Electron and ion thermal forces in complex (dusty) plasmas

0.16 MB·English

by  Sergey A. Khrapak

Electron and ion thermal forces in complex (dusty) plasmas
∗
Sergey A. Khrapak
Max-Planck-Institut fu¨r extraterrestrische Physik, D-85741 Garching, Germany
(Dated: September19, 2012)
Expressions for the ion and electron thermal forces acting on a charged grain, suspended in a
weaklyionized plasma subject totemperaturegradients, arederived. Theseforces can significantly
affectgrain transportinmanyrealistic situations, rangingfrom complexplasmasin laboratory and
microgravity experiments, to charged dust in planetary atmospheres and fusion devices.
2
1 PACSnumbers: 52.27.Lw,52.25.Vy,94.05.Bf
0
2
p The momentum exchange between different species approaches[9–14], numericalsimulations[15–17]andex-
e playsanexceptionallyimportantroleincomplex(dusty) periments [18–23]. Moreover, theoretical results on the
S plasmas–multi-componentsystemsconsistingofcharged ion-grainscattering in the regimeof strong(Yukawa)in-
micron-sized (“dust”) grains embedded in a plasma [1– teraction found their application not only in the field of
8
1 3]. In a weakly ionized plasma, the momentum transfer complex plasmas, but also in the context of quark-gluon
in collisions with the neutral gas “cool down” the sys- plasma [24], scattering of dark matter particles [25], and

] tem, in particular grains and ions, and introduce some conventional strongly coupled plasmas [26]. Regarding
h
damping. In addition, ion and electron flows are usu- the momentumtransferinelectron-graincollisions,it of-
p
- allypresentinplasmasduetoglobalelectricalfields(e.g. ten plays only a minor role due to the small electron
m ambipolar or sheath fields in gas discharges). The mo- mass, although this is not always the case [27].
s mentum transfer from flowingions andelectrons canbe- The relative motion between the grain and other
a come a dominant factor affecting and regulating static plasma species is not the only mechanism which can be
l
p and dynamical properties of the grain component. A re- responsible for the momentum transfer. The latter can
. markable example wherein the momentum exchange in occur even in the absence of net flows of plasma species
s
c ion-grain collisions plays a crucial role is the formation relative to the grain. A relevant example is the ther-
i of a void in the central part of an rf discharge in ex- mophoretic force (terms “thermal force” and “radiome-
s
y periments under microgravity conditions [4–7]. In this ter force” are also employed in the literature [28]). It
h case, the momentum transfer rate from the ions stream- describes the phenomenon wherein small particles, sus-
p ing outwardsexceedsthe electricalforce(pointing to the pended in a gas where a temperature gradient exists
[
center), leaving a void. (but macroscopicflows are absent),experience a force in

1 Normally, the momentum transfer is associated with the direction opposite to that of the gradient. Elemen-
v the relative motion between the grain and other plasma tary consideration of this phenomenon has been already
3 components. The related forces are then called “drag givenbyEinstein[29]andthenmanyothersinvestigated
2
forces” (terms friction and resistance force are also rele- this topic in detail (see e.g. Refs. [30–32] and references
9
3 vant,especiallywhenthegrainmoveswithrespecttothe therein). In the context of complex plasmas, Jellum et
. stationary plasma background). Usually the mean free al.[33]wereapparentlythe firstwhorecognizedthe pos-
9
pathsofallplasmaspecies(neutrals,ions,andelectrons) sibilitytomanipulatetheparticlesingasdischargesusing
0
are very long compared to the grain size [ (µm)], thethermophoreticforce. Sincethen, applyingtheverti-
2
1 whichcorrespondstothelimitoflargeKnuds∼enOnumber. caltemperature gradientsto compensate for the particle
: The neutral drag force in this regime was calculated al- gravity has become a standard technique for controlled
v
mostacenturyagobyEpstein[8]andisoftenreferredto particle manipulationin laboratoryexperiments [34–37].
