Equilibrium emulsification of polymer blends by diblock copolymers

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Equilibrium emulsification of polymer blends by diblock

copolymers

Zhen-Gang Wang, S.A. Safran

To cite this version:

Equilibrium

emulsification of

_polymer

blends

_by

diblock

copolymers

Zhen-Gang Wang

(*)

and S. A. Safran

Corporate

Research Science Laboratories, Exxon Research and

_{Engineering Company,}

Annan-dale, NJ 08801, U.S.A.

(Reçu

le 26

_juillet

1989,

accepté

le 29

_septembre

₁₉₈₉₎

Résumé. 2014

On a étudié

_{l’apparition}

et la

_morphologie

de diverses

phases

à

_{l’équilibre,}

du _type

émulsion, formées par le

mélange d’homopolymères

A et B avec des

_copolymères

diblocs A-B

jouant

le rôle

_d’agents

de

_{compatibilité.}

On s’intéresse, en

_particulier,

au _rapport entre les

longueurs

relatives des blocs et les

_{propriétés élastiques}

de la couche interfaciale

_qui

détermine le

comportement de la

_phase.

On évalue les modules

_élastiques

de courbure K et

_K,

_{ainsi que le}

rayon de courbure

spontané R0.

La situation abordée diffère de celle traitée par

l’approche

habituelle des interfaces de micro-émulsions contenant des surfactants de chaînes courtes _pour

deux raisons :

₍₁₎

la _{surface par}

_{copolymère 03A3}

à l’interface n’est pas fixée, mais déterminée à

partir

de

_{l’équilibre}

entre

_{l’énergie élastique}

de la chaîne et la tension interfaciale ; ainsi 03A3

_dépend

aussi des courbures.

₍₂₎

En

_conséquence

de

_(1),

les coefficients

_élastiques

K, K

et

_R0

sont

interdépendants

et, au contraire de ce

_qui

se _passe avec les

_systèmes

à chaîne courte, le module

K est

_{toujours négatif}

_(voir

le texte _{pour la convention} sur les

signes).

Le

_diagramme

de

_phase,

en

fonction de ces

_paramètres,

est donc différent de celui obtenu à

_partir

des

_descriptions

conventionnelles d’interfaces avec des surfactants à chaînes courtes.

Abstract. 2014 The

onset and

_morphology

of various

_equilibrium

emulsion

_phases

for an A and B

homopolymer

mixture with diblock A-B

copolymers

as the

_{compatibilizer,}

is

investigated.

Attention is focused on the relation between the relative

_lengths

of the blocks and the elastic

properties

of the interfacial

_layer

which determines the

_phase

behavior. The curvature elastic moduli K and

K,

as well as the _spontaneous radius of curvature,

R0,

are obtained. The situation

studied here differs from the conventional interfacial

approach

to microemulsions with short-chain _surfactants, in that

₍₁₎

the area _per

_{copolymer 03A3}

on the interface is not fixed, but is determined

_by

a _{balance between the elastic energy of the chain and the interfacial tension ;} thus

03A3 also

_depends

on the curvatures ;

₍₂₎

as a result of

_(1),

the elastic coefficients K, K and

_R0

are

_{interdependent,}

and in contrast to the short chain _systems, the

_saddle-splay

modulus

K is

_{always negative}

_(see

text for the

_sign

_convention).

The

_{phase diagram}

as a function of these

parameters is therefore different from that obtained

_by

conventional interfacial

_descriptions

for short surfactants.

Classification

Physics

Abstracts

61.25H - 68.10E - 82.70K

(*)

Present address :

_Department

of

_{Chemistry, University}

of California, Los

_Angeles,

CA

90024, U.S.A.

1. Introduction.

Blends of two

_homopolymers

of different chemical constituents often

_phase

_separate,

due to the very low

entropy

of

_mixing

of these macromolecules.

_They

can be

_{compatilized by adding}

some diblock

_{copolymers composed}

of the two

_homopolymers

_{[1] :}

these

_copolymers

are

analogous

to surfactants in

_systems

of small molecules. Like oil-water-surfactant

_systems,

the

morphology

of

_systems

of A and B

_{homopolymer compatibilized}

with A-B block

_copolymer

changes

as a function of the volume fractions of the various

_components

[2].

On a

phenomenological

level,

the

_phase

behavior of such

_systems

can be obtained from an

interface

_description

which focuses on the role of the curvature _energy

_[3-7].

The curvature energy is described

by

a few

_parameters,

such as the

_bending

moduli K

and K

and the

spontaneous

radius of curvature

_Ro

_[8],

which

_depend,

in

_general,

on the

_microscopic

details of the

_systems

in consideration. For the

_polymer

_systems,

the

_dependence

of these

_parameters

on the

_{polymer properties}

_(such

as molecular

_weight)

can be obtained

_{fairly accurately.}

This is because the dominant contribution to the

_polymer

_{free energy} comes from the chain

connectivity,

a feature that is

_{largely independent}

_{of many}

_microscopic

details. Such a

_study

then

_provides

the basis for

_achieving

the desired

_properties

and

_phase

behavior

_{by tailoring}

the

_polymers

to be used.

