Analysis of Pair Interparticle Interaction in Nonideal Dissipative Systems

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1 ISSN , Journal of Experimental and Theoretical Physics, 00, Vol. 0, No. 4, pp Pleiades Publishing, Inc., 00. Original Russian Text O.S. Vaulina, E.A. Lisin, A.V. Gavrikov, O.F. Petrov, V.E. Fortov, 00, published in Zhurnal Éksperimental noі i Teoreticheskoі Fiziki, 00, Vol. 7, No. 4, pp STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS Analysis of Pair Interparticle Interaction in Nonideal Dissipative Systems O. S. Vaulina, E. A. Lisin*, A. V. Gavrikov, O. F. Petrov, and V. E. Fortov Joint Institute for High Temperatures, Russian Academy of Sciences, ul. Izhorskaya /9, Moscow, 74 Russia * ealisin@yandex.ru Received July 8, 009 Abstract A new method is proposed for determining the interaction forces between particles in nonideal dissipative systems with isotropic pair potentials. The method is based on the solution of the inverse problem describing the motion of interacting particles by a system of Langevin equations and allows one to recover the parameters of the external confining potential without referring to a priori information on the friction coefficients of the particles. This procedure was tested by a numerical simulation of the problem in a wide range of parameters typical of experimental conditions in a laboratory dusty plasma. The results of the first experimental approbation of the method as applied to the analysis of the interaction of dust particles in a laboratory high-frequency capacitive discharge plasma are presented. DOI: 0.4/S INTRODUCTION The problem of determining the interaction potential of particles in nonideal dissipative systems is of significant interest in various fields of science and technology (plasma physics, medical industry, physics and chemistry of polymers, etc.) [ 7]. Information on the potential U of interparticle interaction is needed for analyzing various thermodynamic and physical characteristics of systems (such as pressure, internal energy, compressibility, etc.), as well as for calculating various kinetic coefficients (for example, viscosity, heat conductivity, electrical conductivity, etc.) with the use of the well-known Green Kubo formulas [6, 7]. A laboratory dusty plasma is a good experimental model for investigating the properties of nonideal dissipative systems. A dusty plasma represents an ionized gas containing micron-sized charged particles of a substance (dust and macroparticles). This plasma is ubiquitous in nature (in space, in upper layers of the atmosphere, etc.) and is produced in many technological processes [ 5]. Most experiments on the study of a dusty plasma are carried out in various types of gas discharge. Micron-sized dust particles in a gas-discharge plasma may gain considerable negative charge ez (where e is the electron charge) and form dusty structures similar to a liquid or a solid. Depending on the experimental conditions, these structures can either be close to homogeneous three-dimensional systems or have strongly anisotropic quasi-twodimensional character, as, for example, individual dust layers in the near-electrode region in high-frequency discharge [, 4]. The model of screened Coulomb potential (of Yukawa type) [ 5] U = ( ez) exp( l/λ)/l () is the most well-known model for describing pair interaction between identically charged particles in physical kinetics. Here l is the distance between two particles and λ is the screening length. The assumption of screened potential () well agrees with the results of measuring the radial forces of interaction between two particles in a plasma [8] and with the results of calculating the structure of a screening cloud for an isolated dust particle [9] at small distances from the particle l < 4λ D ; here λ D is the Debye radius of the plasma. The screening falls off as the distance from a particle increases, and, for l λ D, the potential behaves as a power-law function: U l [0] or U l []. The above-mentioned studies [8 ] refer to the case of isolated dust particles in a plasma. Presently it is not ultimately clear how the presence of other particles in a dust cloud, gas ionization processes, collisions between electrons (ions) and neutrals of the surrounding gas, and many other factors affect the form of the potential of interparticle interaction [, ]. In addition, the question of the existence of attraction forces in dusty systems has been intensively studied by many authors; however, there is still no convincing experimental evidence for the existence of these forces [, 4]. Thus, the problem of the form of the interaction potential between dust particles in a plasma has no satisfactory solution at present. In the case of isotropic pair interaction U U(l), the equilibrium properties of a nonideal system are completely determined by the temperature T of the 66

2 ANALYSIS OF PAIR INTERPARTICLE INTERACTION 66 particles and the pair correlation function g(l) [6, 7], which can be either measured or obtained by computer simulation of the problem. In the statistical theory of liquids, considerable attention is paid to the methods of recovering the pair potential U(l) by searching for approximate integral equations for the relation between U(l) and g(l). The most popular approximations are simple ones based on the integral equations proposed by Bogolyubov, Born, Green, Kirkwood, and Yvon, as well as a hyperchain approximation and the Percus Yevick equation [6]. A number of recent papers [4 6] have been devoted to the methods for determining the pair interaction potential between dust particles in a plasma. Unfortunately, the existing integral equations involve some simplified assumptions on the relation between the pair correlation function g(l) and the potential U(l) of pair interaction and do not allow one to correctly recover the function U(l) for strongly correlated liquid systems [4]. An additional limitation of these methods is associated with a small spatial