Viscosity calculated in simulations of strongly-coupled dusty plasmas with gas friction
aa r X i v : . [ phy s i c s . p l a s m - ph ] A p r Viscosity calculated in simulations of strongly-coupled dusty plasmas with gas friction
Yan Feng, ∗ J. Goree, and Bin Liu
Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242, USA (Dated: November 7, 2018)A two-dimensional strongly-coupled dusty plasma is modeled using Langevin and frictionlessmolecular dynamical simulations. The static viscosity η and the wave-number-dependent viscosity η ( k ) are calculated from the microscopic shear in the random motion of particles. A recently de-veloped method of calculating the wave-number-dependent viscosity η ( k ) is validated by comparingthe results of η ( k ) from the two simulations. It is also verified that the Green-Kubo relation canstill yield an accurate measure of the static viscosity η in the presence of a modest level of frictionas in dusty plasma experiments. PACS numbers: 52.27.Lw, 52.27.Gr, 66.20.-d, 83.60.Bc
I. INTRODUCTION
Strongly-coupled plasma is a collection of free chargedparticles where the Coulomb interaction with nearestneighbors is so strong that particles do not easily movepast one another. A widely used criterion to determinewhether a plasma is strongly coupled is Γ > >
1, particles move slowly and are trapped by a cageconsisting of a few nearby particles. If they escape thecages gradually, particles in a strongly-coupled plasmacan flow, much like a liquid [3]. However, if Γ ≫ ,nearby particles that form a cage move so little that aparticle inside the cage can seldom escape the cage; thiscondition is like molecules in a solid [4, 5]. If a shearingstress is applied, cages in a solid are elastically deformedbut can restore to their previous state, whereas cages ina liquid are disrupted and a viscous flow can develop.One type of strongly-coupled plasma is dusty plasmaformed in the laboratory. A dusty plasma consists of fourconstituents: micron-size particles of solid matter (dustparticles), electrons, ions, and neutral gas atoms [6–8].The dust particles are strongly coupled amongst them-selves due to a large interparticle potential energy pro-vided by a large particle charge [9, 10]. Several schemeshave been used to confine charged dust particles us-ing natural electric field inside a plasma. One of theseschemes makes use of a radio-frequency plasma [11, 12],with a horizontal electrode that provides a sheath elec-tric field that can confine and levitate dust particles ina cloud with only a few horizontal layers. If experi-menters introduce only a limited number of dust par-ticles, they can settle into just a single layer [9]. In thesesingle-layer clouds, dust particles have negligible verti-cal motion, so that the cloud of dust particles is oftendescribed as a two-dimensional (2D) system [9, 13–16].In this 2D cloud, the interaction between dust particlesis a repulsive Yukawa potential [17]. Due to the largelength scale and the slow time scales [8], dusty plas-mas allow video microscopy to track individual particle motion [18]. In dusty plasma experiments, elasticity insolids [14] and viscosity in liquids [16] has been observedand studied. However, strongly-coupled plasmas cannotalways be classified as purely elastic or purely viscous.Dust particles experience several forces in the ex-periments. The electric force provides strong couplingamongst the dust particles as well as the levitation andconfinement. Gas friction, due to dust particles movingrelative to the rarefied gas, is the primary energy lossmechanism. The gas is usually so rarefied that it rep-resents only a small portion of the mass of the dustyplasma. Gas represents <
10% of the mass of dustin a 3D dusty plasma experiment at 400 mTorr [19],while 2D experiments have even less gas, with a pres-sure <
20 mTorr [9, 20, 21]. There is an ion drag forcedue to a steady flow of ions, arising from the same dcelectric fields that provide levitation and confinement ofdust particles. This ion drag force is parallel to the ionflow. Finally, in some experiments, laser radiation pres-sure forces are used to accelerate dust particles, for exam-ple to create macroscopic flows [12, 16, 21, 22] or simplyraise the kinetic temperature of the dust particles with-out causing a macroscopic flow [9, 20, 23, 24]. This kindof laser heating method is one of several ways that exper-imenters can control Γ so that the cloud of dust particlesbehaves like a liquid or solid [13, 24–27].We assume that the Coulomb interaction amongstcharged dust particles is the dominant mechanism for vis-cosity in laboratory dusty