2. Basic equations

In this research, we considered a low-pressure plasma mixture of two types of positive ions ( ${\text{He}}^{+}\text{ and }{\text{Ar}}^{+}$ ), non-thermal electrons and nano-scale dust particles with a mass density of $2\times 10^{3}$ kg ${\rm m}^{-3}$ , introduced externally into the system (Franklin Reference Franklin2003; Mehdipour et al. Reference Mehdipour, Denysenko and Ostrikov2010). We have analysed the plasma in a one-dimensional (1-D) spatial coordinate within phase space, considering 3-D velocity components $v_{x}, v_{y}$ and $v_{z}$ , as illustrated in figure 1. However, the physical parameters of the sheath are permitted to vary only in the direction perpendicular to the surface, specifically along the z-axis.

Figure 1. Geometrical model for magnetized plasma sheath.

To examine the behaviour of the plasma sheath dynamics and the distribution of ion densities and velocities in the sheath domain, we utilized a multi-fluid model. We assumed that neutral gas particles are evenly dispersed across the sheath zone and remain largely stationary compared with the drift of ions and electrons in both magnetized and un-magnetized plasmas. The collision cross-sections for ${\text{He}}^{+}$ and ${\text{Ar}}^{+}$ ions interacting with neutral particles were determined based on their respective temperatures. When electrons are accelerated to speeds approaching their thermal velocities, electron inertia becomes significant. In these cases, the electron fluid cannot be accurately represented by the Boltzmann relation and instead follows the Cairns (Fouial et al. Reference Fouial, Tahraoui and Annou2016) distribution, with the electron distribution function expressed as (Cairns et al. Reference Cairns, Mamum, Bingham, Bingham, R.O.Dendy, Nairn C.M. and Shukla1995)

(2.1) \begin{equation}f_{e}\left(\nu \right)=\frac{n_{e0}}{(3\alpha +1)\sqrt{2\pi \nu _{the}^{2}}}\left(1+\frac{\alpha \nu ^{4}}{\nu _{the}^{4}}\right)\exp \left( -\frac{\nu ^{2}}{2\nu _{the}^{2}}\right),\end{equation}

where $\nu$ is the electron velocity, n is electron density, $\nu _{the}=\sqrt{{T_{e}}/{m_{e}}}$ is the electron thermal velocity and $T_{e}$ and $m_{e}$ are the electron temperature and mass respectively. The parameter $\alpha$ indicates how much the distribution deviates from a Maxwellian distribution. It is clear that as $\alpha$ approaches 0, the distribution converges to the Maxwellian form, as shown in figure 2. Analysis of figure 2 demonstrates that, as α increases, there is a corresponding rise in the population of high-energy tail electrons. This indicates that the parameter α directly influences the generation of energetic electrons in the suprathermal region of the distribution. From (2.1) we can determine the electron density by integrating it and replacing ${(\nu^{2})}/{(\nu _{the}^{2})}$ by ${(\nu ^{2})}/{(\nu _{the}^{2})}-2(\phi/{{T}_{{e}}})$

(2.2) \begin{align}n_{e} & =\int f_{e}\left(\nu \right)\text{d}\nu =n_{e0}\left[1-b\left(\frac{e\phi }{T_{e}}\right)+b\left(\frac{e\phi }{T_{e}}\right)^{2}\right]\exp \left(\frac{e\phi }{T_{e}}\right),\nonumber\\ b & =\frac{4\alpha }{\left(1+3\alpha \right)}. \end{align}

The parameter α represents the proportion of non-thermal electrons in the system. When b = 0 (i.e. α = 0), (2) corresponds exclusively to the Maxwellian distribution, where $\phi$ and $n_{e0}$ are the potential and electron density at the sheath edge respectively.

Figure 2. Non-thermal electron distribution function $f_{e}(\nu )$ given in (2.1).

The quasi-neutrality conditions are satisfied at the sheath edge and expressed as

(2.3) \begin{equation}n_{10}+n_{20}=q_{d0}n_{do}+n_{e0},\end{equation}

where $n_{10}$ , $n_{20}$ and $n_{do}$ are the ${\text{He}}^{+}$ ion, ${\text{Ar}}^{+}$ ion and charged dust density at the sheath edge, respectively, and q _d0 is the dust charge number at the sheath edge.

