Overview: Understanding nucleation phenomena from simulations of lattice gas models

Low-temperature nucleation rate calculations using the N-Fold way

We present a numerical study to determine nucleation rates for magnetization reversal within the Ising model (lattice gas model) in the low-temperature regime, a domain less explored in previous research. To achieve this, we implemented the N-Fold way algorithm, a well-established method for low-temperature simulations, alongside a novel, highly efficient cluster identification algorithm. Our method can access nucleation rates up to 50 orders of magnitude lower than previously reported results. We examine three cases: homogeneous pure system, system with static impurities, and system with mobile impurities, where impurities are defined as sites with zero interactions with neighboring spins (the spin value of impurities is set to 0). Classical nucleation theory holds across the entire temperature range studied in the paper, for both the homogeneous system and the static impurity case. However, in the case of mobile impurities, the umbrella sampling technique appears ineffective at low mobility values. These findings provide valuable insights into nucleation phenomena at low temperatures, contributing to theoretical and experimental understanding.

A scale-invariant log-normal droplet size distribution below the critical concentration for protein phase separation

Many proteins have been recently shown to undergo a process of phase separation that leads to the formation of biomolecular condensates. Intriguingly, it has been observed that some of these proteins form dense droplets of sizeable dimensions already below the critical concentration, which is the concentration at which phase separation occurs. To understand this phenomenon, which is not readily compatible with classical nucleation theory, we investigated the properties of the droplet size distributions as a function of protein concentration. We found that these distributions can be described by a scale-invariant log-normal function with an average that increases progressively as the concentration approaches the critical concentration from below. The results of this scaling analysis suggest the existence of a universal behaviour independent of the sequences and structures of the proteins undergoing phase separation. While we refrain from proposing a theoretical model here, we suggest that any model of protein phase separation should predict the scaling exponents that we reported here from the fitting of experimental measurements of droplet size distributions. Furthermore, based on these observations, we show that it is possible to use the scale invariance to estimate the critical concentration for protein phase separation.

Finding the differences: Classical nucleation perspective on homogeneous melting and freezing of hard spheres

By employing brute-force molecular dynamics, umbrella sampling, and seeding simulations, we investigate homogeneous nucleation during melting and freezing of hard spheres. We provide insights into these opposing phase transitions from the standpoint of classical nucleation theory. We observe that melting has both a lower driving force and a lower interfacial tension than freezing. The lower driving force arises from the vicinity of a spinodal instability in the solid and from a strain energy. The lower interfacial tension implies that the Tolman lengths associated with melting and freezing have opposite signs, a phenomenon that we interpret with Turnbull's rule. Despite these asymmetries, the nucleation rates for freezing and melting are found to be comparable.

Properties of water and argon clusters developed in supersonic expansions

Using adiabatic molecular dynamics coupled with the fluid dynamics equations, we model nucleation in an expanding beam of water vapor and argon on a microsecond scale. The size distribution of clusters, their temperature, and pickup cross sections in dependence on velocity are investigated and compared to the geometric cross sections and the experiment. The clusters are warmer than the expanding gas because of the time scale of relaxation processes. We also suggest that their translational and rotational kinetic energies are modified due to evaporative cooling. The pickup cross sections determined for the final clusters using molecules of the same kind increase with decreasing velocity, still obeying the (a+bN1/3)2 law.

Mesoscale perspective on the Tolman length

We demonstrate that the multiphase Shan-Chen lattice Boltzmann method (LBM) yields a curvature dependent surface tension σ as computed from three-dimensional hydrostatic droplets and bubbles simulations. Such curvature dependence is routinely characterized, at first order, by the so-called Tolman length δ. LBM allows one to precisely compute σ at the surface of tension R_{s} and determine the Tolman length from the coefficient of the first order correction. The corresponding values of δ display universality for different equations of state, following a power-law scaling near the critical temperature. The Tolman length has been studied so far mainly via computationally demanding Molecular Dynamics simulations or by means of Density Functional Theory approaches playing a pivotal role in extending Classical Nucleation Theory. The present results open a hydrodynamic-compliant mesoscale arena, in which the fundamental role of the Tolman length, alongside real-world applications to cavitation phenomena, can be effectively tackled. All the results can be independently reproduced through the "idea.deploy" framework.

