Scaling theory of polymer thermodiffusion

The motion of a linear polymer chain in a good solvent under a temperature gradient is examined theoretically by breaking up the flexible chain into Brownian rigid rods, and writing down an equation of motion for each rod. The motion is driven by two forces. The first one is Waldmann’s thermophoretic force (stemming from the departure of the solvent’s molecular-velocity distribution from Maxwell’s equilibrium distribution) which here is extrapolated to a dense medium. The second force is due to the fact that the viscous friction varies with position owing to the temperature gradient, which brings an important correction to the Stokes law of friction. We use scaling considerations relying upon disparate length scales and omitting non-universal numerical prefactors. The present scaling theory is compared with recent experiments on the thermodiffusion of polymers and is shown to account for (i) the existence of both signs of the thermodiffusion coefficient of long chains, (ii) the order of magnitude of the coefficient, (iii) its independence of the chain length in the high-polymer limit and (iv) its dependence on the solvent viscosity.