Kinetics of interface formation between weakly incompatible polymer blends

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Kinetics of interface formation between weakly

incompatible polymer blends

J.L. Harden

To cite this version:

Kinetics of

interface formation between

_weakly

_incompatible

polymer

blends

J. L. Harden

Materials

_{Department, College}

of

_{Engineering, University}

of _California, Santa _Barbara, CA 93106 U.S.A.

(ReceiB’ed

on March 7, 1990,

accepted

on _May 11,

1990)

Résumé. 2014 Nous étudions l’évolution

temporelle

de la

_composition

de l’interface entre deux

mélanges

de

polymères

faiblement

_{incompatibles}

à l’aide d’une théorie de

champ

moyen. La

dépendance

en _temps de

_{l’épaisseur}

de l’interface

_présente

deux

_régimes

en loi de

_puissance;

d’une loi de croissance initiale en

t1/2,

on _passe à une loi de croissance en

t1/4

avant de relaxer vers le

_{profil d’équilibre}

de l’interface.

Abstract. 2014 A

mean field model of interface formation between two

_{weakly incompatible}

polymer

blends is used to

_investigate

the time

_development

of interfacial

composition.

The interface width as a function of time exhibits two _power law

_{growth regimes ;}

a

t1/2 growth

law at

early

times crosses over to a

t1/4 growth

law before

_relaxing

towards the

_equilibrium

interfacial

profile.

Classification

Physics

Abstracts 05.70 - 64.75 - 68.22 1. Introduction.

In the last few years, there has been intense and rather successful

study

of interdiffusion of

compatible polymer couples

both

_{experimentally}

_[1]

and

_{theoretically [2].}

However,

there has been somewhat less effort towards an

_{understanding}

of interface formation between

incompatible polymer species.

This case is of

_particular

interest since most

_{polymer couples}

are immiscible over a wide range of

temperatures,

including

those

_typical

of

polymer

processing

conditions. This

immiscibility

is a direct consequence of the

_{inherently large}

size of

polymer

chains which

_strongly

suppresses the

entropy

of

_mixing

of molecules but has little effect on the

_{typically repulsive}

and molecular

_{weight independent enthalpic}

interactions between unlike

_{polymer species [3].}

When bulk

_{incompatible polymer couples}

are

_brought

into contact and heated above the

_glass

transition limited

_{interpenetration}

of unlike chains

occurs,

leading

to an interfacial

_region

of finite extent at

_long

times. The

_{relatively large}

spatial

extent of such

_polymer

interfaces and the slow kinetics of their formation make them ideal model

_systems

for the

_study

of interface formation in non-critical

_binary

mixtures. In

addition,

the kinetics of interface formation in these

systems

may have relevant

technological

implications

to the

_engineering

of

_{polymer alloys}

and

_{polymer welding techniques.}

1778

Very recently,

experimental

techniques

have been

_developed

that allow

_high

resolution studies of interface formation between

_{partially incompatible polymer couples}

_[4-6].

One such

technique, utilizing

an ion beam method based on nuclear reaction

_analysis,

allows direct

measurements of interface

_{composition profiles}

between deuterated and

_protonated

_polymer

blends with _very

_{high spatial}

resolution

_[4].

Mixtures of deuterated and

_protonated

_polymers

are

_usually

_very

_{weakly incompatible [7], allowing}

for the formation of interfacial

_regions

of

finite

_yet

_very

_large

extent. Such a

_technique

allows for detailed

_study

of the time

development

of interface structure in these

_{partially incompatible}

_systems.

_Recents exper-iments

_employing

this method

_[8]

indicate that interface

_development

between

_incompatible

species

is

_quite

unlike that of free interdiffusion between

_{compatible couples}

in which interface width grows

asymptotically

as the _square root of time

_[1].

A theoretical

_description

of the

_{equilibrium properties}

of such interfaces was

_{proposed long}

_ago

[9, 10].

Until now,

however,

theoretical studies of the

_dynamics

of interface formation have focused on

moderately incompatible

mixtures,

which have rather narrow

equilibrium

interfaces

[ 11, 12].

This paper considers the case of interface formation between very

weakly incompatible

polymer couples.

