Surface tension and interface fluctuations in immiscible lattice gases

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Surface tension and interface fluctuations in immiscible lattice gases

Christopher Adler, Dominique d’Humières, Daniel Rothman

To cite this version:

Christopher Adler, Dominique d’Humières, Daniel Rothman. Surface tension and interface fluctu- ations in immiscible lattice gases. Journal de Physique I, EDP Sciences, 1994, 4 (1), pp.29-46.

�10.1051/jp1:1994119�. �jpa-00246889�

Classification Pllysics Abstracts

68.10C 05.70F 47.55K

Surface tension and interface fluctuations in inuniscible lattice gases

Christopher

Adler

(~), Dominique

d'Humières

(~)

and Daniel H. Rothman (~>*)

(~ Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Tech-

nology, Cambridge, MA 02139, ^U-S-A-

(~) Laboratoire de Physique Statistique, C.N.R.S.,

École

_Normale Supérieure, 24 _rue Lhornond, 75005 Paris, ^France

(Recel

vert 3 August 1993, accepted 8 October

1993)

Résumé. partir ^de l'approximation de Boltzrnann, _nous calculons la tension superficielle

en fonction de la densité _{pour un} modèle de _{gaz sur} réseau décrivant à deux dimensions des fluides irnrniscibles _avec conservation de l'impulsion. Les résultats des calculs, qui prédisent _que

la tension de surface disparaît _en dessous d'une densité critique, sont comparés _{aux mesures}

faites à partir de la simulation d'interfaces planes et de bulles; l'accord entre la théorie et les données est qualitativement bon. Les fluctuations à l'équilibre d'une interface plane sont

aussi étudiées. Empiriquement, les fluctuations observées _se comportent de manière classique,

décroissant

comme l'inverse du carré du vecteur d'onde et suivant qualitativement _une relation

d'équipartition de l'énergie de surface.

Abstract. Using _a Boltzmann approximation, _we calculate the surface tension _{as a} function of population density in _a momentum-conserving lattice-gas model of immiscible fluids in two

dimensions. The calculation, which predicts that the surface tension vanishes below _a critical

density, _is compared to measurements made from simulations of flat interfaces and bubbles;

the fit of theory to data _is qualitatively good. Equilibrium fluctuations of flat interfaces

are

also studied. The fluctuations

are empincally observed to be classical, decaying like the _inverse square of wavenumber and obeymg quafitatively the equipartition of surface _energy.

1. Introduction.

Since trie introduction of

lattice-gas

models of

hydrodynamics il,

_2], considerable effort bas been devoted to trie

adaptation

of these models for trie simulation of interfaces in

hydrodynamic

(*) Permanent address: Department of Earth, Atmosphenc, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, U-S-A-

flows,

both for the _case of interfaces

separating

^different

species

of

partides

_[3-6] and interfaces

separating

different

thermodynamic phases

of the _same

species

_[7-10]

Although

ail of these models bave been shown

empirically

to

reproduce

the

physics

of interfacial tension, and various arguments ^{have been}

given

to support ^the presence of surface tension in trie

two-species

models [5,

11],

no theoretical

predictions

of trie surface tension bave yet been made. Here _we present

such _a

study

for _a

particular example

_[3] of _a

two-species,

_or immiscibie

?attice-gas,

model.

A

study

of surface tension in

lattice-gas

models of

hydrodynamics

is

interesting

for several

reasons. First, from _a

practical point

of view, it

gives

_{one a} better

understanding

of llow this

parameter _may ^{be varied for}

applications

and

provides

_a basis for

understanding

_any

anisotropy

in interfacial tension.

Second,

from the

viewpoint

of

theory,

it

gives

additional

understanding

of

phase

transitions in models of this type. ^{As bas been}

previously

discussed

[11-13]

,

immiscible lattice _gases

(ILG'S)

exhibit _a

phase

transition from _a

mixed, one-phase

state, to _an

unmixed, two-phase

state. This transition bas been understood in trie past via _a calculation of trie

diffusivity [11],

but _never

directly

in _terms of surface tension. Our

prediction

of surface tension

as a function of particle

density

shows

dearly

_a

phase

transition from

vanishing

to finite surface tension at a critical

density

de, and confirms

quahtatively

the previous estimate of de made from the calculation of

diffusivity.

To

predict

^{the surface} tension _{m a} lattice _{gas, we} consider

planar

interfaces in _a two- dimensional fluid, onented either

parallel

_or

thirty degrees

from _a lattice direction. The in- terface is

given

finite width, and is described

dynamically by coupled

Boltzmann

equations.

The

magnitude

of the surface tension is then derived from the

stationary

solution _to these

coupled

Boltzmann

equations.

As _is

typical

in trie estimation of transport coefficients _in lat- tice gases, trie Boltzmann

approximation

of _no correlations aids trie calculation

considerably.

