Density-conformation coupling in macromolecular systems: polymer interfaces

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Density-conformation coupling in macromolecular

systems: polymer interfaces

Jean-Pierre Carton, Ludwik Leibler

To cite this version:

Short Communication

Density-conformation coupling

in macromolecular

_systems:

poly-mer

interfaces

Jean-Pierre

_Carton(1)

and Ludwik

_Leibler(2)

(1)

Service de

_Physique

des

_Solides,

et de Résonance

_Magnétique,

CEN

_Saclay,

91191 Gif-sur-Yvette

Cedex,

France

(2)

Laboratoire de

_{Physico-Chimie}

_{Théorique(*)}

ESPCI,

75231 Paris Cedex

_05,

France

(Received May

11, 1990,

accepted in final , form

June

_{13, 1990)}

Abstract. 2014 We define local order

parameters to take into account the orientation of monomers and molecules in

_polymer

_systems. These fields are

_{naturally coupled}

to

_{density gradients.}

Such

_couplings

lead to

_interesting

effects in the

_{neighbourhood}

of interfaces

_{(polymer-solvent}

and

_{polymer-polymer}

cases are

_{considered) :}

macromolecules are

_prolate

on one

_side,

oblate on the other. Classification

Physics

Abstracts 05.90 - 61.40K - 68.10

Polymers

in solution or melt _may

_display

_{spatially inhomogeneous}

_patterns.

_’lypical

_examples

are the interfaces of

_{phase separation}

or the

_mesophases

of block

_copolymers.

From a theoretical

point

of view these situations are

_commonly

described

_{by a}

scalar field i.e. the

_density

or relative

density

difference.

Such a

_description

is

_appropriate

for

_ordinary

_isotropic

fluids. But on the scale of

_spatial

vari-ations

_usually

_involved,

_polymers

cannot be treated as

_point-like

_molecules,

_they

are connected

objects,

and moreover each monomer has a

_spatial

orientation. From this

_point

of view

_they

are

rather akin to

_{liquid crystal}

molecules. In this letter we want to

_emphasize

a

_description

with

more

_complicated

mathematical

_objects,

_e.g. vector or tensor densities. These variables describe the chain conformation and are

_{naturally coupled}

to

_{composition gradients.}

In a recent

_paper

[1],

Szleifer and Widom

_using

a lattice

_model,

_pointed

out

_{interesting polymer}

orientation effects induced at interfaces

_by

a

_phase

_separation

in

_polymer-poor

solvent solutions. Our formalism

proves

useful in

dealing

with such

_problems.

We then treat the case of

_{polymer A -}

_polymer

B interfaces and

_incorporate

orientational interactions and chain

_rigidity

which

_strongly

enhance

concentration-conformation

_couplings.

For these

_{macroscopic separation problems,}

orientational

effects of

_quadrupolar

_type

are

_{interesting perse}

but lead

_only

to corrections in the

_{thermodynamic}

behaviour. The situation is _very different for

_grafted

or block

_copolymers.

In this latter case the

dipolar-type

couplings

have a crucial

_importance

for the

_{microphase separation}

[2].

(* )

Unité de recherches associée au CNRS n° 1382.

1684

1. Orientational order

_parameters.

1.1 MONOMER ORIENTATION PARAMETERS. - A

_geometrical

model for a

_polymeric

chain is

first needed. Let unA be a vector of

_length

a associated with link

(monomer)

n of

_polymer

À. The

hamiltonian will be a function Il

( (un A ) )

of these

_linking

vectors. We also define the centre rn A

of the monomer so that

At the

_macroscopic

level the orientational

_properties

of an

_assembly

of chains will be

_conveniently

described

_by

order

_parameter

densities such as

PM =

Nm

is the total

density

of monomers so that

_fvol

d3x

_jJ(x)

= Vol. For common

_purpose,

higher

rank tensors are not needeed. The tensor

_Q,

a

_{generalization}

of that defined in the

_theory

of

nematics,

is similar to that used

_by

Frederickson and Leibler

_[3]

for semi-flexible chains. In the

_problems

which will be

_explicitly

considered in this

_paper

_Q

will be

_diagonal

in _{x, y,} z and

uniaxial so that

_Qzz

can be identified with the usual scalar

_quantity

S =

(COS2

On -

1/3>.

