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Dusty Plasmas: ���forces
Ed Thomas Auburn University
References • Journals
– TPS – IEEE Transactions on Plasma Science – PoP – Physics of Plasmas – PRL – Physical Review Letters – PRE – Physical Review E – PPCF – Plasma Physics and Controlled Fusion – PSS – Planetary and Space Science
• Textbooks – Low Temperature Plasmas: Fundamentals, Technologies, and Techniques -
Volume 1 - R. Hippler, H. Kersten, M. Schmidt, K.H. Schoenbach (Eds.) ���Ref: Ch. 6 – Fundamentals of Dusty Plasmas – A. Melzer and J. Goree
– Introduction to Dusty Plasma Physics – P. Shukla and A. Mamun – Physics and Applications of Complex Plasmas – S. Vladimirov, K. Ostrikov, and
A. Samarian – Plasma Physics – A. Piel
MORE BACKGROUND INFO
Dusty (complex, fine particle, colloidal) plasmas • Dusty plasmas - four component plasma system
– Ions – Electrons – Neutral atoms – Charged microparticles
• Plasma and charged microparticles - coupled via collection of ions and
electrons from the background plasma.
• Presence of microparticles: – Modifies density and charge distribution – Modifies plasma instabilities – Introduces new dust-driven waves – Ubiquitous in natural and man-made plasmas
• Measurements of dust particles: – Forces – Electrostatic potential – Velocity distributions
Dusty plasmas in terrestrial environments
• Noctilucent clouds (NLC’s) form at extremely high altitudes, about 85 km, that “shine at night”.
• They form in the cold, summer polar mesopause and are believed to be charged ice crystals. • They are believed to be
associated with radar backscatter phenomena (PSME’s) observed during the northern summers.
From: h"p://lasp.colorado.edu/noc1lucent_clouds/
Dust in fusion devices
- Dust in fusion plasmas was often not considered a major issue.
- Recent studies do indicate that uncontrolled dust injection can have detrimental impacts on fusion plasmas.
Dust in fusion devices
Large “dust production” event during a disruption in the Alcator C-mod Tokamak at MIT. Movie courtesy of: Aaron Bader and Robert Granetz (MIT)
Dust contamination in ITER
- Tritium retention in “dust”
- Health and safety hazard
- Reduction of density control
- Degradation of first wall material
- Radiated power losses
From: http://www.iter.org
FORCES ON DUSTY PLASMAS
Summary of dominant forces in dusty plasmas
Name Origin Size dependence Weight force Gravity a3
Neutral drag Streaming neutrals a2
Ion drag force Streaming ions a2
Thermophoresis Temperature gradient a2
Electric force Electric field a1
Magnetic force Magnetic field a1
Adapted from textbook Plasma Physics by A. Piel, Table 10.2 (Springer-Verlag, 2010)
These forces give rise to the majority of the phenomena observed in laboratory and microgravity dusty plasma experiments
Gravitational force
• Relevant for dusty / complex plasmas since the particles need to be suspended in the plasma.
• ���
• Gravitational force can also play an important role in astrophysical environments (e.g., Saturn’s rings).
Fgravity
Image from Auburn University
166 6 Fundamentals of Dusty Plasmas
6.3.1 Electric Field ForceThe governing force on charged particles is the force in the electric field ~E given by
~FE = Qd~E = 4º≤0a¡fl
~E . (6.17)
The force scales linearly with the particle size as predicted by the capacitor model in Sec. 6.2.1.Hamaguchi and Farouki [29] have discussed the question of whether a shielding cloud aroundthe dust particle would lead to a modification of the electric field force. They found that mod-ifications to the electric field force would only have to be considered when the shielding cloudis distorted. In that case, polarization forces on the particles might exist which can be writtenas
~Fdip = ~r(~p · ~E)
where ~p the dipole moment. Dipole moments can either be induced by an external electricfield [30] or by directed charging processes (for dielectric particles) where due to an ion flowthe front side of the particle is charged more positively than the back side [31]. Polarizationforces are usually considered negligible, except for very large particles [32].
