]. In few instances, the functional may reduce to a
function B̂ w (γ ) (caret means a function of another “argument
function,” the argument function is here γ = γ (r )).
The bridge function can be expanded in functional Taylor
series (with the uniform system as the reference) (see, e.g.,
Ref. 27):
1
d2d3 C0(3) (1, 2, 3)δρw (2)δρw (3)
Bw (1) ≡
2!
1
+
d2d3d4 C0(4) (1, 2, 3, 4)δρw (2)δρw (3)δρw (4)
3!
1
+
d2d3d4d5 C0(5) (1, 2, 3, 4, 5)δρw (2)δρw (3)δρw
4!
1
1
··· +
+ ···
5!
6!
= B3 + B4 + B5 + B6 + · · · , respectively.
× (4)δρw (5) +
(2.7)
In the last equality, we have used the order of the DCFs to
define the abbreviations B3 , B4 , and higher order terms. For
example, the third-order term B3 is given by
1
B3 (1) ≡
d2d3 C0(3) (1, 2, 3)δρw (2)δρw (3). (2.8)
2!
In this section, we shall discuss some of the possible simplifications that can be made to h(3) in Eq. (2.4). This is for the
purpose of constructing a usable closure theory. The simplification cannot be too simple so as to lose accuracy, and not too
complicated to be unwieldy. We shall zero in on one specific
approximation and analyze its role in the formulation of future
closure equations. We shall also analyze some of the commonly known closures (such as PY and HNC) in light of OZ3.
The simplest approximation, which has been proposed
for B, is to set C(3) as well as all higher order terms C(n) (n
> 3) to zero (or set their weighted integrals to zero). This in
essence is the Ramakrishnan-Yussouff theory37 in DFT (or the
hypernetted-chain equation):
C (3) (1, 2, 3) ∼
= 0, (HNC).
(3.1)
A more judicious policy may be to select appropriate approximations to the triplet h(3) that appears in Eq. (2.4). We aim
at simplifying Eq. (2.4) while preserving as much physics as
possible in C(3) (and subsequently in B3 ). Several options are
examined below. (We shall drop the subscript “w” without
ambiguity owing to the isomorphism between the expressions
of nonuniform and uniform quantities, especially in light of
the Percus test-particle trick.38 )
First, the common PY closure can be expanded
1
1
B̂(γ ) = ln(1 + γ ) − γ = − γ 2 + γ 3
2
3
1
− γ 4 + − · · · , (PY).
(3.2)
4
To decipher the specific terms in the expansion (3.2), we compare the terms with the exact expression (2.7). For B3 , we can
match the γ 2 -terms from Eqs. (2.8) and (3.2) by making the
specific approximation (3.4) below to C(3) :
1
B3 (1) =
d2d3 C0(3) (1, 2, 3)δρw (2)δρw (3)
2!
1
∼
d2d3 C0(2) (1, 2)C0(2) (1, 3)δρw (2)δρw (3)
=−
2!
1
= − γ (1)2 ,
2
where we have set
C (3) (1, 2, 3) ∼
= −C (2) (1, 2)C (2) (1, 3), (PY).
(3.3)
(3.4)
Observations are as follows: (i) approximation (3.4) in PY is
a fragment of the exact OZ3 expression (2.4). We can identify
the term −C(2) (1, 2)C(2) (1, 3) in Eq. (2.4). (ii) Equation (3.4)
misses many terms and, in particular, two other product terms
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Closure and the third-order OZ relation
J. Chem. Phys. 135, 204706 (2011)
C(2) C(2) , thus becoming “asymmetric” in the indices 1, 2, and
3 (symmetry is a necessary condition for a correct C(3) ). These
are obvious “defects” in the PY theory, as revealed by examining the B3 term. We expect more deficiencies in PY coming
from higher order terms.
Other choice approximations to the triplet h(3) include
(i) the convolution approximation (CA) of Jackson and
Feenberg,39 (ii) the Kirkwood superposition (KS),40 and (iii)
others. The convolution approximation was proposed for calculating the matrix elements of the phonon-phonon interactions in quantum fluid:39
convolution approximation equation (3.5), by switching the
roles of h(3) and C(3) . (iii) This formula improves over the
Percus-Yevick formula (3.4) by symmetrizing the arguments
with three product terms and including a convolution integral.
