6484
J . Phys. Chem. 1984,88, 6484-6487
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 changes at I 1.0. For xenon this means that the density 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 shown for = 0. A direct experimental determination of the 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 is; that is, the correlation length t1 can never exceed 111 by more 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
Statistical Mechanics of Fluid Interfaces direct correlation function, valid up to leading order in curvature dependence, is given in the Appendix. In the two limiting cases of a completely sharp containing field (a hard-wall capillary) and infinitestimal external field (a liquid-vapor interface), where the direct contribution to the surface free energy from the external field vanishes, we consider the possible qualitative nature of the leading order curvature dependence of the surface tension. In section 3 we specialize to the case of a liquid-vapor interface and discuss capillary wave fluctuations of such a surface. On comparison with the results of section 2 we find, in contrast to the situation in spherical symmetry (drops), that the restoring force for capillary wave fluctuations is not given entirely by the surface tension, except in the planar limit. We conclude that in cylindrical symmetry the capillary wave analysis breaks down beyond the planar limit. This statement corresponds to the fact that the leading order curvature dependence of surface tension in cylindrical symmetry is contained in terms of unknown qualitative nature. In particular, in cylindrical symmetry (or in two-dimensional circular symmetry) there is no length corresponding to Tolman’s length describing the curvature dependence of the surface tension of a drop. Naive extension of the thermodynamic theory of drops to a cylindrical surface would describe the leading order curvature dependence of the surface free energy per unit area by a constant term in the total free energy, which of course cannot contribute to the restoring force to distort the interface. Throughout this paper we invoke methods pioneered in the study of planar and spherical surfaces. The work of section 2 is based on an approach contained in ref 1-3. Section 3 obtains results analogous to recent work on the statistical mechanics of drop^,^,^ which in turn was based on previous studies of a planar liquidvapor ir~terface.~,~ We do not consider a virial route to the surface tension of a cylinder because of previous experience with the statistical mechanics of drops, where the virial route to surface tension apparently breaks down beyond the planar limit, due to effects arising from the fact that the pressure tensor is not uniquely defined in an inhomogeneous region.8
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 6485 Q ( R )E -ApnR2L - p1 Y + Qcs)(R)+ K
(1) where Ap is the pressure difference between bulk fluids on opposite sides of the cylindrical surface of length L and radius R (Ap = po - pl) and QW(R)is therefore the curvature-dependent surface grand potential which we choose to define so that QcS) does not contain any terms that are independent of curvature (these terms are described by K ) . The radius of the cylinder is defined by an appropriate but unspecified choice of a Gibbs dividing surface. Here, we generalize Gibbs’s approach by allowing R to be a function of temperature (T), chemical potential ( p ) , and external field (u). In this way we are able to discuss the curvature dependence of the grand potential by using the external field to control the curvature, which can be regarded as including the special case used by Tolman,lo where one imagines making different choices for the position of the Gibbs dividing surface. An important point is that the first two terms on the right side of (1) have coefficients that are defined to be functions of T and p, but not u; Le., po( T,p) and p l ( T , p ) . An appropriate choice of volume terms is usually suggested by the physics, for example, for a system composed of two macroscopic fluids separated by an interface stabilized by an infinitesimal external field, the volume terms would be chosen to correspond to the real bulk fluids. In general, an