6.3.2 GravityThe gravitational force simply is
~Fg = md~g =43ºa3Ωd~g . (6.18)
with the gravitational acceleration ~g and the mass density of the dust particles Ωd. This forceobviously scales as a3. Thus, it is a dominant force for particles in the micrometer range andbecomes negligible for nanometer particles.
6.3.3 Ion Drag ForceIons streaming past a dust particle exert a force on the dust by scattering of the ions in theelectric field of the dust or by collection on the dust surface. In a plasma discharge therealways is a persistent ion stream either due to ambipolar diffusion in the plasma bulk or due tothe electric fields in the plasma sheath. The ion drag force is one of the major forces on dustparticles. It is understood qualitatively, but a complete quantitative description is still missingdue to the complexity of the involved processes.
The ion drag consists of two parts, the collection force ~Fcoll due to ions hitting the dustand the Coulomb force ~FCoul due to scattering of the ions in the electrostatic field of the dust[33, 34, 35, 36, 37]. The total ion drag force is then given by
~Fion = ~Fcoll + ~FCoul . (6.19)
The ions which arrive at the particle not only contribute to ion charging of the dust, butalso transfer their momentum to the dust and thus exert the collection force on the dust. This
Electric force
Felectric
Image from Auburn University
• Because the dust grains are charged, they respond to the internal electric fields within the plasma.
• ���
166 6 Fundamentals of Dusty Plasmas
6.3.1 Electric Field ForceThe governing force on charged particles is the force in the electric field ~E given by
~FE = Qd~E = 4º≤0a¡fl
~E . (6.17)
The force scales linearly with the particle size as predicted by the capacitor model in Sec. 6.2.1.Hamaguchi and Farouki [29] have discussed the question of whether a shielding cloud aroundthe dust particle would lead to a modification of the electric field force. They found that mod-ifications to the electric field force would only have to be considered when the shielding cloudis distorted. In that case, polarization forces on the particles might exist which can be writtenas
~Fdip = ~r(~p · ~E)
where ~p the dipole moment. Dipole moments can either be induced by an external electricfield [30] or by directed charging processes (for dielectric particles) where due to an ion flowthe front side of the particle is charged more positively than the back side [31]. Polarizationforces are usually considered negligible, except for very large particles [32].
6.3.2 GravityThe gravitational force simply is
~Fg = md~g =43ºa3Ωd~g . (6.18)
with the gravitational acceleration ~g and the mass density of the dust particles Ωd. This forceobviously scales as a3. Thus, it is a dominant force for particles in the micrometer range andbecomes negligible for nanometer particles.
6.3.3 Ion Drag ForceIons streaming past a dust particle exert a force on the dust by scattering of the ions in theelectric field of the dust or by collection on the dust surface. In a plasma discharge therealways is a persistent ion stream either due to ambipolar diffusion in the plasma bulk or due tothe electric fields in the plasma sheath. The ion drag force is one of the major forces on dustparticles. It is understood qualitatively, but a complete quantitative description is still missingdue to the complexity of the involved processes.
The ion drag consists of two parts, the collection force ~Fcoll due to ions hitting the dustand the Coulomb force ~FCoul due to scattering of the ions in the electrostatic field of the dust[33, 34, 35, 36, 37]. The total ion drag force is then given by
~Fion = ~Fcoll + ~FCoul . (6.19)
The ions which arrive at the particle not only contribute to ion charging of the dust, butalso transfer their momentum to the dust and thus exert the collection force on the dust. This
• Ion flows can cause the shielding cloud around dust grains to become distorted, leading to a dipole-like charge distributions on grains.