The tensor

notation makes obvious the invariance

_properties

and as a

_consequence

the

_{couplings involving}

Q

which are allowed. It would also be _necessary in situations other than uniaxial.

The

_labelling

index n of a monomer on a chain is not

_uniquely

defined since it could have

been numbered in the

_opposite

sense. Under index-reversal

_operations

the order

_parameters

are

transformed in the

_following

_{way :}

For an

_{ordinary homopolymer,}

_{physical properties}

are

_independent

of the

_labelling

choice so that P = 0 and

only

the

_quadrupolar

field

Q

is

meaningful.

The

_present

work is devoted to this case.

On the

_contrary

if a well defined

_labelling

can be chosen for

_physical

reasons, the vector

_parameter

P can be relevant. The

_simplest

_example

of such a situation is the case of diblock

_copolymers

A - B

[2].

Our choice of the

_{density being}

_peaked

at the middle of the link instead of the

_edge

as in

_[3]

is

somewhat different but make the above

_symmetry

more

_transparent.

It is

_interesting

to

_explicit

the

_relationship

between the two definitions.

_Expanding

_formally

we can write the densities relative to the link

_edges

_(subscript

_±)

in terms of those relative to the

The

_knowledge

of one set of fields

_(i.e.

a

_hierarchy

of

_{tensor) yields}

the other. We heareafter

_only

use densities with

_respect

to the link _centres,

_dropping

the

_subscript

c.

,

1.2 MACROMOLECULE INERTIA TENSOR. - The above définitions deal with the orientation of

monomers at

_point

x whatever the chain

_they

_belong

to. This is not sufficient to describe the

_shape

of the macromolecule. Th do so we introduce the reduced inertia tensor of a chain of N links :

where ga

_{= N}

Ei

rfa is the location of the centre of

_{mass. j}

contains an

_isotropic

_part

_(scalar)

which is the radius of

_gyration

In a

_spatially

_{inhomogeneous}

_system

the overall orientation of a

_polymer

will be

_{given by}

the

anisotropic

part

of the inertia tensor

_specifying

the location of the centre of mass :

where _pp

_{= Np/vol}

is the mean

_density

of

_polymers.

2.

_{Inhomogeneity}

induced orientation.

_Application

to

_{polymer-solvent}

interfaces.

The basic idea is that orientational order

_parameters

are

_coupled

to derivatives of the

_density,

a

property

which is well known for the nematic-smectic transition. As a

_consequence

_if,

for some

reason, the

density

of monomers varies in

_space,

a non zero value of

_Q

for instance is

_expected.

Such a

_phenomenon

has

_recently

been

_reported

_by

Widom and Szleifer

_{[1, 4].}

_They

consider an

_ interface due to

_{phase separation}

between rich and

_poor

_homopolymer

solutions,

expliciting

the statistics of a random walk. It is shown here that this effect is

_quite

_general.

Moreover we are able

to calculate

_analytically

the

_coupling

coefficients between

Q and 0

on

_microscopic

_grounds.

2.1 GENERAL FORMALISM FOR FLEXIBLE CHAINS. - Wu first consider

_completely

flexible

chains,

in which case the link vectors un are

_{statistically independent.}

We construct the free

energy

as a functional of

_{Q and p = 4J -}

_{(~) by}

a

_procedure

which has now become standard

_[5].

In the mean field

_{approximation}

1686

and

_{Fo ( { cp } , {Q})}

is the free _energy of a

_system

of non

_interacting

chains with constrained values of _cp and

_Q

at each

_point.

Introduce U and T as the fields

_{conjugate respectively}

_top

and

Q

and the cumulant

_(or

connected Green’s

_{function) generating}

functional

_{W({U}, {T}).}

Then

Fo

is the

_Legendre

transform of W. W has the

_following

form :

In this shorthand

_notation,

indices have been

_dropped

so that the G functions are to be read as

tensors of convenient rank. All this cumulants for non

_interacting

chains can be

_{computed exactly.}

The

_{calculations,}

which are

_only

sketched

here,

are made for a Gaussian model with

which ensures that (u2nA )

=

a2.

