vapour interfaces

Volume 117, number 2

MCXX-i5

CHEMICAL

CAFWO

SlMULATION

PHYSICS

OF TWO

7 June 1985

LETI’EFtS

INTERACTING

LIQUID/VAPOUR

INTERFACES

Dan W. HAWLEY Micron Teclmology. J.M.D.

ZD 83706, USA

Bose.

MACELROY,

Deporrmem

of Chemical

Rcccived 20 February

J.-C Engineering.

HAJDUK

and X.B.

Unruersiry of MKW~~-

REED Rolla.

Jr.

Roffa, MO

65401. USA

19B5

TWO interactrngEqurd/vapor interfauzsare modelled via a grand canonical Monle Cal;0 simularionof a Lennard-Jon= 12-6 fluid. The resulmg dcnsiry profiles and interracialfree energies suggest tlla~ at a cnrical separation equal IO or greater than four atomic diameters. an inslability develops In rhe interstitial region, and coalescence lakes place.

1. Introduction The intrusion of Lifshitz-van der Waals interactions across the intervening vapor space between two colliding particles or drops is lmown to significantly inlluence the coalescence or coagulation rates of aerosols [12]. However, it is strll not clear as to what extent these interactions may alter the properties of two interacting liquid/vapor interfaces just prior to coalescence. Of interest in this respect is the variatron of both the interfacial free energies and the density profiles of the individual liquid/vapor interfaces as they approach one another within distances of 50 A or less. In the past decade Monte Carlo simulation techruqueshave been employed to investigate the structure and properties of free liquid/vapor interfaces of atomic and molecular fluids [3-71. Each of these works used the petit canonical (NV?) ensemble Monte Carlo (PCEMC) method of Metropolis et al. [S]. This method is surtable for systems 111which fluctuations in density are unimportant. However, for interacting liquid/ vapor interfaces density fluctuations can be quite severe and the appropnate statistical ensemble to consider is the grand canonical @VT) ensemble. We present herein results obtained from an application of the grand canonical ensemble Monte Carlo (GCEMC) method of Adams [9] to the problem of two iuterac154

ting liquid/vapor interfaces. The simulation was performed for a Lennard-Jones 12-6 fluid with the properties of krypton at its triple point, 115.8 K. Although this GCEMC formulation is strictly valid in equilibnum systems, we feel the results presented beIow lend mechanistic implications to coalescence phenomena.

2. Model The fundamental

unit cell is illustrated in fq.

l_

This rectangular box has eight adjacent periodic cells in the x andy

directions. The half-spaces A and A’

are composed of molecularly homogeneous material with a density equal to that of the bulk liqlud phase These haIf-spaces provide the background potential energy required to stabilize the two liquid-like layers formed wrthin the fundamental cell during the course of the simuIatio,r?.In alI runs, L was 5.50 while n was 150, where o is the Lermard-Jones diameter of a particle in the fluid. Subject tc the assumption of painvise additiviw of the particle-particle and particle-“wall” mteractions, the total potential energy for the Lennard-Jones fluid of N particles within the fundamental

0 009-2614/85/S (North-Holland

cell is

03.30 0 Elsevier Science Publishers B.V. Physics Publishing Division)

Volume 117, number 2

CHEMICAL

PHYSICS

LElTERS

7 June 1985

particles. At each step during the realrzation of the Markov chain of events, the long-range correction to the internal energy of the system was taken into account using [ll]

Eq. (2) is valid in the mean field limit where the pair correlation function g(2)(rl, f2) is equal to 1.0. Thu is reasonable beyond the cutoff radius R,. A further approximation which we have employed here is to replace the instantaneous density profile n(r) by the simple step profde

n(z) = nJ_ + (nL - “c)

[W

- lzb2’l) - U(z +

kpl)l

, (3)

in which nL and nG are the bulk liquid and vapor densities and z&r) and zh2) are the instantan~us positions of the centers of the two liquid/vapor interfaces where the number density n = f(nL + nG)_ The free energy per unit area of both tiquidlvapor interfaces was determined using the mechanical strerstensor definition [12] (1)

X [1 - L’((X-

IZil) -R,m)J

and U(x) is the unrt step function The Lennard-Jones force constants used in the simulations were u = 0.3634 nm and e/k = 163.1 K [lo]. The cut-off rd.iusR, was taken as 2.7~ and the wall potentiai E, was set equal to 2~. In all cases studied 250 particles were initially placed next to each of the surface A and A’ (fip. 1) in a simple cubic lattice llu deep. The chemical potential of the system was changed from run to run III such a way that the mean distance between the liquid/ vapor interfaces was varred by addition or removal of

where+ =xi-xPziI=zi-zi. (-Y,i, I was evaluated for those particles with z coordinatesintherange-H+5u
155

Volume

117, number 2

n*(z*)

= rq

CHEMICAL

nz -nE

-

1 - tanh(212~1) *r/r) tanh(--2z&2)*/r) x { 1 - tanh [2(z x tanh[2(z*

-

l

+ lz~“’

Kz(r’s)

= 4

&(r*s)

= (1

+(A3

1)/Z]

zp*Yrl 1 ,

of a profile fitting function for a single free liquid/vapor mterface suggested by Chapela et al. [5] _ I is a measure of the mean thickness of both liquid/vapor interfaces. The above expression was used to obtain the long-

6$ j- i __

7 rlyzg

nf(z;

sinh2(4z’oll)

-z;

t anh(2r*S/l)

])/(A

-A’)

sinh(8z;

/I)

>

- cosh(l6zc/r)

+A)2 cosl-@zgf/l)

The corrected is

range correction to the simulated interfacial ergy. This correction term is given by [S],

-

r-s

,

in which A = exp(4r’sll).

