Quasi-one dimensional classical fluids
We study the equilibrium statistical mechanics of simple fluids in narrow pores. A systematic expansion is made about a one-dimensional limit of this system. It starts with a density functional, constructed from projected densities, which depends upon projected one and two-body potentials. The n...
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| Cite this: | Quasi-one dimensional classical fluids / J.K. Percus, Q.-H. Wang // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 387-394. — Бібліогр.: 10 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1207462025-06-03T16:30:13Z Quasi-one dimensional classical fluids Квазіодновимірні класичні плини Percus, J.K. Wang, Q.-H. We study the equilibrium statistical mechanics of simple fluids in narrow pores. A systematic expansion is made about a one-dimensional limit of this system. It starts with a density functional, constructed from projected densities, which depends upon projected one and two-body potentials. The nature of higher order corrections is discussed Ми вивчаємо рівноважну статистичну механіку простого плину в обмежених порах. Систематичне розвинення виконане в одновимірній границі цієї системи, яке базується на функціоналі густини, по- будованому з проектованих густин, що залежать від одно- і дво-частинкових потенціалів. Обговорюється природа кореляцій вищих порядків. Supported in part by DOE grant No. DE-FG02-02 ER 15292. 2003 Article Quasi-one dimensional classical fluids / J.K. Percus, Q.-H. Wang // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 387-394. — Бібліогр.: 10 назв. — англ. 1607-324X DOI:10.5488/CMP.6.3.387 PACS: 05.20.Jj https://nasplib.isofts.kiev.ua/handle/123456789/120746 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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We study the equilibrium statistical mechanics of simple fluids in narrow
pores. A systematic expansion is made about a one-dimensional limit of
this system. It starts with a density functional, constructed from projected
densities, which depends upon projected one and two-body potentials. The
nature of higher order corrections is discussed |
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Percus, J.K. Wang, Q.-H. |
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Percus, J.K. Wang, Q.-H. Quasi-one dimensional classical fluids Condensed Matter Physics |
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Percus, J.K. Wang, Q.-H. |
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Quasi-one dimensional classical fluids |
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Quasi-one dimensional classical fluids |
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Quasi-one dimensional classical fluids |
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Quasi-one dimensional classical fluids |
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Quasi-one dimensional classical fluids |
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quasi-one dimensional classical fluids |
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Інститут фізики конденсованих систем НАН України |
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2003 |
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Quasi-one dimensional classical fluids / J.K. Percus, Q.-H. Wang // Condensed Matter Physics. — 2003. — Т. 6, № 3(35). — С. 387-394. — Бібліогр.: 10 назв. — англ. |
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Condensed Matter Physics |
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AT percusjk quasionedimensionalclassicalfluids AT wangqh quasionedimensionalclassicalfluids AT percusjk kvazíodnovimírníklasičníplini AT wangqh kvazíodnovimírníklasičníplini |
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Condensed Matter Physics, 2003, Vol. 6, No. 3(35), pp. 387–394
Quasi-one dimensional classical fluids∗
J.K.Percus 1,2 , Q-H.Wang 2
1 Courant Institute, New York University, NY 10012, New York, USA
2 Physics Department, New York University, NY 10003, New York, USA
Received April 23, 2003, in final form July 4, 2003
We study the equilibrium statistical mechanics of simple fluids in narrow
pores. A systematic expansion is made about a one-dimensional limit of
this system. It starts with a density functional, constructed from projected
densities, which depends upon projected one and two-body potentials. The
nature of higher order corrections is discussed.
Key words: classical fluids, one-dimensional limit, narrow pores, higher
order corrections
PACS: 05.20.Jj
1. Introduction
The physical sciences are all about constructing models of reality, and Myroslav
Holovko is one of the expert practitioners of the art [1]. The model may be very close
to reality, so that needed corrections are readily applied, but then the model itself
may be hard to analyze. Or it may be very primitive and easily solved in detail,
in which case everything depends upon systematic, perhaps ingenious, correction
procedures. Fortunately, there are numerous situations that allow the advantages
of both extremes to be combined. One of these, that of molecular systems under
molecule-scale confinement, is assuming an increasing importance as nanoscale con-
trol promises to approach the routine. In this brief communication, we would like to
address a subcategory of this impressive array that ranges from fluids in biological
pores [2] to those in industrial zeolites.
