Oil-Water Interface in the Widom Model: Is It Smooth, Rough or Crumpled?

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Oil-Water Interface in the Widom Model: Is It Smooth, Rough or Crumpled?

Prabal Maiti, Debashish Chowdhury

To cite this version:

Prabal Maiti, Debashish Chowdhury. Oil-Water Interface in the Widom Model: Is It Smooth, Rough or Crumpled?. Journal de Physique I, EDP Sciences, 1995, 5 (6), pp.671-674. �10.1051/jp1:1995108�.

�jpa-00247092�

Classification

Pbysics

Abstracts

68.lom 82.70y

Oil-Water Interface in the Widom Model: Is It Smooth, Rough

or

Crumpled?

Prabal K. Moiti and Debashish

Chowdhury

Department

of

Physics,

Indian Institute of

Technology, Kanpur

208016, India

(Received

23 November 1994, revised 18

January1995, accepted

6

February 1995)

Abstract. Trie interface in trie two-dimensional

Ising

model with

only nearest-neighbour ferromagnetic

interactions is known ta be

rougir

at ail temperatures ^T > o. Trie Widom model for temary microemulsions

is, effectively,

_a

spin-1/2 Ising

model

with,

Dot

Orly nearest-neighbour ferromagnetic interactions,

but aise specific

farther-neighbour antiferromagnetic

interactions.

We

investigate

trie effects of these

farther-neighbour

interactions _m the Widom model _on the

roughening properties

of the interface

by

_carrymg eut Monte Carlo simulations.

Temary

microemulsions _are three component ^{mixtures where}

amphiphilic

molecules act as

surfactants between oil and water

iii.

Several lattice models of these systems ^{bave been} pro-

posed

^and _are similar in

spirit

to trie lattice _gas model for trie

liquid-gas

transition

[2].

^To

our

knowledge,

trie Widom model _[3] is trie

simplest

_among trie lattice models of

ternary

microemulsions. This

is, effectively,

_a

spin-1/2 Ising

model with not

only nearest-neighbour ferromagnetic interactions,

but also with farther

neighbour antiferromagnetic

interactions of

specific _type

descnbed below. Trie

farther-neighbour

interactions _arise from Widom's

prescrip-

tion for

incorporating

trie

bending rigidity

of trie

amphiphilic

membrane

(a monolayer)

formed

by

trie surfactants at the oil-water interface [4]. Trie interface between trie _up-spm and down-

spin

^{domains in trie} two-dimensional

Ising

model with

only nearest-neighbour

interactions is

known to be

rougir

at ail _non-zero

temperatures.

Trie atm of this communication is to

study

trie effects of trie

farther-neighbour antiferromagnetic

interactions _m trie Widom model _on trie

roughening properties

of trie interface. In _{contrast to} trie earlier series

expansion study

for trie three dimensional Widom model

by Kahng

et ai. [5] we carry out Monte Carlo Simulation of trie _two dimensional Widom model.

In trie Widom model trie

amphiphilic molecules,

_as well _as trie molecules of oit and

water,

are

assumed to be located _on trie

nearest-neighbour

bonds of _a

simple

cubic lattice. Classical

Ising spins

are

put

at each of trie lattice sites and _{one uses} trie convention that each bond between

nearest-neighbour (+ +) spin pairs

is

occupied by

_a molecule of

oil,

that between

(--) spin pairs

is

occupied by

_a molecule of

water,

and that between

anti-parallel nearest-neighbour spin pairs

is

occupied by

_a surfactant. Trie Hamiltonian of trie

system

_m ternis of trie

Ising spin

©

Les Editions de

Physique

1995

672 JOURNAL DE

PHYSIQUE

I N°6

variables is

given by

_[3]

