Lifetime of Newton black films. A mean field analysis

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Lifetime of Newton black films. A mean field analysis

A. Benyoussef, A. El Kenz, H. Ez-Zahraouy

To cite this version:

A. Benyoussef, A. El Kenz, H. Ez-Zahraouy. Lifetime of Newton black films. A mean field analysis.

Journal de Physique I, EDP Sciences, 1994, 4 (2), pp.245-252. �10.1051/jp1:1994136�. �jpa-00246901�

Classification Physics Abstracts

05.50 05.70F 75.10

Lifetime of Newton black films. A

_mean

field analysis

A.

Benyoussef,

A. El Kenz and H.

Ez~Zahraouy

Laboratoire de Magnétisme et

Physique

des Hautes Energies,

Département

de

Physique,

B-P-1014, Faculté des Sciences, Rabat, Morocco

(Recetved J9 Jaly J993, revised10 October J993, accepted 20 October J993)

Résumé. En utilisant

l'approximation

du

champ

_{moyen, nous avons} calculé le temps ^{de vie des} bicouches qui sont

thermodynamiquement

instables pour une configuration initiale

particulière

dans le modèle de Widom pour les micro-émulsions. Pour des configurations initiales identiques

de _ce modèle de bicouches, _nous étudions la dépendance du temps ^de vie en fonction de la

magnétisation initiale de la couche centrale, de la température, et l'élasticité de la membrane amphiphilique.

Abstract.

Using

a mean field approximation we calculate the lifetime of thermotlynamically unstable

bilayers

of _a

specific

initial conformation in the Witlom mortel of microemulsions. For itlentical initial conformations of these mode[

bilayers,

_we study ^the tlepentlence ^{of the lifetime} on

the initial

magnetization

of the central layer, the temperature. ^{antl the elastic} curvature of the

amphiphilic

membrane.

1. Introduction.

Membrane, foam and emulsion

bilayers

are

interesting examples

^of matter

organized

in _two

dimensions. These

bilayers

_comprise two

monolayers

^of

amphiphile

molecules

(Iipids

or

synthetic

surface~active agents ^called

surfactants,

for

example)

which stick fast _to each other.

Amphiphile bilayers generally

form _at the

boundary

between _two

contacting phases

which may be

equwalent

_or different. In systems ^with

highly developed

interfaces, for instance foams,

emulsions and suspensions, ^contact between the bubbles

(in foams),

the

droplets (in

emulsions)

and the sohd

particles (in suspensions)

^is made

through

thin

hquid

films which _are stabilized with surfactants. These films _can

gradually

become thinner until

they

transform into very thin

liquid

films which _are called _« black _» because

they

reflect

virtually

no

light.

^The

black films _are thinner than about

20nm,

and in _many cases

they

_can consist of

just

monolayers

of surfactant molecules. The

stability

of

disperse

systems

depends

on the

stabihty

of thin

liquid films,

and this may

explam why

these films have been _an important

abject

of

study.

The structures of

amphiphilic

membranes have been

an active field of _outrent

research

[Il.

These membranes _consist of either

«monolayers»

or

«bilayers»[2]

of

246 JOURNAL DE PHYSIQUE I N° 2

amphiphilic

molecules.

Monolayers

of

amphiphilic

molecules appear, for

e~ample.

in microemulsions

[3].

whereas

bilayers

of these molecules form _most of the

biological

cells

[4].

Significant

_progress has been made _over the last few _years in

understanding

the

therrnodynamic phases

of

complex

fluids

containmg amphiphilic

membranes

[5].

Part of this work has been

possible by using

_a class of models defined _{on a} lattice

[6].

The

simplest

lattice model of ternary microemulsions

(ail plus

water

plus amphiphiles)

is the Widom model

[7].

The

phase diagram

of this model

[8]

and _some

generalized

versions

[9]

of its _{are now} well known. The

structures of such systems ^{confined in model} _pores

by rigid

walls have also been

mvestigated loi.

There _are several well-known

regimes

of parameters m the Widom model where the

bilayers

_are

thermodynamically

unstable. Theoretical and

experimental

results _on the

stabihty

and

permeability

of membrane, foam and emulsion

bilayers

_are

given by

Exerowa et ai.

