Table Of ContentMobility in a strongly coupled dusty plasma with gas
Bin Liu and J. Goree
Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242
(Dated: January 31, 2014)
Themobilityofachargedprojectile inastronglycoupleddustyplasmaissimulated. Anetforce
F,opposed bya combination of collisional scattering and gas friction, causes projectiles to drift at
amobility-limited velocityu . Themobility µ =u /F oftheprojectile’s motionisobtained. Two
p p p
regimes depending on F are identified. In the high force regime, µ ∝ F0.23, and the scattering
p
4 cross section σ diminishes as u−6/5. Results for σ are compared with those for a weakly coupled
s p s
1 plasmaandfor two-bodycollisions inaYukawapotential. Thesimulation parameters arebasedon
0 microgravity plasma experiments.
2
n PACSnumbers: 47.55.D-,47.60.-i,47.20.Ky,63.22.-m
a
J
I. INTRODUCTION coefficients,suchas mobility,to be differentaswell. The
0
3 velocity relaxation rate, which is related to the mobil-
AprojectiledrivenbyanetforceFthroughamedium ity, has been studied in ultracold plasmas with an ionic
] oftargetparticleswillcollidewiththem, anditwilldrift Coulomb coupling parameter Γ of order unity [26, 27].
h
p in the direction parallel to F at an average velocity up. The mobility and drift motion have also been studied in
- This motion is described by the transport coefficient for several two-dimensional strongly coupled Coulomb sys-
m mobility tems, which are not plasmas but have similar Coulomb
collisions; these include colloidal crystals [28], and elec-
s
a µp =up/F. (1) trons[29–31]andions[32]onthesurfaceofliquidhelium.
l
p Thetargetparticlescanbeinanystateofmatter. Re- Tothebestofourknowledge,mobilityhasnotbeenstud-
. search on mobility and diffusion of electrons and ions ied much in strongly coupled plasmas with liquid-like
s
c began over 100 years ago for gases [1], and later for conditions Γ > 10, three-dimensional Yukawa systems,
si solids [2, 3] and weakly coupled plasmas [4, 5]. or dusty plasmas. Other transport processes including
y Here, the target we investigate is a strongly coupled diffusion [33–35], viscosity [36–39], and thermal conduc-
h plasma, in which the potential energy exceeds the ki- tivity [40] have been studied for dusty plasmas. We ex-
p netic energy, so that particles self-organize into a liquid- pectmobilityinadustyplasmatobe determinedbytwo
[ like or solid-like structure [6]. Strongly coupled plasmas effects experienced by the dust particles: Coulomb colli-
1 in nature include neutronstar crusts [7], giantplanetin- sions (which in dusty plasmas are modeled by a Yukawa
v teriors, and white dwarf interiors [8]. Strongly coupled potential)andfrictionaldragontheambientneutralgas.
4 plasma can be realized in the laboratory using a dusty The conditions we investigateareat a moderate value of
7 plasma, which is a four-component mixture of electrons, Γwherethestronglycoupledplasmaisinadenseliquid-
9
ions, neutral gas, and micron-size particles of solid mat- like state.
7
. ter [9–22]. The solid particles, which we call dust parti- There are at least two regimes of projectile transport,
1 cles, become strongly coupled due to their largecharges. depending on the driving force F. In what we term the
0 Weinvestigateasystemthathindersaprojectile’smo- low regime, F is small so that projectiles are near ther-
4
tionbytwotypesofcollisions: Coulombcollisionsamong mal equilibrium with target particles. In what we term
1
: strongly coupled dust particles, and the friction due to the high regime, F is so large as to cause a considerable
v collisions of gas atoms with the projectile. The latter is departure from the thermal equilibrium.
i
X modeled asa simple dragterm, whichdoes not requirea The literature for ions in gases is well developed, and
r particledescriptionofthegasatoms. Thegasfrictionhas many experiments have been reported [41–43]. It is
a been reviewed in [20, 23], and binary Coulomb collisions known for that system that the transport in the high
for an isolated pair of dust particles are reviewedin [24]. regime is different from the low regime: the mobility is
Inastronglycoupledplasma,Coulombcollisionsaredif- notconstantbutvarieswithF ina waythatdepends on
ferentfromcollisionsofanisolatedpair (i.e., binary)be- the scattering potential [42, 44].
cause the target particle in a strongly coupled plasma For the denser physical system of liquids instead of
does not move freely as it recoils. Instead it collides im- gases,whileitispossibletopropelasmallprojectile,the
mediately with other target particles, which collide with target’shighdensityposesagreatdifficultyforattaining
others in a chain of collisions. In this way, the Coulomb asuperthermalspeedfor the projectile. Consequently,it
collisional process is collective and not binary [25]. To is difficult to performexperiments to study mobility in a
simulate this system, we require a model that represents liquid in a high regime. This difficulty can be overcome
dust particles as discrete particles. by using a dusty plasma as a model system for a liquid
Since the collisions are so different in weakly and because adusty plasma hasa smallvolumefraction[45].
