Simulation of membranes, micelles and interfaces with asymmetric surfactants

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Simulation of membranes, micelles and interfaces with asymmetric surfactants

Naeem Jan, Dietrich Stauffer

To cite this version:

Naeem Jan, Dietrich Stauffer. Simulation of membranes, micelles and interfaces with asymmetric surfactants. Journal de Physique I, EDP Sciences, 1994, 4 (3), pp.345-350. �10.1051/jp1:1994110�.

�jpa-00246909�

Classification Physics Abstracts

05.50 68.10 82.35 82.65

Short Communication

Simulation of membranes, micelles and interfaces with

asymmetric surfactants

Naeem Jan and Dietrich Stauffer

Physics Department, St. F-X- University, P-O- Box 5000, Antigomsh, N-S-, Canada B2G 2W5

(Received

22 December 1993, ^received in final form 13 January 1994, accepted 17 January

1994)

Abstract We show that _a model recently introduced to describe _a microemulsion is capable of capturing nlost of trie essential features observed experimentally. In particular _our Monte

Carlo results show trie expected variation of trie critical micelle concentration with length of surfactants, trie formation of micelles, the accumulation of surfactants at _an interface and aise trie variation of the denlixing temperature ^{with surfactant} length.

Microemulsions,

systems

composed

of water, oil and

surfactants,

show _a ricin

variety

of

phases

and self

assembly

of

complex

structures such _as

micelles,

bicontinuous

remous,

etc.

il,

_2]. A

simple

model introduced

by

Larson _[3] and Stauffer _et al. _{[4] seems} to capture ail the essentials of these

complex

_{systems, as seen} from the results of Monte Carlo simulations ai limite temperatures. The essential difference between this model and earlier models is _m the

description

of the surfactant.

The solvent is modelled

by

_a

spin 1/2 Ising

model where spm up

corresponds

to water and

spin

down to oil. Chains

of,

in

general,

_a

length

of six lattice

spacings

represent ^the surfactants.

If _{we use}

only

two mstead of three components, then all solvent

spins

_{are up} and these _are conserved the _two different components model surfactants in _water. The surfactant chairs have fixed spins. ^We use three different models for

surfactants; A,

B and C. In model A the chains have spins +2, 1, 0, -1, -1, -1, but head-head

(2-2)

interactions between two surfactants _are

repulsive

to take into account iomc

repulsion.

All orner interactions _are attractive. Model B has the _same head

repulsion,

but the

spins

_{on a} chain _are 2 _ai the

head,

-2 at the rail and

zero for ail

spins

_m between. Model C has _no head-head

repulsion,

and

spins

1, 0,

0,

0, 0, -1.

Additional _cases like

spins

1, 1, 1, -1, -1, ^-1 without _{any zeros were} simulated

by Larson, mainly

to

investigate

the

periodic

ordered

phases

of

three-component

models which _can also be found in

simpler

lattice models without chains [5]. ^{Our earlier work used models B} and C

whereas _{now we}

emphasize

model A and _use B and C

mainly

for

comparison.

We _use

always

a

simple

cubic lattice with L _x L _x Lz _spins;

typically

Lz is 50 whereas L

346 JOURNAL DE PHYSIQUE I N°3

varies up ^to 230. Our time t is the number of iterations where each iteration consists of _one Glauber sweep

through

the solvent

spins

which is followed

by

ter

slithering

snake attempts

to move each surfactant chain. In the twc-component case, the solvent

spins

do flot

flip,

and thus _an iteration consists of _ten attempts to move each chain. If _a chain _moves

by

_one unit, the

spin

ai the _new site to which the chair end _moves is put onto the site freed

by

the other end of the chain. This is the standard reptation

approach

to trie

dynamics

of _a

polymer

in _an

interacting

medium. We refer the reader _to

[loi

for details of the Monte Carlo method and also for _a

description

of the

slithefing

snake

algorithm.

With three components, oit, _water, and surfactants, _we look at interfaces and membranes.

