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Resonance Frequency

Mechanisms of Efficient Dynamic Catalysts

Dynamic catalysts that undergo forced variation of free energy surface during a catalytic reaction are called 'programmable catalysts.' Perturbation of the active site with strain, light, or condensed charge modulates the binding energy of adsorbates and transition states to alter the rate, selectivity, and conversion of a catalytic reaction. New opportunities exist with dynamic catalysts and oscillating free energy surfaces not possible with conventional static active sites. However, these opportunities require energy input to modulate the catalyst, raising the issue of efficiency of a programmable catalyst.^[1]

A programmable catalyst oscillating between strong and weak binding energies exhibits positive scaling between reaction intermediates; B* and A* both weaken and strengthen in binding energy simultaneously. Under strong binding conditions, A* readily reacts over the transition state to form B*. For weaking catalyst binding conditions, B* readily desorbs to form B(g), as A(g) immediately adsorbs as A* to restart the catalytic cycle. An efficient programmable catalyst converts molecules from reactants to products with every oscillation of binding energy of the active site, such that most active sites on the catalyst surface produce a product molecule for every catalytic oscillation cycle. Of key importance is the height of the transition state barrier in the weak-binding catalyst state; a high barrier creates a ratchet mechanism, whereby B* is prohibited from reacting backwards to A*.^[2]

The efficiency of a programmable catalyst can be determined by the metric of the turnover efficiency (η_TOE). The turnover efficiency compares the difference between the time-averaged dynamic turnover frequency of the reaction (TOF_dyn) and the steady state turnover frequency (TOF_ss) to the applied catalytic oscillation frequency, f_app.

\eta _{TOE}={\frac {TOF_{dyn}-TOF_{ss}}{f_{app}}}

Highly efficient programmable catalysts will exhibit turnover efficiencies close to unity (η_TOE ~ 1), indicating that there is parity between the applied frequency and the catalytic turnover frequency.^[3]

Of fundamental importance are mechanisms that lead to a reduction in the turnover efficiency. One such catalytic mechanism is the leaky programmable catalyst mechanism. On these dynamic free energy surfaces, the adsorbed surface reactant A* readily reacts to form surface product B* in the strong-binding catalyst state. However, in the weak-binding catalyst state, B* readily reacts backwards to reform A* rather than desorbing to form B(g) in the gas phase. This yields an inefficient programmable catalyst, whereby most input energy to modulate the catalyst between states results in heat generation as molecules interconvert between A* and B*. Programmable catalysts exhibiting the leaky ratchet phenomenon exhibit time-averaged turnover frequencies far from parity with the applied catalytic oscillation frequency, resulting in low turnover efficiency.^[4]

An alternative programmable catalytic mechanism leading to reduced turnover efficiency derives from the extent of the surface that participates in the overall reaction. When the programmable catalyst switches to the strong-binding state, A* reacts to B* and equilibrates. However, for systems with comparable free energy of A* and B* in the strong binding state, only a fraction of reactant A* is converted to B*. In this case, only a fraction of surface active sites desorb B* to form B(g) when the catalyst switches to the weak-binding state. Adsorbates of A* that change in binding energy between modulating catalyst states consume energy and release heat without completing the catalytic cycle, yielding an inefficient programmable catalyst.^[5]

High turnover efficiency is critical for efficient use of a programmable catalyst. The two key mechanisms leading to lower efficiency, the leaky ratchet and low participating surface mechanisms, can significantly reduce the time-averaged catalytic rate, even orders of magnitude lower than the applied catalyst oscillation frequency.^[6]

Characteristics of Dynamic Surface Reactions

Catalytic reactions on surfaces exhibit an energy ratchet that biases the reaction away from equilibrium.^[7] 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).^[8] 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})}}

^[9]

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}}}

^[10]

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)}}

^[11]

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 }}

^[12]

