Dynamics of the diffuse gas-liquid interface near the critical point

Physica 48 (1970) 541-560 © North-Holland Publishing Co.

DYNAMICS OF T H E D I F F U S E GAS-LIQUID I N T E R F A C E NEAR T H E CRITICAL POINT B. U. F E L D E R H O F Instituut voor Theoretische Fysica der Universiteit, Utrecht, Nederland

Received 26 J a n u a r y 1970

Synopsis The dynamics of the diffuse liquid-gas interface near the critical point is studied. Consistent with the theory of van der Waals and Fisk and Widom for the equilibrium interface we derive hydrodynamic equations which are valid for fluid states in the two-phase region. On the basis of these equations the dispersion equation for surface waves is derived and it is shown that the surface tension appearing in the dispersion equation is correctly given by van der Waals's expression for the surface free energy of t h e equilibrium interface.

1. Introduction. The detailed nature of the interface separating two coexisting phases still represents an outstanding problem in statistical mechanics. The early theories of capillarity have been reviewed by Bakker 1). Laplace and Gauss regarded the interface as a mathematical surface of discontinuity. Poisson, Maxwell and Rayleigh took the point of view that the interface must be regarded as diffuse. Van der Waals 2) gave a detailed theory of the liquid-gas interface based on his famous equation of state. He showed that at the critical point the thickness of the interface becomes infinite. At the same time the surface free energy per unit area of interface tends to zero. The corresponding critical exponent was calculated by van der Waals but his result was larger than the experimental value. Van der Waals's ideas have been successfully applied to other phase equilibria3). An essential feature of the van der Waals theory is the assumed existence of thermodynamic functions for homogeneous fluid states in the two-phase region. Recently Fisk and Widom 4) have replaced the classic analytic behaviour of the extended thermodynamic functions near the critical point by a scaling hypothesis corresponding to similar scaling in the one-phase region. The predicted value of the surface free-energy critical exponent is in good agreement with experiment. Also the detailed shape of the density profile predicted by Fisk and Widom is in excellent agreement with light reflection experiments on critical fluid mixtures S). 541

542

B.U. FELDERHOF

In the van der Waals and Fisk-Widom theories the surface free energy is calculated for the state of maximum probability in an ensemble of possible density functions. This method has been put into doubt on the ground t h a t near the critical point the fluctuations are very large so that the state of m a x i m u m probability m a y not be representative6). In a theory by Buff, Lovett and Stillinger 7) the fluctuations themselves are held responsible for the diffuseness of the interface. Experimentally the fluctuations of the interface are observed by light scattering S). In the spectrum there appear characteristic peaks corresponding to the fact that on the average the fluctuations propagate as damped surface waves. The frequency shifts and linewidths are in agreement with the dispersion equation of surface waves as calculated from hydrodynamics when the interface is treated as infinitesimally thin with a surface tension equal to the surface free energy of the diffuse interface. The latter is found from light-reflection experiments with the aid of the van der Waals theory. From a theoretical point of view it seems desirable to study the dynamics of the diffuse interface in greater detail. For a diffuse interface there is no a priori reason why the surface tension appearing in the dispersion equation should be identical to the surface free energy of the static planar interface. In this article we show that the two are in fact identical but the proof requires a rather elaborate exercise in hydrodynamics. At the same time this answer thes questions 6) about the validity of the van der Waals method of calculation. The present approach also allows one to treat the light scattering and light reflection from a unified point of view, but we shall leave this topic for a future article. In sec. 2 we reformulate the van der Waals-Fisk-Widom theory2,4) in a form suitable for our purposes. We consider the complete set of hydrodynamic variables, viz., density, average velocity and energy density, rather than just the density. In see. 3 we study the equilibrium situation, i.e., the state of maximum probability. In sec. 4 we consider isothermal motions. The equations of motion are derived from the Lagrangian formalism. This leads to hydrodynamic equations valid for states in the two-phase region and consistent with the van der Waals theory of equilibrium. For simplicity we neglect dissipative effects, but these could easily be included. In sec. 5 we study small isothermal motions in the neighbourhood of equilibrium. In sec. 6 it is shown that the surface tension appearing in the dispersion equation for surface waves is indeed given by the expression for the surface free energy derived by van der Waals. In sec. 7 we argue that in the absence of dissipation the motions of the fluid are adiabatic rather than isothermal. The correct equations of motion are derived. In sec. 8 it is shown that in the limit of long wavelengths the dispersion equation is not affected by adiabaticity. The article is concluded with a discussion.

