Quantum field theory approach for multistage chemical kinetics in liquids

TL;DR

The paper tackles the breakdown of mass-action kinetics in diffusion-influenced multistage reactions due to microscopic spatial correlations. It develops CMET, a complete modified encounter theory derived from a second-quantization quantum-field-theory framework, yielding coupled differential equations for concentrations $C_i(\mathbf r,t)$ and pair distributions $p_{ij}(\mathbf r_1,\mathbf r_2,t)$ that incorporate environment-induced correlations beyond prior theories. CMET reproduces known kinetic regimes and exact long-time fluctuation asymptotics (e.g., $t^{-3/2}$) for both irreversible and reversible multistage networks, as demonstrated across multiple case studies including geminate recombination and $A+B\leftrightarrow C+D$. The approach provides a robust, computationally efficient first-principles framework for modeling complex reaction-diffusion systems in liquids and is complemented by the public Tegro software for practical kinetic modeling.

Abstract

Reaction-diffusion processes play an important role in a variety of physical, chemical, and biological systems. Conventionally, the kinetics of these processes are described by the law of mass action. However, there are various cases where these equations are insufficient. A fundamental challenge lies in accurately accounting for the microscopic correlations that inevitably arise in bimolecular reactions. While approaches to describe microscopic correlations in many specific cases exist, no general theory for multistage reactions has been established. In this article, we apply the quantum field theory approach to derive kinetic equations for general multistage reactive systems termed CMET (complete modified encounter theory). CMET can be formulated as a set of coupled partial differential equations that can be easily integrated numerically, thereby serving as a versatile tool for investigating reaction-diffusion processes. Across multiple case studies, we demonstrated that CMET reproduces the kinetics predicted by many other theories within their respective scopes of applicability.

Figures (8)

• Figure 1: Time dependence of the concentration $C_{D^*}(t)$ for the reaction $D^* + A\rightarrow D + A$, calculated using CMET Eqs.(\ref{['eq:MasterCW']}) and (\ref{['DiffpC']}-\ref{['S_CMET']}) (color lines), the SM approach Eqs.(\ref{['PCD*']}) and (\ref{['eq:DET']}-\ref{['eq:n']}) (dashed lines), and formal kinetics Eq.(\ref{['eq:exp']}) (thin black lines) at various concentrations of $A$ particles. The inset gives a more detailed view of the initial stage of the reaction. The parameters of the reaction are: $D_{D}=0,\, D_A = 5\times 10^{6}\ \text{cm}^2/\text{s}$, the distance dependent reaction rate $W(r)$ is given by Eq.(\ref{['Forster']}) where $W_c = 10^{10}\ \text{s}^{-1}$, the contact radius is $b= 5\AA$. Concentration of $A$ is $C_A= 0.1\, M$ (red), $C_A= 0.05\, M$ (cyan), $C_A= 0.025\, M$ (orange), $C_A= 0.01\, M$ (green).
• Figure 2: (top) Acceptor concentration dependence of the inverse luminescence quantum yield ( Stern - Volmer plots) calculated within CMET (blue circles) and the SM approach Eq.(\ref{['phi']}) (red circles). The dotted line represents the linear Stern-Volmer dependence Eq.(\ref{['Stern']}). (bottom) Time dependence of the $D^*$ concentration under CW excitation for different $A$ concentrations calculated within CMET Eqs.(\ref{['eq:MasterCW']}) and (\ref{['DiffpC']}-\ref{['S_CMET']}) (solid lines) and the SM approach Eq.(\ref{['CDt']}) (dashed lines) with colors indicating the concentration of $A$ particle: $1$ M (red), $2$ M (indigo), $3$ M (green), and $4$ M (blue). The remaining parameters of the reaction are: $\tau = 10^{-9}\ \text{s}$, $D_A = 5\cdot 10^{-6}\ \text{cm}^2/\text{s}$, $q = 10^7 \ \text{s}^{-1}$, the distance dependent reaction rate $W(r)$ is given by Eq.(\ref{['Forster']}), where $W_b = 10^{10}\ \text{s}^{-1}$, the contact radius is $b= 5\AA$.
• Figure 3: Time dependence of the $D^*$ concentration for the multistage reaction Eq.(\ref{['Dorf']}), calculated using UT Eq.(\ref{['CD*UT']})) (dashed line) and CMET Eqs.(\ref{['eq:MasterCW']}) and (\ref{['DiffpC']}-\ref{['S_CMET']}) (cyan line). These lines are multiplied by the factor of $0.1$ for clarity. The time dependence of the $D^+$ concentration, calculated using UT Eqs.(\ref{['CD+']}-\ref{['eq:p']}) (dashed line) and CMET with the initial concentration $C_{D^*}(0)$ being equal to $10^{-3}\ \text{M}$ (red line), $10^{-4}\ \text{M}$ (blue line), and $10^{-5}\ \text{M}$ (turquoise line). The parameters of the reactions are $W_i^0=10^{12} s^{-1} , W_r^0=10^{11} \text{s}^{-1}, L=1\AA, b=5 \AA, D_A=5\times10^{-6}\ \text{cm}^2/\text{s}, C_A=10^{-2}\ \text{M}$.
• Figure 4: Time dependence of the reaction functions for $A+B\leftrightarrow C+D$ reaction, calculated within CMET Eqs.(\ref{['eq:MasterCW']}) and (\ref{['DiffpC']}-\ref{['S_CMET']}) (solid lines). The asymptotic dependencies Eqs.(\ref{['eq:GS0']}-\ref{['eq:GS2']}) are shown by dash-dotted lines. The distance dependent reaction rate $W_f(r)$ is given by Eq.(\ref{['Cont']}), where $W_{\delta}= 10^{10}\ \text{s}^{-1}$. The equilibrium constant is $K_e = 1$. The initial concentrations and diffusion coefficients for different colors are given in Table \ref{['t:pars']}.
• Figure 5: Time dependence of the $A$ particle concentration for $A+B\leftrightarrow C+D$ reaction, calculated within CMET Eqs.(\ref{['eq:MasterCW']}) and (\ref{['DiffpC']}-\ref{['S_CMET']}) (solid lines with the colors corresponding to the $K_e = 10^{-1}$ for the red, $K_e = 10^{-3}$ for the orange, $K_e = 10^{-5}$ for the green, $K_e = 10^{-7}$ for the blue line). Initial parameters of the reaction are: $C_A(0) = C_B(0) = 0.1\ \text{M}$, $D = 5\times 10^{-6}\ \text{cm}^2/\text{s}$, the distance dependent reaction rate $W_f(r)$ is given by Eq.(\ref{['Cont']}), where $W_{\delta}= 10^{10}\ \text{\AA}/\text{s}$. The cyan line represents the $A$ particle concentration of the irreversible reaction $A+B\rightarrow C+D$, calculated within the CMET. The dashed black line represents a solution of Eq. (\ref{['eq:Benson']}). The solid black line represents the asymptotic dependence Eq.(\ref{['eq:Zeld']}).