Driven Colloidal Monolayers: Static and Dynamic Friction

Abstract

Trapping and dragging colloidal monolayers in two-dimensional optical lattices is offering the possibility to mimic friction between crystals (or even quasicrystals) visualizing directly the intimate mechanisms of sliding friction, with the additional possibility to change parameters freely, and to compare directly experiment with theory. Realistic simulations, which we review here, make a number of predictions about static features and dynamic sliding and reproduce well recent observations. Together, they provide a first demonstration of the potential impact of colloid dynamics in nanotribology.

Appendix: Static Configurations

Fig. 19.9

The static initial configuration for $U_0=0.1$, $F=0$, $a_\mathrm{las}=0.95$, i.e. $\rho =0.95$ (antisoliton incommensurate pattern: AI). Darker dots indicate colloidal particles sitting at a repulsive point of the corrugation landscape, namely with $W({\mathbf r})>-U_0/2$. This configuration is essentially unique, since an extremely similar configuration is retrieved at the end of a long relaxation, regardless of the initial condition. The central portion of this figure is represented in Fig. 19.3a

Fig. 19.10

The static initial configuration for $U_0=0.1$, $F=0$, $a_\mathrm{las}=1.00$, i.e. $\rho =1.02$ (commensurate: CO). Darker dots  have the same significance as in Fig. 19.9. At this nearly matched value $\rho =1$, this configuration is the lowest-energy state, on the blue-circle curve in the phase diagram of Fig. 19.7. The metastable high-energy state of the red-square curve in Fig. 19.7 is qualitatively similar to the one depicted in the subsequent Fig. 19.11

Fig. 19.11

The static initial configuration for $U_0=0.1$, $F=0$, $a_\mathrm{las}=1.05$, i.e. $\rho =1.05$ (soliton incommensurate pattern: SI). Darker dots have the same significance as in Fig. 19.9. At this comparably large mismatch value $\rho =1.05$, this configuration is the lowest-energy state, on the red-square curve in the phase diagram reported in Fig. 19.7. The metastable high-energy state of the blue-circle curve looks very similar to the one represented in the previous Fig. 19.10

Fig. 19.12

A typical initial configuration for a stronger ($U_0=0.5$) corrugation potential. The other parameters ($F=0$, $a_\mathrm{las}=0.95$, AI) and the color notation are the same as in Fig. 19.9. By comparison with the weaker corrugation, here antisolitons are much narrower, intersecting and isolating well-faceted in-registry regions. Note that the pattern formed by the center of the antisoliton lines is the same in both figures

Fig. 19.13

The effects of thermal fluctuations. A typical $F=0$ snapshot of a Langevin simulation at $k_\mathrm{B} T=0.04$, corresponding to room temperature in model units. The parameters ($U_0=0.1$, $F=0$, $a_\mathrm{las}=0.95$, AI) and the color notation are the same as in Fig. 19.9. By comparison with the $T=0$ configuration, the antisoliton pattern is only marginally affected by thermal noise. A small thermal expansion is responsible for a slight reduction of $\rho$, producing a visibly denser antisolitonic pattern than at $T=0$

Figures 19.9, 19.10, 19.11 display three overall views of the static fully relaxed $F=0$ configurations for different values of $a_\mathrm{lub}$ (or, equivalently, of $\rho$). These pictures represent the lowest-energy configurations of the three regions in the phase diagram—Fig. 19.7. Soliton/antisoliton patterns are highlighted by coloring colloids differently for different positions relative to the potential profile of Fig. 19.2: dark, bolder colloids occupy locally unfavorable repulsive regions for the corrugation profile $W$. Figure 19.12 illustrates the antisoliton pattern for a larger amplitude of the corrugation $U_0$, to be compared with Fig. 19.9 obtained with smaller corrugation. Note that the antisoliton lines are narrower in Fig. 19.12, but they form the same pattern as in Fig. 19.9.

Finally, Fig. 19.13 is to be compared with Fig. 19.9 to appreciate the effect of the random thermal motions characteristic of $300$ K: (i) Brownian fluctuations smear the boundaries between in-registry and antisolitonic regions and (ii) a small thermal expansion is marked by a reduction in $\rho$, and therefore in the separation between antisolitons.