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Plasma Sources Sci. Technol. 27 (2018) 065004 (14pp) https://doi.org/10.1088/1361-6595/aac6b9 Test cases for grid-based direct kinetic modeling of plasma flows
Kentaro Hara1 and Kyle Hanquist2
1 Texas A&M University, College Station, Texas, United States of America 2 University of Michigan, Ann Arbor, Michigan, United States of America
E-mail: [email protected]
Received 19 January 2018, revised 1 May 2018 Accepted for publication 22 May 2018 Published 18 June 2018
Abstract Grid-based kinetic models are promising in that the numerical noise inherent in particle-based methods is essentially eliminated. Here, we call such grid-based techniques a direct kinetic (DK) model. Velocity distribution functions are directly obtained by solving kinetic equations, such as the Vlasov equation, in discretized phase space, i.e., both physical and velocity space. In solving the kinetic equations that are hyperbolic partial differential equations, we employ a conservative, positivity-preserving numerical scheme, which is necessary for robust calculations of problems particularly including ionization. Test cases described in this paper include plasma sheaths with electron emission and injection and expansion of neutral atom flow in a two-dimensional configuration. A unifying kinetic theory of space charge limited sheaths for both floating and conducting surfaces is presented. The improved theory is verified using the collisionless DK simulation, particularly for small sheath potentials that particle-based kinetic simulations may struggle due to statistical noise. For benchmarking of the grid-based and particle-based kinetic simulations, hybrid simulations of Hall thruster discharge plasma are performed. While numerical diffusion occurs in the phase space in the DK simulation, ionization oscillations are well resolved since ionization events can be taken into account deterministically at every time step.
Keywords: kinetic simulation, plasma sheaths, plasma instability, nonlinear plasma waves, Hall thruster, space charge limited sheath, Vlasov simulation
1. Introduction particle methods are easy to implement, the statistical noise due to the use of macroparticles in particle simulations may Understanding time-varying (dynamic) and kinetic (non- become problematic, particularly if the high frequency signals Maxwellian) effects is important when investigating tempo- are altered by the statistical noise. Hence, the DK method can rally and spatially small-scale plasma phenomena and their serve as an alternative to the particle methods. effects on macroscopic dynamics. Some examples include Since first-principles gas kinetic equations are a multi- plasma-wave, plasma-material, and plasma-beam interactions, dimensional first-order hyperbolic partial differential equation where the velocity distribution functions (VDFs) may become (PDE), the numerical methods for advection problems non-Maxwellian. Such plasma dynamics are relevant in low developed in the computational fluid dynamics (CFD) com- temperature plasmas, including ion acoustic waves in cathode munity can be used effectively for DK methods. Development plumes [1–3], plasma waves in cross-field devices [4–6], and of high-order accurate Vlasov simulation techniques has also plasma sheath dynamics [7–11]. Although particle methods been a popular research topic, mainly in the applied mathe- have been widely used for low temperature plasmas [12–14], matics community. The first Vlasov simulation developed an alternative method is to directly solve the kinetic was a finite difference scheme using cubic spline interpolation equations, which we call a grid-based direct kinetic (DK) [15]. Since then, many researchers have developed other method. This approach is also called a continuum kinetic methods including semi-Lagrangian methods [16, 17], method, Vlasov method, and discrete velocity method. While Weighted Essentially Non-Oscillatory (WENO) schemes
0963-0252/18/065004+14$33.00 1 © 2018 IOP Publishing Ltd Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist [18], Discontinuous Galerkin methods [19], and finite volume ordinary differential equations (ODEs). The macroscopic methods using Runge-Kutta (RK) methods for time integra- quantities are obtained by sampling the particle information. tion [20]. Collision processes are often taken into account probabil- While the DK methods have popularly been used in istically using random numbers. There exists statistical noise collisionless plasma physics, the noiseless kinetic simulation in the system that may affect estimation of the collision int- can be applied to low-temperature plasmas and other fluid egral and alter high frequency plasma oscillations. On the dynamics. It is, however, important to test a new kinetic other hand, grid-based DK methods eliminate the statistical model via verification, validation, and benchmarking [21]. noise inherent in particle simulations, as the VDFs are directly First, verification refers to checking whether a numerical code solved for in discretized phase space. The collision integral is correctly implemented. This is often done by comparing the can be calculated from the discretized VDFs. numerical results with analytic (exact) solutions. In addition Memory requirement and computational cost has been a to the code verification, a theory can also be verified by drawback for grid-based kinetic simulations since the phase numerics. Second, benchmarking of multiple codes can serve space needs to be discretized in all dimensions. Let us assume as an important testing. Different solvers can be used to solve that the discretization in physical coordinate is identical in the exact same equations to investigate the performance of a grid-based and particle-based kinetic simulations (N and d [ ] x x numerical method 22 . For instance, comparing grid-based are the number of cells and dimensions in physical space). and particle-based kinetic methods is essential in under- dv The total number of velocity bins per one physical cell is Nv , standing the similarities and differences of different kinetic where N is the number of velocity bins and d is the number [ ] v v models 23, 24 . Finally, validation of a computer simulation of dimensions in the velocity space. For particle simulations, can be performed by comparing the results with measure- each macroparticle has (dx+dv)-dimensional information, so ments or experimental observations. It is important to keep in the total number of degrees of freedom per one computational mind that experiments also include uncertainties through cell is (dx+dv) Np, where Np is the number of computational measurements, making them not exact, which is the differ- particles per cell. The computational memory required is fi ence between validation and veri cation. directly associated with the resolution of