Network analysis of 3D complex plasma clusters in a rotating electric field

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by  Ingo Laut

Network analysis of 3D complex plasma clusters in a rotating electric field
I. Laut,1,∗ C. Ra¨th,1 L. W¨orner,1 V. Nosenko,1 S. K. Zhdanov,1
J. Schablinski,2 D. Block,2 H. M. Thomas,1 and G. E. Morfill1
1Max-Planck-Institut fu¨r Extraterrestrische Physik, D-85741 Garching, Germany
2Christian-Albrechts Universita¨t zu Kiel, D-24118 Kiel, Germany
(Dated: February 21, 2014)
Network analysiswas usedtostudythestructureandtimeevolution ofdriventhree-dimensional
4 complex plasma clusters. The clusters were created by suspending micron-size particles in a glass
1 box placed on top of therf electrode in a capacitively coupled discharge. Theparticles were highly
0 chargedandmanipulatedbyanexternalelectricfieldthathadaconstantmagnitudeanduniformly
2 rotatedinthehorizontalplane. Dependingonthefrequencyoftheappliedelectricfield,theclusters
rotated in the direction of the electric field or remained stationary. The positions of all particles
b
were measured using stereoscopic digital in-line holography. The network analysis revealed the
e
interplay between two competing symmetries in the cluster. The rotating cluster was shown to be
F
more cylindrical than the nonrotating cluster. The emergence of vertical strings of particles was

0 also confirmed.
2
PACSnumbers: 52.27.Lw89.75.Hc89.75.Fb

]
h
p I. INTRODUCTION Driving particle clusters with external forces adds an
-
extra degree of flexibility to the experiments. One way
m
of manipulating a particle cluster is the “rotating wall”
s Complex plasmas exist in various forms, from small
technique, where an externalrotating electric field is ap-
a clusterstolargeextendedsystems[1,2]. Two-andthree-
l pliedtothecluster[6,10]. InRef.[6],3Dparticleclusters
p dimensional(3D)clustersarepopularobjectstostudyfor
hadaspheroidalshape,yetacompetingcylindricalsym-
. two main reasons. First, they areused as model systems
s metrywasalsopresent. Thelatteriscausedbystreaming
c tostudygenericphenomenasuchasself-organizationand
ions [20, 21], which create an anisotropy in the interpar-
i transport, at the level of individual particles [3–7]. Sec-
s ticle potential and lead to vertical particle strings. To
y ond, clusters can be used for diagnostic purposes, e.g.
properly study this complicated structure, new analysis
h probing the plasma parameters at the position of parti-
methods are clearly required.
p cles [8–10].
[ Thedustparticlesconstitutingacomplexplasmaclus- An advanced method of studying large systems is
2 ter are highly charged and therefore have to be exter- (complex) network analysis. Originating from “classi-
v nally confined. For instance, in Ref. [8] the particles cal”networkssuchaspowergrids[22]ortheWorldWide
0 were suspended in a short open glass tube placed on Web[23],complexnetworkanalysis[24]hasbeenadopted
3 top of the rf electrode in a gas discharge. The au- for a wide range of systems. Common to all these appli-
5 cationsistheprocedureofassociatingtheconstituentsof
thors suggested that the nearly isotropic particle clus-
3
terconfinementwasmainly determinedbygravitational, a system (current generators or routers) with the nodes
.
1 of a network, and their interactions (transmission lines
electric and thermophoretic forces. A contribution from
0 the ion drag force due to streaming ions was not ob- betweengeneratorsorconnectionsbetweenrouters)with
4
edges connecting them. While one strength of the net-
served. Dust particles suspended in a glass box can
1
: exhibit various structures, such as isolated single linear work analysis is without doubt the access to vast and
v intricateobjects [25], one otherpossible applicationis to
chains[11], verticalstrings[6], zigzagstructures[12], he-
Xi licalstructures[7,13],andCoulombclusterswithonion- small systems: By keeping track of all individual inter-
like shells [4, 8, 14]. actionsthroughouttheanalysisprocedure,thisapproach
r
a mayremainapplicablewheremanyothertoolsrelyingon
Sincethefirstobservationofcrystallinestructurein3D
somekindofaveragingfailbecauseoftooweakstatistics.
complex plasmas [15], the structural properties of such
systems(consistingofabout104 particles)havebeenex- In this paper, we apply network analysis to driven 3D
amined thoroughly. For instance, domains of different complex plasma clusters that were observed in Ref. [6].
lattice geometries were found[16–18]. Complex plasma In these spheroidal clusters, vertical strings were iden-
clusters containing up to a hundred particles allowed an tified by introducing a certain fixed threshold (about a
analysis at the level of individual particles and were ex- plasmascreeninglength)totheirtransverseextent. This
ploited to study, e.g., the interaction force between the simpleapproachhasitsevidentadvantagesbutalsolimi-
particles [19]. tationssuchaserroneouslyincludingpassingbyparticles
and a somewhat arbitrary threshold. With the help of
multislice networks[26], we now find strings in a natural
way,and resolve them throughout the whole time series.
