Hexagonal and lamellar mesophases induced by shear

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Submitted on 1 Jan 1990

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Hexagonal and lamellar mesophases induced by shear

C.M. Marques, M. E. Cates

To cite this version:

Hexagonal

and lamellar

_mesophases

induced

_by

shear

C. M.

_Marques

_{(1, 2)}

and M. E. Cates

₍₁₎

(1)

Cavendish

_{Laboratory, Madingley}

Road,

Cambridge

CB3 0HE, G.B.

(2)

Institut Charles Sadron, 67083

_Strasbourg

_Cedex, France

(Received

on _{January 15, 1990,} revised on

_April

_{10, 1990,}

_accepted

on

_April

_26,

₁₉₉₀₎

Abstract. 2014 The role of a shear flow in the formation of an

_{hexagonal mesophase}

is

_investigated

and the transition between the

isotropic,

lamellar and

_{hexagonal phases}

described in a three dimensional

_{phase diagram [03C4, u, 0394] where}

03C4 is a

_{temperature-related}

_parameter,

_u

a measure

of the intrinsic _asymmetry of the system and 0394 the

strength

of the shear rate. At zero shear rate the

_{microphase separation}

transition

(M.S.T.)

is dominated

by

fluctuations but it

acquires

a

strong mean-field character as 0394 is increased. We find a

_region

in the

_{phase diagram}

where the same

_{isotropic phase}

_may

_undergo

a transition to a lamellar or an

_hexagonal

structure

_according

to

_whether

is cooled or sheared. The relevance of our results for diblock

_copolymers

and for

microemulsions is discussed. Classification

Physics

Abstracts

64.60H - 61.25H - 64.70M

1. Introduction.

Blends

_{[1, 2]}

or solutions

_[3]

of diblock

_copolymers

are a classic

_{example [4]}

of

_systems

undergoing

a

_{microphase separation}

transition

_(M.S.T.).

At

_high

_temperatures

the

_mixing

entropy

of the two blocks is

_enough

to overcome the natural

_tendency

for

_separation

as

dictated

_by

the

_positive

heat of

_mixing

of monomers A and B. Thus the

_system

is

_isotropic

for

such

temperatures.

Under

_cooling,

a

_point

is reached where

_separation

is favoured over

mixing

and the two blocks tend to

_stay

as far

_apart

as

_possible

in order to reduce the number of A-B contact

_points.

_However,

since

_they

are

_chemically

bonded,

a

_{macroscopic phase}

separation

is not

_possible

as it would be in the case of blends or solutions of two

_chemically

different

_polymers.

The

_{connectivity implies}

that the structures which the

_polymers

are able

to form have a

_typical

size of the order of the radius of

_gyration

of the chains

RG. Consequently,

the Fourier transform of the two

_point

correlation

function,

S(k),

shows a

well-pronounced peak

at k

_{= k}

_{[ = ko -}

_{RG 1,}

where it can be

approximated by

T

being

is a

_{temperature-like}

control variable.

Several different ordered structures can result from the

M.S.T.,

depending

upon the

asymmetry

of the diblock

_{copolymer (usually}

measured

_by

the ratio of the

chemical lengths

of the

_{blocks, f}

=

NA/NB).

Very asymmetric

copolymers

tend _to form

spherical objects,

packed

in a three-dimensional

_body-centre

cubic lattice. For smaller

_asymmetries

the

tendency

is to form

_{cylinder-like}

_aggregates,

_packed

in a two-dimensional

_{hexagonal lattice,}

or one dimensional lamellar structures. More exotic

_phases,

such as bicontinuous ordered

structures, have been also

_{reported [5].}

Many

other

_systems,

like

_{lyotropic liquid crystals [6-9]}

or microemulsions

_{[ 10-12] undergo}

a

microscopic mesophase separation

as well. Here the

_asymmetry

is a more

_complicated

function of the

_{system parameters.}

For

microemulsions,

for

_example,

the intrinsic

_asymmetry

is controlled not

_{only by}

the

_spontaneous

curvature of the surface of an ordered domain but also

_by

the ratio of water and oil contents. In suitable

systems,

hexagonal liquid crystalline

phases,

as well as lamellar

_(smectic)

ones, are seen

_[13].

The onset of the

_ordering

at finite values of wave vector has also turned out to be a useful tool in

_explaining

the

_phase

transitions observed in colloidal

_{crystals [14]}

or even in convective

_systems

_{[15] (Rayleigh-Bénard}

instabilities).

In all these cases the transition is second order or

_weakly

first order and this has motivated a

description

of the M.S.T. based upon a

_{Landau-Ginsburg free-energy,}

which is

_expected

to

correctly

account for the

_{phase diagram}

in the

_regions

close to the transition lines. At the level of the mean-field

_{approximation}

the

_{isotropic-lamellar}

transition is

_predicted

to be second order

_{[ 16, 2]}

for

_symmetric

_systems

_(for

_instance,

for diblock

_copolymers

which have blocks of

equal lengths

or for microemulsions with no

_spontaneous

curvature and

_equals

amounts of

water and

_{oil). Taking}

into account the role of fluctuations at the Hartree level

_[17]

leads to a

negative

shift into the

temperature

at which the transitions occur

_{[ 17, 1] :}

under

_cooling,

the fluctuations

_delay

the

_ordering.

Furthermore the transition becomes first order even for

symmetric

systems

and the

_{spinodal point}

is shifted

_formally

to an infinite

_negative

temperature

T.

