Density waves at the interface of a binary complex plasma

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by  Li Yang

Density waves at the interface of a binary complex plasma
Li Yang,1 Mierk Schwabe,2 Sergey Zhdanov,2 Hubertus M. Thomas,2 Andrey M Lipaev,3
Vladimir I Molotkov,3 Vladimir E Fortov,3 Jing Zhang,1 and Cheng-Ran Du1,∗
1College of Science, Donghua University, 201620 Shanghai, PR China
2Institut fu¨r Materialphysik im Weltraum, Deutsches Zentrum fu¨r Luft- und Raumfahrt (DLR), 82234 Weßling, Germany
3Joint Institute for High Temperatures, 125412 Moscow, Russia
7 Densitywaveswerestudiedinaphase-separatedbinarycomplexplasmaundermicrogravity con-
1 ditions. For the big particles, waves were self-excited by the two-stream instability, while for small
0 particles,theywereexcitedbyheartbeatinstabilitywiththepresenceofreversedpropagatingpulses
2 of a different frequency. By studying the dynamics of wave crests at the interface, we recognize a
“collisionzone”anda“mergerzone”beforeandaftertheinterface,respectively. Theresultsprovide
r
a a generic picture of wave-wave interaction at theinterface between two “mediums”.
M
7 Wavebehaviorattheinterfaceconsistsofvariousinter- mixed and form a glassy system [35]. Other phenom-
2 esting phenomena including transmission, reflection and ena suchas phase separation[36, 37] and lane formation
refraction, etc. [1] The early study of refractions can be [38] can also be studied in such systems. Recently it

] dated back to the year 984, when Snell’s law was first was discovered that phase separation can still occur due
h
p proposed by a scientist, Ibn Sahl, in Baghdad [2]. Up- to the imbalance of forces under microgravityconditions
- to-date studies of waves at an interface include not only despite the criteria of spinodal decomposition not being
m
light waves [3], but also acoustic waves in solids [4] and fulfilled [39]. An interface between two different types of
s fluids [5], capillary waves in liquids [6], electromagnetic particles emerges in such binary systems.
a
l waves [7], charge density waves [8], and spin waves [9].
p These studies result in wide applications such as dental 20
.
s diagnostics [10] and study of seismic wavesin geoscience
c 15
[11]. Recent progresses in granular matter [12] and col-
i
s loidalphysics[13]enableustostudythewavebehaviorat m)
y m 10
h aninterfacewithresolutionofindividualparticles. How- y (
p ever,rigidcontactsindense granularmattersandstrong
5
[ dissipation in solvent in colloids prohibit studies at the
(a)

kinetic level.
2 0
v Dust density waves in complex plasmas provide a 20
0
unique opportunity to study various aspects of wave
1
propagation. A complex plasma is a weekly ionized 15
2
7 gas containing electrons, ions, neutral atoms and small m)
0 macroscopic particles [14, 15]. Such a system allows ex- m 10
1. perimental studies of various physical processes occur- y (
5
0 ring in liquids and solids at the kinetic level [16]. Since
(b)
7 the first observation of self-excited dust acoustic waves,
1 0
dust density waves have drawn much attention [17–20]. 0 5 10 15 20 25 30 35
: x (mm)
v StreamingionsgenerateaBuneman-typeinstability[21].
Xi Thus, dust density waves can be self-excited if inter- FIG. 1. (Color online) Snapshot of a binary complex plasma
nal sources of free energy exist [18]. They can also at gas pressure 20 Pa (a) and 10 Pa (b). Five consecutive
r
a be triggered by the heartbeat instability [22, 23]. The imagesareoverlaidwithdifferentcolorsfrombluetored. The
particle dynamics can be recorded by video microscopy interface is highlighted by a grey curve, where big particles
are confined in the left area and small particles in the right
[24]. While some propertiesofwavessuchas growthand
area. The ROI for the periodgram in Fig. 2 is marked in
clustering can be studied by recording the dynamics of
blue. Because of high velocity, microparticle images have an
wave crests alone [25–28], tracking individual particles
elongated shape, shown in the raw image in the insets. The
in waves reveals the wave-particledynamics in much de- centerof thevacuum chamberis marked bya red cross.
