COMPUTATIONAL PHYSICS

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Molecular dynamics simulation of a two-dimensional dusty plasma

Istv!an Donk!o,Faculty of Informatics, E€otv€os Lor!and University, P!azm!any P!eter stny. 1/c, H-1117 Budapest, Hungary

Peter Hartmann, and Zolt!an Donk!oa)

Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Konkoly-Thege Miklos str.29-33, H-1121 Budapest, Hungary

(Received 23 July 2019; accepted 20 September 2019)

Two-dimensional dusty plasmas can be realized experimentally and are examples of a classical manybody system with a screened Coulomb interaction. After discussing experimental approaches, wepresent the basics of molecular dynamics simulations of dusty plasmas. A web-based platform isdeveloped that allows users to perform molecular dynamics simulations, visualize the system forselected system parameters, and obtain results for the pair correlation function and the dispersionrelation of waves in the system. VC 2019 American Association of Physics Teachers.

https://doi.org/10.1119/10.0000045

I. INTRODUCTION

Dusty plasmas are systems where millimeter to nanometersized particles are suspended in a partially ionized gaseousenvironment consisting of neutral atoms/molecules, elec-trons, and ions. The initial interest in dusty plasmas origi-nated in the fields of space physics and astrophysics,1,2

where it was realized that dust particles are naturally presentin these environments. More recent developments in gas dis-charge physics and microelectronics3 have led to the recog-nition that dust contamination is naturally present inindustrial plasmas due to plasma etching effects as well asdust particle growth through chemical reactions.4

A recent experimental breakthrough has been the abilityto optically track individual dust grains suspended in dis-charge plasmas, which provides the possibility of examiningthe dynamics at the level of individual dust particles.5–7

In addition to its relevance to numerous topics in plasmaphysics, including dust charging, dust shielding, ion dragforces, and plasma–surface interactions, dusty plasmas playan important role as analogue systems for investigating com-plex cross-disciplinary phenomena. For example, experimen-tal studies of nonlinear dynamics and long-range interactionsin strongly correlated systems are often challenging becausethese systems generally require extreme conditions such asvery low temperatures (ultracold ion/atom systems) or highdensities (high energy density physics). Dusty plasmas allowthe formation and study of analogs to such strongly coupledsystems, where the potential energy dominates the kineticenergy.8 Such studies can be conducted under laboratoryconditions at close-to-room temperatures and easily attain-able pressures.9 Dusty plasma analogues have recently beenused to model crystallization dynamics in two dimensions,10

quantum dot excitation,11 nonlinear and shock waves,12 elec-trorheological fluids,13 and viscoelastic materials.14

Strongly coupled dusty plasmas were initially establishedin radio frequency excited noble gas discharges using spheri-cal monodisperse solid particles made of glass,15 melamine-formaldehyde,16 silica,17 or titanium-dioxide.7 In these sys-tems, the dust particles acquire a negative electric charge tobalance the electron and ion currents flowing to their surfa-ces, and become confined by the sheath electric fields thatform near the electrodes and other surfaces surrounding thegas discharge plasma. In the presence of gravity, thousandsof micrometer sized particles can arrange themselves in aquasi-two-dimensional (2D) layer.

In Sec. II, we discuss experimental systems in more detail.We then discuss the “one component plasma” model and itsmain parameters in Sec. III. This model has successfullybeen applied to 2D dusty plasma systems. The basics of themolecular dynamics method, which is one of the main toolsfor the simulation of dusty plasmas, are presented in Sec. IV,along with illustrative numerical results. Section V explainsthe implementation of this technique using a web-based toolthat is freely available and supplements this article.Suggested problems are given in Sec. VI.

II. EXPERIMENTAL SYSTEMS

A schematic of an experimental setup is shown in Fig. 1.A low-pressure capacitively coupled radio frequency plasmais established within a vacuum chamber (not shown)between the powered and grounded electrodes. When smalldust grains (with a diameter typically in the micrometerrange) are introduced into this plasma, they acquire a nega-tive charge18 of typically 103–104 elementary charges.