THE GAS-LIQUID INTERFACE NEAR THE CRITICAL POINT

543

2. The probability distribution o/ hydrodynamic variables. We consider a single-component fluid of particles of mass m located in a domain of volume V in the presence of a gravitational potential ~)(r). We assume that the system is in equilibrium with a heat 'bath at temperature T and in contact with a particle reservoir with chemical potential fro. The probability of finding the system in a particular microscopic state is then given by a grand canonical probability distribution with parameters (fro, T). We shall, however, use a coarse-grained description in which the state of the fluid is characterized by hydrodynamic variables, namely the Fourier components of the number density, the local average velocity and the energy density. The microscopic definitions of these variables in terms of the coordinates and momenta of the particles read

/

exp[--ik.rj],

t~k= Z ( p d m ) exp[--ik, rj],

(2.1)

i

(p~/2m) expE--ik.rj] + ½ ~ 9(rjz) expE--ik.rj], J

i~g

where we have assumed a central two-body interaction ~0(r). The probability that the variables take the values {n k, v~, ek} for the set of wavevectors k = 2rcm/V~, where (rex, my, mz) are integers, is given by P{nk, v~, ek) = [ II' c52(n~ -- ¢~) c56(vk -- vk) 62(e~ -- e~)]A,,(,0,T), Ikt < k ....

(2.2)

where the subscript on 8 denotes the dimensionality of the delta-function and where the average is over the grand canonical ensemble. We have restricted ourselves to wavenumbers less than some maximum value kmax in accordance with our intention to deal with macroscopic variables. The prime on the product sign indicates that from each pair (k, --k) only one is present. We define the coarse-grained local number density as follows n(r) ---. V - 1

~

n k exp[ik.r]

(2.3)

Ikl < km~x

and similarly the local velocity v(r) and energy density e(r). We write the probability measure for the set of functions {n(r), v(r), e(r)}, as given by (2.2), in the form P{n(r), v(r),

e(r)}

= exp[-//~gfn(r), v(r), e(r)}~/.AG

(2.4)

where dff is a normalizing factor and /5 = 1/kBT with kB Boltzmann's constant. We shall call the functional f2, defined by (2.4), the thermodynamic potential. Correspondingly we define an entropy functional St by ~2 = E -- T S t -- ffoN,

(2.5)

B.U. FELDERHOF

544

where E = I e(r) dr,

N = [. n ( r ) d r ,

V

(2.6)

V

are the total energy and the total number of particles, respectively. In the so-called square gradient approximation the thermodynamic potential (2.5) is further specified b y the following definition of the internal energy density u(r) e(r) ~-- u(r) + ~mnv z + mnqb + ~-A[co(Vn) 2 -- (1 -- Co) nV2n],

(2.7)

where A and Co are constants. Usually the dimensionless constant Co is taken equal to unity, but there is no compelling reason to do this, and in the present local theory it is advisable to leave the choice of Co open. Van der Waals himself chose co equal to zero. The last term in (2.7) evidently takes into account the contribution to the energy density arising from the presence of density inhomogeneities. In homogeneous subregions (2.7) reduces to the usual definition of the internal energy density 9). In the Fisk-Widom scaling theory A is found to be a slowly varying function of the heat-bath temperature T. As our basic postulate we shall assume that for a suitable value of A the entropy functional St defined by (2.4)-(2.7) is given to a good approximation b y an integral of a local function of n(r) and u(r) S t = S at(n(r), u(r)) dr.

(2.8)

V

From thermodynamic fluctuation theory for homogeneous subsystems it follows that the function at(n, u) coincides with the thermodynamic entropy per unit volume for values (n, u) in the one-phase region. In the van der Waals theory2) at(n, u) for values (n, u) in the two-phase region was the analytic continuation from the one-phase region. Such an analytic continuation does not necessarily exist. Fisk and Widom 4) hypothesized that near the critical point in the two-phase region the free energy per unit volume

~t = u -- T a t

(2.9)

is a scaling function with the same characteristic exponents as in the onephase region. The superscript ~ serves as a reminder that in the two-phase region the functions thus defined differ from the thermodynamic functions as calculated from the statistical mechanics of homogeneous systems in the thermodynamic limit, from which one automatically finds a linear combination of the values on the liquid and gas side of the coexistence curve. For the extended functions in the two-phase region we shall simply use the thermodynamic nomenclature. We note that in the present formulation the existence of the extended thermodynamic functions follows simply from the probability ensemble

THE GAS-LIQUID INTERFACE NEAR THE CRITICAL POINT

545

by considering spatially homogeneous subregions. Thus we do not agree with Tolman 1°) and Widom 11) who regard the assumed existence as an unsatisfactory hypothesis. In basing ourselves on a probability ensemble we follow closely van Kampen's theory 12) of the van der Waals gas. It is worth remarking that the complete van der Waals loop in the pressurevolume isotherm has recently been observed in a computer experiment by Hansen and Verlet 13) in which the states of the fluid were constrained to be homogeneous.

3. The equilibrium interlace. The equilibrium interface is defined as the state of maximum probability of the probability distribution (2.4). Minimizing the exponent in (2.4) with respect to the functions {n(r), v(r), u(r)} one finds the conditions

½mvo(r)2 -

T ( ~at ~ ° \ ~-n-n /u

- - #0 + m~b(r) - - A V 2 n o = O,

(3.1 a)

mno(r) vo(r) ~- O,

(3.1b)

1 -

(3. lc)

r k~-u/.

