J. Phys. Chem. 1984,88, 6479-6484 where
w
E
(-l)"n'ar+l a,d+l Q1/2 [(2n
+ 1)(2n' + 1)(2L + 1 ) ] - l l 2 ('414)
and Q E
(n
( L + M ) ! ( L- M)! + m)!(n- m)!(n+ m')!(n' - m')!
(-415)
the hard-core surface of radius 2a. The two surface integrals have the same magnitude but opposite sign, and A . = 0, A = 0. The symmetry argument that A . 0: 1 and A . = 0 because tr A . = 0 seems a bit weaker. The integrand in (5.18) is short ranged: R13 is restricted to small values by the 1 - g(2)factor, and if R 1 2grows large, the integrand decays as 1 / R 1 2 6 .We therefore impose a cutoff R,, of arbitrarily large magnitude on R I 2and RZ3.The d2)Kfactors are given a Fourier integral representation
In eq A13 L and M a r e not summation variables; they have the values L=n+n'
M=m+m'
6479
g(2)(R)K(R) =
1 G exp(ik.R)
dk
(B3)
where
A fair number of bookkeeping rearrangements revolving around the relations between the real and imaginary parts of A(i,n,m), and the behavior of A(i,n,m) under the transformation m -m, are advantageous in programming. However, the analysis seems too lengthy and pedestrian to record here.
and
Appendix B A few details concerning the evaluation of the integrals A and B (eq 5.17 and 5.18) are given here. The integrals over R2 and R3 in eq 5.17 can be done in sequence
j l ( x ) is a spherical Bessel function. Substitution of this representation into eq 5.18 gives, in the limit R,, m
-
G = (3ekek - l ) f k
(B4)
-
B = ( ~ ~ / T ) ~ ~ X - ~dx~ ~ ( X ) ] (B6) '
A = 1 g ( 2 ) ( 1 2 tr ) K(12)-A0(2)d2
(B1)
Ao(2) = 1 g ' 2 ) ( 2 3 )K(23) d3
(B2)
McQuarrie states the value of the integral to be 51r/192 and ascribes the result to Katsura.12 A numerical integration was thought to provide the quickest verification, and the value was confirmed. The corresponding value of B is E = 5/16.
For the problem at hand g@)(R)is unity if R > 2a and 0 if R < 2a. The integral in (B2) can therefore be converted to a surface integral over the spherical surface enclosing the system and over
(12) McQuarrie, D. A. "Statistical Mechanics"; Harper and Row: New Yofk, 1976; Chapter 12. The integral is ascribed to: Katsura, S. Phys. Reu. 1959, 115, 1417.
where
Gravity Effects on Critical Fluctuations in Gases J. V. Sengers* Institute f o r Physical Science and Technology, University of Maryland, College Park, Maryland 20742
and J. M. J. van Leeuwen Laboratorium voor Technische Natuurkunde, Technische Hogeschool Delft, 2600 GA Delft, The Netherlands (Received: May 23, 1984; In Final Form: July 30, 1984)
The theory of Ornstein and Zernike for critical fluctuations in gases can be extended to include the effects of gravity. In the presence of a gravitational field the order parameter correlation function becomes anisotropic and the correlation length remains finite in all directions. Quantitative estimates of the predicted effects are discussed.
1. Introduction
A system near a critical point exhibits large fluctuations in the order parameter associated with the critical-point phase transition. For a fluid near the gas-liquid critical point the order parameter is the density; for a binary liquid near the critical point of mixing the order parameter is the concentration. The critical fluctuations in turn cause the medium to scatter electromagnetic radiation rather strongly, a phenomenon known as critical opalescence. Professor Debye was very much interested in this phenomenon, and he and his collaborators published many research papers on the subject.'-I2 (1) Debye, P. J. Chem. Phys. 1959, 31, 680. (2) Debye, P. Makromol. Chem. 1960, 35, 1. (3) Debye, P.; Coll, H.; Woermann, D. J . Chem. Phys. 1960, 32, 939. (4) Debye, P.; Woermann, D.; Chu, B. J. Chem. Phys. 1962,36, 851; J . Polymn. Sei., Part A 1963, 1 , 255.
