Multiple charge carrier species as a possible cause for triboelectric cycles

TL;DR

The paper addresses how triboelectric series and cycles can arise in insulator contact electrification and tests whether a single or multiple charge carriers can explain both phenomena. It develops an equilibrium framework in which charge carriers reside in finite-depth surface wells and transfer upon contact, analyzed via a canonical ensemble with a two-well partition function $Z$, yielding a sign rule for charge transfer. The key result is that a single carrier produces TE series but forbids cycles, whereas introducing a second carrier enables cycles, including higher-order cycles through material insertions. This work reframes TE series and cycles as diagnostic tools for the underlying charge carriers and mechanisms, showing cycles do not require non-equilibrium dynamics in this toy model. While the model is simplified, it provides a rational basis for interpreting observed series and cycles and highlights new experimental directions to identify the active carriers in CE.

Abstract

The tendency of materials to order in triboelectric series has prompted suggestions that contact electrification might have a single, unified underlying description. However, the possibility of triboelectric cycles, i.e. series that loop back onto themselves, is seemingly at odds with such a coherent description. In this work, we propose that if multiple charge carrying species are at play, both triboelectric series and cycles are possible. We show how series arise naturally if only a single charge carrier species is involved and if the driving mechanism is approach toward thermodynamic equilibrium, and simultaneously, that cycles are forbidden under such conditions. Suspecting multiple carriers might relax the situation, we affirm this is the case by explicit construction of a cycle involving two carriers, and then extend this to show how more complex cycles emerge. Our work highlights the importance of series/cycles towards determining the underlying mechanism(s) and carrier(s) in contact electrification.

Figures (4)

• Figure 1: (a) The triboelectric series of Wilcke Assiss.2010, where materials are ordered by their tendency to charge plus/minus. (b) Sketch of what (a) may have looked like in matrix form. The color represents the sign of the charge transferred to the column material. (c) Shaw and Jex's triboelectric 'cycle' Shaw.1928. Zinc charges positive to silk (at the top of the series) but negative to washed glass (at the bottom). (d) Sketch of what (b) may have looked like in matrix form.
• Figure 2: In our 'toy model', we assume charge carriers inhabit energetic wells of depth, $\varepsilon$. In the figure, the green curve corresponds to the well of the charge carrier on surface 1, the purple curve the well on surface 2, and the gray curve the sum of the two. When the two surfaces make contact, the wells 'merge,' allowing the carrier to transfer between them. At zero temperature, the carrier would always move toward the deeper well; however as we explain in the text, at non-zero temperature and when many carriers are involved, some fraction of carriers will move 'uphill', and a thermodynamic picture is appropriate.
• Figure 3: (a) We explicitly construct a cycle with two carriers and three materials. Assuming positive carriers $A$ and $B$, $A$ alone would give the series $\{i,j,k\}$, while $B$ alone would give $\{k,i\}$. (b) In combination, and with appropriately chosen neutral numbers, a 3-cycle $\langle i,j,k\rangle$ is possible. (c) Matrix representation of the $3$-cycle. (d) Arrow diagram to visualize the relations between surfaces when cycles are involved. An arrowhead pointing from $k$ to $i$ implies that charge transfers in that direction, or equivalently, $k$ charges negative and $i$ charges positive.
• Figure 4: Creation of more complex cycles through insertion of new materials (a-c) before the existing cycle, (d-f) after, and (g-i) in-between materials. Figures (a),(d), & (g) show the matrix representation of the material insertion process for a $3$-cycle, while the respective arrow diagrams (b),(e) & (h) provide visual aid to understand the relation between surfaces in each case. The process can be extended to introduce an arbitrary number of new materials, as shown in diagrams (c), (f), & (i).