We shall call (3.7b) the CA-C3 approximation. In cluster diagrams we can see the “symmetry” of terms
C(3) (1, 2, 3)(CA-C3) =
h(3) (1, 2, 3) ∼
= h(2) (1, 2)h(2) (1, 3) + h(2) (2, 1)h(2) (2, 3)
+ h(2) (3, 2)h(2) (3, 1) +ρ d4 h(2)
× (1, 4)h(2) (2, 4)h(2) (3, 4)
(CA-h3).
(3.7c)
(3.5)
After some algebra, it turns out that Eq. (3.5) implies also that
C(3) = 0, similar to the HNC approximation. We shall call the
Jackson-Feenberg equation (3.5) the CA-h3 approximation.
The Kirkwood superposition40 is given by
g (3) (1, 2, 3) ∼
= g (2) (1, 2)g (2) (1, 3)g (2) (2, 3).
(3.6a)
By applying the definitions of h(3) and g(3) , we obtain
h(3) (1, 2, 3) ∼
= h(2) (1, 2)h(2) (1, 3) + h(2) (2, 1)h(2) (2, 3)
+ h(2) (3, 2)h(2) (3, 1)
+ h(2) (1, 2)h(2) (2, 3)h(2) (3, 1)
(Kirkwood).
(3.6b)
Comparing Eqs. (3.5) and (3.6b), we see that the two equations are similar except that the convolution integral (last term
in Eq. (3.5)) is replaced by the cyclic product in Eq. (3.6b).
The expression of KS-based C(3) can be easily obtained by
substitution of Eq. (3.6b) into Eq. (2.4) (details not shown).
Abe41 and Salpeter42 have further extended KS to higher order terms. Bildstein et al.43 have analyzed the performance of
Eq. (3.6b) for h(3) .
Aside from these common formulations, other approximations can be made for h(3) . Following the example of Jackson and Feenberg,39 we also propose a convolution approximation but with a variation: instead of setting C(3) = 0 as
Jackson and Feenberg did, we set
h (1, 2, 3) ∼
= 0.
(3)
(3.7a)
Thus, Eq. (2.4) simplifies to
C (3) (1, 2, 3) ∼
= −C (2) (1, 2)C (2) (1, 3) − C (2) (2, 1)C (2) (2, 3)
− C (2) (3, 2)C (2) (3, 1) +2ρ d4
× C (2) (1, 4)C (2) (2, 4)C (2) (3, 4)
(CA-C3).
(3.7b)
(i)This expression is “symmetric” in its arguments (i.e., it
maintains permutation invariance in 1, 2, and 3). (ii) This
is a counterpart, in mirror image, of the Jackson-Feenberg
(where a single bond represents the C(2) function; filled circle
is an integrated field point).
The ansatz h(3) = 0 may seem unjustified. However, after propagation through the OZ3 equation (2.4), the value of
the outcome remains to be weighed. Just as the assumption,
C(3) = 0, seemed contrived at first, the effect of the JacksonFeenberg equation is quite salutary for quantum fluid. It was
pointed39 out that the convolution approximation satisfies the
recursion and normalization conditions of the distribution
functions, while the Kirkwood superposition does not. We
shall reserve judgment on CA-C3 before it has a chance of
being fully tested.
The third choice is the triple product formula due to Barrat, Hansen, and Pastore (BHP):44
C (3) (1, 2, 3) ∼
= t(1, 2)t(2, 3)t(3, 2),
(3.8)
where t(1,2) is an unknown function and is to be determined
by the sum rule on the derivatives of DCF. The validity of
Eq. (3.8) has been investigated in Ref. 44. The validity of
Eq. (3.7b) has been shown partly in Ref. 45. Other approximations exist in literature,46–56 and the promising ones will
be investigated.
A. Implications of the convolution approximation
on the B3-term
We substitute the CA-C3 expression (3.7b) into Eq. (2.8)
for B3 and obtain after some algebra
γ (1, 2)2
− ρ d3 C (2) (1, 3)h(2) (3, 2)γ (3, 2)
B3 (1, 2) ∼
=−
2
+ ρ d3 C (2) (1, 3)γ (3, 2)2 .
(3.9)
This equation expresses B3 both as a function and as a functional of the icf γ . Comparison of Eq. (3.9) with the PY expansion (3.2) shows that they share one common term – γ 2 /2.