However,

because surface tension arises in part from trie correlations

themselves,

_we obtain

only

_a

qualitative,

rather thon

quantitative,

match between trie surface tension obtained via

the Boltzmann

approximation

and that observed in simulations.

As _a kind of

application

of what _we bave learned about

lattice-gas

surface tension from _our Boltzmann

approximation

^and

empirical

measurements, we condude _our

study

with _a brief

investigation

^of interface fluctuations.

(This subject

has been

previously

^discussed in Ref. [14]

in trie context of _a

lattice-gas

model with

explicitly

constructed interfaces. We find that trie

equilibrium

fluctuations of ILG interfaces

decay

like the inverse of wavenumber

squared,

and

are thus,

by

virtue of _an

assumption

of

equipartition

of surface _energy, in

qualitative

agreement with dassical

theory (e.g.,

_[15]

).

In what

follows,

_we first

precisely

define _a

microdynamical

equation ^{for trie} evolution of the ILG. We then

approximate

the

dynamics

of this

equation by neglecting

correlations

(the

Boltzmann

approximation),

and seek the

stationary

solution of _{a set} of these

equations

_cou-

pled

_{across an} interface. The

stationary

solution _is obtained

numerically,

and is

compared

to

measurements obtained

directly

from ILG

simulations,

from both bubbles and

plane

interfaces.

The

study

of interface fluctuations then follows.

2.

Microdynamics.

In the usual _manner, the two-dimensional lattice _gas

il,

_2] is defined _{on a}

triangular

lattice with vertices

pointing

in directions

given by

^trie unit vectors

c~ =

(cos 27ri16,sin 27ri16),

₌ 1,.

,

6.

(1)

Thus the lattice is oriented with _one of the _axes

parallel

to the horizontal direction, _as shown in

figure

1.

~ ~

~2 ~l

~ ~

c~ c~

~ ~

~4 ~5

Fig.

1. Correspondence of trie vectors _c, to trie vertices _on the hexagonal lattice.

In trie immiscible lattice _gas [3], ^{each site} on this two-dimensional lattice contains trie 14-bit state defined

by

s =

(r, b)

=

(ro,

_{ri r6 bo} bi b6

(2

where the bits _ri and _b~ represent ^the presence

(1)

_or absence

(0)

of _a "red"

partide

and "blue"

particle, respectively,

_moving with

velocity

_one in the direction _c~, <1 < 6, or with

velocity

zero in the direction _{c0 =} 0 when

= 0. Since _r~ and _b~ cannot both

equal

_{one, we may} define the additional Boolean variable

n~ = ri + _b~

(3)

to mdicate the _presence of either _a red _or blue

particle moving

with

velocity

_c~.

The collisions of

particles

at time t and their motion from sites located _at

position

x to sites at x + c~ are

expressed symbolically by

the

coupled microdynarnical equations

r~(x

₊ c~,t +

ii

=

r((x,t) (4)

b~(x

+ ci,t +

ii

=

b[(x,t), (Si

where the result of _a

collision,

_{ri ~}

r[,

_{b~ - b[,} is

given by rl(x,t)

=

C[(S(x,t), f#) (6)

bj(x,t)

=

c)js(x,t), _{f~). ii)}

The collision operator C( ^E

(0,1)

takes _as

input

the 14-bit _state

six)

and the

discrete,

scalar-

valued,

color-field

angle f~

and

gives

_as output the state of the ith element of _species

j

E

(r, b)

after collisions have occurred. Trie discrete color-field

angle f~

enters the

dynamics

in the

followmg

_way. Trie collision operators

C[

and

Cl

_are constructed such that collisions maximize trie flux of

color,

6

qlr', b')

=

LCz(rl bl), (8)

z=i

in trie direction of trie

gradient

f of

color,

_{or, m} other

words,

such that q f is maximized.

(If

the choice of r' and b' _is

non-unique,

then trie result of _a collision _is chosen

randomly

from this

set.)

The color

gradient

f is

given by (within

_an irrelevant

constant)

6

f(X)

"

~

C~#j,

(9)

i=1

where the relative color

density

#~ at the site located at

position

_x + c~ is

~ ⁼

~ ÉÎrJ(x +c~)

b J=o

J x + c~)j

(io~

=

fi4(S(x+Cz)). (ii)

Here the

color-counting look-up

table fi4 has been

implicitly

defined. Since #~ E

(-7,

-6,

,

6,

7),

there _are 15~

possible

color distributions

(#~ )~=i,...,6

^Because 1) many of these distributions

are

redundant; 2) only

trie orientation, not the

magnitude,

off determines the outcome of _a

collision;

and

3)

small differences _m orientation _are

insignificant

for trie creation of surface tension, f is

typically

discretized further _{to a set} of discrete field

angles,

denoted

by

trie

angle

code

f~, uniformly

distributed from 0 to 2x.