In a steady-state scenario, where ionization, ion loss and collisions are present, the behaviour of positive thermal ions is described by the continuity and momentum equations

(2.4) \begin{equation}u_{i}\frac{\text{d}n_{i}}{\text{d}z}+n_{i}\frac{\text{d}u_{i}}{\text{d}z}=\nu _{iz}n_{e}-\nu _{il}n_{i},\end{equation}

(2.5) \begin{equation}{u_{d}}\frac{\text{d}n_{d}}{\text{d}z}+n_{d}\frac{\text{d}v_{d}}{\text{d}z}=0,\end{equation}

(2.6) \begin{align}m_{i}n_{i}(u_{i}.\boldsymbol{\nabla })u_{i} & =-en_{i}\boldsymbol{\nabla }\phi +en_{i}(u_{i}\boldsymbol{\cdot} B)-\boldsymbol{\nabla }p_{i}-m_{i}n_{i}u_{i}\nu _{in}-m_{i}u_{i}\nu _{iz}n_{e}\nonumber\\& \quad -m_{i}n_{i}\left(u_{iz}-u_{\text{d}z}\right)\nu _{id},\end{align}

(2.7) \begin{align}m_{d}n_{d}u_{d}\boldsymbol{\cdot}\boldsymbol{\nabla }u_{d}=-{q_{d}}n_{d}\boldsymbol{\nabla }\phi +{q_{d}}n_{d}V_{d}\boldsymbol{\cdot} B+\sum _{i=1, 2}F_{id}+F_{nd},\end{align}

where i = 1, 2 correspond to the lighter ion ( ${\text{He}}^{+})$ and heavier ion ( ${\text{Ar}}^{+})$ respectively. Also, $m_{{i}}$ , $m_{d}$ , $n_{i} , n_{d}$ , $u_{i}$ and $u_{d}$ are the mass of ions and dust, density of ions and dust and velocity of ions and dust species, respectively, B is the magnetic field that makes an angle $\theta$ with the z axis $B=B_{0}(\hat{z}\cos \theta +\hat{x}\sin \theta )$ , as shown in figure 1, $\nu _{iz}$ and $\nu _{il}$ are the ionization frequency and ion loss frequency (Ivlev & Morfill Reference Ivlev and Morfill2000), respectively and $\nu _{in}$ and $\nu _{id}$ represent the ion–neutral and ion–dust collision frequencies, respectively. In the theoretical study of solitary waves in collisional dusty plasmas, the unperturbed particle densities adhere to the condition $\nu _{iz}n_{e0}-\nu _{il}n_{i0} = 0$ (Misra & Chowdhury Reference Misra and Chowdhury2004). However, to examine the impact of ionization, we assumed the ionization frequency was higher than the ion loss frequency. The parameter $ p_{i}=K_{B}T_{i}n_{i}$ , represents the partial pressure of positive ions, where, $K_{B}$ is the Boltzmann constant, $T_{{1}}$ is the ${\text{He}}^{+}$ ion temperature and T ₂ is the ${\text{Ar}}^{+}$ ion temperature.

Using Poisson’s equation, we relate the number densities of ions, electrons and dust to the electric potential under equilibrium conditions

(2.8) \begin{equation}\boldsymbol{\nabla} ^{2}\phi =-\frac{e}{\varepsilon _{0}}\left(n_{1}+n_{2}-q_{d}n_{d}-n_{e}\right).\end{equation}

Using magnetic field inclination, the above equation (2.6) and (2.7) can be written as

(2.9) \begin{equation}m_{i}u_{iz}\frac{\text{d}u_{ix}}{\text{d}z}=eB_{0}\cos \theta u_{iy}-m_{i}n_{i}u_{i}\nu _{in}-m_{i}u_{i}\nu _{iz}n_{e},\end{equation}

(2.10) \begin{equation}m_{i}u_{iz}\frac{\text{d}u_{iy}}{\text{d}z}=eB_{0}\left\lceil {\sin\theta }u_{\mathit{iz}}-{\cos\theta }u_{\mathit{ix}}\right\rceil -m_{i}n_{i}u_{i}\nu _{in}-m_{i}u_{i}\nu _{iz}n_{e},\end{equation}

(2.11) \begin{align}m_{i}u_{iz}\frac{\text{d}u_{iz}}{\text{d}z}& =-e\frac{\text{d}\phi }{\text{d}z}-eB_{0}\sin\theta u_{iy}-\frac{T_{j}}{n_{1}}\frac{\text{d}n_{1}}{\text{d}z}-m_{i}n_{i}u_{i}\nu _{in}-m_{i}u_{i}\nu _{iz}n_{e}\nonumber\\[4pt]& \quad -m_{i}\left(u_{iz}-u_{dz}\right)\nu _{id},\end{align}

(2.12) \begin{equation}m_{d}u_{\text{d}z}\frac{\text{d}u_{\text{d}x}}{\text{d}z}=q_{d}B_{0} \cos \theta u_{\text{d}y}+ F_{id}+F_{nd},\end{equation}

(2.13) \begin{equation}m_{d}u_{\text{d}z}\frac{\text{d}u_{\text{d}y}}{\text{d}z}=q_{d}B_{0}\left\lceil {\sin\theta }u_{{\text{d}z}}-{\cos\theta }u_{{\text{d}x}}\right\rceil ++F_{id}+F_{nd}+F_{G}+F_{EM},\end{equation}

