Statistical field theory for liquid vapor interface
A statistical field theory for an inhomogeneous liquid, a planar liquid/vapor interface, is devised from first principles. The grand canonical partition function is represented via a Hubbard-Stratonovitch transformation leading, close to the critical point, to the usual φ4 scalar field theory which...
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Інститут фізики конденсованих систем НАН України
2010
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| Цитувати: | Statistical field theory for liquid vapor interface / V. Russier, J.-M. Caillol // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23602: 1-15. — Бібліогр.: 47 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859484569156190208 |
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| author | Russier, V. Caillol, J.-M. |
| author_facet | Russier, V. Caillol, J.-M. |
| citation_txt | Statistical field theory for liquid vapor interface / V. Russier, J.-M. Caillol // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23602: 1-15. — Бібліогр.: 47 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | A statistical field theory for an inhomogeneous liquid, a planar liquid/vapor interface, is devised from first principles. The grand canonical partition function is represented via a Hubbard-Stratonovitch transformation leading, close to the critical point, to the usual φ4 scalar field theory which is then rigorously considered at the one-loop level. When further simplified it yields the well-known capillary wave theory without any ad hoc phenomenological parameter. Internal coherence of the one-loop approximation is discussed and good overall qualitative agreement with recent numerical simulations is stressed.
|
| first_indexed | 2025-11-24T15:33:25Z |
| format | Article |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 2, 23602: 1–15
http://www.icmp.lviv.ua/journal
Statistical field theory for liquid vapor interface
V. Russier1∗, J.-M. Caillol2†
1 ICMPE, UMR 7182 CNRS and Université Paris Est 2–8 rue Henri Dunant, 94320 Thiais, France
2 Laboratoire de Physique Théorique, UMR 8627 CNRS and Université de Paris Sud, bat. 210, 91405 Orsay
Cedex, France
Received January 15, 2010, in final form February 26, 2010
A statistical field theory for an inhomogeneous liquid, a planar liquid/vapor interface, is devised from first
principles. The grand canonical partition function is represented via a Hubbard-Stratonovitch transformation
leading, close to the critical point, to the usual φ4 scalar field theory which is then rigorously considered at
the one-loop level. When further simplified it yields the well-known capillary wave theory without any ad hoc
phenomenological parameter. Internal coherence of the one-loop approximation is discussed and good overall
qualitative agreement with recent numerical simulations is stressed.
Key words: field theory, gas-liquid interface, surface tension
PACS: 61.30.Hn, 64.70.F, 68.03.Cd
1. Introduction
The structure in the interface between two fluid phases at coexistence plays a key role in many
specific situations, such as, for instance, the wetting transition and related phenomena [1–7]. An
understanding and proper description of this structure is undoubtedly a challenge that presents a
great interest in itself from a theoretical point of view. For simple fluids, one of the key features
of interfaces that separate subcritical phases at coexistence is the presence of thermally activated
capillary waves [8] coupled with density fluctuations in the bulk. Since the pioneering work of
Buff et al. [8] capillary waves have been widely studied and are usually described through the
introduction of an ad hoc effective surface Hamiltonian, written as a functional of the interface
height, zint(~s), ~s being the coordinate parallel to the interface in the case of a planar interface
as considered henceforth. H [zint(~s)] can be obtained through different ways and is most generally
deduced from phenomenological arguments taking into account the interplay between gravity and
surface tension effects. It is noteworthy that most works in this area are devoted to the wetting
transitions where the Hamiltonian includes an external potential, say W (zint(~s)), which pins the
interface to a solid substrate.
The Hamiltonian H [zint(~s)], can be treated in the framework of statistical field theoretical
methods [2, 4], including renormalization group (RG) [9] techniques. For instance the hierarchical
reference theory (HRT) has been recently generalized to inhomogeneous cases in order to deal with
the wetting transition [7]. In the case of the liquid/vapor interface, when H [zint(~s)] is treated at
the gaussian level, the capillary wave theory (CWT) is recovered, leading to the well known 1/q2
behavior for the height-height structure factor S∆z ∆z(q). Moreover, 3D φ4 model in which the
stochastic variable φ does not coincide with the height zint(~s) was studied in refs [5, 6, 10, 11].
Alternative ways of deducing an effective Hamiltonian from a density functional theory (DFT)
including the effect of gravity [12] or the local curvature of the interface [13, 14] have been also
considered. Recently [15], a model for the density profile based on an extension of a displaced profile
approximation, i.e. in which the profile is written as a function of (z − zint(~s)), was considered.
∗E-mail: russier@icmpe.cnrs.fr
†E-mail: jean-michel.caillol@th.u-psud.fr
c© V. Russier, J.-M. Caillol 23602-1
http://www.icmp.lviv.ua/journal
V. Russier, J.-M. Caillol
In addition, the model takes into account, together with the surface fluctuations described by the
height zint(~s), bulk phase fluctuations.
The purpose of the present work, which is an extension of a preliminary work [16], is to provide
a simple description of the liquid / vapor interface of fluids in the framework of an exact statistical
field theory. The latter is obtained from a Kac-Siegert-Stratonovich-Hubbard-Edwards (KSSHE)
[17–22] transformation devised to rewrite the grand canonical partition function (GCPF) of the
inhomogeneous fluid as the partition function of a statistical field theory involving the stochastic
real field φ. The method is an alternative to the so-called collective variables method developed
notably by I. Mryglod and coworkers [23, 24], that was mainly used for homogeneous fluids in
[23–25] and is considered here in studying a planar interface. It is important to mention that the
present work is not based on the determination of an effective surface hamiltonian, conversely to
[26, 27]. The theory is studied at the one-loop level where everything can be achieved analytically.
The links between correlations of the field and density correlations are established as well as the
expressions for the surface tension and the surface structure factor are presented. Here the surface
tension is not assimilitated to the stiffness factor of the effective surface Hamiltonian gradient
term [26] but conversely is the surface contribution of the logarithm of the GCPF. Technically,
this is done via the determination of the eigenvalues and eigenfunctions of the second functional
derivative of the mean field KSSHE Hamiltonian, which can be done exactly, as well known, for the
one-dimensional kink [28]. The whole spectrum of eigenstates being taken into account proves to
be of the outmost importance in obtaining the correct result. Thus we obtain a satisfying picture
of the liquid vapor interface, at least qualitatively, even at this simple level of description. One of
the salient points of our work is that we recover both the CWT and its first extension, namely
the appearance of the bending rigidity factor k and the coupling between surface and bulk density
correlations in S(q), without invoking any ad hoc phenomenology.
