Extension of discrete tribocharging models to continuous size distributions

Triboelectric charging, the phenomenon by which electrical charge is exchanged during contact between two surfaces, has been known to cause significant charge separation in granular mixtures, even between chemically identical grains. This charging is a stochastic size-dependent process resulting from random collisions between grains. The prevailing models and experimental results suggest that, in most cases, larger grains in a mixture of dielectric grains acquire a positive charge, while smaller grains charge negatively. These models are typically restricted to mixtures of two discrete grain sizes, which are not representative of most naturally occurring granular mixtures, and neglect the effect of grain size on individual charging events. We have developed a model that predicts the average charge distribution in a granular mixture, for any continuous size distribution of dielectric grains of a single material. Expanding to continuous size distributions enables the prediction of charge separation in many natural granular phenomena, including terrestrial dust storms and industrial powder handling operations. The expanded model makes predictions about the charge distribution, including specific conditions under which the usual size-dependent polarity is reversed such that larger grains charge negatively.

Figure 1

Diagram of physical system considered. Grains with a normalized size distribution given by $g\left(R\right)$ are mixed in a container of volume $V_c$. Mixing is performed by moving the container with average speed $v_r$, such that the grains inside also move with average speed $v_r$. The grains are then assumed to move in random directions with average speed $v_r$ as well. When a grain of radius $R_i$ collides with a grain of radius $R_j$, some contact area is formed in region $A_{ij}$. High-energy electrons in this region are capable of transferring between the two grains and settling into stable low-energy states.

Figure 2

Nondimensional charge distribution $Q^{*}=Q\left(R\right)/4e\pi {\rho }_0{R}_1^2$ for a specified size distribution function. (a) Normalized particle size distribution function $g\left(R\right)$ in the form of Eq. (27), with $k_2/k_1=8$, $a_1=a_2=0.005\mu {\mathrm{m}}^{-2}$, $R_1=100\mu \mathrm{m}$, and $R_2=50\mu \mathrm{m}$. (b) Nondimensional charge distribution function corresponding to size distribution in (a) (solid line, magnified $\times 10$ for clarity). For comparison, the distribution using $A_{ij}=\mathrm{constant}$ is overlayed (dashed line). Note the large difference in magnitude, and the additional peak in the distribution for our model at low values of $R$.

Figure 3

Effect of variation in peak width of size distribution on charge polarity, for nondimensional charge $Q^{*}=Q\left(R\right)/4e\pi {\rho }_0{R}_1^2$. (a) Normalized particle size distribution functions $g\left(R\right)$ in the form of Eq. (27), with $k_2/k_1=8$, $R_1=100\mu \mathrm{m}$, and $R_2=50\mu \mathrm{m}$. (b) Nondimensional charge distribution functions corresponding to the size distributions in (a). Vertical lines at $R=R_1$ and $R=R_2$ provided as reference.

Figure 4

Nondimensional net charge $Q_1^{*}=Q_1\left(R\right)/4e\pi {\rho }_0{R}_1^2$ on grains of radius $R_1=100\mu \mathrm{m}$ for a size distribution of the form of Eq. (27), with $a=0.005\mu \mathrm{m}$. As the ratio of the peak heights ($k$) increases, the size ratio ($d=R_2/R_1$) below which large grains charge negatively decreases. Note that the charge predicted by the continuous distribution is very similar to that predicted by the discrete model, with some variation due to the effect of the nonzero peak width [as seen in Fig. 3].