Influence of Xenon–Fluorine–Sulfur Hexafluoride (Xe⁺–F⁻–SF6⁻) and Argon-Fluorine-Sulfur Hexafluoride (Ar⁺–F⁻–SF6⁻) Streaming on Dust Surface Potential (DSP) That Has Cairn–Tsallis Distributed Plasmas

Abstract

The dust grain surface potential is examined analytically and numerically in dusty plasmas containing negative/positive ion species by using the Cairn Tsallis (non-Maxwellian) dusty plasma. The equations for the dust-charging process are derived to solve the current balance equation for the xenon–fluorine–sulfur hexafluoride and argon–fluorine–sulfur hexafluoride plasmas. The charging process affected by plasma properties such as spectral indices $\alpha$ and q, in addition to positive ion streaming ($U_{A{r}^{+}}$ and $U_{X{e}^{+}}$) and negative ion streaming ($U_{F^{-}}$ and $U_{S{F}_6^{-}}$) of both types of plasmas, is examined. Our findings suggest that considering a wide range of $X{e}^{+}-F^{-}-S{F}_6^{-}$ and $A{r}^{+}-F^{-}-S{F}_6^{-}$ masses is critical for understanding plasma physics, specifically multi-component plasmas.

1. Introduction

During the last few decades, a notable improvement has been made regarding the dusty (complex) plasmas having negatively/positively charged dust grains, ion, electron, and neutral plasmas components. The examples of dusty plasma in astrophysics and space systems are asteroid regions, interstellar media, interplanetary spaces, dark molecular clouds, accretion disks, earth’s surroundings, and industry plasmas [1,2,3,4]. Inertia is produced by the dust particle mass, and the restoring force is provided by the pressure of electrons/ions in the dust-acoustic (DA) mode. Rao et al. [5] studied a dusty plasma (unmagnetized) and theoretically predicted the low-phase-velocity dust acoustic (DA) waves (in comparison to both electron as well as ion thermal velocities). The laboratory experiments [6] demonstrated some nonlinear features of dust acoustic waves. Dust grains are embedded in a surrounding plasma and radiative background under typical experimental settings. Because of the charge they carry, they interact with electrons and ions. Photoelectric emission, due to collision with electrons/ions, and secondary emission are some of the competing processes that charge dust grains, depending on local conditions [6]. Massive dust particles can accumulate a huge number of plasma electrons with high mobility, resulting in a dust particle with a negative charge. Consequently, the positive ion current increases and the electron current decreases as a result of the negative dust-surface potential. The steady-state of the electron current $\left(I_e\right)$ and ion current $\left(I_i\right)$ is $|I_e|=|I_i|,$ which establishes the equilibrium dust charge. The bombardment of electrons and positive ions was postulated to be the cause of dust charge in 1994 [7], when DSP was theoretically and experimentally investigated. These studies have been further expanded [8,9] to emphasize the importance of negatively charged ions in Maxwellian/non-Maxwellian-distributed plasmas.

There are several plasma events in space, such as the interstellar medium, ionosphere, thermosphere, planetary magnetosphere, ionosphere, and magnetosheaths, and the non-Maxwellian velocity distribution is frequently observed [10,11,12]. A novel distribution, developed to describe super-thermal particles detected by the Viking [13] and Freja-satellite [14], has been presented by Cairns et al. [15]. It has been expressed as a parameter alpha ($\alpha$) that calculates the variance from the Maxwellian to non-Maxwellian. As a result, it has been observed that in the presence of non-thermal electrons, the characteristics of ion-sound-solitary waves can alter, resulting in nonlinear solitary waves with either an augmented or depleted density. The Cairns model has since been used in a number of studies [16,17,18,19]. Verheest and Pillay, for example, studied large-amplitude dust-acoustic (DA) solitary structures in negatively dusty plasmas, in either non-thermal ions or electrons [20]. Later, [21] further looked into large ion-acoustic (IA) solitary structures and double layers (DL) that have positive ions as well non-thermal electrons plasmas. Mamun and colleagues have also looked at the combined effect of positively dust grains and non-thermal electrons in dust–IA solitary waves and the presence of non-thermal ions in DA waves [16]. In recent years, statistically non-extensive events using the Boltzmann–Gibbs–Shannon (BGS) entropic dimension deviation have received much attention. In statistical mechanics, large thermodynamical quantities are proportional to the mass of matter present, e.g., the particle number $\left(N\right)$, the volume $\left(V\right)$, the magnetization $\left(M\right)$, the internal energy $\left(E\right)$, and so on. The temperature $\left(T\right),$ pressure $\left(P\right),$ and chemical potential $\left(\mu \right)$ are non-extensive quantities that are not dependent on the mass of matter. Furthermore, the ratio of the extensive variables yields a non-extensive variable, for example, the number density = (particle number/volume), which is the proportionality of the number of particles to the system’s volume. Renyi proposed and Tsallis [22] researched a suitable BGS entropy, statistically under equilibrium. The nonlinear and non-extensive situation of the entropies was continued by Tsallis; the entropic (spectral) index q depicts the level of non-extensivity of the assumed framework. There are two major physical zones that can be identified: the $-1<q\le 1$ region, which has all velocities, and $q\ge 1$, in which the plasma species distribution-function capacity indicates a thermal cut-off on the most extreme quality considered: the particles’ velocity [23].