the VDFs. As In this paper, we summarize the test cases suitable for computational cost is linearly proportional to the degrees of kinetic simulations relevant to low temperature plasmas. The freedom in the calculation, grid-based DK methods are often test cases discussed in this paper include electron emission more computationally expensive than particle methods. from plasma-immersed materials and a multidimensional The most critical comparison may be the error associated collisionless flow that accounts for injection, wall reflection, with the kinetic methods. Truncation error is a type of error and expansion of neutral atoms. Benchmarking of a grid- that corresponds to omitting high order terms when numeri- based and particle-based kinetic model in a Hall thruster cally solving a discretized differential equation [25]. The discharge [23] is also discussed, with a particular emphasis on + local truncation error is O(h p 1) for a p-th order method, the time-varying ion VDFs. In addition to development of DK where h is the discretization step, e.g., time step or grid size. method, a general theory of plasma sheaths in the presence of The global truncation error is the accumulative error through strong electron emission from a surface is proposed. The improved theory takes into account both floating and con- multiple integration steps. For a total length of L, the total number of integration steps will be L/h. Hence, the global ducting materials and is consistent with existing theories. p truncation error, òT,ofap-th order accurate scheme is O(h ). In table 1, the numerical error in the grid-based method is shown to be proportional to O()N-p as the cell size h can be 2. Particle-based versus grid-based kinetic method v given by h=Lv/Nv for a given system size Lv. The error level can be reduced by choosing a large number of grid The first-principles gas kinetic equation is given by: points or a high-order accurate numerical method. ¶f ¶ff¶ + v ··+ a = S,1 ()The numerical error in particle-based methods is asso- ¶t ¶x ¶v ciated with the statistical noise, òS, due to the use of discrete [ ] where f is the velocity distribution function (VDF), v is the macroparticles 26 . The statistical noise arises when sam- velocity, x is the physical space, t is time, a is the accelera- pling the particle information. The error level, i.e., -12[ ] tion, which can be written as a=q(E+v×B)/m for S ~ Np 27 , can be reduced by increasing the number of nonrelativistic plasmas, q is the charge, m is the mass, E is the particles for the particle simulations, but will increase the electric field, B is the magnetic field, and S is the collision computational cost. For a steady-state calculation, when the term. In the plasma community, the collisionless Boltzmann flow is static, a time averaging technique is often employed equation is often called the Vlasov equation: S=0in for particle sampling, which will reduce the statistical noise equation (1). since macroscopic quantities are calculated by averaging over Table 1 summarizes the comparison of grid- and particle- multiple time steps. This allows usage of a small number of based kinetic simulations. In particle-based simulations, such macroparticles in the domain. For an unsteady calculation, as particle-in-cell (PIC) and Monte Carlo collision (MCC) where a large sampling window cannot be employed, the methods, the equations of motion are solved for each mac- number of macroparticles per cell at each time step must be roparticle. The equations of motion are a set of two first-order sufficiently large. Hence, the degree of freedom in particle
2 Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist
Table 1. Comparison of grid-based and particle-based kinetic simulations. Grid-based Particle-based Model Solve for VDFs in phase space Solve for motions of macroparticles Differential equation Hyperbolic PDE ODE Specification Eulerian Lagrangian Collision rates Integral of collision operator Collision probability ( dvxd ( dx ( Computational memory N and d are the number of cells Nv Nx Nv: number of grid points in velo- ()ddNNxvp+ x Np: number of and dimensions; subscripts x and v denote physical city space) particles per cell) and velocity space) -p Primary source of numerical error Global truncation error: Tv~ ON[( ) ] (p is Statistical -12 the order of accuracy of a numerical scheme) noise: Sp~ ON[( ) ]
methods (i.e., number of macroparticles) may become com- in physical and velocity space, subscript j=x, y, z denotes parable to or more restrictive than that in a grid-based method the dimension, and C0 is the CFL number. For an explicit (i.e., number of velocity bins). In addition, a particle method scheme, the time step must be chosen small enough to satisfy using constant particle weights suffer from resolving the small equation (2). The CFL number is dependent on the time VDFs, e.g., the high energy tail, unless variable particle integration schemes. For Strang’s time splitting method, the weights or merge-split techniques are used [28]. For instance, CFL condition, C0=1, must be satisfied independently in to resolve a VDF that is three orders of magnitude smaller each direction [29]. than the bulk VDF, the required minimum number of particles is 1000 per cell. The total number of macroparticles can also 3.2. Spatial discretization significantly increase when the fluctuation in plasma density fi is large in the presence of ionization oscillations [23]. A nite volume method is chosen for the DK method due to the following reasons. As negative VDFs are not physical, the solver must preserve positivity of the VDFs. In particular, it [ ] 3. Numerical method was found in 23 that negative VDFs may make simulations unstable in the presence of ionization. Thus, total variable For developing a new kinetic method, it is important to verify a diminishing scheme that does not generate numerical oscil- numerical solver. It is often difficult to derive an analytic lations is preferred. Additionally, the total number of particles fi solution including source terms (collisions) in the kinetic fra- is conserved in the system using nite volume methods mework. Therefore, 1D1V (one dimension in physical space because they are inherently conservative. ∂ + and one dimension in velocity space) collisionless Vlasov- Assume a one-dimensional hyperbolic PDE, tu ∂ ( )= = ( ) Poisson problems can be employed to verify grid-based kinetic xF u 0, where u u x, t is the conserved quantity and F ( ) fl = fl methods. A 2D2V collisionless DK simulation and a 1D1V u is the ux. Note that F Vu, where V is a constant ow fi collisional DK simulation are also developed and employed. velocity for a linear advection equation. Using a nite volume method, first-order forward Euler time integration, and equidistance cells, the discretized equation can be written as 3.1. Time integration n+1 n UU- jj+-1 - 1 For the time integration on the left hand side in equation (1),a j j =- 2 2 ,3() Runge-Kutta (RK) method or Strang’s time splitting techni- DtxD que is often used. A higher order time integration scheme can where U is the cell-averaged quantity, is the discretized be used accordingly with a high-order numerical scheme for flux at cell interface, superscript n denotes the index of time spatial discretization. For instance, second order accuracy is step, and subscript j represents the index of cell center. Note sufficient for time integration if the fluxes are second-order subscript j + 1 and j - 1 are at cell interfaces. 