Electrification in granular gases leads to constrained fractal growth

doi:10.1038/s41598-019-45447-x

. 2019 Jun 21;9(1):9049.

doi: 10.1038/s41598-019-45447-x.

Electrification in granular gases leads to constrained fractal growth

Chamkor Singh^{1

2}, Marco G Mazza^{3

4}

Affiliations

¹ Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077, Göttingen, Germany.
² Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077, Göttingen, Germany.
³ Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077, Göttingen, Germany. [email protected].
⁴ Interdisciplinary Centre for Mathematical Modelling and Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom. [email protected].

PMID: 31227758
PMCID: PMC6588598
DOI: 10.1038/s41598-019-45447-x

Electrification in granular gases leads to constrained fractal growth

Chamkor Singh et al. Sci Rep. 2019.

. 2019 Jun 21;9(1):9049.

doi: 10.1038/s41598-019-45447-x.

Authors

Chamkor Singh^{1

2}, Marco G Mazza^{3

4}

Affiliations

¹ Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077, Göttingen, Germany.
² Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077, Göttingen, Germany.
³ Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077, Göttingen, Germany. [email protected].
⁴ Interdisciplinary Centre for Mathematical Modelling and Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom. [email protected].

PMID: 31227758
PMCID: PMC6588598
DOI: 10.1038/s41598-019-45447-x

Abstract

The empirical observation of aggregation of dielectric particles under the influence of electrostatic forces lies at the origin of the theory of electricity. The growth of clusters formed of small grains underpins a range of phenomena from the early stages of planetesimal formation to aerosols. However, the collective effects of Coulomb forces on the nonequilibrium dynamics and aggregation process in a granular gas - a model representative of the above physical processes - have so far evaded theoretical scrutiny. Here, we establish a hydrodynamic description of aggregating granular gases that exchange charges upon collisions and interact via the long-ranged Coulomb forces. We analytically derive the governing equations for the evolution of granular temperature, charge variance, and number density for homogeneous and quasi-monodisperse aggregation. We find that, once the aggregates are formed, the granular temperature of the cluster population, the charge variance of the cluster population and the number density of the cluster population evolve in such a way that their non-dimensional combination obeys a physical constraint of nearly constant dimensionless ratio of characteristic electrostatic to kinetic energy. This constraint on the collective evolution of charged clusters is confirmed both by our theory and our detailed molecular dynamics simulations. The inhomogeneous aggregation of monomers and clusters in their mutual electrostatic field proceeds in a fractal manner. Our theoretical framework is extendable to more precise charge exchange mechanisms, a current focus of extensive experimentation. Furthermore, it illustrates the collective role of long-ranged interactions in dissipative gases and can lead to novel designing principles in particulate systems.

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Conflict of interest statement

The authors declare no competing interests.

Figures

**Figure 1**
Snapshots of the aggregating charged granular gas at different non-dimensional times: (a) $t^{⁎} = 19$ , (b) $t^{⁎} = 59$ , and (c) $t^{⁎} = 99$ . Here clusters containing 10 particles or more ( $N_{c l} \geq 10$ ) are shown, and each color represents a different cluster. Clusters are identified on the basis of the monomer distances: if the centers of two particles are separated by a particle diameter, or less, they belong to the same cluster. See Methods for non-dimensional time $t^{⁎} = t v_{ref} / d_{0}$ , Coulomb force strength $K$ , and other reference scales.

**Figure 2**
(a) When charged particles collide (particle group I), two possible pathways are considered in the kinetic theory depending on their velocities and charges: (i) a typical inelastic collision (dissipation with velocity dependent coefficient of restitution) with charge exchange (particle group II) if the relative collision speed is above a threshold speed $\sqrt{b}$ , and (ii) a collision that leads to the aggregation (with coefficient of restitution equal to zero) and merging of charges (particle group III), if the relative collision speed is below $\sqrt{b}$ and the particles are oppositely charged. The latter event introduces a size distribution of aggregates, which we further simplify in our kinetic theory by adjusting the size of all the particles (particle group IV). Thus, the particle size is assumed to remain monodispersed during the course of aggregation. Primed variables represent post-collision or post-aggregation values. (b) Coefficient of restitution $ϵ (v_{i j})$ in the present kinetic theory. Notice that the threshold $\sqrt{b} = \sqrt{\frac{2 k_{e} | q_{i} q_{j} |}{m d}}$ also evolves with time due to charge-exchange and aggregation events, and the coefficient of restitution is zero only if $Θ (- q_{i} q_{j}) Θ (b^{1 / 2} - v_{i j}) = 1$ . The above kinetic approach is compared with the MD simulations where the size distribution of aggregates is polydisperse.

**Figure 3**
Evolution of (a) temperature $T$ of cluster population, (b) charge variance $⟨ δ q^{2} ⟩$ of cluster population, and (c) number density $n$ of cluster population, for different monomer filling fractions $φ$ and charge strength $K$ . (d) The granular temperature, charge variance and average size of the cluster population during aggregation evolve in such a manner that their non-dimensional combination $ℬ (t) = k_{e} ⟨ δ q^{2} ⟩ / (T d) \leq 1$ (see also Fig. 4). Both temperature and charge variance of cluster population decay as power laws (with exponents marked as legends). The number density evolution, however, is highly dynamic and exhibits a non-monotonic behavior due to emergence of mesoscopic flow (see Tables 2 and 3 for $φ$ , $K$ , $v_{ref}$ , $d_{0}$ ).

