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Thermodiffusion in Polymer Solutions Jutta Luettmer-Strathmann Department of Physics, The University of Akron, Akron, OH 44325-4001, USA. T A. Introduction Thermodiffusion in polymer solutions Single polymer chain in an incompressible solvent Incompressible two chamber system - PowerPoint PPT Presentation
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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
2
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
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
14
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
17
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
18
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
21
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
23
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
24
Comparison with experiment
25
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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