Negative ions’ impact in space areas [24] and in the laboratory plasmas [25] was discovered in the earth’s mesosphere, the ionosphere’s D zone, and the sun’s photosphere. The dynamics of IA dispersive waves have also been observed with negative ions [26]. Ichiki et al. [27] looked at the parameters of IA waves in positive-/negative-ion plasma elements and found primary modes. They have demonstrated that in $X{e}^{+}-F^{-}-S{F}_6^{-}$ plasma, two modes exist considering the values of the electron temperature and ion temperature, and they also occurr in one mode in $A{r}^{+}-F^{-}-S{F}_6^{-}$ plasma, regardless of the multi-ions temperatures. Ichiki et al. [28] studied the slow mode of IA waves with various features utilizing the $X{e}^{+}-F^{-}$ plasmas in two distinct DP devices. Later research looked at the IAWs in $X{e}^{+}-F^{-}-S{F}_6^{-}$ and $A{r}^{+}-F^{-}-S{F}_6^{-}$ plasma species, revealing that this work will help with the analysis of dusty plasmas with negative ions. By using $A{r}^{+}-X{e}^{+}$, the heavy and light ion modes have been studied in lab plasma [29]. In addition to the negative ion plasma employed in the labs [30], sulfur hexafluoride $\left(S{F}_6\right)$ gas has been employed to generate negative ions. It has been also discovered using mass spectrometry in $Ar/S{F}_6$ plasma created in DP devices in which $A{r}^{+}$, $F^{-},$ and $S{F}_6$ were present for the sufficiently high negative ion to positive ion density ratios [27], and experiments were also conducted using such plasmas [27]. In the current paper, we will analyze the $X{e}^{+}-F^{-}-S{F}_6^{-}$ and $A{r}^{+}-F^{-}-S{F}_6^{-}$ dusty plasma and do numerical computations using the static negative dust particle. These negative ions are a form of plasma species that can arise naturally or be infused from an outside source in the space plasma and lab plasmas [31]. The control of an electrode in deposition and etching plasmas [32], the development of voids [33], and the extraction of negative ion electrode sources [34] are all dependent on negative ions. The DSP was previously [35] calculated using non-Maxwellian currents due to multi ions for dust particles that are negatively charged. We have studied the$X{e}^{+}-F^{-}-S{F}_6^{-}$ and the $A{r}^{+}-F^{-}-S{F}_6^{-}$ plasmas. To investigate the influence of plasma characteristics, we used numerical analysis on the negatively charged DSP. We noted the effects of the spectral indices $\alpha$ and $q,$ positive ion streaming ($U_{A{r}^{+}}$ and $U_{X{e}^{+}}$), and negatively ion streaming ($U_{F^{-}}$ and $U_{S{F}_6^{-}}$) on DSP for both types of plasmas.

The rest of this paper is divided into the following sections: we investigate the generalized Cairn–Tsallis $(\mathit{q},\mathit{\alpha })$-distribution function in Section 2. In Section 3, we calculate the DSP for $X{e}^{+}-F^{-}-S{F}_6^{-}$ plasma. In Section 4, we have derived the DSP for $A{r}^{+}-F^{-}-S{F}_6^{-}$ plasma utilizing the same method of analysis. The Numerical results and discussion for the type of plasmas are studied in Section 5. Finally, in Section 6, a summary of the study’s significant findings is presented.