Equation (3.9) is obviously a correction to PY by including
the two extra integrals. If Eq. (3.9) can be considered as an
improvement over PY at order n = 3, what improvements
shall come from the higher order terms (n > 3)? We formally
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204706-6
Lloyd L. Lee
J. Chem. Phys. 135, 204706 (2011)
consider the fourth-order Ornstein-Zernike equation (OZ4)
below.
B. Consideration of the OZ4 Formula
The OZ4 equations31 can be obtained from the third functional derivatives of the form δ 3 (.)/δ(.)δ(.)δ(.) with differentials from the singlet densities ρ w and the intrinsic chemical
potential W. There are six members of OZ4. One of the OZ4
members is derived from the following chain rule:
−
δ 3 W (1)
δρw (2)δρw (3)δρw (4)
· F3 (1, 2, 3)
δ 2 ρw (6)
δW (7) δW (1) δ 2 W (5)
= d5d6d7
δρw (3) δρw (6) δρw (2)δρw (3) δW (5)δW (7)
δ 2 ρw (6)
δW (5) δW (1) δ 2 W (7)
+ d5d6d7
δρw (4) δρw (6) δρw (2)δρw (3) δW (5)δW (7)
δ 2 ρw (6)
δW (5) δW (7) δ 2 W (1)
+ d5d6d7
δρw (4) δρw (3) δρw (2)δρw (6) δW (5)δW (7)
δW (5) δW (8) δW (7) δW (1)
+ d5d6d7d8
δρw (4) δρw (2) δρw (3) δρw (6)
δ 3 ρw (6)
.
×
δW (5)δW (7)δW (8)
(3.10)
After substituting the known formulas for the functional
derivatives (see, e.g., Eq. (2.3)), Eq. (3.10) will give a lengthy
expression involving C(4) and h(4) , plus integrals of lower order correlations. We shall not write out explicitly the details,
except by noting that (i) it contains product terms of C(2)’ s
which shall yield the γ 3 -term shown in PY equation (3.2); (ii)
it will contain additional integrals of C(n) and h(n) with forms
more or less similar to those in Eq. (2.4), except being lengthier. In principle, OZ4 can be used to obtain the exact term B4
in the bridge function (2.7). We would have traded B4 as a
function in C(4) to a function in h(4) . Unless we already knew
h(4) , the exercise is not of much practical utility. Thus, instead,
we shall consider, in the following, a resummation scheme.
C. Resummation of the higher-order bridge terms
We start with the BHP (Ref. 44) formula (3.8). We notice that in Ref. 44, the t-function obtained resembles in
many ways, in shape and in range, to the pair DCF C(2) (see
Fig. 1 in Ref. 44, 1988). We conjecture that a superposition of
three pair DCF’s C(2) would probably be equally as effective.
Namely (k being a constant to be determined),
C (3) (123) ≡ kC (2) (12)C (2) (13)C (2) (23),
(3.11)
or more generally, by applying a “modification” function
F3 (1,2,3) such that
C (123) ≡ C (12)C (23)F3 (1, 2, 3),
(3)
(2)
sented before.57 We shall include here for completeness. We
examine the B3 term:
ρ2
B3 (10) ≡
d2d3 C (3) (123)h(20)h(30)
2!
ρ2
=
d2d3 F3 (1,2,3)[C (2) (12)h(20)] [C (2) (13)h(30)]
2!
ρ2
=
d2 [C (2) (12)h(20)] · d3 [C (2) (13)h(30)]
2!
(2)
(3.12)
F3 (1,2,3) is considered as defined by Eq. (3.12) in terms of
C(3) and C(2) . Certainly, F3 (1,2,3) must be such that it satisfies the symmetry requirements of C(3) (invariant under permutations of the three arguments 1, 2, and 3). With this definition of F3 , we shall develop a resummation formula for
the bridge function. Some of these arguments have been pre-
= (apply mean-value theorem to F3 )
1
1
F̄3 [h(10) − C (2) (10)]2 = F̄3 γ 2 (10). (3.13)
2!
2!
To make progress, we have applied the mean-value theorem
to F3 (1,2,3) in the last equality. This gives a mean valueF̄3 .