(5)

where n* = no3 and zL = z/a. Eq_ (5) is a simple generalization

g-R =

7 June 1985

PHYSICS LmERS

inter-facial free energy per unit area

fmally computed

as

Y=(ysh)++R-YC,

where yc is the contribution to Cysi, ) arising from the artificial cut-off planes at 2 = %(JY - 50). All simulations reported here were performed on an FPS-164 computer. The storage requirements were less than 400k, and the number of events which could

free en-

be executed per hour was 1.75 X 105_ Ensemble averages were obtained over 7.5 X lo6 events in each run.

-r-s)

-1 R;

3. Results X 1 - 3s2 rq4

dr*dsdz;,

where Rz = R,/rz and r* = r/u_ The integration over z; may be performed

Three simulations were conducted in which chemical potentials (see table 1) were chosen so as to provlde adsorbed Iayers at least 100 deep on each of the background walls A and A’ of fig. 1. The interfacial

using

eq_ (5) with the result

properties

?LR =

(6)

8cosh2(4z;/r)(rz;

1-f talh2(q/l)

to the properties

of normal liquid/vapor

In runs 1 and 2 ensemble averages for the properties of the system were obtained over the last six miilion events in the MC chains. In run 3 averages were obtained over the last 3.5 X lo6 events. The results

where

K, =

for such systems may be assumed to closely

correspond interfaces_

- rr;)*

for @J>, (u*) and (ysim 1 are given m the third, fourth and fifth c~lurnns of table 1. The long-range corrections to y given in the sixth column were obtained

’

Table 1 a)

RtUl number 1 2 3

(N?



<7ti)

-10.780

522.5

-6.179

20.4

9.2

-10.766 -10.750

550.6 738.4

-6.224 -6.488

21-l

9.9

11.8

W

7LR

-

b)

7

b,

17 8 19 2

-

a) p* and tu*) are the reduced chemical potential and internal energy per particle in units of l/e where E = 2.252 X lo-l4

palticlc. b) In dyne/cm 156

erg/

Volume

117, number 2

CHEMICAL

PHYSICS

and the ensemble averaged density profiles. The correction term -yc was obtained from run 3. This last run underwent a jump to filling during the course of the simulation as will be illustrated below. The density profiles for runs 1 and 2 are shown in fig. 2. Beyond 50 from both waIis “ringing” has essentially died away and the residual structure is due to the comparatively short length of the Markov chains generated. In light of similar results obtained m earlier studies of free liquid/vapor interfaces [4,5] which employed background stabilizing walls, we beheve that this residual structure does not seriously using eq. (6)

affect

the estimates

for y given in table

1. The pro-

file for run 3, which IS not shown here, displayed “rrnging” effects very similar to those shown in fg. 2 The bulk liquid densrtres obtamed in each of the runs (n;. = 0.831, 0.841, and 0 836 for runs 1, 2, and 3 respectively) are in good agreement with the experrmental value of 0.845 for krypton [ 131. The interfacial profiles shown in fig. 2 are asymmetric to a small extent. The reason for this is uncertain at present. One possible cause which we are currently investigating is that a rather slow surface oscil-

Fig. 2. Den&y profiles for runs 1 and 2 (a) N* = -10.780; (b) U* = -10.766.

7 June 1985

LEJTERS

Iation may be superimposed on the rapid density fluctuations which were occurring very close to the interfaces during the course of a CCEMC simulation. A further

point

to note

from

these profiles

is that

mean inter-facial thicknesses (Z in eq. (5)) in both runs are approximately 14-20% greater than the corresponding interfacial thickness of 1.54 (in units of I/u) obtained by Chapela et al. [5] for a PCEMC simulation of the free liquid/vapor interface of argon at p = 0.701. On the basis of the principle of corresponding states we suggest that the inter-facial broadening observed in the present studies is due to cooperatrve forces acting between the two Interfaces. The interfacial free energies given in table 1 also appear to ref,zct this effect. In the limit as the mean distance between the two interfaces approaches infinity, the inter-facial free energy per unit area reduces to the surface tension of the liquid in contact with rts saturated vapor. Thus, again invoking the principle of correspondmg states, the predicted surface tension for Lennard-Jones krypton at 115.8 K arising from the result given by Chapela et al. [5] would be 22.2 dyne/ cm for a PCEMC simulation of a free liquid/vapor interface with the fundamental cell dimensions used here. This result is about 20% higher than the results obtained in the present studies. On the basis of statistical variations in the ensemble averages of (ysti ) given in table 1, we estimate that a~ error of the order of SO% 1s involved in the values of-y. Therefore the significantly lower values for -y in table 1 result from a cooperative interaction between the two interfaces which lowers the potential field inhomogeneities normally present in free liqurd/vapor interfaces. The mean distance between the two interfaces, defmed as was found to be 8 430 in run 1 (I#)1 + l#)l>=h, and 7.29a in run 2. Thus, in view of the above comments, one would expect a lowering of-y from run 1 to run 2. The fact that this is nor actually observed is believed to be due to statistical fluctuations involved in the simulations. In run 3 a significant jump in the density of the par-hcles within the system occurred at an early stage in the simulation. With only a marginal increase in the chemical potentA from run 2 to run 3, the ensemble average number of.particles in the cell rose from 550.6 to 738.4. This is to be compared with the small change in UV) found for runs 1 and 2 for a similar change in chemical potential. In fig. 3 we illustrate the course

