Eur. Phys. J. D (2013) 67: 210 DOI: 10.1140/epjd/e2013-40277-0 Regular Article T HE EUROPEAN P HYSICAL JOURNAL D Charging kinetics of dust particles in a non-Maxwellian Lorentzian plasma Sanjay K. Mishra 1, a , Shikha Misra 2 , and Mahendra Singh Sodha 2 1 Institute for Plasma Research (IPR), 382428 Gandhinagar, India 2 Centre for Energy Studies (CES), Indian Institute of Technology Delhi (IITD), 110016 New Delhi, India Received 30 April 2013 / Received in final form 4 July 2013 Published online 18 October 2013 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2013 Abstract. Charging kinetics of uniformly dispersed spherical dust particles in a non-Maxwellian plasma, characterized by a Lorentzian (κ) distribution function of electrons/ions has been developed; the formu- lation is based on the uniform potential theory, applicable to the dust particles characterized by a size distribution function. Owing the openness character of the complex plasmas, the charging kinetics has been developed on the basis of number and energy balance of the plasma constituents along with the charge balance over the dust particles; the neutrality of the complex plasma is a consequence of the number bal- ance of electrons/ions and charge balance on the dust particles. A more rigorous approach, proposed by Mott-Smith and Langmuir [Phys. Rev. 28, 727 (1926)] has been adopted to derive the expressions for the electron/ion accretion current over the dust surface and corresponding mean energy in a non-Maxwellian Lorentzian plasma. Further the formulation has been implemented to determine the secondary electron emission (SEE ) from the spherical dust particles in such plasmas. The departure of the results for the Lorentzian plasma, from that in the case of Maxwellian plasma has been graphically illustrated and dis- cussed. It is seen that the Lorentzian nature of the plasma and the inclusion of the collective effect of the dust particles significantly affects the dust charge and other plasma parameters; the formulation and understanding of the charging kinetics in a Lorentzian plasma have implications for both the physics (e.g. grain growth and disruption) and the dynamics of dust in laboratory and space environment, when the dimension of the plasma are much larger than the diffusion length. 1 Introduction The kinetics of plasmas with dust suspension, commonly known as dusty or complex plasma is of considerable in- terest on account of the relevance of such studies to phe- nomena [1–18] in nature, laboratory, space and industrial applications, in which complex plasmas play a significant part. An important parameter in the complex plasma is the charge on the dust particle, which is determined [1–15] by the net electron/ion flux on the surface. In dark and cold complex plasmas, the dust gets negatively charged on account of the larger flux of accreting electrons than that of accreting ions; however significant emission (viz. thermionic, photoelectric, electric field, light induced field and secondary) of electrons from the surface of the dust particles may lead to a positive charge on the dust parti- cles. Numerous theoretical [2–5] and experimental [19–22] studies on the determination of the dust charge and its effect on the plasma environment have been made but most of these are based on the assumption of a Maxwellian distribution of electron/ion velocities in the plasma. It is well-known that in non-equilibrium complex plasmas the electrons/ions can be characterized by a non-Maxwellian energy distribution function [23–26]; this is a frequent a e-mail: [email protected]feature in astrophysical and space plasmas where the con- stituent plasma electrons/ions get characterized by gener- alized Lorentzian (κ)[25,26] and shifting Maxwellian [27] distribution functions of energy. Even in laboratory plas- mas under the influence of an RF electric field the electron energy distribution function (EEDF) gets modified from Maxwellian to drifting Maxwellian or bi- Maxwellian (in case of low pressure RF discharges) distribution of en- ergy [28]. Therefore, it is of interest to explore the effect of non-Maxwellian electron/ion energy distribution func- tions on the complex plasma kinetics and other plasma parameters. The departure of the energy distribution function from Maxwellian to non-Maxwellian significantly alters the electron/ion collection current over the dust particles and consequently the particle charge and other plasma pa- rameters. The first attempt to include the non-Maxwellian distribution in the evaluation of the dust charge in space plasmas was made by Meyer-Vernet [29] who considered the super-thermal electrons described by a power law dis- tribution of energy; the analysis was based on the charge balance, taking secondary electron emission from and electron/ion accretion on the dust surface into account. The analysis was extended by Rosenberg and Mendis [23] to take account of the electrons/ions, governed by the
Transcript
Eur. Phys. J. D (2013) 67: 210DOI: 10.1140/epjd/e2013-40277-0
Regular Article
THE EUROPEANPHYSICAL JOURNAL D
Charging kinetics of dust particles in a non-MaxwellianLorentzian plasma
Sanjay K. Mishra1,a, Shikha Misra2, and Mahendra Singh Sodha2
1 Institute for Plasma Research (IPR), 382428 Gandhinagar, India2 Centre for Energy Studies (CES), Indian Institute of Technology Delhi (IITD), 110016 New Delhi, India
Abstract. Charging kinetics of uniformly dispersed spherical dust particles in a non-Maxwellian plasma,characterized by a Lorentzian (κ) distribution function of electrons/ions has been developed; the formu-lation is based on the uniform potential theory, applicable to the dust particles characterized by a sizedistribution function. Owing the openness character of the complex plasmas, the charging kinetics has beendeveloped on the basis of number and energy balance of the plasma constituents along with the chargebalance over the dust particles; the neutrality of the complex plasma is a consequence of the number bal-ance of electrons/ions and charge balance on the dust particles. A more rigorous approach, proposed byMott-Smith and Langmuir [Phys. Rev. 28, 727 (1926)] has been adopted to derive the expressions for theelectron/ion accretion current over the dust surface and corresponding mean energy in a non-MaxwellianLorentzian plasma. Further the formulation has been implemented to determine the secondary electronemission (SEE ) from the spherical dust particles in such plasmas. The departure of the results for theLorentzian plasma, from that in the case of Maxwellian plasma has been graphically illustrated and dis-cussed. It is seen that the Lorentzian nature of the plasma and the inclusion of the collective effect ofthe dust particles significantly affects the dust charge and other plasma parameters; the formulation andunderstanding of the charging kinetics in a Lorentzian plasma have implications for both the physics (e.g.grain growth and disruption) and the dynamics of dust in laboratory and space environment, when thedimension of the plasma are much larger than the diffusion length.
