Anomalously high kinetic energy of charged macroparticles in a plasma A. P. Nefedov, A. G. Khrapak, S. A. Khrapak, O. F. Petrov, and A. A. Samaryan Scientific Research Center for the Thermal Physics of Impulsive Interactions, Russian Academy of Sciences, 127412 Moscow, Russia

~Submitted 23 January 1997! Zh. E´ksp. Teor. Fiz. 112, 499–506 ~August 1997!

A number of recent experimental investigations of nonideal plasmas containing macroparticles have revealed an anomalous increase in that part of the kinetic energy of these macroparticles that corresponds to their random motion. In this paper a model is proposed for the dynamic motion of charged macroparticles that explains this phenomenon. Calculations based on this model are compared with experimental results. © 1997 American Institute of Physics. @S1063-7761~97!00808-1#

1. INTRODUCTION

Systems consisting of a plasma plus solid charged particles with sizes ranging from below a micron to several microns are commonly encountered in space ~planetary rings, interstellar clouds, comet tails, the ionosphere of the Earth, etc.!. Macroparticles are also observed in the lowtemperature laboratory plasmas used in plasma sputtering and etching, in the manufacture of microelectronic components, etc. The spontaneous generation of ordered structures made up of macroparticles observed in recent experiments on various types of laboratory plasmas1–7 implies that such systems can be used as macroscopic models of real microscopic crystalline structures. The obvious advantage of using a plasma with microparticles ~compared, e.g., with colloidal solutions! to model microsystems is the relative ease of obtaining and studying such systems, as well as the smallness of the times they require to relax to an equilibrium state and respond to changes in external parameters. These facts explain the recent surge of interest in studying the properties of plasmas with macroparticles. One question that has been intensely studied recently concerns the dynamic behavior of the macroparticles. In a low-temperature weakly ionized laboratory plasma, macroparticles having a large surface area efficiently exchange energy with atoms of the neutral component. Therefore it is customary to assume that the macroparticles are characterized by the temperature of the neutral gas, and that their motion is subject to the laws of Brownian motion. A convenient way to clarify the nature of the behavior of these particles qualitatively is to introduce the autocorrelation function of the particle velocity F( t )5 ^ v (0) v ( t ) & , where the angle brackets denote averaging over an ensemble. The simplest theory of Brownian motion that does not include hydrodynamic interactions ~which are important only in extremely viscous media! predicts exponential decay of F~t! with a characteristic time t s , 8 defined as the time for slowing down of a macroparticle due to the viscosity of the medium: t s 5M / b ~here M is the mass of the macroparticle and b is its coefficient of friction in the medium!. Thus, F ~ t ! 5 ^ v 2T & exp~ 2 t / t s ! .

~1!

Here ^ v 2T & 53T n /M is the mean square thermal velocity of 272

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the particle ~where T n is the temperature of the neutral gas in energy units!. The mean square displacement of the particle within a time t, in accordance with Ref. 9, is given by the expression

^ Dr 2 ~ t ! & 52

E

t

0

~ t 2t ! F ~ t ! dt

H F S D GJ

52 ^ v 2T & t s t 1 t s exp 2

t 21 ts

.

~2!

From Eq. ~2! it is easy to determine the value of the mean square displacement of a macroparticle in two limiting cases. For t @ t s we have

^ Dr 2 ~ t ! & 52 ^ v 2T & t s t 56D 0 t ,

~3!

where D 0 5(1/3) ^ v 2T & t s 5T n / b is the diffusion coefficient for a Brownian particle. In the opposite case t ! t s

^ Dr 2 ~ t ! & 5 ^ v 2T & t 2 .

~4!

