Influence of dust particles on ionization equilibrium in partially ionized plasmas

A chemical model is proposed for a dusty plasma consisting of electrons, ions, neutrals, and positively charged dust particles all being at thermodynamic equilibrium. An expression is derived for the Helmholtz free energy, which comprises the ideal part, taking into account the charge of dust particles, and the excess part, evaluated in the framework of the self-consistent chemical model [Phys. Rev. E 83, 016405 (2011)]. The ionization potential depression for a dust-free partially ionized hydrogen is analytically evaluated for weakly and strongly ionized states to consistently account for the presence of charged and neutral components. An ad hoc interpolation of the ionization potential depression, valid across the whole ionization region, is put forward and subsequent solution of the generalized Saha equation is found to be in a perfect agreement with exact calculations. Minimization of the Helmholtz free energy of dusty plasmas provides the number densities of free electrons, free ions, neutrals, and the dust electric charge as well. Based on consideration of weakly and strongly ionized states, a straightforward comparison is made of the ionization equilibrium in a partially ionized plasma with and without dust particles to demonstrate that at thermal equilibrium positively charged dusts are responsible for an increase in the number density of free electrons and a decrease in the number density of free ions. It is analytically proved that nonideality effects result in a growth of the number densities of free electrons and ions by obtaining the so-called electron and proton ionization potential depressions. Electric charge of dust particles is systematically studied as a full plasma component rather than considering a detailed balance of the electron and ion fluxes on the surface of a solitary dust grain.

Figure 1

Ionization degree $\alpha$ of a partially ionized hydrogen plasma as a function of $\mathrm{\Gamma }$ at $r_s=40$ and as a function of $r_s$ at $\mathrm{\Gamma }=0.2$ for the inset. Squares: [37]; solid lines: formulas (33) and (34); dashed lines: formulas (27) and (28); dotted lines: formulas (30) and (31).

Figure 2

Base-10 logarithms of the electron ionization degree ${\alpha }_e$ and of the proton ionization degree ${\alpha }_p$ for the inset as functions of $\mathrm{\Gamma }$ at $r_s=200$, $\gamma =0.0001$, $\eta =1.25\times {10}^{-4}$, and $W=1\text{eV}$. Solid lines: present exact result; dashed lines: approximations (52) or (53); dotted lines: approximations (50) or (51).

Figure 3

Base-10 logarithms of the electron ionization degree ${\alpha }_e$ and of the proton ionization degree ${\alpha }_p$ for the inset as functions of $\mathrm{\Gamma }$ at $\gamma ={10}^{-4}$, $\eta =1.25\times {10}^{-4}$, and $W=1\text{eV}$. Solid lines: $r_s=200$; dashed lines: $r_s=300$; dotted lines: $r_s=400$.

Figure 4

Base-10 logarithms of the electron ionization degree ${\alpha }_e$ and of the proton ionization degree ${\alpha }_p$ for the inset as functions of $\mathrm{\Gamma }$ at $r_s=200$, $\gamma ={10}^{-4}$, and $W=1\text{eV}$. Solid lines: $\eta ={10}^{-3}$; dashed lines: $\eta =1.25\times {10}^{-4}$; dotted lines: $\eta =1.5625\times {10}^{-5}$.

Figure 5

Base-10 logarithms of the electron and proton ionization degrees ${\alpha }_e,{\alpha }_p$ of dusty plasmas as functions of $\mathrm{\Gamma }$ at $r_s=200$, $\gamma ={10}^{-3}$, $\eta =1.25\times {10}^{-4}$, and $W=1\text{eV}$. Dashed line: ${log}_{10}{\alpha }_e$; dotted line: ${log}_{10}{\alpha }_p$; solid line: ${log}_{10}\alpha$ from formulas (33) and (34).

Figure 6

Base-10 logarithm of the electric charge of the dust particles $Z$ as a function of $\eta$ at $\mathrm{\Gamma }=0.1$, $r_s=200$, and $W=1\text{eV}$. Solid line: $\gamma ={10}^{-3}$; dashed line: $\gamma ={10}^{-4}$; dotted line: $\gamma ={10}^{-5}$.

Figure 7

Base-10 logarithm of the electric charge of the dust particles $Z$ as a function of $\gamma$ at $\mathrm{\Gamma }=0.1$, $\eta =1.25\times {10}^{-4}$, and $W=1\text{eV}$. Solid line: $r_s=200$; dashed line: $r_s=300$; dotted line: $r_s=400$.

Figure 8

Base-10 logarithm of the electron and proton ionization degrees ${\alpha }_e,{\alpha }_p$ of dusty plasmas as functions of $\gamma$ at $\mathrm{\Gamma }=0.1$, $\eta =1.25\times {10}^{-4}$, and $W=1\text{eV}$. Dashed lines: ${\alpha }_e$; dotted lines: ${\alpha }_p$; solid lines: $\alpha$ from formulas (33) and (34). Upper lines for $r_s=300$ and lower lines for $r_s=400$.