We have also used the OZ relation (2.1) for the convolution
of h(r) and C(r) to obtain the icf γ (r). Similarly, we factorize
C(4) in terms of an F4 (1,2,3,4) function:
=
C (4) (1234) ≡ C (2) (12)C (2) (13)C (2) (14)F4 (1, 2, 3, 4).
(3.14)
Applying the mean value theorem to F4 (1,2,3,4), the fourthorder term in Eq. (2.7) becomes
ρ3
B4 (10) ≡
d2d3d4 C (4) (1234)h(20)h(30)h(40)
3!
ρ3
=
d2d3d4 F4 (1, 2, 3, 4) [C (2) (12)h(20)]
3!
× [C (2) (13)h(30)] [C (2) (14)h(40)]
(apply mean-value theorem to F4 )
1
1
F̄4 [h(10) − c(2) (10)]3 = F̄4 γ 3 (10). (3.15)
3!
3!
Repeated applications of the modification functions Fn to
higher order C(n) s, and applications of the mean value theorem result in the infinite series
=
B(10) ≡
B(r) ≡
F̄3 2
F̄4
F̄5
γ (10) + γ 3 (10) + γ 4 (10) + . . . ,
2!
3!
4!
F̄3 2
F̄4
F̄5
γ (r) + γ 3 (r) + γ 4 (r) + · · · .
2!
3!
4!
or
(3.16)
The mean valuesF̄3 , F̄4 , . . . are now functions of T and ρ;
and r = |r0 – r 1 | is the inter-particle distance. Let us examine the PY expansion (3.2). Matching terms of Eq. (3.2) with
Eq. (3.16) give the mean values F̄i for PY:
F̄3 = −1, F̄4 = 2, F̄5 = −6, . . . etc. (PY). (3.17)
Thus, in the PY approximation, the mean values F̄i are constants, independent of the state condition. As to the Verlet
closure58 (α being a constant = 0.8)
γ2
2(1
+ αγ )
γ2
= − [1 − αγ + α 2 γ 2 − α 3 γ 3 + α 4 γ 4 − + · · ·]
2
(Verlet).
(3.18a)
B̂(r) ∼
=−
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204706-7
Closure and the third-order OZ relation
J. Chem. Phys. 135, 204706 (2011)
Comparison with Eq. (3.16) gives
F̄3 = −1,
F̄4 = 3α,
F̄5 = −12α , . . . etc. (Verlet).
2
(3.18b)
It is obvious that the expansion of Verlet differs from the PY
closure. Verlet’s closure has been shown to be far superior, for
hard spheres, to the PY closure. Another closure of interest is
the so-called ZSEP closure59, 60 (the zero-separation closure,
i.e., a closure that satisfies the zero-separation theorems for
the cavity functions) (α, ϕ, and ζ are adjustable parameters
and are functions of T and ρ. They can be determined by enforcing the thermodynamic consistencies.):
ζ 2
φ
∼
B̂ZSEP (r) = − γ (r) 1 − φ +
.
2
1 + αγ (r)
=−
γ2
ζ [1 − φαγ + φα 2 γ 2 − φα 3 γ 3 + − . . .]
2
(ZSEP).
(3.19)
This closure reduces to the Verlet closure upon taking ϕ = 1
and ζ = 1. Comparison with Eq. (3.16) gives the following
interpretation of the mean values:
F̄3 = −ζ, F̄4 = 3αφζ, F̄5 = −12α 2 φζ, . . . etc.
(ZSEP).
(3.20)
We shall adopt this ZSEP form as one part of the resummation formula for B (the part that is a function of γ , see
Eq. (3.9). For the other parts involving the integrals (i.e., as
a functional of γ in Eq. (3.9)), we shall retain the same format of the integrals while adding a multiplicative factor ψ in
order to modulate their values so as to include higher order
contributions. The justifications are based on the following
assumptions: (i) the B-expansion (2.7) is fast convergent after
the third-order term n > 3; (ii) the modification Fn -function
expansion does not include all high-order effects, thus the
residual is to be accounted for by the convolution integrals
in Eq. (3.9); and (iii) the higher-order integrals can scale proportionally as the third-order integrals in Eq. (3.9). The final
resummed form we propose is
φ
ζ γ (1, 2)2
1−φ+
B(1, 2) ∼
=−
2
1 + αγ (1, 2)
− ψρ d3 C (2) (1, 3)h(2) (3, 2)γ (3, 2)
+ ψρ
(2)
d3 C (1, 3)γ (3, 2)
2
(CA-OZ3).