Equations ~3! and ~4! reflect the diffusive character of the macroparticle motion at large times and its essentially ballistic character for small times. The most recent investigations of an ideal plasma with macroparticles show that under certain conditions the temperature can exceed the temperature of a neutral gas by factors of a hundred and even a thousand ~here and in what follows we understand the macroparticle temperature to mean the temperature corresponding to the kinetic energy of their random motion!.7,10–12 This phenomenon is observed at low pressures and has not been explained. The temperature itself is determined in the following way: the motion of the macroparticles in a horizontal plane is recorded by a video camera, and then the appropriate processing of the video image is carried out. Based the projection of the mean square motion of the particle in a prespecified direction within the time t between frames, the mean-square velocity is calculated to be ^ v 2x & 5(Dx 2 )/ t 2 . The temperature is introduced through the relation 2(T mes /2)5(M /2)( ^ v 2x & 1 ^ v 2y & ). Of course it goes without saying that this procedure itself needs an appropriate justification. Thus, even for ordinary noninteracting Brownian particles the interpretation of the experi-

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mental results depends on the ratio of the braking time of a particle in the medium t s to the time interval between frames t ~see Eqs. ~3! and ~4!!. In this paper we present a model of the dynamic behavior of charged macroparticles in a plasma that can explain the considerable departure of the temperature of dust particles from that of the neutral component as the pressure in the system falls. We will compare the results obtained in the framework of this model with the experimental results available at the present time. The conditions for its applicability are also discussed.

2. DYNAMIC BEHAVIOR OF CHARGED MACROPARTICLES IN A PLASMA

Consider a system consisting of negatively charged macroparticles with a charge Z ~in units of the electron charge!, electrons, singly charged positive ions, and neutral gas. Let n p ,n e ,n i ,n n be the corresponding concentrations of macroparticles, electrons, ions, and neutral gas atoms. The system is assumed to be quasineutral, so that Zn p 1n e 5n i .

~5!

Moreover, we will assume that the plasma is weakly ionized, i.e., n n @n e ,n i . We further assume that each of the subsystems of electrons, ions, and neutral gas are in a state of thermodynamic equilibrium with temperatures T e , T i , and T n respectively. Since the mobility of electrons and ions considerably exceeds that of the macroparticles, the distribution of electrons and ions in a quasi-uniform plasma will adiabatically follow the distribution of macroparticles. When a small perturbation in the macroparticle density n p (r) arises, an electric field E52¹ w appears that attempts to return the system to a uniform distribution. Let us write Boltzmann distributions for the electrons and ions:

F G F G

n e ~ r ! 5n 0e exp

ew~ r ! , Te

n i ~ r ! 5n 0i exp 2

~6!

ew~ r ! . Ti

~7!

Combining Eqs. ~6! and ~7! with the condition of quasineutrality ~Eq. 5!, we obtain

S

Z n 0i n 0e E5 1 e Ti Te

D

21

¹n p .

~8!

The macroparticle flux is written in the form ZeE n p 2D 0 ¹n p . I p 52 3

F

I p 52D 0 11

Z 2 n 0p

G

Ti ¹n p . T n ~ 11T i /T e ! n 0e 1Zn 0p

JETP 85 (2), August 1997

where the diffusion coefficient for the charged macroparticles has the form

F

D p 5D 0 11

G

Z 2 n 0p Ti . T n ~ 11T i /T e ! n 0e 1Zn 0p

~10!

~12!