2 2 accurate. Let us assume a time-dependent equation: For the numerical flux in equation (3), Monotonic du = Lu(), where L(u) is an operator for the quantity u. The ( ) dt Upwind Scheme for Conservation Laws MUSCL is mainly time step must satisfy the Courant-Freidrich-Lewy (CFL) used [30]. Godunov’s theorem states that no linear numerical condition in order to achieve numerical stability. For a RK scheme that is better than first-order accuracy (p 2) can integration, preserve monotonicity. The use of nonlinear flux limiter functions in the MUSCL framework can limit numerical ⎛ ⎞ ⎜max∣∣vtj D max ∣∣atj D ⎟ extrema, i.e., undershoot and overshoot, while achieving the å + C0 ()2 fi j ⎝ Dxj Dvj ⎠ global order of accuracy higher than rst order. For the DK simulation, it was first observed in [23] that cubic spline where v is the characteristic velocity, a is the characteristic interpolation produces too large numerical oscillations and acceleration, Δt is the time step, Δx and Δv are the cell size made the simulations unstable particularly in the presence of
3 Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist collision terms After several limiter functions were tested, a large-scale simulation with a large number of cores is modified Arora-Roe limiter [31] is chosen due to the pre- reserved for future work. servation of the zero values of VDFs, i.e., f=0 due to no particles. The MUSCL framework is compared with a fifth- order Weighted Essentially Non-Oscillatory (WENO) scheme [18, 32] in collisionless plasma wave simulations [33]. While 4. Plasma sheaths with emitted electrons from wall: both methods are inherently conservative due to finite volume improved theory and DK simulations methods, MUSCL framework is chosen, particularly for the advection in physical space, for the reminder of the test cases Collisionless plasma sheaths in the presence of emitted due to its robustness for injection boundary conditions [34] to electrons from plasma-immersed materials serve as a good achieve flux conservation in the system and the positivity- test case since analytic solutions exist [35, 36]. From a preserving feature. practical perspective, plasma sheaths play a critical role in determining the bulk plasma [9, 37–40]. Various theories of transition from bulk plasma (presheath) to sheath have also 3.3. Parallelization been proposed [41, 42]. Kinetic theories of non-magnetized sheath [43–48] and magnetized sheath [49–51] play an Message-Passing Interface (MPI) is used for parallel com- important role in understanding the loss mechanisms of bulk putation. Cartesian coordinate is used for the phase space plasmas. discretization and partitioned among multiple processors. The The collisionless DK simulations are applied to both ions total numbers of cells in the computational domain in the x and electrons, and were verified against theories of nonlinear and v directions are first specified, from which the cell size is [ – ] determined, i.e., Δx and Δv. plasma wave 33, 52 54 as well as trapped particle [ ] For a collisionless 1D1V DK method, the domain is instabilities 55, 56 . Recently DK methods have been applied decomposed in both x and v directions, which is found to be for inverse sheath including some collisional effects by [ ] best for the scale up of parallel computing. Note that such Campanell and Umansky 57, 58 . Particularly, the present domain decomposition is applied for collisionless cases while DK method has been successfully used for trapped particle [ ] it also works in some collisional cases, such as the Bhatnagar- bunching instability 59 and ladder climbing of electron [ ] Gross-Krook (BGK) operator, charge exchange, and ioniz- plasma waves 60 . ation. Nevertheless, integration across the velocity space may Here, we present sheath simulations and a unifying the- lead to additional computational overhead, which requires ory for plasma-immersed materials in the presence of electron further attention when the computational size becomes larger emission from the material. A general sheath theory is con- and performing multidimensional calculations. For MUSCL, structed where, for instance, the plasma sheath of floating each processor possesses 2 ghost cells including the parti- material is a special subset of conducting materials. The tioned phase space discretization. For the ghost cells, physical theory for floating surfaces has been proposed by Hobbs and quantities using a Dirichlet or Neumann condition are speci- Wesson [43] and later corrected by Sheehan et al taking into fied for the boundaries of the computational domain or the account the non-Maxwellian effects of primary and emitted quantities from neighboring processors are exchanged for the electrons [44]. Theories of emitted electrons for conducting interior regions of the computational domain. For a 2D2V DK materials have been proposed by several researchers. In simulation, domain decomposition is performed in the phy- Takamura and Ye’s model [45, 46], cold ions, Boltzmann sical space while all processors possess the same two- primary electrons, and half-Maxwellian emitted electrons are dimensional velocity space. This way, macroscopic quantities considered. Din further updated the model by accounting for (e.g., density, mean velocity, and temperature) can be inte- the truncated Maxwellian for primary electrons in the theory grated locally on each processor, and they can be commu- but no verification was shown [47]. Cavalier et al extended nicated to the neighboring processors. Takamura’s model considering a more realistic condition by Linear scalability of the parallelized DK simulation is including elastically reflected electrons on the surface while observed up to 60–100 processors for the 1D case and 40 the effects of truncated Maxwellian for primary electrons are processors for the 2D case. Note that there are several factors not fully taken into account [48]. In this paper, additional fi ( ) that affect the parallelization ef ciency. i Global commu- kinetic corrections for primary electrons are made to Taka- nication may be required for calculating macroscopic quan- mura’s original theory and a good agreement between the ( tities by integrating the VDFs when domain decomposition is improved theory and DK simulations is shown. Our results ) ( ) performed also in the velocity space ; ii The Poisson solver are also consistent with Sheehan’s theory when the surface is is typically the main factor that affects scalability for kinetic at floating potential, i.e., the net current density is zero. simulations since the Poisson’s equation is an elliptic partial differential equation that requires global communications with other processors; (iii) Computational time may not be 4.1. Improved sheath theory equally distributed among cores since some processors pos- sess finite VDFs while others may only have velocity bins Consider an ion attracting sheath, in which x=0 is the wall without any particles (VDF is zero). Improving scalability for and x=L is the sheath edge. From the ion continuity and