i
X asEpsteindrag. The iondragforceis considerablymore In complex plasmas, the thermophoretic force has its
r complicated issue. First, in addition to direct ion-grain counterparts associated with the charged electron and
a collisions,elasticionscatteringinthe electricalpotential ioncomponents,providedthecorrespondingtemperature
around the grain has to be taken into account. Second, gradients are present. We will refer to these forces as
theion-graininteractionundertypicalexperimentalcon- the electron and ion thermal forces to distinguish them
ditions is not weak, which implies that the conventional from the conventional drag forces. To the best of our
theory of Coulomb scattering is inapplicable to describe knowledge, these forces have not yet been considered in
elastic ion-grain collisions [9, 10]. This explains why the theliteratureinanyreasonabledetail. Themainpurpose
problem of the accurate determination of the ion drag ofthisLetteristofillthisgap. Inthefollowingwederive
force in complex plasmas has received considerable at- the expressions for the ion and electron thermal forces
tention in the last decade, including various theoretical acting on a highly charged grain immersed in a weakly
ionized plasma subject to temperature gradients. We
analyzethedirectionsandthemagnitudesoftheseforces
and show that they exhibit interesting and non-trivial
∗AlsoatJointInstituteforHighTemperatures, Moscow,Russia behavior. Comparison with other forces indicates that
2
the thermal forces can provide important contributions where the remaining integration is over the reduced ve-
to the net force balance in complex plasmas. locity x=v2/2v2 . Let us first apply this ansatz to cal-
Tα
Thegeneralexpressionfortheforceassociatedwiththe culatethethermalforceactingonanon-chargedparticle
momentumtransferfromalightspeciesαtothemassive in a non-ionized atomic gas – thermophoretic force. In
grain at rest is thiscasethemomentumtransfercrosssectionisvelocity-
independent, σ (x) = πa2, and the integration yields
n
F =m vvσ (v)f (v)d3v, (1) F = (8√2π/15)(κ a2/v ) T ,whereaistheparti-
α αZ α α Tn − n Tn ∇ n
cle radius. This coincides with the celebrated expression
where m , f (v) and σ (v) are the corresponding mass, by Waldmann [30]. The force pushes the particles into
α α α
velocity distribution function, and (velocity dependent) the region with lower gas temperature. Physically, this
momentum transfer cross section (α = n,i,e for neu- is because hotter atoms transfer more momentum to the
trals,ions,andelectrons,respectively). The netmomen- grain than the colder ones.
tum transfer occurs when the velocity distribution has It is convenient to write the generic expressionfor the
some asymmetry (e.g. relative motion). Assuming weak thermal forces in complex plasmas as
asymmetry,wewrite f (v) f (v)+f (v), wherethe
α α0 α1
symmetric component fα0 i≃s taken to be Maxwellian F = 8√2πκαa2∇TαΦ . (4)
Tα α
− 15 v
f =n (m /2πT )3/2exp( m v2/2T ). Tα
α0 α α α α α
−
Here nα and Tα are the density and temperature (in en- For the neutral component (thermophoresis) Φn = 1.
ergyunits)ofthe species α. The asymmetriccomponent The factors Φi(e) account for the electrical interactions
f , which gives contribution to the integral in (1), de- betweentheions(electrons)andthechargedgrain. They
α1
pends on the nature of the anisotropy. In the case of depend on the shape of the plasma electrical potential
subthermal drifts with relative velocity u (u <v ) it aroundthegrain. WeusetheconventionalDebye-Hu¨ckel
α α Tα
reduces to fα1 ≃ fα0(vuα/vT2α), where vTα = ∼Tα/mα (gYrauskianwcah)afrogrema,nφd(rλ)≃is (tQhe/rp)laexsmp(a−src/rλe)e,nwinhgerleenQgthis.thIne
is the thermal velocity. This ansatz corresponpds to the
low-temperature gas discharges the dominant charging
conventional calculation of the neutral, ion and electron
processisthecontinuousabsorptionofelectronsandions
drag forces and has been thoroughly investigated ear-
on the grain surface. In this case the charge is negative
lier[2,3,8,9,11,27]. Ifthedriftisnotsubthermal,non-
and proportional to the product of the grain radius and
linearized shifted Maxwellian distribution function [14]
the electron temperature, viz. Q = z(aT /e), where z
or another distribution function, appropriate for the sit- − e
isthecoefficientoforderunity,whichdependsonplasma
uation considered [13, 38, 39] has to be used in Eq. (1).
parameter regime [41]. The screening comes from redis-
The focus of this Letter is on a complementary situa-
tribution of plasma electrons and ions in the vicinity of
tion when relative drifts are absent, but the momentum
thechargedgrain. Intheabsenceofsubstantialionflows
transfer do occur due to the net momentum flux caused
andstrongnonlinearitiesinion-graininteraction[41–43],
by the temperature gradients. In this case, the asym-
the screening length is approximately given by the ion
metric part of the velocity distribution function of the
Debye radius λ λ = T /4πe2n , providedT T .
component α can be approximated as ≃ Di i i e ≫ i
Althoughitiswellrecognpizedthatthelongrangeasymp-
m κ f m v2 tote of the electrical potential can be modified (by e.g.
f α α α0 1 α v T , (2)
α1 ≃ n T2 (cid:20) − 5T (cid:21) ∇ α continuous plasma absorption on the grain surface [44–
α α α
47]or plasma ionization/recombinationeffects [48]), this
where κ is the thermal conductivity of the species α.