As mentioned

_above,

when the

_system

_approaches

the transition the fluctuations of the order

_parameter

become

_large.

Since the

_{microscopic mobility}

remains

_{essentially unchanged,}

this means that the lifetime of such

_{fluctuations,}

also becomes

_large

near the transition. It is

then

_interesting

to

_investigate

the effect of a

_{dynamical perturbation imposed}

on the

system,

because the relaxation

_frequencies

become

_{experimentally}

accessible

_{[18, 19].}

The

_typical

example

is that of an

_applied

shear flow

(for example

in a Couette

_geometry)

which

_imposes

an _average

velocity

flow ux =

(Dy,

0,

0 ).

For small

enough

shear rates

D,

the time for a

fluctuation to be convected away from the critical shell

( 1 k 1

- ka )

is

longer

than its

spontaneous

relaxation

time,

and the flow has

_only

a weak effect on the

phase

transition.

However,

for

_large

shear rates the fluctuations are convected away before

they

would

otherwise have time to

_relax,

and are less and less effective in

suppressing

the

phase

transition

which,

at

_high

shear rates, therefore

_acquires

a

_strong

mean-field character.

In a recent letter

_[20]

Cates and Milner

_investigate

the role of a shear-flow on the

isotropic-lamellar

_phase

transition. Their results show that the transition

_temperature

increases with shear rate

_and,

for

_{large enough}

shear rates,

approaches exponentially

its mean-field value.

Hence,

close to the static transition

_temperature,

there is a

region

where the disordered

homogeneous phase

becomes unstable to a lamellar

phase

at

_{high enough}

shear rate. Cates and Milner

_argued

that the lamellar

_phase

should order in the direction which better avoids the constraints of the

flow, i.e.,

with the wavevector

_{perpendicular}

to both the flow and its

gradient.

Obviously,

two dimensional structures are also

good

candidates for

phases

that can arise

under shear flow conditions. In this paper we

study

the formation of a two-dimensional

The role of the shear field on the I-H and H-L

_phase

transitions is discussed in section 3. In

section 4 we outline the

_practical

relevance of this work for diblock

_copolymers

and

microemulsions. Section 5 summarizes our results and

_presents

our conclusions.

2. The

_{microphase séparation}

transition.

A second order or

_weakly

first order

_phase

transition can be described

_{[21] by}

the

Landau-Ginsburg expansion

for the

_{free-energy density}

as a function of the order

_parameter

rb

(r) (in

units of

_{kB T) :}

where

and

_S(k)

is

_{approximated by ( 1.1 ).}

The order

parameter

is chosen to vanish on _{average in the}

homogeneous

disordered

_phase

and has a non-zero value in the ordered

_phases.

In this

formulation the coefficients of the third and

_quartic

order terms are taken as constants which

corresponds

to a local

_{approximation}

for the vertex functions in the

_expansion

of the free

energy as a function of the order

_parameter

_[1].

This

_{approximation}

has so far

proved

accurate

_enough

for the

_systems

of interest in the

_present

work. The third order

coefficient »

is a measure of the intrinsic

_asymmetry,

_being

zero for

completely symmetric

_systems.

The

Landau-Ginsburg free-energy

is invariant under the

_{transformation {cp -+ - cp ; J.L -+ - J.L }.}

Thus,

we assume, without lost of

generality,

that »

is

positive. Throughout

this section we

shall discuss the

_predictions

of the

_{Landau-Ginsburg}

_free-energy

from its

_general

_formulation,

we

_specify

the

_{physical meaning}

of the order

_parameter

and of the several coefficients when

analyzing

the diblock

_copolymer

and microemulsion cases, in section 4.

2.1 THE MEAN-FIELD APPROXIMATION. In the limit of

_high

shear rates the transition is

expected

to become mean-field in

_character,

we therefore first recall the

_predictions

of the

mean-field

_theory

for the formation of the lamellar and

_{hexagonal mesophases.}

The one

dimensional lamellar

_{phase along}

the z-axis is described

_by

and for the two dimensional

_{hexagonal phase}

in the Y-Z

_plane

we have

Inserting

these forms of the order _parameter in the

_{Landau-Ginsburg}

free energy

(2.1 )

one

obtains

for the lamellar structure and

for the

_{hexagonal phase.}

The

_{isotropic-lamellar (I-L), isotropic-hexagonal (I-H)}

and

lamellar-hexagonal (L-H) phase

transitions are determined

_by

the minimization of these free

_energies,

The

_phase

diagram

may be

displayed

in a convenient _way in the

_[T/A,

_IL

_/A

_{] plane (Fig. 1 ) .}

As

_expected

we have an

_homogeneous

disordered

_phase

at

_high

_{temperatures.}

For

_symmetric

systems

(IL

=

0)

there is a second-order

phase

transition from the

_isotropic

to the lamellar

phase

at T = 0. For

asymmetric

_systems

(u = 0)

the transitions are first order. Under

_cooling

the

_{hexagonal phase}

appears

first,

at

_temperatures

_{given by [2]}

T =

(4/45)

u 2/ À =

0.089 IL

2/ À.

At lower

temperatures

there is a first order

_{phase-transition}

from

_hexagonal

to lamellar. The H-L line satisfies the

_equation

Fig.