2. Cairns–Tsallis $(\mathit{q},\mathit{\alpha })$-Distribution Function

The 3-D $(\mathit{q},\mathit{\alpha })$-distribution function, which is particularly the combination of the Cairns–Tsallis distribution, is defined as [36]

$$f_{CT}\left({\mathbf{V}}_j\right)=D_{q\alpha }{\left(\frac{2\pi }{1-q}\right)}^{-\frac{3}{2}}{\left(\frac{T_j}{m_j}\right)}^3\left(1+\alpha \frac{m_j^2{V}_j^4}{T_j^2}\right){\left[1-(q-1)\frac{m_j{V}_j^2}{2T_j}\right]}^{{(q-1)}^{-1}},$$

(1)

where $T_j,$$m_j,$ and $V_j$ are the temperature, mass, and velocity of jth plasma particles (j$=e$ is for the electrons; $X{e}^{+}$ and $A{r}^{+}$ are for the positive ions; $S{F}_6^{-}$ and $F^{-}$ are the negative ions (and the letter d stands for negative charge dust grains); q and $\alpha$ are the spectral indexes, associated with non-extensivity and the quantity of non-thermal electrons, and positive/negative ions, respectively. Here, $D_{q\alpha }$ is a constant of normalization expressed by [37].

$$\left.\begin{array}{c}{C}_{q\alpha }=\left[\frac{{\left(1-q\right)}^2\Gamma \left(\frac{1}{1-q}\right)}{\Gamma \left(\frac{1}{1-q}-\frac{5}{2}\right)\left[3\alpha +{\left(1-q\right)}^2\left(\frac{1}{1-q}-\frac{5}{2}\right)\left(\frac{1}{1-q}-\frac{3}{2}\right)\right]}\right]\mathrm{for}-1<q\le 1,\\ C_{q\alpha }=\left[\frac{{\left(1-q\right)}^2\Gamma \left(\frac{5}{2}+\frac{1}{1-q}\right)\left(\frac{3}{2}+\frac{1}{q-1}\right)\left(\frac{5}{2}+\frac{1}{q-1}\right)}{\Gamma \left(1+\frac{1}{q+1}\right)\left[3\alpha +{\left(q-1\right)}^2\left(\frac{3}{2}+\frac{1}{q-1}\right)\left(\frac{5}{2}+\frac{1}{q-1}\right)\right]}\right]\mathrm{for}q\ge 1,\end{array}\right\}$$

(2)

$\Gamma$ is called the Gamma function. Both values of $C_{q\alpha }$ correspond to the Cairns distribution in the limit $q\to 1$ [15], and the $(q,\alpha )$-distribution simplifies to a pure Tsallis distribution for $\alpha =0$ [22]. The $(q,\alpha )$-distribution reduces to the Maxwellian distribution exactly in the case of $\alpha =0$ and $q\to 1$. It is watched that the real Tsallis (q)-distribution function carries on distinctively in the limits, $q>1$ and $-1<q<1$ [23]. As in the previous extent, the real T-sallis distribution function is non-zero for the complete velocities limit $-\infty$ to $+\infty$, and it shows the high energy tail because of an abundance of super-thermal species. These standards, on a basic level, are like a power-law or $\kappa -$distribution function.

3. Dust-Surface Potential (DSP) with ${\mathit{Xe}}^{+}-{\mathit{F}}^{-}-{\mathit{SF}}_{\mathbf{6}}^{-}$ Plasma

For xenon–fluorine–sulfur hexafluoride $\left(X{e}^{+}-F^{-}-S{F}_6^{-}\right)$ plasmas, two modes exist depending on the temperatures of the electrons and ions. We investigate a non-Maxwellian complex plasma that has electrons, positive ions ($X{e}^{+}$), two negative ions ($S{F}_6^{-}$ and $F^{-}$), and negatively charged dust particles as constituents. While in balance, the charge-neutrality precondition requires $n_e-Z_{X{e}^{+}}{n}_{X{e}^{+}}+Z_{F^{-}}{n}_{{}_{F^{-}}}+Z_{S{F}_6^{-}}{n}_{S{F}_6^{-}}-q_d{n}_d/e,$ where $n_s$ is the sth species of the unperturbed number density (s$=e$ stands for electrons; $X{e}^{+}$ is the Xenon ions; $F^{-}$ and $S{F}_6^{-}$ stand for fluorine and sulfur ions, respectively; and d stands for charged (negatively) dust particles). $Z_{X{e}^{+}},$$Z_{F^{-}}$, and $Z_{S{F}_6^{-}}$ stand for the charging state of the argon, fluorine, and sulfur hexafluoride, respectively, and $q_d$ stands for the charge on the dust grains. When $q_d=$ constant, the electrons, the positive ion, and the two negative ions in the $(q,\alpha )$-distributional currents may be represented, [38]

$$I_e^{q\alpha }+I_{X{e}^{+}}^{q\alpha }+I_{F^{-}}^{q\alpha }+I_{S{F}_6^{-}}^{q\alpha }=0,$$

