Types of photophoresis
If the accommodation coefficient α of the surface of the
particles is α > 0 and the particle is heated by
radiation, a photophoretic force can act on the particle. Two different idealized
extremes can be distinguished: (a) Variation in temperature on the surface of the
particles (ΔT force) and (b) variation of accommodation and
constant temperature (Δα force):
Case (a): Constant accommodation coefficient but variation in surface
temperature (ΔTphotophoretic force)
Let us assume a spherical particle of diameter d consisting of a
lightabsorbing material with a thermal conductivity
k_{p} and immersed in a gas with pressure
p, viscosity η, and molecular weight
M and illuminated by visible or infrared light. The surface
facing the radiation source is warmer than the backside (see Fig. 3); due to accommodation, the
molecules on the warmer side leave the surface faster, resulting in a force away
from the light source (positive photophoresis). This was found to take place,
e.g. for particles consisting of Au, Ag, Hg, Cd, K, Na, Mg, soot produced by the
combustion of turpentine, or camphor (Ehrenhaft, 1918).
View Details
 Fig. 3
Explanation of positive and negative photophoresis (Rubinowitz, 1920). The surface of a strongly
absorbing particle is hotter on the side facing the radiation, thus
photophoresis is in the direction of the radiation. A particle that
slightly absorbs light focuses the light to the rear side, which
consequently is hotter, thus the particle moves in the opposite
direction.

For a slightly absorbing particle, a force in the opposite direction has been
observed. The maximum absorption may be at the backside, causing the reversal of
the force. The early interpretation is the lens effect, see Fig. 3: The convex surface of the
particle acts like a lens, concentrating the rays on the backside (Rubinowitz, 1920), which becomes
hotter than the surface facing the radiation, resulting in a motion towards the
light source (negative photophoresis). For particles comparable to the
wavelength of light, the lens effect as a simplification may not be permitted,
but strictly applying Maxwell’s equation to an absorbing spherical
particle leads to the same results (Dusel et al., 1979; Figure 4.31 of Barber and Hill, 1990). Negative photophoresis has
been observed, e.g. for spheres of S, Se, J, Bi, Th, P, Pb, Te, As, Sb, for
tobacco or wood smoke particles (Ehrenhaft, 1918).
The direction of the photophoretic force is determined by the direction of the
radiation and is almost independent of the orientation of the particle, thus the
force is called spacefixed, the movement is called longitudinal photophoresis
and since the temperature difference causes the force, it is called
ΔT force. Although all particles perform Brownian
rotation, the rotation is slow compared to the time needed to establish the
temperature gradient within the particle (Fuchs, 1964, p. 62). The “ideal”
particles would have a low thermal conductivity and heat capacity.
In the free molecular regime, calculation of the force is simple: Molecules
impacting on the particle are reflected with a larger velocity due to
accommodation. Using considerations similar to the radiometer which was
considered above, Rubinowitz
(1920) determined the force on a spherical particle with diameter
d at gas pressure p as:
F
free
=
π
24
⋅
α
⋅
d
2
⋅
p
⋅
Δ
T
T
=
π
12
⋅
α
⋅
d
3
⋅
p
⋅
gradT
T
 (3) 
For the derivation in the continuum regime one has to consider the thermal creep
flow around the particle and the resulting viscous forces. In contrast to the
free molecular regime, the gas molecules around the particle have temperatures
similar to those on the surface of the particle (see Fig. 4). A gas molecule impacting on the surface from the
left (hot) side thus has a larger velocity than the molecule impacting from the
colder side. After accommodation, the molecules have a symmetric velocity
distribution with respect to the perpendicular direction of the surface,
therefore the warmer molecule coming from the left side transfers to the
particle a larger momentum to the right than the colder molecule to the left.
This results in a net force on the particle which directs from the hot to the
cold side. The gas loses this momentum, thus a creep flow around the particle
from the cold side to the hot side occurs simultaneously which has its maximum
flow at a distance to the mean free path of the gas (Knudsen, 1910a, b; Hettner, 1924; Fuchs, 1964,
p.57). Solving the hydrodynamic equations (similar to Stokes Law), the
following formula for the photophoretic force is obtained (Hettner, 1926)
F
cont
=
3
π
2
⋅
η
2
⋅
d
⋅
R
⋅
gradT
p
⋅
M
 (4) 
with
R the gas constant,
η the viscosity of the gas,
M the
molecular weight of the gas molecules,
p the gas pressure and
d the diameter of the particle.
