Conformation of a polymer chain dissolved in a critical fluid

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Conformation of a polymer chain dissolved in a critical fluid

Thomas Vilgis, Anne Sans, Gérard Jannink

To cite this version:

Thomas Vilgis, Anne Sans, Gérard Jannink. Conformation of a polymer chain dissolved in a critical fluid. Journal de Physique II, EDP Sciences, 1993, 3 (12), pp.1779-1786. �10.1051/jp2:1993229�.

�jpa-00247937�

Classification Phj,sics Abstiacts

05.40 05.70J 36.20

Conformation of

_a

polymer chain dissolved in

_a

critical fluid

Thomas

Vilgis (I),

Anne Sans

(2)

and Gerard Jannink

(2)

(')

Max-Planck-Institut for

Polymerforschung,

Ackermannweg IQ, 55021 Mainz, Germany

(2) L-L-B-,

CEA-Saclay,

91191 Gif-sur-Yvette, France

(Re~eii>ed 11 J~lfie J993, rei'ised 29 J~llj, J993, accepted 3J August J993)

Abstract. The

problem

of _a polymer ^chain immer~ed in _a critical fluid, _{e-g- a} mixture of low molecular

weight

solvent close _to its critical point, is studied and reformulated in _a field theoretic framework. The starting

point

is _a

path

integral formulation of the

polymer

chain which interacts with the solvent molecules. lithe critical solvent is

approximated

by _a Gaussian model all classical results _are recovered, I-e- the

polymer

chain collapses ^close to the critical

point.

These results become modified _at the critical

point

where non-classical exponents ^determine the

precise

form of the fluctuation induced interaction

potential

between two monomers of the chain, and _so its

conformation.

1. Introduction.

The conformational behaviour of _a

polymer

chain dissolved in _a mixture of _two solvents and 2

(both

_are assumed to be

good

solvents for the

chain)

has been much attended in the past

[1-4].

The reference

by

^{Brochard and de Gennes}

[3]

reviews the

experimental

situation and _uses

scaling

_arguments to account for effects which go far

beyond

the _mean field

predictions

I

].

To

our

knowledge

no further

rigorous

treatment of this

problem

has been

presented.

A

path integral

formulation allows for _a detailed theoretical

description

and _recovers all classical

predictions II

and non-classical features

[3]

^of this

problem.

The _common feature is that the chain

collapses

before the critical

point

^{of the mixture} solvent and forms _a

globule

of size

R N ^"~ where N is the chain

length

and d the space dimension. The conformational behaviour

of the chain is discussed in detail below.

Another

noteworthy

reason for this

study

is

completely

different. In several papers the effective

potential

between _{two monomers} of _a

tagged

chain in _a critical

polymer

blend _or other

comparable

situations such as block

copolymers

have been calculated

[5-7], they

^suffer from the fact that these

potentials

have been calculated in Gaussian

approximation.

Their final form

contains an

unphysical singularity

near the critical temperature T~ of the form

U

(r) ldd~

_T ^e'~~

^{f (k}

^g

^(T)

T~ +

(bk

_)~ (T _{T )~}

'

1780 JOURNAL DE PHYSIQUE II N° 12

where the exponent a

depends

_on the function

f(k).

In the _case of blends g

(T) changes sign

before T~, ^of a temperature ^T _> T~ for a

tagged

chain and

(I. yields

the

unphysical

conclusion that U(r) becomes

infinitely

attractive, I.e, the chain

collapses

to a

globule,

^which is

clearly

in

contradiction to the obvious result R N ^"~ in the two

phase region

for domain sizes

larger

^than R.

The

study

^of one

polymer

chain in _a critical fluid will suggest a solution of the blend and

copolymer problem,

since it

provides

a

general

formulation which is

applicable

also for temperatures ^{below the}

Ginzburg

temperature ^{where the Gaussian model}

(RPA)

fails

completely.

^{This will become the issue of} a more detailed

study.