[1]
M.A. Ardagh; Turan Birol; Q. Zhang; O.A. Abdelrahman; P.J. Dauenhauer (2019). "Catalytic Resonance Theory: superVolcanoes, catalytic molecular pumps, and oscillatory steady state". Catalysis Science & Technology. doi:10.1039/C9CY01543D. S2CID 182444068.
[2]
M.A. Ardagh; O.A. Abdelrahman; P.J. Dauenhauer (2019). "Principles of Dynamic Heterogeneous Catalysis: Surface Resonance and Turnover Frequency Response". ACS Catalysis. doi:10.1021/acscatal.9b01606.
[3]
Shetty, Manish; Walton, Amber; Gathmann, Sallye R.; Ardagh, M. Alexander; Gopeesingh, Joshua; Resasco, Joaquin; Birol, Turan; Zhang, Qi; Tsapatsis, Michael; Vlachos, Dionisios G.; Christopher, Phillip; Frisbie, C. Daniel; Abdelrahman, Omar A.; Dauenhauer, Paul J. (2020). "The Catalytic Mechanics of Dynamic Surfaces: Stimulating Methods for Promoting Catalytic Resonance". ACS Catalysis. 10 (21). American Chemical Society: 12666–12695. doi:10.1021/acscatal.0c03336.
[4]
Omar A. Abdelrahman; P.J. Dauenhauer (2023). "Energy Flows in Static and Programmable Catalysts". ACS Energy Letters. doi:10.1021/acsenergylett.3c00522.
[5]
Jesse R. Canavan; Justin A Hopkins; Brandon L. Foley; Omar A. Abdelrahman; Paul J. Dauenhauer (2024). "Catalytic Resonance Theory: Turnover Efficiency and the Resonance Frequency". ChemRxiv. doi:10.1021/acsenergylett.3c00522.
[6]
Jesse R. Canavan; Justin A Hopkins; Brandon L. Foley; Omar A. Abdelrahman; Paul J. Dauenhauer (2024). "Catalytic Resonance Theory: Turnover Efficiency and the Resonance Frequency". ChemRxiv. doi:10.1021/acsenergylett.3c00522.
[7]
M.A. Ardagh; Turan Birol; Q. Zhang; O.A. Abdelrahman; P.J. Dauenhauer (2019). "Catalytic Resonance Theory: superVolcanoes, catalytic molecular pumps, and oscillatory steady state". Catalysis Science & Technology. doi:10.1039/C9CY01543D. S2CID 182444068.
[8]
M.A. Murphy; Sallye R. Gathmann; Rachel Getman; Lars Grabow; O.A. Abdelrahman; P.J. Dauenhauer (2024). "Catalytic Resonance Theory: The Catalytic Mechanics of Programmable Ratchets". Chemical Science. doi:10.1039/D4SC04069D. S2CID 271629800.
[9]
M.A. Murphy; Sallye R. Gathmann; Rachel Getman; Lars Grabow; O.A. Abdelrahman; P.J. Dauenhauer (2024). "Catalytic Resonance Theory: The Catalytic Mechanics of Programmable Ratchets". Chemical Science. doi:10.1039/D4SC04069D. S2CID 271629800.
[10]
M.A. Murphy; Sallye R. Gathmann; Rachel Getman; Lars Grabow; O.A. Abdelrahman; P.J. Dauenhauer (2024). "Catalytic Resonance Theory: The Catalytic Mechanics of Programmable Ratchets". Chemical Science. doi:10.1039/D4SC04069D. S2CID 271629800.
[11]
M.A. Murphy; Sallye R. Gathmann; Rachel Getman; Lars Grabow; O.A. Abdelrahman; P.J. Dauenhauer (2024). "Catalytic Resonance Theory: The Catalytic Mechanics of Programmable Ratchets". Chemical Science. doi:10.1039/D4SC04069D. S2CID 271629800.
[12]
M.A. Murphy; Sallye R. Gathmann; Rachel Getman; Lars Grabow; O.A. Abdelrahman; P.J. Dauenhauer (2024). "Catalytic Resonance Theory: The Catalytic Mechanics of Programmable Ratchets". Chemical Science. doi:10.1039/D4SC04069D. S2CID 271629800.

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