View Details
 Fig. 4
Photophoretic force on a particle in the continuum regime. The particle
is hotter on the left side and in the immediate vicinity of the surface,
the gas molecules have the same temperature as the surface. Molecules
impacting on the particle from the left are faster than those coming
from the right, transferring a momentum to the right onto the particle.
This momentum is lost in the gas, thus a thermal creep flow of the gas
molecules to the left occurs.

Formula (3) is valid for the free
molecular range, i.e. the particles must be considerably smaller than the mean
free path of the gas molecules (λ = 66 nm at
standard conditions, with decreasing pressure the mean free path increases),
whereas formula (4) is only valid
for the continuum range, i.e. for particles much larger than
λ. Magnitudes influencing the force are the diameter of
the particle and the temperature gradient both proportional to the force in
formulas (3) and (4), the accommodation coefficient only
in (4) and the pressure in (3) and (4). The force is proportional to the pressure in the free
molecular regime (3) and
inversely proportional to it in the continuum regime (4), i.e. a maximum force occurs in
between, which is the most important range for photophoresis. Similar to
Stokes’ Law, an interpolation is used for the intermediate range, as
proposed by Hettner (1926):
1
F
=
1
F
free
+
1
F
cont
, or in detail:
F
=
π
4
⋅
d
2
⋅
η
⋅
α
⋅
R
M
⋅
T
⋅
gradT
p
p
0
+
p
0
p
with
p
0
=
6
η
d
RT
M
α
or
d
0
=
6
η
p
RT
M
α
 (5) 
The pressuredependent force for a given particle size has a maximum for
p = p_{0}, or for a given
pressure the maximum photophoretic force is for particle diameter
d_{0}. The pressure for the maximum photophoretic force
depends on the gas properties, the particle size and the accommodation
coefficient. For a pressure of 1 bar the optimum particle size is 229 nm for
α = 0.8 and 375 nm for
α = 0.3. For a pressure of 5500 Pa (20 km
a.s.l.), the corresponding diameters are 6.8 and 4.1 μm.
The viscosity of the gas, η, is obviously decisive for
the photophoretic force in (4).
The viscosity of a gas is mainly determined by the mean free path,
λ, of the gas molecules and the pressure. Using the
relation
η
=
8
M
9
π
RT
⋅
p
⋅
λ
(see, e.g. Hinds, 1982). Inserting this in (4), the size for the maximum
photophoretic force is obtained as
d
0
=
4
⋅
2
π
⋅
1
α
⋅
λ
=
3.19
α
⋅
λ
, therefore optimal photophoretic effects will be
found if the particle size is in the order of the mean free path of the gas
molecules.
Denoting the maximum force at p_{0} with
F_{0}, (5) can be written as
F
F
0
=
2
p
p
0
+
p
0
p
.
 (6) 
This formula has been tested by many authors. Various particle materials,
particle sizes, gases and pressures have been used. Usually the accommodation
coefficient and the particle size are not known exactly, whereas the temperature
gradient is completely unknown. The experimental procedure is quite elaborate,
since the motion of one and the same particle is measured for pressures varying
over three orders of magnitudes, lifting the particle up in between. Formula (6) contains only magnitudes
that are experimentally accessible. Since the size of the particle in relation
to the mean free path of the surrounding gas is important, it is also possible
to make photophoretic analogy experiments with centimetersized particles, as
long as the mean free path of the particles is large enough. This requires a
high vacuum but offers the advantage that the particle can be fixed, and the
force can be measured with a torsion balance. In this way many data points can
be gathered, without the fear of losing the particle during pressure change
(Schmitt, 1961). Compared to
the experiments with submicrometersized particles, the distance to the walls
of the container is much smaller, and wall effects cannot be neglected. A
summary of various experiments is shown in Fig. 5. Particles consisting of Te, Se (Mattauch, 1928), Se (Reiss, 1935), CdS (Arnold et al., 1980) and also 1cm
polystyrene particles at correspondingly low pressures (Schmitt, 1961), all show the same behavior for gases
such as H_{2}, A, O_{2}, N_{2}, CO_{2}. All data
demonstrate the maximum photophoretic force at an intermediate pressure.
