Thermodiffusion-induced traveling and shock waves in a colloidal solution

The formation of chemical waves in a nonlinear spatially extended system is one of the most fascinating far-from-equilibrium phenomena. An externally imposed thermal gradient in a liquid mixture may induce a concentration gradient generating a thermodynamic cross-flow, which is known as thermal diffusion or the Ludwig-Soret effect. The motion of the components of the mixture is governed by a nonlinear, partial differential equation for the density fraction in space and time. Here, we show that under an externally imposed constant thermal gradient, a traveling wave can emerge in a solution of self-propelled neutral colloid. An exact analytic solution of the spatially extended system is presented in one dimension for a constant thermal gradient to show the time development of a traveling wave. We analyze the effect of a small finite relaxation time of flux, which takes care of the finite inertia of the dispersing colloidal species. While the wave speed remains unaffected, the wave shape is significantly modified by the presence of the finite relaxation time of flux. Our result demonstrates that the traveling wave may reduce to a shock wave provided the product of the square of the wave speed and the relaxation time exactly balances the mass diffusion coefficient. This condition can be achieved by suitably adjusting the velocity of the emergent traveling wave by tuning the value of the constant thermal gradient and maintaining the appropriate boundary condition.

Figure 1

Plot of the traveling wave $U\left(z\right)$ as a function of collective coordinate $z$ for different values of imposed constant temperature gradients for the parameters $v_0=1.0, \alpha =1.0, D=1, D_T=1.0, u_1=2.0,u_2=1.0.$ The arrow represents the direction of the traveling wave.

Figure 2

Plot of speed of the traveling wave as a function of temperature gradient for the parameters $v_0=1.0, \alpha =1.0, D=1, D_T=1.0, u_1=2.0,u_2=1.0$.

Figure 3

Plot of the speed of the traveling wave as a function of temperature gradient for the parameters $v_0=1.0, \alpha =1.0, D=1, D_T=-1.0, u_1=1.0,u_2=2.0$.

Figure 4

A plot of the traveling waves $U\left(z\right)$ as a function of collective coordinate $z$ for different values of imposed temperature gradients for the parameters $v_0=1.0, \alpha =1.0, D=1, D_T=-1.0, u_1=1.0,u_2=2.0$. For $\mathbf{\nabla }T=0.5$, the wave speed is zero. Arrows represent the direction of the movement of the traveling waves.

Figure 5

A plot of the traveling waves $U\left(z\right)$ as a function of collective coordinate $z$ for different values of the finite relaxation time of flux $\tau$ of the dispersing colloidal particles. The other parameters are $v_0=1.0, \alpha =1.0, D=1, D_T=1.0, u_1=2.0,u_2=1.0$, and $\mathbf{\nabla }T=1.0$. The arrow refers to the direction of propagation.

Figure 6

Plot of externally applied constant thermal gradient vs boundary condition in the limit when shock thickness is zero. The other parameters are $v_0=1.0, \alpha =1.0, D=1, D_T=1.0, \tau =0.25$.