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Unusual Thermal Diffusion in Polymer Solutions

Berend-Jan de Gans,1,*Rio Kita,1,†SimoneWiegand,1,2,‡and Jutta Luettmer-Strathmann3,x

1Max Planck Institut fu ¨r Polymerforschung, Ackermannweg 10, D-55128 Mainz, Germany

2Forschungszentrum Ju ¨lich GmbH, IFF-Weiche Materie, D-52428 Ju ¨lich, Germany

3Department of Physics, University of Akron, Akron, Ohio 44325-4001, USA

(Received 10 August 2002; published 9 December 2003)

Thermal diffusion forced Rayleigh scattering results on thermal diffusion of poly(ethylene oxide)

(PEO) in ethanol/water mixtures are presented. In water-rich solvent mixtures, PEO is found to migrate

towards regions of lower temperature. This is typical for polymer solutions and corresponds to a

positive Soret coefficient of PEO. In solvent mixtures with low water content, however, the polymer is

found to migrate towards higher temperatures, corresponding to a negative Soret coefficient of PEO in

ethanol-rich solutions.To our knowledge, this is the first observed sign change of the Soret coefficient of

a polymer in solution. We also present a simple lattice model for the polymer solvent system and

calculate Soret coefficients with statistical mechanics methods. The calculated values agree qualita-

tively with the experimental results.

DOI: 10.1103/PhysRevLett.91.245501PACS numbers: 61.25.Hq, 61.43.Bn, 66.10.Cb

A temperature gradient applied to a fluid mixture gen-

erally induces net mass flows which lead to the forma-

tion of concentration gradients. This effect is known

as thermal diffusion or the Ludwig-Soret effect [1,2]. In

the stationary state where the mass flows vanish, the

magnitude of the effect is described by the Soret co-

efficients ST;i:

ST;i? ?

1

ci0?1 ? ci0?

rci

rT;

(1)

where ciis the mass fraction of component i, ci0is its

equilibrium value, and where T is the temperature. ST;i

is positive if component i moves to the low temperature

region. Since the massfractions add up to unity,P

a K-component mixture has K ? 1 independent Soret

coefficients.

Typically, the Soret coefficient of the heavier compo-

nent of a binary liquid mixture is positive. This is not

always the case, however, and the Soret coefficients in

some low molecular weight liquid mixtures are known to

change sign [2]. A sign change of the Soret coefficient

was also observed in very recent thermophoresis experi-

ments on protein solutions [3]. While there is no theory

which reliably predicts the sign of the Soret effect in

liquid mixtures, approaches based on the ‘‘heat of trans-

fer’’ concept [2] suggest that molecular interactions play

an important role. In recent experiments, Debuschewitz

and Ko ¨hler [4] identified two distinct contributions to the

Soret coefficients of isotope substituted liquid mixtures

of benzene and cyclohexane. The first contribution, due

to differences in the molecules’ masses and moments of

inertia, was found to be independent of composition of

the mixture. The second, reflecting chemical differences

of the molecules, was found to vary with composition and

change sign, inducing a sign change of the total Soret

coefficients.

ici? 1,

With only two known exceptions, polymers in solution

conform to the rule that the heavier component migrates

to the colder regions of the fluid. Giglio and Vendramini

[5] found a negative Soret coefficient for poly(vinyl al-

cohol) in water.Very recently, we reported our first ther-

mal diffusion results for poly(ethylene oxide) (PEO) in

ethanol-rich ethanol/water mixtures which indicated that

the polymer migrates to the warmer region in these

mixed solvents [6]. In this work, we investigate system-

atically the Soret effect of PEO in ethanol/water mix-

tures. We expect interesting thermal diffusion properties

for this system, since the interactions between the poly-

mer and the two solvents are very different. Hydrogen

bonding makes water an excellent solvent for PEO at

room temperature while PEO is insoluble in ethanol.

Because of the biocompatibility of PEO, this system is

also relevant for biological applications (cf. [7]).

Experiment.—We measured Soret coefficients with a

holographic grating technique called thermal diffusion

forced Rayleigh scattering (TDFRS) with heterodyne

detection and active phase tracking [8]. The principle of

TDFRS is analogous to ordinary forced Rayleigh scatter-

ing: An intensity grating is created by the interference of

two laser beams. A trace amount of inert absorbing dye

added to the sample converts the intensity grating within

microseconds into a temperature grating, which is con-

verted into a concentration grating within milliseconds

by the effect of thermal diffusion. Both temperature and

composition grating contribute to a refractive index grat-

ing readout by Bragg diffraction of a third laser beam.