View Details
 Fig. 5
Experimentally determined photophoretic forces as a function of pressure.
All data show a maximum photophoretic force at an intermediate pressure
as postulated by theory. The dark lines are results of model
measurements with cmsized spheres in vacuum, the points result from
measurements made with individual microor nanometersized particles
consisting of various materials, the gray line is theory (6).

Once the size and the photophoretic force of the particle is known, equation (4) can be used to estimate
the temperature difference between the hot and cold side of the particle. Ehrenhaft (1918) and Parankiewicz (1918) have measured
the photophoretic forces: They are in the order of 10^{−16} to
10^{–15} N for particle radii ≤ 100 nm of various
materials in N_{2} and Ar. The estimate for the temperature difference
across the particles is 10^{−4} to 10^{−2} K
(Hettner, 1926).
For calculation of the photophoretic force in (3) and (4), the
temperature gradient in the particle is needed. For this, both the inhomogeneous
heat production within the particle, the heat conduction to the surface, and the
momentum transfer at the surface has to be solved. Fuchs (1964, p. 58) remarks: “The
main difficulty in calculating the radiometric force on a particle is the
determination of the temperature gradient in the particle
itself.” For simple shapes such as spheres this appears
possible (see below), but many particles showing photophoresis are not spheres.
A first attempt for spheres has been made by Rubinowitz (1920) yielding
gradT
=
S
2
(
k
p
+
k
g
)
, with S the flux density of the
illumination and k_{p} and
k_{g} the thermal conductivities of the material
forming the particle and the gas. In the meantime, several investigators have
treated the problem in depth: Reed
(1977) for low Knudsen numbers, Yalamov et al (1976), Arnold and Lewittes (1982) close to the continuum
range, Akhtaruzzaman and Lin
(1977) for the free molecular range, Chernyak and Beresnev (1993) and Mackowski (1989) for the whole range
of Knudsen numbers.
Using all available theories, Rohatschek (1995) formulated an empirical model. For the free molecular
and the continuum regime, the force on a particle with diameter d illuminated by
a flux density S is obtained as
F
free
=
D
⋅
p
p
*
⋅
d
2
⋅
α
⋅
J
1
4
k
p
⋅
S
and
F
cont
=
D
⋅
J
1
2
k
p
⋅
p
*
p
⋅
d
2
⋅
S
 (7) 
with
D
=
π
2
π
3
κ
⋅
c
¯
⋅
η
T
and
p
*
=
1
d
3
π
k
⋅
c
¯
⋅
η
=
6
π
⋅
D
⋅
T
d
⋅
 (7a) 
With k_{p} the thermal conductivity of the particle,
J_{1} the asymmetry parameter for light absorption
(½ for opaque particles, J_{1} has the opposite
sign if absorption is mainly at the backside, e.g. for negative photophoresis),
D a factor only containing the gas properties, thus independent of particle size
and pressure, κ the thermal creep coefficient
(κ = 1.14 for a fraction of 0.1 of the
specularreflected molecules, Bakanov,
1992), and p* a reference pressure. (The
pressure for the maximum force is obtained by
p
0
=
2
/
α
⋅
p
*
) Combining the forces for the two regimes (7) as above, a formula for the
ΔT force is obtained for the entire pressure range:
1
F
=
1
F
free
+
1
F
cont
.
As an example, the ΔT
forces on particles at various pressures are plotted in Fig. 6. The following properties of
the particles have been assumed: Accommodation coefficient
α = 0.7, density
ρ_{p} = 1000 kg/m^{3}, thermal
conductivity k_{p} = 1
Wm^{−1}K^{−1}.
The pressures p* for each particle size are indicated
as data points on the curves. The maximum force and the decrease both for lower
and higher pressures are clearly visible. The thermal conductivity appears in
the denominator and thus strongly influences the photophoretic force. The value
chosen for the graph may be representative, but it depends greatly on the
particle’s material. For example, crystalline graphite (Pedraza and Klemens 1993) has a thermal conductivity of 1200
Wm^{−1}K^{−1}
parallel to the layer planes, it thus reduces the temperature difference in the
particle considerably as well as the photophoretic force. On the other hand
perpendicular to the layer plane, the conductivity is only 6
Wm^{−1}K^{−1}.