(3.21)
This closure relation (3.21) will be called CA-OZ3 (the convolution approximation-based OZ3). Written out for nonuniform fluids,
φ
ζ γw (r )2
1−φ+
.
Bw (r ) = −
2
1 + αγw (r )
+ ψρ0 d r C0(2) (|r − r |)γw (r )[γw (r ) − hw (r )]
(CA-OZ3).
(3.22)
We shall choose this CA-OZ3 equation (3.22) as the test closure in this work. Other sophisticated schemes of approximation to h(3) , as discussed earlier or found in literature,46–56 can
be developed along the same line and will be taken up at a
later date.
IV. NUMERICAL CALCULATION FOR THE
LENNARD-JONES FLUID/HARD WALL SYSTEM
The Lennard-Jones fluid/hard wall system has been
extensively studied in the literature.30, 61, 62 The fluid-fluid
potential is u(r) = 4ε [(σ /r)12 – (σ /r)6 ] (ε is the energy
parameter and σ is the size parameter). Balabanic et al.29
and Lutsko 30 have carried out MC simulations for the singlet
density profiles at T* = 1.35 and the three densities ρ *
= (i) 0.50, (ii) 0.65, and (iii) 0.82. At these densities, judging
from the chart presented in Fig. 1, cases (i) and (ii) belong to
“depletion” adsorption, while case (iii) belongs to “adhesion”
adsorption. We have shown earlier27, 63 that at ρ * = 0.82 (the
adhesion case), the density profile ρ w (z) of MC is well reproduced by the ZSEP closure (Eq. (3.19)). See also Eq. (4.4)
below). For the depletion cases (ρ * = 0.50 and 0.65),
the ZSEP closure (3.19) is less successful (see Fig. 12 of
Ref. 27). The present closure (4.22) is proposed to close this
gap. In using Eq. (3.22), it is imperative that one “renormalizes” the argument function,59, 60, 64 i.e., changing the icf γ w
in such a way as to extract out the long-range contributions.
A. Renormalization of the base function γ w
Instead of Eq. (1.6) for γ w which uses the bulk LennardJones pair DCF C0 (2) as the kernel
(4.1)
γ w (r ) ≡ d r C0(2) (|r − r )δρ w (r ),
we construct a new base function γ H (r) (a renormalized
function27 ) by using the short-range “core” part (i.e., CH (2)
defined below) of C0 (2) to get
(4.2)
γ H (r ) ≡ d r CH(2) (|r − r )δρ w (r ),
where
CH(2) (r) ≡ C0(2) (r), for r ≤ σ ; and CH(2) (r) ≡ 0, for r > σ.
(4.3)
Namely, the renormalization takes the core part (r ≤ σ ) of the
bulk DCF, and discards all function values of C0 (2) beyond
the core. To illustrate the advantage of the core-based γ H , we
renormalize the ZSEP closure (3.19) as
2
φ
ζ γH (r)2
Renorm
∼
1−φ+
BZSEP (r) = −
. (4.4)
2
1 + αγH (r)
We shall construct the Duh-Haymet65 plots for BZSEP (r) and
γ H (r) (see Ref. 27). The MC data of Balabanic et al. were
used to calculate (i) γ H according to Eq. (4.2) and (ii) the
inverted bridge function according to Eq. (2.6). The DuhHaymet diagram is shown in Fig. 4 for the case ρ * = 0.65
(T* = 1.35). We observe that the curve for the pair Bw -γ H is a
one-to-one function and well-behaved; while the curve Bw -γ w
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204706-8
Lloyd L. Lee
J. Chem. Phys. 135, 204706 (2011)
g0
2.5
0
-0.5
0
0.5
1
1.5
2
2.5
-1
3
3.5
4
4.5
5
γ
5.5
vs.γ H
2
1.5
Bw
-2
-3
1
vs.γ w
0.5
r
0
0.5
1
1.5
2
2.5
3
3.5
4
-4
FIG. 4. The Duh-Haymet plot for Bw vs. γ w (lower curve) and Bw vs. γ H
(upper curve) at ρ * = 0.65. The lower curve shows multiple function values
and a strange “hook” near the origin.