• ���
166 6 Fundamentals of Dusty Plasmas
6.3.1 Electric Field ForceThe governing force on charged particles is the force in the electric field ~E given by
~FE = Qd~E = 4º≤0a¡fl
~E . (6.17)
The force scales linearly with the particle size as predicted by the capacitor model in Sec. 6.2.1.Hamaguchi and Farouki [29] have discussed the question of whether a shielding cloud aroundthe dust particle would lead to a modification of the electric field force. They found that mod-ifications to the electric field force would only have to be considered when the shielding cloudis distorted. In that case, polarization forces on the particles might exist which can be writtenas
~Fdip = ~r(~p · ~E)
where ~p the dipole moment. Dipole moments can either be induced by an external electricfield [30] or by directed charging processes (for dielectric particles) where due to an ion flowthe front side of the particle is charged more positively than the back side [31]. Polarizationforces are usually considered negligible, except for very large particles [32].
6.3.2 GravityThe gravitational force simply is
~Fg = md~g =43ºa3Ωd~g . (6.18)
with the gravitational acceleration ~g and the mass density of the dust particles Ωd. This forceobviously scales as a3. Thus, it is a dominant force for particles in the micrometer range andbecomes negligible for nanometer particles.
6.3.3 Ion Drag ForceIons streaming past a dust particle exert a force on the dust by scattering of the ions in theelectric field of the dust or by collection on the dust surface. In a plasma discharge therealways is a persistent ion stream either due to ambipolar diffusion in the plasma bulk or due tothe electric fields in the plasma sheath. The ion drag force is one of the major forces on dustparticles. It is understood qualitatively, but a complete quantitative description is still missingdue to the complexity of the involved processes.
The ion drag consists of two parts, the collection force ~Fcoll due to ions hitting the dustand the Coulomb force ~FCoul due to scattering of the ions in the electrostatic field of the dust[33, 34, 35, 36, 37]. The total ion drag force is then given by
~Fion = ~Fcoll + ~FCoul . (6.19)
The ions which arrive at the particle not only contribute to ion charging of the dust, butalso transfer their momentum to the dust and thus exert the collection force on the dust. This
Gravitational and electric forces: ���zero-order equilibrium
• For most ground-based experiments: Fgravity ~ Felectric
• Defines zero-order equilibrium
• Typical values: - a = 1.5 µm - md = 2.8 x 10-14 kg - Zd = 4600 electrons - E = 3.8 V/cm
Fgravity
Felectric
Image from Auburn University
Ion drag force - origin • Arises from the momentum
transfer from ion-dust interactions.
• Two components: direct collection of ions and the scattering of ions in the electrostatic field of the dust particle.
• Fion-drag = Fcollection + Fcollision
• Critically depends upon the screening length of the dust particle.
collection
Coulomb collision
Refs: M. Barnes, et al., PRL, 68, 313 (1992) S. Khrapak, et al., PRE, 66, 046414 (2002) A. Ivlev, et al., PRE, 71, 016405 (2004) I. Hutchinson, et al., PPCF, 48 185 (2006) S. Khrapak, et al., IEEE TPS, 37, 487 (2009)
Ion drag force – collection force
collection
Coulomb collision
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Arises from the direct collection of ions by the dust grain.
Ion drag force – collection force
collection
Coulomb collision
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Arises from the direct collection of ions by the dust grain.
Ion drag force – collection force
collection
Coulomb collision
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Arises from the direct collection of ions by the dust grain.
bc – collec1on impact parameter
Ion drag force – collection force
collection
Coulomb collision
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Number of ion collisions/time
Arises from the direct collection of ions by the dust grain.
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
bc – collec1on impact parameter
Ion drag force – collection force
collection
Coulomb collision
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Number of ion collisions/time
Force is from the rate of momentum transfer due to ion collection.
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Arises from the direct collection of ions by the dust grain.
bc – collec1on impact parameter
Ion drag force – Coulomb collision force
collection
Coulomb collision
Arises from the deflection of ions in the electrostatic sheath that surrounds the dust grain.