It is a pleasure for the authors to dedicate this paper to Ihor Mryglod to celebrate his fiftieth
anniversary. One of the authors (JMC) acknowledges many interesting discussions with Ihor, about
science and other topics, either in Orsay University or L’viv ICMP on the occasion of many meetings
and collaborations of French and Ukrainian teams.
Bon anniversaire Ihor et longue vie à toi !
2. KSSHE transform and mean-field approximation
The important steps leading to the appropriate statistical field theory for a liquid are outlined
in Appendix A; the interested reader is referred to [18, 23–25] for deeper details and to [29] for
an alternative formulation. We consider a simple fluid whose pair interaction potential includes a
hard sphere (HS) repulsive part and a soft attractive part, denoted by v(r). Let w(r) = −βv(r) as
usual, and suppose that only purely attractive potentials are considered (w is a positive definite
operator, i.e. w̃(k) > 0). We work in the grand canonical ensemble, namely at constant chemical
potential ν = βµ, and we consider the grand canonical partition function (GCPF) related to the
grand potential Ξ = exp(−βΩ). Following the notations of appendix A we obtain for Ξ
Ξ[ν] =
1
Nw
∫
Dφ exp(−H [φ]),
H =
1
2
φ · w−1 · φ− ln(ΞHS[ν̄ + φ]), (1)
where φ is a real scalar field, Dφ a functional integration measure, ν̄ = ν − w(0)/2 and, finally,
ΞHS[ν] is the hard sphere grand partition functional of the local chemical potential ν(~r). A Landau-
Ginzburg form is obtained from a functional Taylor expansion of ln(ΞHS) around a conveniently
chosen reference chemical potential of the hard sphere fluid, ν0 (see Appendix A). The propagator
is expanded up to order k2 in Fourier space and we are left with
H [φ] = H0 +
∫ (
K2
2
(
∂φ
∂~r
)2
+
K0
2
φ2 + V (φ) −Bφ
)
d~r, (2a)
23602-2
Statistical field theory for liquid vapor interface
V (φ) =
∑
n>3
un
n!
φn, (2b)
B(x) =
∫
dy w−1(x, y)∆ν(y) + ρHS [ν0] (x) with ∆ν = ν̄ − ν0 , (2c)
where the coupling constants K0,K2 and un depend only on the equation of state of the hard
sphere fluid and the potential w (see below). This formulation permits an exact mapping between
the densities and their correlations on the one hand and the mean value and the correlations of
the field on the other hand, as exemplified in equation (A-13) in the appendix. For instance at
coexistence, the densities ρl and ρg of the two phases correspond to the two values 〈φ〉l and 〈φ〉g .
The value of ν0 is chosen in such a way that u3(ν0) vanishes and the coexistence condition between
the liquid and vapor phases in the absence of external field is B = 0. K0 is related to the deviation
from the critical temperature: t = (Tc − T )/Tc ∝ (−K0).
In the inhomogeneous system, we assume a special realization of the two phases coexistence: we
explicitly impose the occurrence of a bulk liquid and a bulk gas, at densities ρl and ρg separated
by a planar surface located at z = z0. The system is supposed to be bounded in the direction
z according to (z0 − L) < z < (z0 + L), L being macroscopically large though formally finite.
The localization of the interface is obtained through the introduction of a small valued external
field hext(z − z0), odd in (z − z0). Since the mean field profile takes constant values in both bulk
phases, so is the case for hext which therefore plays the same role as the truncated gravitational
field used in [12]. This leads to a constraint on the averaged field profile although this method
differs from that of [27] since we do not seek a collective variable because we still deal with the
GCPF of the whole system. The advantage of our choice will be shown below; we emphasize that
we do not aim to study gravitational effects but rather to fix the location of the interface at z = z0.
The inhomogeneous mean field theory or saddle point equation, δH [φ]/δφ = hext(z − z0), is solved
by exploiting the fact that we look for a monotonous solution for the mean field profile φc(z) =
φMF(z)/φb, where φb is the value of φMF(z) when z → z0 +L, and by expanding hext as a function
of φc(z), namely hext(z) = hext(φc(z)). The mean field equation then reads
K2
∂2φMF
∂z2
= K0φMF +
∂V
∂φ
(φMF) − hext(φc) . (3)
Since we consider a local approximation for the n > 3 HS kernels (see equation (A-14)), the r.h.s. of
equation (3) is a polynomial in φMF(z − z0). Limiting V (φ) to the first non-vanishing term, namely
V (φ) = (u4/4!)φ4, the analytical form for φMF is not modified by the introduction of an external
field if the latter expands as h(φc) = h1φc + (h3/3!)φ3
c (h1, h3 constants), as already noted by
Zittartz [30] when h(φc) = h1φc (This result holds in a more general way for a potential V (φ) even
in φ and an external field odd in φc where the highest power of V (φ) is that of h(φc) plus one).
The solution of the mean field equation in the presence of external field, φMF(K2,K0, u4, hext),
coincides with the solution in the absence of external field, with modified values of the coupling
constants K0 and u4.
φ
(K2,K0,u4;hext)
MF (z) = φ
(K2,K′
0
,u′
4
;0)
MF (z), K ′
0 = K0 − δK0 , u′4 = u4 − δu4
with δK0 = h1
(
u′4
−6K ′
0
)1/2
and δu4 = h3
(
u′4
−6K ′
0
)3/2
. (4)
The solution of equation (3) is φMF(z) = φbφc((z−z0)/l) with φc(x) = tanh(x), φb = (−6K ′
0/u
′
4)
1/2
and l = (2K2/−K ′
0)
1/2 is the bulk correlation length which also coincides with the intrinsic interface
width. Contrary to the situation encountered in wetting problems, as for instance in [27], we do
not need to approximate the mean field solution in our case. We emphasize that we deal only with
subcritical temperatures where K ′
0 < 0. We first have to check the mapping between the physical
external field seen by the liquid, say Ψ(z) = βVext(z), and the external field which enters the
23602-3
V. Russier, J.-M. Caillol
effective Hamiltonian of equation (2a). Thus we have to solve
B(1) = B0 −
∫
w−1(1, 2)Ψext(2)d2 ≡ hext(1). (5)
B0 given by equation (2c) is the constant external field appearing in the effective Hamiltonian
when hext vanishes. The coexistence condition at constant chemical potential still reads B0 = 0.