= 0,

where we have used subscripts 0 to denote the equilibrium situation. The superscript 0 on the partial derivatives indicates that the derivative is to be evaluated at the local values {n0(r), uo(r)). The condition (3. lb) implies that the equilibrium velocity Vo vanishes. Condition (3. l c) states that the temperature is constant throughout the system and equal to that of the heat bath. Using these facts we may write the first condition

#*(no(r), T) + mq~(r) -- AV2no -----#o.

(3.2)

For the gravitational potential ~b(r)= gz the equilibrium has planar symmetry and (3.2) becomes

A ~

\ ~n ),¢ + mgz -- #o.

(3.3)

In the limit of vanishing potential (g ~ 0) this equation has a first integral

(clno ~

½ A \ dz ) + / ~ ° n ° -

%btCno, T)

=

P0,

(3.4)

where po(T) is a constant. The absolute minimum of the thermodynamic potential is reached when dno/dZ -~ 0 as z -+ ± oo. In that case Po is the pressure in the bulk gas or liquid phase. Eliminating/~0 from (3.4~ with the aid of (3.2) one m a y write (3.4) in the form

rl(d 0y

p*E~0(z), T] + A L 2 \ dz / -- n°--dYz2 J : P0,

(3.5)

546

B.U.

FELDERHOF

where p*

n~* -

¢*.

(3.6)

At the m i n i m u m the c o n t r i b u t i o n of the interface to the t h e r m o d y n a m i c p o t e n t i a l (2.5) per unit area of cross section is given b y co

=

p*E,~0(z),"1"]+ .A ~ c , , | - > - I

po-

(

JL

\,z

/

+ (1 + co),,,,- T:=,--r/d2, (3.7) (re-

.it

oo

where we have s u b t r a c t e d the c o n t r i b u t i o n from the homogeneous bulk regions. Using (3.5} and integrating b y parts one finds for the surface tension oo

f(a 0

<

j\~T/ --oo

which is the expression derived b y van der Waals") and used also b y Fisk and W i d o m 4). In the van der Waals t h e o r y the free energy is analytic in the neighbourhood of the critical point. E x p a n d i n g to f o u r t h order in the deviations from the critical density one m a y write

#(~, T) ~ -p0(m) + ~0(T),, + b

~

2

- \ - - -2-- ] J '

(3.9) where b is a c o n s t a n t and where nt(7/') and rig(T) are the limiting values of the density in the bulk liquid and gas phase. In this case eq. (3.4) m a y be solved explicitly and one finds ~(n~ + rig) -- ~(ni -- rig) tanh(Kz/2),

no(Z)

(3.10)

where K e =- 2 b A - 1

(nl

--

/.tg) 2

= lira A - 1 (aa¢,/~n2) ~..

(3.11)

Inserting (3.10) into (3.8) and using the fact t h a t nl -- *ag -- 2a-~ (Te -- T)a,

(3.12)

where a is a c o n s t a n t and/5 = ½ in the van der Waals theory, one finds t h a t the surface tension behaves as ~

(Te -- T)<

with # = a.

(3.13)

THE GAS-LIQUID INTERFACE NEAR THE CRITICAL POINT

547

The more complicated density profile derived by Fisk and Widom 4) agrees better with light reflection experiments 5) on critical fluid mixtures than the classical profile (3.10). They have also derived a scaling relation for the exponent # in terms of other critical exponentsll). Experimentally 1.22 ~
L -~ 2K -- ~,

(4.1)

where K{n(r), v(r)} is the kinetic energy of macroscopic flow K{n(r), v(r)} -~ ½m ~ n(r) v(r) ~ dr

(4.2)

v

and where £2{n(r),v(r),T) is the thermodynamic potential (2.5) in which, in accordance with the assumption of isothermal motion, we have eliminated the internal energy density u in favour of the temperature T as given by (3. l c). Following a standard procedure 14) one derives by a transformation from Eulerian to Lagrangian variables the equations of motion of the fluid. In Euler form the equations read ~n

~-7- + V.(nv) = o,

mn

Dr)

Dt

--

T constant,

vp* -- mnVq~ + AnV(VZn),

(4.3)

where we have made use of the thermodynamic relation (3.6), and where D/Dt is the substantial derivative D/Dt =- ~]St + (v, V). We emphasize that the set of eqs. (4.3) is consistent with our definitions of internal energy and entropy density only in the limit of infinite specific heat. In sec. 7 we shall derive a consistent set of equations. The equilibrium situation studied in the preceding section is a time-

548

B. U. FELDERHOF

independent solution of the equations of motion. The static equilibrium must satisfy Vp*o + m n o V ~

-- A n o V ( V 2 n o )

(4.4)

- O,

which is identical with the equilibrium condition (3.2) on account of the thermodynamic relation

(OP*/O~*)r =

n.