Il Ii

The lifetime of such unstable model

bilayers

has been studied

using

Monte Carlo

simulations

by Chowdhury

and Stauffer

[12].

The rupture mechamsm of Newton black films

(NBFS)

is described

by Chowdhury

and Stauffer

[13] using

Monte Carlo simulations. The purpose of this paper is to

study

the

dependence

of the lifetime of such unstable model

bilaj,ers

on the initial

magnetization

of the central

layer.

the temperature and the _curvature

elasticity

of the

amphiphilic

membrane. The

dynamical

_aspects of the Widom model of microemu[~ions _is studied

using

the _mean field

approximation.

The paper is

organized

_as follows _in section 2 _we

descnbe the model and method, results and discussion _are

presented

in section 3.

2. Model and method.

2,1 MODEL. In Widom's model the molecules _are identified with the bonds between two

nearest

neighbors

_on the lattice. Thus _two

neighboring

_{up spms}

ⁱⁱ correspond

to a water

molecule _in the

original mterpretation,

two

neighbonng

down spins

Î Î

_to

an ail molecule, and

a

pair

of

antiparallel neighbors

to an

amphiphilic

molecule. In the latter _case, the

hydrophihc

part ^{of the}

amphiphile

points in the direction of the up

spin,

and the

hydrophobic

_part to down spin.

By

construction, this model does net allow for où molecules dissolved in _water without

amphiphiles surrounding

them. TO allow for

complex phases

and low interface

tensions,

Widom's model _consists of three

types

^of interactions _on the square or

simple

cubic lattice.

Nearest

neighbors

_are

coupled ferromagnetically

with an

exchange

_energy J~ 0, 1.e. two

neighbonng

_spins

prefer

to be

parallel.

This facilitates the creation of like

neighbors

and hence

the creation of _{water or} ail molecules. In addition, Widom's model has

antiferromagnetic

interactions 2 M

~ 0 to next-nearest

neighbors,

which thus tend to be

antiparallel.

Third, it has

half _as strong

antiferromagnetic

interactions M~0 _to the

neighbors

which _are 2 lattice distances away. On the square lattice these

neighbors

are the third~nearest

neighbors,

have

distance _, 3 and _{are not}

coupled directly.

Since both

in-plane

and

eut-of-plane

^bonds

correspond

to molecules in Widom's model there _are ? L

layers

of molecules

corresponding

to L lattice

planes containing

the spms. Widom's

prescription

^for computing ^the curvature

energy

[14]

of the

amphiphilic

membranes _on discrete lattices

gives

_ose to

non-~anishing

interaction between the second- and the

fourth-neighbor

spm

pairs

on a

simple-cubic

^{lattice in} addition _to those between the

nearest-neighbor

ones. The total interaction energy is given

by

H=-JjjS,S~-2M jjS~Sj-M jjS,Si Il)

,, 2NN 4,,

where the summation in the first, second, and the third terms are to be carried eut over,

re~pectwely,

the

nearest-neighbor (NN), second-neighbor (2 NN),

and the

fourth-neighbor (4 NN) spin

_pairs on a

simple-cubic

^{lattice. S} ₌

corresponds

to up spins, S

= to down

spins. The interaction J

~ 0 is

ferromagnetic

whereas M _~ 0 ii

antiferromagnetic.

We

neglect

a

magnetic

field _term which

corresponds

to a chemical

potential

difference between oit and

water _; m our

calculations,

_up and clown spms have

equal rights.

The _nonzero interaction M is

essential to

assign bending

_energy to the

amphiphilic

membranes in Widom's mortel

(Gaussian

curvature energy is not included

[15]

in

(1)).

On the cubic

lattice,

each site has 6 _nearest

neighbors (J),

12 next nearest

neighbors (2 M),

and 6 distance

= 2

neighbors (M).