plasma experiments. Viscoustransport of momentum occurs when the dust particlesmoving relative to one another in a shearing motion col-lide, causing some of their momentum to be transferredacross the flow. We expect that collisions involving gasatoms will contribute less to the viscosity. Although theforce of gas friction is effective in diminishing momen-tum of dust particles in the direction of their motion,there are two reasons it has little effect in transferringmomentum across a flow of dust particles. First, thegas is rarefied so that it can carry much less momentumthan a viscous solvent in a colloidal suspension [28, 29],for example. Second, in a 2D experiment like [20], a gasatom that is struck by a dust particle is usually knockedinto a direction out of the dust layer, so that there is lit-tle opportunity for a dust particle to push another dustparticle indirectly through collisions with a gas atom.Here, we will refer to the viscosity as the static viscosity η to distinguish it from viscoelasticity. In the literature ofdusty plasmas, the static viscosity η has been measuredexperimentally [16] and quantified in simulations [30–33].There are two ways to quantify the static viscosity. Ifthere is a macroscopic velocity shear, the static viscositycan be calculated from the velocity flow profile [12, 16,32, 33]. On the other hand, if there is no macroscopicvelocity shear, the microscopic shear associated with therandom motion of particles can be used to calculate thestatic viscosity using the Green-Kubo relation [31, 33,34].Viscoelasticity is a property of materials that exhibitboth liquid-like viscous and solid-like elastic characteris-tics [35]. Most materials in reality are viscoelastic, suchas wood, synthetic polymers, and human tissue [35]. Vis-cous effects correspond to energy dissipation, while elas-tic effects corresponds to energy storage. In general, liq-uids exhibit mostly viscous effects at large spatial andtemporal scales, but they exhibit some elastic effects atsmall spatial and temporal scales [36].To characterize viscoelasticity quantitatively, it is com-mon to use either the frequency-dependent viscosity η ( ω )or the wave-number-dependent viscosity η ( k ). The lattercharacterizes materials at different length scales, and wasintroduced by theorists performing simulations [37–40].The static viscosity η is the hydrodynamic limit of thewave-number-dependent viscosity η ( k ) when k →
0. Inconsidering this limit, the relevant characteristic lengthscale for k is the interparticle distance, which is oftenmeasured as the lattice constant b of a perfect crystal.Viscoelasticity of strongly-coupled plasmas has beenstudied theoretically [41–43] and experimentally [20, 44,45]. The few experiments that have been reported forviscoelasticity of dusty plasma include a descriptive pre-sentation [44] and a characterization using a correlationfunction of the microscopic motion of dust particles [45].In our recent 2D experiment [20], a single horizontallayer of electrically charged dust particles was levitatedin a glow-discharge plasma. The kinetic temperature ofthe dust cloud was raised by laser heating [20, 24]. View-ing from above, we recorded movies of particle motion,then calculated particle positions and tracked them tocalculate their velocities. Based on the trajectories ofparticles, the wave-number-dependent viscosity η ( k ) ofthe 2D dusty plasma was quantified using an expressionwe derived that accounts for gas friction.In simulations, the viscoelasticity of both 2D [20] and3D strongly-coupled plasmas [43] have been studied re-cently. In this paper, we carry out further simulationsfor two purposes: to validate the η ( k ) calculation methodtaking into account gas friction, as presented in [20], and to assess the accuracy of the Green-Kubo relation fordusty plasmas with a modest level of gas friction.Simulations of strongly-coupled plasmas usually usethe molecular dynamical (MD) method [20, 43]. Eachparticle is tracked individually, unlike the case of particle-in-cell (PIC) simulations, where aggregations of particlesare simulated by a hypothetical super-particle [7]. Track-ing individual particles is suitable because otherwise thedominant effects of strong particle-particle Coulomb in-teractions would be lost. Another difference is that inMD simulations, as compared to PIC simulations, Pois-son’s equation is not solved. The only equation that issolved is the equation of motion for each particle, whichis integrated to track particle trajectories. The result ofthe MD simulation is a record of all particle positionsand velocities, which is the