strongly coupled plasmas, one would expect transport Motion of projectile dust particles through a cloud of
2
target dust particles has been observed in recent micro- force has an amplitude set by the fluctuation-dissipation
gravity dusty plasma experiments [46–52]. For these ob- theorem, hζ(t)ζ(0)i = 2νmk T δ(t). We integrate the
B gas
servations, the target and projectile particles generally equationsofmotionusinganalgorithmthatincorporates
have different sizes. Here we simulate drifting motion as thefrictionandthe fluctuatingforce[57]. Toaccountfor
in the experiments of [46–48], except that we consider particleheatingmechanismsinadditiontogas-atomcol-
individual projectiles, not dense beams of projectiles, in lisions, we augment the Markovian fluctuating force by
order to determine a projectile’s mobility coefficient due a multiplier γ [58, 59], which would be unity for ther-
tocollisions,withoutanycooperativemotionamongpro- mal equilibrium. The terms in Eqs. (5) and (6) with
jectiles. A projectile drifts througha targetdue to a net gradients are the electric force due to particle-particle
force F; this net force could be due to an imbalance of interaction −∇φ and confinement −∇Φ. To simulate a
electric and ion drag forces, as can happen for different 3D dusty plasma with a uniform spatial distribution, we
dust particle sizes. Due to their different sizes, a projec- choose a confining potential Φ that is mostly flat, with
tileparticledrifts,whilethetargetparticlesareinaforce a rising parabola at the edge. Projectiles introduced at
equilibrium and do not drift. This situation is possible theedgearespacedsufficientlysothattheyinteractonly
because of different scalings of forces with a particle’s with target particles and not with other projectiles, as
size [53]. demonstrated in Appendix A.
Inthis paper,our mainresultsare: (1) acharacteriza- ThenetforceF canarisephysicallyfromanimbalance
tion of two regimes of projectile transport, (2) an eval- of the ion drag force and other forces, because the ion
uation of mobility coefficient µ for projectiles, and (3) dragforcedependsonparticlesize[60]. Inthispaper,we
p
a determination of the scattering cross section σ as a treat F simply as an adjustable input parameter, which
s
function of the drift velocity u . wevaryoverawiderangebracketingthevaluesweexpect
p
in an experiment.
We use simulation boxes of two sizes. A larger force
II. SIMULATION F requires the larger box since the projectiles move
a greater distance. We verified the simulation gener-
Weperformathree-dimensional(3D)Langevinmolec- ates the same results with both box sizes in the range
ular dynamics simulation of dust particle motion includ- 6.8<F <10. Theboxdimensionsare: 132×81×69.3λ3D
ing Coulomb collisions. Dust particles also experience for the smaller and 263× 122× 104 λ3D for the larger
frictional drag on the gas atoms. Due to their charge boxes. Boundary effects, such as the initial acceleration
Q, dust particles also repel one another with a Yukawa of the projectile when it is released, are avoided by ana-
potential, φ(r) = Q2e−r/λD/4πǫ r, where the screening lyzing data only in the central volume that excludes the
0
length λ due to electrons and ions reduces the interac- edges. Further details of the simulation method are in
D
tion at a large distance of r. This many-particleYukawa Appendix A.
system is described by dimensionless parameters Our simulation parameters are motivated by ground-
based[61]andmicrogravity[48,62]experimentswiththe
Γ=Q2/4πǫ ak T , (2) PK-4 instrument. The polymer particles have a density
0 B t
1.51 g/cm3. The projectiles have a radius 0.64µm while
wherek T isthekinetictemperatureofthesystem,and the targets are 3.43 µm radius with mass m = 2.55×
B t t
10−13kg. For neon at 50 Pa pressure, the ion and gas
κ=a/λD, (3) temperaturesareassumedtobe0.03eV,andtheelectron
densityandtemperatureareestimatedas2.4×108 cm−3
where and 7.3 eV [58], so that λ = (λ−2 +λ−2)−1/2 = 8.3×
D De Di
a=(3/4πn )1/3 (4) 10−3 cm. OurprojectileparticlechargeisQp =−1590e,
t
based on Fig. 7(a) in [61], and our targetparticle charge
isQ =−8520e. Thegasfrictioncoefficients[20,63]are
is the Wigner-Seitz radius, and n is the number density t
t ν = 273 s−1 and ν = 51 s−1. The characteristic time
of dust particles. For microgravity experiments, typical p t
parametersarent =5×104 cm−3 [54]anda=0.017cm. for collective motion in the target is ωt−1, where
We integrate the equations of motion [55, 56]
ω = Q2n /ǫ m , (7)
t t t 0 t
mmptxx¨¨ji == −−ννtpmmtpxx˙˙ij++γγζζtip(jt()t)−−Pkk∇∇φφikjk−−∇∇ΦΦ+F((56)) which has a value of 157qs−1.
