Coupling parameter for low-temperature plasma with condensed phase
The conditions of formation of ordered structures in low-temperature plasma with the condensed disperse phase are studied. Various modifications of the coupling parameter for polydisperse systems of condensed grains are proposed.
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Інститут фізики конденсованих систем НАН України
2007
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| Цитувати: | Coupling parameter for low-temperature plasma with condensed phase / V.I. Vishnyakov, G.S. Dragan // Condensed Matter Physics. — 2007. — Т. 10, № 2(50). — С. 201-208. — Бібліогр.: 23 назв. — англ. |
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irk-123456789-1183772017-05-31T03:05:02Z Coupling parameter for low-temperature plasma with condensed phase Vishnyakov, V.I. Dragan, G.S. The conditions of formation of ordered structures in low-temperature plasma with the condensed disperse phase are studied. Various modifications of the coupling parameter for polydisperse systems of condensed grains are proposed. Дослiджено межi застосування параметра неiдеальностi та його вираз для опису фазових переходiв в упорядкованих структурах заряджених конденсованих часток в плазмi з конденсованою дисперсною фазою. Показана доцiльнiсть використання виразу для параметру неiдеальностi iз застосуванням значення потенцiалу на границi комiрки Вiгнера-Зейтца, отриманого з розв’язку електростатичної задачi, що вiдповiдає виду i складу плазми. Отримано вираз параметра неiдеальностi для полiдисперсної конденсованої фази з електростатичною енергiєю мiжфазної взаємодiї у виглядi добутку узагальненого потенцiалу плазми i об’ємного заряду газової фази плазми в дебаєвськiй сферi. 2007 Article Coupling parameter for low-temperature plasma with condensed phase / V.I. Vishnyakov, G.S. Dragan // Condensed Matter Physics. — 2007. — Т. 10, № 2(50). — С. 201-208. — Бібліогр.: 23 назв. — англ. 1607-324X PACS: 52.27.Gr, 52.27.Lw DOI:10.5488/CMP.10.2.201 http://dspace.nbuv.gov.ua/handle/123456789/118377 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The conditions of formation of ordered structures in low-temperature plasma with the condensed disperse
phase are studied. Various modifications of the coupling parameter for polydisperse systems of condensed
grains are proposed. |
| format |
Article |
| author |
Vishnyakov, V.I. Dragan, G.S. |
| spellingShingle |
Vishnyakov, V.I. Dragan, G.S. Coupling parameter for low-temperature plasma with condensed phase Condensed Matter Physics |
| author_facet |
Vishnyakov, V.I. Dragan, G.S. |
| author_sort |
Vishnyakov, V.I. |
| title |
Coupling parameter for low-temperature plasma with condensed phase |
| title_short |
Coupling parameter for low-temperature plasma with condensed phase |
| title_full |
Coupling parameter for low-temperature plasma with condensed phase |
| title_fullStr |
Coupling parameter for low-temperature plasma with condensed phase |
| title_full_unstemmed |
Coupling parameter for low-temperature plasma with condensed phase |
| title_sort |
coupling parameter for low-temperature plasma with condensed phase |
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Інститут фізики конденсованих систем НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/118377 |
| citation_txt |
Coupling parameter for low-temperature plasma with condensed phase / V.I. Vishnyakov, G.S. Dragan // Condensed Matter Physics. — 2007. — Т. 10, № 2(50). — С. 201-208. — Бібліогр.: 23 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT vishnyakovvi couplingparameterforlowtemperatureplasmawithcondensedphase AT dragangs couplingparameterforlowtemperatureplasmawithcondensedphase |
| first_indexed |
2025-07-08T13:47:37Z |
| last_indexed |
2025-07-08T13:47:37Z |
| _version_ |
1837086760493907968 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 2(50), pp. 201–208
Coupling parameter for low-temperature plasma with
condensed phase
V.I.Vishnyakov, G.S.Dragan
Mechnikov Odessa National University, Odessa 65026, Ukraine
Received March 6, 2006, in final form May 18, 2006
The conditions of formation of ordered structures in low-temperature plasma with the condensed disperse
phase are studied. Various modifications of the coupling parameter for polydisperse systems of condensed
grains are proposed.
Key words: low-temperature plasma, condensed grains, plasma crystal
PACS: 52.27.Gr, 52.27.Lw
1. Introduction
The detection of plasma crystals in the dusty plasma [1,2], and the ordered structures of the con-
densed grains in the plasma of combustion products (smoky plasma) [3,4] stimulated the research
of mechanisms of the charged grains interaction in the low-temperature plasma and definition of
the criteria parameters of phase transition. Thus it has been found out that the formation of the or-
dered structures of dust grains occurs in strongly coupled (nonideal) plasma. It has been suggested
to use a Coulomb coupling parameter Γ, defined as the ratio of the potential energy of Coulomb
interaction between neighboring particles and their kinetic temperature, in order to distinguish the
weakly coupled plasma and the strongly coupled plasma [5–11].
