The dust-charged particles together with Maxwellian/non-Maxwellian electrons as well as ions have been embedded in the dusty plasmas. The typical orderings for such a plasma incorporating the length scale with the conditions ${L}\gg {{\lambda }}_{{\rm{Def}}}$ and α$\,\gg \,{{r}}_{{d}}$ have been considered, where L, ${{\lambda }}_{{\rm{Def}}},$ α and ${{r}}_{{d}}$ explicitly represent the plasma dimension, the effective Debye radius, the distance between two grains, and the dust particle radius, respectively [1, 2]. They often consist of ice particles, dielectric, and metallic such as amorphous carbon, graphite, silicate, magnetite, etc. The masses of dust-charged particles are different from a million to a billion times that of an ion while their size ranges from nanometers to millimeters. The charge caused by the dust particles fluctuates with regard to space and time rather than being constant. As a result, a unique dust-associated mode is introduced by dust constituents and associated dust charge perturbations that dramatically change the pre-existing joint modes and their instabilities [1]. This is regardless of the fact that dust-acoustic and dust-ion-acoustic modes [3, 4] have been identified as the deepest collective modes in the complex plasmas [5]. These modes are further verified by numerous experimental laboratory investigations [6–10]. Barkan et al [7] that dealt with the charging of micron-size dust grains experimentally, and these dust grains were also merged into completely ionized, steady-state, and magnetized plasma, comprising electrons and ${{\rm{K}}}^{+}$ ions (both at a temperature of 0.2 eV). Nevertheless, measurements using a Langmuir probe were carried out to ascertain the negative charge in the plasma, which was then broken down into free electrons and dust grains. It is generally known that electron–ion currents flowing over the surface of dust grains cause the dust particulate to become negatively charged. However, dust particles can also have a positive charge in some physical processes such as secondary emission, thermionic emission, light emission, etc [1]. Physically, this is due to the fact that for the electron as well as ion fluxes to be balanced in the equilibrium state, most of the electrons must be reflected by the potential barrier situated in the particle regime and ambient plasma. The coefficient is determined in the particular regime for electron and ion fluxes to the particle regime that is achieved.

The orbital motion limited (OML) assumption is one of the most effective approaches for describing the electron and ion fluxes obtained by the dust particles [11, 12]. Using this method, the cross-sections may be calculated from the energy and angular momentum conservation for collisionless electron as well as ion paths near a small individual dust particle. The dust particulates largely become negatively charged in low-temperature plasmas since the electrons travel considerably more quickly than heavier ions due to their small masses. The dust has a negative potential, which causes the enhancement in ion current and reduces the electron current. This process will have remained till the electron and ion current magnitude become equal, such as $\left|{{I}}_{{i}}\right|=\left|{{I}}_{{e}}\right|,$ where ${{I}}_{{i}}({{I}}_{{e}})$ is the ion (electron) current that lies over the dust surface. Approximately two decades ago, the positive ions and Maxwellian electrons [7] were taken into consideration when investigating the dust potential both theoretically and experimentally [7]. Subsequently, numerous other researchers expanded on the notion of dust surface potential [13–15], examining the effects of ${q}$-distributed and κ-distributed non-Maxwellian ions and electrons [13, 14]. The currents and the dust potential relevant to both $q$-distributed and $\kappa $-distributed ions and electrons were numerically demonstrated to be dramatically different from the scenario in which ion and electron species adopt the Maxwellian velocity distribution function. Mamun and Shukla [16] specifically investigated the potential of dust grains using the dust-charging model. They further investigated the currents for thermal and streaming (Boltzmann-distributed) ions carrying negative charges when there were Boltzmann-distributed positive ions and negative electrons.

