General Features and Solutions of Master Equation for equilibrium structures in complex plasmas

摘要

1. Drag and diffusion coefficients for non-linear dust screening. The balance of forces in self-organized in Master Equations are using dust-ion interaction cross-section with exact formulation of non-linear screening in most developed model [1,2]. For typical experiments in laboratories and on board of International Space Station (ISS) the individual dust screening is non-linear, i.e the parameter B, the ratio of electrostatic energy of ion-grain interaction at the distance of Debye radius to the ion average kinetic energy T_i, is large β>1 (typically β ≈ 20 - 30). Dusty plasmas are considered to. be partially ionized and the ion neutral collisions, determined by ion-neutral mean free path λ, are taken into account together with ion-dust collisions. Normalization of forces F, distances r, dust sizes a, densities and Havnes parameter. P is the following: F → Fλ/T_i;r→ r/λ, a →a/λ, n→ n_i4πe~2λ~2/Ti;n→n_e4πe~2λ/TiP → Z_dn_dn4πeλ/T_i;z → Z_de~2 /aλT_e;τ→ T_i/T_e. Plasma flux and Φ ion drift velocity ware normalized with respect to ion thermal velocity v_(Ti): Φ →ΦT_i//2vTie~2λ~2;u→ui//2vTi. For this normalization non-linearl parameter β the drag coefficient fdr and diffusion coefficient D are determined by expressions: β = za√n/τ, Fdr= Zdfdru/√n;Φ= nu -D(dn/dr). The transport cross-sections for scattering of ions on grains is calculated taking into account both large angle scattering and reflection from potential barriers. It enters both in f_(dr) and D. Linear drag coefficient (for β l)is f_(dr) ∞ β1n(l/β) and cannot be extrapolated to large β and is much larger than the non-linear screening coefficient, but the absolute value of calculated non-linear coefficient is larger, than the maximum possible linear one estimated for β≈ 1. The numerical calculation of f_(dr) as function of uand β have shown that after some increase with β the drag coefficient is decreasing with an increase of β. The maximum drag is 2-3 times larger than that calculated in [3] and is close to that obtained in some numerical simulation of [4]. Contrary to [4] the calculations are able to give f_(dr) in broad range of u(-4 < u < 4) and β (3 < β < 90. The results are numerical continuous functions of f_(dr) an D appropriate for solutions of Master Equations for structures. An example of these results for f_(dr) are presented on Fig. 1. Both ion-neutral and ion dust collisions are taken into account in diffusion coefficient which was found as numerical continuous function of 3 parameter u, β, p = P/2√n and an examples of these results is presented on Fig. 2.