external field contributes directly to Qcs), for example, note the generalized Laplace formula given by eq 2 and 5. However, even in the presence of a strong external field, the “surface” grand potential defined by ( l ) , together with an appropriate choice of po, pi, and R, will behave as a surface quantity, provided only that the external field does not induce any contributions to the grand potential that vary more strongly than R . For our purpose, the equation of state for the volume terms may be left unspecified, because the only property of po and pi that one requires in order to derive correlation function expressions for QcS) is that p o and pi be independent of u. In particular, although first-order changes in curvature involve a volume term
2. Correlation Function Expressions for the Surface Grand Potential in Cylindrical Symmetry
by going to second-order one can subtract out volume contribuThe equilibrium theory of surfaces was pioneered by G i b b ~ . ~ tions: In particular, he noticed that the formidable problems involved in obtaining a quantitative understanding of the nature of interfaces of microscopic width can be avoided by a simple but powerful piece of lateral thinking; instead of attacking the surface Our approach is based on eq 3, that is, on considering second-order problem directly one concentrates on the uniform regions of the changes in curvature, and is thus analogous to the compressibility system, since the theory of uniform fluids is well-developed, and route to the equation of state of uniform fluids. then what is left over is by definition a surface quantity. This To calculate the left side of (3) directly from the partition approach enables us to obtain statistical mechanical expressions function, let us introduce a particular type of scaling, defined by for surface properties that are completely rigorous, since their the class of external fields (relevant to cylindrical symmetry) derivations involve nothing more than mathematical transforu(s;R) = U(S - R ) (4) mations of the partition function. The understanding of the microscopic nature of surfaces then follows by comparison with where s is the variable denoting the distance from the axis of our results based on partial knowledge of interfacial structure. cylindrical system. Then, we may use the functional identity, Following Gibbs, we write the grand potential of our cylindrical (6Q/6~),,, = p (the density profile), that follows directly from the system in the form definition of Q in terms of the partition function, to obtain the left side of (2): (1) Henderson, J. R. Mol. Phys. 1983, 50, 741. (2) Henderson, J. R.; van Swol, F. Mol. Phys. 1984, 5J, 991. (3) Henderson, J. R. In “Fluid Interfacial Phenomena”; Croxton, C . A., Ed.; Wiley: New York, in press. (4) Henderson, J. R.; Schofield, P. Proc. R. SOC.London, Ser. A . 1982, 380, 211. ( 5 ) Hemingway, S. J.; Henderson, J. R.; Rowlinson, J. S.Faraday Symp. Chem. SOC.1981, 16, 33. Hemingway, S.J.; Rowlinson, J. S.; Walton, J. P. R. B. J . Chem. Soc., Faraday Trans. 2 1983, 79, 1689. (6) Triezenberg, D. G.; Zwanzig, R. Phys. Rev. Lett. 1972, 28, 1183. (7) Lovett, R. A.; DeHaven, P. W.; Vieceli, J. J.; Buff, F. P. J. Chem. Phys. 1973, 58, 1880. (8) Schofield, P.; Henderson, J. R. Proc. R. SOC.London, Ser. A . 1982, 379, 231. Walton, J. P. R. B.; Tildesley, D. J.; Rowlinson, J. S.Mol. Phys. 1983, 48, 1357. Rowlinson, J. S. Physica B+C (Amsterdam), in press. (9) Gibbs, J. W. Trans. Conn. Acad. 1875-1878, 3, 108 343. “The Collected Works of J. Willard Gibbs”; Longmans, Green: New York, 1928; Vol. 2.
(”) dR
= -Jdr
u‘(s - R ) p(s;R)
T.P
(5)
Continuing to differentiate and integrating by parts gives
where the first term on the right side involves the curvature (10) Tolman, R. C. J . Chem. Phys. 1949, 17, 333.
Henderson and Rowlinson
6486 The Journal of Physical Chemistry, Vol. 88, No. 26, 1984
shown from density functional theory” to be given by the expansion
dependence of the density profile: (7) From use of a second functional identity, -kT(Gp(r)/Gu(r’)) = N(r,r‘), which follows straightforwardly from the statistical mechanical definition of the one-body density, we may express (7) in terms of the density-density distribution function N(r,r’): aP(sl;R)
2!