P
for target and projectile particles, respectively. A con-
stant net force F = Fxˆ acts only on the projectile. The III. TARGET CONDITIONS
first two terms on the right hand side are the frictional
force with a coefficient ν and the Markovian fluctuating Since transport can vary with temperature, we per-
forceζ(t);bothoftheseareduetocollisionsofgasatoms formsimulationsfortwotargettemperatures,T =10T
t m
of temperature T with dust particles. The fluctuating and 2T , correspondingto Γ=62 and 310,respectively.
gas m
3
2 (a)
(a) G = 310 10
T = 2T
t m
a
z / 0
1
g(r)
-10
F = F x
-20 -10 0 10 20
x / a
0 60
2 (b)
x
(b) G = 62
Tt = 10Tm nt / a40
e
m
e20
c
g(r) 1 displa y
0
z
0 20 40 60 80
tw
t
0
0 5 10 15
r / l FIG. 2: (Color online) (a) A typical projectile trajectory
D
shown as a curve projected onto the x−z plane, from a run
atT =10T . Alsoshownisasnapshotoftargetparticlepo-
FIG.1: Characterization ofsimulationconditionsforκ=2.4 t m
sitions within a slab of thickness ∆y =1.7a. (b) Time series
attwotemperatures: (a)Γ=310orT =2T and(b)Γ=62
t m of displacements of a representative projectile, showing drift
or T = 10T . The pair correlation functions shown here
t m in the xˆ direction and random walk or diffusion in the yˆand
indicate that thetarget has a liquid-likestructure.
zˆ directions. Data shown are for F = 3.8. The time series
duration corresponds to 610 msin physical units.
Here, T is the melting point [64]. These two kinetic
m
temperatures, which are T = 8.3 and 1.66 eV in physi-
t
cal units, are achievedby selecting the multiplier γ =16
IV. RESULTS
and 7, respectively. For all our simulations, κ = 2.4,
corresponding to n =3×104 cm−3 and a=0.02 cm.
t
To characterize the target, we performed a simulation We present our results in dimensionless units. We
without projectiles. Figure 1 shows the pair correlation normalize distance, time, velocity, force, temperature,
function g(r) from our simulation for these two condi- and mobility by a, ω−1, aω , m ω2a, m (ω a)2, and
t t p t t t
tions. (m ω )−1, respectively.
p t
The 3D structure of the target,for T =2T , can also
t m The projectile motion, Fig. 2(a), reveals the drift par-
be viewed from a movie which we include in the Sup- allel to F = Fxˆ, and random scattering in the perpen-
plemental Material [65]. This movie shows a still image
dicular direction. In Fig. 2(b), the projectile’s drift is
of the three-dimensional structure, viewed from a time-
seen in the time series for the displacement x, which has
varying angle.
a slope that corresponds to the drift velocity. The per-
As the projectile moves through the target there is a pendicular displacements y and z exhibit only a random
shear motion on a microscopic scale, i.e., a scale anal- walk.
ogous to the molecular scale in a simple liquid. If the
shear motionwere insteadon a macroscopicor hydrody- Wecalculatetheperpendicularrandomvelocityvp⊥ =
namic scale, with a gradient length of at least a dozen (x˙2 + y˙2)1/2, and we calculate the parallel drift veloc-
interparticlespacing[66],the target’scollectivebehavior ity u by fitting the x displacement as in Fig. 2(b) to
p
could be described by its viscosity. We determined this a straight line. Results for vp⊥ and up are presented in
viscosity, using the standard Green-Kubo method [67], Fig. 3 and Fig. 4(a), respectively. These velocity results
to have a value 0.065 and 0.044 n m a2ω for T = 2T are presented using log-log axes so that we can identify
t t t t m
and 10T , respectively. In physical units, these viscosi- power-law scalings. We will next use the magnitude of
m
ties are 3.1×10−9 and 2.1×10−9 g mm−1 s−1. Later vp⊥ to identify regimes of the projectile motion, and af-
wewillmakeuseofthe ideathattheviscosityislowerat terthatwewillusethedriftvelocityu todeterminethe
p
higher temperatures. mobility µ and the scattering cross section σ .
p s
4
3
low regime 10Tm high 10
2
T = 10T
t m
/war va^ pt1 NN == 1527 860000 /wcity ua pt 1 Tt = 1NN0 T==m 1527 860000 slope = 1.23
cul elo
erpendi Tt = 2Tm drift v Tt = 2NT m= 12 800
p N = 12 800 N = 57 600
N = 57 600 0.1
low regime high slope = 1.0 (a)
2T
m
0.2
1 10 0.7
F / mpw t2a 0.6 expected maximum due to gas w t /np
0.5 10Tm
FIG.3: (Coloronline)Characterization ofregimesusingpro- low regime high
aimljnaercttaeiteloiirslznseeeese’dscntwrihaaoenenrnreddseoptombhfeyertfshotpωreretamaeand,essydiwvtmpihwo⊥pinicttihohnbtehetttshawwes(oeddeaasniirsvzehteachesltdueioNmelni3ni⊥fs1eo.sri4)Fd.te.mhnSTetmpiwfien/eeousdd.mreibbSsgyeiinmmrtohueores--f mwbility mppt 00..34 slo p e = 0.2 3
o
target particles. m
0.2
A. Characterization of regimes
As our first chief result, we will identify the transi- low regime 2T high (b)
m
tion between regimes of the projectile’s motion. In the 0.1
1 10
high regime, the perpendicular random velocity vp⊥ in- F / m w 2a
creaseswithF,asprojectilesgainsignificantrandomen- p t
ergy from the acceleration corresponding to F, while in
FIG.4: (Coloronline)(a)Projectilespeedu inthedirection
the low regime vp⊥ has a constant value, Fig. 3. kF. This drift velocity scales as u ∝F1.01p±0.12 in the low-
p
We identify the transition between regimes as the in- forceregimeandu ∝F1.23±0.02 inthehigh-forceregime. (b)
p
tersectionofasymptotesinFig.3. Theforceatthetran-
MobilitydependencewithF. Inthehighregime(largeF),we
sitionisfoundtobeF ≈2or3,asmarkedwitharrowsin find µ ∝ F0.23. We expect the maximum mobility limit to
p
Fig.3, for Tt =2Tm or10Tm, respectively. We note that be (mpνp)−1, as indicated by the dashed line, corresponding
these values for the transition coincide with the condi- to the gas drag on a projectile without Coulomb collisions.