We refer the reader to reference

[7],

for _a

description

and

justification

of this mortel. We work with the

parameters

j

₌

J/KB T,

_m

=

M/KB T,

and _r

=

m/j

₌

M/J,

where T is the temperature ^and

KB is the Boltzmann _constant. The disordered fluid

phase corresponds

to the

paramagnetic phase

^whereas the oil~rich and the water-nch

phases correspond

to the

ferromagnetic _phases

of the

spin

system ^with

positive

and

negative magnetization, respectively.

It is _now well

established that in the parameter regime 0.1

~ r _~ 0 the

equilibrium

structure of Widom's model is oit nch

(water drops

in

ail)

or water nch

(ail draps

_in

water) provided j

is

larger

than _a critical value

j~ [12].

Therefore if _a

bilayer

is created

artificially

_m the system in this parameter

regime, it would

certainly

be unstable.

Using

_mean field

theory

we have studied the

dependence

of the lifetime of such unstable

bilayers

on the initial

magnetization

of the central

layer m~(0) (1.e.

the concentration of the

amphiphihc molecules)

_as well _{as on} T and _m.

We _now

specify

the initial structure and conformation of the mortel

bilayers [12].

The system consists of

Ising

_{spms on a}

simple

cubic lattice. A fraction

Co

of the

spins

m the central

plane

are

mitially

in the clown state

(1.e.

l _<

m~(0)

_<

0),

whereas ail the other spms in the system are

mitially

m the up state

(see Fig. l).

The _nearest

neighbor

bonds between the

spins

m

the

layer (z=3)

and the

Iayer (z=2)

and those between the

spins

_m the

Iayer (z

=

2)

and the

layer (z =1) together

represent a

bilayer

with _an initial concentration

Co

of the

amphiphilic

molecules. There _{is a}

single layer containing

water m between the two

monolayers

of

amphiphiles constituting

the

bilayer.

The spms m the two upperrnost ^{and the}

two lowermost lattice

planes

_are

kept

_« frozen _» in the up ^state

throughout

the

temporal

evolution of the system ⁱⁿ our mean field

approximation.

As _a consequence of the Glauber

dynamics

^{the number of}

amphiphilic

molecules _m the model

bilayer

_is net conserved

dunng

its

temporal

evolution. This _{case is} examined

by Chowdhry

and Stauffer

il 3]

_usmg Monte Carlo simulations.

t t t t t t z=5

t t t t t t z=4

111111 _z=3

1 1 1 1 1 1 z=2

111111 _z=1

t1t t t t z=0

« « « « « « z=-1

Fig.

I. A schematic

representation

of the _cross section of the initial

configuration

of the

spin

system ⁱⁿ the XZ plane. This spin

configuration corresponds

to a

bilayer

with

Co

= Ii,e, _mi (0)

=

m310)

= 1, and

m~(0

) 1). The symbols antl Î _represent the spms which _are

up-dated,

whereas the spms representetl by ( are kept frozen in their initial state.

2.2 MEAN FIELD APPROXIMATION. -We consider the

spins

of the

system

^descnbed

by

the

Hamiltoman

(1)

which is

represented by

the spm variable

ii ). They

also interact with _a

large

heat bath. at a constant temperature ^{which will} not be treated

explicitly.

The heat bath

248 JOURNAL DE PHYSIQUE I N° 2

functions

only

m

giving

rise to spontaneous

flips

of

spins by exchanging

the energy.

Using

the

mean field method, ^the equations ^of motion are given

by

:

yfi=-m,+tanh (flJ[i(m- ~+m,,,)+(1+8r)(m-

_j+m,

j)+(4+12r)1>1])

~Î

2

with ₌

= 1,

2,

3 and the initial conditions

m~(t

=

0

=

m=(0),

and the

boundary

conditions m_ _i

(t)

=

mo(t)

=

m~(t)

=

m~(t)

₌ 1.

The functional free energy of the system ^is given

by

_:

-~(1'(1l1-

m-

~+m,~?)+(Î+81')(1l1- ~+1l1-~j)+(4+Î2i)1l1) (3)

2 ^~

At the

global

^minimum of

3),

the magnetization _per site in the

layer

_z is

(S~)

₌ ^mf"~ and the free energy of the system ix F

=

F

[m?'~].