In this case the

_equilibrium

interface is many radii of

gyration

in extent, and

experimental

observation of interface

_growth

kinetics is

feasible,

allowing

theoretical

predictions

of

_{dynamical scaling}

laws to be tested.

2. Model.

Consider two semi-infinite blocks

_composed

of A and B

_polymer

mixtures in the

_glassy

state

brought

into contact to form an

_{initially sharp planar}

interface. For

_simplicity,

attention will

be restricted to

_{incompressible}

mixtures of

_equal

molecular

_weight

_polymers

(NA

=

NB

=

N). Subsequently,

the

_system

is heated above the

glass

transition to a

Fig.

1. - Initial interface

configuration.

Two semi-infinite blocks

_composed

of

_polymer

mixtures

~1

and ~2

from the bulk coexistence curve at the intended

annealing

temperature

To (near

the critical

temperature

To

near the critical

_point

of

_de-mixing

of the A-B

_system,

_tg

_{TO:5 T,.}

The

composition

of the blocks on either side of the interface is chosen from the bulk coexistence curve for mixtures of

_polymers

A and B at the intended

_annealing

_temperature

To,

as shown in

_figure

1. Such a choice is made in order to eliminate the

_complication

of bulk convective

_transport

of

_polymer

across the interface.

_Upon

_heating,

the interface

_undergoes

a

healing

process that smears the

_sharp

interface to a narrow but finite

_region

some fraction of a

radius of

_gyration

in extent. This

_healing

process is

complicated by

constraints

imposed by

chain folds near the initial interface and is rather sensitive to the initial

_density

of chain ends at

the

interface,

a

_quantity

that

_{depends strongly}

on the method of

_{sample preparation.}

The

healing

of

_sharp

interfaces is a

_topic

of active

_{investigation}

that we will not address here

[13,

14]. Rather,

the

analysis

of this paper will consider the time

development

of the interface

as it

_expands

from the

pre-healed

initial width

_Wo

towards the final

_equilibrium

width

Weq,

as sketched in

_figure

2.

Fig.

2. - Sketch of the

range of interfacial

growth

to be considered. The

_analysis

considers the

development

of interfacial structure from an

_initially

broadened

_{profile ~0(z)}

of width

_Wo

towards the final mean field

equilibrium

_{profile ~ eq (z)}

of width

_{Weq ~}

_{N 1/2 a (X / Xc - 1)- 1/2.}

We assume that the free energy per unit area of the

_system

can be described

_by

the well

known

_{Flory-Huggins}

free energy

expression,

extended to account for slow

_spatial

variations in

_polymer

concentration normal to the interface

_{[15-17] :}

where 0

is the volume fraction of monomers of

_species

A, a

is the monomer

_size,

and

f[~J(z)]

is the local free energy per

kB

T as

_{given by}

the usual

_{Flory-Huggins expression}

[3, 15] :

1780

while the last term _represents the

_enthalpy

of

_mixing

as

_{gauged by}

the

_Flory

interaction

parameter

X (T).

One obtains the local chemical

_{potential g}

_(z)

of monomers of

_species

A

_by

functional differentiation of the free _energy

_{expression given by equations (1)}

and

₍₂₎

with

_respect

to

41

(z) : 1£ (z )

oc

_{d F / d ~ .}

This local chemical

_potential

drives the

growth

of the interface wi th

time ;

the

_system

will redistribute the A

_species

in order to _{relax any} non-zero

_gradient

of

J.L. The

growth

of the interface may be characterized

_by

the time

_dependence

of the A

monomer concentration

_profile

across the interface. _{This may be} determined from a

one-dimensional

_{continuity equation :}

where

_{J(z, t )}

is the local current of

_species

A. Within the context _{of linear response,}

J(z, t )

may be related to the

_gradient

of the chemical

_potential

via a non-local

_transport

coefficient

[ 15] :

where

_{A (z - z’ )}

is a

generalized

non-local

_Onsager

coefficient and _IL

_{(z’ )}

is the local chemical

potential

per

kB

T. The

non-locality

of A is a _consequence of the

_{large degree}

of

connectivity

of the monomers in a

_polymer

medium. After

_{integration by}

_parts

and omission of the

vanishing boundary

terms,

equations (3)

and

_{(4) yield}

an

_equation

of motion for

~ (z, t ) :

:

Equation (5) together

with

_boundary

_{conditions, ~ = ~1}

1 at z = + oo

and l/J = l/J 2

at

z _{= - 00,} and a

_specified

initial

_{profile, ~/0(z, 0) = ~o(z),}

constitute a

_well-posed

initial-boundary

value

_problem.