range of correct identification of the function U(l); the effective range of the method is limited by interparticle distances of from l ~ l p / to l l p, where l p is the average distance between two particles [4]. A wide variety of methods for determining the potentials of interparticle interaction and of the charges of dust particles are based on the measurements of the dynamic response of these potentials to various external (for example, periodic) perturbations followed by the analysis of this response with the use of the equations of motions of individual dust particles in the field of known external forces [, 4, 7 0]. One of the drawbacks of these diagnostic methods is that they need a priori information on the electric fields and external forces that act on a dust particle in a plasma. This information can be obtained either from additional measurements or from numerical simulation of the experimental conditions. Other limitations are related to the possibility of determining the interaction force only between two isolated particles [8, 7] and/or to the presence of external perturbations of the system [8 0], which may lead to a significant variation in the parameters of the surrounding plasma and of the dust particles. The aim of the present study is to develop a method for recovering the parameters of plasma dust systems in a laboratory plasma by solving the inverse problem describing the motion of dust particles by a system of Langevin equations. The specific feature of the problem is that the Langevin equations are irreversible in the sense that they include the action of random forces. Therefore, even if the potential of interparticle interaction is defined by a certain parametric function, the correct recovery of the unknown parameters of the inverse problem requires the analysis of the dynamics of the system during a certain (rather long) time interval in order to avoid random errors associated with the stochastic (thermal) motion of particles. In the present paper, the recovery of unknown parameters is based on the best fitting of the solution of the direct problem on the motion of particles to the information on the coordinates and displacements of these particles, which is easily recorded in both numerical and real experiments. This method takes into account the friction forces acting on the particles of the system and allows one to recover both the pair interaction forces and the parameters of the external confining potential.. RESULTS OF NUMERICAL SIMULATION AND THEIR DISCUSSION.. Simulation of the Dynamics of Interacting Particles Calculations were carried out by the Langevin method of molecular dynamics, which is based on the solution of a system of differential equations (consisting of N p equations of motion, where N p is the number of particles) with a Langevin force F ran, which is a source of stochastic (thermal) motion of particles with given kinetic temperature T. This method takes into account the pair interparticle interaction forces F int and external electric forces F ext from a trap that keeps the cloud of particles in the steady state. The method of simulation is described in detail in []. The equations of motion are expressed as M d l k F dt int ( l kj ) l k l = j l kj j () dl + F ext Mν fr ---- k + F ran. dt Here, F int (l) = U/ l, l kj = l k l j is the interparticle distance, U is the potential of pair interaction, M is the particle mass, and ν fr is the coefficient of friction of dust particles due to their collisions with neutrals of the surrounding gas. For correct simulation, the integration step Δt must satisfy the condition Δt /max{ν fr, ω*}, where ω* = (U''/πM) / is the characteristic collision rate of charged particles with each other and U'' is the second-order derivative of U at the point of average interparticle distance l p []. In our calculations, the integration step was chosen to be Δt = /0max{ν fr, ω*}, and the duration of a numerical experiment varied from 0/min{ν fr, ω*} to 000/min{ν fr, ω*}. The potentials of interparticle interaction used in the calculations were isotropic and represented various combinations of power-law and exponential functions, which are most frequently used for simulating repulsion in the kinetics of interacting particles []: U U c c ( l p /l) n = [ exp( κl/l p ) () + c ( l p /l) n c ( l p /l) n + ]. Here κ, c i, and n i are variable parameters; κ = l p /λ; and U c = (ez) /l is the Coulomb potential of interparticle interaction. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 0 No. 4 00

3 664 VAULINA et al. Γ* = 7.5 Γ* = 7.5 (c) Γ* = 80 Γ* = 80 The problem was solved in the two-dimensional statement for a monolayer of particles confined in a linear electric field F ext = eze(r), E(r) = αr, with radial symmetry. Here, r is the distance from a particle to the center of the trap, and α is the gradient of the electric field. The displacement of particles in the vertical direction (in the direction perpendicular to the dust monolayer) was neglected, and the forces acting on the particles in this direction were not taken into consideration. To analyze the dynamics of particles interacting with potentials (), we solved equations of motion () for different effective parameters introduced by analogy with the parameters obtained for large quasi-twodimensional systems of particles interacting with an isotropic pair potential, namely, for the effective nonideality parameter [, ] Γ* =.5l pm U'' ( l pm )/T (4) and the scaling parameter ξ U'' ( l pm ) / ( πm) / = ν fr. (5) Here, l pm is the most probable distance between the particles of the system for a crystalline structure that is sought by constructing pair correlation functions g(l). (The position of the first maximum of the function g(l) for a crystalline structure with Γ* 0 corresponds to the value of l pm.) Note that the spatial correlation of macroparticles is completely determined by the parameter Γ* (for Γ* > 0), provided that the following empirical condition is satisfied [4,, ]: π > U' ( l p ) l p / U( l p ). As mentioned above (see the Introduction), in strongly correlated liquid systems, this fact presents an (d) Fig.. Arrangement of (a, b) N p = 50 and (c, d) N p = 500 particles in a dust cloud for ξ = 5 and potential U = U c exp( ) for different Γ*. The dashed lines show the regions of the solution to the inverse problem for a part of the dust cloud, and the solid lines show the region of the central cell. obstacle to the correct recovery of the function U(l) on the basis of approximate integral equations for the relation between U(l) and g(l) [4, 6]. The effective parameter Γ* was varied from 5 to 80, while the scaling parameter ξ was varied from 0. to 5.0, which are typical values for the experimental conditions in a gas-discharge dusty plasma. The number of interacting particles in the system varied from N p = to 500. Note that the number of particles observed in a dust monolayer in a laboratory high-frequency discharge plasma depends on the characteristic size of the trap design, in particular, on the size of the ring, and may vary from to 000 particles. However, for the correct spatial resolution of the trajectories of dust particles, one usually restricts the field of view of the video systems used for diagnostics to a fragment of a dust cloud containing no more than particles. The arrangement of 50 and 500 particles (N p = 50, 500) in a simulated dust cloud is illustrated in Figs. a and b for ξ 5 and various values of Γ*. The pair correlation functions g( ) are shown in Figs. a and b for various values of the potential and the parameter Γ*... A Method for Solving the Inverse Problem At each integration step Δt = t m + t m, the solution of the direct problem () was expressed as an array of coordinates of all the particles at time instant t m (the velocities and accelerations of the particles were not recorded). For solving the inverse problem, we defined the velocity V k and the acceleration a k of an individual (kth) particle at t m as dl V k ( t m ) k l ---- k ( t m ) l k ( t m ) = , dt Δt (6) d l a k ( t m ) k V k ( t m + ) V k ( t m ) = dt Δt Thus, the data of numerical experiments used for solving the inverse problem were analogous to the data that are usually recorded in real-time laboratory experiments. To recover the force F F int of pair interparticle interaction (and hence the pair potential U), we used expansions in the form of various combinations of power-law and exponential functions: I p ( i + ) F a i l b i l i κl = + exp ---. (7) l p i = Here a i, b i, and κ are unknown coefficients, and I p is the number of terms in the expansion. Thus, the total force F pp acting on the kth dust particle from other particles of the dust cloud was expressed as k F pp = N p j =, j k l p i = κl a i + b i l exp kj kj l p l k l j i + l kj (8) JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 0 No. 4 00

4 ANALYSIS OF PAIR INTERPARTICLE INTERACTION 665 g g Γ* = 80 Γ* = Fig.. Pair correlation function g( ) for N p = 50 and 500 for various potentials U /l (symbols) and U = U c exp( ) (lines). The forces of pair interparticle interaction (F F int ) were also approximated by splines of the form ψ n () l = c ni l i, l [ l n ; l n + ), i = 0 n =,,, I s, (9) where I s is the number of partition intervals. At the endpoints of the partition intervals, the functions F(l) were continuous and smooth: ψ n ( l n + ) = ψ n + ( l n + ), ψ n ( l n + ) = ψ n + ( l n + ), while their length l n + l n varied from 0.5l pm to.5l pm in various numerical experiments. As an approximating function for the force F pt acting on the kth particle from the external field of the trap, we used polynomials of the form k F pt = I t r k d i r k i i = (0) Here, d i are expansion coefficients, r k is the distance from the kth particle to the center of the trap, and I t is the number of terms in the expansion. Thus, the inverse problem consisted in searching for a list of unknown variables κ, a i, b i (or c ni ), d i, and v fr of the system of equations of motion recorded for each analyzed particle at different time instants t m during the entire numerical experiment, km Ma km = ν fr MV km + F pp + F km pt. () Here, V km V k (t m ) and a km a k (t m ) are the velocity and the acceleration of particles at time instant t m, that are determined by analyzing the displacements of the particles (see Eq. (6)), (t m, d, d,, d It ) is km k an F pt F pt. approximating function for the force (0) acting from the trap, and km F pp k F pp ( t m, κ, a, a,, a Ip, b, b,, b Ip ) is an approximation of the pair interaction force in the case when this force is defined by function (8), or km F pp k F pp ( t m, c 0,, c Is 0,, c,, c Is ) is an approximation of pair forces by splines (9). The coefficients were determined by the best fitting of the experimental data on the positions of particles (V km, km km a km ) to the approximating functions ( F pp, F pt ) that enter Eqs. (), with the use of the standard procedure of minimization of the rms deviation S so that the value of all the deviations satisfies the condition N p N c km km S ( Ma km + ν fr MV km F pp F pt ) = min. k = m = () Here, N c = texp/δ c t is a quantity equivalent to the number of analyzed recording pictures in real-time laboratory experiments, t exp is the total time (duration) of a numerical (or laboratory) experiment, and Δ c t is the time step with which the variation of the coordinates of individual particles is analyzed... Results of the Solution of the Inverse Problem and Their Analysis We solved the inverse problem for various systems (N p = 500, Γ* = 5 80, and ξ = ) of dust particles that interact with different pair potentials U(l). During the solution, we recovered the forces F (l) of pair interparticle interaction, electrostatic force of the trap F pt (r) (0), the friction coefficient of the particles ν fr, and two additional unknown parameters (x o, y o ), JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 0 No. 4 00

