Crumpling and second sound in lyotropic lamellar phases

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Crumpling and second sound in lyotropic lamellar phases

T.C. Lubensky, Jacques Prost, S. Ramaswamy

To cite this version:

Crumpling

and second sound

in

_lyotropic

lamellar

_phases

T. C.

_Lubensky

_(1),

J. Prost

₍₂₎

and S.

_Ramaswamy

₍₃₎

(1)

Department

of

Physics, University

of

_{Pennsylvania, Philadelphia, Pennsylvania}

19104,

U.S.A.

(2)

Ecole

_Supérieure

de

_Physique

et de Chimie Industrielles, 10 Rue

_Vauquelin,

75231 Paris Cedex 05, France

(3)

Dept.

of

Physics,

Indian Institute of Science,

Bangalore

560012, India

(Reçu

le 14 novembre 1989, révisé le 23

_janvier

1990,

accepté

le 1 er

_{février 1990)}

Resume. 2014 Nous montrons que les fluctuations de concentration dans une

_{phase smectique}

formée de lamelles de surfactant

d’epaisseur

w,

séparées

de couches fluides

isotropes,

sont

dominées par le

froissage

des lamelles

_lorsque

_{(w/l)2 ~}

1,

(l

periodicite

de la

_structure).

Nos résultats sont fondés sur une

_{généralisation}

de la théorie d’Helfrich, dans

laquelle

le taux de

froissage

k-2,

les densités 03C1s et 03C1b du surfactant et du fluide sont

susceptibles

de varier. Dans la

limite

_{« incompressible »}

où 03C1s, 03C1b et w sont

fixés,

les fluctuations de concentration à l constant sont déterminées entièrement _par

_k-2,

et le rapport du module de

_compression

des couches à concentration constante

_(mesuré

à « haute

_{fréquence »)}

sur celui à

potentiel chimique

constant

(mesuré

à « basse

_{fréquence »)}

est d’ordre

_(03BA/T)2

_{(l 2014}

_w)2/l2

où 03BA est le module de courbure des lamelles et T la

température.

Nous montrons _que les corrections à la limite «

_{incompressible}

_»,

liées aux fluctuations de w sont d’ordre

_(w/l)2

_(03BA/U)

où U est une

_énergie

moléculaire.

Abstract. 2014 We

study

relative concentration fluctuations in two _component lamellar smectic

liquid crystals consisting

of surfactant

layers

of width w

_{separated by}

a

_background

fluid and show that these fluctuations are dominated

_{by crumpling}

fluctuations of the surfactant

_layers

when

(w/l)2 ~

1 where f is the average

layer spacing.

Our results are based on

_{generalizations}

of the Helfrich

_theory

in which the

_crumpling

ratio

_k-2,

densities _03C1s

_and

_03C1b of the surfactant and

background

fluid, and w as well as l are allowed to _{vary. In} the

_{incompressible}

limit with

03C1s, 03C1b and w fixed, concentration fluctuations at constant f are determined

_{entirely by}

fluctuations

in

k-2.

In this limit, the ratio of the

_{high-frequency,}

constant concentration

_{compressibility}

modulus B to the

_{low-frequency,}

constant chemical

_potential

modulus B is of

order

(03BA

/T)2

(l2014

03C9)2/l2

where k is the

_{layer bending rigidity}

and T is the _temperature. We show that corrections to the

_{incompressible}

limit

arising

from fluctuations in w are of relative order

(03C9/l)2

(03BA/U)

where U is a molecular energy.

Classification

Physics

Abstracts

61.30 - 62.90 - 82.70D

1. Introduction.

Lyotropic

smectics

_{[1] are}

lamellar

_{phases consisting}

of stacks of

_{regularly spaced fluctuating}

fluid membranes

separated

by

a

_background

fluid of oil

_{(or of water).}

The

_separation

between

membranes can be as

_large

as hundreds or even thousands of

_angstroms

_[2-10].

The

membranes consist of

_bilayers

of surfactant

_molecules,

_{possibly enclosing}

a thin

_layer

of water

(or

oil).