In

_{principle, given u ( ~ )}

and

A,

we _may solve

_{equation (5)}

for

0 (z, t ).

In

_practice,

the

_{resulting integro-differential equation}

is

_strongly

non-linear and not

amenable to

_simple

solution. Chemical

_{potential gradients computed}

with the full free _energy functional as

given by equations (1)

and

₍₂₎

are non-linear in the monomer concentration

variables.

Furthermore,

recent theoretical work

_suggests

that

_Onsager

coefficients for

transport

in

_{spatially inhomogeneous polymer}

blends are often a

_complicated

functional of

the monomer concentration

_{profiles [12].}

However,

we _may

_exploit

the weak

_{incompatibility}

of the

_{interpenetrating}

bulk mixtures under consideration to make several

_{simplifying approximations.}

First,

the

_Onsager

coefficient may be estimated from calculations of

single

chain

_transport

in a melt of

non-interacting

Gaussian chains

_{[16, 17].}

_Second,

the chemical

_potential

may be linearized around the critical

_{composition l/J c}

=

1/2.

It is convenient _to write the

resulting simplified equation

of

motion in terms of the

_gradient

of the

_{concentration, 0}

=

a ~ / az.

Assuming A (z - z’)

is

independent of 0 (z)

as discussed above and

_using

the linearized chemical

_potential,

the

equation

of motion

_{for 0 (z, t) given by equation (5)}

can be recast as a linear

integro-differential

_equation

of motion

_{for gi (z,}

_{t ).}

The solution to this

_equation

must also

_satisfy

boundary

conditions, 03C8 =

0 at z = ± oo, and an initial

_{condition, 03C8 (z, 0 )}

=

dl/J

o/dz.

The _new

equation

of motion and

_boundary

conditions are

_easily

solved in Fourier space. The modes of

where 03C8(q,

t ),

03A8o{q)

and

_A(q)

are the

_spatial

fourier transforms of

_{03C8 (z, t ),}

_03C8o(z)

and

A (z ), respectively,

and where _xc =

y (Tc)

=

2 /N

within the context of

Flory-Huggins

mean

field

_{theory [3].}

The wavevector

_{dependent Onsager}

coefficient for monomer

_transport

in

_weakly

incompat-ible

_polymer

blends may be estimated from the response of a

_single

chain in a melt of

non-interacting

Gaussian

_polymers

to an

_{applied spatially periodic}

chemical

potential gradient

[16].

The

_{resulting expression}

for

_{A (q )}

has the form

where

_{Rg ~ N 1/2 a}

is the radius of

_gyration

of a

_polymer

coil in melt

conditions,

L oc N is the

_length

of the constraint tube

enclosing

a

_{typical polymer}

coil as created

_by

its

entanglements

with

_{neighboring polymer}

chains,

and where

_llo

is an effective monomer

diffusion constant

_proportional

to the

_microscopic

monomer

_mobility

_{go :}

_{Ao - kB Tuo.}

In the

_{small q}

_limit,

_{qRg 1,}

A

_approaches

the

_{q-independent}

tube diffusion constant,

Dt ’"

llo/N,

and chains

_{respond by ordinary}

diffusion. In the

_large

q

limit,

_{qRg »}

1,

ll scales

_{as A (q) ~ q - 2,}

_reflecting

the

_increasing

_{inhibition of chain response} to

_{perturbations}

that vary

rapidly

on the scale of a chain size.

_{Equations (6)}

and

_{(7) yield}

an

_explicit

form for

the interfacial relaxation

modes ;

modes

_{of tb}

_decay

_{exponentially}

with time constant

Tq

given by

The direct space solution to the

_equation

of motion and

_boundary

conditions is obtained

_by

inverse Fourier transformation

_{of 03C8 (q,t ) :}

with

_Tq

as

given

in

_equation

_(8).