Since the propagator has been expanded up to order k2 (see appendix A), w−1 is given in real
space, at the same order of approximation, by
w−1(1, 2) =
1
w̃0
[
1 +
w̃2
w̃0
∆1
]
δ(1, 2), (6)
where w̃0 and w̃2 denote the coefficients of the expansion at order k2 of the Fourier transform,
w̃(k), of w. By restricting ourselves to functions depending only upon z, we have
∫
w−1(1, 2)Ψext(2)d2 =
1
w̃0
[
1 +
w̃2
w̃0
∂2
∂z2
1
]
Ψext(1) = hext(z1) . (7)
We introduce the bulk correlation length, l and we look for a solution depending on t = φc(z/l)
for both hext(z1) and Ψext(z1). Equation (7) then reads
(
(1 − t2)2
∂2
∂t2
− 2t(1 − t2)
∂
∂t
+ l2
w̃0
w̃2
)
Ψ = − w̃
2
0
w̃2
l2hext . (8)
A one to one mapping with hext limited to the third power in φc is obtained when ∂2Ψ/∂t2 = 0
and we get
Ψ(z) = Ψ1φc(z); hext(z) = h1φc(z) +
h3
3!
φc(z)
3,
h1 = −Ψ1
1
w̃0
(
1 − 2
l2
w̃2
w̃0
)
,
h3 = −Ψ1
2
l2
w̃2
w̃0
. (9)
Formally l is finite but clearly the present approach should be valid only but in the vicinity of the
critical point where l → ∞ and we can therefore neglect the term h3.
The mean field surface tension follows from the identity ΩMF[ν] = H [φMF(z)] which is easily
calculated by using both the equation satisfied by φMF and the expression of the external field
contribution which takes a simple form with the particular choice for hext, with the result
βσ2γMF = 4σ2(−2K2K
′3
0 /u
′2
4 )1/2 + 2σ2
(
3K2
u′4
)1/2
h1
(
1 +
h3
3h1
)
which is similar to the expression given by Brilliantov [11] in the absence of the external field and
differs from that given by Zittartz [30] due to a different choice for the external field as a function
of φc. When h3 = 0 and h1 → 0 at finite K0 we get
βσ2γMF(K0,K2, h1 → 0) → βσ2γMF(K0,K2, h1 = 0) +
σ24(K2)
1/2φb
(−3K0)1/2
h1 . (10)
For t → 0, one recovers the mean field exponent, γMF ∼ t3/2.
23602-4
Statistical field theory for liquid vapor interface
3. One loop equations
In order to go beyond the mean-field approximation, we expand the Hamiltonian about φMF(z)
and consider the Gaussian approximation. Hence, we drop the cubic term in the external field (i.e.
we set h3 = 0) and, in order to unclutter notations, we define h̄1 = h1/φb. Without any loss of
generality, we also choose z0 = 0. The first correction stems from the second order term and we
have (z0 = 0),
H [φ = φMF + χ] ' H [φMF] +
1
2
∫
χ(1)H(2)(1, 2)χ(2)d1d2. (11)
The operator H(2) = δ2H/δφ(1)δφ(2) is diagonalized, after a Fourier transform parallel to the
surface, in the set of eigenfunctions ϕλ solution of the eigenvalues equation (ελ = K2(q
2 + h̄1 +
ωλ/l
2))
K2
(
− ∂2
∂z∗2
+ 6(tanh2(z∗) − 1) + (4 − ωλ)
)
ϕλ(z∗) = 0, z∗ = z/l (12)
the solutions of which are known [5, 30, 31]. We emphasize that keeping only the linear term in the
external field leads to the same Schrodinger-like equation (12) as in the absence of external field
and the only change is a shift of the eigenvalues ελ of h̄1. The spectrum of eigenfunctions includes
two bound states, ϕ0 = C0/ cosh2(z/l) and ϕ1 = C1 sinh(z/l)/ cosh2(z/l), with ω0 = 0, ω1 = 3
respectively and a subset of unbounded states, or continuum spectrum ϕk(z), with kl =
√
ωk − 4
which behave as plane waves in the bulk phases, i.e. far from the interface, and are given by
ϕk(z) = C(k)eikz
[
2 − k2l2 − 3i kl tanh(z/l) + 3 (tanh2(z/l)− 1)
]
. (13)
Notice that the shift of eigenvalues due to the external field is of crucial importance especially
for the lowest bounded state ϕ0 the energy of which ε0 tends to a finite value, K2h̄1 when q → 0.
As we shall see in the sequel this corresponds to a pinning of the interface by the field. It is
important to note that ϕ0(z) ∝ ∂φMF(z)/∂z. The constants C0, C1 and C(k) are determined in
order to normalize the ϕλ on the interval [−L,L]. The whole spectrum of eigenstates should then
be orthogonalized. For the bound states this is a direct consequence of the eigenvalues equation
(12) in the limit L/l → ∞, i.e. disregarding terms or order exp(−2L/l). On the other hand the
orthogonalization of the subset of unbounded eigenstates is no more a consequence equation (12)
but rather follows from the boundary condition
[ϕkϕ
?