0022-3654/84/2088-6479$01.50/0
The classical theory of critical fluctuations was developed by Ornstein and Zernike.13J4 The theory was further developed by Debye, in particular with the aim of interpreting criticalopalescence experiment^.'*^^^^^^ With the subsequent refinement (5) Debye, P.; Chu, B.; Woermann, D. J . Chem. Phys. 1962, 36, 1803. (6) Debye, P.; Chu, B.; Kaufmann, H. J . Chem. Phys. 1962, 36, 3378. (7) Debye, P. In "Electromagnetic Scattering"; Kerker, M., Ed.; Macmillan: New York, 1963; p 393. (8) Debye, P.; Caulfield, D.; Basham, J. J . Chem. Phys. 1964, 41, 3051. (9) Debye, P.; Bashaw, J.; Chu, B.; Tan Creti, D. M. J. Chem. Phys. 1966, 44, 4302. (10) Debye, P. Pure Appl. Cheni. 1966, 12, 23. (1 1) Debye, P.; Gravatt, C. C.; Leda, M. J . Chem. Phys. 1967,46,2352. (12) Balta, Y.; Gravatt, C. C. J . Chem. Phys. 1968, 48, 3839. (1 3) Zernike, F.Proc. Acad. Sci. Amsterdam 1919, 17, 793; Arch. NZerl. Sci. Exactes Nat., Ser. 3A 1917, 4 , 74. (14) Ornstein, L. S.; Zernike, F. Phys. Z . 1918, 19, 134; 1926, 27, 761.
0 1984 American Chemical Society
6480 The Journal of Physical Chemistry, Vol. 88, No. 26, 1984
of experimental techniques light-scattering experiments have yielded a considerable amount of detailed quantitative information about critical phenomena in fluids as e.g. reviewed by Chu,lS Beysens,I6 Sengers,I7 and Goldburg.I8 In applying the theory of critical phenomena, one usually treats the fluid as a spatially homogeneous system in thermodynamic equilibrium. Near the critical point the properties of a fluid become very sensitive to small changes in the ordering field which for fluids is the chemical potential. Since a gravitational force couples with the ordering field, fluids become in practice strongly inhomogeneous near a critical point. Gravity induces density gradients in fluids near the gas-liquid critical and concentration gradients in fluid mixtures near the critical mixingg point.21 Nevertheless, in the interpretation of light-scattering experiments it is assumed that the properties of the fluid can be identified with those of a spatially homogeneous fluid with a density corresponding to that of the location of the scattering volume. This assumption will cease to be valid very close to the critical point. In view of the increased sophistication in attaining accurate temperature control, it becomes of interest to investigate the limitations of the locally homogeneous assumption on the actual behavior of fluids in a gravitational field close to the critical point. The experimental work of Debye and co-workers was concerned with critical opalescence in binary liquids near the critical point. However, in evaluating the effects of gravity, we focus here our attention on one-component fluids near the gas-liquid critical point. First, much more detailed information concerning the behavior of the thermodynamic properties near the critical point is available for fluids near the gas-liquid point. Second, and more importantly, in liquid mixtures one usually encounters very long relaxation times for approaching true thermodynamic equilibrium and most experiments near the critical mixing point are performed in a time scale during which the gravity-induced gradients have not yet fully developed.21 We shall proceed as follows. In section 2 we review the scaling laws that characterize the temperature and density dependence of spatially homogeneous fluids near the critical point. In section 3 we discuss the density profiles induced by gravity in fluids near the critical point. In section 4 we examine the influence of this inhomogeneity on the order parameter correlation function and, in particular, the effects of gravity on the correlation length. In section 5 we elucidate the magnitude of the predicted gravity effects using xenon as an example. In section 6 we conclude the paper with a discussion of some consequences for critical-phenomena experiments in gases. 2. Critical Power Laws and Scaling Laws Let p be the density, T the temperature, P the pressure, p the chemical potential, and x 3 ( a p / a p ) , the response function which is related to the isothermal compressibility KT ~ - ‘ ( a p / a Pby) ~ x = pZKT.The order parameter is Ap p - p c and the ordering field AM3 p f p , 7 ‘ ) - p(pc,T),where pc is the critical density. We prefer to make these quantities dimensionless with the aid of the critical parmaeters p c . T,, and P,. Specifically we define T - .Tc P - Pc Ap* = -, A T * = Pc TC
(15) Chu, B. Ber. Bunsenges. Phys. Chem. 1972, 76, 202. (16) Beysens, D. In “Phase Transitions: CargEse 1980”; Levy, M.,Le Guillou, J. C., Zinn-Justin, J., Eds.; Plenum Press: New York, 1982; p 25. (17) Sengers, J. V. In “Phase Transitions: CargEse 1980”; Levy, M., Le Guillou, J. C., Eds.; Plenum Press: New York, 1982, p 95. (18). Goldburg, W. I. In “Light Scattering Near Phase Transitions”;
Cummms, H. Z., Levanyuk, A. P., Eds.; North-Holland Publishing Co.: Amsterdam, 1983; p 531. (19) Hohenberg, P. C.; Barmatz, M. Phys. Rev. A 1972,6, 289. (20) Moldover, M. R.; Sengers, J. V.; Gammon, R. W.; Hocken, R. J. Rev. Mod. Phys. 1979, 51, 19. (21) Greer, S.C.; Block, T. E.; Knobler, C. M. Phys. Rev. Lett. 1975,34, 250.