∗Electronicaddress: [email protected] Furthermore, the global structure of the clusters is ana-
2
lyzed in great detail by comparing network measures of a
the experimental data with null models. As we demon-
strate in this paper, the network analysis is a powerful
tooltoanalyzethe structure ofcomplex plasmaclusters. b
The paper is organizedasfollows. In Sec.II we briefly
describe dynamically driven clusters. In Sec. III we ex-
amine particle strings ofthe clustersand their globalge-
ometry with the aid of network analysis. In Sec. IV we
examine the particle confinement in the clusters by as- beam
expanders
suming a dynamicalforcebalance. Finally, in Sec.V, we
conclude with a summary of our results.
II. DYNAMICALLY DRIVEN CLUSTERS
FIG.1: Experimentalsetup. A3Dclusterofmicron-sizepar-
ticles is suspended in the glass box mounted on top of the
Particle clusters which we analyze in this paper were
rf electrode in a capacitively coupled discharge. The parti-
observed in the course of experiment of Ref. [6]. Be-
cles are charged negatively and are manipulated by applying
low, we briefly outline the experimental procedure used
voltagesontheconductivesideplatesofthebox. Byshifting
in Ref. [6]. sinusoidal signals on the adjacent plates by π/2, a rotating
3D clusters of micron-size particles were suspended in electricfieldiscreatedatthepositionofparticles. Expanded
a glass box mounted on top of the rf electrode in a ca- laser beams at 532nm illuminate the particles from two per-
pacitively coupled discharge in argon (see Fig. 1). Si- pendicular directions. The diffracted light is attenuated by
nusoidal voltages were individually applied to the sides neutral density filters (NDF) and registered by CCD cam-
erasAandB.Insetsshowtheinterferencepatterns(a)ofthe
of the box, which are coated with indium tin oxide and
whole cluster and (b) of an individualdust particle.
arethereforeconductiveyettransparent. Thephaseshift
betweenneighboringsideswassettoπ/2,whichresulted
in the electric field at the position of the particles that
The main result of Ref. [6] is the formation of verti-
had a constant magnitude and rotated in the horizontal
calparticlestringsindrivenclusters. Acomparisonwith
plane.
dustdynamicssimulations[29],wherethe verticalorder-
The particle coordinates were measured using a 3D
ingparameterdependedonthe thermalMachnumberof
imaging method, stereoscopic digital in-line holography
flowing ions, showed reasonable agreement.
[27]. In two identical channels, expanded laser beams il-
Structuralchangesintheglobalclustersymmetrywere
luminate the particle cluster from two perpendicular di-
alsoobservedinRef.[6]. Drivenclustershadelementsof
rections. The two channels use the same 532-nm laser
competing symmetries that changed from one type to
with its output beam split in two parts. The diffracted
the other while the cluster rotated. This topic was not
light is registered directly by two CCD cameras operat-
pursued further in part for the lack of proper analysis
ing at 50 frames per second over a time interval of 10s.
technique.
In the resulting images, each particle is represented by
a system of concentric circles (see insets in Fig. 1). The
depth information is encoded in the intercircle spacing.
III. NETWORKS DERIVED FROM DISTANCE
Depending on the frequency of the applied electric
MATRICES
field, the clusters rotated in the direction of the elec-
tricfield(albeitwithmuchlowerfrequency)orremained
stationary. Twomechanismsofclusterrotationwerepro- A. Characterizing Particle Clusters
posedinRef. [10]. One mechanismis basedonthe inter-
action of the induced dipole moment of the cluster with In order to analyze the cluster structure, we adopted
the applied field, the other on the ion drag force. In the a network approach similar to those which have re-
present paper, we analyze a rotating cluster driven at cently been successfully applied to (nonlinear) time se-
5kHz and a stationary one driven at 1kHz. ries [30, 31]. There, a scalar time series of T steps is
Whetherrotatingornot,theclustersweresignificantly embeddedintoanm-dimensionalphasespace. Thecorre-
compressed in the radial direction by the applied field. sponding (T −m)×(T −m) recurrence matrix R has
ij
The specific mechanism of this compression is not clear. unityentriesifthedistancebetweentheithandjthpoint
The ponderomotive force [28], a naturally expected can- inphasespaceisbelowacertainthreshold,andzerooth-
didate,isevidentlyofminorimportanceintheconditions erwise. Interpreting the recurrence matrix as the adja-
ofthisexperiment. Indeed,theclustersdrivenatthefre- cency matrix of a network, one can investigate the time
quenciesof1and5kHzwereactuallyofthesamesize,see seriesbymeansofnetworktheory. Theresultingnetwork
Fig. 8, whereas the ponderomotive force has an inverse is called unweighted, as all connections (links) between
quadratic dependence on frequency. the network nodes have the same strength.
