Transport Coefficients of Ag–SiO₂ Plasmas

Abstract

Calculated values of viscosity, thermal and electrical conductivities of plasma formed in mixtures of silver (Ag) and silica (SiO₂) are presented. The calculations, which assume local thermodynamic equilibrium, are performed for three pressures (1, 10 and 30 atm) in the temperature range from 4,000 to 30,000 K. All the data for the potential interactions and the necessary formulations to obtain values of transport coefficients are given in details. For atmospheric pressure, five mixtures (100% Ag, 75% Ag and 25% SiO₂, 50% Ag and 50% SiO₂, 25% Ag and 75% SiO₂, 100% SiO₂) in weight percentage are studied. In order to analyse the pressure influence on the transport coefficients, three samples of Ag–SiO₂ mixtures (100% Ag, 50% Ag and 50% SiO₂, 100% SiO₂) in weight percentage are discussed for pressures of 1, 10 and 30 atm. In addition for the test case of oxygen plasma, we compare the computation code results with values obtained by other authors: discrepancies are found and explained.

Appendices

Appendix A. Collision Integrals

The reduced collision integral is given by:

$$ \Omega ^{\ast(l,s)}\left({T^{\ast}} \right)=\frac{1}{(s+1)!}\frac{1}{\left({T^{\ast}} \right)^{s+2}}\int_0^\infty {e^{-E/T^{\ast}}E^{s+1}Q^{l}\left(E \right)}\,{\rm d}E $$

with

$$ Q^{l}(E)=\frac{2}{\left[ {1-\frac{1}{2}\frac{1+\left({-1} \right)^{l}}{1+l}} \right]}\int_0^\infty {\left\{ {1-\cos ^{l}\left({\chi \left({E,b} \right)} \right)} \right\}b\,{\rm d}b} $$

and χ (E, b) the scattering angle:

$$ \chi \left({E, b} \right)=\pi -2b\int_{r_{\rm m}}^\infty {\frac{{\rm d}r}{r^{2}\left[ {1-\frac{b^{2}}{r^{2}}-\frac{V(r)}{E}} \right]^{1/2}}} $$

where r _m is the minimum relative distance between the two atoms following the chosen potential V(r). b is the impact parameter. E is the relative total energy.

The reduced temperature T ^* is given by : T ^* = kT/ɛ where T is the temperature, k is Boltzmann constant and ɛ is the depth potential (Table 1).

Then the integral collision can be calculated from the reduced integral collision Ω ^*(l,s) _ij (T ^* ) versus the reduced temperature: $\overline{\Omega}_{_{ij} }^{(l,s)} \left(T \right)=\sigma ^{2}\overline{{\Omega }}_{ij}^{\ast(l,s)} \left({kT/\varepsilon } \right)$.

With σ the value of r for which V(r) = 0.

Appendix B. Formulation of Transport Coefficient

Electrical Conductivity

To calculate the electrical conductivity σ we used the third-order approximation of the Chapman–Enskog method [60, 61]:

$$ \sigma _{\rm elec} =\frac{3 e^{2}}{2}n_1^2 \sqrt{\frac{2\pi }{m_1 k T}}\frac{\left| {{\begin{array}{ll} {q^{11}}& {q^{12}} \\ {q^{12}}& {q^{22}} \\ \end{array} }} \right|}{\left| {{\begin{array}{lll} {q^{00}}& {q^{01}}& {q^{02}} \\ {q^{01}}& {q^{11}}& {q^{12}} \\ {q^{02}}& {q^{12}}& {q^{22}} \\ \end{array} }} \right|} $$

(1)

where e is the electronic charge, n ₁ the electron number density, k the Boltzmann constant, m ₁ mass of an electron and T is the temperature. The parameters q ^ij are given in [60, 61].

