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On the thermophoresis in dense dust structures in neon plasma
Physics Letters A 383 (2019) 125853
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Physics Letters A www.elsevier.com/locate/pla
On the thermophoresis in dense dust structures in neon plasma V.V. Shumova a,∗ , D.N. Polyakov a , E.K. Mataybaeva b , L.M. Vasilyak a a b
Joint Institute for High Temperatures of Russian Academy of Sciences, Izhorskaya 13 Bldg 2, Moscow 125412, Russia Moscow Institute of Physics and Technology, Institutskiy Pereulok 9, Dolgoprudny, Moscow Region 141701, Russia
a r t i c l e
i n f o
Article history: Received 18 March 2019 Received in revised form 10 June 2019 Accepted 24 July 2019 Available online 30 July 2019 Communicated by F. Porcelli Keywords: Glow discharge Neon Dusty plasma Hollow dust structures Void Thermophoresis
a b s t r a c t Numerical study of the effect of dust particle concentration on the thermophoretic force acting on a dust particle inside a dust structure in plasma has been carried out. The experimental data on the formation of voids in dust structures formed by 2.55 μm dust particles in a glow dc discharge in neon have been used. The simulation has been performed using the diffusion-drift model with taking into account joule heating of discharge. The dependence of the thermophoretic force acting on a dust particle in a dust structure on the ratio of atom mean free path to the distance between the adjacent particles in the dust structure has been obtained. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Micron-sized particles with coatings having specified properties gain ever increasing prospects for application in various industrial technologies and medicine [1–4]. Plasma of gas discharges is employed for surface modification and for deposition of oriented coatings onto particles of micron and submicron sizes [2, 5,6], along with other techniques [7,8]. To improve technologies imploying plasma with dust particles (dusty plasma), the development of conceptions considering the processes determining the shape and position of dust structures in technological devices is required. Shape, position of dust structures, and concentration of microparticles maintained in a plasma are self-consistently determined by parameters of a potential determined by a superposition of forces acting on microparticles in a plasma trap. In the presence of a thermal field, the shape and position of the dust structures are influenced by the thermophoretic force determined by temperature gradients in a plasma. Thermal fields can be applied for diagnostics and calculation of force fields acting on dust particles [9,10], the charge of dust particles and the magnitude of the electric field [11]. An increase in heat generation in a discharge causes a change in the shape of dust structures [12]. This can lead to the formation of dust structures with cavities (hollow dust structures) – dust voids [12–15].
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Corresponding author. E-mail address: [email protected] (V.V. Shumova).
https://doi.org/10.1016/j.physleta.2019.125853 0375-9601/© 2019 Elsevier B.V. All rights reserved.
With an increase in concentration of microparticles in dust structures, the existing conceptions considering the forces acting on the microparticles (including the thermophoretic force), seem to be insufficient to describe the shapes and properties of dense dust structures in plasma [16,17]. In [17], the transition from solid dust structures to hollow ones in a glow discharge in neon was experimentally studied and represented as the transition line in coordinates of “neon pressure – discharge current”, I ( P ). Experiments were performed with 2.55 μm and 4.14 μm spherical particles. With larger size of dust particles, the transition to hollow structures occurred at lower discharge current. The dependence of the shape of dust structures on the discharge parameters was simulated using the diffusion-drift model of a homogeneous positive column of a glow discharge in neon with dust particles. The model considered joule heating of the discharge and plasma energy dissipation on the discharge tube walls and the dust particles. The simulation indicated that the experimentally obtained dependence of the dust structure shapes on the discharge current is associated with an increase in heat release in the discharge and, accordingly, with an increase in the contribution of the thermophoretic force in the net force acting on dust particles in the radial direction. It was found that satisfactory accuracy of the combined description of experimental data on the boundary of a transition to hollow dust structures for dust particles of various sizes can only be achieved when a thermophoretic coefficient for 4.14 μm particles is lower than for 2.55 μm particles (at equal pressure). The simulation indicated that the thermophoretic force acting on the microparticles inside the dust structure is re-
2
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lated to the size of the microparticles and to the parameters of dust structures in a more complicated manner than the classical theory predicts for a single particle in unbounded plasma. It was found that for a microparticle inside the dust structure, the neighboring microparticles produce a shadowing effect for the transfer of momentum in the plasma. This leads to a change in the thermophoretic coefficient in the dust structure and requires additional examination. The aim of the present study is the numerical investigation of the influence of dust particle concentration on the magnitude of the thermophoretic force within the framework of the verification and further study of the shadowing effect inside dust structures. For the simulations, we have used the experimental data obtained in [17] for 2.55 μm dust particles. 2. Numerical model The simulations have been carried out on the basis of the diffusion-drift model of a uniform positive column of a glow discharge in neon with dust particles, with taking into account joule heat release and plasma energy dissipation on the discharge tube walls and on the dust particles. The basic points of the diffusiondrift approximation were discussed in [18]. The model of uniform positive column of a glow discharge with constant longitudinal electric field and uniform gas density in the longitudinal direction was implemented. The electric field was represented as a superposition of applied longitudinal component E l and self-consistent radial component E r determined by radial gradient of plasma potential. Neon plasma was considered to be composed of atoms, ions, electrons and metastable atoms as described in [19]. Within this approach, the flow densities of electrons and ions Γe,i are the sums of drift and diffusion terms, and the flow density of metastable neon atoms Γm is governed by diffusion only:
Γe = −μene E − D e ∇ ne ,
(1)
Γi = μini E − D i ∇ ni ,
(2)
Γm = − D m ∇ nm ,
(3)
where ne,i,m are concentrations of electrons, ions and metastable atoms, and μe,i and D e,i,m are their mobility and diffusion coefficients respectively. Ion and electron flows Γi and Γe in the regime of ambipolar plasma are equal and submit to the equation of continuity, as well as metastable atom flow Γm :
∇Γe,i,m = qe,i.m .