(2.14) \begin{equation}m_{d}u_{\text{d}z}\frac{\text{d}u_{\text{d}z}}{\text{d}z}=-q_{d}\frac{\text{d}\phi }{\text{d}z}-q_{d}B_{0}\sin\theta u_{dy}+F_{id}+F_{nd}+F_{G}+F_{EM},\end{equation}

where $ F_{{EM}}$ , $ F_{G}$ , $F_{nd}$ and $F_{{id}}$ are the electromagnetic, gravitational, neutral drag and ion drag forces, respectively. To determine the dust charge $q_{{d}}$ (which corresponds to the dust surface potential, $\varnothing _{d}$ ) in the equations mentioned above, we presumed that the dust charging time is shorter than the characteristic time of dust motion since the charging process is a localized aspect (Franklin Reference Franklin2003). When a neutral dust particle enters the plasma, it progressively gains charge by collecting electrons and ions, following the governing equation

(2.15) \begin{equation}\frac{\text{d}q_{d}}{\text{d}t}=\sum _{i=1, 2}I_{i}+I_{e}.\end{equation}

The electron and ion currents to the dust particles, denoted as $I_{i}$ and $ I_{e}$ , can be determined using orbit motion limited theory (Tang & Delzanno Reference Tang and Delzanno2014). In this study, we assume that the distance between dust particles is greater than the ion Debye length, which allows us to apply dust charging theory. Additionally, we consider a maximum magnetic field strength of approximately 1.5 T and a dust radius of $4 \times 10^{-8}$ m. Under such magnetic conditions, the electron gyro-radius $(\rho_e = v_{te}/\Omega_e, \Omega_e = eB_{0/m_e} )$ is calculated to be $2.249 \times 10^{-6}$ m. Since the electron gyro-radius is significantly larger than the dust radius $ (\rho _{e}\gg r_{d})$ (Foroutan Reference Foroutan2010), the magnetic field has a negligible effect on the dust charging process and is therefore ignored in our analysis. To achieve charge equilibrium, we set ${({\rm d}q_{d})}/{({\rm d}t)}$ = 0 in (2.14), resulting in the steady-state potential and charge

(2.16) \begin{equation}I_{e}=-\pi r_{d}^{2}e\left(\frac{8T_{e}}{\pi m_{e}}\right)^{{1/2}}n_{e}\exp \left(\frac{eq_{d}}{r_{d}T_{e}}\right),\end{equation}

(2.17) \begin{equation}\sum _{i=1, 2}I_{i}=\pi r_{d}^{2}en_{j}v_{j}\left(1-\frac{2eq_{d}}{r_{d}m_{j}v_{j}^{2}}\right),\end{equation}

for (dq _d)/(dt) and

(2.18) \begin{equation}I_{e}=-\pi r_{d}^{2}e\left(\frac{8T_{e}}{\pi m_{e}}\right)^{{1/2}}n_{e}\left(1+\frac{eq_{d}}{r_{d}T_{e}}\right),\end{equation}

(2.19) \begin{equation}\sum _{i=1, 2}I_{i}= \pi r_{d}^{2}en_{j}v_{j} exp\left(\frac{2eq_{d}}{r_{d}m_{j}v_{j}^{2}}\right),\end{equation}

for $q_{d}\gt 0$ .

Near the sheath edge, positive ions accelerate toward the wall, reaching a directed velocity that may be comparable to their thermal velocity. To apply the orbital motion limited theory in this region, the ion velocity must be replaced with the effective mean velocity of ions moving toward the dust particles. For a spherical dust particle, the charge $q_{d}$ can be expressed in terms of the dust surface potential as $q_{d}=r_{d}\phi _{d}$ , where $\phi _{d}$ represents the dust surface potential. The dust charge $q_{d}$ is determined by using the dust particle’s current balance equation, $I_{e}=\sum _{i=1,2}I_{i}$ .

Once the dust becomes charged, it experiences different forces due to electricity being charged and its own mass. The net force on the charged dust particle can be expressed as

(2.20) \begin{equation}m_{d}\frac{{\rm d}v_{d}}{{\rm d}t}=F_{EM}+F_{G}+F_{D}+F_{T}+F_{p}.\end{equation}

The electromagnetic force arises from the combined effects of electric and magnetic forces. The ion drag force on dust particles primarily results from momentum transfer during ion–dust interactions, whereas the neutral drag force stems from interactions between neutral particles and dust. The ion drag force influencing dust particles can be divided into two components: (i) direct ion collection by dust particles and (ii) ion scattering within their electrostatic field. The expressions for these forces are given below

(2.21) \begin{equation}\sum _{j=1,2}F_{\text{collecton}}=\pi r_{d}^{2}n_{j}m_{j}v_{sj}\widetilde{v_{j}}\left(1-\frac{2eq_{d}}{r_{d}m_{j}v_{sj}^{2}}\right),\end{equation}

(2.22) \begin{equation}\sum _{j=1,2}F_{\text{coulomb}}=2\pi b_{0}^{2}n_{j}m_{j}v_{sj}\widetilde{v_{j}} ln\left(\frac{b_{0}^{2}+\lambda _{D}^{2}}{b_{0}^{2}+b_{c}^{2}}\right).\end{equation}