Coefficient of Viscosity

To calculate the viscosity assuming that it is due to the heavy chemical species, we used the formulation given in [10, 62]:

$$ \eta =\frac{\left| {\begin{array}{llll} {H_{11} }& {\cdots}& {H_{1\upsilon}}& {x_1} \\ {\cdots}& & & \\ {H_{\upsilon1}}& {\cdots}& {H_{\upsilon\upsilon} }& {x_\upsilon } \\ {x_1 }& {\cdots}& {x_\upsilon }& 0 \\ \end{array} } \right|}{\left| {\begin{array}{lll} {H_{11} }& {\cdots}& {H_{1\upsilon} } \\ {\cdots}& & \\ {H_{\upsilon1} }& {\cdots}& {H_{\upsilon\upsilon} } \\ \end{array} } \right|} $$

(2)

where

$$ \begin{aligned} H_{ii} &=\frac{x_i^2 }{\eta _i }+\sum_{\begin{array}{l} k=1 \\ k\neq i \\ \end{array}}^\upsilon {\frac{2 x_i x_k }{\eta _{ik} }\frac{M_i M_k }{(M_i +M_k)^{2}}} \left({\frac{5}{3A_{ik}^\ast }+\frac{M_k }{M_i }} \right)\\ H_{ij} &=-\frac{2x_i x_j }{\eta _{ij} }\frac{M_i M_j }{\left({M_i +M_j } \right)^{2}}\left(\frac{5}{3A_{ij^\ast }-1} \right)\quad (i\neq j) \end{aligned} $$

with $A_{ij}^\ast =\frac{\overline{\Omega}_{i j}^{(2,2)}}{\overline{\Omega}_{i j}^{(1,1)} }$, $\eta _i =\frac{5}{16}\frac{1}{\overline{\Omega}_{i,i}^{(2,2)} }\sqrt\frac{k T}{N_a \pi }\sqrt{M_i }$, $\eta _{ij} =\frac{5}{16}\frac{1}{\overline{\Omega}_{i,j}^{(2,2)} }\sqrt{\frac{k T}{N_a \pi }}\sqrt{\frac{2 M_i M_j }{M_i +M_j }}$

x _i and M _i are the molar fraction and molar mass of i chemical species, N _a is the Avogadro number, $\overline{\Omega}_{i j}^{(l,m)}$ are collision integrals given in Appendix A and υ is the number of heavy chemical species.

Total Thermal Conductivity

The total thermal conductivity λ _tot can be separated into four terms with a good accuracy [60, 63]:

$$ \lambda _{\rm tot} =\lambda_{\rm tr}^e +\lambda _{\rm tr}^{\rm h} +\lambda _{\rm int} +\lambda _{\rm react} $$

(3)

where λ ^e_tr is the translational thermal conductivity of electrons, λ ^h_tr the translational thermal conductivity of heavy species particles, λ_int the internal thermal conductivity and λ_react the chemical reaction thermal conductivity.

Thermal conductivity of electrons at the third approximation order is given by [60, 61]:

$$ \lambda _{\rm tr}^e =\frac{75}{8}k n_1^2 \sqrt{\frac{2\pi RT}{M_1 }}\frac{q^{22}}{q^{11}q^{22}-\left({q^{12}} \right)^{2}} $$

(4)

The translational thermal conductivity due to the heavy species in the second-order approximation can be written as [62, 64]:

$$ \lambda _{\rm tr}^{\rm h} =4\frac{\left| {{\begin{array}{llll} {L_{11} }& {\cdots}& {L_{1\upsilon} }& {x_1 } \\ {\cdots}& & & \\ {L_{\upsilon1}}& {\cdots}& {L_{\upsilon\upsilon} }& {x_\upsilon } \\ {x_1 }& {\cdots}& {x_\upsilon }& 0 \\ \end{array} }} \right|}{\left| {{\begin{array}{lll} {L_{11} }& {\cdots}& {L_{1\upsilon} } \\ {\cdots}& & \\ {L_{\upsilon1} }& {\cdots}& {L_{\upsilon\upsilon} } \\ \end{array} }} \right|} $$