For _an

interface,

half of the lattice _is

initially occupied by

water, trie other

by oil,

and then chains _were distributed

throughout

the whole volume. Model A shows the _same

tendency

^of

the chains to accumulate _near trie interface _as did mortel C in _our earlier work and _as do real surfactants. As _{seen m}

figure

1 trie surfactants concentrate at the interface and this _is followed

by

_a

depletion region

^which

slowly

fills in lime. If the chains

initially

_are

placed

in _a

completely

ordered

single layer

of L _* L

parallel chains,

with heads

adjacent

to water and tails to oil, then this ordered

layer

is broken _up, and the _rms width of the surfactant

layer

_increases with time.

At

moderately high

temperatures

(T

= 3.0

J/k)

the final

configuration

is

independent

of the initial

configuration

after

100,000

iterations. Ai lower temperatures

although

_we ^observe the dissolution of the

layer by

the diffusion of chains _away from the

layer

and the increase of the

rms width there is still _a

recognisable layer

after 100,000 iterations. We have flot

observed,

within the limits of _our simulations, the formation of _an ordered

layer starting

from _a random

configuration

of surfactants. These properties _were observed for A and C.

ail, Water and surfactants _at T 2.2, L 73

6000

~

D 8 o

~ D 8 ^D

~ D _D

~ ô _o

4000 _{o o} ° °

~

~

$c ⁸

Î ₂₀₀₀ ~~ ~

~

~ ~

~

~

~ ~ ~ ~

l

~ ~

~ ~ x x x t

~ x u + ~ ^~ +

~

~ 0 * ~ ~ ~ Î ~ + ( 4 ^{~ * ~ ~ ~ ~} x t _~

c ~

j

Î

~

f -2000

z c

~ o D

-4000 _{~ ~}

o D

~ ^u n o

resUlts _at t 1°°° and 25°°° MOS

o 9 ^{~ ~ 8} _à

-6000

0 5 10 15 20 25 30

Layer

Fig. 1. Interface profile for trie three-component model A. Diamonds and squares refer ta ail _con- centrations after 1000 and 25000 iterations,

(+)

and

(x)

ta surfactant concentration; the rest is water.

Another type ^{of interface} is formed

by

^surfactant chains which _{are as}

long

_as the lattice and which _are

initially

put

parallel

to each other to _cover the whole oit-water interface.

Agoni,

for

long enough

limes this ordered _{structure seems to} break _{up, as} shown

by

the _rms thickness of the surfactant

layer

_m mortel A. We also

investigated monolayers

^of

length 16,

1-e-

2,

1,

0, -1,

-1. These

layers

also show the _same properties e-g- the

tendency

to dissolve _as those with the Shorter surfactants of

length

6.

Membranes _are studied for both the

three-component

and the twc-component case.

Initially

the whole lattice is water, and then _a

completely

ordered

lipid bilayer

_is

placed

in the cerner of the

lattice, completely separating

the two water halves from each other.

Hydrophilic

^heads

point

outwards to the water, and

hydrophobic

rails inside _are next to each other. We observe _a

slow _mcrease in the _rms width of the

bilayer

ai intermediate temperatures ^and a fast increase

ai

higher

temperatures. This

instability

of the membrane is observed for models

A, B,

and

C,

even in the _case where in model C the

spins

_on head and tait have twice the

strength.

At _a temperature ^{of T} = 1.5

J/k

the

bilayer

remained constant for the duration of the simulation.

We cannot exdude that ai very low temperatures, ^{where the}

growth

is very

slow,

the

decay

would non stop

eventually.

Also,

_one lattice

plane

of water _was left between the two

planes

of rail ends to test the

ability

of the membranes for

self-organization.

With two components, the tail ends _move towards each other very fast, whereas with three components, the lattice

plane

between the rail ends is

immediately

filled with off, and the rail ends _move towards each other

only slowly.

The

instability

mentioned above _may be masked

by

the contraction of the membrane _to tilt the _gap.

2-component model A, kT/J=2.5, t=10~4 to 10~5, 78~3 _or 143~3 o-1

'larmic.fig' o

o

o o.oi

o

o

0.001 o

o

0.0001

5 6 7 8 9 10

chain length

Fig. 2. Exponential decay ^{of trie} CMC with chair length for the two-comportent ^{model A. See} text for details.

348 JOURNAL DE PHYSIQUE I N°3

In all _cases we found that the

bilayers

_are unstable above T

= 2.5

J/k. (In

the twc-

component case of model

A,

without head

repulsion,

when

initially

the heads _were inside and the tails

outside,

the

tendency

of the membrane _to dissolve _was

particularly weak.)