Sengers and van Leeuwen At the coexistence boundary the order parameter varies with the temperature as Ap* = &BlAPl@,at the critical temperature the ordering field varies with density as Ap* = *DlAp*I6, and at the critical density above T , the response function diverges as x* = I’lAT*lr, where y = p(6 - 1). At arbitrary densities and temperatures near the critical point the ordering field AM*and the response function x* satisfy scaling laws of the form22-24
A ~ = * A D I A ~ * ~ ~ ( ~ )
(2.2)
x*ll = D IAp*lr/@X(u )
(2.3)
where u = A.T*/x0(Ap*I’/fl with xo = E’/@. The scaling functions h(u) and X ( u ) are related by X ( u ) = 6h(u) - F l u dh(u)/du. The intensity of scattered light is proportional to the structure factor x ( k ) which is the Fourier transform of the order parameter correlation function. According to the Ornstein-Zernike theory for a spatially homogeneous system, this structure factor has the form X
x(k) = where k is the wavenumber and 5 the correlation length.25 In reality,2s (2.4) is only strictly valid for k5 > 1, x ( k ) becomes proportional to Ilkz-. The exponent TJ is small, and we neglect in this paper the small deviations from the OrnsteinZernike form of the correlation function. At the critical density above Tc the correlation length diverges as ,$ = folAT*l-”,where u = p(6 + 1)/3. At arbitrary densities near the critical point 5 satisfies a scaling law analogous to (2.2) and (2.3) for the thermodynamic properties
5 = [O~A.P*I-”@Z(U) (2.5) The critical exponents 0, y, 6, u and the scaling functions h(u), X ( u ) , Z ( u ) are the same for all systems within a universality class. Fluids near the critical point belong to the universality class of 3-dimensional Ising-like systemsnZ6The critical exponent^^'*^* and the scaling functionsz9 are known for this universality class with considerable accuracy. The amplitudes xo and D are system-dependent quantities; the amplitudes r and lo are related to xo and D by the universal amplitude relations3s32
where kB is Boltzma‘nn’s constant and where Rr N 1.I4 and Rt 0.83. For further details the reader is referred to reviews in the literature.17,20~24*33
3. Gravity-Induced Density Profiles A theory of the thermodynamics and structure of nonuniform systems was introduced by van der Waals in which the effect of the inhomogeneity is taken into account by a term proportional to the square of the density gradient in the expression for the local free energy.34 The squared-gradient theory leads to a differential (22) Widom, B. J. Chem. Phys. 1965, 43, 3898. (23) Griffiths, R. B. Phys. Rev. 1967, 158, 176. (24) Sengers, J. V.; Levelt Sengers, J. M. H. In “Progress in Liquid Physics”; Croxton, C. A., Ed.; Wiley: New York, 1978; p 103. (25) Fisher, M. E. J . Math. Phys. 1964, 5 , 944. (26) Levelt Sengers, J. M. H.; Hocken, R.; Sengers, J. V. Phys. Today 1977, 30 (12), 42. (27) Le Guillou, J. C.; Zinn-Justin, J. Phys. Rev. B: Condens. Matter 1980, 21, 3976. (28) Albert, D. 2. Phys. Rev. B Condens. Matter 1982, 25, 4810. (29) Wallace, D. J.; Zia, R. K. P. J . Phys. C 1978, 7, 3480. (30) Aharony, A.; Hohenberg, P. C. Phys. Rev. E Solid State 1976,13, , 3081. (31) Hohenberg, P. C.; Aharony, A,; Halperin, B. I.; Siggia, E. D. Phys. Rev. B: Solid State 1976, 13, 2986. (32) Bervillier, C. Phys. Rev. B Solid State 1976, 14, 4964. (33) Greer, S . C.; Moldover, M. R. Annu. Rev. Phys. Chem. 1981,32,233.