(4)
The source terms of species qi,e,m are determined by rates of collision processes in a plasma bulk and by rates of plasma losses on dust particle surface: qe = ki na ne + kim nm ne + kmm n2m − nd J de , qi = ki na ne + kim nm ne + kmm n2m − nd J di , and qm = kexc na ne − kim nm ne − 2kmm n2m − kqa nm na − kqe nm ne − kr nm ne − nd J dm . Here, ki and kim are the rate coefficients of ionization from the ground and metastable states of neon correspondingly, kmm the rate coefficient of chemi-ionization, kexc the rate coefficient of excitation of metastable state from the ground, kqe and kqa the rate coefficients of the metastable atom quenching in collisions with electrons and atoms, and kr the rate coefficient of the metastable atom quenching to resonant state, tabulated in [20], na and nd are concentrations of atoms and dust particles correspondingly. J di and J de are the flows of ions and electrons to the dust particle surface, determining the dust particle equilibrium charge Z d , calculated using the collision enhanced collection (CEC) model for dust particle charging in weakly collisional plasma [21–23]; J d is the free molecular flow of metastable atoms to the dust particle surface [24].
Ion, electron and metastable atom distributions meet the zero boundary condition at the discharge tube wall, and the boundary condition following from the symmetry of the discharge on the axis: (dni,e,m /dr )|r =0 = 0. The self-contained system of equations is completed with equation of dusty plasma quasi-neutrality, and the integral equation for the total discharge current I = 2π e r Γel dr, where Γel = μe ne E l . The mean electron energy and transport coefficients were obtained using the electron Boltzmann equation solver BOLSIG+ [25]. The radial distribution of dust particles nd (r ) was defined by the axisymmetric flat profile with slight end blurring nd (r ) = nd,0 for r ≤ rd , and nd (r ) = nd,0 exp{(rd − r )/0.1R } for r > rd , where nd,0 is a dust particle concentration on the axis of a discharge tube with radius R. This type of distribution was chosen for reasons of maximum identity with the distributions observed in our previous experiments [13,18]. The exponential blurring at the boundary of the dust structure was chosen to provide smooth distribution, as it is commonly accepted in this type of simulations [26]. Solving this boundary problem, we found radial distributions of plasma components, dust particle charge Z d , radial E r and longitudinal E l components of the electric field. The radial temperature profile of a gas T (r ) in the discharge tube was calculated by solving the stationary one-dimensional heat conduction equation [15,17] with an assumption that the power supplied to the discharge goes to gas heating:
∇(r κ (∇ T )) = − Q r ,
(5)
where κ is the thermal conductivity of neon, with boundary conditions specified on the discharge axis (d2 T /dr 2 )|r =0 = (dT /dr )|r =0 = 0, and on the discharge tube wall, T ( R ) = 295 K. Heat release Q in discharge was determined by the longitudinal current density as:
Q = eΓel E l .
(6)
Net force F res acting on a dust particle in a radial direction, is a sum of electric field force F e , ion drag force F i and thermophoretic force F th . The electric field force is F e = Z d e E r . The ion drag force was calculated using expression from [27] as:
F i = −mi ni
vv f i (v)[σc ( v ) + σs ( v )]dv,
(7)
where v is the ion velocity and mi is the mass, f i (v) is the ion velocity distribution function, σc ( v ) and σs ( v ) are the velocity dependent momentum-transfer cross-sections for the ion collection and scattering, respectively, calculated in [27]. The thermophoretic force F th was calculated in the approximation of a continuous medium for the heat flux to the wall and the free molecular mode for the flux to the microparticle surface as:
F th = −C (kb d2 /4σtr )∇ T ,
(8)
where kb is the Boltzmann constant, d is the dust particle diameter, σtr is the transport cross-section of neon [28], and C is the variable coefficient discussed below. Expression (8) was taken from [29], where C was constant, C = 3.33. In our simulations, the formation of a minimum on the potential energy profile of a dust particle was taken as a criterion for the formation of an internal cavity in dusty structures [12]:
U =−
[ F e + F th + F i ]dr .