α is expected to affect merely grain-graininteractions, but
This form ensures that the self-consistent density gradi-
not the momentum transfer from the ions and electrons.
ent and/or electric field (for charged species), build up
Electrons. Normally, the grainradius in complex plas-
in response to the temperature gradient,resultin no net
mas is much smaller than the plasma screening length.
flux: j = vf d3v =0 (i.e., u =0). In addition, Eq.
α α1 α In this case, the electron-grain interaction can be called
(2) obviousRlysatisfies the Fourier’s law for heattransfer: “weak”,inthe sensethatitsrange–the Coulombradius
qα = (mαv2/2)vfα1d3v = κα Tα. Equation(2)isan Re za – is much smaller than the screening length
approxRimation,whichisexac−ton∇lyforthespecialcaseof λC[3]∼. ThisimpliesthattheCoulombscatteringtheoryis
r−4 interactions (for the rigorous mathematical treat-
appropriate to describe electron-grain elastic collisions.
∝
mentofnon-uniformgasessee Ref.[40]). Itsaccuracyis,
The general momentum transfer cross section for scat-
however,more than acceptable for our presentpurposes,
tering in the Coulomb potential Q/r is
especially in view of further simplifications involved in
treating electron- and ion-grain collisions. R2(v)+ρ2 (v) 1/2
Substituting Eq. (2) into (1) we get after integration σs(v)=4πR2(v)ln α max , (5)
α α (cid:20)R2(v)+ρ2 (v)(cid:21)
over the angles in spherical coordinates α min
∞
FTα = 1156√κα2∇πvTTαα Z0 x2(52 −x)exp(−x)σα(x)dx, (3) wmhuemreimRpαa(vc)t p=ar(a|Qm|eet/emr αρvm2a)x aisndneαce=ssaer,yi.toTahveoimdatxhie-
3

InFigure1thecurveseparatingthepositiveandnegative
values of Φ is plotted in the plane (a/λ, z). For most
e
0.8 experimental conditions we expect Φ <0, although the
e
transition line does not seem unreachable [50].
Ions. The Coulomb radius of ion-grain interaction
z
< 0 Ri zaτ isnotnecessarysmallcomparedtotheplasma
0.6 e scCre∼ening length due to the presence of a (normally)
large factor τ = T /T – electron-to-ion temperature ra-
e i
> 0
e tio [3]. The interactionrangecanexceedλandconsider-
ableamountofmomentumtransfercanoccurforimpact
0.4
0.01 0.1 parameters beyond λ. This implies that the standard
a/
Coulomb scattering approach is inappropriate [9, 10]. It
FIG. 1: The transition curve separating regions of positive
makessensetoconsidertworegimesofionscatteringsep-
and negative electron thermal force in the plane of reduced
arately. In the regime of moderate ion-grain interaction,
grain size a/λ and charge z.
an extension of the standard Coulomb scattering theory
ispossiblebytakingintoaccountthemomentumtransfer
from the ions that approach the grain closer than λ [9].
logarithmic divergence of the cross section. In the stan-
dard Coulomb scattering theory ρmax = λ, and this This results in ρmax = λ 1+2Ri(v)/λ. This approxi-
choice is appropriate for the weak electron-grain inter- mation demonstrates goopd accuracy for Ri(v) < 5λ and
action [3, 27]. The minimum impact parameter ρmin is reducestoρmax =λinthelimitofweakion-grai∼ninterac-
zero in the standard Coulomb scattering theory. In the tion [Ri(v) λ]. The impact parameter corresponding
≪
consideredcase, however,some electrons are falling onto to the ion collection is ρ (v)= a 1+2R (v)/a and the
c i
thegraininsteadofbeingelasticallyscatteredinitselec- corresponding collection cross secption is σc(v) = πρ2c(v).