Gravity Effects on Critical Fluctuations in Gases
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 6481 z
equation for the density gradient of the form
2
+2
+4
-4
-2
+2
0
14
(3.1) where z is the height, taken to increase in the direction opposite to the gravitational field, and g the gravitational acceleration constant. The coefficient A is to be identified with
+0.5
t0.5
-0.5
-0.5
la‘
0 -10
- 1.5
The quantities p(p(z)), t(p(z)), and x(p(z)) are those of a spatially homogeneous system with uniform density p = p(z) at the given temperature T . Without the gravitational term gz in (3.1), the theory was used by van der Waals, and later by Cahn and Hilliard,35to determine the density profile of the interface below the critical temperature, in conjunction with a mean-field equation of state. Subsequently, Fisk and W i d ~ mincorporated ~ ~ the nonclassical scaling laws in applying the equation. With the gravitational term the differential equation can also be used to determine the gravity-induced profiles in the one-phase region. We identify the reference level z = 0 with the level where the density equals the critical density pc. In terms of the dimensionless quantities introduced in (2.1), eq 3.1 can be written as
t2 d2Ap* - Ap* + -2g*
HO
X* dz2
+05 -0 5
-I 0
-I 0
-I 5
-I 5
-
AT=5.0
Here g* = g/go,where go = 9.8 1 m/s2 corresponds to the earth’s gravitational field, and Ho is a gravity scale height defined as Ho = Pc/p,go. The ordering field Ap* is an antisymmetric function of the density Ap*. As a consequence, the density profile Ap*(z) will be an antisymmetric function of the height z , and it is sufficient to consider the differential equation for positive z only. Substitution of the scaling laws (2.2), (2.3), and (2.5) in (3.3) yields
+ I .O
]+0.5 +0.5
la‘
1%
0
- 0.5 - 1.0
-0.5 - 1.0
4.5-
%+?yT?$-15
-4
z
(3.3)
la“
0 -05
z
Figure 1. Scaled density difference
as a function of the scaled height 2 a t various values of the scaled temperature difference
I
-4l
I
- 2l
/
2 0I
I
/
0.31
,
003 -
, , , ,
,
/
-2 l
-4 I l
i 0I l+ 2I +4 I I
-
l
l
-
I
4 1.0
AT=0.3
\
0.03
,
+ 2I / c 4I
z.
,
,
1
1
1
1
41,
1]003
I
where
G(u) =
P ( 0 ) X(0)
P ( u ) X(U)
(3.5)
1,
, , , , ,
,
,
,
,
, ,
, ,
4003
and
(3.6) It is possible to eliminate the explicit dependence on the strength of the gravitational field by rescaling the density, temperature, and height as Ap* =
AG,
AT* = TAT, z =
(3.7)
with scale factors A = Ao8+/2g*8+,
=
xdo4/2g*+,
{ = DH&08s6/2g*-”+ (3.8)
and
4=-
1
ps
+v
(3.9)
The differential equation (3.4) then reduces to d2Ap - - [lApI8h(u) - 2]G(u)lZlqY/@ di2
(3.10)
(34)van der Waals, J. D.; Kohnstamm, Ph. “Lehrbuch der Thermodynamik”; Maas en van Suchtelen: Amsterdam, 1908;Vol. I. (35) Cahn, J. W.; Hilliard, J. E. J . Chem. Phys. 1958, 28, 258. (36) Fisk, S.;Widom, B. J. Chem. Phys. 1969, 50, 3219.
0031, -4
, , , , , , -2
0 I
+2
,
11,
+4
,
-4 - 2
, , , , 0
,
4003
+2 r 4
z
Figure 2. Scaled density gradient I d p / d t l as a function of the scaled - height i at various values of the scaled temperature difference AT.
- _
with u = AT/lAp(’/P. The critical exponents and the scaling functions are universal. Hence, the scaled density 4p is a universal function of the scaled height z and the scaled temperature A solution of the universal differential equation (3.10) for the density profile was presented in a previous p~blication.~~ In Figure 1 we show & as a function z for a few representative temperatures AT; in Figure 2 we show the corresponding profiles for the density gradient l d z / d z l . In the traditional literature the gravity-induced density profiles are usually calculated in the locally homogeneous approxima-
E.
(37)Sengers, J. V.;van Leeuwen, J. M. J. Physicu A (Amsterdam) 1982, 116A. 345.