(9)
The value of the current, at which the resultant force F res (0) near the discharge axis becomes negative value, was considered to be the boundary value for the transition from solid dust structure to
V.V. Shumova et al. / Physics Letters A 383 (2019) 125853
Fig. 1. Line of transition from homogeneous to hollow dust structures for 2.55 μm exp particles: experimental data (triangles); interpolation of experimental data I b ( P ) (dashed line); simulation of transition boundary I bsim ( P ) (solid line). I – the region of homogeneous dust structures, II – the region of dust structures with void.
hollow one. At this value, a minimum on the profile of the potential energy of dust particle displaced from the center of the discharge to its wall, is formed. In this model, we consider initial moment of void formation on the discharge axis, supposing that the dust particles are still immobile, and the dust particle density gradient is zero due to the homogeneity of the dust structure. The latter assumption was based on experimental observations [12,13, 18]. In this case, the equation of force balance does not include the term associated with the gradient in the dust density. The simulations were carried out with the experimentally obtained discharge parameters (neon pressure and discharge current) and concentration of microparticles equal to the experimentally measured one. In addition, the concentration of microparticles was varied in order to numerically study its effect on the magnitude of the thermophoretic force. 3. Results and discussion In [17], it was found that an increase in the discharge current at constant pressure leads to a change in the shape of the dust structure, i.e. to an increase in its radial and a decrease in its axial size. In the radial direction, dust particles displace to the walls of the discharge tube, and a cavity (void) is formed in the center of the structure. The solid dust structure undergoes an internal discontinuity on the discharge axis. In [17], the transition boundary from solid to hollow dust structures was experimentally obtained in the “gas pressure – discharge current” coordinates for 2.55 μm spherical particles, and their experimentally obtained concentration nd,0 in dust structures was 2 × 104 cm−3 . The experimental data obtained in [17], are indicated in Fig. 1 with triangles. For convenience, let us denote the interpolation function of the experimental data on the transition exp exp boundary as I b ( P ). At I < I b ( P ), the dust structures were hoexp
mogeneous, while at I > I b ( P ) they were hollow. According to our previous observations [12,13,17,18], the dust structures, both below and above the transition boundary, represent specific analogues of solids as free motion of individual dust particles was impossible. However, such dust structures can not be characterized as crystals because of the absence of a long-range order. Fig. 1 also represents the simulated transition boundary I bsim ( P ), obtained with the model parameters discussed below. The simulation confirmed that with increasing pressure, the transition to hollow structures appeared at a lower value of the discharge current. The simulation also demonstrated that this was caused by an increase in joule heating of the discharge and, accordingly, by an
3
Fig. 2. Radial distribution of the potential energy of dust particles at P = 0.6 Torr, nd,0 = 2 × 104 cm−3 for different values of discharge current I : 0.6 mA (1), 2.0 mA (2), 3.0 mA (3).
increase in the contribution of the thermophoretic force to the balance of forces acting on dust particles in the radial direction. Fig. 2 displays the potential energy profiles U (r ) of dust particles for three increasing values of discharge current I . The transformation of U (r ) with I visualizes the formation of a void. Dust particles occupy spatial positions corresponding to the minimum in their potential energy. The value of I , at which the solid dust structure loosed the continuity on the discharge axis (r = 0) and a void was formed, corresponds to the transition of the potential energy of dust particles below zero. Starting the analysis of the thermophoresis inside the dust structure, we have analyzed the ratio between the magnitudes of forces acting on dust particles. In the studied range of pressures, dust particle concentrations and at the discharge currents of about exp I b ( P ), the electric force is directed to the center of the discharge, and the ion drag and thermophoretic forces act oppositely. Inside the dust structure, the radial electric field is lower than that in the discharge without dust particles [18,19,26], and this significantly decreases the ion drift; hereupon, in the vicinity of the discharge axis F th (r ) is higher than F i (r ). Simulations show that, at every P , the thermophoretic force prevails in our conditions, being a few times higher than the ion drag force. The ratio F th (r )/ F i (r ) varies over the radius of dust structure within the factor of two, as the thermophoresis increases faster. For instance, closely to the axis of the discharge, F th (r )/ F i (r ) changes from 2.15 to 3.3 upon increasing the pressure from 0.35 to 0.9 torr. It should be noted that in other conditions the relationship between these forces can be different. For instance, in RF discharges, an inverse relationship is usually observed (F i > F th ), especially when the distance between electrodes is small. To study the effect of the microparticle concentration on the magnitude of the thermophoretic force, we simulated an experimentally obtained transition boundary with a variable value of the concentration of microparticles in the dust structure. The experiexp mental data on I b ( P ) transition boundary, represented in Fig. 1, were simulated with concentrations of dust particles of 2 × 104 , 5 × 104 and 105 cm−3 . For each nd,0 , the coefficient C in equation (8) was varied to bring the simulated transition line I bsim ( P ) into exp
coincidence with the experimental one, I b ( P ). Since the discrepancy in the calculations was insignificant the simulation results are presented in Fig. 1 by a single dashed line. The obtained values of C are presented in Fig. 3. In Fig. 3 one can see that even at non-zero concentration of dust particles, coefficient C exceeds the corresponding value taken in the approximation of electrically neutral spherical particle in a
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Fig. 3. The dependence of coefficient C in (8) on pressure at various concentrations of microparticles nd,0 : 2 × 104 cm−3 (1), 5 × 104 cm−3 (2), 3–105 cm−3 (3).