Thus,

unlike

_thermotropic

_smectics,

_lyotropic

smectics are

_necessarily

multi-component systems

with at least two conserved masses and

_{correspondingly}

at least one

hydrodynamic

mass diffusion mode. In this paper, we will

_generalize

Helfrich’s

_{theory [11, 12]}

of

_sterically

stabilized lamellar

_phases

to include

_density

as well as

_layer

fluctuations in a two

component

smectic. Our

_principal

new result is that

_changes

in the

_degree

of

_crumpling

of membranes

_{(depicted schematically}

in

_{Fig. 1)}

at constant

_density

of material in both the

background

fluid and surfactant

_layer

are the dominant cause of relative

_density

fluctuations

at constant

_{layer spacing}

when the ratio

_{w If}

of the width w of the surfactant

_layer

to l

is small. Such

_changes

in surfactant volume fraction can be described

_{mathematically}

in terms of the

_crumpling

ratio

_{[13] specifying}

the inverse ratio of the area of a membrane surface

element to its area

_projected

onto a

_{plane parallel}

to _{the average}

_layers.

Fig.

1. - Schematic

_{representation}

of how

changes

in the

degree

of

crumpling

of a surfactant

layer

lead

to

_changes

in the relative surfactant

_density.

The

_crumpling

ratio

_{k2 equation (2.1)}

is the ratio

AB /A

of the

_projected

area

_{A B}

of a membrane to its total area A.

_(a)

shows a surfactant

layer

with

k2

near one.

_(b)

shows a similar

_layer

but with _greater total area and thus smaller

k2.

The relative

surfactant concentration in

(a)

is

larger

than in

(b).

The conserved variables of a two

_component

smectic

_{[ 14-16]}

are _{the energy}

_density

_E, the

density

Pb

of the

_background

fluid

_(water

or

_oil),

the

_density

_fis

of the

_surfactant,

and the momentum

_density

_g. In

addition,

the strain

_Vzu

=

dl/l

expressing

the

change

8 Q

in

layer

spacing f

from

_equilibrium

is a broken

_symmetry

elastic variable. The variables

entering

naturally

into the

_{two-component}

smectic

_{hydrodynamics}

are a,

ozu,

the total mass

_density,

and the relative

density,

To obtain a

_{complete description}

of all

_hydrodynamic

_modes,

one needs to know the free energy to second order in all of these variables. At low

_frequencies,

one _may assume that both

the

_background

fluid and the surfactant

_layers

are

_{incompressible,}

_i.e.,

that the local

_density

p b of the

background

fluid and the local

density

p

and

width w of the surfactant

layers

are

Here z is the direction normal to the smectic

_{layers, X}

is the relative concentration

susceptibility, Ce

a strain-concentration cross

_coupling,

and B the

_{layer compression}

modulus

at constant relative concentration which determines the

_velocity

of second

_sound,

c2 =

(B/p) 1/2.

In _{our present}

thermodynamic analysis,

_{we may assume} that

u(x)

is

independent

of _{coordinates x1}

_{perpendicular}

to the z

axis,

and we

_ignore

the

_bending

term

KI

(01

u )2/2

in

_f.

This term

_is,

or _course, needed to describe

_spatially

nonuniform distortions of the smectic. Fluid membranes are characterized

_{[ 12] by}

a

_bending

modulus K with units of

energy. All of the coefficients in

equation (1.3)

can be calculated in a controlled

expansion

in

the ratio

_{T/ K}

of the

_temperature

T to the

_bending

modulus. To lowest order in

T/ K ,

we find

B is related to the static or constant chemical

_potential

modulus,

calculated in the Helfrich

_{theory [6,11-14,17]}

via the

_{thermodynamic identity,}

Thus

_{B = B - XC2c; is}

determined

_by

corrections to the

_leading

order terms in

_{T/ K}

in

B,

X, and

Cc. Equations (1.4)

and

(1.5) imply

Our

_{calculations-yield}

A =

256 / [3 - (12/ 03C02)] _100.

