Interface between two incompatible polymers treated as a random potential problem

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Interface between two incompatible polymers treated as a random potential problem

C. Brosseau

To cite this version:

C. Brosseau. Interface between two incompatible polymers treated as a random potential problem.

Journal de Physique II, EDP Sciences, 1993, 3 (3), pp.279-286. �10.1051/jp2:1993130�. �jpa-00247831�

Classification Physics ^Abstracts

64.75 68.90 82.90

Short Commtlnication

Interface between two incompatible polymers treated

_{as a}

random

potential problem

Chrhtian Brosseau

Laboratoire de Spectrom4trie Physique

(*) (CERMO),

Universit4 Joseph Fourier, B-P. 87, 38402 Saint-Martin-d'H+res Cedex, France

(Received

6 November 1992, accepted in final form 8

January1993)

Abstract. A method based _{on a} two-level system interacting with _a bath is outlined to describe the interface between two immiscible polymers in _a melt. This method is built _upon the analogy existing between the

configurations

of _a flexible chain and possible paths of _a

quantum-mechanical particle. In the limit of _zero coupling to the bath

(coherent regime),

_we

refind the results of the Helfand-Tagami theory giving the size and the _energy of the interface.

As the coupling to the bath increases, the incoherent contribution dominates the interfacial properties. A polymer blend in _a confined geometry illustrates also this method.

1. Introduction.

Studies of

polymer

blends _are

currently attracting

_a wide research interest. In this respect, the

subject

of

compatibilization

of these blends is of

primary importance.

The _{reasons are not}

only technological (e,g. coextrusion)

but also

imply

fundamental

questions (e,g,

interracial

adhesion).

The

experimental

^situation has been reviewed in details

by

Wu

ill

who

gives

_many

references to the literature. The

difficulty

of

finding compatible polymer

mixtures stems from their

positive enthalpy

of

mixing.

As _a

result,

it is

thermodynamically

unfavorable for most

polymers

to form

homogeneous

mixtures at the molecular level. From _a

thermodynamical point

of

view,

the

phase diagram

of _a

binary

AB

symmetric (NA

_" NB _"

N,

where N is the

degree

^of

polymerization) polymer

mixture is under the

dependence

of the parameter

Nx(T, #)

where _x denotes the

Flory-Huggins monomer(A)-monomer(B)

interaction parameter, ^{T the} temperature ^and

#

the relative concentration. It

hardly

needs to be said that _x is known

only

for _a small number of

polymer couples

and _even less is known about its

dependence

on

temperature

^and concentration. When the interaction between the _two

polymers

is

strongly

(*)

Laboratoire assoc16 _au C.N.R.S.

280 JOURNAL DE PHYSIQUE II N°3

repulsive (e,g.

_{x =}

10~~), they

_can form _a stable interface. Anastasiadis et al. [2] have studied the interface formation in _a melt of

poly(styrene)-poly(methylmethacrylate)

mixtures

using

a

pendant drop technique.

These

experimental

observations _gave the

following

orders of

magnitude

^{for the} interfacial _{energy 7 =}

dyncm~~

and the interface width _e

= 3 _nm at

T ₌ 400

K,

N

being equal

to 10~.

Different theoretical

approaches

have been

proposed

to describe the interfacial

properties

at

equilibrium

between two immiscible

polymers,

I-e- the self-consistent mean-field

theory by

Helfand and

Tagami

[3,

4],

^{the mean-field} Cahn-Hilliard

theory

and random

phase approxima-

tions

by

Broseta et al. [S].

Recently Tang

and Freed [6] have also

analyzed

^this

question

in

terms of local

density

functional

theory.

As _an alternative to these

methods,

the

objective

of

this _paper is to show that this

problem

_can be

analyzed

as a random

potential problem.

We utilize _a

procedure

which has been

pioneered by

Edwards [7] and others

[8-10].

The main idea is to

recognize

that the

configurations

of _a flexible chain _can be viewed _as the

possible paths

for _a quantum ^mechanical

particle.