3
Wewilladoptthis approachinSec. IIICtodetermine 8
a)
theglobalstructureofclusters: Then×nadjacencyma- 6 rotating
trixthenconnectstwoparticleswhosedifferenceincylin- η4
drical radius is below a predefined threshold, where n is
2
the number of particles. nonrotating
0
We will also use a modified approach, where the com-
0 1 2 3 4 5 6 7 8 9 10 11 12
ponentsoftheadjacencymatrixaredifferentfromzeroor γ
one. For such a weightednetwork,we will consider com-
b) γ=3
munities, i.e., sets of nodes which are more connected to 8000 rotating
each other than to the rest of the network, in order to nts6000 nonrotating
u
find vertical strings in the cluster. co4000
2000
0
c) γ=4
B. Detection of Strings 8000
s
nt6000
u
o4000
Before we focus on the global structure of the clus- c
2000
ters,webrieflyoutlinethepossibilityforstringdetection
0
using community assignment [32] in networks. A pop-
d) γ=5
ular method to quantify communities is by the quality 8000
s
function modularity that compares the number of intra- nt6000
u
community edges to the expectation value of a network o4000
c
with randomized links [33, 34]. 2000
0
Withthetoolofmultislicenetworks[26,35]itispossi- 1 2 3 4 5
bletofindcommunitiesinsystemsthatevolveovertime. Number of particles per community
InadditiontoadjacencymatricesA foreachtimestept
ijt
FIG. 2: Dependency of detected communities on resolution
there is aninterslice coupling parameterC connecting
jtr parameterγ. (a)Ratioη ofintercommunitydistancetocom-
node j at time t to itself at time r. The multislice mod-
munitysizefordifferentvaluesofγfortherotating(triangles)
ularity [26] then reads
and nonrotating (squares) cluster. The maximum values are
plottedassolidsymbols. Thestandarddeviationsareplotted
kitkjt aserror barsand areconcealed bythesymbolsin thecase of
Q∝ A −γ δ +δ C δ(g ,g ),
ijt tr ij jtr it jr
2m thenonrotatingcluster. (b)–(d)Histogramsofthenumberof
ijtr(cid:20)(cid:18) t (cid:19) (cid:21)
X particles percommunity for different valuesof γ.
(1)
whereγ is afree resolutionparameter,k = A the
jt i jit
strength of an individual node, mt = jkjt, and git is Theresolutionparameterγ isconsideredoptimalwhen
P
the community assignment of particle i at time t. the averageextent of a community is small compared to
P
We choose the slices Aijt to be proportional to the the intercommunity distance, i.e., when
element-wise inverse distance matrix ofthe projectionof
the particles on the xy plane at a given time step, η =hdt /dt i (3)
i,next i,same
is maximized. Here, dt is the horizontal distance of
∆/d for i6=j i,next
A = ij (2) particle i to the next particle of another community at
ijt
(0 otherwise. time t, dt is the average horizontal distance to the
i,same
particles in the same community if the particle is in a
Here, dij = (xi−xj)2+(yi−yj)2 is the horizontal communitywithmorethanoneparticle,andinfinityoth-
distance between two particles at time t and ∆ the min- erwise. Thebracketsdenotetheaverageoverallparticles
p
imalhorizontaldistance in the time series suchthat Aijt and time steps.
