Processing math: 45%

2022.12.01 — simple model for ¹⁸O → ¹⁶O substitution

consider an

$\oxya$ -precovered surface with an influx of

$\waterb$ . a simple model to describe the rise and decay of the

$\oha$ RAIRS signal consists of the following processes:

the reaction of an impinging $\waterb$ with a surface $\oxya$ to produce a pair $\paira$ on the surface.
the "disproportionation" of 18OH16OH surface pair, which leads to either:
1. the desorption of a $\waterb$ molecule and a surface $\oxya$ atom, or
2. the desorption of a $\watera$ molecule and a surface $\oxyb$ atom.
the reaction of an impinging $\waterb$ with a surface $\oxyb$ to produce a pair $\pairb$ on the surface.
the disproportionation of a $\pairb$ surface pair, leading to the desorption of a $\waterb$ molecule and a surface $\oxyb$ atom.

Assigning

$k$ as the disproportionation reaction rate constant, and

$f$ for the

$\waterb$ flux, we obtain the following set of differential equations governing the reaction dynamics:

$\begin{align} & \frac{d}{dt}\conc{\oxya} & = & -f\conc{\oxya} & & +\frac{k}{2} \conc{\paira} & \\ & \frac{d}{dt}\conc{\oxyb} & = & & -f\conc{\oxyb} & +\frac{k}{2} \conc{\paira} & +k\conc{\pairb} \\ & \frac{d}{dt}\conc{\paira} & = & +f\conc{\oxya} & & -k \conc{\paira} & \\ & \frac{d}{dt}\conc{\pairb} & = & & +f\conc{\oxyb} & & -k\conc{\pairb} \\ \end{align}$ or, in matrix form:

$\begin{equation} \frac{d}{dt} \underbrace{ \begin{bmatrix} \conc{\oxya} \\ \conc{\oxyb} \\ \conc{\paira} \\ \conc{\pairb} \end{bmatrix} }_{\bm{c}} = \underbrace{ \begin{bmatrix} -f & & +\frac{k}{2} & \\ & -f & +\frac{k}{2} & +k \\ +f & & -k & \\ & +f & & -k \\ \end{bmatrix} }_{M} \begin{bmatrix} \conc{\oxya} \\ \conc{\oxyb} \\ \conc{\paira} \\ \conc{\pairb} \end{bmatrix} \end{equation}$ or, symbolically:

$\begin{equation} \dot{\bm{c}} = M \bm{c} \end{equation}$ given some initial surface coverage condition

$\bm{c_o} = (c_1^o,c_2^o,c_3^o,c_4^o)$ and a time

$t_o$ , the coverages

$c'_i$ at a later time

$t' = t_o + T$ are given by:

$\begin{equation} c'_i = \sum_{j,k=1}^4 S_{ij}e^{\lambda_j T}S^{-1}_{jk}c^o_k \end{equation}$ where

$S_{ij}$ and

$\lambda_i$ is the eigenvector decomposition of

$M$ in the following sense:

$\begin{equation} \sum_{k=1}^4 M_{ik} S_{kj} = S_{ij} \lambda_j \end{equation}$ for

$i,j = 1,2,3,4$ . as an example, I plot below the time evolution of the surface coverages assuming

$f=1$ and

$k=2$ and initial surface coverages of

$c_1^o = \conc{\oxya}(t=t_o) = 1$ ,

$c_2^o=c_3^o=c_4^o=0$ :

**above:** model predictions for a surface with an initial precoverage of $\oxya$ . in addition to the coverages, we plot also the expected RAIRS signal for the $\oha$ (blue) and $\ohb$ (red) stretches. the $\oha$ RAIRS signal is presumed to be proportional to $\conc{\paira}$ , while the $\ohb$ RAIRS signal is presumed to be proportional to $\conc{\paira} + 2 \conc{\pairb}$ . the python code used to calculate and plot these results can be found at here. the code can easily be adapted to investigate the model for arbitrary values of $k$ and $f$ .

2022.12.01 — simple model for 18O → 16O substitution

2022.12.01 — simple model for ¹⁸O → ¹⁶O substitution