The distribution of plasma species, for instance, electrons and ions in numerous space plasma systems, for example, the interstellar medium, planetary magnetosphere, thermosphere, solar winds, etc, greatly deviates from the Maxwellian distribution. In velocity space, they frequently exhibit a superthermal or high-energy tail that follows an inverse power law. Numerous studies on the $\kappa $-distribution have already been conducted to explicate the collective behaviors in plasma environments [17–19]. In this regard, Vasyliunas [20] addressed the high-energy plasma particle in the tail plasma distribution curve by establishing a non-Maxwellian function that contained a spectral index. Therefore, a more generalized distribution function ${{f}}_{{\boldsymbol{\beta }}}\left({r},{q},{{v}}_{{\beta }}\right)$ in the power law that forms with distinct indices ${r}$ and ${q}$ is necessary for describing the flat shoulders and high-energy tails of the plasma species (electrons and ions) both in space and laboratory plasmas [21]. Here, it is essential to point out that the generalized power-law $\left({\rm{r}},{\rm{q}}\right)$-distribution function can be evolved into a Maxwellian distribution function when the parameters ${r}=0$ and ${q}={\rm{\infty }}$ are considered [20]. The Maxwellian distribution function is inadequate to describe the turbulent plasmas, in terms described by the non-thermodynamic equilibrium states. Therefore, the generalized $\left({\rm{r}},{\rm{q}}\right)$-distribution function is the most effective way to model the non-thermal plasma particles since it accounts for the flat shoulders in addition to the high-energy tails in the distribution function. The natural space environments (such as the solar wind [22], Earth plasma sheet [23] and near plasma shock, [24]) are also taken into account, resulting in a more precise match to empirical data collected in the laboratory under different conditions. It is crucial to note that the generalized $\left({\rm{r}},{\rm{q}}\right)$-distribution function [21] is more appropriate than any other non-Maxwellian distribution when analyzing solar wind data. In the recent past, (r,q)-distribution successfully fitted the observed distributions from space plasmas as well as qualitatively and quantitatively interpreting the space observation [25–27]. However, this is commonly used to describe the behavior of non-thermal plasma particles in low-pressure discharges [28, 29], and it has the same similar spectral index ${r},$ which correlates to a more generalized $\left({\rm{r}},{\rm{q}}\right)$-distribution function in the mathematical description for the bulk electron distribution function. It is concluded here that the Maxwellian distribution function does not remain valid in all circumstances due to the existence of highly energetic superthermal electrons [30, 31]. Therefore, the investigation of dust surface potential via generalized $\left({\rm{r}},{\rm{q}}\right)$-distribution becomes more physical and appropriate to address the non-thermal particle populations with highly energetic tails on the flat shoulders in multi-ion dusty plasmas. These non-thermal particle populations could not be modeled simultaneously either by kappa distribution [20] or by Druyvesteyn–Davydov distribution [30]. In addition, the dispersion relations for left/right electromagnetic waves in a hot magnetized plasma have been determined using a generalized $\left({\rm{r}},{\rm{q}}\right)$-distribution function by Qurashi et al [21]. They have elaborated on the unique characteristics of slow/fast Alfven waves. Mushtaq and Shah [32] have used the $\left({\rm{r}},{\rm{q}}\right)$-distribution function to model the electron–ion plasma and they further examined the propagation features of supersonic(subsonic) ion-acoustic solitons along with trapped electron effects. They have pointed out that the presence of high energy on tails is caused by the modification of the spectral indexes ${q}$ and ${r}$ that contributed to the presence of high energy on the flat shoulders in the distribution function profile. Kiran et