The

key

issue resides in the choice of

potential

_seen

by

the

particle.

The

binary

character of the

problem

leads _us to consider _a

potential

with _two minima. As _a

result,

the

description

of the interface between the two

polymer species

has been translated into the

problem

of the _escape of _a

particle

from _a double-well

potential coupled

to its environment. It is worth

noting

at the outset that _we shall confine _our attention to the

unentangled

situation.

The

organization

of the _paper is _as follows. Section 2 introduces the quantum

description

of _a twc-level system

interacting

with _a bath and the main

assumptions

that _are needed in

our derivation. The calculation of the interfacial characterhtics of _a

phase-separated binary

blend _are

given

in section 3. Then in section 4, _an

example

^{of the} effect of confinement

(molten polymer

^blend in _a

pipe)

_on ^the interfacial

properties

^of _a

polymer

blend is studied

by

this

method. Some

concluding

remarks _are

presented

in the final section.

2. Formulation of the

problem.

The

starting point

of _our

analysis

relies _on the strong

analogy

between the

equation

of

segmental

diffusion

(

~

"

(V~GN

+

fIUGN> (I)

(fl

₌

~)

,

of the propagator GN for _a

single

flexible chain of N

rigid

lattice segments

(a being

the

Flory-Huggins

lattice

parameter)

and the

Schr6dinger equation -jh '~

=

~ V~J# + V~#,

(2)

for _a non-relativistic

particle

of _mass m, without

spin

_[8]. In this

analogy,

the

potential

_energy V

plays

_a similar role _as the seft-consistent field U further assumed _{to vary}

slowly

_on the

atomic scale. A characteristic feature is also that the

degree

of

polymerization

is

replaced by

the

imaginary

time

it.

If the

potential

U does not

depend

_on

N,

the

propagator

can be written

using

the

spectral expansion

GN

" a~

£ u(ukexp (-NEk)

,

(3)

k

where u; and

E;

_are

respectively

the

eigenvectors

and the

eigenvalues (whose

spectrum ^is _~2

bounded from below

E;

> ED) ^{of the} operator

-V~

₊

flU. Clearly,

the contribution of 6

bound states and of the smallest

eigenvalue

ED

(I,e, ground

state

dominance)

to GN will dominate the

spectral expansion.

One introduces at the start _a

symmetric

double-well

potential

of the form shown in

figure

I

coupled dissipatively

to its environment: this is the

quantum-mechanical analog (QMA)

of the

problem

under

investigation.

We consider _a one-dimensional

problem

to

simplify

_our task. The basic idea behind the model is that

particles (chains)

have available _two accessible

potential

minima and may be

subjected

to

tunneling (diffusion) through

the barrier. The

description

of the interfacial

properties

at

equilibrium

between two immiscible

polymers

in the molten state

becomes _a

problem

of the escape of _a

particle

from _a

potential

well

(I,e. A)

to the other

(I,e.

B)

under the influence of _a random force

(Kramers problem).

We elaborate _on this aspect below. For

consistency

_{we assume} that the two

monodispersed

linear

polymers

A and B have the _same

properties:

^identical

number-average

of _monomers per chain

NA

_"

NB

"

N,

identical

statistical segment

length

_{aA " aB " a,} identical friction _{qA " nB}

" n and

equal

stillness

[I Il.

This mixture is

governed by simple

van der Waals interactions between dissimilar segments. ^In

addition,

_we restrict _our

analysis only

where the condition _xN » I

(strong segregation limit) holds, making

the results

independent

of N. In other

words,

this condition is _{on a par} with the low temperature ^{limit for the}

QMA.

Energy

EA

~ Es

-e 0 _e

Posifion

Fig. 1. Schematic

diagram

of the double-well potential considered. EA and EB _are the lowest two

eigenstates of the double-well potential. The _zero of energy is chosen _as the _mean of the two ground

state energies. Vo ^{denotes the} height of the barrier separating the two ground states.