(3)

with

$${\sigma }_j^d=\pi r_d^2\left(1-\frac{2q_j{\varphi }_d}{m_j{V}_j^2}\right),$$

(4)

$$I_j^{q\alpha }=q_j\int V_j{\sigma }_j^d{f}_{CT}\left({\mathbf{V}}_j\right)d^3{\mathbf{V}}_j.$$

(5)

${\sigma }_j^d$ stands for the cross-section for charging interactions among dust particles and plasmas constituents of charge $q_j$, and ${\varphi }_d$ is the DSP of radius $r_d$. To continue, we integrate over the $f_{CT}\left({\mathbf{V}}_j\right)$ to simplify Equation (2). In spherical-coordinates, write the volume element as $d^3{\mathbf{V}}_j$=$V_j^{3}d{V}_{j}d\mu d\varphi$ to obtain the currents owing to electrons, positive ions, and the two negative ions due to negative charge dust particles [38], respectively.

$$I_e^{q\alpha }=-\sqrt{2^3}\pi r_d^{2}exp\left(\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\frac{e{\varphi }_d}{m_e{V}_{Te}^2}\right)V_{Te}{n}_e{\tau }_{q\alpha }C,$$

(6)

$$I_{X{e}^{+}}^{q\alpha }=\sqrt{2^3}\pi r_d^2\left\{1-\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\frac{e{\varphi }_d}{m_{X{e}^{+}}{V}_{X{e}^{+}}^2}\right\}{Z}_{X{e}^{+}}{V}_{X{e}^{+}}e{n}_{{}_{X{e}^{+}}}{\tau }_{q\alpha }C,$$

(7)

and

$$I_{F^{-}}^{q\alpha }=-\sqrt{2^3}\pi r_d^2\left\{1+\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\frac{Z_{F^{-}}e{\varphi }_d}{m_{F^{-}}{V}_{F^{-}}^2}\right\}{Z}_{F^{-}}{V}_{F^{-}}e{n}_{{}_F^{-}}{\tau }_{q\alpha }C,$$

(8)

$$I_{S{F}_6^{-}}^{q\alpha }=-\sqrt{2^3}\pi r_d^2\left\{1+\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\frac{Z_{S{F}_6^{-}}e{\varphi }_d}{m_{S{F}_6^{-}}{V}_{S{F}_6^{-}}^2}\right\}{Z}_{{}_{S{F}_6^{-}}}{V}_{S{F}_6^{-}}e{n}_{{}_{S{F}_6^{-}}}{\tau }_{q\alpha }C,$$

(9)

and

$$C={\left(\frac{1}{1-q}\right)}^{-\frac{3}{2}}\left[\frac{{\left(1-q\right)}^2\Gamma \left(\frac{1}{1-q}\right)}{\Gamma \left(\frac{1}{1-q}-\frac{5}{2}\right)\left[3\alpha +\left(\frac{1}{1-q}-\frac{3}{2}\right)\left(\frac{1}{1-q}-\frac{5}{2}\right){\left(1-q\right)}^2\right]}\right],\mathrm{for}-1<q\le 1,$$

where ${\tau }_{q\alpha }=\left[(6+24\alpha -17q+12q^2)/\left(-6q+29q^2-46q^3+24q^4\right)\right]$ and ${\chi }_{q\alpha }=1/q+8\alpha /$ $(2q-7q^2+6q^3).$ The DSP of a negatively charged dust particle can be written as by substituting Equations (6)–(9) into Equation $\left(3\right)$.