B /É

should be

_{approximately}

indepen-dent

of f for i »

w. In

addition,

B /B

can be

_{quite large ( ~ 400 )}

even for

_{experimentally}

realizable values of

_{K / T}

of order 1.5 or 2.0.

We have also calculated the contributions to

B,

X, and

_{Ce arising}

from a nonzero

compressibility

of surfactant

_{layers (i.e., by}

variations in w considered in Ref.

_[16]).

We find when

wu

that these are smaller than those

_arising

from steric

_repulsion

and

_{crumpling by}

a

factor of order

_{(w If)2}

_{(K 1}

_{U) «}

1 where U is a molecular energy that is

_expected

to be of order K or

_greater.

This paper contains four sections in addition to this one. Section 2 reviews the

ther-modynamics

of stacks of membranes. Section 3

_presents

the calculations of

B,

Y, and

Ce using

the Helfrich

_theory.

Section 4 outlines how the Helfrich

_theory

can be

_generalized

to include variations of

_parameters

such as the densities of the surfactant and of the

_background

fluid and calculates the corrections to

B,

X, and

Ce

arising

from a nonzero

_layer

compressibility.

Section 5 reviews our results and discusses their relevance to the

hydrodyn-amic mode structure discussed and measured in reference

_[16].

2.

_{Thermodynamie potentials.}

A B (as

shown in

Fig. 1).

A measure

of

the

_degree

to which a membrane is

_crumpled’

is

provided by

the

_crumpling

parameter

[13]

The surfactant and

_background

volume fractions are

_respectively

w (k -2/ f)

and

[1 -

w(k - 2/f)]

]

so that the mass

_density

of the surfactant and the

_background

fluid are,

respectively,

Note that the average mass densities

_Ps

S and

Pb

differ from the local densities p s and

p b, and

they

can

_change

at constant _{p S, p b, and w in response} to

_changes

in

k- 2

and

f.

The relative concentration is

so that

at constant _{pg, Pb, and w where a} =

p

Pb/p-’2.

Thus,

in this

limit,

changes

in c are determined

entirely by

the

_ratio,

of the inverse

_crumpling

ratio to the

_{layer spacing.}

The function

_{f(Vzu, c )}

is most

_directly

obtained

_by

_calculating

first the

_{thermodynamic}

potential

that is a function of the variable a-

_conjugate

to c’ and then

_{Legendre transforming.}

We

_{begin by reviewing}

definitions of the various

_{thermodynamic potentials}

_[13]

that can be

used to describe a

_{fluctuating nearly}

flat membrane. A

_nearly

flat membrane is characterized

by

two extensive

_{parameters :}

its total area A and its

_projected

area

_{A B.}

The

_{thermodynamic}

potential

satisfies

_(at

constant

f)

where

is the surface tension and h is the field

_conjugate

to

_{A B.}

_{The energy per} unit volume of a stack

The free energy

f (Eq. (1. 3»

is the

_part

of

_f

1 harmonic

in

dl

and 8 c = aw8 c’. Note the effects

of a finite width of the membrane are described

by

replacing f

by

f - w

in

_{l/J 1.}

Other

membrane

_potentials

can be obtained

_{from 0}

₁ via

Legendre

transformation. In

particular,

Changes in l/J2 at

constant f

_satisfy

so that

Thus,

the surface tension cr, is the variable

conjugate to k- 2.

In

analogy

with

equation (2.9),

we introduce the free energy

_density,

which satisfies

where _{a i} is the force

_conjugate

to the

_{layer spacing}

which is zero in

equilibrium.

Thus,

fi (f,

c’ )

is the

_Legendre

transform

_{Of f2:}

1 (V zU, 8 c )

is

_easily

obtained from

_{equations (2.15)}

and

(2.5).

In the next

_section,

we will calculate

_{12(f, u)}

to lowest order in

_{T/ K}

_using

the

_Monge

gauge for the fluid surface. As discussed in reference

[13],

the short

length

cutoff is treated

more

_consistently

in a _gauge in which A rather than

_{A B}

is fixed. This leads more

_naturally

to the

_potential

_{13(f, h) = 4J3/(fA)}

rather than

_{12(f, u).}

The latter

_{is, however,}

easier to calculate and suffices for our _purposes.