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Ion drag force – Coulomb collision force
collection
Coulomb collision
Arises from the deflection of ions in the electrostatic sheath that surrounds the dust grain.
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Impact parameter for 90˚ scattering
Ion drag force – Coulomb collision force
collection
Coulomb collision
Arises from the deflection of ions in the electrostatic sheath that surrounds the dust grain.
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Impact parameter for 90˚ scattering
Ion drag force – Coulomb collision force
collection
Coulomb collision
Arises from the deflection of ions in the electrostatic sheath that surrounds the dust grain.
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Impact parameter for 90˚ scattering
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Number of ion collisions/time
Ion drag force – Coulomb collision force
collection
Coulomb collision
Arises from the deflection of ions in the electrostatic sheath that surrounds the dust grain.
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Impact parameter for 90˚ scattering
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Number of ion collisions/time
Two challenges: 1. Role of ion-neutral collisions 2. Correct screening length (λD)
Ion drag force – Coulomb collision force
collection
Coulomb collision
Arises from the deflection of ions in the electrostatic sheath that surrounds the dust grain.
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Impact parameter for 90˚ scattering
6.3 Forces on Particles 167
force is given by [34]
~Fcoll = mivsni~uiºa2
µ1° 2e¡fl
miv2s
∂. (6.20)
Here, vs = (u2i +v2
th,i)1/2 is the geometric mean of the ion drift velocity ui and the ion thermal
velocity vth,i. Upon impact, the ions transfer the momentummiui. Further,
æcoll = ºb2c = ºa2
µ1° 2e¡fl
miv2s
∂
is the collection cross section and bc is the maximum impact parameter for ion collection (Thisis related to the charging model in Sec. 6.2.1). Then, ni~vsæ is the number of ions hitting thedust surface per unit time.
The Coulomb force is due to those ions which are deflected in the local electric field in thesheath surrounding the dust grain. It is given in an early formulation by Barnes et al. [34] as
~FCoul = mivsni~ui4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
, (6.21)
where bº/2 = Qde/(4º≤0miv2s ) is the impact parameter for 90± scattering and
æCoul = 4ºb2º/2 ln
√∏2
D + b2º/2
b2c + b2
º/2
!1/2
is the cross section for Coulomb scattering [38]. In this formulation, only ion scattering in theDebye sphere of radius ∏D is considered. Perrin et al. [36] and Khrapak et al. [39, 40] haveshown that due to the high charge on the dust, ion scattering outside the Debye sphere willdecisively increase the strength of the ion drag force.
Ion-neutral collisions can substantially influence the ion drag. Recent simulations [41]have indicated that ion-neutral collisions can even lead to a force opposite to the ion motion.A kinetic approach including ion collisions [42] showed a modification to the ion drag force,but not a reversal of the ion drag force. Generally speaking, the experiments are in a rangethat is not fully covered by theoretical calculations that have been developed so far.
Experimentally, the ion drag force has been measured by dropping particles through anRF plasma (see Fig. 6.2). There the particles experience oppositely directed electric force andion drag. Depending on the plasma power, the relative strength of the two forces can be varied[43]. From that experiment, good agreement was obtained with the Barnes formulation of theion drag force. In a similar experiment the ion drag force was measured in a DC discharge[44]; the force has been explained with reasonable accuracy by the Khrapak model [40] thatincludes ion scattering outside the Debye sphere.
6.3.4 ThermophoresisThe thermophoretic force appears due to a temperature gradient in the neutral gas. In a sim-plified picture, neutral gas atoms from the “hotter” side that hit the dust grain transfer a larger
Number of ion collisions/time
Ion drag force: voids in space
• In rf plasmas in microgravity experiments, a void appears.
• Enhanced ionization at the center: provides a source of ions and outward electric field.
• Boundary determined by the force balance: Felec = Fion-drag!
Fion-drag
Felectric
E
Image from Max Planck Institute, PK-3
Ion drag force: voids in the lab
• Control size of the void region using different potentials on a probe tip.