trical potential. Consequently, ρ should be set equal Combiningthecontributionsfromcollectionandscatter-
min
tothemaximumimpactparametercorrespondingtoelec- ing yields:
troncollectionbythegrasin,ρ . Theorbitalmotionlim-
c Φ =1 1zτ z2τ2Λ , (8)
ited (OML) theory yields [49] ρc(v) = a 1 2Re(v)/a i − 2 − i
−
for 2Re(v) < a and ρc = 0 otherwise (spufficiently slow where Λi is the (modified) ion-grain Coulomb logarithm
electrons cannot be collected even in head-on collisions
∞
2x(λ/a)+zτ
with the grain due to electrical repulsion). The momen- Λ = h(x)ln dx. (9)
i
tum transfer cross section for electron collection is sim- Z0 (cid:20) 2x+zτ (cid:21)
ply σc(v) = πρ2(v). Expressing now velocity in terms
e c In typical complex plasmas with λ a, τ 1, and
of x = v2/2v2 , substituting σ (x) = σs(x)+σc(x) into ≫ ≫
Te e e e z 1, the Coulomb logarithm can be roughly estimated
Eq. (3) and performing the integration with the appro- as∼Λ ln(1+β−1), where β = β(v ) = (a/λ)zτ and
priate limits yields: i ≃ T T Ti
β(v) = R (v)/λ is the ion scattering parameter [9]. The
i
Φ =(1+ 3z+z2)exp( z) z2Λ , (6) approach is reliable up to βT < 5. Since the product zτ
e 2 − − 2 e is normally quite large, zτ ∼(102), it follows that (i)
∼ O
thescatteringprovidesdominantcontributiontothemo-
where Λ is the electron-grainCoulomb logarithm
e
mentum transfer; (ii) Φ < 0, i.e. the grains are pushed
∞ ∞ i
Λ = h(x)ln 1+ 4λ2x2 dx 2 h(x)ln 2x 1 dx.into the region with higher ion temperature. The physi-
e Z0 (cid:16) a2 z2(cid:17) − Zz (cid:0) z − (cid:1) calreasonisagainfastdecreaseofthescatteringmomen-
(7) tum transfer cross section with the ion velocity, so that
The function h(x) is defined as h(x)= 5 x exp( x). cold ions transfer more momentum to the grain. This is
2 − −
Equations (4), (6) and (7) constitute th(cid:0)e exp(cid:1)ression for notauniqueexamplewhentheforceactingonacharged
the electron thermal force. grain is directed oppositely to the net ion momentum
For an uncharged particle (z = 0) we have Φ = 1, as flux. Another example is related to the sign reversal of
e
expected. In this case the electron thermal force pushes the ion drag force acting on an absorbing particle in the
the grains in the direction of lower electron tempera- highlycollisional(continuum)limit[41,51–54],although
ture,similarlytothethermophoreticforce. Accordingto the detailed physics is different.
Eq. (6) the contributions from collection and scattering In the regime of very strong ion-grain interaction, the
are directed oppositely to each other. For a sufficiently scattering is characterized by the formation of a poten-
high charge, the scattering part becomes dominant and tial barrier for ions with impact parameters above the
the force reverses direction. The physical reason is that criticaloneρ∗,whichconsiderablyexceedsλ. Forρ<ρ∗
the (Coulomb) scattering momentum transfer cross sec- scattering with large angles occurs, which gives the ma-
tionquicklydecreaseswithvelocity,sothatthecoldelec- jorcontributiontothemomentumtransfer. Relativeim-
trons are more effective in transferring their momentum portance of momentum transfer from distant collisions
upon scattering. In this regime (Φe < 0), the thermal with ρ > ρ∗ decreases rapidly with the increase in ion-
forceactsinthedirectionofhigherelectrontemperature. graininteractionstrength. The detailed considerationof
4
the momentum transfer in this regime can be found in wellasfromnumericalmodeling[61,62]allrevealedgra-
Refs.[10,11]. InthelimitR (v) λ(i.e. β(v) 1[55]) dients in T of the order of (eV/cm). The spatial dis-
i e
≫ ≫ O
thetotalmomentumtransfercrosssection(collectionand tribution of T generally depends on discharge geome-
e
scattering)canbeapproximatedasσ (v) πρ2(v),with try, plasma parameters,and can be affected by the pres-
Σ ∗
ρ2∗ λ2 ln2[β(v)]+2ln[β(v)] . The integ≃ration yields ence of grains [61]. Let us therefore assume Te = 1
≃ ∇
(cid:8) (cid:9) eV/cm and restrict ourselves to the comparison of the
Φ λ 2 x∗x2h(x) ln2 βT +2ln βT dx. absolute magnitudes of the electron thermal force and
i ≃−(cid:0)a(cid:1) Z0 h (cid:16)2x(cid:17) (cid:16)2x(cid:17)i other forces acting on a small grain in the bulk of a
(10) gas discharge. We take the plasma parameter set from
The upper limit of integration can be chosen as x∗ = Ref. [14] used to estimate relative importance of the
β /2toavoidunphysicalregimeofnegativecrosssection electrical and ion drag forces: argon gas at a pressure
T
in this approximation. However, due to the presence of p = 10 Pa (n 2 1015 cm−3), n = n = 3 109
n e i
exponentiallysmallterminh(x),takingx∗ = willnot cm−3, Te = 1 ≃eV, ×Ti = Tn = 0.03 eV, a = ×1 µm
∞
produce big errors for β 1. Note that the sign of Φ (a/λ 0.04),andz 3(estimated fromthe collisionless
T i
≫ ≃ ≃
is changed from negative to positive upon increasing β OML theory). From Eqs. (6) and (7) we get Φ 22.