is at most one. The link between two particles is thus As can be seen in Fig. 2(a), η varies rather weakly in
stronger, the nearer they are to each other, and one the range of resolution parameter γ from 2 to 6 around
can consider the communities of the resulting network the maxima for the clockwise rotating and nonrotating
as strings of the cluster. clusters. The number of particles per community is very
In order to resolve the communities in time, the com- sensitivetoγ [seeFigs.2(b)–(d)]. Forthethreevaluesof
ponents of the multislice adjacency matrix connecting a γ considered here, the algorithm finds more small com-
particle with itself at the consecutive time step were set munities of one or two particles in the case of the non-
to C =0.1δ . rotating cluster, while larger communities with at least
jtr t,r+1
The algorithm used [26] is an adaptation of the Lou- three particles are more likely for the rotating cluster.
vain method [36] that aggregates nodes locally to small Byconsideringthecommunitiesastheverticalparticle
communities which are the nodes of a new network at a stringsinthecluster,onethusobtainsstringsofdifferent
later iteration step until Q is maximized. sizes depending on γ, in contrast to a more traditional
4
2a) b) 60 a) b) 60
50 50
1 er
b
m] um40 40
y [m0 cle n30 m]21 30
−1 parti20 z [m−01 −2 20
−1
−2 100 −2−y2 [m−m1]0 1 2 21x [0mm] 100
2 0 10 20 30 40 50 60
c) d)
60 c)
1 er50
b
m] um40
y [m0 cle n30
arti20
−1 p
10
−2 0
−2 −1 0 1 2 0 2 4 6 8 10
x [mm] t [s]
FIG. 4: Method for generating an unweighted network from
FIG. 3: Time evolution of detected strings for resolution pa- the data. (a) 3D plot of the clockwise rotating cluster. The
rameter γ = 4. (a) Projection of the rotating cluster on the 60th particle is plotted as a blue triangle and the particles
xy plane at t = 0. Particles which were found to be in the j satisfying |ρ60−ρj| < ǫ are plotted as red squares. (b)
same string are grouped together and have the same color. The corresponding adjacency matrix. The nodes connected
(b)Stringaffiliation ofparticlesovertime. Atransition from to the 60th node can be read off the 60th column (or line)
onestring toanother appears as achangein color of theline of thematrix. (c) Representation of thenetwork. The nodes
corresponding to theparticle. (c),(d) Same for thenonrotat- representing theparticles havethesame markersas in (a).
ing cluster.
C. Global Structure of the Clusters
approachofgroupingparticles whosedistance is below a
certain value as in Ref. [6]. The global structure of the clusters is analyzed by
means of unweighted networks. The adjacency matrix
The strings for γ = 4 and their evolution in time for
connects two particles whose difference in cylindrical
therotatingandnonrotatingclustersareshowninFig.3.
radiiissmallenough. TheprocedureissketchedinFig.4:
It is evident that in the case of the nonrotating cluster
In (a), the 60th particle of the cluster is plotted as a
there are fewer transitions between the strings.
blue triangle and the particles with comparable cylin-
The strings are quite robust against encounters with
drical radii as red squares. The latter can be read off
only occasionally passing by particles: During their pas-
the 60th column of the corresponding adjacency matrix,
sage, these roaming particles are not considered to be
(b). A representation of the resulting network can be
part of the string. The cases where the horizontal dis-
seeninFig.4(c). Atthis time step,the networkconsists
tancebetweentheparticlesissmall,whilealargevertical
of two main components (groups of connected nodes),
distance prohibits physical correlation, do not persist in
which correspond to the two ring regions of the cluster,
time. These events are thus not considered as strings
and various smaller components.
by the community-finding algorithm. Indeed, the mean
The network thus obtained is analyzed using network
vertical distance between particles of the same string
measuresandthe results arecomparedto thoseofa net-
is 0.79mm (rotating cluster) and 0.86mm (nonrotating
workfromanullmodelwhereacertainfractionofpoints
cluster),andtheeventswherethisdistanceislargerthan
is in perfectly spherical order,and the rest in cylindrical
1.2mmarenotfrequent(lessthan3%)forbothclusters.
order. The ratio that shows the best agreement with
By inspection of Fig. 3(a) alone, it is not obvious why,
the experimental data will be considered as the ratio of
for example, the particle near the center belongs to the
thecompetingsphericalandcylindricalgeometriesofthe
highlighted longish community. In order to understand
cluster.
this, the whole time series has to be considered, as this
We define the adjacency matrix as
roaming particle quickly joins the two remaining parti-
cles and forms a persistent string. This special feature Acyl(ǫ)=Θ(ǫ−|ρ −ρ |)−δ , (4)
ij i j ij
ofcommunityassignmentinnetworksmaybeapowerful
tool within a wide range of possible applications. where Θ(·) is the Heaviside function, ρ = x2+y2 is
i i i
p
5
Local connectivity Clustering coefficient Average path length
the cylindrical radius, and ǫ an appropriate threshold.