the

157

Volume 117, number 2

CHEMICAL PHYSICS LErrERS

?-June 1985

#J., shown in the figure; Over the short period between& = 2.2 X lo6 and NE = 2.5 X 106 more than 100 particles (“-4 layers) were-added to the system. The jump is more clearly ilhrstrated in fig. 3b where the partial ensemble average density at the centerline, z = 0, is plotted as a function of NC. For NE < 1.9 X 106 the densityisseen to fluctuate about the saturated vapor density for krypton at 115.8 IC. Beyond this point the vapor at z = 0 momentarily reaches a supersaturation level of lOO%, and over the natrow range 2.2 X lo6 < NE < 2.5 X 106, the central density increases by two orders of ma8rr&ude. For NE > 2-S X 106 the density finally relaxes to a value closely corresponding to the saturated liquid density of krypton at 115 8 IL

Fig. 3. (a) Partial ensemble avowe number of particles and (b) centerline density for NU 3 with p* = --10.750.

of events which took place before, during and after the period of instability. In fig. 3a we have plotted the partial ensembIe average number of particles in the system as a function of the total number of events NE from NE = 0 to NE = 4 X lo6 _ These partial averages were obtained from cumulative averages over a range *l@ events about a given value 0fNE Within the first 0 5 X 10S events approximately 50 particles were added to the cell. This roughly corresponds to the additron of a single layer of particles to each interface. In the next 0.75 X lo6 events two more layers were added to the system_ At this point the system appeared to settle out with all properties remaining relatively constant up to NE = 2.2 X 106_ Beyond 2.2 X 106 events however, density fluctuations within the narrow space between the two interfaces resulted in the rapid increase in 158

These results suggest that as two semi-mfinite bodies of liquid are reversibly brought into contact, the cooperative forces of interaction between the two bodies initially broaden the interfacial profrtes and lower the inter-facialfree energy. At a certain critical separation greater than zero, an tistability develops in the iuterstitial region and coalescence takes place_ It is also clar that these results have important imphcations for multilayer adsorption in microporous media. For example, one form of the Brunauer-Emmett-Teller (BET) adsorption theory 1141predicts that the thickness of the adsorbate layers on opposing surfaces in lamellar pores increases gradually up to filling. The present results show that capillary condensation, which is known to occur in cylindrical pores [1 11,also takes place in slit-shaped pores. Aclmowhdgement We wish to thank the UMR computer center for their assrstancein this work.

References [l] W.H. Marlow. J. Chem. Phys. 73 (1980) 6288. (21 & Schmidtcltt and H. Burtscher, J. Colloid Interface SC%89 (1982) 353. 431J-X, Lee. J-A. Barker and GM_ Pound. J_ Chem. Phyr 60 (1974) 1976..

Volume

117, number 2

CHEMICAL.

PHYSICS

[41 KS. Liu. J. Chem. Phys 60 (1974) 4226. [S] GA. Chapela. G. Sawlle. S.M. Thompson and J.S. Row&son, 3. Chem. Sot. Faraday Trans. I1 73 (1977) 1133. [6] M. Rao and B.J. Bcmc, Mol. Phys 37 (1979) 455. [7] S-M. Thompson and K.E. Gubbins. J. Chem. Phys. 74 (1981) 6467. [BIN- hfetropozis A-W. Rosenbluth, M-N. Rosenbluth. A-H. Teller and E. Teller. J. Chem. Phys 21(1953) 1087. [9] DJ. Adams, Mol Phys 28 (1974) 1241; 29 (1975) 307.

LE’ITERS

7 June 1985

[lG] KS.C. Freeman and 1-R. NcDonaId, MoL Phys. 26 (1973) 529. [ll] D. Nicholson and N.G. Parsonage,Computer simulation and the statistica mechanics of adsorption (Academic Press, New York, 1982). [ 121 O.K. Rice, Statistical mechanics, thermodynamics and _Enem (F~man. San Frandsco, 1967$ 1131 R-H. Davies, A-G Duncan, G. Saviile and LX Staveley, [14]

Trans. Faraday Sot. 63 (1967) 855. S. Brunauer, P.H. Emmett and E. Teller, J. Am. Chen SOC 60 (19383 309.

159