1. - Mean-field

phase diagram

of the

isotropic (1), hexagonal (H)

and lamellar

_(L)

structures. T is a

temperature related parameter and IL the measure of the intrinsic _asymmetry. A is the fourth coefficient of the

_{Landau-Ginsburg free-energy}

and accounts for the relative

importance

of the _spontaneous fluctuations. The I-H and H-L lines are

_parabolas

of

_equations

T =

(4/45)

_IL

2/ À =

0.089 _IL

2/ À

and

respectively [2].

2.2 THE HARTREE APPROXIMATION. - The role of the fluctuations on the

_phase

transition

has been

_previously

considered

_by

Brazovskii

_[17]

who shown

that,

within a self-consistent

Hartree

_{approximation,}

the

_{thermodynamic potentials}

for the lamellar and

_{hexagonal phases}

are

where a =

k20/ (4

TT).

The renormalized inverse

susceptibilities

of the

isotropic (ro),

lamellar

The difference to the mean-field

_picture

resides in the presence of the u-terms and it is easy to

check that

_{by formally setting a}

=

0,

the

potentials (2.7)

reduce to their mean-field form.

These terms are the

_explicit

contributions of the

fluctuations,

self-consistently

calculated

from

where

_{S(k) = [ri}

+

(k -

k o )2 ] -1.

The

_{phase diagram}

can be

_obtained,

as in the

_preceding

section,

by

minimization of these three

_{thermodynamic potentials}

and

by

comparison

of their values at the minimum

_[1].

We summarize in

_figure

2 the results

plotted

in the’

_plane

T, u

where T =

_{Ta - 2/3 À - 2/3, fi = À - 5/6 a - 1/3.}

There are two main differences with the mean-field

_{phase-diagram.}

_First,

the

isotropic-lamellar

_phase

transition is shifted to

_negative

values of the

_temperature

_{(Tc = - 2.03 )}

and the transition is first order even for

_symmetric

_systems.

This can be

_easily

seen from the

equation

for the renormalized

_{susceptibility}

in the

_{homogeneous phase.}

Indeed,

the

_spinodal

line,

given by

ro = 0 is shifted to T = - oo . In the

symmetric

case the lamellar structure first

appears from the

isotropic phase

with a finite

_{amplitude ai}

= 1.45 a

1/3 À - 1/6.

The second difference is the existence of a finite

_region

of

_{small g}

for which the

_hexagonal

phase

is never stable.

Only

for

_{asymmetries larger}

than

_fic

= 0.564 does this structure _appear.

Fig.

2. - Hartree

_{phase-diagram}

of the

_{isotropic (I), hexagonal (H)}

and lamellar

_(L)

structures. The coordinates of the

_{triple point}

are

_{Te = -}

2.308,

uc

= 0.564. For

high

values

of /à7

the transition lines

approach

their mean-field values.

3. The

_{microphase séparation}

under a shear flow

We now

_investigate

how an

_applied

shear flow modifies the M.S.T. described above. We

closely

follow a

previous

letter of Cates and Milner

_[20],

and the works of Frederickson

_[22]

and Onuki

_[23].

The

starting point

is the convective

_{Langevin equation}

for the order

where 0

(r, t )

is the usual random noise source and

K (r - r’ )

the kernel for the

_driving

force. The main

_assumption

in this

_equation

_{is that the flow field ux} =

(Dy,

0,

0 )

is fixed

_by

_an

externally imposed perturbation.

This

_{completely neglects}

any reaction of the flow field to

local fluctuations in the order

_parameter,

but it is a

_{good approximation}

for the disordered state and

_presumably

also in

_weakly

ordered

_{phases (since}

the order

_parameter

is small

everywhere).

The

_{Langevin equation (3.1)}

is

_equivalent

to the

_following

Fokker-Planck

_equation

for the

probability

of the k Fourier

_component

of the order

_parameter

_[24]

if the

_{Onsager mobility}

_{coefficient (k) = k2 K (k)}

is

_{approximated by}

its

value C

at

ko.

To

_completely

solve this

_equation

in

_steady

state

_{(ap {4> (k, t)} lat = 0)}

is a rather

formidable

_task,

since the functional derivative of the

_free-energy

include terms _up to third order. Two main

_{approximations}

will be used to deal with this

_strong

_{nonlinearity.}

3.1 THE EFFECT OF SHEAR FLOW ON THE FLUCTUATIONS. - First we remark

that,

for a flow

field

_along

the

x-axis,

all the fluctuations

_{with kx =1=}

0 will be

_strongly

convected.

_Indeed,

any

ordered structure

_involving

a wavevector with

_{non-zero kx}

cannot

_exist,

in

_steady

state, at

nonzero shear rate D

_[20].

This is because we have assumed a uniform shear field that is

imposed externally

and does not react to the

_{ordering :}

a finite

_{amplitude density}

wave with

kx =1=

0 is

_simply

_{convected away}

_from

_{[ k 1}

=

ko

and _so cannot be

_part

of a

time-independent

solution. This rules out

_{triply-periodic}

_structures, for

_example

the

_body

centered cubic

_phase

(BCC),

even

_though

these are

_present

on the static

phase diagram. (Since

it cannot

_persist

under

flow,

this

_phase

was omitted from our

previous discussions.)

Lamellar and

hexagonal

states can

exist,

but

_only

with

wavevector(s)

transverse to the flow. In the

_hexagonal

case, this

means that the

_symmetry

axis lies

along

the streamlines.