A traditional activity in theoretical chemical physics is that of deriving the prop-
erties of bulk fluids in thermal equilibrium. When surfaces abound, imposed or self-
generated, naive thermodynamic approaches fail in detail, even with simple fluids,
and more powerful techniques are required. Well-established integral equation meth-
ods [4] have no difficulty in principle in dealing with these systems, but they become
quite cumbersome. Density functional approaches [5] – often much more of an art
than a science – while fairly ancient in both classical and quantum versions, have
∗Supported in part by DOE grant No. DE-FG02-02 ER 15292.
c© J.K.Percus, Q-H.Wang 387
J.K.Percus, Q-H.Wang
more and more filled in this area. Their aim has generally been to emphasize struc-
tural accuracy of the model employed, rather than to develop effective correction
procedures, and almost all rely conceptually upon piecing together suitably localized
bulk fluids. When the model system is sufficiently specialized, e.g. classical pure hard
cores in equilibrium, impressive accuracy is obtained, as in Rosenfeld’s extension [6]
of the structure suggested by Percus [7]. But when 3-dimensional bulk provides a
poor picture of a system at all levels of resolution, one can do better. Here, as men-
tioned, we will deal with classical fluids confined to a pore-like space, so that they
are best termed quasi-one-dimensional, and literal one-dimensional space becomes
the conceptual reference. Indeed, as we will show, it is more than just conceptual.
2. The one-dimensional fluid
The term quasi-one dimensional loses its vagueness when confinement is so tight
that the particles involved must maintain their order, i.e. they cannot pass each
other, and so remain in single file. And for sufficiently tight confinement, if the in-
terparticle forces have a small enough range in comparison with that of the assumed
hard core, this is equivalent to having only next neighbor interactions. Under these
circumstances, one can imagine the limiting model of next-neighbor interacting par-
ticles in a common one-dimensional space. If one wants to allow for an arbitrary
external potential and arbitrary interactions – restricted as above – this is not a
trivial situation to analyze, but it can be done. In the one species case, it requires
the introduction of a single auxiliary field Λ(x) to accompany the observable particle
density field u(x), and results [8] in the (1-D) grand ensemble excess Helmholtz free
energy – at reciprocal temperature β – concisely written as
βF ex
1 [n, Λ, w] =
∫
n ln(Λ/1 + wΛ) −
∫
(n ln n − n) − ln(1 +
∫
Λ). (2.1)
Here w(x, x′) = e−βφ(x′
−x)θ(x′ − x) is the one-sided next neighbor interaction Boltz-
mann factor, and in (wΛ)(x) ≡
∫
w(x, x′)Λ(x′)dx′, it acts as an integral operator.
The local chemical potential µ(x) = µ − u(x) for external potential u(x) is then
obtained as usual:
if βµ(x) = ln n(x) + βµex(x),
then βµex(x) = δβF ex/δn(x)|Λ = ln(Λ(x)/n(x)) − ln(1 + (wΛ)(x)), (2.2)
and the implied condition on (2.1) that βF ex is stationary with respect to Λ yields
0 = δβF ex/δΛ(x)|n =
n(x)
Λ(x)
−
(
wT n
1 + wΛ
)
(x) −
1
1 +
∫
Λ(x′)dx′
, (2.3)
here wT is the transpose of w.
In the special case of hard rods of diameter a, Λ can be eliminated from (2.1)
via (2.3), resulting in
βF ex =
∫
nσ(x) ln(1 − anτ (x))dx, (2.4)
388
Quasi-one dimensional classical fluids
where
nσ(x) =
1
2
(n(x + a/2) + n(x − a/2)), nτ (x) =
1
a
∫ a/2
−a/2
n(x + x′)dx′.
(which generalizes without difficulty to hard core mixtures as well). The fact that
(2.4) represents a local free energy density that is a function of only the linearly
weighted densities nσ(x) and nτ (x), results in a uniform fluid direct correlation
function which is bilinear on a vector space of only two dimensions as the bulk
density varies. Since the bulk direct correlation function for a PY-approximated
hard sphere fluid can similarly be constructed from a vector space of dimension
five [7], this suggests that its non-uniform generalization be constructed from the
corresponding weighted average densities, which is the genesis [9] of the Rosenfeld
approach, among others. We will be interested in seeing whether such a structure
persists under tight confinement.