$$\begin{array}{cc}\hfill & \sqrt{U_{X{e}^{+}}}-\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\frac{Z_{X{e}^{+}}\psi }{\sqrt{U_{X{e}^{+}}}}-{\epsilon }_1\left(1-{\Delta }_1\frac{Z_{F^{-}}}{Z_{X{e}^{+}}}-{\Delta }_2\frac{Z_{S{F}_6^{-}}}{Z_{X{e}^{+}}}+Z_{X{e}^{+}}\psi P_d\right)exp\left(\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\psi \right)\hfill \\ \hfill & ={\Delta }_1{\eta }_1\sqrt{U_{F^{-}}}\frac{Z_{F^{-}}}{Z_{X{e}^{+}}}\left(1+\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\frac{Z_{F^{-}}\psi }{U_{F^{-}}}\right)+{\Delta }_2{\eta }_{2^{-}}\sqrt{U_{S{F}_6^{-}}}\frac{Z_{S{F}_6^{-}}}{Z_{X{e}^{+}}}\left(1+\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\frac{Z_{S{F}_6^{-}}\psi }{U_{{}_{S{F}_6^{-}}}}\right),\hfill \end{array}$$

(10)

$${\Delta }_1=\frac{n_{F^{-}}}{n_{X{e}^{+}}},{\Delta }_2=\frac{n_{{}_{S{F}_6^{-}}}}{n_{X{e}^{+}}},{\eta }_1={\left(\frac{m_{{}_{X{e}^{+}}}}{m_{F^{-}}}\right)}^{1/2},{\eta }_{{}_2}={\left(\frac{m_{{}_{X{e}^{+}}}}{m_{{}_{S{F}_6^{-}}}}\right)}^{1/2},$$

(11)

$$U_{F^{-}}=\frac{m_{{}_{F^{-}}}{V}_{F^{-}}^2}{2T_e},U_{{}_{S{F}_6^{-}}}=\frac{m_{{}_{S{F}_6^{-}}}{V}_{{}_{{}_{S{F}_6^{-}}}}^2}{2T_e},U_{X{e}^{+}}=\frac{m_{{}_{X{e}^{+}}}{V}_{X{e}^{+}}^2}{2T_e},\mathrm{and}{\epsilon }_1={\left(\frac{m_{X{e}^{+}}}{m_e}\right)}^{1/2}.$$

(12)

Normalized DSP is denoted by $\psi =e{\varphi }_d/T_e$, and the normalized dust number density by $P_d=4\pi n_d{r}_d{\rho }_0^2$ with ${\rho }_0=\sqrt{T_e/\left(4\pi n_{X{e}^{+}}{Z}_{X{e}^{+}}^2{e}^2\right)}.$ It has been observed that for significant conditions for $\alpha =0$ and $q=1$, ${\chi }_{q\alpha }=1$ and ${\tau }_{q\alpha }=1$, and by using one species of negative ion, Equation (10) agrees to the previous Equation (6) of Ref. [8], which shows that the DSP in a Maxwellian dusty plasma has negative ion streaming. Equation (10) is numerically solvable to investigate the relationship between $\psi$ and $P_d$.

4. Dust Grain Surface Potential (DSP) with ${\mathit{Ar}}^{+}-{\mathit{F}}^{-}-{\mathit{SF}}_{\mathbf{6}}^{-}$ Plasmas

When we use different forms of plasmas $A{r}^{+}-F^{-}-S{F}_6^{-}$, the $A{r}^{+}$ current can be stated as follows:

$$I_{A{r}^{+}}^{q\alpha }=\sqrt{2^3}\pi r_d^2\left\{1-\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\frac{e{\varphi }_d}{m_{A{r}^{+}}{V}_{A{r}^{+}}^2}\right\}{Z}_{A{r}^{+}}{V}_{A{r}^{+}}e{n}_{{}_{A{r}^{+}}}{\tau }_{q\alpha }C,$$

(13)

Using the same technique as in Section 3, when we substitute Equations $\left(6\right)$, $\left(8\right)$, $\left(9\right)$, and $\left(13\right)$ into $\left(3\right),$ we obtain

$$\begin{array}{cc}\hfill & \sqrt{U_{A{r}^{+}}}-\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\frac{Z_{A{r}^{+}}\psi }{\sqrt{U_{A{r}^{+}}}}-{\epsilon }_2\left(1-{\Delta }_3\frac{Z_{F^{-}}}{Z_{A{r}^{+}}}-{\Delta }_4\frac{Z_{S{F}_6^{-}}}{Z_{A{r}^{+}}}+Z_{A{r}^{+}}\psi P_d\right)exp\left(\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\psi \right)\hfill \\ \hfill & ={\Delta }_3{\eta }_3\sqrt{U_{F^{-}}}\frac{Z_{F^{-}}}{Z_{A{r}^{+}}}\left(1+\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\frac{Z_{F^{-}}\psi }{U_{F^{-}}}\right)+{\Delta }_4{\eta }_{4^{-}}\sqrt{U_{S{F}_6^{-}}}\frac{Z_{S{F}_6^{-}}}{Z_{A{r}^{+}}}\left(1+\frac{{\chi }_{q\alpha }}{{\tau }_{q\alpha }}\frac{Z_{S{F}_6^{-}}\Psi }{U_{S{F}_6^{-}}}\right),\hfill \end{array}$$

