Diffusion of Collisional Plasma by the Example of a High-Current Arc in He: Binary and Trinary Ionized Mixtures

Abstract

In this study the applicability of the binary mixture model utilizing the first-order gas-kinetic Chapman–Enskog theory is substantiated for describing diffusion processes at different degrees of ionization of a single-temperature simple gas plasma consisting of three components: atoms, ions and electrons. On the same bases, the obtained expressions for a trinary mixture are applicable to a plasma with a fourth component that is difficult to ionize. The thermal diffusion relations of a trinary mixture are derived, whose peculiarity is the electronic component, which does not affect the diffusion flows of atoms and ions. It is shown that in a highly ionized He arc plasma with developed diffusion and ionization nonequilibrium, thermal diffusion is insignificant. It is noted that when thermodynamic equilibrium is violated, the diffusion coefficients may not change at all or decrease by half at most.

APPENDIX

THERMAL DIFFUSION RATIO IN A TRINARY MIXTURE: IONIZED SIMPLE GAS—HELIUM PLASMA

Thermal diffusion is a second-order transport phenomenon [8–10], which greatly complicates the calculation of its coefficient. The thermal diffusion ratio relates it to the first-order diffusion coefficient and, in general, to determine K_i both of these coefficients are required. However, in the first approximation, it can be obtained directly [8]. To do this, we will need to introduce two types of coefficients similar to (10) (linear combinations of integral brackets), only of the second order, which we will take from the detailed monograph [8], remaining, as before, within the framework of the hard sphere model. Finally, we consider its applicability.

We start with the coefficients connecting the first and second orders (the order n is indicated by superscripts equal to n – 1):

$$\begin{gathered} {{\Lambda }}_{{ij}}^{{01}} = \frac{{{{m}_{i}}}}{{{{m}_{j}}}}{{\Lambda }}_{{ji}}^{{01}} = C{{x}_{i}}{{x}_{j}}{{{{\sigma }}}_{{ij}}}\sqrt {\frac{{{{m}_{j}}}}{{{{m}_{1}}}}} {{\left( {\frac{{{{m}_{i}}}}{{{{m}_{i}} + {{m}_{j}}}}} \right)}^{{1.5}}}, \\ {{\Lambda }}_{{ii}}^{{01}} = - \mathop \sum \limits_j {{\Lambda }}_{{ji}}^{{01}},\,\,\,\,i \ne j. \\ \end{gathered} $$

(A.1)

Here C = 32/75${{v}_{1}}\sqrt \pi$ is constant [8], depending only on T; and ${{v}_{1}}$ is the velocity of a particle with energy ε = T and mass m₁ = m_e, in this case an electron. The mass ratio on the right side gives unity at j = 1 and $\sqrt {\frac{m}{{8{{m}_{{\text{e}}}}}}}$ at i, j $\ne$1. Note that ${{\Lambda }}_{{23}}^{{01}} = {{\Lambda }}_{{32}}^{{01}}$ due to the equality of masses of ions and atoms m₂ = m₃ = m.

In the second order, the coefficients become more complicated:

$$\begin{gathered} {{\Lambda }}_{{ij}}^{{11}} = {{\Lambda }}_{{ji}}^{{11}} = - 13.5\frac{{{{m}_{j}}}}{{{{m}_{i}} + {{m}_{j}}}}{{\Lambda }}_{{ij}}^{{01}},\,\,\,\,i \ne j, \\ {{\Lambda }}_{{ii}}^{{11}} = 2Cx_{i}^{2}{{{{\sigma }}}_{i}}\sqrt {\frac{{{{m}_{j}}}}{{{{m}_{{\text{e}}}}}}} \\ - \,\,\mathop \sum \limits_j {{\Lambda }}_{{ij}}^{{11}}{{\left( {30\frac{{{{m}_{i}}}}{{{{m}_{j}}}} + 13\frac{{{{m}_{j}}}}{{{{m}_{i}}}} + 16} \right)} \mathord{\left/ {\vphantom {{\left( {30\frac{{{{m}_{i}}}}{{{{m}_{j}}}} + 13\frac{{{{m}_{j}}}}{{{{m}_{i}}}} + 16} \right)} {27}}} \right. \kern-0em} {27}}. \\ \end{gathered} $$

(A.2)

Here σ_i is the collision cross section in its own gas (see Table 1).