• Estimate the void size, x0, using different electric field estimates. ������Dashed: 25 V/cm���Solid: 17 V/cm
E. Thomas, Jr., et al., PoP, 11, 1770 (2004)
Thermophoretic force • Arises from the momentum
transfer from neutral-dust interactions.
• Neutral atoms from “hot” side provide more momentum than those from “cold” side.
•
vth,n – neutral thermal velocity Tn – neutral gas temperature kn – thermal conductivity
168 6 Fundamentals of Dusty Plasmas
Figure 6.2: Particles of 4.8 µm diameter dropping through a cylindrical plasma discharge operated at apower of 2 W(a) and 5 W(b). At lower power the electric field force FE exceeds the ion drag Fion andthe particles are focused to the center of the discharge. At higher power the particles are pushed awayfrom the center due to the dominance of the ion drag. After [43].
momentum to the dust than atoms from the “colder” side. Consequently, a force towardsregions of colder gas is established. From gas kinetic theory this force is [45, 46]
~Fth = °3215
a2kn
vth,n
~rTn (6.22)
whererTn is the temperature gradient in the neutral gas and kn the thermal conductivity of thegas. This formula holds if the mean free path of the gas molecules ∏mfp is much larger than theparticle radius. The action of the thermophoretic force has been demonstrated by Jellum et al.[46]. In his experiments particles were driven towards cooled electrodes by thermophoresisor have even been removed from the discharge since the particles are collected at a liquidnitrogen cold finger placed near the plasma. Using heated electrodes the thermophoretic forcehas also been exploited to levitate particles (see below) [47, 48].
6.3.5 Neutral Drag ForceThe neutral drag is the resistance experienced by a particle moving through a gas. For particlesmuch smaller than the mean free path aø ∏mfp and particle velocities much smaller than thethermal velocity of the gas (vd ø vth,n) Epstein’s expression [49] is appropriate
~Fn = °±43ºa2mnvth,nnn~vd . (6.23)
Here, mn, nn and vth,n are the mass, the density and the thermal velocity of the neutral gasatoms, respectively. The parameter ± depends on the microscopic mechanism of the collisionbetween the gas atom and the particle surface. In Epstein’s model, ± can have a value in
Thermophoretic force and Voids
• Dusty plasma of 3.4 µm melamine formaldehyde particles • Applied temperature gradient of ~1200 K/m • ∆Tn = 25 K over 20 mm • Lower electrode is heated using a Peltier element
H. Rothermel, et al., PRL, 89, 175001 (2002)
∆Tn = 0 K ∆Tn = 25 K
30
Thermophoretic force and Coulomb Balls
effect of thermophore1c force
O. Arp, et al, Phys. Plasmas, 12, 122102 (2005)
Neutral drag force
• Model for the motion of small particles through a gas.
• Presented by P. Epstein [1924].
• Valid for:���a << λmfp �vd << vth,n"
• δ is a coefficient that depends upon the type of collision: ���δ = 1 for specular reflection���δ = 1.442 for diffuse reflection
168 6 Fundamentals of Dusty Plasmas
Figure 6.2: Particles of 4.8 µm diameter dropping through a cylindrical plasma discharge operated at apower of 2 W(a) and 5 W(b). At lower power the electric field force FE exceeds the ion drag Fion andthe particles are focused to the center of the discharge. At higher power the particles are pushed awayfrom the center due to the dominance of the ion drag. After [43].
momentum to the dust than atoms from the “colder” side. Consequently, a force towardsregions of colder gas is established. From gas kinetic theory this force is [45, 46]
~Fth = °3215
a2kn
vth,n
~rTn (6.22)
whererTn is the temperature gradient in the neutral gas and kn the thermal conductivity of thegas. This formula holds if the mean free path of the gas molecules ∏mfp is much larger than theparticle radius. The action of the thermophoretic force has been demonstrated by Jellum et al.[46]. In his experiments particles were driven towards cooled electrodes by thermophoresisor have even been removed from the discharge since the particles are collected at a liquidnitrogen cold finger placed near the plasma. Using heated electrodes the thermophoretic forcehas also been exploited to levitate particles (see below) [47, 48].