6482 The Journal of Physical Chemistry, Vol. 88, No. 26, 1984
tion,19~20J8 i.e. by neglecting the second derivative in (3.3) and identifying Ap* with -g*z/Ho. This procedure yields the correct density profiles except for temperatures and densities close to the critical point. We find that the actual density profiles differ significantly from those calculated in thelocallyhomogeneous approximation at temperatures for which AT 4 ATmax= 6 and at levels for which -2, < ?, < ZmX with Zmx = 5. In the locally homogeneous approximation the density gradient dAp*/dz is proportional to the response function x* which diverges at the critical point. From Figure 2 we note that the actual density gradient remains finite. At “the critical point”, Le. at the layer z = 0 ( p = pc) for = 0, it reaches a maximum value of Id&/dilc = 0.96. For further details the reader is referred to the previous p ~ b l i c a t i o n . ~ ~
+
e
Sengers and van Leeuwen geneous system leading to a structure factor of the form (4.6) withf0AO;z) = x ( p ( Z ) ) . The actual local correlation length will differ from the correlation length E(p(Z)) in the locally homogeneous approximation. Furthermore, gravity causes the correlation function to become anisotropic and we distinguish between an actual local correlation length EIl(2)in the direction parallel to the gravitational field and an actual local correlation length El(Z) in the direction perpendicular to the gravitational field. We define these correlation lengths as
4. Gravity Effects on the Fluctuations Having determined the density profiles in a fluid near the critical point, we are ready to consider the nature of the critical fluctuations in the presence of a gr_av$ational field. For this purpose we consider a density p(?-R;R) which is proportional to the conditional probability+of finding a molecular a t 7 = (x,y,z) if a molecule is Er5sent at R = (X,Y,Z). In a spatially homogeneous system p(%R,R) depends onli 0,“ the distance IFRI; in our spatially inh_omogeneoussystem p(%R,R) depen_dson the relative position FR and also explicitly on the position R of the reference patticle through the vertical coordinate 2. For distances large compared to the intermolecular distance we can again apply the squaredgradient theory to obtain39 AV2p(+&;&) = p(p(%&;&)) - p ( p ( 0 ) ) + gz (4.1) where Vz = d2/dxZ d2/_dyz dZ/dz2. At large distance 17 - RI, :he density p(%j?;j?) will become independent of the location R of the reference particle and approach the local density p(z). We thus define a (dimensionless) correlation function by
+
+
-
f(>j?;&) = [p(7-&;&) - p(z)] / p c so that f(%j?;j?) approaches zero as 17 - dl (3.1) from (4.1) yields the equation
~ 3 .
(4.2) Subtracting
pJ VZf(+E;d) = p(p(+&;j?)) - p(p(z)) (4.3) For distances where (4.3) will be valid+we_may linearize the right-hand sid? ,of (4.3) as p ( p ( F R ; R ) ) - p(p(z)) = x-’(p(z))pJ(F-R;R). With (3.2) we thus deduce from (4.3) r
The correlation length [(p(z)) appearing in the left-hand side of (4.4) is again the correlation length of a spatially homogeneous system with uniform denjity p = p(z). In order to obtain the , first need to evaluate ((p(z)) correlation function f ( F R ; R ) we associated with the density profiles presented in the previous section and then solve the differential equation (4.4). We note that this equation does not depend explicitly on the gravity g but only implicitly via the induced density profile p(z). In order to make contact with scattering experiments, we prefer to copider a structure factor which is the Fourier transform of f(+R;R)
f(i;&) = I d 7 &FRN7-&;&)
(4.5)
If we were to identify E in (4.4) with the correlation length [ ( p ( Z ) ) in the locally homogeneous approximation corresponding to the density p ( 2 ) at the position of the reference particle but independent of z, then (4.4) reduces to the well-known differential equation of the Ornstein-Zernike theory for a spatially homo(38) Levelt Sengers, J. M. H. In “Experimental Thermodynamics”; Le Neindre, B., Vodar, B., Eds.; Butterworths: London, 1975, Vol. 111, p 657. (39) Rowlinson, J. S.;Widom, B. “Molecular Theory of Capillarity”; Clarendon Press: Oxford, 1982.