H

=

-J£ _S~Sj

2M

£ _S~Sk

_M

£ Szsi (1)

where trie summations in trie

first,

second and trie third ternis _on trie

right

hand side _are to be carried _over the

nearest-neighbour,

the

second-neighbour,

and the fourth

neighbour spin

_pairs,

respectively,

_{on a}

simple

cubic lattice. The interaction J is

positive (ferromagnetic),

whereas M is

negative (antiferromagnetic);

the latter _anse from _a

particular prescription, suggested by Widom,

for

taking

into account the

bending

_energy of the

amphiphilic

membrane at trie

interface between oil and water. Trie M

= 0 limit of the Hamiltonian

(1)

is identical with the

ordinary Ising

^{Model with} nearest

neighbour

interactions _{on a}

simple

cubic lattice. The interface between domains of _up and down

spins

is known to be

rough

at all T

#

0. The _aim of this communication _is to

investigate

the effects of

non-vanishing

M _on the

roughness

of this

interface. We restrict _our

investigation

to space dimension d

= 2

only.

In _space dimension d

=

2,

the Hamiltonian for the Widom model [6] is

given by

trie _same form _as

(1) except

^that the summations in the third term _on the

nght

hand side _is to be carried ont over trie

third-neighbour spin pairs,

instead of trie

fourth-neighbour spin pairs.

Trie

phase diagram

of this model is

usually given

^{in trie}

(j, m) plane,

where _j

=

J/kBT

and _m

=

M/kBT,

both in d

= 2 [6] and d = 3

[7,8].

We bave focused attention

only

_on that part of the

phase diagram

on the

(j, m) plane

where the

droplet phase (1.e.,

^the

ferromagnetic phase

_m trie

spin language)

is

thermodynamically

stable.

We have used the standard

technique

of Monte Carlo

(MC)

simulation. The

system

consists of

L~

x

(L~

+

4)

lattice

where, by

convention X and Y denote the horizontal and vertical

directions, respectively.

Each _site of the lattice is

occupied by

_a classical

Ising spin

(S~ =

+1).

Penodic

boundary

condition _is

applied

in the X-direction whereas _a fixed

boundary

condition

is

applied

in the

Y-direction;

the

spins

in the two uppermost

layers

remain "frozen"

m the

up

(+)

_state, while those in the two lowermost

layers

remain "frozen" in the down

(-)

state

throughout

the simulation. In _some of _{our runs we} used _an alternative

boundary

condition

where,

instead of fixed

boundaries, anti-periodic boundary

condition _was

applied

in the Y-

direction.

However,

within the _accuracy of _our

computation,

the data for these two sets of

boundary

conditions _were

indistinguishable. Therefore,

almost all of _our

production

_runs

were made with fixed

boundary

condition in the Y-direction.

Application

of these

boundary

conditions _are used

routinely

_m

creating

interfaces in the

Ising

model where

majority

of the

spins

m the _Upper half _{are m} the _up state, whereas

majority

of the

spins

_m the lower half _are

m the down state. For each set of values of

j,

_m,

L~

and

L~

we

began

with _an initial state _m which all the

spins

in the Upper half of the

system

were in the _up state whereas those _in the lower half _were in the down

state;

^{this initial condition}

helps

_us in faster

equilibration

of the

system.

Dur

computations

were carned out on DEC ALPHA PB22H-CX and

HP9000/735

and the total CPU time

required

for the

production

_{runs were} 200 hours and 80 hours with

spin updating speeds

of 20 million

updates

and 1 million

updates

_per

second, respectively.

If the two nearest

neighbour

_{spms are}

antiparallel

_we call trie bond

connecting

these two

spms ^to be _a "broken" _one. After the

equilibration

of each

configuration,

_we measured the number of broken bonds

Nb(1/)

as a function of the row-index y; m the

droplet phase (ferro- magnetic phase),

_one expects a maximum m

Nb(v)

_near

L~/2, provided

accurate data have been

generated by averaging

_{over a}

sufficiently large

number of

configurations. Moreover,

_one

can calculate the width of the interface

by measuring

the full width at half _maximum

(FWHM)

m trie

plot

of

Nb(v).