1 Introduction
The kinetics of plasmas with dust suspension, commonlyknown as dusty or complex plasma is of considerable in-terest on account of the relevance of such studies to phe-nomena [1–18] in nature, laboratory, space and industrialapplications, in which complex plasmas play a significantpart. An important parameter in the complex plasma isthe charge on the dust particle, which is determined [1–15]by the net electron/ion flux on the surface. In dark andcold complex plasmas, the dust gets negatively chargedon account of the larger flux of accreting electrons thanthat of accreting ions; however significant emission (viz.thermionic, photoelectric, electric field, light induced fieldand secondary) of electrons from the surface of the dustparticles may lead to a positive charge on the dust parti-cles. Numerous theoretical [2–5] and experimental [19–22]studies on the determination of the dust charge and itseffect on the plasma environment have been made butmost of these are based on the assumption of a Maxwelliandistribution of electron/ion velocities in the plasma. It iswell-known that in non-equilibrium complex plasmas theelectrons/ions can be characterized by a non-Maxwellianenergy distribution function [23–26]; this is a frequent
feature in astrophysical and space plasmas where the con-stituent plasma electrons/ions get characterized by gener-alized Lorentzian (κ) [25,26] and shifting Maxwellian [27]distribution functions of energy. Even in laboratory plas-mas under the influence of an RF electric field the electronenergy distribution function (EEDF) gets modified fromMaxwellian to drifting Maxwellian or bi- Maxwellian (incase of low pressure RF discharges) distribution of en-ergy [28]. Therefore, it is of interest to explore the effectof non-Maxwellian electron/ion energy distribution func-tions on the complex plasma kinetics and other plasmaparameters.
The departure of the energy distribution functionfrom Maxwellian to non-Maxwellian significantly altersthe electron/ion collection current over the dust particlesand consequently the particle charge and other plasma pa-rameters. The first attempt to include the non-Maxwelliandistribution in the evaluation of the dust charge in spaceplasmas was made by Meyer-Vernet [29] who consideredthe super-thermal electrons described by a power law dis-tribution of energy; the analysis was based on the chargebalance, taking secondary electron emission from andelectron/ion accretion on the dust surface into account.The analysis was extended by Rosenberg and Mendis [23]to take account of the electrons/ions, governed by the
generalized Lorentzian (κ) energy distribution function;however the theory was solely based on the current bal-ance over the dust particle and the effect of dust parti-cles on surrounding plasma environment was ignored. Ina recent investigation, Tribeche and Shukla [30] have an-alyzed the dust charging in a plasma characterized by anon extensive electron distribution function. Despite ac-cepting the fact that the dust charge significantly affectsthe surrounding plasma environment, it is interesting tonotice that the above mentioned studies [27–30] are basedonly on the charge balance over the dust particles; the col-lective effect of dust particles on the surrounding plasmaenvironment was ignored.
For the understanding of the physical phenomena anddesign of experiments and applications, a knowledge ofthe basic plasma parameters and good understanding ofthe kinetics of the complex plasma is necessary. With duedeference to the openness character [17] of the complexplasma, some recent investigations [31–35] have taken intoaccount the number and energy balance of all the con-stituents; in particular the production and annihilation ofelectrons and ions in the gaseous component has also beenaccounted for. In a recent investigation Mishra et al. [35]have analyzed the charge distribution over dust particlesin a meso thermal plasma, where the ions are describedby a shifting Maxwellian distribution; the analysis is alsoof relevance to the charging of satellites spacecrafts, me-teors and meteorites in the ionospheric and near spaceplasmas [2].