In the disordered

(I)

state, we can under flow

expect

a

_{strongly anisotropic S(k ).}

We now

calculate this fluctuation

_spectrum

_by

_writing

the « Hartree version » of

_{equation (3.2)}

in the Gaussian form

which

_gives

an

_equation

for the flow-distorted

_{S(k) [22, 23]}

The method of characteristics

_gives

a formal solution

_{[19, 22]}

which reads

with

_{k ( t ’)2 = k2x}

+

_(ky

+ Dt’

kX/2)2

+

k2Z.

This shows

_clearly

the

_origin

of the distortion in the

correlation function

_{S(k ).}

The

_equation

can be understood

_by

_noting

that a

randomly

excited

Fourier

_component

at wavevector

_{k ( t), injected}

at

time - t,

is convected to a

_present

(time

(instantaneous)

_{rate - £ (r}

+

_{[k ( t’ ) -}

kO]2},

_appropriate

to the

_{thermodynamic}

force

_acting

on it at this intermediate time. The

_{susceptibility}

_S(k)

at time zero can be viewed as the

superposition

of a _sequence of random Fourier

_components

_{(supplied by}

thermal

_noise),

which have been

_injected

at _past time - t and convected to the

_present

wavevector k. Note

that,

in accordance with our

_previous

_discussion,

_only

Fourier

_components

with

_kX

= 0 escape

convection and it is

_only

in these

_components

that order

_(i.e.

finite values of the

_amplitude)

can

_develop.

We now consider the

_possibility

of ordered states. Since those with

_{kx =F}

0 cannot

_persist

in

a flow

field,

any ordered states must be

_steady

state solutions of

_{(3.2) with kx}

= 0.

However,

setting kx

= 0 in that

equation

_{removes any} direct effect of the flow field

through

the

convective

_(D)

term

_[20].

We are thus left with an

equivalent

static

problem,

whose

solution,

within Hartree

_theory,

is

_{straightforward (since}

the

_steady

state of a static Fokker-Planck

equation

is

_{given by}

the Boltzmann

_{distribution).}

Of course, the flow field does enter

indirectly,

because the Hartree renormalization

_{parameter r}

= T + cr

_depends

(self-consist-ently)

on the flow-distorted

_{S(k ),}

and hence on the flow rate. Nonetheless

_[20],

the search for ordered states under flow is reduced to that of

_{minimizing self-consistently}

an effective free

energy that contains the

flow-dependent

parameter

To determine a it is

_possible,

in

_principle,

to find a

_complete

numerical solution for the

equation (3.5).

However,

the most relevant information may be obtained more

directly,

at

least in the limit of small and

_large

shear rates. In the limit of small shear rates we obtain the

perturbative

correction to the fuctuation

_{integral (2.9),}

which now

reads,

to the

_leading

order in D and small r

where _{T-c = -} 2.03

(a À

)2/3

and

D * = ,À a 1/2.

The

_{negative sign}

of this contribution shows

that,

as

_expected,

the shear flow reduces

thé

fluctuations. The functional form

(3.6)

for

u(D)

is as

_{given by}

Cates and

Milner,

who did not,

however,

calculate the

_prefactor.

In the limit of

_large

shear rates, since we are interested in the solutions with

_vanishing

kX’

it is not

_permissible

to treat

_{(3.3) perturbatively}

in

_{1/ D.}

_Instead,

_{the memory kernel in}

(3.5)

may be

approximated by

an « effective lifetime » for

_injected

Fourier

_components,

which is

_dependent

on both k and D

_{[20, 23]. Using}

this

_approach

one finds for the

susceptibility S(k)

the

_{following interpolation}

formula

_{[20, 23]}

with b a constant of order

_unity.

This leads to the fluctuation

_{integral [20] (at}

the

_leading

order

in r)

with

_f3

_{=| T c 1|-1}

1

_{(D / D * )2/3.}

This

_integral

vanishes with

_increasing

shear rates, even in the limit r --> 0.

_Accordingly

the behaviour

_approaches

that of mean-field

_theory,

as

_{D /D *}

approaches infinity.

The crossover from the Hartree to the mean-field

_description

is

_roughly

at D = D *.

hexagonal

phases

along

with the

_isotropic

_phase.

_{Triply-periodic}

structures are excluded since

they

are unstable under an

imposed

uniform shear

flow,

as discussed above. Also excluded

are, for

example,

distorted

near-hexagonal phases

in which the three Fourier

components

have different

_amplitudes.

These are

_possible

in

_principle,

since the shear flow

ux =

(Dy,

0,

0 ) fully

breaks rotational

_symmetry,

as is reflected in the form of

_{S(k) (3.7).}

However,

when kx

=

0,

(3.3)

_recovers rotational

symmetry

in the

_ky,

_{k, plane.}

In other

words,

although S(k)

is

_{highly anisotropic}

in

_general,

it is

_isotropic

in the

_kx

= 0

subspace

of

wavevectors at which order can

_{actually develop [25].}

Of course, this

assertion,

along

with the rest of our results for the

_phase

_diagram

under flow

(presented below)

are

_only

true within the limits of the

_{approximations}

we have made. The

main ones are :

_locality

of the third and fourth order terms in the free energy

expansion

(2.1 ) ;

conditions such that the Hartree treatment holds in the static

_{regime ;}

and the existence of a

convective

_velocity

field that is

_{imposed externally (with}

no local fluctuations that

_might

be

coupled

to those of the order

_parameter).

