Nonequilibrium Phase Research Articles

Motility-induced phase separation of soft active Brownian particles

Motility-induced phase separation (MIPS) is the hallmark of non-equilibrium phase transition in active matter. Here, by means of Brownian dynamics simulations, we determine the phase behavior and the critical point for phase separation induced by motility of a two-dimensional system of soft active Brownian particles, whose interaction is modeled by the generalized purely repulsive Weeks–Chandler–Andersen potential. We refer to this model as soft active Brownian particles. We determine and analyze the influence of particle softness on the MIPS and show that the liquid–gas coexistence region is wider, the softer the interparticle interactions becomes. Moreover, the critical value of the self-propulsion velocity at which diluted and dense phases start to coexist also increases; as a consequence, the softer the particle interaction is, the bigger self-propulsion velocities are needed in order to observe a MIPS.

A continuous phase-evolution model for cold and strain-induced crystallization in semi-crystalline polymers

The dynamic crystallization during extreme thermomechanical history under producing or service strongly influences the dimensional stability of the semi-crystalline thermoplastic polymers. To produce high-precision structural components, the proper thermomechanical history needs to be carefully designed to eliminate the crystallization-induced shrinkage. On the contrary, the rapidly developed four-dimensional (4D) printing technology tries to amplify the crystallization-induced shrinkage to directly produce the complex curvatures without supporting materials. However, the crystals formed at different deformation states might have different orientations, leading to an anisotropic residual deformation. Therefore, tracing of all crystals formed with different initial configurations is the key issue to ensure the geometric accuracy of both the high-precision and the deformable components. In this study, we developed a continuous phase-evolution model for both the cold and the strain-induced crystallization in semi-crystalline thermoplastics. The irreversible cold crystallization is analogized as the raindrop falling into the pool, and the reversible strain-induced crystallization is analogized as the nonequilibrium liquid-gas phase transformation. Using the continuous phase-evolution concept, the crystal growth is described by a series of continuously formed crystal phases sequentially added into the initial amorphous medium. Each newly formed crystal phase is in a stress-free state at the formation moment, and therefore the crystallization history coupled with the whole deformation history can be memorized. By introducing the oriented growth tensor representing the stress-free state of all formed crystals, the deformation-history dependent anisotropic crystallization and the corresponding residual deformation can be traced. The developed model is validated by comparing with the experimental data of the dynamic crystallization in amorphous poly l-lactide polymers under thermomechanical loading cycles. Finally, the predictive capability of the model is illustrated by several demonstrations, to show the influences of deformation-history dependent crystallization orientations and the corresponding anisotropic residual deformation.

Phase change and solute mixing in multicomponent metal additive manufacturing: A new numerical approach

Creation of new alloys through in-situ alloying in additive manufacturing is being actively explored to improve the properties and performance of manufactured parts. Multiphysics models are often used to investigate the underlying physics controlling AM processes using high-fidelity simulations of individual powder particle melting within the meltpool. In this paper, we present a novel multiphysics-multiphase model for simulating multicomponent powder beds during the AM process. A convection–diffusion formulation for component mass fraction, in conjunction with the Navier–Stokes equations and enthalpy form of the energy equation, enable us to predict the phase evolution in multicomponent beds and the dispersion of solute metals in the melt pool during the process. A non-equilibrium phase diagram (NEPD) model that incorporates the associated phase diagram is introduced to obtain the mass fraction in both solid and liquid phases as well as local temperature and liquid fraction by efficiently solving a set of nonlinear equations locally at each computational cell. This work introduces a numerical method for multicomponent Al-Zr binary material system for light-weight aluminum alloys, specifically demonstrating its efficacy and accuracy for eutectic solidification of this multicomponent.

Analysis of the Nonequilibrium Phase Change Behaviors of the Cryoprotectant Solutions for Cryopreservation of Human Red Blood Cells with Low-Concentration Glycerol.