(4.5)

The momentum balance equation in (4.3) may also be written Dv mn Dt

(4.6)

V ' P -- m n V q ) ,

where the pressure tensor P is found to be of tile form

I

[ ~n "~21 "

~2n

k

~x~ 0xy

where cl is an arbitrary dimensionless constant. The second line in (4.7) does not contribute to the divergence of the pressure tensor. For the planar equilibrium situation the off-diagonal components of the pressure tensor vanish identically whereas the diagonal components are given by P'~x = I",~,, -- pt(no(z), T) + ,4

[ (½ ~:- el) \

[ , f dn0 ~"

d:' o l

dz ] + Ctno dz 2 .], d=n0 -]

PL : pt(~o(~), T) + A U \ d~ ] -- ~o ~ y J .

(4.8)

These expressions were derived from the van der Waals theory of long range attraction b y Fuchs la) with c l ½. For the one-phase region an expression similar to (4.7) with A proportional to the density n and with Cl = --1 was derived by Fixmanl6). From (3.5) and (4.8) it follows that we may write the expression (3.7) for the surface tension in the form oo

= I

[P;'o

-- P.x(Z)] o dz.

(4.9)

--¢o

This suggestive expression was derived in general fashion by BakkerlV). 5. I s o t h e r m a l m o t i o n s n e a r e q u i l i b r i u m . For small motions in the neighbourhood of the static equilibrium solution one writes n(r, t) = no(r) + n l ( r , t),

(5.1)

THE GAS-LIQUID INTERFACE NEAR THE CRITICAL POINT

and linearizes the eqs. (4.3) to first order in equations of motion read

nl(r, t) and v(r, t). The linearized

--t- + V. (noV) -- O, mno ~v ~t --

549

(5.2a)

V {(~_n )i ,nl } -- mnlV¢ + A[nlV(V2no) + noV(V2nl)l. (5.2b)

For small deviations from a spatially homogeneous gas or liquid state in the one-phase region the last term in (5.2b) is just the Fixman correction termlS). Thus we have generalized the Fixman correction to spatially inhomogeneous situations and to states in the two-phase region. On making use of the equilibrium condition (4.4) one m a y rewrite (5.2b) in the form ~V mno

noV f l / ~ *[_~__/° ' ~ ) ~ _ _nl ~ + AnoV(V2nl), (no \ ~n /T )

~t

(5.3)

so that ~V

m ~--

V(~nl),

(5.4)

where ~- is a linear operator given by

1 ( ~pt ~0 o~" = ~n00\~n-n/T -- A V2.

(5.5)

We introduce the usual scalar product =

g*(r)/(r) dr

(5.6)

in the space of square integrable functions/(r). Evidently #- is hermitian in this scalar product. We note that its expectation value with respect to density variations nl(r), =

f [ 1 (~n )°Tn~(r) + a(Vnl)2]dr,

(5.7)

precisely equals twice the second-order contribution to the thermodynamic potential (2.5) for isothermal deviations from static equilibrium, on account of the identity

(02~t/~n2)~, = n -1 (~pt/~n)T.

(5.8)

From (5.4) it follows that we m a y look for solutions for which the motion

550

B.U. FELDERHOF

is curl free, so that the fluid displacements may be derived from a potential,

g = vT,, v--

~t -

vq

(5.9)

For this class of motions the equations become nl = --V. (n0VT),

(5.10a)

8`2T m ~t2 --

(5.10b)

~nl.

If nl(r, t) is regarded as a field density and mkO(r, t) as its conjugate momentum density these equations may be derived from the Hamiltonian H = ½(~, J~f~) + 1
(5.11)

where ~ is a linear operator defined by 9 U ~ -- --rnV . (noVgJ).

(5.12)

Hence ~ ( T , J U T ) : ½m S no(r)(VT)`2 dr

{5.13)

is just the kinetic energy K to second order in the displacements. From the equations of motion (5.10) it follows that for oscillatory motions in which nl and 7 t vary harmonically in time with frequency ~o, the frequency is given by al 2 =

(5.14)

( n l , "o~'¢41)/(~, ~tfl/J),

which m a y be regarded as a form of Rayleigh's principle 19). Expressing nl in terms of T by (5.10a) one may use it for a variational calculation of the frequencies of eigenmodes. Since the kinetic energy (5.13) is positive definite, it follows that the equilibrium is stable for the class of motions under consideration provided o~- is a positive definite operator. Slightly generalizing a calculation by van der Waals2) we write the density variation in the form nl : ~" Vno,

(5.15)

where ~(r) is an arbitrary vector field. Hence (nl, f f n l )

= -- I (~" Vq))(t]" Vno) dr

+ S (v.~)'2 (Vno)Z dr,

(5.16)

where we have made use of the equilibrium condition (4.4). The second term in (5.16) is always positive. For the gravitational potential q~ -~ gz and planar equilibrium the first term equals --

f

(rl. Vq~)(rl. Vno) d r :

--g

faro

r12~-~Z-Z dr,

(5.17)

THE GAS-LIQUID INTERFACE NEAR THE CRITICAL POINT

,551

which is always positive provided one considers a monotonically decreasing solution no(z) of (3.3).