The

following assumptions

_are made

concerning

the

QMA:

(I)

^{without loss} of

generality

_we have chosen the _zero of energy as the _mean of the two

ground

state

energies (it)

_we limit _our considerations to _a well for which the

assumption

of

ground-state

donfinance for each

species

A and B is

appropriate,

then only the lowest two quantum ^states of the double well _are

relevant;

and

(iii)

_our

d~~~ussion

^is

phrased

in terms of the harmonic

approximation.

We have also set A

=

~~fi

^~ ₊

(6x)"~

^~ for notational convenience.

h 2 2a

Although

there is _a wide _range of notation in the

literature,

the twc-level quantum system

282 JOURNAL DE PHYSIQUE II N°3

ofone-dimensional _space

interacting

with _a

bath, displayed

in

figure I,

_can be described within the

non-diagonal spin representation by

the Hamiltonian

[12]:

H = -Ka~ + Hb +

Hc(A), (4)

where K is the

overlap

_{parameter, a~ =}

~l, ^Hb

^{is the bath}

Hamiltonian,

H~ is the

coupling

Hamiltonian

depending

on the

coupling

parameter and which is

supposed

to be

sufficiently

weak to

give

_a linear _response. For _{our purpose we} do not need to _use the

explicit

form of A, ^however from the

prediction

of Harris and

Silbey

_[12] and

arguments

similar to those of Caldeira and

Leggett

[13]

concerning

the

dissipative

quantum ^{coherence effects in} systems described

by

the

spin-boson

Hamiltonian

(4)

_{we may}

regard

this parameter as a

dissipative

coefficient. The

overlap

parameter is central to the

analysis

and _can be evaluated for the _case of harmonic

potentials

_{as [12]:}

It = 2

(~fl~

^~~ exp

(- (~(fi)

^~~~

))

_,

^(S)

where Vo is the minimum _energy barrier between the two wells. In _our

notation, equation (S)

can be rewrittten _as:

It =

2xhA~"~exp(-A), ₍₆₎

which is _a

decreasing

function of the parmeter ^A.

Now _a system ^described

by

the Hamiltonian of

equation (4)

exhibits _a

competition

between friction

(localization)

and

tunneling (delocalization).

The former arises in the limit

=

0,

while in the latter the

eigenstates

_are the

symmetric

and

antisymnJetric

combinations of the delocalized _states J#A,

llB

^with

energies

+ It. It is known

[12-14]

^{that in the limit of} _zero

damping,

the system ^oscillates from _one localized state to the other with _a

frequency equal

to 2K. In the intermediate

regime,

the system _can be

represented

_{as a}

damped

oscillator.

3. Interfacial

properties

between two

strongly incompatible

blends.

We _{now move} from quantum mechanics to

polymer

^chain statistics. A few

qualitative

remarks

are of

importance.

From the results of section 2, two _{cases can occur.} The first arises in the limit of

zerc-damping (I.e.

£t

0):

the chain oscillates in the interfacial _{zone as}

displayed

in

figure

2a. A

particle (chain) initially

confined _to the

right-hand

side _can tunnel

(diffuse) through

the barrier

(interface)

into the left-hand

side,

then back _to the

right-hand side,

and

so on. The

equilibrium

interfacial thickness is

e =

2a(6x)~~'~ (7)

as

expected

from the value of the

overlap

K which is of the order

xh

_{(Et Vo)} for A

~- I. This

value stems from: the fact that the diffusion _rate of _a double-well

potential

^{which is shallow}

compared

to kT is

mainly

driven

by energies

close to the barrier

height,

the _very definition of the

Flory-Huggins

_parameter for systems ⁱⁿ contact with _a thermal bath

(the

bath Hamiltonian Hb does not affect the

ground

state energy

[12])

at temperature T _{x "}

(

%

flU

and the

A B

A B

A+B A+B

(a) (b)

Fig. 2. Cartoons of two physical situations representing _a polymer chain at the interface.

(a)

Zero

damping

situation

(coherent state).

Chain configurations are considered _as paths normal _to the interface. The _case of _a single chain A is illustrated.

