Interface motion in semidilute polymer solutions

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Interface motion in semidilute polymer solutions

Akira Onuki

To cite this version:

Akira Onuki. Interface motion in semidilute polymer solutions. Journal de Physique II, EDP Sciences,

1993, 3 (9), pp.1299-1303. �10.1051/jp2:1993201�. �jpa-00247906�

Classification Physics Abstracts

46.608 64.70J

Short Communication

Interface motion in semidilute polymer solutions

Akira Onuki

Department ^of Physics, Kyoto University, Kyoto 606, Japan

(Received

17 May 1993, accepted in final form 19 July

1993)

Abstract Linear viscoelastic effects _are examined in semidilute polymer solutions in _macro-

scopic two-phase coexistence. If the temperature T _or the _{pressure p is} oscillated in time, the interface motion exhibits gel-like behavior at

high

frequencies ^{and fluid-like} behavior at low frequencies. We also study transient relaxation after _a step-wise change of T _or p. We propose such experiments to check the validity of fundamental hypotheses _on viscoelasticity in polymer

solutions.

Considerable theoretical efforts have been made _on the form of

dynamic equations

in semidi- lute

polymer

^solutions in connection with shear-induced

phase separation [1-5].

It is _now estab- lished that there is _a

dynamic coupling

between stress and diffusion in such

entangled polymer

systems

including polymer

blends [5]. ^That

is,

imbalance in the network

stress$i

can induce relative motion between

polymer

and

solvent,

v~ v~ =

(1 ~)(-i(-v«

₊ v

.m)

,

(1)

where vp and _{v~ are} the

polymer velocity

and the solvent

velocity, #

is the

polymer

volume fraction assumed to be in the semidilute

region,

and _r is the osmotic pressure. The

(

is the friction constant between

polymer

and solvent and is of order

~~fp~,

^where _~~ is the solvent

viscosity

and

fb

is the blob size. The first _term (cc

Vr)

in

(I)

is the usual _term in fluid

binary

mixtures without

entanglements

^{while the} second term (cc V

.$i)

has

long

been known in

gels

^with permanent

crosslinkages

_[6]. We _are

assuming

that

#

is

considerably larger

than the

overlapping

volume fraction

#*

and

entanglement

effects _are crucial in

dynamics,

so that the solution

viscosity

_{~ is} much

larger

than _j and the

dynamic coupling

_can be

important.

In shear flow _a relative

velocity proportional

to

I(3~/3#)V#

arises in

(I)

from V

.$i

_{due to}

the strong concentration

dependence

of _~,

i being

the shear rate. It

gives

rise to dramatic enhancement of the concentration fluctuations in theta solvent under shear

[7-9].

^{On the other}

hand,

this

coupling

_can

explain nonexponential decay

of the time correlation function

S(q, t)

of the concentration fluctuations around

equilibrium is,

_10]. It _can be

significant

in theta solvent and _was

originally predicted

in reference [11].

1300 JOURNAL DE PHYSIQUE II N°9

We then expect ^{that the above}

dynamic coupling

should

profoundly

influence

phase

_separa-

tion processes in

entangled polymer

systems. To

investigate

^this aspect,

however,

_we further-

more need _to

specify

the interfacial

boundary

condition in addition _to the bulk relation

(I).

To examine nucleation in

entangled polymer

solutions the present ^author has

recently proposed

_a modified Gibbs-Thomson relation at the interface

[12],

r-x~x-an+'t~=0

,

(2)

where _r~x (Gt

0)

is the osmotic pressure on the coexistence curve, 't is the surface

tension,

and

~ is the curvature of the interface. The _an is the normal component of the network stress at

the

interface,

an=n.$i.n

,

(3)

where _n is the normal unit vector. The normal _stress in

(2)

_serves to decelerate the

growth

rate of _a

droplet consisting mostly

of solvent when its radius R is smaller than _a

long

viscoelastic

length fve

^defined

by

fve

_"

(((~~il)~/~

^~

fb(illils)~/~ (4)

This

length

is also of order

(Dor)~/~,

where Do is the collective diffusion constant

(see (lo) below)

and _r is the

rheological

relaxation time. It has been introduced in various contexts [4, 5,

13].

For R «

fve,

R-R~

changes exponentially

with _a small

growth

rate r

=

~t/2~Rc

+~

r~~A,

where _~t is the surface tension, Rc is the critical

radius,

and A is the supersaturation smaller than I. For R _»

fve,

R

obeys

the usual

Lifshitz-Slyozov equation.

On the other

hand,

viscoelasticity

in

spinodal decomposition

is very

complicated

and has not yet been well studied.

Instead,

the aim of this letter is to examine the

implications

^of

(I)

and

(2)

in a

relatively simple situation,

in which _a semidilute

region (#

>

#*)

and _a

nearly

_pure solvent

region (#

lit

0)

_are

separated macroscopically by

_a

planar

interface. Such _a state will appear as a

final _state after

quenching

the fluid into metastable _or unstable

regions.