Obviously, the mechanism we are discussing is analogous to ambipolar diffusion in an electron–ion plasma. Note that, whereas in an ordinary low-temperature plasma the coefficient of ambipolar diffusion can be significantly larger than the diffusion coefficient of ions only in the presence of hot electrons, in the present case we may expect a significant increase in D p compared to D 0 under certain conditions even for an isothermal plasma (T e 5T i 5T n ) due to the large value of the charge Z. It is interesting to consider various limiting expressions for the diffusion coefficient of charged macroparticles in a plasma. Setting Z50, we have D p 5D 0 , as we should expect for uncharged macroparticles. Reducing n p to zero, we also obtain D p 5D 0 ; this shows that the increase in the macroparticle diffusion coefficient is essentially a collective effect. It is also easy to obtain an expression for D p in an isothermal system consisting of positively charged macroparticles, electrons emitted by them, and neutral gas. This expression was discussed in Refs. 13 and 14 ~in this case we simply need to replace Z by 2Z!. In complete agreement with the results of Refs. 13 and 14 we obtain D p 5D 0 (11Z). It is well known that D p can exceed D 0 in dilute suspensions of interacting particles,9 which in most ways are analogous to systems of plasmas with macroparticles. The dynamic interactions of macroparticles can be represented approximately in the following way: let us write the velocity of each macroparticle in the form of a sum v p 5 v T 1 v d , where v T is the velocity connected with imbalance in collisions with atoms of the surrounding medium ~the ‘‘Brownian’’ force!, while v d is the velocity caused by the interaction between the macroparticles ~the drift velocity in the field that arises as the macroparticle density is perturbed!. Let us also introduce the the drift velocity correlation time t I . Under the assumption that v d and v T are not correlated, we can write by analogy with Eq. ~1!9 F ~ t ! 5 ^ v 2T & exp~ 2 t / t s ! 1 ^ v 2d & exp~ 2 t / t I ! .

~13!

For t @ t I , Eq. ~2! implies the following expression for the mean square displacement of the macroparticles: ~14!

where 1 D c 5 ^ v 2d & t I 3

After taking the divergence of both sides of Eq. ~10! and using the equation of continuity for the macroparticle density, we finally obtain a diffusion equation 273

~11!

^ Dr 2 & 56 ~ D 0 1D c ! t 56D p t , ~9!

Substituting ~8! into ~9! we obtain

]np 1D p ¹ 2 n p 50, ]t

~15!

is the so-called coefficient of collective diffusion, whose appearance is explained by the interaction between the macroparticles. For another limiting case t ! t s we have

^ Dr 2 & 5 ~ ^ v 2T & 1 ^ v 2d & ! t 2 .

~16! Nefedov et al.

273

It is obvious that correctly defining the time t I presents some problems. Here we define it as the time required to travel the average distance a between macroparticles in the plane of observation while moving at the drift velocity:

t 2I ~ ^ v 2dx & 1 ^ v 2dy & ! 5a 2 .

~17!

In suspensions of interacting particles the difference between t s and t I is very large, four to five orders of magnitude for particles of micron size.9 Therefore, although D c can greatly exceed D 0 , the following condition usually holds:

^ v 2d & ! ^ v 2T & .

~18!

In this case the average kinetic energy of the macroparticles K5

M ^ v 2d & 3 T n1 2 2

~19!

is determined exclusively by the temperature of the surrounding medium. In a plasma with macroparticles no such difference between t s and t I can arise, so that Eq. ~18! may not be satisfied and the subsystem of macroparticles will not be in thermodynamic equilibrium with the neutral component. If the temperature of the subsystem of macroparticles is defined through the relation ¯ K 53T p /2, after rewriting Eq. ~12! in the form D p 5D 0 1D c , using Eq. ~15!, and the definition of t I in Eq. ~17!, we obtain T p 5T n 1

S D

2M D 20 D c a2 D0

2

,

~20!

where in agreement with Eq. ~12! the ratio D c /D 0 is defined by the expression Z 2 n 0p Dc Ti 5 . D 0 T n ~ 11T i /T e ! n 0e 1Zn 0p

~21!

Under certain conditions the second term on the right side of Eq. ~20! can exceed T n . This also will correspond to a difference between the temperatures of the macroparticles and the neutral component. This difference arises from the internal fields that appear in a low-density system due to the spatial separation of charges. As the system pressure decreases, D 0 increases, which qualitatively explains the phenomenon of increased macroparticle kinetic energy observed in Refs. 7 and 10–12. 3. COMPARISON OF THE MODEL WITH EXPERIMENT

