J. Plasma Fusion Res. SERIES, Vol.2 (1999) 173-176

Simulation Studies on Grain
Charging / Coagulation in Dusty Plasmas
WATANABE Kunihiko, NISHIMURA Kazumi, ISHIGURO Seiji and SAT0 Tetsuya
National Institute for Fusion Science, Toki' 509-5292 Japan(Received: 24 December 1998 / Accepted: 15 April 1999)

Abstract
We have developed a new particle simulation code for grains which are dynamically charged by thebackground plasma. Here, not only the regular attachment effect of the electrons and ions, but also theeffects of the secondary electron emission and the coagulation of grains are taken into account.
Simulation results show that all the grains change their charge randomly from negative to positive, orpositive to negative in the way of "flip-flop" as the time passes by. The flip-flop effect is outstanding
when the radius of the grains is of the order of 10nm, and such grains with opposite charges coagulate

with each other to grow.

Keywords:
dust plasma, simulation, charging, coagulation, flip-flop effect

1. lntroductionof the ordered structure formation from the viewpoint of

The dust grains are found in planetary plasmas.
They are considered to play an important role in theformation of the structure, e.g., the rotating radial

developed a dynamical charging scheme and a new dust

"Self-organization". For this purpose, we haveparticle simulation code in which a dynamic charging
process of the grains including the secondary electron
emission effect and the coagulation effect between thegrains are taken into account.

spokes of the Saturn's rings. The dust grains also
become important and interesting in various processing
plasmas.
The dust grains are usually considered to be

2. Simulation Schemecharged negative because electrons of the backgroundplasma attach to the grains more frequently than ions

The average number of attachment per unit time isgiven by the background plasma distribution function.

due to the difference between their thermal speeds.
Meyer-Vernet, however, pointed out that it is possible
for the grains to change their charges from negative topositive or from positive to negative in the way of "flipflop", when the secondary electron emission effect fromthe grains is taken into account [1]. His theory is staticone, but it can be extended to the dynamic one. In thisstudy, we are investigating the dynamic charging
process of the dust grains and the detailed feature of the

if -q"Q> 0,
No=44td'nokTo
21T moll - q"0l kT"l, (1)if -qoQ< O,

No=4naz no{#"

exp r-eoOtkr"t, Q)

where { and a are the surface potential and the radius ofcharging and coagulation processes.
Our final goal is to clarify the physical mechanism@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchC o r r e sp o ndin g autho r' s e - mai I : kuni @ nifs. ac.j pr73

watanabe K. et al., Simulation studies on Grain charging / coagulation in Dusty plasmas

the grain, respectively, and a indicates electron or ion,Q1/r' exp {-r/rol.

fra, ea, To, nra are the density, charge, temperature and(s)mass of o species, respectively. Here, the backgroundplasma distribution is assumed to be Maxwellian.

the j-th grain, respectively.

The attachment of electrons and ions to dust grains
is considered to occur according to Poisson process [2].

the center of the simulation box randomly without

Here, Q, and 16 are the charge and the Debye length of
The dust grains, which are initially located around

Therefore, using above average number, namely, theaverage probability for Poisson process, the timeintervals of the attachment to each grain are determinedvelocity and charge, are charged due to the backgroundplasma and moved by the electrostatic potential field inby the random number at every time step in thepasses by.

simulation.

The flow chart of the simulation procedure isshown in Fig. 1, where r(i), v(i), Q(i), tjll6 and are,respectively, the position, velocity, charge and the time
when the electron or ion attachment to the j-th grain
occurs next, and Nlon and N"1" are the total number of theions and electrons of the background plasma.

the 2-dimensional open boundary system, as the time

In addition to the above regular attachment effect,
the secondary electron emission is taken into accountwhen the kinetic energy of the colliding electron islarger than a certain critical value, Es. The number ofthe secondary electrons is given by [3]1g..=61r) -A E expt-2{EtEM),(3)

3. Simulation Results

if Q> 0,N.. = d(E)exp' t!4 t[.\t -g4 \.

kr,kr,lThe temporal charge evolutions of 6 dust grains in
the typical simulation run are shown in Fig. 2. Eachpanel corresponds to the each grain, and the charge istaken in the vertical axis from -2OO e to +2OO e.
It is clearly seen that the charges of all the grainschange randomly from negative to positive, or positive
to negative in the way of "flip-flop" as the time passes
by. The lowest (negative) charge is determined by thebalance between the attachment of the electrons andions, while the highest (positive) charge is determined(4)

where E is the kinetic energy of the colliding electron; Aand .EM are constants; Z" is the temperature of thesecondary electrons.

Thus, the number of the electrons of the
background plasma decreases due to the attachment tograins, and increases due to the secondary electronemission from the grain.

by the balance between the electron attachment and thesecondary electron emission.
When the grain size is small enough, all the grains

When the distance between the dust grains with
opposite charge becomes less than a critical length, r., it
is considered in this simulation model that they makecoagulation and the change of the mass and radius of thecoagulated grain as well as the change of the charge will
occur. Namely, the mass and charge of such a grain isgiven by the summation of the coagulating grains, andthe radius becomes large as to be proportional to 1/3
powers of the coagulated grain's mass.
It should be noticed that the rates of electron (ion)attachment, Eqs. (l) and (2), are depending on thestay negatively charged as is shown in Fig. 3. The
reason is as follows. Since the grain size is small, there

are not so many electrons which attach to the grains.