Recently, we proposed a low-glycerol cryoprotectant formulation (consisting of 0.4 M trehalose and 5% glycerol) for cryopreservation of human red blood cells (RBCs), which greatly reduced the concentration of glycerol, minimized intracellular ice damage, and achieved high recovery. Although this study was successful in cellular experiments, the nonequilibrium phase transition behaviors of the cryoprotective agent solution have not been systematically analyzed. Therefore, it is essential to provide reliable thermodynamic data to substantiate the viability of this cryopreservation technique. In this study, the phase change behaviors and thermal properties of typical trehalose and/or glycerol solutions quenched in liquid nitrogen were investigated using differential scanning calorimetry and cryomicroscopy. It was found that the glass transition temperatures of both the trehalose aqueous solution (<1.0 M) and glycerol aqueous solution (<40% w/v) did not vary apparently with the concentration at low concentrations, while they increased significantly with increasing concentration at high concentrations. Moreover, it was revealed that the inhibitory effect of trehalose on ice growth was affected by glycerol. We further found that the addition of low concentrations of glycerol facilitates the partial glass transition of trehalose solutions at low concentrations. The results of this work provide reliable thermodynamic data to support the cryopreservation of human RBCs with unusually low concentrations of glycerol.

Cu-based metastable microstructural template and its role in the heat flux application

The present paper focuses on developing an optimised strength-conductive alloy in Cu–Fe–Si system. We present the effect of a cost-effective rapid-casting route on modifying the microstructure. The non-equilibrium phase transformation events are tracked and the evolution sequences of metastable phases in the microstructure are elucidated. The metastable phase transformation of the precipitates from DO19- to L12-ordered structure and its role in enhancing thermal conductivity are presented. A density functional theory (DFT)-based calculation also complements the experimental evidence of the relative stability of the evolved metastable phases. One of the key findings is the evolution of L12-ordered precipitates via face centred cubic (FCC)-based interfacial matrix template.

Non-equilibrium dynamics and phase transitions in Potts model and interacting Ehrenfest urn model

We show that the recently proposed interacting Ehrenfest M-urn model at equilibrium (Cheng et al., 2020) can be exactly mapped to a mean-field M-state Potts model. By exploiting this correspondence, we show that the M-state Potts model with M≥3, with transition rates motivated by the non-equilibrium urn model, can exhibit rich non-equilibrium spin dynamics such as non-equilibrium steady states and non-equilibrium periodic states. Monte Carlo simulations of the 3-state Potts model are performed to demonstrate explicitly the first-order transitions for the equilibrium and non-equilibrium steady states, as well as the far-from-equilibrium periodic states.

Nonequilibrium phase transitions in a Brownian p-state clock model.

We introduce a Brownian p-state clock model in two dimensions and investigate the nature of phase transitions numerically. As a nonequilibrium extension of the equilibrium lattice model, the Brownian p-state clock model allows spins to diffuse randomly in the two-dimensional space of area L^{2} under periodic boundary conditions. We find three distinct phases for p>4: a disordered paramagnetic phase, a quasi-long-range-ordered critical phase, and an ordered ferromagnetic phase. In the intermediate critical phase, the magnetization order parameter follows a power-law scaling m∼L^{-β[over ̃]}, where the finite-size scaling exponent β[over ̃] varies continuously. These critical behaviors are reminiscent of the double Berezinskii-Kosterlitz-Thouless (BKT) transition picture of the equilibrium system. At the transition to the disordered phase, the exponent takes the universal value β[over ̃]=1/8, which coincides with that of the equilibrium system. This result indicates that the BKT transition driven by the unbinding of topological excitations is robust against the particle diffusion. On the contrary, the exponent at the symmetry-breaking transition to the ordered phase deviates from the universal value β[over ̃]=2/p^{2} of the equilibrium system. The deviation is attributed to a nonequilibrium effect from the particle diffusion.

