Response of Soft Continuous Structures and Topological Defects to a Temperature Gradient

Thermophoresis, which is mass transport induced by a temperature gradient, has recently attracted considerable attention as a new way to transport materials. So far the study has been focused on the transport of discrete structures such as colloidal particles, proteins, and polymers in solutions. However, the response of soft continuous structures such as membranes and gels to a temperature gradient has been largely unexplored. Here we study the behavior of a lamellar phase made of stacked surfactant bilayer membranes under a temperature gradient. We find the migration of membranes towards a low-temperature region, causing the increase in the degree of membrane undulation fluctuations towards that direction. This is contrary to our intuition that the fluctuations are weaker at a lower temperature. We show that this can be explained by temperature-gradient-induced migration of membranes under the topological constraint coming from the connectivity of each membrane. We also reveal that the pattern of an edge dislocation array formed in a wedge-shaped cell can be controlled by a temperature gradient. These findings suggest that application of a temperature gradient provides a novel way to control the organization of soft continuous structures such as membranes, gels, and foams, in a manner essentially different from the other types of fields, and to manipulate topological defects.

Figure 1

(a) Temporal change in the fluorescence intensity $I(t)/I(0)$ for $\varphi =12\mathrm{wt}\%$ in the HT (circles), TG (squares), and LT (triangles) regions under the temperature gradient. (b) Temporal change of $I(t)/I(0)$ in the relaxation process to the equilibrium state at $27°\mathrm{C}$. After $t=1800\min$ under the temperature gradient, the temperature of the HT stage is set to the same as that of the LT stage.

Figure 2

Behavior of the edge dislocation array in a wedge-shaped cell under the temperature gradient ($\nabla T\perp \overrightarrow{s}$) for $\varphi =7\mathrm{wt}\%$. We show phase contrast microscopy images at $t=0\min$ (a), 300 min (b), and 600 min (c) after heating the left stage from $25°\mathrm{C}$ to $32°\mathrm{C}$ while keeping the right stage at $25°\mathrm{C}$. The size of each image is $1\times 1\mathrm{mm}$. The yellow dotted curves in (a)–(c) are drawn to follow the same edge dislocation line. See also Supplemental Material movie 1 [25], which shows the above process from 0 to 1200 min. (d) Temporal change of the $y$ coordinate change of the defect line, $\mathrm{\Delta }s$, caused by the temperature gradient. Circles and squares correspond to $\mathrm{\Delta }s$ at the locations pointed by the left and right arrows in image (a), respectively. For a reference, we also show $\mathrm{\Delta }s$ (open triangles) after homogeneously changing the temperature of the whole system from 25 to $32°\mathrm{C}$.

Figure 3

Behavior of the edge dislocation array in a wedge-shaped cell for $\varphi =12\mathrm{wt}\%$ under the temperature gradient ($\nabla T//\overrightarrow{s}$) after heating the left stage from $25°\mathrm{C}$ to $32°\mathrm{C}$ at $t=0\min$. We schematically show the cross sections of the system for two setups: (a) setup A and (b) setup B. (c) and (d) show trajectories of edge dislocations in the TG region for setup A and B, respectively. We can see that $\lambda$ becomes broader (narrower) for setup A (setup B). See also Supplemental Material movies 2 and 3 [25], which show, respectively, the above processes of setup A and B from 0 to 1200 min. (e) and (f) show the $x$ dependence of the line defect velocities, $v$, at $t=200\min$ for setup A and B, respectively $.\partial v/\partial x$ is positive (negative) for setup A (setup B).