For a more detailed comparison of this model with experimental results we will use the paper by Melzer et al.,10 which contains the most complete information on the parameters of our system. In this paper the authors observed the formation of crystal-like macroparticle structures in an rf He plasma at low pressures. The crystal structure forms near the boundary of the cathode sheath, where the force of gravity is balanced by an electric field. The macroparticles used in these experiments were spherical monodispersed particles of radius R54.7 m m and mass M 56.7310210 gm. Melzer et al. observed a phase transition from a crystalline state to a gaslike state as the pressure of the gas dropped from 120 to 40 Pa at constant discharge power. They explained the phase transition by invoking an increase in the effective macropar274

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FIG. 1. Macroparticle temperature as a function of system pressure: d—values obtained in experiment; s—calculated using the model given here

ticle temperature, which in turn decreases the interaction parameter G, defined as the ratio of the electrostatic interaction energy between neighboring particles to their kinetic energy: G5Z 2 e 2 /aT p . For values of the interaction parameter larger than a critical value G c the system should be in the crystalline state, while as G decreases the system of macroparticles enters a liquid state. Debye screening in the plasma causes the value of G c to depend on the ratio of the averaged interparticle distance to the Debye radius: k5a/l D . The case k 50 corresponds to the simplest model of a single-component plasma. In this model G c .170. Our scheme for measuring the particle temperature was described in the Introduction. In our model, this approach is correct only in the limit t ! t s , in which case the real kinetic energy of the macroparticles is measured according to Eqs. ~16! and ~19!. The time between frames in these measurements was t 520 ms. Figure 1 shows the temperature of the macroparticles measured in this way for various values of the pressure. The horizontal straight line in Fig. 1, which corresponds to a temperature of ; 0.7 eV, is the smallest value of macroparticle temperature that can be determined by this video apparatus. This instrumental limit is connected with the discrete structure of the video image.10 Let us now discuss our choice of system parameters to be substituted into Eq. ~20!. We take T e @T n 5T i 5300 K. The average distance between macroparticles is practically independent of the pressure and equals a5450 m m, 10 which corresponds to a macroparticle density n p 5a 23 '104 cm23. In the range of pressures under study, the mean free path of He atoms greatly exceeds the radius of the macroparticles. Correspondingly, the coefficient of resistance of the medium is determined ~under the assumption of complete accommodation! by the expression15

b5

S D A

4p p 2 41 R P 3 2

mn , 2pTn

~22!

where P is the neutral gas pressure and m n is the mass of the He atom. Now, by substituting into Eqs. ~20! and ~21! the macroparticle charge and ion concentration obtained in experiment at various pressures, we obtain the dependence of the macNefedov et al.

274

roparticle temperature on pressure ~within the framework of our model! that is applicable to the conditions of the experiment in Ref. 10. This dependence is also shown in Fig. 1. The quantity t I 'a 2 /2D c obviously is a minimum at the lowest pressure in the system, and comes to ; 0.2 s at 40 Pa, while t s in this range of pressures varies from 20 to 60 ms. Thus, the condition t s . t is fulfilled; consequently, the procedure applied to measure the kinetic energy of the macroparticles is correct. The results of calculations using our model are found to be in satisfactory agreement with the experimental results. It is unreasonable to expect more precise quantitative agreement for a large number of reasons. First of all, we note that the discussion given in this paper of the dynamic behavior of macroparticles is correct when the parameter G is not too large ~i.e., for gaslike and liquidlike states of the macroparticle system!, which corresponds to the region of low pressures in the experiments under study. In this sense, the use of Eq. ~20! for all ranges of pressure may be considered to be an extrapolation of the model outside the limits of its region of applicability. The good qualitative agreement of the calculated macroparticle temperature with the experimental results at low pressures and its merely qualitative agreement at higher pressures ~see Fig. 1! is also a reflection of this fact. Some of the discrepancy between the model and experiment can also be explained by experimental error in determining the macroparticle charge and ion concentration. For example, uncertainties in the charge measurements due to the method used in these experiments can be up to 50 leads to still larger uncertainties in the macroparticle temperature calculated using Eq. ~20!. The error corresponding to this is shown in Fig. 1 for one of the calculated points. Finally, we cannot avoid pointing out that there is a certain arbitrariness in the definition of t I . Besides the considerations mentioned above, the following circumstance suggests that certain refinements are needed. Our observations during the particle motion are made near the boundary of the cathode sheath ~a layer of positive bulk charge and electric field!. In this region it is possible that the quasineutrality condition Eq. ~5! is not strictly satisfied. Moreover, in this region T i can exceed T n somewhat due to ion drift towards the cathode in the electric field. Figure 2 shows the dependence of the interaction parameter G on pressure in a system with temperature T p calculated according to Eq. ~20!. This function clearly demonstrates that, despite some increase in the interaction between dust particles as the neutral gas pressure decreases ~their charge increases while the screening decreases!, the interaction parameter rapidly decreases due to the increase in macroparticle kinetic energy. This behavior of the parameter G is in complete agreement with the melting of the quasicrystalline structure observed in the experiments as the pressure falls in the discharge from 120 to 40 Pa.

