Molecular thermodiffusion (thermophoresis) in liquid mixtures
Semen N. Semenov
1
and Martin E. Schimpf
2
1
Institute of Biochemical Physics RAS, 117977 Moscow, Kosygin Street 4, Russia
2
Department of Chemistry, Boise State University, Boise, Idaho 83725, USA
Received 20 April 2005; published 18 October 2005
Thermodiffusion thermophoresis in liquid mixtures is theoretically examined using a hydrodynamic ap-
proach. Thermodiffusion is related to the local temperature-induced pressure gradient in the liquid layer
surrounding the selected molecule and to the secondary macroscopic pressure gradient established in the
system. The local pressure gradient is produced by excess pressure due to the asymmetry of interactions with
surrounding molecules in a nonuniform temperature field. The secondary pressure gradient is considered an
independent parameter related to the concentration gradient formed by volume forces, calculated from the
generalized equations for mass transfer. Values of Soret coefficients for mixtures of toluene and n-hexane are
calculated using parameters in the literature. When the molecules are assumed to be similar in shape, the
calculated Soret coefficients are lower than the empirical values found in the literature. However, by introduc-
ing an asymmetry parameter, which is calculated from independent measurements of component diffusion in
the literature, very good agreement is obtained.
DOI: 10.1103/PhysRevE.72.041202 PACS numbers: 66.10.Cb
I. BACKGROUND
When a liquid mixture is placed in a temperature gradient,
there is movement of the constituent components, generating
a concentration gradient. This coupling between temperature
and concentration gradients is known as thermodiffusion or
the Ludwig-Soret effect. Since its discovery by Ludwig 1
and the first systematic investigations of thermodiffusion in
liquid mixtures by Soret 2, the effect has been subject to
numerous experimental and theoretical studies. Investiga-
tions of the Ludwig-Soret effect in liquid mixtures typically
involve measurements of a so-called thermal diffusion coef-
ficient, an ordinary mass diffusion coefficient, or a ratio of
the two that is referred to as the Soret coefficient. Tools used
to measure the Soret coefficient in liquid mixtures include
the thermogravitational column 3 and thermal field-flow
fractionation TFFF4. Thermal diffusion coefficients are
measured by forced Rayleigh scattering TDFRS5 and
beam deflection methods 6. A variety of techniques are
used to measure mass diffusion coefficients, including
TDFRS and dynamic light scattering 7. Data provided by
these methods have been used to compare values obtained by
molecular dynamic MD simulations 8,9. A summary of
the information obtained by these methods can be found in
Refs. 10,11.
TFFF has proven to be particularly adept at measuring
the Soret coefficients of dissolved polymers and suspended
particles. In one of the more comprehensive studies on
polymer thermodiffusion 12 by TFFF, the independence
of thermodiffusion on chain length and branching configura-
tion was demonstrated, as predicted by scaling
considerations 13,14.
In a later study of polymers 15 dissolved in binary
solvent mixtures, the dependence of thermodiffusion on the
relative concentration of solvent components in specific
mixtures was found to be nonlinear. The curvature in these
relationships, which can be either positive or negative,
provides evidence for the existence of multiple forces
affecting the thermodiffusion of dissolved solutes in liquid
mixtures.
The independence of polymer thermodiffusion on chain
length and branching configuration means that the polymer
chain moves at the same velocity as that of the individual
monomer units mers, at least for homopolymers. Therefore,
we have modeled polymer thermodiffusion using a similar
approach as that used for particle thermophoresis 16. This
hydrodynamic approach, which has been used to explain the
thermodiffusion of hydrophobic homopolymers in pure sol-
vents 17, considers the molecules surrounding a selected
mer as a continuous medium, while the selected mer is con-
sidered a solid particle suspended in that medium. The flow
of liquid around the particle is caused by a local pressure
gradient in the surface layer of the particle, as defined by the
Navier-Stokes equation
u =-
loc
+ f
loc
, 1
where u is the velocity of the liquid,
loc
is the local pressure
distribution around the particle due to its interaction
with molecules of the liquid, is the dynamic viscosity
of the liquid, and f
loc
is any local volume force in the liquid
around the particle. In a temperature or concentration gradi-
ent, the local pressure distribution is not uniform due to
asymmetry in the distribution of molecules around the
particle. The same asymmetry also causes a local volume
force on the particle. The validity of this approach does not
depend on the size of the particle, provided the size is com-
parable or larger than the solvent molecules, because the
model is based on the hydrodynamic motion of liquid in the
layer surrounding the particle. However, the model’s validity
does depend on the frequency of intermolecular collisions
being great enough to consider the solvent as a continuous
medium.
PHYSICAL REVIEW E 72, 041202 2005
1539-3755/2005/724/0412029/$23.00 ©2005 The American Physical Society 041202-1