Here,

as elsewhere [3], we allow for 36

possible angles,

and therefore 37

possible

values of

f~. (Trie

extra

angle

code allows for trie _case f

=

0.)

More _severe

discretizations, however,

_are

possible.

Trie transformation of trie caler distribution

(#~)

to trie discrete

angle f~

is

symbofically represented by

trie operator

(or look-up table)

T such that

y(ji~j)

₌

f~. (12)

We note that other forms of trie immiscible lattice _gas do not approximate f

by equations (11)

^and

(12),

but instead _use colored "holes" in addition to colored

partides,

and obtain _an estimate of trie

gradient

^from trie information

propagated by

both trie

partides

and hales [4-6].

3. Boltzmann

approximation.

To obtain _an estimate of surface tension, _we first need to express

equations (4)

and

(5)

in terms of the evolution of _a

probability

field. In _an ensemble of

systems prepared

^with different initial

conditions but

subjected

to the _same extemal constraints, _we define the average

quantities

R~ ₌

jr~i,

B~ ₌ jb~i,

j13)

and also

N~ ₌ (n~) = R~ +

B~, (14)

which _are,

respectively,

the

probability

of

observing

_r~

= 1, _{b~ =} 1, and _n~

= 1. We then

make the assumption that ail

partides

_are uncorrelated with trie others. This is the Boltzmann approximation ^{of molecular}

chaos,

well-founded for

estimating lattice-gas

_transport coefficients such _as viscosity

[16],

but of

questionable validity

for the estimation of surface tension _smce trie surface tension itself arises in part ^from correlations. Nevertheless such _an

approximation

allows _us to make _progress, and _{serves as a} useful reference for better

approximations

that may follow.