@~S;‘;;S, - R)(1 - 1,.S2) N(r1,r2); /3 = (kT)-’ ( 8 )
N(r1~2)E P ( S I ) 6(ri
- rz) + P ( S I ) ~ ( 8 2 N d r 1 ~ 2-) 1)
where g(rl,r2)is the pair distribution function. Combining eq 3, 5, 6, and 8 we obtain our desired result
6(r
- r’)
In particular, consider a small amplitude capillary wave fluctuation,I3 confined to the interfacial region, defined by
-
(T) -
l d r l d r ’ 6p(r) 6p(r’)
6Pcw,/(r)
-P’(s)rCw,/(4)
+ W2)
(14a)
CCw,/(4) €Re‘@
(14b) Here, r i s the coordinate of the fluctuations of the dividing surface, i.e., the change in area involved is
_-
A 2 So, the work done per unit change in area by the fluctuation (14), in the limit e 0, is
-
In the Appendix we show that up to leading order in curvature dependence the right side of (9) may be rewritten in terms of the two-body direct correlation function C(r,r’): (cos 012 - cos ( h ) l C(rl,rd (16)
e,, = 4, - +2; where d is a measure of the range of the direct correlation function in the surface region (presumed to be of the order of a bulk correlation length, or the surface thickness). From (lo), it follows that the first term on the right side of (9) is the contribution arising from the intermolecular forces.’ In a two-phase region (loa) remains significant even in the limit of very weak external field, due to the diverging contribution to surface integrals of N(rl,r2) from capillary wave correlations! The last term of (9) is the direct contribution to the surface grand potential from the external field. In the two extreme limiting cases of a hard-wall capillary and a liquid-vapor interface in an infinitesimal stabilizing external field, there is no direct external field contribution. For the remainder of this paper we restrict discussion to this field-independent limit. In particular, introducing the surface tension, y, and a function 6(R)describing the curvature dependence of 7,via the definitions
cos e,, = i1.i2
where the transformation of the first term on the right side of (16), involving the 6 function, is equivalent to an equation for p’(s) that follows from the functional identity -/?(6u(r)/6p(r’))T,, = p-I(r) 6(r - I-‘) - C(r,r‘), applied to the particular fluctuation that results V p , 6u Vu) in an from an infinitesimal translation ( 6 p infinitesimal external field. Further, we may make use of the fact that the density gradients appearing in the integrand of (16) confine the contributions to those from particles in the surface region, to expand about cos 012 = 1, which is equivalent to expanding in the square of the ratio of the range of the direct correlation function to the radius of the cylinder:
-
1 - cos eI2= COS
Is1 - s212 -
(SI
2SlSZ
-
- S2l2
(17a)
(14,)= COS e,, - (1 - COS e12)(12 - 1) + o ( i - COS e,,), ( 17b)
This expansion leads to the result YCW
(7..the planar surface tension) we have shown that
We immediately note two points of contrast between (12) and the corresponding result in spherical symmetry: (i) beyond the planar limit the right side of (12) will depend on the choice of Gibbs dividing radius, and (ii) a curvature independent 6 cannot contribute to (12) because it corresponds to a constant term in the grand potential. In particular, a leading-order curvature expansion of the right side of (12) implies a logarithmic variation of 6(R),indicating the absence of a Tolman lengthlo in cylindrical symmetry. Further discussion is left until after section 3.
3. Fluctuations of a Cylindrical Liquid-Vapor Interface The work done by an isothermal surface fluctuation, 6 p , of a liquid-vapor interface in an infinitesimal external field can be
= (&)I
where I is given by the second form of (lo), and ( 18) breaks down in the high curvature region at the same order as (lob). On comparison with (12), note that ynvis equal to the thermodynamic surface tension in the planar limit only. An alternative approach to the fluctuation theory discussed above is to calculate the second-order change in free energy via the first-order fluctuation in the pressure tensor. The pressure tensor, pa@,is defined to within an arbitrary path integral by the condition for mechanical equilibrium* V@psp”@(r) = -p(r) Vad(r) (19) and in cylindrical symmetry may be rewritten in terms of a normal and transverse component
( 1 1 ) Mermin, N. D. Phys. Reu. [Secr.] A 1965, 137, A1441. (12) Fisher, M. E. Physics 1967, 3, 255. (13) Lord Rayleigh, Proc. R. SOC.London 1879, 29, 71, Appendix I.
The Journal of Physical Chemistry, Vol. 88, No. 26, 1984 6487
Statistical Mechanics of Fluid Interfaces The effect on the interfacial structure due to application of a small additional external field, confined to the surface region, is described by integrating the change in pressure tensor across the interface:
A6p = 6 x m d ss-l(pds)- P A S ) )+ x m d sp ( s ) Sv’(s)
(21)
where 6v is related to the density distortion, via the functional identity -b(6v(r)/6p(r’))T,, = p-I(r) 6(r - r’) - C(r,r’); Le.
