Diffusion accelerates and enhances chirality selection

The diffusion effect on chirality selection in a two-dimensional reaction-diffusion model is studied by Monte Carlo simulations. The model consists of achiral reactants $A$ which turn into either of the chiral products $R$ or $S$ in a solvent of chemically inactive vacancies $V$. The reaction contains a nonlinear autocatalysis as well as a recycling process, and the chiral symmetry breaking is monitored by an enantiomeric excess (ee) $\varphi$. Without dilution a strong nonlinear autocatalysis ensures chiral symmetry breaking. By diluting a diffusionless system, the ee $\varphi$ decreases and a racemic state is recovered below a critical concentration $c_c$. When the diffusion is allowed, the steady value of $\varphi$ increases and $c_c$ decreases. As for the relation between the ee $\varphi$ and the concentration $c$, a formula interpolating between the diffusionless $(D=0)$ and the homogeneous $(D=\infty )$ limits is proposed by incorporating the diffusional enhancement of the concentrations of chiral species. Diffusion also accelerates the development of the chiral order, and the time required to establish the order in a system of a size $L^2$ is inversely proportional to the diffusion constant $D$ as $L^2∕D$.

Figure 1

Various processes with transition probabilities: (a) nonautocatalytic, (b) linearly autocatalytic, (c) nonlinearly autocatalytic reactions with recycling, and (d) a vacancy diffusion. $X$ in the diffusion process (d) represents chemically active species $A$, $R$, or $S$.

Figure 2

Configuration evolution of (a) $R$ and (b) $S$ molecules in a diffusionless system. Times are $k_{0}t=50,250,1750$ from left to right. Parameters are $k_0=\lambda$, $k_1=0$, $k_2=100k_0$ with $c=1$, and the system size is ${100}^2$.

Figure 3

Enantiomeric excess $\mid \varphi \mid$ versus $k_2∕k_0$ at the full occupation $c=1$ in a semilogarithmic plot. A curve represents the temperature dependence of the magnetization [21] of a square Ising model, where a temperature is related to the chemical reaction rates as $k_2∕k_0=\exp (4J∕k_{B}T)-1$.

Figure 4

Analogies of the state transition in the reaction model (above) and the Ising ferromagnet (below). (a) $n=2$ and (b) $n=3$.

Figure 5

Enantiomeric excess $\mid \varphi \mid$ versus $c$ in a diffusionless system with a strong nonlinear autocatalysis $k_2∕k_0=100$.

Figure 6

(a) Enantiomeric excess $\mid \varphi \mid$ versus $c$ at various values of the diffusion coefficient $D∕k_0$. The reaction parameters are $k_2∕k_0=100$, $k_1=0$, and $\lambda =k_0$. A solid curve represents relation (9) for a homogeneous system. Dashed curves represent interpolation (10) at various diffusion coefficients. (b) Diffusion coefficient dependence of the concentration $c$ where the order parameter takes the value $\varphi =0.6$. A curve represents the interpolation (10) with a parameter $A=0.235$. The dotted line represents the homogeneous case with $c(0.6;D=\infty )=0.118$.

Figure 7

Time evolution of the order parameter $\varphi$ starting from an achiral initial condition with only $A$ species with a concentration $c=0.40$. (a) For a system with a size $L=100$ but with different diffusion constant $D$ and (b) for systems with different $D$’s and sizes $L$. The time is scaled as $Dt∕L^2$ in (b).