2. The model.

The

polymer

chain of

configuration R(s)

is

conveniently

modelled

by

_an Edwards-Wiener functional

[9]

d

N dR(s)

^~

~ v~m

j~

^ds

j~

^{dS' ~ ~~~~~ ~~~~~~ ~~}

~

~~( jR(S))

)

@

~ dS

o °

where _s is the dimensionless _contour

variable,

f the Kuhn step

length,

d the space dimen~ion and v

~~~

the bare monomer-monomer excluded volume

(assumed

to be short

ranged),

N is the

degree

of

polymerization.

The critical fluid is modelled

by

a

binary

mixture of

good

solvents, which may

undergo

a

phase separation

at a certain temperature. ^The Hamiltonian for the fluid

can be

given

in terms of three

binary

interaction _terms of short range

potentials

between the different solvent molecules

PHj ~j ~j ~~~(~$ () (~-2)

U. T i,j

where _{«, r}

= 1, 2 and

I, j

label the molecule~.

The interaction between the chain and the fluid is described _as

pH

= v

6(R(s) r")

(2.3)

'

~j ~j

m<T

~i~

where _v~~ is the short range excluded volume between the _monomers and the different

species

of the fluid. It is essential that _v~j # v~~~, I,e. the solvent

quality

is

slightly

different. In

essence it _means that the chain is

likely

to be surrounded

by

^solvent

2,

when _v~j

> v~~~

(preferential adsorption).

The

problem

can be

greatly simplified

when collective

density

fields for the fluid _are

introduced,

I-e-

p

"(r

=

jj

6 (r

rl') (2.4)

which

~ati~fy

the

incompressibility

constraint _p

'(r)

₊

p

~(r)

= po, p~

being

the _mean

density.

Note that the

incompressibility

condition is _not

complete

since the presence of the chain is

neglected.

To lowest order this will

bring

_a shift of the critical temperature ^{of the order of the}

total

polymer

^{volume fraction} _~b~. ^Since _~b~ ^{is small} we

neglect

this effect, _see also

reference

[7].

The

complete

effective Hamiltonian _can be calculated in _a standard way

[10].

^It reads

pH~rj

=

~ j~

^ds

?

^~

+ v~~

j~

^ds

j~

^{ds' 6}

^(R(s)

^R

^is'))

⁺

2 _o S o o

N

+ (v~~j

v~~) jj

^ds e~~~~~' _p

(k)

+

pHr( (p) (2.5)

L 0

PHr( (p

is the fluid Hamiltonian in _terms of the

density

variable _p

(r

for the

incompressible

fluid

given by

PHr( lpi

=

d~r( ^Vp

^{~ +}

rp~(r))

₊

(

p

~(r)) ^(2.6)

where the _mass

r is

given by

_T

= X~ X where Xs and _{X are} the

Flory-Huggins

interaction

parameters. Equation (2.6)

follows from e.g.

(2.2) by introducing microscopic

fields

(2.4)

and

taking

into _account the entropy ^{of the fluid} system

[10].

To this end the

preferential adsorption

term is _most

important,

so that for the _case v~j = v~~ the fluid and the chain _are

decoupled.

Equation (2.6)

is the effective Hamiltonian for the order parameter p~.

A is _a

coupling

con~tant. X, is the _mean field

spinodal

_X~

=

I/(~bj

_~b~) where

~bj,

~b~ are the volume fraction of the solvent I and 2

respectively,

and _X the

Flory-Huggins

parameter of the fluid 2 _y

=

? vj~

(vjj

+

v~~).

^Note that the Hamiltonian

proposed by equations (2.5)

and

(2.6)

^is _very similar to the free energy

proposed by

Brochard and de

Gennes,

apart ^{from the}

microscopic

details in

(2.5).

The _use of the chain

density

C~ N

=

ds e'~~~~~

m

C

(2.7)

0

transforms

equation (2.5)

into the Brochard de Gennes

equation

in the _mean field

approxi-

mation and for the _{wave vector zero.} The

preferential adsorption

which is

predicted by

equations (2.5-2.7)

is

compared

to the result of reference

[3]

and is _not

changed

in the k

= 0 limit since the variation of

PH~r~

^with respect to the fluid

density

fluctuation in the k ₌ 0 limit is

given by

j

N

pk ~

~

(v

j _~

v~~~)

ds e'~~~~~

(2.8)

T +

(bk)

₀

or for k

=

0

p =

~~ N

(2.9)

which is the result of Brochard and de Gennes. This

simple

form reflects the Gaussian nature of the

approximation.

Note that

equation (2.8) predicts

the fluctuations of the _mean

density

_p in

contrast to reference

[3]

which discusses

only

the k

=

0 _term.

This model is in its

generality

_very difficult _to handle and

approximations

on several levels will be discussed in the

following

sections.

3. The _mean field

approximation (A

=

0).

Far above the critical

point (Xs

w X the

quartic

term in the Hamiltonian

(2.6)

_can be omitted.

This is the classical Gaussian

approximation.

In this _case the _p

(r)~field

_can be

integrated

_out

exactly

which

yields

the effective chain Hamiltonian

PH

-

fi _{II II}

_~dS

+

i II

^dS

II

^{dS v}

(k) e~<~<'

^~<~'~' _~3.

'i

with the effective _mean field

potential

P(k)

_{= vmm}