If _we define the function

f(m-)

_as fol[ows,

f(m-)

=

tanh

(flJ[i

(m-

~ + m- ~) + l ₊ 8 _r )(m, + m, + (4 ₊ 12 r)

m-j

(4j

equations (2)

become

n':(1) ^dm=

t

,~ joj ^m,

f (m-)

_Y ^~~~

Using

the

general

_trapezes formula

[17],

the

magnetizations

_{are given}

by

_: Ni

m-(t)

=

m,(0

) ^~ ₌ l 2, 3 (6)

m-(0) + m,(t ~ ' i

+

jj

2

,

«-

-f(,<,=)

m~(t

m,(0)

where .t.,,

=m,(0)+1

and N _is the subdivision number of the interval

N

[m~(0),

m-(t

Ii. Equations (61

_are solved

by

iteration. Different initial guesses can lead _to two solutions,

namely

mj(t)

_~

0, m~(t)

_~

0, mi(t)

_~ 0 _or

mj(t)

_~ 0, m~(t) _~ 0, mi(t) _~ 0.

the _one which makes the variational functional

(3)

smallest _is selected _as the

global

minimum.

3. Results and discussion.

The

equilibrium configuration

of the model

(1)

under the

boundary

^{conditions mentioned in}

figure [12]

is an off nch

fluid(ferromagnetic

m the spin

terminology)

^for

j higher

than _a

cntical value

j~

in the range 0

~ i ~ 0.1. Therefore, the

bilayer

created in the sy~tem

by

the choice of the initial spm

configuration

is unstable. We _are interested in the lifetime of such

unstable

bilayers.

We

implement

the time evolution of the system

by updating

the

magnetizations

of the three central

layers according

to equations (6). As a consequence of the Glauber

dynamics

the number of

amphiphilic

molecules _in the mortel

bilayer

is not conserved

during

its

temporal

evolution. However the rupture ^{of the}

bilayer

is defined in _{our mean} field

approximation by

the transition of the

magnetization

of the central lattice

plane

from _a

negative

to a

positive

value. Hence _we define the lifetime

(r)

of the

bilayers by

the

following properties,

ÎÎÎÎ ÎÎÎ

=

ÎÎÎÎÎÎ Î ^{~~ ÎÎÎÎÎÎ} Î

~ÎÎ Î

_~~~

In order to

study

the Iifetime of the model

bilayers

_{as a} function of

m~(0)

at several temperatures we have

computed

_r ^for

m~(0)

= 0.I to

m~(0)

= 1.0

by varying j

^and m

along

the fine _r

= const, where the numerical value of the _{constant was} chosen

arbitranly.

We compare our results _to those of the Monte Carlo simulations of reference

[13].

For

this,

_i is chosen _to be In

figure

2a the Iifetime ris shown _{as a} function of _m~

(0)

for several values

15

of

j. Higher j (and m) corresponds

to a Iower temperature. ^At any given finite temperature, ^ris small for small and intermediate values of

m~(0)

^but increases

sharply

_near

m~(0)

=

1.0.

The lifetime of the mortel

bilayer

is _{maximum at}

m~(0)

=

1.0.

Moreover,

for _a given initial

magnetization m~(0), r(T)

decreases when

increasing

temperature m agreement with Monte Carlo simulations results

[12].

The lifetime calculated with _mean field

approximation

is greater than that of Monte Carlo simulations for _a fixed temperature ^T.

As

explained

in reference

[12],

_nonzero M

distinguishes

an

amphiphilic

membrane from _an interface _m the

Ising

mortel. In order _to show that _{we are}

observing dynamical properties

of

membranes and _not those of interfaces in the

Ising

model _we have

represented

_{ras a} function of

m~(0)

for the _same numencal values of

j

_as those of

figure

2 _a but

always keeping

m=0

(Ising model).

We find that both the

qualitative

_as well _as

quantitative

features of

r(m~(0))

data for the model

(1)

_are

quite

different from those for _a

simple Ising

model but _m agreement ^{with Monte Carlo} simulations. When _m

= 0, _r

(m~(0))

is much smaller thon check for _m ~é

0,

and for _a

given m~(0),

_v is almost

independent

of T for aII

m~(0)

smaller than _a

certain initial

magnetization

of the central lattice

plane (see Fig. 2b).