In

Figure

1 we

plot Nb(v)

_{as a} function of _y for trie parameter ^values

j

₌ 1.0 and _m

=

0;

^this

corresponds

to a

two-dimensiona1Ising

model with

only nearest-neighbour

interactions. The

30 CO 500 00

L, 2000 L, 20000

400 00 60 00

~ ^{300 00}

j 40 00 £

200 00

20 où

ioo oo

~)

^{~~ ~~ ~~ b '°° Où 150 00}

~ ~~

y

Fig.

l. Two

typical plots

of

Nb(y)

for j ₌ 1-o, m = o-o _are shown. L~

= 2000 in

la)

and 20000 in

(b). Ly

₌ 200 in both

(a)

and

(b).

Trie data _m

(a)

and

(b)

bave been obtained

by

_{averaging over} 20 and 6

configurations, respectively.

18 oo

1300

~

~fi

8 00

/

3

0 00

~ i/z

2.

interactions

1.e., = o) are _plotted against _LÎ/~ for

three

different values

of j.

figures

1(a)

_and

(b) respond

to

L~

=

2000

and 20000, respectively. Similar _plots ere

obtained

for

j

= 1.25,

m

=

o-o and j

= 1.6,

m

=

_o-o- Since

two-dimensional Ising dels are known to vary as fi~ _ith

trie length ^L~ we show m _Figure

2

that

^our data are, deed,

consistent with

this

behaviour. ^oreover,

note

that for trie

saine

L, trie _smaller trie _value

of

j trie larger trie

width

of trie nterface, as _it

ext, in Figure 3, we _plothe width

of

the interface as a

function

_{of 4~} for trie

of

j

as m

Figure

2,

but now for

non-zero values _{of m.} Clearly, trie ^nterface is still _rougir in

spiteof the on-zero

values

of the

farther-neighbour interactions. Moreover,

note

_that, for _given L _and j,

trie interface

is

wider

Ising _interface for

which

^{m =}

0.

This anses

from

trie fact _that, for

given j, trie

system

is doser

674 JOURNAL DE

PHYSIQUE

I N°6

1800 J=10 m=-002

O

~

~1300

W10,

~

Î

j

^O

~l

6 m=-0 08

8 00

i~' 6, m=0 0

3 00

30

~,/2

Fig.

3. Trie width of the interface of trie Widom model (1.e., J

#

ù, _m

# ù)

_are

plotted against LÎ/~

_{for two different values of}

_j;

_the

corresponding plots

for _m

= 0 _are also

given

for

comparison.

to trie critical

point

for

larger

value of _{m as}

long

_as trie

sj,stem

remains in trie

ferromagnetic phase.

We conclude that

although

the

farther-neighbour

interactions _m the Widom mortel bave

crucially important

^effects on trie bulk

phase diagram

of trie

system,

these

bave, however,

no

observable elfect _on trie

roughening properties

of trie

interface,

_except that trie interface is wider when _m

#

0 than when _m

= 0 because of trie _doser

proximity

to trie cntical

point

in

case of _{non-zero m.}

Acknowledgments

We thank D. Stauifer for

suggesting

the

problem

durin~ a short visit of _one of _us

(DC)

tu

Cologne;

the visit _was

financially supported by

SFB341 and tue Richard Nixon fund. We also thank D. Stauffer for useful discussions and comments _on the

manuscript.

References

iii

Micellar Solutions and

Microemulsions,

S. H. Chen and R.

Rajagopalan,

Eds.

(Springer, Berlin, 199ù).

[2]

Gompper

G. and Schick M., Modem Ideas and Problems

m

Amphiphile Science,

W. Gelbert, ^D.

Roux and A.

Ben.Shaui,

Eds.

(Springer 1993).

[3] Widom B., J. Chem.

Phys.

84

(lQ86)

6943.

[4] ^Helfrich

W.,

Z.

Naturforsch.

C 28

(1973)

~93.

[Si

Kahng B.,

Rerera A. and Dawson

K.A.,

Phys. Rev. A 42

(199ù)

6ù93.

[fil Morawietz D.,

Chow~dhury

D., Vollmar S. and Stauffer D.,

Physica

A 187

(1992)

126.

[7] ^{Jan N. and Stauffer}

D.,

J.

i~hys.

France 49

(1988)

623.

[8] ^Dawson K-A-, Lipkin M.D. and lN'idom B., ^J. Chem. Phys. 88

(1988)

5149.