In view of the relevance of non-Maxwellian plasmasto space/astrophysical environment and absence of ap-propriate theoretical basis for the kinetics for such plas-mas, this paper aims to develop a comprehensive theo-retical model to analyze the kinetics of a complex plasmacomprising of electrons/ions described by non-MaxwellianLorentzian (κ) distribution function of energy and thespherical dust particles characterized by the MRN (Mathiset al.) power law of the size distribution. The expressionsfor the accretion current on the surface of the dust parti-cles and corresponding mean energy of the electrons andions characterized by κ-distribution function of energyhave been derived; in contrast to usual orbital motionlimited (OML) approach the derivation is based on theformulation of Mott-Smith and Langmuir [36]. Followingthe modified theory for the secondary electron emission(SEE) [37] from the spherical particles, this formulationhas been applied to estimate the SEE yield from the dustparticles corresponding to non-Maxwellian κ-distributionof electron energy. In view of the openness [17] character ofthe complex plasma, the present formulation incorporatescharge balance over the dust particles and the numberand energy balance of electrons, ions and neutral species;the accretion of electrons and ions on and the emission ofelectrons from the dust surface have been considered asthe principal charging mechanisms. The charging kineticshas been developed on the basis of the uniform potentialtheory [38–40], applicable to uniformly dispersed spher-ical dust particles in the complex plasma characterizedby a size distribution function. The plasma is assumed
to get maintained on account of the ionization of neutralspecies (to produce electron-ion pairs) and electron-ionrecombination; as a simplification the temperature of theneutral species is considered to be constant on account ofthe large heat capacity. Further the contribution of theelastic collision between electron, ion and neutrals in theenergy balance equations has been ignored on account oftheir small magnitude in comparison to the energy loss byelectrons/ions due to accretion over dust particles. It maybe mentioned that the electrical neutrality of the dustyplasma is inherent in the number balance and need not beseparately accounted for in the computations.
Thus, in contrast to the earlier studies, the presentinvestigation takes the following aspects into account:
(i) a realistic situation in the complex plasmas where thedust particles are characterized by a size distributionfunction (viz. MRN power law) has been consideredherein; further the plasma electrons/ions are assumedto be non-Maxwellian, characterized by generalizedLorentzian (κ) distribution function of energy;
(ii) the expressions for the accretion current and corre-sponding mean energy of the electrons/ions accretingover dust particles subjected to non-Maxwellian κ- dis-tribution of electrons/ions has been derived; in con-trast to OML approach the derivation is based on therigorous formulation of Mott-Smith and Langmuir [36]which also takes account of oblique incidence of elec-trons/ions accreting over dust particles. Further theeffect of the κ-distribution of electron energy on thesecondary electron emission has also been estimated;for this purpose the modified approach proposed byMishra et al. [37] is utilized;
(iii) owing the openness nature of the complex plasma,number and energy balance of the plasma parti-cles characterizing the effect of dust particles on thesurrounding plasma environment (i.e. density and tem-perature) has been taken into account; the tempera-ture dependence of the charging mechanisms (i.e. elec-tron/ion accretion) has been taken care of;
(iv) to describe the steady state charging characteristicsof the dust particles in a complex plasma uniform po-tential theory [38], applicable to the dust particles hav-ing size distribution has been implemented;
(v) to appreciate the physical basis of the theory, theeffect of dust particle density and spectral index(κ, characterising the plasma electrons/ions) on thesteady state dust surface potential and other relevantplasma parameters has been analyzed and illustratedgraphically.
It is necessary to point out that the theory is strictly appli-cable to the situation where the dimension of plasma sys-tem is large enough in comparison to the diffusion lengthand inter particles distance. Specifically in laboratory ex-periments, in the presence of the large number density ofthe dust particles the shielding distance becomes compa-rable or even larger than the inter particle distance; undersuch conditions the OML (Orbital Motion Limited) theoryand consequently our simple expressions for nec and nic
are not valid. Hence the observations made under theseconditions might not be explained by the present theory.
For the numerical computations, the kinetics corre-sponding to dark plasmas (i.e. absence of the electronemission from dust) has been discussed; the accretion ofelectrons/ions on the dust surface is the mechanism con-sidered for the charging of the dust particles. The steadystate surface potential on the particle and electron/iondensity and temperature dependence on the dust densityand other plasma parameters have been illustrated in theform of curves which has been interpreted in the sectionon “Numerical results and discussion”. The results ob-tained in the case of non-Maxwellian plasma have beencompared with Maxwellian plasma results and presentedgraphically. A summary of the outcome of this study hasbeen presented in the concluding section.