3. Quasi-one dimensional expansion
In a way, our objective is to look at a pore-confined fluid at low resolution, so that
we are projecting onto a one-dimensional fluid, and dimension-reducing projection
methods have a long history. But we will want to be very explicit in carrying out
this reduction. From a density functional viewpoint, the path is clear: we have a
density pattern n(x, y), where x denotes longitudinal direction (actually, it need not
be a Cartesian axis) and y encompasses the transverse coordinates. In the situation
of interest, we can imagine starting with a basic pattern n0(x, y) and then squashing
it down transversely to n(x, y) by contracting the confining space via an external
field. The result would then appear as
n(x, y) = n0(x, y/a)/aD−1, (3.1)
where a measures the contraction ratio. As a → 0, the observed density pattern is
strictly one-dimensional (D denotes the actual spatial dimensionality), conserving,
however, the density per unit length
n0(x) =
∫
n(x, y)dD−1y =
∫
n0(x, y)dD−1y. (3.2)
To make use of (3.1) and (3.2) systematically, one can first write down an explicit
expression, e.g. a diagrammatic expansion [10] for
βF ex[n(x, y), f(x, y; x′, y′)],
where f = e−βφ − 1 is the pair interaction Mayer function, assuming only pair (and
singlet) potentials. Then, rewrite this as
βF ex[n0(x, y/a)/aD−1, f(x, y; x′, y′)],
389
J.K.Percus, Q-H.Wang
and finally make the transformation y → ay in all integrals. The result is clearly
that
βF ex[n(x, y), f(x, y; x′, y′)] = βF ex[n0(x, y), f(x, ay; x′, ay′)], (3.3)
which can then be expanded directly in a, or to higher accuracy with no additional
labor, expanded in the difference
∆f(x, ay; x′, ay′) = f(x, ay; x′, ay′) − f0(x; x′), (3.4)
where f0(x; x′) = f(x, 0; x′, 0). The leading order in the expansion of (3.3) is at
∆f = 0, i.e.
βF ex[n(x, y), f(x, y; x′, y′)] = βF ex[n0(x, y), f0(x, ; x′)] + · · · ,
which, transforming back, is just βF ex[n(x, y), f0(x, x′)]+ · · · . But the y integrations
can then be done at once, requiring only
∫
n(x, y)dD−1y = n0(x). We conclude that
the leading order is given by
βF ex[n, f ]|f=f0
= βF ex
1 [n0, f0], (3.5)
the strictly one-dimensional free energy in which only the full projected density and
the interaction at y = y′ = 0 appears: the obvious projection is correct at this order.
The interest, of course, is in what happens at higher order.
The diagrammatic background in which (3.5) was derived is crucial, but details
are not necessary until one tries to be explicit. So we can start a full expansion by
just carrying out a formal Taylor expansion with respect to f around f0:
βF ex[n, f ] =
∞
∑
s=0
1
s!
∫
· · ·
∫
δsβF ex[n, f ]
δf(x1, y1; x′
1, y
′
1) · · · δf(xs, ys; x′
s, y
′
s)
∣
∣
∣
∣
f=f0
×
s
∏
j=1
∆f(xj, yj; x
′
j, y
′
j)
s
∏
j=1
(dxjdx′
jd
D−1yjd
D−1y′
j). (3.6)
The zeroth order term is known from (3.5), but the first order term is already
significant: the combination
∆f(x, y; x′, y′)δβF ex[n, f ]/δf(x, y; x′, y′)|f=f0
clearly replaces all f -bonds but one in βF ex by f0 and converts that one to
∆f(x, y; x′, y′); the y integrations other than dD−1ydD−1y′ then replace all n’s by
n0’s except for n(x, y) and n(x′, y′). But let us set
n(x, y) = n0(x)ρ(x, y) (3.7)
and similarly, of course, for n(x′, y′). Then all nodes are n0’s, all f ’s but one are f0’s
and that one becomes, on y, y′ integration
∆f0(x, x′) =
∫∫
ρ(x, y)∆f(x, y; x′, y′)ρ(x′, y′)dD−1ydD−1y′. (3.8)
390
Quasi-one dimensional classical fluids
Only the x-integrations remain, and we conclude that
(
∫∫
∆f(x, y; x′, y′)
δ
δf(x, y; x′, y′)
dxdx′dD−1ydD−1y′
)
F ex[n, f ]|f=f0
=
=
(
∫∫
∆f0(x, x′)
δ
δf0(x, x′)
dxdx′
)
F ex
1 [n0, f0]. (3.9)
If equation (3.9) were to apply to products of Taylor operators as well, we would
conclude that
F ex[n, f ] = F ex
1 [n0, f0 + ∆f0] = F ex
1 [n0, f̄0], (3.10)
where
n0(x) =
∫
n(x, y)dD−1y
and
f̄0(x, x′) =
∫∫
ρ(x, y)f(x, y; x′, y′)ρ(x′, y′)dD−1ydD−1y′,
and it is indeed this first approximation that we will examine in detail. One should
note of course that the same result would be obtained by expanding about an un-
known f̄0(x, x′) rather than f0(x, x′) and requiring the first correction to vanish.