Thus, specifically,

for the evolution of the red

partides

_we write

~~~~

_{~ ~~~} ~ ~~ =

sÎ, ~~~~~'~"

~*~~'~~~~>

t)Q(f#;

_x,

t),

_~~~~

and for evolution of the blue

particles

we have

B~(x

+ c~, t +

1)

=

~j _{b[A(s, s', f~ )P(s;}

x,

t)Q(f*

_ix,

t). (16)

s,s>, f.

Here the _{sums are} taken _over ail

possible

states s that may ^enter a collision, ail

possible

states s' that _may result from _a

collision,

and all

possible

discrete field

angles f~.

The factor

A(s, s',

f~

represents ^the

probability

of

obtaining

state s' when state s enters a collision. The

probability

that state s

actually

enters the collision at time t at the site located at position _x is

given by

6

P(s;

_x,

t)

=

fl Rl'Bl'(1 _N~)~~~'~~' (17)

1=0

The

probability

that the discrete field

angle

is

f~

is

6

Q(f#; x)

=

L fl W(#z;x)

,

(18)

iw,iT(iw,ii=f. i=1

where the relative color

density

_{#~ was} defined in

equation (10).

Here the _sum is taken _over all

possible

color distributions

(#~)

that

correspond

to

f~,

and the

product

is taken _over the

probabilities W(#~)

of

observing

the relative color

density

_#~ at the ith

neighbor. Specifically, W(#~)

is

given by

tlle _sum of tlle

probabilities

of ail _states that

yield

the caler

density

#~:

W(#~; x)

=

~j _P(s;

x + c~).

(19)

S=fi4(S)=4,

4. Surface tension calculation.

To calculate the surface tension, we note that in the

vicinity

of _an interface the _{pressure is}

locally

anisotropic, since the pressure in the direction

parallel

to the interface is reduced

by

the tension _on the interface itself. For the _case of _a flat interface

perpendicular

to the

z-axis,

the surface tension _a is given

by

the

integral

_{over z} of the difference between the component FN of pressure normal to the interface and the component PT transverse to the interface

[17]:

a =

/°° ^{p~(z) p~(z)dz. (20)}

In mechanical

equilibrium

_one has

PN(z)

=

P,

the

(isotropic)

_pressure far from the interface;

at this juncture however _we retain the

z-dependence

for

generality. Equation (20) gives

the surface tension _{as a} function of the _pressure. The _pressure in lattice gases is

given

in tensorial form

by

_[2]

6

~afl

_"

~

CmC~pfil~

(21)

1=0

where

a and

fl

_are tensor indices. Prediction of the surface tension is thus _a

problem

of

predicting

the distribution of the

populations

_{N~ near an} interface.

To determine surface tension _m the

ILG,

_we

study

two _{cases: one} in which the interface _is

parallel

to _a lattice direction

(say,

c6)> ^{and the other} in which the

perpendicular

to the interface

is

parallel

to a lattice direction. See

figure

2. The former _{case is} called the

"0-degree interface,"

while the latter _is called the

"30-degree interface,"

where the number of

degrees

refers to the smallest

angle

the interface makes with _a lattice

axis.

Below,

_we describe the surface-tension

calculation for the

0-degree

interface for the

simplest

_{case m} which it is

composed

of

only

two

layers.

An extension of this calculation to _an interface which is 4

layers

thick is given in

Appendix

A. The

equations

for the

30-degree

interface _are

given

in

Appendix

B.

As shown _m

figure

2a, the _center of the

0-degree

interface is taken _to be between and

parallel

to two

(horizontal)

lattice hnes. The upper line _is labeled _yi and the lower line _is labeled _y-1.

interface

Yi

4- interface _x~

Y-i

(ai

jbi Fig.

2. a) The 0-degree interface. b) The 30-degree interface.

We _{assume an} average of id

particles

_per site for from the

interface;

in

equilibrium,

therefore,

we must have

partides arriving

at interface _sites with

probability d, independent

of time. This fixes the

boundary

^conditions

N~(yi, t)

= d, ₌ 4, 5 Vi

(22)

at all sites in

layer

_yi and

N~(y-i t)

= d, ₌ 1,2 Vi

(23)

at all sites in

layer

_y-i These

boundary

conditions

require

that the

populations

N~ be symmet-

nc across the the center of interface after rotation

through

180

degrees.

Thus for the moving

partides

N~(yi,t)

=

N~+3(Y-i>t),

₌ 1,.

,

6,

(24)

where the circular shift 1+ 3

=

j

such that cj = -c~,

j

₌ 1,.

.,

6,

while for the _rest

partiales No(Yi>t)

=

No(Y-i>t). (25)

Thus m this two

layer

_case, the

dynamics

of the interface _can be

completely

determmed

by solving only

for the

populations

in

layer

_yi Within this

layer, requirements

of symmetry ^and

mechanical

stability

^{further reduce the}

remaining

rive

populations

to

only

two

mdependent populations. Specifically,

mechanical

stability requires

that the _pressure be

divergence-free,

and therefore that FN

" P

= 3d. This

gives, by

virtue of the

boundary

condition

(22),

Ni"N2"N4"Ns"d.

(26)

In addition, since there is _no current

parallel

to the interface

(or, equivalently, by

symmetry with respect to the

perpendicular

to the

interface),

_we have

N~ ₌

N6 (27)

Thus the two free

population

variables _are the

rest-partide population,

No and _one of the

laterally moving populations,

_say

N3.

These

populations

evolve

according

to

N~(yi>t +1)

=

N((yi,t),

1= 0,3,6,

(28)

where the

post-collision

state is denoted

by

in which _we have also used

n[

₌

r[

+ b[.

The evolution of color _must also be

specified.

We work in _terms of the red concentration [ ₌

R~/N~.

In addition to the symmetry

given by equations (24)

and

(25),

_we also have

Ù~(yl>t)

_" ^Î

Ùj+3(Y-1>t),

# Î,...,6, (3Ù)

and

Ùo(Yi,t)

"

Ùo(Y-i,t). (31)

Additional

symmetries

and

stationarity

of the interface allow the _seven concentration vari- ables in layer _vi to be reduced to three

independent

variables. First, _we note that symmetry with respect to the

perpendicular

to the interface

gives

63 ₌ 66, ôi

= 62, 64

= ôs

(32)

Together

with the concentration ôo for the

rest-partiale population,

the first of these three pairs evolve

according

to

ô~(gi,t

_{+ 1)}

=

à((yi>t),

= 0, 3,6

(33)

where _we have used

à[

=

R[/N].

The evolution of the second

pair

of concentrations is deter- mined

by partides

that _cross the

interface; using equation (30),

_we obtain

ô~(yi,t +1)

=

à[~~(yi>t),

₌ 1,^2.

(34)

Since the

stationarity

of the interface requires that _no net concentration _crosses it, in

steady

state we must have

~~~~~~ ~~~~~~

Î'