Table 1.   Transport cross sections for elastic collisions in He plasma, 10^–15 cm²

We move on to the equations for thermal diffusion relations

$${{K}_{i}} = \frac{5}{2}\mathop \sum \limits_{hj} {{\Lambda }}_{{ih}}^{{01}}{{X}_{{hj}}}{{x}_{j}},\,\,\,\,i,j,h = 1,2,3,$$

(A.3)

where X_hj are the inverse ${{\Lambda }}_{{ih}}^{{11}}$ coefficients determined by equations

$$\mathop \sum \limits_h {{\Lambda }}_{{hi}}^{{11}}{{X}_{{jh}}} = {{\delta }_{{ij}}}.$$

(A.4)

Solving three systems of linear equations (A.4) with three unknowns in each, we obtain

$$\begin{gathered} {{X}_{{11}}} = {{\left[ {{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - \left. {{{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - \left. {{{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}} \right)} \right]} B}} \right. \kern-0em} B}, \\ {{X}_{{22}}} = {{\left[ {{{{\left( {{{\Lambda }}_{{13}}^{{11}}} \right)}}^{2}} - \left. {{{\Lambda }}_{{11}}^{{11}}{{\Lambda }}_{{33}}^{{11}}} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {{{{\left( {{{\Lambda }}_{{13}}^{{11}}} \right)}}^{2}} - \left. {{{\Lambda }}_{{11}}^{{11}}{{\Lambda }}_{{33}}^{{11}}} \right)} \right]} B}} \right. \kern-0em} B}, \\ \end{gathered} $$

(A.5)

$$\begin{gathered} {{X}_{{33}}} = {{\left[ {{{{\left( {{{\Lambda }}_{{12}}^{{11}}} \right)}}^{2}} - \left. {{{\Lambda }}_{{11}}^{{11}}{{\Lambda }}_{{22}}^{{11}}} \right)} \right]} \mathord{\left/ {\vphantom {{\left[ {{{{\left( {{{\Lambda }}_{{12}}^{{11}}} \right)}}^{2}} - \left. {{{\Lambda }}_{{11}}^{{11}}{{\Lambda }}_{{22}}^{{11}}} \right)} \right]} B}} \right. \kern-0em} B}, \\ {{X}_{{23}}} = {{X}_{{32}}} = {{\left( {{{\Lambda }}_{{11}}^{{11}}{{\Lambda }}_{{23}}^{{11}} - {{\Lambda }}_{{12}}^{{11}}{{\Lambda }}_{{13}}^{{11}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{\Lambda }}_{{11}}^{{11}}{{\Lambda }}_{{23}}^{{11}} - {{\Lambda }}_{{12}}^{{11}}{{\Lambda }}_{{13}}^{{11}}} \right)} B}} \right. \kern-0em} B}, \\ \end{gathered} $$

(A.6)

$$\begin{gathered} {{X}_{{12}}} = {{X}_{{21}}} = {{\left( {{{\Lambda }}_{{12}}^{{11}}{{\Lambda }}_{{33}}^{{11}} - {{\Lambda }}_{{13}}^{{11}}{{\Lambda }}_{{23}}^{{11}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{\Lambda }}_{{12}}^{{11}}{{\Lambda }}_{{33}}^{{11}} - {{\Lambda }}_{{13}}^{{11}}{{\Lambda }}_{{23}}^{{11}}} \right)} B}} \right. \kern-0em} B}, \\ {{X}_{{13}}} = {{X}_{{31}}} = {{\left( {{{\Lambda }}_{{13}}^{{11}}{{\Lambda }}_{{22}}^{{11}} - {{\Lambda }}_{{12}}^{{11}}{{\Lambda }}_{{23}}^{{11}}} \right)} \mathord{\left/ {\vphantom {{\left( {{{\Lambda }}_{{13}}^{{11}}{{\Lambda }}_{{22}}^{{11}} - {{\Lambda }}_{{12}}^{{11}}{{\Lambda }}_{{23}}^{{11}}} \right)} B}} \right. \kern-0em} B}, \\ \end{gathered} $$

(A.7)

where

$$\begin{gathered} B = \left[ {{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - {{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}} \right]{{\Lambda }}_{{11}}^{{11}} + {{\Lambda }}_{{22}}^{{11}}{{\left( {{{\Lambda }}_{{13}}^{{11}}} \right)}^{2}} \\ + \,\,{{\Lambda }}_{{33}}^{{11}}{{\left( {{{\Lambda }}_{{12}}^{{11}}} \right)}^{2}} - 2{{\Lambda }}_{{12}}^{{11}}{{\Lambda }}_{{13}}^{{11}}{{\Lambda }}_{{23}}^{{11}}. \\ \end{gathered} $$

