Nonequilibrium solid-solid phase transition in a lattice of liquid jets

Solid-solid phase transitions are commonly encountered at the atomic scale in alloys and in superatomic mesoscopic systems of colloidal particles. Here we investigate a solid-solid phase transition occurring at the macroscopic scale between lattices of liquid jets with different symmetries generated by convection in a horizontal layer of a binary liquid mixture. In the absence of a shear stress, upwelling and downwelling jets arrange into two staggered square lattices with a spacing of approximately 3 mm. Applying a shear stress triggers a phase separation of the square patterns into two centered-rectangular lattices drifting into opposite directions, each lattice being made either by upwelling or downwelling jets. This structural phase transition is reversible. The macroscopic nature of the system allows us to investigate the kinetics of the transition by direct visualization with shadowgraphy. The mechanism of the transition depends on the path followed. It occurs through a nucleation and growth mechanism when the shear stress is imposed, and through a martensitic transformation of the lattice when the stress is removed.

Figure 1

Square and centered-rectangular lattices formed by liquid jets during convection in a binary liquid mixture. (a) Schematic setup of the shadowgraph optical diagnostics. The collimated light coming from a led source crosses the sample in the vertical direction and is deflected by the liquid jets, producing a shadow image of their configuration onto a charged coupled device sensor. The sample is sandwiched between a hot and a cold sapphire window whose temperature is set at $T_0+\mathrm{\Delta }T/2$ (bottom) and $T_0-\mathrm{\Delta }T/2$ (top). (b) Square lattice obtained with a temperature difference $\mathrm{\Delta }T=8.0\text{K}$, in the absence of inclination. Dark (bright) spots represent upwelling (downwelling) jets. (c) Centered-rectangular lattice obtained with a temperature difference $\mathrm{\Delta }T=9.0\text{K}$ and an inclination of 24 mrad. (d) Soret mass flow determines the formation of two unstable boundary layers at the top and bottom of the cell. Velocity fluctuations in the boundary layers are amplified by gravity and give rise to upwelling and downwelling jets crossing the sample in the vertical direction. The presence of an inclination of the sample gives rise to an in plane component $g\text{sin}\left(\alpha \right)$ of the acceleration of gravity, which drives the large scale flow of the boundary layers (black arrows). The top boundary layer drifts downwards along the top boundary, and the bottom boundary layer drifts upwards along the bottom boundary.

Figure 2

Time evolution of the transition between the square and centered-rectangular lattice. (a)–(e) Transition from centered-rectangular to square lattices (Supplemental Material movie 1 [39]). (f)–(j) Transition from square to centered-rectangular lattices (Supplemental Material movie 2 [39]). Time $t=0$ corresponds to the instant where an inclination of 24 mrad was either removed or imposed from the layer of liquid to trigger a shear stress. A stationary temperature difference of 10.1 K was applied to the sample during the measurement. Scale bar: 5 mm.

Figure 3

Relation between the primitive lattice vectors of the square and centered rectangular lattices. (a) Stationary lattices obtained under different shear stress conditions. In the presence of inclination, patterns align in the direction of shear. Scale bar: 2 mm. (b) Spatial autocorrelation functions obtained under the conditions shown in (a), averaged over sequences of 400 images recorded at a frame rate of one image per 30 s. a and b represent the primitive vectors and $\theta$ the angle between them. (c) Shear rate plotted as a function of the inclination angle $\alpha$. (d) Moduli of the primitive vectors a (hollow) and b (full), (e) angle between the primitive vectors, and (f) ratio $b/a$ plotted as a function of the inclination angle $\alpha$ for the square (squares) and centered rectangular (diamonds) lattices. The dotted lines in (e) and (f) show the predictions for a triangular phase, while the dashed lines correspond to the predictions for a centered-rectangular one. (g) Martensitic transformation from a centered-rectangular to a square lattice, where alternate rows of jets move coherently into opposite directions to form the new phase. The rectangular unit cell of the centered-rectangular lattice is twice as large as the square unit cell of the square lattice.