~~~~ ~~~~

(3.2)

r +

(bk)-

1782 JOURNAL DE

PHYSIQUE

II N° 12

where b is _a

length

^of the order of the diameter of the solvent molecules. If _{a mean} field correlation

length

^for the critical mixture is introduced i'ia

f

₌ ^{fir- "2}

(3.3)

equation (3?)

_can be rewritten _a~

(V~j

~m?~~

~~ ~~

P(k)

_vmm

j

~j

~~

m+ ^~

f~

This is

(in

the limit k

-

0) basically

the result of

Flory

and Schultz

[I

and Brochard and de Gennes

[3].

Indeed

equation (3.4)

_agrees

exactly

with the results derived in

[I]

_as will be

shown in the

forthcoming

extended paper

[I I]

when v~~ and _Tare i~

expres~ed by simple

calculations in terms of _v,,~. Such detail~ will become

important

in _more

general

situations

compared

to those discussed in this paper.

This dominant effect in

equation (3.2)

_comes from the limit k

- 0 when P (k ₌ 0 is

given by

v

- vmm (AU i?

?

(3.5>

(Au _{= v~~~j} v~~~) ^and v becomes

negative

when

(f/h

>

~~

(3.6)

v

and the chain is

expected

to

collap~e,

since the net

potential

on every monomer

along

^the

polymer

chain is attractive. The Hamiltonian

(3, contains,

however, attractive and

repulsive

interactions with different

spatial

_ranges. ^Therefore the

collapse

is

partial.

This is the

regime

discussed in detail in reference

[3],

and further

generalizations

_are

given

below.

To thi~ end it has been shown that the results of references

(1,

3

j

_can be derived

by using path integral

methods. The

theory presented

_so far _uses the Gau~sian

approximation

for the

critical fluid and it is well-known that it fails

completely

near the critical

point

and, _moreover,

cannot make

predictions

for ~ituations below the critical

point,

I-e- _r

~ 0. Another

point

to be made clear i~ the appearance of

using

type exponents as a result of the p^~

theory

for the

binary

solvent. The

following

section will

firstly

confirm result~ considered

by

Brochard and de Gennes, and

secondly give

rise _{to an} estimate of _a

new blob ~ize for this

problem.

4. The critical

region.

The critical

regime

below the

Ginzburg

temperature ^is more subtle and _not

easily

accessible

~ince the

p-field

cannot be

integrated

out due _to the presence of the

quartic

term in the Hamiltonian

(?.6).

To avoid _a detailed RG

study

the Hamiltonian

(2.6)

can be

replaced by

_an

effectively

renormalized

expression

which will

provide

the correct

scaling

^for the correlation

iunction

(p (r)

_p

(0))

=

G(r).

At the critical

point G(r)

is

given by

G~(r)

= ~

(4.1)

~. --+~

where A is the

amplitude

and

7~ the usual critical exponent for the correlation

function,

which

was 7~ # 0 in the _mean field _case, I-e- the Gaussian

approximation

of section 3. In terms of

renormalized

density

fields the fluid Hamiltonian _can be written _as

pH)

=

ld~

^{r d~ r' p}

^{(r) G~(r r')}

^{p jr') (4.2)}

at the critical

point.