6. Sur/ace waves. We consider the gravitational potential qb _--gz and the planar equilibrium no(z) corresponding to the absolute minimum of the thermodynamic potential. We shall study the motions of the interface in the limit g --> 0. We also neglect boundary effects. Thus we consider a solution no(z) of (3.4) centered about z = 0 which tends to the bulk gas density rig(T) as z -~ + oo and to the bulk liquid density n~(T) as z -+ --oo. Small isothermal motions of the fluid are described by the linearized equations derived in the previous section. On account of the translational invariance in the lateral directions we m a y look for solutions of (5. I0) with (x, y) dependence given by a plane wave factor expEi(kxx + kvy)J. Assuming also harmonic variation in time one finds that the z dependence of such solutions follows from the equations

(iz

A

no~

dz~

--k2noT:--nl,

k 2 nl -- no \ ~n ]T nl = --mo2~/j,

(6.1)

where k ~ = k~2 + kv2. A qualitative insight into the types of solution we may expect is obtained from a comparison with macroscopic hydrodynamics. If the interface is treated as an infinitesimally thin boundary with surface tension ~, and the gas and liquid on either side are treated as uniform fluids we can easily find the eigenmodes and eigenfrequencies. There are two types of solution: The first type represents sound waves being reflected and partly transmitted by the interface. Corresponding to the different angles of incidence for fixed wavevector k = (kz, ky) along the interface there is a continuous spectrum of eigenfrequencies in the range

m-l(~p/~n)~, k 2 < co2 < oo,

(6.2)

where (~p/~n)aT is the value of the derivative on the gas side of the coexistence curve. The second type of solution has the character of a bound state. The z dependence of nl and T is given by decaying exponentials

nl(k, z) = n~: expE--q. Izll, kV(k, z) = T~ exp[--q~ IzL~.

(6.3)

with q± real, rather than by oscillating functions (on at least the gas side) as in the case of sound waves. In the limit of long wavelengths (k -+ 0) the

552

B. U. FELDERHOF

frequency of this surface wave is given by the dispersion equation ~o~ --

O(

m(ng + nl)

k3[l + O(k)].

(6.4)

One also has q± = k[1 + •(k)], ~

= :i:: (¢/k)[~ +

(6.5)

e(k)j,

where ~ is a constant amplitude factor. Hence in the limit k ~ 0 the motion in the z direction is a simple uniform displacement. The equations of motion (6.1) cannot be solved exactly but we can show that in the long-wavelength limit (k ~ 0) the asymptotic behaviour for [zl --->co and the frequency of the surface waves agree with the macroscopic treatment. Moreover we show that the value of ~ is given by the expression (3.8) derived from the static equilibrium calculation. The spectrum of sound waves again lies in the range (6.2). For the approximate solution of eqs. (6.1) it is useful to study the linear operator o~ defined in (5.5) in greater detail. For simplicity we shall assume that the interface is symmetric in the sense that the limiting values of nol(~pt/~n) °, as z -> :c co are equal, as it is the case in the van der Waals theory 2) and in the Fisk-Widom theory4). We shall denote the limiting value by AK 2, as in (3.1 1), so that we may write

no \ ~n ,&, =- .4[,:2 + V(z)l,

(6.6)

where

V(z) ~-~ exp[--K [zi]

as

Iz] -~ co

(6.7)

as follows from (3.4). The eigenvalue equation for the hernfitian operator in the subspace of functions with plane wave dependence along the interface reads explicitly A I -- ddz--~2-~ 4- (k'* 4- K") q~v 4- V (z) c?~1 ~- ~.vc?v.

(6.8)

This equation has the form of the SchrSdinger equation for a particle moving in the potential V(z). From the equilibrium condition (4.4) it follows that we m a y also write

dano/dz 'a - K" + v(z). dno/dz

(6.9)

Hence for the classical profile (3.10) the potential V(z) is given by

V(z) = --~K 2 [cosh(Kz/2)1-2.

(6.10)

THE GAS-LIQUID INTERFACE NEAR THE CRITICAL POINT

553

More generally the potential will have the same qualitative features. For the classical profile the eigenvalue equation (6.8) m a y be solved exactly, as has been noted b y Langer2°). There are two bound states with eigenvalues

~o ~ A k z,

~1 = A(~K 2 + k 2)

(6.11)

and corresponding eigenfunctions ~0o(Z) = (3,
(6.12)

Furthermore there is a continuum of scattering states with eigenvalues in the range A(~¢ 2 + k 2) < ;t < oo.