2 Current and mean energy of electrons/ionsaccreting over dust surface immersedin a non-Maxwellian Lorentzian plasma
2.1 Approach
For the evaluation of the current and the mean en-ergy of the electrons/ions, accreting on the dust particlesin a Lorentzian plasma the formulation by Mott-Smithand Langmuir [36] for Maxwellian plasma has beenadopted. Consider a spherical dust particle of radius aina plasma, with electron/ion density n, charge q (±e referto ions/electrons, respectively), temperature T and massm with suffixes e and i referring to electrons and ions re-spectively. Like Mott-Smith and Langmuir [36] it is con-venient to consider a sphere of radius b(�a), concentricwith the dust particle so that the potential energy of theelectron/ion is zero for r ≥ b and Vs on the surface of dustparticle r = a.
Let u and v denote the radial and tangential com-ponents of the electron/ion velocity and u be taken aspositive when directed towards the centre of the particle.The number of electrons/ions, having radial and tangen-tial velocity components between u and (u+du) and v and(v + dv), which cross the surface r = b, per unit time is:
Sbunf(u, v)dudv, (1)
where nf(u, v)dudv is the number of electrons/ions perunit volume having radial and tangential velocity compo-nents between u and (u+du) and v and (v+dv), at r = b,Sb(=4πb2) is the surface area of the sphere at r = b andf is the distribution function of electrons/ions.
If ua and va denote the radial/tangential componentsof the electron/ion velocity at the surface of the particler = a, the conservation of energy and angular momentumrequires
Vs + (m/2)(u2a + v2
a) = (m/2)(u2 + v2) (2a)
andava = bv. (2b)
From equations (2a) and (2b) one obtains
u2a = u2 − (b2/a2 − 1)v2 − (2Vs/m). (3)
It may be noted that only those electrons/ions will reachthe surface of the particle i.e. r = a for which u > 0and u2
a > 0. Using equation (3), the minimum valueof u corresponding to u2
a > 0 and v = 0, is given byum = (2Vs/m)1/2 and um = 0 corresponding to Vs > 0and Vs ≤ 0, respectively. For a given value of u, the max-imum value of v2 viz. v2
1 corresponds to u2a = 0; using
equation (3) one gets
v21 = [u2 − (2Vs/m)](b2/a2 − 1)−1. (4a)
Without detriment to the evaluation of the rate of accre-tion of electrons and ions one can choose a large value ofb � a such that
v21 = (a2/b2)[u2 − (2Vs/m)]. (4b)
This also reflects v1 � u. From equation (1), the numberof electrons/ions incident on the surface of the particle perunit time or the accretion current is given by:
nc = Sbn
∞∫
um
v2=v21∫
v2=0
uf(u, v)dudv. (5)
It is natural to equate b to the screening length of thecharge on the particle. Hence the present derivation isvalid when the screening length is much larger than theradius of the particle and less than the inter-particle dis-tance; the nature of screening i.e. the function Vs(r) is notimportant, only the surface potential Vs matters becauseVs = 0 at r = b (screening length).
2.2 Maxwellian κ-distribution
The κ-distribution [25,26] of the plasma particles (i.e. elec-trons/ions) having radial (u) and tangential (v) velocity,can be expressed as:
fκ(u, v)dudv = 2πβ(πγv2T )−3/2v
× [1 + (u2 + v2)/γv2
T
]−(κ+1)dudv, (6)
where vT = (2kBT/m)1/2 is the thermal speed of theplasma electrons/ions, γ = (κ−3/2), β = [Γ (κ+1)/Γ (κ−1/2)] and κ is the spectral index.
Substituting for fκ(u, v) from equation (6), equa-tion (5) reduces to
To illustrate the difference between the two distribu-tions viz. κ- and Maxwellian distribution of speeds of theplasma particles, one can define the ratio of correspondingaccretion rates and mean energies, as follows:
σ = (nκc /nm
c ) = γ1/2 exp(vs)(
Γ (κ − 1)Γ (κ − 1/2)
)
×(
1 +vs
γ
)1−κ
, for vs > 0 (12a)
σ = (nκc /nm
c ) = γ−1/2
(Γ (κ − 1)
Γ (κ − 1/2)
)
×(
γ − (κ − 1)vs
1 − vs
), for vs ≤ 0 (12b)
ε = (εκc /εm
c ) =(
2γ + vsκ
(κ − 2)(2 + vs)
)for vs > 0 (12c)
and
ε = (εκc /εm
c ) =(
γ
(κ − 2)
) (1 − vs
2 − vs
)
×(
2γ − (κ − 2)vs
γ − (κ − 1)vs
)for vs � 0. (12d)
It is interesting to notice that for the large values of spec-tral index (i.e. κ → ∞), the ratios tend to unity i.e. resultsfor the κ-distribution approach the results for Maxwelliandistribution as κ → ∞. Another interesting feature ofequation (12a) is that σ displays a minimum correspond-ing to vs = 1/2 for all value of κ > 3/2 and it always takesa value smaller than unity around vs = 1/2 (as shown inFig. 1a).