T e
≃−
(in the considered approximation this happens at β The electron thermal conductivity in a weakly ionized
T
≃
120; the exact value is rather sensitive to the functional plasma is κ (5n T /2m ν ) with ν n σ v
dependence of the cross section on the ion velocity and, (σ 10−1e6 c≃m2 foer Te 1e eeVn in argonen). ≃ThisnreesnulTtes
en e
thus, is subject to significant uncertainty). Physically, in F ∼ 2 10−8 dyne.∼This force is more than three
Te
≃ ×
thesignreversaloccursbecausescatteringintheYukawa timeslargerthantheforceofgravity,experiencedbythe
potential in the limit of strong interaction tends to that grain of this size (and material density 1.5 g/cm3) in
on a hard sphere with the radius ρ∗, which only weakly ground-based experiments. The equivalent electric field
(logarithmically) depends on the ion velocity. E∗,forwhichFTe QE∗,isE∗ 5V/cm. Suchafield
≃| | ≃
Having derived the expressions for the ion and elec- wouldproduceasignificantplasmaanisotropy,character-
tron thermal forces, let us discuss their role in practi- ized by superthermal ion flows (Fig. 4b from Ref. [14]).
cal situations. As a first example, consider creating a Finally, this magnitude is comparable to the maximum
gradient of the neutral gas temperature to compensate value of the ion drag force the grain can experience in
forgraingravityinlowionizedlaboratorygasdischarges subsonic ion flows for this set of parameters( 5 10−8
∼ ×
(thermophoretic levitation)[33, 34]. Inthis case,the ion dyne according to Fig. 4a from Ref. [14]).
temperature is likely coupled to the neutral gas temper- Observationof large grains,trapped in standing stria-
ature, T T . The neutral and ion thermal force tions of a stratified dc glow discharge [63], provides an-
i n
∇ ≃ ∇
are directed in the opposite directions (in the regime of other example where the electron thermal force can play
weak and moderate ion-grain interaction). Their ratio a significant role. The electron temperature is known to
is F /F (n /n )(σ /σ )z2τ2Λ , where we take increase considerably in the head of the striation, which
Ti Tn i n nn in i
| | ∼
intoaccountthatκ (n /n )(v /σ )inweaklyion- can explain that even very massive grains can be con-
α ∼ α n Tα αn
ized plasmas (σ and σ are the transport cross sec- finedthere[63]. Othersituationswhereinplasmathermal
nn in
tions for neutral-neutral and ion-neutral collisions, re- forces can play significant role include charged aerosols,
spectively). Assumingσ σ [56]andΛ 1wefind dustinplanetary(e.g.,Earth)atmospheresandinfusion
in nn i
thatF dominatesoverF ∼forn /n >(zτ)∼−2 10−4. devices. In the latter case, however, the regime of fully
Ti Tn i n
For a more typical (in laboratory gas d∼ischarges)∼ioniza- ionized magnetized plasma, considered recently [64], is
tion fractionn /n 10−6, the ion thermal force is only apparently more relevant.
i n
∼
slightly reducing the effect of thermophoretic force. Altogether,there is a certainconfidencethat the ther-
Theelectronthermalforceisexpectedtoplaymoreim- malforces(especiallyfromelectrons)canhavesignificant
portant role. Spatially resolved probe measurements of impact on the particle transport in many realistic situ-
the electron energy distribution function in various gas ations. This finding should be properly addressed when
discharges have previously shown that significant elec- developing new (and updating existing) numerical codes
trontemperaturegradientscanexist[57,58]. Fordevices to model particle transport in various plasma environ-
used in complex plasma research,the results from probe ments. Expressions derived in this Letter can serve as a
measurements[59],opticalemissionspectroscopy[60],as theoretical basis for further detailed studies.
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