Effect of Shear Flow on Polymer Demixing- the

unanswered questions

H. GERARD, J. T. CABRAL, J. S. HIGGINS

Department of Chemical Engineering

Imperial College, London

Thermodynamics

Flory-Huggins lattice theory

mmm STHG

Combinatorial entropy

Enthalpy

ABBABB

BA

A

A

r

m lnN

lnNRTn

G

Polymer miscibility

1

Phase separation

2

0 1B' B''

S' S''

Composition

Spinodal

Binodal

To

T

To

Free Energyof mixing

0

LCSTConcentration fluctuations

o+

-

Metastable:nucleation & growth

Unstable: spinodal decomposition

G 0

Free Energyof mixing

CompositionC 0

B'

B''T0

Spinodal decomposition

3

EARLY LATER FINAL

distance

C0

B'

B''80 m

Optical microscopy

4

Cahn-Hilliard

t

MG

Mk 2

22 42

Cahn-Hilliard linearised theory

equation of motion concentration fluctuations

-DappM: diffusional mobility f(D1,D2)

~o

R q MG

q Mkq( )

2

22 42

qG

km

1

2

1

2

& qc

Growth rate of Fourier components

t-indep

diffusionSharp interfaces

Qm

Qc

R (s-1)

growth

relaxation

Q (nm-1)

0

Scattering

photodiode array

Heating block

He-Ne 5mW laser

)0()()( QQQ ttS

m: characteristic length of phase separation ~ 1m (LS)

Structure factor

oK

1K

solid angle: d

dEd

d2

S(Q,)

LS schematic7

1.36

E-0

3

4.60

E-0

3

7.80

E-0

3

1.09

E-0

2

1.39

E-0

2

1.67

E-0

2

0

100

200

300

400

500

Time (s)0 300 600 900 1200 1500

Inte

nsity

(au)

0

5

10

15

20

25

30q=0.00124 A-1

deep quench

shallowquench

LS

0

2000

1000

time (s)

q (nm-1)

I (au)

Light scattering

8

TMPC/PS 50:50Tjump=240.6oC

TMPC/PSd 70:30 MM254 oC

Q (A -1)

0.002 0.004 0.006 0.008 0.010

Intensity (cm)

-1

0

50

100

150

200

250

300

350180 s

x 15 s

TMPC/PSd 70:30258oC

Time (s)0 200 400 600

Intensity (au)

0

2

4

6

8

LS: Q=0.00152 A -1

Light scattering

Q (A-1)0.000 0.004 0.008 0.012 0.016

R (s

-1)

0.00

0.01

0.02

0.03 TMPC/PSd 70:30 MM254

oC

QC

QM

Time (s)

0 100 200 300

Intensity

e1

e2

e3

e4Q=4.9x10

-3 A

-1

TMPC/PSd 70:30T=252 o C

t)Q(R2QQ e)0(S)t(S

Early stages (SANS)

11

In 1996 we had observed apparent SID at temperatures well below the quiescent LCST for a number of amorphous blends.

Shifts in LCST ranged from 40K to less than 5K and seemed to correlate with differences in the component rheological behaviour.

We had not followed the kinetics of SID, and there was no theoretical development to describe such kinetics

We believed that observing the kinetics might answer the question of SID (ie a true thermodynamic phase separation) v enhanced concentration fluctuations

Theoretical ApproachesWolf

Add a stored energy term to the Gibbs free energy:

with

May dramatically influence

2Gm/ 2

Good qualitative description of the shear behaviour of PS/PVME, SMA/PMMA and

SAN/PMMA blends.

But no description of the anisotropy

equilibrium thermodynamics applied to such a case?

Clarke & McLeishThey use, following Doi and Onuki, the two-fluid model considering the visco-elastic behaviour of both components.