tail [24, 29, 30]. Apart from self-excited waves, external
excitation can also sustain continuous waves [31, 32] or
Despiteplentyofstudiesondustacousticwavesinpast
trigger solitary waves in complex plasmas [33, 34].
years,itisnotclearhowtwowavesofdifferentoriginsin-
A complex plasma consisting of two differently sized teractataninterface. Inthisletter,westudythedensity
microparticles is known as binary complex plasma. Un- waves at the interface of a phase-separated binary com-
der certain conditions, two types of particles can be plex plasma under microgravity conditions. The emer-
2
genceoftwofrequenciesduetodifferentexcitationorigins x (mm) Ueff (V)
and respective features in terms of kinetics of individual 100 4 8 12 16 20 24 2820.5 21 21.5
particles are reported. A detailed study of the period-
gram reveals characteristic zones in the vicinity of the 8
interface where waves in two “mediums” strongly inter-
act. 6
TheexperimentswereperformedinthePK-3Pluslab- t (s)
oratory on board the International Space Station (ISS). 4
Technical details of the setup can be found in [31, 40].
Anargonplasmawasproducedbyacapacitively-coupled 2
(a) (b)
radio-frequency (rf) generator in push-pull mode at
0
13.56 Hz. We prepared a binary complex plasma by
5
sequentially injecting melamine formaldehyde (MF) mi- z)
H (c) (d)
croparticlesoftwodifferentsizesintheplasmadischarge. f (
The mass density of the microparticles is 1.51 g/cm3. 00 4 8 12 16 20 24 28 0 1 2 3 4 5
x (mm) FFT
The small particles have a diameter d = 6.8 µm while
s 80
the big ones have a diameter db = 9.2 µm. With video nt 60 (e) (f)
microscopy[40],acrosssectionofthelefthalfofthepar- ou 40
c 20
ticle cloud (illuminated by a laser sheet) was recorded
0
1 2 3 4 5 0.1 0.2 0.3
with a frame rate of 50 frames-per-second (fps) and a λ (mm) T (s)
spatial resolution of 0.05 mm/pixel.
FIG. 2. (Color online) (a) Periodgram of horizontally prop-
As we see in Fig. 1(a), initially at a pressure of 20 Pa,
agating waves from the void to the edge of particle cloud.
microparticlesformeda 3-dimensional(3D)cloudwith a
(b) Evolution of effective voltage. (c) Spectrum distribution
particle-free region in the center (void) [41, 42]. Two along x-axis obtained from FFT applied on the periodgram.
particle species were phase-separated with a clear in- (d) Spectrum of the effective voltage. Histograms of wave-
terface (highlighted by a grey curve) due to the follow- length λ (e) and period T (f) determined from the period-
ing two mechanisms: First, the disparity of particle size gram: Red histograms (left) correspond to the big particles
(∆d/d¯≈ 0.3) was larger than the critical value of spin- [red (left) region in (a)] while blue ones (right) correspond
to the small particles [blue (right) region in (b)]. The curves
odal decomposition (0.25) [36]. Second, both particle
in the histograms are Gaussian fits to the distribution. The
species were subjected to two forces under micrograv-
interface is located at x≈13 mm.
ity conditions, namely ion drag force (directed outwards
[43]) and the electric field force (directed inwards). The
thepublishedresults[22,40,47]. Theestimatedelectron
total force acting on two particle species had a subtle
densityis n ∼1.3×109 cm−3,the electrontemperature
difference depending on the particle diameter [39]. The e
is T ∼ 3 eV and the ion temperature T equals to the
synergisticeffectofspinodaldecompositionandforcedif- e i
roomtemperature. Basedon the plasma parameters,we
ference led to the instantaneous phase separation. Par-
estimatetheparticlechargeandthesoundspeedforboth
ticularly the second effect drove the small particles into
small and big particles based on three theoretic models
the inner part of the particle cloud and left big particles
[44–46]. The results are listed in Table. I.
outside, as shown in Fig. 1(a).