In this paper, we establish the

_relationship

between the relative

_lengths

of the two blocks in the

_copolymer

and the

_morphologies

of the emulsified

_phases.

Such a

_study

offers a

comparison

between the results obtained here on a more fundamental level and those

obtained

_{by phenomenological approaches.}

The

_{physical origin}

of the

_emulsifying

_{power of the diblock}

_copolymers

comes from the

exceedingly

low

_solubility

of the diblocks in either solvent due to the

_positive

_{Flory-Huggins}

parameter

and the

_large

molecular

_weight

of the chains.

_By

_migrating

to the A-B

interface,

this energy cost is

_partially

_relieved,

at the cost of both a loss of translational

_entropy

(due

to the localization at the

_interface)

and an increase in the

_stretching

_{free energy. When this}

competition

is balanced - this is achieved when the volume fraction is above

some critical

value -

a

_macroscopic

number of domains of A and B

_{homopolymer separated by}

a

_single

interfacial layer

of diblock

_{copolymer spontaneously}

form. In what

_follows,

we calculate the

interfacial energy of the surfactant

(diblock copolymer)

layer

and use statistical

ther-modynamics

to _{obtain the sequence of}

_morphologies

and

_phase

_equilibria.

The relation of the elastic constants to the molecular

_properties

in short-chain surfactants is discussed in references

_[9-11].

The curvature _{energy of}

_{grafted homopolymer}

chains have

recently

been calculated in reference

_[12].

Cantor

_[13]

has also studied the interfacial

properties

for diblock

_copolymers

adsorbed at the interface of two

_incompatible

small-molecule

_liquids.

His work was concemed with the

_{scaling properties}

of

_polymers

in

_good

solvents,

as

_opposed

to our

_present

_study

of

_polymer

melts. In a recent _{paper, Leibler}

_[14]

has treated the

_problem

of the

_instability

of the random

_mixing

of A and B

_{homopolymer by}

a

small amount of A-B block

_copolymer.

_However,

that work did not

_apply

to the case where a

macroscopic

number of interfaces

_{(globules, domains)}

exist and does not

_predict

the

dependence

of the

_morphologies

on the

_properties

of the diblock

_copolymer.

A

_simplifying

feature in the

_problem

we _treat, as

_opposed

to

_systems

of diblock

_copolymer

with small-molecule

solvents,

is that the

_penetration

of the

_homopolymers

to the interfacial

copolymer layer

is rather small. This is because the

_copolymers

at the interfacial

_layer

are

interfacial

_properties

can be

_regarded

as _{those of the pure diblock}

layer only.

This allows us to make use of some of the known results obtained for the latter to the

present

problem.

The

_{interfacial layer}

is

_{depicted schematically}

in

_figure

1. It consists

_{predominantly}

of the diblock

_copolymers

in their melt

condition,

with very little

homopolymer penetration.

This melt state

_implies

_space

_filling.

For short

surfactants,

such a condition

_implies

a more or less

fixed surface coverage ; but for

polymers,

space

filling imposes

no restriction on the area _per

copolymer

molecule,

1. In

fact, 1

is determined

_{by balancing}

the

_stretching

_{energy which} favors

_large

values of

X,

with the surface tension energy which favors small values of £. For the flat

layer,

this balance leads to the well-known 2/3 power

scaling

D ~

N 2/3

_[18-20]

for the

equilibrium layer

thickness,

D,

as a function of the molecular

_weight,

N. In the case of a

curved

interface,

the area _per

_chain,

£, depends

on the curvature. This curvature

_dependence

of Z makes a contribution to the elastic moduli K

and K

that is

_proportional

to _{the square of} the

_spontaneous

curvature, as discussed below.

Fig.

1. - A schematic _cross section illustration of

a

_spherical

_{(cylindrical)}

block

_copolymer

shell for the calculation of the

_bending

_{free energy. The dashed} curve _represents the location of the

_{junction points}

between the two blocks.

No-penetration

of

homopolymers,

strong

segregation

of the A and B _domains,

and

_{incompressibility}

relate the

_layer

thickness of each block to the radius R and the area per

copolymer

£via: NA=

d-’£R[l -

(1 - LA/R)d],

and

NB

= d-l

XR [(l

₊

LB/R)d-1]

] (d

= 2 for _a

cylinder

and

d = 3 for _a

sphere).

The

_organization

_{of the paper is} as follows : iI1 section

_2,

the elastic

_properties

of the diblock

_layer

are derived from the

_stretching

_{free energy of the}

_system.

The

_bending

elastic

modulii, K

and K

and the

spontaneous

radius curvature,

Ro,

are obtained. In section

3,

the

thermodynamics

of the

_system

is

_analyzed

and the

_phase

_diagram

of the

_system

_indicating

the

morphology

(spheres,

cylinders,

lamellae)

is obtained as a function of the volume fractions

and the relative

_lengths

of the A-B

_regions

in the diblock

_copolymer.