5 666 VAULINA et al. 0 8 g 0 9 Γ* = Fig.. Results of the recovery of the pair interaction force F and pair correlation function g( ) for N p = for various U and ξ: (, ) U /l and ξ = 0.; (, ) U = U c exp( ) and ξ 5; and (, ) U = U c exp( ) and ξ 0.. The filled symbols correspond to Γ* = 0, and the open symbols, to Γ* = 80. The heavy lines in Fig. a represent preassigned pair forces. which determine the position of the center of the trap. (The last two unknown parameters should be taken into account in the situation when the center of the trap does not coincide with the center of the system of particles.) Here we used different approximations for the pair forces F(l). As an illustration, we present four different approximations of F(l), namely, (A) approximation by inverse polynomials [formula (7), I p = 4 and b i = 0], (B) approximation by a nonlinear exponential function [formula (7), I p = 4 and a i = 0], (C) approximation by a combined function [formula (7), I p = 4, a i 0, and b i 0], and (D) approximation by splines [formula (9), l n + l n = l pm ]. Note that the first results of the numerical analysis of the inverse Langevin problem with the use of the approximation of pair interaction forces by inverse polynomials (case (A)) were reported in [4]. Nevertheless, such features of the correct solution of the inverse problem as the minimal duration of an experiment and the temporal and spatial resolution of the motion of particles that is necessary to determine the possibilities and limitations of the method in real laboratory conditions were not considered in [4] even for this approximation. The solution of the inverse problem for two interacting particles (N p = ) in an external linear electric field (F pt = d r) is illustrated in Fig. a for various parameters U, Γ*, and ξ. The use of different approximations ((A) (D)) for F(l), as well as an increase in the number of terms (I p > 4), did not appreciably improve the results of recovery of the unknown parameters of the system. The substitution of the approximation of pair interaction for expansion (A) by means of a Taylor series showed that, for the same accuracy of recovery of the unknown parameters of the inverse problem, the inversion requires at least ten terms (I p 0) and, accordingly, a larger number of equations (longer measurements). Moreover, it turned out that the Taylor expansion is hardly applicable to the recovery of the parameters of systems consisting of many particles (N p > 0). The analysis of the solution of the inverse problem for two interacting particles showed that the forces F (and, consequently, the pair potentials U) are in good agreement with the initially defined functions within the analyzed trajectories of particles. The deviation of the interparticle distance of two interacting particles from their most probable value is shown in Fig. b in the form of pair correlation functions for various nonideality parameters of the systems. Spatial range. The solution of the inverse problem in the case of N p = particles is shown in Figs. 4 7 for various approximations (A) (D) and various parameters Γ* and ξ. The particle displacements obtained by the direct solution of problem () were analyzed for each instant of time t m at each integration step Δ c t Δt. (Here Δt is the time step of solving the direct problem and Δ c t is the recording step of the particle coordinates in the numerical experiments.) In all these cases (when Δ c t = Δt), the error in determining the friction coefficient of particles and the parameters of the field of the trap was less than 5%. Figure 8 illustrates the recovery of the electrostatic force F pt of the trap (N p = 50, U /l ), which was initially defined by a linear function and a cubic polynomial. The recovered values of the pair forces correspond to the initially defined functions F(l) to within 5% in the spatial range l min l l max, where the lower bound l min JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 0 No. 4 00

6 ANALYSIS OF PAIR INTERPARTICLE INTERACTION ξ = 5 U l 0 0 ξ = 0. U l 0 ξ = 5 U l e ξ = U l ξ = 0. U l e ξ = U l e Fig. 4. Results of the recovery of the function F( ) by polynomials (A) for N p = 50 and 500 for various U, ξ, and Γ*: ( ) Γ* = 7.5 and ( ) 80. The lines represent preassigned pair forces. 0 9 ξ = U l 0 0 ξ = 0. U l 0 0 U l e ξ = 5 U l e Fig. 5. Results of the recovery of the function F( ) for N p = 50 and 500 for Γ* = 80 and various U and ξ by different approximations: ( ) A, ( ) B, and ( ) C. The lines represent preassigned pair forces. is determined by the condition g(l) 0, and l max depends on the numerical error of searching for the minimum rms deviation between the solutions of the direct and inverse problems and is bounded by the condition F(l pm )/F(l max ) With an appropriate choice of the approximating function (which admits a good extrapolation of F(l)), the spatial range of correct recovery of the spatial dependence of pair forces can be substantially higher than l min and l max (see Figs. 4, 5). Duration of experiment. The results of the recovery of pair forces by approximations (A) and (B) are close to those obtained in the case of (C). However, compared with (A) and (B), the use of approximation (C) for the correct recovery of the function F(l) in the same spatial range (l min l l max ) required a much larger number N c of equations of motion for each individual particle in a dust cloud. Here, N c = t exp /Δ c t and t exp is the total time (duration) of a numerical experiment. Note once again that, in real laboratory experiments, N c is equivalent to the number of pictures in a video JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 0 No. 4 00

