In Part 1 of the companion paper, we have investigated lane formation dynamics of driven three-dimensional (3-D) pair-ion plasmas (PIP) in the presence of parallel external forcing using extensive Langevin dynamics (LD) simulation. In continuation of the work, in this Part 2, we investigate lane formation dynamics in the presence of non-parallel external forcing, and the effect of both constant and time varying forces are studied. In our model, positively charged PIP particles are pushed into the field direction by an external force $\boldsymbol{F}_{A} = E_A\cos \omega _At$ while the negatively charged PIP particles are pulled by an external force $\boldsymbol{F}_{B} = E_B\cos \omega _Bt$ in a direction perpendicular to the external force $\boldsymbol{F}_{A}$ . We show that in the case of non-parallel forces, the lanes are observed with an orientation (characterised by angle of inclination $\theta$ ) tilted in the direction of the force difference vector ( $\boldsymbol{F}_{A} - \boldsymbol{F}_{B}$ ). The instantaneous lane order parameter, order parameter with gradient of angle of inclination ( $\phi (\theta )$ ) and distribution function of $\Delta \theta$ have been implemented to characterise the phase transition. A spontaneous formation and breaking of lanes are observed when under the influence of time-varying forces. The effect is further investigated for three different situations: first, when $\omega _A \neq 0$ and $\omega _B = 0$ ; second, when $\omega _A = \omega _B \neq 0$ ; and third, for $\omega _A \neq \omega _B \neq 0$ . Our study reveals that for the first case, a periodic oscillation of angle of inclination is observed. If oscillating forces of the same frequency are applied, the oscillation in angle of inclination disappears, and spontaneous formation and breaking of lanes is observed. However, in the presence of forces with different frequencies, flipping of lane inclination between the positive and negative domain of $\theta$ is observed. Further, some aspects of the lane formation dynamics of a PIP system is also studied in the presence of an external magnetic field where the lane formation dynamics is found to be accelerated. 

Lane formation in 3D driven pair-ion plasmas: II Non-Parallel External Forcing

In Part 1 of the companion paper, we have investigated lane formation dynamics of driven three-dimensional (3-D) pair-ion plasmas (PIP) in the presence of parallel external forcing using extensive Langevin dynamics (LD) simulation. In continuation of the work, in this Part 2, we investigate lane formation dynamics in the presence of non-parallel external forcing, and the effect of both constant and time varying forces are studied. In our model, positively charged PIP particles are pushed into the field direction by an external force $\boldsymbol{F}_{A} = E_A\cos \omega _At$ while the negatively charged PIP particles are pulled by an external force $\boldsymbol{F}_{B} = E_B\cos \omega _Bt$ in a direction perpendicular to the external force $\boldsymbol{F}_{A}$. We show that in the case of non-parallel forces, the lanes are observed with an orientation (characterised by angle of inclination $\theta$) tilted in the direction of the force difference vector ($\boldsymbol{F}_{A} - \boldsymbol{F}_{B}$). The instantaneous lane order parameter, order parameter with gradient of angle of inclination ($\phi (\theta )$) and distribution function of $\Delta \theta$ have been implemented to characterise the phase transition. A spontaneous formation and breaking of lanes are observed when under the influence of time-varying forces. The effect is further investigated for three different situations: first, when $\omega _A \neq 0$ and $\omega _B = 0$; second, when $\omega _A = \omega _B \neq 0$; and third, for $\omega _A \neq \omega _B \neq 0$. Our study reveals that for the first case, a periodic oscillation of angle of inclination is observed. If oscillating forces of the same frequency are applied, the oscillation in angle of inclination disappears, and spontaneous formation and breaking of lanes is observed. However, in the presence of forces with different frequencies, flipping of lane inclination between the positive and negative domain of $\theta$ is observed. Further, some aspects of the lane formation dynamics of a PIP system is also studied in the presence of an external magnetic field where the lane formation dynamics is found to be accelerated.

Lane formation dynamics of driven two-dimensional pair-ion plasmas is investigated in under-damped cases where the effect of particle inertia cannot be neglected. Extensive Langevin dynamics simulations using an OpenMP parallel program are carried out to analyse the effect of obstacle and geometric aspect ratio on lane formation dynamics previously reported in Sarma et al. (Phys. Plasmas, vol. 27, 2020, 012106) and Baruah et al. (J. Plasma Phys., vol. 87, 2021, 905870202). Lanes are found to form when like particles move along or opposite to the applied field direction. Lane order parameter, cumulative order parameter and distribution of the order parameter have been implemented to detect phase transition. The effect of geometric aspect ratio on the stability of lanes is systematically determined in both the presence and absence of an obstacle. Here, a specular reflective boundary condition is implemented to mimic an obstacle. We demonstrate that an obstacle promotes the merging of lanes, and the system gradually transitions to a partially mixed phase with higher value of aspect ratio. The occurrence of lane mixing phenomena at the separation boundary of two oppositely flowing lanes at higher value of aspect ratio is observed. In the presence of an oscillatory electric field, the lane merging tendency is reduced to a large extent as compared to the system where a constant electric field is applied. Furthermore, in the presence of both space- and time-varying electric fields, an appearance of a void is observed on either side of the obstacle. The study finds that the presence of an external magnetic field promotes acceleration of the phase transition process towards the lane mixing phase; it also reveals the existence of electric field drift in the system. Our findings may prove to be useful in understanding the nature of lane dynamics in naturally occurring pair-ion plasma systems as well as their relevance to technological applications that exploit or mitigate self-organization.

Vishal K. Prajapati

Papers

Lane dynamics in pair-ion plasmas: effect of obstacle and geometric aspect ratio