6 Structure of a Liquid-Vapor Interface
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M. RAO and B. J. BERNE Columbia University, New York, NY 10027
A l i q u i d c o e x i s t s with its vapor i n e q u i l i b r i u m below the critical temperature Tc. The two w e l l defined phases are separated by a small t r a n s i t i o n region c a l l e d the i n t e r f a c e . Recently there has been much t h e o r e t i c a l e f f o r t devoted t o an understanding of p r o p e r t i e s o f the interface(1,2,3,4). I t i s a l s o p o s s i b l e now to shed some l i g h t on the s t r u c t u r e and dynamics o f the i n t e r f a c e s using computer simulation(5,6,7,8). With t h i s information one can begin t o evaluate v a r i o u s t h e o r e t i c a l approximations and provide a q u a n t i t a t i v e framework f o r the microscopic phenomenology o f the l i q u i d - v a p o r i n t e r f a c e . In this paper we present some r e s u l t s on the l i q u i d - v a p o r i n t e r f a c e obtained from a Monte Carlo s i m u l a t i o n . Our model system c o n s i s t s o f 2 0 4 8 p a r t i c l e s , i n t e r a c t i n g via a classical mechanical Lennard-Jones (6,12) p o t e n t i a l (Mr)
= V(r) - V ( r )
0
< r < r
=0
r
Q
Q
where V(r) = 4 e [ ( ^ )
1 2
6
- (^) ] and r
Q
Q
< r
= 2.5a.
The p a r t i c l e s are
placed i n a f u l l y p e r i o d i c box o f s i z e 1 4 . 7 a * 1 4 . 7 a * 2 5 . 1 a . The d e t a i l s o f c r e a t i n g an inhomogeneous system without any ex t e r n a l f i e l d i n t h i s p e r i o d i c box are presented elsewhere (5^) . The s t a r t i n g c o n f i g u r a t i o n thus generated i s a two sided f i l m with l i q u i d density n^ i n the middle o f the box and vapor d e n s i t y n^ on e i t h e r s i d e .
Using the standard Metropolis
scheme the f i l m
i s e q u i l i b r a t e d a t a temperature o f 1 1 0 K. A step s i z e o f 0 . 2 a i s used f o r the random walk which gives an acceptance r a t i o o f 0.5.
At e q u i l i b r i u m the vapor d e n s i t y has an average value o f 0 . 0 5 and the l i q u i d d e n s i t y has an average value o f 0 . 6 5 ( r e duced u n i t s are used throughout). The symmetrized d e n s i t y pro f i l e i s shown i n f i g u r e 1 with dots. The o r i g i n i s chosen t o be
0-8412-0463-2/78/47-086-072$05.00/0 © 1978 American Chemical Society Lykos; Computer Modeling of Matter ACS Symposium Series; American Chemical Society: Washington, DC, 1978.
Lykos; Computer Modeling of Matter ACS Symposium Series; American Chemical Society: Washington, DC, 1978.
1
X
o
-6.0 -5.0
1
X
o
1 -4.0
X
O
x
°
-3.0
1
X
1
X
0
-2.0
o
-1.0
1
O X
z
C)
0.2 -
O 0.4 -
0.6 -
X
0.8 •
1.0 -
1.2 -
o
X
1.0
1
o
X
0
1 2.0
X X X
o o o 1 1 3.0 4.0
X
X
o 1 5.0
Figure 1. Density and structure factor profiles of a planar sheet of Lennard-Jonesium fluid in equilibrium with its own vapor at 110 K. The origin is chosen to be the Gibbs equimolecular dividing surface. The curves are obtained from Monte Carlo simulation using one million configurations. Circles denote the density profile and the crosses denote the structure factor, r.
