Stochasticity in Single-Entity Electrochemistry

Abstract

Most electrochemical processes are stochastic and discrete in nature. Yet experimental observables, e.g., i vs E, are typically smooth and deterministic, due to many events/processes, e.g., electron transfers, being averaged together. However, when the number of entities measured approaches a few or even one, stochasticity frequently emerges. Yet all is not lost! Probabilistic and statistical interpretation can generate insights matching or superseding those from macroscale/ensemble measurements, revealing phenomena that were hitherto averaged over. Herein, we review recent literature examples of stochastic processes in single-entity electrochemistry, highlighting strategies for interpreting stochasticity, contrasting them with macroscale measurements, and describing the insights generated.

Keywords: Brownian motion; Randomness; activated processes; probability; statistics.

Figure 1.

Stochastic collision electrochemistry of particles and atoms. (a) Simulated random walk and (b) experimental i-t trace of a Ag nanoparticle stochastically colliding with a Au microelectrode. Each collision results in partial electrooxidation and dissolution. (c) Schematic of two-step experiment and (d) voltammograms for characterizing proton reduction on Pt_n (1≤n≤9). (e) Fraction of experiments with Pt_n deposited (orange) vs prediction from Poisson distribution (blue). Particles/atoms not to scale. Reprinted adapted with permission from references [17] and [25]. Copyright 2017 American Chemical Society.

Figure 2.

Quantification of the stochastic nucleation of a single H₂ nanobubble at a nanoelectrode. (a) Schematic of experiment. (b) Characteristic current and voltage transient measuring the induction time for a single bubble to nucleate, t_ind. (c) Repeated nucleation trials at a 41 nm radius Pt nanoelectrode indicating stochasticity and sensitivity on applied current (−30 nA, left; −33 nA, right). (d) Exponential fits to cumulative nucleation probabilities (7 applied currents) quantify nucleation rate vs concentration. (e) Measured geometry of a H₂ nanobubble critical nucleus. Reprinted adapted with permission from references [36] and [38]. Copyright 2018 & 2019 American Chemical Society.

Figure 3.

Single-molecule electrochemical STM break junction monitoring the redox state of a ferrocene (Fc) linked between a Au tip and Au substrate. (a) Schematic of experiment. (b-d). Conductance during electrochemical potential sweep differentiates Fc/Fc⁺ (=0.03/0.09 G/G₀, respectively). Individual molecules (single traces) exhibit different switching behaviors. (e) Probability of oxidation (red) and reduction (blue) as a function of overpotential. (f-h) Simulated transitions between O and R at different k₀. Reproduced with modification from Li et al.[40]

Figure 4.

Measuring the kinetics of base flipping and protonation in a single DNA molecule through ion conductance measurements. (a) Schematic of a single DNA molecule (blue) with a C:C base pair mismatch (red) captured in α-hemolysin (α-HL). (b) Hidden Markov model (HMM) for the protonation/deprotonation and base flipping of a C:C mismatch. (c) Experimental i-t traces at different pHs show switching between two current levels. (d) Simulated i-t traces using the HMM. (e) log-histogram of duration of intra-helical (red) vs extra-helical (blue) states at pH 7.5 (bars are experiments; lines are simulations). Reprinted adapted with permission from reference [44]. Copyright 2018 American Chemical Society.