4 Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist energy conservation equations, the ion density is described as equations (5) and (7) to obtain - 1 n ⎛ F ⎞ 2 pe0 2 ⎜⎟2 = ,9() nni =-0 1 ,4() ⎝ M 2 ⎠ n0 (∣∣)(∣∣)1erf+F+-FwwG 1erf t Φ= f/ nee0 2expG ()F wt where n0 is the plasma density at the sheath edge, e = ,10() (kB Te) is the normalized potential and M=u0/cs is the ion n0 (∣∣)(∣∣)1erf+F+-FwwG 1erf t Mach number. Here, e is the elementary charge, k is the B where G =F-sttexp[( 1 )] . Note that Takamura’s Boltzmann constant, T is the electron temperature, u is the w e 0 theory has neglected the kinetic effects of the primary elec- ion velocity entering the sheath, and csBei= kT mis the ion trons, namely, equation (5). In the limit of F-¥, acoustic velocity. The ion flux is constant J =n u =n Mc w i 0 0 0 s Takamura’s theory can be recovered: n = n exp F within the sheath as there are no collisions. Note that the f pe pe0 (x=L)=0 is assumed at the sheath edge and the wall because erf F-Fw 1. In the presence of excessive electron emission from the potential is negative, i.e., f (x=0)=f <0. It was also w wall, a space charge limited (SCL) sheath will form [43, 44]. confirmed that equation (4) is valid for an accelerating ion Note that inverse sheath [57, 58] is not considered for this test VDF even when comparing with the kinetic formulation. case. Using equations (4), (5), (7), (9), and (10) and assuming The number density and flux of primary electrons can be zero electric field at the sheath edge, the Poisson’s equation calculated assuming a Maxwellian VDF in the bulk plasma. can be integrated as, In the presence of a potential drop towards the wall, the electron VDF inside the sheath becomes a truncated Max- ⎛ ⎞2 ⎛ ⎞ 1 ⎜⎟dF 2 2F wellian: fxv( ˜˜,exp22)[=-+F n v˜212( x˜)] (p ) for =++HHMpe ee ⎜ 1 - - 1,⎟ () 11 e pe0 2 ⎝ dx˜ ⎠ ⎝ M 2 ⎠ vv˜˜ c and fxve ( ˜˜,0) = for vv˜˜> c, where npe0 is an electron density for the truncated Maxwellian, xx˜ = lD is the normalized position, λ is the Debye length, where Hpe and Hee are given in the appendix, see D ( ) ( ) vvv˜ ==F-F2()is the normalized cutoff velocity equations 21 and 22 . The transition from normal to SCL ccthe, w = of the truncated electrons due to the potential drop in the sheaths occurs when dF=dx˜ 0 at the wall, x 0. Thus, the SCL condition can be obtained, when the left hand size of sheath, and Φw<0 is the normalized wall potential. The equation (11) is 0 and Φ=Φw is considered: electron number density, npe(x), and flux, Jpe(x), can be cal- culated by taking the moment of the truncated electron VDF: D EGSCL t 0,12= + + Hion () npe 1 BGC+ SCL BGC+ SCL =+F-FF()()1erfw exp, 5 npe0 2 where GSCL is the critical value of G that results in SCL sheath, B =+1erf,∣∣ FwwC =- 1erf ∣∣ Ft , D = 1 8kTBe Jn=F=exp() const. . () 6 2 2 pe pe0 w expF+ww(∣∣) 1 F-B, E =Fexp()(wwtt 1 - ∣∣) F- 4 pme p p C,andH =-F-MM22()12 1. Thus, from equation A half-Maxwellian is considered for the emitted electrons ion w (12), the critical SEE rate is given by from the wall. The emitted electron temperature is assumed to be in equilibrium with the wall temperature, --DBHion exp[(F-w t 1 )] Tw. The VDF of emitted electrons can be described sSCL = ,13() 212CHion + E t t as fxvee ( ˜˜,exp22)([=--F-F nee0 tt v˜ ( w )]) ( p ) < n for vv˜˜c and fxvee ( ˜˜,0) = for vv˜˜c, where ee0 is the τ f number density of secondary electrons at the wall and which is a function of , w,andM. The condition for M,e.g., the Bohm condition, is shown in appendix. In the kinetic τ=Te/Tw is the ratio between primary and emitted electron temperature. The number density and flux of secondary simulations presented here, M varies depending on the simula- tion setup and may not represent the actual sheath condition electrons, nee(x) and Jee(x), are without taking the presheath into account. As the so-called nee 1 =exp[( F-Fww )tt ][ 1 - erf ( F-F ) ] , ( 7 ) Bohm condition is also dependent on the presheath, the trans- nee0 2 ition from bulk plasma to the sheath must be resolved to obtain
1 8kTBw M. The length of presheath is observed to be on the ion mean JJee== ee,0 wall n ee ()() =const. . 8 free path [61]. However, for the verification of the kinetic 4 pme simulation shown in this paper, we obtain M from the simulation Note that primary and emitted electron current densities are results and insert it into the theory. constant in space, as shown in equations (6) and (8). Here, A special case of interest is the plasma sheath of floating σ ∣JJee∣∣= s pe∣ at the wall, where is the secondary electron surfaces where net current is zero within the sheath [44]. ( ) emission SEE rate. Using the relation between the primary Hence, J −J =J , leading to (1−σ)J =J . Therefore, fl ( ) ( ) pe ee i pe i and secondary electron uxes in equations 6 and 8 the sheath potential for a floating wall is given by gives nee00=Fstn peexp () w . A quasineutral plasma is assumed at the sheath edge, ⎛ ⎞ ⎜ npe0 1 - s ⎟ n (x=L)+n (x=L)=n . To calculate the electron F=-w ln ⎜ ⎟.14() ee pe 0 ⎝ n ⎠ number densities, Φ=0 at the sheath edge is inserted in 0 Mmm2p ei
5 Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist
Figure 1. Total current of conducting plasma sheaths as a function of G = JJemit,surfacei ,0 for Vd=−2.5Te. Dashed lines represent the linear dependence of emitted electrons and SCL regime to help visualize the differences between two phenomena. Figure 2. SCL sheath obtained from DK simulation (symbols), ( ) ’ ( ) improved theory solid line , and Takamura s original theory In the limit of F-¥w , equation 14 becomes (dashed line) for τ=10. Insert is a zoom up in logarithmic scale. F=-weiln[( 1 -sp ) 2 mm ], which is consistent with Theory curves shown are for M = 1.2 and 2, which are consistent Hobbs and Wesson’s theory [43]. Thus, the kinetic correction with the DK results. is particularly important for small sheath potentials. assumed. The computational wall time is approximately 14 h 4.2. Transition from normal to space charge limited sheath using 9 processors. A sheath setup similar to Schwager’s sheath simulations The total current through the plasma boundary, = − − =− fi [44, 62] is employed. A source sheath occurs at the plasma Jtot Ji Jpe Jemit, for Vd 2.5Te is shown in gure 1. injection boundary (x=L) while the wall is located at x=0. In the absence of emitted electrons (Γ=0), the sheath is A plateau is observed in the middle of the computational electron attracting. The total current decreases linearly as Jemit domain, where we define as the sheath edge. The voltage drop increases, and the sheath becomes electron emitting. At a Γ=Γ ∼ between the wall and the plasma, Vd=fw, is chosen as an certain point, e.g., SCL 38, the total current saturates input condition. Here, Vd<0, i.e., the wall potential is lower which indicates that the electron emission is space charge with respect to the plasma potential to form an ion attracting limited. Note that the simulation results show slight increase sheath. For the DK module, the total number of charged in the current, which is likely due to the presence of virtual species in the domain is kept constant by reinjecting the ions cathode. A slightly increased sheath potential contributes to and electrons that move out of the domain through the wall decreased primary electron flux (see equation (6)) and results and plasma boundaries. This way, the spatially averaged in an increase in total current. plasma density is constant. The flux of electrons emitted from the surface is kept constant in time. The ratio of the emitted 4.3. Comparison with theory electron flux to a reference ion flux is given by From σ=Jemit/Jpe and equation (6), the ratio between the Jemit,surface fl G= ,15() uxes of SCL emitted electrons and ions is Ji,0 J 1exps ()F G=emit,SCL = SCL w .16() where Ji,0=n0cs and Jemit,surface is the flux emitted from the SCL JMi 2pmm surface, which is different from the flux that reaches the ei sheath edge and plasma boundary Jemit. From a mathematical ¥ It can be seen from equations (13) and (16) that JJemit,SCL i is a perspective, Jvfxvdvemit,surface ==()0, , while Jemit = ò0 Φ τ ¥ function of w, , and M. vf() x= 0, v dv. If there are no electrons reflected ò-¥ Figure 2 shows good agreement between the DK simu- back to the wall, i.e., if the SCL sheath does not lation and improved theory while Takamura’s original theory form, JJemit,surface= emit. fails to capture the simulation trend at ∣F
6 Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist
Figure 3. Steady-state potential profiles of SCL sheaths for −1Φw<0. The absence of statistical fluctuations makes the DK simulations enable to model such small sheath potential cases. two-stream instability on the sheath theory are discussed later Φ =− in section 4.5. tion. Figure 4. SCL sheath properties for w 0.14 case, when Vd=−Te. Shown are (a) primary electron VDF, (b) emitted electron At large ∣Fw∣, Takamura’s theory is accurate since the VDF, (c) potential profile, and (d) ion VDF at 0<x/λD<15. primary electron VDF is truncated at high vc, resulting in a Figure 4(c) is the same as the black line (Vd/Te=−1) in figure 3. primary electron distribution close to a Maxwellian. Taka- ’ mura s theory assumed Boltzmann electrons for simplicity, half-Maxwellian, which are consistent with what is con- which is correct for large ∣Fw∣ cases, but the kinetic effects of sidered in the theory. It is therefore worth noting that the primary electrons play an important role at small ∣Fw∣.In source sheath does not alter the shape of the electron VDFs ( ) ’ = equation 13 , Takamura s theory can be recovered if B 2 considered in the theory within the wall sheath region. On the and D =F-exp()w 1. Moreover, in the present DK simu- other hand, the presence of the source sheath affects the ion lation, the ion velocity entering the sheath is not prescribed dynamics, since ions are accelerated to a velocity faster than and it was found that M increases as ∣∣Vd increases. The M the ion acoustic speed predicted by Bohm’s sheath theory, as fi values shown for the theory curves in gure 2 are consistent shown in figure 4(d). This means that the source sheath does with those observed in the DK simula. not capture the correct presheath physics that determines the ion velocity at the sheath edge. Nevertheless, the proposed sheath theory holds given that the ion velocity at the sheath 4.4. Small sheath potential cases edge is known, which is why a good agreement between Figure 3 shows the steady-state potential profiles obtained for simulation and theory is obtained in figure 2. Vd/Te=−1, −1.5, and −1.75, which correspond to sheath potentials, Φ =−0.14,−0.51, and −0.71, respectively. w 4.5. Electron emission induced instability The cases shown employ the critical SEE rates that yield SCL sheath, which can be seen from the curved potential profile A strong electron emission from a plasma-immersed material near x=0. The potential at the plateau, i.e., at x=15λD, is shown to excite instabilities [64].In[46], one remaining varies for different Vd, which is due to the source sheath [62]. question was the effect of the two-stream instabilities induced PIC simulations may not resolve small ∣Fw∣ when the by the emitted electrons when comparing the numerical fluctuation in electrostatic potential due to the statistical noise results with Takamura’s theory. The excitation of instability is comparable to or larger than the sheath potential. Here, the and its effects were not discussed. In addition, the results grid-based DK simulation shows stable results at sheath shown for comparison with theory (see figure 2) employ time- potential as low as Φw=−0.1. Note that the results shown averaged results in the presence of instabilities and oscilla- are instantaneous results, i.e., not time averaged. Comparison tions. Therefore, it is important to investigate whether the with PIC simulations is reserved for future work. oscillations affect the agreement between the kinetic simula- Figure 4 shows the potential profile and the VDFs of tion and theory. primary electrons, emitted electrons, and ions for the Figure 5 shows the time evolution of the ratio of emitted Φw=−0.14 case. In figure 2, the sheath potentials are electron and ion fluxes (Γ=Jemit/Ji) using different com- defined as the potential difference between x=0 and putational domain size, i.e., L=30λD and 60λD, for x=15λD. It can be seen from figure 4(a) that the primary Vde=-4T . It can be seen that the oscillations are more electrons exhibit a truncated Maxwellian distribution and pronounced in the larger domain case. The oscillation ampl- from figure 4(b) that the emitted electrons are accelerated itude can be up to 35% of the time averaged value when
7 Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist
Figure 5. Time evolution of Jemit/Ji for different domain size for Vd=−4Te and τ=10. Red: L=30λD; Black: L=60λD. For the results in figure 2, this signal is time-averaged for over ωpe t=100.
