Thermophoresis
In a temperature gradient , particles migrate toward the cold or the warm side, and their stationary concentration is not uniform. Thermophoresis is usually described as a drift that competes with diffusion. The zero-flux condition then gives
where the Soret coefficient is the ratio of the thermodiffusion and diffusion coefficients. Particles accumulate on the cold side when and on the warm side when . Equation (1) is phenomenological: it does not say which properties of the particles set the sign of .
Internal states
Let the particles have internal states : conformations of a polymer, isomers of a molecule, or the free and bound forms of an enzyme. The probability of state obeys the reaction–diffusion equation
where is the diffusion coefficient of state , here independent of temperature, and is the rate from to . The rates obey local detailed balance, , so the reactions alone would relax to the Boltzmann distribution at the local temperature. Experiments measure the total .
Particles linger where they are slow
When the reactions are much faster than diffusion, every point relaxes to local equilibrium. The total then diffuses with the equilibrium average , and zero flux requires . Particles therefore linger where they diffuse slowly, and Eq. (1) holds with the chemical Soret coefficient
The temperature dependence of comes from the average over the internal states, not from the . Thermophoresis therefore requires that the diffusion coefficients of the states differ.
Energy and diffusion
For thermally activated rates, the chemical Soret coefficient is a covariance over the local equilibrium:
When high-energy states diffuse faster, particles accumulate on the cold side; when they diffuse slower, on the warm side. The panel of states traces the equilibrium mean from the cold end to the warm end. Warming always raises the mean energy, and the mean moves toward faster or slower states with the sign of the covariance.
Slower reactions
Equation (2) can be solved for any rates. For two states, zero flux of alone, with currents of the states allowed, gives
with . Thermophoresis persists away from the fast-reaction limit, as long as the diffusion coefficients of the states differ. The phenomenological Eq. (1) is the fast-reaction limit, which corresponds to local equilibrium.
Hidden currents carry heat
Zero flux holds for only. The states keep flowing, and the reactions close their currents. Heat is absorbed on the warm side and drives particles to high-energy states. Diffusion moves them to the cold side, where they release heat and return to low-energy states. These currents, hidden in Eq. (1), are a thermodynamic necessity for heat transport from warm to cold. For two states with fast reactions, the heat current is
Velocities are internal states too
A particle without internal states responds as well, because its diffusion coefficient follows the Einstein relation . When velocities relax much faster than positions, the Kramers equation for reduces to , and
The velocity plays the role of the internal state, and its energy, the kinetic energy, grows with transport. The currents now close in velocity space: particles slower than drift toward the warm side and faster ones toward the cold side. The energy current they carry, , recalls Fourier's law.
A change of sign
When every state obeys the Einstein relation, , the two contributions add:
Take a polymer whose folded state has energy 0 and whose unfolded state has energy , diffuses slower, , and has times more microstates. With ,
The chemical part is negative and dominates only in a range of temperatures. As the temperature rises, the Soret coefficient changes sign from positive to negative and back.
Dimers
Interactions between particles allow complexes. For dimer formation, , the monomer and dimer concentrations obey
In the fast-reaction limit is again uniform, with the average weighted by mass, . Because the fraction of dimers depends on the concentration, so does the Soret coefficient. Dimers are larger than monomers, so when warmth breaks them apart the total concentration is higher on the cold side.