But agglomerates of carbon particles are more likely. For soot the thermal
conductivity has values of 3, or even below 1
Wm^{−1}K^{−1}.
Thus the photophoretic force very much depends on the microstructure of the
particle. A few data points for sulfur spheres are added to the graph on the
right side. The values have been measured by Parankiewicz (1918) for a pressure of 100 kPa;
unfortunately she did not specify the flux density of the illumination. Sulfur
has a thermal conductivity of 0.205
Wm^{−1}K^{−1},
thus the force should be five times higher than in the graph. On the other hand,
sulfur only slightly absorbs light and thus shows negative photophoresis,
consequently the force is considerably less since only a small fraction of the
light causes a temperature increase. So these data points can be considered
merely as an indication, nevertheless they still show the rapid drop of the
relative force for increasing particle size.
View Details
 Fig. 6
Photophoretic ΔT force as a function of pressure
for various particle diameters. The force is given relative to the
weight of the particle. The data points in the right graph are for
sulfur spheres (Parankiewicz,
1918)

Case (b): Variable accommodation coefficient on the surface but constant
temperature (Δαphotophoretic force)
As a simple model we assume a spherical particle of diameter d consisting of a
lightabsorbing material immersed in a gas with pressure p, and temperature T
(mean molecular velocity c̄). The particle’s
surface shall have a variation in the accommodation coefficient, e.g. be caused
by a difference in surface roughness or by different materials forming the
particle. For this simple model, one half of the sphere’s surface shall
have the accommodation coefficient α_{1}, the
other α_{2} (see Fig. 7). At the location of larger accommodation
coefficients, the molecules are reflected with a higher average velocity,
resulting in a thrust on the particle from the location of the higher
accommodation coefficient to the location of the lower one. This can be
considered similar to the jet of an airplane: this thrust has a fixed direction
relative to the airplane, approximately in the plane of symmetry of the
airplane; if the airplane changes direction the thrust vector changes as well.
Likewise the direction of the force on the particle is determined by the
orientation of the particle and is independent of the direction of the
illumination (the direction of the force is thus given in a body axis system, we
call it bodyfixed or particlefixed force). Upon a change of orientation of the
particle, the direction of the force likewise changes. Since all particles
perform Brownian rotation, the direction of the force is distributed randomly,
and the motion of the particles can be considered as a strongly enhanced
Brownian motion (see Fig. 1d).
This has been observed by various authors, e.g. Ehrenhaft (1907) and Steipe (1952), and was named the “trembling
effect”. It only occurred if no orienting field was present, therefore,
e.g. for iron particles, the earth magnetic field had to be compensated
completely. Obviously the average (net) force is zero, due to Brownian
rotation.
View Details
 Fig. 7
Photophoretic Δα force. A particle,
hotter than the surrounding air, has a larger accommodation coefficient
on the lower hemisphere. Consequently the molecules reflected by the
lower hemisphere are faster than the ones on the upper hemisphere,
resulting in an upwarddirected thrust.

The instantaneous Δαphotophoretic force on the
particle in the direction from higher to lower accommodation coefficients
amounts to (Rohatschek, 1995)
F
=
1
2
⋅
c
¯
⋅
γ
−
1
γ
+
1
⋅
1
1
+
(
p
/
p
*
)
2
Δ
α
α
¯
⋅
H
,
or
for
air
:
F
=
1
12
⋅
c
¯
⋅
1
1
+
(
p
/
p
*
)
2
Δ
α
α
¯
⋅
H
 (8) 
with Δ
α =
α_{2} –
α_{1},
α
¯
=
1
2
(
α
1
+
α
2
)
,
γ =
c_{p}/
c_{v} the ratio of the
specific heats of the gas, and
H the net energy flux
transferred by the gas molecules. This flux is obtained by an energy balance for
the particle: Absorbed light flux
A.S plus infrared
E_{abs} must equal the power
H
transferred to the molecules due to accommodation plus the emitted infrared
radiative flux
E_{emitt}. For pressures above 100 Pa,
there are sufficient molecules available to dissipate the absorbed radiation
thus
E_{abs} and
E_{emitt} can
be considered equal, and
A.S =
H. At
lower pressures, the particle’s temperature increases due to a lack of
sufficient energy transfer by the reflected gas molecules undergoing
accommodation, and thus
E_{emitt} rises, therefore
H <
A.S. To determine the absorbed
light flux, the absorption crosssection
A can be obtained,
e.g. for spherical particles by the absorption efficiency factor
Q_{a} of the Mie theory as
A
=
1
4
⋅
d
2
⋅
Q
a
(
Bohren
and Huffman, 1983).