(6.13)

For a general interface no(z) only the ground state can be found exactly. Its eigenvalue and eigenfunction are given by

~o : A k 2,

99o(Z) -= (A/a)½ dno/dz,

(6.14)

where ~ is defined in (3.8). There is again a continuous spectrum of eigenvalues in the range (6.13). In between the eigenvalue 40 and the continuous spectrum there m a y be one or more discrete states. The physical significance of the ground state (6.14) is clear. It corresponds to a simple displacement of the interface in the z direction, with an amplitude modulated along the interface with wavevector k = (kx, kv). The second bound state in (6.12) corresponds to squeezing of the interface. Returning to eqs. (6.1) we now seek from the class of admissible solutions the one with minimum value of ~o2. We m a y anticipate that when the corresponding density variation is expanded in terms of the orthonormal set of eigenfunctions of ,~, the ground state (6.14) is represented with nonvanishing amplitude. Thus we may write

nl(k, z) ~- --~[dno/dz + v(k, z)],

(6.15)

where ~ is a constant, and where v(k, z) is a linear combination of the higher eigenstates. Hence v(k, z) is orthogonal to the ground state oo

(6.16)

(dnoldz) v(k, z) dz -----0. --co

Eqs. (6.1) may now be written d2v dz 2 d dz

(k 2 + K2 + V(z)) v

k2 . [- dno

dn0 dz

moo2 T, A~

(6.17a) (6.17b)

554

B . U . FELDERHOF

Multiplying the first e q u a t i o n b y d n o / d z and integrating over all z we have oo

J\

oo

dz ]

--

A~ J

--oo

dz

dz.

(6.13)

--c~

Multiplying (6.17b) b y T ( k , z) and integrating co

¢o

"°L\ dz /

dz=:(Tdn0d J dz

--oo

oo

+:

--oo

T, dz.

(6.19)

--~

If for large Jz[ we assume asymptotic behaviour of the form (6.3) we find t h a t the a s y m p t o t i c rate of d e c a y is given b y - - k2)(q~ - - k 2 - - K2)rig, l = m e , 2.

A(q~

H e n c e we m a y look for a solution for which , ; ' -÷ 0 as k q2± = k 2 + ¢(02).

(6.20) .~ 0 with (6.21)

We shall anticipate t h a t ¢o2/k 2 = ~0(k) and show t h a t this leads to a consistent solution. If we split the real z axis into three regions z < --d, - - d < z < d, and d < z we m a y conclude from (6.21) t h a t for the integral on the left of (6.19) and for the second one on the right the c o n t r i b u t i o n s from the two a s y m p t o t i c regions relative to t h a t of the center region are of order k -1. F r o m (6.17a) and (6.21) it follows t h a t --K2v ~-, (m~o2/A~) T

as

Jz] ~ 0% k > 0.

(6.22)

H e n c e the second t e r m on the left in (6.19) and the second term on the right are of relative order (9(co2/k2) = (~(k). Similarly, if we i n t e g r a t e (6.17b) over all z we have oo

- - k 2 ~ n o t dz = $(ng -- nl)[1 + (¢(k)],

(6.23)

--oo

where we have again used (6.22). Combining this with the fact t h a t in the limit of long wavelengths the m o t i o n in the z direction must be a uniform displacement, we conclude from (6.21) and (6.23) T ± = ~ (~/k)[1 + ~(k)~.

(6.24)

F r o m (6.19) we finally have oo

f T dn0 J dz dz ~ --oo

- - ~ ' ( n g -[- ?$1) k-:l [1 + ~(k)l ,

(6.25)

THE GAS-LIQUID INTERFACE NEAR THE CRITICAL POINT

555

which in combination with (6.18) yields Oo

02

Aka f(dno'~ z dz[1 + g)(k)]. "m(n, + nil J \-dz-z]

=

(6.26)

To leading order in k this is precisely the dispersion equation for surface waves (6.4) with surface tension ~ given by (3.8). It is easily seen that the continuous spectrum corresponding to reflected sound waves is still given by (6.2). Only the value of the reflection coefficient is determined b y the detailed nature of the interface. From Rayleigh's principle (5.14) it follows that the second bound state found in (6.12) does not give rise to a mode with vanishing frequency in the limit k -> 0. It m a y make itself felt as a resonance in the reflection coefficient of sound waves.

7. Adiabatic motions. As we have remarked in sec. 4, when dissipation is neglected the motion of the fluid is adiabatic rather than isothermal. In this section we shall derive the correct equations of motion and in the next section we shall show that the effect on the dispersion equation of surface waves is negligible to lowest order in the wavenumber. For adiabatic motion we again postulate (4.1) as the Lagrangian of the fluid with the additional constraint that during the motion of each fluid element the entropy per particle st ---- at/n remains constant. Thus in (2.5) we eliminate the internal energy density u in favour of st b y means of the thermodynamic definition u ---- u(n, st). The equations of motion m a y again be derived from the Lagrangian formalism. In Euler variables the equations read ~n

--

+

~t

V" ( n v ) =

Dv

mn . . . . Dt DsT Dt

O,

Vp* -- mnVqb + AnV(V2n),

-- O,

(7.1a) (7.1b)

(7. lc)

where we have used the thermodynamic relation

The static equilibrium is again a solution of these equations. The momentum balance in equilibrium is still given b y (4.4). From (3. Ic) it follows that in equilibrium the temperature T is constant. This determines the solution

556

B.U. FELDERHOF

no(r) of (4.4). The density profile no(r) and the constant temperature T together determine the equilibrium entropy profile sto(r). In order to show that the equations of motion (7.1) are consistent with our definition (2.7) of the internal energy we first derive the equation describing conservation of energy. Multiplying (7.1b) by v we obtain the balance equation for the kinetic energy of macroscopic flow Dv 2 ½ran Dt --

v. Vp+ -- m n v . Vq) + A n v .

