Molecular dynamics simulations of strongly coupled plasmas

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1 Molecular dynamics simulations of strongly coupled plasmas Workshop on Dynamics and Control of Atomic and Molecular Processes Induced by Intense Ultrashort Pulses - CM0702 WG2, WG3 meeting September 2011, Debrecen, Hungary Zoltán Donkó Research Institute for Solid State Physics and Optics of the Hungarian Academy of Science, Budapest, Hungary and Physics Department, Boston College, Chestnut Hill, MA, USA

2 Molecular dynamics simulations of strongly coupled plasmas Results obtained in collaboration with: G. J. Kalman - Boston College, USA P. Hartmann - RISSP Budapest, Hungary K. I. Golden - University of Vermont, USA J. Goree - University of Iowa, USA Supported by: NSF / MTA (HU Academy of Sciences) / OTKA (HU Sci. Research Fund)

3 Outline Why do we need simulations? Systems of interest Basics of Molecular Dynamics (MD) simulations What do we learn from MD? Structural & thermodynamic properties Localization and transport Collective excitations

4 Why do we need simulations? Experiment Simulations are useful for checking theoretical results Simulation Very much desired interaction Theory for cases where no theoretical results are available for understanding experimental observations Simulations allow: identification of important processes visualization of the system Most dramatic advance of resources is experienced in the field of simulations

5 Dramatic advance of resources Pioneering MD simulations in the 1970s-80s (OCP, BIM, statics, dynamics, transport, etc.)

6 Dramatic advance of resources... where does this go? "Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs 30 tons, computers in the future may have only 1,000 vacuum tubes and weigh only 1.5 tons." [Popular Mechanics, 1949]

7 Systems of interest Classification of plasmas from the American Physical Society fusion chart Is there really not much here to look for???

8 Plasmas. a better phase diagram Consider the interaction between a single type of particles (ion-ion) VPOT Γ= VKIN STRONGLY COUPLED PLASMAS R. Redmer, Phys. Reports 282, 35 (1997)

9 The one-component plasma (OCP) model OCP model: only one type of species is considered explicitly, the presence and effects of other species are accounted for by the potential Coulomb Coulomb potential: Characteristic energies (Coulomb): Φ(r) = 1 Q 4πε 0 r V POT = Q2 4πε 0 a V KIN = kt non-polarizable background a : Wigner-Seitz radius Coupling parameter: Debye-Hückel / Yukawa Γ = V POT V KIN = 1 4πε 0 Q 2 akt D-H / Yukawa potential & screening parameter: Φ(r) = 1 Q exp ( r/λ D ) 4πε 0 r, κ = a λ D polarizable background

10 Molecular Dynamics (MD) basics (one-component plasma strongly interacting classical many-body system) Equilibrium & non-equilibrium MD We let the system evolve according to interactions Perturb the system and measure response

11 Molecular Dynamics (MD) simulation basics Equilibrium MD SIMULATION CORE + MEASUREMENTS Time evolution of phase space trajectories of an ensemble of N particles Calculate quantities of interest from phase space coordinates Example: finite system with external confinement: m r i = i j F i,j (t) + F ext (t) mηv i (t) + R F i,j = φ(r ij) r F ext = fr 2 (e.g.) Friction Brownian randomly fluctuating force (Langevin force)

12 Molecular Dynamics (MD) simulation basics Integration of the equation of motion ( leapfrog scheme ) i ( v i t t ) 2 r i (t) ( v i t + t ) 2 r i (t + t) ( v i t + t ) ( = v i t t ) + F i(t) 2 2 m t ( r i (t + t) = r i (t) + v i t + t ) 2 time m r i = i j F i,j (t) + F ext (t) mηv i (t) + R How to calculate Fi,j (t)?