k′ ]
L
−L = 0 for k 6= k′ (14)
from which we get the non-trivial dispersion relation leading to the density of states (x = k l)
n(k) =
L
π
− l
π
[
1
(1 + x2)
+
2
(4 + x2)
]
=
L
π
+ l f(x). (15)
The closure relation then follows
+∞∫
−∞
dk
2π
(
n(k) |ϕk(z)|2 − 1
)
+ ϕ2
0(z) + ϕ2
1(z) = 0. (16)
It is important to note that
∫∞
−∞
f(x)dx = −2 which proved useful in all calculations. Zittartz [30]
has already got this density of states but without explicitly mentioning the need of the orthogo-
nalization. In [32] Evans outlined the way to deal with the liquid –vapor interface in the framework
of statistical field theory, also at the one loop level; however, the continuous spectrum was only
qualitatively evoked and no particular attention was paid to the orthogonalization problem which
should be performed on the formally finite system. In [33] a similar kind of dispersion relation was
obtained in the modeling of the charge density profile of electrolytes in the framework of another
23602-5
V. Russier, J.-M. Caillol
field theory. We emphasize that the orthogonalization of the eigenstates is of crucial importance
for the calculation of gaussian functional integrals. In order to calculate ln(Ξ) and the correlation
functions, we write the fields χ(~r) in the basis which diagonalizes the operator H (2), namely a
Fourier transform in the ~s-direction and a projection on the ϕλ(z) where {λ} = {n = 0, 1; k}
denotes the whole spectrum of eigenstates. From usual gaussian functional integrals [34] we get
ln(Ξ) = −H [φMF] +
V
2
∫
d~k
(2π)3
ln
(
ŵ−1(k)
εk(q)
)
− S
2
∫
d~q
(2π)2
(
∑
n
ln (εn(q)) +
∫
dkf(kl) ln (εk(q))
)
, (17)
where the εk coincide with the eigenvalues of the second functional derivative of the Hamiltonian
for the φ4 homogeneous model taken at φ = φMF(± L). Therefore, the volume term of ln(Ξ) in
(17) is nothing but βpV at the one loop approximation [25]. We finally write the surface term of
ln(Ξ) in terms of the surface tension γ = [ln(Ξb)−ln(Ξ)]/S = γMF+γ(1). As already mentioned, this
expression for γ differs from the definition as the stiffness factor of the effective surface Hamitonian
(see for instance [26]) although the two routes coincide at the mean field level. We obtain (γ(1∗) =
βγ(1)σ2)
γ(1∗) =
1
8π
[
(t− 1) ln(1 − 2√
t+ 1
) + (t− 4) ln(1 − 4√
t+ 2
) − 6
√
t
]a+(qM l)2
a+(qml)2
, (18)
where qm and qM are the lower and the upper bound of the integral over q and a = (4 + h̄1),
respectively. A similar result was obtained in [30] for a spin model. This result differs from that
obtained in [4] where only the n = 0 eigenstate is kept. In the limit h̄1 → 0, we get
γ(1∗)(h̄1) → γ(1∗)(h̄1 = 0) − (1/8π)h̄1 ln(h̄1). (19)
We now consider the calculation of the field two-body correlation functions,Gφ(1, 2) = 〈χ(1)χ(2)〉H .
More precisely, we focus on the Fourier transform parallel to the surface, Gφ(z1, z2, q). To this end
we have to calculate the sum
∑
λ
ϕλ(z1)ϕ
∗
λ(z2)
ελ(q)
=
∑
n=0,1
ϕn(z1)ϕn(z2)
εn(q)
+
∫
ρ(k)
ϕk(z1)ϕ
∗
k(z2)
εk(q)
dk.
A contribution of the integral over k exactly cancels the direct contribution of the two bound states,
which shows, once again, that the approximation consisting in keeping only the ϕ0(z) eigenmode
is not sufficient. The result is (z>,< = sup, inf(z1, z2))
Gφ(z1, z2, q) =
9l
2K2
exp(− |z12| (x2 + 4)1/2/l)
x2(x2 + 3)(x2 + 4)1/2
[
1 + x2/3 + (x2 + 4)1/2 tanh(z>/l) + tanh2(z>/l)
]
×
[
1 + x2/3− (x2 + 4)1/2 tanh(z</l) + tanh2(z</l)
]
, (20)
where x2 = (q2 + h̄1)l
2. Notice that formally this expression coincides with that obtained in the
absence of the external field, the only difference being the definition of x which takes a finite value,
h̄1 when q → 0.
4. Results and discussion
4.1. Capillary behavior and surface structure factor
We started from the expansion of the effective Hamiltonian on the basis of the eigenstates
ϕλ(z). The first eigenstate, ϕ0(z) is proportional to the derivative of the mean field result φc(z), the
23602-6
Statistical field theory for liquid vapor interface
proportionality constant being determined by normalization. If we keep only this first eigenstate,
the expansion of χ reads χ(~s, z) = ξ(~s)ϕ0(z). Hence the field takes the form φ(z) = φc(z −
zint(~s)) and only the linear term in the expansion of φ with respect to zint. remains. This indeed
corresponds to the so-called rigidly displaced profile approximation, where zint = −aξ(~s) represents
the fluctuating location of the interface. The corresponding contribution to H is a functional of
zint(~s) which defines an effective surface Hamiltonian given by
H(0)
s [zint(~s)] =
K2
2a2
∫
d~s
[
(∂~s(zint(s))
2 + h̄1zint(s)
2
]
, (21)
where we have used ε0 = K2(q
2 + h̄1) since ω0 = 0. This Hamiltonian pertains to the capillary
wave models [12, 35–37]. In general, the capillary wave Hamiltonian is obtained either from phe-
nomenological arguments [35] or by restricting the fluctuations beyond the mean field level to those
deduced from a rigidly displaced mean field profile [36]. In [12] a formulation based upon the direct
correlation function of the liquid state theory is used. In [38] a supplementary zint(s)
4 term in the
surface Hamiltonian (21) was introduced in order to account for curvature effects of the interface;
this additional term should vanish with the external field, [37] and in any case cannot be generated
in the framework a one loop scheme. From the normalization of ϕ0 and from the analytic form of
the mean field profile we deduce K2/a
2 =σ2βγMF − σ2lφ2
b h̄1 and accordingly we rewrite (21) in
the form
H(0)
s =
1
2
[
βγMF − σ2lφ2
b h̄1
] ∫
d~s
[
(∂~s(zint(~s)))
2 + h̄1zint(~s)
2
]
(22)
which exactly coincides with the usual effective surface Hamiltonian HCWT[zint(~s)] of the CWT
theory for the free surface, when the external field vanishes. Therefore, in the limit h1 → 0 we ob-
tain the CWT as the lowest approximation beyond the mean field approximation without invoking
phenomenological arguments. The structure is characterized by the height-height correlation func-
tion, 〈zint(~s1)zint(~s2)〉 or its Fourier transform parallel to the surface which defines the surface
structure factor, S∆z∆z(q), where zint(~s) is the location of the interface relative to its mean value.