tionsthatyieldadriftvelocitycomparabletotheequilib- The power law scaling of the mobility is the same for two
riumthermalvelocityoftheprojectile,u ≈ k T /m . temperatures we simulated.
p B t p
The latter finding is comparable to the casepfor ion pro-
jectiles in a gas [43].
projectile,themobilitywouldbelimitedonlybygasfric-
tionanditwouldhavealimitingvalueof0.58(m ω )−1,
p t
B. Evaluation of mobility coefficient asindicatedbythedashedline. Allourdatapointsfrom
the simulation lie below this limiting value due to the
combinationofCoulombcollisionsandgasfriction,which
To determine the mobility µ = u /F, which is our
p p
second chief result, we divide the drift velocity u in both retard the projectile’s motion in response to the
p
force F.
Fig. 4(a) by the force F, which is the horizontal axis
in that graph. The resulting mobility data are presented A power-law scaling for the mobility can be found by
in Fig. 4(b). The mobility typically has a value in the noting that data lie mostly on straight lines, in the log-
range 0.16 to 0.5 (m ω )−1, for the target temperatures log plots of Fig. 4. By fitting, we find that u varies
p t p
and range of forces that we consider. In physical units, as ∝ F1.23±0.02 in the high regime, where nonequilib-
this rangecorrespondsto6.1×108 to 1.92×109 g−1sfor rium effects become significant, as comparedto the scal-
the PK-4 parameters listed in Sec. II. If there were no ing F1.01±0.12 for the low regime. Correspondingly, the
Coulomb collisions to retard the motion of the drifting mobility u /F is essentially constant in the low regime,
p
5
while it has an exponent of 0.23, i.e., µ ∝ F0.23, in the
p
(a) T = 2T
high regime. Expressing the scaling in terms of drift ve- 1000 t m
locity instead of force, we find µ ∝ u0.19 in the high
p p
regime. 2
D
We expectthat these scalinglaws forthe mobility will l
failatevenhigherforcesbecausethemobilitycannotex- p/ s
cvoeafel0ud.e6t4hiseµ(mlmimpriνatpdin)i−ugs1v,ianwluhaeic5dh0ueiPstao0N.5ge8ao(snmfrpgiωcatsti.)o−nT1.hfTiosrhliiasmlpiitmariistt,iicnilnge section 100
s
effect, a third regime, which we did not explore because s
iltarwgoeuilnderxepqueririmefeonrtcsessutchhatawsePeKx-p4e.ctHtoowbeeveurn,awtetaeinxpabeclyt cros 10 slope = -
ananalogouslimitmustoccurinacolloiddueto friction 1.2
1
on the solvent, and that limit might be easily attained
becauseofthestrongerfrictioneffectforaliquidsolvent,
as compared to the rarefied gas in a dusty plasma.
1
The target temperature is found not to have an effect
on the mobility in the high regime. This result is seen
(b) T = 10T
by the overlapping data points in the right hand side of 1000 t m
Fig. 4(b), where the mobility obeys the same µ ∝F0.23
p
power law for both temperatures.
2
D
Temperature does, however, affect the constant value l
of the transport coefficients in the low regime. This is p/ s
s0e.1en6±on0.t0h1efolerftTsid=e2oTf F,igw.h4ic(bh),iswdhiffereerewnteffironmd µµp == sn 100
t m p o
0.29±0.02 for Tt =10Tm. ecti
Wecanspeculatewhy,inthelowregime,µpislowerfor s s
ourcoldertemperature. Asmentionedearlier,thedistur- os slo
bancecreatedamongstthetargetparticlesbythemoving cr 10 pe = -
projectileislikeashearmotionwithamicroscopicscale. 1.1
7
If it instead had a macroscopic scale, the shear motion
could be described by a hydrodynamic equation where
shear motion is opposed by dissipation characterized by
1
ashearviscosity. Itiswellknown[68]thatforastrongly 0.1 1 10
coupledplasmatheshearviscosityvariesoppositelywith drift velocity u /w a
p t
T whenT isonlyamodestmultipleofT asitisinour
t t m
case. Even though we can not apply the hydrodynamic FIG.5: (Coloronline)Scatteringcrosssectionσ fortargetat
s
equations to the microscopic shear in our target, we ex- differenttemperatures(a)Tt =2Tm and(b)Tt =10Tm. The
pectthesametendencyoftheshearmotiontoexperience scattering cross section is calculated from Eq. (8) using the
a greater dissipative resistance at a colder temperature. resultsinFig.4(b)forthemobility,whichincludestheeffects
due to gas friction. The cross section exhibits a power law
This expected tendency agrees with our finding that µ
increases with T . p scaling, which approaches σs ∝ u−p6/5 at large drift velocity,
t
for both temperatures we simulated.