(un-renormalized) shows bizarre “kinks” with multiple function values. Bw -γ w does not satisfy mathematically the definition of a function (not single-valued). In other words, it is not
possible to have a function relation between Bw and γ w (in
fact, Bw is a functional, not a function, of γ w ). By extracting
out the extra-functional contributions as was done with the
base function γ H , Bw is made into a function Bw = B̂(γ H ).
This is the advantage of using the Duh-Haymet plots for detecting function relationships. The above observations were
derived entirely from MC data, no theories were used.
The CA-OZ3 based closure (3.22) can now be written
after renormalization as
φ
ζ γH (r )2
1−φ+
Bw (r ) = −
2
1 + αγH (r )
+ψρ0 d r CH(2) (|r − r |)γw (r )
×[γw (r ) − hw (r )] (CA-OZ3 renormalized).
(4.5)
It contains four parameters α, ϕ, ζ , and ψ that are to be determined by known theoretical principles. One of the criteria
chosen is the hard wall sum rule equation (1.1). At this rodage
stage, we let the other three parameters float while using the
MC data to guide the choice of their values (In the future,
we shall use other known sum rules28, 66 to decide their values). To solve the EL equation (1.4) for the nonuniform density profiles, one needs as input the pure Lennard-Jones pair
DCF C0 (2) . For T* = 1.35 and ρ * = 0.50, 0.65, and 0.82 we
employed an accurate integral equation theory59, 60 (the ZSEP
theory, Eq. (3.19)) to supply these inputs. To check the accuracy of the ZSEP results, we have carried out new molecular dynamics simulations for pure LJ fluid in this work. Since
MD does not directly produce the direct correlation functions,
we calculated instead the radial distribution functions (RDF)
g(2) (r) for the three density states. (Note that the DCF can be
obtained from the RDF through the OZ2 equation.) The theoretical ZSEP-calculated g(2) (r) are compared with the MDgenerated RDF. We verify that there is excellent agreement
FIG. 5. Comparison of the radial distribution functions g0 (r) of the bulk
(pure Lennard-Jones) fluid obtained from the ZSEP method (Eq. (3.19)), (the
three lines) and the new molecular dynamics data (the symbols). This is for
testing the accuracy of the ZSEP integral equation and, consequently, the bulk
DCF values. Conditions: T* = 1.35 and ρ * = 0.50, 0.65, and 0.82.
between the two (Fig. 5), lending credence to the DCF C0 (2) ’s
used here.
B. External potential, w(z)
The hard wall external potential w(z) is taken to be
σ
σ
w(z) = ∞, if z < ; and = 0, if z ≥ ; (4.6)
2
2
where σ is taken to be the same as the LJ size parameter. z
is the distance of the center of a LJ molecule perpendicular
to the wall. We shall use the LJ σ as the unit of length in the
figures and in the following.
C. Depletion adsorption of the Lennard-Jones fluid
For density ρ * = 0.50, the parameters (α, ϕ, ζ , and ψ)
determined are listed in Table I. The EL equation (1.4) was
solved by numerical iterations using Picard’s method with
relaxation.27 The grid size z = 0.005σ and the grid number N = 4096. This gave an integration range of 20.48 σ .
For a number of cases, the range was doubled to 40.96 σ
with 8192 grid points. Cauchy’s absolution convergence was
checked with the γ w -function. Convergence criterion is δ
= 0.00001. Double precision for all the variables was
adopted. For details, see Ref. 27.
Figure 6 shows six curves at the density ρ * = 0.50: the
singlet densities ρ w (z): one from the CA-OZ3-based closure
TABLE I. Parameters used in the CA-OZ3 closure (4.5) for the LJ/HW system at T* = 1.35.