V(V2n).

(7.3)

For the gravitational energy we have Dq) mn Dt -- m n v . Vq).

(7.4)

Furthermore one derives ~-½(Vn) 2 = - - n v . V(Ven) + V.(nvV2n) -- V . [ ( V . n v ) Vn],

(7.5a)

--nV2n~t

(7.5b)

~t

(Vn) 2 - - V . [ n V ( V . n v )

+ (V.nv) Vn],

where we have made use of the continuity equation (7.1a) and of the identities V2(/g) = / V 2 g + 2V/. Vg + gV'2/,

(7.6) lV2g _

gv21 =

v.(IVg

-

gVl)

as applied to n and V.nv. Finally we postulate that the internal energy density u(r, t) satisfies the balance equation ~u -~-t + V.(uv) = - - p t V . v . (7.7) From (7.3)-(7.5) and (7.7) it follows that the energy density e(r, t), given by the sum (2.7), satisfies the conservation equation ~e }~ + V "je ---- O,

(7.8)

with the energy-current density given by je = (u + ~mnv~ + mnqb + p* -- AnV2n) v + ½A[(l + co)(V.nv) Vn -- (1 -- Co) nV(V.nv)J.

(7.9)

Hence the postulated balance equation for internal energy (7.7) is consistent with conservation of energy. Eq. (7.7) m a y also be written in the form D(u/n) ---Dr - - -

pt Dn n 2 Dt '

(7.10)

T H E G A S - L I Q U I D I N T E R F A C E NEAR T H E CRITICAL P O I N T

557

where we have used the continuity equation (7. l a). Comparing (7.10) with the thermodynamic relation T dst -----d(u/n) + pt d(n-1),

(7.11)

we conclude that (7. lc) is valid. Hence we have shown that the equations of motion (7.1) are a consistent set. 8. Adiabatic mo'tions near equilibrium. In this section we show that to leading order in the wavenumber the dispersion equation of surface waves is not affected b y the fact that the motion is adiabatic rather than isothermal. The spectrum of sound waves does of course change due to the change in sound velocity in the bulk gas or liquid. For small motions in the neighbourhood of static equilibrium we write

n(r, t) = no(r) + nl(r, t),

(8.1)

st(r, t) -----s~(r) + s~(r, t),

and linearize the equations of motion (7.1) in {hi, v, s*l}. The linearized equations read 8---t- + V. (nov) = 0,

mno 8l --

V

(8.2a)

nl +

S*l -- m n l V ¢

+ A[noV(VZnl) + nlV(i7Zno)l,

(8.2b)

+ v. VS*o =

(8.2c)

o.

Introducing the fluid displacements ~(r, t) b y v = ~ / S t one may integrate (8.2a) and (8.2c) over time and write nl ---- -- V. (no~),

(8.3a)

s*~ =

(8.3b)

--~. Vs*o.

Using (8.3b) and the equilibrium condition (4.4) one m a y rewrite (8.2b) in the form 8z~ -mno 8t z

noV(#~'nl) + V

°

1 noV'~

L ( \ ~ - n ] ~ -- \ - ~ n ]T )

]

.

(8.4)

This is of the usual Rayleigh form i8) mno 8t z --

~s~,

(8.5)

B.U.

558

FELDERHOF

where A s is a linear operator, which m a y be written

(8.6)

A s = A T + #o,

where A T is defined by the first term on the right-hand side of (8.4) and is the Rayleigh operator for isothermal motion. The three operators As, A T and <9° are hermitian in the scalar product <~',

A~> = Y, I ~* (~tS~Js) dr.

(8.7)

i,j

The frequencies m of the normal mode solutions to (8.5) are given by the ratio ~2 =

<~, ~8~>l
~>.

(8.8)

This is Rayleigh's principle and it may be used for a variational calculation of the eigenfrequencies. The denominator is twice the second-order kinetic energy and is identical to (5.13) for curl-free motions. From (8.4) and (8.6) it follows that <#, ~ g >

= <~, ~ T ~ > + <~, se~>

tk e~ ). -\-~n-/,rJ On using the fact that the static equilibrium is isothermal so that - I--)

b wo = -

Vs~

(8. ~o)

1 one m a y show that ~<~, As~> precisely equals the second-order contribution to the thermodynamic potential (2.5) for deviations from equilibrium {nl(r), o, sl(r)} for which sl(r) is given by (8.3b). From the thermodynamic identity

,-

~

T =-~-\~r-J,dt, Vf-/,,'