single particle regime [30]. This difference can be a consequence of the fact that dust particles are contained in a plasma-dust structure. The dust structure represents an object, in which dust particles are not completely independent, so the single-particle regime is not applicable. In this study, we imply that C is a certain effective coefficient, reflecting the thermophoretic force acting on a strongly charged dust particle located inside the dust structure. There are several studies on dusty plasmas, in which the results for thermophoretic force with C also exceeding that for a solitary microparticle were obtained. In the article by Rothermel et al. [29], the value of C = 3.33 was obtained, and it was indicated that it was almost two times less than that in the study by Havnes et al. [32]. Let us note that in our simulations, the calculation of the dust particle charge can be a source of uncertainty for the obtained coefficient C , since under our conditions the thermophoretic force is less than the electric one. In Fig. 3 one can see that, with an increase in the concentration of dust particles, the coefficient C decreases, and in the pressure range under study, the C ( P ) dependence is stronger at smaller nd,0 (that is, at larger distance between dust particles). One can also see that the C ( P ) dependence vanishes with increasing pressure, i.e. with decreasing mean free path of neon atoms. The obtained dependence of the thermophoretic force on the concentration of dust particles in the dust structure suggests that the thermophoretic force acting on a single dust particle located in an unbounded plasma differs from that acting on a dust particle located inside a dense dust structure. Starting from the fact that, for every nd,0 , the C ( P ) dependencies vanish with pressure, one can conclude that the reason is the variation in the ratio between the atom mean free path, λ, and the distance between the adjacent particles in the dust structure, L. This ratio can be considered as the analogue of the dimensionless parameter related to the geometry of the system, Knudsen number, K nL = λ/ L [30,31]. Indeed, at a pressure of about 0.3 torr, λ is about 300 μm, and L at nd,0 = 2 × 104 cm−3 is about 370 μm (that is, λ slightly exceeds L), while at nd,0 = 105 cm−3 , the latter is about 220 μm (that is, L becomes less than λ). With an increase in K nL due to an increase in concentration of dust particles in the structure, the effect of mutual shadowing of dust particles enhances. This affects the process of momentum transfer in the direction opposite to a temperature gradient in a gas. One can see that the effect of increasing dust particle concentration seems to be similar to the effect of reducing the thermophoretic force acting on a dust particle located close to the walls in dusty plasma, which was theoretically considered in [32]. Taking into account the closeness of the plasma walls, the authors of [32] analyzed the perturbation of the velocity distribution
function, formed in an “infinite” gas with a temperature gradient, resulting from atom collisions with a wall. The reflected atom temperature was supposed to be a wall temperature. They found that, in comparison with the results calculated from the “standard” formula, the thermophoretic force reduced out to many neutral gas molecular collision lengths from the wall. The similar effect was found in [17], where it was produced by increase of dust particle diameter. The effect of dust particle shadowing observed here, resembles those in another phenomena taking place in dusty plasmas with plasma particle flows. The example can be the ion drift in a strong electric field, which generates a perturbed region of plasma density around the dust particle due to the downstream focusing of ions – the so called “plasma wake” [33–35]. Recently, the influence of ion mean free path length on plasma polarization behind a dust particle in an external electric field was studied in [36]. The dependencies of the distributions of the space charge density and potential, as well as the dependence of the dipole moment of the ion cloud on the ion mean free path, were presented. Another example is the shadowing effect arising between dust particles due to the anisotropy in ion-impact momentum deposition. For a single dust particle in a stationary plasma, the ion flow to the dust particle surface is spherically symmetric. But in the vicinity of another dust particle, the ion flow may be partially intercepted by it. As a result, this causes an attractive force between the neighboring dust particles, known as the shadowing force [35]. In cases mentioned above [33–35], the phenomena are governed by interactions between charged particles, while in the case of thermophoresis, the effect of shadowing manifests for neutral plasma component. 4. Conclusion The simulation of experimental data on the formation of void in dust structures in a glow discharge in neon has allowed us to numerically study the effect of the dust particle concentration on the thermophoretic force acting on a dust particle inside the dust structure. The shadowing effect of neighboring dust particles inside the dust structure have been numerically demonstrated. It has been found that with an increase in the concentration of dust particles (that is, with a decrease in the ratio of the distance between adjacent dust particles in the dust structure to the mean free path of atoms in neon) the thermophoretic force decreases. The dependence obtained confirms the assumption that the thermophoretic force for a single dust particle in plasma differs from that for a dust particle inside the dust structure. One can conclude that the thermophoretic force acting on a dust particle in a dust structure depends on the ratio of the atom mean free path to the distance between the adjacent particles in the dust structure. Acknowledgement The financial support of the Program of basic research of the Presidium of RAS No. 13 “Condensed matter and plasma at high energy densities” is gratefully acknowledged. References [1] I.K. Herrmann, R.N. Grass, W.J. Stark, High-strength metal nanomagnets for diagnostics and medicine: carbon shells allow long-term stability and reliable linker chemistry, Nanomedicine 4 (7) (2009) 787–798. [2] T.M. Vasil’eva, Experimental study of the synthesis of supramolecular complexes in hybrid dusty plasma, High Energy Chem. 45 (1) (2011) 66–72. [3] M.N. Vasiliev, A.H. Mahir, Synthesis and deposition of coatings in the electronbeam plasma, Surf. Coat. Technol. 180–181 (2004) 132–135. [4] Y. Xu, et al., Cobalt nanoparticles coated with graphitic shells as localized radio frequency absorbers for cancer therapy, Nanotechnology 19 (2008) 435102, https://doi.org/10.1088/0957-4484/19/43/435102.