al [33] established a correlation between theory and observational data for slow solar wind streams. Furthermore, they have calculated the power dissipated through oblique Alfven waves by employing the generalized $\left({\rm{r}},{\rm{q}}\right)$-distribution function, particularly for parallel proton heating in solar wind plasma. Aziz et al [34] recently explored the properties of adiabatically trapped electrons in accordance with the $\left({\rm{r}},{\rm{q}}\right)$-distribution function. They have considered two potential theories to examine the nonlinearly coupled kinetic Alfven-acoustic waves. The electromagnetic waves were discussed in regard to the terrestrial magnetotail with flat-topped electron-distributed plasma [35, 36]. Numerous studies have been conducted over four decades to examine the ion-acoustic waves whose Debye wave number ${{K}}_{{e}}$ is much greater than its wave number ${K}$ in a multi-ion plasma [37]. They have studied the multi-component ${\rm{Ar}}\mbox{--}{\rm{He}}$ plasma theoretically and found numerous new phenomena that may not be observed using single-ion plasma. These phenomena are as follows: (i) the wave damping rate is greatly increased with a small amount of light-ion impurity. Previously, the enhanced Landau damping was investigated experimentally by Alexeff et al [38], and (ii) for ${{T}}_{{e}}$/${{T}}_{{i}}$ greater than 20, the phase velocity becomes larger than the acoustic speed of ${\rm{Ar}}$-ions compared to the ${\rm{He}}$-ions when more ${\rm{He}}$-ion concentration is considered, where ${{T}}_{{e}}$/${{T}}_{{i}}$ is the electron-to-ion temperature ratio. The goal of this experiment was to observe the existence of two ion-acoustic waves in ${\rm{Ar}}\mbox{--}{\rm{Ne}}$ and ${\rm{Ar}}\mbox{--}{\rm{He}}$ plasmas under certain conditions of ${{T}}_{{e}}$/${{T}}_{{i}}=10$ to ${{T}}_{{e}}$/${{T}}_{{i}}=15.$ Moreover, Nakamura et al [39] investigated the ion-acoustic wave propagation in two-ion plasmas theoretically and experimentally by considering the ${{T}}_{{e}}$/${{T}}_{{i}}=10$ to ${{T}}_{{e}}$/${{T}}_{{i}}=15.$ They have observed the two modes in an ${\rm{Ar}}\mbox{--}{\rm{He}}$ plasma, whereas single mode is detected in an ${\rm{Ar}}\mbox{--}{\rm{Ne}}$ plasma. The Bohm criterion is re-examined in the context of 1D static and unmagnetized low-pressure Ar–He plasma mixture containing dust particles and non-thermal electrons [39]. It have been identified that Bohm velocities are strongly correlated and have shifting peaks in their profiles. These two gases could be combined to produce both supersonic ${\rm{He}}$ and subsonic ${\rm{Ar}}$ Bohm velocities. Recently, the current equations for two negative ions, ${{\rm{Xe}}}^{+}\mbox{--}{{\rm{F}}}^{-}\mbox{--}{{\rm{SF}}}_{6}^{-}$ and ${{\rm{Ar}}}^{+}\mbox{--}{{\rm{F}}}^{-}\mbox{--}{{\rm{SF}}}_{6}^{-}$, within the framework of kappa-distributed dusty plasma, are derived analytically [40]. They have obtained the dust grain surface potential that greatly depends on kappa, temperature and density ratios. They have shown in their findings that the dust grain surface potential displays non-monotonic behavior for both plasmas using the different values of kappa, temperature and density ratios. Moreover, the temperature ratio slightly affects the surface potential. The dust particle surface potential for Ar–He plasma in the context of negatively charged dust particles with a power-law $\left({r},{q}\right)$-distribution dusty plasma has not yet been explored.