As an example, the photophoretic Δα force (formula (8)) for particles having
Δα = 0.1 and α = 0.7,
consisting of an absorbing material with a refractive index m
= 1.6 – 0.66 i and a density of 2000
kgm^{−3}, illuminated by 1000
Wm^{−2} at 550 nm at a temperature of 293 K, is
shown in Fig. 8 (left). For
better comparison, the force is given as a ratio to the weight of the particle.
Similar to the ΔT force, it decreases with increasing
pressure, but there is no decrease for low pressures. The force is considerable,
for particles of 0.2 μm it is more than 10 times the weight, causing,
e.g. the trembling motion as observed, e.g. by Steipe (1952).
View Details
 Fig. 8
Photophoretic Δα force as a function of
pressure for various particle sizes. The force is given relative to the
weight of the particle. Left side: Force on a particle, which is assumed
to be fixed in space. Right side: average force for a particle oriented
by a magnetic field, but disturbed by Brownian rotation.

But this force continuously changes direction due to the Brownian rotation of the
particle. The mean square angle of rotation
θ
2
¯
of a spherical particle with diameter
d in time t is (Fuchs, 1964, p.245):
θ
2
¯
=
2
π
⋅
kT
η
d
3
⋅
t
, with k the Boltzmann constant and
T the temperature. A few values of average revolutions per
second are given below:
Diameter
[μm] 
0.1 
0.2 
0.5 
1 
2 
Average revolutions in 1 second
[
1
2
π
⋅
θ
2
¯
]

61. 
21 
5.44 
1.92 
0.68 
Considering the vigorous rotation, it is immediately evident that none of the
above particles will perform a reasonably straight motion.
The situation changes immediately if the particle orientation is forced in a
certain direction. For our considerations we will assume the particle is located
in a magnetic field B and has some remanent magnetism characterized
by the magnetic momentum m (Fig. 9). The particle orients itself such that its
magnetic momentum points in the direction of the magnetic field. The direction
of the photophoretic force does not usually coincide with the direction of the
magnetic moment and the angle between the two is δ. The
particle can still freely rotate around the axis of the magnetic moment,
therefore all force vectors are located on a cone with an opening angle of
2·δ. But Brownian rotation is stochastic, thus
the photophoretic force will cause a quasihelical motion (see insert of Fig. 9). The averaged force on the
direction of the field is obtained by multiplying the instantaneous force (8) with cos
δ. Brownian rotation is not only around the axis of the
magnetic momentum, but it will also disturb the orientation of the force vector,
which we have so far assumed to be at angle δ to the magnetic field
line. The stronger the magnet, the less deviations from this direction will be
possible. The competition between orienting torque by the magnetic field and the
disorientation by thermal disturbance is very similar to the orientation of
magnetic dipoles in the theory of paramagnetism (see, e.g. Brand and Dahmen, 2005, pp. 290–294): The
magnetic dipole with magnetic moment m in the magnetic field
B has a potential energy, m·B,
its thermal energy is kT. The ratio
x
=
m
⋅
B
k
⋅
T
of the maximum potential energy
E_{p} =
m·B and the thermal energy
E_{th} =
k·T is a measure for the strength of
orientation. No orientation means no magnetic field or moment, or
T→∞, which implies x =
0. Perfect orientation would be for
m·B→∞ or
T→0 signifying x→∞.
The theory of paramagnetism yields for the magnetization M the expression
M
=
M
^
(
coth
x
−
1
x
)
=
M
^
⋅
L
(
x
)
(Brand
and Dahmen, 2005, p. 293), with M̂ the saturation
magnetization and
L
(
x
)
=
1
tanh
x
−
1
x
the Langevin function. The asymptotic behavior of
L(x) is:
L
(
x
)
=
1
3
x
for x << 1, i.e.
almost no orientation for small magnetic field and/or moment, and
L
(
x
)
=
1
−
1
x
for x >> 1 which
means perfect orientation for a strong dipole and/or field.