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Physics Letters A www.elsevier.com/locate/pla
On the thermophoresis in dense dust structures in neon plasma V.V. Shumova a,∗ , D.N. Polyakov a , E.K. Mataybaeva b , L.M. Vasilyak a a b
Joint Institute for High Temperatures of Russian Academy of Sciences, Izhorskaya 13 Bldg 2, Moscow 125412, Russia Moscow Institute of Physics and Technology, Institutskiy Pereulok 9, Dolgoprudny, Moscow Region 141701, Russia
a r t i c l e
i n f o
Article history: Received 18 March 2019 Received in revised form 10 June 2019 Accepted 24 July 2019 Available online 30 July 2019 Communicated by F. Porcelli Keywords: Glow discharge Neon Dusty plasma Hollow dust structures Void Thermophoresis
a b s t r a c t Numerical study of the effect of dust particle concentration on the thermophoretic force acting on a dust particle inside a dust structure in plasma has been carried out. The experimental data on the formation of voids in dust structures formed by 2.55 μm dust particles in a glow dc discharge in neon have been used. The simulation has been performed using the diffusion-drift model with taking into account joule heating of discharge. The dependence of the thermophoretic force acting on a dust particle in a dust structure on the ratio of atom mean free path to the distance between the adjacent particles in the dust structure has been obtained. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Micron-sized particles with coatings having specified properties gain ever increasing prospects for application in various industrial technologies and medicine [1–4]. Plasma of gas discharges is employed for surface modification and for deposition of oriented coatings onto particles of micron and submicron sizes [2, 5,6], along with other techniques [7,8]. To improve technologies imploying plasma with dust particles (dusty plasma), the development of conceptions considering the processes determining the shape and position of dust structures in technological devices is required. Shape, position of dust structures, and concentration of microparticles maintained in a plasma are self-consistently determined by parameters of a potential determined by a superposition of forces acting on microparticles in a plasma trap. In the presence of a thermal field, the shape and position of the dust structures are influenced by the thermophoretic force determined by temperature gradients in a plasma. Thermal fields can be applied for diagnostics and calculation of force fields acting on dust particles [9,10], the charge of dust particles and the magnitude of the electric field [11]. An increase in heat generation in a discharge causes a change in the shape of dust structures [12]. This can lead to the formation of dust structures with cavities (hollow dust structures) – dust voids [12–15].
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Corresponding author. E-mail address: [email protected] (V.V. Shumova).
https://doi.org/10.1016/j.physleta.2019.125853 0375-9601/© 2019 Elsevier B.V. All rights reserved.
With an increase in concentration of microparticles in dust structures, the existing conceptions considering the forces acting on the microparticles (including the thermophoretic force), seem to be insufficient to describe the shapes and properties of dense dust structures in plasma [16,17]. In [17], the transition from solid dust structures to hollow ones in a glow discharge in neon was experimentally studied and represented as the transition line in coordinates of “neon pressure – discharge current”, I ( P ). Experiments were performed with 2.55 μm and 4.14 μm spherical particles. With larger size of dust particles, the transition to hollow structures occurred at lower discharge current. The dependence of the shape of dust structures on the discharge parameters was simulated using the diffusion-drift model of a homogeneous positive column of a glow discharge in neon with dust particles. The model considered joule heating of the discharge and plasma energy dissipation on the discharge tube walls and the dust particles. The simulation indicated that the experimentally obtained dependence of the dust structure shapes on the discharge current is associated with an increase in heat release in the discharge and, accordingly, with an increase in the contribution of the thermophoretic force in the net force acting on dust particles in the radial direction. It was found that satisfactory accuracy of the combined description of experimental data on the boundary of a transition to hollow dust structures for dust particles of various sizes can only be achieved when a thermophoretic coefficient for 4.14 μm particles is lower than for 2.55 μm particles (at equal pressure). The simulation indicated that the thermophoretic force acting on the microparticles inside the dust structure is re-
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lated to the size of the microparticles and to the parameters of dust structures in a more complicated manner than the classical theory predicts for a single particle in unbounded plasma. It was found that for a microparticle inside the dust structure, the neighboring microparticles produce a shadowing effect for the transfer of momentum in the plasma. This leads to a change in the thermophoretic coefficient in the dust structure and requires additional examination. The aim of the present study is the numerical investigation of the influence of dust particle concentration on the magnitude of the thermophoretic force within the framework of the verification and further study of the shadowing effect inside dust structures. For the simulations, we have used the experimental data obtained in [17] for 2.55 μm dust particles. 2. Numerical model The simulations have been carried out on the basis of the diffusion-drift model of a uniform positive column of a glow discharge in neon with dust particles, with taking into account joule heat release and plasma energy dissipation on the discharge tube walls and on the dust particles. The basic points of the diffusiondrift approximation were discussed in [18]. The model of uniform positive column of a glow discharge with constant longitudinal electric field and uniform gas density in the longitudinal direction was implemented. The electric field was represented as a superposition of applied longitudinal component E l and self-consistent radial component E r determined by radial gradient of plasma potential. Neon plasma was considered to be composed of atoms, ions, electrons and metastable atoms as described in [19]. Within this approach, the flow densities of electrons and ions Γe,i are the sums of drift and diffusion terms, and the flow density of metastable neon atoms Γm is governed by diffusion only:
Γe = −μene E − D e ∇ ne ,
(1)
Γi = μini E − D i ∇ ni ,
(2)
Γm = − D m ∇ nm ,
(3)
where ne,i,m are concentrations of electrons, ions and metastable atoms, and μe,i and D e,i,m are their mobility and diffusion coefficients respectively. Ion and electron flows Γi and Γe in the regime of ambipolar plasma are equal and submit to the equation of continuity, as well as metastable atom flow Γm :
∇Γe,i,m = qe,i.m .