TtohaevKoirdonseelcfk-leoroδpss.ets the diagonalterms to zeroin order unit] R = 0 R = 0 R = 0
The threshold ǫ was chosen to be a fraction α of the b.
ar
mean difference in cylindrical radius: ǫ=αh|ρi−ρj|i. s [
The brackets denote the average over all particles and nt
u
o
time steps. Throughout this study we used α=0.1. c
The null models are artificial structures with a pre-
defined ratio R=nsph/ncyl of the number of particles nit] R = 2 R = 2 R = 2
in a perfect spherical structure to the number of parti- u
b.
cles in a perfect cylindrical structure. The total num- ar
ber of particles n=nsph+ncyl of the null models is s [
nt
equal to the number of particles in the experimental u
o
data. We use two-shell null models with different ra- c
tios R = 0, 1/3, 1/2, 1, 2, 3, ∞. Each model is con-
structed as follows. The cylindrical structure consists nit] R = ∞ R = ∞ R = ∞
of two concentric cylinders with the same (cylindrical) b. u
radii as the main components of the experimental data ar
(ρ = 1.0mm, ρ = 1.6mm). For a given ncyl, the ratio s [
1 2 nt
of the particle number in the inner cylinder to that in u
o
c
the outer cylinder is chosen to be equal to ρ /ρ . The
1 2
0 0.1 0.2 0.3 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8
twoshellsofthesphericalstructurehave(spherical)radii ν Cν Lν
r = 1.1mm and r = 1.7mm. A uniformly distributed
1 2
random noise of amplitude 0.15mm is added to the po-
FIG. 5: Comparison of network measures for the clockwise
sitions of all particles. It compensates the simplifica-
rotatingclusterwiththemeasuresfornullmodelsofdifferent
tion made by choosing the two-shellmodel—the clusters
ratios R. The histograms (for all particles and time steps)
areapparentlythree-shellstructured[see,e.g.,Fig.4(c)]. of measures on networksof theclockwise rotating cluster are
The presence of the innermost shell, consisting of only plotted in black. They are identical in all three rows. The
a few particles, does not change the results significantly, null models are plotted in red (grey) with R = 0 (top row),
andthe two-shellmodel, being properlyadjusted, agrees 2(middlerow), and∞(bottomrow), whereRistheratioof
well with the experiment (see Fig. 5, middle row). See particlesfromaperfectlysphericalgeometrytoparticlesfrom
Figs.6(c)and(d)fortheprojectionsofsuchanullmodel a cylindrical geometry. First column: local connectivity κν,
with R=2 on the ρz and xy plane. secondcolumn: clusteringcoefficient Cν,thirdcolumn: aver-
As a first network measure, the degree centrality k age path length Lν. The differences are marked as hatched
ν areas. The good agreement of all three network measures in
countsthenumberofnodesthatareconnectedtonodeν.
themiddle row is clearly visible.
It is defined as [30]
n
path length L is then calculated by averaging over all
k = A . (5) ν
ν ν,i
nodes i that are in the same component as ν:
i=1
X
Normalizingitbythe maximumpossiblevalueyieldsthe Lν =hlν,ii (7)
local connectivity κ =k /(n−1). Taking into account
ν ν
only the immediate neighborsof the node [42], this mea- Foreachtimestepoftheexperimentaldata,thecorre-
sure may be regardedas the local particle density of the spondingnetworksofthedataandofanullmodelwitha
cluster. givenratio R arecreatedandanalyzedwith the network
measures at hand. The histogramsof these measures for
The clustering coefficient C [30] acts on intermedi-
ν
ate scales. It evaluates the number N∆ of links between allparticles and alltime steps ofthe data arecompared,
ν
andtheratioRofthenullmodelwiththebestagreement
neighbors of a given node versus the maximum possible
isconsideredtobetheratioofsphericaltocylindricalor-
number k (k −1)/2:
ν ν
der of the cluster.
n In Fig. 5, the results for Acyl of the clockwise rotating
2 1
C = N∆ = A A A cluster are shown and compared to the null models with
ν k (k −1) ν k (k −1) ν,i i,j j,ν
ν ν ν ν i,j=1 ratiosR=0(purelycylindricalstructure,top),2(small-
X
(6) estdeviation,middle)and∞(purelysphericalstructure,
Finally,theaveragepathlength[30]isconsidered. For bottom). Given the good agreement for R = 2, one can
every node ν of the network, the minimum numbers l argue that the spherical geometry of the cluster is two
ν,i
of edges that have to be traversed to get to any other times more pronounced than the cylindrical geometry.