We then

produce

_a small

homogeneous change

in the temperature T _or the pressure p and follow the motion of the interface position x =

x;(t)

and the evolution of the concentration deviation

b#(x,t)

in the semidilute

region

x <

x;(t).

We

neglect inhomogeneities

of T and _p. Note that _{a pressure}

quench

^method has

usually

been used in

phase separation experiments

of near-critical

binary

mixtures

[14],

in which T and _{p are}

changed adiabatically

to

bring

the fluids into metastable _or unstable

regions.

After such _a

change

in _p and T the interface _{moves as a} result of

absorption

or

desorption

of solvent _across the interface because the

polymer density

is _so small in the dilute

region

_x >

x;(t).

This _process is

nearly

the _{same as} in

swelling

_or

shrinking

of

gels

immersed in solvent [15]. ^{It is} very instructive to compare the _process with those in fluids and

gels.

In addition _we comment that the excluded volume interaction is weak and the chains _are

nearly

^Gaussian in the temperature

region

under consideration.

We _assume that all the deviations

depend

_{on space}

only

^{in the} x direction which is taken

to be

perpendicular

to the interface. For

simplicity

_{we assume} that the total _mass

density

iS _a constant and the _average

velocity

v =

#vp

+

(I _#)v~

vanishes. It follows that _v~

=

-(1 #)~~ #vp.

We introduce the dimensionless

density

deviation

o(x,t)

=

ii(x,t)/i

,

(s)

which

obeys

to linear order in the deviations

=

-t~p ₍₆₎

where

#

= 30

lat.

The network stress is related to the

gradient

of the network

velocity

_{vp [3,} 5],

ai>

(x, t)

=

/~ ^{dsG(t s)} (£vp>(x, ^s)

⁺

£vpi(x, s) (iii

^V _VP

^{(x, )j}

,

(7)

-m ^I J

where

G(t)

is the _stress relaxation function and its one-sided Fourier

(Laplace)

transformation is the

complex viscosity,

~

~*(uJ) =

/ dte~'"~G(t) (8)

o

The relaxation time

r(+~ ~/G(0))

of

G(t)

is very

long

due _to

entanglement

effects. It is

highly

nontrivial

that$i

_{is determined}

by Vvp,

but this

assumption just

leads to the

gel-like

behavior at

high frequencies. Using (1, 6, 7)

^and up; =

bmvp,

up may be

expressed

in terms of

0,

and

(6)

becomes _a diffusion

equation

in the

region

_x <

z,(t),

fi2

R(X>

t)

_{" DO}

~QIX> ^{t) (9)}

with

Do

=

(i -1)2(-iK

a

(-iK

,

(io) Q(x, t)

=

o(x, t)

₊

(K-1 /~ ^{dsG(t s)o(x, s)}

,

(ii)

where K

=

#(3r/3#)Tp

is the osmotic bulk modulus. The Do is the diffusion _constant in the absence of

viscoelasticity,

while

Dg

₌

Do (1+ ~~1/K)

is the diffusion _constant in

gels

_[6],

~1=

G(0) being

the shear modulus.

Next _{we set up} the

boundary

conditions at _{x =}

x,(t).

Note that the deviation of _{r r~x} in the semidilute

region

consists of the

inhomogeneous

part K0 and the

homogeneous

part

(br)o

G~

(3r/3T)pjbT

+

(fir /3p)Tjbp.

In experiments, ^bT and

bp

_are the control parameters.

In

particular,

if

bp

is

changed adiabatically

_as in near-critical fluids

[14],

we have

(br)o

"

(3r/3p)~jbp.

It is convenient to introduce

Ro(t)

=

-(br)o/K.

Then the modified Gibbs- Thomson relation

(2)

becomes

Q(x;(t), t)

=

00(t) (12)

The _{curvature term} vanishes in _{our one} dimensional _case. The interface

velocity

I; ₌

8zi13t

is related _to the diffusion _{current at z}

= z; from the conservation of

polymer

_{mass, so} that it ₌

-DOQ/(xi(t),t)

,

(13)

where

Q~(x,t)

=

3Q(z,t) lax.

Here _we stress _our two basic

assumptions:

_I) We _are

assuming

that the interface is infinites-

imally

thin and that

(12)

holds

instantaneously

after _a

change

of

(br)o.

This is

justified

when

we are interested in _space and time variations much slower than the interface thickness _(+~ the blob size

fb)

and the characteristic diffusion time

f(/Do

within _a blob. On time scales

longer

than

f(/Do

the concentration

profile

at the interface is almost

equal

to that in

equilibrium

determined

by

₀₀

(t)

and there remains

slowly varying,

concentration variations far from the in- terface

(the

distance

being

^much

larger

^than

fb). ii)

^If

00(t)

> 0, the fluid becomes metastable and

00(t)

coincides with the so-called

supersaturation

[12]. ^For

00(t)

<

o,

_a

sharp

interface still exists _as

long

_as T <

T~(p),T~(p) being

the critical temperature

dependent

_{on p.} We

assume that

00(t)

is

sufficiently

small _so that there _{appears no}

appreciable

nucleation in the

1302 JOURNAL DE

PHYSIQUE

II N°9

fluid within the observation _time. Under this condition the

following

relations _are valid both for positive and

negative

00.