4. Application
To make use of (3.10), we need the strictly one-dimensional solution, with inter-
action Boltzmann factor
w0(x, x′) = (1 + f̄0(x, x′))θ(x′ − x). (4.1)
Since (4.1) has in general no special properties aside from those emanating from
the short range of f̄0, we have little choice but to avail ourselves of the exact equa-
tion (2.1), reading here
βF ex[n, Λ] =
∫
n0(x) ln
Λ(x)
1 +
∫
∞
x
(1 + f̄0(x, x′))Λ(x′)dx′
dx
−
∫
(n(x) ln n(x) − n(x))dx − ln(1 +
∫ +∞
−∞
Λ(x′)dx′), (4.2)
where
δβF ex/δΛ(x) = 0.
The complicating factor is the necessity of solving for Λ(x) to insert into βF ex. But
since (4.2) is stationary in Λ, a viable option is to simply insert a reasonable ansatz
for Λ, such as its form when f̄0(x, x′) is taken simply as f0(x, x′) = f(x, 0; x′, 0), and
this form is indeed known in the hard core case. Results along these lines will be
reported elsewhere.
391
J.K.Percus, Q-H.Wang
If our standards are lowered a bit, we may be content with the first order correc-
tion arising from (3.9). Then, the required input information is only that obtained
from the strictly 1-D Mayer factor. Let we see how this goes. What we need specif-
ically is δβF ex
1 [n0, f0]/δf0(x, x′), but since βF ex
1 is stationary with respect to Λ, we
can just differentiate (2.1) at constant Λ, resulting at once in
δβF ex
1 [n0, f0]/δf0(x, x′) = −
n0(x)Λ0(x
′)θ(x′ − x)
1 + (w0Λ0)(x)
. (4.3)
Here our first order expansion becomes the very accessible
βF ex[n, f ] = βF ex
1 [n0, f0] −
∫∫
x′>x
n0(x)
1 + (w0Λ0)(x)
∆f0(x, x′)Λ0(x
′)dxdx′. (4.4)
For hard spheres, where the one-dimensional reference is that of hard rods, Λ0 is
indeed known, and so (4.4) is completely explicit. An immediate consequence, inci-
dentally, is that there will be no F ex
1 local density that is a function only of finitely
many weighted densities.
5. Higher corrections
To see why approximation (3.10) is incomplete, one can develop the higher deriva-
tive extensions of (3.9). It’s simplest to go directly to the standard Mayer expansion
of the excess free energy F ex[n, f ]. In this form, one has a weighted sum of integrat-
ed diagrams, each of which consists of nodes n(x, y) joined by bonds f(x, y; x′, y′).
Thus, F ex[n, f0+∆f ] will consist of the same diagrams with f0(x, x′) bonds, a certain
subset of which has been replaced by ∆f(x, y; x′, y′) bonds. These subsets can be
decomposed into connected subsets joined to the rest of the diagrams by f0 bonds.
One can now set n(x, y) = n0(x)ρ(x, y), as in (3.7), and carry out all y-integrations.