nodeiofthesamecomponentiscalculated. Theaverage The projections of the null model with R=2 on the ρz
6
inf a)
(cid:1)(cid:2) (cid:3)(cid:2)
3
R2
clockwise rotation
1
counterclockwise rotation
nonrotating cluster
1/2
3 b)
2
R1
(cid:4)(cid:2) (cid:5)(cid:2)
1/2
1/3
0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
noise amplitude [mm]
FIG. 7: Ratio R = nsph/ncyl vs. noise amplitude of null
models for Acyl (a) and Asph (b). The results are averaged
over40 calculations, and the standard deviations are plotted
as error bars. The results in Table I correspond to a noise
amplitudeof 0.15mm.
resultsinthe casesofclockwiseandcounterclockwisero-
tation,whilethevalueofRisgreaterforthenonrotating
cluster.
FIG. 6: Comparison of experimental cluster (top) to null
ThedependenceofthebestratioRonthenoiseampli-
model(bottom). Projectionsoftheclockwiserotatingcluster
tudeofthenullmodelsisshowninFig.7. Noticeably,de-
ontotheρzplane(a)andxyplane(b). Thetwomaincompo-
nentsofAcylareplottedastriangles(squares)andcorrespond creasingthe noise amplitude generallyshifts the optimal
totheinner(outer)ringregionsofthecluster. Forthesakeof ratios R to higher values for Acyl but towards lower val-
clarity the network edges are only plotted in panel (a). The ues for Asph. This can be understood as in the first case
network is identical to the one in Fig. 4, only the positions the difference in the cylindrical radii mostly determines
of the nodes are chosen differently: While in Fig. 4 a more whether nodes are connected or not. Hence, in order to
abstractrepresentationofthenetworkwaschosentoshowits findthebestagreement,moreparticlesofsphericalstruc-
structure,herethepositions correspond totheprojections of turehavetobeaddedwhenthenoiselevelisreduced,in-
theparticles. (c),(d)Thesamefor thenullmodel whosenet- creasing nsph and thus increasing R = nsph/ncyl. In the
work measures showed the best agreement with the cluster: secondcase of Asph, decreasing the noise amplitude is to
The ratio of particles in a spherical arrangement to particles
be compensated by adding more cylindrically structured
in a cylindrical arrangement is R=2.
particles, yielding a lower value for R.
The results depend rathersensitively onthe noise am-
and xy planes are plotted in Fig. 6 (bottom) and com- plitude of the null models (see Fig. 7). The exact values
pared to the clockwise rotating cluster (top). of the ratios R have thus to be treated with caution,
In order to test our approach, the analysis has been but they are a convenient tool for comparing structures.
repeated, this time comparing the spherical radii of the Considering Fig. 7, it is not possible to name the exact
particles. The adjacency matrix now reads valueoftheratioofthecompetingsphericalandcylindri-
cal geometries of the clusters, but it is evident that the
Asph(ǫ)=Θ(ǫ−|r −r |)−δ (8) nonrotating cluster systematically shows a higher value
ij i j ij
of R. The geometry is thus more sphericalin the case of
in the same notation as in Eq. 4 where r is now the the nonrotating cluster, while the rotating clusters have
spherical radius and ǫ=αh|r −r |i with α=0.1. a more pronounced cylindrical geometry.
i j
The results of the analysis of the rotating and nonro- Another possibility to examine the structure of a clus-
tating clusters for both adjacency matrices are summa- ter is to merely plot the particle z positions vs. cylin-
rized in Table I. Even though the results for the cluster drical radius ρ [3, 4, 37] (see Fig. 8). In this projection,
geometry are not the same for Acyl and Asph, the gen- spheres appear as semicircles and cylindrical structures
eralobservationremainsthesame: Onefindscomparable as vertical lines. Comparing the projections in Fig. 8,
7
TABLE I: Ratio R = nsph/ncyl of the competing geome-
tries (spherical and cylindrical) and anisotropic confinement
parameters Ωρ,z of theclockwise (CW)and counterclockwise
(CCW) rotating, and nonrotating clusters. The values of R
shownhereprovidedthebestagreementfornetworkmeasures
onthecylindrical(Acyl)andspherical(Asph)propertiesofthe
clusters. Theresultswereaveragedover40calculations, with
the standard deviation indicated as error. The confinement
parametersΩρ,z [Eq.(12)]werecalculatedfromthelinearfit-
ting parameters of the projections of the Yukawa forces (see
Fig. 9).