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Because of their high electric charge, the dust particles arelevitated above the powered electrode due to the balance ofgravity and the electrostatic force that originates from thesheath electric field.19 The horizontal confinement of the dustsuspension is ensured by the radial electric field which arisesdue to the charging of a confining element, for example, a glasstube surrounding the system (not shown in Fig. 1).

Images of the particle layer are recorded by a video cam-era. The observation is aided by illuminating the dust layerwith an expanded laser beam with a specific wavelength.The spectral filter placed in front of the camera transmits thiswavelength but blocks most of the radiation from the plasmathereby significantly increasing the contrast. A sequence ofrecorded images can be used to identify and trace individualparticles, providing the position and velocity coordinates ofeach particle. The length and time scales allow relativelyeasy observation of the dynamics.

Figure 2 shows a snapshot of the particles recorded in ourexperimental setup. Due to the strong interaction between

the particles, most of the particles are located at sites of a tri-angular lattice. The crystalline ordering is a consequence ofthe strong electrostatic interaction of the highly charged dustparticles which creates a potential energy that is muchgreater than the kinetic energy of the system. The particlescannot form a perfect lattice due to the boundary conditionsin the experimental system and their density slightly variesspatially. Therefore, we observe lattice defects, predomi-nantly dislocations, which appear as bounded pairs of siteswith 5 and 7 particles in their nearest neighbor shells, deter-mined by performing a Delaunay triangulation.20 The signifi-cance of the different types of lattice defects and theirconsequences on the topological phase transitions in 2Dmaterials was discovered by Kosterlitz and Thouless.21 Industy plasmas, the static and dynamical properties of thesesystems have been investigated in detail.22,23

III. THE ONE-COMPONENT PLASMA MODEL

We will discuss the properties of the dust particle systemin the plasma environment within the framework of the “onecomponent plasma” model. This model considers only oneof the species of the system (the dust particles in our case).As we have discussed, dust particles acquire a large negativecharge in the plasma environment. In the absence of othercharged species (the electrons and ions of the surroundingplasma), the dust particles would interact via the r–1

Coulomb potential. However, the presence of electrons andions modifies this potential due to an important property ofplasmas, Debye screening, which results in deviations of theelectron and ion densities from their average density aroundthe dust particles, and leads to the screening of the potentialcreated by the dust particles. It can be shown that, as a conse-quence, the Coulomb potential is modified by an exponentialfactor, resulting in a dust particle pair potential of the form24

/ðrÞ ¼ Q exp ð$r=kDÞ4pe0r

; (1)

where Q is the charge of the dust particles, kD is the screen-ing (Debye) length, and e0 is the permittivity of free space.The form of the potential in Eq. (1) is known as theDebye–H€uckel or Yukawa potential. If the particles interactvia the potential in Eq. (1), the model is often referred to asthe “Yukawa one component plasma” model.

The main parameter of the system is the couplingparameter

C ¼ Q2

4pe0akBT; (2)

which is the ratio of the Coulomb potential energy at a char-acteristic particle separation aws to the thermal energy kBT(equal to the mean kinetic energy per particle in 2D), whereT is the dust temperature and kB is the Boltzmann constant.The Wigner-Seitz radius is defined as

aws ¼ ðpn0Þ$1=2; (3)

where n0 is the areal number density of the particles.The strength of the screening of the Coulomb interaction

can be expressed in terms of the dimensionless parameter

j ¼ aws=kD: (4)

Fig. 1. Schematic of a dusty plasma experiment.

Fig. 2. Experimentally recorded snapshot of a dusty plasma layer for a largevalue of the coupling parameter C. The configuration is highly ordered withan underlying triangular lattice (identification is aided by the lines in a partof the snapshot). Although most particles have six neighbors, lattice defectsresult in exceptions that are indicated by circles (five neighbors) and trian-gles (seven neighbors).