We consider
∫
χ(~s, z)dz/∆φb, where ∆φb = φMF(L)−φMF(−L), as a measure of the instantaneous
location of the surface at ~s, which amounts to define the location of the surface from a constraint on
the integral of χ, as is done in [15] for the density profile. We are then led to identify S∆z ∆z(q) =
(∆φb)
−2
∫
Gφ(z1, z2, q)dz1dz2, which corresponds to the Sic used in [15]. It is important to notice
that the coupling with the bulk fluctuations is included in the present formulation through the
eigenstates of the continuum. We also introduce the effective surface width, or surface corrugation,
σeff =
√
〈zint(~s1)zint(~s1)〉. Henceforth we consider only small values of h̄1 and more precisely we
relate the value of h̄1 to an effective lateral size of system, according to h̄1 ∼ L−2
x as it will be
justified from the analysis of the capillary wave limit. The behavior of S∆z ∆z(q) is analyzed from
the function g̃(q) =
∫
dz1
∫
dz2Gφ(z1, z2, q). From (20) for the leading term of Gφ(z1, z2, q) when
q → 0 we get
Gφ(z1, z2, q → 0) ' 1
K2(q2 + h̄1)
ϕ0(z1)ϕ0(z2) +
3l
2K2
h̄1
q2 + h̄1
e−2|z12|/l [f(z1, z2)] . (23)
The first term of the r.h.s. of equation (23) corresponds to the CWT limit, especially when h̄1 = 0,
and the correction, which vanishes at h̄1 = 0, plays a role only far from the interface where
ϕ0(z1)ϕ0(z2) is negligible and moreover when the two particles are on the same side of the interface,
i.e. when e−2|z12|/l is not negligible. Thus we can rewrite the q → 0 behavior of Gφ(z1, z2, q) as
Gφ(z1, z2, q → 0) ' 1
K2(q2 + h̄1)
ϕ0(z1)ϕ0(z2) +
l
2K2
f e−2|z12|/l , (24)
where f takes a constant value, f = 1, when both |z1|, |z2| � l and are of the same sign. The
second term of the r.h.s. of (24) is nothing but the limit at q → 0 of the bulk correlation function
and is not to be included in the capillary limit of the model. In any case, for sufficiently small
23602-7
V. Russier, J.-M. Caillol
values of h̄1, the bulk contribution to equation (24) becomes negligible at q = 0. This is made more
precise hereinafter concerning the behavior of S∆z ∆z. Let us denote by GCW the first term of the
r.h.s. of equation (24) and by g̃CW(q) its integral over z1 and z2. It is easy to show from the relation
already used between the normalization of ϕ0(z) and γMF that
1
(∆φb)2
g̃CW(q → 0) → 1
(γMF − σ2lφ2
b h̄1)(q2 + h̄1)
. (25)
In the limit h̄1 = 0 this is exactly the CWT behavior, leading to the well known logarithmic
divergence of the squared surface corrugation for which one gets (σeff)2 = (4πγMF)−1 ln(q2M/q2m).
The effect of the external field is to make the integral over q entering the determination of σeff
finite at q → 0 with the result
(σeff )2 =
1
4πβγMF
ln
(
(2π)2
σ2h̄1
)[
1 +O(σ2h̄1)
]
, (26)
where we have assimilated qM to the inverse of the molecular size, 2π/σ. Since the result for the
CW at the free interface in the absence of external field is a logarithmic divergence with the lateral
size, ∼ ln(σ/Lx) [8], this shows that, as we have already mentioned, the effect of the external
field is to pin the interface in such a way that the capillary wave is restricted to a lateral scale
Lx ∼ h̄
−(1/2)
1 . It can be shown from (20) that the contribution to g̃(q) diverging with the system
size, due to the bulk correlations, is exactly (2L) times the integral of Gb(z12, q),the correlation
function of the bulk phase, over z12. If we keep only these two terms we get
S∆z ∆z(q) '
2L
∆Φ2
∫
Gb(q, z12)dz12 +
1 + (σ2lφ2
bγMF)h̄1
γMF(q2 + h̄1)
+O(h̄2
1) , (27)
where ∆Φ = Φl−Φg denotes the difference between the mean-field values in the two phases. Given
that the bulk term leads to a constant when q → 0, we see that (27) presents a cross-over like
behavior in terms of wave vector q, where a threshold value qs naturally appears, separating the
capillary wave behavior at small values of q from the bulk like behavior at q > qs, with qs given by
(q2s + h̄1) ∼ ∆Φ2
2L βγMFK2
∫
Gb(q, z12)dz12
. (28)
We see that this cross-over behavior holds only if h̄1 is negligible compared to the r.h.s. of equa-
tion (28) or in other words
h̄1 l
2 <
l
2L
16φ2
blK2
βγMFl2
and confirms the above mentioned separation of the capillary behavior from the bulk correlations
when this condition is fulfilled. We emphasize that h̄
1/2
1 can be understood as the inverse of an
effective lateral size pinning the interface. The cross-over like behavior resulting from (28) is in
agreement with that of [15, 39, 40]; however, in the present formulation the bulk fluctuations come
out of the calculation through the continuum subset of eigenstates and have not to be added to
the field profile. Furthermore, from (27) we can drop exactly the bulk contribution, and doing this
we define a purely interfacial contribution to S∆z ∆z(q) (see figure 1).
Sint(q) = S∆z ∆z(q) − (2L/∆φ2
b)
∫
Gb(q, z12)dz12 . (29)
Then the departure of Sint(q) from its 1/q2 behavior allows us to isolate the deviations from
the capillary wave like behavior of S∆z ∆z(q). At this point, since we focus on the deviation form
the capillary-like behavior, we can consider h̄1 = 0. We define β(q) = (q2Sint(q))
−1 − γMF. β(q)/q2
is found nearly constant and thus leads us to define a bending rigidity of the interface,
κ = lim
q→0
((1/q2)β(q)) ' ((1/q2)β(q)) (30)
23602-8
Statistical field theory for liquid vapor interface
10-1
100
101
102
103
104
105
0.01 0.1 1
g(
q)
*
q l
Figure 1. Log-log plot of g̃∗(x) versus x = ql for h̄1 = 0 and L/l = 100, 80, 60, 40 and 20 from
top; interfacial contribution ∆φ2
bS
∗
int(x), bottom; CWT limit, straight line.
which takes a positive value, as it should be for the stability of the interface [14], as is also found
from the simulation results of [40] after a separation between bulk density correlations and interface
fluctuations. In other words, the small q behavior of Sint(q) is
Sint(q) ' 1/[γMFq
2 + κq4].
Here we find κ = κ∗(γMFσ
2)/(σ/l)2, with the reduced value κ∗ = 0.288.
