J. Plasma Fusion Res. SERIES, Vol.4 (2001) 559-563

surface Tension in Plasmas Related to Double LayerFormation
POPESCU Sebastianx and LOZNEANU Erzilia
Department of Plasma Physics, "Al.I.Cuza" ()niversity

Blvd. Copou No.l1, RO-6600 lasi, Romania(Received: 5 December 2000 / Accepted:24 August 2001)

Abstract
Self-organized space charge configurations bordered by electric double layers appear in plasma as
the result of the transition into a state characterized by local minimum of the free energy. Considering theself-assemblage process of such a complex well-confined space-charge configuration in plasma, known

by the name of ball of fire, as a nucleation process, it becomes possible to define an equivalent surface
tension for the double layer that covers the core of the ball of fire and to make some predictions for its
surface tension coefficient and capacitance.

Keywords:
self-organization, ball of fire, double layer, nucleation, surface tension

1. Introduction
An external constraint (e.g. an electric field)
applied on a gaseous conductor, acting on a large

tension, is able to balance the kinetic pressure ofelectrons, positive ions and neutrals from the welllocated volume delimited by it.
Because of the resemblances between the behavior
of BF and a liquid drop we may see BF as a result of anumber ofparticles can, in appropriate conditions, givebirth to new phenomena enjoying self-organization
related to long-range correlations [1-4]. In other words,

nucleation process. More precisely, we will consider thestationary stable BF a "droplet" of a new ordered phase,the constraint leads to a "new state of the same matter",more ordered than the former one, following a "non-spontaneously formed in a uniform non-equilibrium

equilibrium phase transition" [5]. The term of..nonequilibrium phase transition" is used here concerning

plasma for a critical value of the control parameter (i.e.

the potential applied on the anode) and, consequently,

self-organization phenomena leading to the appearanceof ordered space charge arrangements. Such a complex
space charge arrangement bordered by an electrical
double layer (DL) (Fig.1), known as ball of fire (BF), isself-assembled in a plasma diode in front of the anodewhen the external constraint surpasses a critical value.

After its appearance the BF behaves as a structure
whose self-consistence and stability are related tosurface tension. These BF properties are determined bya self-consistent DL, present at its border. The DL,Fig. 1 Photograph of a stationary stable ball of fire.

acting as a membrane characterized by a certain surface*@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Research

C o rre sp ondin g autho r' s e -mail : s eba @ uaic. ro559

Popescu S. et al., Surface Tension in Plasmas Related to Double Layer Formation

covers the core of BF.

we will define an equivalent surface tension for the DL
that surrounds its core.
The aim of this paper is to estimate the surface
tension coefficient and the specific capacitance (i.e. thecapacitance per unit of surface) associated with the selfconsistent DL that surrounds the core of the stationary

3. Surface Tension of Plasma Double Layers
It is experimentally proved that self-organizedstructures of BF-type exist in spite of electrons'
recombinations with positive ions. This proves that ir
dynamical equilibrium is established between thestable BF.

processes of generation and accumulation andrespectively recombination and diffusion of the two

2. Why Should Double Layers Have Surface

kinds of charges. More specifically, those electrons fromthe surrounding plasma that have enough kinetic energy
to surpass the negative part of the DL will beTension?
Based on the studies on laser-plasma interaction,
Hora asserted that the existence of electric fields at theaccelerated in the DL's electric field, so they will reachboundary between two plasmas is an argument forconsidering the surface tension concept in order toanalyze the stability of the boundary region [6,7]. He
introduced the surface tension for characterizing the DL
as an interface between two plasmas with different
properties. Contrary to surface tension in liquids, thecohesion forces work as long as the current ensures theflux of electrons necessarv for the continuous selfassemblage of the DL.
There are many arguments for considering BF as
the result of a phase transition (i.e. nucleation) process.
Both BF and the liquid drop appear spontaneously andcoexist with the phase from which they arose; both ofthem are protected from the surroundings by a quasispherical "membrane" that is essential in maintaining
them. After Lenard [8] the liquid drop membrane is infact an electric DL, like in BF's case. Moreover, in thethe core with sufficient energy for producing neutrals'ionization. After the ionization processes a part ofelectrons' kinetic energy remains as thermal energy. As
a result of thermal diffusion these will leave the core. A

part of the reminders will recombine with the positive
ions, ensuring in this way the neutrals' concentration
required for maintaining a constant ionization rate.
Under such conditions, the net positive charge in theBF's core will remain constant. When the number ofrecombinations in unit time interval will be equal to thenumber of ionizations in the same time interval, we may
consider BF in a stationary stable state.
In collisional plasma at relatively high pressures'
DL becomes able to sustain its own existence only byinternal processes, or with other words, DL becomes
self-consistent. This happens when the potential drop
across it reaches the ionization potential Vr of thecase ofthe anode-electrolyte interface [9], the DL theory

was applied first to the theoretical analysis of theworking gas [3].
In agreement with experimental results [3'4'll]' ina very simple model, we will consider the stationary
stable BF as a spherical structure, electrically neutral asa whole, being composed by a core with radius R'
uniformly charged in volume with positive charge andcovered by a DL having thickness 5. The external partof the self-consistent DL is negatively charged with thecharge (-Q) uniformly distributed on the surface [seeFig. 2(a)1. In the theory ofelectricity, a DL is defined asa structure composed by two adjacent space charge
layers with the same superficial charge density, but withproperties of interfaces (e.g. surface tension, differential
capacitance). In the case of a liquid-liquid vapors
interface the forces responsible for the surface tension