Here, $v_{sj}=\sqrt{(\check {v}_{j}^{2}+8T_{j}/\pi m_{j})}$ represents the mean ion velocity, and $\tilde{v}_{j}=(v_{j}-v_{d})$ is the ion velocity relative to the dust particles. The impact radius for a $90^{0}$ deflection is given by $b_{0}=(r_{d}T_{e}/(m_{j}v_{j}^{2}))$ , while the direct collision impact parameter is $b_{c}=r_{d}\sqrt{(1-2b_{0}/r_{d})}$ . The Debye length of the plasma is expressed as $\lambda _{D}=\lambda _{De}\lambda _{Dj}/\sqrt{(\lambda _{De}^{2}+\lambda _{Dj}^{2})}$ and the Debye lengths for electrons and ions are ( $\lambda _{De}=(\varepsilon _{0}T_{e}/n_{e0}e^{2})^{1/2})$ and $\lambda _{Dl}=({\varepsilon _{0}}T_{j}/n_{j0}e^{2})^{1/2}$ ( $j=1,2$ ), respectively.

For a typical dust radius, $r_{d}\ll \lambda _{mfp}$ and the velocity $v_{d}\ll v_{th,n}$ , where $v_{th,n}=\sqrt{T_{n}/m_{n}}$ (with $T_{n}$ and $m_{n}$ being the temperature and mass of neutral gas, respectively), and $\lambda _{mfp}=1/\sqrt{2\pi r_{d}^{2}n}$ . The neutral drag force can be expressed as follows:

(2.23) \begin{equation}F_{nd}=-\chi \frac{4}{3}\pi r_{d}^{2}m_{n}n_{n}v_{thn}v_{d}.\end{equation}

Here, $\chi = 1$ corresponds to specular reflection, while $\chi$ = 1.442 applies to diffuse reflection. The parameter $v_{thn}$ represents the thermal velocity of the neutral gas, and $n_{n}$ denotes the neutral gas density. The gravitational force acting on the dust species is given by

(2.24) \begin{equation}F_{G}=m_{d}g=\frac{4}{3}\pi r_{d}^{3}\rho _{d }g,\end{equation}

where $\rho _{d }$ represents the mass density of the dust species and $g$ represents the acceleration due to gravity.

Additionally, the electromagnetic forces $F_{EM}$ acting on moving charged dust particles with charge $q_{d}$ is the combined effect of the electric field and the Lorentz force, expressed as follows:

(2.25) \begin{equation}F_{EM}=F_{E}+F_{M},\end{equation}

(2.26) \begin{equation}F_{E}=q_{d}\boldsymbol{\cdot}E,\end{equation}

(2.27) \begin{equation} F_{M}=q_{d}v_{d}\boldsymbol{\cdot} B. \end{equation}

Here, E denotes the electric field, B represents the magnetic field.

To validate the dusty plasma sheath, the dust radius $(r_{d}$ ), impact radius ( $b_{0}$ ) and electron Debye length ( $\lambda _{De}$ ) must follow $\lambda _{De}\gt b_{0}\gt r_{d}$ . The conventional values estimated are $ \lambda _{De}=3.36\times 10^{-4}\, \mathrm{m}$ , $b_{0}=1.2\times 10^{-7}\,\mathrm{m}$ for $ r_{d}=4\times 10^{-8}\,\mathrm{m}$ . This confirms that the sheath assessment in dusty plasma is satisfied. For a standard dust oscillation frequency of 15 Hz (Kong et al. Reference Kong2016) and taking into account the dust acoustic speed of 20 cm s⁻¹, we calculated the dust acoustic wavelength to be 0.19 cm, indicating that the phase velocity is comparable to the dust acoustic speed. Additionally, we estimated the dust thermal speed to be 2.4 cm s⁻¹. Therefore, we conclude that the fluid approximation is valid within our specified range of parameters.

3. Dimensionless equations

For a better understanding of the momentum and continuity equations for the ions and dust, the following dimensionless variables are introduced: ${N}_{1}={n}_{1}/{n}_{10}$ , $ {N}_{2}={n}_{2}/{n}_{20}$ , $N_{d}=n_{d}/n_{d0}$ and $N_{e}=n_{e}/n_{e0}$ , $\psi =-e\phi /t_{e}$ , $T_{\mathrm{i}1}=T_{1}/T_{e}$ , $T_{\mathrm{i}2}=T_{2}/T_{e}, \mu =\mathrm{m}_{1}/\mathrm{m}_{2}$ , $\mu _{1}=\mathrm{m}_{1}/\mathrm{m}_{d}, \mathrm{U}_{i,x,y,z}=\mathrm{u}_{i, x,y,z}/\mathrm{c}_{\mathrm{si}}$ , ${U}_{d, x,y,z}={u}_{d, x,y,z}/{c}_{{sd}}$ , $\nu _{{Li}}={(\nu _{Li}\lambda _{Di})}/{(C_{si}\nu _{id})}$ , $\sigma _{i}={(\nu _{iz}\lambda _{Di})}/{(C_{si})}$ , $\varepsilon _{i}={(\nu _{in}\lambda _{Di})}/{(C_{si})}$ , $\lambda _{Di}=({(T_{e})}/{(n_{io}e^{2})})^{{1}/{2}}$ , $Z_{d}=q_{d}/{e}$ , $\upsilon _{id}={(\upsilon _{id}\lambda _{Di})}/{(C_{sd})}$ , $\gamma = \lambda _{{De}}/m_{d}c_{sd}^{2}$ , $\beta =\omega _{ci}/{\omega} _{pi}$ , $\omega _{ci}=eB_{0}/m_{i}$ and, $\omega _{pi}=(n_{i}e^{2}/\varepsilon _{0}m_{i})^{1/2}$ , $c_{sd}=(Z_{d}T_{e}/m_{d})^{1/2}$ , $\zeta =z/\lambda _{De}$ . Using above dimensionless parameters in (3.1)–(3.14), we may write