The first

_assumption

is not _{very severe,} and has

_(for

all static

_{purposes) proved}

to be accurate

_enough

for the

_systems

of interest here. The Hartree

factorization,

if valid in the static case, holds even better under flow since this

_only

reduces the fluctuations

_[20].

The third

_{assumption (passive convection)}

should hold for

systems

that

order

_{sufficiently weakly}

since the

_density

variations that arise are then

_unlikely

to

_perturb

the

flow field. In contrast, under conditions of

_strong

_ordering,

structures that we have ruled out

(such

as

BCC)

can survive under flow

_{by completely distorting}

the

flow-lines ;

this

_regime

presumably requires

an

_entirely

different

_approach,

which we do not

_attempt

here.

3.2 THE PHASE DIAGRAM UNDER A SHEAR FLOW. - As described in the

_preceding

section,

the Hartree and mean-field

_{phase-diagrams}

can be

_represented

in a two-dimensional space of

reduced

_parameters.

These

_parameters

are related to the coefficients of the

_{Landau-Ginsburg}

free-energy :

the

_temperature

T, the

asymmetry

IL, the

strength

of the fluctuations A

and,

additionally

for the Hartree

_{representation,}

the wavevector

_ko

and c, the curvature of the

susceptibility

at its maximum. In order to have a common

_{representation}

for both

phase

diagrams

we chose the Hartree set

_[T,

_u

_{] (Sect. 2.2)}

which is

simply

related to the mean field

representation by

[ T

=

(À 1/2 a - 1 )2/3 T

/A,

_{9 =}

(À 1/2 a - 1 )1/3 IL / À]

] (notice

that the

mean-field transition lines

_{T / À}

= _{const. x}

(u

/ À)2

keep

the same form in this

_{representation).}

Following

the above

_scheme,

in the _presence of a dimensionless flow rate A =

_{D /D *,}

the

phase diagram

for the

_{crystallization}

of the lamellar and

_hexagonal

structures is still obtained

by

minimization of the « static »

_{thermodynamic potentials}

of

_{equations (2.7),}

where the contributions from the fluctuations to the renormalized

_{susceptibilities (Eqs. (2.1)}

and

_(2.9))

are now

_{flow-dependent (Eqs. (3.6)}

and

_(3.8)).

In the

_regime

of small shear rates we solve

these

_{equations by}

a

_perturbation

_method,

to first order in

A2.

For

_large

shear rates we

indicate the

_general

trends of the

solutions,

which may be determined

largely by applying

topological

arguments

to the

_{phase diagram.}

The new

_{phase diagram}

which accounts for the role of the shear flow is set in a

three-dimensional _space

_{[f, u, A].}

The A = 0 and A = _oo

planes correspond, respectively,

to the

static Hartree

_{(Fig. 2)}

and mean-field

_{(Fig. 1) phase-diagrams.}

The

_g-

= 0

plane (Fig. 3) corresponds

to the situation

previously analyzed by

Cates and

Milner

_[20].

This situation arises for

_completely

_symmetric

systems,

and the

_only

two

_possible

phases

are the lamellar and the

_isotropic.

The I-L transition

_temperature

f

_departs

continuously

from

_{Te = -}

2.03 for zero shear rate, to reach the infinite shear rate value of

Te

= 0. in the

region

of small flow _rate the first correction _to the

_temperature

is

quadratic

in

Fig.

3. - Role of the shear

rate in the

_{isotropic-lamellar phase}

transition for

_symmetric

_systems

/à7 = 0. As the shear rate â goes to

large

values the I-L line

exponentially approaches

the

spinodal

Ts.

The transition remains first order

_[20].

For

large

values of d the transition line

_approaches

_{exponentially}

the

_spinodal

line TS : :

where

_TS,

_{the solution of ro} =

0,

is

given by

Since the transition line

_always

lies above the

_spinodal,

the I-L

_phase

transition for

_symmetric

systems

remains first order.

However,

all the relevant

_quantities,

such as the

_amplitude

or the

transition

_temperature,

are

_{exponentially}

close to their values on the

_spinodal

and thus _may

be,

at

_{high enough}

_shear,

_{indistinguishable}

from a second order transition for

_practical

purposes.

When u =

0 the

_{planes [T, A )}

_]

at

_{constant fi}

are

_{qualitatively}

different for values of

fi

larger

or smaller than

_{uc =}

0.564.

In the

_large

_/I

case

_(g

_uc)

three different

_{phases already}

exist at zero shear rate : the

homogeneous

disordered

_phase

at

_high

_{temperatures,}

the

_{hexagonal phase}

at intermediate

temperatures

and the lamellar

_phase

at lower

temperatures.

Thus,

by increasing

the shear rate at constant

_/I

the

_only

effect is to shift each of the two

_{correspondent}

transition lines

_{(Fig. 4).}

Fig. 4.

Schematic

phase diagram

for the transition between the

isotropic, hexagonal

and lamellar

Again,

for small shear rates there is a

parabolic departure

from the Hartree values :

where

_To(u)

are the I-L and H-L transition

_temperatures

for a

_given

_g

at zero shear. We list some values of the coefficients

_{C (û )}

in table I.

_CHL

is smaller than

_CIH

and

_thus,

on

increasing

values of the

_asymmetry

_the

Hartree H-L transition line

_approaches

its

mean-field limit much faster than the I-L line. For

_large

shear rates each of the transition lines

approaches

its

_respective

mean-field value

_following

an

_equation

similar to

_(3.11).