For a binary mixture, the heterodyne signal intensity

reflects a single concentration mode whose time constant

and amplitude are related to the mass-diffusion and Soret

coefficient of the mixture, respectively [8]. For a polymer

in a mixed solvent, there are two concentration modes

with widely separated time constants. The normalized

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heterodyne signal intensity ?het?t? for the time after the

intensity grating has been switched on is derived for the

PEO/ethanol/water system in Ref. [9] and given by

?het?t? ? 1 ?

X

i?0;1

?@n

@T

??1

i

?@n

@c

?

iSTici?1 ? ci?

? ?1 ? e?q2Dit?;

(2)

where t is the time, n is the index of refraction, and q

is the grating wave number. Diand ST;i, i ? 0;1, denote

the mass-diffusion and Soret coefficients, respectively,

where i ? 0 refers to the polymer in the mixed solvent

while i ? 1 refers to the solvent mixture. The contrast

factors multiplying the exponential terms contain deriva-

tives of the index of refraction, which are determined

independently with a Michelson interferometer operating

at a wavelength of 632.8 nm. Hence, mass-diffusion and

Soret coefficients of the ternary PEO/ethanol/water mix-

tures can be determined from the time constants and

amplitudes of the normalized heterodyne signal.

The PEO in our experiments had a molecular weight

of Mw? 2:65 ? 105gmol?1with a polydispersity PD ?

1:1. All samples were semidilute solutions containing

5:0 ? 0:1 gL?1PEO. Fifteen different solvent mixtures

were studied, the water content varying between 5% and

100% by weight. A trace amount of dye (quinizarin,

Sigma-Aldrich) wasaddedtothesamples (opticaldensity

1–2 cm?1). As for PEO in pure water basantolyellow 215

(BASF) was used. Adsorption of the dyes on PEO does

not occur as their absorption spectrum remains un-

changed [6,9].

Figure 1 shows typical normalized heterodyne signals

as a function of time obtained for three different solvent

compositions. The inset shows the two clearly separated

decays for the 50 wt% mixture on a logarithmic time

scale. Analysis of the amplitude and the decay time of the

fast mode leads to the same values for the Soret coeffi-

cient of water as obtained from measurements of binary

ethanol/water mixtures [9]. This confirms that the fast

mode reflects the thermal diffusive behavior of the binary

solvent mixture while the slow mode corresponds to the

establishment of a concentration gradient of PEO in the

solvent mixture. For long times, the heterodyne signals in

Fig. 1 reflect the slow mode, which is seen to decrease

with time for solvent compositions of 15:02 wt% and

50 wt% water. This indicates a negative Soret coefficient

of PEO in mixtures with low water content. In contrast,

the signal of PEO in pure water shows an increase with

time, corresponding to a positive Soret coefficient of PEO

in water. Figure 2 shows the Soret coefficient of PEO in

ethanol/water as a function of water weight fraction. We

observe a sign change of STof PEO in ethanol/water

mixtures at a weight fraction of 83%. To our knowledge,

this is the first observation of a sign change of the Soret

coefficient of a polymer in solution. In pure ethanol/water

mixtures withup to 25 wt%water,the signal contribution

of the Soret effect is negligible compared to the thermal

contribution [6,10]. Neglecting the solvent contribution to

the thermal diffusion leads to 6%–10% smaller values for

STof PEO in ethanol/water mixtures, which is just barely

outside the error bars [6]. In the middle concentration

range, the influence of the solvent mixture is much

stronger and the deviations exceed 30% and more [9].