Pressure-tensor method evaluation of the interfacial tension between Gay-Berne isotropic fluid and a smooth repulsive wall

The interfacial properties of a confined thermotropic liquid crystalline material are investigated using a molecular dynamics simulation technique. The pairwise interaction among the soft ellipsoidal particles is modeled by the Gay-Berne (GB) potential. The GB ellipsoids are confined by two soft, smooth, repulsive walls defined by the Weeks-Chandler-Andersen (WCA) potential. The aperiodic confinement due to walls makes the system mechanically anisotropic. Hence using the pressure-tensor method, the interfacial tension of an interface between the bulk isotropic (I) phase and WCA wall at various number densities (ρ) is calculated. From the pressure tensor and orientational order profiles, the arrangement of ellipsoids in the bulk and the vicinity of the wall is determined. The effect of system size and the wall-particle interaction strength (ε_W) on is also analyzed by varying the system size and ε_W.

Nucleation rate in the two dimensional Ising model in the presence of random impurities

Nucleation phenomena are ubiquitous in nature and the presence of impurities in every real and experimental system is unavoidable. Yet numerical studies of nucleation are nearly always conducted for entirely pure systems. We have studied the behaviour of the droplet free energy in two dimensional Ising model in the presence of randomly positioned static and dynamic impurities. We have shown that both the free energy barrier height and critical nucleus size monotonically decreases with increasing the impurity density for the static case. We have compared the nucleation rates obtained from the Classical Nucleation Theory and the Forward Flux Sampling method for different densities of the static impurities. The results show good agreement. In the case of dynamic impurities, we observe preferential occupancy of the impurities at the boundary positions of the nucleus when the temperature is low. This further boosts enhancement of the nucleation rate due to lowering of the effective interfacial free energy.

Spinodal-assisted nucleation in the two-dimensional q-state Potts model with short-to-long-range interactions

We study homogeneous nucleation in the two-dimensional q-state Potts model for q=3,5,10,20 and ferromagnetic couplings J_{ij}∝Θ(R-|i-j|) by means of Monte Carlo simulations employing heat bath dynamics. Metastability is induced in the low-temperature phase through an instantaneous quench of the magnetic field coupled to one of the q spin states. The quench depth is adjusted, depending on the value of temperature T, interaction range R, and number of states q, in such a way that a constant nucleation time is always obtained. In this setup, we analyze the crossover between the classical compact droplet regime occurring in the presence of short-range interactions R∼1 and the long-range regime R≫1 where the properties of nucleation are influenced by the presence of a mean-field spinodal singularity. We evaluate the metastable susceptibility of the order parameter as well as various critical droplet properties, which along with the evolution of the quench depth as a function of q,T and R are then compared with the field theoretical predictions valid in the large R limit to find the onset of spinodal-assisted nucleation. We find that, with a mild dependence of the values of q and T considered, spinodal scaling holds for interaction ranges R≳8-10 and that signatures of the presence of a pseudospinodal are already visible for remarkably small interaction ranges R∼4-5. The influence of spinodal singularities on the occurrence of multistep nucleation is also discussed.

Weak scaling of the contact distance between two fluctuating interfaces with system size

A pair of flat parallel surfaces, each freely diffusing along the direction of their separation, will eventually come into contact. If the shapes of these surfaces also fluctuate, then contact will occur when their centers-of-mass remain separated by a nonzero distance ℓ. An example of such a situation is the motion of interfaces between two phases at conditions of thermodynamic coexistence, and in particular the annihilation of domain wall pairs under periodic boundary conditions. Here we present a general approach to calculate the probability distribution of the contact distance ℓ and determine how its most likely value ℓ^{*} depends on the surfaces' lateral size L. Using the Edward-Wilkinson equation as a model for interfaces, we demonstrate that ℓ^{*} scales weakly with system size, i.e., the dependence of ℓ^{*} on L for both (1+1)- and (2+1)-dimensional interfaces is such that lim_{L→∞}(ℓ^{*}/L)=0. In particular, for (2+1)-dimensional interfaces ℓ^{*} is an algebraic function of logL, a result that is confirmed by computer simulations of slab-shaped domains formed under periodic boundary conditions. This weak scaling implies that such domains remain topologically intact until ℓ becomes very small compared to the lateral size of the interface, contradicting expectations from equilibrium thermodynamics.