7 668 VAULINA et al Fig. 6. Results of the recovery of the function F( ) Γ* = 80 and U = U c exp( )) by polynomial (A) for various N p, ξ, and N c : ( ) N p = 50, ξ = 5, and N c = 500; ( ) N p = 500, ξ = 5, and N c = 500; ( ) N p = 50, ξ = 5, and N c = 500; ( ) N p = 50, ξ = 0., and N c = 500; and ( ) N p = 50, ξ = 0., and N c = 500. The line represents a preassigned pair force. recording. The solution of the inverse problem (Γ* = 80, N p = 50, U = U c exp( )) for various values of N c is demonstrated in Figs. 6 and 7 for the function F(l) approximated by polynomials (A) and splines (D), respectively. In all the cases (A) (D), the analysis of the solution of the inverse problem shows that the number of recorded pictures necessary for the correct recovery of pair forces F(l) in the spatial range l l max must satisfy the condition ( l N c 0N max /l pm ) un , min{ ; ξ} where N un is the number of unknown parameters. The maximal length l max of recovery of pair forces, as well as the necessary number of pictures N c and the duration t exp of an experiment for the video recording rate of f vr = 00 s for various approximations of F(l) ((A) (D)) used when solving the inverse problem are shown in Table. One can see that, in the case of approximations (C) and (D), the necessary recording time t exp can be crucial for carrying out real experiments under stationary (stable) conditions in a laboratory plasma. Note that the necessary duration t exp of an experiment can be reduced by 0 0% if the center of the trap coincides with the geometrical center of the analyzed system of particles (i.e., if the coordinates x o and y o are known). We also considered other features of application of the method proposed to the diagnostics of plasma dust systems in laboratory experiments; namely, we considered the temporal and spatial resolution of the motion of particles, the visualization of a part of a dust 0 9 F pt, 0 dyn 9 6 ξ = Fig. 7. Results of the recovery of the function F( ) (N p = 50, Γ* = 80, and U = U c exp( )) by splines (D) for various ξ and N c : ( ) ξ 5 and N c = 000, ( ) ξ 5 and N c = 5000, ( ) ξ 5 and N c = 5000; ( ) ξ 0. and N c = 5000, and ( ) ξ 0. and N c = The line represents a preassigned pair force. 0 4 Fig. 8. Results of the recovery of the electrostatic force of the trap F pt (N p = 50 and U /l ) defined by ( ) a linear function and ( ) a cubic polynomial for various parameters ξ. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 0 No. 4 00

8 ANALYSIS OF PAIR INTERPARTICLE INTERACTION 669 Table. The maximal length l max of recovery of pair potential U(l) and the necessary number N c of pictures and the duration t exp of an experiment for a frame repetition rate of video recording of f vr = 00 s for various approximations of the function U(l) ((A) (D)) used when solving the inverse problem U(l) l l l l e l e l e 4 l max /l pm N c (A), N un = N c (B), N un = N c (C), N un = N c (D), N un = t exp (A), s t exp (B), s t exp (C), s t exp (D), s cloud, and the effect of an additional degree of freedom (displacement of particles in the direction perpendicular to the dust monolayer) on the recovery of the unknown parameters of particles. Temporal resolution. To analyze the effect of the frame repetition rate f vr of the video camera used for recording the motion of particles in laboratory experiments on the recovery of the unknown parameters of a dust cloud, we solved the inverse problem for the particle coordinates recorded in time periods of Δ c t = mδt, where m = 60. The accuracy of recovery of the o o friction coefficient ν fr /ν fr (here ν fr is the recovered value of the coefficient) decreased as the accuracy of resolution of their motion decreased (as Δ c t = mδt increased) by about from 5% (Δ c t = Δt) to 60 00% (Δ c t = 40 60Δt) (see Fig. 9). The accuracy of recovery of the pair interaction force and of the gradient of the trap field remained virtually unchanged (the change was less than 0%) for Δ c t max {ν fr, ω*}. A further increase in Δ c t to max { ν fr, ω* } max { ν fr, 0.5ω* } led to an increase in the error in the recovery of pair forces F(l) from 0% to 40% (see Fig. 0). Thus, the minimal value of the frame rate of video recording for min correct laboratory measurements is f vr = max{ν fr, ω*}. As a preliminary estimate for the characteristic frequencies ν fr and ω*, we can use either well-known theoretical models or independent methods of experimental diagnostics of the parameters of dust particles, for example, a method based on the analysis of masstransfer in small observation times [, ], or the measurement of the pair correlation functions and the diffusion coefficients of particles []. Spatial resolution. The effect of the spatial resolution on the results of measurements in a laboratory dusty plasma is associated with incorrect determination of the center of mass of the analyzed particles. Under typical experimental conditions, the spatial resolution δl ranges from 0.005l pm (small cluster systems) to 0.0l pm (large dust clouds with N p ). To estimate the necessary spatial resolution, we can use the relation δl T --- / < f M vr ω*l pm f π vr /. 4Γ* For a fixed spatial resolution δ l = δ, we can obtain the maximal value of the frame rate for the correct analysis of the motion of particles in experiments: f vr max < f vr = ω*δ π l /. 4Γ* Then, for ξ and δ l = 0.005, the range of frame rates min max is defined as [ ; ] = [ω*; ω*] for Γ* = 7.5 ν o fr /ν fr..6.0 f vr f vr Δ c t o Fig. 9. Errors in the recovery of the friction coefficient ν fr /ν fr (N p = 50) as a function of Δ c t for ( ) ξ = 5.0 and ( ) 0.. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 0 No. 4 00