In

fact,

we observed that

even for m~

(0

=

0.5,

_ris almost

independent

of temperature when

j

is vaned over an order of

magnitude.

Thus

r(m~(0)

=

0.5

=

0.44 for

j

=

4 whereas

r(m~(0)

=

0.5

= 0.74 for

j

=

0.5. This _must be contrasted with

r(m~(0)

=

1)

=

9.45 for

j

=

0.8 and

r

(m~ (0)

=

1)

- 2.475 for

j

=

0.5. In order to compare

quantitatively

our results, obtained

by

the _mean field

approximation,

with those of the Monte Carlo simulations _we give the

lifetime of _some initial concentration for

j

₌ 0.4 and _m

= 0.

r(m~(0)

₌ 0.95

w 2.085 for

mean field

approximation,

whereas

r(m~(0)= 0.95)

w1.86 for Monte Carlo _simu-

lations

[12].

In order _to

study

the

dependence

^of Ton m within the _mean field

approximation,

we hâve

kept j

^fixed at an

arbitranly

chosen value

j

=

III.5 and

computed

_r ^{for several} values of _m

(see fig. 3).

The

dependence

of the lifetime _Ton the temperature, ^{for several values of the}

initial central

magnetization m~(0),

and fixed value of _i~

= is

represented

in

figure

4, 15

from which it is clear that the lifetime _{r increases} with

increasing

temperature. ^In agreement

with the results of Monte Carlo simulations, we find that _{r increases} when increasing

(m(. Thus, (m(,

tends to stabilize the mortel

bilayer

(1).

4. Conclusion.

Using

_a

simple

_mean field

theory

^{that is} a

good

approximation to Monte Carlo results _we hâve studied the

dependence

of the lifetime _on the initial

magnetization

of the central Iattice

plane

of

thermodynamically

unstable

bilayers

of _a

specific

initial conformation _m Widom's mortel of

microemulsions. The Iifetime of this

bilayer

is an mcreasing function of the initial

250 JOURNAL DE PHYSIQUE I N° 2

200

- 2.50

- 2.00

- .80

- 1.70

-- l.50

Z _- 1.35

2

₁₀₀

1

o

-i o o,5 o,o

a)

20

- .20

--- .25

- .35

- 2.00

- 2.50

w

E

w

~

o

Initial

b)

ioo

-- 8

-- 20

80 _- 30

- 40

~

- 60

E ⁶⁰ ---c--- oo

j

à

40

20

o

-i,o o,5 o,o

In Ill ai _ma g ⁿ et iz ai ion

Fig. ^{3. The hfetime} T of the

bilayer

depentlence _on the initial magnetization _m~ (0) _{at a} fixent value of temperature (T/J ₌ ⁵). ^{The number}

accompanying

each _{curve is} the value of (- 1/J.).

400

-- -1 .o

- -0.9

---o--- -O.8

---*--- -O.7

300 _- -O.6

---OE-- -O.5

- -O.4

C

iÎ

~

₂₀₀

oo

o

,

O 1, 5

2,

0

2,

5

Temperature

Fig. 4. The lifetime _r of the bilayer

tlepentlence

on the temperature for _a fixent value of the ratio

i = -1/15. The number accompanymg each _curve tlenotes the value of the initial magnetization

m~(O).

252 JOURNAL DE PHYSIQUE I N° 2

concentration, and decreases with

increasing

temperature. ^{Dur results} are in agreement ^with

experiments

of Exerowa et ai. Il and Monte Carlo simulations

[12].

Acknowledgements.

Une of the authors A.

Benyoussef

is

grateful

to Prof. Stauffer for

helpful

discussion~. A.

Benyoussef acknowledges

the

hospitality

of the Institut für Theoretische

Physik

der Universitàt Zu KôIn, where this work _was finished.

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[16] Su?uki M. and Kubo R., J Pfiys ^Soc. Jpfi 24 (1968) 51.

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