(4)
The source terms of species qi,e,m are determined by rates of collision processes in a plasma bulk and by rates of plasma losses on dust particle surface: qe = ki na ne + kim nm ne + kmm n2m − nd J de , qi = ki na ne + kim nm ne + kmm n2m − nd J di , and qm = kexc na ne − kim nm ne − 2kmm n2m − kqa nm na − kqe nm ne − kr nm ne − nd J dm . Here, ki and kim are the rate coefficients of ionization from the ground and metastable states of neon correspondingly, kmm the rate coefficient of chemi-ionization, kexc the rate coefficient of excitation of metastable state from the ground, kqe and kqa the rate coefficients of the metastable atom quenching in collisions with electrons and atoms, and kr the rate coefficient of the metastable atom quenching to resonant state, tabulated in [20], na and nd are concentrations of atoms and dust particles correspondingly. J di and J de are the flows of ions and electrons to the dust particle surface, determining the dust particle equilibrium charge Z d , calculated using the collision enhanced collection (CEC) model for dust particle charging in weakly collisional plasma [21–23]; J d is the free molecular flow of metastable atoms to the dust particle surface [24].
Ion, electron and metastable atom distributions meet the zero boundary condition at the discharge tube wall, and the boundary condition following from the symmetry of the discharge on the axis: (dni,e,m /dr )|r =0 = 0. The self-contained system of equations is completed with equation of dusty plasma quasi-neutrality, and the integral equation for the total discharge current I = 2π e r Γel dr, where Γel = μe ne E l . The mean electron energy and transport coefficients were obtained using the electron Boltzmann equation solver BOLSIG+ [25]. The radial distribution of dust particles nd (r ) was defined by the axisymmetric flat profile with slight end blurring nd (r ) = nd,0 for r ≤ rd , and nd (r ) = nd,0 exp{(rd − r )/0.1R } for r > rd , where nd,0 is a dust particle concentration on the axis of a discharge tube with radius R. This type of distribution was chosen for reasons of maximum identity with the distributions observed in our previous experiments [13,18]. The exponential blurring at the boundary of the dust structure was chosen to provide smooth distribution, as it is commonly accepted in this type of simulations [26]. Solving this boundary problem, we found radial distributions of plasma components, dust particle charge Z d , radial E r and longitudinal E l components of the electric field. The radial temperature profile of a gas T (r ) in the discharge tube was calculated by solving the stationary one-dimensional heat conduction equation [15,17] with an assumption that the power supplied to the discharge goes to gas heating:
∇(r κ (∇ T )) = − Q r ,
(5)
where κ is the thermal conductivity of neon, with boundary conditions specified on the discharge axis (d2 T /dr 2 )|r =0 = (dT /dr )|r =0 = 0, and on the discharge tube wall, T ( R ) = 295 K. Heat release Q in discharge was determined by the longitudinal current density as:
Q = eΓel E l .
(6)
Net force F res acting on a dust particle in a radial direction, is a sum of electric field force F e , ion drag force F i and thermophoretic force F th . The electric field force is F e = Z d e E r . The ion drag force was calculated using expression from [27] as:
F i = −mi ni
vv f i (v)[σc ( v ) + σs ( v )]dv,
(7)
where v is the ion velocity and mi is the mass, f i (v) is the ion velocity distribution function, σc ( v ) and σs ( v ) are the velocity dependent momentum-transfer cross-sections for the ion collection and scattering, respectively, calculated in [27]. The thermophoretic force F th was calculated in the approximation of a continuous medium for the heat flux to the wall and the free molecular mode for the flux to the microparticle surface as:
F th = −C (kb d2 /4σtr )∇ T ,
(8)
where kb is the Boltzmann constant, d is the dust particle diameter, σtr is the transport cross-section of neon [28], and C is the variable coefficient discussed below. Expression (8) was taken from [29], where C was constant, C = 3.33. In our simulations, the formation of a minimum on the potential energy profile of a dust particle was taken as a criterion for the formation of an internal cavity in dusty structures [12]:
U =−
[ F e + F th + F i ]dr .