(3.1) \begin{equation}N_{e}=\left[1-b\left(\psi \right)+b\left(\psi \right)^{2}\right]{\rm exp} \left(\psi \right),\end{equation}

(3.2) \begin{equation}N_{1}\frac{{\rm d}U_{1z}}{{\rm d}\zeta }+U_{1z}\frac{{\rm d}N_{1}}{{\rm d}\xi }=\sigma _{1}N_{e}-\nu _{L1}N_{1},\end{equation}

(3.3) \begin{equation}N_{2}\frac{{\rm d}U_{2z}}{{\rm d}\zeta }+U_{2z}\frac{{\rm d}N_{2}}{{\rm d}\zeta }=\sigma _{2}N_{e}-\nu _{L2}N_{2},\end{equation}

(3.4) \begin{equation}N_{d}\frac{{\rm d}U_{{\rm d}z}}{{\rm d}\zeta }+U_{{\rm d}z}\frac{{\rm d}N_{{\rm d}}}{{\rm d}\zeta }=0,\end{equation}

(3.5) \begin{equation}U_{1z}\frac{{\rm d}U_{1x}}{{\rm d}\zeta }=\beta U_{1y}{\cos } \theta -\frac{\sigma _{1}U_{1x}}{N_{1}}\left(N_{e}+\frac{\varepsilon _{1}}{\sigma _{1}}N_{1}\right),\end{equation}

(3.6) \begin{equation}U_{1z}\frac{{\rm d}U_{1y}}{{\rm d}\zeta }=\beta \left[\sin\theta U_{1z}-\cos\theta U_{1x}\right]-\frac{\sigma _{1}U_{1y}}{N_{1}}\left(N_{e}+\frac{\varepsilon _{1}}{\sigma _{1}}N_{1}\right),\end{equation}

(3.7) \begin{equation}U_{1z}\frac{{\rm d}U_{1z}}{{\rm d}\zeta }=\frac{{\rm d}\psi }{{\rm d}\zeta }-\frac{T_{i1}}{N_{1}}\frac{{\rm d}N_{1}}{{\rm d}\xi }-{\beta }\sin\theta U_{1y}-\frac{\sigma _{1}U_{1z}}{N_{1}}\left(N_{e}+\frac{\varepsilon _{1}}{\sigma _{1}}N_{1}\right)-\left(U_{1z}-U_{{\rm d}z}\right)\upsilon _{1d},\end{equation}

(3.8) \begin{equation}U_{2z}\frac{{\rm d}U_{2x}}{{\rm d}\zeta }=\mu \beta U_{2y}{\cos } \theta -\frac{\sigma _{2}U_{2x}}{N_{2}}\left(N_{e}+\frac{\varepsilon _{2}}{\sigma _{2}}N_{2}\right),\end{equation}

(3.9) \begin{equation}U_{2z}\frac{{\rm d}U_{2y}}{{\rm d}\zeta }=\mu \beta \left[\sin\theta U_{2z}-\cos\theta U_{2x}\right]-\frac{\sigma _{2}U_{2y}}{N_{2}}\left(N_{e}+\frac{\varepsilon _{2}}{\sigma _{2}}N_{2}\right),\end{equation}

(3.10) \begin{align}U_{2z}\frac{{\rm d}U_{2z}}{{\rm d}\zeta } & =\mu \frac{{\rm d}\psi }{{\rm d}\xi }-\mu \frac{T_{i2}}{N_{2}}\frac{{\rm d}N_{2}}{{\rm d}\zeta }-{\mu }{\beta }\sin\theta U_{2y}\nonumber\\[5pt]& \quad -\frac{\sigma _{2}U_{2z}}{N_{2}}\left(N_{e}+\frac{\varepsilon _{2}}{\sigma _{2}}N_{1}\right)-\left(U_{2z}-U_{{\rm d}z}\right)\upsilon _{2d},\end{align}

(3.11) \begin{equation}U_{{\rm d}z}\frac{{\rm d}U_{{\rm d}x}}{{\rm d}\zeta }=Z_{d}\mu _{1}{\beta }\cos\theta U_{dy}+\gamma \left(\sum _{i=1, 2}F_{id}+F_{nd}\right),\end{equation}