A unified friction factor formulation: Bridging laminar and turbulent friction factor with critical points analysis

The friction factor is widely recognized as a pivotal parameter in the analysis of fluid–boundary interactions; however, a comprehensive grasp of friction mechanics remains elusive. This investigation revisits measurements from the benchmark Nikuradse measurements, furnishing indirect evidence of two critical points in pipe turbulence. It underscores that friction factors within laminar and turbulent regimes are intimately interconnected, bearing significant relations to subcritical and critical phenomena. The two critical points directing the laminar–turbulent transition consist of a standard non-equilibrium phase transition and a fully matured turbulent regime, accompanied by an extensive crossover to its asymptotic scaling. Relying on a mathematical model, the scaled friction factor for rough pipes converges into a unified curve. New formula of friction factor of pipe flow is derived, and it was illustrated that it can be derived as a geometric weighted parameter, bridging the laminar and turbulent friction factors. Conclusively, the proffered model was juxtaposed with pipe experimental data from the antecedent study and more contemporaneous transitional pipe data to authenticate the aptness of the suggested model, and it united the friction factor in all three regimes.

Microstructural changes near the interface of eleven-layered AZ31/AA1050 composites fabricated by single-shot explosive welding

The evolution of the microstructure of reaction regions in as-welded and annealed states was thoroughly investigated by SEM and TEM in the eleven-layered AZ31/AA1050 composite plates fabricated via a single-shot explosive welding process. With the detonation velocities used, only the first and second interface is wavy, while the others are flat. Near all interfaces, local melting and rapid solidification processes lead to the formation of reaction regions composed of phases of different chemical compositions and structures. Even though 2 equilibrium phases of γ-Mg17Al12 and β-Mg3Al2 have been found a large part of the solidified melt region consists of non-equilibrium phases that exhibit an amorphous or ultrafine-grained structure. Further heat treatment of the multilayer composite resulted in rapid nucleation and growth of the γ-Mg17Al12 and β-Mg2Al3 phases in the form of hard layers. Within the pre-existing reaction regions, the systematic transformation of phases of different chemical compositions into one β-Mg2Al3 phase was observed. It was found that using a pressure of 3 MPa is very important in preventing delamination of the clad during heating, but it is not able to block the formation of the linear cracks in the β-Mg3Al2 phase, which crucially reduces the composite formability.

Фазообразование в реакционных порошковых смесях (AL-Fe-Fe2O3) при наличии конкурирующих стадий

The paper presents the results of the experimental studies and theoretical analysis using a kinetic model of the reaction interaction of a multicomponent powder system (Al-Fe-Fe2O3) with different ratios of components, one of which is a nonequilibrium powder product of processing steel swarf as an analogue of the composition (Fe-Fe2O3). The use of aluminum in a powder mixture with a composite product of processing steel swarf as an analogue of (Fe-Fe2O3) involves the parallel-sequential occurrence of several reactions during heating, including reduction reactions (aluminothermy) and reactions of intermetallic compounds formation. The phase and structural changes of the studied powder mixtures were analyzed after heat treatment under the conditions of vacuum sintering and volumetric synthesis up to 1000 °C. It has been shown that in the powder compositions (Al-Fe-Fe2O3), the interaction of the components during heating is realized through competing stages, which have a noticeable impact on the nature of the reactions, including their sequence and degree of conversion. As a result, a set of nonequilibrium phases, which are the reaction products, and the elemental residues of the interacting components is formed.

Fluctuations and Shape Dependence of Microphase Separation in Systems with Long-Range Interactions.

The combination of phase separation and long-ranged, effective, Coulomb interactions results in microphase separation. We predict the sizes and shapes of such microdomains and uniquely their dependence on the macroscopic sample shape which also affects the effective interfacial tension of fluctuations of the lamellar phase. These are applied to equilibrium salt solutions and block copolymers. Nonequilibrium phase separation in the presence of chemical reactions (e.g., cellular condensates) is mapped to the Coulomb theory to which our predictions apply. In some cases, the effective interfacial tension can be ultralow.

Open-Source and Multi-Fluid Numerical Modeling of Proton Exchange Membrane Fuel Cell

Numerical modeling is a powerful tool for the virtual design and optimization of the next-generation proton exchange membrane fuel cells (PEMFC). While almost all of them are not open-source, which causes inconvenience and limits. In this paper, a three-dimensional, multi-component, multi-fluid, open-source model developed in the commercial CFD package (CFX) is used to investigate the species’ transport in a proton exchange membrane fuel cell. The model includes a complete fuel cell with both the seven-layer membrane-electrode-assembly and the gas flow channels and bipolar plates, which account for all major transport phenomena. In the porous medium, liquid water transport is dominant by the capillary pressure gradient; momentum loss describes the Darcy equation; and mass transfer between phases by a nonequilibrium phase change model. Multi-component gas phases are governed by convection and diffusion. Moreover, diffusion is the predominant transport mechanism for dissolved water between the membrane and catalyst layers. Consequently, the code of this model is shared on GitHub to provide a starting point and inspire further development and optimization.