_Actually

the Gaussian

_{approximation}

amounts to neglecting terms

of order

_higher

than

a2.

Since we are interested in a

_gradient

_expansion

the

_following

results are

given

to lowest order in the

_long

_wavelength

limit

The

_Legendre

transform of W contains a

_Q

_independent

contribution

_{Fo1 ({Q}),}

which will not

be used

here,and

a

_Q

_dependent

_part

_{Fo2( {p},}

_{{Q}) :}

Fig. 1. -

as a function of Z =

z/03BE

associated with a

_{polymer-solvent}

interface of

_profile

p = -6 tanh

Z,

calculated

’for 5

=0.3.

This term of

_{purely entropic origin}

accounts for the

_density

induced behaviour of

_Q.

It is

impor-tant to stress that it exists even in the absence of

_anisotropic

interactions

_(which

will be introduced in

_sec,3)

and

_{only originates}

from the

_connectivity

of the chain as it involved in the

_entropy.

As an

illustration,

assume we are concerned with a

_{density profile,}

a situation which occurs when

segre-gation

takes

_place

between a

_high

and low

_{density phase.}

This is the _very

_example

considered in Ref.

_[1].

In the absence of

_anisotropic

interactions in

_(2),

i.e. V2 =

0,

the variation

principle

with

respect

to

_Q

-gives

for a

_z-dependent

situation

Qzz

is thus of order

(f) 2

_{where e}

is the width of the interface and is

_very

small. It can be shown that the interface

_profile

is still

_given

_by

the classical form :

so that

Th first order in 6 we find that

_Q

is an odd function of z,

Qzz

0 for z

0,

Qzz

> 0 for z > 0. This

means that the bonds are

_{preferentially parallel}

to the interface on the rich side and

_{perpendicular}

on the _poor side. Further terms in

S2

introduce an

_assymetry

in the variation of

_Q(z)

as shown in

1688

2.2 SHAPE OF THE MOLECULE. - As

_{already recognized by}

Szleifer

_[4],

the orientation of a

molecule as a

_whole,

_{quantitatively given}

_by

_J,

is a relevant

_parameter.

We have shown that a

collective variables formalism can be set

_up

for J as we did for

_Q.

We restrict ourselves to the lowest order

_coupling

between J

_{and V,}

i.e. the

_{term JaBâ2Ba}

This

approach

is

_{meaningful only}

if the

_spatial

scale of variation is

_larger

than the

_polymer

extension

RG.

The cumulants of interest are

These leads to a free _{energy contribution}

and a linear

_response

to the

_{density profile}

7ivo conclusions are to be drawn :

i)

the variation of J

_along

the z axis is similar to that of

_Q.

Thus the

_polymer

is oblate

_(parallel

to the

_interface)

on the

_polymer

rich

side ;

it is

_{prolate (normal}

to the

_interface)

on the low

con-centration side

_(Fig.2).

This result has also been obtained

_by

Szleifer

_using

numerical calculations

on a lattice model

[4].

ii) According

to definition

_(1),

J is a measure of the

_departure

from a

_{spherical shape}

of radius

RG.

Eq.(5)

through

the

_{N2 dependence}

shows that the

_{spatially varying density}

causes a

_stretching

parallel

or

_{perpendicular}

to the interface.

-For an interface in the

_{demixing problem}

the

_strength

of the

_stretching,

of order

1

also

dépends

on N

_through

e.

For instance in the

_{scaling regime}

_[1],

if

N1/2

_{(T - Tc)}

«

1, 03BE ~

N 1/4

and Jzz ~

N3/2.

Note that similar behaviour is

_{expected (for}

_Q

and

_J)

in a

_polymer

solution close to a hard wall:

oblate

_{(resp. prolate)}

conformation in the

_{repulsive (resp. attractive)}

case.

Furthermore our formalism can be used for non

_planar

interfaces

_(in

which case the

_symmetry

can be non

_longer

_uniaxial).