2.5 Secondary electron emission
As seen in the earlier section that the non-Maxwelliannature of the plasma (described by κ-distribution) sig-nificantly affects the accretion current of the electronson the dust particle; therefore it is of interest to discussthe secondary electron emission from dust particles im-mersed in such plasmas. In a recent investigation, Mishraet al. [37] have modified the theory for secondary elec-tron emission from the spherical particles, in a Maxwellianplasma by considering the appropriate path traversed byincident electron inside the dust material. In their anal-ysis, the efficiency (δ) of the secondary electron emissionfrom spherical particle has been evaluated as a function ofthe energy of the incident beam of mono-energetic primaryelectrons. Utilizing this relation viz. δ(εr) and followingthe approach similar to that used in reference [37], the netelectron flux associated with secondary electron emissioncan be obtained by substituting κ-distribution function[fκ(u, v)] of electrons for non-Maxwellian plasma in placeof Maxwellian distribution function [fm(u, v)]; thus
Jκs = ne
(kBTe
2πmeγ3
)1/2
β
∞∫
vs
δ(εr)(εr − vs)
× [1 + (εr/γ)]−(κ0+1)dεr for vs > 0 (13a)
and
Jκs = ne
(kBTe
2πmeγ3
)1/2
β
∞∫
0
δ(εr)(εr − vs)
× [1 + (εr/γ)]−(κ0+1)dεr, for vs ≤ 0. (13b)
In writing the above expressions the inequalityεr � ε1 > εt has been used. A detailed discussion ofthe mechanism and approach can be seen in the paperby Mishra et al. [37]. Following the analysis in theearlier section and utilizing equations (8a) and (8b), thenet electron flux incident on the dust particle can beexpressed as:
Jκp = (nκ
ec/4πa2) = ne
(γkBTe
2πme
)1/2
×(
Γ (κe − 1)Γ (κe − 1/2)
)(1 +
vs
γ
)1−κe
, for vs > 0,
(14a)
and
Jκp = (nκ
ec/4πa2) = ne
(γkBTe
2πme
)1/2
×(
1γ
Γ (κe − 1)Γ (κe − 1/2)
)[γ − (κe − 1)vs], for vs ≤ 0.
(14b)
The secondary electron emission yield corresponding toκ-distribution of the plasma electrons can be expressedas δκ = (Jκ
s /Jκp ); the effect of the non-Maxwellian κ-
distribution of electron energy on the secondary emissionyield and deviation from the results of Maxwellian distri-bution has been graphically illustrated.
3 Charging kinetics
Consider a non-Maxwellian plasma consisting of neutralatoms/molecules, singly charged ions and electrons char-acterized by Kappa distribution of electron/ion velocitiesalong with uniformly dispersed spherical dust particles,characterized by a size distribution function. The ambi-ent plasma (i.e. in the absence of dust) is characterized bythe ambient electron (ne0, Te0), ion (ni0, Ti0) and neutral(n00, T0) density and temperature, respectively. Ioniza-tion of neutral atoms and recombination of electrons/ionsin the plasma and the accretion of electrons/ions on thedust particles and electron emission (in case of illumi-nated/thermal plasmas) from the dust surface are themechanisms considered for electron/ion production andannihilation.
In the context of present investigation and the sim-plicity in the analysis, the dust particles in the plasma areassumed to be characterized by the MRN power law ofsize distribution [38]; this can be expressed as:
where f(a)da is the number of particles per unit volumehaving radii between a and (a + da), s is a constant andthe parameter A is a normalization constant which maybe determined by integrating f(a)da within suitable radiispace (a1 ≤ a � a2) and equating it to the number densityof dust particles nd. The two useful parameters, which aredetermined from the size distribution viz. the mean andthe root mean square values of radius of the dust grains,are given by:
a =
⎛⎝
a2∫
a1
af(a)da
⎞⎠
/⎛⎝
a2∫
a1
f(a)da
⎞⎠ = am
and
a2 =
⎛⎝
a2∫
a1
a2f(a)da
⎞⎠
/⎛⎝
a2∫
a1
f(a)da
⎞⎠ = a2
rms,
where a1 and a2 denote the extreme limits of radii in theregion characterized by the size distribution function f(a).
In further analysis the authors have exploited the factthat particles of the same material having a size distri-bution in a complex plasma, exhibit a uniform electricpotential [38–40] in the steady state, provided that thedependence of the absorption efficiency of light by theparticles on size is ignored; a detailed discussion aboutthis aspect may be seen in the paper by Sodha et al. [38].Utilizing the concept of uniform potential theory the basiccharging kinetics equations can be written as follows.
