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Thermodiffusion in Polymer Solutions

Jutta Luettmer-StrathmannDepartment of Physics, The University of Akron, Akron, OH 44325-4001, USA

• Introduction

• Thermodiffusion in polymer solutions

• Single polymer chain in an incompressible solvent

• Incompressible two chamber system

• Lattice model for polymer in a compressible mixed solvent

• Application to poly(ethylene oxide) in ethanol/water mixtures

• Results for static properties and thermodiffusion

• Discussion

TB

TA

Condensed Matter Colloquium, Physics Department, Ohio University, September 12, 2002

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Thanks to Mike Boiwka for performing Monte Carlo simulations

3

Thanks to Simone Wiegand, Berend Jan de Gans, and Rio Kita from the Max Planck Institut für Polymerforschung in Mainz for sharing their experimental data.

4

Thermodiffusion — Ludwig-Soret Effect

side warm theto

side cold theto

migrates 2component

negative is

positive is

2component of

:2component offraction mass theis

and re temperatu theis where

)1(

1:2component oft coefficienSoret

T

2211

22

T

S

mNmN

mNxx

T

T

x

xxS

1 2

Fluid mixture with uniform temperature T

under a temperature gradient

• There is no microscopic theory that (reliably) predicts the sign of the Soret coefficient.

• Typically, the heavier component migrates to the cold sideThot Tcold

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Heat of Transfer

The heat of transfer Qa* , introduced by Eastman and Wagner (1926, 1930)

T’, P’, V, Na-1, NbT, P, V, Na, NbQa

*T, P, V’, Na-1, Nb

constant gas ideal theis R where R

:solutions idealFor

0:Groot de

2

*b

*a

T

*bb

*aa

T

QQS

QxQx

Wirtz (1943) and Denbigh (1951) estimate Qa*- Qb

* from two energy contributions, the energy to detach a molecule from its neighbors and the energy to create a hole.

Prigogine et al. (1950) consider a free energy for detaching a molecule to describe associated solutions

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Thermodiffusion in polymer solutions

J. Rauch and W. Köhler, Phys. Rev. Lett. 88, 185901 (2002)

Dilute solutions:

Soret coefficient is independent of concentration, increases with chain length (ST ~ M0.53)

Concentrated solutions:

ST is independent of chain length, decreases with concentration (ST ~ (c/c*)-0.73)

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In solution, the polymer migrates almost always to the cold side, with only two known exceptions

poly(vinyl alcohol) in water, Giglio and Vendramini, Phys. Rev. Lett. 38, 26 (1977)

poly(ethylene oxide) (PEO) in ethanol/water mixtures with low water content,B.-J. de Gans, R. Kita, and S. Wiegand (to be published)

The Soret coefficient of PEO changes sign!

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Single chain on a simple cubic lattice - exact enumerations

pair contact with

interaction energy

For a chain of Np beads, ( Np-1 bonds), on a simple cubic lattice generate all conformations so that no two beads overlap.

Determine the number c(m) of conformations with m pair contacts.

Determine the mean radius of gyration for conformations with m pair contacts.

mm

m

m

m

m

mRpR

Z

mcpm

TkmcZ

)()( :eraturegiven temp afor gyration of radius Average

e)()( :contacts on with conformatifor y Probabilit

1,e)(:functionPartition

2g

2g

isolated

Bisolated

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Single chain in an incompressible solvent

T

ZTkmcmE

ZU

mcNTZ

mNNz

mNmmE

m

mE

m

mE

s

lne)()(

1:energy average

e)()(:functionpartition

))224(2

1

2()224()(:energy Internal

2B

)(pol

)(pol

ssppsppp

energyn interactiosolvent -solvent

energyn interactiosolvent -polymer

energyn interactiopolymer -polymer

sites lattice ofnumber total

sitessolvent ofnumber

chain in the beads ofnumber

ss

ps

pp

sp

s

p

NNN

N

N

ss

pp

ps

10

Np=51, N=250*Np, Ns=N-Np, T*=0.01

10

15

20

25

1 3 5 7 9 11

T*

Rg

2 in

latt

ice

un

its

sq

ua

red

ps= -0.5, pp=ss=0

ps= 0.5, pp=ss=0Rg

2()

11

ss

pp

ps

ss

nopAe)(

lattice cubic simple for the 62

Anop

nopssnop

ETZ

z

UNz

E

T

ZTkU

TZ

mE

pol2Bpol

Bpol

ln

)(

)(

Chamber A, temperature TA Chamber B, temperature TB

)()(functionpartition total chambers ginteractin-non BpolAnop TZTZ

12

TTTTTkT

TTqq

qq

TZTZTZTZ

TZTZq

TZTZTZTZQ

TT

BA

ABBA

BA

ABBA

)(,/

with offunction a as /)(

:Plot

1

)()()( )(

)( )(

:A chamber,warmer

in thepolymer thefind y toProbabilit

)()()( )(

:states of Sum

be? likely to moreit is where

chambers,between move

tofree ispolymer theif and If

:Question

B

coldhot

hotcold

noppolnoppol

noppolhot

noppolnoppol

BA

Np=51, N=250*Np, Ns=N-Np, T*=0.01

-8

-6

-4

-2

0

2

4

6

8

1 3 5 7 9 11

T*

(qh

ot-q

cold

)/

T*

ps=0.5, pp=ss=0

chain more likely in hot chamber

ps= -0.5, pp=ss=0

chain more likely in cold chamber

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polnophot

polnop2B

hot

polnop2B

noppol

noppol

noppol

noppol

noppol

hot

for 2

1

exp1

1

)ln()ln(

)( )(

)()(ln

)( )(

)()(1

1

:Note

UUq

UUTk

Tq

UUTk

T

T

Z

T

ZT

TZTZ

TZTZ

TZTZ

TZTZ

q

BA

AB

BA

AB

Hence, the difference in internal energy between two boxes at the same temperature, one with and one without polymer, determines the probability to find the polymer in the warmer of two boxes at different temperatures “heat of transfer”