(b)

Non-zero damping ^{situation. Different} possible configurations of _a chain obtained by decreasing ^the coupling parameter A.

choice of the _zero of energy. It is

possible

to make _an estimate of the interfacial _energy

by

means ofthe

uncertainty principle

_[15]:

lia) ~~~

_"

~'

_~~~

hence the result

~~

7 ^£~

$(6x)~~~i (9)

one recovers the results obtained

by

Helfand and

Tagami

_{[3, 4]} and others [5]. A

comprehensive

discussion of the

order-of-magnitude

dimensional argument

leading

to

equation (8)

is

given by

Ballentine

jib].

Note that in the references [3, 4] ^{the number 6}

appearing

in

equation (7, 9)

is reminiscent _to the number of _nearest

neighbors

of the cubic lattice

approximation

used.

Whenever _one considers _{a non-zero}

damping

situation

(I,e. # 0),

_a number of

configurations

may arise _as

displayed

in

figure

2b

depending

_on the value of the parameter ^{which is} respon- sible for

dissipation.

These results _concern

long

flexible chains in the strong

segregation

limit.

It _turns out that this

analysis

becomes

approximate

for short chains because of the effect of

chain ends. The

"tunneling" amplitude

decreases _as the

dissipation increases,

the extreme _case

leading

to the

suppression

of the

amplitude

of oscillation

by

the sc-called

"tunneling

friction"

[12-14],

^I,e, spontaneous symnJetry

breaking leading

to a localized

regime.

The

question

of

just

when the

coupling

is of sufficient

strength

to observe this localization transition has been discussed in the context of quantum

tunneling

in

macroscopic

systems

by Bray

and Moore

[14].

A different but related diffusion

problem

is the _one associated with the

solubility #

of A into B. The

penetration

of chains into the interfacial _zone is

important

in itself for

understanding

certain aspects ^{of adhesion of}

polymer

^blends _{[1, 16].} In _our

model,

^the concentration becomes the _square of the

amplitude

of the _wave function _J# which

decays

to zero with _some character- istic

length

L.

Introducing

_a Rouse friction _qR

~w

x~~, Brochard-Wyart

et al. [17] were able to calculate this

length by

dimensional arguments ^{and found L} _~w

aN~x"~ _[17]. Consequently

#

is of the order of exp

(- _~)

% exp

(-N~x~~~)

This

expression parallels

the

exponential

a

284 JOURNAL DE PHYSIQUE II N°3

decrease of the

tunneling

rate found

by

Caldeira and

Leggett

[13] ^{in the} quantum mechanical

problem.

4.

Polymer

^blend i~~ _a confined

geometry.

The

problem

now shifts _to the

application

of this method to _a

polymer

mixture confined in _a

capillary.

The effect of confinement of _a

binary polymer

mixture to _narrow pores or films is

a

subject

of

ongoing

interest [18] ^and may find

applications

to

problems concerning polymer processing.

The situation of _a melt of two

polymer

chains confined in _an infinite

pipe

of diameter

D,

with _no

adsorption

_on the wall and with identical monomer-wall interactions for the two

species

_was

previously

considered

by

^Brochard and de Gennes

[19],

^further

analyzed by Raphail

_[20] and

numerically investigated

in _a recent

study by

Cifra et al.

[21].

A lucid discussion _was also

given by

Edwards

[22,23].

^{A few remarks} _are of

importance.

If _we

ignore dissipation

two

segregation regimes

are observed

[20].

On the _one

hand,

when ^~ _>

N~'~

_the

a

chains _are ideal with size

a

N~'~

_{and the interface}

overlaps

in _a

region

of thickness _e

~w

ax~~'2.

The chains _are not affected

by

the confinement and the above results _are still valid. This

picture

^{is however}

inadequate

when _< ^~ _<

N~/~

_{because the} confinement induces _a

spatial

a

~

rearrangement: the chains _are

spatially segregated

such that each chain

occupies

_a

length ~$

of the tube. Hence the above

dependence

of the interfacial characteristics

(Eqs. (7, 9))

_{are no}

longer

valid because the chain has much less orientational freedom than when it is in the bulk state; _or, put ^another _way, ^the confinement of chains induces _a loss of translational entropy.

