S. Hamaguchi

Thermodynamics of strongly‐coupled Yukawa systems near the one‐component‐plasma limit. II. Molecular dynamics simulations

Molecular dynamics simulations are employed to study the equilibrium thermodynamics of strongly‐coupled systems of particles interacting through the Yukawa potential. Such systems serve, under the Debye–Huckel approximation, as a model for the physical behavior of plasma or colloidal suspensions of charged particulates. The thermodynamics may be characterized in terms of two dimensionless parameters—the ratio κ of the mean interparticle distance to the Debye length, and an approximate measure Γ of the interparticle potential energy in units of the thermal kinetic energy. Employing an accurate representation of infinite periodic boundary conditions, we focus on the regime of weak Debye screening (κ ≲ 1) and strong coupling (Γ≫1). Excess internal energies measured at many points (κ,Γ) are fitted to simple functional forms for the fluid and solid phases, representing extrapolations of the classical one‐component plasma (OCP) limit, κ=0. Quantitative expressions for the Helmholtz free energy and the ‘‘equatio...

Thermodynamics of strongly‐coupled Yukawa systems near the one‐component‐plasma limit. I. Derivation of the excess energy

The excess energy for a system of charged mesoscopic particles or ‘‘particulates’’ immersed in a neutralizing background medium is derived analytically, and is shown to approach that of the classical one‐component plasma in the limit of high background temperatures. Examples of such systems, which are known as Yukawa systems due to the form of the interparticle pair potential, include dusty plasmas and colloidal suspensions. The expression for the excess energy allows thermodynamic properties of Yukawa systems to be determined from Monte Carlo or molecular‐dynamics simulations.

Effects of sheared flows on ion‐temperature‐gradient‐driven turbulent transport

Previous studies of the ion‐temperature‐gradient (ITG) driven turbulence are expanded to include the effect of sheared E×B flows in sheared magnetic fields. The radial eigenmodes are shown to substantially change character by shifting the modes off the rational surface. The new mode structure and growth rate directly affects the transport of both thermal energy and momentum in the sheared flows. The growth rate first increases with small shear flow and then decreases. The theoretical correlation of the shear flow with the thermal transport is important with respect to the transitions observed in tokamaks of a low (L mode) to a high (H mode) thermal confinement state as a function of the poloidal rotation velocity in the shear flow layer. The three‐dimensional nonlinear simulations show that the anomalous ion thermal diffusivity is reduced significantly when dvE/dx≂2(cs/Ls) ((1+ηi)Ti/Te)1/2. This condition is thought to be satisfied in a boundary layer in tokamaks with shear flow.

Phase diagram of Yukawa systems near the one-component-plasma limit revisited

Transition inverse temperatures ~or G values! at the fluid‐solid phase boundary of Yukawa systems near the one-component-plasma ~OCP! limit have been evaluated by molecular dynamics simulations. These values are systematically smaller than those obtained in an earlier study by Farouki and Hamaguchi @J. Chem. Phys. 101, 9885 ~1994!#. The discrepancy is attributed to the fact that, in the earlier study, the harmonic entropy constants were approximated by that of the OCP, whereas the new results are based on more accurate harmonic entropy constants obtained from lattice-dynamics calculations. The new molecular dynamics simulations also confirm that the bcc‐ fcc phase transition curve is in good agreement with that of the quasiharmonic theory in the regime k<1.4, where k is the ratio of the Wigner‐Seitz radius to the Debye length. Examples of Yukawa systems include dusty plasmas and colloidal suspensions.

A shock-tracking algorithm for surface evolution under reactive-ion etching

A new algorithm that determines the evolution of a surface eroding under reactive‐ion etching is presented. The surface motion is governed by both the Hamilton–Jacobi equation and the entropy condition for a given etch rate. The trajectories of ‘‘shocks’’ and ‘‘rarefaction waves’’ are then directly tracked, and thus this method may be regarded as a generalization of the method of characteristics. This allows slope discontinuities to be accurately calculated without artificial diffusion. The algorithm is compared with ‘‘geometric’’ surface evolution methods, such as the line‐segment method.

Phase transitions of dense systems of charged dust grains in plasmas

The behavior of ‘‘dust’’ grains (particulates) in microelectronics process plasmas has been studied using N‐body simulations. Grains are assumed to be negatively charged and interact through a screened Coulomb potential Φ(r). The dimensionless parameters κ=a/λ and Γ=Φ(a)/kBT characterize the thermodynamics of the particulate system, where λ is the ion Debye length, a=(3/4πnD)1/3 is the mean intergrain distance, and nD and T are the dust density and temperature. The simulations exhibit a transition between ‘‘fluid’’ and ‘‘solid’’ phases at a critical value Γc that depends on κ and weakly on the system history (i.e., whether it ‘‘melts’’ from an ordered state or ‘‘freezes’’ from a random one).