However,

for

large 00,

resultant

phenomena

_{are very} different

depending

_on the

sign

^of

00.

Now

(9-13)

constitute _a Stefan

problem

^with _a

moving boundary

and their solutions _{are gen-}

erally complicated.

In this letter _{we assume}

00(t) («

I and solve the

equations

in

expansions

with respect to

00(t).

First let _us consider _an

oscillatory

_case,

00(t)

= 00

cos(uJt) (14)

To

leading

order in

00, (9)

_may be solved

by setting z;(t)

= 0 to

give

°(z, t)

=

°oRe l~~ _exP(~z

+

uJt)j

,

(is)

~ ~

3~°~~~~°~~

where

Re[.

denotes

taking

the real part, ~*(uJ) ^{is the}

complex viscosity

defined

by (8),

and

~~

=

iuJDp~K / K

+

~iuJ~*(uJ)

3

% iuJ

/

_{[Do +}

iuJf(~~*(uJ) /~]

,

(16)

where

fve

is defined

by (4).

We define _~ such that its real part is

positive.

The

length 1/

_~

is the thickness of the

perturbed region.

The interface value

0(z,, t)

%

0(0, t)

tends to

(14)

in the low

frequency

case uJr < I

,

while it behaves in _a

complicated

manner for

uJr Z _{I. Here}

r is the

rheological

^relaxation time. Insertion of

(15)

into

(13) yields

4 1/2

zj(t)

=

-00Re

e'~~

DoKliuJ

K ₊

-iuJ~*(uJ)

,

(17)

3

~

which is valid to first order in 00. The behavior of

z,(t)

is also

singular

for _uJr iS _{I. If}

uJ <

K/~

+~

l/r,

the network _can

fully reorganize

itself

against

the

perturbation

and the viscoelastic effect becomes weak. The _reverse holds for _uJ iS _I

IT.

Secondly

we

change 00(t)

in _a

step-wise

_manner,

~°~~~

loo

"

~onst. ^{~~ ~}

^~~~~

In this _case the

equations

_can be solved

using

the

Laplace

transformation with respect to time.

If

G(t)

is assumed to relax

exponentially [10,

11] as

G(t)

₌

/1e~~l'

,

(19)

The interface value

0(z;,t) approaches

to 00

exponentially

_as

0(z,, t) loo

^{" I I +}

)

exp

-t/r

1+

~) ⁽²⁰⁾

l~ 3

~

The interface

position z,(t)

_{grows as}

z;(t)

Gt

(4Dt/r)~/~00

for t » _r and

z,(t)

Gt

[4Dt/r(1+

4p/3K)]~/~00

for _{t « r.} In

rheological experiments, however,

it is known that

G(t) decays algebraically

_as ^t~~+P for t < r or

~*(uJ) ^Gt

A(iuJ)~P

for _{uJr >} 1

,

(21)

where

p

G£ 0.8

(as

the so-called Cox-Merz law

suggests)

_[7,

_16].

This leads to behavior in the short time

region

t 1~ _r of the

forms,

0(z,, t) loo

Gt

(3K/4r(2 p)A]t~~P

,

(22) z,(t)

Gt

-00(3D/4A)~/~r(2 ^)~~t~~P/~

,

(23)

where A is the coefficient in

(21)

^and

r(z)

is the Gamma function. Here _we have assumed that

(21)

holds _even for _{uJ - cc, so} that it follows that

0(z;,t)

_- 0 _as t _- 0.

However,

iuJ~* (uJ) can tend to _a well-defined limit _pm for _uJ greater ^than a crossover

frequency

_uJo. Then

0(z,,t) loo instantaneously

_{grows up} to

K/(K

₊

)~lm)

ⁱⁿ a short time of order uJp~ ^{In the}

single

relaxation

approximation (19)

we find _~lm

= ~t and _uJo

+~ I

IT.

In fact

(20)

shows that

0(z,, t) loo

_-

K/(K

+

p)

_as t _- 0. In usual fluids it

approaches

to I in _a

microscopic

time far from

criticality

and in the order parameter relaxation time _near

criticality.

In summary, we have examined the viscoelastic behavior of the interface motion _on the basis of the three fundamental

hypotheses (1, 2, 7),

which _are those _on the

dynamic coupling,

the

boundary condition,

and the network _stress. Future

experiments especially

in the

oscillatory

case should be _very informative _to

unambiguously

confirm the theories. We also _propose the

same kind of

experiments

in

polymer

blends [5], ^{which will be} very informative because the counterpart ^of

(2)

remains unknown in blends.

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