The application of such a form of coupling parameter for the description of the condensed grains
interactions in the smoky plasma is not always comprehensible, which is caused by a number of its
specific properties. The principal distinction of the smoky plasma from discharge dusty plasma is its
thermal collision nature. The displacement of ionization equilibrium in the plasma gas phase, caused
by the presence of a noncompensate volume charge, is formed in plasma due to the interphase
interaction [12]. The condensed grains are formed by homogeneous and heterogeneous condensation
in the smoky plasma at burning. Therefore, the condensed phase contains a multimode distribution
function for sizes of grains of submicron fraction. This enables us to consider the dusty and smoky
plasma as two different kinds of plasma with a condensed disperse phase.
Thus, multimode distribution functions for the mentioned grain sizes and the displacement of
ionization equilibrium do not make it possible to use the coupling parameter, obtained both for
the dusty and smoky plasmas. The present paper is devoted to the development of a coupling
parameter applicable to the description of polydisperse systems such as the smoky plasma.
2. The state of the problem
The Coulomb coupling parameter was suggested [5,6] as the ratio of average electrostatic en-
ergy to the average kinetic energy in the spatially homogeneous one-component plasma (OCP),
consisting of a single species of charged particles embedded in a uniform background of neutralizing
charges,
ΓOCP =
(eZ)2
4πε0RWT
, (1)
c© V.I.Vishnyakov, G.S.Dragan 201
V.I.Vishnyakov, G.S.Dragan
where Z is the charge of particles, RW = (3/4πN)
1/3
is the Wigner-Seitz radius, N is the number
density of the particles.
Such a system can be realized, for example, within a highly evolved star, where the strongly
degenerate electron gas creates a neutralizing background to the Coulomb interaction of ions. Then
strength of the ionic interaction can be characterized by parameter equation (1). It is generally
accepted to call the system nonideal, if Γ ≈ 1 and strongly nonideal or strongly coupled at Γ � 1.
The dusty plasma contains electrons, ions and the charged dust grains. In the plasma any charge
is screened by free charges of other sign. Therefore, one-component approach is incorrect. Ikezi [7]
suggested considering the screening of charges in the equation for coupling parameter, using the
Debye-Hückel or Yukawa potential for calculation of the potential energy of grains interaction.
Then the coupling parameter for the dust grains subsystem can be presented as follows:
Γ∗ =
(eZd)2
4πε0RWTd
exp (−κ) , (2)
where κ = RW/rD is the structural parameter, rD is the screening length, Td is the temperature
of dust grains. Plasma crystals are formed at value Γ > Γc where Γc is some critical value of the
coupling parameter. Numerical calculations for one-component plasma give the value Γc ∼ 170.
The phase diagram of crystallization in coordinates Γ − κ, was gained in the papers [8–11] by
means of numerical modelling (figure 1). Thus it is noted that the coupling parameter in the form
of equation (2) provides a better description of a phase transition.
Figure 1. The phase diagram of the dust grains crystallization; Γ∗
c1 = Γc exp(−κ), Γ∗
c2 – equation
(3) (data from [9,10]).
The results of experimental research regarding the formation of plasma crystals [13] showed the
necessity of modernizing the coupling parameter equation (2) to the following form
Γ∗ = ΓOCP (1 + κ) exp (−κ) .
The further researches [14–17] showed the necessity of improving the coupling parameter [17],
Γ∗ = ΓOCP
(
1 + κ + κ2/2
)1/2
exp (−κ) , (3)
Let us note that the requirement of crystallization is satisfied at values Γ∗ > 170 and κ < 7.
Apparently from figure 1, calculation under the equation (3) gives a smaller change of critical
value Γ∗
c .
3. Plasma with the condensed disperse phase
The grains in the combustion plasma are usually of different types and sizes, and thus of
different charges. The question remains unanswered: for which subsystem the coupling parameter
202
Coupling parameter for low-temperature plasma
equation (3) is realizable in such a system? As it has been shown in [18], the electrostatic interaction
in thermal plasma is missing at the distances exceeding 8 screening lengths. Therefore, if the spatial
distribution of smoke grains satisfies this requirement, all grains are electroneutral one to another
due to strong screenings. Moreover, we should note that numerical modelling of the crystallization
phase diagram [8–11] does not take the grain sizes into consideration. In real systems the structural
parameter cannot accept zero value. The Wigner-Seitz radius should be larger than the radius of
dust grain RW > a, and the screening length has a terminating value. Hence, there is some minimal
value of the structural parameter κa = a/rD, corresponding to the closely-packed structure of the
dust grains.
Let us calculate the charge of grains corresponding to the line of fusion of plasma crystal
in figure 1, using equation (1). The dependence of the grains charge critical value on structural
parameter is presented in figure 2.
