Erasing no-man's land by thermodynamically stabilizing the liquid-liquid transition in tetrahedral particles

Abstract

One of the most controversial hypotheses for explaining the origin of the thermodynamic anomalies characterizing liquid water postulates the presence of a metastable second-order liquid-liquid critical point [1] located in the "no-man's land" [2]. In this scenario, two liquids with distinct local structure emerge near the critical temperature. Unfortunately, since spontaneous crystallization is rapid in this region, experimental support for this hypothesis relies on significant extrapolations, either from the metastable liquid or from amorphous solid water [3, 4]. Although the liquid-liquid transition is expected to feature in many tetrahedrally coordinated liquids, including silicon [5], carbon [6] and silica, even numerical studies of atomic and molecular models have been unable to conclusively prove the existence of this transition. Here we provide such evidence for a model in which it is possible to continuously tune the softness of the interparticle interaction and the flexibility of the bonds, the key ingredients controlling the existence of the critical point. We show that conditions exist where the full coexistence is thermodynamically stable with respect to crystallization. Our work offers a basis for designing colloidal analogues of water exhibiting liquid-liquid transitions in equilibrium, opening the way for experimental confirmation of the original hypothesis.

Figure 1. Tetramer model

Each tetramer consists of five hard spheres of diameter σ: a core at the centre and four “arms” oriented along a tetrahedral geometry. The centres of the arms are located a distance L from the centre of the core. The vector connecting the centre of the core and each arm is allowed to fluctuate within a variable angle ϕ from its ideal tetrahedral position. Additionally, there is an attractive patch on the surface of each arm characterized by an angular width cos θ = 0.95 and a bond range δ = 0.251σ, pointing directly away from the core. a) A two-dimensional cartoon of the model, showing a central sphere with three (out of four) arms, indicating the relevant angles and distances. b) The average configuration (solid) and typical fluctuations (partly transparent) for a tetramer with cos ϕ = 0.9.

Figure 2. Conditions for liquid-liquid phase separation: arm length

Diagram showing the region in which LL phase separation and crystallization occur as a function of arm length L and temperature T for cos ϕ = 0.9. The state points where LL phase separation was observed are denoted by red crosses, while the blue dots indicate points where this was not the case. The solid line indicates the LL critical temperature T_c, as determined from the SUS simulations. The dashed line shows the equilibrium crystallization temperature T_x at ρ_c, obtained from free-energy calculations performed for L/σ = 0.5, 0.625, 0.75, 0.8375, and 1. The gray circles indicate spontaneous crystallization into a BCC crystal in the density region sampled by the LL critical fluctuations.

Figure 3. Conditions for liquid-liquid phase separation: bond flexibility

Diagram showing the phase behaviour at ρ_c, the LL critical density, in the bond flexibility (cos ϕ) vs. temperature (T) plane for L = 0.5σ. The indicated regions are the same as those shown in Fig. 2. The red crosses and blue dots indicate points where liquid-liquid phase separation was or was not detected in the SUS simulations, while the gray circles indicate spontaneous crystallization. The dashed and solid black lines are based on free-energy calculations performed at cos ϕ = 0.8, 0.825, 0.8375, 0.85, 0.875, 0.9, and 1. The four vertical dotted lines denote the bond flexibility values for which the phase diagrams in Fig. 4 are drawn.

Figure 4. Effects of bond flexibility on the phase behaviour

Phase diagrams for different values of the bond flexibility cos ϕ, for L = 0.5σ. Symbols indicate state points where phase coexistences were calculated, with red symbols denoting metastable phase coexistences. The light blue regions indicate a stable LL coexistence, and the red shaded region in panel (d) shows the metastable liquid-liquid coexistence region. The dotted lines in (c) and (d) delimiting the BCC region at high T indicate the expected behaviour of the fluid-BCC coexistence. Crystal structures with a higher density than the BCC phase were not investigated, but are not expected in the density range where the liquid-liquid phase separation occurs. Note that in panel (b) no crystal phases compete with the LL phase separation, at any temperature.