In the steady state, this equation yields a unique solutionfor the dust surface potential Vs(=Ze/a); since this pa-per aims to analyze the steady state (t → ∞) only, thissimplification does not lead to any error. The steady stateresults are independent of the initial boundary conditions.
where βi represents the coefficient of ionization andαr(Te) = αr0T
−μe cm3/s is a typical coefficient of recom-
bination [41] of electrons and ions.
The first two terms on the right hand side of equa-tions (17) and (18) correspond to the net gain in electronand ion densities due to ionization of neutral species andthe recombination of electrons and ions in the plasma. Thelast term in both the equations represents the net electronand ion currents, accreting on the surface of the dust par-ticles. Equations (17) and (18) are valid ideally for infiniteplasma, or in practice when the dimensions of the plasmaare large compared to the diffusion length; in many labo-ratory situations this is not valid. However in space it isvalid.
3.3 Electron and ion energy balance equations
d
dt
(32kBneTe
)= βin0εe − αrneni(3kBTe/2)
− πa2rmsnd(fecεec − feeεee) − Qe (19)
and
d
dt
(32kBniTi
)= βin0εi − αrneni(3kBTi/2)
− πa2rmsndficεic − Qi. (20)
The first two terms in equations (19) and (20) refer tothe net gain in the mean energy of the electrons and theions due to the ionization of neutral atoms and the re-combination of electrons and ions in the plasma; all theelectron/ion production and annihilation mechanisms inthe plasma have been lumped in the parameters βi andαi, respectively. The next term in both the equations cor-responds to the net loss in energy of electrons and ions,due to (i) their accretion on and (ii) electron emissionfrom the dust surface. Last term in both equations refersto the net power loss of electrons and ions due to elas-tic collisions between electron, ion and neutral speciesand their expressions can be seen from classic book byShkarofsky et al. [42]. However, the energy exchange inthe elastic collision has been ignored for computationalpurpose on account of small magnitude in comparison toother terms in energy balance equation viz. plasma ioniza-tion/recombination and electron/ion accretion over dustparticle surface.
It may be mentioned that the electrical neutrality ofthe complex plasma is inherent in the number balanceequations and can easily be derived as [38]:
∫(aVs/e)f(a)da + (ni − ne) = 0. (21)
The energy exchange due to elastic collisions between elec-trons, ions and neutral atoms has been ignored in thisanalysis on account of its small magnitude as comparedto the exchange of energy in the accretion, recombinationand ionization processes. On account of the large ther-mal capacity of the neutral atoms and dust particles andefficient energy exchange between them, it is a good ap-proximation to assume their temperatures to be the same
and constant. The value of the ionization coefficient maybe determined by applying the electron kinetics in dustfree plasmas, thus
βin00 = αr(Te0)ne0ni0 = αr(Te0)n2e0. (22a)
Further the mean energy of electrons and ions produceddue to ionization may be obtained by imposing the ini-tial conditions for dust free plasma nd = 0 in the energybalance equations and making use of equation (19); thus
εe = (3/2)kBTe0 and εi = (3/2)kBTi0. (22b)
For a chosen set of initial parameters, one can obtain thesteady state potential on dust grains and other parame-ters viz. ne, ni, nn, Te and Ti by simultaneous numericalintegration of equations (16)–(20) along with appropriateexpressions for other relevant parameters as t → ∞; thesteady state is indeed independent of the initial conditions,since as t → ∞, (d/dt = 0) the set of differential equationsreduces to a set of algebraic equations, independent of thevalues at t = 0.
4 Numerical results and discussion
A theoretical investigation of the charging kinetics of thedust particles, characterized by a size distribution func-tion (viz. MRN power law) in a non-Maxwellian plasmadefined by the κ-distribution function of electron/ion ve-locities, has been made; the characteristic charging (viz.accretion and emission) currents corresponding to the dustparticles depend on the plasma parameters (density, tem-perature and composition), grain potential and inherentdust properties (photoelectric yields, work function andsize distribution function). In this course an expressionfor the accretion rate and corresponding mean energy ofthe plasma particles (electrons/ions) characterized by theκ-distribution function, has been derived; further the for-mulation of the derivation for the electron accretion cur-rent has been applied to evaluate the efficiency of thesecondary electron emission from the dust particles byconsidering the appropriate path traversed by primaryelectrons inside the particle. In view of the openness [17]of the complex plasma and the temperature dependenceof the charging processes, the formulation incorporatescharge neutrality, the balance of number density and en-ergy of plasma species and the ionization/recombinationprocesses in the gaseous component; the charge neutralityis inherent in the number balance and charging equationsand it is hence not considered separately in the analysis.The kinetics has been developed on the basis of uniformpotential theory applicable to plasmas consisting of dustparticles characterized by a size distribution function; thistheory predicts that the surface potential over the dustgrains of the same material is independent of size in thesteady state and hence all the dust particles carry chargesof the same sign with magnitude proportional to the ra-dius of the particles. The accretion of electrons/ions on thedust particles and emission of electrons from dust grains
are considered as basic charging mechanism; the plasmamaintenance is taken care of by an external agency sourcethrough the ionization of neutral atoms and recombinationof electrons and ions. As seen in the earlier section thatthe non-Maxwellian character of the plasma only affectsthe accretion current of electrons/ions on the dust grainsand thus it is enough to consider the accretion mechanismto investigate the effect of κ-distribution of the plasma onthe dust charging kinetics and its deviation from the re-sults based on the Maxwellian distributions; the electronemission from the dust particles has not been taken intoaccount for the computation purpose.