Derivation of an exact, nonequilibrium framework for nucleation: Nucleation is a priori neither diffusive nor Markovian

We discuss the structure of the equation of motion that governs nucleation processes at first order phase transitions. From the underlying microscopic dynamics of a nucleating system, we derive by means of a nonequilibrium projection operator formalism the equation of motion for the size distribution of the nuclei. The equation is exact, i.e., the derivation does not contain approximations. To assess the impact of memory, we express the equation of motion in a form that allows for direct comparison to the Markovian limit. As a numerical test, we have simulated crystal nucleation from a supersaturated melt of particles interacting via a Lennard-Jones potential. The simulation data show effects of non-Markovian dynamics.

Temperature and initial composition dependence of pattern formation and dynamic behavior in phase separation under deep-quenched conditions

Phase separation of SCN-H2O ([CH2CN]2-H2O) transparent solutions is simulated in two dimensions and the effects of quenching temperature and initial composition on the pattern formation and dynamic behavior of the second phase are examined via Minkowski functionals. The simulation is based on model H where the molar free energy of the SCN-H2O solution is obtained by the CALPHAD approach. We find that the composition and temperature do not affect the exponent in the domain growth law, where the average domain size with time yields R(t)-t n . However, they influence the pattern formation and dynamic behavior of the second phase in phase separation. Lower temperature leads to a finer bicontinuous structure in spinodal decomposition and promotes the nucleation rate, which accelerates the phase separation and results in more liquid droplets with smaller size. As the initial composition diverges from the critical value, the spatial patterns change gradually from bicontinuous into a droplet-like structure. When the initial composition is closer to the critical value, for spinodal decomposition, the diffusion-driven growth lasts for a longer time and the average domain size of liquid droplets is larger. For nucleation-driven growth, in contrast, the single phase separates more quickly and the average size of liquid droplets is smaller.

Universal scaling in first-order phase transitions mixed with nucleation and growth

Matter exhibits phases and their transitions. These transitions are classified as first-order phase transitions (FOPTs) and continuous ones. While the latter has a well-established theory of the renormalization group, the former is only qualitatively accounted for by classical theories of nucleation, since their predictions often disagree with experiments by orders of magnitude. A theory to integrate FOPTs into the framework of the renormalization-group theory has been proposed but seems to contradict with extant wisdom and lacks numerical evidence. Here we show that universal hysteresis scaling as predicted by the renormalization-group theory emerges unambiguously when the theory is combined intimately with the theory of nucleation and growth in the FOPTs of the paradigmatic two-dimensional Ising model driven by a linearly varying externally applied field below its critical point. This not only provides a new method to rectify nucleation theories, but also unifies the theories for both classes of transitions and FOPTs can be studied using universality and scaling similar to their continuous counterpart when nucleation and growth are properly taken into account.

Aspects of nucleation on curved and flat surfaces

We investigate the energetics of droplets sourced by the thermal fluctuations in a system undergoing a first-order transition. In particular, we confine our studies to two dimensions with explicit calculations in the plane and on the sphere. Using an isoperimetric inequality from the differential geometry literature and a theorem on the inequality's saturation, we show how geometry informs the critical droplet size and shape. This inequality establishes a "mean field" result for nucleated droplets. We then study the effects of fluctuations on the interfaces of droplets in two dimensions, treating the droplet interface as a fluctuating line. We emphasize that care is needed in deriving the line curvature energy from the Landau-Ginzburg energy functional and in interpreting the scalings of the nucleation rate with the size of the droplet. We end with a comparison of nucleation in the plane and on a sphere.

