Scaling of submicrometric Turing patterns in concentrated growing systems

Abstract

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 solution leads to decreasing the wavelength by two. The results could be used to design materials with controled submicrometric properties in chemical engineering. They could also be considered as a possible interpretation of proportion preservation of embryos in morphogenesis.

Termination Mechanisms of Turing Patterns in Growing Systems

“…2, the decrease in the maximum value µ max of the eigenvalue as the diffusion coefficient ing, a local increase of temperature could be used to induce a local increase of the rate constants. However, local increase of confinement or temperature is susceptible to simultaneously affect several dynamical parameters [13,22,27,28,29,30,31,32]. Whereas a decrease of D B is desired to destabilize the Turing pattern while decreasing its wavelength, a simultaneous decrease of D A would be detrimental.…”

“…In other words, a Turing pattern emerges by local self-activation and lateral inhibition [14]. The concepts developed to model living systems provide a source of inspiration in chemical engineering [15,16,17,18,19,21,22]. However, standard models of Turing patterns generate structures in infinite systems and the question of the termination of a striped structure in a finite system arises in a perspective of biomimetism in material science.…”

“…Specifically, it is desirable to find experimentally achievable conditions creating a finite-size structure, whose growth stops by itself with decreasing oscillation amplitude and respects the decrease of the wavelength during the termination process. To this aim, we use an as simple as possible reaction-diffusion model [23] admitting a Turing structure developing behind a propagating wave front and examine the effect of all parameters on both the stability and the wavelength of the structure [5,22]. We already used a stochastic approach to a Turing pattern [23] and showed that, contrary to intuition, the internal fluctuations may have a constructive effect and stabilize the structure outside the domain of stability predicted by a deterministic description.…”

DSMC simulations of Turing patterns in concentrated growing systems

“…Indeed, considering the step B + S 2S would lead to the correct term proportional to −BS but to an erroneous term proportional to S 2 instead of a constant in the rate equation of species B. Eventually, the chemical scheme is modified as follows [6] A + S…”

“…We use a trick to simulate ternary reactions that already proved to be satisfactory [4,5]: The trimolecular interaction is decomposed into a standard binary collision and reaction is accepted depending on the presence of the third reactive species in the neighborhood of the colliding pair. We refer to reference [6] for a detailed presentation of the adapted DSMC algorithm. As a result, the diffusion coefficients of the components and the rate constant of the ternary reaction depend on the total density and consequently the departure from dilute conditions, implying a scaling of the simulation results before the 31st International Symposium on Rarefied Gas Dynamics AIP Conf.…”