Theory.—In order to investigate the origin of the un-

usual experimental findings, we have developed a simple

lattice model for a polymer chain in a mixed, compress-

ible solvent [11]. We consider a simple cubic lattice of N

sites. The polymer chain occupies N0contiguous sites,

while the two types of solvent particles occupy N1and N2

single sites, respectively, and N3sites remain unoccupied,

N ?P3

neighbor sites are described by interaction energies ?ij,

i;j 2 f0;1;2g. In aqueous solutions, hydrogen bonding

between PEO and water plays an important role

(cf. [12]). In order to account for these specific interac-

tions, each elementary cube representing water is as-

sumed to have one special face. If this face is exposed

to a polymer segment, the interaction energy is ?01;s

(strongly attractive); otherwise it is ?01;n(nonspecific).

i?0Ni. Interactions between occupied nearest-

FIG. 1.

a solution of PEO in ethanol/water [15:02 wt% H2O (?),

50wt% (4), 100wt% (?)]. The inset shows the two decays for

50 wt% mixture ethanol/water versus logarithm of the time.

Typical normalized heterodyne diffraction signal for

FIG. 2.

function of water weight fraction.

Soret coefficient ST of PEO in ethanol/water as a

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From an exact enumeration of all self-avoiding random walks of length N0? 1 on a simple cubic lattice (cf. [13]), we

determine the number c?m? of chain conformations with m segment pair contacts and the average chain dimensions as a

function of m. In this work N0? 17. Under a random mixing approximation for all but the polymer contacts, the

canonical partition function for the system can be written as

Z ?N

X

m

c?m?

X

?n1?

6N1?n1

?nn

n1

??N ? nn? N0

N1? n1

?

?

X

?n2?

?nn? n1

n2

??N ? nn? N0? ?N1? n1?

N2? n2

?

? e???m?00?n2?02??5e???01;n? e???01;s?n1e??Er;

where ? ? 1=kBT, kB is Boltzmann’s constant, nn?

4N0? 2 ? 2m is the number of nearest-neighbor sites of

the polymer which are occupied by ni, i 2 f1;2;3g, sol-

vent particles and voids. The square brackets around the

summation indices indicate that the summation is per-

formed consistent with the available nearest-neighbor

sites and the total filling of the lattice. The energy Er

denotesthe contribution tothetotalenergyduetosolvent-

solvent interactions evaluated in random mixing approxi-

mation [14]. In the absence of the polymer, Z reduces to

the lattice fluid partition function of a compressible bi-

nary mixture of single site particles [14]. By performing

partial summations over the terms in Eq. (3), we compute

the average radius of gyration R2

used to characterize solvent quality (see Fig. 3). The

pressure P of the system is calculated from ?vP ?

?@lnZ=@N?Ni;i?3, where v is the volume of one lattice

site. In this work, we retain the system-dependent pa-

rameters determined in Ref. [11] except for ?ws, the en-

ergy of ethanol-water interactions. Originally, ?wswas

estimated from the geometric mean approximation. Here,

we determined a value of ?ws? ?3600 J=mol from a

comparison with tabulated values for the density of

ethanol-water mixtures [16], weighted to ensure a good

(3)

gof the chain, which is

fit at high water concentrations. In Fig. 3 we present

graphs for the radius of gyration of the PEO chain. The

chain expands (solvent quality improves) with increasing

water content of the solution, in qualitative agreement

with the experimental data [6] presented in the inset. For

PEO in ethanol, the chain dimensions increase with in-

creasing temperature while they decrease with tempera-

ture for PEO in water, in agreement with observed

changes in solvent quality cf. Ref. [7].

In order to investigate the Soret effect, we consider a

system divided into two chambers of equal size that are

maintained at slightly different

Particles are free to move between the chambers, which

do not otherwise interact. If the pressure difference be-

tween the chambers is small enough to be neglected, the

Soret coefficient can be determined from the difference

in composition of the solutions in the two chambers [1].

Consider a single-chain system that is divided into two

chambers, A and B, with slightly different temperatures,

TA> TB. Under the assumption that the chambers are

noninteracting, the partition function of thewhole system

is a product of the partition functions of the individual

chambers, ZAZB. The chambers are represented by latti-

ces with Na, a 2 fA;Bg sites and occupation numbers Na

i 2 f0;...;3g, where N ? NA? NBand NB

The temperature difference employed here, ?T ?

10?4K, is sufficiently small to neglect thermal expansion

and we set NA? NB? N=2. If the particles are allowed

to move freely between the chambers, the sum Q of all

two-box configurations is given by

temperatures [11].

i,

i.

i? Ni? NA

Q ?

X

?NA

i?

ZA?fNA

ig?ZB?fNi? NA

ig? ? Q0;A? Q0;B;

(4)

where the square brackets have the same meaning as

previously. Q0;Aand Q0;Brepresent the sums of states

with the polymer in chambers A and B, respectively.