Now, knowing the cross sections, we can calculate the thermal diffusion ratios (A.3) for any three gases. In the case of an ionized mixture of a simple gas, the problem is extremely simplified. Let us consider, as before, the ratios of coefficients that reach very large values:¹

$$\begin{gathered} \frac{{{{\Lambda }}_{{23}}^{{11}}}}{{{{\Lambda }}_{{13}}^{{11}}}} = \frac{{{{{{\sigma }}}_{{23}}}}}{{2{{{{\sigma }}}_{{13}}}}}{{\left( {\frac{m}{{2{{m}_{{\text{e}}}}}}} \right)}^{{1.5}}} \approx {\text{ }}5 \times {{10}^{5}}, \\ \frac{{{{\Lambda }}_{{23}}^{{11}}}}{{{{\Lambda }}_{{12}}^{{11}}}} = \frac{{{{{{\sigma }}}_{{23}}}{{x}_{3}}}}{{2{{{{\sigma }}}_{{12}}}{{x}_{1}}}}{{\left( {\frac{m}{{2{{m}_{{\text{e}}}}}}} \right)}^{{1.5}}}\sim {{10}^{4}}n{\text{/}}{{n}_{{\text{e}}}}^{1}. \\ \end{gathered} $$

(A.8)

It follows from this that only the coefficients ${{\Lambda }}_{{23}}^{{11}}$, as well as${{\;\Lambda }}_{{22}}^{{11}}$, ${{\Lambda }}_{{33}}^{{11}}$, and ${{\Lambda }}_{{11}}^{{11}}$ need to be taken into account, since they include ${{\Lambda }}_{{23}}^{{11}}$ and the factor m/m_e at ${{\Lambda }}_{{12}}^{{11}}$ and ${{\Lambda }}_{{13}}^{{11}}$ (see (A.2)). Then the inverse coefficients (A.7) and the three terms in the numerators (A.5), (A.6) can be neglected without affecting the accuracy significantly, and the value of B can be radically simplified:

$$B = \left[ {{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - {{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}} \right]{{\Lambda }}_{{11}}^{{11}}.$$

In this case, the first coefficient from (A.5) takes the form

$${{X}_{{11}}} = 1{\text{/}\Lambda }_{{11}}^{{11}}.$$

The remaining significant coefficients are also greatly simplified:

$$\begin{gathered} {{X}_{{22}}} = - {{{{\Lambda }}_{{33}}^{{11}}} \mathord{\left/ {\vphantom {{{{\Lambda }}_{{33}}^{{11}}} {\left[ {{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - {{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}} \right]}}} \right. \kern-0em} {\left[ {{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - {{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}} \right]}}, \\ {{X}_{{33}}} = - {{{{\Lambda }}_{{22}}^{{11}}} \mathord{\left/ {\vphantom {{{{\Lambda }}_{{22}}^{{11}}} {\left[ {{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - {{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}} \right]}}} \right. \kern-0em} {\left[ {{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - {{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}} \right]}}, \\ {{X}_{{23}}} = - {{{{\Lambda }}_{{23}}^{{11}}} \mathord{\left/ {\vphantom {{{{\Lambda }}_{{23}}^{{11}}} {\left[ {{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - {{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}} \right]}}} \right. \kern-0em} {\left[ {{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - {{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}} \right]}}. \\ \end{gathered} $$

Another simplification is provided by the ratios between ${{\Lambda }}_{{i1}}^{{01}}$ and ${{\Lambda }}_{{1i}}^{{01}}$, which, in accordance with (A.1), are also very large (m/m_e). It only applies to electrons K₁ = K_e, since the sum of (A.3) contains small parameters ${{\Lambda }}_{{1i}}^{{01}}$ only when i = 1. Excluding them, we get only one significant term:

$$\frac{2}{5}{{K}_{1}} = {{x}_{1}}{{\Lambda }}_{{11}}^{{01}}{{X}_{{11}}} = {{{{x}_{1}}{{\Lambda }}_{{11}}^{{01}}} \mathord{\left/ {\vphantom {{{{x}_{1}}{{\Lambda }}_{{11}}^{{01}}} {{{\Lambda }}_{{11}}^{{11}}}}} \right. \kern-0em} {{{\Lambda }}_{{11}}^{{11}}}}.$$