_{The paper concludes} with a discussion in section 4.

2. Elastic

_properties

of a diblock

_{copolymer layer.}

In the classical

_theory

of curvature

_elasticity,

the

_properties

of a sheet

(layer)

are

characterized

_by

three

_{phenomenological}

_parameters

_{[8] :}

the

_splay

modulus

_K,

the saddle

splay

modulus

_9,

and the

spontaneous

radius of curvature

_Ro.

_Thus,

the _{local free energy per} unit area

f

for a deformation of curvature radii

_R,

and

R2

is written as

For a

_layer

of diblock

_copolymer,

where the area _{per molecule is} not

_fixed,

it is more

meaningful

to _{consider the free energy per}

_copolymer

_molecule,

h.

_However,

we still assume

that the elastic

_description

takes the form of

_equation

_(1),

_although

the dimension of K and

K

in h will differ from those

_appearing

in

_equation

_(1).

In

addition,

we take all

_energies

in

units of kT.

The _{free energy per chain for}

_polymer

chains

_grafted

on a

_slightly

curved surface

_(polymer

brush),

can be obtained

_by

_using

the Alexander-de Gennes

_argument

_{[21, 22]}

for the

stretching

energy

together

with the

assumption

of

incompressibility

of the melt.

Recently,

Milner and Witten

_[12]

have gone

beyond

the

_simplified

_description

of references

_{[21, 22]}

and have calculated the curvature elastic

_properties

of the stretched

_polymer

brush. The

_stretching

energy for a

_{copolymer layer}

involves a

_{straightforward generalization}

of these

_{results ;}

one

adds the

_{stretching energies}

of the two diblock sections.

We denote the

_{total length}

of the

_copolymer

chain as

_N,

with

_NA

and

NB

the

_lengths

of the A and B

blocks,

respectively. Defining

an

_{asymmetry parameter,}

E, as e = 1/2 -

_{N A/N,}

the

stretching

energy per chain for a uniform

(spherical

or

_cylindrical)

deformation of radius R at fixed area _{per molecule}

X,

can be shown to be

where d = 2

applies

to a

_cylinder

and d = 3

applies

to a

_sphere.

The dimensionless _curvature, c, is defined as c =

(1/2) Nv / (£R),

and a and v _are,

respectively,

the

length

and volume of _a

monomer.

In

_equation

_(2),

we have assumed that the two blocks of the

_copolymer

have identical

microscopic

features, a and v ;

these

_parameters

are set to

_unity

in

_subsequent

discussions for

simplicity.

The

_asymmetry

in the

_microscopic

structures of the diblock sections can be

_easily

incorporated

when

_comparing

with

_experimental

data,

with

_only

a

_slight

modification

_[23].

For a more

_general

deformation,

assuming

the local form of free energy is still

valid,

equation (2)

is

_{replaced by}

where cl and c2 are the local

(dimensionless)

curvatures.

The total free energy of the

layer

consists of the

_stretching

_{energy and the} interfacial

tension,

the latter

_{being linearly proportional}

to the area of contact between A and B

segments

which must occur at the surface

_dividing

the diblock. In the

_strong

_segregation

_limit,

we have

where y is the interfacial tension between A and

B,

and is related to the

_{Flory-Huggins}

parameter

by

’Y -- X 1/2 av-1

[24].

Equation

(4)

contains two

_competing

terms : the interfacial tension term which tends to

The value of the free energy

for £ = £ *

is

h *,

given by

To obtain the correct dimensions for Z * and

_{h *,}

we

_{merely replace}

N with

Nv21a2 and

use the

dimensional y

(’Y -- X 1/2 av - 1 ).

For a

_slightly

curved

_layer,

we seek an

_expansion

of X _up to second order in the curvatures cl and c2. After

collecting

terms, we obtain the

following expansion

of the free energy per

chain,

where the

_Ei’s

are defined as

ti = (1/2) NI(£* Ri),

i =1, 2,

and

where £*

and

h * are the values derived for the flat

layer

in

_{equations (5)}

and

_(6).

Comparing equation

(7)

with

_equation

_(1),

we

_identify

the

_splay

and saddle

splay

moduli as

and

The

_spontaneous

radius of curvature is

Another

_commonly

used definition of the

_bending

moduli results from

_writing

the free energy in the form

[7, 11]

This definition has the

_advantage

that the

_spontaneous

radius is that of a

_sphere

which

minimizes the free energy for

any k >

0. The conversion from

_K,

K,

and

_Ro

to

_k,

k,

and

_Ro

is

_{straightforward,}

and we

_find,

and

Throughout

the entire range of E, the saddle

splay

modulus

K(k)

is

always

negative

(positive) ;

thus a saddle

_shaped

deformation is disfavored

_{energetically}

[25].

It is often of interest to consider the ratio K, of the saddle

splay

to the

_splay

modulus. This

ratio,

for k and

and is seen to have a limited range of variation

during

the range of e between 0 and 1/2. The

latter has its consequence in the

phase diagram

in these

_parameters.