The movement of linear ion-acoustic modes in multi-ion plasma was investigated in the upper ionosphere [39]. They have observed that a wave propagates along with acoustic velocity that is determined from the effective temperature when the temperature ratio of electron-to-ion was substantially higher than one. The effective temperature is one at which the ion-acoustic wave grows significantly. However, in a multi-ion plasma, the number of modes is influenced by both ${{T}}_{{e}}/{{T}}_{{i}}$ and heavy-to-light ion mass ratio [41]. They have further examined the ${\rm{Ar}}\mbox{--}{\rm{He}}$ plasma with few novel findings that were not found in single-ion plasma. Numerous researchers have explored the decay in magnetosonic waves and lower hybrid waves by considering the two-ion plasma species [42–44]. He ions were added to an Ar-ion plasma to find the critical value of a parametric decay in a Langmuir wave [45] and also create collisionless shock-wave turbulence in electromagnetic fields [46]. In the electrostatic plasma, the charged particles are composed of electrons, positive ${\rm{Ar}}$ and He ions as well as spherical dust grains. The dusty plasma structures created in the mixing of He $\left({\rm{He}}\sim 89 \% \right)$ and Ar $\left({\rm{Ar}}\sim 11 \% \right)$ at high pressure $\left({\rm{P}}=0.009{\rm{Torr}}\right)$ are being studied [47]. In this study, we present details of the dust particle surface potential in an Ar–He plasma mixture having non-thermal distribution. Motivated by the above investigations in a variety of plasmas, here, we explore the dust-charging model modified by the generalized $({\rm{r}},{\rm{q}})$-distribution function. We derive the equations for currents due to non-thermal electrons and ions, including the He and Ar ions as well as dust grain surface potential. The presence of various currents on the dust grain’s surface strongly modifies the collective phenomena and instabilities. The dust grain surface potential is evaluated in a broad range of the dust number density where the parameters ${r}$ and ${q},$ Ar ion-to-electron temperature ratio, He ion-to-Ar ion number density ratio and He ion-to-electron temperature ratio significantly affected the dust grain surface potential. The remainder of the paper is organized as follows: the generalized distribution function incorporating ${r}$ and ${q}$ as spectral indices are pointed out in section 2. In section 3, the equations for the currents are derived that involve the positive ions ${{\rm{Ar}}}^{+}\mbox{--}{{\rm{He}}}^{+}$, and electrons in the generalized $\left({\rm{r}},{\rm{q}}\right)$-distribution function are also employed to obtain the equations for the dust particle potential. Section 4 involves discussion of numerical investigations and concludes with a concise summary.

A generalized distribution function with indices r and q is employed to examine the deviation of the thermal equilibrium and graphically displays a flat shoulder with large energy tails, which measure the deviation from thermal equilibrium that depends on the spectral indices ${q}$ and ${r}.$ The $\left({\rm{r}},{\rm{q}}\right)$-distribution for the ${\beta }{\rm{th}}$ plasma species can be written as [21, 47, 48],withwhereand

Note that ${{\theta }}_{{\beta }}$ is the real effective thermal speed comprising the standard thermal speed ${{\boldsymbol{v}}}_{{\boldsymbol{T}}{\boldsymbol{\beta }}}={\left({{T}}_{\beta }/{{m}}_{\beta }\right)}^{1/2}$(where ${{T}}_{{\beta }}$ and ${{m}}_{{\beta }}$ are the temperature and mass of the plasma species, respectively), where ${{n}}_{{\beta }}{{\rm{and}}{v}}_{{\beta }}$ are the plasma number density, and ion/electron speeds, respectively. Here, Γ stands for gamma function. In addition, it should be noted that the spectral indices r and q should satisfy the inequality conditions $1\lt {q}$ and $5/2\lt \left({q}+{qr}\right)$, which comes from the definition of the temperature [49]. The distribution function ${{f}}_{{\beta }}\left({r},{q},{{v}}_{{\beta }}\right)$ also normalized unity, i.e. $\frac{1}{{n}}\int {{f}}_{{\beta }}\left({r},{q},{{v}}_{{\beta }}\right){d}{{v}}_{{\beta }}=1.$ The number of highly energetic plasma particles on the top shoulder of the distribution curve was enhanced with the increase in the ${r}$ index. On the other hand, the high-energy particles on the tail of the distribution function increase with ${q}.$ Furthermore, it has been investigated that $\left({\rm{r}},{\rm{q}}\right)$-distribution exhibits the behavior of κ-distribution under certain conditions of ${r}=0$ and ${q}={\kappa }+1$ [20] and reduces to the Maxwellian distribution by applying the limits ${r}=0$ and ${q}\to {\rm{\infty }}$ [21].