4.2. Surface tension
In this section we focus on the behavior of the one loop contribution to the surface tension γ
in the case h̄1 = 0, since its dependence on h̄1 vanishes (see equation (19)). The contribution γ(1),
given by (18), depends on the two bounds qm and qM of the wave vector q which are related to
the relevant parameters of the interface: on the one hand lqM = 2πl/σ where l/σ is the intrinsic
width of the interface in unit σ, lqM ∈ [1,∞]. On the other hand, lqm = (2πl/Lx where Lx is the
system actual lateral size; hence lqm = (lqM )(σ/Lx). Thus lqM appears as a natural parameter,
with lqM ∈ [2π,∞] and 1/(lqM ) ∝
√
t. We can now re-write γ(1∗) in a more convenient form
(σ/Lx ∈ ]0, 1])
γ(1∗) =
π
2(lqM )2
[
γ̃
(
4 + (lqM )2
)
− γ̃
(
4 + (lqM
σ
Lx
)2
)]
, (31)
(Lx/σ) is either the actual lateral system size or the scale at which γ is measured, for instance in a
numerical simulation (see [41]), but in any case does not depend on t. We can note that whatever
the values of σ/Lx or (lqM ), σ2γ(1) remains finite and more precisely
γ
(1∗)
Lx→∞(lqM → ∞) ∼ − 6π
lqM
(32)
which → 0 when t → 0 as t1/2. The results for σ2γ(1∗) are displayed in figure 2. We interpret
σ2γ(1∗)(σ/Lx) as the q−dependent contribution to γ with q = (2πσ/Lx). This means that the
flat interface corresponding to the mean field approximation is obtained when no fluctuations at
all are taken into account, namely for q ∼ qM . This differs from what is done in [14] (see also [42])
where the contribution to γ due to the surface fluctuations vanishes at q → 0. The small q behavior
23602-9
V. Russier, J.-M. Caillol
of γ(1∗) is easily obtained from (31) and yields
γ
(1∗)
q→0 ' γ(1∗)(0) +
1
8π
[
(qσ)2[a+ ln(σ/l)] − 2(qσ)2 ln(qσ)
]
with a = 3.48491
(33)
from which we can deduce a crossover value of q given by q0σ = (σ/l)ea/2, separating the q2 behavior
from the (qσ)2 ln(qσ) behavior obtained for q < q0 and q > q0 respectively. Firstly, it is important
to note that we always get an increasing γ(1)(q) and secondly that since q0 is proportional to l−1,
we get a plateau (corresponding to the q2 dependence) at small values of σ/Lx only when lqM
takes small values, i.e. for the lowest temperatures (see figure 2). This behavior is in qualitative
agreement with the simulation results of [41]. The term proportional to q2 in the variation of γ(1)
with q can be interpreted as resulting from the energy necessary to bend the interface, and should
be related to κ obtained from the behavior of Sint(q) (see equation (30). This is not a priori the
case since we do not expect a full coherence in the framework of a loop expansion between the
energetic and structure quantities.
-1
-0.95
-0.9
-0.85
-0.8
-0.75
-0.7
-0.65
-0.6
0 0.1 0.2 0.3 0.4 0.5
γ(
1
)
σ/Lx
Figure 2. γ(1)(σ/Lx)/
∣∣∣γ(1)(0)
∣∣∣ in terms of σ/Lx for lqM/(2π) = 1; 2; 5 and 50 from bottom
to top.
4.3. Density profile and density correlation function
Now we go back to the mapping between the averaged field, φ, and the density in the fluid, ρ
on the one hand and between the field and the two body density correlation function G
(2)T
ρ on the
other hand. Formally we have from equation (A-13))
ρ(1) = ρHS[ν0](1) − hext(1) +
∫
w−1(1, 2)φMF(2)d2,
G(2)T
ρ (1, 2) =
∫∫
w−1(1, 1′)w−1(2, 2′)Gφ(1′, 2′)d1′d2′ − w−1(1, 2). (34)
We recall that G
(2)T
ρ (1, 2) is ρ(1)ρ(2) h(1, 2)+ρ(1)δ(1, 2) where h(1, 2) is the usual total correlation
function of liquid state theory[43]. We easily get
ρ(1) = ρHS(ν0) + ∆ρ1φc(z1) + ∆ρ3φc(z1)
3, (35)
23602-10
Statistical field theory for liquid vapor interface
where
∆ρ1 = φb
[
1
w̃0
− w̃2
w̃2
0 l
2
− h̄1
]
,
∆ρ3 = φb
w̃2
w̃2
0 l
2
. (36)
If we limit the expansion to the term linear in h1 we recover the usual tanh-like profile (ρl +ρg)/2+
tanh(z− z0)(ρl − ρg)/2. Concerning the two body correlation function, since we limit the effective
Hamiltonian to a φ4 model, the bulk field correlation function is the same in the two phases and as
a result of equation (34) this is also the case for the density correlation function G
(2)T
ρ (1, 2). This is
of course a drawback of the simplicity of the present model for the description of the liquid vapor
interface. To overcome this drawback, we may have to consider a true inhomogeneous reference
fluid, which imposes to chose two different values for the hard sphere reference chemical potential,
ν0. This is beyond the scope of the present work.
5. Conclusion
We have shown in this article that the one-loop approximation of the KSSHE statistical field
theory of simple liquids gives a coherent and qualitatively exact description of a planar liquid/vapor
interface. The mathematical treatment of the kink is subtle and requires to take into account the
whole spectrum of eigenvalues and eigenstates of the Gaussian Hamiltonian and not, as often and
incorrectly proposed in the literature, to restrict oneself to its fundamental state. Moreover,without
invoking any phenomenological description of the interface, we were able to recover the usual CWT
surface Hamiltonian as an approximation of the one-loop calculation. We also show that we can
extract a rigidity bending factor which takes a positive value, in agreement with the requirement
for the stability of the interface with respect to fluctuations. On the other hand, we clearly control
the limit of the model by computing the two-body density correlation functions which appear to
coincide in the liquid and vapor bulk phases, a drawback due to the identity of the two phases in
the field φ representation.
Nevertheless, qualitatively, our results are in good agreement with numerical simulations. We
stress that no parameter is involved in the theory which can be worked out for any pair potential. A
quantitative agreement is expected only for long range pair potentials which we plan to investigate
by numerical simulations in future work.