-7.0
1
X
°
O 0 n (z) XX s(k,z
14 -
74
COMPUTER
the Gibbs equimolecular L z
d i v i d i n g surface given
MODELING OF
MATTER
by
(n - n )
9_
= 2
g
(n
-
£
n) g
where i s the length of the box i n Z d i r e c t i o n and n i s the average d e n s i t y (=N/V). The two components of the pressure ten sor ( Z ) (the l o n g i t u d i n a l component) and P (Z) (the transverse P
N
T
component) are determined using 1 m i l l i o n c o n f i g u r a t i o n s gen erated during the random walk. The surface t e n s i o n y i s obtained from the pressure tensor L
Z/2
Y = J
[P (Z) - P ( Z ) ] d Z N
T
" Z/2 L
The value of y obtained from the s i m u l a t i o n i s 0 . 4 2 reduced u n i t s . The s u r f a c e t e n s i o n y i n curved i n t e r f a c e s i s r e l a t e d to the y i n a plane i n t e r f a c e £y Tolman(j)) Y
= Y/d
+
26/r)
c where r i s the r a d i u s of the curved i n t e r f a c e and 6 i s the cur vature dependence d i s t a n c e . The d e t a i l s o f e s t i m a t i n g 6 from computer s i m u l a t i o n are presented e l s e w h e r e ( 1 0 ) . The value obtained f o r 6 i s 1 . 0 a . In f i g u r e 1 we a l s o present the transverse s t r u c t u r e f a c t o r S (k,Z) p r e v i o u s l y s t u d i e d ( 6 ) i n a s i m i l a r system a t lower temp e r a t u r e . S (k,Z) i s defined as T
T
N(AZ)
S (k,Z) = < T
I e x p [ - i k - ( r . - r .) ] >/ i,j=l 1
where k =
0)
and
(0,
3
p a r a l l e l to the s u r f a c e , i and
j
running over the N ( A z ) atoms i n a s l i c e of volume L x L * A z centered at Z. The enhancement of S (k,Z) i n the i n t e r f a c e r e g i o n i s a t t r i b u t e d to the c a p i l l a r y waves that are thermally a c t i v a t e d . The genesis of the s i n g u l a r low-k transverse be h a v i o r has been d i s c u s s e d r e c e n t l y by Wertheim(_2) , Weeks (1) and Kalos, Percus and R a o ( 6 ) . The emerging p i c t u r e of the i n t e r f a c e from these analyses i s t h a t the t r a n s i t i o n from l i q u i d to vapor i s q u i t e abrupt i n the i n t e r f a c e — of the order of one d i a meter — but t h i s i n t e r f a c e f l u c t u a t e s markedly i n space and time. I t i s these f l u c t u a t i o n s that broaden the i n t r i n s i c den s i t y p r o f i l e which i s q u i t e sharp. T h i s broadening a l s o depends on the s i z e of the system. Such a broadening has r e c e n t l y been observed i n computer s i m u l a t i o n ( 8 ) . T
Lykos; Computer Modeling of Matter ACS Symposium Series; American Chemical Society: Washington, DC, 1978.
6.
RAO A N D BERNE
Structure of a Liquid-Vapor
Interface
75
I t i s now p o s s i b l e t o simulate not only plane i n t e r f a c e s but a l s o s p h e r i c a l d r o p l e t s i n e q u i l i b r i u m with vapor(11). The s t r u c t u r e and dynamics o f these i n t e r f a c e s should throw some l i g h t on the phenomenon o f n u c l e a t i o n .
Abstract
The structure of the interface of an argon like fluid in equilibrium with its own vapor at 110 K is studied using the Monte Carlo method. The geometry of the interface is chosen to be planar and both longitudinal and transverse correlations are investigated. The longitudinal density profile shows no significant structure. From the measurement of the pressure tensor, the surface tension γ and its curvature dependence distance δ are determined. The transverse correlations exhibit very long range order in the interface suggesting a capillary wave-like behavior. Literature Cited 1.
Weeks, J. D., J. Chem. Phys. (1977) 67, 3106.
2.
Wertheim, M. S., J . Chem. Phys. (1976) 65, 2337.
3.
L o v e t t , R. A., DeHaven, P. W., Viecelli, P., J. Chem. Phys. (1973) 58, 1880.
4.
Singh, Y. and Abraham, F. F., J. Chem. Phys. (1977) 67, 537.
5.
Rao, M. and Levesque, D., J. Chem. Phys. (1976) (65, 3233.
6.
Kalos, M. H., Percus, J . K. and Rao, M., Jour. S t a t . Phys. (1977) 17, 111.
7.
Miyazaki, J . , Barker, J . A. and Pound, G. M., J. Chem. Phys. (1976) 64, 3364.
8.
Chapela, G. A., S a v i l l e , G., Thompson, G. M. and Rowlinson, J . J . , J. Chem. S o c i e t y , Faraday T r a n s a c t i o n s I I (1977) 73, 1133.
9.
Tolman, R. C., J. Chem. Phys. (1949) 17, 333.
J.
J.
and Buff, F.
10. Rao, M. and Berne, B. J. (to be p u b l i s h e d ) . 11. Rao, M., Berne, B. J. and Kalos, M. H., J. Chem. Phys. (1978) 68, 1325. RECEIVED
August 15, 1978.
Lykos; Computer Modeling of Matter ACS Symposium Series; American Chemical Society: Washington, DC, 1978.