We note that the

_bending

modulus K for the lamellar

_phase

of

_{strongly segregated}

diblock

copolymers

has also been calculated

_by

Kawasaki and Ohta

_[26].

Their

result,

obtained

_by

using

a

_density

functional

_theory

_{[27, 28],}

however,

has a rather sensitive

_dependence

on the

fractional length

of a block

f ( f

= 1/2 + e in our

_{presentation).}

In

_fact,

even the

_equilibrium

layer spacing

calculated

_by

the same authors

_[28],

has a

_strong

_dependence

on

_f,

whereas on

physical ground

it is

_expected

to

_{depend only}

on the

_{total length}

of the chain in the

_strongly

segregated regime.

3.

_{Thermodynamics}

and

_phase

behavior.

In this

_section,

we discuss the

_stability

of various

_equilibrium,

emulsion

_phases,

where domains of A and B

_homopolymer

are

_{separated by}

internal interfaces

_consisting

of A-B diblocks. This can be achieved

_{by using}

the results of the

_previous

section for

_K,

R

and

_Ro.

However,

for the uniform

_shapes

_(lamellae,

_cylinders,

and

_spheres)

considered in this paper, we

_prefer

to use the interfacial energy in the form of

_{equation (4),}

with the

stretching

energy g

given by

equation

(2).

In

_3.1,

_{the emergence of the emulsion}

_phase,

with a

macroscopic

number of internal

interfaces,

is discussed and the critical diblock volume fraction is derived. In

_3.2,

the _sequence of microstructures obtained

_{by varying}

the volume

fractions is

_analyzed

and the

_{phase diagram}

of the

_system

is

_presented.

The essential

_physics

are described in the discussion

_{preceding equation}

(14),

and the

_phase

_diagram

is described at the end of section

_{3.2 ;}

the

_intervening

material

_represents

the mathematical details which may be omitted

by

the

general

reader.

3.1 EMERGENCE OF THE EMULSION PHASE. - When two

incompatible

fluids are

_brought

together they phase

separate,

with a minimal interfacial area,

_usually

a

_flat,

_macroscopic

meniscus. Surfactant-like

_molecules,

_randomly

dissolved in the two

_incompatible

_fluids,

are

attracted to the interface. In this

_section,

we

_study

the transition from dilute solutions of diblock

_copolymers

in A and B to the

spontaneous

formation of a

_macroscopic

number of internal interfaces. We assume that the two

_homopolymers

A and B are

_completely

immiscible,

but that either of them can dissolve a limited amount of the

_copolymer

C molecules.

We first

_study

the transition from the

_{completely phase-separated}

to the lamellar

_phase

as

this is the

_simplest.

The reference state is taken to be _{that of pure}

A,

B,

and

C,

the latter assumed to be in its lowest free energy state, which we take to be a lamellar

mesophase

(see

discussion

_Sect.).

_{The free energy of}

_mixing

_{per unit}

_volume,

with volume fractions

cJ> A’ cJ> Band cJ> c,

respectively,

is

where

(i

=

A,

B )

with OA and 0 Bbeing

the concentration

(volume fraction)

of the

copolymers

in A

homopolymers

and B

_{homopolymers,}

_{respectively, S}

is the total interfacial area, and V is the

total volume. The terms in F

_represent

the free _{energy of}

_mixing

_[29]

of C and A

_(B),

while

the two terms in the brackets in

_equation

₍₁₂₎

are _{the interfacial free energy of the}

_copolymer

incompressible,

namely, P A + PB + Pc = 1.

For

_simplicity,

we also assume that both A and

B have the same molecular

_weight

N as the

_copolymer.

_{Defining 0 s}

= SN

/ ( ZV )

which is the

total volume fraction of

_copolymers

in the interfacial

_layer,

the conservation law for the

copolymers

can be written as,

This conservation

_law,

_together

with the free energy

equation

(12)

completely

determine the

thermodynamic

states of the

_system.

In

_writing

_{down the free energy}

_equation

_(12),

we have

_neglected

the contribution

_coming

from the

_entropy

of

_mixing

of the surfaces. Such a contribution is

_vanishingly

small for the

lamellae and

_cylinders,

since

_they

are one and two dimensional

_{respectively.}

The

_entropy

of

mixing

is also

_unimportant

for the

_spheres,

_except

when the

_spheres

are small and their

concentration is low. Another defect of

_equation

₍₁₂₎

is the

_neglect

of the fluctuations of the surfaces and the interactions _{among them}

_[30].

These _{factors may}

_{quantitatively modify}

the

phase diagrams,

but are

_beyond

_{the scope of the}

_present

treatment.

_They

are not

_expected

to have

_{major qualitative}

effects on the

_aspects

we are _{concerned with in this paper.} Under these

approximations

and

_assuming

no

_{homopolymer penetration}

of the diblock

layer,

the

interfacial layer

should then behave the same as an isolated one, as will be seen below.

The minimum of the free energy

subject

to the

constraint,

equation

(13),

can be obtained

by

using

the method of

_{Lagrange multiplier,}

i.e.