(A.9)

Sums (A.4) for ions and atoms K₂ and K₃ contain five significant terms

$$\begin{gathered} \frac{2}{5}{{K}_{2}} = {{x}_{1}}\frac{{{{\Lambda }}_{{21}}^{{01}}}}{{{{\Lambda }}_{{11}}^{{11}}}} \\ + \,\,{{\Lambda }}_{{23}}^{{01}}\frac{{{{x}_{2}}\left( {{{\Lambda }}_{{23}}^{{11}} + {{\Lambda }}_{{33}}^{{11}}} \right) - {{x}_{3}}\left( {{{\Lambda }}_{{23}}^{{11}} + {{\Lambda }}_{{22}}^{{11}}} \right)}}{{{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - {{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}}}, \\ \end{gathered} $$

(A.10)

$$\begin{gathered} \frac{2}{5}{{K}_{3}} = {{x}_{1}}\frac{{{{\Lambda }}_{{31}}^{{01}}}}{{{{\Lambda }}_{{11}}^{{11}}}} \\ - \,\,{{\Lambda }}_{{23}}^{{01}}\frac{{{{x}_{2}}\left( {{{\Lambda }}_{{23}}^{{11}} + {{\Lambda }}_{{33}}^{{11}}} \right) - {{x}_{3}}\left( {{{\Lambda }}_{{23}}^{{11}} + {{\Lambda }}_{{22}}^{{11}}} \right)}}{{{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - {{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}}}. \\ \end{gathered} $$

(A.11)

As ${{\Lambda }}_{{11}}^{{01}} = - {{\Lambda }}_{{21}}^{{01}} - {{\Lambda }}_{{31}}^{{01}}$ (see (A.1)), condition (3) for (A.9)–(A.11) is satisfied.

The difference between these thermal diffusion relations and the case of a binary mixture lies in the appearance of electronic components in ions and atoms—these are the first terms of the right-hand sides of (A.10), (A.11)

$$\begin{gathered} {{K}_{{{\text{ei}}}}} = - {{\chi }_{2}}{{K}_{1}} = 2.5{{x}_{1}}\frac{{{{\Lambda }}_{{21}}^{{01}}}}{{{{\Lambda }}_{{11}}^{{11}}}} \\ = \frac{{{{{{\sigma }}}_{{12}}}{{x}_{1}}}}{{({{{{\sigma }}}_{{12}}} + {{{{\sigma }}}_{1}}{\text{/}}2.3){{x}_{1}} + {{{{\sigma }}}_{{13}}}{{x}_{3}}}}{{x}_{1}}{\text{/}}2.6, \\ {{K}_{{{\text{ea}}}}} = \left( {{{\chi }_{2}} - 1} \right){{K}_{1}} = 2.5{{x}_{1}}\frac{{{{\Lambda }}_{{31}}^{{01}}}}{{{{\Lambda }}_{{11}}^{{11}}}} \\ = \,\,\frac{{{{{{\sigma }}}_{{13}}}{{x}_{3}}}}{{({{{{\sigma }}}_{{12}}} + {{{{\sigma }}}_{1}}{\text{/}}2.3){{x}_{1}} + {{{{\sigma }}}_{{13}}}{{x}_{3}}}}{{x}_{1}}{\text{/}}2.6, \\ \end{gathered} $$

(A.12)

where χ₂ is determined by relation (19). Their values are of undoubted interest. In the considered highly ionized plasma (α ≫ α₀), the electronic component of atoms is much smaller than that of ions. This is evident from their ratio $\frac{{{{\Lambda }}_{{21}}^{{01}}}}{{{{\Lambda }}_{{31}}^{{01}}}}$, also determined by expression (23). Therefore, it can be neglected in (A.9) (it will be shown later that it can also be neglected in (A.11)). This means that at high degrees of ionization the electronic component of ions K_ei $\gg$ K_ea practically coincides with K₁ = K_e = ‒K_ei ‒ K_ea. Let us compare it with the second term on the right-hand sides of (A.10), (A.11), which remains unchanged when moving to binary mixtures. This is the main one; let us call it binary K_b, constituting K₂ and K₃