C. Determination of the scaling of σ
s
Results for σ are presented in Fig. 5 as a function
s
As our third chief result, we find the slowing-down of the drift velocity u . The cross section diminishes
p
cross section σ , which is also often called a momentum with u , and in the log-log plots the data fall mostly
s p
transfercrosssection[41]. Weusetheforcebalanceequa- on a straight line, indicating that σ obeys a power law.
s
tion ν m u = F = u /µ for a projectile moving at a The power law scalings, obtained by fitting the data in
pt p p p p
constant drift velocity u , where ν =n σ u is the col- the high regime, are σ ∝ u−1.21±0.02 for T = 2T and
p pt t s p s p t m
lision frequency for projectiles to slow down. Combining σ ∝ u−1.17±0.02 for T = 10T . The exponent in both
s p t m
these equations with Eq. (4) yields an expression for σs casesis ≈−6/5. We willnextcomparethis exponentfor
ourmany-bodycollectivesystemtotheexponentfortwo
4πa2 aω 1
σ = t , (8) binary systems.
s
3 (cid:18) u (cid:19)m ω µ
p p t p
Forthefamiliarbinarysystemofafastprojectilescat-
which we will use to obtain σ from our results for u tering in a 1/r Coulomb potential, which is the case
s p
and µ . for a weakly coupled plasma, the exponent is −4, i.e.,
p
6
σ ∝ u−4. Our exponent of −6/5 is a much weaker de- in a Yukawa potential in two ways. First, the cross sec-
s p
pendence. The system we simulate is different in three tion for our dusty plasma is generally larger than that
ways. Insteadofthebinarysmall-anglecollisionsthatare of the two-body collision. Second, our data tend to ex-
typical of a weakly coupled plasma, we have large angle hibit a distinct power-law scaling for σ vs β, unlike the
s
scattering and collective effects among the target parti- two-body case, where σ does not follow a single power
s
cles, which collide with one another as they recoil. Our law scaling with β. These differences can arise from two
scattering potential is Yukawa instead of 1/r. Finally, effects that are present in the dusty plasma but not the
our system includes dynamical friction with gas atoms. binary Yukawa case: gas friction and collective effects
in the collisions in a strongly coupled plasma system, in
which the motion of a recoiling particle is hindered by
103 many body strongly coupled
interactions with neighboring target particles.
dusty plasma (from Fig. 5):
T = 2T
t m
T = 10T
102 t m V. SUMMARY
2
D
l
p/ s Insummary,weinvestigatedachargedprojectiledrift-
sn 10 icnegsstehsrtohuagthaaredusisgtnyipfilcaasnmtain,teaxkpinegriimnteontasc:coCuonutlotmwobpcrool--
o
cti lisions in a many-body strongly coupled dusty plasma,
e
s s 1 and gas friction. We determined the mobility for the
os projectileandcharacterizedthetworegimesofprojectile
cr two body Yukawa collisions: motion. For this strongly coupled plasma, the scaling of
Lane and Everhart
µ withF inthe highregime indicates ascattering cross
0.1 Hahn et al p
−6/5
section σ ∝u in the range of force we studied. Our
s p
resultsforσ arelargerthanthatfortwo-bodycollisions
s
0.1 1 10 102 103 104 in a Yukawa potential in the absence of gas. We antici-
scattering parameter, b: pate that mobility-limited drift of an isolated projectile
many-body b (QpQt / 4ep 0lD) / mptup2 throughatargetofstronglycoupleddustyplasmacanbe
two-body b (QpQt / 4ep 0lD) / mptv2 observedinfuturedustyplasmaexperimentsusingvideo
imaging. The experiment would require that the projec-
FIG. 6: (Color online) Comparison of the scattering cross tile has a different size from the target, so that there is
sectionforastronglycoupleddustyplasma(ourdataforTt = a net force that can drive the projectile while the target
2Tm and 10Tm) with that for classical two-body collision in particles remain in a non-drifting equilibrium.
a Yukawa potential by Lane and Everhart [69] and Hahn et Remaining issues that could be addressed in future
al [70]. In therange of 0.2<β <50, the cross section in our
workincludethe dependence ofprojectilemotionontar-
strongly coupled many-bodydustyplasma isgenerally larger
getparameterssuchasΓandκ,therelationshipbetween
than that for the two-body collision; it also exhibits a single
various transport coefficients, and the possibility of ex-
power-law scaling with β.
tending ourworkto other systemssuchas aYukawa one
component plasma (YOCP) [71–73].