_~~~~

and

therefore, by equation (34),

~~~~~~ ~~~~~~

Î'

_~~~~

It remains

only

to

specify

the red concentration coming m from afar. Since _m

equilibrium

the concentration that leaves the interface _must be

equal

to the concentration that enters

it,

_we

set

ô~(yi,t

+ 1) =

à(~~(yi>t),

=

4,

5.

(37)

Thus two of the three free concentration variables may be taken to be ôo and 63, which enter via equation

(33),

while the third may be taken to be 64> which ^enfers equation

(37)

above.

To

complete

the

specification

of the

problem

_we need _an

expression

for

Q( f~),

the

probability

of trie discrete field

angle f~.

^From

equation (18),

_{one sees} tllat all tllat is

required

is

required

is

knowledge

of

W(#~),

for

=

1,...,6.

Tllese

quantities

_may each be obtained from the

symmetries

and

boundary

conditions of the

problem. Noting

that

W(#o)

is the

probability

distribution for relative color

density

for the interface site in

loyer

_{yi, one} finds

w(i~)

=

w(i~)

=

w(io) (38)

for the

neighboring

sites in

loyer

_yi> and

Wl#4)

₌

Wl#s)

₌

Wl-#o) (39)

for the

neighbonng

sites _across the interface in

layer

_y-i For the sites _on the

boundary,

_one has,

assuming

that the chosen site in

layer

_yi is at position _xi,

N~

(xi

+ cj,

t)

=

d,

=

0,.

.,

6,

j

= 1,2

(40)

for the

populations,

and

ô~(xi

+ cj,

t)

=

ôj +3(x, t),

= 0,..., 6,

j

₌ 1,2

(41)

for the concentrations.

W(#i

and

W(#2)

_may then be calculated

directly

from

equations (17)

and

(19).

The

dynamics

^{of the}

two-layer 0-degree

^interface is thus

fully specified by equation (28)

for the two free

populations

No ^and

N3, equations (33)

and

(37)

for the three free concentrations

ôo, 63,

and 64, ^and equations

(38)-(41)

for the determination of the color field

angle.

Note that the

remaining populations

and concentrations _m the two

layers,

23 variables _m

all,

_may eacll be obtained

directly

from trie considerations of symmetry,

stability,

and

stationanty

detailed

above.

To determine trie surface

tension,

_{one may}

simply

seek trie fixed point, or

steady

state, of trie à-variable _{map given}

by

trie free

population

and concentration

equations. However,

since trie

use of collision tables mandates that the calculation be

performed numerically (to

evaluate the

sums and

products

_m

Eqs. (15)-(19)),

_we instead choose to seek the

steady

state of trie entire 14-variable system

(1.e.,

one

layer),

and then check to _see that trie relevant

symmetries

_are satisfied

by

trie calculation.

Specifically,

after

mitializing

trie calculation with N~

=

d, [

= 1

for ₁

= 0,.

,

6, we

successively

evaluate trie evolution

equations

until trie time t when

É(iNi(Yi,t) Ni(Yi,t -1)i~

₊

_iÙi(Yi,t) Ùi(Yi,t -1)i~)

_<

E,

(42)

where _we choose _e

=

10~~°

The surface tension may then be calculated from

equations (20)

and

(21).

In trie present

case, we bave

~~ ~~~

~

~~

~

~~

~

~~~

_~~~~

and

~~

~~

~

~~

_~

Î~~

~

~~

~

~~

~

~~~

_~~~~

where

NI

denotes the

steady-state post-collision populations.

The surface tension is then

given by

°

~ÎÎ~~~ ^~~~

~~~~

where the factor of 2 _comes from the symmetry ^of the interface and tlle factor of

và/2

_anses

from the lattice geometry.

5. Results: Boltzmann

theory

_vs. simulation.

We first

provide

_a discussion of the results of the theoretical calculation.

Then,

after

providing

an

empirical

demonstration of

Laplace's law,

_{we compare} the theoretical

predictions

with results from simulations. The simulation results _are obtained from both bubbles and flat

interfaces.

5.1 THEORETICAL _cuRvEs.

Figure

3 summarizes the results from the theoretical calcula- tion of surface tension. There _are two sets of _curves. The first set

(smooth curves)

is from the

prediction

^of _a for the

0-degree interface,

both for _two

layers

and four

layers.

One _sees that

0.S ;.---.-.,

,/ 30,J. ", 30,6." ,/"' ",".

0A

g

0.3 e

)

©

~ ^0.2

W

o-1

~~0.0 _{0.2 0.4 0.6 0.8 1-o}

reduced density

Fig.

3. Theoretical prediction of surface tension _{as a} function of reduced density d in the _irn- miscible lattice gas. Smooth _curves: Boltzmann approximation for 0-degree, 2-layer interface (0, 2)

and 0-degree, 4-layer interface (0, 4). Dotted _curves: Boltzmann approximation for 30-degree, 4-loyer

interface

(30, 4)

and 30-degree, 6-loyer interface (30, 6). In each _case, the thicker interface results in the greater surface tension.

the thicker interface

always yields

_a

larger

surface tension,

though

the two curves

effectively

converge for d > o-1- Thicker interfaces

yield

greater surface tension because the difference between P and PT _is not forced _{to go to zero as}

rapidly.

At

high densities, however,

there is

virtually

_no

mixing

of red and

blue,

and thus the interface

dynamics

is

adequately

descnbed

by just

two

layers.

One _sees

essentially

the _same behavior for the second set of _curves, which represent

predic-

tions for

30-degree interfaces,

for four

layers

and six

layers, respectively.

We note,

however,

that the

o-degree

surface tension and the

30-degree

surface tension _are in

general

not

equal,

thus showing that the surface tension _is

anisotropic.