$${{K}_{2}} = {{K}_{{{\text{ei}}}}} - {{K}_{{\text{b}}}},\,\,\,\,{{K}_{3}} = {{K}_{{{\text{ea}}}}} + {{K}_{{\text{b}}}},$$

which we will also simplify by using a larger value of ${{\Lambda }}_{{22}}^{{11}}$ at high degrees of ionization. In the considered plasma with n/n_e = x₃/x₁ ≈ 0.5‒2, it is 11‒3 times higher than ${{\Lambda }}_{{33}}^{{11}}$ and 27‒8 times higher than ${{\Lambda }}_{{23}}^{{11}}$. Within these limits, with an accuracy of ~5%, we obtain

$$\begin{gathered} \frac{2}{5}{{K}_{{\text{b}}}} = - {{\Lambda }}_{{23}}^{{01}}\frac{{{{x}_{2}}({{\Lambda }}_{{23}}^{{11}} + {{\Lambda }}_{{33}}^{{11}}) - {{x}_{3}}\left( {{{\Lambda }}_{{23}}^{{11}} + {{\Lambda }}_{{22}}^{{11}}} \right)}}{{{{{\left( {{{\Lambda }}_{{23}}^{{11}}} \right)}}^{2}} - {{\Lambda }}_{{22}}^{{11}}{{\Lambda }}_{{33}}^{{11}}}} \\ \approx - \frac{{{{x}_{3}}{{\Lambda }}_{{23}}^{{01}}}}{{1.2{{\Lambda }}_{{33}}^{{11}}}}. \\ \end{gathered} $$

(A.13)

Then the ratio of the first term to the second on the right side of (A.10) is

$$\begin{gathered} \gamma = - {{K}_{{{\text{ei}}}}}{\text{/}}{{K}_{{\text{b}}}} \approx \frac{{1.2{{x}_{1}}{{\Lambda }}_{{21}}^{{01}}}}{{{{x}_{3}}{{\Lambda }}_{{11}}^{{11}}}}\frac{{{{\Lambda }}_{{33}}^{{11}}}}{{{{\Lambda }}_{{23}}^{{01}}}} \\ \approx 1.9(0.23 + {{x}_{1}}{\text{/}}{{x}_{3}}) \approx 4{-} 1.4\,\,\,{\text{at }}n{\text{/}}{{n}_{{\text{e}}}} \approx 0.5{-} 2, \\ \end{gathered} $$

i.e., the electronic component approximately triples the rate of thermal diffusion of ions. Moreover, in accordance with (23), the first term in (A.11) is unimportant: –K_b ~ K_ei $\gg$ K_ea and K₃ ≈ K_b. In addition, in accordance with (A.8), members ${{\Lambda }}_{{13}}^{{11}}$ and ${{\Lambda }}_{{12}}^{{11}}$ can be neglected in Eqs. (A.2) for ${{\Lambda }}_{{33}}^{{11}}$ and ${{\Lambda }}_{{22}}^{{11}}$. This means a complete analogy with a binary mixture. The thermal diffusion ratio for it, identical to expression (A.13), is given in [8]:

$${{K}_{{\text{b}}}} = ({{S}_{3}}{{x}_{3}} - {{S}_{2}}{{x}_{2}}){\text{/}}({{Q}_{3}}{{x}_{3}}{\text{/}}{{x}_{2}} + {{Q}_{2}}{{x}_{2}}{\text{/}}{{x}_{3}} + {{Q}_{{23}}}),$$

where for an ionized simple gas

$$\begin{gathered} {{S}_{3}} = {{\sigma }_{3}}{\text{/}}{{\sigma }_{{23}}} - 1 \approx - 0.57,\,\,\,\,{{S}_{2}} = {{\sigma }_{2}}{\text{/}}{{\sigma }_{{23}}} - 1 \approx 9, \\ {{Q}_{3}} = 5.9{{\sigma }_{3}}{\text{/}}{{\sigma }_{{23}}} \approx 2.6,\,\,\,\,{{Q}_{2}} = {\text{ }}5.9{{\sigma }_{2}}{\text{/}}{{\sigma }_{{23}}} \approx 59,\, \\ {{Q}_{{23}}} = 3.2{{\sigma }_{3}}{{\sigma }_{2}}{\text{/}}\sigma _{{23}}^{2} + 8.6 \approx 22.5, \\ \end{gathered} $$

and indices 2 and 3 correspond to indices 2 and 1 in [8]. In the numerical expression for He we get

$$\begin{gathered} {{K}_{{\text{b}}}} \approx - ({{x}_{2}} + 0.063{{x}_{3}}){\text{/}}6.6({{x}_{2}}{\text{/}}{{x}_{3}} \\ + \,\,0.043{{x}_{3}}{\text{/}}{{x}_{2}} + 0.38). \\ \end{gathered} $$