Direct Molecular Dynamics Simulation of Nucleation during Supersonic Expansion of Gas to a Vacuum

We develop a methodology for direct molecular-level simulation of adiabatic expansion of gas through a small orifice to a vacuum. The gas attains supersonic speeds, cools, and nucleates. The proposed approach combines equations of frictionless fluid dynamics with molecular dynamics simulation in an expanding periodic box. There are two key components of the proposed algorithm: (i) a time-reversible integrator tailored to an expanding system, and (ii) an iterative procedure employed to satisfy the condition of steady flow. For a conical nozzle (opening angle of 60°), the simulations with argon and water vapor predict cluster sizes in agreement with the experiment. Clusters of irregular shapes observed in the experiment [J. Lengyel et al. Phys. Rev. Lett. 2014, 112, 113401] are not reproduced. The role of friction, turbulence, and sonic boom originating at the sharp nozzle edge is discussed.

Heterogeneous nucleation of a droplet pinned at a chemically inhomogeneous substrate: A simulation study of the two-dimensional Ising case

Heterogeneous nucleation is studied by Monte Carlo simulations and phenomenological theory, using the two-dimensional lattice gas model with suitable boundary fields. A chemical inhomogeneity of length b at one boundary favors the liquid phase, while elsewhere the vapor is favored. Switching on the bulk field H_b favoring the liquid, nucleation and growth of the liquid phase starting from the region of the chemical inhomogeneity are analyzed. Three regimes occur: for small fields, H_b<H_b^crit, the critical droplet radius is so large that a critical droplet having the contact angle θ_c required by Young's equation in the region of the chemical inhomogeneity does not yet "fit" there since the baseline length of the circle-cut sphere droplet would exceed b. For H_b^crit<H_b<H_b^*, such droplets fit inside the inhomogeneity and are indeed found in simulations with large enough observation times, but these droplets remain pinned to the chemical inhomogeneity when their baseline has grown to the length b. Assuming that these pinned droplets have a circle cut shape and effective contact angles θ_eff in the regime θ_c < θ_eff < π/2, the density excess due to these droplets can be predicted and is found to be in reasonable agreement with the simulation results. On general grounds, one can predict that the effective contact angle θ_eff and the excess density of the droplets, scaled by b, are functions of the product bH_b but do not depend on both variables separately. Since the free energy barrier for the "depinning" of the droplet (i.e., growth of θ_eff to π - θ_c) vanishes when θ_eff approaches π/2, in practice only angles θ_eff up to about θ_eff^max≃70° were observed. For larger fields (H_b>H_b^*), the droplets nucleated at the chemical inhomogeneity grow to the full system size. While the relaxation time for the growth scales as τ_G∝H_b^-1, the nucleation time τ_N scales as lnτ_N∝H_b^-1. However, the prefactor in the latter relation, as evaluated for our simulations results, is not in accord with an extension of the Volmer-Turnbull theory to two-dimensions, when the theoretical contact angle θ_c is used.

Free-energy barriers for crystal nucleation from fluid phases

Monte Carlo simulations of crystal nuclei coexisting with the fluid phase in thermal equilibrium in finite volumes are presented and analyzed, for fluid densities from dense melts to the vapor. Generalizing the lever rule for two-phase coexistence in the canonical ensemble to finite volume, "measurements" of the nucleus volume together with the pressure and chemical potential of the surrounding fluid allows us to extract the surface free energy of the nucleus. Neither the knowledge of the (in general nonspherical) nucleus shape nor of the angle-dependent interface tension is required for this task. The feasibility of the approach is demonstrated for a variant of the Asakura-Oosawa model for colloid-polymer mixtures, which form face-centered cubic colloidal crystals. For a polymer to colloid size ratio of 0.15, the colloid packing fraction in the fluid phase can be varied from melt values to zero by the variation of an effective attractive potential between the colloids. It is found that the approximation of spherical crystal nuclei often underestimates actual nucleation barriers significantly. Nucleation barriers are found to scale as ΔF^{*}=(4π/3)^{1/3}γ[over ¯](V^{*})^{2/3}+const with the nucleus volume V^{*}, and the effective surface tension γ[over ¯] that accounts implicitly for the nonspherical shape can be precisely estimated.