3. The harmonic free _energy.

In order to

_{calculate ’02(0-,}

Q ),

we follow references

_{[ 18]}

and

_[13]

and

_replace

the hard wall

constraint

_by

a

_global

constraint on

f.

In the

_Monge

gauge where the membrane

position

is R =

(x1,

h (x1

))

where x1 -

(x, y),

the membrane Hamiltonian is

where and n

= gaz2(-

_{01 h,}

_{1 )}

is the unit normal to the surface. The

xy

plane

and remains fixed in this calculation. The free energy

density

associated with H

is,

therefore,

The mean _square

_height

is

where » is a

phenomenological

_parameter

introduced

by

Helfrich

[1].

The free energy

P2(u, f)

is the

_Legendre

transform

_{of ~ 2 :}

1

To

_{obtain ~ 2}

at low

_temperature

(small

T/ K ),

wu can

expand H

to harmonic order in h.

Higher

order terms in h

_give

rise to subdominant corrections in

_{T/ K ,}

which for

_example

lead to a

_softening

of the effective

_bending

modulus

_{[19-23]. They}

will not concern us here. In this

case,

and

where A is the ultraviolet wavenumber cutoff and where

KA 4>

y,

uA 2

and

Ky

> a

2.

_Using

equations (3.7), (3.3),

and

_(3.4),

it is

_{straightforward}

to derive

_(apart

from an irrelevant

constant)

where

and

In the absence of external stresses,

8 fj /ô t

=

0,

and _we find

Using equations (3.10), (3.12),

and

_(1.3),

we obtain

It is

_{straightforward}

to

_{verify using equation (1.6)}

that

in

agreement

with references

_[6]

and

_{[11]. Equations (3.14)}

and

_{(3.13a) yield equation (1.7)}

for

_BIÉ.

Note that

_jS/2?

is

_independent

of the

_{phenomenological}

_parameter

g. Note also that

BX

=a 2 w 2 (rl t )2

to lowest order in

_T/K.

Thus,

to this

order, BX

=

c 2

if w

f.

4. Generalization to more variables.

In the last two

_sections,

we treated

_changes

in relative concentration

_brought

about

_by

changes

in

_{layer spacing}

and the

_crumpling

ratio. In this

_section,

we will outline a

generalization

of Helfrich’s

_theory

which can take into account the effects of variations in

P S, Pb,

W, f,

and

temperature

T. This

_theory

can be used to calculate all

_{thermodynamic}

derivatives

_appearing

in the linearized

_{hydrodynamics}

of the lamellar

_phase.

As a

_particular

application

of this

_approach,

we will outline how the results of the

_previous

sections can be

corrected for a nonzero

_{layer compressibility.}

When w «

f,

these corrections are of relative

order

_{(K / U) (w / f)2}

to lowest order in

_{(T / K),}

where U is a molecular _{energy which is of}

order K or

_{larger. They}

are thus subdominant

_compared

to

_{already neglected}

corrections of relative order

_{( T/ K )2}

when

_{( w/Q )2 ( T/ K )2.}

_They

_may,

_however,

become

_important

when

(w /f ) =

(T/K )

as _may be the case in

_systems

such as those studied

_by

_{Safinya et}

al.

_[6]

in

which the Helfrich

_theory

_appears to be valid for ratios

_w/Q

as

_large

as

_2.9/5.8

= 0.5.

There are four

_hydrodynamic

variables,

p b, P s, ,,

and f,

that are even under time reversal.

The

_{susceptibility}

matrix

_involving

all of these variables can be obtained from a _{free energy}

density g

that is a function of the

_background

fluid and surfactant chemical

_potentiels,

a b and as, the

temperature T,

and the force _ai

_conjugate

to

f.