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Another important quantity is the plasma frequency

x0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinQ2=2awse0m

p; (5)

which governs the dynamics of the system. Here, m is the massof the particles. (Note that the plasma frequency for a 2D systemdiffers from the 3D plasma frequency x0;3D ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin3DQ2=e0m

p.)

IV. MOLECULAR DYNAMICS SIMULATIONS

Molecular dynamics (MD) simulations produce particletrajectories in phase space by accounting for the interactionsbetween the particles and for the effects of additional forcesoriginating from an external potential (for example, a trap-ping potential) and/or the effect of an embedding environ-ment (for example, friction25). The case that we discuss hereis one of the simplest:

(1) We consider a microcanonical (“NVE”) ensemble, wherethe number of particles, the volume of the system, andthe total energy is conserved.

(2) We use point-like charged particles which have onlytranslational degrees of freedom. (Molecules may haveadditional (for example, rotational) degrees of freedomand their pair interaction may depend on their mutualorientation. The mutual orientation may also becomeimportant if the particles have a magnetic moment,which also contributes to inter-particle forces.)

(3) The particles interact via an isotropic pair potential. Inexperiments, the particle charging is anisotropic due tostreaming ions. Neglecting this anisotropy is well justified aslong as the particles arrange themselves into a single layer.

(4) The motion the particles is assumed to be frictionless, whichis a good approximation for low gas pressures (!1 Pa).

(5) Although particles are confined within a layer by a trap-ping potential in experiments, we idealize the system byassuming that the motion of the particles is restricted to aplane.

(6) We consider a system consisting of a single type of spe-cies, although in reality there exists a broad range ofmulti-component systems.

Our simulations are relevant to the central domain of alarge experimental system where the density can be consid-ered to be homogeneous.

The system consists of N identical particles confined to asquare, H%H simulation cell. Given the simplifications wehave discussed, the form of the equations of motion for parti-cle i is

m€ri ¼X

i 6¼j

Fðri;jÞ; (6)

where F (ri,j) is the force between particles i and j, whichdepends only on their separation ri,j. The summation goesover the particles that make a non-negligible contribution tothe total force acting on particle i.

Equation (6) can be solved by using a numerical integra-tion method. We use the velocity-Verlet algorithm26

rðtþ DtÞ ¼ rðtÞ þ vðtÞDtþ 1

2aðtÞðDtÞ2; (7)

vðtþ DtÞ ¼ vðtÞ þ 1

2aðtÞ þ aðtþ DtÞ½ (Dt: (8)

Because the acceleration appears both at t and t þ D t inEq. (8), the computation proceeds as follows:

(1) The particle positions are updated according to Eq. (7),based on the values of the relevant quantities at time t.

(2). From Eq. (8), we compute the intermediate velocityv) ¼ vðtÞ þ ð1=2ÞaðtÞDt.

(3) We calculate the force and aðtþ DtÞ based on theupdated positions of the particles.

(4) Finally, we calculate vðtþDtÞ¼v)þð1=2ÞaðtþDtÞDt.

These steps are repeated many times to obtain the particletrajectories.

In the simulations, we apply periodic boundary conditions,which eliminate any surfaces so that a particle that leaves thesimulation cell is introduced into the cell at the oppositeside, without any change of its velocity (see Fig. 3). Thiscondition ensures a homogeneous system and makes itunnecessary to use a confining potential.

The solution of the equations of motion requires the com-putation of the total force acting on each particle. For apotential that decays sufficiently quickly, for example, theYukawa potential, we can define a cutoff distance rc beyondwhich the pair interactions are negligible. As illustrated inFig. 4(a), the force acting on a particle can be obtained bysumming over neighboring particles that are situated withina circle of radius rc, centered at the given particle. For par-ticles near the edges of the simulation cell, the circle, whichencloses the neighbors, may be partly outside the cell, asshown in Fig. 4(b). In this case, neighbors also have to besearched for in the replicas of the simulation cell (which arecopies of the simulation cell shifted by the cell length in oneor both directions and which surround the primary simula-tion cell).