(3.12) \begin{equation}U_{{\rm d}z}\frac{{\rm d}U_{{\rm d}y}}{{\rm d}\zeta }=Z_{d}\mu _{1}\beta \left[\sin\theta U_{{\rm d}z}-\cos\theta U_{{\rm d}x}\right]+\gamma \left(\sum _{i=1, 2}F_{id}+F_{nd}\right),\end{equation}

(3.13) \begin{equation}U_{{\rm d}z}\frac{{\rm d}U_{{\rm d}z}}{{\rm d}\zeta }=Z_{d}\mu _{1}\frac{{\rm d}\psi }{{\rm d}\xi }-\beta \mu _{1}\sin\theta U_{{\rm d}y}+\gamma \left(\sum _{i=1, 2}F_{id}+F_{nd}\right),\end{equation}

(3.14) \begin{equation}\frac{{\rm d}^{2}\psi }{{\rm d}\zeta ^{2}}=\left[N_{1}+N_{2}-Z_{d}N_{d}-N_{e}\right].\end{equation}

4. Numerical results and discussion

In this section, we presented the findings from a numerical simulation of the coupled nonlinear differential equations that describe a magnetized plasma sheath containing two types of positive ions, non-thermal electrons and negatively charged dust particles. We have estimated the forces acting on the dust species as follows: collection force, $7.89\times 10^{-9}$ N; collisional force, $ 4.83\times 10^{-10}$ N; neutral drag force, $7.03\times 10^{-16}$ N; electromagnetic force, $5.38\times 10^{-18}$ N; and gravitational force, $5.2518\times 10^{-18}$ N. Both gravitational and electromagnetic forces are balanced to enable dust levitation. In our analysis, these forces are relatively small and thus neglected. Instead, we focus on the drag force, which is the most significant for charged dust species.

At the start of this section, we note that weakly ionized plasmas imply the presence of finite numbers of neutral particles in the model, necessitating the consideration of ion–neutral collisions, ion loss and ionization effects. Using the fourth-order Runge–Kutta method in MATLAB and an appropriate initial value, (3.1)–(3.14) are solved with the step size 0.0003. For our calculations, we used the initial values. The densities of electrons, $He^{+}$ ions, $\text{Ar}^{+}$ ions and dust are $n_{e}=1\times 10^{15} $ , $n_{1}=0.6\boldsymbol{\cdot} n_{e}$ , $n_{2}=0.4\boldsymbol{\cdot} n_{e}$ and $n_{d}=1\times 10^{11}\,\mathrm{m}^{-3}$ , respectively. The temperature of electrons, ions and dust are, $T_{e}=3 \text{}$ , $T_{1}=0.13 \text{}, T_{2}=0.05 \text{}$ and $T_{d}=0.003\, \text{eV}$ (Mand & Mashayek Reference Mand and Mashayek2005), respectively. The ion–neutral collision cross-sections of ${\text{Ar}}^{+}$ and ${\text{He}}^{+}$ ions (Foroutan Reference Foroutan2010) are $5\times 10^{-15}$ and $1.6\times 10^{-15}\,\mathrm{cm}^{2}$ respectively. The initial normalized velocities of ${\rm He}^{+}$ ions, $\text{Ar}^{+}$ ions and dust particles along the z-axis are $\mathrm{U}_{1,z}(\xi =0)=1.0$ , $\mathrm{U}_{2,\mathrm{z}}(\xi =0)=0.2$ and $\mathrm{U}_{d,\mathrm{z}}(\xi =0)=0.054$ , other velocity components of $He^{+}$ ions, ${\rm Ar}^{+}$ ions and dust particles along the x and y axes are taken as zero. At the sheath edge ( $\xi =0),$ the potential is zero ( ${\psi}$ = 0), however, to avoid numerical divergence we assume that ${({\rm d}\psi (\xi =0))}/{({\rm d}\xi)}$ takes an infinitesimal value of 0.001.

4.1. Ion density and velocity with and without charged dust and ionization effect

Our model considered a composition consisting of a ratio of 60 % ${\text{He}}^{+}$ and 40 % ${\text{Ar}}^{+}$ ions concerning the electron plasma density inside the sheath vicinity. Figures 3, 4, 5, 6 and 7 illustrate the effects of electron impact ionization in the presence and absence of charged dust particles. Figures 3 and 4 precisely depict the normalized ion densities of ${\text{He}}^{+}$ and ${\text{Ar}}^{+}$ ions, respectively, as a function of the normalized distance from the sheath edge. It is evident from both figures that, as the normalized ionization frequency ( $\sigma )$ increases, the ion density of both species also increases. However, the amplitude of the density peaks differs between the ions due to the higher ionization rate of ${\text{Ar}}^{+}$ compared with ${\text{He}}^{+}$ ions, which can be determined from their respective ionization rate equations: $R_{{\rm He}}=k_{{\rm He}0}n_{e}n_{n}\exp \left({(-24.6)/}{(k_{b}t_{e})}\right)$ for ${\rm He}^{+}$ and $R_{{\rm Ar}}=k_{{\rm Ar}0}n_{e}n_{n}\exp \left({(-15.76)}/{(k_{b}t_{e})}\right)$ for ${\rm Ar}^{+}$ ions, respectively.

