DYNAMICS OF THE DIFFUSE GAS-LIQUID ... - Science Direct

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


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,