Discrete stochastic charging of aggregate grains

Dust particles immersed in a plasma environment become charged through the collection of electrons and ions at random times, causing the dust charge to fluctuate about an equilibrium value. Small grains (with radii less than 1 μm) or grains in a tenuous plasma environment are sensitive to single additions of electrons or ions. Here we present a numerical model that allows examination of discrete stochastic charge fluctuations on the surface of aggregate grains and determines the effect of these fluctuations on the dynamics of grain aggregation. We show that the mean and standard deviation of charge on aggregate grains follow the same trends as those predicted for spheres having an equivalent radius, though aggregates exhibit larger variations from the predicted values. In some plasma environments, these charge fluctuations occur on timescales which are relevant for dynamics of aggregate growth. Coupled dynamics and charging models show that charge fluctuations tend to produce aggregates which are much more linear or filamentary than aggregates formed in an environment where the charge is stationary.

Figure 1

A two-dimenisonal representation of the open lines of sight to three points on the surface of an aggregate. The open lines of sight are determined by checking many test directions $\widehat{t}$ to see if they intersect other monomers in the aggregate. The angle between the surface normal vector and the test direction is ${\gamma }_t$.

Figure 2

First 2500 time steps of the charging history and (insets) probability distribution of the fluctuating charge (for 70 000 time steps after equilibrium is reached). (a) A 20nm grain with resolution $n=20$ in LAB plasma conditions, (b) 100-nm grain with resolution $n=10$ in a PPD plasma.

Figure 3

Percent difference between the equilibrium charge calculated with DSC and the predicted value as a function of the patch size. Points are shown for grain radii ranging from 10 nm to 1 μm using resolutions of 10, 20, 40, and 90 surface patches. The two grains shown in Fig. 2 are marked with stars.

Figure 4

Characteristic times for growth (right arrows) and dissipation (left arrows) of fluctuations. The symbols are calculated from the charging time history of a 20-nm grain in a LAB plasma and a 100-nm grain in a PPD plasma. The lines are analytic results for a sphere.

Figure 5

Sample aggregates from the three populations, shown to scale. (Left) Aggregates with monomer radius $a=500\mathrm{nm}$, (center) aggregates with monomer radius $a=1\mu \mathrm{m}$, (right) aggregates with monomer radii ranging from 0.5 to 10 μm. Superimposed on each aggregate is a sphere with radius equal to $R_{\sigma }$, as described in the text, and a sphere with radius R (centered at the aggregate's center of mass) which just circumscribes the aggregate.

Figure 6

Average charge (a) and standard deviation of charge as a fraction of the predicted charge (b) for aggregates of spherical monomers with radius $a$, as indicated in the legend. Data calculated for spheres using the DSM are shown by filled circles. The lines in (a) and (b) are theoretical predictions for spherical grains, and insets show the charge mean and standard deviation normalized by the predicted values. All data shown above were generated for PPD plasma conditions.

Figure 7

(a) Characteristic times for growth and dissipation of charge fluctuations on aggregate grains consisting of ten monomers with radius $a=500\mathrm{nm}$. The dashed lines are analytic results using ${\tau }_f$ calculated for an equivalent sphere, while the solid lines are from the analytic fits using ${\tau }_c$ determined from the point where the growth and dissipation curves intersect. (b) Ratio of predicted and calculated fluctuation timescales for aggregates in LAB plasma conditions (red, open symbols) and PPD plasma conditions (blue, filled symbols). The larger, darker symbols are for aggregates with $a=1\mu \mathrm{m}$; smaller, lighter symbols are for aggregates with $a=500\mathrm{nm}$.

Figure 8

Contour lines for the electrostatic potential. Green lines show the potential contours for a point charge centered at the origin, black lines show the potential contours calculated for charge located at each patch point. (a) Grain charged in PPD plasma condition where the average grain charge is $4.7e^{-}$. Here five of the patches, designated by the blue dots, have one electron. The patches indicated by the red dots have no charge. (b) Grain charged in LAB plasma conditions with an average charge of $214e^{-}$. The patch charges range from $15e^{-}$ (red) to $30e^{-}$ (dark blue).

Figure 9

Four different charging cases considered in the aggregate dynamics. (a) All particles are uncharged; (b) the time-averaged charge is applied to each spherical monomer; (c) the total charge on each monomer fluctuates in time; (d) the charge on each patch fluctuates in time. For the spheres, red indicates a large (negative) charge and blue indicates a small (negative) charge. For the patches, red indicates the largest negative charge, and blue is the largest positive charge.

Figure 10

Compactness factor as a function of the number of monomers for (a) LAB plasma and (b) PPD plasma.

Figure 11

Aggregate size as a function of the number of monomers N for (a) LAB plasma, and (b) PPD plasma. The upper curves designate the average maximum radial extent, R, measured from the center of mass, while the lower curves show the average equivalent radius $R_{\sigma }$.

Figure 12

Ratio of the maximum radial extent to the equivalent radius as a function of the number of monomers for (a) LAB plasma, and (b) PPD plasma.

Figure 13

Comparison of aggregates built using the different charging models in PPD plasma conditions. All aggregates have 25 monomers. (a) Patch charge with DSC, (b) constant Average charge, (c) Neutral. The inner sphere shows the extent of $R_{\sigma }$, which is approximately equal for the three aggregates, while the outer sphere indicates the maximum radial extent.

Figure 14

Cumulative number of misses in building aggregate to N monomers for (a) LAB plasma and (b) PPD plasma.