Finding the neighbors efficiently is aided by the “chainingmesh” technique: The simulation cell is divided into sub-cells, and at each time step the particles are assigned to listscorresponding to one of these cells.27 The edge length of thesub-cells h is chosen such that h > rc, so that it is sufficientto search for neighbors only in the sub-cell where the givenparticle resides and in neighboring sub-cells.

Molecular dynamics simulations can be performed at con-stant temperature, constant pressure, or constant energy. Thechoice depends on the nature of the effects that are

Fig. 3. Illustration of periodic boundary conditions. Particles that leave thesimulation cell are re-introduced at the opposite side, with the samevelocity.

988 Am. J. Phys., Vol. 87, No. 12, December 2019 Donk!o, Hartmann, and Donk!o 988

investigated. The results discussed in this section wereobtained from simulations where the energy is conserved.The interactive MD tool that supplements this article, on theother hand, applies an Andersen thermostat26 to ensure thatthe system equilibrates within a reasonable time at the givenvalues of C and j values set by the user.

In the following, we illustrate some of the results that canbe obtained from the simulations. Figure 5 shows particlesnapshots obtained from the simulations for different valuesof C and j ¼ 1. For C ¼ 10, we observe a weak order in theparticle positions, which is significantly enhanced when thecoupling is increased to C ¼ 100. Ordering is a consequenceof the dominance of the interparticle potential energy overthe kinetic energy of the particles.

One measure of the degree of regularity of the particleconfigurations is the pair correlation function g(r). Todetermine g(r), we count, around a given particle, the num-ber of particles situated within concentric rings as illus-trated in Fig. 6(a), and normalize this number by therespective area of these domains and by the average particledensity. We then average over all particles. For a randomarrangement of the particles (for which the most importantexample is the ideal gas) the resulting value is gideal(r) ¼ 1.The pair correlation function g(r) thus expresses the rela-tive density of neighbors around a given particle. Byassuming isotropy of the system, which is an inherent char-acteristic of the fluid phase, the pair correlation function

has a single scalar argument, the separation of the particles.If the system crystallizes, the pair correlation function has avector argument.

The pair correlation function for the Yukawa fluid at dif-ferent values of C is plotted in Fig. 6(b). The function g(r)exhibits a “correlation hole” at small separations as a conse-quence of the repulsion between the particles, which pre-vents them from getting close to each other due to their lackof sufficient kinetic energy. At a distance corresponding tothe most probable first neighbor distance, g(r) exhibits apeak, which marks the first coordination shell. This peak isfollowed by a depleted region and a sequence of other coor-dination shells, which appear as smaller amplitude peaks ing(r) due to the decay of correlations over longer distances.The spatial order of fluids extends over a finite distance suchthat within this distance the fluid “inherits” the structure ofthe lattice, which is why the peak positions of g(r) changeonly slightly with increasing C in Fig. 6(b).

Fig. 4. (a) For short-range potentials, it is sufficient to consider neighbor-ing particles within a circle of radius rc to compute the total force actingon a particle. (b) When periodic boundary conditions are applied, theseneighbors might also need to be searched for in replicas of the primarysimulation cell.

Fig. 5. Snapshots for (a) C ¼ 10 and (b) C ¼ 100 at j ¼ 1. The spatialcoordinates are normalized by the Wigner-Seitz radius. Note the less regu-lar particle arrangement for C ¼ 10 and the more ordered configuration forC ¼ 100.

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The prominent structure of the system for large C gives riseto quasi-localization of the particles, which means that theparticles oscillate in local potential wells of the potential sur-face created by other particles. For strong coupling8 (C* 1),the change of the potential surface due to the diffusion ofthe particles is much slower than the time scale of theseoscillations. This behavior is the basis of the quasi-localizedcharge approximation where the dynamical properties arederived from the pair correlation function and the interactionpotential.28,29

An intrinsic property of many-body systems is that they cansupport collective excitations; for example, gases and liquidssupport sound waves, which are propagating compressionaldensity perturbations. Waves are not only generated by anexternal perturbation of the system, but can also be thermallyexcited, where the motion of the particles of the system gener-ates a (usually low amplitude) density perturbation. The wave-length and the frequency of the waves are determined by thestructure of the system and the interparticle interaction.

