A nice discussion of radiometers and photophoresis is found in the book:
The Kinetic Theory of Gases With an Introduction to Statistical Mechanics
by E.H. Kennard MCGraw Hill 1938.
The following material is from the text of section 188 ; The Radiometer and
Photphoresis. The Figures and equations are not included . If anyone
wants to see them I can send to individual persons the Pict files of these
items.
SEC. 188] PROPERTIES OF GASES AT LOW DENSITIES 333
188 The Radiometer and Photophoresis.
It was discovered by Fresnel in 1825 that a small body suspended in a gas
is sometimes set into motion when light falls upon it. The effect was
exhaustively studied by Crookes (1874-1878) and has formed the subject of
numerous recent investigations. Often the body in question takes the form
of a wheel carrying vanes blackened on one side, which revolves
continuously when illuminated; or, to measure the force, a light vane may
be mounted on one end of a crossbar, with a counterpoise or another vane at
the other end, and the bar may then lie suspended from a torsion fiber so
that its deflection can be read with mirror and telescope. An analogous
motion of microscopic particles suspended in a gas was observed in 1914 by
Ehrenhaft and was called by him "photophoresis." Some particles move toward
the light, others against it.
The laws governing all of these phenomena appear to be substantially the
same. The force is found to be strictly proportional to the intensity of
illumination, so that mechanical devices of the sort described can be used
to measure a beam of radiation; for this reason thcy have come to be called
radiometers. With increasing pressure, the force rises to a maximum at a
pressure of about 1 mm Hg in the case of a disk of ordinary size, or at
several hundred millimeters of Hg in the case of photophoresis, and then
decreases rapidly.
It has been pretty well established that in all cases the light acts by
heating the suspended body; the same effects can, in fact, be produced by
establishing in the gas such a temperature gradient as will give rise to
the same temperature differences at the surface of the suspended body. The
general rule is that hot surfaces hehave as if repelled by the gas. The
movements toward the light which are often observed in photophoresis are
ascribed to greater heating of the far side of a transparent particle; this
latter effect has been successfully imitated with a radiometer carrying
disks of molybdenite, one of whose surfaces was fresher than the other.
Various opinions have been expressed from time to time as to the origin of
the radiometric force. A tempting hypothesis at first sight is that it is
due to the reaction from gaseous molecules rebounding with higher
velocities from a hot surface than from a cold one; but this is quickly
seen to be untenable when we reflect that such molecules, upon reentering
the gas, must drive it back and thereby thin it out until uniformity of
pressure is reestablished, whereupon the force on the hot surface will
become the same as on the cold one and no radio-metric action can occur.
The cause must, therefore, be sought in some secondary action. The effect
has very commonly been regarded as occurring at the edge of the radiometer
disk, where conditions in the gas must be far from uniform; experiments
designed to show that it is simply proportional to the length of the
perimeter failed, however, to yield this result. Recent theoretical and
experimental studies have now made it pretty clear that most, if not all,
radiometric phenomena are due, in one way or another, as Maxwell suggested
in 1879, to the thermal creep of the gas over an unequally heated solid (or
liquid) surface, as described in Sec. 184 above.
=46IG. 78.-Production of the radiometer effect.
It can be seen easily that this creep must give rise to forces on the
surface whenever the resulting flow of gas is hindered in any way. In a
simple two-vane radiometer, for example, the gas will creep around the
edges toward the centers of the blackened and therefore heated surfaces,
and must then flow out and around somewhat as suggested in Fig. 78(a), in
which the vanes VV are supposed to be transparent but blackened on one
side; this circulatory motion is then hindered by viscosity, and
consequently the gas accumulates somewhat over the blackened surfaces and
exerts a slightly increased pressure on transparent particle; this latter
effect has been successfully imitated with a radiometer carrying disks of
molybdenite, one of whose surfaces was fresher than the other.
Various opinions have been expressed from time to time as to the origin of
the radiometric force. A tempting hypothesis at first sight is that it is
due to the reaction from gaseous molecules rebounding with higher
velocities from a hot surface than from a cold one; but this is quickly
seen to be untenable when we reflect that such molecules, upon reentering
the gas, must drive it back and thereby thin it out until uniformity of
pressure is reestablished, whereupon the force on the hot surface will
become the same as on the cold one and no radio-metric action can occur.
The cause must, therefore, be sought in some secondary action. The effect
has very commonly been regarded as occurring at the edge of the radiometer
disk, where conditions in the gas must be far from uniform; experiments
designed to show that it is simply proportional to the length of the
perimeter failed, however, to yield this result. Recent theoretical and
experimental studies have now made it pretty clear that most, if not all,
radiometric phenomena are due, in one way or another, as Maxwell suggested
in 1879, to the thermal creep of the gas over an unequally heated solid (or
liquid) surface, as described in Sec. 184 above.
It can be seen easily that this creep must give rise to forces on the
surface whenever the resulting flow of gas is hindered in any way. In a
simple two-vane radiometer, for example, the gas will creep around the
edges toward the centers of the blackened and therefore heated surfaces,
and must then flow out and around somewhat as suggested in Fig. 78(a), in
which the vanes VV are supposed to be transparent but blackened on one
side; this circulatory motion is then hindered by viscosity, and
consequently the gas accumulates somewhat over the blackened surfaces and
exerts a slightly increased pressure on these and so pushes them back, thus
tending to produce revolution about the suspension S. If, on the other
hand, the vanes are alike on both sides but are given a cup-shaped form, as
in Fig. 78b, or are fitted with points, the edges of the cups or the points
are observed to move toward the light. Presumably these parts are more
effectively cooled by the gas than are other parts, so that a circulation
is set up as suggested in the figure, and those eddies which reach out to
the surroundings are effective by reaction in moving the vanes. Most
practical cases can readily be understood in this way.
The existence of such streams in the gas as we have here postulated was
shown directly by Gerlach and Sch=FCtz. [Gerlach & Sch=FCtz, Zeits. Physik, =
78,
43, 418 (1932); 79, 700 (1932); 81, 418 (1933)].
They suspended a tiny vanelet near the radiometer and observed that it
became deflected, presumably by the action of the streaming gas, in the
right direction. Another interesting experiment pointing to the same
conclusion is that of Czerny and Hettner, [Czerny and Hettner, Zeits.
Physik, 30, 258 (1924)].
who mounted a movable disk parallel to another disk along which a
temperature gradient was maintained (Fig. 79).
=46ig. 79. - A force F due to thermal creep
They observed that a tangential force acted on the movable disk in such a
direction that it could be explained as arising from viscous drag by the
gas as it creeps along the unequally heated disk.
189. The Quantitative Theory of Radiometer Action. Qualitatively the creep
theory of radiometric action is completely successful. A quantitative
calculation of the force, however, presents, unfortunately, a difficult
problem. One has first to solve the thermal problem in order to find the
distribution of temperature, which is determined, under the given
conditions of illumination or of boundary temperature, by the conduction of
heat through the gas and through the disk itself; allowance must also be
made, if accuracy is desired, for convection of heat by the creep motion
itself. Then one has to solve the hydro-dynamical problem of the streaming
as determined by the velocity of creep as a boundary condition; and,
finally, from this the total pressure on the disk is found by integration.
The complete problem has been solved only for the ideal case of an
ellipsoidal disk, circular in principal outline but of elliptical cross
section, which, if thin, should present some approximation to a flat disk.
The best treatment is that of Epstein [Epstein, Zeits. Physik, 54, 537
(1929)].