Early-stage aggregation in three-dimensional charged granular gas

Neutral grains made of the same dielectric material can attain considerable charges due to collisions and generate long-range interactions. We perform molecular dynamic simulations in three dimensions for a dilute, freely cooling granular gas of viscoelastic particles that exchange charges during collisions. As compared to the case of clustering of viscoelastic particles solely due to dissipation, we find that the electrostatic interactions due to collisional charging alter the characteristic size, morphology, and growth rate of the clusters. The average cluster size grows with time as a power law, whose exponent is relatively larger in the charged gas than the neutral case. The growth of the average cluster size is found to be independent of the ratio of characteristic Coulomb to kinetic energy, or equivalently, of the typical Bjerrum length. However, this ratio alters the crossover time of the growth. Both simulations and mean-field calculations based on Smoluchowski's equation suggest that a suppression of particle diffusion due to the electrostatic interactions helps in the aggregation process.

Figure 1

(a) The evolution of the granular temperature $T_{\mathrm{g}}$ for a purely repulsive dilute granular gas with monopolarly charged particles and constant coefficient of restitution $\epsilon =0.85$. We study the dependence on the ratio of characteristic Coulomb to kinetic energy $\mathcal{K}$. The $\mathcal{K}=0$ curve corresponds to a neutral granular gas. At very short times, the granular gas follows Haff's law $[T_{\mathrm{g}}\left(t\right)\sim t^{-2}]$ in the homogeneous cooling state. The repulsive electrostatic interactions among the particles reduce the collision frequency and thus result in a slower decay of $T_{\mathrm{g}}$ as time progresses [$T_{\mathrm{g}}\left(t\right)\sim 1/ln(t/t_c)$, also shown analytically in Ref. [18], where $t_c$ is the characteristic time separating power law from inverse logarithmic decay]. As $\mathcal{K}$ increases, the deviation from Haff's law is more pronounced and occurs earlier in time. The solid line represents the theoretical prediction of Haff's law for a neutral granular gas with $\epsilon =\text{const.}$, and the dashed line is a theoretical prediction for monopolarly charged granular gases. (b) Same as (a) but for early stage of evolution of the viscoelastic ($\epsilon \ne \text{const}.$) granular gas with charge exchange. The dashed line represents the theoretical prediction of Haff's law for a neutral viscoelastic granular gas.

Figure 2

Snapshots of the granular gas showing the time evolution of the neutral system of viscoelastic particles (left column) and charged viscoelastic particles (center and right columns). Here time $t^{*}=t{v}_{\mathrm{ref}}/d$, particle number $N=50016$ and the particle filling fraction in the system $\varphi =0.076$. As the ratio of characteristic Coulomb to kinetic energy $\mathcal{K}$ increases, the characteristic time for the emergence of clustering decreases, however, their growth rate is unchanged (see also Fig. 5). The clusters exhibit a relatively compact morphology in the charged system.

Figure 3

Sticking of particles during clustering in the charged system. We observe a mechanism similar to collide-and-capture events observed in experiments on a falling granular stream in Ref. [14]. Particles stick together in clusters and exhibit pronounced persistence in each cluster over a considerable duration of time. This mechanism is not observed in the neutral system, where instead particles collide and separate. (a) Specific particles and their first neighbors are shown at different times. (b)–(d) The evolution of the number of contacts, $N_{\mathrm{stick}}$, with time for the same particles shown in panel (a). The occasional fragmentation results in the fluctuation of $N_{\mathrm{stick}}$ with time.

Figure 4

(a) The evolution of mean absolute charge, $\overline{q}=\frac{1}{N}{\sum }_{i=1}^N\left|q_i\right|$, for the viscoelastic model. The mean charge saturates as a consequence of continuous reduction in the granular temperature. The solid line is the theoretical prediction in Eq. (8). (b–d) Comparison of the cluster size distribution $N_s\left(s\right)$ for charged ($\mathcal{K}=5.0$) and neutral granular gases at different times: (b) $t{v}_{\mathrm{ref}}/d=50$; (c) $t{v}_{\mathrm{ref}}/d=100$; (d) $t{v}_{\mathrm{ref}}/d=150$, from an average over twenty independently initialized simulation runs. The dashed lines are the corresponding best fits. During the evolution of the granular gases, the slope of the distributions, or equivalently, the Fisher exponent decreases. However, the charged gas exhibits a more rapid decrease of the slope indicating an enhanced cluster growth (see also Fig. 5).

Figure 5

Temporal dependence of the average cluster size $S\left(t\right)$, from an average over 20 independently initialized simulation runs and corresponding best fits. Granular gas with collisional charging exhibits faster growth of clusters than for neutral system. (Inset) Saturation of the growth exponent $z$ as $1/t\to 0$ using the method of local slope. A change in the ratio of characteristic Coulomb to kinetic energy $\mathcal{K}$ does not alter the growth exponent $z$ and only influences the crossover time of initiation of clustering.

Figure 6

Comparison of the average cluster size $S\left(t\right)$ predicted by the Smoluchowski equation with the MD results. At $\beta =2$ (dashed line) the mean-field solution agrees reasonably well with the neutral gas, while as $\beta \to 3$ (solid lines) the growth rate for the charged granular gas is recovered. (Inset) Comparison of the MSD of particles between the charged and the neutral gas. The sub-diffusion due to dissipation is further suppressed by the electrostatics, which in the mean-field approximation is modeled by increasing $\beta$. Here $S_c$ and ${\tau }_c$ are factors used to shift the curves and plot them close to each other for the sake of comparison of the slopes.

Figure 7

(a) The relaxation of the scaled velocity ($\tilde{v}=v/{\langle v^2\rangle }^{1/2}$) distribution function toward the Maxwellian for neutral viscoelastic particles. This result from our simulations for neutral viscoelastic particles is consistent with the Sonine expansion for the time dependent distribution function [50], which depicts that the distribution relaxes back toward the Maxwellian in the long time limit. (b) The time evolution of the distribution $f\left(\tilde{v}\right)$ for a charged system, however, shows a behavior opposite to the neutral case: the distribution does not relax back to the Maxwellian. (Insets) Same results as in (a) and (b) but on a linear scale to highlight the shift of the most probable velocity for the charged granular gas.

Figure 8

Scatter plot of the particles' charges and velocities at different times. As time progresses, a subpopulation of high velocity, nearly neutral particles develops (highlighted in the dashed ellipse), which contributes to the exponential tails in the velocity distribution function (see Fig. 7). This subpopulation is expected to cause the occasional fragmentation of the local aggregates (see Fig. 3). Here, $\tilde{q}=q/{\langle q^2\rangle }^{1/2}$ and $\tilde{v}=v/{\langle v^2\rangle }^{1/2}$.