3.

_{Polymer-polymer}

interface.

The same

_concept

_applies

to an

_amorphous

blend of 2

_{polymer species}

A and B. In this case it

is natural to take into account the

_anisotropic

interactions V2 which exists between all

types

of

polymer :

Fig.

2. - Conformation and orientation with

respect to the interface

_{(vertical plane).}

The

_{high density}

side is z 0

(left

hand

side).

The

ellipsoids picture

the

general

conformation of the chain

(as given

by

J)

and

the rods the

_preferred

orientation of monomers

_{(as given by}

_Q).

if the total monomer densities of A and B are

_{equal (pMA}

= _PMB =

PM/2).

Concerning

interactions

_{v2 A, V2}

_{v2 B}

we _argue that

_they

are of the same order of

_magnitude

as

_respectively

vo

@ vo ,

vô AB .

A and B contribute

_{independently}

to the

_entropy

the terms

Assume for

_simplicity

that

_v:A

=

vBB.

Then

_Q

=

_QA

₊

QB

_represents

the local order of ail

species

and is

_{given by}

.

Here and is assumed such

_{that 1 -}

V > 0 which means that there is no

spon-taneous

orientational

order.

Q

varies

_{symmetrically}

about

the interface. It becomes

_assymetric

if

v2AA =

vBB.

4.

_{Introducing flexibility}

and

_{persistence length.}

The

_phenomenon

described above is remarkable because it is not due to _any intrinsic stiffness

1690

persistence length.

As a model we take a

_gaussian

distribution of the UnÀ vectors with hamiltonian

which ensures correlations of the form

The

_{persistence length}

À is defined as À = la. The results are

_{naturally expressed}

in terms of a

and À rather than the

_parameters

in

_Ho.

The main feature is seen on

_{GQcp :}

for

_large

_{N ( N »}

_À)

and"B » a. The conclusions of Sec 2 -1 remain valid

replacing

a

_by

2"B. There is a further

_dependence

on a

_through

the

_spatial

scale

_e.

To illustrate

_this,

consider for

_simplicity

the case of the A - B interface without orientational interactions

_{( 2}

=

v2 B

=

2

=

0)

and

with

NA

=

NB.

Th lowest order in the concentration variation 6 one has

for the classical

_profile

Here,

not too far from the critical

_{point, e}

is

_[6]

so that the order of

_magnitude

of

_Q

is

The orientational effect is thus enhanced

_by

a

factor §,/a

a fact which is

_important

for

_experimental

verifications.

5.

_Concluding

remark.

The

_properties

associated with the orientation of a

_polymer

chain can be

_interpreted

as in the Landau

_{theory using}

order

_parameters

and

_{symmetry arguments.}

Our formalism which relies on a

_microscopic

formulation and _{may include} chain

_rigidity

and orientational interactions

_(which

are

_{always present)}

is rather _{realistic. It may} be useful for

_quantitative

_{interpretation}

of

experi-ments on

_polymer

interfaces which are now able to

_probe

the concentration

_{profile (e.g.}

_optical

reflectivity methods).

We are indebted to B. Widom for

_illuminating

comments and to I. Szleifer for

_sending

us Ref.

References

[1]

SZLEIFER I. and WIDOM

_{B., J.}

Chem.

_Phys.

90

₍₁₉₈₉₎

7524.

[2]

CARTON

J.P.,

FREDERICKSON

_G.H.,

LEIBLER

L.,

to be

_published.

[3]

FREDERICKSON

_G.H.,

LEIBLER

_L.,

Macromolecules 23

₍₁₉₉₀₎

531.

[4]

SZLEIFER

I.,

preprint.

[5]

LEIBLER

L.,

Macromolecules 13

₍₁₉₈₀₎

_{1602 ;}

OHTA

T.,

KAWASAKI

K.,

Macromolecules 19

₍₁₉₈₆₎

2621.

[6]

NosE

T.,

Polymer

J. 8

₍₁₉₇₆₎

_{96 ;}

JOANNY

_J.-F,

LEIBLER

_{L., J.}

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₍₁₉₇₈₎

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