ldPf(Fj?;&)[(x
EIZ(Z)=
4
- a2+ 0,- V 2 ]
SdPf(?-j?;d)
(4.8)
The gravity also causes the correlation functionflit$?;&) to become asymmetric in the +z and -z directions. As a consequence, the structure factor defined by (4.5) becomes complex. Since light-scattering measurements probe the real part of the structure factor, we restrict ourselves here to a discussion of this real part which for small wavenumbers approaches the form
+
where kL2= k,Z ky2 and kllz= .k: Far away from the critical point the locally homogeneous approximation will be adequate and (4.9) will reduce to the Ornstein-Zernike form (4.6) with L ( Z ) = 51(2) = F(P(Z)). The correlation lengths Eil(Z) and El(Z) can be determined from an asymptotic solution of the differential equation (4.4). Our procedure for obtaining this solution will be published elsewhere.@ Here we restrict ourselves to a discussion of the principal results. The differential equation (4.4) becomes universal if we scale all distances with the scale factor f introduced in (3.7). The scaled correlation lengths Ell = Ell/t and = El/{ are then universal functions of the scaled height 2 and the scaled temperature In Figures 3 and 4 we show Ell and as a function of Z at the same temperatures for which the density profiles were shown in Figure 1. The dashed curv~sin these figures represent the scaled hypothetical correlation length $ ( p ( Z ) ) in the locally homogeneous approximation. We first consider the correlation length Ell in the direction parallel to the gravitational field shown in Figure 3. At levels Z far away from the central layer, Ell(2) is larger than the correlation length t ( p ( 2 ) ) calculated in the locally homogeneous approximation. For sufficiently large values of 2 the layers z > 2 are less critical and therefore less correlated than the layer z = Z , while the layers -2 < z < Z are more critical and, hence, more correlated than the layer z = Z. The latter effect dominates due to the rapid increase of the correlation upon approaching p = po and as a result, gravity enhances the fluctuations associated with the layer at 2. Closer to the central layer, where p = pc, the correlation length ,&II(Z)becomes much smaller than the correlation length t(p(Z))calculated in the locally homogeneous approximation; Le., gravity suppresses the fluctuations at these levels. It is interesting to note that at a given temperature the maximum correlation length is not attained at the layer 2 = 0 where p = p c but at levels slightly above and below this layer. The reason is that the second moment of the correlation function is not determined by the properties of the system at the position Z of the reference particle but at positions for which z - Z is larger
e
El El
E.
(40) van Leeuwen, J. M. J.; Sengers, J. V. submitted for publication in Physica A (Amsterdam).
Gravity Effects on Critical Fluctuations in Gases
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 6483
z -2
-4
+4
1.0
Ell
0.5 0.2
1
I
1.0
\
I
\
0.2
4
-2.0
a'i.20
-4
0 +2 +4
-2
-
;:E
-02 -4
-2
0 +2 +4
z
z
Figure 3. Scaled correlation length EL parallel to the gravitational field as a function of the scaled height 2 at various values of the scaled temperature difference E.The dashed curves represent the scaled correlation length F ( p ( 2 ) ) evaluated in the locally homogeneous approximation.
z -4 3.0r,
-2 I
0
I
I
0.5
5
/
+4
+2
, \E=o
I : I I I
I
I
c
I
I
I
+ 2 +4 3,0
0 I
I
I
I
I
I
I
I
I
,
I
02 13.0
'
0.2
2.0
-2
-4 1 1
-
AT=0.6
,
, , , ,
4t
ri=I,O 12.0
1.0 0.5
1.0 0.5
0.3
/
\
AT=5.0
t i t
12.0
-c
0,2 -4 -2
0 + 2 +4
-4 -2
0
-
+2
r = 0.055
3.0 2.0
ET.l.0
m=0.6
TABLE I: Critical-Region Parameters for Xenon critical parameters P, = 5.840 MPa p, = 1110 kg/m3 T, = 289.72 K 0 = 0.325 critical exponents (universal) y = 1.240 6 = 4.815 Y = 0.63 critical amplitudes XO 0.339
+4
0.2
f 2 Figure 4. Scaled correlation length perpendicular to the graviational field as a function of the scaled height 2 at various values of the scaled temperature difference E.The dashed curves represent the scaled correlation length E(p(Z))evaluated in the locally homogeneous approximation. or smaller than zero. For a purely exponentially decaying correlation function the integrand determining the second moment yields its maximum contribution where z = Z f 2&. Indeed it appears that the maximum value of the correlation length tilis encountered when z N 0, i.e. when 2 N 2tll(2). The presence of the gravitational field also modifies the range of the correlation function in the directions perpendicular to the field as shown in Figure 4. In the layers sufficiently away from the central layer the gravity again enhances the correlation length, but the effect is less pronounced than in the direction parallel to
D = 1.9 fo = 1.89 pm
TABLE 11: Gravity Effects in Xenon near the Critical Point gravity scale X factors T
1.14
X
0.356 X 10" l 1.86 pm critical layer Idp*/dzl, 6 mm-' properties &, 1 pm L C 1.7 pm range of lA71msx 0.6 mK intrinsic 22,,, 0.02 mm gravity effects