For low shear rates, in the y,z plane:

quiescent part shear part

where =((A’/A)-(B’/B))/(A’ + B’),

i’ being the frictional drag per monomeric volume associated with component i, related to the monomeric friction coefficient per volume by:

i’= i (Ni/Nei) 0i’

smm EGG

)( 2e0BBAAs JVVE

)M()R( 220app

2 AqDqq

iii ,,,

k

MGf

T3

v4A

B

m

Shear Light Scattering Experiments1D LS Shear Experiments:

samples sheared in plate-plate geometry, the scattered light (He-Ne Laser, =632.8 nm, incident beam // to the velocity gradient direction) being collected along the vorticity direction

For blends 1 and 2 (critical composition) and for a shear rate , scattered intensity increases with time after a delay time d:

Mw

(kg.mol.-1)

Mw/Mn 30/70% w/wblend Ts (ºC)

PS1 270 4.7 101.8 .5

PS2 283 1.35 103. .4

d-PS3 303 1.6 141.5 1.

PVME 89 2.5

Diode array

c

0

5

10

15

20

0 40 80 120 160 200 240

shearing time (s)

d

q = 10.43 10-3

nm-1

q = 3.37 10-3

nm-1

q = 8.46 10-3

nm-1

q = 5.43 10-3

nm-1

q = 13.29 10-3

nm-1

Inte

ns

ity

(a

.u.)

Blend 1: T - Ts = - 17.2 K, 6.5 s-1

20qapp

q

qlimD

)R()(

I(q)=I(q,d)exp(2R(q)(t- d))

“early stage” R(q)

0

0.1

0.2

0.3

0 0.003 0.006 0.009 0.012 0.015 0.018

q (nm-1

)

R(q

) (s

-1)

Velocity Gradient

Vorticity

Characteristics of the three 30/70% w/w PS/PVME blends studied

.

- D

app (

10-1

3 cm

2 s-1)

0.1

1

10

100

1000

0.1 1 10shear rate (s-1)

Blend 1, -22.3 K

Blend 1, -17.2 K

Blend 1, - 8. K

Blend 2, -24.6 K

Blend 2, -14.9 K

Clarke and McLeish:

extracted from shear LS experiments

Decent fit for low shear rates:

LS Results and Theoretical Predictions

appD

220app

2 q2ADqq MR Blends 1 and 2 : - Dapp0 from quiescent and shear LS experiments

1E-14

1E-13

1E-12

1E-11

75 80 85 90 95

T (ºC)

Blend 1 quiescent LS experiments

Blend 1 shear LS experiments

Blend 2 quiescent LS experiments

Blend 2 shear LS experiments

(cm2s-1)

Dapp0

May also be obtained from “remixing” quiescent LS experiments:

Influence of molecular deformation

on blends thermodynamics?

Estimation of shear rate?

Did we miss the “early stage”?

- Dap

p (1

0-1

3 c

m2s-1

)

0.1

1

10

100

1000

0.1 1 10shear rate (s-1)

Blend 1, -22.3 K

Blend 1, -17.2 K

Blend 1, - 8. K

Blend 2, -24.6 K

Blend 2, -14.9 K

Small Angle Neutron Scattering – the aim is to look at much smaller size scale and “catch” the early stages

One component is deuterated to give contrast - this may shift the LCST.

We had no shear cell for SANS and had to use quenched samples.

The neutron beam is much larger than the laser beam so we were averaging over a range of shear rates

  Mw

(kg/mol.)Mw/Mn

 

Tg (ºC) blend Tg

(ºC)

spinodal temperature (Ts) (ºC)

PVME 89 2.5 -32    

PS1 270 4.7 106.6 -15.5 ± 4. 101.8 ± .5

PS2 283 1.35 108.7 -19.4 ± .5 101.5 ± .4

PS3 (d-PS)

303 1.6 108.6 -19.2 ± .5 141.5 ± 1.

PS4 1020 1.15 110.6 -18.4 ± 3. 90.2 ± 2.

shear-quenched SANS

q (A-1)0.01 0.1

1

10

100

1000

0 s-1

4 s-1

4 s-1

(highest shear rate)

ConclusionsOur first SANS results seem to partially confirm what was first observed through light scattering in similar (from a rheological point of view) protonated blends.

For low q : the rise of S(q) for high shear rates may be due to an enhancement of concentration fluctuations in accordance with our LS results

For high q : higher shear rates seem to reduce concentration fluctuations, a feature not explained by two-fluid models inspired approaches such as Clarke & McLeish’s one.