Another binary system for comparison is a projectile
thatisscatteredbyanisolatedtargetwhichhasaYukawa
potential. This was also studied long ago [69, 70], with-
Appendix A: Simulation method
out gas. In Fig. 6, we replot our cross-section data to
compare with the binary-Yukawa data from Table II of
Here we provide further details of the simulation
Ref. [69] and Table I of Ref. [70]. As in Ref. [24], we
method.
normalize the cross section by πλ2 , and the horizontal
D
axis represents the scattering parameter,
1. Confinement
Q Q 1
p t
β(v)= , (9)
4πǫ λ m v2
0 D pt
We model a small portion of a 3D dusty plasma by
where m =m m /(m +m ) is the reduced mass, and confining particles in a finite rectangular volume. The
pt p t p t
v is the relative velocity before collisions. For our data, confining potential is flat in most of the volume, and a
we replacethe relativevelocityv (for the binarysystem) rising parabola at the edge, i.e.,
withthedriftvelocityu (whichissuitableforthemany-
p Φ=ψ(x,b)+ψ(y,c)+ψ(z,d), (A1)
body target).
where
Based on the comparison in Fig. 6, we find that the
scattering cross section for our strongly coupled dusty 0, |x|<b
ψ(x,b)= (A2)
plasma differs from that of classical two-body collisions (cid:26)mtωe2(|x|−b)2/2, |x|≥b
7
andsimilarlyforyandz. Themainvolume,wherewean- tiles are sufficiently separated to avoid cooperative mo-
alyzeourresults,hasaflatpotential,ψ =0,withawidth tion among projectiles: after injecting one projectile, we
2b, 2c, and 2d along the x, y, and z axes, respectively. wait for a time delay of 4.7ω−1 before injecting the next
t
Here, ω is a constant that characterizes the parabolic projectile, and we inject the next projectile from a dif-
e
confinement at the edge. The design of this confining ferent site separated by a distance >8a.
potentialhelps provideanumber densitythat isuniform
everywhere except within 7λ of the edge, according to We now present a simple estimate that demonstrates
D
our simulation test, with the constant ω chosen to be that a separation > 8a provides orders of magnitude of
e
Q2/4πǫ m λ3 . To avoid any boundary effects, in our suppression of any cooperative effects. There are two
t 0 t D
panalysis we will use data only from the central portion possible mechanisms for interaction among projectiles:
of the simulated volume, i.e., |x | ≤ 0.84b, |y | ≤ 0.86c, direct via pairwise repulsion and indirect via a wake-like
i i
and |z | ≤ 0.86d. We perform our simulation with two disturbance of the target medium. Pairwise repulsion is
i
system sizes, N = 12 800 and 57 600 target particles, so small at a distance >8a that it does not evensurvive
and we found no significant size effect. ourcutoffradius,mentionedabove. Thewake-likedistur-
bance of the targetmedium is conveyedby soundwaves,
the fastest of which is the longitudinal wave. This wave
2. Potential truncation willdiminish with distance fortwo reasons: a1/r2 effect
and an exponential decay due to wave damping. The
For efficiency, we truncate the Yukawa potential at a wave damping can be estimated from the sound speed
large cutoff radius of 13.25λD. At this distance the po- ≈0.33ωta, which we determine by analyzing the phonon
tential is five orders of magnitude smaller than at the spectrum for both temperatures,and a damping rate es-
distance of a nearest neighbor. timated as ωi ≥ νt = 0.32ωt. Combining these two val-
ues, we estimate that a planar longitudinal sound wave
is damped by a factor of 1/e after a distance of < 1.0a.
3. Initial configuration Using these values, we canestimate that ata distance of
> 8a, the wake-like disturbances of the medium will di-
We perform four simulation runs for each value of the minishbytwoordersofmagnitudedue tothe1/r2 effect
force F. Each run is done with a different initial con- and at least three orders of magnitude due to damping
figuration of the target particles. For each initial config- for a total of at least five orders of magnitude. Our use
uration, we record time series of particle positions and ofalaunch-siteseparationof>8aalsohelpstoeliminate
velocities for a duration of 480 ω−1. anylong-lasting“lane”effects[46–52]thatcoulddevelop
t
if one projectile were launched from the same site as the
previous one.
4. Integration
We do not use periodic boundary conditions because
doingsocouldleadtoprojectileswanderingtoocloseto-
We numerically integrate the equations of motion,
gether. By using a finite simulation box, we can assure
Eqs.(5)and(6),usingtheLangevinintegratorof[57]. To
that projectiles are always separated by a large multiple
accountfordisparatetimescalesforthelighterprojectile
ofa. Ifinsteadweusedperiodicboundaryconditions,as
andheaviertargetparticles,weuseamultiple-time-scale
a projectile departed on the right side it would be intro-
method [74].