View Details
 Fig. 9
Particle containing a magnetic dipole in a magnetic field. The particle
is oriented such that the magnetic dipole moment points in the direction
of the magnetic field. Rotation around the axis is possible, and the
photophoretic force vectors are positioned on a cone with an opening
angle of 2δ (right). Due to the stochastic manner of Brownian
rotation, the particle performs a quasihelical motion (left).

Using the formalism of paramagnetism, the projected force is multiplied with the
Langevin function L(x), i.e. the average
projected force in the direction of the magnetic field using (8) is
F
=
1
12
⋅
c
¯
⋅
1
1
+
(
p
/
p
*
)
2
Δ
α
α
¯
⋅
H
⋅
cos
δ
⋅
L
(
x
)
,
with

x
=
m
⋅
B
kT
⋅
(see, e.g.
Preining, 1966). This has been done for the
considered particles, assuming the material to have a volume magnetization of
0.02
T/
μ_{0}, and a magnetic
field of 7.10
^{−5} T (earth’s magnetic
field). The mean projected force is shown in
Fig. 8 (right side). Besides the factor cos 45°,
there is little difference to the instantaneous force for the particles of 5 and
2 μm in diameter since the magnetic moment is sufficiently large to
orient the particles. For the smaller sizes disorientation is essential, e.g.
for 0.2μm particles, the averaged force only is 25% of the
instantaneous one. Nevertheless the averaged photophoretic force can be larger
than gravity, as is the case for the 0.2 to 1μm particles, thus
levitation by sunlight in the earth’s magnetic field is possible.
Fig. 10 shows the size dependence of
the relative force for various air pressures. For particles smaller than 0.1
μm, the Brownian rotation prevents levitation, for particles larger than
1 to 5 μm, depending on pressure, orientation would be optimal, but the
weight increases ∝
d^{3} whereas the
photophoretic force ∝
d^{2}, thus the relative
force decreases.
View Details
 Fig. 10
Relative photophoretic Δα force: Left:
Magnetic orientation of the particle. Right: Orientation by gravity. For
conditions above the horizontal line, levitation in the atmosphere is
possible.

The Δα force can be channeled around one
direction if an orienting torque exists. This has been shown above for magnetic
orientation, but is also possible by electric fields or by gravity. A typical
everyday example of orienting torque by gravity would be a fishing float which
usually stands upright in the water, when disturbed it will return to its
initial position due to the uneven distribution of mass. An inhomogeneous
distribution of mass will orient the particle in the same way. This can be seen
in Fig. 11: Assume a particle
having different accommodation coefficients on the two hemispheres causing a
photophoretic force to upwards right. (a) The center of gravity is displaced
from the center of the photophoretic force by a vector q. Gravity
and photophoretic force vectors jointly cause a motion of the particle to the
right and slightly upwards; (b) the motion of the particle causes a friction
force which is added to the photophoretic force. Now gravity and the other
forces are balanced and a stationary state is reached; but, due to the offset of
the center of gravity and the center of photophoretic plus friction force, a
torque acts on the particle, rotating it counterclockwise until the center of
photophoretic force is above the center of gravity. This results in an upward
motion, since the photophoretic force is larger than the weight (c).
View Details
 Fig. 11
Orientation of a particle by gravity: (a) The center of mass
of the particles is displaced by a vector q. The
photophoretic force (due to different accommodation coefficients) added
to the weight of the particles causes a motion to the right.
(b) The friction is a vector to the left (b),
adding friction and photophoretic force yields a vertical vector. Since
the two vertical vectors are anchored at different points, a torque on
the particle forces it to rotate (c) until the two anchor
points are above each other (c).

The orientation of the particle is disturbed by Brownian rotation, the important
magnitude is the ratio of the potential energy E_{p}
=
m·g·q to the
thermal energy E_{th} =
k·T, i.e. the value given by eq. (8) has to be multiplied by
L(x) with
x
m
⋅
g
⋅
q
k
⋅
T
. Using the same procedure as above, the average
force for particles of various sizes and at different pressures has been
calculated and results can be seen in Fig. 10 (right). The assumed density is
ρ_{p} = 1000 kgm^{−3}
and the center of gravity is displaced by q =
0.2·d. The only difference to magnetic orientation is
the smaller force and the larger diameter at which the relative force peaks.