are directed towards the liquid interior, because theliquid has a grater density than its vapors. This situation
is formally encountered in the BF's case. The
electrostatic forces acting on electrons are directed
towards the inner positive sheet of the DL that covers
the core of BF. We have considered the forces acting onelectrons because they are responsible for the selfassemblage of BF, as well as for the ionization

opposite signs [12,13]. From this we may conclude thatthe total charges from each layer, disregarding the sign.
are not equal, excepting the planar case. It follows thatthe inner positive layer has a smaller total charge thanthe outer negative one, meaning that the positive charge
difference is located in the core of BF. The above

processes inside the core that make the core density tobe greater than that of the outside plasma' The
experiments in which BFs arise reveal a hysteresis
phenomenon [3] for which appearance the DLs areresponsible. Because the hysteresis phenomenon is alsopresent in nucleation processes [10], for whichassumption is supported by experimental results [4'll]'which clearly show that in the core the positive ions'appearance surface tension is responsible, this is anotherreason for associating a surface tension to the DL that560

Popescu S. et aI., Surface Tension in Plasmas Related to Double Layer Formation

concentration is greater than in the outside plasma.

potential energy of the DL:

IConsidering the above said we can analyze thespace charge configuration of a stationary stable BF

w"P= ! p,4n R2 v,

2'"with the methods of electrostatics. Under thesecircumstances, using Gauss law, the potential drop on
the self-consistent DL will be:o .t
vi- 47ttoR

R+d^

.^\2

-27reov7V:!1@)'RR*2

where p, = Q/[42(R + 612] is the surface charge densityof the DL.
In order to evaluate the surface tension coefficient(l)o associated with the DL that covers the BF's core wehave estimated the change of the free energy for theincrease with AS of the BF's surface without changing
its volume. This change of the free energy in isothermal

where ee is the vacuum permitivity, Q = en*.4nR313 is
the total electric charge of the BF's core, n+ being its
charge concentration and e the electron charge. As aresult of the above mentioned dynamical equilibrium weconditions is just the potential energy variation of theDL:can consider n* having a constant value. The radial
distribution of the potential over BF [Fig. 2(b)] isLWo = oA,S(5)

similar to that obtained from experiments [4,] l].
From Eq. (1) it will follow the dependence of thethe surface of the BF increases under the above

DL's thickness don the core radius R:

conditions can be expressed as:'RR*,-R' - R*'

Consequently, the potential energy variation whenQ)

where R* has the expression

From (5) and (6) it follows that:

3eoVien.

(3)o=Since 6 must have a positive and finite value it results[see Eq. (2)] that R > R*. This means that R* is a criticalvalue of the BF's core radius R.
Because the surface tension of a quasi-gaseous
object such as BF may only result owing to electric€ov? ( n' -n*') ( :n'+ n *')2R*'Rta)This result sustains the above observation, which statesthat the DL exists only for R > R*. For R = R* thesurface tension coefficient becomes zero making the DLto lose its role of "membrane" that maintains the BFexistence.

charges [14], we have calculated the electrostatic

4. Specific Capacitance of Plasma DoubleLayers

As experimentally proved (Fig.3) between thecurrent and the potential of a gaseous conductor,+ ,++?

++ T'++:*+i*;i +.i+i-+:f'-_:

containing a BF there exist a phase difference showing-iRR
Fig. 2 (a) Ball of fire schematic representation with thecore charge uniformly distributed in volume.
R is the core raduis and 6 the DL thickness;{b) Potential profile of the stationary stable ball offire.

Fig.3 Phase difference between i(t) anO u(t) for aplasma diode in which a ball of fire is formed.

561

Pooescu S. et al., Surface Tension in Plasmas Related to Double Layer Formation

It is a matter of experimental measurements tothat the DL which border the BF plays a capacitive role.
This allows us to compute the specific capacitance ofthe DL. For this purpose let us apply an ac tension overthe dc one between the anode and the cathode of theprove the convergence of these two expressions for thespecific capacitance of a quasi-spherical DL, this being
the subject of a forthcoming paper.

plasma diode, after the stable BF's formation:u(t) = ryo so"

5. Gonclusions

(8)

''

In the phase transition theory [15] it is stated thatthe approaching of the critical point is accompanied by
the thickness growing of the superficial layer between
the two phases taken into consideration. Also, at thecritical point, the surface tension goes to zero. In thepresent case the critical point corresponds to R = R*
because the decreasing of the core radius towards R*
makes the DL's thickness to approach an extremely
large value (the DL practically vanishes at this core

where Us is the ac tension amplitude. The frequency a/
2n and the amplitude of the applied ac tension musthave such low values that the current distribution inside
the DL, as well as outside it, to be the same as in thecase of a constant electric field.
Because the DL is a capacitive element of circuit,between the tension and the current there will be a phasedifference rp, such that:i(t)= Io cos((Dt + E)

radius). The superficial tension coefficient, as well as
the specific capacitance go to zero when R tends to R*.
We notice that all the above mathematical(e)where /6 is the ac current amplitude.

evaluations, in spite of their simplicity, are valid for astationary stable BF. The essential novelty of this paper
is that BF is considered as a result of a phase transition

The instantaneous power will be:p (t ) = (Jo I o (cos2 at cos (p- sin @/ cos arl sin rp) (1 0)

produced in plasma after self-organization.

The first part of Eq. (10) is the power dissipated byJoule effect. The second represents the supplementary

Consequently, it is stressed that the DL that covers thecore of the BF plays the role of a "membrane" that splits
the plasma in two regions with different characteristics.

power accumulated into the DL structure, due to thesuperposition of the ac tension over the dc one:P".,(t) = Uo Io sin r0l cos alt sin I

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