(5)

where

$$ \begin{aligned} L_{ij} &=\frac{2 x_i \;x_j M_i M_j }{\left({M_i +M_j } \right)^{2}A_{ij}^\ast k_{ij} }\left({\frac{55}{4}-3B_{ij}^\ast -4A_{ij}^\ast } \right)\quad \hbox{for}\,i\neq j\\ L_{ii} &=-4\frac{\left({x_i } \right)^{2}}{k_{ii} }-\sum_{\begin{array}{l} k=1 \\ k\neq i \\ \end{array}}^\upsilon{\frac{2 x_i x_k \left({\frac{15}{2}M_i^2 +\frac{25}{4}M_k^2 -3B_{ik}^\ast M_k^2 +4A_{ik}^\ast M_i M_k } \right) } {\left({M_i +M_j } \right)^{2}A_{ik}^\ast k_{ik} }} \end{aligned} $$

with $k_{ij} =\frac{75}{64}k^{3/2}\sqrt{\frac{N_a T}{\pi }}\sqrt{\frac{M_i +M_j }{2 M_i M_j }}\frac{1}{\overline{\Omega}_{ij}^{(2,2)} }$, $A_{ik}^\ast =\frac{\overline{\Omega}_{ik}^{(2,2)} }{\overline{\Omega}_{ik}^{(1,1)} }$ and $B_{ik}^\ast =\frac{5 \overline{\Omega}_{ik}^{\left({1,2} \right)} -4 \overline{\Omega}_{ik}^{(1,3)} }{\overline{\Omega}_{ik}^{(1,1)} }$

The internal conductivity due to the effect of internal degrees of freedom is taken into account with the Eucken correction [10, 61, 62]:

$$ \lambda _{\rm int} =\sum_{i=1}^N {\lambda _{{\rm int} i} \left({\sum_{j=1}^N {\frac{D_{ii} \left(1 \right)}{D_{ij} \left(1 \right)}\frac{x_j }{x_i }} } \right)} ^{-1} $$

(6)

with the internal conductivity of the i chemical species $\lambda _{{\rm int} i} =\frac{PD_{ii} \left(1 \right)}{T}\left({\frac{C_{p i} }{R}-\frac{5}{2}} \right)$ and the use of the first approximation for the binary diffusion coefficients:

$$ D_{ij} \left(1 \right)=\frac{3}{8}\left({kT} \right)^{3/2}\frac{1}{P\overline{\Omega}_{ij}^{(1,1)} }\sqrt{\frac{Na\left({M_i +M_j } \right)}{2\pi \left({M_i M_j } \right)}} $$

(7)

The formulation of the chemical reaction thermal conductivity was developed by Butler and Brokaw [65] and is written as:

$$ \lambda _{\rm react} =-\frac{1}{R T^{2}}\frac{\left| {{\begin{array}{llll} {A_{11} }& {\cdots}& {A_{1\mu } }& {\Delta H_1 } \\ {\cdots}& & & \\ {A_{\mu 1} }& {\cdots}& {A_{\mu \mu } }& {\Delta H_\mu } \\ {\Delta H_1 }& & {\Delta H_\mu }& 0 \\ \end{array} }} \right|}{\left| {{\begin{array}{lll} {A_{11} }& {\cdots}& {A_{1\mu } } \\ {\cdots}& & \\ {A_{\mu 1} }& & {A_{\mu \mu } } \\ \end{array} }} \right|} $$

with $A_{ ij} =\sum_{k=1}^{\upsilon-1} {\sum_{l=k+1}^\upsilon {\frac{R T}{ P D_{kl} \left(1 \right)}} } x_k x_l \left({\frac{a_{ik} }{x_k }-\frac{a_{il} }{x_l }} \right)\left({\frac{a_{jk} }{x_k }-\frac{a_{jl} }{x_l }} \right)$ i reaction can be written as $\sum_{k=1}^N {a_{ik} B^{k}} =0$ where B ^k is the symbol of the k chemical species. The μ reaction that we have to take into account must be linearly independent. For i reaction the enthalpy variation ΔH _i is ΔH _i = ∑ ^ν _j=1 a _ij H _j where H _j is the specific enthalpy of the j chemical species.

Appendix C. Chemical Composition

Fig. a

Chemical composition versus temperature at two pressures (1, 30 atm) for a plasma formed of 100% Ag

Fig. b

Chemical composition versus temperature at two pressures (1, 30 atm) for a plasma formed of 50% Ag and 50% SiO₂ (wp)

Fig. c

Chemical composition versus temperature at two pressures (1, 30 atm) for a plasma formed of 100% SiO₂