Ion-temperature-gradient-driven transport in a density modification experiment on the Tokamak Fusion Test Reactor

Tokamak Fusion Test Reactor (TFTR) profiles from a supershot density‐modification experiment [Zarnstorff et al., Plasma Physics and Controlled Nuclear Fusion Research, 1990, Proceedings of the 12th International Conference, Washington (IAEA, Vienna, 1991), Vol. I, p. 109] are analyzed for their local and ballooning stability to toroidal ηi modes in order to understand the initially puzzling results showing no increase in χi when a pellet is used to produce an abrupt and large increase in the ηi parameter. The local stability analysis assumes that k∥=1/qR and ignores the effects of shear, but makes no assumption on the magnitude of k∥vti/ω. The ballooning stability analysis determines a self‐consistent linear spectrum of k∥’s including the effect of shear and toroidicity, but it expands in k∥vti/ω≤1, which is a marginal assumption for this experiment. Nevertheless, the two approaches agree well and show that the mixing length estimate of the transport rate does not change appreciably during the density mod...

Flux considerations in the coupling of Monte Carlo plasma sheath simulations with feature evolution models

The evolution of two features, a 2-d trench and a circular via exposed to a 3-v ion flux is calculated. The surface flux is calculated using the particle distribution function, obtained from a Monte Carlo sheath simulation. While Monte Carlo sheath simulations are 1-d/2-v (axisymmetric), shadowing on a substrate surface due to topography breaks the axisymmetry and necessitates a 2-d/3-v surface flux calculation. The authors show a methodology for the flux calculation in long trenches and circular vias which adds very little to computational expense over that incurred for a 2-d/2-v flux calculation. Comparison between the trench and the via shows that, as expected, the geometry of a via blocks out more ion trajectories than that of a trench leading to a smaller etch rate in a via. If the flux is calculated in 2-d/2-v phase space, the resulting feature shape is shown to be an artifact of the calculation method. It is also shown that the angular distribution of the particle as well as energy fluxes and the resulting shape of the feature are largely insensitive to the electric field profile assumed for a given sheath voltage. >

Ion energetics in collisionless sheaths of rf process plasmas

Ion energy distribution functions in collisionless radio‐frequency (rf) sheaths are discussed from the viewpoint of kinetic theory. Effects of rf fields on ion density and velocity profiles in plasma sheaths are also derived, based on ion fluid equations. It is shown that the ponderomotive force due to rf modulation of the magnitude of the sheath electric field exerts a retarding effect on the ion motion that counteracts the dc‐bias field when the ratio of the ion transit frequency ωtr to rf modulation frequency ω is small but finite. Consequently, the time‐averaged ion density is higher and the time‐averaged ion fluid velocity is lower in rf sheaths by the order of (ωtr/ω)2 than those in corresponding dc sheaths. The influence of an oscillating plasma/sheath boundary on the ion energetics is also considered. Under suitable conditions, this induces rapid ‘‘quasiperiodic’’ variations in the ion energy distribution as the rf ω is increased.

Hydrodynamic analysis of electron motion in the cathode fall using a Monte Carlo simulation

The exact mass, momentum, and energy conservation equations for electron transport in a dc glow are derived from the Boltzmann equation. A Monte Carlo particle simulation is used to explicitly calculate the individual terms of the moment equations, and to gain insight into the behavior of the electron distribution function (EDF) moments such as density and average velocity. Pure forward scattering and isotropic scattering are considered as two limiting scattering mechanisms. When forward scattered, the electron fluid shows the maximum change in properties and in transport mechanisms at the field transition point between the cathode fall (CF) and the negative glow. Isotropic scattering, however, results in property changes a short distance inside the sheath. Diffusion of the low‐energy, high‐density, bulk plasma electrons into the CF causes dilution of the low‐density, high‐energy beam from the CF before the beam actually arrives at the low‐field region. The applicability of commonly used closure relations which yield a fluid description of the system is evaluated. Use of fluid equations to characterize this system with no a priori knowledge of the EDF is limited by kinetic effects, such as heat flow against the temperature gradient, especially in the forward‐scattered case where the EDF is very anisotropic. The description of inelastic rates by Arrhenius kinetics is found to be surprisingly accurate with both scattering mechanisms. However, while temperature is an adequate gauge of the characteristic energy under isotropic scattering, the energy of the bulk electron motion must be included under forward scattering. Also, Arrhenius kinetics sometimes produce a spurious double peak in the inelastic rate profile which is not reproduced by the Monte Carlo simulation. The anisotropy of the EDF under the forward‐scatter assumption makes it difficult to justify the use of the mobility and heat conduction closure relations. Under isotropic scattering, however, electron inertia is negligible. In that case, under the discharge conditions used here, the drift‐diffusion approximation to the flux is good to within a factor of 2. Classical heat conduction theory overestimates the heat flux by a factor of 4 at the sheath edge.