Scaling of submicrometric Turing patterns in concentrated growing systems

The wavelength of a periodic spatial structure of Turing type is an intrinsic property of the considered reaction-diffusion dynamics and we address the question of its control at the microscopic scale for given dynamical parameters. The direct simulation Monte Carlo method, initially introduced to simulate particle dynamics in rarefied gases, is adapted to the simulation of concentrated solutions. We perform simulations of a submicrometric Turing pattern with appropriate boundary conditions and show that taking into account the role of the solvent in the chemical mechanism allows us to control the wavelength of the structure. Typically, doubling the concentration of the solute leads to decreasing the wavelength by two. The results could be used to design materials with controlled submicrometric properties in chemical engineering. They could also be considered as a possible interpretation of proportion preservation of embryos in morphogenesis.

Figure 1

(a) Scaled positive eigenvalue ${\mu }_{+}/k_2$ of the linear stability operator given in Eq. (23) versus square of the scaled Fourier mode $D_A{q}^2/k_2$. Red dotted line: dispersion relation in the high dilution limit ($\delta =0$). Black solid line: dispersion relation for $\delta =0.05$. The square of the most unstable modes $q_{\max }^2$ are indicated by vertical dashed lines. (b) Maximum value of the scaled positive eigenvalue ${\mu }_{\max }/k_2$ versus the deviation $\delta$ from the dilution limit. The parameters take the following values: $\frac{k_1}{k_2}=2.9\times {10}^4, \frac{k_3}{k_2}=2.2\times {10}^4, \frac{k_{-3}}{k_2}=8.8\times {10}^6, d=10$.

Figure 2

Concentration profiles deduced from the numerical solution of Eqs. (6) and (7) (blue) and number of particles per cell in simulations (red) versus scaled spatial coordinate $x\sqrt[]{\frac{k_2}{D_A}}$ for Eqs. (1, 2, 3, 4), for a given deviation $\delta =0.05$ from the dilution limit, at scaled time $k_{2}t=6986$. Snapshot of the profiles for species $A$ (a) and $B$ (b). The value of the other parameters is given in Sec. 3.

Figure 3

Macroscopic prediction of the steady state concentrations $A^T$ (solid line) given by Eq. (A24) and $B^T$ (dotted line) deduced from Eq. (A25) versus the deviation $\delta$ from the dilution limit. Squares (triangles, respectively) give the mean number of $A$ ($B$, respectively) particles per cell in the region of the simulated Turing pattern. For $\delta \ne 0$, the mechanism given in Eqs. (1, 2, 3, 4) is used whereas the mechanism given in Eqs. (11)–(13) is used for $\delta =0$. The value of the other parameters is given in Sec. 3.

Figure 4

Macroscopic prediction of the scaled wavelength of the Turing pattern given by Eqs. (24), (25), (27), and (31) (solid line) and scaled wavelength deduced from the simulations (squares) versus deviation $\delta$ from the dilution limit. For $\delta \ne 0$, the mechanism given in Eqs. (1, 2, 3, 4) is used, whereas the mechanism given in Eqs. (11)–(13) is used for $\delta =0$. The value of the other parameters is given in Sec. 3.

Figure 5

Same as described in the caption of Fig. 4 but without scaling of the wavelength.