Might explain the discrepancy between the apparent diffusion coefficients obtained from quiescent experiment and deduced from )( appD

Due to the effect of molecular deformation on the thermodynamics of the system?

The Unanswered Questions

Is the D from SID really different from the D obtained in re-mixing experiments?-experiments first and if confirmed theory needs some thought.

Would a more sophisticated statistical mechanics description of the free energy help? We have been having considerable success in quiescent systems using a version of BGY which includes compressibility and non-random mixing.

What happens in the early stages of SID? – can we find a system where deuteration does not have such a large effect on LCST? – or can we use another technique, eg AFM on quenched samples?

All these Qs aimed at the “big” one –is this SID or just enhanced concentration fluctuations?

Parameters which characterize the pure fluid

iiri v

strength of nearest-neighbour interaction number of contiguous

lattice sites per molecule

volume per mole of lattice sites

…and the mixture

g = ij/(ii jj)1/2

characterizes the deviation from the geometric mean approximation

Lattice Born-Green-Yvon (BGY) theory links the microscopic

character of a polymer/blend to its thermodynamic properties

Macromolecules 36, 2977 (2003)

Formal definition of p(i,j),pair distribution function

approximations

p(i, j) i j exp( ij )

l exp( il )h

l1

m

• non-randomly mixed

• local connectivity• compressible

A few remarks about the theory

Sum over lattice of pair contributions to internal energy

E (T, ii, ij, i) internal energy

Various partial derivativeslead successively to …

A (T, ii, ij, i, V)P (T, ii, ij, i, V) (T, P, ii, ij, i)

Helmholtz

equation of state

chemical potential

Polymer Blend Parameters

Blend11

(J/mol)22

(J/mol)

(11 22)(J/mol) gexp-1 Tc,exp (K) notes

PS/PB -2062.0 -1988.3 73.7 -0.00480 362.9 pure PVT & Tc

PE/PEP -1977.5 -2000.0 22.5 -0.00031 421.0 pure PVT & Tc

PP/hhPP -2040.7 -2027.8 12.9 -0.00015 303.0 pure PVT & Tc

PMB/PEB -2239.6 -2285.9 46.3 -0.00050 313.0 pure PVT & Vmix

PpMS/PS -2282.6 -2251.1 31.5 -0.00022 413.0 pure PVT, NS & Tc

PEMS/PDMS -1808.2 -1744.2 64.0 -0.00019 334.0 pure PVT & Tc

PS/PVME -2251.1 -2000.9 250.2 0.00230 437.1 NSPS/TMPC -2105.5 -2209.5 104.0 0.00028 508.4 NS

PS/PCS -2228.2 -2409.5 181.3 0.00241 423.0 pure PVT & Tc

PS/PPO -2131.1 -2477.6 346.5 0.00380 miscible blend PVT

gexp=exp/(

When (gexp-1) is negative/positive the geometric mean is over/underestimating the strength of the 1-2 interaction

Ornstein Zernicke Formulae

BB

B

wB

zB

AA

A

wA

zA

ABS v

a

N

N

v

a

N

Nv

2202

~~36

221

)0()(

q

SqS AB

AB

q (A-1)0.01 0.1

I coh

(cm

-1)

1

10

100

1000

q2 (A-2)

0.0005 0.0010 0.0015

1/I (cm)

0.1

0

47oC

88oC

110oC

122oC

135oC

135oC

47oC

q (A-1)0.01 0.1

I coh

(cm

-1)

1

10

100

1000

q2 (A-2)

0.0005 0.0010 0.0015

1/I (cm)

0.1

0

47oC

88oC

110oC

122oC

135oC

135oC

47oC

q (A-1)0.01 0.1

I coh

(cm

-1)

1

10

100

1000

q2 (A-2)

0.0005 0.0010 0.0015

1/I (cm)

0.1

0

47oC

88oC

110oC

122oC

135oC

135oC

47oC

q (A-1)0.01 0.1

I coh

(cm

-1)

1

10

100

1000

q2 (A-2)

0.0005 0.0010 0.0015

1/I (cm)

0.1

0

47oC

88oC

110oC

122oC

135oC

135oC

47oC

q (A-1)0.01 0.1

I coh

(cm

-1)