A. KSSHE Transform
In this appendix we review the so-called Kac-Siegert-Stratonovich-Hubbard-Edwards (KSSHE)
[17–22] transformation devised to rewrite the GCPF of a classical fluid as the partition function
of a bosonic statistical field theory; more details are given in refs [23–25].
We consider the case of a simple three dimensional fluid made of identical hard spheres of
diameter σ with additional isotropic pair interactions v(rij ) (rij = ‖xi − xj‖, xi position of particle
“i”). Since v(r) is arbitrary in the core, i.e. for r 6 σ, we assume that v(r) has been regularized in
such a way that its Fourier transform ṽ(q) is a well behaved function of q and that v(0) is a finite
quantity. We denote by Ω ⊂ R
3 the domain of volume V occupied by the molecules of the fluid. The
fluid is at equilibrium in the GC ensemble and we denote by β = 1/kT the inverse temperature (k
Boltzmann constant) and by µ the chemical potential. For the sake of generality the particles are
subject to an external potential ψ(x) and ν(x) = β(µ − ψ(x)) is the dimensionless local chemical
potential. We stick to notations usually adopted in standard textbooks on the theory of liquids
(see e.g. ([43])) and denote by w(r) = −βv(r) the negative of the dimensionless pair interaction.
Moreover, we restrict ourselves to attractive interactions, i.e. such that their Fourier transform
w̃(q) > 0 is definite positive for all q.
23602-11
V. Russier, J.-M. Caillol
In a given configuration C = (N ;x1 . . . xN ) of the GC ensemble the microscopic density of
particles reads
ρ̂(x|C) =
N∑
i=1
δ(3)(x− xi) , (A-1)
and the GC partition function Ξ [ν] can thus be written as
Ξ [ν] =
∞∑
N=0
1
N !
∫
Ω
d1 . . . dn exp
(
−βVHS(C) +
1
2
ρ̂ · w · ρ̂+ ν · ρ̂
)
, (A-2)
where ρ · w · ρ is a short hand notation for
∫
ρ(1)w(1, 2)ρ(2)d1d2, i ≡ xi and di ≡ d3xi. For a given
volume V , Ξ [ν] is a function of β and a convex functional of the local chemical potential ν(x)[44, 45].
In equation (A-2) exp(−βVHS(C)) denotes the hard sphere contribution to the Boltzmann factor
in a configuration C and ν = ν + νS where νS = −w(0)/2 is β times the self-energy of particles.
From our hypothesis on w(r), νS is a finite quantity which depends, however, on the regularization
of the potential in the core.
We now recognize the Gaussian integral [46, 47]
exp
(
1
2
ρ̂ · w · ρ̂
)
=
1
Nw
∫
Dϕ exp
(
−1
2
ϕ · w−1 · ϕ+ ρ̂ · ϕ
)
, (A-3)
where ϕ is a real scalar field, Dϕ is a functional measure, w−1(1, 2) is the inverse of w(1, 2) in the
sense of operators and the normalization Nw reads as
Nw =
∫
Dϕ exp
(
−1
2
ϕ · w−1 · ϕ
)
= exp
(
−V
2
∫
d3k
(2π)3
ln w̃(k)
)
. (A-4)
It follows that the GCPF Ξ [ν] can be re-expressed as the functional integral
Ξ [ν] = N−1
w
∫
Dϕ exp (−H [ϕ, ν]) , (A-5)
where the KSSHE action (or effective Hamiltonian) is expressed as
H [ϕ, ν] =
1
2
ϕ · w−1 · ϕ− ln {ΞHS [ν + ϕ]} . (A-6)
The action H [ϕ, ν] is non-canonical in the sense that the coupling between the field ϕ and the
external source ν is non-linear as in usual Landau-Ginzburg actions [47]. In order to get a canonical
field theory we perform the translation φ = ϕ+ ∆ν with ∆ν = ν − ν0 where ν0 is some arbitrary
uniform reference chemical potential to be conveniently chosen later. The Jacobian of the transfor-
mation is obviously equal to one and then, performing a Taylor functional expansion of the grand
potential ln {ΞHS [ν0 + φ]} about ν0 we obtain a standard scalar field theory [23–25], characterized
by the partition function
Ξ? [B] = N−1
w
∫
Dφ exp
(
1
2
φ · ∆−1 · φ− V [φ] +B · φ
)
, (A-7)
where the propagator ∆(x, y) and the interaction V [φ] are given by
∆−1(x, y) = w−1(x, y) +G
(2) T
HS (ν0|x, y) , (A-8a)
V [φ] = −
∞∑
n=3
1
n!
∫
dx1 . . . dxn G
(n) T
HS (ν0|x1, . . . , xn)φ(x1) . . . φ(xn) , (A-8b)
where the G
(n) T
HS are the truncated n-body correlation functions of a HS fluid at chemical potential
ν0 [43]
G
(n) T
HS (ν0|x1, . . . , xn) =
δn ln ΞHS [ν0]
δν0(x1) . . . δν0(xn)
, (A-9)
23602-12
Statistical field theory for liquid vapor interface
which are supposed to be exactly known, and where, finally, the external source or “magnetic”
field B(x) of the theory is related to the local chemical potential through the linear relation
B(x) = ρHS(ν0) + w−1(x, y)∆ν(y) , (A-10)
where Einstein convention of summation over repeated indices – here continuous variable y –
was adopted. The free energies of the non-canonical and canonical theories differ essentially by a
quadratic form, i.e.
ln Ξ [ν] = ln ΞHS [ν0] −
1
2
∆ν · w−1 · ∆ν + ln Ξ? [B] , (A-11)
which allows us to easily relate the correlations of the density to that of the field. With the
definitions
G(n) T
ρ (ν|x1, . . . , xn) =
δn ln Ξ [ν]
δν(x1) . . . δν(xn)
, (A-12a)
G
(n) T
φ (B|x1, . . . , xn) =
δn ln Ξ? [B]
δB(x1) . . . δB(xn)
, (A-12b)
one finds [23–25]
ρ(x) = w−1(x, y) {〈φ〉(y) − ∆ν(y)} = w−1(x, y)〈ϕ〉(y) , (A-13a)
G(2) T
ρ (x, y) = −w−1(x, y) + w−1(x, x
′
)w−1(y, y
′
)G
(2) T
φ (x
′
, y
′
) , (A-13b)
G(n) T
ρ (x1, . . . , xn) = w−1(x1, x
′
1) . . . w
−n(x1, x
′
n)G
(n) T
φ (x
′
1, . . . , x
′
n) for n > 3 . (A-13c)
Note that the truncated n-body correlation functions of the fields φ and ϕ coincide for n > 2 since
the two fields differ by a simple additional function.