2
Closure: Eq. (4.5) Bw (r ) = − ζ γH2(r ) 1 − φ + 1+αγφ (r )
H
(2)
+ψρ0 d r CH (|r − r |)γw (r )[γw (r ) − hw (r )]
ρ*
0.50
0.65
0.82
α
2.0
5.0
0.51
φ
1.0
0.95
1.0
ζ
1.215
3.05866
0.87
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ψ
0.155
0.0079
0
204706-9
Closure and the third-order OZ relation
J. Chem. Phys. 135, 204706 (2011)
3.5
5
γ
2.5
γ
1.5
0.5
ρ
4
γ
3
γ
H
H
w
2
w
ρ
w
1
w
0
-0.5 0.5
1
1.5
2
B
2.5
w
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
Z
-1.5
-1
0.5
1
1.5
2
2.5
3
3.5
4
4.5
z
Bw
-2
5
-3
-2.5
FIG. 6. Results of calculations with the new closure (CA-OZ3, Eq. (4.5)) at
ρ * = 0.50. We show six curves: γ H = top curve; γ w = next lower curves;
next three curves are for ρ w . They are (i) from CA-OZ3, (ii) from MC of
Balabanic et al.,29 and (iii) from MC of Lutsko.30 These three curves are
very close and indistinguishable at the scale shown. The lowest curve = Bw
(obtained from the present closure CA-OZ3).
(black line); and two curves from the two different sources of
MC – (♦) = Lutzko30 and (×) = Balabanic29 ; the icf γ w (z)
(un-renormalized, second curve from the top); the renormalized γ H (z) (the top curve); and the bridge function Bw (z) (the
lowest curve). We notice that the renormalized γ H (z) is higher
than the original γ w (z), as the core-valued CH (2) will enhance
the function values. γ H (z) is also less oscillatory than γ w and
stays mostly in the positive territory. The ρ w (z) values from
CA-OZ3 and the two MC sources are indistinguishable on
this scale. Figure 7 magnifies the y-axis. The density profile
ρ w (z) from the CA-OZ3 matches closely the MC curves. The
theoretical curve stays well within the statistical errors of the
MC simulations. This agreement confirms that the new closure theory is successful for highly depleted absorption (in
contrast to the PY and HNC theories (Fig. 2) where large deviations from the MC data were incurred).
For density ρ * = 0.65, the state is near the border between depletion and adhesion (see Fig. 1). Examination of
FIG. 8. Results of calculations with the new closure (CA-OZ3, Eq. (4.5)) at
ρ * = 0.65. We show five curves: γ H = top curve; γ w = next lower curve;
then two curves for ρ w : (i) from CA-OZ3 and (ii) from MC of Balabanic
et al.29 The lowest curve = Bw (obtained from the present closure CA-OZ3).
the MC density profile shows that there is some “depletion”
taking place outside the wall (the location is shifted to a larger
distance r ∼ 1.2σ ). At contact, ρ w (σ /2) is about 0.60 (compared to the bulk density of 0.65). So we have depletion. Conventional closure theories (PY or HNC) failed to give quantitative description for this case. We apply the CA-OZ3 closure to this state. The results are shown in Fig. 8 and the parameters used are listed in Table I. Five curves are given in
Fig. 8: the CA-OZ3-calculated singlet density ρ w , the icf γ w ,
the renormalized γ H , the bridge function Bw , and the MC data
ρ w .30 We witness again that the values of γ H are higher than
γ w . The singlet density ρ w is well predicted by CA-OZ3. The
last curve is blown up in Fig. 9. The CA-OZ3 curve shows
a slightly “exaggerated” structure near the wall (between 0.6
< r* < 1.1), but settles down to the correct values at larger r.
Overall, the agreement is excellent.
D. Adhesive adsorption of the Lennard-Jones fluid
For the first two cases with depletion (ρ * = 0.50 and
0.65), the CA-OZ3 is shown to perform accurately. For the
1
0.8
ρ
1.4
w
1.2
1
0.6
ρw
0.8
0.4
0.6
0.4
0.2
Z
0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
FIG. 7. Enlarged view (from Fig. 6) of the singlet density ρ w (z) at ρ *
= 0.50. Comparison of the new closure CA-OZ3 result with Monte Carlo
data. Diamond = MC (Lutsko30 ); Cross = MC (Balabanic29 ); Line = CAOZ3 closure.
0.2
Z
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
FIG. 9. Enlarged view (from Fig. 8) of the singlet density ρ w (z) at ρ *
= 0.65. Comparison of the new closure CA-OZ3 result with Monte Carlo
data. Diamond = MC (Lutsko30 ); Cross = MC (Balabanic29 ); Line = CAOZ3 closure.