Thewaveswereexcitedastheneutralgaspressurewas
lowered below a critical value (15 Pa) [24]. As we see in (cid:24)(cid:18)(cid:3)(cid:4) (cid:17)(cid:8)(cid:18) (cid:17)(cid:15)(cid:18)
Fig.1(b), at the pressureof10Pa the particle cloudwas (cid:23)
compressedverticallybytheexpansionofthesheath,and (cid:21)(cid:22)(cid:11)(cid:17)
(cid:12)
thewavespropagatedthroughtheentirecloudfromright (cid:13)(cid:10) (cid:2)
(cid:20)
to left. The wave crests had a convex shape due to the (cid:19)(cid:16)(cid:10)
configuration of the 3D particle cloud. In this letter, we (cid:4)
(cid:6)(cid:5)(cid:4) (cid:6)(cid:3)(cid:4) (cid:4) (cid:3)(cid:4) (cid:5)(cid:4) (cid:6)(cid:5)(cid:4) (cid:6)(cid:3)(cid:4) (cid:4) (cid:3)(cid:4) (cid:5)(cid:4)
focus on the waves in the region of interest (ROI) with (cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:10)(cid:16)(cid:11)(cid:17)(cid:14)(cid:14)(cid:6)(cid:3)(cid:18) (cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:10)(cid:16)(cid:11)(cid:17)(cid:14)(cid:14)(cid:6)(cid:3)(cid:18)
a vertical width of 2 mm [marked by a blue square in
FIG. 3. Highly anisotropic particle velocity fluctuation spec-
Fig. 1(b)] around the middle plane where the wave front
tra of the binary complex plasma. Panels (a) and (b) show
can be approximated as flat plane. The average number
the oscillations in the big and small particle sub-clouds, re-
density of the microparticles can be estimated by the
spectively. Positive wave numbers correspond to the wave
paircorrelationfunction. Themeanvalueoftheeffective propagation to the left in Fig. 1. Note thepresence of sound
currentwasabout14mAandthatoftheeffectivevoltage wave traces (continuum spectrum) and heartbeat oscillation
(U ) was about 21 V with an oscillation frequency of harmonics (discrete spectrum).
eff
1.1 Hz, see Fig. 2(b,d). The plasma parameters were
not measured directly but could be estimated based on As we can see in Fig. 1(b), the interface smeared out
3
TABLEI.Parametersincludingaveragenumberdensityhni,chargeQandsoundspeedC estimatedforsmallandbigparticles.
Among these parameters, the charge and the sound speed are estimated based on three theoretic models: The drift motion
limited theory (DML) [44], the modified orbital motion limited theory (mOML) [45] and the orbital motion limited theory
(OML) [46].
d hni QDML QmOML QOML CDML CmOML COML
[µm] [mm−3] [104e] [104e] [104e] [mm/s] [mm/s] [mm/s]
6.8 240 1.0 0.9 1.4 10.5 9.5 12.9
9.2 60 1.7 1.4 1.9 7.0 5.9 7.6
as the waves developed. However, the position of the
interface can still be extrapolated during the process of 0.8 1.0 1.2 1.4 1.6 1.8
log < v2 > (u.a.)