If the motion of the particles is confined to a plane, twotypes of waves may develop: longitudinal and transversewaves, as illustrated in Fig. 7(a). (If the motion is notrestricted to a plane, but is confined by a parabolic potentialthat allows particles to move perpendicularly to the plane,there are three different types of waves: Two of them are thesame as here; the third, which is also a transverse mode withparticle oscillations perpendicular to the plane, has an opticcharacter due to the confining potential.) For longitudinal or

compressional waves, the direction of oscillations is in thesame direction as the wave propagation [represented by thewave vector k in Fig. 7(a)]. For transverse or shear waves,the direction of oscillations of the particles is perpendicularto the wave vector k. Transverse waves are unusual in gasesand liquids, which cannot sustain shear. The presence ofstrong correlations enables the existence of shear waves,23,30

which are strongly damped and consequently have a shorterlifetime compared to compressional waves.

Longitudinal waves propagating in the x–direction can becharacterized by the dynamical structure function S(kx,x),

Fig. 6. (Color online) (a) Illustration of the measurement of the pair correla-tion function g(r). (b) The pair correlation function of the Yukawa liquid fordifferent values of C and j ¼ 1.

Fig. 7. (Color online) (a) Longitudinal and transverse waves in a plane; k isthe wave vector, which points in the direction of wave propagation, and thearrows indicate the oscillations of the particles. (b) The current fluctuationfunction L(kx,x) for C ¼ 100 and j ¼ 1. The argument kx of L implies thatdata were collected along the x axis. The frequency and the wave numberare normalized, respectively, by the plasma frequency and the Wigner–Seitzradius. Large values of L(kx,x) identify waves in the system, and the loci ofthese values yield the dispersion relation xðkxÞ. Because the system is in thefluid phase, the loci is not a sharp line. The dispersion relation of Yukawafluids is quasi-acoustic; that is, it is linear for small wave numbers andacquires a square root form at intermediate wave numbers. The slope ofxðkxÞ at small k defines the longitudinal sound speed (Ref. 23).

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which gives the spectrum of the density fluctuations. Thepeaks of S(kx,x) identify the collective excitations. If S(kx,x)is computed for a series of wave numbers, the dispersionrelation x(kx) of the longitudinal wave can be obtained fromthe simulations by identifying the x values where S has apeak as a function of kx. This procedure is easy to carry outfor C * 1 for which S exhibits sharp peaks, but is moreambiguous when the peaks of S become broad due to thehigher damping or decreasing lifetime of the modes at lowerC values.

The computation of S(kx,x) proceeds by decomposing theparticle density in terms of its Fourier components

qðkx; tÞ ¼XN

j¼1

e$ikxxjðtÞ: (9)

Given a time series of q(kx,t), another Fourier transform (intime) is performed to obtain q(kx,x), from which thedynamic structure function is computed as31,32

Sðkx;xÞ ¼1

2pN

1

T0qðkx;xÞq)ðkx;xÞ½ (; (10)

where q)ðkx;xÞ is the complex conjugate of q(kx,x) and T0

is the duration of the recording of q(kx,t).In addition to the spectrum of density fluctuations, we

can compute the spectrum of the current fluctuations,L(kx,x). The longitudinal current fluctuation spectrum isobtained in a similar way as S(kx,x) from Eq. (10) byreplacing q(kx,t) by

kðkx; tÞ ¼X

j

vj;xe$ikxxjðtÞ; (11)

where vj,x is the x component of the velocity vector of thej-th particle. We note that S(kx,x) and L(kx,x) are related byLðkx;xÞ ¼ ðx=kxÞ2Sðkx;xÞ.33 Because L(kx,x) is usuallyobtained with better signal to noise ratio in simulations, wecompute L(kx,x). (In case of sharp peaks in Sðkx;xÞ the posi-tions of the corresponding peaks in L(kx,x) are very nearlythe same, but in general, the x2 factor in the above relationmay shift the peak positions to some extent.)