The initial interfacial

_profile

is as

_yet

_{unspecified. Actually,}

the

_precise

form

_{of CPo(z),}

and hence of

_03C8o(q),

turns out to be of little _{conséquence ; any} initial

_profile

that

_{smoothly interpolates}

between bulk values

_{of .0}

over a distance

corresponding

to the initial extent

_Wo

of the interface will

_{yield qualitatively}

similar results. For

convenience,

we obtain

_{03C8o (q ) by}

_{choosing 00}

of the form

_{[CPo(z) - CPc] ~}

tanh

_{(z/ Wo).}

3. Results.

Widths of

_{experimentally}

determined

_profiles

are often estimated from the inverse of the

maximum rate of

_change

of monomer concentration across the interfacial

_{region ;}

in terms of the difference in bulk values of monomer concentration on either side of the

interface,

A’k = 0 2

- ~ 1,

the width is

_{approximately given by}

In the above

_analysis,

this

_{approximation}

is

_equivalent

to

_estimating

the width

_by

the

1782

Although

not

_integrable

in closed

_form,

this is

easily

evaluated

_{numerically by treating}

time as a

_parameter.

Before

_doing

_so, it is instructive to consider the

_limiting

behavior of

W(t)

at short and

_long

times. To facilitate this

_analysis,

we assume for the moment that

f/lo(q)

is

_{approximately}

constant and that Xc - X is

negligible.

This

corresponds

to the idealized case of a _very

_sharp

initial interface

separating

two _very

_{nearly compatible}

bulk

phases.

In this case, the

integral

at

_early

times is sensitive to the

_{large q}

behavior of the

integrand

which

_{asymptotically approaches}

the Gaussian form

_{exp ( - x2)}

with

x2 ~

_{tq 2.}

Consequently,

the interface grows as

W(t) ~ t1/2

at

_early

times. On the other

_hand,

the

integral

at late times is dominated

_by

the

_{small q}

behavior of the

_integrand

which

asymptotically

approaches

the form

_{exp ( - x4)}

with

_{x4 ~ tq4.}

Thus,

the interface _grows as

W(t) -

t 1/4 at late times. At

sufficiently

late

times, however,

the

_growth

of the interface must

slow as the

_equilibrium

with

_{Weq -}

N 1/2 a (X / Xc - 1)- 1/2

is

approached ;

in the late

_stages

of

interface

_{formation, X - X c}

is no

_{longer negligible}

and we

_expect

the

_growth

to

_eventually

deviate from the

t 1/4 law.

The linear

_analysis

breaks down at this

_{point ;}

in order to

_correctly

describe the final

_approach

to

_equilibrium,

non-linear terms in the chemical

_potential

must be retained.

As mentioned

earlier,

the

_healing

of a

_sharp

initial interface is a

_complicated

_{process that is}

outside the scope of this

analysis,

and hence we are limited to consideration of

_pre-healed

interfaces.

However,

we

_expect

the

_general

case of an

initially

diffuse interface to behave in a

qualitatively

similar manner

_except

at _very

_early

times. Profiles

_{computed numerically}

confirm this

_expectation.

Consider,

for

_example,

the time

development

of an interface

pre-healed to an initial width

_Wo

= 0.25

Rg

and with an

_equilibrium

width

_Weq

= 10

Rg.

The results of numerical

_integration

of

_{equation (11)}

in this case are shown in

_figure

3.

Following

a

period

of internal redistribution of material within the

_initially

diffuse

interface,

interfacial

growth rapidly

falls onto a

t1/2 growth

law. Growth kinetics

subsequently

cross-over to a

t 1 /4 growth

law before

_{finally relaxing}

towards

_equilibrium.

The cross-over

_{region separating}

the two _power law

_regimes

occurs at a time t * _{that scales with the time}

required

for a

polymer

chain to diffuse its own diameter in a melt of identical chains :

_{t * ~- R2g/ Ds}

s where

Fig.