9 670 VAULINA et al ξ = 5 U l 0 0 U l ξ = 0 0 ξ = 0. U l e ξ = 5 U l e 0 0 Fig. 0. Results of the recovery of the function F( ) (N p = 50 and Γ* = 80) for various U and ξ for ( ) Δ c t = 7Δt, ( ) 0Δt, ( ) 0Δt, and ( ) 5Δt. The lines represent preassigned pair forces. 0 0 Fig.. Results of the recovery of the function F( ) by polynomial (A) for a part of the dust cloud (see Figs. c and d) for U and ξ and ( ) Γ* = 7.5 and ( ) 80. The lines represent preassigned pair forces. min max and [ f vr ; f vr ] = [ω*; ω*] for Γ* = 80. When the frame rate of video recording satisfies the inequality max f vr > f vr, one should restrict the processing of experimental data to pictures that follow with a period of max min Δ c t > / f vr, which is obviously less than Δ c t = / f vr. Visualization of a part of a dust cloud. In real experiments with a dusty plasma, one often encounters a situation when only a part of a dust cloud falls into the field of view of a video camera. To simulate this situation, we used calculations performed for a dust cloud consisting of 500 particles. The analyzed part of the cloud contained from 0% to 0% of the total amount of particles. The fragments of the solution of the inverse problem for a part of a dust cloud are shown by dashed lines in Figs. c and d. The selected domain was partitioned into nine equal parts; the interaction between particles was considered only for the central cell (under the assumption that the interaction potential between the particles of the dust cloud falls off sharply enough at a distance of L c = L/, where L is the length of the side of the distinguished square domain). The solutions of this problem obtained with approximation (A) for various potentials and parameters ξ and Γ* are illustrated in Fig.. As a result of simulation, we found that the pair forces F recovered in this situation correspond to the initially defined functions in the regions l L c to within less than 5%. The presence of an additional degree of freedom. In conclusion, recall that (see Subsection.) the direct problem in the two-dimensional statement was solved for a monolayer of particles confined in an external electric field with cylindrical symmetry. Since the displacement of particles in the vertical direction (in the direction perpendicular to the monolayer) does not appreciably affect the transport characteristics of the particles, the degree of freedom in the vertical direction may lead to additional random error associated with incorrect determination of the distance between dust particles in the plane of the dust layer. These errors are analogous to errors due to the chaotic thermal displacement of particles and are also suppressed by solving a redundant system of equations. Note that, in the case of a multilayer dust system, which is often observed in laboratory experiments, one usually applies two-dimensional diagnostics, which detects the motion of particles within a single dust layer. In this case, the two-dimensional approach to the recovery of the parameters of the system leads, first of all, to the distortion of the parameters of the external field of the trap, because the action of the forces from the particles of a neighboring dust layer gives rise to an additional electric field in the radial direction.. EXPERIMENT.. Description of the Experimental Setup and the Results of Measurements A simplified scheme of the setup for testing the method proposed in a laboratory dusty plasma in the near-electrode area of high-frequency discharge is shown in Fig.. During the experiment, the vacuum chamber was filled with a noble gas, and voltage from a high-frequency oscillator with a carrier frequency of JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 0 No. 4 00

10 ANALYSIS OF PAIR INTERPARTICLE INTERACTION 67 Video camera HF (c) Particles Laser Ring Fig.. Simplified scheme of the setup for experiments in capacitive high-frequency discharge (HF)..56 MHz was applied to the electrodes; this generated a glow discharge between the electrodes in the argon atmosphere. Dust particles of various sizes and of different materials were injected from a special container into the discharge area through an aperture in the upper electrode. Having had entered the discharge area, the particles gained negative charge and hovered in the near-electrode region. To prevent the escape of particles in the horizontal direction, metal rings of different diameters D r and with a height of about 0. cm were mounted on the lower (grounded) electrode; these rings formed a potential trap for the cloud of dust particles. We used rings with various diameters (D r =.4 and 5.0 cm). This allowed us to diagnose both small-size clusters of particles (D r =.4 cm, N p = 6 0, and a 6.7 μm, where a is the radius of a dust particle) and rather large quasi-two-dimensional dust systems (D r = 5 cm, the number of particles in a monolayer N p 000, and a.755 μm). The spatial resolution δ l = δ ranged from δ l (for clusters) to δ l 0.0 (for large systems). The experiments were carried out in argon with pressure of P = Torr for a discharge power of W 0 W. As the dust component, we used monodisperse plastic spheres with density of ρ p.5 g cm and radii of a.75 and 6.7 μm. To visualize the process, we illuminated the dust cloud by a flat beam of a He Ne laser (λ = 6 nm; the characteristic thickness of the beam waist region was about 00 μm). The position of dust particles was recorded by a high-speed CMOS video camera (with a frame rate of f vc = s ). Each experiment lasted 8 4 s under stationary conditions. Video recording was processed by a special software, which allowed us to identify the positions of individual dust particles in the field of view of the video system. The observed dust structures represented crystalline or liquid systems with an average interparticle distance l p from 750 to 00 μm. Most experiments were carried out on monolayer (two-dimensional) systems. The processing of video data yielded the kinetic temperature T, the velocities and accelerations of the dust particles, their concentrations and pair correlation Fig.. Arrangement of dust clusters in the experiments; (a, b) dust clusters, (c) a fragment of a large dust structure (experiment, Table ). The solid lines show variants of the central cell inside the region of solution of the inverse problem for a part of the dust cloud. The symbol indicates the recovered position of the trap center. functions g(l), as well as the mass transfer functions D(t) = Δl /4t, where Δl is the rms displacement of dust particles. The arrangement of particles in different clusters (N p =, 9) is illustrated in Figs. a and b. Figure c