To construct g, we first

construct the

_layer

_{free energy}

_density,

Here

_Ibis

the free _energy

_density

of the bulk

_background

fluid and

_{w 1 s is}

the free energy per unit area of the surfactant

_{layer. f}

_is

the Helfrich free energy, calculated in the last section

(Eq. (2.9))

but with variations in w and

temperature

dependence

of K

permitted.

The free

over _{p b, Ps’} w,

c’,

and .l . ’

As an

_application

of the above

development,

we consider

incompressible

surfactant and

background

fluids

(p

S and

p b

fixed)

at constant

temperature

and calculate corrections to

_{X, B}

and

_Cc

arising

from variations in w. For each

_T,

_{p b, Ps,} there is a

_{preferred layer}

width

wo, and we can

_expand

_[16]

_{w f S}

as

The

_parameter

D has units of

_{(energy)/(length) [4]}

and

should, thus,

be of order

U jw4

where U is a molecular energy that should be of order or

_greater

than K. To

_calculate

B,

and

_Cc,

we consider the function

where a =

(ac, a f), TI = (w,

c’,

f )

and g

is defined in

equation (4.2).

Let t =

(c, l ).

Then

the

_{susceptibility}

matrix is

where summation over IL, JI =

1, 2,

3 is understood and where

When f >

w, it is

straightforward

to show that

where

The coefficients x,

Cc,

and B are obtained from

_Xij

calculated

_according

to the above

prescription

from

where the

_quantities

at D = _ao are those evaluated at fixed w in the

_preceding

section and where

We have retained

_only

the lowest order terms in

_{T/ K}

in both

_{equations (4.9)}

and

_(4.10).

The molecular _{energy U} should be of order T so that s « 1 if

_{w /Q}

1 even if

T/ K

approaches

one.

5.

_Summary.

In _{this paper,} we have considered

_coupling

between fluctuations in relative surfactant concentration c =

Psi P

and the strain

Vzu

in lamellar

lyotropic liquid crystals

stabilized

by

the

Helfrich steric

_repulsion

between surfactant

layers

when the densities _p S and

Pb of

the

surfactant and

_background

fluid and the width w of the surfactant

_layers

are fixed. In this

incompressible approximation,

relative concentration

_changes

are controlled

_{by changes}

in

k-

_2/j@

the inverse

_crumpling

ratio divided

_by

the

_{layer spacing.}

We

_generalized

the Helfrich

theory

to treat fluctuations in both

k- 2

and

_f,

and we calculated both the

low-frequency,

constant chemical

_potential

and the

_{high-frequency,}

constant relative concentration compres-sion

moduli, B

and B. The ratio

_B/B

is

_proportional

to

(K 1 T)2 (f - w )2 1 f2

with a

proportionality

constant of order 100.

Thus,

in this

_{incompressible}

limit,

_B/B

is much

_greater

than one and

depends

only weakly

on

f

when w «

f.

We also

_generalized

the Helfrich

theory

to allow for

_changes

in _{pg, p b,} and w as well as in

k - 2

and

f.

We then fixed

p

and

p b and calculated the contributions to B

and È

arising

from the finite

compressibility

of

w. We found these contributions to

_provide

corrections to the

_{incompressible theory}

of relative order

_(W/f)2

( K

1 U)

where U is a molecular _energy.

_Thus,

_crumpling

fluctuations are

the dominant source of concentration fluctuations at

constant f

so

_long

as

(Wl)2 « 1

if

U -K.

_Layer

_compression

can become

_important,

however,

when

(w If)2

is not small

compared

to one. The Helfrich

theory

provides

a

_{good description}

of the

systems

studied

_by

Safinya

et al.

_[6]

for values of

_{(w If)2}

as

_large

as

_1/4.

Such

systems

may,

therefore,

be in a crossover

_region

between

_crumpling

dominated and

_layer

_{compressibility}

dominated

concen-tration fluctuations.

_Layer

_{compressibility}

should be

_unimportant

in

large

layer

spacing

systems

such as those studied

_by

Porte et al.

[6].

Nallet et al.