Levitation is only possible at lower pressures. Since the force is directly
proportional to the flux density of the radiation, levitation at atmospheric
pressures is possible at flux densities which can be achieved easily by lasers
or by intense illumination.
The Δαphotophoresis needs an orienting field,
thus it is called field photophoresis, specifically gravito, magnetoor
electrophotophoresis. For optimum
Δαphotophoresis, the particles should have a
considerable variation of the accommodation coefficient on the surface and a
high thermal conductivity, in order to have a homogeneous temperature on the
surface.
(c) Other types of photophoresis
In the two previous sections we have assumed the particle to be spherical and
having a rather symmetric distribution of the accommodation coefficients on the
surface. It is hard to imagine that particles formed by combustion, coagulation,
condensation on existing surfaces, mechanical abrasion or interstellar dust
particles fulfill these requirements. It can be assumed that particles
illuminated by light have a temperature gradient on their surface (needed in
ΔT force), but it is unrealistic for a particle to have
the same temperature everywhere on an inhomogeneous surface, as we assumed for
the Δα force. Thus
Δα and ΔT
photophoresis will most likely occur simultaneously.
An inhomogeneous surface of the particles will also cause a photophoretic torque.
The radiometer itself (Fig. 2)
is an example. Two arrangements where a photophoretic torque will act are shown
in Fig. 12. The shape
(a) is the “Spitzenradiometer” (pointed radiometer)
theoretically investigated by Hettner
(1924). The pointed ends of the body will release more thermal energy,
thus be colder. The photophoretic force is in the direction of the temperature
gradient, resulting in a counterclockwise torque. The particle (b)
with an inclusion on the surface having a different accommodation coefficient
will have a clockwise torque, if the accommodation coefficient of the surface of
the inclusion is smaller than for the remaining surface. Simultaneously, an
upwards Δα force acts on this particle.
View Details
 Fig. 12
Examples for particle shapes which cause photophoretic torque.
(a) The “Spitzenradiometer” (Hettner, 1924). The points
dissipate more thermal energy and are thus colder. Due to the
temperature difference, a counterclockwise torque arises.
(b) A particle with an impurity on the surface having a
different accommodation coefficient. Both a torque and a
Δα force arise.

In addition to the torque and the Δαforce, this
particle with the inclusions (Fig.
12b) will also have a ΔT force. In the
stationary state, the orientation of the photophoretic torque will be parallel
to the direction of light, resulting in a helical motion around the direction of
the illumination. The direction of the photophoretic
Δα force need not agree with the direction of
the photophoretic torque (both vectors are particlefixed). Depending on their
angle to each other, the resulting motion can be in the direction of the light
or opposite (for theoretical details, see Preining, 1966, p. 130, for observations, see Lustig and Söllner,
1932).
For the Δα force, an inhomogeneous distribution
of the accommodation coefficient on the surface is needed. In reality this will
not be two homogeneous hemispheres, as assumed above, but a particle with a
complicated surface structure. Therefore in addition to the
Δα force, a photophoretic torque will make the
particle rotate around a particlefixed axis. This rotation has to be added to
the linear motion caused by the Δα force.
Therefore the simple picture of Fig.
9 has to be replaced by a helical movement around the field line with a
constant pitch, obviously with Brownian disturbances superimposed. The average
velocity or the force in the direction of the magnetic field is not altered. The
same is true for gravitophotophoresis.
In an inhomogeneous horizontal illumination, oscillations along a vertical axis
are possible. Consider a particle performing gravitophotophoresis with an
additional photophoretic torque. The particle will have a helical movement
upwards and out of the horizontal beam of light. With lower illumination, both
the gravitophotophoretic force and the torque decrease and after some time for
temperature adjustment, the particle follows a helical downward movement with a
different pitch and then resumes the upward motion, see Fig. 1f.
In two inhomogeneous light beams in the opposite direction, complicated closed
loops in the horizontal direction are possible, in rotating magnetic fields the
particles have even more impressive orbits, for details see Preining (1966).