_{by letting}

_{aF/asi -}

À

_aG/aSi

=

0,

where

Si

= 0 A@ 4, B@

OSc

and o-

_(o-

À is the

_{Lagrange multiplier,}

and F = N

A . Taking

derivatives in the above

_order,

we find

The last two

_equations

lead

_directly

to a = 1 and À = h = h *. These results

imply

that the

chemical

_potential

_(the

_Lagrange

_{multiplier) À}

is

_equal

to _{the minimized surface free energy}

when the amount of surface is free to _{vary. The}

_{concentrations OA and cpBC}

can be obtained

_by

susbtituting

À back into

_equations

_(14a)

and

_(14b),

and we

_find,

where h * is a function of the surface tension y and the

polymerization

index N

(Eq. (6)).

The critical concentration

_{0 c *}

for the onset of

_spontaneous

formation of

_surfaces,

i.e. when

os

=

0, is,

from

_equation

(13),

with c/AC, BC

given by equations

(15a)

and

_(15b),

_{respectively.}

These

_equations

have also been

given by

Leibler

_{[14]. For c/Jc::> c/Jê,}

a finite fraction of the

_copolymer

molecules will be in the

For e ~

0,

we

_expect

_drops

of a

_particular

curvature to form instead of flat sheets. In this paper we use the convention that A is

_always

the interior

_phase

so that E > 0. We start from the situation

_depicted

in

_figure

2,

where we have

_drops

of A

_suspended

in B in coexistence with a

_phase

of excessive A

_{homopolymers.}

Because there is a definite relation between the volume and the area of the

_spheres,

the total amount of C in the surface

_layers

is related to the radius of the

_spheres

and the number of

_spheres.

If the volume fraction of A in the interior of the

_spheres

is

_{Ô A,}

then the volume fraction of

_copolymers

on the surfaces is

where the function

_Wd

arises from the

_{(dimensionless)}

surface to volume ratio and is related to the

_{(dimensionless)}

curvature c

_by

Fig.

2. - A

schematic illustration of the

_{phase equilibrium}

considered in section 3. A, where emulsified

droplets

of A

_homopolymers

coexist with excessive A

_{homopolymers.}

In

_equation

_(19),

the second term in the denominator results from the finite thickness of the

layer.

We have used

_Wd

to allow for simultaneous treatment of

_spheres

and

_{cylinders :}

d = 2 for

cylinders

and d = 3 for

spheres.

In what follows _we sometimes

drop

the

subscript

d

for notational

_simplicity.

It should be

_kept

in

_{mind, however,}

that the relation between W and

c is different for

_cylinders

than for

_spheres

_(see

_Eqs.

₍₁₉₎

and

_(24)).

If the

_solubility

of the

copolymer

is very low in either

homopolymers,

W is

_{approximately}

the ratio of the volume fraction of the

_{copolymer layer}

over the volume fraction of the interior A

_homopolymers

_(cf.

Eq.

(18)).

The conservation law for the surfactants now becomes

and the free energy of the

(multiphase)

system

is

where the function

_F(iC

_E)

is the same as is

_given

below

_equation

_(12).

The surface free energy

hd ( U , c )

applies

to

_spheres

_(d

=

3 )

or

_cylinders

_(d

=

2)

_at fixed radius Rand fixed

where

_Sd(c)

is

with

In terms of

_W,

and

_Sd(c)

becomes,

It should be noted that

_equation

₍₂₃₎

was obtained as an

_expansion

for small curvatures,

keeping

terms to

_only

second order in the curvature ;

therefore,

Sd(W)

should also be

expanded

to the same order.

_However,

since we will be interested in the full range of

W,

and since

_equation

₍₂₄₎

_implies

that the curvature, c, cannot be

_large

even for

_large

W

(except

when e is very close to

_1/2),

we

_keep

the form of

_equation

(25)

and will assume it is valid for all W. For

_large

_W,

_{Sd (W )}

is still

_expected

to be

_{qualitatively}

correct ; for small

W,

the difference

between an

_expansion

and

_equation

₍₂₅₎

is not

_important

_anyway.

Equilibrium

is obtained when the free energy is minimized with

respect

to all five variables

cP¿, oBe Ug

W

_{and A’}

_subject

to the conservation condition

_equation

_(20).

_Again,

we

introduce the

_{Lagrange multiplier}

À.

_Taking

the

_partial

derivatives in the order indicated

above,

we

obtain,

The last

_{equation yields}

À = h

(or, W),

whereas

equations

(26c)

and

(26d)

_are none other than conditions for a minimum of

h (o-, W).

The u and W values that minimize

h (o-, W)

are,

and

It can be shown that

_{for e ~}

0

_{(e >}

0

_by

our

_{convention), W3}

_W2*,

and

_h3*

_h2

h*.

where

Again,

if an

_experiment

is

_{performed by adding}

C while

_keeping

the amount of A and B

fixed,

the volume fraction of C in the interfacial

_layers

_{for Pc::> pê}

will be

and the amount of A in the interior of the

_droplets

is

_(from

_Eq.