Then, the high energy electrons which can makesecondary electron emission seldom collide with thegrarns.

In contrast with the small grain case, when thegrain size is large enough, many electrons collide withthe grains. Under the simulation parameters used here,
the grains are positively charged which is shown in Fig.

potential and radius of each grain as well as thebackground plasma density. Therefore, depending on theAconditions both of the grains and of the backgroundplasma due to the effects of the attachment, secondary
electron emission, and coagulation, the probability of
the attachment itself is modified, In other words, thecharging process is a nonlinear feedback system inIn the small and large grain cases, the grains, all ofwhich are charged in negative or positive respectively,
are repulsing each other to be scattered away from thesimulation box, and thus, the simulation is ended at theearlier time.

When the grain size is appropriate, both thepositively and negatively charged grains can exist

itself.
The electrostatic potential field given by each grain
at the distance of r is proportional tosimultaneously. Thus, such grains gather to coagulatet74

Watanabe K. et al., Simulation Studies on Grain Charging / Coagulation in Dusty Plasmas

other grain, the coagulated grain plays a role of a
nucleus for more coagulation. In the final stage of thisdue to the attractive force between them.

In Fig. 5, the temporal evolutions of the spatial
distribution of grains are shown. In the earlier stage, thegrains which are located around the center of thesimulation box are going toward the boundary (leftpanel). As the time goes, the grains are moving slowly
towards the outer region of the system (center panel)'and in the final state of the simulation run (right panel)'the grains, especially, those around the center of thesystem, coagulate with each other. The position of thegrains are indicated by "o" (the circle becomes large as
the coagulation proceeds, but does not correctly reflect
the size of the grain, which is much smaller).
Since the mass of the grain becomes larger thanthose of the others when the grain coagulates with thesimulation run, the largest grain consists of about 50grains. Thus, the mass becomes 50 times larger than theoriginal one, while the grain size becomes 3.7 times
larger.

4. Summary
We have developed a new particle simulation codefor dust grains which are dynamically charged by thebackground plasma. Here, not only the regularattachment effect of the electrons and ions' but also theeffects of the secondary electron emission and thecoagulation of grains are taken into account.
If the grain size is appropriate all the grains changer(j), Y0), Q(i), t.rchu) Nt t = t0

l=t+Atl(instic Energy Er

lectf0n 0l

Secondary El€ctrors ilsr

l!ctnc ti!ld: Q(J) + (il$-l) 'll.r.=l{.,.+il5E-lllert AttNchreni Tine t.,.r(j)f I r(i)-r{n l<r

Fig. 1 The flow chart of the simulation procedure

Fig. 2 The temporal charge evolution of the grainst75

watanabe K. et al., simulation studies on Grain charging / coagulation in Dusty plasmas

their charge randomly from negative to positive, or
positive to negative in the way of ..flip-flop" as the time
passes by. Such oppositely charged grains make aa significant role in this simulation, the precise treatment

is quite important. For the number of secondary
emission electrons, we employed the formula given bycoagulation to grow. In the typical simulation

Sternglas [3]. It is pointed out by Chow that theparameters, the "flip-flop" effect, and hence, coagulation

secondary electron emission effect would be enhanced
more than that given by Sternglas (Eqs. (3) and (4)),

is outstanding when the grain size is of the order ofl0nm. When the grain becomes large enough, the grainstops growing due to the following reasons. First, since

when the grain size is of the order of l0nm as is treatedhere [4]. If this enhancement is taken into account, thethe mass of the large grain is large, the coagulation
between the large grains is almost impossible, and thecoagulation between the small and large grains can notchange much the size of the large grain. Second reason

flip-flop effect shown in the present work becomes more
active for the wider range of grain parameters.

is that the attachment of single electron to the large
grain does not affect much on the surface potential of
the grain, and the "flip-flop" effect can nor play at1l N. Meyer-Vernet, Astron. Astrophys l0S, 98(r98 .

References

l2l C.Cui and J. Goree, IEEE Trans. Plasma Phvs. 22.

significant role.

r5r (1994\.

Therefore, there is a suitable grain size for the rapid
growth of the grains. This fact shows a good agreement[3] E.J. Sternglas, Sci. Pap. lTT2,Westinghouse Res.

with the experiments, in which grains grow farstest

t4l V.W. Chow, Dust-Plasma Interactions, University

when the grain size is 10nm, and stop growing when thegrain size becomes 100 - 200nm [5].

of Califomia, San Diego (1996),
t5l Y. Watanabe et al., J. Vac. Sci. Tech. Al4. 540

Lab., Pittsburgh (1957).

Since the secondary electron emission effect plays(l996).

Fig. 3 The temporal charge evolution of the small grains

Fig. 4 The temporal charge evolution of the large grains

t= 1X lOtt-

t= IX l0e

5X l0tFig.5Thespatial distributionof grainsatt=0,t=5x108 a-|",andt=1xlOe a'|".fhecirclebecomeslargeasthe
coagulation proceeds.

r76