Cluster rotation
Parameter CW CCW Nonrotating
R for Acyl 2.0±0.2 2.0±0.2 3.0±0.2
R for Asph 1.3±0.5 1.0±0.2 2.0±0.2 FIG. 8: Projection of particle positions on the ρzplane for
Ωρ [s−1] 19±2 12±15 16.2±0.8 clockwise (a) and counterclockwise (b) rotating, and nonro-
Ωz [s−1] 33±1 36±2 32.0±0.3 ttaottinhgec(yc)lincldursitcearlss.tVruecrttuicraelolifntehsearnodtasteimngiccilrucsletserasraenaddjtuosttehde
spherical structure of the nonrotating clusters, respectively,
inordertoguidetheeye. Particlepositionsateachtimestep
correspond to a transparent marker, leading to lines in the
one can see that in the case of rotation [Figs. 8(a) and
eventofastringtransitionandsolidstructuresiftheparticle
(b)]theparticlesatthebottomandthetopofthecluster stays in the same position throughout thetime series.
havemovedoutward,while the particles atz ≃0appear
tohavemovedinwardcomparedtothe nonrotatingclus-
ter, Fig. 8(c). The rotating clusters thus appear more λ=0.4mmisthescreeninglengthasproposedinRef.[6].
cylindrical,whilethespheresaremorepronouncedinthe The radial and vertical projections Frep of the Yukawa
ρ,z
nonrotating case, but there is neither strong visual evi- forcecomputedfromthe experimentaldataareshownin
dence nor a numerical result as in the case of network Fig. 9. The figure also contains the best-rms fits intro-
analysis. duced by the relations
Frep =A ρ+B ρ2, Frep =A z, (11)
ρ ρ ρ z z
IV. CONFINEMENT ANISOTROPY
with A ,A ,B the parameters and ρ the cylindrical ra-
ρ z ρ
dius. In the case of counterclockwise rotation, the par-
The difference in structure between rotating and non-
ticles occasionally approacheach other very closely (this
rotatingclustersstemsfromtheparticularsofthemutual
may be an imaging issue), leading to a wide spread of
particle interactions and the forces confining the clus-
the estimated Yukawa forces. Considering the large un-
ter. Even though it is a rather difficult task to explore
certainty of the parameters in Fig. 9(b), this approach
the forces controlling the cluster dynamics in detail, it
may not be applicable there.
is instructive to assume that globally the system is in
The friction force is Ffr = −Mγ v, where M =
quasiequilibrium. The latter is determined by the bal- Eps
anceofrepulsionFrep viathe Yukawaforces,neutralgas 1.1×10−12 kg is the particle mass [6], γEps is the Ep-
friction Ffr, inertial forces Fin, and confinement Fconf stein drag coefficient [38] and v is the particle velocity.
The value of the friction force at the cluster periphery
provided by all other forces [43]:
was estimated in Ref. [6] to be smaller than 10fN. Com-
hFrepi+hFfri+hFini+hFconfi=0. (9) pared to the Yukawa forces in the same region of the
cluster, which are on the order of 0.5–2pN, the friction
Here, the stochastic averaging is assumed to be per- force is of minor importance and canbe neglected in the
formed. balance of Eq. (9).
TheYukawaforces,governingthemutualrepulsionbe- The inertial forces in the rotating frame of the clus-
tween the particles, can be directly calculated from the ter consist of the centrifugal force Fcentri and the Cori-
data: olis force FCor [39]. The former is readily estimated as
Fcentri = Mω2ρ < 1fN, where ω ≃ 0.4s−1 is the rota-
Friep =−(Ze)2∇ri N exp(−|r|ir−i−rjr|j|/λ), (10) titioiens,sfipveeedcoonfstehceutcilvuestferra.mTeos wesetrime aavteertahgeedp,aartsictlheevjeitlotecr-
j6=i
X on some particles due to imaging processes complicates
where r is the position of particle i, Ze = 50000e is the calculation of the instantaneous velocity. The Cori-
i
the particle charge (e is the elementary charge) and olis force, which is maximal during transitions of a par-
8
stronger confinement, increasing the cylindricity of the
rotating cluster, is in good agreement with our findings
from the network analysis of Sec. III.
From our estimate of the centrifugal and the Coriolis
force it follows, furthermore, that the structural changes
intherotatingclusterarenotdue tothe clusterrotation
per se. It must rather be inferred that the electrostatic
confinement of the whole cluster changes with the ap-
plied frequency, allowing for string formation within the
rotating cluster.