In the

_{intermediate iî}

case

₍₀

_g

_{uc), only}

two

_phases

are

_present

in the static Hartree

phase diagram,

but three

_phases

do exist in the mean field limit. Therefore the I-L transition

line must

_split

in the two I-L and H-L lines for some value of d

_(Fig.

_5).

_{Correspondingly,}

the

triple point

located at

_{[t c’, u c’ A = 0] in the}

static case moves towards its

[f

=

0,

u =

0, à

= _oo

]

position

in the mean-field

diagram.

We sketch in

figure

6a the

g

coordinate of the

_{triple point}

as a function of L1.

_Figure

6b shows this line calculated

perturbatively

for small shear rates : curves are shown for both the intersection of the

_1-L/H-L

and of the

_1-L/I-H

sheets ;

these must of course coincide in _any exact calculation. We can see

from the extension of the

_region

where the lines almost coincide that the

_perturbation

method

works very well up to shear rates of A = 0.1.

Table I. - Some numeric values

_for

the

_{coefficients of}

the temperature

shift

with shear rate.

Fig.

5. - Schematic

_{phase diagram}

for the transition between the

isotropic, hexagonal

and lamellar

mesophases

for _systems with small _asymmetry

_{(0 g}

_uc

=

0.564).

This

diagram implies

that a

isotropic

system

which,

under

_cooling,

would form a lamellar

phase

may nonetheless

crystallize,

upon

Fig.

6. -

a)

Schematic

representation

of the I-H-L

triple

line.

b)

The

triple

line calculated from the intersection of the

_I-H/1-L

and

_H-L/I-L

surfaces. The

_region

where the two lines

_reasonably

coincide

gives

a measure of the accuracy of the

perturbation

method.

The

_special

_geometry

of the

_{phase diagram}

near the static

_{triple point}

allows for a very

interesting

situation where the same

_isotropic

_system

can transform either into the lamellar or

into the

_{hexagonal phase according}

to whether is cooled or sheared. The

_{optimal position}

in

the static

_{phase diagram (A}

=

0 )

_to observe this

particular

behaviour is for

à

smaller than but

close to the

_{triple point}

value

_(uc

=

0.56),

and for

_temperatures

such that f is

_greater

than

but close to its I-L value

_{Te = -}

2.03.

4.

_Applications

to

_copolymer

and surfactant

_systems.

The results

_presented

above determine the effect of shear on

_systems

whose

_phase

transitions

are well described

_by

a

_{Landau-Ginsburg expansion}

with small

_quartic

_nonlinearity

A. In this

sense

_they

can be

_{straightforwardly applied}

to blends of diblock

_copolymers

and,

with some

extra

_input

from

_experimental

_results,

to microemulsions. We believe also that the

explanation

of flow induced

_striped

_phases

in colloidal

_{crystals [14]}

_may well be related to the

present

work,

but a detailed discussion of this is

_beyond

our _{scope. The} same remark

_applies

to the

_possibility

of

_hexagonal

_crystalline

order

_being

induced

_by

shear in wormlike or rodlike

micellar

_systems

_[26].

4.1 THE M.S.T. OF DIBLOCK COPOLYMERS UNDER A SHEAR FLOW. - We follow in this

subsection _{the definitions and notation used in the paper}

_by

Fredrickson and Helfand

_[1]

_]

hereafter denoted

_by

FH. The

_system

consists on a blend of AB diblock

_copolymers

with

polymerization

index N. The A block is a _sequence of

f N

monomers and the B block has

(f - 1 )

N

_segments.

The

_unperturbed

radius of

_gyration

of the chains is R = b

J N / 6

where b is the statistical size of both monomers A and B. The interaction between monomers A and

B is

_expressed

_by

the

Flory

interaction

_parameter

X.

Combining

our results and

equations

(4.3), (4.4)

and

(2.16)

of FH one

_gets

the

following

shear induced shift for the values

of X

at

the I-L transition line

It is

_already

clear from the FH paper

(see

also Refs.

_[22,

27])

that the

_region

around the

triple point

is

_easily

accessible from the

_{experimental point}

of

view,

its

_particular

_position

in the

_{phase diagram being essentially}

a function of the

_{ratio f}

and of the

_{polymerization}

index

N. The last relevant

parameter

to be determined is then the crossover

_frequence

D* =

_{(À a 1/2}

_(related

to the dimensionless shear rate

_{by A}

=

D /D *).

From its definition

a 1/2 iS

_proportional

to the dominant wavevector

_ko

and thus

_proportional

to

N- 1/2 .

The coefficient A is of order of

N -1

and,

assuming

that the

_dynamics

of

the individual chains is

governed by reptation,

the

_Onsager

_{mobility C}

is

_proportional

to the

_{reptation frequency [27,}

28].

This last can

already

be,

for

large enough

chains,

of the order of 1

s-1,

and thus well in

the range of the Couette flow

experiments.

Note that because of the

enhancement

factor

N-

_{1/2 (D* ~ N - 3/2 03BE)}

a flow _{induced transition may arise} at flow rates small even

_compared

to

the

_reptation

_{frequency [22, 27].}

Numerical

_predictions

for D* can be calculated

_by

combining

equation (2.10)

of reference

_[27]

and the results of table 1 in FH

_(note

that in this table À is N times

_larger

than À in our

_paper).

4.2 THE ORDERING OF MICROEMULSIONS UNDER A SHEAR FLOW. - We consider in this

paragraph

mixtures of

_oil,

water and surfactant which form

_{thermodynamically}

stable

_phases.