6.3.5 Neutral Drag ForceThe neutral drag is the resistance experienced by a particle moving through a gas. For particlesmuch smaller than the mean free path aø ∏mfp and particle velocities much smaller than thethermal velocity of the gas (vd ø vth,n) Epstein’s expression [49] is appropriate
~Fn = °±43ºa2mnvth,nnn~vd . (6.23)
Here, mn, nn and vth,n are the mass, the density and the thermal velocity of the neutral gasatoms, respectively. The parameter ± depends on the microscopic mechanism of the collisionbetween the gas atom and the particle surface. In Epstein’s model, ± can have a value in
Radiation pressure force
• In many experiments, lasers have been used to manipulate the particles in a dusty plasma.
• This force is proportional to the intensity of the laser. ���
•
6.3 Forces on Particles 169
the range between 1.0 and 1.442, where ± = 1.0 is for specular reflection of all impingingmolecules and ± = 1.442 for diffuse reflection with accommodation, i.e. for a perfect thermalnonconductor. Recent experiments [50] with laser-driven polymer microspheres yielded ± =1.26± 0.13. In connection with equations of particle motion we will often use the drag forcein the form
~Fn = °mdØ~vd Ø = ±8º
p
aΩdvth,n. (6.24)
Here, Ø is the friction coefficient and linearly depends on the gas pressure p. The frictioncoefficient is inversely proportional to the particle radius a which means that in relation totheir mass smaller particles experience stronger damping than larger particles.
6.3.6 Radiation Pressure ForcesRadiation pressure is the momentum per unit area and unit time transferred from photons toa surface. If a beam of photons strikes the particle, some photons will be reflected and otherswill be transmitted or absorbed. All three processes contribute to the radiation pressure. Ingeneral for a laser beam of intensity Ilaser, the radiation pressure force is [51]
Frad = ∞ºa2Ilaser
c, (6.25)
where c is the speed of light and ∞ is a dimensionless factor that is determined by the reflection,transmission or absorption of the photons on the particle. If all photons are absorbed ∞ will be1, and for complete reflection ∞ = 2. Experimental investigations report this parameter to be∞ = 0.94± 0.11 for polymer microspheres [50].
Lasers are widely used in dusty plasmas to manipulate dust particles without disturbingthe plasma environment. For example, they are used to measure forces or excite waves, Machcones and resonances, see e.g. Fig. 6.19 and [52].
6.3.7 Particle Interaction PotentialsAfter the discussion of the plasma-related forces onto the dust particles we now will dwell onthe interaction among dust particles.
6.3.7.1 Particles in Isotropic Plasmas
In isotropic plasmas, the particles are usually treated as point charges that are shielded by theambient plasma background. Thus, the interaction is described by a Debye-Huckel or Yukawapotential energy
U(r) =Z2
de2
4º≤0rexp
µ° r
∏D
∂(6.26)
with the screening length ∏D. In a plasma with flowing ions, this model is known to beaccurate only in the plane perpendicular to the ion flow. Experimentally, the interaction can
Laser beam injected from the top, pushes particles out of the dust cloud. (Auburn University)
Radiation pressure force
• Example: laser excited Mach cone 194 6 Fundamentals of Dusty Plasmas
Figure 6.19: Mach cones in dusty plasmas. a) Scheme of the experimental setup for the excitation of acompressional Mach cone, b) gray scale map of the particle velocities in the compressional Mach cone.After [115].