0.41 X
0.15 X
1.53 X lo-* 6.57 X
13.5 prn
98.1 prn 0.01 mm-'
0.3 mm-' 7 pm 12 pm
87 pm
27 pK 0.1 mm
1 pK 1 mm
50 pm
the field. Near the central layer the gravity again causes a reduction of the correlation length. The correlation length [ ( p ( Z ) ) in the locally homogeneous approximation would become infinite at the critical point, i.e. at the critical layer z = 0 for = 0. The actual correlation length remains finite at this critical layer both in the direction parallel to the gravitational field with Ell, = 0.515 and in the direction perpendicular to the gravitational field with = 0.889. We note that represents the maximum attainable correlation length in a fluid in the presence of gravity. The maximum attainable correlation length in the direction parallel to the gravitational field is 1.42EII,, = 0.731 encountered at 2 = Et1.57.
E
5. Application to Xenon To discuss the magnitude of the predicted gravity effects, we consider xenon as a representative example. The reason for this choice is that Moldover et al. have earlier selected xenon to review the magnitude of the gravity effects when evaluated in the locally homogeneous approximation.20 Here we consider the intrinsic gravity effects close to the critical point where the locally homogeneous approximation breaks down. The parameters that characterize the behavior of the equation of state and the correlation length of xenon without gravity eff e c t ~ ~are~ presented , ~ ~ ) ~ in~ Table I. The parameters that characterize the magnitude of the intrinsic gravity effects in xenon near the critical point are presented in Table 11. Included in this table are the gravity scale factors A, r, {, the critical layer properties (Le. at the layer where p = pc at T = T,) Idp*/dzl,, Ell,, El,,, and the temperature range AT,,,,, and the height range 22,, where the locally homogeneous approximation ceases to be valid. These quantities are shown for three different values of the magnitude of the gravitational field: g* = 1 corresponds and g* = lod to the earth's gravitational field, while g* = correspond to the gravitational levels that one can reach in the space lab project under the worst and optimum condition^.^^ The maximum density gradient Idp*/dzl, in xenon in the earth's gravitational field is 0.6%/pm. (The maximum density gradient Idp*/dzl, is a factor of 10 smaller than the values inadvertently quoted in Tables I and I1 of ref 28.) The maximum attainable correlation length in xenon in the earth's gravitational field is 1.7 pm. From (3.8) it follows that this maximum attainable correlation length scales with the strength of the gravitational field as (41) Sengers, J. V.; Moldover, M. R. Phys. Lett. A 1978, 66A, 44. (42) Giittinger, H.; Cannell, D. S. Phys. Reu. A 1981, 24, 3188. (43) Morrell, R. Conremp. Phys. 1977, 18, 1.
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g*-”4 and is thus inversely proportional to g*0.3. The intrinsic gravity effects in the earth’s gravitational field become important at temperatures within 600 pK from the critical temperature to be compared with a currently attainable level of temperature control to within 10 pKW7 they are present at levels within fO.O1 mm around the layer where p = pc. 6. Experimental Consequences From the information presented in Figures 1 and 2 we note that the actual density profiles become insensitive to temperature I 1.0. For xenon this means that the density changes at profiles change little at temperatures within 0.1 mK from the critical temperature. The theory described in section 3 can also be applied to determine the gravity effects on the density profile, including the interface, at temperatures below the critical temp e r a t ~ r e . ~ *Again one approaches smoothly the limiting profile = 0. A direct experimental determination of the shown for critical temperature is normally based on the observation of the temperature a t which the vapor-liquid meniscus disappears. Within a range of about *O. 1 mK no sharp profile change will be observed, causing a gravitationally induced intrinsic limit on the accuracy with which the critical temperature can be observed. The predicted intrinsic gravity effects will be enhanced when one considers critical phenomena experiments in an ultracentrifuge, where one can produce reduced gravitational fields g* of the order of lo4 or lo5 while the profile can be observed with Schlieren phot~graphy.~~,~~ (44) Hocken, R.; Moldover, M. R. Phys. Rev. Left. 1976, 37, 299. (45) Sarid, D.; Cannell, D. S.Rev. Sci. Instrum. 1974, 45, 1082. (46) Dratler, J. Rev. Sci. Instrum. 1974, 45, 1435. (47) Kopelman, R. B.; Gammon, R. W.; Moldover, M. R. Phys. Rev. A 1984, 29, 2048. (48) van Leeuwen, J. M. J.; Sengers, J. V., in preparation. (49) Starobinets, S.; Yakhot, Y.; Esterman, L. Phys. Rev. A 1979, 20, 2582. (50) Salinas, R.; Huang, H. S.; Winnick, J. Ado. Chem. Ser. 1979, No. 182, 271.