In Eq. (9), Frep was assumed to consist of Yukawa in-
teractionswithconstantparticlechargeZeandscreening
length λ. It is straightforward to see that different val-
ues of λ and Z only change the scale of the profiles in
Fig. 9, and not the general shape. A dependency of the
parameters on the vertical position of the particles, as
can be expected in the sheath electric field, will change
the force profiles, but one can guess that this will lead
to the same result of a stronger radial confinement in
FIG. 9: Radial profiles of the repulsive Yukawa-type inter-
the case of cluster rotation. While Yukawa interaction
actions of the particles for clockwise (a) and counterclock-
is valid and widely used in two-dimensional systems, an
wise (b) rotating, and nonrotating (c) clusters. The cylindri-
calprojectionoftheYukawaforcesisplottedvs. thecylindri- additionalcontributionfromthewake-fieldinteractionis
calradiusρ. Thesolidlinesarethebest-rms-fitparabolas[Eq. not negligible in 3D [2, 20]. An adequate description of
(11)] to the force profiles. The insets show the mean values the wake field interaction is not straightforwardand the
of Aρ in pN/mm and Bρ in pN/mm2 which were calculated subject of current research. Here, these anisotropic ef-
for each time step, with thestandard deviations indicated as fects are included in the confinement force Fconf, which
errors. The vertical lines are the same as in Fig. 8. (d)–(f) is a mixture of external confinement and internal forces.
The same for the z profiles of the Yukawa forces, here, the
The latter may be a significant contribution, as they re-
few particles suspended below thecluster at z≃−2mm (see
sultintheformationofparticlestrings. Equation(11)is
Fig. 8) were left out. The insets show the parameter Az in thusafittotheglobalforceprofile,wherethemodulation
pN/mm of thelinear fit tothe profiles.
due tothe mutualwake-fieldinteractionsisnotresolved.
Nonetheless,oursimplifiedmodeloftheYukawa-typein-
teraction gives a hint for the origin of the differences in
ticle from an inner to an outer string or vice versa, then
yields FCor = 2M|ω ×v| < 3fN. Hence, compared to symmetryrevealedbynetworkanalysis. Amoredetailed
analysis should take into account the mass dispersion of
the Yukawaforces,the contributionofinertialforcescan
theparticlesaswellastheanisotropyintheinterparticle
be neglected. The confinement force profiles can thus be
estimated as Fconf ≃−Frep. interaction.
ρ,z ρ,z
Given the values of the fitting parameters A intro-
ρ
duced above, the radial confinement near the center of
the cluster can be estimated as
V. SUMMARY
Fconf ≃−MΩ2ρ, MΩ2 ≡A , (12)
ρ ρ ρ ρ
To conclude, the data obtained with the help of a
where Ω is the cylindrical confinement parameter. The fully 3D holographic particle tracking diagnostic [6, 27]
ρ
values of Ω as well as the values of the vertical confine- allowed us to thoroughly explore the statistical as well
ρ
ment parameterΩ (obtained similarly with the relation as dynamical properties of particle clusters driven exter-
z
MΩ2 ≡A ) are shown in Table I. nally by rotating electric fields. The data was subjected
z z
It is not surprising that the vertical confinement force tonetworkanalysiswhichdemonstratedasignificantdif-
is systematically stronger than the radial one, as can ferencebetweenthewell-studiedcaseofnonrotatingclus-
be naturally expected due to the stronger forces of the tersanddynamicallydrivenclusters. Whilethestructur-
sheath electric field and gravity compressing the cluster ing is more spherical in the case of nonrotating clusters,
vertically[40]. Note that a strongerverticalconfinement the dynamically drivenclusters have a more pronounced
was also found for “dust molecules” consisting of only cylindricalstructure. Thisdifferenceisinagreementwith
two particles [41]. the estimate of the radial confinement with the aid of a
Neglecting the case of counterclockwise rotation due dynamical force balance. As for string detection, com-
to the wide spread of data points, this approach yields munityassignmentinnetworksprovedavaluabletoolfor
a stronger radial confinement for the clockwise rotating finding structures (without setting a predefined thresh-
cluster than for the nonrotating one (see Table I). This old) in systems that evolve over time.
9
Acknowledgments the European Unions Seventh Framework Programme
(FP7/2007-2013)/ERCGrantagreement267499,andhas
The research leading to these results has received been financed by DFG under grant No. SFB-TR24,
funding from the European Research Council under project A3.
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