The gross features of the _curves that

they

vanish below d

=

de

m

0.2,

rise to _a

single

maximum, and then

dropoff

towards _zero at d

= I.o is

perhaps

of the greatest ^{interest. The} appearance of the critical

density

de _is the signature ^of the ILG'S

phase

transition from the mixed state

(d

<

de)

to the

unmixed, two-phase

state

(d

_>

de). Broadly speaking,

there is _no surface tension for d _< de because the discrete model offers insufficient

degrees

of freedom to

create a stable interface at low densities.

(That

the theoretical estimate of de decreases _as the interface becomes thicker results from the additional

degrees

of freedom available to form _a thicker

interface.)

The situation at d

= I.o is somewhat

analogous:

in this _case, the fact that N~ ₌ ^{I.o for} all offers _no

possibility

for _an anisotropic _pressure. The theoretical maximum

surface tension at d _m o-S thus _occurs at _a

density

that

optimally provides

_an anisotropic pressure while

minimizing

_any

mixing

between the

phases.

We note that the

phase

transition at de _m 0.2 _was

previously

located in two different _ways [11].

First,

_a Boltzmann calculation and

empirical

measurement of the ILG diffusion coefficient

showed that the

diffusivity changes sign

from

positive

to

negative

as the

density

is increased past de.

Second,

simulations initialized _as

homogeneous

mixtures of concentration à

= o-S _were

observed to

spontaneously undergo phase separation

above de but not below.

5.2 LAPLAcE'S _LAW. Before _companng the results from the Boltzmann

approximation

and

simulation, we first

provide

_an

empirical

demonstration that surface tension exists.

Figure

4

is a

graph

of the difference between the _pressure inside _a

bubble, fin,

and the _pressure outside

a bubble, P~ut> as a function of the inverse of the bubble's radius R, for

density

d

= 0.7.

Laplace's

law [17] ^{states that}

fin P~ut _" ^°

(46)

R

m two dimensions. The pressure is calculated from

equation (21), which, sufficiently

far from the

interface, gives

P

=

Papô~p

₌

3p/7,

where _p is the average number of partides

(of

either

color)

_per site. To obtain fin and P~ut> _we locate the center of the bubble

by computing

the average location of each

partide

of the bubble's

color,

and then define fin to be the average

pressure within 0.7Ro of the bubble's center, ^and P~ut to be the average pressure ^{at sites} further than 1.3Ro from the bubble's center, where

Ro

_is the

input

radius. Bubbles with radii _rangmg from Ro

" 4 to Ro _" 64 were

placed

in boxes with

penodic boundary

conditions and lmear

O.io

o.08

w

(

_0.06

~ i

ce E .Oz

0.00

0.00 1/R

Fig.

4. Venfication of Laplace's law _m trie immiscible lattice gas. Bubble radii R range from

4 to 64 lattice umts. An estimate of surface tension _is given by the slope of the best fitting fine thon passes through trie engin. Error bars

(based

on bubble _size and number of lime steps ^{used for} averagmg) _mcrease finearly with 1/R, ^but _are always less than 10~~, and thus srnaller than the _size of the symbols.

dimension of at least

4Ro.

After

being

^allowed 1000 time steps to

relax,

AP

= fin P~ut _was

averaged

_over the

following

2 _x 10~ time steps.

(Two independent

measurements were

averaged

for the _cases Ro

" 4 and RD _"