Table 2. Dependence of domain size on Jemit/Ji for τ=10.
Vd=−2.5Te
Ji Jemit Jemit/Ji
L=30λD 1.466 38.61 26.34 L=60λD 1.353 35.08 25.93 (−1.5%)
Vd=−4Te
Ji Jemit Jemit/Ji Figure 6. Small computational domain: L=30λ for V =−4T = λ D d e L 30 D 1.543 56.26 36.5 and τ=10. (a) Ions; (b) Primary electrons; (c) Emitted electrons; = λ (− ) L 60 D 1.333 46.53 34.91 4.4% (d) Electric field and potential profile.
electrons at v˜ < 0 oscillate near x=L, travel towards the L=60λD. The oscillation is due to two-stream instability wall, and get reflected back towards the right boundary. The induced by a large emitted electron current from the surface. positive electric field near x=0 clearly shows that the elec- A similar run was also performed at Vd=−2.5Te and tron emission is space charge limited. oscillations are not observed when L=30λD but large- Large-amplitude plasma oscillations are excited when amplitude oscillations are observed when L=60λD or larger. using a larger domain, as shown in figure 7. Particle trapping Table 2 summarizes the results of different domain size due to plasma wave is observed for primary and emitted / =− − for Vd Te 2.5 and 4, which corresponds to sheath electrons. In addition, signature of ion trapping can be also Φ =− − fi potential of w 1.45 and 2.8, respectively, in gure 2. seen although a full enclosed orbit in phase space is not / One important observation is that Jemit Ji are within a few formed. The wavelength of the oscillation obtained from the percent difference with and without the oscillations. Hence, it DK simulation is approximately (10–20)λD. This is consistent can be considered that the kinetic sheath theory is not affected with a standard two-stream instability in which the maximum fi by the two-stream instability and gure 2 remains qualita- growth rate occurs approximately at kvc=ωpe, which corre- tively similar regardless of the oscillations, which was not sponds to one wavelength being conclusively shown in [46] and [63]. Figure 6 shows the VDFs of ions, primary electrons, and 2p L0 ==2,plv˜()cD 17 emitted electrons, as well as the electric field and potential for k Vd=−4Te with different domain size, L=30λD and 60λD. These results are instantaneous snapshots at ωpe t=300. A If the emitted electrons achieve a velocity of v˜c = 2.4 at the source sheath occurs at the plasma boundary to accelerate the sheath edge assuming a sheath potential of Φw=−2.8, a ions above the ion acoustic speed. Here, M≈1.4, which standard two-stream instability theory predicts a wavelength satisfies the condition for a stable sheath formation (see of L0≈15λD. Hence, the domain size L>L0 is needed for equation (24) in appendix). Although small amplitude oscil- large-amplitude plasma waves to be fully excited, particularly lations can be observed, the ions are relatively close to a to see the response to the ions. Additionally, due to the dc steady-state solution. It can be seen that a large wavelength discharge at constant voltage, the sheath potential fluctuates oscillation exists in primary and emitted electrons, but there is during the oscillations, so vc also fluctuates. It is likely no feature of particle trapping. Figure 6(b) shows that the that the fluctuation in emitted electron velocity generates
8 Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist profile of ion velocity in the presheath, which may change the instability. Second, collisional damping, e.g., inelastic collisions between electrons and neutrals, Coulomb colli- sions, etc may damp the energy the plasma wave [70].
5. Two-dimensional neutral atom flows
A two-dimensional DK simulation is developed to model the plasma flow in a Hall thruster like configuration. Here, a two-dimensional kinetic equation is solved. ¶f ¶f ¶f + ååvj + aj = S,18() ¶t j ¶j j ¶vj where subscript j denotes the dimension, e.g., j= xyz,,, a =+´q [(EvB )] is the acceleration term, and S is j m jj the collision term. Strang’s splitting is applied to decouple the physical and velocity advection. Now for a multi- dimensional kinetic equation (e.g., 2D2V), each advection contains multidimensional advection equations. Thus, the individual advection equations are solved using MUSCL with a second-order RK method.