Figure 3. Variation of normalized ${\text{He}}^{+}$ ion density with normalized distance for different values of $\sigma$ , for $\beta = 0.5$ , $\theta =10^{\circ}$ , $T_{e}=3 \text{eV}$ , $T_{1}=0.13 \text{eV}, T_{2}=0.05 \text{eV}$ , $T_{d}=0.003 \text{eV}$ .

Figure 4. Variation of normalized ${\text{Ar}}^{+}$ ion density with normalized distance for different values of $\sigma$ , and the other physical variables are the same as figure 3.

Figure 5. Normalized current profile function of normalized distance.

Figure 6. Variation of normalized ${\text{He}}^{+}$ ion velocity with normalized distance for different values of $\sigma$ , and the other physical variables are the same as figure 3.

Figure 7. Variation of normalized ${\text{Ar}}^{+}$ ion velocity with normalized distance for different values of $\sigma$ , and the other physical variables are the same as figure 3.

When charged dust enters the sheath, the ${\text{He}}^{+}$ ion density decreases, whereas the ${\text{Ar}}^{+}$ ion density increases. This occurs because the magnitude of the dust charge near the sheath edge is significantly higher when ${\text{He}}^{+}$ ions are dominant. This is due to the fact that ${\text{He}}^{+}$ ions, with their smaller size, yield a larger dust charging current compared with ${\text{Ar}}^{+}$ ions, resulting in a greater absolute dust charge in the vicinity of the sheath edge. Additionally, the ion loss frequency of ${\text{He}}^{+}\,\text{ions is higher than that of }{\text{Ar}}^{+}$ ions. The ion current ratio profile (figure 5), which compares the ${\text{He}}^{+}$ ion current with the ${\text{Ar}}^{+}$ ion current, demonstrates that the ${\text{He}}^{+}$ ion current decreases rapidly from the sheath edge to the wall, whereas the ${\text{Ar}}^{+}$ ion current declines more gradually. Therefore, in the presence of charged dust, the density of ${\text{He}}^{+}$ ions decreases while the density of ${\text{Ar}}^{+}$ ions increases.

The accumulation of ions near the sheath edge arises from a partial breach of the quasi-neutrality condition within the sheath. An electric field, that is dependent on the sheath’s thickness, forms and grows more substantial from the sheath edge toward the wall. This field repels negatively charged dust particles, increasing the dust density near the sheath boundary. The higher dust density may also contribute to the observed rise in ion concentration at the sheath edge. Figures 6 and 7 show the corresponding normalized velocities of ${\rm He}^{+}$ ion (figure 3) and ${\rm Ar}^{+}$ ions (figure 4) as a function of normalized distance. The velocity profiles exhibit an inverse relationship to the density profiles, ensuring flux conservation.

4.5. Magnetic field and orientation impact on ion density

Figures 17 and 18 illustrate the normalized ion density variations of ${\text{He}}^{+}$ and ${\text{Ar}}^{+}$ ions as a function of normalized distance for constant magnetic field angle, respectively. Both figures show that, with an increase in the magnetic field, the ion densities for both species exhibit pronounced peaks and narrow distributions. With the increase in magnetic field strength, the gyro-radius of the ions decreases, as the gyro-radius is inversely proportional to the magnetic field strength ( $\rho _{i}\propto \beta ^{-1})$ , hence the sharpness of the density peak is observed towards the sheath edge. For the magnetic fields of 0.5 and 1.5, the gyro-radii of ${\text{He}}^{+}$ and ${\text{Ar}}^{+}$ ions are calculated. For the magnetic field strength (β) of 0.5, the gyro-radii of ${\text{He}}^{+}$ and ${\text{Ar}}^{+}$ ions are 0.120 and 0.235 mm, respectively. When $\beta$ increases to 1.5 the gyro-radii of ${\text{He}}^{+}$ and ${\text{Ar}}^{+}$ ions decrease to 0.0401 and 0.0786 mm, respectively. This reduction indicates that, as the B-field strength increases, the gyro-radius becomes smaller, implying that the space-averaged ion density tends to be enhanced. Consequently, ions are confined to tighter circular orbits around the B-field lines, resulting in more constrained gyro-motion. The shifting of peaks towards the sheath edge can be explained by figure 19.

Figure 8. Variation of normalized ${\text{He}}^{+}$ ion density with normalized distance for different values of $\epsilon$ , and the other physical variables are the same as figure 3.

Figure 9. Variation of normalized ${\text{Ar}}^{+}$ ion density with normalized distance for different values of $\epsilon$ , and the other physical variables are the same as figure 3.

Figure 10. Variation of normalized sheath potential with normalized distance for different values of $b$ , and the other physical variables are the same as figure 3.