pressure decrease. Obviously, the wavelength λ is not 10 x,fast
constant over the entire dust cloud. The right part has 0.5
a larger wavelength than the left part. This is clear in 0.4
the periodgram in Fig. 2(a) (see also the video in sup- s) 0.3
plementalmaterials). Theplottakes500consecutiveim- t ( 0.2
ages registered from the experiment. Each vertical col- 0.1 (a)
umn was obtained by averaging over the vertical axis
0.0
of the ROI in one frame [see the area enclosed by blue
300
square in Fig. 1(b)] [24]. Clearly the periodgram is di- 3) 250 (b)
vided into left and right parts, corresponding to the big −mm 200
and small particles. By linearly fitting the wave crests n (c 150
(bright pixel clusters) for the two parts separately, we 100
50
obtain histograms of the wave length and period from s)
the spatial resp. temporal distances of the wave crests, m/ 40
sthheowdnatianbFyigfi.t2t(ine,gf)a. rWobeudsetteGramuisnseiatnhetomtehaensvmaoluoetsheodf l v l (mx,c 123000 (c)
5 10 15 20 25
histogram. Forthe wavespropagatinginthe smallparti- x (mm)
cles,wehavewavelengthλ =4.2±0.3mm,waveperiod
s
T =0.28±0.01s,frequencyf =3.6±0.1Hz,andphase FIG. 4. (Color online) Kinetic periodgram (a). The number
s s
velocityνs =14.2±1.0mm/s. Forbigparticles,wehave density nc (b) and speed |vx,c| (c) at the wave crests along
thex-axis. Theinterfaceismarkedbyverticalreddashedline
λ =1.8±0.3 mm, T =0.19±0.02 s, f =5.4±0.5 Hz,
b b b
and the two characteristic regions close to the interface are
and ν =9.9±1.8 mm/s.
b marked by vertical blue (left) and black (right) dash-dotted
The fluctuation spectrum of the binary complex lines. The green [left in (c)] and orange [right in (b,c)] solid
plasma is shown in Fig. 3. The discrete spectrum sug- lines guide theeyes.
gests the presence of heartbeat oscillation harmonics,
while the waves in the continuum are identified as the
dust acoustic waves [48] [49].
To further study the dynamics of individual particles
Thecharacteristicfrequenciesofthe self-excitedwaves inthewaves,weplotthekineticperiodgraminFig.4(a),
can also be obtained by directly applying Fast Fourier in which the intensity is logarithmly proportional to the
Transformation (FFT) on the image intensity evolution mean square speed of the most mobile particle fraction
of the periodgram (representing the number density). (fastest 25%). Because of the limited frame rate of the
The results are shown in Fig. 2(c). For big particles cameras in the PK-3 Plus laboratory on board the ISS,
(x < 10 mm), we see one characteristic frequency of particle velocity cannot be deriveddirectly from particle
5.4 Hz, in agreement with the analysis of the period- tracking for such fast moving and dense particle clouds.
gram. At the interface (x≈13 mm), the peak at 5.4 Hz However, we can still estimate the particle speed by di-
is suppressedwhile anothercharacteristicpeak at3.6Hz viding the length of the elongated shape of individual
starts to emerge. The later corresponds to the eigenfre- particles [see the inset in Fig. 1] by the exposure time
quency of small particles. Inside the sub-cloud of small of each image. Considering the frequencies of waves of
particles(x>16mm),oneseesbothpeaksat3.6Hzand small and big particles,we select the least commonmul-
5.4 Hz. For the edge of the void, both the fundamental tiple of the periods (≈0.6 s) as the duration to perform
frequency at 1.8 Hz and its harmonics are visible. anaveragingprocessover3sforbettervisualization[50].
4
For big particles, the kinetic periodgram exhibits a left- We measured the particle number density n and ab-
c
wards propagating feature with a frequency of 5.4 Hz, solute speed |v | at wavecrests (ignoringthose at wave
x,c
corresponding to the result of FFT analysis. However, throughs) and plot them along the x-axis in Fig. 4(b,c).
forsmallparticlesweseea cross-shapedtrellis. The left- This number density is relatively low close to the void
wardspropagatingwavehas a frequency of3.6Hz, while andtheedge[51]. Aswecanseeinthefigure,n islower
c
the right-wards propagating pulses have a frequency of in big particles than in small particles. The drop starts
5.4 Hz, which is already exhibited in the FFT map in at x ≈ 16 mm and reaches a stable level at x ≈ 13 mm,
Fig.2(c). The maximalparticlespeedin the right-wards wheretheinterfaceisroughlylocated(markedbythered
propagating pulses can reach ∼ 35±5 mm/s while that dashed line). We highlight this drop by the orange solid
in the left-wards propagating waves is ∼27±4 mm/s. line. Fortheabsolutespeedofparticles,weseeadistinct
valley at the interface, where the left shoulder starts at
Several effects contribute to the origin of the waves:
x ≈ 10 mm (blue dash-dotted line) and right shoulder
There is a heartbeat vibration visible in the electric
starts at x≈16 mm (black dash-dotted line).