(A.14)

Here, the small numerical coefficients have atomic components. They can be neglected at high degrees of ionization with α > 0.75. Then the dependence on the Coulomb cross sections in (A.14) practically disappears, as in the thermal diffusion relation for electrons (A.9), mutually canceling out in the numerator and denominator. In a weakly ionized plasma, at α < α₀ = 0.02, the Coulomb collisions can be generally neglected. Their cross sections require consideration only in the range of degrees of ionization α ≈ 0.02‒0.75 (x₂ ≈ 0.02‒0.43), having the greatest impacton the thermal diffusion ratio (A.14) at n/n_e = S₃Q₂/S₂Q₃ ≈ 1.5 (x₂ ≈ 0.29, α ≈ 0.41). In this case, a change by a factor of 2 in σ₁, σ₂, and σ₁₂ leads to an error (A.14) of only ~10%. Consequently, the hard sphere model is also applicable for describing thermal diffusion in ionized simple gases.

So, the difference from the binary model lies in the presence of an electronic component of thermal diffusion, comparable in magnitude to the binary component. It accelerates the diffusion of ions and slows down the diffusion of electrons, but is insignificant for the entire electron-ion gas included in a binary mixture with atoms; thus, in Eqs. (20) and (27), the electronic components of ions K_ei and atoms K_ea are not taken into account.

Figure 2 shows the dependences of all thermal diffusion ratios of helium plasma considered here on the mole fraction of electrons (ions) x₁ = x₂, to which expressions (A.9)‒(A.14) are reduced by the substitution x₃ = 1 ‒ 2x₁. Let us consider the main features of these dependences.

The main binary component of the thermal diffusion of atoms and ions is large at high degrees of ionization and has a maximum at α ≈ 0.31 (in Fig. 2 this corresponds to x₁ ≈ 0.24). In the limit of strong ionization α > 0.7, x₁ $\gg$ x₃ (but not more than 10³x₃, see (18)), the expressions for thermal diffusion relations are simplified:

$$\begin{gathered} {{K}_{{\text{b}}}} \approx {{S}_{2}}{{x}_{3}}{\text{/}}{{Q}_{2}} \approx - {{x}_{3}}{\text{/}}6.6 \to 0, \\ {{K}_{{{\text{e}}}}} \approx \frac{{{{\Lambda }}_{{21}}^{{01}}}}{{{{\Lambda }}_{{12}}^{{11}}}}\frac{{3.6{{x}_{1}}{{m}_{{\text{e}}}}}}{m} \approx - {{x}_{1}}{\text{/}}3.7 \to - 1{\text{/}}7.5. \\ \end{gathered} $$

As can be seen from Fig. 2, the binary component becomes insignificant and in Eqs. (20), (27) thermal diffusion can be neglected without significantly affecting the accuracy. The electronic component reaches a maximum when the ionization is complete. The numerical values, as above, are given for He.

In weakly ionized plasma (α < α₀ ~ 0.02), all thermal diffusion ratios, as can be seen from Fig. 2, become very small. The ions and atoms in a certain sense change places, since the electronic component is significant only for atoms in (A.11) (the first term on the right side), but unlike (A.10), with the opposite sign for K_b. This happens at large x₃ ~ 1 and small x₁ < 0.02, when K₂ ≈ –K_b, and the numerator and denominator (A.14) are dominated by terms with small numerical coefficients. At the same time, in He

$${{K}_{{\text{b}}}} \approx {{S}_{3}}{{x}_{2}}{\text{/}}{{Q}_{3}} \approx - {{x}_{1}}{\text{/}}4.5,\,\,\,\,{{K}_{{\text{e}}}} \approx - {{x}_{1}}{\text{/}}2.6.$$

Thus, K₃ = K_b ‒ K_e > 0 becomes a small positive value. Thus, with a decrease in the degree of ionization, the thermal diffusion of both atoms and charged particles practically stops due to the small mole fractions x₁.