5.) By assuming

that AP is due

entirely

to _an increase of the

pressure inside the

bubble,

_we calculate the

equilibrium

radius R of the bubble from

~~

Po ÎAP~~'

~~~~

where Po

" 3d.

(Though

_more

appropriate, using

R instead of Ro

changes

trie estimate of

the surface tension

by

less thon 1Yo.)

Figure

4 shows excellent agreement witll tlle

Laplace

law of

equation (46).

Here the

best-fitting

^line

passing through

the

origin gives

^the

empirical

estimate _a

= o.382 + 0.col for the surface

tension(~).

Note that _one expects

departures

from the

Laplace

law _as R _- o, in part ^{because trie} finite-state automaton cannot support

large

pressure contrasts.

5.3 COMPARISON _{OF SIMULATION AND THEORY.}

Figure

5 compares trie

0-degree, 2-loyer

Boltzmann

approximation

and the

30-degree, 4-layer

Boltzmann

approximation

to two types ^of

empirical

_measures of surface tension. The first

empincal

_measure,

given by

the

circles,

_comes

from measurements made from bubble tests as described above.

(For

densities d

#

o-1the fit

to

Laplace's

law _was

performed

with 4 points,

using

radii RD " 8, 12, 24, and

64).

The second

set of measurements are made from flat

interfaces, by empincally computing

trie

integral

_given

o.5

~'~

;"

~

à,

C .~

° O

0.3

~©

)

_O.Z

ce

o-1 ^".

~'~0.0 _{0.2 0.4 0.6 0.8 1.0}

reduced denslty

Fig.

5. Companson of Boltzmann approximations for 2-layer, 0-degree interface

(smooth curve)

and 4-layer, 30-degree interface

(dotted curve)

with three empincal _measures of surface tension. Cir- des: bubble tests. Squares: measurements from flan interfaces oriented at 0 degrees. Triangles:

measurements from flan interfaces oriented _at 30 degrees. Errer bars

are smaller than the

size of the

symbols.

(~) ^Details of this measurement, and therefore the results, dioEer from the bubble tests descnbed in reference [3].

o.5

o

Îf 0.5 OE

<

Oé 2.5

~

ogl

0

cycles/latt>ce unit)

Fig.

^6. Comparison of interface fluctuations ([Aq[~) obtained from simulations

(circles)

and those

predicted from classical theory

(straight

fine, given by Eq.

(54)).

by

equation

(20),

which _now takes the form

a =

Ii

~

L

~~

(CII

^lii

ni(zi>

_zii

(48)

~+<~ <_

~i~À<~

Here _c~i and cqj are the components of _c~

perpendicular

and

parallel

to the

interface,

_respec-

tively, (.)

indicates

averaging

over time, Li and Ljj represent the

length,

_m lattice units, of the lattice _m the directions

perpendicular

and

parallel

to the

interface, respectively,

and _xi and zjj ^{are the}

physical

coordinates of lattice sites.

Specifically,

_a computational box of size Li = 128ai

by

_{Ljj =}

64ajj

was mitialized with _a flat interface _m the center, where _ai and ajj

are the lattice

spacings

in the

perpendicular

and

parallel directions, respectively (e.g.,

_ajj

equals

v5/2

for

30-degree

interfaces and unity ^for

o-degree interfaces). Using no-slip

"bounce-back"

boundanes _on the two walls

parallel

to the interface and

penodic boundary

conditions in the other

direction,

_we allowed the interface 1000 time steps to

relax,

after which _we

computed

(n~) over 2.5 x 10~ time steps.

The fit of the Boltzmann calculation to the

empirical

measurements is

qualitatively,

but not

quantitatively, good. Qualitatively,

_{one sees} that the

empincal

and theoretical _curves have

approximately

the _same

shape

and their maxima _are at

approximately

the _same

density.

Moreover the anisotropy

predicted by

the Boltzmann

theory

is

evident,

with the _same

sign

and

approximately

the _same

magnitude,

m the measurements made from

interfaces,

while the

measurements made from bubbles fall

approximately

within the measurements made from 0-

degree

and

30-degree

interfaces. The poor

quantitative

fit _can be attributed to several factors.

First, and most

obviously,

the

neglect

of correlations _m the

theory

is _{a severe}

deficiency,

not the least because the interfaces themselves _are created

by

correlations.

Second,

because interface fluctuations

(see below)

_cause the true values of _c~i and

c~jj to differ from the assumed values used in equation

(48),

measurements made from interfaces _are

necessarily

approximate.

Lastly,

_we note that _companson of the

empincal

data with the Boltzmann calculations made from the thicker interfaces is somewhat less

good

at low densities than the companson with the Boltzmann calculation made from the thinner interfaces.

6. Interface fluctuations.

Having investigated

^{above the}

physics

of

stationary

interfaces _m immiscible lattice _{gases, we}

now tum to the

question

^of

fluctuating

interfaces. Interface fluctuations in ILG'S _are

interesting

for several _reasons.

First,

for

applications,

_one would like _to know if

quantitative

aspects of ILG interface fluctuations _are in agreement ^with theoretical

predictions

for dassical systems.

Second,

the

question

of agreement ^{with dassical}

theory

is _an

interesting

issue in and of itself.

Because the

microdynamics

of the ILG lacks semi-detailed

balance,

the model is

microscopically time-irreversible,

and therefore lacks _a dassical

description

in which fluctuations _are distributed

m

equilibrium according

to _a Gibbs distribution. Thus it cannot be

expected

_a priori ^{that the} dassical

theory

would hold.

In the

following,

_we first review what is

expected

for interface fluctuations in dassical sta- tistical systems ⁱⁿ

equilibrium,

and then _compare these

predictions

with simulations of ILG interfaces.

6.1 CLASSICAL EQUILIBRIUM THEORY. Classical interface fluctuations _may be understood

m terms of fluctuations of surface _energy [15]. ^For a one-dimensional interface _{m a} two-

dimensional _space, the _energy,

H,

of the interface _is

proportional

to its

length

L. Assume that the interface has

length

Lo when it is flat. If

h(z)

is the

single-valued

interface

height

_as a

function of

position

_z, then the

length

of the interface is

given by

~ ~ j

L =

/

^~ _dz

+

()) ₍₄₉₎

o 1

For

sufficiently

small fluctuations such that

dh/dz

< 1,

L _m Lo +

/ ^~

^~ _dz

_{~)) (50)}

1

~

Now express the interface _as the _sum of Fourier modes of

amplitude

_(Aq(

corresponding

to wavenumber _q:

h(z)

₌

~j

_Aqe~~~~

₍₅₁₎

q

Substituting

equation

(51)

into equation (SO)

above,

the interfacial _energy H

= aL is, to

leading

order,

H _m aLo +

°Lo ~ _{q~ÎAq(~, (52)}

2

q

where _{a is}

again

the surface tension.

By

the

equipartition theorem,

_one has

aLoq~ÎA~(~

=

(kT, ⁽⁵³⁾

or

(Aq(~ =