(14)

where

$${\Delta }_3=\frac{n_{F^{-}}}{n_{A{r}^{+}}},{\Delta }_{{}_4}=\frac{n_{{}_{S{F}_6^{-}}}}{n_{A{r}^{+}}},{\eta }_{3^{-}}={\left(\frac{m_{A{r}^{+}}}{m_{F^{-}}}\right)}^{1/2},{\eta }_4={\left(\frac{m_{{}_{A{r}^{+}}}}{m_{{}_{S{F}_6^{-}}}}\right)}^{1/2},$$

(15)

$$U_{A{r}^{+}}=\frac{m_{{}_{A{r}^{+}}}{V}_{A{r}^{+}}^2}{2T_e},\mathrm{and}{\epsilon }_2={\left(\frac{m_{A{r}^{+}}}{m_e}\right)}^{1/2}.$$

(16)

Equation (14) depicts a relationship between the DSP and $P_d$ in the presence of $A{r}^{+}$ ions is dipicted in Equation (14). By applying a limit on spectral indices $\alpha =0$ and $q=1,$ the value of ${\chi }_{q\alpha }=1$ and ${\tau }_{q\alpha }=1$, and Equation $\left(14\right)$, Equation (6) of Ref. [8] uses only one species of negative ion, which shows that the DSP in a Maxwellian dusty plasma has negative ion streaming. Equation (14) can be numerically solved to investigate the relationship between the DSP and the $P_{\mathbf{d}.}$

5. Numerical Results and Discussion

We have displayed the $(q,\alpha )$-distribution [15] in Equation (1) for various values of spectral index $\alpha$ and q for numerical demonstration. The spectral indices q and $\alpha$ represent the influence on the tail of super thermal particles and the shoulder of the velocity-disturbed graph, respectively. Figure 1a,b are plotted for condition $-1<q\le 1.$ The effect of different values of the spectral index $\alpha (0.1,0.15,0.2)$ for the fixed value of q (=0.7) is shown in Figure 1a. $\alpha$ illustrates the high-energy plasma elements on the shoulder of the graph. The distribution function $f_{CT}\left(V_j\right)$ as a function of $V_j$ is plotted for different values of $\alpha (0.1,0.15,0.2)$ at a fixed value of q (=0.9), as shown in Figure 1b. This figure shows that $(q,\alpha )$-distributed reduces to Maxellian at $q=0.9$ and $\alpha =0$. Figure 1c,d depict the $(q,\alpha )$-distribution for different values of non-extensive spectral index $q\ge 1$. It is observed that for increasing values of $\alpha ,$ the prominent effects of distribution are shown on the top shoulders, while the magnitude of the velocity distribution curves are nearly the same on the initial and end regions. Figure 1c,d show the values of the non-extensive spectral index q. It is discovered that as the value of increases, the prominent effects of plasma species are observed on the top shoulders but have no effect on the tails of velocity distribution curves.

Figure 1. (Color online) normalized $(q,\alpha )$-distribution function (a) $\alpha (0.1,0.15,0.2)$ with $q=0.7$ (b) $\alpha (0.0,0.1,0.2)$ with $q=0.9$ (c) $\alpha (0.0,0.3,0.6)$ with $q=2.5$ and (d) $\alpha (0.0,0.3,0.6)$ with $q=4$.