signals [1.1 Hz, see Fig. 2(b,d)]. The particle move-
ment in the cloud at this frequency is shifted by fric- To better understand the subtle change of frequency
tion (≈ 10 s−1), so that only harmonics of 1.8 Hz are andwavelength,wezoomintotheareaclosetotheinter-
excited as waves in the particle cloud. In the small par- facemarkedbyayellowsquareinFig.2(a)andshowthe
ticlesub-cloudclosetothevoid,the wavesexcitedatap- details in Fig. 5(a). At x≈16 mm the wave crests start
proximately3.6Hzalsoshowreversedpropagatingpulses to bend and we can see the emergence of a new wave
with a frequency of 5.4 Hz. The later pulses correspond front between two existing wave crests at x ≈ 13 mm.
to the contracting phase of the heartbeat and suggests We highlight five typical wave crest pairs by different
that this wave was excited by the heartbeat instability. colorsinFig.5. Thetransmittingwavecrestsfromsmall
In the big particle sub-cloud, the waves didn’t transmit particles to big particles appear in solidcurves while the
through the interface [suppression of 5.4 Hz peak at the self-excitedwavecrestsstartingattheinterfaceappearin
interface,showninFig.2(c)]. Thissuggeststhatthewave dashed curves. Due to the “collision” of the wave crests
was self-excited by the two-stream instability. However, (with high concentration of heavily charged particles),
the heartbeat (generally affecting the entire dust cloud the phasespeed|ν| decreasesforthe transmittingwaves.
[22, 23]) may synchronize the frequency to its third har- Accordingly, |ν| for the self-excited waves increases, as
monics. clearly demonstrated in Fig. 5(b). As both waves prop-
agate towards the cloud edge, they can either merge to
3.0 one wave crest (orange-red and cyan-green) or coexist
and propagate further (green-yellow). This explains the
2.5 transitionofthe frequency of3.6Hz in smallparticlesto
5.6 Hz in big particles.
2.0 Togainbetterstatistics,wetrackedallthewavecrests
s)
t ( close to the interface in Fig. 2, calculated the phase
1.5
speed and show the results with errors (1σ deviation)
in Fig. 5(c). The drop of the phase speed of the trans-
1.0
mitting wave is still clearly visible while the collision of
(a)
thewavescrestssmearsoutduetotherandomnessofthe
0.5
s) 20 merge. Combining with the analysis on nc and vx,c, we
m/ 15 are able to define the region 13 mm < x < 16 mm as
m
ν l ( 10 (b) “collision zone” and the region 10 mm < x < 13 mm as
l 5 “merger zone”.
25
In summary, we have presented the first experimen-
s) 20 tal realizationof wavepropagationacrossan interface in
m/ 15
m complex plasmas. The experiments were performed in
νl l ( 150 (c) a binary complex plasma under microgravity conditions
0 on board the ISS. The small particles and big particles
6 8 10 12 14 16 18
x (mm) were phase separated with an interface in between due
to spinodal decomposition and the difference of electric
FIG. 5. (Color) Periodgram close to the interface (a) and
force and ion drag force. For the big particles, waves
phase speed |ν| for five typical wave crest pairs (b) and all
crests(c). Thetransmittingwavecrestsappearinsolidcurves were self-excited by the two-stream instability, while for
in (a,b) and in red symbols (c) and self-excited wave crests small particles, they were excited by heartbeat instabil-
appear in dashed curves in (a,b) and in black symbols (c). ity with the presence of reversedpropagating pulses of a
Colors in (a) and (b) correspond to each other. The vertical different frequency. By studying the dynamics of wave
dashedlinesanddash-dottedlinesarethesameasinFig.4. crests at the interface, we recognize a “collision zone”
5
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