Thermophoresis

:: Section 2

Thermophoretic Force and Velocity

II. Large Particles

For particles whose diameters are larger than the gas mean free path (0.066 μm), the mechanism is different from that of the smaller ones. It is more complicated because a temperature gradient is established within the particle. This gradient (within the particle) affects the temperature gradient in the gas immediately surrounding the particle. Both gradients (within the particle and surrounding the particle) are influenced by the thermal conductivity of the particle, k_p, and the thermal conductivity of the air, k_a. The net result is that the particle still receives more momentum from the gas molecules on the hot side than on the cool side, and therefore the net force is still in the direction of decreasing temperature. However, the particle with a higher thermal conductivity will experience lower forces since the temperature gradient between the particle and surrounding air is reduced due to the high thermal conductivities. In other words, the energy received from the hot side of a conducting particle can be transferred to the cold side quickly and warms up the gas molecules on the cold side, thus minimizing the overall temperature gradient of the air and decreasing the effect of thermophoresis.

The thermophoretic force of the larger particle can be expressed as:

where H is a coefficient that accounts for the differences in thermal conductivity and size effect (λ/d_p):

For a large particle (d_p >> λ), H can be further reduced down to the ratio of the conductivities, k_a/k_p. That is:

As you can see, the thermophoretic effect strongly depends on the conductivity of the gas and the aerosol.

If the thermophoretic force is equated to the Stokes's drag force, the thermophoretic velocity can be obtained as:

where C_c is the gas slip correction factor.

1. Which kind of material will experience a larger thermophoretic force? A copper particle or a wood dust particle? 2. How can we make a "large particle" of a given size and conductivity move faster by thermophoresis?