Next we consider some consequences for critical-opalescence experiments. As pointed out by Debye,’ssl a major triumph of the Ornstein-Zernike theory is that it accounts for the observed angular dissymmetry of critical opalescence, i.e. the dependence of the scattered light near the critical point on the scattering angle 0. However, for a spatially homogeneous system this angular dissymmetry, apart from polarization corrections, will be independent of whether the scattering is observed in a horizontal or a vertical plane. Due to the anisotropy of the correlation function induced by the gravitational field, FI # til in (4.9) and the scattering intensity will also depend on whether one probes fluctuations with wavevectors in the direction of the gravitational field or perpendicular to the graviational field. The fact that gravity suppresses the correlation length (,I in the direction of the gravitational field at the critical point, as compared to the correlation length 6 of a spatially homogeneous system which would diverge at the critical point, is immediately obvious. Less obvious is the fact that gravity also strongly suppresses at the critical point the correlation length in the direction perpendicular to the gravitational field. Hence, our conclusion that the scattering function becomes anisotropic is no surprise. Perhaps more surprising is that the scattering function remains as isotropic as it by more is; that is, the correlation length t1 can never exceed 111 than 75%. It will be difficult to obtain the required resolution to determine the predicted anisotropy of the scattering function experimentally. Experimental verification of our prediction that the correlation length of gases in the earth’s gravitational field can never exceed a value of about 2 pm in any direction may be a more realistic goal.
Acknowledgment. We acknowledge stimulating discussions with M. R. Moldover and R. K. P. Zia. The research at the University of Maryland was supported by National Science Foundation Grant DMR 82-05356. Our collaboration was supported by NATO Research Grant 008.8 1. (51) Debye, P. In “Non-Crystalline Solids”; Frechette, V. D., Ed.; Wiley: New York, 1960; p 1.
Statistical Mechanics of Fluid Interfaces in Cylindrical Symmetry J. R. Henderson and J. S. Rowlinson* Physical Chemistry Laboratory, Oxford University, Oxford OX1 3QZ. England (Received: May 30, 1984)
We discuss the curvature dependence of the surface tension of a cylindrical fluid interface. Density functional theory is used to derive correlation function expressions for curvature derivatives of the surface grand potential. The restoring force for fluctuations of a cylindrical liquid-vapor interface is also considered. However, in contrast to previous work on drops, we do not find a well-defined length describing the leading order curvature dependence of the surface tension. Instead, the geometry of a cylindrical system is such that the thermodynamic approach, based on an assumed expansion of the grand potential in powers of the curvature, breaks down beyond the planar limit of surface tension.
1. Introduction
Inhomogeneous fluids with cylindrical symmetry arise naturally when fluids are absorbed by porous solids possessing capillary shaped channels. In general, the fluid-wall forces play a major role, stabilizing the cylindrical symmetry and controlling the wetting behavior on the capillary wall. In special cases, involving complete wetting in a capillary of macroscopic radius, it should even be possible to observe liquid-vapor interfaces with cylindrical symmetry, when an incipient phase capable of coexistence with the bulk fluid is preferentially adsorbed on the capillary wall. The phase next to the wall can be either liquid or vapor according to 0022-365418412088-6484$01.50/0
the nature of the intermolecular forces between the fluid and the wall. This paper derives correlation function expressions for the surface free energy, or surface tension, of a cylindrical fluid surface. The fluid-wall forces are modeled by a one-body external potential so that we can use density functional theory based on a free energy functional of one-body functions. The general result links the surface grand potential, as a function of the curvature of the fluid surface, to an expression involving the density profile, the external field, and the pair distribution function (section 2 ) . A transformation to a companion result involving the two-body 0 1984 American Chemical Society