Our time steps, 2.3×10−4 ω−1 and 4.5×10−6 ω−1 duced again on the left side, possibly with a separation
t t from the nearest projectile that is ≪ 8a due to the cu-
for the target and projectile particles, respectively, were
mulative effects of diffusion. We avoid this problem by
selectedbyperformingaconvergencetest. Intheconver-
using finite boundary conditions.
gence test, we solved mx¨ = −∇φ −∇ψ for a system
i ij
consistingofonlytwoparticles. Aprojectilewasdirected
toward a stationary target particle with zero impact pa-
rameter. Because of the confinement ψ, these particles
repeatedly collided. We calculated the discrepancy in a
particle’spositionandvariedthetimestepdownwardun-
til the discrepancy was <0.4% over an observation time
480 ω−1, the same as for our main simulation.
p
Acknowledgments
5. Projectile injection
This work was supported by NASA and NSF. We
The projectiles are introduced individually, one after thank S. D. Baalrud, W. D. S. Ruhunusiri, and F. Skiff
another. We take two steps to assure that two projec- for helpful discussions.
8
[1] J.J.Thomson,ConductionofElectricityThroughGases, Phys. Plasmas 17, 034502 (2010).
2nd ed. (UniversityPress, Cambridge, 1906). [35] K. N. Dzhumagulova, T. S. Ramazanov, and
[2] F. Seitz, Phys.Rev.73, 549 (1948). R. U.Masheeva, Contrib. Plasma Phys.52, 182 (2012).
[3] C. Kittel, Introduction to Solid State Physics, 5th ed. [36] K. Y. Sanbonmatsu and M. S. Murillo, Phys. Rev.Lett.
(John Wiley & Sons,New York,1976). 86, 1215 (2001).
[4] L. Spitzerand R. H¨arm, Phys.Rev.89, 977 (1953). [37] V. E. Fortov, O. F. Petrov, O. S. Vaulina, and
[5] S.I.Braginskii, Transport Processes in a Plasma, in Re- R.A.Timirkhanov,Phys.Rev.Lett.109,055002(2012).
views of Plasma Physics, edited by M. A. Leontovich [38] Z. Donko´, J. Goree, P. Hartmann, and K. Kutasi, Phys.
(Consultants Bureau, New York,1965). Rev. Lett.96, 145003 (2006).
[6] S.Ichimaru, Rev.Mod. Phys.54, 1017 (1982). [39] Z. Donko´ and P. Hartmann, Phys. Rev. E 78,
[7] C.J.Horowitz,D.K.Berry,andE.F.Brown,Phys.Rev. 026408 (2008).
E 75, 066101 (2007). [40] Z. Donko´ and P. Hartmann, Phys. Rev. E 69,
[8] G. J. Kalman, K. B. Blagoev, and M. Rommel (eds.), 016405 (2004).
Strongly Coupled Coulomb Systems (Plenum Press, New [41] E. A. Mason and E. W. McDaniel, Transport Properties
York,1998). of Ions in Gases (John Wiley & Sons, New York,1988).
[9] W.T.JuanandLinI,Phys.Rev.Lett.80,3073 (1998). [42] L. A. Viehland and E. A. Mason, At. Data. Nucl. Data
[10] P. K. Shukla and A. A. Mamun, Introduction to Dusty Tables 60, 37 (1995).
Plasma Physics (Instituteof Physics, Bristol, 2002). [43] G. H.Wannier, Phys.Rev. 87, 795 (1952).
[11] O.Ishihara, J. Phys. D:Appl.Phys. 40, R121 (2007). [44] J. Dutton,J. Phys. Chem. Ref. Data 4, 577 (1975).
[12] A. Melzer and J. Goree, in Low Temperature Plasmas: [45] Y. Feng, J. Goree, and B. Liu, Phys. Rev. Lett. 109,
Fundamentals, Technologies and Techniques, 2nd ed., 185002 (2012).
edited by R. Hippler, H. Kersten, M. Schmidt, and K. [46] K. R. Su¨tterlin et al., Phys. Rev. Lett. 102,
H.Schoenbach (Wiley-VCH,Weinheim, 2008), p. 129. 085003 (2009).
[13] G. E. Morfill and A. V. Ivlev, Rev. Mod. Phys. 81, [47] K. R. Su¨tterlin et al., IEEE Trans. Plasma Sci. 38,
1353 (2009). 861 (2010).
[14] V. E. Fortov and G. E. Morfill, Complex and Dusty [48] M.A.Fink,M.H.Thoma,andG.E.Morfill,Micrograv-
plasma: From Laboratory to Space in Series in Plasma ity Sci. Technol. 23, 169 (2011).
Physics (CRC Press, New York,2009). [49] D. Caliebe, O. Arp, and A. Piel, Phys. Plasmas. 18,
[15] M. Bonitz, C. Henning, and D. Block, Rep. Prog. Phys. 073702 (2011).
73, 066501 (2010). [50] O.Arp,D.Caliebe,andA.Piel,Phys.Rev.E83,066404
[16] A.Piel, Plasma Physics, (Springer,Heidelberg, 2010). (2011).