(8.11)

it follows that the sign of the integrand in (8.9) is determined by the sign of the specific heat at constant volume. In the one-phase region the positivity of the specific heat follows from thermodynamic stability. For the extended definition of the thermodynamic functions in the two-phase region one must relax the requirement of thermodynamic stability. Inside the spinodal curve the compressibility is in fact negative. From (8.8) it follows, though, that the specific heat must be positive throughout the two-phase region since otherwise the right-hand side of (8.8) could be made negative and the static equilibrium would be dynamically unstable. In the van der Waals theory the positivity of the specific heat is implied. More generally

THE GAS-LIQUID INTERFACE NEAR THE CRITICAL POINT

559

it is an additional condition on the nature of the extended thermodynamic functions in the two-phase region. On the assumption that the specific heat at constant volume is everywhere positive the quadratic form is always positive (or at least nonnegative) so that St is a positive definite operator. Multiplying this operator in (8.6) b y a parameter , between 0 and 1 we can go continuously from isothermal to adiabatic motions. From the positive-definiteness of 5¢ it follows that ~o~ for an eigenmode of ~8 is larger than or at most equal to the value of ~o2 for the corresponding eigenmode of AT. As a trial solution for the calculation of the frequency of surface waves for adiabatic motion we can use the solution ST for isothermal motion. From the variational nature of Rayleigh's principle it follows that the actual increase of the frequency is bounded by

Ao~2 ~ /KmnoS~, ST>.

(8.12)

Using the estimates of sec. 6 on the right-hand side we conclude that in the limit of long wavelengths Ao)2 = d~(ka).

(8.13)

Hence to leading order the fact that the motion is adiabatic does not influence the dispersion equation (6.26).

9. Discussion. We have derived hydrodynamic equations valid for fluid states in the two-phase region and have demonstrated their usefulness. The equilibrium liquid-gas interface of the van der Waals-Fisk-Widom theory appears as a static solution of these equations. The limits of validity are essentially those inherent to the square-gradient approximation. Thus close to the critical point and on a sufficiently coarse-grained scale of description we m a y expect the equations to be valid. We have completely neglected dissipative effects and it would be of interest to take these into account. It would also be useful to extend the theory to binary fluid mixtures, in which case experiments are easier. The present theory also allows one to give a unified treatment of light scattering and light reflection. Any density fluctuation m a y be uniquely decomposed into an orthonormal set of normal modes which are statistically independent. We hope to return to these matters in future work.

REFERENCES

1) Bakker, G., KapillaritA* und OberflAchenspannung, Handbuch der Experimental° physik, Vol. VI (Akademische Veflagsgesellschaft, Leipzig, 1928). 2) Van der Waals, J. D., Z. physik. Chemie 13 (1894) 657. 3) ]31och, F., Z. Phys. 74 (1932) 295;

560

4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)

T H E G A S - L I Q U I D I N T E R F A C E N E A R T H E CRITICAL P O I N T Ginzburg, V. L. and Landau, L. D., Soviet P h y s i c s - J E T P 20 (1950) 1064; Cahn, J. W. and Hilliard, J. E., J. chem. Phys. 28 (1958) 258. Fisk, S. and Widom, B., J. chem. Phys. 50 (1969) 3219. Huang, J. S. and Webb, W. W., J. chem. Phys. 50 (1969) 3677. Landau, L. D. and Lifshitz, E. M., Statistical Physics, Pergamon Press (London, 1958) p. 458. Buff, F. P., Lovett, R. A. and Stillinger, F. H., Phys. Rev. Letters 15 (1965) 621. Huang, J. S. and Webb, W. W., Phys. R e v . Letters 23 (1969) 160. ; Bouchiat, M. A. and Meunier, J., Phys. R e v . Letters 23 (1969) 752. De Groot, S. R. and Mazur, P., Non-Equilibrium Thermodynamics, NorthHolland Pub1. Cy. (Amsterdam, 1962) Chap. II. Tolman, R. C., J. chem. Phys. 17 (1949) 118. Widom, B., J. chem. Phys. 43 (1965) 3892. Van Kampen, N. G., Phys. Rev. 135 (1964) 362. Hansen, J.-P. and Verlet, L., Phys. Rev. 184 (1969) 151. See e.g. Van Kampen, N. G. and Felderhof, B. U., Theoretical Methods in Plasma Physics, North-Holland Publ. Cy. (Amsterdam, 1967) Chap. VII. Fuchs, K., Wiener Sitzungsber. Nov. 1889, p. 1; see also ref. 1, p. 378. Fixman, M., J. chem. Phys. 47 (1967) 2808. Bakker, G., Z. physik. Chemie 33 (1900) 482. Fixman, M., J. chem. Phys. 33 (1960) 1357; Felderhof, B. U., J. chem. Phys. 44 (1966) 602. Lord Rayleigh, The Theory of Sound, 2nd. ed., Macmillan (London, 1929) Vol. I, Chap. IV. Langer, J. S., Ann. Physics 41 (1967) 108.