same kind of data that areproduced in dusty plasma experiments. The interpar-ticle interaction that is assumed in MD simulations ofstrongly-coupled dusty plasmas is a repulsive Yukawa po-tential [17], φ i,j = Q (4 πǫ r i,j ) − exp( − r i,j /λ D ) , (1)where Q is the charge on dust particles, λ D is the Debyelength, and r i,j is the distance between the i th and j thparticles.We list here additional parameters for the dustyplasma cloud. Because the dust cloud is 2D, we usean areal number density n and an areal mass density ρ = mn for the cloud, where m is the dust particlemass. We note that while the units for mass densityand viscosity are different in 2D and 3D, the units arethe same for the kinematic viscosity [16], η/ρ . Distancesbetween dust particles are characterized by both the lat-tice constant b for a crystal or the 2D Wigner-Seitz ra-dius a = ( nπ ) − / [46]. Time scales for collective mo-tion are characterized by the nominal 2D dusty plasmafrequency [46] ω pd = ( Q / πǫ ma ) / . Gas friction ischaracterized by the damping rate ν f , which is the ratioof the gas friction force and the dust particle’s momen-tum.We will discuss how to calculate η and η ( k ) in Sec. II.In Sec. III, we will discuss our two MD simulation meth-ods, Langevin and frictionless. In Sec. IV, we will re-port new simulation data for η ( k ) of 2D strongly-coupleddusty plasmas. We will validate our analysis method [20]for calculating η ( k ) in 2D strongly-coupled plasma withgas friction. We will also test the accuracy of the Green-Kubo relation with a modest level of gas friction as inour experiment. II. METHODS FOR CHARACTERIZINGVISCOSITYA. Static viscosity η The Green-Kubo relation is widely used for calculatingthe static viscosity η , based on the random motion of par-ticles. This method is used when there is no macroscopicvelocity shear. The Green-Kubo approach assumes linearmicroscopic fluctuations and equilibrium fields in the sys-tem [31]. The assumptions of this approach are similarto those for the fluctuation-dissipation theorem [47, 48].Previously, the Green-Kubo relation was generally usedwith data from frictionless simulations [30, 31, 33, 34, 43].To calculate the static viscosity, first we calculate thestress autocorrelation function (SACF) C η ( t ) = h P xy ( t ) P xy (0) i , (2)where P xy ( t ) is the shearing stress P xy ( t ) = N X i =1 mv ix v iy − N X j = i x ij y ij r ij ∂φ ( r ij ) ∂r ij , (3)where i and j are indices for different particles, N is thetotal number of particles of mass m , r i = ( x i , y i ) is theposition of particle i , x ij = x i − x j , y ij = y i − y j , r ij = | r i − r j | , and φ ( r ij ) is the interparticle potential. Second,we calculate the static viscosity η from the Green-Kuborelation [31], η = 1 V kT Z ∞ C η ( t ) dt. (4)Here, V is the simulation volume, which is replaced bythe area of the simulation box for 2D simulations likethose reported here.The Green-Kubo relation, Eq. (4), is intended for usein equilibrium systems, but in this paper we will assesswhether it can also be used in systems with a modestlevel of gas friction as in our experiment [20]. The dustparticles in an experiment experience gas friction, in ad-dition to collisions amongst themselves, whereas only thelatter are modeled in the Green-Kubo relation. We willcarry out simulations, with and without friction, and ver-ify that Eq. (4) yields the same result in both cases. B. Wave-number-dependent viscosity η ( k ) The wave-number-dependent viscosity η ( k ) character-izes viscous effects at different length scales. A method ofcalculating η ( k ) from the trajectories of random motionof molecules in liquids has been developed [39, 40]. In cal-culating η ( k ) using this method, one starts with particletrajectories, such as x i ( t ) and the perpendicular velocity v iy ( t ) for the i th particle. These are used to calculate thetransverse current, j y ( k, t ) = P Ni =1 v iy ( t ) exp[ ikx i ( t )].The normalized transverse current autocorrelation func-tion [39, 40] (TCAF) is then calculated as C T ( k, t ) = h j ∗ y ( k, j y ( k, t ) i / h j ∗ y ( k, j y ( k, i , (5)where the wave vector k is parallel to the x axis. (Here, k serves only as a Fourier transform variable, and isnot intended to characterize any waves.) The wave-number-dependent viscosity of frictionless systems is cal-culated [39, 40] using η ( k ) /ρ = 1 / (Φ k ) , (6)where Φ is a time integral representing the area underthe TCAF after normalizing the TCAF to have a value ofunity at t = 0. Generally, η ( k ) diminishes gradually as k increases, meaning that viscous effects gradually diminishat shorter length scales.In [20], we generalized this expression as η ( k ) /ρ = [(1 / Φ) − ν f ] /k (7)to account for the friction of gas drag ν f acting on dustparticles. As in Eq.