The electric charge of dust particles is highly contributive in assessing plasma in laboratory experiments, in ionosphere layer, and in its interplanetary state. It is assumed that the dust particles are initially charge free, while electrons and ions eventually collide with the dust surface and have a high chance of sticking to them, thus they are being charged. Some factors like photoemission, secondary electron emission, thermionic emission, and electromagnetic fields can contribute to electrically charged dust particle count17,18. Orbital motion limited (OML) is considered a common method of tracking the motion direction of electrons and ions, while can influence different forces within the plasma, determine the collisional cross-sections, and compute the electric charge of the dust at equilibrium19,20.
Here, it is assumed that dust particle is of a conductive spherical shape, so the electric potential of the dust surface, φswhich depends on the electric charge to the capacitance ratio of the conductive sphere, φs = Q/C, where, \({\text{C}} = 4\pi \varepsilon_{0} {\text{r}}_{{\text{d}}}\) is the spherical dust capacity21. Most electrons, due to their low mass and high temperature, are exposed more to dust particles in relation to ions, producing a negative charge on the dust particle (φs < 0). The net charge can be positive on the dust, and φs > 0 by considering other factors like the emission of electrons from the surface of the dust due to light emission. Solving the motion equations for the electrons and ions reveal the intensity of ion and electron flow towards the dust, provided φs < 0 hold true22:
$$I_{i} = I_{0i} \left( {1 – \frac{{z_{i} e\varphi_{s} }}{{k_{B} T_{i} }}} \right),$$
(1)
$$I_{e} = I_{0e} \exp \left( { \frac{{e\varphi_{s} }}{{k_{B} T_{e} }}} \right),$$
(2)
otherwise, φs > 0 for positive dust:
$$I_{i} = I_{0i} exp\left( { \frac{{ – z_{i} e\varphi_{s} }}{{k_{B} T_{i} }}} \right),$$
(3)
$$I_{e} = I_{0e} \left( {1 + \frac{{e\varphi_{s} }}{{k_{B} T_{e} }}} \right),$$
(4)
where zi is the ionization degree, Ti is the ion temperature, and Te is the electron temperature. The symbols, kB and I0a are the Boltzmann constant and the intensity of the initial current of electrons and ions, respectively:
$$I_{0\alpha } = 4\pi r_{d}^{2} n_{\alpha } q_{\alpha } \left( {\frac{{kT_{\alpha } }}{{2\pi m_{\alpha } }}} \right)^{1/2} ,\quad \alpha = e, i,$$
(5)
where ne and ni are the electron and ion count per unit volume, respectively, while ma is either e or i of mass, and qa is either e or i of charge. The radius of the dust particles, rdis usually just a few μm, and the charge of the dust particles leads to a balance between electron and ion count.
$$\frac{dQ}{{dt}} = I_{e} + I_{i} .$$
(6)
By inserting Eq. (5) in Eq. (6), The negative and positive potentials are yield, respectively:
$$\frac{dQ}{{dt}} = 4\pi er_{d}^{2} \sqrt {\frac{{k_{B} }}{{2\pi m_{e} }}} \left\{ { – n_{e} \sqrt {T_{e} } exp\left( { \frac{eQ}{{k_{B} CT_{e} }}} \right) + n_{i} z_{i} \sqrt {T_{i} } \left( {1 – \frac{{z_{i} eQ}}{{k_{B} CT_{i} }}} \right)} \right\},$$
(7)
$$\frac{dQ}{{dt}} = 4\pi er_{d}^{2} \sqrt {\frac{{k_{B} }}{{2\pi m_{e} }}} \left\{ { – n_{e} \sqrt {T_{e} } \left( {1 + \frac{{e\varphi_{s} }}{{k_{B} T_{e} }}} \right) + n_{i} z_{i} \sqrt {T_{i} } exp\left( { \frac{{ – z_{i} eQ}}{{k_{B} CT_{i} }}} \right)} \right\}.$$
(8)
Both the Eqs. (7) and (8) are the time evolution of the electric charge of dust particles23.
To assess the collisions between dust particles and electrons or ions, The Monte-Carlo method is applied. The electrons and ions have cross-sections of σe and σi and energies Ee and Eirespectively computed through Eqs. (9) and (10) and the immobile dust particles with charge Qd and radius rd are modeled according to OML theory19:
$$\sigma_{e} = \pi r_{d}^{2} \left( {1 + \frac{{Q_{d} }}{{4\pi \varepsilon_{0} r_{d} E_{e} }}} \right),$$
(9)
$$\sigma_{i} = \pi r_{d}^{2} \left( {1 – \frac{{Q_{d} }}{{4\pi \varepsilon_{0} r_{d} E_{i} }}} \right)$$
(10)
where Ee and Ei are the electron and ion energy in eV. The cross-sections are subject to the momentum and energy conservation of electrons and ions interacting with dust particles, therefore, the cross-sections are valid for electrons and ions as to they being absorbed or rejected by the dust particles24.
The electron–ion collision cross-sections applied in this model are resemble that of25. The Coulomb cross-section, σ, for electron and ion scattering by immobile dust particles are extracted from26:
$$\sigma = \frac{{\pi \left( {e_{\alpha }^{2} e_{\beta }^{2} } \right) ln\Lambda }}{{\left( {\mu v^{2} /2} \right)^{2} }} = \frac{{\pi \left( {e_{\alpha }^{2} e_{\beta }^{2} } \right) ln\Lambda }}{{16\pi^{2} \varepsilon_{0}^{2} \left( {\mu v^{2} /2} \right)^{2} }} = \frac{{Q_{d}^{2} ln\Lambda }}{{16\pi \varepsilon_{0}^{2} E_{\alpha }^{2} }},\quad \alpha ,\beta = e, i,$$
(11)
where α and β are the interacting particles, μ is their reduced mass, close to the electron or ion mass because of the existence of large dust-particle mass, lnΛ is the Coulomb logarithm ~ 10, ea and eb are the particle charges, Qd is the dust particle charge, and Ea is the electron or ion energy in eV.