Figure 2. The value of the grains charge, corresponding to line of fusion, at the temperature
T = 0.2 eV.
Let us take advantage of expression [19]
Zd =
8πε0a
2T
e2rD
sinh
(
eφs
2T
)
(4)
in order to define the superficial potential of grains corresponding to the charges in figure 2. We set
the grain radius as a part of the Wigner-Seitz radius a = RW/m, which is equivalent to a relation
κa = κ/m.
Figure 3. The values of the surface potential of the grains, corresponding to the fusion line at
temperature T = 0.2 eV, rD = 0.3 µ (data from [11,14]).
In figure 3 the dependences of the surface potential of the condensed grains, corresponding to
the fusion line, on the structural parameter are presented at various values of m. Here we can see
203
V.I.Vishnyakov, G.S.Dragan
that all dependences have a minimum in the region of values κ ∼ 2−3. Proceeding from aspiration
of the system to a minimum of energy, it is possible to consider such a value of structural parameter
as optimum for a plasma crystal.
Figure 4. The dependence of the surface potential of grains on the relative Wigner-Seitz radius
1: for κ = 0.6; 2: for κ = 1.3; 3: for κ = 2.6; and 1′, 2′, 3′ is the eφs/mT .
The dependence of the surface potential on the grains radius (figure 4, curves 1–3) is obvious,
because with diminution of the grains radius, the distance between the surfaces of the grains should
increase at constant value of κ. Hence, a larger magnitude of the potential is necessary in order to
maintain the force of interaction between grains. Zero value of the surface potential must correspond
to the value κa = κ, since in this case RW = a and the grains adjoin the surfaces, forming a
closely-packed arrangement. The existence of nonzero potential for κa = κ testifies to essential
restrictions of applicability of the coupling parameter in the form of equations (1)–(3) to the real
systems containing grains of terminating size. The surface potential, providing the interaction in
the crystal, increases with the increase of m = κ/κa. However, the ratio eφs/mT = aeφs/RWT
(curves 1′ − 2′, figure 4) changes a little and aspires to unity with increase of m.
It is possible to note, that in equation (1) the charge of particle with radius a defines the
Coulomb potential at a distance RW from the particle surface φ(RW) = eZ/4πε0RW. Hence, the
parameter equation (1) can be written down as follows:
Γ =
eZdφ(RW)
T
. (5)
When the screening effect occurs it is also possible to use equation (5). However, it is necessary
to calculate the potential proceeding from the solution of the Poisson equation. For example,
definition of the surface potential of the large grains (a � rD) should use equation (4), which can
also be dilated for the case of fine grains (a < rD) [20]:
φs = 2
T
e
asinh
(
e2Zd
8πε0TrDκa(κa + 1)
)
. (6)
Following Ikezy it is impossible to use the Debye potential being the potential of the grain
surface that corresponds to fusion line. The analysis of the possible solutions of the Poisson equation
[21] has shown that at larger values of the surface potential (φs → ∞) the curvature of the grain
appears incidental, and the potential spatial distribution around the grain can be described by the
following expression [19,20]
φ (RW) =
2T
e
ln
1 + tanh
(
eφs
4T
)
exp (κa − κ)
1 − tanh
(
eφs
4T
)
exp (κa − κ)
. (7)
204
Coupling parameter for low-temperature plasma
The dependences of the coupling parameter, calculated using the equations (5)–(7), on the
structural parameter along a fusion line of the plasma crystal for various values of m = RW/a =
κ/κa are presented in figure 5.
Figure 5. The dependence of the coupling parameter on the structural parameter; 1: for m = 2;
2: for m = 3; 3: for m = 4; 4: for m = 5.
From the presented dependences it follows that the coupling parameter calculated using the
formulas of equations (5)–(7) coincides well with the asymptotics of equation (3) when RW ∼ 2a.
The further magnification of coefficient m leads to the impairment of dependence of the coupling
parameter on the structural parameter. The relation RW ∼ 4a (curve 3) may be optimum, because
in this case the coupling parameter changes a little with the change of the structural parameter.
Thus, the Coulomb coupling parameter, introduced for the one-component plasma, is applicable
to the complex plasma if it is used in the form of equation (5). In this case, the value of potential
on the boundary of the Wigner-Seitz cell should be taken from the solution of the electrostatic
task corresponding to the type and composition of plasmas.
4. The polydisperse systems
So far this was a question to be resolved in a monodisperse system of dust grains. When plasma
contains the grains of different size or different chemical composition, equation (5) can be applied
only to the interaction of the neighbouring grains of two sorts. If there are more than two sorts
of grains it is necessary to introduce the coupling parameters for interaction of each sort of grains
with all others sorts. For example, if plasma contains the dust grains of three sorts [20,22] it is
necessary to use three different coupling parameters, which makes the use of coupling parameter
inconvenient as a measure of formation of the ordered structures in the plasma.