1

10

100

1000

q2 (A-2)

0.0005 0.0010 0.0015

1/I (cm)

0.1

0

47oC

88oC

110oC

122oC

135oC

135oC

47oC

delay time d (s)

1

10

100

1000

10000

100000

0.1 1 10 100shear rate (s-1)

Blend 1, -22.3 K

Blend 1, -17.2 K

Blend 1, - 8. K

Blend 2, -24.6 K

Blend Rheology

Rheological Experiments:

The Clarke & McLeish approach has been developed for the weak shear regime ( < 1 where is the longest relaxation time of the blend). It also assumes that both components present different relaxation times in the blend.

Rheological experiments were performed on a Paar Physica UDS 200 Rheometer to check both assumptions:

Relaxation time and for blend 1

Blend 1: T - Ts = - 17.2ºC

1000

10000

100000

1000000

0.01 0.1 1 10 100 1000 (Hz)

G',

G''

(Pa)

G'

G''

d

-1

Blend 1: relaxation and "critical" time (s)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

350 355 360 365 370

T (K)

"critical" time

relaxation time

weak shear regime if we assume that corresponds to the intercept of G’ and G’’.

in our frequency range, only one relaxation time may be detected in the blend.

c

cCritical time = -1

T d (s) critical shear rate (s-1)

-22.3 1.25 ± .06 .689

-17.2 .87 ± .04 .725

- 8. .43 ± .02 .747

Concentration Fluctuations in Polymer Blends

Effect of shear flow:• shear (dispersed phase) droplet break-up (Taylor) • influence on thermodynamics (Wolf)• stress/concentration fluctuations coupling (Doi, Onuki)

T

one phaseDapp < 0

two-phase(unstable)Dapp > 0

spinodal Ts BinodalDapp < 0

glass transition Tg

Concentration fluctuations enhancement (for the early stage inside the spinodal line) and decays (for the “late stage” in the one phase region) may be described Cahn-Hilliard theory, giving an expression for their growth rate R(q):

with the apparent diffusion coefficient

)()R( 2app

2 qM2Dqq

2

m2

app

GMD

Small Angle Neutron Scattering

Ornstein-Zernike at high q:

S(0)-1

( from 100 to Å

2q1

0q

)(

)S()S(

2m

2 G

S(q

) (c

m-1

)

0.1

1

10

0.01 0.1 1q (Å-1)

0.04

0.38

0.8

2.22

2.96

4.02

Ornstein-Zernike

Structure Factor S(q) for blend 3 sheared at T=86.6ºC then quenched in liquid N2

Beam centre shear rate

These features are not described by the available theoretical models

S(0

)-1 (

cm)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6

shear rate (s-1)

maximal shear rate 5.2/s

maximal shear rate 2.1/s

Enhancement of Concentration Fluctuations in the (x,z) Plane

q (m-1)

0

5

10

2D LS patterns for blend 1 at T=84.6ºC a) before shear and b) after 26.5 min. of shearing with = 5 s-1

a)

b)

Optical micrograph of the bulk of a quenched sample after 25 min. shear at T = -17.2 K and

= 1.4 s-1 for two different magnifications.

10 m

100 m

flow

vorticity

Small Angle Neutron ScatteringBlend 3: deuterated PS/PVME blend sheared at T = 86.6ºC ( T = - 54 K)

Two samples with maximal (2.1 and 5.2 s-1) but similar maximal strain ( Rheological steady state is reached with no change in LS patterns

The shearing is then stopped and the sample quenched in liquid N2

SANS (on D22 under cryostat at the ILL, Grenoble):

No obvious anisotropy in our q range (7. 10-3 to 1.1 10-1 Å-1)

High q (> 4 10-2 Å-1): S(q) with

reduction of small wavelength concentration

fluctuations with shear

S(0)-1 obtained from high q fits is increasing with “Shear Induced Mixing”

Low q (< 4 10-2 Å-1): S(q) then with

intermediate scale structure growing (as seen in PS/DOP)? “Shear Induced Demixing”?

But similar low q high scattering for unsheared blend!

Effect of quench?

But we are scanning local flow directions:

incident beam