T, Unop T, Upol TA> TB TB

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Np=51, N=250*Np, Ns=N-Np, T*=0.01

10

15

20

25

1 3 5 7 9 11

T*

Rg

2 in

latt

ice

un

its s

qu

ared

ps= -0.5, pp=ss=0

ps= 0.5, pp=ss=0

Np=51, N=250*Np, Ns=N-Np, T*=0.01

-8

-6

-4

-2

0

2

4

6

8

1 3 5 7 9 11

T*

(qh

ot-q

cold

)/

T*

ps=0.5, pp=ss=0

chain more likely in hot chamber

ps= -0.5, pp=ss=0

chain more likely in cold chamber

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Poly (ethyleneoxide) in ethanol/water Poly (ethyleneoxide) in ethanol/water

H O CH2 CH2 OHn

E.E. Dormidontova, Macromolecules, 35 (2002), 987

H2O

Ethanol: not a good solvent at room-temperature

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0.0 0.2 0.4 0.6 0.8 1.0-0.5

0.0

0.5

1.0

1.5

2.0S

T / K

-1

weight fraction water

PEO in ethanol/waterPEO in ethanol/water

PEO moves to hot side

PEO moves to cold side

TDFRS results

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light scattering results

The addition of water expands the chains

5 10 15 20 2522

24

26

28

30

32

34

36

RG

2

weight fraction of water

aggregation

guide for the eye

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Observations regarding PEO in ethanol/water

ethanol/water highly miscible

PEO in ethanol immiscible at room temperature, chains collapsedUCST phase diagram

PEO in water miscible at room temperature, chains highly extendedLCST phase diagramspecific interactionspressure dependence

PEO in ethanol/water

solubility increases (chains expand) with water content

for low water concentrations, ethanol is preferentially adsorbedat a water concentration of 19% by weight, a transition to preferential adsorption of water takes sets in

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Lattice model for PEO in ethanol/water

simple cubic lattice

Np = number of contiguous sites for polymer

Ns = number of solvent sites

Nw = number of water sites

Nv = number of void sites

Interaction energies:

pp , ss , ww from pure component PVT propertiesws geometric mean approximationps PEO/ethanol, poor solvent conditionpw,n pw,s PEO/water, non-specific (poor solvent) specific (very attractive)

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Canonical Partition Function

rspwnpwpsspp

w

Ewnm

s s

wpnn

w

pn

m w

nwN

sN

wNNnN

s

n

wN

NnN

w

nmcNZ

eee5e

)(6)(

,,)(

][][

ionapproximat mixing randomin nsinteractiosolvent -solvent todueenergy

ethanol

waterby occupied sitesneighbor nearest ofnumber

chain theof sitesneighbor nearest available ofnumber 224

r

pn

E

s

w

mNn

shell n.n. in the water of conc. avg.shellneighbor nearest in the sites

water with state afor y probabilit

contactspolymer -polymer with

on conformatichain afor y probabilit

summations Partial

site latticeper volume,ln

from Pressure

,1

2

,

nnw

gm

NN

cw

p

Rm

p

vN

ZPv

ws

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T = 293 K

P 0.1 Mpa

5g/L of PEO

Np = 17

Lattice model calculations reproduce:

Chains expand with increasing water content.

Preferential adsorption changes from ethanol to water at 19 % water wt

Note: thermodynamic properties of the pure components, solvent quality of the solution, and preferential adsorption are used to determine the system-dependent parameters.

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Chamber A, temperature TA Chamber B, temperature TB

Chambers are non-interacting ZAZB = partition function for given configuration

Set T = 10-3 K and NA = NB = N/2

Q

Q

wspiQQQQNNZNZQ

A

BABAAii

B

N

Ai

A

Ai

0

00in polymer in polymer ][

A)(chamber chamber hot in thepolymer thefind y toProbabilit

},,{,})({})({

:states of Sum

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Lattice model results for the probability to find the polymer in the warmer/colder chamber

properties onalconformation basedt coefficienSoret for estimate

)1(

1 PEOm,

PEOm,PEOm,LatticeT,

TS

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Comparison with experiment

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Discussion• In general, the better the solvent quality the higher the probability to find the polymer

on the cold side.

• PEO moves to the cold side in ethanol/water with high water content

• PEO moves to the hot side in ethanol/water with low water content

• PVA moves to the hot side in water (Giglio and Vendramini, 1977)

• also seen in calculations of the Soret coefficient of PEO in pure water and ethanol

• In model calculations, the trend is reversed if the polymer-polymer interactions are very attractive

• Preferential adsorption is an important indicator for the behavior of the Soret coefficient

Acknowledgements:The authors would like to thank Mark Taylor and Simone Wiegand for many helpful discussions. Financial support through the National Science Foundation (DMR-013704), the Ohio Board of Regents, the Research Corporation (CC5228), and the Petroleum Research Fund (#36559 GB7) is gratefully acknowledged.

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