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204706-10
Lloyd L. Lee
J. Chem. Phys. 135, 204706 (2011)
tion equation:67
2.5
∂σ
= −,
∂μ
2
ρw
1.5
1
0.5
Z
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
FIG. 10. Comparison of the singlet density profile, ρ w (z), at ρ * = 0.82 from
(i) Diamonds = MC (Balabanic29 ) (ii) Line = CA-OZ3 (or ZSEP) calculation. This is an “adhesion” case.
adhesion adsorption at ρ * = 0.82, this is not a “difficult” case.
Most closure-based DFT can give quantitative prediction for
its profile. For completeness, we show in Fig. 10 the result of
ρ w using the CA-OZ3 closure (4.5). The agreement is again
very close at this high density. The parameters are given in
Table I.
V. CONCLUSIONS
Starting from the exact relation of the third-order
Ornstein-Zernike equation (2.4), we are able to formulate a
viable approximation to the triplet direct correlation function C(3) in Eq. (3.7b) (the CA-C3). This approximation is
modeled after the convolution approximation of Jackson and
Feenberg. Substitution into the functional expansion of the
bridge function (2.7) enables us to obtain in closed form an
expression (3.9) for the third-order term B3 of the bridge function. Using the B3 equation as a basis, we generalize to a full
closure equation (the CA-OZ3 equation (4.5)) for the bridge
function. This CA-OZ3 equation is shown to be an improvement over the classical Percus-Yevick closure. The prominent
difference is that it is at the same time a function and a functional of the indirect correlation function γ w , with convolution integrals involving also other correlation functions.
We test this new closure by calculating the density profiles of the Lennard-Jones fluid adsorbed on a planar hard
wall. The calculations prove successful in predicting the depletive adsorption cases as well as the adhesive adsorption.
Traditionally, depletion has been treated by other density
functional theories (such as the fundamental measure-based
theories5–7, 61, 62 ). We show here by “updating” the closure
equation with the aid of fundamental theories (in this case the
OZ3), we can achieve improvements with the closure-based
theories.
As the formulation stands, the present approach is not
self-contained. There are four parameters in the closure equation (4.5). To determine their values, we need at least four
theoretical conditions to fix α, ϕ, ζ , and ψ. One of the conditions that has been used is the hard-wall sum rule (Eq. (1.1)).
In future developments, we shall also use the Gibbs adsorp-
(5.1)
which links the surface tension σ to the adsorption density .
In addition, the sum rules proposed by Henderson66 can be
employed to ensure self-consistency.
The CA-OZ3 formulation is only one of many other possible improvements. Other viable theories on the triplet correlations, some of which we have alluded to here, will be examined in the future. The positive outcome also opens the door
to the study of a variety of other fluid systems (such as the
Yukawa fluids, Coulomb fluids, and some of the soft matter
potentials). Our derivations were based on “bulk fluid” (uniform system) statistical mechanics. However, we note that the
closure relation proposed here can equally well be applied to
nonuniform as well as uniform systems through Percus’ prescription of the source particles that puts the nonuniform fluids on equal footing with the uniform fluids.38
ACKNOWLEDGMENTS
We are thankful to the OSCER Supercomputing Center
of the University of Oklahoma for allocation of computer
times.
APPENDIX: THE EXACT THIRD-ORDER BRIDGE
TERM B3 BASED ON THE OZ3 EQUATION
We give the exact third-order bridge term B3 in terms of
h(3) as based on the OZ3 equation (3.4) (using simplified notation for a uniform fluid):
2
− 2ρ d3 C31 γ30 h30
2B3 (10) = ρ 2 d2d3 h20 h30 h123 − γ10
−ρ
3
d4d5d6 C14 γ50 γ60 h456
+ ρ3
d3d5d6 C15 γ60 h30 h356
− ρ3
d2d3d5 C15 h20 h30 h235
+ ρ 3 d2d4d5 C15 γ40 h20 h245
− ρ d3d6 γ60 h30 h136 + ρ
2
2
d4d6γ40 γ60 h146
2
− ρ 2 d2d4 γ40 h20 h124 + 2ρ d4 C14 γ40
.
(A1)
For nonuniform fluids, replace B3 (1,0) by Bw3 (1), ρh30 by
δρ w (3), γ 30 by γ w (3), and h456 will be the uniform fluid
h0 (3) (4,5,6).
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