The fraction Q0;A=Q is the probability for finding the

polymer in chamber A. A Taylor expansion shows that,

to first order in the temperature difference ?T, this proba-

bility is determined by a difference in internal energy

Q0;A?1

2’ ?1

4

hUnopi ? hUpoli

kBT

?T

T:

(5)

Here hUnopi and hUpoli are the internal energies of two

chambers at the same temperature, averaged over all

FIG. 3.

in ethanol/water mixtures. The left panel shows calculated

chain dimensions as a function of solvent composition at

temperature T ? 293 K, pressure P ? 0:1 MPa, and a PEO

concentration of 5 gL?1. Experimental data from static light

scattering [6] are presented in the inset. The right panel shows

the temperature variation at constant pressure of the calculated

chain dimensions of PEO in the two pure solvents ethanol and

water. The dashed line indicates the chain dimensions, R2

of the isolated 17 bead chain at the ? temperature of the infinite

chain [15].

Radius of gyration squared, R2

g, of short PEO chains

g????

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configurations of particles, where the polymer is confined

to one of the chambers. Results for Q0;A=Q ? 1=2 calcu-

lated from the sum of states are presented in Fig. 4. For

low water concentrations, the polymer is more likely to be

found in the higher temperature chamber while the op-

posite is true for high water concentrations. The inset

of the figure shows lattice model estimates, ST;Lattice, for

the Soret coefficients of PEO obtained by monitoring the

composition of each chamber during calculation of the

sum of states, Eq. (4). Our simple lattice model, which

takes only conformational contributions to thermodiffu-

sion into account, uses short chains with parameters to

emulate large chains, and neglects hydrogen bonding

between solvent molecules, cannot be expected to yield

quantitatively correct results. However, the calculated

values for ST;Lattice initially decrease with increasing

water content of the solvent, go through a minimum,

and exhibit a change in sign of the Soret coefficient at

high water content of the solution, in qualitative agree-

ment with the experimental data.

Discussion.—Our experimental and theoretical inves-

tigation of PEO in ethanol/water mixtures gives some

insight into conditions that lead to unusual thermodiffu-

sion in polymer solutions. The experimental data pre-

sented in Figs. 1 and 2 and the calculated results

presented in Figs. 3 and 4 illustrate the role of solvent

quality. Positive Soret coefficients are observed for PEO

in pure water, which is an excellent solvent, while nega-

tive Soret coefficients are observed for low water content

of the solution, that is under poorer solvent conditions.

The negative Soret coefficient reported [5] for poly(vinyl

alcohol) in water at 25?C agrees with our findings since

the system is very close to poor solvent conditions [17].

Our investigation leads us to expect other systems to

yield changes in the sign of polymer Soret coefficients.

For example, a solution of a copolymer in a single sol-

vent may change sign as a function of composition of

the copolymer, if the chains are composed of two types

of segments, one with highly attractive and one with

net repulsive segment-solvent interactions. For the PEO/

ethanol/water system studied here, hydrogen bonding

plays a most important role. The Soret coefficient of

PEO changes sign at a solvent composition, where large

structural changes occur in binary mixtures of ethanol

and water [18].We are currently investigating this aspect

in more detail.

The authors would like to thank Cindy Leppla,

Thomas Wagner,PetraRa ¨der,

Christine Rosenauer for assistance with the experiments

and Florian Mu ¨ller-Plathe and Mark Taylor for helpful

discussions. Financial support through the National

Science Foundation (DMR-0103704), the Ohio Board of

Regents (R5413), the Petroleum Research Fund (36559-

GB7), and the Research Corporation (CC5228) is grate-

fully acknowledged.

Beate Mu ¨ller, and

*Electronic address: bjdegans@gmx.de

†Electronic address: kita@mpip-mainz.mpg.de

‡Electronic address: s.wiegand@fz-juelich.de

xElectronic address: jutta@physics.uakron.edu

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FIG. 4.

in the higher-temperature chamber as a function of water

content of the solution for T ? 293 K, ?T ? 10?4K. The inset

shows the lattice model predictions for the Soret coefficient and

the experimental data reported in this work.

Excess probability Q0;A=Q ? 1=2 to find the polymer

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