In the framework of Cairn–Tsallis-distributed dusty plasmas, the $\left(10\right)$ and $\left(14\right)$ are numerically solved, considering the typical properties of laboratory plasmas. In recent work, we evaluated the DSP ($\psi ),$ versus dust-density parameter $P_d,$ for species $X{e}^{+}-F^{-}-S{F}_6^{-}$ and $A{r}^{+}-F^{-}-S{F}_6^{-}$. Furthermore, the mass of ions associated with $X{e}^{+}$ and $A{r}^{+}$ is composed of $131m_P$ and $40m_P$, respectively, where $m_p$ is mass of a proton. The mass ratios for $X{e}^{+}-F^{-}-S{F}_6^{-}$ and $A{r}^{+}-F^{-}-S{F}_6^{-}$ are ${\eta }_1,$ ${\eta }_2,$${\eta }_3,$ and ${\eta }_4$ are ${\left(131/19\right)}^{1/2},$ ${\left(131/146\right)}^{1/2},$${\left(40/19\right)}^{1/2}$, and ${\left(40/146\right)}^{1/2}$, respectively. Previously, the IA waves in plasmas (multi-ions) were explored numerically and experimentally [18], with the two kinds of negative ions. Figure 2 illustrates the DSP versus dust-density parameter $P_d$ for the various values of spectral index $\alpha (0.0,0.01,0.02)$; the fixed values of spectral index q and the values of other parameters include $Z_{Xe}=Z_F=Z_{A{r}^{+}}=Z_{SF}=1,$ $U_{X{e}^{+}}=U_{A{r}^{+}}=U_{F^{-}}=U_{S{F}_6^{-}}=U_{X{e}^{+}}$$=0.5,$ ${\Delta }_1={\Delta }_2={\Delta }_3={\Delta }_4=0.4,$${\epsilon }_1={\left(131/0.00055\right)}^{1/2},{\epsilon }_2={\left(40/0.00055\right)}^{1/2}.$ It is shown in Figure 2a that the magnitude of DSP varies directly with the $\alpha ,$ by taking into account the $X{e}^{+}-F^{-}-S{F}_6^{-}$ plasmas (using Equation (10)). The same effect spectral index $\alpha$ is observed in $A{r}^{+}-F^{-}-S{F}_6^{-}$ plasmas (using Equation (14)). Furthermore, the dust surface potential in $X{e}^{+}-F^{-}-S{F}_6^{-}$ is frequently larger than in $A{r}^{+}-F^{-}-S{F}_6^{-}$, owing to the higher mass of $A{r}^{+}$. For spectral index $\alpha$ with other parameters held constant, the dust $DSP$ demonstrates almost monotonic behavior.

Figure 2. (Color online) $\psi$ versus $P_d$ for $\alpha (0.0,0.01$,$0.02)$ with $q=0.9;$ (a) xenon–fluorine–sulfur hexafluoride as given in Equation (10) (b) for argon–fluorine–sulfur hexafluoride plasma as given in Equation (14).

Using Equations (10) and (14), we show the profiles of the $DSP$ vs. dust parameter $P_d$ for both types of plasmas with different values of spectral index q (=0.90.05,1.0) and a fixed value of alpha (=0.01) for both types of plasmas in Figure 3a,b, respectively. Both Figure 3a,b show that as the value of q increases, the dust-surface potential drops.

Figure 3. (Color online) $\psi$ versus $P_d$ for $q(0.9,0.95$,$1.0)$ with $\alpha =0.01;$ (a) xenon–fluorine–sulfur hexafluoride as given in Equation (10) (b) for argon–fluorine–sulfur hexafluoride plasma as given in Equation (14).

In Figure 4a,b, we illustrate the profiles of the DSP versus dust parameter $P_d$ with different values $(0.1,0.4,0.8)$ of Xenon ion streaming velocity [$U_{X{e}^{+}}=m_{{}_{X{e}^{+}}}{V}_{X{e}^{+}}/2T_e]$ and Argon ion streaming velocity [$U_{A{r}^{+}}=m_{{}_{A{r}^{+}}}{V}_{A{r}^{+}}/2T_e]$ with a fixed value of spectral indices $\alpha$$(=0.02)$ and $q(=0.95)$ by using Equations (10) and (14), respectively. We observed that by increasing the values of $U_{X{e}^{+}}(0.1,0.4,0.8)$ and $U_{A{r}^{+}}(0.1,0.4,0.8)$, the magnitude of DSP increases, while the value of negative streaming velocity stays the same. In Figure 5a–d, the DSP curves are plotted as a versus $P_d$ for different cases of negative ions ($U_{F^{-}}$ and $U_{{}_{S{F}_6^{-}}}$) for both types of plasma.