13 Molecular Dynamics (MD) simulation basics Short range interaction potentials Interaction is considered only between closely-separated pairs of particles (cutoff radius) F i (t) = r ij <r C F i,j (t) Finite system Infinite system PERIODIC BOUNDARY CONDITIONS Primary simulation cell Image cells

14 Molecular Dynamics (MD) simulation basics Long range interaction potentials (e.g. Coulomb): Not possible to find cutoff radius, tricks are needed F i (t) = cell+images F i,j (t) Possible solutions: Primary simulation cell (yellow) Ewald summation Particle-Particle, Particle-Mesh (PPPM, P3M) method (Hockney & Eastwood)

15 Molecular Dynamics (MD) simulation basics The PPPM method uses finite size charge clouds ( ρ(r) = ρ 0 1 r ) R R 0 ρ(r)dv = Q Fourier transform is band-limited, the interaction between clouds can be represented on a mesh in k-space, images are included (PM) if r R : if r < R : F ( ) = F ( ) F ( ) = F ( ) + F corr (r) Hockney R W and Eastwood J W 1981 Computer Simulation Using Particles (New York: McGraw-Hill) Correction force, to be applied for closely separated neighbors only (PP, chaining mesh)

16 Molecular Dynamics : What do we see? Γ=120, κ=1 Γ=5, κ=1 2D frictionless Yukawa liquids

17 Molecular Dynamics : What do we learn? Phase space coordinates (r i, v i ) i = 1... N Correlation functions Identification of collective modes Calculation of transport parameters Structure Thermodynamic quantities

18 It s real: experimental realization of 2D dusty plasma Dust particles dispersed in a glow discharge plasma acquire a charge of 10 4 q e Plasma V Φ(r) = 1 Q exp ( r/λ D ) 4πε 0 r, κ = a λ D Plasma crystal Sheath Dust layer is levitated due to the balance between electrostatic force and gravity Interaction: screened Coulomb (Yukawa) potential Crystallization at high Γ Quasi-2D confinement Extensive experimental work from early 1990s (Morfill, Thomas, Goree, Fortov, Piel, et al., ) in the crystal and liquid phases

19 Experimental realization of 2D dusty plasma melamine-formaldehyde microspheres Dusty plasma experiment in RISSP, Budapest (P. Hartmann) - + Plasma - Ii - Ie + + particle rd u s t - + h

20 Experimental realization of 2D dusty plasma Laser manipulation to measure complex viscosity P. Hartmann, M. Cs. Sándor, A.-Zs. Kovács, Z. Donkó: Phys. Rev. E 84, (2011).

21 Structural and thermodynamic properties

22 Pair correlation & thermodynamic properties Pair correlation function e.g. 3D Coulomb OCP (one-component plasma) Energy: E N = 3 2 k BT + n 2 Pressure: 0 ϕ(r)g(r) 4πr 2 dr Strong correlation, liquid-like structure at high coupling k B T p = nk B T n2 6 ( n p Isothermal compressibility: ) T = 1 + n 0 0 ϕ(r) r g(r) 4πr3 dr [g(r) 1] 4πr 2 dr

23 Phase transitions: 3D Coulomb / Yukawa systems Coulomb (Monte Carlo) S. G. Brush, H. L. Sahlin and E. Teller, J. Chem. Phys. 45, 2102 (1966). Yukawa S. Hamaguchi, R.T. Farouki and D.H.E. Dubin, Phys. Rev. E 56, 4671 (1997). Γ 125 E. L. Pollock and J. P. Hansen Phys. Rev. A 8, 3110 (1973) G. S. Stringfellow, H. E. DeWitt and W. L. Slattery, Phys. Rev. A 41, 1105 (1990). Γ 175

24 Transport phenomena Diffusion, shear viscosity and thermal conductivity in 3D systems ( although 2D is more interesting )

25 Measurements of transport coefficients Equilibrium Molecular Dynamics: Measure correlation functions D = 1 2 η = 1 V kt λ = 1 V kt C v dt C η dt C v v(t) v(0) C η P xy (t)p xy (0) C λ dt C λ J Qx (t)j Qx (0) VACF SACF EACF Non-Equilibrium Molecular Dynamics: Perturb the system and measure the response

26 Diffusion coefficient Coulomb Yukawa D = D a 2 ω p Universal scaling: H. Ohta and S. Hamaguchi, Phys. Plasmas 7, 4506 (2000) D = α(t 1) β + γ α, β, γ : functions of κ T = T/T M J. Daligault, Phys. Rev. Lett. 96, (2006)