(6), the integral Φ is a function of k . Our derivation of Eq. (7) was provided in the supple-mentary material of [20]. In this paper, we will carry outsimulation tests to validate the use of Eq. (7) for a widerange of k . This validation test will be performed for themodest level of gas friction ν f in our experiment [20].The TCAF measures the memory of transverse cur-rent, which reflects the decay of microscopic velocityshear. The shear decay can be caused by several mecha-nism in 2D dusty plasma clouds, such as Coulomb colli-sions amongst dust particles and the friction due to gasdrag. We will study how gas friction affects the TCAFlater. III. SIMULATION METHODS
In order to test the effects of gas friction, we will com-pare the results of two simulations: a Langevin MD sim-ulation with friction, and a frictionless equilibrium MDsimulation.Our two simulation methods are the same in many re-spects. Both use a binary interparticle interaction with aYukawa pair potential. In both simulations, particles areonly allowed to move in a single 2D plane. Conditions re-mained steady during each simulation run. For both sim-ulations, the parameters we used were N = 4096 particlesin a rectangular box with periodic boundary conditions.The box had sides 64 . b × . b . The integration timestep was 0 . ω − pd , and simulation data were recordedfor a time duration of 68 000 ω − pd after a steady statewas reached. Both of our simulations were performed atΓ = 68 and κ = 0 .
5, which are the same values as in ourexperiment [20].Our Langevin MD simulation takes into account thedissipation due to gas friction. The equation of motionthat is integrated in the Langevin simulation is [36, 43,49–53] m ¨ r i = −∇ X φ ij − ν f m ˙ r i + ζ i ( t ) , (8)where ν f m ˙ r i is a frictional drag and ζ i ( t ) is a randomforce. There is no thermostat to adjust the tempera-ture; instead the temperature is established by choosingthe magnitude of ζ i ( t ). Here, we chose the experimentalvalue ν f = 0 . ω pd [20]. Note that this gas friction levelis modest, i.e., the dust particle motion is underdamped,since ν f ≪ ω pd .Our frictionless equilibrium MD simulation [43, 50, 54]has no gas friction in the equation of motion m ¨ r i = −∇ X φ ij . (9)A Nos´e-Hoover thermostat is applied to maintain a de-sired temperature [50, 54].Trajectories r i ( t ) are found by integrating Eq. (8) or(9) for all particles. An example is shown in Fig. 1 fromthe frictionless MD simulation. IV. RESULTSA. Hydrodynamic and viscoelastic regimes
Comparing the results from the two simulations, Fig. 2,we can see how friction speeds the loss of memory ofthe system’s microscopic shearing motion. The memoryof the shearing motion is indicated by the decay of theTCAF.As expected [20], in the typical hydrodynamic limit oflong length scales, as shown in Fig. 2(a), the TCAF isjust a monotonic decay from unity to zero without anyoscillations [20]. We find that at the same hydrodynamiclength scale, the TCAF decays much faster with frictionthan without, indicating that in experimental dusty plas-mas gas friction plays an important role in shear decayin large length scales.When the wave number k is slightly larger, in the inter-mediate regime between the hydrodynamic and viscoelas-tic regimes, Fig. 2(b), the difference in TCAF betweenfrictional and frictionless is smaller. The integral of thefrictional TCAF is about a half of that for the friction-less TCAF, as seen in the inset of Fig. 2(b). This integralcorresponds to Φ, as in Eq. (6) or Eq. (7).When the wave number k is even larger, in the vis-coelastic regime, the TCAF oscillates around zero af-ter its decay due to the elastic effects, Fig. 2(c). Inthis viscoelastic regime, there is little difference between the TCAF from the two simulations, indicating that atsmaller length scales, gas friction does not contributemuch to shear decay. The friction plays a larger role inTCAF at larger length scales than smaller length scales.The calculation of η ( k ) using Eq. (6) or (7) requireschoosing an upper limit in the time integral of TCAF C T ( k, t ). An infinite time is of course impractical forboth experiments and simulations, so for a finite valuewe chose t I , the time of the first upward zero crossingof the TCAF [20], as shown in Fig. 2(c). This choice issuitable for two reasons: first, it is sufficiently long toretain both viscous and elastic effects; second, we foundthat contributions to the integral after t I are negligible,for a TCAF that is not noisy. The calculation resultfor η ( k ) is not very sensitive to the chosen upper limit.Extending the limit to a higher value would only cause alimited effect on the value of the integral. B. Validating the generalized η ( k ) expression Results for the wave-number-dependent viscosity η ( k )are presented in Fig. 3(b) and (c) for both simulations.We find an agreement in the values of η ( k ) for thefrictionless and Langevin simulations. This agreementcan be seen by comparing the circles in Fig. 3(b) for thefrictionless simulation with Eq. (6), and the triangles inFig. 