₍₁₉₎₎

In the

_{phase-coexistence region,}

the chemical

_potential

À is fixed at the value of

h3 ;

hence the concentrations of C in A and B remain

_{unchanged ; only}

the amount of each

phase changes,

as

_{given by equations}

₍₃₂₎

and

_(33).

Such a coexistence terminates when all the A

_homopolymers

are in the interior of

_spheres,

_{i. e. ,}

when $A

_{= cP A.}

If an

_experiment

is

conducted

_starting

from a blend of A and B

_{by adding copolymers}

C,

then 0 A=

cp(l - CPc),

where

_{0 0}

is the volume fraction of A before

_adding

any

copolymers.

From

_equations

₍₃₂₎

and

_(33),

we

_easily

find the volume fraction of the

_copolymer

surfactants above which all A

_homopolymers

are in the

_droplets,

to be

or in terms of

For cf>c : cf>c,

there are not

_{enough copolymers}

to

_emulsify

the blends. This value of

cf> c

is called the emulsification failure in references

_{[7, 11].}

In the emulsification failure

regime,

the microemulsion coexists with a

_phase

of excessive A

_{homopolymers.}

Above the

emulsification

_{failure A}

_{= cf> A’}

so that it is no

_longer

a

_quantity

free to _{vary. Then}

_equation

(26e)

no

_{longer applies.}

The chemical

_potential

now becomes

where a is still

_{given by o-}

=

Sd1/3(W).

The concentrations

OA

and

cpBC

are then

Notice that the concentration in the interior

_phase

is no

_longer

determined

_by

the chemical

potential

alone,

but has an additional

_piece

W2

_{ah (a , W),}

which does not vanish in

_general.

3.2 EQUILIBRIUM EMULSION PHASES. - We

now consider the case of very

_incompatible

A

and B

_{homopolymers,}

where

yN » 1,

so that the concentration of the

_copolymer

in either

homopolymer

solvent is

_exceedingly

_{low ;}

_essentially

all the

_copolymers

are found at the A-B

consider

only

the interfacial

_layers.

The conservation law in the case of

_only

one emulsion

phase

reduces

_{simply to OC = O A}

W,

so that the volume fraction ratio of

_copolymers

to A

homopolymers

is

_{W = Pei PA.}

In what

follows,

we will use the variable W to

_study

the various

_phase

transitions. For a

_given

value of

W,

the free energy per

copolymer

on the

surface is obtained after

_minimizing

h(a, W)

with

_respect

to as. This process

_yields

u =

Sd1/3(W)

(Eq. (27))

and

This free energy for e = 0.1 for the three different

morphologies

considered,

i.e.

spheres,

cylinders

and

_lamellae,

is shown in

_figure

3. As the volume fraction ratio W

_increases,

first the

spheres

have the _{lowest free energy, but when}

spheres

and

_cylinders

have the same _{free energy. For W}

_larger

than the value

_{given by}

equation

(39),

cylinders

become the favored

_morphology

in terms _{of the free energy, until}

they

are overtaken

_by

the lamellae at

Fig.

3. -

The three branches of

_layer

_{free energy}

_{(unit arbitrary)}

as a function of the volume fraction

ratio the

_copolymer

to the

_(interior)

A

_homopolymer

W

_(see

_Eq.

₍₁₈₎

for the definition of

W).

The letters L, C and S stands for lamellar,

cylinder

and

sphere, respectively.

Of course, when

_considering

first-order

_phase

_transitions,

the correct construction is the

common

_tangent

construction

_[31].

This construction is

_equivalent

to the

_equality

of the chemical

_potentials

and the

_equality

of

_{pressure-like}

_variables,

as we now discuss.

The free energy per

copolymer,

in the case of two

_{coexisting phases,}

can be

_written,

neglecting

contributions from the

_solution,

as

phases.

The

variable e

is the fraction of

_copolymers

in the

_{cylinder phase.}

The conservation condition is then

_{given by}

Minimizing f with

respect

to _ui,

_Wi,

and

e,

we find

Equation

(43a)

yields

the

relation,

0’i =

SJ/3(Wi),

whereas

equation (43b)

gives

Combining

this with

_equation

_(43c)

_yields

If we define a

_Legendre

transform

_by

then

_equations

₍₄₄₎

and

₍₄₅₎

become

where the chemical

_potential

_1£ i is defined as

Recalling

the standard

_{thermodynamic}

relations

_[32],

we see that

Q

is a

_{pressure-like}

variable

conjugate

to the

_density-like

variable

W,

and that h

_plays

the role of the Helmholtz free energy, whereas 1£

_plays

the role of the Gibbs free energy

(the

chemical

potential).

It should be noted that the derivative

_ahlaw

is a

_partial

derivative at fixed a.

From

_{equations (47), (48)}

with

_{equations (46)}

and

_(49),

the values of

_Wi

can be

obtained,

and

_using

the relation

_{equation (42), § can}

be obtained. The case

_involving

coexistence with the lamellar

_phase

is a bit trickier.

