Multiple timescale contact charging

Contact charging between insulators is one of the most basic, yet least well understood, of physical processes. For example we have no clear theory for how insulators recruit enough charge carriers to deposit charge but not enough to discharge. In this paper we note that charging and discharging kinetics may be distinct, and from this observation we develop a mathematical model. The model surprisingly predicts that charging can decrease as contact frequency increases: We confirm this prediction experimentally and propose future steps.

Figure 1

Example of contact charge pattern, from Ref. [3]. Inset summarizes experiment in which a celluloid ball mounted on a wooden post is contacted with a synthetic rubber ball on a wooden post. Main plot shows charge patterns made visible by exposing the celluloid ball to a cloud of bipolar toner [2]: the red (black) toner sticks to negatively (positively) charged areas. Note that charge remains on the ball centimeters away from the contact point, and that point retains minimal charge. This suggests different spatial and temporal scales for charging and discharging.

Figure 2

Simulations of repeated collisions between mirror-symmetric charged particles. (a) Inset: depiction of polarizations and charges on each particle; these generate an electric field on its partner, modeled in Eqs. (1) and (2). Here the polarizations are identical, and the net charges are equal and opposite; we describe the field on either particle due to its mate in Eq. (3). Main plot: integration of Eqs. (2, 3, 4) using $A=0.825$, $B=1$, $\chi =1$, $\alpha =1$, $\beta =0.5$, $p_b=1000$, and when $p>p_b$, $r_b=0.03.$ (b) Exemplar growths in polarization on semilog axes for $f=0.3$, 0.6, and 0.9: Notice that as the frequency $f$ of particle collisions increases, the mean asymptotic polarization, $〈p〉$, decreases. Solutions shown are obtained using mathematica’s multistep Adam's method solver, NDSolve. Complete code used is included as Supplemental Material [16].

Figure 3

Experiment using vibrated bed to evaluate predicted decrease in charging with an increase in contact frequency. (a) Vibrated bed of glass beads; distance between probe and top of bed held fixed as described in text. (b) Plot of experimental measurements of voltage vs vibration frequency confirming the predicted decrease in charge generation with an increase in frequency. Broken line is copied and rescaled from Fig. 2 between $f=0.4$ and 1.