3(c) for the Langevin simulation with Eq. (7). Thereis not only a qualitative agreement in the downward trendas the wave number k increases, but also a quantitativeagreement. This quantitative agreement is most easilyseen by fitting the calculated η ( k ) to the Pad´e approx-imant of [20, 39] and comparing the fit parameters, asindicated in Fig. 3 for the smooth curves.This agreement leads us to our first chief result: avalidation of Eq. (7) for computing η ( k ) in the presence ofgas friction. Since the two simulations were performed forthe same values of Γ and κ , an agreement indicates thatEq. (7) is valid. If there had been a discrepancy betweenthe circles in Fig. 3(b) and the triangles in Fig. 3(c),we would question whether Eq. (7) is valid. We gainconfidence in the validity of Eq. (7) by the lack of anysignificant discrepancy in the two results.The importance of correcting for friction, in Eq. (7), isdemonstrated in Fig. 3(c). If we use Eq. (6) instead, thepresence of friction leads to an exaggerated value for η ( k ),as seen by comparing the two sets of data in Fig. 3(c).This exaggeration is most extreme at small wave numbers(where the effect of friction is greatest, as we found inSec. IV A for the TCAF). C. Testing the Green-Kubo relation for staticviscosity in the presence of friction
To determine whether the Green-Kubo relation,Eq. (4), still provides an accurate calculation of staticviscosity η of a 2D Yukawa liquid, in the presence of amodest level of gas friction, we performed a test of Eq. (4)comparing η computed from our frictional Langevin sim-ulation and our frictionless simulation. These resultsfor the normalized kinematic static viscosity are η/ρ =(0 . ± . a ω pd for the frictionless simulation, and η/ρ = (0 . ± . a ω pd for the Langevin simulationwith friction. These values are also shown in Fig. 3(b)and (c) as star symbols. Noting that these results are inagreement within the uncertainty, we conclude that theGreen-Kubo relation remains accurate, at least with amodest level of gas friction, for a 2D Yukawa liquid atthe value Γ = 68 and κ = 0 . η ( k ) in Fig. 3(c). We notean agreement of η ( k ) as k → η from the Green-Kubo relation. This agreement is significant because η ( k )is computed from the TCAF, which is unrelated to theGreen-Kubo relation used to compute η .We can provide two intuitive suggestions to explainthe accuracy of the Green-Kubo relation in the presenceof a modest level of gas friction. First, we note thatthe gas friction that we have considered is so small that ν f /ω pd < .
1. This inequality demonstrates that fric-tional effects will in general be much smaller than effectsarising from particle charge as measured by ω pd . Sec-ond, the TCAF in Fig. 2 showed us that gas friction hasthe least effect on motion at small length scales, and dy-namical information at these small length scales are alsoreflected in the Green-Kubo relation because it is basedonly on fluctuations of individual particle motion.We cannot rule out the possibility that friction will af-fect the static viscosity computed using the Green-Kuborelation in other parameter regimes. In fact, for a 3DYukawa Langevin simulation at a much lower Γ = 2,Ramazanov and Dzhumagulova found that η computedusing the Green-Kubo relation diminishes as the frictionwas raised to a very high level [55]. IV. SUMMARY
Motivated by experiments with 2D clouds of chargeddust particles suspended in a plasma, we carried out twotypes of simulations, with and without gas friction. Wevalidated the newly-introduced Eq. (7) for calculating η ( k ) as a measure of viscoelasticity, in the presence ofgas friction. We also verified that the Green-Kubo rela-tion can accurately measure the static viscosity η of the2D collection of charged dust particles even when they experience gas friction. The level of gas friction we con-sidered was at a low level ν f /ω pd < .
1, and the couplingwas moderate with Γ = 68 and κ = 0 .
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