Then

contrasting

with the bulk

equilibrium result,

the interfacial _{energy must} be

proportional

to the

Flory

_{parameter as} discussed

by Raphail

_[20]

7 Ef

~(X. (10)

When ^~ _»

l,

_so that the diffusion

equation (I) applies,

^the

uncertainty principle

is still valid

a

and

permits

to evaluate _e: from

equation (8),

_we

predict

that _e scales _as

e ~-

ax~~> (II)

in

sharp

contrast with the bulk

dependence.

^{The confined} state _can force the interfacial thick-

ness to

elongate significantly.

This

point

_was

recently

confirmed

numerically

in _a Monte Carlo lattice simulation

[18]. Experimental

verification of these results represents _an

enticing

chal-

lenge

since it is

certainly

difficult to achieve the measurement of _e and 7 ^at molten temperatures in such _a confined geometry. ^{This situation} can

presumably

be realized

experimentally

_e-g-

by using

zeolite structure materials.

5. Discussion.

An attractive feature of the

approach presented

here is that it _comes under the

general

^umbrella

of twc-level behavior for which there exists _an extensive

body

of literature in the _context of quantum ^{mechanics. The twc-level}

description corresponding

to the double-well

coupled

to its environment is

adequate

for

describing

the interface between _two dissimilar

polymers.

A

central aspect ^{of this} formulation is that the

configurations

^of _a ^{flexible chain} _are

analogous

to the

possible paths

of _a quantum

particle.

It is _not clear _to what extent it is

possible

to retain such _a

picture

when

entanglements

_are

present

^{in the} interfacial _zone.

Brochard-Wyart

et al. [17] ^have

suggested

that the existence of

some

entanglements

in the interface with

probability

_exp

(-xN~)

should _suppress the

slippage

whenever

xN~

> In

(N~N7~~~)

^N~

^being

^{the number of monomers per}

entanglement

in bulk A et B.

Taking

into account

topological

constraints for the chain is _a

complicated problem

^that

is outside the _scope of this _paper. However _we note that similar

topological problems

arise in

(2 +1)-quantum

field

theory (e.g,

^connected

configuration

_space ^of

quantal particles)

_[9].

There is _{one more}

point

to be noted about the

possiblility

of

applying

similar arguments ^such

as those

presented

here _to treat the _case of block

copolymer

mixtures

by multiple potential

barriers

[24].

Work

along

these lines is

currently

in _progress.

References

ill

WU S., Polymer Interface and Adhesion,

(Dekker,

New-York,

1982).

[2] ANASTASIADIS S-H-, GANCARZ I., ^KOBERSTEIN J-T-, Macromolecules 21

(1988)

2980.

[3] ^HELFAND E., TAGAMI Y., J. Polym. Sci., ^Part 89

(1971)

741.

[4] ^HELFAND E., TAGAMI Y., J. Chem. Phys. 56

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3592.

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132.

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6307.

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613.

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[9] ^WIEGEL F., Introduction _to

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Prog.

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[11] Asymmetry in the statistical segment lengths _{or a} large difference in flexibility induces unusual properties such _as the multiple-ordered phases recently observed by

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K-A-, BATES F.S., MORTENSEN K., Macromolecules 25

(1992)

1743. It is also interesting

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7330.

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See also LEGGETT A-J-, CHAKRAVORTHY S., DORSEY A.T., FISHER M.P.A., GARG A.,

ZWERGER W., Rev. Mod. PJ~ys. ⁵⁹

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1545.

[15] ^BALLENTINE L-E-, Quantum Mechanics,

(Prentice

Hall, Englewood Cliffs,

1990),

especially chapter lo.

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13.

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(1990)

i169.

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5229.

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399.

[20]

RAPHAiL

_{E., J. PJ~ys. France 50}

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803.

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1401.

286 JOURNAL DE PHYSIQUE II N°3

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(Gordon

^and Breach, London,

1975)

_p, 153.

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492.