6.6.2.4 Transverse Dust Lattice Waves
Finally, transverse dust lattice waves will be demonstrated, here. In this wave mode, theparticle motion is perpendicular to the wave propagation, and also out-of-plane, i.e. in thevertical direction for typical plasma crystals. Such out-of-plane elongations are only stabledue to the presence of a confinement potential. Otherwise the particles would just move awayfrom each other due to their Coulomb repulsion. The dispersion of the transverse DLW isreadily obtained as [119]
!2 + iØ! = !20 °
1º
!2pd
MX
`=1
e°`∑
`3(1 + `∑) sin2
µ`qb
2
∂. (6.42)
Here, the influence of many neighboring particles is included in the sum over `. Here,M = 1includes the interaction of nearest neighbors only, M = 2 also includes the interaction ofnext-nearest neighbors etc. One can see that the influence of the vertical confinement !2
0 isnecessary to yield a stable dispersion relation. It is interesting to note that this wave is abackward (@!/@q < 0) and “optical” wave, i.e. ! ! !0 for q ! 0.
Misawa et al. [120] investigated vertical oscillations which propagate along a 1D chain ofparticles. The authors have determined part of the dispersion relation where a finite frequencyis found for q ! 0 and the dispersion also has a negative slope, as expected for the transverseDLW. However, the overall agreement of the measured and the theoretical dispersion was notsatisfying.
6.6.2.5 Natural Phonons
Meanwhile, image analysis techniques are advanced enough to determine the dispersion rela-tion of waves from the naturally excited lattice oscillations (phonons). The particles in a dustyplasma exhibit a Brownian motion that is excited by thermal or electrostatic fluctuations inthe plasma. This small-amplitude motion is sufficient to reconstruct the wave spectrum in thesystem [121]. Thereby it is assumed that the Brownian motion is a random superposition of all
A. Melzer, S. Nunomura, D. Samsonov, and J. Goree, Phys. Rev. E 62, 4162 (2000).
Inter-particle interactions
• Particles interact through a screen Coulomb interaction (also known as a Yukawa potential)
• ���
6.3 Forces on Particles 169
the range between 1.0 and 1.442, where ± = 1.0 is for specular reflection of all impingingmolecules and ± = 1.442 for diffuse reflection with accommodation, i.e. for a perfect thermalnonconductor. Recent experiments [50] with laser-driven polymer microspheres yielded ± =1.26± 0.13. In connection with equations of particle motion we will often use the drag forcein the form
~Fn = °mdØ~vd Ø = ±8º
p
aΩdvth,n. (6.24)
Here, Ø is the friction coefficient and linearly depends on the gas pressure p. The frictioncoefficient is inversely proportional to the particle radius a which means that in relation totheir mass smaller particles experience stronger damping than larger particles.
6.3.6 Radiation Pressure ForcesRadiation pressure is the momentum per unit area and unit time transferred from photons toa surface. If a beam of photons strikes the particle, some photons will be reflected and otherswill be transmitted or absorbed. All three processes contribute to the radiation pressure. Ingeneral for a laser beam of intensity Ilaser, the radiation pressure force is [51]
Frad = ∞ºa2Ilaser
c, (6.25)
where c is the speed of light and ∞ is a dimensionless factor that is determined by the reflection,transmission or absorption of the photons on the particle. If all photons are absorbed ∞ will be1, and for complete reflection ∞ = 2. Experimental investigations report this parameter to be∞ = 0.94± 0.11 for polymer microspheres [50].
Lasers are widely used in dusty plasmas to manipulate dust particles without disturbingthe plasma environment. For example, they are used to measure forces or excite waves, Machcones and resonances, see e.g. Fig. 6.19 and [52].
6.3.7 Particle Interaction PotentialsAfter the discussion of the plasma-related forces onto the dust particles we now will dwell onthe interaction among dust particles.
