Role of nonlinearities in off-critical quenches as described by the Cahn-Hilliard model of phase separation

Sequence determinants and solution conditions underlying liquid to solid phase transition

Life consists of numberless functional biomolecules that exist in various states. Besides well-dissolved phases, biomolecules especially proteins and nucleic acids can form liquid droplets through liquid-liquid phase separation (LLPS). Stronger interactions promote a solid-like state of biomolecular condensates, which are also formerly referred to as detergent-insoluble aggregates. Solid-like condensates exist in vivo physiologically and pathologically, and their formation has not been fully understood. Recently, more and more research has proven that liquid to solid phase transition (LST) is an essential way to form solid condensates. In this review, we summarized the regions in the sequence that have different impacts on phase transition and emphasized that the LST is affected by its sequence characteristics. Moreover, increasing evidence unveiled that LST is affected by various solution conditions. We discussed solution conditions like protein concentration, pH, ATP, ions, and small molecules in a solution. Methods have been established to study these solid phase components. Here, we summarized low-throughput experimental techniques and high-throughput omics methods in the study of the LST.

Beyond Cahn-Hilliard-Cook ordering theory: early time behavior of spatial-symmetry-breaking phase transition kinetics

We extend the early time ordering theory of Cahn, Hilliard, and Cook (CHC) so that our generalized theory applies to solid-to-solid transitions. Our theory involves spatial-symmetry breaking (the initial phase contains a symmetry not present in the final phase). The predictions of our generalization differ from those of the CHC theory in two important ways: exponential growth does not begin immediately following the quench and the objects that grow exponentially are not necessarily Fourier modes. Our theory is consistent with simulation results for the long-range antiferromagnetic Ising model.

Nonlinear relaxation patterns in the Cahn–Hilliard equation: An exact solution

We consider the 1-D Cahn-Hilliard equation with the order parameter v and derive an equation for a modified order parameter g such that g''=v'. The new equation allows for separation of variables. This yields exact solutions for v expressed in terms of generalized hypergeometric functions. These solutions have an infinite gradient at their zeros and the first three derivatives of zero at their extrema. The amplitude of these patterns decreases as the inverse square root of time. It is suggested that the phenomenon of compartmentalization of evolving structures typically observed in evolutionary models of the Cahn-Hilliard type is a manifestation of relaxation patterns similar to those derived in this paper.

Phase ordering with a global conservation law: Ostwald ripening and coalescence

Globally conserved phase ordering dynamics is investigated in systems with short range correlations at t=0. A Ginzburg-Landau equation with a global conservation law is employed as the phase field model. The conditions are found under which the sharp-interface limit of this equation is reducible to the area-preserving motion by curvature. Numerical simulations show that, for both critical and off-critical quench, the equal-time pair correlation function exhibits dynamic scaling, and the characteristic coarsening length obeys l(t) approximately t(1/2). For the critical quench, our results are in excellent agreement with earlier results. For off-critical quench (Ostwald ripening) we investigate the dynamics of the size distribution function of the minority phase domains. The simulations show that, at large times, this distribution function has a self-similar form with growth exponent 1/2. The scaled distribution, however, strongly differs from the classical Wagner distribution. We attribute this difference to coalescence of domains. A theory of Ostwald ripening is developed that takes into account binary coalescence events. The theoretical scaled distribution function agrees well with that obtained in the simulations.

Two-dimensional model of phase segregation in liquid binary mixtures

The hydrodynamic effects on the late stage kinetics of phase separation in liquid mixtures is studied using the model H. Mass and momentum transport are coupled via a nonequilibrium body force, which is proportional to the Peclet number alpha, i.e., the ratio between convective and diffusive molar fluxes. Numerical simulations based on this theoretical model show that phase separation in low viscosity, liquid binary mixtures is mostly driven by convection, thereby explaining the experimental findings that the process is fast, with the typical size of single-phase domains increasing linearly with time. However, as soon as sharp interfaces form, the linear growth regime reaches an end, and the process appears to be driven by diffusion, although the condition of local equilibrium is not reached. During this stage, the typical size of the nucleating drops increases like t(n), where 1/3< n <1/2, depending on the value of the Peclet number. As the Peclet number increases, the transition between convection- and diffusion-driven regimes occurs at larger times, and therefore for larger sizes of the nucleating drops.