_However,

it can be shown that the condition for

coexistence with the lamellar

_phase

with chemical

_potential

_{J.L 1 and surface free energy}

hl,

respectively,

is

where m

is

the chemical

_potential

of the other

_phase

_(cylinder

or

_sphere)

defined

_through

equation

(49).

The value of

_W1

is obtained

_{by using}

the

_{thermodynamic}

_relation,

i.e.

_1/W1

= 0 or

_W1

= oo. This

_suggests

that the

_coexisting

lamellar

_phase

is the pure state

consisting only

of block

_copolymer

sheets.

This result should not be

_surprising

because,

in our

model,

the pure lamellar

mesophase

and the emulsion lamellar

_phase

are

_{energetically equivalent}

since interactions between the

sheets are

_neglected.

This artifact can be corrected

_{by incorporating,}

for

_example,

the Helfrich

_{entropic repulsive}

interactions

_[30]

between the sheets. The

_{implementation}

of this calculation is

_beyond

_{the scope of the}

present

work.

The

_preceding

results are summarized in

_figure

4 which shows the

_complete

(mean-field)

phase diagram

in the £-W

_plane,

where e

_{= 1/2 - N AIN}

and W =

_{cp ci cp A. (The}

exact relation is

_Eq.

_(18)).

Fig.

4. - Phase

diagram

in the £-W

plane

where e = 1/2 -

_{N A/N,}

which defines the _asymmetry in the

lengths

of the two blocks of the

_copolymer,

and where

_{W ~ b C/ 0 A,}

which is the volume fraction ratio of the

_copolymer

to the

_(interior)

A

_homopolymer.

_Regions

I, II, III, IV and V

_correspond,

respectively,

to

_spheres

+ excessive A,

spheres, spheres

+

_{cylinders, cylinders,}

and

_cylinders

+ lamellae.

The sequence of the appearance of various

phases

is as described above : When

oc

::> cP t,

first there is a two

_{phase region}

of excessive A

_homopolymers

in coexistence with

spherical

droplets

of A

_suspended

in B. As

_W,

the volume fraction

ratio,

increases

past

W3 ,

all the A

_homopolymers

will be in the interior of the

_{spherical drops.}

This

_one-phase

region

continues until the first appearance of the

cylinder phase

whence we have a

_two-phase

coexistence of

_spheres

and

_cylinders.

As W increases

_further,

the

_{spheres disappear, leaving}

a

single phase

of

_cylinders.

For

_{sufficiently large}

_W,

the lamellar

_phase

_appears as in

coexistence with the

_{cylinder phase.}

Because the

_coexisting

lamellar

_phase

has W = 00 in our

calculation,

a

_{single phase}

of lamellae is not

_{predicted ;}

on

_{physical grounds,}

however,

we

expect

a

_{single phase}

of lamellae at small values of e.

To compare our results with the

_{predicted morphologies}

of short-chain surfactant

microemulsions,

where the area _{per molecule is held}

_fixed,

it is of interest to

_re-plot

the

_phase

diagram using

the variables r and K, where r =

W3*/W,

and K =

k/k ;

this is shown in

figure

5. This

_{phase diagram}

_{appears rather different from the} one in reference

_[11]

for short

Fig.

5. - Phase

diagram

in the r-K’

_plane.

r =

W3*/W

and K’ =

_(13/2)

K where W is the volume

fraction ratio of the

_copolymer

to the

_(interior)

A

_homopolymer,

_W3

is the value of W which minimizes the free energy of the

sphere, corresponding

to the spontaneous radius

_(cf.

_Eq.

_(28)),

and

K is the ratio between the two

_bending

moduli k

and k

_given

_{by equation}

_(12).

_Regions

I, II, III, IV and

V

_{correspond, respectively,}

to

_spheres

+ excessive A,

spheres, spheres

+

cylinders,

cylinders,

and

cylinders

+ lamellae. The

_{région K ’ >}

280/73

_(the

limiting

value for K’ at e =

1/2)

is

physically

inaccessible.

regions.

(It

should be

_{noted, however,}

that the «

phase diagram »

in reference

[11]

is not a

rigorous

one in that it is obtained

_{by comparing}

the free

energies

of different branches instead

of

_by

the correct common

_tangent

construction.

Nevertheless,

the difference in the

interdependence

among k, k

and

_Ro,

in the two cases, has a

_major

effect in the difference

between these two

_phase

_diagrams.)

Notice that in

_figure

4,

for e >

0.45,

the

cylinder phase disappears.

This

disappearance

of

the

_{cylinder region}

is known also for short surfactant

_systems

for

_{large, positive}

values of the saddle

_{splay, k}

and

_hence,

K

_values,

and is a result of our free energy in the form of

_equation

(23).

However,

our

_predictions

for values of c close to 1/2 are

_{only qualitative}

because of

possible complications

of the

_{phase diagram by,}

_{e.g., the formation of}

micelles,

a

_possibility

which we have

_neglected

so far but is discussed in the next section. 4. Discussion.