For propagation along the x direction, information can beobtained for waves whose wavelength fits integer times(1; 2;…) into the simulation cell, that is, for which the wavenumbers obey kx ¼ nkmin ðn ¼ 1; 2;…Þ, where kmin ¼ 2p=His the minimum wave number. It is conventional to introducethe normalized wave number #k ¼ kminaws. The normalized

minimum wave number, #kmin ¼ 2ffiffiffipp

=ffiffiffiffiNp

(using Eq. (3)with n0 ¼ N=H2) decreases with increasing density.

Figure 7(b) shows an example of the longitudinal current-current fluctuation spectrum, L(kx,x), for C ¼ 100 and j¼ 1.The region with high amplitude indicates that longitudinalwaves are present at the given frequency and wave number.At low wave numbers, kx aws ! 1, the peaks of L(kx,x) as afunction of x (at a given kx) are quite sharp, but at higherwave numbers the peaks broaden due to the decrease in thelifetime of collective excitations with increasing kx. At smallwave numbers, the dispersion relation xðkxÞ is linear, and itsslope defines the sound speed. With increasing wave number,the mode frequency starts to deviate from this linear depen-dence on the wave number, and the mode is said to have aquasi-acoustic character. At even higher values of kx, the

mode frequency saturates and subsequently starts to decrease.This behavior is reminiscent of the behavior of a mode in asolid, where the frequency is a periodic function of the wavenumber in the principal directions.

The transverse waves can be analyzed similarly to longitu-dinal waves by computing the transverse current fluctuationspectrum T(kx,x) from the transverse current

sðkx; tÞ ¼X

j

vj;ye$ikxxjðtÞ; (12)

by replacing vj;x by vj;y in Eq. (11), where vj;y?kx. As men-tioned previously, transverse waves are unusual in the liquidphase and can exist in the present system only because of thestrong correlations at high C. Because these waves arestrongly damped for C* 1, their spectrum is quite broadand the mode frequency is more difficult to identify com-pared to the longitudinal modes.23 Therefore, we consideronly the longitudinal modes in our simulation tool.

V. IMPLEMENTATION

Our MD simulation follows N¼ 500 particles in a plane.34

Our desire to obtain results and visualize the system in realtime limits the value of N. For research purposes, muchlarger particle numbers are routinely used. The simulationuses a 5% 5 chaining mesh, which, with the choice ofrc ¼ H=5, provides sufficient accuracy for the forces actingon the particles even for the smallest value of the screeningparameter, j ¼ 1, for which the decrease in the potential isthe slowest.

The particles are initially placed randomly within the squaresimulation cell with their initial velocity vectors sampled froma Maxwell–Boltzmann distribution. Due to the randomness ofthe initial configuration, the system needs to be thermalized toreach equilibrium, which is achieved by rescaling the veloci-ties of the particles at each time step to match the desired sys-tem temperature. This procedure ensures a fast thermalizationof the system, but creates a non-Maxwell–Boltzmann distribu-tion from the initial Maxwell–Boltzmann distribution, andtherefore no data are collected during this initial thermalizationperiod.

After turning on data collection at later times, theAndersen thermostat35 is applied to the system: Randomlychosen particles interact with a defined frequency with a heatbath that has a temperature defined by the value of C. Uponinteraction with the heat bath, the particles acquire newvelocity vectors that are randomly sampled from aMaxwell–Boltzmann distribution at the desired temperature.

During the simulation, the particle positions are displayedas a function of time, while the actual value of C, the paircorrelation function g(r), and the wave behavior are moni-tored in separate panels.