In particular, let 6v be a localized external field applied to bend a portion of the interface into a portion of a thinner cylinder, so as to increase its curvature by 6( 1/ R ) : 6p(r) = 1/s2 sin2 OIz p’(s) 6(1 / R )
(23)
Then the left side of (21) is zero and the first term on the right side describes the change in that part of the surface grand potential arising from intermolecular forces:
a In R term in the grand p ~ t e n t i a l Le., , ~ expansions of the correlation function expressions in powers of the curvature predict logarithmic terms in the grand potential. However, such expansions may well breakdown before yielding the constant term in the grand potential and thus the status of the logarithmic term remains unclear. The existence of a logarithmic term has been suggested previously, in the context of cluster partition functions (see, e.g., ref 12 and references therein). The problem with cylindrical symmetry (or a circular interface in two dimensions) is that the lowest-order curvature dependence of the surface tension (1 1) is of uncertain qualitative nature. In particular, it is not appropriate to assume an expansion of y ( R ) in powers of 1 / R , since the leading-order curvature-dependent term would then correspond to a constant term in the grand potential, which cannot enter into the restoring force acting against changes of curvature. The same breakdown of the thermodynamic approach occurs in the theory of spherical surfaces, but at one order higher in curvature expansion because for a drop there is an extra power of R in the grand potential. Appendix: Proof of Eq 10b Insertion of the identity discussed after eq 16
That is, (24) may be regarded as the change in ( A P ) ~that , ~ would stabilize the new curvature in the absence of an external field (see (2) and (5) applied to the case of a liquid-vapor interface; v 0). Combining eq 21-24, we have
-
p-( E ) a(l/R)
aR
= - 2 a L x - d s p ( s ) 6v’(s)/6(1/R) =
T,&
Pv’(s1) = -Jdr2
N-Y~IJJE into eq 10a gives I = kTJdr,
cos
? r L k T x m d s lp’(sl)1dr2 p ’ ( ~ , ) ssinZ , ~ 8,, C(rl,r2) (25) where we have integrated by parts before substituting (22) and (23). Expanding about cos 812 = 1, as before, we obtain
P’Wcos 812 W ’ ( r 1 ~ 2 1
P-W6(rl - r2) - C ( r l , ~ )
Jdr2 Jdr3 Jdr4 p’(s3) p’(s4) cos 824
813 X
(1 - cos B I 2 ) N-’(r1,r3) N(rl,rz) N-’(r>r4) (A2)
Let us use the variables dr, = dx, ds, s, d41; 0 I x , I L, 0 I s, I m,
0 I 4, 5 271. (‘43) and expand the density-density distribution function, N , as
N(rl,r2) = CNh2,s1,s2)eJ”~2
(‘44)
I
s22
p’(sz)-(l
(AI)
XI2 x2 - XI, 4 2 = 41 - 4 2 and similarly for its inverse, N-I, and we can make use of the identities
- i1.i2) C(rl,r2)(l + O ( d / R ) z )=
s1
x2‘d0 eil8 = 27dOl, x z r d 8 cos 8 eils = ~ ( 6 _+~ 6,,) ,
(A5)
Note that (26) differs from (9) and (lob), beyond the planar limit, unless (23) is restricted to the particular choice of dividing surface such that the term involving (sz - R ) on the right side of (26) is identically zero. 4. Discussion The result (9) links the thermodynamic surface tension to a statistical mechanical correlation function expression. Equation 9 is valid for arbitrary curvature but beyond the planar limit it yields a surface tension, (1 l), that depends on an arbitrary choice of Gibbs dividing surface. This result contrasts with that for drops, where the leading-order curvature dependence of the surface tension is described by a well-defined length, Tolman’s length,5J0 which is independent of the choice of dividing surface. Similarly, attempts to obtain a unique number describing the restoring force for fluctuations of a cylindrical liquid-vapor interface break down beyond the planar limit. The key point to note is that if the quantity 6(R),appearing in (1 l ) , were a constant it could not contribute to the results (12) and (26). For example, the second term on the right side of the final version of (26) and a corresponding term in an expansion in curvature of the right side of (12) correspond to 6(R) In R:
(A
-
- R$)6(R)
-
1
=+6(R)
-
In R
(27)
Interestingly, the same procedure applied in the case of a drop, in particular, to the equation corresponding to (12), also yields
. SdX3
Sds3 #3
N~1(x13,sl,s3)
.
.
Nl(x32,s3,s2)
(A9)
The result (lob) follows directly from (A7) and (A8), since (17a) implies that J is of order I(d/R)*,where d measures the range of the direct correlation function.