(9)
The value of the current, at which the resultant force F res (0) near the discharge axis becomes negative value, was considered to be the boundary value for the transition from solid dust structure to
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Fig. 1. Line of transition from homogeneous to hollow dust structures for 2.55 μm exp particles: experimental data (triangles); interpolation of experimental data I b ( P ) (dashed line); simulation of transition boundary I bsim ( P ) (solid line). I – the region of homogeneous dust structures, II – the region of dust structures with void.
hollow one. At this value, a minimum on the profile of the potential energy of dust particle displaced from the center of the discharge to its wall, is formed. In this model, we consider initial moment of void formation on the discharge axis, supposing that the dust particles are still immobile, and the dust particle density gradient is zero due to the homogeneity of the dust structure. The latter assumption was based on experimental observations [12,13, 18]. In this case, the equation of force balance does not include the term associated with the gradient in the dust density. The simulations were carried out with the experimentally obtained discharge parameters (neon pressure and discharge current) and concentration of microparticles equal to the experimentally measured one. In addition, the concentration of microparticles was varied in order to numerically study its effect on the magnitude of the thermophoretic force. 3. Results and discussion In [17], it was found that an increase in the discharge current at constant pressure leads to a change in the shape of the dust structure, i.e. to an increase in its radial and a decrease in its axial size. In the radial direction, dust particles displace to the walls of the discharge tube, and a cavity (void) is formed in the center of the structure. The solid dust structure undergoes an internal discontinuity on the discharge axis. In [17], the transition boundary from solid to hollow dust structures was experimentally obtained in the “gas pressure – discharge current” coordinates for 2.55 μm spherical particles, and their experimentally obtained concentration nd,0 in dust structures was 2 × 104 cm−3 . The experimental data obtained in [17], are indicated in Fig. 1 with triangles. For convenience, let us denote the interpolation function of the experimental data on the transition exp exp boundary as I b ( P ). At I < I b ( P ), the dust structures were hoexp
mogeneous, while at I > I b ( P ) they were hollow. According to our previous observations [12,13,17,18], the dust structures, both below and above the transition boundary, represent specific analogues of solids as free motion of individual dust particles was impossible. However, such dust structures can not be characterized as crystals because of the absence of a long-range order. Fig. 1 also represents the simulated transition boundary I bsim ( P ), obtained with the model parameters discussed below. The simulation confirmed that with increasing pressure, the transition to hollow structures appeared at a lower value of the discharge current. The simulation also demonstrated that this was caused by an increase in joule heating of the discharge and, accordingly, by an
3
Fig. 2. Radial distribution of the potential energy of dust particles at P = 0.6 Torr, nd,0 = 2 × 104 cm−3 for different values of discharge current I : 0.6 mA (1), 2.0 mA (2), 3.0 mA (3).
increase in the contribution of the thermophoretic force to the balance of forces acting on dust particles in the radial direction. Fig. 2 displays the potential energy profiles U (r ) of dust particles for three increasing values of discharge current I . The transformation of U (r ) with I visualizes the formation of a void. Dust particles occupy spatial positions corresponding to the minimum in their potential energy. The value of I , at which the solid dust structure loosed the continuity on the discharge axis (r = 0) and a void was formed, corresponds to the transition of the potential energy of dust particles below zero. Starting the analysis of the thermophoresis inside the dust structure, we have analyzed the ratio between the magnitudes of forces acting on dust particles. In the studied range of pressures, dust particle concentrations and at the discharge currents of about exp I b ( P ), the electric force is directed to the center of the discharge, and the ion drag and thermophoretic forces act oppositely. Inside the dust structure, the radial electric field is lower than that in the discharge without dust particles [18,19,26], and this significantly decreases the ion drift; hereupon, in the vicinity of the discharge axis F th (r ) is higher than F i (r ). Simulations show that, at every P , the thermophoretic force prevails in our conditions, being a few times higher than the ion drag force. The ratio F th (r )/ F i (r ) varies over the radius of dust structure within the factor of two, as the thermophoresis increases faster. For instance, closely to the axis of the discharge, F th (r )/ F i (r ) changes from 2.15 to 3.3 upon increasing the pressure from 0.35 to 0.9 torr. It should be noted that in other conditions the relationship between these forces can be different. For instance, in RF discharges, an inverse relationship is usually observed (F i > F th ), especially when the distance between electrodes is small. To study the effect of the microparticle concentration on the magnitude of the thermophoretic force, we simulated an experimentally obtained transition boundary with a variable value of the concentration of microparticles in the dust structure. The experiexp mental data on I b ( P ) transition boundary, represented in Fig. 1, were simulated with concentrations of dust particles of 2 × 104 , 5 × 104 and 105 cm−3 . For each nd,0 , the coefficient C in equation (8) was varied to bring the simulated transition line I bsim ( P ) into exp
coincidence with the experimental one, I b ( P ). Since the discrepancy in the calculations was insignificant the simulation results are presented in Fig. 1 by a single dashed line. The obtained values of C are presented in Fig. 3. In Fig. 3 one can see that even at non-zero concentration of dust particles, coefficient C exceeds the corresponding value taken in the approximation of electrically neutral spherical particle in a
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Fig. 3. The dependence of coefficient C in (8) on pressure at various concentrations of microparticles nd,0 : 2 × 104 cm−3 (1), 5 × 104 cm−3 (2), 3–105 cm−3 (3).