represents a fragment of a video picture of a large dust cloud. (The total number of particles recorded by the video camera was about 550; an approximate number of particles in the entire dust layer was 000.) The pair correlation functions for various experiments are shown in Fig. 4. The parameters of the dust systems analyzed (a, l p, and V T = T/M) are shown in Table. This table represents the results of measurements of the friction coefficients of dust particles ν fr, the characteristic collision rate ω c = ( U'' ( l /πm) / p ) of particles, and the effective nonideality parameter Γ*, measured by a method based on the analysis of mass transfer D(t) for small observation times [, ]... Results of Recovery of Pair Interaction Forces and Their Discussion To determine the pair interaction forces F(l), we used four different approximations (A) (D) of the interparticle interaction forces, which are described in detail in Subsection.. A criterion for the correctness of the recovery was the coincidence of the functions F(l) obtained with the use of all the four approximations within the analyzed spatial range l. The results of the recovery of F(l) for various experiments are illustrated in Fig. 5. Figure 5a shows the results of recovery for small (cluster) systems performed with different approximations (A) (C) of pair interaction forces (symbols), and Fig. 5c shows similar results for large dust clouds for approximations (A) and (B). One can see that the results of the recovery of F(l) by different approximations differ by at most 5 0%. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 0 No. 4 00

11 67 VAULINA et al. g 5 g l/l p 0 4 l/l p Fig. 4. Pair correlation functions for different experiments (see Table ); for clusters and for large dust structures. To analyze how the behavior of the functions F(l) corresponds to the assumption of screened Coulomb interaction between dust particles in a plasma, we approximated the experimental data (shown by symbols in Fig. 5) by the curves f exp( κl/l p )( + κl/l p )l with different values of the distance l. We found that, for small cluster systems, the recovered potential well corresponds to the Coulomb interaction between dust particles (F(l) l ). A similar asymptotic behavior was obtained for large dust systems at distances of l > l p. The behavior of the recovered function F(l) at distances less than or about the average interparticle distance (l < l p ) is well described by the function f exp( l/l p )( + l/l p )l, which corresponds to the screened Coulomb potential with the screening parameter κ l p /λ =. Note that the power-law asymptotic behavior (F(l) l ) of interparticle interaction forces either can be explained by weak screening under the experimental conditions or confirm the Wigner Seitz-cell model, which assumes the Coulomb interaction between some effective charges of dust particles in the plasma at distances l greater than l p for nonideal dusty plasma systems in which Γ = (ez) /l p T [5, 5]. The characteristic frequency ω c = ( F '( l πm) / p ) obtained from the recovered functions F(l) is represented in Table. It is easily seen that this characteristic frequency ω c is in good agreement (within experimental errors of 0 5%) with the value of ω c obtained by an independent method [, ]. An exception is given by the measurements of a multilayer dust cloud (experiment 4 in Table ). In this case, due to the weak correlation of particles (Γ* 4.5), the particles of a different layer may fall into the field of view of the video system (see Fig. 4b). This situation does not tell on the results of the analysis of mass transfer D(t); however, in the case of the method proposed, this could lead to a significant distortion of the functions F(l) in the range of small interparticle distances l. The values of the characteristic frequencies (ω c and ω c ) were used to determine the charges Z of dust particles. In experiments,, and 4, it was assumed that the interaction of particles follows the Coulomb law Table. Parameters of particles for different experiments; () a cluster (N p = and P = 0.0 Torr), () a cluster (N p = 9 and P = 0.06 Torr), () part of a dust cloud, a monolayer (P = Torr), and (4) part of a dust cloud, a multilayer structure (P = 0.0 Torr) No. a, μm, mm s l p, mm ν fr, s ω c, s Γ* ω c, s Z/000 α, s α, s V T ± ± ± ± 0..5 ± ± ± ± ± ± ± ±.4 ± ± ± ± ± ± ± ± JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 0 No. 4 00

12 ANALYSIS OF PAIR INTERPARTICLE INTERACTION 67 F /M, dyn/g l, mm F /M, dyn/g l, mm 4 F /M, dyn/g (c) 4 l, mm Fig. 5. (a, b) Results of the recovery of the function F /M for various experiments (see Table ) by different approximations of interparticle interaction forces: ( ) A, ( ) B, and ( ) C. (c) Expanded fragment of Fig. 5b. Thin lines represent approximations of the experimental data by the curves f l, and the heavy line, by the curve f exp( l/l p )( + l/l p )l. The open and filled symbols for curves and 4 in Fig. 5c correspond to the solution of the inverse problem for various regions of the dust cloud (see Fig. c). (U''(l p ) = (ez) F '( l p ) / l p ), while, in experiment, the pair forces were defined in the Yukawa approximation: F () l ez ---- l --- κl = exp κl. To verify the assumption of pair interaction between dust particles in the experiments (i.e., to verify the potential character of the forces F(l)), we carried out the measurements for various samples of particles. For example, the inverse problem on a dust layer was solved for various numbers N p * of particles in the central cell of the analyzed picture of a video recording F pt /M, dyn/g l p l p l, mm Fig. 6. Results of the recovery of the electrostatic force of the trap F pt /M for experiments with clusters (see Table ) for linear (lines) and cubic (symbols) approximations (0). ( N p * ~ 0 50; Fig. c). In all the cases, the difference between the recovered functions F(l) was less than 5 7% (see Fig. 5b). The results of measuring the radial forces F pt /M acting on the