Figure 11. Variation of normalized ${\text{He}}^{+}$ ion density with normalized distance for different values of $b$ , and the other physical variables are the same as figure 3.

Figure 12. Variation of normalized ${\text{Ar}}^{+}$ ion density with normalized distance for different values of $b$ , and the other physical variables are the same as figure 3.

Figure 13. Variation of normalized ${\text{He}}^{+}$ and ${\text{Ar}}^{+}$ ion density with normalized distance in presence and absence of charged dust for a constant values of $b$ , and the other physical variables are the same as figure 3.

Figure 14. Normalized electron density as a function of normalized distance with varying $\sigma$ , and the other physical variables are the same as figure 3.

Figure 15. Variation of normalized sheath potential with normalized distance for different values of $\sigma$ , and the other physical variables are the same as figure 3.

Figure 16. Variation of space charge with normalized distance for different values of $\sigma$ , and the other physical variables are the same as figure 3.

Figure 17. Variation of normalized ${\text{He}}^{+}$ ion density with normalized distance for different values of $\beta$ , and the other physical variables are the same as figure 3.

Figure 18. Variation of normalized ${\text{Ar}}^{+}$ ion density with normalized distance for different values of $\beta$ , and the other physical variables are the same as figure 3.

Figure 19. Net ion ( ${\text{He}}^{+}+{\text{Ar}}^{+})$ density profile as function of normalized sheath thickness with varying $\theta$ and the other parameters are same with figure 3.

Figure 19 presents the net ion density variation inside the sheath as a function of normalized distance for the constant magnetic field. We vary the magnetic field incidence angle is in the range $\theta =5^{\circ}-55^{\circ}$ keeping the magnetic field strength the same ( $\beta =1$ ). It is observed that the density peak shifts towards the sheath edge with the increase of the angle of incidence. The magnetic field is allowed to intercept the wall at an angle $\theta$ heaving horizontal components ( $B_{z}=B_{0} \cos\theta \hat{z}$ ) and vertical components $(B_{x}=B_{0} \sin\theta \hat{x})$ . At $\theta =0^{\circ}$ , the loss of ions towards the wall is maximum. The vertical component of the magnetic field increases, followed by the decrease in the horizontal component with the increase in $\theta$ . Thus, the ion loss decreases along the horizontal direction, therefore, the ion density begins to peak in the vertical direction, and its amplitude tends to increase proportionally to the magnetic field strength.

5. Summary

The behaviour of a collisional magnetized plasma sheath with two positive ion species, non-thermal electrons and negatively charged dust is studied under electron impact ionization, ion–neutral collisions and an external magnetic field. The dynamics of helium and argon ion densities, with and without charged dust, is analysed for varying ionization frequencies. Sheath potential, electron density, ion flux and space charge are evaluated, along with the effects of non-thermal electrons and the B-field on electron and ion densities.

1. i. With increased electron impact ionization, ${\text{He}}^{+}$ and $\text{Ar}^{+}$ ion densities rise due to enhanced ion production, increasing the sheath ion density. Charged dust in the sheath reduces the ${\text{He}}^{+}$ ion density but increases the ${\text{Ar}}^{+}$ ion density. The $ \text{He}^{+}$ ion decline is due to a stronger dust charge near the sheath edge, as lighter ${\text{He}}^{+}$ ions generate higher dust charging currents, consuming more ions. Increased ion loss frequency further decreases helium ions.

2. ii. As σ increases, sheath potential decreases, indicating stronger ionization and narrowing of the sheath. Higher ionization produces more positive ions moving toward the negatively charged wall, steepening the electric potential and confining ions to a smaller region.

3. iii. Density peaks appear with increasing ion–neutral collision frequency and narrow with stronger collisional forces. Ion–neutral collisions restrict ion motion toward the wall and hinder acceleration by the electric field. This effect is more pronounced for ${\text{Ar}}^{+}$ ions, which have a larger collision cross-section than ${\text{He}}^{+}$ ions, causing more collisions and greater ion bunching.

4. iv. Higher non-thermal electrons (b) leads to more energetic electrons reaching the wall, making it more negative and increasing the sheath potential. This also extends the sheath region, as sheath thickness depends on electrostatic potential, plasma density, and ion-entering velocity.

5. v. Ion density peaks with an increase in B-field due to a smaller ion gyro-radius, confining ions to tighter orbits around the B-field lines. The shift in ion peaks is due to the magnetic field’s inclination.

6. vi. In this study, we adopted a 1-D simplified sheath model, assuming an arbitrary electric field at the sheath edge and neglecting transverse ion velocities ( $u_{x}$ and $u_{y}$ ). However, as noted by Valentin (Reference Valentin2000), the electric field at the sheath edge should have a well-defined value rather than an arbitrary one. In our future work, we plan to extend the study to a 3-D model by considering non-zero transverse ion velocities and a definite value of the electric field.

The results of our research hold significant technological value for numerous applications involving low-temperature plasmas where dust particles are commonly found.