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Characteristics of Dynamic Surface Reactions

Catalytic reactions on surfaces exhibit an energy ratchet that biases the reaction away from equilibrium. In the simplest form, the catalyst oscillates between two states of stronger or weaker binding, which in this example is referred to as 'green' or 'blue,' respectively. For a single elementary reaction on a catalyst oscillating between two states (green & blue), there exists four rate coefficients in total, one forward (k₁) and one reverse (k_-1) in each catalyst state. The catalyst switches between catalyst states (j of blue or green) with a frequency, f, with the time in each catalyst state, τ_j, such that the duty cycle, D_j is defined for catalyst state, j, as the fraction of the time the catalyst exists in state j. For the catalyst in the 'blue' state:

$D_{B}={\frac {\tau _{B}}{\tau _{total}}}$

The bias of a catalytic ratchet under dynamic conditions can be predicted via a ratchet directionality metric, λ, that can be calculated from the rate coefficients, k_i, and the time constants of the oscillation, τ_i (or the duty cycle). For a catalyst oscillating between two catalyst states (blue and green), the ratchet directionality metric can be calculated:

$\lambda _{\ }={\frac {k_{1,blue}D_{B}+k_{1,green}(1-D_{B})}{k_{-1,blue}D_{B}+k_{-1,green}(1-D_{B})}}$

For directionality metrics greater than 1, the reaction exhibits forward bias to conversion higher than equilibrium. Directionality metrics less than 1 indicate negative reaction bias to conversion less than equilibrium. For more complicated reactions oscillating between multiple catalyst states, j, the ratchet directionality metric can be calculated based on the rate constants and time scales of all states.

$\lambda _{\ }={\frac {\sum _{j}\tau _{j}k_{1,j}}{\sum _{j}\tau _{j}k_{-1,j}}}$

The kinetic bias of an independent catalytic ratchet exists for sufficiently high catalyst oscillation frequencies, f, above the ratchet cutoff frequency, f_c, calculated as:

$f_{c}={\frac {k_{II}D_{II}}{4({\sqrt {2}}-1)}}$

For a single independent catalytic elementary step of a reaction on a surface (e.g., A* ↔ B*), the A* surface coverage, θ_A, can be predicted from the ratchet directionality metric,

$\theta _{A}={\frac {1}{1+\lambda }}$