In ail the

above,

except where ii _was otherwise

stated,

the surfactant consists of six _monomers.

Longer

surfactants ₍₁

=

16)

showed the _same behaviour _as the shorter chains. The

ilistability

of the membranes _may be related to interface

roughening

and membrane

crumpling

in three dimensions.

The cfitical micelle distribution

(CMC)

is defined _as that surfactant

concentration,

ai which half of the chairs remain isolated and half accumulate in clusters

containing

at least two chains.

Figure

2 shows for the two-component model

A,

that the CMC

decays exponentially

with

increasing

chain

length; roughly,

each additional molecule added to the chain halves trie CMC.

This _is in

good

agreement ^with

experiment

_[6].

Two components avg, micelle distribution, L 83, T 2.30

20

18

o

o o Ô

16 Ô o o

Ô °

o o

~ ~oÔ

14

o o

~Ô o

12 °ooÔ

c ~ Ôo

Î 10

OE ~

f _o

~ 8 oo o

o

Ô Ôo _o

6

~

°

o ~

o

4 o oo

°

~ o

ooo

2 ^° o~o

o~o

60,000 Mcs initialisation

0

0 10 20 30 40 50 60 70

Size

Fig. 3. Equilibrium micelle size distribution for the two-component model A.

The Shell _group [7, 8] studied trie

hquid-vapor equihbrium

of surfactant chairs with molec- ular

dynamics.

^We _may

interpret

_ouf two-component models

(A

and

B)

_as chains in _vacuum

instead of in _a solvent. Trie

liquid polymer density

in trie

region

of concentrated

surfactants,

is difficult to determine

numerically,

but trie _vapor densities

equilibrate

to values which increase with chain

length.

^Ouf simulations of the twc-component systems showed that the conden- sation temperature of the surfactants to form

large

^{clusters varied}

linearly

with chain

length.

For surfactants of

length

6 and model A the condensation temperature is 2.3, 3.0 for

length

8, 4.0

(10),

à-o

(12)

and 7.0 for

length

16. Most of _our computer ^time was spent to determine

these five transition temperatures.

We bave also looked _at the micelle size distribution for surfactants of

length

6

(model A)

in _a two component system at _a temperature of 2.3. Here _we define _a micelle _{as a} set of surfactants

DD H H HO

*O HO

HO*OTOO OOOOOO

OOOOOOH OOOOOTOO

HOOTOO HO *TTOTO

*OOTTH *OTOOTT H

DD H OOOTTO

O *OOOTO O *TO

DD O *OO

O O

H H H

OH H

OOOOO OOOOOO

DD *TOTOT

DD *OO *OOTTT

*TOC DDT OOH H *OO

OOTTO H HOHHO OOO DD

OOOTOO H OOOOOT *OOO

*OOTTO O H *TTOOO OTTOTT

*OOOTT DD *O *TOOTOOOOH OOOOTOOO

*OOTOO *TOOTTT *OTOTOOH OOTOOTOOOOH

*TOOOO *OTTTOOH *OT *OOOOO

DD O *OOTTOO *OO *OTOOH

H OOOOO OH DD *OO

HH *O H OH

HH DD HHO HO

OTOT OOOOO

OOOOT H OOOOOO O H

*TOOOOO O OOOTTOOO DD

*OOTOOO DDT *OOTTTOHO H*OTOOO

OOOTTOT *T *OTTTOH HOOOOTOOOOH

*TTTOT OOOOOOOO O *OTOOO OOOTOTTOO

*DODO H*OTOTTOOO DD O *TOOOTTOOH

OOOHO *TTOOOTT H H HOOOOTO

DD H *OOOTTOOH HH

OH *DOT DD

OH

H H

OOOOT

*DODO OOOH *H OOOTOH

OOOOOOO*OO OHOOTOOO

H *TOTOO H OOOOOOTTOTOH

OOOOOO *HOOTOOOTTO

O OOOOH H *OOOOOOOOOH

H OOO O H OOOHOOTT

H *TOTH HH

*DODO H OHO

*TOOOO DD H

OOOT CHOC H

OOOTOT O *OTO

OTTOOOH H*OO

*OOTO DD DD

OOO O O O H

H O

T = 2,30000

52083.53u 13.33s 28:56:42 49% 0+0k 5+446io 0pf+0w

Fig. 4. Example of shces

through

three-dimensional micelles showing rather compact ^clusters with heads H mostly outside and tails T mostly inside. Trie other symbols (#) and

(O)

denote trie interior of trie surfactant chairs _in this two-comportent model A.

which bave _ai least _{one nearest}

neighbour

contact with each other. Trie

equilibrium

distribution

is shown _m

figure

3. Note that there is _a

peak

in trie distribution at 40 surfactants. This distribution _was obtained from 40

samples

after _an initial 60,000 iterations for

equihbration.