27 Shear viscosity: methods (1) Equilibrium MD Nonequilibrium (transient perturbation) MD y time P xy = N [ mv ix v iy 1 2 i=1 η = 1 V kt 0 N j i P xy (t)p xy (0) dt ] x ij y ij φ(r ij ) r ij r ij ( ) 2πyk W (y k ) = W M0 sin L ( 2πy W (y, t) = W M0 sin η = ρ τ L ( L 2π v x t = η ρ ) ( exp t t 0 τ ) 2 2 v x y 2 ) Z. Donkó and B. Nyíri, Phys. Plasmas 7, K. Y. Sanbonmatsu and M. S. Murillo, Phys. Rev. Lett. 86,

28 Shear viscosity: methods (2) Reverse Molecular Dynamics Homogeneous Shear Algorithm y B External momentum transfer y v x x v x A x η dv x(y) dy = p 2t sim S F. Müller-Plathe, Phys. Rev. E 59, 4894 (1999). dr i dt = p i m + γy iˆx η = lim t P xy (t) γ d p i dt = F i γ p yiˆx α p i D. J. Evans and G. P. Morriss, Statistical mechanics of nonequilibrium liquids (Academic Press, 1990)

29 Shear viscosity of 3D Coulomb liquids Kinetic Potential P xy = N [ mv ix v iy 1 2 i=1 N j i ] x ij y ij φ(r ij ) r ij r ij η = η mnω p a 2

30 Thermal conductivity: MD methods Reverse molecular dynamics Exchange of momenta time Spatial temperature modulation Heat flux T t = λ cρ 2 T x 2 T ( x ) Cold Hot λ = cρ τ H ( ) 2 L 2π x λ = E 2St sim T x F. Müller-Plathe, J. Chem. Phys. 106, 6082 (1997). Z. Donkó, B. Nyíri, L. Szalai, and S. Holló, Phys. Rev. Lett. 81, 1622 (1998).

31 Thermal conductivity of 3D Coulomb liquids λ = λ nkω p a 2

32 Collective excitations

33 Collective excitations in 3D plasma liquids Longitudinal wave Transverse wave k k Microscopic density fluctuations: N ρ(k, t) = exp [ ikx j (t) ] j=1 S(k, ω) = 1 2πN Dynamical structure function: lim T 1 T ρ(k, ω) 2 ρ(k, ω) = F [ ρ(k, t) ] Microscopic current fluctuations: N λ(k, t) = v jx (t) exp [ ikx j (t) ] j=1 N τ(k, t) = v jy (t) exp [ ikx j (t) ] j=1 Longitudinal and transverse current-current fluctuation spectra L(k, ω) T (k, ω)

34 Collective excitations in 3D plasma liquids Coulomb: L : const. freq. T : acoustic Yukawa: L : quasi-acoustic T : acoustic

35 Collective excitations in 3D liquids: MD vs. theory QLCA theory: Cutoff of T-mode at finite k: liquid M.S. Murillo Phys. Rev. Lett. 85, 2514 (2000) Ω 2 L(k) = Ω 2 0(k) + ω 2 0,3D Λ 3D (x, y) = 2 e y x k Λ 3D ( k r, κ r ) h( r)d r Ω 2 T(k) = ω 2 0,3D [ (1 + y + y 2 ) ( sin(x) x Θ 3D (x, y) = cos(x) [ e y k 2 2 x 2 x y2 0 k Ω 2 0(k) = ω0,3d 2 2 k 2 + κ 2 Θ 3D ( k r, κ r ) h( r)d r 3 sin(x) x 3 ( 1 sin(x) x ) y2 Z. Donkó, G. J. Kalman & P. Hartmann, J. Phys. Cond. Matter 20, (2008) ) ( sin(x) 6 x ] Λ 3D (x, y) + 12 cos(x) x 2 EXP: J. Goree COMPRESSIONAL MODE SHEAR MODE 12 sin(x) ) ] x 3

36 Summary Simulation studies aid the understanding of theoretical and experimental results Simulations are suitable for a wide variety of strongly coupled many-particle systems Equilibrium / non-equilibrium Molecular Dynamics simulations can be used to study structural & thermodynamical properties localization and transport properties collective excitations... THANK YOU FOR YOUR ATTENTION