_[16]

have derived the

_dispersion

relations for the low

_frequency

modes of an

incompressible lyotropic

smectic and measured the

_frequency

of one of these modes. The

nature of this mode

_depends

on the orientation of its wave vector _q =

(qx,

q y,

qz)

relative the

layer

normal

_along

the z-axis.

_Assuming

_temperature

fluctuations relax

_quickly,

there are four

other low

_frequency

modes. If q =

(qx,

0,

0 ),

the four modes are two shear modes with

frequency

ws = 2013

1 ( q /#) qjj an

undulation mode with

frequency

w u

= - 1 (K / q )

qx,

and a

relative

_density

mode with

_frequency

_{{ù c = - i ( a .1 1 p2 X) q;}

where q

is a

viscosity,

K = K

/f

is the

splay

elastic _constant, and

à 1

is a

_dissipative

coefficient. In the relative

density

mode,

variations in c are

_{decoupled completely}

from those of u and the transverse

velocity.

The structure of these modes is

_independent

of _any detailed model for the smectic. Nallet et al.

_identify

the relative

_density

mode as a membrane

peristaltic

mode

[24]

in which

the thickness of the surfactant

_bilayer

is modulated in time. We

find, however,

that relative

density

variations are dominated

_by

variations in

_{k- 21 e}

rather than

_{by bilayer}

thickness variations. Since at _qz =

0,

there are no variations in

_{V zu}

=

8f If

in the relative

density mode,

density

fluctuations relax via modulation of the local

_crumpling

ratio. When bôth

qx and q, are nonzero, there is a second sound mode with

positive

and

negative frequencies

:t (B 1 p) 1/2 q x q z

and a

_density

or baroclinic mode with

frequency

_{W b =}

- i ( a 1 / p 2 ) (B/XB ) qx

when

_{q, « qz.}

_Again

these

_dispersion

relations are

_independent

of

any

particular

model for the lamellar

phase.

In the model

presented

in reference

[ 16]

in which

density

fluctuations are dominated

by

fluctuations in

bilayer

thickness, X B

=

c 2.

The same

relations

_applies

to the

_crumpling

ratio dominated

_theory

_presented

here when

_{( w /l )}

1 and

T/ (4

’W K ) «

1 as discussed after

equation (3.14).

The

experiments presented

in reference

_[16]

verify

that both _{co u} and _(JJb are

_proportional

to

qx

In

_addition,

_they

show that both

w u and (JJb vary as

f-l

at fixed _q. The first result is in accord with an

f-independent

_viscosity

and K =

_{K /t.}

The second follows from

_{BX =}

_{c 2}

1 the Helfrich

expression

for

B,

and the

prediction by

Brochard and de Gennes

_{[ 15]}

that

_{a / (p-C) 2 _}

_{(f - w)2.}

Thus the

_experimental

results of reference

_[16]

are in

_agreement

with the

theory presented

here,

which

predicts

BX =

c 2

to a

_good

_{approximation. They}

do not,

however,

provide

independent

measurements

of

_B,

X, and

Ce

which would be needed for a verification of our

_predictions.

It would be of some interest to _carry out further

_experiments

to

_verify

the

_predictions

made here.

_Perhaps

the most direct check would be

_{provided by}

a measurement of the

_frequency

(w c = - i (a .1.

1 p2 X)

qx

of the

_density

mode at _{q z} = 0. Then the ratio of the coefficients of

qx

in the

_density

mode at _qz = 0 and the baroclinic mode at

q2x q2Z

is

simply

the

ratio,

B/B,

which in the

_present

_theory

is

_{given by equation (1.7). Alternatively,}

of course, second

sound measurements

_{with q}

of order

102 cm-1

1 should

_provide

a direct measurement of B.

Acknowledgements.

We are

grateful

to D. Roux for

_helpful

conversations

_regarding

this work. We are

particularly

indebted to F. Nallet for a critical

_reading

of an

_early

version of this

manuscript

which lead us

. to discover and correct

an error. T.C.L.

_acknowledges

financial

_support

from NSF under

grants

No. DMR 88-15469 and No. DMR 85-19059 and from C.N.R.S. and S.R. from the

University

Grants

Commission,India.

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