Any node not belonging to a ∆f -bond will, since
∫
ρ(x, y)dD−1y = 1, reduce to
n0. A connected ∆f subset will, however, integrate to an entangled object which, if
comprising γ nodes, will be recognized as the Mayer f -function of a γ-particle inter-
action. In order of complexity, there will be 2-particle factors, 3-particle factors,· · · :
∆f2(x, x′) =
∫∫
ρ(x, y)∆f(x, y; x′, y′)ρ(x′, y′)dD−1ydD−1y′,
∆f3(x, x′, x′′) =
∫∫∫
ρ(x, y)∆f(x.y; x′, y′)ρ(x′, y′)
× ∆f(x′, y′; x′′, y′′)ρ(x′′, y′′)dD−1ydD−1y′dD−1y′′,
∆f3′(x, x′, x′′) =
∫∫∫
ρ(x, y)∆f(x, y; x′, y′)ρ(x′, y′)
× ∆f(x′, y′; x′′, y′′)ρ(x′′, y′′)∆f(x′′, y′′; x, y)dD−1ydD−1y′dD−1y′′,
· · · (5.1)
and systematizing these contributions is not difficult. Let us just start the process.
392
Quasi-one dimensional classical fluids
One wants to compare the one-dimensional free energy expansion generated by
the sequence (5.1) with what would have been obtained for a system governed by a
sequence of 2-body, 3-body, · · · interactions. In the latter case, the Mayer expansion
would be that of Π(1 + fij)Π(1 + fijk) · · · , and the next correction we would want
corresponds to the 3-body cluster that would appear in the form (1+fij)(1+fjk)(1+
fik)fijk. Thus, adding up the 4 connected 3-body terms of (5.1), we have in effect
appended a 3-particle interaction with f -function
f3(x, x′, x′′) =
[
∫∫∫
ρ(x, y)(1 + ∆f(x, y; x′, y′))ρ(x′, y′) (1 + ∆f(x′, y′; x′′, y′′))
× ρ(x′′, y′′)(1 + ∆f(x′′, y′′; x, y))dD−1ydD−1y′dD−1y′′ − 1
]
× [(1 + f̄0(x, x′))(1 + f̄0(x
′, x′′))(1 + f̄0(x
′′, x))]
−1
. (5.2)
Actually, complexity only arises as one departs further and further from the nearly
one-dimensional domain that we are interested in. For example, if ∆f(x, y; x′, y′)
vanishes (for accessible y and y′) except in a narrow range of |x − x′|, as would
be the case for hard objects with small limited transverse motion, there will be no
contribution from any subdiagram with loops. The remaining tree diagrams can be
handled by more routine methods.
6. Conclusion
We conclude that in our chosen arena, effective systematic methods exist for
carrying out the low resolution reduction that is perhaps the qualitative objective
of any quantitative theory. The general strategy is quite simple, and extensions are
being carried out; that to equilibrium mixtures is direct, as is extension to curved
pores, but the time-dependant situation much less so. To be sure, as we depart more
and more from one-dimensional reference, the “correction terms” must dominate,
and this particular path is not appropriate. However, along the way, we expect to find
the corrections come in one bunch at a time, as it becomes necessary to track larger
and larger sets in what started out as strictly single file (see [8] for a discussion of the
beginning of the sequence, in a somewhat different context). In other words, each
bunch signals an analytic break – but weaker and weaker – in the thermodynamic
behavior. Whether or not the increased complexity merits a detailed study is not
obvious, but such an investigation is indeed under way.
References
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branes. New York, Wiley NY, 1987.
3. Karger J, Ruthven D.M. Diffusion in Zeolites and Other Microporous Solids. New
York, Wiley NY, 1992.
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4. Henderson D. Fundamentals of Inhomogeneous Fluids. New York, Dekker NY, 1992.
5. Percus J.K. Density functional theory in the classical domain. // Th. and Comp.
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6. Rosenfeld Y., Schmidt M., Lowen H., Tarazona P. Fundamental measure free energy
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Квазіодновимірні класичні плини
Й.К.Перкус 1,2 , К.-Х.Ванг 2
1 Інститут Куранта, Нью-Йоркський університет,
10012 Нью-Йорк, США
2 Фiзичний факультет Нью-Йоркський університету,
10003 Нью-Йорк, США
Отримано 23 квітня 2003 р., в остаточному вигляді – 4 липня
2003 р.
Ми вивчаємо рівноважну статистичну механіку простого плину в об-
межених порах. Систематичне розвинення виконане в одновимір-
ній границі цієї системи, яке базується на функціоналі густини, по-
будованому з проектованих густин, що залежать від одно- і дво-
частинкових потенціалів. Обговорюється природа кореляцій вищих
порядків.
Ключові слова: класичні плини, одновимірна границя, обмежені
пори, кореляції вищих порядків
PACS: 05.20.Jj
394
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