Figure 1 illustrates the dependence of the parameterσ(=nκ
c /nmc ) and ε(=εκ
c /εmc ) on the spectral index (κ) as
a function of vs. Figure 1a indicates the following facts:(i) for vs > 0, σ displays a minimum corresponding tovs = 1/2 for all possible values of κ, (ii) σ is less thanunity (i.e. σ < 1) and decreases with increasing κ nearvs = 1/2, (iii) σ monotonically increases with increasingvs (usually for vs > 1), (iv) σ approaches unity for largeκ viz. when non-Maxwellian distribution of the plasmaapproaches the Maxwellian regime, (v) for vs � 0, σ dis-plays a weak dependence on vs and κ while it approachesunity for large values of κ. Physically this nature is thereflection of the presence of a high-energy particle tail inκ-distribution function. Figure 1b suggests that the meanenergy associated with accreting electrons/ions increaseswith increasing vs while it decreases with increase in κ;with increasing spectral index (κ) of the κ-distribution, εapproaches unity (Maxwellian regime).
The effect of non-Maxwellian plasma characterized byκ-distribution on secondary electron emission from thespherical dust particle has been illustrated in Figure 1c forthe following set of parameters a = 50 nm, kBTs = 3 eV,kBTe = 100 eV, ηse = 0.1, α = 1012 (eV)2/cm,K = 0.01/ eV and β0 = 106/cm; here a is the radius ofthe dust particle, Ts is the temperature of secondary elec-trons, ηse is the sticking coefficient of electrons on theparticle surface, α and K are the dust material dependentconstants and β0 is the attenuation constant for the sec-ondary electrons in the substance. The parameters α, Kand β0 have been chosen from Chow et al. [43] and book byShukla and Mamun [15] while kBTs has been taken fromGrasp et al. [44] (originally estimated by Sternglass [45]);the other parameters viz. a, kBTe and ηse has arbitrarilychosen for the illustration purpose and are good enoughto describe the SEE effect considered herein. The under-lined physics of the modified approach for the secondaryelectron emission from the spherical particles can be seenfrom the paper by Mishra et al. [37]. The figure reflectsthe fact that the secondary electron emission yield (δ) in-creases with increasing dust potential vse(=Vs/kBTs) anddecreasing κe; however the increase is slower in the caseof vs > 0 than that in case of vs ≤ 0 where δ increasessharply with increasing vs. This nature can be understoodin terms of larger the mean energy of electrons accretingover dust surface in a non-Maxwellian plasma; with in-creasing spectral index (κ) the yield approaches the valuecorresponding to a Maxwellian distribution of electrons.
Fig. 1. Dependence of σ(a) and ε (b) on; the labels p, q, r, s,t, u and v refer to κ = 3, 4, 5, 7, 10, 20 and 100, respectively.(c) Dependence of δκ on for the standard set of parametersstated in the text. The labels p, q, r, s, t and u refer to κe = 3,4, 5, 7 and 10, respectively, while broken curve correspond toMaxwellian plasma.
Fig. 2. Dependence of v0s on nd for the standard set of param-eters stated in the text; the labels p, q, r and s refer to (κe, κi) =(2.5, ∞), (2.5, 2.5), (∞, ∞) and (∞, 2.5) respectively.
For a numerical appreciation of the effect of the non-Maxwellian nature of the plasma characterized by theκ-distribution function on the charging kinetics of dustparticles in a plasma, computations have been made fora dark complex plasma (i.e. in the absence of electronemission from dust particles) corresponding to followingstandard set of parameters; the effect of various param-eters on the dust potential and other plasma parame-ters in complex plasma has been studied by varying oneand keeping others the same. ne0 = ni0 = 1010/cm3,n00 = 103ne0, a1 = 10 nm, a2 = 10 μm, s = 1.0,Te0 = 1000 K, Ti0 = 400 K, T0 = 300 K, κe = κi = 3,m0 ≈ mi = 30 a.m.u., μ = 1.0 and αr0 = 5 × 10−7cm3/s.
The paper aims to develop the physical understandingof the dust charging kinetics in a Lorentzian plasma andthe choice of numbers are essentially illustrative.