[17] H.Thomas et al.,Phys. Rev.Lett. 73, 652 (1994). [51] M. Schwabeet al., Europhys.Lett. 96, 55001 (2011).
[18] J. H. Chu and L. I, Phys.Rev.Lett. 72, 4009 (1994). [52] D.I.Zhukhovitskiietal.,Phys.Rev.E86,016401(2012).
[19] A. Melzer, A. Homann, and A. Piel, Phys. Rev. E 53, [53] D.SamsonovandJ.Goree,Phys.Rev.E59,1047(1999).
2757 (1996). [54] O. Arp et al.,IEEE Trans. Plasma Sci. 38, 842 (2010).
[20] B. Liu, J. Goree, V. Nosenko, and L. Boufendi, Phys. [55] S. Ratynskaia, G. Regnoli, K. Rypdal, B. Klumov, and
Plasmas 10, 9 (2003). G. Morfill, Phys.Rev.E 80, 046404 (2009).
[21] Y. Feng, J. Goree, and B. Liu, Rev. Sci. Instrum. 78, [56] B. Klumov et al., Plasma Phys. Control. Fusion 51,
053704 (2007). 124028 (2009).
[22] T. M. Flanagan and J. Goree, Phys. Rev. E 80, [57] W.F.VanGunsterenandH.J.C.Berendsen,Mol.Phys.
046402 (2009). 45, 637 (1982).
[23] P.Epstein, Phys.Rev.23, 710 (1924). [58] J.Goree, YanFeng,andBinLiu,Plasma Phys.Control.
[24] S.A.Khrapak,A.V.Ivlev,andG.E.Morfill,Phys.Rev. Fusion 55, 124004 (2013).
E 70, 056405 (2004). [59] An examination of Eqs. (5) and (6) shows that using
[25] S. D. Baalrud and J. Daligault, Phys. Rev. Lett. 110, themultiplierisequivalenttoaddinganotherMarkovian
235001 (2013). forceinadditiontothatduetothegas,andtheresulting
[26] G. Bannasch, J. Castro, P. McQuillen, T. Pohl, and motionwillhavetheattributesofthermalmotionbutat
T. C. Killian, Phys. Rev.Lett. 109, 185008 (2012). higher temperature.
[27] T.C.Killian,T.Pattard,T.Pohl,andJ.M.Rost,Phys. [60] S. A. Khrapak, A. V. Ivlev, G. E. Morfill, and
Rep.449, 77 (2007). H. M. Thomas, Phys. Rev.E 66, 046414 (2002).
[28] C. Reichhardt and C. J. Olson Reichhardt, Phys. Rev. [61] S. A. Khrapak et al.,Phys. Rev.E 72, 016406 (2005).
Lett.92, 108301 (2004). [62] V.Fortov,G.Morfill,O.Petrov,M.Thoma,A.Usachev,
[29] P.Glasson et al., Phys. Rev.Lett.87, 176802 (2001). H. Hoefner, A. Zobnin, M. Kretschmer, S. Ratynskaia,
[30] Kimitoshi Kono, J. Low Temp. Phys.126, 467 (2002). M. Fink, K. Tarantik, Y. Gerasimov, and V. Esenkov,
[31] H.Ikegami, T. Matsumoto, and K. Kono, J. Low Temp. Plasma Phys.Control. Fusion 47, B537 (2005).
Phys.171, 159 (2013). [63] These friction constants were calculated using a leading
[32] C. F. Barenghi et al., Phil. Trans. R. Soc. Lond. A 334, coefficient of 1.26 in theEpstein drag formula [20].
139 (1991). [64] S.Hamaguchi,R.T.Farouki,andD.H.E.Dubin,Phys.
[33] O. Vaulina and S. V. Vladimirov, Phys. Plasmas 9, Rev. E56, 4671 (1997).
835 (2002). [65] See Supplemental Material at XXXXXX for a movie
[34] S. Ratynskaia, G. Regnoli, B. Klumov, and K. Rypdal, showing the3D structureof thetarget.
9
[66] P. Tabeling, Introduction to Microfluidics, (Oxford Uni- 14, 278 (1971).
versity Press, Oxford,UK, 2005). [71] H. Ohta and S. Hamaguchi, Phys. Plasmas 7,
[67] J.P.HansenandI.R.McDonald,Theory of SimpleLiq- 4506 (2000).
uids, 2nd ed. (Academic Press, San Diego, 1986). [72] J. Daligault, Phys. Rev.E 86, 047401 (2012).
[68] T. Saigo and S. Hamaguchi, Phys. Plasmas 9, [73] Y.Rosenfeld,J.Phys.: Condens.Matter11,5415(1999).
1210 (2002). [74] M. E. Tuckerman, B. J. Berne, and A. Rossi, J. Chem.
[69] G.H.LaneandE.Everhart,Phys.Rev.117,920(1960). Phys. 94, 1465 (1991).
[70] H.-S.Hahn,E.A.Mason,andF.J.Smith,Phys.Fluids