275

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FIG. 2. Dependence of the interaction parameter G between macroparticles on system pressure

4. CONCLUSION

In this paper we have described a model for the dynamic behavior of macroparticles in a plasma that explains the anomalous increase in macroparticle kinetic energy observed in experiments at low pressures of the neutral component. The reason for this increase in temperature is the drift motion of dust particles in the internal fluctuating fields. The model described here is in satisfactory agreement with the experimental results. Possible reasons for certain discrepancies between calculated and experimental points have been discussed. This paper was partially supported by the Russian Fund for Fundamental Research ~project No. 97-02-17565!. J. Chu and I. Lin, Phys. Rev. Lett. 72, 4009 ~1994!. H. Thomas, G. E. Morfill, V. Demmel et al., Phys. Rev. Lett. 73, 652 ~1994!. 3 A. Melzer, T. Trottenberg, and A. Piel, Phys. Lett. A 191, 301 ~1994!. 4 Y. Hayashi and K. Tachibana, Jpn. J. Appl. Phys., Part 1 33, 804 ~1994!. 5 V. Fortov, A. Nefedov, O. Petrov et al., Phys. Lett. A 219, 89 ~1996!. 6 V. E. Fortov, A. P. Nefedov, O. F. Petrov et al., JETP Lett. 63, 187 ~1996!. 7 V. E. Fortov, A. P. Nefedov, V. M. TorchinskiŽ et al., JETP Lett. 64, 92 ~1996!. 8 P. A. Weitz, D. J. Pine, P. N. Pusey et al., Phys. Rev. Lett. 63, 1747 ~1989!. 9 P. N. Pusey and R. J. A. Tough, in Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy, R. Pecora ed. ~Plenum Press, New York, 1985!, p. 85. 10 A. Melzer, A. Homann, and A. Piel, Phys. Rev. E 53, 2757 ~1996!. 11 H. M. Thomas and G. Morfill, Nature ~London! 379, 806 ~1996!. 12 J. B. Pieper and J. Goree, Phys. Rev. Lett. 77, 3137 ~1996!. 13 D. I. ZhukovitskiŽ and I. T. Yakubov, Teplofiz. Vys. Temp. 23, 842 ~1985! @in Russian#. 14 I. T. Iakubov and A. G. Khrapak, Sov. Technol. Rev. B: Therm. Phys. 2, 269 ~1989!; D. I. ZhukovitskiŽ, A. G. Khrapak, and I. T. Yakubov, in the book: Plasma Chemistry, Vol. II, p. 130 @in Russian#, B. M. Smirnov ed. ~Energoizdat, Moscow, 1984!. 15 E. M. Lifshits and L. P. PitaevskiŽ, Physical Kinetics, p. 86 ~Pergamon Prtess, Oxford, 1981! @Russ. original Nauka, Moscow, 1979#. 16 Th. Trottenberg, A. Melzer, and A. Piel, Plasma Sources Sci. Technol. 4, 450 ~1995!. 1 2

Translated by Frank J. Crowne

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