Emergent spatiotemporal instabilities in reactive spatially extended systems by thermodiffusion

Thermodiffusion or thermophoresis or Soret effect, i.e., mass-transport induced by thermal gradient, has immense application in segregation of species in two or multicomponent gaseous, liquid, or colloidal mixtures. Here, we show that an external thermal gradient can be effectively utilized in creation and modification of patterns in spatially extended systems. We consider Brusselator and chlorine-dioxide iodine malonic acid (CDIMA) reaction-diffusion systems, which follow activator-inhibitor kinetics subjected to an external thermal gradient. We find that the conspicuous interaction of emergent thermodiffusive flux with reaction kinetics and diffusion can lead to various spatiotemporal instabilities in these two models. Specifically, our result reveals formation of Turing-like spatial patterns even for equal diffusivities of the activator and inhibitor components in the Brusselator model under the influence of differential thermodiffusion, whereas formation of such stationary patterns in the CDIMA system from a homogenous stable steady state, which is also stable under differential diffusion, requires the same sign and magnitude of Soret coefficients. However, with equal diffusivities of the components of the CDIMA system and without starch in the medium, our result identifies formation of drifting spiral waves which finally disappears at longer times under the influence of thermodiffusion. We also show formation of propagating patterns of spotlike or stripelike heterogeneity in both the model systems under appropriate conditions. Our study provides a route to pattern formation beyond Turing space and reveals remarkable influence of thermodiffusion to modify the pattern types just by employing an external thermal gradient which also opens up the possibility to set up new related experiments.

Figure 1

Bifurcation diagram obtained from linear stability analysis of Eq. (5) and Eq. (6) with Brusselator kinetics in the absence of thermal gradient ($\mathbf{\nabla }T=0$) and for $D_v=5.0$ and $a=3.0$. Hopf (blue) and Turing (red) curves divide the ($b-D_u$) parameter space into a homogeneous stable region, a homogeneous oscillatory region, and a region of inhomogeneous stationary Turing patterns. The homogeneous stable steady states are denoted by points O1, O2, and O3 on the bifurcation diagram. O1 lying below both the Hopf and Turing curves is diffusionally stable and O2 and O3 inside the Turing region show stationary patterns.

Figure 2

Formation of Turing-like stationary patterns for equal diffusion coefficients of the two components ($D_u=D_v=5.0$): a three-dimensional dispersion diagram for thermodiffusion induced instability is obtained from linear stability analysis for the state O1 for the Soret coefficients $S_{T_u}=0.1,S_{T_v}=0.5$. Shown here are the plots of the real part of the growth rate $\mathrm{Re}(\lambda$) as a function of $k_x$ and $k_y$ for different values of the imposed temperature gradient: (a) $\mathbf{\nabla }T=0$, (b) $\mathbf{\nabla }T=4.0$, and (c) $\mathbf{\nabla }T=5.0$. All the other parameters are the same as mentioned in the main text. The corresponding numerically simulated spatial patterns of concentration of the activator $u$ are depicted in bottom figures (d), (e), and (f), respectively.

Figure 3

Modulation of Turing patterns under thermodiffusion: plot of numerically simulated concentration patterns of activator $u$ corresponding to the parameter set designated by point O2(1,7) are shown by the images (a)–(c) and point O3(1,9) are depicted in images (d)–(f) for different values of the imposed temperature gradient $\mathbf{\nabla }T=0.0, \mathbf{\nabla }T=2.0$, and $\mathbf{\nabla }T=4.0$, respectively. The Soret coefficients are taken as $S_{T_u}=S_{T_v}=0.1$.

Figure 4

Bifurcation diagram obtained from linear stability analysis of Eq. (16) and Eq. (17) in the absence of thermal gradient ($\mathbf{\nabla }T=0$) and for $d=1.6$ and $\sigma =10.0$. Hopf (blue) and Turing (red) curves divide the ($a-b$) parameter space into a homogeneous stable region, a homogeneous oscillatory region, and a region of inhomogeneous stationary Turing patterns. The two homogeneously stable steady states are denoted by points P1 and P2 on the bifurcation diagram. P1 lying above the Turing curve is diffusionally stable and P2 inside the Turing region is unstable.