5.1. 2D2V DK method For testing the two-dimensional DK solver, a collisionless neutral atom flow is modeled, where aj=0 and S=0. The channel length is L =4 cm and the width is W =2 cm. Figure 7. Large computational domain: L=60λD for Vd=−4Te ch ch and τ=10. (a) Ions; (b) Primary electrons; (c) Emitted electrons; Neutral atoms are injected in the domain from a 2mm slit at (d) Electric field and potential profile. the channel centerline. The mass flow rate considered is 5mg/s, the gas species is xenon, and a half-Maxwellian is assumed for the injection. The plume that is extended 4cm in plasma waves of different amplitude and wavenumber, which the z (parallel to the injection) direction and 3cm in both ±y explains the complex multi-mode signal shown in figure 5. (perpendicular to the injection) direction is also considered. Electron emission induced two-stream instability pro- The y=0 plane assumes a reflection boundary condition duces an ion acoustic wave. The phase space of the oscilla- while the other boundaries use an outflow boundary condition tions are on the order of the ion acoustic speed around the ion where no particles are injected back into the domain. bulk velocity, which is close to v/vth,e≈0. This a phenom- A VDF can be also used to model particle injection from enon relevant for near-cathode physics, such as hollow outside the domain to study the facility effects [71]. The DK cathodes [2, 65, 66]. As can be seen from figure 7(a), a large- method has been tested against several particle methods amplitude plasma wave can excite high energy ions that may [72–74]. The minimum mean free path is on the order of cm contribute to sputtering of the cathode materials [67].At in this configuration, as the maximum neutral density is on the 21 −3 larger sheath potential (∣F>w∣ 4), while a larger domain size order of 10 m . Hence, elastic collisions between neutral is required to fully capture the wavelength of the two-stream atoms may occur near the anode but overall the wall reflection instability (see equation (17)), it is observed that the oscilla- is the dominant collision mechanism. Thus, a collisionless tions are present in our DK simulations, using both L=30λD calculation is performed. and L=60λD. The effects of the plasma length on the At the channel walls and outer walls, particles are instability in a bounded system has also been discussed reflected as either specular or diffuse reflection. For a specular in [68]. reflection, the VDFs of reflected particles are determined by Standard two-stream instability theory typically assumes incoming particle VDFs. For a diffuse reflection, the reflected a Maxwellian distribution for primary electrons and an elec- particle VDFs follow a half-Maxwellian based on the wall tron beam, for which the dispersion relation does not predict temperature. Hence, the VDFs of reflected particles follow any growth rate at k w v . For a more complete disper- pe c fv()n 01,19=+-aa f () f () sion theory, an incomplete dispersion relation due to a trun- refl. diff. spec. cated electron VDF [69] and the effects of emitted electron where vn is the velocity normal to the wall, α is the accom- temperature must be taken into consideration. Note that the modation factor, fdiff. is a half-Maxwellian, and fspec. present simulation model does not consider a presheath is the mirror reflection of incoming particles: fspec. and collisions. First, the presheath determines the spatial (vn)=fplasma(−vn). Here, fplasma is the VDF of incoming
9 Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist
Figure 8. Steady-state neutral atom density profile in a 2D −1 configuration. No time-averaging is used. (a) Δvy=5ms and −1 (b) Δvy=2ms .
fl particles from the plasma to the wall. The re ected particle ( ) ( )=( fl fl Figure 9. Neutral atom VDFs a near the injection slit z, y 0m, ux must be identical to the incoming particle ux so that 0.04 m), (b) 1mm inside the channel exit (z, y)=(0.039 m, 0.04 particle conservation is satisfied. Hence, m), and (c) downstream at (z, y)=(0.08 m, 0.04 m).
¥ 0 vfnnnrefl. () v dv= ∣∣ v n fplasma () v nn dv.20 ( ) distribution of particles injected from the slit. For instance, the òò0 -¥ particles that are injected from (z, y)=(0 m, 0.04 m) and The amplitude of fdiff. is determined from equation (20). reach (z, y)=(0.08 m, 0.04 m) are the ones only with Δ = ( = ) −1 The grid size is z 0.67 mm Nz 60 inside the vy=0ms . Figure 8(b) shows that such artificial features channel and 0.83mm (Nz=48) in the plume, Δy=0.5mm can be reduced when using finer velocity bins, since the beam (Ny=40) at 0.03m < y<0.05 m and Δy=0.75 mm distribution emitted from the slit is better resolved along the −1 otherwise. The velocity space discretization is Δvz=5ms , channel centerline. −1 −1 and two different velocity bins Δvy=5ms and 2ms are Despite its simplicity, this test case therefore illustrates a − used for comparison. The velocity space is [−1200 m s 1, fundamental challenge for grid-based DK simulations to −1 1200 m s ] for both vz and vy. The time step is Δt=0.25 μs. resolve such beam distribution, particularly in a multi- Thesimulationrunsuptot=3ms, thus the number of time dimensional setup. The main advantage of the DK method is steps is 12,000. Here, a fully diffuse reflection is assumed on that the results are noiseless and no time sampling is needed, the walls. For parallelization, the physical domain is split into while particle methods typical require time averaging to four subsections, namely, the channel, the plume outside the reduce the numerical noise. channel from y=0.03 m to y=0.05 m, and the two regions above (y>0.05 m) and below (y<0.03 m).Withineach subsection, the domain is split in the z direction. The compu- 5.3. Neutral atom velocity distribution functions tational wall time is 30 hours using 24 processes in total. Instantaneous neutral atom VDFs at several locations in the computational domain are shown in figure 9. The results 5.2. Neutral atom density shown are along the channel centerline, i.e., y=0.04 m, near Figure 8 shows the instantaneous neutral atom density profile the injection slit, channel exit, and downstream. The neutral at 3ms. It is shown that the neutral atom density is smooth atom trajectory can be used as a verification of the kinetic inside the channel. The two figures show good agreement method. except for z>0.05 m, y=0.04 m, i.e., on the centerline for Figure 9(a) shows the particle VDFs near the injection slit. the lower resolution case in figure 8(a). This is because the The particles with vz>0 particles correspond to the particles −1 velocity discretization is not sufficient to resolve the beam injected from the slit, while particles with vz<0ms exist at