**Figure 4**
The granular temperature, charge variance and average size of the cluster population during aggregation evolve in such a manner that their non-dimensional combination $ℬ (t) = k_{e} ⟨ δ q^{2} ⟩ / (T d) \leq 1$ . This is not captured in the kinetic theory if only restitutive (no aggregation) collisions are considered (dashed line). The granular MD simulations (symbols) confirm the analytical results. (inset for different monomer filling fraction $φ$ ).

**Figure 5**
(a–c) The scaling between cluster mass $m$ and their radius of gyration $R_{g}$ , $m \sim R_{g}^{D_{f}}$ at different times, and (d) the time evolution of $⟨ D_{f} ⟩$ , thus obtained, for different filling fractions. The average fractal dimension in the aggregating charged gas varies between the average values reported for ballistic cluster-cluster aggregation (BCCA, $⟨ D_{f} ⟩ ≃ 1.94$ ) and diffusion-limited particle-cluster aggregation (DLPCA, $⟨ D_{f} ⟩ ≃ 2.46$ )^,. $t^{⁎} \equiv t v_{ref} / d_{0}$ .

**Figure 6**
The scaled charge distribution $f (\tilde{q})$ of individual particles (monomers) obtained from typical MD simulation runs (dots) in the aggregated granular gas. The solid line is a Gaussian fit. Here $\tilde{q} = q / {⟨ δ q^{2} ⟩}^{1/2}$ .

**Figure 7**
Individual comparisons of (a) temperature of cluster population $T$ , (b) charge variance of cluster population $⟨ δ q^{2} ⟩$ , and (c) number density of cluster population $n$ , with the theoretical predictions. The granular temperature, charge variance and average size of the cluster population during aggregation evolve in such a manner that their non-dimensional combination $ℬ (t) = k_{e} ⟨ δ q^{2} ⟩ / (T d) \leq 1$ (main text). The number density evolution, however, is highly dynamic and exhibits a non-monotonic behavior due to emergence of mesoscopic flow, marked by a transition in the rate of increase of Mach number Ma, shown in (c) inset. The numbers in (c) inset are the power law exponents of the time dependence of Ma. The short-hands *res* and *agg* denote restitution and aggregation respectively.

**Figure 8**
Growth of the average cluster size, and the size of the largest cluster.

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References

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[1] Castellanos A. The relationship between attractive interparticle forces and bulk behaviour in dry and uncharged fine powders. Advances in Physics. 2005;54:263–376. doi: 10.1080/17461390500402657. - DOI

[2] Castellanos A. The relationship between attractive interparticle forces and bulk behaviour in dry and uncharged fine powders. Advances in Physics. 2005;54:263–376. doi: 10.1080/17461390500402657. - DOI

[3] Schwager T, Wolf DE, Pöschel T. Fractal substructure of a nanopowder. Physical Review Letters. 2008;100:218002. doi: 10.1103/PhysRevLett.100.218002. - DOI - PubMed

[4] Schwager T, Wolf DE, Pöschel T. Fractal substructure of a nanopowder. Physical Review Letters. 2008;100:218002. doi: 10.1103/PhysRevLett.100.218002. - DOI - PubMed

[5] Wesson PS. Accretion and electrostatic interaction of interstellar dust grains; interstellar grit. Astrophysics and Space Science. 1973;23:227–255. doi: 10.1007/BF00647661. - DOI

[6] Wesson PS. Accretion and electrostatic interaction of interstellar dust grains; interstellar grit. Astrophysics and Space Science. 1973;23:227–255. doi: 10.1007/BF00647661. - DOI

[7] Harper JSM, et al. Electrification of sand on titan and its influence on sediment transport. Nature Geoscience. 2017;10:260. doi: 10.1038/ngeo2921. - DOI

[8] Harper JSM, et al. Electrification of sand on titan and its influence on sediment transport. Nature Geoscience. 2017;10:260. doi: 10.1038/ngeo2921. - DOI

[9] Brilliantov N, et al. Size distribution of particles in saturn’s rings from aggregation and fragmentation. Proceedings of the National Academy of Sciences USA. 2015;112:9536–9541. doi: 10.1073/pnas.1503957112. - DOI - PMC - PubMed

[10] Brilliantov N, et al. Size distribution of particles in saturn’s rings from aggregation and fragmentation. Proceedings of the National Academy of Sciences USA. 2015;112:9536–9541. doi: 10.1073/pnas.1503957112. - DOI - PMC - PubMed

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Electrification in granular gases leads to constrained fractal growth

Affiliations

Electrification in granular gases leads to constrained fractal growth

Authors

Affiliations

Abstract

Conflict of interest statement

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