dust particles in the electrostatic trap are presented in Fig.6 and in Table as α = d /M. We used both linear approximations of the function F pt (r) and higher order power-law polynomials I t = (0) for the recovery. We did not observe any noticeable effect of the degree of a polynomial on the results of determining the pair forces. The comparison of the values of α obtained here and the values of α, which determine the relation between the number of particles in a dust monolayer and the gradient of the field of the linear trap [6], π( ez) α , () ( N + )Ml p shows that these values are in good agreement (see Table ). Here, N R/l p (N p /π) / is the number of interparticle distances that lie within the radius R of the layer. To estimate α in a large dust monolayer (experiment ), we used the values of N p = CONCLUSIONS We have proposed a new method for determining pair interparticle interaction forces (and, accordingly, the pair potential). The method is based on the solution of the inverse problem describing the motion of interacting particles by a system of Langevin equations and allows one to recover the parameters of the external confining potential without referring to a priori information on the friction coefficients of the particles. The procedure proposed has been tested by numerical simulation of the problem in a wide range of conditions that are typical of experiments in a labora- JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 0 No. 4 00

13 674 VAULINA et al. tory dusty plasma. The results of the first experimental approbation of the method are presented as applied to the analysis of the interaction of dust particles in a laboratory capacitive high-frequency plasma. In contrast to earlier developed methods, the present method does not introduce perturbations into the system of particles, it is not based on any additional assumptions on the external forces or the relations between spatial correlation functions and the pair interaction potential, and it can be applied both to weakly correlated and strongly nonideal systems that consist of two or more interacting particles. In addition, we have considered some features of application of the method proposed to the diagnostics of dusty plasma systems in real laboratory experiments; these features are associated with the technical parameters of the video observation systems employed, such as the duration of an experiment, visualization of a part of a dust cloud, as well as the temporal and spatial resolution of the motion of particles. We have obtained empirical relations for determining the limitations and operation conditions of the method. In conclusion, we stress once again that, under the above-mentioned conditions, the method does not require any other information except the information on the coordinates and displacements of particles, which can easily be recorded in both numerical and real experiments. ACKNOWLEDGMENTS This work was supported in part by the Russian Foundation for Basic Research (project nos and ), by a CRDF grant (project no. RUP-89-MO-07), by the NWO (project no ), by the State Corporation on Atomic Energy Rosatom, by the Federal Agency on Science and Innovations (state contract no ), and by a program of the Presidium of the Russian Academy of Sciences. REFERENCES. A. A. Ovchinnikov, S. F. Timashev, and A. A. Belyi, Kinetics of Diffusion-Controlled Chemical Processes (Khimiya, Moscow, 986; Nova Science, Singapore, 989).. Photon Correlation and Light Beating Spectroscopy, Ed. by H. Z. Cummins and E. R. Pike (Plenum, New York, 974).. V. E. Fortov, A. V. Ivlev, S. A. Khrapak, A. G. Khrapak, and G. E. Morfill, Phys. Rep. 4, (005). 4. S. V. Vladimirov, K. Ostrikov, and A. A. Samarian, Physics and Applications of Complex Plasmas (Imperial College, London, 005). 5. I. T. Yakubov and A. G. Khrapak, Sov. Tech. Rev. B: Therm. Phys., 69 (989). 6. N. P. Kovalenko and I. Z. Fisher, Usp. Fiz. Nauk 0, 09 (97) [Sov. Phys. Usp. 5, 59 (97)]. 7. N. H. March and M. P. Tosi, Introduction to Liquid- State Physics (World Sci., Singapore, 995). 8. U. Konopka, L. Ratke, and H. M. Thomas, Phys. Rev. Lett. 79, 69 (997). 9. J. E. Daugherty, R. K. Porteous, M. D. Kilgore, and D. B. Graves, J. Appl. Phys. 7, 94 (99). 0. J. E. Allen, Phys. Scr. 45, 497 (99).. D. Montgomery, G. Joyce, and R. Sugihara, Plasma Phys. 0, 68 (968).. G. E. Morfill, V. N. Tsytovich, and H. Thomas, Plasma Phys. Rep. 9, (00).. S. V. Vladimirov and M. Nambu, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 5, R7 (995). 4. O. S. Vaulina, O. F. Petrov, A. V. Gavrikov, and V. E. Fortov, Fiz. Plazmy (Moscow) (4), (007) [Plasma Phys. Rep. (4), 78 (007)]. 5. V. E. Fortov, A. V. Gavrikov, O. F. Petrov, and I. A. Shakhova, Phys. Plasmas 4, (007). 6. V. E. Fortov, O. F. Petrov, and O. S. Vaulina, Phys. Rev. Lett. 0, (008). 7. V. E. Fortov, A. P. Nefedov, V. I. Molotkov, M. Y. Poustylnik, and V. M. Torchinsky, Phys. Rev. Lett. 87, 0500 (00). 8. V. E. Fortov, O. F. Petrov, A. D. Usachev, and A. V. Zobnin, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 70, (004). 9. J. B. Pieper and J. Goree, Phys. Rev. Lett. 77, 7 (996). 0. A. Homann, A. Melzer, and A. Piel, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 59, 85 (996).. O. S. Vaulina, O. F. Petrov, V. E. Fortov, A. V. Chernyshev, A. V. Gavrikov, I. A. Shakhova, and Yu. P. Semenov, Fiz. Plazmy (Moscow) 9 (8), 698 (00) [Plasma Phys. Rep. 9 (8), 64 (00)].. O. S. Vaulina and K. G. Adamovich, Zh. Éksp. Teor. Fiz. (5), 09 (008) [JETP 06 (5), 955 (008)].. O. S. Vaulina, K. G. Adamovich, O. F. Petrov, and V. E. Fortov, Zh. Éksp. Teor. Fiz. 4 (), 67 (008) [JETP 07 (), (008)]. 4. O. S. Vaulina and E. A. Lisin, Fiz. Plazmy (Moscow) 5 (7), 66 (009) [Plasma Phys. Rep. 5 (7), 58 (009)]. 5. J. M. Ziman, Models of Disorder (Cambridge University Press, New York, 979). 6. O. S. Vaulina, X. G. Koss (Adamovich), and S. V. Vladimirov, Phys. Scr. 79, (009). Translated by I. Nikitin JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 0 No. 4 00