The system parameters C and j can be adjusted using thegraphical interface shown in Fig. 8. Following a change,velocity rescaling is conducted for 1000 time steps, and thenaccumulation of data for g(r) and L(kx,x) at the new parame-ter values begins.

The simulation is implemented in JavaScript with the lay-out and interactions defined using HTML and CSS, whichruns either in a browser or as a self-contained Electron36

package with the open source Chromium browser bundledwith the contents. The rendering of the particles, graphs, andthe wave dispersion map use the Web Canvas technology.37

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To increase performance, the application is divided intofour kinds of threads as illustrated in Fig. 9.

• The Main thread handles the front-end, while the threeothers running in the background handle the bulk of thecomputation. The results of these worker threads are thensent back to the main thread and displayed on theinterface.

• The Simulation thread provides the necessary data for thevisualization and other calculations.

• The Buffer thread preprocesses the data it receives, thusproviding buffer contents for the Wave dispersion thread.There are two instances of this thread running at the sametime, one for handling each of the dimensions in thesimulation.

• When enough information is accumulated, the Wave dis-persion thread executes the final steps needed to producethe “heat map” displayed on the interface.

Further details about the implementation can be found inthe Readme file, which is attached to the source code of theapplication, that is available at: https://github.com/Isti115/dusty-plasma-molecular-dynamics. The online version canbe accessed by visiting: https://isti115.github.io/dusty-plasma-molecular-dynamics/. The packaged executables aredownloadable from the AJP servers.34

VI. SUGGESTED PROBLEMS

(1) In a hexagonal lattice, the six nearest neighbors are situ-ated at a distance of r1 ¼ b, and the second and third

nearest neighbors are at r2 ¼ffiffiffi3p

b and r3 ¼ 2 b, respec-tively, where b is the lattice constant, which is related to

the Wigner-Seitz radius by b¼aws

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3p

=2pq

ffi1:905aws.

The first peak of g(r) occurs at r ¼ r1 for C*1. In thefluid phase, the coordination shells at r2 and r3 are

usually not resolved in g(r). Find the value of C for j ¼1, which results in distinct peaks in g(r) for the secondand third coordination shells. Does this value of Cbelong to the fluid or the solid phase? (The transitionbetween the two phases occurs at C,186.22)

Fig. 8. (Color online) Screenshot of the output of the simulation program. The simulation cell on the right is indicated by the square. When the switch“Periodic display” is on, the particles situated in the replicas are also visible.

Fig. 9. Schematic representation of the program’s structure and the commu-nication between its parts.

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(2) Observe the domain coarsening process after a tempera-ture quench by setting the screening parameter to a lowvalue j ¼ 1, and quickly changing C from a low to ahigh value (for example, from C ¼ 10 to 500). The sys-tem should change from a fluid to a crystalline phase.This process is not instantaneous, and some time isneeded for crystalline order to form. There are in princi-ple different pathways for this process, such as crystalli-zation front propagation, simple diffusion, and domaincoarsening, where in the beginning several small crystal-lites form, and then later merge into a few bigger ones,which eventually form a single ordered structure.10

(3) Observe how the peak value of the frequency of the longi-tudinal waves depends on C and j. Which of these param-eters results in a strong dependence and which in a weakdependence? The slope of the xðkÞ dispersion relation atsmall k defines the sound speed. How does this speedchange with the dust-dust interaction potential? Note thatthe potential depends on j, but not C [see Eq. (1)].

(4) Similar values of g(r) can be obtained for certain combi-nations of C and j. Find the value of C for j ¼ 1; 2;3; 4; 5 that results in the same amplitude, for example,2.0, of the first peak of g(r). Observe how the L(kx,x)spectrum differs for the different ðC; jÞ pairs, eventhough the structure of the system is very nearly thesame. Recall that the interaction potential between theparticles softens with increasing j, which then slowsdown the dynamics.

ACKNOWLEDGMENTS

This work was supported by the National Office forResearch, Development and Innovation (NKFIH) via GrantNos. 119357 and 115805.

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