Although

no

_complete

derivation of a

_{Landau-Ginsburg free-energy}

from a

_microscopic

Hamiltonian is

available,

we _{may estimate the} coefficients on which our

_{phase diagram}

depends by combining

phenomenological expressions

of the

_free-energy

in the

_isotropic

state

with

_{existing experimental}

results. A convenient form for the

free-energy

per unit volume of

the

isotropic

state

_{G (~f, ~s’ Xo,d ),}

has been obtained

_by

Andelman et al.

_[12]

within a

random

_mixing

_{approximation.}

This

_{free-energy depends}

on four

_{independent parameters : 0}

the volume fraction of

_{water; os}

_the

volume fraction of the surfactant

_(which

resides in the

surface

_dividing

the water and oil

_{domains) ; 6}

_{=03BEk/a where ek}

is the

_{persistence length}

of that surface and a its

_thickness,

_{and Xo}

which is the ratio of the

_{persistence length}

to the

spontaneous

radius of curvature of the interface.

_{By minimizing}

this

_free-energy

the authors

obtain a

_{phase diagram}

which

_{qualitatively explains}

the main features of microemulsions. We now concentrate on the

_region

of the

_{phase diagram}

where

_only

one

_isotropic

_phase

is

_present

(avoiding

the

_complexity

of

_coexisting

oil-rich and water-rich

_{phases). Specializing}

to the role of the fluctuations in the

_water/oil

contents we

_expand

the

_free-energy

_{G (Q, Qs , Xo, d) in}

powers of the order

parameter

03C8 = ~ 2013 ~

and its

_gradients

where

_f

₌

_{a) (I)G ( cf> ) 1 cfJ =~.}

_{Here 0- is}

the average value of the water contents in the

homogeneous

disordered

_phase

_{and 0}

its

_spatially

_dependent

value in an ordered

phase.

The

coefficients

_f3

and

_f4

can be identified with

_{parameters IL}

and À of

_equation

_{(3 .1 ).}

Parameters

A and B can be eliminated in favour of

_{experimentally}

determined

_quantities,

_namely’

the maximum of the

_{susceptibility (at}

the static

_transition)

_S(ko),

the wavevector

_ko

for which it

occurs and the

_{compressibility}

f 2 = S-1 ( o ) .

It is thus easy to estimate the

asymmetry

parameter

g

where

_/3

=

S(0 ) /S(ko),

a coefficient which for microemulsions is of order

unity [29].

Within

7TCPs/(6 (1 - ~))».

We

_display

in

_{figure 7}

the values of

_g

as a function

of ~

for

f3 =

0.5, ~s

=

0.2,

8 = e 10 and _xfl = 0

(see

Ref.

[12]).

The

asymmetry parameter

can

_easily

be

in the

_region

of

_9,

even for

_quite

small

_asymmetries

of the

_water/oil

contents. This does not

contradict the

_{original assumption}

of Cates and Milner

_[20]

_{that 1£}

=

0, since

the

_systems

studied in that paper

(namely

«

_{spongelike » bilayer phases)}

have under suitable conditions a

special

symmetry

(inside/outside

symmetry

[30])

that ensures the

_vanishing

of a cubic term in the

_analogue

of

_{equation (4.4).}

To

_investigate

the

_possibility

of a microemulsion

_exhibiting

a shear-induced

_hexagonal

or

lamellar

_phase,

we need to estimate the crossover shear rate D *. This was done

_by

Cates and

Milner

_[20]

on the basis of a

_hydrodynamic

estimate of the relaxation time for

_piece

of

surfactant film of the characteristic size

_(related

to the

_{persistence length).}

However,

their estimate

_neglects

the fact that full structural relaxation

_{requires topological}

reconnections of the

film,

for which there is a

_significant

_energy barrier

_{[31]. Incorporation}

of this effect _may

allows a reduction of D* from rather

_high

rates of order

104-106 s-1

toward

values more

readily

accessible

_{experimentally.}

In

_particular

_{for suitable sponge}

_{phases [6]}

the I-L transition appears to be induced

_{by simple agitation}

of the test

_{tube ;}

while this is an extreme

case, D * _{values of order}

a few hundred for normal microemulsions

_{might easily}

be attainable.

Fig.

7. - Variation of the

asymmetry parameter « as a function

of 0

in a microemulsion. Here

CPs

= 0.2, 5

= e 10

and xfl = 0

(see

Ref.

[12]).

The value of il _can be

easily

in the

_region

of

gc

even for

relatively

small

asymmetries

of the

water/oil

contents.

5. Conclusions.

In this paper we have studied the effect of shear on the transition from

_{isotropic (I)}

to

hexagonal (H)

and lamellar

_{(L) phases}

in

_{weakly ordering}

_systems,

such as block

_copolymers,

microemulsions,

and

_lyotropic

surfactant. The work differs from earlier studies

_[20]

in

including

a cubic term in the

_appropriate

Landau

_expansion,

thus

_allowing

_systems

to be described that do not have reflection

_symmetry

of the order

_parameter.

In such

systems

hexagonal phases

can arise as well as lamellar ones.

_(Cubic

order can also arise in the static case but

is,

within the

_present

theoretical

_framework,

unable to survive even an infinitésimal

Our work confirms the earlier conclusion of Cates and Milner

_{[20] that,}

_{by suppressing}

nonlinear fluctuation

_effects,

the role of shear is to move the

system

away from Hartree and toward mean-field behaviour. For the case of the I-L

_phase

transition this leads to a transition

temperature

that increases with flow rate ; thus for suitable

_temperatures

a shear induced I-L transition is

_predicted.