~~~

_~.

⁽⁵⁴⁾

oaq

Thus for _a dassical interface _m

equilibrium,

_{one expects} the _average

amplitude

of fluctuations to be

inversely proportional

to the _square of the wavenumber.

6.2 RESULTS _FROM SIMULATIONS. TO _compare the

prediction

of

equation (54)

with the fluctuations of ILG

interfaces,

the ILG _was initialized with _a flot interface of

length Lo

₌

Nv5/2 _dividing

the red fluid from trie blue fluid in _a box of

height

N, where N

= 256 and

the

partide density

d

= o-1- The

boundary

conditions _were

periodic

in the direction

parallel

to the

interface,

while walls _were

placed

above and below trie

interface,

with

no-slip boundary

conditions. After

allowing

trie system ^looo time steps to relax _to

equilibrium,

trie _power spectrum

11_~ 2

lAql~(~) "

j _{£ h(~,~)} ~~P(~iq~/N) (55)

~=0

was

computed

from measurements

h(z, t)

of the interface

heights

at each time step t and lateral location _z, and then

averaged

_over 10~ time steps. ^{The interface}

height

_was determined at each discrete value of _z

by searching

from above

(the

red

half)

for the

height

of the location of the first site

occupied by only

blue

partides,

then

searching

from below

(the

blue

half)

for the

height

of the first site

occupied by only

red

partiales,

and then averaging these two

heights.

If any

spikes

of

height

greater than _or

equal

to rive lattice units _were found in any interface

h(z, t),

this interface _was eliminated from the

averaging

to avoid _any measurement errors due to the small dissolution of each

phase

in trie other. This aspect ^{of the data}

processing

ehminated

approximately

^1.5% of trie interfaces from the

averaging.

Figure

^{6 shows}

log((Aq(~)

as a function of

log

_q,

compared

to the theoretical _curve obtained from equation

(54).

To compute ^the theoretical _{curve, we} bave used _a

= o.403, trie prediction obtained from the Boltzmann

approximation

for the

o-degree, 2-loyer

interface. For _an estimate of

kT,

_we have chosen the

equal-time

momentum-momentum correlation function

L _L(lez«(nz Nz)llcjp(nj _NJ

_III

=

3d(1 dlô«p,

(561

or,

equivalently,

the vanance of momentum fluctuations _{at a} site [18].

Comparing

trie two curves, one finds that the

slope

of the empirical curve is indeed

approximately

-2 for _wavenum- bers below _a

high-wavenumber cutoff,

and that the

prefactor (and

thus _our estimate of kT from trie

equipartition theorem)

is

close,

within about 25%. We

condude, then,

that ILG interface

fluctuations _are

qualitatively

in accord with the statistical mechanics of dassical interfaces.

That _we have recovered the relation (Aq_(~ oc

q~~ probably only

mdicates that ILG interfaces

are

"rough,"

in the _sense that their

slopes

_are random and uncorrelated. In other words,

h(z)

is

just

a random walk in _one

dimension,

and (Aq(~ oc q~~ results from the usual considerations of diffusive _processes.

However,

that the

prefactor

in equation

(54)

is also

approximately

recovered in the simulations is

relatively

remarkable. This agreement ^indicates net

only

that

our estimate of kT from equation

(56)

^is

approximately

_correct, but that the

equipartition

of

energy aise _appears to be honored by the simulation. Such _a result

implies

that the states of

trie system are distributed with Gibbs probabilities, which itself

implies

that the microscopic

dynamics effectively

^behaves as if it _were time-reversible. Such _a behavior could

possibly

^be

due _to the fact that the noise arises

mostly

_m the bulk

phases,

which _are time-reversible, while the irreversible _noise

generated

at the interface _is bounded

by

the finite thickness of the

interface,

and is thus

possibly negligibly

small.

7. Conclusions.

The most important result of this paper is contained in

figure

5. There _we have

compared

the

predictions

of _a Boltzmann

approximation

for surface tension m an immiscible lattice gas with