single particle regime [30]. This difference can be a consequence of the fact that dust particles are contained in a plasma-dust structure. The dust structure represents an object, in which dust particles are not completely independent, so the single-particle regime is not applicable. In this study, we imply that C is a certain effective coefficient, reflecting the thermophoretic force acting on a strongly charged dust particle located inside the dust structure. There are several studies on dusty plasmas, in which the results for thermophoretic force with C also exceeding that for a solitary microparticle were obtained. In the article by Rothermel et al. [29], the value of C = 3.33 was obtained, and it was indicated that it was almost two times less than that in the study by Havnes et al. [32]. Let us note that in our simulations, the calculation of the dust particle charge can be a source of uncertainty for the obtained coefficient C , since under our conditions the thermophoretic force is less than the electric one. In Fig. 3 one can see that, with an increase in the concentration of dust particles, the coefficient C decreases, and in the pressure range under study, the C ( P ) dependence is stronger at smaller nd,0 (that is, at larger distance between dust particles). One can also see that the C ( P ) dependence vanishes with increasing pressure, i.e. with decreasing mean free path of neon atoms. The obtained dependence of the thermophoretic force on the concentration of dust particles in the dust structure suggests that the thermophoretic force acting on a single dust particle located in an unbounded plasma differs from that acting on a dust particle located inside a dense dust structure. Starting from the fact that, for every nd,0 , the C ( P ) dependencies vanish with pressure, one can conclude that the reason is the variation in the ratio between the atom mean free path, λ, and the distance between the adjacent particles in the dust structure, L. This ratio can be considered as the analogue of the dimensionless parameter related to the geometry of the system, Knudsen number, K nL = λ/ L [30,31]. Indeed, at a pressure of about 0.3 torr, λ is about 300 μm, and L at nd,0 = 2 × 104 cm−3 is about 370 μm (that is, λ slightly exceeds L), while at nd,0 = 105 cm−3 , the latter is about 220 μm (that is, L becomes less than λ). With an increase in K nL due to an increase in concentration of dust particles in the structure, the effect of mutual shadowing of dust particles enhances. This affects the process of momentum transfer in the direction opposite to a temperature gradient in a gas. One can see that the effect of increasing dust particle concentration seems to be similar to the effect of reducing the thermophoretic force acting on a dust particle located close to the walls in dusty plasma, which was theoretically considered in [32]. Taking into account the closeness of the plasma walls, the authors of [32] analyzed the perturbation of the velocity distribution
function, formed in an “infinite” gas with a temperature gradient, resulting from atom collisions with a wall. The reflected atom temperature was supposed to be a wall temperature. They found that, in comparison with the results calculated from the “standard” formula, the thermophoretic force reduced out to many neutral gas molecular collision lengths from the wall. The similar effect was found in [17], where it was produced by increase of dust particle diameter. The effect of dust particle shadowing observed here, resembles those in another phenomena taking place in dusty plasmas with plasma particle flows. The example can be the ion drift in a strong electric field, which generates a perturbed region of plasma density around the dust particle due to the downstream focusing of ions – the so called “plasma wake” [33–35]. Recently, the influence of ion mean free path length on plasma polarization behind a dust particle in an external electric field was studied in [36]. The dependencies of the distributions of the space charge density and potential, as well as the dependence of the dipole moment of the ion cloud on the ion mean free path, were presented. Another example is the shadowing effect arising between dust particles due to the anisotropy in ion-impact momentum deposition. For a single dust particle in a stationary plasma, the ion flow to the dust particle surface is spherically symmetric. But in the vicinity of another dust particle, the ion flow may be partially intercepted by it. As a result, this causes an attractive force between the neighboring dust particles, known as the shadowing force [35]. In cases mentioned above [33–35], the phenomena are governed by interactions between charged particles, while in the case of thermophoresis, the effect of shadowing manifests for neutral plasma component. 4. Conclusion The simulation of experimental data on the formation of void in dust structures in a glow discharge in neon has allowed us to numerically study the effect of the dust particle concentration on the thermophoretic force acting on a dust particle inside the dust structure. The shadowing effect of neighboring dust particles inside the dust structure have been numerically demonstrated. It has been found that with an increase in the concentration of dust particles (that is, with a decrease in the ratio of the distance between adjacent dust particles in the dust structure to the mean free path of atoms in neon) the thermophoretic force decreases. The dependence obtained confirms the assumption that the thermophoretic force for a single dust particle in plasma differs from that for a dust particle inside the dust structure. One can conclude that the thermophoretic force acting on a dust particle in a dust structure depends on the ratio of the atom mean free path to the distance between the adjacent particles in the dust structure. Acknowledgement The financial support of the Program of basic research of the Presidium of RAS No. 13 “Condensed matter and plasma at high energy densities” is gratefully acknowledged. References [1] I.K. Herrmann, R.N. Grass, W.J. Stark, High-strength metal nanomagnets for diagnostics and medicine: carbon shells allow long-term stability and reliable linker chemistry, Nanomedicine 4 (7) (2009) 787–798. [2] T.M. Vasil’eva, Experimental study of the synthesis of supramolecular complexes in hybrid dusty plasma, High Energy Chem. 45 (1) (2011) 66–72. [3] M.N. Vasiliev, A.H. Mahir, Synthesis and deposition of coatings in the electronbeam plasma, Surf. Coat. Technol. 180–181 (2004) 132–135. [4] Y. Xu, et al., Cobalt nanoparticles coated with graphitic shells as localized radio frequency absorbers for cancer therapy, Nanotechnology 19 (2008) 435102, https://doi.org/10.1088/0957-4484/19/43/435102.
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