The critical temperature ^{itself becomes} renormalized

by

the usual _maw shift and the presence oi the

polymer [12].

Using

the renormalized Hamiltonian

(4.2)

the effective _monomer

potential P~(k)

_at the critical

point

reads

i~(k)

~ v~~

~~[~~ ^(4.3)

,Ak- ^~

or

P~(r)

= v~~ 6

(r) ~~

_~~

(4.4)

,/A ^{i~ + ~}

The attractive fluctuation induced component ^{scales in the} same way as the correlation

function. The

scaling

form derived

by

de Gennes

[3]

_can be recovered if inside the coil

(r

=

f) (4.4)

is

replaced by

i

(Av)2 ~~j

~~~~

/

^~ ~~

,,

i

^~

^~~'~~

[note

^{that 6}

^(r) _f~

^{and with f - hi ~ the}

^scaling

^limit

Pc

v mm

~~~~

T~ ~~~ _~

« v

mm

~~~ _~~

~

(4.6)

,, A

; A r

which agrees

perfectly

with the

conjectured

result

given by

Brochard and de Gennes

[3],

and proves the natural appearance of the non-classical value for _y

(m1.25).

Moreover another

regime

can be

analyzed

since the

spatial dependence

of the

repulsion

and the attraction of the

potential

in

equation (4.5)

is different. The

Flory

type Hamiltonian of the

chain _near

critically

_can be written _as

pF

= + v~~

~j

~~)

^~~

~~j~

~

+

v(~ ~~ (4.7)

R , A R R

where _a three

body

term has been added for convenience. The balance between the bare

excluded volume and the attractive interaction

provides

an

analytic expression

of the relevant blob size of this

problem [3]

_:

<blob ~

'~ )

(v~~

_v

~j?

' ' ~'2

m~

~ _v

(4,8j

For distances _r

~ f~j~~ ^{the chain behaves} as a

self-avoiding

walk, whereas for _I

>

f~j~~

^the chain is

collapsed.

It is

interesting

to note that the correlation function exponent ^{determines the} blob size _near the critical

point.

1784 JOURNAL DE PHYSIQUE II N° 12

Fig. I. This figure illustrates the result of e,g. (4.8). The chain is self-avoiding inside the blob of radius f~~~~ but

collapsed

_on larger ^{scales. The size of the blob becomes infinite when} Au

= 0 lone solvent). The exponent ^{of the blob size variation with Au is} non-classical due _to the appearance of the correlation function exponent.

Discussion.

We

presented

_a field theoretical model for the

study

of the behaviour of _a

single polymer

coil in

a critical

fluid,

which _was modeled

by

_a

binary

mixture of _two

(good)

solvents _near their consolute

point.

The Gaussian

approximation

rederived all classical results

published

earlier.

This model fails _near the critical

point

when the solvent

phase

^transition falls into the class of the

Ising

model. The use of _a renorrnalized

(non-Laplacian)

field

theory

^for the

density

field

allowed _{us to}

proceed

with the

analysis

of the critical

region.

Moreover the effective

potential

which consists of two parts, one hard-core

repulsion

and _one attraction gave rise _{to an} estimate of the blob size for the

polymer chain,

I-e- at scales below the blob size the chain is _a

SAW,

whereas for scales

larger

than the blob size the chain is attractive. Such conclusions II are also in accord with simulation;

by Magda

et al. (4].

To be _more

specific

and to summarize the results the radius of

gyration

of the

polymer

^chain is considered in _more detail. The

picture

which emerges from the calculations above is shown in

figure

2.

Rg

xg xG

xL xc ^x

Fig. 2. The radiu, ot gyration oi _a chain disolved in _a binary mixture (see text).

At low values of _X, I-e- far away from the critical

point

_Xc the fluid fluctuations _are very small and do _not affect the chain very much. Therefore the chain is swollen. When the

phase separation

of the fluid is

approached

the fluctuations become