6.3.7.1 Particles in Isotropic Plasmas
In isotropic plasmas, the particles are usually treated as point charges that are shielded by theambient plasma background. Thus, the interaction is described by a Debye-Huckel or Yukawapotential energy
U(r) =Z2
de2
4º≤0rexp
µ° r
∏D
∂(6.26)
with the screening length ∏D. In a plasma with flowing ions, this model is known to beaccurate only in the plane perpendicular to the ion flow. Experimentally, the interaction can
170 6 Fundamentals of Dusty Plasmas
Figure 6.3: Measured horizontal interaction potential from collisions of two dust particles. The curves Aand B reflect two different discharge conditions. From the experiment, Zd = 13 900 and ∏D = 0.34mm(case A) and Zd = 17 100 and ∏D = 0.78 mm (case B) are deduced. After [53].
be precisely determined from the dispersion of waves, a method that will be presented inSec. 6.6.2. Alternatively, the dust interaction has been probed by the analysis of collisionsbetween dust particles [53]. There, two dust particles were forced to collide at different speedsand the collision dynamics has been analyzed. From the collisions the interaction has indeedbeen determined as a purely repulsive Yukawa potential with a screening length ∏D that iscompatible with the electron Debye length ∏De, see Fig. 6.3.
6.3.7.2 Particles in the Plasma Sheath
Often, the dust particles are trapped in the plasma sheath where strong electric fields and adirected ion motion towards the electrode prevail. This strongly non-neutral, non-equilibriumenvironment drastically alters the particle-particle interaction. An obvious sign is the unex-pected vertical alignment of dust particles in the sheath. There the particles are located directlyon top of each other forming vertical chains [5, 54] which definitely is not a minimum energyconfiguration for purely repulsive particle interaction (see also Fig. 6.6).
Analytical models of the particle interaction in the plasma sheath have been put for-ward e.g. in [55, 56, 57, 31, 58, 30, 32, 59]. First detailed simulations and experiments ofSchweigert et al. and Melzer et al. [60, 61, 62, 63] have revealed that the force between dust
Inter-particle interactions • Strong ion flows can lead to a modification of the interaction
potential.
• This can be observed in the asymmetric response of upper/lower particles trapped near the plasma sheath. 6.3 Forces on Particles 171
Figure 6.4: Simulated ion density distribution around dust particles in the plasma sheath when a) thedust particles are vertically aligned (±x = 0) and b) when they are shifted horizontally by a quarter of theinterparticle distance (±x = 0.25b). c) Horizontal component of the attractive force on the lower particleas a function of displacement from the vertically aligned position. The symbols denote experimentalvalues from laser manipulation and the solid line is derived from the simulations. The inset shows asketch of the experimental setup of the force measurement. After [62, 61]
particles in the sheath is a very peculiar nonreciprocal, attractive force: dust particles locatedlower in the sheath are attracted by upper particles, but the upper particles experience a repul-sive force from the lower.
The origin of this non-reciprocal attraction lies in the ion streaming motion in the sheathregion between the plasma and an electrode (see Fig. 6.4a,b). The ions have entered thesheath with Bohm velocity and stream towards the electrode through the particle arrangement.The electric field of the upper dust particle deflects ions so that they are focused beneath theparticle. This yields a region of enhanced positive space charge which provides the attractionfor the lower particles. In other words, the shielding cloud around the dust particles is distorteddownwards due to the ion streaming motion. This is often term a “wake effect”. Since theions move at supersonic speed in the sheath the attractive force can only be communicateddownwards and not upwards, which results in the peculiar situation that only the lower particleexperiences attraction, but there is no reaction on the upper particle. The attraction by the ioncloud is stronger than the repulsion from the upper particle since the ion cloud is located closerto the lower dust particle.
The nonreciprocal attraction has been directly demonstrated using laser manipulation tech-niques [60, 61, 64] in a system with two vertically aligned particles. The lower particle waselongated from the vertical position by the radiation pressure of a laser beam. From the forcebalance of laser pressure and attractive force the attraction is determined (see Fig. 6.4c). Thesimulations and the experiments suggest that the distorted ion cloud beneath the dust particlecan be modeled by a single positive point charge of charge Q+ located at a position d ° d+