10 Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist the injection plane because of the diffuse (random) reflection The electron fluid model is identical to the one proposed by from the channel walls. Here, it can be seen that the particles Boeuf and Garrigues [76], which neglects the effect of electron with 0 >>-=-vvz∣∣ y L ch ( W ch 2 ) 4 only reach z=0m pressure, i.e. diffusion. This model is not physical because because of the geometry, i.e., consider a particle originating anode diffusion cannot be captured. However, it is chosen to from the lower channel wall at (z, y)=(0.04 m, 0.03 m) that ensure electron fluid model does not produce any oscillations reach (z, y)=(0m,0.04m). since recent calculations show numerical instabilities occur near In figure 9(b), the particles with vvzy∣∣>-0.1 can only the anode region when electron pressure is included [77].The exists at the location 1mm inside the channel exit, while some details of the problem setup, electron fluid model, and transport finite VDFs can be observed in -0.25<<-vvzy∣∣ 0.1 which coefficient models are described in [23]. The boundary condition seems to suggest that numerical error exists during the advection for potential is f=300 V at the anode, i.e., x=0cm, and > of the VDFs. For particles with vz 0,itcanbeseenthatthe f=0 V at the channel exit, also assumed to be the cathode VDFs at 0 <
11 Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist
Figure 10. Breathing oscillation results obtained from hybrid-PIC Figure 11. Breathing oscillation results obtained from hybrid-DK simulation. Ion VDFs and plasma properties of the three time simulation. Ion VDFs and plasma properties of the three time snapshots in (a) are shown: (b, c) for time x, (d, e) for time y, and snapshots in (a) are shown: (b, c) for time x, (d, e) for time y, and (f, g) for time z. Ion VDFs are calculated as binning the particles in (f, g) for time z. velocity bins similar to the DK simulation. be relevant to low temperature plasmas. The collisionless test Δò=2.7 eV for v=20km s−1. While the numerical diffusion cases include nonlinear electron plasma waves, plasma may be problematic for sputtering rate calculation, such ion sheaths in the presence of emitted electrons for plasma- VDF diffusion does not affect the thruster performance, i.e., immersed materials, and a multidimensional neutral atom thrust and specific impulse, as long as the mean velocity is solver. All such collisionless problems have analytic solu- obtained correctly. In addition, note that the DK simulation uses tions, thus serving as good verification tests. In addition to the MUSCL scheme for both physical and velocity advection. This code development, Takamura’s model of space charge limited is because the simulations become unstable when using a fi scheme that does not preserve positivity. Hence, positivity-pre- sheath for conducting materials is modi ed so that the kinetic serving DK simulation is recommended for collisional phe- effects of truncated primary electron VDFs are taken into nomena that involve ionization. account. It is shown that the improved sheath theory shows One interesting feature that occurs in the two hybrid good agreement with the DK simulation results, particularly simulations is the generation of slow ions shown in at the small sheath potentials where the kinetic effects of figures 10(a) and 11(a). Under a strong ionization oscillation, primary electrons become important. The improved sheath if the ion density becomes too small, then the electric field theory is consistent with the space charge limited sheath will locally be large in the ionization region (upstream of the theory for floating materials. To illustrate the collisional DK acceleration region, typically where magnetic field is largest) simulation, a 1D Hall thruster discharge is modeled using a so that the ionization rate increases. Recent experimental hybrid simulation where kinetic methods are used for ions and results obtained using laser induced fluorescence observed a simplified fluid model is used for electrons. Particle-based such slow ion population in the acceleration region [80, 81]. and grid-based kinetic methods are benchmarked and the results show good agreement while illustrating the advantages and disadvantages of both methods. 7. Conclusions The noiseless grid-based kinetic simulations possess an advantage over particle-based kinetic methods, particularly in A 1D1V and 2D2V direct kinetic (DK) simulations are dynamic problems where plasma oscillations and instabilities developed and verified with several different theories that can occur. For a steady-state solution, particle methods typically
12 Plasma Sources Sci. Technol. 27 (2018) 065004 K Hara and K Hanquist require time-averaging of data or usage of a large number of A.2. Modified Bohm condition computational particles to eliminate the statistical noise, while A modification to the Bohm condition can also be derived. the DK simulations do not require time-averaging of under- Linear perturbation of n (equation (4)), n (equation (5)), and resolved VDFs. Additionally, ionization events can be mod- i pe n (equation (7)) with respect to Φ are assumed. Thus, taking eled at every time step in the DK method, enabling a smooth ee the Taylor expansion in the limit of F , plasma profile. The multidimensional DK simulation illus- ∣ ∣ 1 trates well that fine velocity bins are needed to resolve a beam n F i =+1, 2 distribution. However, the main advantage lies in the thermal nM0 distributions where particle methods suffer in resolving a npe npe0 wide range of velocity space with a finite number of com- =+[]1erf -F+Fw N1 , n0 2n0 putational particles. nee npe0 =--F+FGN[]1erf wt 2 , n0 2n0 Acknowledgments where
exp Fw The authors acknowledge fruitful discussions with I D Boyd, N1 =+1erf -F+w , H C Dragnea, V I Kolobov, I D Kaganovich, J P Sheehan, p∣∣Fw ⎡ ⎤ R L Berger, J W Banks, M J Kushner, and B vanLeer. The exp()F t N =--F-tt⎢ w ⎥ simulations were performed on high performance computing 2 ⎢1erf w ⎥. ⎣ pt∣∣Fw ⎦ clusters in Texas A&M University and University of Michigan. Thus, the linearized Poisson’s equation can be written as
2 ⎡ ⎤ d F 1 npe0 =-F⎢ -()NGN + ⎥.23 () 22⎣ 12⎦ Appendix dx˜ M 2n0 In order to obtain a monotonically decreasing potential profile A.1. The integral in Poisson’s equation towards the wall, dE/dx 0, i.e., d2f/dx20. Since Φ<0, ( ) ( ) In deriving equation (11), the Poisson’s equation is integrated. from equations 9 and 23 , The integral of equation (5) can be written as ()()1erf+-F+--FG 1erf t M 2 ww,24() F¢=F npe0 1 NGN12+ Hpe =ò []1erf + F¢ - Fw exp F¢d F¢ n0 2 F¢=0 which is a modified Bohm condition for a stable sheath for- 1 mation. This is an equation modified compared to equation = ( ) [ ] (∣∣)(∣∣)1erf+F´+-FG 1erf t 33 in 46 that accounts for the kinetic correction of the ww = ⎧⎡ ⎤ primary electrons. It can be seen that M 1 can be recovered ⎨ 2 = ´⎢exp F() 1 + erf F-Fwww - F-Fexp F ⎥ when G 0 and F-¥w . ⎩⎣ p ⎦ ⎡ ⎤⎫ 2 ⎬ -+⎢(∣∣)∣∣1erf F-www Fexp F⎥ , ⎣ p ⎦⎭ ORCID iDs ()21 Kentaro Hara https://orcid.org/0000-0002-1816-165X and that of equation (7) is
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