The effect of the non-Maxwellian nature of the plasmacharacterized by κ-distribution function on the dust charg-ing has been illustrated in Figure 2. It is seen that in a darkcomplex plasma having κ-distribution of electrons, thedust surface potential v0s (=Vs/kBTe0, hence the charge)is larger than that in case of Maxwellian plasma whilethe opposite is true when ions are characterized by κ-distribution for moderate values of nd; however the trendgets reversed for large values of nd. These facts can be at-tributed to the dependence of the electron/ion accretioncurrent on vs and κ. The monotonic decrease of v0s withincreasing nd can be attributed to smaller availability ofthe electrons/ions per particle for the accretion over thedust particles. Further it is of interest to comment on roleof such non-Maxwellian character of electrons/ions in thecrystal structure formation in dusty plasmas which usuallycharacterized by coupling parameter. Since the couplingparameter directly proportional to the charge on the par-ticle, as a consequence of Figure 2 the coupling parameterdisplays nature similar to that v0s dependence and signif-icantly departs from usual Maxwellian distribution. Theeffect of varying spectral index on the dust surface po-tential and other plasma parameters in a non-Maxwellian
Fig. 3. Dependence of v0s and ne on nd for the standard set ofparameters stated in the text with κi → ∞; the labels p, q, r,s, t and u refer to κe = 3, 4, 5, 10, 20 and ∞ respectively. Solidand broken lines refer to left and right hand scales, respectively.(b) Dependence of Te and Ti on nd; the parameters and labelsare the same as in Figure 3a.
plasma having electrons and ions described by κ- andMaxwellian distribution respectively, has been illustratedin Figure 3. It is noticed that the surface potential de-creases with increasing κe and approaches the Maxwellianresults for large κe values; this nature has been shown inFigure 3a. Corresponding electron/ion temperatures havebeen displayed in Figure 3b which increase with increasingκe and tend to approach the value corresponding to theMaxwellian plasma as κe → ∞. This can be ascribed tothe decrease in the mean energy of accreting electrons/ionsand corresponding accretion current with increasing κ.The dependence of electron density (ne) on the spectralindex (κe) and nd is the reflection of the electron/iontemperature dependence on κe and nd. Correspondingion density can be obtained by using the quasi chargeneutrality of the complex plasma (i.e. Eq. (21)), which isinherently valid throughout the process. The set of Fig-ure 4 displays the dependence of the surface potentialand other plasma parameters in a non-Maxwellian plasma
(a)
(b)
Fig. 4. Dependence of v0s and ne on nd for the standard setof parameters stated in the text with κe → ∞; the labels p, q, rand s refer to κi = 3, 5, 10 and ∞, respectively. (b) Dependenceof Te and Ti on nd; the parameters and labels are the same asin Figure 4a.
characterized by the κ-distribution of ions and Maxwelliandistribution of electrons. It is seen that the surface poten-tial increases with increasing κi and approaches the valuecorresponding to Maxwellian regime, as displayed in Fig-ure 4a; this behaviour can be explained on the basis ofthe ion accretion current dependence on vs and κi. Theelectron/ion temperature does not show significant devi-ation from the Maxwellian results in this case; this hasbeen shown in Figure 4b. Thus it is seen that the non-Maxwellian nature of the plasma significantly affects thecharge on the particle and hence the plasma environment;however the effect is more pronounced when the electronsin a complex plasma are characterized by κ-distributionfunction of energy.
5 Conclusions
A theoretical basis for the understanding of the chargingkinetics of the dust particles in a non-Maxwellian plasmacharacterized by a κ-distribution function of energy hasbeen presented herein; the analysis is based on the uniform
potential theory, applicable to complex plasma with uni-formly dispersed spherical dust particles characterized bya size distribution function. Owing to the openness [17] ofthe complex plasma, the annihilation/formation of elec-trons/ions have been taken care of through number andenergy balance of all the plasma constituents; collectivebehaviour of the dust particles has also been taken into ac-count. The expressions for the electron/ion accretion cur-rent on the dust particles and corresponding mean energyhas been derived and graphically illustrated as a functionof surface potential and spectral index. It is interesting tomention that the accretion current exhibits a minimumat vs = 1/2 corresponding to vs > 0, this significantlyaffects the dust charging kinetics; however the mean en-ergy monotonically increases with vs. It is noticed thatthe results obtained for the non-Maxwellian plasma ap-proaches the results corresponding to Maxwellian plasmaas the spectral index increases (i.e. κ → ∞). Furtherthe dependence of the dust surface potential and otherplasma parameters on the dust density has been evalu-ated and is graphically illustrated. The formulation andresults are useful in understanding the dust charging innon-equilibrium laboratory and space dusty plasmas.
One of the authors (S.K.M.) is grateful to Department of Sci-ence and Technology (DST) for financial support and to Prof.M.P. Verma for helpful discussion.
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