We study the spherical dust particles of finite size in unmagnetized plasmas containing electrons, Ar and He ions. The negatively charged dust particles are considered due to the current of electrons and ions (Ar and He) within the framework of $\left({r},{q}\right)$-distributed plasma that can be formulated by the current equation as $\partial {{q}}_{{\rm{d}}}$/$\partial {t}={{I}}_{{\rm{e}}}^{{\rm{r}},{\rm{q}}}+{{I}}_{{\rm{Ar}}}^{{\rm{r}},{\rm{q}}}+{{I}}_{{\rm{He}}}^{{\rm{r}},{\rm{q}}},$ where ${{\rm{q}}}_{{\rm{d}}}$ is the dust charge. Furthermore, we will assume the dust charge constant $\left(\partial {{q}}_{{\rm{d}}}/\partial {t}=0\right),$ and then the current equation of electrons, Ar and He ions can be written [1, 14, 16], as,where the electron, Ar and He currents may take the following forms [1]:

${{\sigma }}_{{\beta }}^{{d}}={\pi }{{r}}_{{d}}^{2}\left(1+2{e}{{\rm{\phi }}}_{{d}}/{{m}}_{{\beta }}{{v}}_{{\beta }}^{2}\right)$ is the corresponding cross-section of dust particles where ${{\rm{\phi }}}_{{d}}={{q}}_{{d}}/{{r}}_{{r}}$ depicts the dust potential with dust radius ${{r}}_{{d}}$ and negatively charged dust grains ${{q}}_{{\rm{d}}}=-{{\rm{Z}}}_{{\rm{d}}0}{\rm{e}}.$ Using the volume element in the spherical polar coordinates system and $\left({\rm{r}},{\rm{q}}\right)$-distribution function for the ${\beta }{\rm{th}}$ species, the equations for ${{\rm{I}}}_{{\rm{e}}}^{{\rm{r}},{\rm{q}}},{{\rm{I}}}_{{\rm{Ar}}}^{{\rm{r}},{\rm{q}}}$ and ${{\rm{I}}}_{{\rm{He}}}^{{\rm{r}},{\rm{q}}}$ are also obtained by solving equations (3) and (1): andwhere where ${{\theta }}_{{\rm{e}}}={\left({{\rm{\epsilon }}}_{{r},{q}}\right)}^{1/2}{{\rm{v}}}_{{\rm{Te}}},$ ${{\theta }}_{{\rm{Ar}}}={\left({{\rm{\epsilon }}}_{{r},{q}}\right)}^{1/2}{{\rm{v}}}_{{\rm{TAr}}},$ and ${{\theta }}_{{\rm{He}}}\,={\left({{\rm{\epsilon }}}_{{r},{q}}\right)}^{1/2}{{\rm{v}}}_{{\rm{THe}}}$ are the thermal speeds of electrons, Ar and He ions, respectively. The substitution of equations (4)–(6) to (2), leads to the generalized equation for dust particle surface potential as, In equation (7), the remaining parameters are defined as ${{\rm{\Lambda }}}_{{\rm{r}},{\rm{q}}}={{x}}_{{\rm{r}},{\rm{q}}}/{{\rho }}_{{\rm{r}},{\rm{q}}},$ ${\beta }={\left({{m}}_{{\rm{Ar}}}/{{m}}_{{\rm{He}}}\right)}^{1/2},$ γ$\,=\,{{T}}_{{\rm{He}}}/{{T}}_{{e}},$ ${\sigma }={{T}}_{{\rm{Ar}}}/{{T}}_{{e}}$ and μ$\,=\,{\left({{m}}_{{\rm{Ar}}}/{{m}}_{{e}}\right)}^{1/2}.$ The normalized dust potential is shown by the parameter ${{V}}_{{\rm{d}}}={e}{\phi }_{{\rm{d}}}/{{T}}_{{e}}.$ At equilibrium, the condition of charge neutrality ${{n}}_{{e}}={{Z}}_{{\rm{Ar}}}{{n}}_{{\rm{Ar}}}+{{Z}}_{{\rm{He}}}{{n}}_{{\rm{He}}}-{{q}}_{{\rm{d}}}{{n}}_{{\rm{d}}}/{e},$ where ${{n}}_{{e}},{{n}}_{{\rm{Ar}}},{{n}}_{{\rm{He}}}$ and ${{n}}_{{\rm{d}}}$ are the unperturbed number density of electrons, Ar, He and dust, respectively. The terms ${{\rm{Z}}}_{{\rm{Ar}}}$ and ${{\rm{Z}}}_{{\rm{He}}}$ are the charging states of Ar and He where α$={{n}}_{{\rm{He}}}/{{n}}_{{\rm{Ar}}}.$ Using the charge quasi-neutrality condition, we obtain the equation ${{n}}_{{\rm{e}}}/{{Z}}_{{\rm{Ar}}}=1+{\alpha }\left({{Z}}_{{\rm{He}}}/{{Z}}_{{\rm{Ar}}}\right)+{{Z}}_{{\rm{Ar}}}{{P}}_{{\rm{d}}}{{V}}_{{\rm{d}}}$ and then substitute in equation (7) to obtain the following generalized equation:

${{P}}_{{d}}=4{{\lambda }}_{0}^{2}{\pi }{{n}}_{{d}}{{r}}_{{d}},$ is the dust number density with a characteristic length of ${{\lambda }}_{0}={\left({{T}}_{{e}}/4{\pi }{{Z}}_{{\rm{Ar}}}^{2}{e}{{n}}_{{\rm{Ar}}}\right)}^{1/2}.$ Equation (8) is now notably changed by the ${r}$ and ${q}$ indices of the $\left({r},{q}\right)$-distribution incorporating electrons, Ar and He ions and is considered to be the generalized form for the dust surface potential. At ${r}=0$ and for a larger value of q, the ratio ${{\rm{\Lambda }}}_{{\rm{r}},{\rm{q}}}$ approaches unity. Therefore, equation (8) reduces to equation (6) of [16], where Maxwellian electrons, one positive and one negative ion are accounted for.

Using the typical values of the spectral indices ${r}$ and ${q},$ we demonstrate the 3D plots of the electron, He ion and Ar ion current from equations (4), (5) and (6), respectively. The parameters used here are consistent with low-temperature laboratory plasma ${{T}}_{{e}}=2.0{\rm{eV}},$ ${{T}}_{{\rm{Ar}}}={{T}}_{{\rm{He}}}=0.1{\rm{eV}}$ and the ion number density is ${10}^{9}/{{\rm{cm}}}^{3}.$ It is observed from figure 1(a) that the spectral index ${r}=0.0-2$ increases the electron current while the electron current varies inversely with the spectral index ${q}=2.6-10.$ The value of the current is minimum at ${r}=2$ and ${q}=2.6$ because, at this point, plasma particles are fully non-Maxwellian, and repulsion between negatively charged dust particles and electron current occurs. The energetic particles increase on the broad shoulder of the velocity distribution linear curves with spectral index ${r}$ while the supernormality on the tail, as well as the width of velocity distribution curves, increases by decreasing the spectral index ${q}$ [50]. Opposite effects have been observed in figures 1(b) and (c) at ${r}=2$ and ${q}=2.6$ when the He ion and Ar ion current is maximum due to attraction between negative dust and positive current. It is concluded that the electron current decreases at a particular point, whereas the ion current increases at that point, which in turn justifies the OML method and satisfies the charge quasi-neutrality condition. The magnitude of current decreases in the ion case due to its large mass and smaller thermal velocity compared to the electron thermal velocity.

Within the framework of the $\left({r},{q}\right)$-distribution function in dusty plasmas, equation (8) is numerically solved by considering the typical values of laboratory plasma parameters for Ar–He ${{\rm{Ar}}}^{+}\mbox{--}{{\rm{He}}}^{+}$ plasma [51]. Furthermore, the mass of ions associated with ${{\rm{Ar}}}^{+}$ and ${{\rm{He}}}^{+}$ are $40{{\rm{m}}}_{{\rm{p}}}$ and ${4{\rm{m}}}_{{\rm{p}}}$, respectively, (where ${{m}}_{{\rm{p}}}=1.67\times {10}^{-27}{\rm{kg}}$ is the mass of a proton). Previously, the acoustic waves have been explored numerically and experimentally by Ar–He plasmas [50], with the two kinds of positive ions. Figures 2(a) and (b) illustrate the ${{V}}_{{d}}$ versus ${{P}}_{{d}}$ for the spectral index ${r}{=}0,0.5,1.0$ at fixed ${q}=2.6$ and spectral index ${q}=2.6,5.0,10$ with ${r}=1,$ respectively. The remaining parameters used here are α$\,=\,0.1,{\gamma }={\sigma }=0.5$ and ${{Z}}_{{{\rm{He}}}^{+}}={{Z}}_{{{\rm{Ar}}}^{+}}=1.$ It is observed that the magnitude of ${{V}}_{{d}}$ varies inversely with spectral indices ${r}$ and ${q}$ in an Ar–He plasma. This is because, when spectral index ${q}$ increases the distribution of plasma species on the tail of the distribution curve, it can lead to Maxwellian behavior.

