Effect of grain size on the forces governing the dynamic behavior of electrostatically driven powder media

The aim of this paper is to identify the forces that govern the dynamic behavior of polydisperse granular systems under high electric fields in air media. In a previous paper, we reported the changes in the morphology and dynamics of grain structures under a frequency and amplitude variable voltage and summarized them in a diagram [L. M. Salvatierra et al., Chem. Phys. Lett. 481, 194 (2009)]. In the present article, we investigate the influence of the grain size distribution and the speed of voltage increase during tests. We explain the main transitions observed in the diagram in terms of the change in grain size and the frequency-dependent electrical response, related to electrophoretic and dielectrophoretic interactions.

Figure 1

Voltage amplitude-frequency diagram experimentally constructed in Ref. [1]. PI blue (dark gray): precipitated granular solid phase; PII yellow (light gray): constructive behavior; and PIII (gray): destructive behavior. The diagram was constructed by inspection of a video recording of the experiments performed at the explored frequencies (which are labeled on the horizontal axis). In one experiment, the electric field is raised from 0 V at constant rate and at constant frequency. Constructive behavior in Phase II was characterized in Ref. [1] by measuring the maximum height reached by the filaments, normalized to the interelectrode distance $H_{\max }$ as a function of the applied voltage $V_p$.

Figure 2

Dependence of the Phase I-Phase II boundary on the grain size and speed of voltage increase. The violet curve (gray) with filled circles, black curve with filled squares, and orange (light gray) curve with filled triangles are S, M, and L groups at $0.3\mathrm{kV}/\mathrm{s}$, respectively. The dashed black curve with the plus symbol is M at $0.15\mathrm{kV}/\mathrm{s}$. The dotted black curve with empty triangles is M at $0.6\mathrm{kV}/\mathrm{s}$.

Figure 3

Snapshots showing the different morphologies and dynamic regimes of the system at different frequencies and voltages (typical for PI, PII and PIII) for the three grain size groups. (a) L grain size group, (b) M grain size group, and (c) S grain size group. Dotted lines show the phase limits. Background color code (grayscale code) is the same as in Fig. 1: Phase I is blue (dark gray), Phase II is yellow (light gray), and Phase III is gray. Peak voltages are in kilovolts (${10}^3\mathrm{V}$).

Figure 4

Growth mechanism of: (a) one L filament in the LFZ (sequence at $\Delta t=0.2\mathrm{s}$) and (b) one L structure in the HFZ (sequence at $\Delta t=1\mathrm{s}$) at $0.3\mathrm{kV}/\mathrm{s}$.

Figure 5

(a) Dependence of $H_{\max }$ on the applied voltage and grain size. In the LFZ (blue labels and dashed lines), circles: $\mathrm{S},1\mathrm{kHz}$; squares: $\mathrm{M},750\mathrm{Hz}$; and triangles: $\mathrm{L},500\mathrm{Hz}$. In the HFZ (red labels and full lines), circles: $\mathrm{S},6\mathrm{kHz}$; squares: $\mathrm{M},6\mathrm{kHz}$; and triangles: $\mathrm{L},6\mathrm{kHz}$. (b) Dependence of $H_{\max }$ on the voltage speed increase at 750 Hz (filled labels) and 6 kHz (empty labels), for M particles at $0.15\mathrm{kV}/\mathrm{s}$ (black line and circles), $0.3\mathrm{kV}/\mathrm{s}$ (red line and squares), and $0.6\mathrm{kV}/\mathrm{s}$ (blue line and triangles).

Figure 6

Photographs of trajectories at different frequencies in Phase III. (a) $100\mathrm{Hz}\times \frac{1}{15}\mathrm{s}=6.66\mathrm{cycles}$, (b) $200\mathrm{Hz}\times \frac{1}{30}\mathrm{s}=6.66\mathrm{cycles}$, (c) $300\mathrm{Hz}\times \frac{1}{60}\mathrm{s}=5\mathrm{cycles}$, (d) $300\mathrm{Hz}\times \frac{1}{60}\mathrm{s}=5\mathrm{cycles}$, (e) $500\mathrm{Hz}\times \frac{1}{60}\mathrm{s}=8.33\mathrm{cycles}$, and (f) $750\mathrm{Hz}\times \frac{1}{30}\mathrm{s}=25\mathrm{cycles}$.

Figure 7

Phase III, Fig. 6d with modeled trajectories. $\Delta t=\mathrm{experimental}\mathrm{time}=1/30\mathrm{s},f=300\mathrm{Hz},V_p=1000\mathrm{V},2R=30\mu \mathrm{m},d=4.13\mathrm{mm},\varphi =4.35\times {10}^{-5}\mathrm{C}/{\mathrm{m}}^2$. T1: $(x_0,z_0)=(0,2)\times {10}^{-3}\mathrm{m},v_{z0}=0.12,v_x=0.12\mathrm{m}/\mathrm{s}$. T2: $(x_0,z_0)=(6,3)\times {10}^{-3}\mathrm{m},v_{z0}=0,v_x=0.12\mathrm{m}/\mathrm{s}$.

Figure 8

$V_p$ vs $f$, needed to move a grain S (circles), M (squares), or L (triangles) a distance of $A_0=2R$ due to an EP force [see Eq. (8) with $d=4.13\mathrm{mm}$].

Figure 9

Critical frequency $f_c$ against $V_p$ to oscillate a grain S (circles), M (squares), or L (triangles) a $z$ distance of $2A_0=d=4.13\mathrm{mm}$, where $d$ is the interelectrode distance.