Possibly, in such cases there is no need to describe the interaction of the condensed grains among
themselves, but concentrate on the interaction of grains with plasma. So, in the smoky plasma which
is of thermal collision nature, the interaction of grains is defined by the ion interface pressure in
the thin layer of plasma at the grain surface [20,23]. These forces are caused by nonhomogeneous
ionization of a gas phase which is described by the modernized Saha equation [20–23]
neni
na
= KS exp
(
−eϕpl
T
)
, (8)
where KS is the Saha constant, ϕpl is the bulk plasma potential.
The bulk plasma potential ϕpl characterizes electrostatic energy of the volume of plasma [18]
ϕplQpl = ε0
∫
V
E2dV ,
205
V.I.Vishnyakov, G.S.Dragan
where E is the electric field, Qpl is the charge of the plasma volume. Thus the potential φ is
calculated with respect to the bulk plasma potential.
Then the coupling parameter can be defined as the ratio between electrostatic energy in the
Wigner-Seitz cell ϕplQpl and the thermal energy. The volume of the cell will be calculated with
respect to the screening length rD:
Γpd =
Qplϕpl
(RW/rD)3
1
T
. (9)
In equation (9) the value Qpl/R3
W is equal to the charge density of a gas phase in the Wigner-
Seitz cell. Hence,
Qpl
R3
W
r3
D = e(ni − ne)r
3
D = e (Ni − Ne)D
represents the volumetric charge of the Debye sphere. Thus, the coupling parameter can be pre-
sented in the following form
Γpd = (Ni − Ne)D
eϕpl
T
. (10)
As it follows from the above expression, the coupling parameter does not depend on the grains
number densities, because equation (10) does not contain the Wigner-Seitz radius. On the other
hand, equation (10) contains the bulk plasma potential, which is the average value for the whole
plasma volume. Therefore, parameter Γpd can be used as a measure of nonideality of polydisperse
system of condensed grains in the plasma with condensed disperse phase. Equation (10) can also
be used for the flat electrode, such as a probe, in the plasma.
Figure 6. The dependence of the coupling parameter on the structural parameter; 1 – equa-
tion (3), 2 – equation (10).
The lines of fusion of the plasma crystal are similar to those presented in figure 5, but calculated
using equation (3) and equation (10) in figure 6. In the diagrams it is seen that the parameter Γpd
is more preferable for the monodisperse grains as well because its value to a less degree depends
on the structural parameter.
5. Conclusion
Thus, we can draw a conclusion that the coupling parameter in equation (10) is applicable to the
description of polydisperse systems of condensed grains in complex plasma. This form of coupling
parameter contains the average electrostatic energy of the system as a product of the average charge
of a gas phase into the Debye sphere and the bulk plasma potential. Therefore, equation (10) is
independent of the size and charge of the single grains, but depends on the average charge of all
the grains.
206
Coupling parameter for low-temperature plasma
In order to describe the system of the monodisperse grains in the complex plasma it is possible
to use the coupling parameter in equation (10) or the coupling parameter in equation (5). This
form is similar to the coupling parameter for the one-component plasma in equation (1), but the
potential on the boundary of the Wigner-Seitz cell should be taken as a solution of the Poisson
equation corresponding to the type and composition of the plasma.
In the one-component plasma, the potential on the boundary of the Wigner-Seitz cell is the
Coulomb potential, and therefore equation (5) is transformed into equation (1).
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207
V.I.Vishnyakov, G.S.Dragan
Параметр неiдеальностi для низькотемпературної плазми
з конденсованою фазою
В.I.Вишняков, Г.С.Драган
Фiзичний факультет Одеського нацiонального унiверситету iм. I.I. Мечникова, 65026, Одеса, Україна
Отримано 6 березня 2006 р., в остаточному виглядi – 18 травня 2006 р.
Дослiджено межi застосування параметра неiдеальностi та його вираз для опису фазових переходiв
в упорядкованих структурах заряджених конденсованих часток в плазмi з конденсованою диспер-
сною фазою. Показана доцiльнiсть використання виразу для параметру неiдеальностi iз застосува-
нням значення потенцiалу на границi комiрки Вiгнера-Зейтца, отриманого з розв’язку електроста-
тичної задачi, що вiдповiдає виду i складу плазми. Отримано вираз параметра неiдеальностi для
полiдисперсної конденсованої фази з електростатичною енергiєю мiжфазної взаємодiї у виглядi
добутку узагальненого потенцiалу плазми i об’ємного заряду газової фази плазми в дебаєвськiй
сферi.
Ключовi слова: плазма, конденсована дисперсна фаза, параметр неiдеальностi
PACS: 52.27.Gr, 52.27.Lw
208
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