At this point we introduce some approximations. First we adopt the “point of view of Sirius”
and consider the kernels G
(n) T
HS (ν0|x1, . . . , xn) to be short-range functions of their arguments when
compared with the range of variations of field or density correlations near the critical point. More
precisely, we assume that, for n > 3, one has
G
(n) T
HS (ν0|x1, . . . , xn) ≈ βP
(n)
HS (ν0)δ(xn, x1) . . . δ(x2, x1) , (A-14)
where P
(n)
HS (ν0) denotes the n-th derivative of the HS pressure with respect to the chemical potential
ν0. Moreover, neglecting high-order contributions to V [φ] we get the somehow sketchy interaction
V [φ] ≈ u3
3!
∫
dx φ3(x) +
u4
4!
∫
dx φ4(x) , (A-15)
where un = −βP (n)
HS (ν0) = −ρ(n−1)
HS (ν0), which should be valid but very close to the critical point.
Note that one should have u4 > 0 for the action to be bounded from below at large fields so that
the theory is well-behaved.
A similar approximation is devised for the propagator for which we adopt a gradient expansion
up to the second order, more conveniently written in Fourier space as
∆̃−1 (q1, q2) = (2π)3δ(3) (q1 + q2) ∆̃−1 (q1) , (A-16a)
∆̃−1(q) = w̃−1(q) + C̃(2)
HS(q) = K0 +K2q
2 + O(q4) , (A-16b)
where the Ornstein-Zernicke relation C̃(2)
HS(q) = −1/G̃
(2) T
HS (q) defines the Fourier transform of the
HS two-body direct correlation function (the definition of which includes an ideal gas contributions
so that C̃(2)
HS(q) = c̃HS(q) − 1/ρ, c̃HS(q) being a usual direct correlation function [43].) A short
calculation gives
K0 =
1
w̃(0)
− ρ
(1)
HS(ν0) , (A-17a)
K2 =
1
2
{
− w̃2
w̃(0)2
+ c̃HS ,2 ρ
(1)
HS(ν0)
2
}
, (A-17b)
23602-13
V. Russier, J.-M. Caillol
where w̃2 and c̃HS ,2 are the second derivatives of w̃(q) and c̃HS(q) with respect to q at q = 0,
respectively. The quantities entering the coupling constants K0 and K2 in equations (A-17) require
the knowledge of the equation of state and pair correlation functions of the HS fluid, which can be
done in the framework of Percus-Yevick theory, for instance [43].
At this point we choose ν0 such that u3 = −ρ′′HS(ν0) = 0 so that the action S of field φ reduces
exactly to that of a φ4 Landau-Ginzburg model, i.e.
S = −φ · B +
∫
dx
{
1
2
K0 (∇φ)
2
+
1
2
K2φ
2 +
u4
4!
φ4
}
. (A-18)
A last remark is in order. As discussed in [25] the chemical potential ν0 ≈ −0.025 is uniquely
defined (a consequence of the expected – and satisfied – convexity of the function ν 7−→ ρ
′
HS(ν))
and such that ρHS(ν0) ≈ 0.25, ρ
(1)
HS(ν0) ≈ 0.09 and u4 > 0.
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Статистико-польова теорiя мiжфазної границi рiдина-пара
В. Русьєр1, Ж.-М. Кайоль2
1 Iнститут хiмiї i матерiалiв, UMR 7182 CNRS i Унiверситет Парi-ЕСТ, Францiя
2 Лабораторiя теоретичної фiзики, UMR 8627 CNRS i Унiверситет Парi-Сюд, Францiя
З перших принципiв розвинуто статистико-польову теорiю для неоднорiдної рiдини, плоскої мiжфа-
зної границi рiдина/пара. Велика статистична сума представляється через перетворення Габбарда-
Стратоновича, що приводить поблизу критичної точки до звичної скалярної теорiї поля φ4, яка потiм
строго розглядається на однопетльовому рiвнi. Пiсля подальшого спрощення це дає добре вiдому
теорiю капiлярної хвилi без жодного ad hoc феноменологiчного параметра. Обговорюється внутрi-
шня узгодженiсть однопетльового наближення i наголошується на доброму якiсному узгодженнi з
недавнiми числовими симуляцiями.
Ключовi слова: теорiя поля, мiжфазна границя газ-рiдина, поверхневий натяг
23602-15
Introduction
KSSHE transform and mean-field approximation
One loop equations
Results and discussion
Capillary behavior and surface structure factor
Surface tension
Density profile and density correlation function
Conclusion
KSSHE Transform
|
| id | nasplib_isofts_kiev_ua-123456789-32093 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-11-24T15:33:25Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Russier, V. Caillol, J.-M. 2012-04-08T15:45:34Z 2012-04-08T15:45:34Z 2010 Statistical field theory for liquid vapor interface / V. Russier, J.-M. Caillol // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23602: 1-15. — Бібліогр.: 47 назв. — англ. 1607-324X PACS: 61.30.Hn, 64.70.F, 68.03.Cd https://nasplib.isofts.kiev.ua/handle/123456789/32093 A statistical field theory for an inhomogeneous liquid, a planar liquid/vapor interface, is devised from first principles. The grand canonical partition function is represented via a Hubbard-Stratonovitch transformation leading, close to the critical point, to the usual φ4 scalar field theory which is then rigorously considered at the one-loop level. When further simplified it yields the well-known capillary wave theory without any ad hoc phenomenological parameter. Internal coherence of the one-loop approximation is discussed and good overall qualitative agreement with recent numerical simulations is stressed. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Statistical field theory for liquid vapor interface Статистико-польова теорія міжфазної границі рідина - пара Article published earlier |
| spellingShingle | Statistical field theory for liquid vapor interface Russier, V. Caillol, J.-M. |
| title | Statistical field theory for liquid vapor interface |
| title_alt | Статистико-польова теорія міжфазної границі рідина - пара |
| title_full | Statistical field theory for liquid vapor interface |
| title_fullStr | Statistical field theory for liquid vapor interface |
| title_full_unstemmed | Statistical field theory for liquid vapor interface |
| title_short | Statistical field theory for liquid vapor interface |
| title_sort | statistical field theory for liquid vapor interface |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32093 |
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