larger f~~

_(X~

X and the

effective monomer-monomer

potential

becomes

significantly

altered

by

^{the fluid} fluctuations,

as

long

_as

preferential adsorption

is present, ^I-e- _v

~

# _{v ~}

~.

The radius of

gyration

shrinks

[1, 3]

and detailed calculations will be

presented

in

[I I].

The dashed line in

figure

2 is

predicted by

the Gaussian

approximation

which fails _at the non-universal

Ginzburg Criterion, defining

XG in the usual way

[10].

The theta temperature _Xo ^{is also} non-universal and differs from system to system. ^{Below the}

Guizbury-XG

non-classical exponents come into

play

and the A

p~(r)

_term in the Hamiltonian _matters

significantly.

At this

point

the renormalized Gaussian

model is used, ^I-e-

equation (4.2)

for the fluid. R~ ^{decreases more, but the}

(n

= I

Ising

exponents ^{determine the variation of}

R~.

The

important point

_near the critical

region

is when the size of the fluctuations exceeds the radius of

gyration

of the chain. I-e-

f>R~

or

(Xs-X)~~>N~~, _v~m3/5.

The chain is _more

likely

to stay ⁱⁿ a « one

phase

fluctuation _{» as}

depicted

in

figure

3.

-R-

Fig. ^{3. The size of the chain and the size of the clusters} (determined by D determine the actual behaviour of R~ near the critical point.

In such a case the chain does no

longer

_« feel _» the eifect of the mixed solvent and it

expands again

to its size

given

in pure solvent A, _say. Therefore the blob size is identical _to

R~

^and

R~ N~'~

_{When the size of the} fluctuations is of the order of

R~,

I-e-

1786 JOURNAL DE PHYSIQUE II N° 12

j

X~XL~N~

_-Xs

the chain is

expected

to

expand

to its natural size. The latter

depends strongly

on the size of the chain, therefore the

precise

location of the minimum is difficult _to

predict

and non universal

[I il.

^To see this consider

P(R)

where R is the

typical

size of the chain. Whenever R «

f

the last term can be

neglected

and

v ~~

will be the dominant effect. If the

physical picture

derived in this paper

(see Fig. 2)

is

compared

to reference

[4]

_agreement of the simulation data

can be found.

It is worth

mentioning

that the mixed solvent _near its critical

point

^is a very

exciting

version of the _« troubled solvent

» considered

by Duplantier ([13]

and references

therein).

In _our

study

the essential features _come from the

critically

of the solvent whereas in

[13]

the main effect

comes from the disorder.

Acknowledgment.

The authors

acknowledge stimulating

discussions with R.

Borsali,

A. Sans _was

supported by

the Max-Planck-Institut

during

her visit in Mainz where this work has been initiated and carried out

partly.

References

[ii Schultz R. C.,

Fiory

P. J., J. Polym. Sci. IS (1955) 23i.

[2] Dondos A., Benoit H., Makiomol. Chem. 133 (1970) l19.

[3] Brochard F., de Gennes P. G., Ferroelectrics 30 (1980) 33.

[4]

Magda

J. J., Fredrickson G. H., Larson R. G., Helfand E., Macr(Jmolecfile.< 21(1988) 726.

[5] Brereton M. G.,

vilgis

T. A., J. Ph_vs. Fiance 50 (1988) 245.

[6]

vilgis

T. A., Borsali R., Macromolecule~< 23 (1990) 3172.

[7] Weyer~berg A., vilgis T. A., Ph_>..<. Ret E 48 (1993) 377.

j8] Sariban A., Binder K., Macromolecules 21 (1988) 389.

[9j Doi M., Edwards S. E., The Theory ^of Polymer Dynamics (Oxford

University

Press, Oxford, 1976).

[10] Negele J. F., Orland H., Quantum

Many-Body

Systems (Addison Wessley, N-Y-, 1991).

ii I] Sans A., Vilgis T. A., Jannink G., to be

published.

j12] Amit D. J., Field Theory, Critical Phenomena and Renormalization (World Scientific,

Singapore,

1985).

j13]

Duplantier

B., Phys. Rev. A 45 (1988) 4851.