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. 2023 Aug 28;13(1):14010.
doi: 10.1038/s41598-023-39741-y.

A new approach for simulating inhomogeneous chemical kinetics

Georgia Bradshaw  1 , Mel O'Leary  2   3 , Arthur S F Purser  2 , Balder Villagomez-Bernabe  2   3   4 , Cyrus Wyett  2   3 , Frederick Currell  2   3 , Marcus Webb  5
Affiliations

A new approach for simulating inhomogeneous chemical kinetics

Georgia Bradshaw et al. Sci Rep. .

Abstract

In this paper, inhomogeneous chemical kinetics are simulated by describing the concentrations of interacting chemical species by a linear expansion of basis functions in such a manner that the coupled reaction and diffusion processes are propagated through time efficiently by tailor-made numerical methods. The approach is illustrated through modelling [Formula: see text]- and [Formula: see text]-radiolysis in thin layers of water and at their solid interfaces from the start of the chemical phase until equilibrium was established. The method's efficiency is such that hundreds of such systems can be modelled in a few hours using a single core of a typical laptop, allowing the investigation of the effects of the underlying parameter space. Illustrative calculations showing the effects of changing dose-rate and water-layer thickness are presented. Other simulations are presented which show the approach's capability to solve problems with spherical symmetry (an approximation to an isolated radiolytic spur), where the hollowing out of an initial Gaussian distribution is observed, in line with previous calculations. These illustrative simulations show the generality and the computational efficiency of this approach to solving reaction-diffusion problems. Furthermore, these example simulations illustrate the method's suitability for simulating solid-fluid interfaces, which have received a lot of experimental attention in contrast to the lack of computational studies.

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Conflict of interest statement

The authors declare no competing interests.

Figures

Figure 1
Figure 1
Flow chart showing order of processes for a single iteration of the Strang split-step method. One loop represents a single complete timestep of length Δt for the reaction-diffusion equation (two diffusion steps of length Δt/2 and one reaction step of length Δt). The graphs in each box are a schematic to represent whether the concentrations are stored as values or coefficients at that point in the process.
Figure 2
Figure 2
Function f(x)=exp-(x-0.5)20.02 displayed in value space [left] and coefficient space [right]. Left: the function f(x) and its values on a grid of 25 equally spaced points. Right: the coefficients cn for n=0,1,…,24 obtained by taking a discrete transform of the 25 values on the left figure. The right figure demonstrates how only the first few terms of the spectral sum are important, because the coefficients rapidly decay to zero (a consequence of the smoothness of f). This means that an accurate approximation can be obtained with relatively few coefficients. Applying an inverse discrete transform on the coefficients in the figure on the right will produce the 25 values of the left figure.
Figure 3
Figure 3
Simulations of incident alpha radiation on thin water layers. Left: a demonstration of the inhomogeneous reaction system reaching equilibrium. The system models incident α radiation with a dose rate of 10-2 Gy ns-1 uniformly in a thin film of water of thickness L=0.5 nm, with the dynamics tracked at x=0.5nm (water gas boundary) (see Supplementary Fig. S1 online for sample simulations at varying time steps). Right: results from 160 simulations of incident alpha radiation, with various dose rates, in a thin film of water of different thicknesses ranging from typical dose rates up by over 12 orders of magnitude to illustrate the wide parameter space available. The plotted quantity is the concentration of H2 at the water-gas boundary at equilibrium, as a function of the thickness of water layer for various dose rates. The black crosses show the data points gathered from the simulations, and the gradient triangle in the bottom right verifies that the gradients of the plots show a quadratic dependence between yield of H2 at equilibrium and water thickness L∈ [0.25 nm, 5 nm]. G-values were converted to reaction rate coefficients for the zeroth order reactions used to simulate the constant input of chemical species as a result of irradiation in the physical stage (see Supplementary Table S1 in Supplementary Data online).
Figure 4
Figure 4
Plots of the concentration of H2 at the water-gas boundary at equilibrium, as a function of the thickness of water layer for various dose rates for Spinks and Woods [left], and Kreipl et al. [right]. The black crosses show the data points gathered from the simulations, and the gradient triangle in the bottom right shows that for gamma radiation, the quadratic dependence between yield of H2 at equilibrium and water thickness L does not hold universally. Reaction rate coefficients for the zeroth order reactions used to simulate the constant input of chemical species as a result of irradiation in the physical stage were taken from the G-values from: Table 7.4 in Spinks and Woods using ‘γ and electron radiation with 0.1–20 MeV energies, pH 3–11’ for the left figure, and Table 5 in Kreipl et al. using the bottom row for the right figure. Diffusion coefficients, reaction network and reaction rates are taken from Tables 2 and 3 in Kreipl et al., respectively.
Figure 5
Figure 5
Early spatially inhomogenous time evolution of solvated electrons around an ionising spur. Left: rapid depletion close to the origin due to a fast fourth-order recombination reaction second order in eaq·- from a Gaussian initial condition based on Schwarz’ modified prescribed diffusion model, leading to a central minimum forming at 50 ps. Right: distribution with a prescribed central minimum due to a prechemical ballistic transit leading to a slower decay, later proposed by Trumbore. Similar behaviour is seen here using spectral methods as in Burns et al.’s FEM implementation. Simulations were run assuming an ionising spur of energy 100 eV, with 40 coefficients in the radially symmetric basis functions of the form in Eq. (6), over 4606 timesteps. Diffusion coefficients and reaction rate constants were taken from Burns et al.. Simulations were completed in 43 s on a commercial laptop.
Figure 6
Figure 6
Evolution of relative G of ·OH, the principle mediator of radiation damage in water radiolysis, from 0-3 ns as a result of initial spatial distributions. Deviation from experiment partly arises due to the breakdown of continuous distributions at the limit of low initial radical numbers per spur noted in and failure to account for zero-time reaction. Our simulations in the figure above agree with the results of a more intensive but lower resolution FEM implementation of the same models (Ref., Fig. 21). Parameters used as in Fig. 5.
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