Figure 4. (Color online) $\psi$ versus $P_d$ (a) for $U_{X{e}^{+}}(0.1,0.4.0.8)$ with $q=0.95$ and $\alpha =0.02,$ for xenon–fluorine–sulfur hexafluoride as given in Equation (10) (b) for $U_{A{r}^{+}}(0.1,0.4.0.8)$ with $q=0.95$ and $\alpha =0.02$ for argon–fluorine–sulfur hexafluoride plasma as given in Equation (14).

Figure 5. (Color online) $\psi$ versus $P_d$, (a) for $U_{F^{-}}(0.1,0.4.0.8)$ with $q=0.95$ and $\alpha =0.02,$ (b) $U_{{}_{S{F}_6^{-}}}(0.1,0.4.0.8)$ with $q=0.95$ and $\alpha =0.02$ for xenon–fluorine–sulfur hexafluoride as given in Equation (10) (c) for $U_{F^{-}}(0.1,0.4.0.8)$ with $q=0.95$ and $\alpha =0.02,$ (d) $U_{{}_{S{F}_6^{-}}}(0.1,0.4.0.8)$ with $q=0.95$ and $\alpha =0.02$ for argon–fluorine–sulfur hexafluoride plasma as given in Equation (14).

Figure 5a shows the variation of the DSP against the $P_d$ of $X{e}^{+}-F^{-}-S{F}_6^{-}$ plasma for different values of $U_{F^{-}}(=0.1,0.4,0.8)$, with $\alpha =0.02$ and $q=0.95$ kept the same. It is observed that the DSP increases by increasing the values of streaming fluorine ion. Figure 5b examines the streaming effect of $U_{{}_{S{F}_6^{-}}}$ on dust potential with $U_{F^{-}}$, and the remainder of the parameters are the same as in Figure 2a. The streaming influence of $U_{{}_{S{F}_6^{-}}}$ on DSP is the same as $U_{F^{-}}$, but the difference between the curves of $U_{{}_{S{F}_6^{-}}}$ is smaller than $U_{F^{-}}$. The effects of $U_{F^{-}}$ and $U_{{}_{S{F}_6^{-}}}$ on DSP for $A{r}^{+}-F^{-}-S{F}_6^{-}$ plasmas are shown in Figure 5c,d. The non-monotonic change in dust potential can also be seen in Figure 5c,d, with comparable consequences to those shown in Figure 5a,b. The physical significance for the observed trend is mainly due to the positive/negative ion streaming that gradually increases the relative potential ($\psi )$ of ions as well as the negative charge dust particles. The dust-surface potential increases with the increase in ion velocity but decreases with the collisions of ions with dust particles.

6. Conclusions

In the present work, we have studied a DSP variation by taking into consideration the positive/negative ion streaming in the presence of Cairns–Tsallis distributed plasmas. For the $X{e}^{+}-F^{-}-S{F}_6^{-}$ and $A{r}^{+}-F^{-}-S{F}_6^{-}$ plasma species, the DSP for the negatively charged dust particles is calculated numerically and analytically. In the contest of Cairns–Tsallis plasma, the equations for the DSP from the current balance equation are determined analytically by taking into consideration the electrons, negative ions ($F^{-}$ and $S{F}_6^{-}$), and positive ions ($X{e}^{+}$ and $A{r}^{+}$) for the charged dust grains. (i) The magnitude of dust-surface potential $\left(\right|\Psi \left|\right)$ increases with an increase in the spectral index $\alpha$, positive ion streaming velocity $(U_{Ar+}$ and $U_{Xe+})$, and negative ion streaming velocity $(U_{F^{-}}$ and $U_{S{F}_6^{-}})$. (ii) The magnitude of dust-surface potential $\left(\right|\Psi \left|\right)$ diminishes and approaches the Maxwellian case when the value of the spectral index q is increased. The key parameters have a significant impact on the DSP, for example, the spectral indices $\alpha$ and q, positive ion streaming ($U_{A{r}^{+}}$ and $U_{X{e}^{+}}$), and negative ion streaming ($U_{F^{-}}$ and $U_{S{F}_6^{-}}$) of both types of plasmas. For the $X{e}^{+}-F^{-}-S{F}_6^{-}$ and $A{r}^{+}-F^{-}-S{F}_6^{-}$ plasma species, the numerical outcomes show both monotonic as well as non-monotonic profiles of the DSP. Finally, we come to the conclusion that the results of our investigations should be important for understanding dust-charging processes, computing dust-charge variation, and comparing dust-surface potential in low-temperature plasma species that contain electrons as well as positive and negative ions and by using the Cairns–Tsallis distribution function.