The samples were taken 1000 iterations apart. We _were able to

visually display

the micelles _as

350 JOURNAL DE PHYSIQUE I N°3

a three-dimensional

object.

We observe that the micelles _are compact, with _a somewhat

rough

surface. Two-dimensional slices

through

_our lattice confirm that the heads of the surfactants

are on the surface of the micelle whilst the rails _are in the interior. This is shown in

figure

4. We have _non studied the distribution of the micelles _{as a} function of the

length

of the surfactant.

In this work _we have obtained further

properties

of the microemulsion from the

simple

model introduced in [3, 4]. ^{The model}

gives

_a distribution of micelles _as observed

by experimentalists

[6, 9] ^and we are now in _a

position

to

study

how

properties

of trie micelles vary as a function of chain

length

and

strength

of interactions. Ii _seems that _our

simple

model is

incapable

of

describing

stable membranes for trie parameters we bave considered. Our model is close to that of the Shell _group and _we have confirmed _many of the results

reported

from their molecular

dynamics

with _our

simpler

Monte Carlo model which allows the

study

of

longer

limes than

molecular

dynamics.

Thus _we avoid metastablities _as the diffusion of surfactants slows the

equilibration

process.

D. Andelman has

suggested

'~bolas" _as

good

candidates for stable

layers

at

high

tempera- tures.

Preliminary

results confirm the

stability

of these membranes. A "bola" _consists of two

hydrophilic

"heads" _{one ai} each end of the chair and _a

hydrophobic

interior.

Acknowledgements.

We thank D.

Andelman,

D.G.

Marangoni,

R-B-

Pandey,

M.

Sahimi,

M. Telo da Gama and J.

Andrea for discussions, and

NSERC,

the Canada

Council,

and the German Israeli Foundation for support.

References

[1] ^W-M- Gelbart, D. Roux and A. Ben-Shaul Eds., Micelles, Membranes, Microemulsions, and Monolayers

(Springer

Verlag, Heidelberg, 1994);

Degiorgio V. and Corti M., Physics of Amphiphiles: Micelles, vesicles and microemulsions

(North

Holland, Amsterdam,

1985).

[2] Gompper G, and Schick M., Phase Transition and Critical Phenomena, vol. 16, C. Domb and

JL. Lebowitz Eds.

(Academic

Press, New York,

1994).

[3] ^Larson R-G., J. Chem. Phys. 96

(1992)

7904 with earher literature.

[4] Staulfer D., Jan N, and Pandey R-B-, Physica A 198

(1993)

401;

Staulfer D., Jan N., He Y., Pandey R-B-, Marangoni D.G. and Smith-Palmer T., prepnnt ^for J.

Chem. Phys.

[5] Widom B.~ J. Chem. Phys. 84

(1986)

6943.

[fil Lindman B, and Wennerstrom H., Top. Curr. Chem. 87

(1984)

1.

[7] Siepmann J-I-, Karaborm S, and Smit B., Nature 365

(1993)

330.

[8] Smit B., Essehnk K., Hilbers P-A-J-, _van Os N-M-, Rupert L.A-M, a~ld Szleifer I., Langmuir 9

(1993)

9;

Karaborm S., _van Os N-M-, Essehnk K. and Hilbers P-O-J-, La~lgmuir 9

(1993)

1175;

Smit B., Hilbers P-A-J- and Esselink K., I~lt. J. Mort. Phys. C 4

(1993)

393.

[9] Simiarczuk A., Ware W-R- and Liu Y.S., J. Phys. Chem. 97

(1993)

8082 with earlier literature.

[10] ^K. Binder Ed., Applications of trie Monte Carlo Method in Statistical Physics

(Springer-Verlag,

Berlin, 1994).