We have

_presented

results of a mean-field calculation of the

_{thermodynamics}

of different

morphologies

in

_systems

_consisting

of two

_incompatible

_homopolymer

mixtures and diblock

copolymers

made of these

_{homopolymers.}

The

_phase

_equilibria

of isolated

_{spheres, cylinders}

and lamellar sheets have been

_analyzed.

_{The interfacial free energy is modeled rather}

accurately by adapting

the recent results of Milner and Witten for

_{polymer layers}

of

_high

molecular

_weight.

Elastic

_properties,

such as the two

_bending

moduli K and

_K,

and the

_spontaneous

curvature radius

_Ro,

are calculated

_explicitly

as functions of the molecular

_weight

of each block. The ratio

_K/K

_{changes only weakly}

as the

_{asymmetry parameter}

e is varied between 0 and

1/2,

and remains

_negative

even when the

_sign

of the

_spontaneous

curvature is reversed. In

spontaneous curvature. These features are in marked contrast to the

_{phenomenological}

theories where

K,

K,

and

_Ro

are treated as

_independent

_parameters

and to other

« microscopic

calculations »

_{[9-11, 33]}

of these

_parameters.

A more

_{complete analysis}

of

molecular

_origin

of these

_{phenomenological}

coefficients and their

_{interrelationship}

will be

presented

in a future

_publication

[34].

The

_{phase diagram}

calculated

_using

the free energy forms

equations

(22)

and

_(23),

_predicts

the

_{morphological change}

in the emulsion

_phases

as functions of e and the volume fraction

ratio,

W

_{= cf> el cf> A’}

where e is the

_asymmetry

of the two block

_lengths

and

_Oc

and

cf> A

are the volume fractions of the block

_copolymer

and the A

_type

homopolymer

respectively.

For e *

0,

the

_sphere

is

_always

the first emulsion

_phase

to _appear as

W increases. Below

_W3

_(cf.

_Eq.

_(28)),

the radius is

_{unchanged by adding}

more

_copolymer,

if

the

_entropy

of

_mixing

is

_neglected.

This radius is determined

_by

the surface tension between A and

B,

and the molecular

_weight

of each

block,

which

_suggests

a

_possible

means for

measuring

the interfacial tension

_{by using equation}

_(10c).

In our

_calculation,

we have

_{completely neglected}

the

_possibility

of

_{micellization,}

since we

always

assume that the

_system

forms domains of A and B

_homopolymer

with a

_monolayer

of

copolymer

at the interface. This

_monolayer

can coexist with

(a

_{very small number}

of)

isolated

copolymer

chains solubilized in the A and B

domains,

but we have

_neglected

_{any additional}

equilibria

with

_copolymer

micelles within these domains. It is

_expected

that when the diblock

copolymer

is very

asymmetric,

or if the volume fraction of one of the

_homopolymers

_{is very}

low,

such micelle formation will be

_important,

and indeed could

_{possibly preclude}

the

emulsion

_phases

_[14].

Thus,

our calculation is most

_appropriate

when the molecular

_weights

of the two blocks and the volume fractions of the two

_{homopolymers,}

are

_comparable.

That in these cases the formation of micelles can be

_neglected

can be

_argued

as follows : In the

strongly segregated

limit,

the lowest free energy state _{for the pure diblock}

_copolymers,

is the lamellae

_phase

_{up until E} =

0.28,

according

to _{a recent} calculation

by

Semenov

[15].

Below

this

_value,

the

spherical

and

_{cylindrical mesophases}

both have

_higher

_{free energy than the} lamellar

_phase.

_Furthermore,

Semenov shows that the

_spherical

and

_{cylindrical mesophases}

both have lower free energy than their

respective single

micelles. It follows then that these micellar

_phases

must be a

_higher

free energy state _{than the pure lamellar}

_phase.

_Thus,

for

e

_0.28,

these

_higher

_{free energy} states can be

_{safely neglected,}

since the lamellae

_phase

is the last one to _{appear in} our calculation. For - >

0.28,

the

phase diagram predicted

in this

paper becomes

doubtful,

as micellization can no

_longer

be

_ignored.

Formation of micelles has also been discussed

_by

Leibler

_[14],

who obtained a different numerical value for

e

_{by using}

an Alexander-de Gennes

_{type argument}

for the

_stretching

_energy.

The _present calculation has also

_neglected

the effects of thermal fluctuations of the

interfacial

_copolymer

film as well as interactions between the

_droplets.

These effects can

modify

the

_{simple picture presented}

in _{this paper. The effects of film fluctuations and}

_globular

interactions have been treated in the context of microemulsions of short-chain surfactants

[35].

A similar

_analysis

can be used in the

_present

case. The focus of this paper has been to derive the curvature

_{elasticity description}

of the

_{copolymer layer}

from a more fundamental

description

of the free energy of the film and to

_point

out some

_important

differences between

the

_polymer

_system

and the short surfactant

_systems.

The model and calculations

_presented

in this paper suffice to serve this

end,

and the

_important

features that are

_primarily

associated with the

_polymeric

nature of the

_systems

in

_study

should

_persist,

even if more

_rigorous

approaches

are used.

Acknowledgments.

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