Moreover, it is found that spectral indices have a significant impact when ${\rm{Log}}{{P}}_{{d}}$ is less than $-1$ but have no impact when ${\rm{Log}}{P}$ is greater than $-0.5.$ The dependence of Ar ion-to-electron temperature ratio ${\sigma }\,=\,0.1,0.5,1.0$ and He ion-to-electron temperature ratio γ$=0.1,0,5,1.0$ on ${{V}}_{{d}}$ and ${{P}}_{{d}}$ are shown in figures 3(a) and (b), respectively. It is evident from the plot that the magnitude of dust surface potential increases by increasing the Ar temperature. Whereas the opposite effect is observed in the dust grain surface potential by increasing the temperature of He ions due to them having less mass than Ar. Figures 4(a) and (b) signify the effect of the number density ratio of He to Ar α$={{n}}_{{\rm{He}}}/{{n}}_{{\rm{Ar}}}$ on the dust grain surface potential for non-Maxwellian for ${r}=1$ and ${q}=2.6$ and Maxwellian for ${r}=0$ and ${q}=20,$ respectively. The positive ion number density would lead to an increase in the dust surface potential in both cases due to the increase in positive ion density that significantly increases the electrons to achieve the charge neutrality condition.

In this paper, the ${{\rm{Ar}}}^{+}\mbox{--}{{\rm{He}}}^{+}$ plasma is studied both analytically and numerically through the current balance equation by attaining the charge quasi-neutrality condition. We have evaluated the ${{\rm{Ar}}}^{+}\mbox{--}{{\rm{He}}}^{+}$ currents as a function of a generalized distribution function with indices ${r}$ and ${q},$ which leads to the dust grain surface potential. This surface potential is evaluated in a wide range of key plasma parameters containing the spectral indices of the distribution function ${r}$ and ${q},$ He ion-to-Ar ion number density ratio, Ar ion-to-electron temperature ratio, and He ion-to-electron temperature ratio that affect the dust grain surface potential significantly. Recently, Abdullah Khan et al [40], carried out analytical and numerical investigations on dust surface potential for the negatively charged dust particulates in ${{\rm{He}}}^{+}$ and ${{\rm{Ar}}}^{+}$ plasmas. In the context of kappa-distributed plasma, they have analytically derived the dust grain surface potential equations from the currents (by taking into consideration the electrons, ions, as well as negative ions for dust particulates). Furthermore, they have elaborated on the monotonic and non-monotonic behavior of the dust surface potential for both plasmas and their findings qualitatively resemble previous works by Mamun and Shukla [16] and Barkan et al [7]. In this paper, we use a similar choice of current equations to see the effect of the $({\rm{r}},{\rm{q}})$-distribution function on the dust grain surface potential. To conclude, we examined a dust-charging model that considered the electrons and multiple positive ions ${{\rm{Ar}}}^{+}\mbox{--}{{\rm{He}}}^{+}$ on a dust-charged particle’s surface potential in the existence of more generalized $({\rm{r}},{\rm{q}})$-distributed plasmas. The current equations are derived for the electron and two positive ions. We have elucidated how the dust surface potential changes with ${\rm{r}}$ and ${\rm{q}}$ together with other dusty plasma variables such as the temperature ratio (He ion-to-electron temperature and Ar ion-to-electron temperature) and the density ratio (the Ar-to-He ions). However, significant modifications in the dust potential are measured by increasing the parameters γ as well as ${\sigma }$ and the potential reductions as α value arises. Finally, we emphasize the importance of our research in understanding dust surface potential mechanisms and calculating dust charge in low-temperature multi-ion plasmas species [51–54] that satisfy the $({\rm{r}},{\rm{q}})$-distribution function.