3. -

Log-log plot

of interface width

W(t)

vs. time t for the case of an interface with initial width

J-fô

= 0.25

Rg

and

_equilibrium

width

_Weq

= 10

Rg.

Width is measured in units of

_Rg,

while time is measured in units of t * ~

_{R g2/Ds,}

the time

_required

for self-diffusion of a

_{polymer by}

one

Ds -- kB

T/N 2

is the self diffusion constant of a

_polymer

chain with

_degree

of

_{polymerization}

N. For a

_given

choice of bulk

_polymer

mixtures this cross-over time is

_universal,

i.e.

independent

of the width and detailed features of the initial interfacial structure. 4. Discussion.

The novel feature of these results is the appearance of two _{power law}

_{growth regimes.}

The

early

time

_dynamics

in

_particular

is

_unique

to macromolecular

_systems,

_being

a direct

consequence of the

strong

non-locality

of

_transport

coefficients on submolecular

length

scales.

At

_early

times,

interface width is narrow on the scale of a

_polymer

_coil,

_leading

to

_rapidly

varying

chemical

_potential

_gradients

within the interfacial

_region.

As discussed

previously,

response to such

_{rapidly varying}

chemical

_{potential gradients}

is

_strongly

_inhibited,

_leading

to

different

_growth

kinetics at

_early

times than at late times when the interface is broad and

transport

coefficients are

_effectively

local. In contrast, the

_application

of this

_analysis

to

ordinary

small molecule

_binary

mixtures would lead to a

single

_exponent

governing growth

kinetics ;

transport

in such

_systems

is not

_strongly

non-local on _any reasonable

_length

scale

[18].

It is instructive to contrast the

_physics

of interface formation between immiscible

_mixtures,

which we have considered in this paper, with that of miscible

_systems.

One

_expects

the

differences in interfacial

_growth

kinetics in these two cases to arise from the differences in the free

_energies

that drive these _{processes. In} the case of immiscible

_systems,

interfacial

_growth

is

_entirely

driven

_{by composition gradients}

across the interface. This manifests itself in the

t1/4

_growth

behavior at late times. In miscible

_systems,

_however,

the

_{enthalpy of mixing}

favors

growth

and both the bulk and interfacial free energy terms drive the process. As such an

interface

_expands,

_however,

_{composition gradients}

weaken and

_growth

is driven

_increasingly

by

the bulk free _energy terms. Hence in miscible

_systems,

an

ordinary

diffusion

_equation

governs the

growth

of the

_interface,

_leading

to

_growth

that scales

_{asymptotically}

with time as

t 1/2

The results

_presented

here are in

_qualitative

_agreement

with recent

_experimental

data on

the kinetics of interface formation between mixtures of

_protonated

and deuterated

poly-styrenes

[8].

The measured

_profiles

are consistent with the two

_{growth regime}

behavior described above.

_However,

further refinement both of

_experimental

resolution and of the theoretical model are _necessary before detailed

_comparisons

can be made. Current

theoretical work is

_extending

the model to

_asymmetric

_{mixtures, NA # N B,}

the situation most

relevant to current

_experiments

and most often encountered in

_{practice. Forthcoming}

are

detailed results of interfacial

_{composition profiles}

for mixtures of

_asymmetric

chains,

as well

as a _proper

_analysis

of the kinetics of the final

_approach

to the

_equilibrium

interfacial

structure.

Acknowledgements.

This work was initiated

_during

a

_stay

with the group of Jacob Klein in the

_polymer

research

department

of the Weizmann Institute of Science in

_Rehovot,

Israel. 1 am

grateful

to the US-Israel Binational Science Foundation and the US

_Department

of

_Energy

under contract # DE-FG03-87ER45288 for financial

_support,

and to the

_faculty,

staff,

and students of the

polymer

research

_department

for their

_{gracious hospitality.}

In

_addition,

1 would like to thank the Institute for Theoretical

_Physics

at the

_University

of

California,

Santa Barbara for

1784

kinetics. 1 also benefitted from very

stimulating

discussions with D.

Andelman,

G.

Fredrickson,

P.-G. de

_Gennes,

A.

_Halperin,

K.

_Kawasaki,

S.

_Milner,

and P. Pincus.

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Ed. M.

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