Figure 5

Three-dimensional dispersion diagram for thermodiffusion induced instability in the presence of diffusive transport: plot of real part of the growth rate $\mathrm{Re}(\lambda$) as a function of $k_x$ and $k_y$ obtained from linear stability analysis for different values of the imposed temperature gradient : (a) $\mathbf{\nabla }T=0$, (b) $\mathbf{\nabla }T=3.0$, (c) $\mathbf{\nabla }T=4.0$, and (d) $\mathbf{\nabla }T=6.0$. The parameters' values are corresponding to point P1 with Soret coefficients taken as $S_{T_u}=S_{T_v}=0.1$.

Figure 6

Thermodiffusion induced stationary patterns for different values of imposed constant temperature gradients. Numerically simulated spatial patterns for the parameter values corresponding to point P1 for (a) $\mathbf{\nabla }T=0$, (b) $\mathbf{\nabla }T=4.0$, (c) $\mathbf{\nabla }T=6.0$, and (d) $\mathbf{\nabla }T=10.0$. A modification of stationary pattern occurs from spot to stripe with increasing values of $\mathbf{\nabla }T$. All other parameter values are kept fixed as mentioned in the main text. Red refers to the higher values of concentration of iodide $I^{-}$.

Figure 7

Thermodiffusion induced transition of stationary patterns from stable spots to stripelike heterogeneous patterns for the parameter values $a=18.0,b=1.4$ with the increase in the applied temperature gradient: (a) $\mathbf{\nabla }T=0.0$, (b) $\mathbf{\nabla }T=2.0$, (c) $\mathbf{\nabla }T=3.0$, and (d) $\mathbf{\nabla }T=5.0$. All the other parameters are kept the same as described in the main text. Red refers to the higher values of concentration of iodide $I^{-}$.

Figure 8

Thermodiffusion induced spatiotemporal instability: moving patterns for the parameter values $a=18.0,b=1.7,S_{T_u}=0.1,S_{T_v}=-0.1$ for applied temperature gradient of values: (a) $\mathbf{\nabla }T=2.0$, (b) $\mathbf{\nabla }T=3.3$, and (c) $\mathbf{\nabla }T=5.0$. Red refers to the higher values of concentration of iodide $I^{-}$. All the other parameters are kept the same as described in the main text.

Figure 9

Thermodiffusion induced spatiotemporal instability: moving patterns for the parameter values $a=18.0,b=1.4,S_{T_u}=0.1,S_{T_v}=-0.1$ with the increase of the applied temperature gradients of values: (a) $\mathbf{\nabla }T=1.0$ (moving spots); (b) $\mathbf{\nabla }T=3$ (moving stripes). Red refers to the higher values of concentration of iodide $I^{-}$. All the other parameters are kept the same as described in the main text.

Figure 10

Spatiotemporal instability for equal diffusivity: (a) plot of $b-a$ bifurcation diagram when diffusion coefficients of activator and inhibitor ions are equal and starch is absent in the medium, i.e., $d=1,\sigma =1$. (b) Snapshot of numerically simulated concentration profile of iodide $\left(I^{-}\right)$ in the absence of thermal gradient for the parameter values $a=18.0,b=1.7,S_{T_u}=0.1,S_{T_v}=0.1$ and simulated in a domain of size ($200\times 200$) with spatial grid $\mathrm{\Delta }x=\mathrm{\Delta }y=0.5$. Successive snapshots of concentration profiles in the presence of applied temperature gradients ($\mathbf{\nabla }T=1.0$) are shown in successive figures from (c) to (h). In this case, we consider a domain size ($200\times 200$) with a larger spatial grid $\mathrm{\Delta }x=\mathrm{\Delta }y=1.0$. All the other parameters are kept the same as described in the main text. Red signifies high concentration of iodide ions.

Figure 11

Thermodiffusion induced spiral waves for opposite signs of Soret coefficients: shown is the snapshot of spiral patterns for the parameter values $a=18.0,b=1.7, \mathbf{\nabla }T=1.0$ and the Soret coefficients are (a) $S_{T_u}=0.1,S_{T_v}=-0.1$; (b) $S_{T_u}=-0.1,S_{T_v}=0.1$. The domain size is ($200\times 200$) with spatial grid $\mathrm{\Delta }x=\mathrm{\Delta }y=0.5$. Red refers to high concentration of iodide ions $I^{-}$. All the other parameters are kept the same as described in the main text.