Exciton reactions on carbon nanotubes: an experimental testbed for critical dynamics

The one-dimensional coalescing random walk ($X+X\to X)$ is a paradigmatic reaction-diffusion system due to both its exact solvability [1,2] and the experimental observation of nonclassical kinetics at asymptotically long times [3]. The solvability rests on the anticommutative property of intersecting trajectories of particles that react \textit{instantly} and at \textit{short range}: however, the validity of these assumptions in real systems has not previously been tested by experiment. We have shown that exciton-exciton recombination (fusion) on carbon nanotubes provides a platform for quantitative studies of critical kinetics in a simple non-equilibrium system [4]. Under high excitation density we observed a crossover in the exciton density $n$ between regimes of classical $(n\propto t^{-1})$ and anomalous $(n\propto t^{-1/2})$ scale invariance as predicted by renormalization group [5] and approximate [1] theories, arising from a finite reaction probability of $\approx 0.2$ per encounter. At long times the exciton population per nanotube exponentially approaches unity (i.e. a finite size effect), allowing calibration of the exciton density and hence a demonstration of universality extending over both classical and critical regimes. Under low excitation, the early kinetics followed a Smoluchowski-Noyes form ${dn} \mathord{\left/ {\vphantom {{dn} {dt}}} \right. \kern-\nulldelimiterspace} {dt}\propto n^{2}t^{-1/2}$ rather than the asymptotic ${dn} \mathord{\left/ {\vphantom {{dn} {dt}}} \right. \kern-\nulldelimiterspace} {dt}\propto n^{3}$, providing direct evidence for the spatial self-ordering that precedes critical scale invariance. We studied the re-emergence of microscopic detail at the classical-nonclassical crossover, which is abrupt and nonmonotonic due to competition between temporal and spatial averaging of critical fluctuations (i.e. finite reaction rate and range). It appears that real-world experiments will require more complete descriptions of the interactions than is available in existing models.\\[4pt] [1] D. Ben-Avraham and S. Havlin, \textit{Diffusion and Reactions in Fractals and Disordered Systems} (Cambridge University Press, Cambridge, 2000).\\[0pt] [2] G. M. Sch\"{u}tz, Phase Transitions and Critical Phenomena 19, 1 (2001).\\[0pt] [3] J. Prasad and R. Kopelman, Phys. Rev. Lett. 59, 2103 (1987).\\[0pt] [4] J. Allam et al., Phys. Rev. Lett. 111, 197401 (2013).\\[0pt] [5] U. C. T\"{a}uber et al., \textit{J. Phys. A: Math. Gen.} 38, R79--R131 (2005).