In the more

general

case considered

here,

the same

_tendency

to

suppress fluctuations can lead to more subtle effects on the

phase diagram.

When the cubic coefficient

_Ji

is

_{large enough,}

a

hexagonal phase

is stabilized relative to lamellae at all flow

rates ; its stabilization

_(relative

to the

_{isotropic phase) by}

a shear flow is

_{qualitatively}

similar

to that found

_previously

for lamellae.

However,

the most

_interesting

case arises when there is

a

_relatively

small cubic term. Under these conditions a mean field calculation

(valid

at

_high

flow

_rate)

show that the

_{hexagonal phase}

is stable whereas the Hartree results for the static

case show that the lamellar

_phase

is

_preferred.

This results in the

_{phase diagram}

of

figure

5,

which shows that

_hexagonal

order may be induced under shear in

systems

that have no

_phase

of this

_symmetry

in their static

_{phase diagram.}

Whereas the observation of static

_hexagonal

order in some microemulsion

_forming

_systems

[13]

provides ample

motivation for our

study,

the above comments show that the results are

relevant to _many other

_systems

for which

_only

lamellar ordered

phases

are seen

_statically.

For surfactant

_systems

the characteristic flow rates

_required

to induce a

_change

of

_symmetry

_may

depend strongly

on the _energy barriers for

topological

reconnection of the surfactant

layers

[31],

but flow rates of order a hundred

s-1

should be sufficient to see a transition in many

cases. For block

_copolymer

_systems

the flow rates

_required

are _very much smaller

(at

least for

long

chains) although

for

large

molecular

weights

the

_temperature

range in which the flow-induced transition can occur

_(namely

the range between the Hartree and mean-field

predictions

of the static transition

_temperature)

becomes

_increasingly

narrow.

Finally

we

_point

out that in the low flow rate

_regime

our

_perturbative

results should be

amenable to

_{quantitative experimental}

test.

_For,

_although

there are several free

_parameters

(il,

À,

c,

etc.),

most of these can be fixed

_{by performing}

suitable static measurements

_(e.g.

of

S(k)

at more than one

_{temperature).}

The

_Onsager

coefficient £

(which

via D * sets the time scale for

_dynamics)

is estimable

_{theoretically,}

as indicated

above,

but it should be

possible

in

any case to eliminate this

_parameter

between two different

_dynamics

measurements

_(for

example

the

_dependence

on flow rate of the structure factor

[22],

and that of the transition

temperature)

so as to obtain a

quantitative

test of the

_{theory. (The}

measurement of a

_dynamic

structure factor in the absence of flow would also

_give

the needed

information.)

The accuracy

of our

_approach

of course

_depends

on the correctness of the Hartree

_{approximation ;}

_roughly

speaking

this is valid near the transition to whatever extent there is a

_{large peak}

in the

structure factor

_{(S(ko) IS(O) =}

_{f3 -1 >}

_{1 ).}

_By

this measure, we

_expect

close

agreement

for

block

_{copolypers [1]}

but

_{only qualitative validity}

in the microemulsion case, for which

f3

values of order 0.5 are more

_typical.

Acknowledgements.

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BRAZOVSKII S., Sov.

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JETP 41

₍₁₉₇₅₎

85.

[18]

DE GENNES P. _G., Mol.

_{Cryst. Liq. Cryst.}

Lett. 34

₍₁₉₇₆₎

91.

[19]

KAWASAKI K., ONUKI A.,

Dynamic

Critical Phenomena and Related

Topics,

Ed. C. P. Enz

(Springer,

New

_York)

1979.

[20]

CATES M. E., MILNER S. T.,

Phys.

Rev. Lett. 62

₍₁₉₈₉₎

182.

[21]

See, e.g., Critical Phenomena, Ed. F.J.W. Hahne

(Springer,

New

York)

1976.

[22]

FREDRICKSON G. _H., J. Chem.

_Phys.

85

₍₁₉₈₆₎

5307.

[23]

ONUKI A., J. Chem.

_Phys.

87

₍₁₉₈₇₎

3693.

[24]

See, e.g., ZINN-JUSTIN J.,

Quantum

Field

Theory

and Critical Phenomena

(Clarendon, Oxford)

1989.

[25]

Hence the flow does

_nothing

to favour a distorted

hexagonal

phase

over an

_ordinary

one, and the distorted

_phase

is not

_{expected (unless}

for some reason it is

_already

_present in the static

_phase

diagram).

[26]

See CATES M. E., TURNER M. S.,

accepted

for

_publication

in

Europhysics

Lett., and references therein.

[27]

FREDRICKSON G. H., HELFAND E., J. Chem.

Phys.

89

₍₁₉₈₈₎

5891.

[28]

KAWASAKI K., SKIMOTO K.,

Dynamics

of

_Ordering

Processes in Condensed Matter, Ed. S.

Komura

_(Plenum,

New

_York)

1988.

[29]

MILNER _S., SAFRAN S. A., ANDELMAN _D., CATES M. _E., ROUX _D., J.

_Phys.

France 49

₍₁₉₈₈₎

1065.

[30]

ROUX D., CATES M. _E., OLSSON U., BALL R. _C., NALLET _{F., BELLOCQ} A.-M.,

Europhys.

Lett.

(in press).