Kibble–Zurek scaling of nonequilibrium phase transition in barium titanate

Far-from-equilibrium phase transition dynamics is one of the grand challenges in modern materials science. A theoretical landmark is the Kibble–Zurek (KZ) scaling to describe the relationship between the temperature quenching rate and the resulting defect density in the vicinity of symmetry-breaking phase transformations. Despite the confirmation of the KZ scaling in ferroic perovskite materials and macroscopic simulations, its atomistic mechanisms remain elusive. Here, we demonstrate the KZ scaling using all-atom molecular dynamics simulations for a prototypical ferroelectric perovskite, barium titanate, with the scaling exponent corresponding to the theoretical prediction for rapid quenching. Simulated diffuse neutron scattering data are presented to guide future experiments.

Experimental investigation and thermodynamic assessment of phase equilibria in the GaTe–AgGa5Te8–Te system below 600 K

Equilibrium T–x space of the Ag–Ga–Te system in the GaTe–AgGa5Te8–Te part was divided below 600 K into three-phase regions Ga2Te5–AgGa5Te8–Te, Ga2Te3–AgGa5Te8–Ga2Te5, Ga7Te10–AgGa5Te8–Ga2Te3, Ga3Te4–AgGa5Te8–Ga7Te10, and GaTe–AgGa5Te8–Ga3Te4 by the electromotive force (EMF) method. To accomplish accurate experimental data, the following electrochemical cells (ECs) were assembled: (−)IE|NE|SSЕ|R{Ag+}|PЕ|IE(+), where IE is the inert electrode (graphite powder), NE is the negative electrode (silver powder), SSE is the solid-state electrolyte (glassy Ag3GeS3Br), PE is the positive electrode, R{Ag+} is the region of PE that contact with SSE. At the stage of cell preparation, PE is a nonequilibrium phase mixture of the well-mixed powdered compounds Ag2Te, GaTe, Ga2Te3, and tellurium, taken in ratios corresponding to two or three different points in each of the mentioned regions. The equilibrium set of phases was formed in the R{Ag+} region at 600 K for 48 h with the participation of the Ag+ ions. Silver cations, displaced for thermodynamic reasons from the NE to the PE of the ECs, acted as catalyst, i.e., small nucleation centers of equilibrium phases. The spatial position of the established three-phase regions relative to the silver point was used to assign the overall potential-determining reactions of synthesis of the ternary AgGa5Te8 and binary Ga2Te5, Ga7Te10, Ga3Te4 compounds. For the first time, the values of the standard thermodynamic functions (Gibbs energies, enthalpies, and entropies) of these compounds were determined based on the temperature dependences of the EMF of the ECs.

Microstructural feature-driven machine learning for predicting mechanical tensile strength of laser powder bed fusion (L-PBF) additively manufactured Ti6Al4V alloy

The rapid solidification inherent in laser powder bed fusion (L-PBF) additive manufacturing (AM) introduces segregation phenomena and formation of non-equilibrium phases in duplex titanium alloy components, thereby impeding their suitability for high-reliability engineering applications. Consequently, heat treatment becomes indispensable for optimizing both the microstructure and mechanical properties to meet application requirements. This study aims to investigate the influence of varied annealing temperatures on the evolution of L-PBF-built Ti6Al4V alloy microstructure, subsequently elucidating their impact on tensile properties by analyzing of L-PBF process parameters, building orientations, and annealing temperatures. The findings reveal that annealing at 850 °C for 2 h facilitates the transformation of brittle martensite into a ductile lamellar (α + β) microstructure, thereby conferring excellent tensile properties upon the L-PBF-built Ti6Al4V alloy. Furthermore, to accurately predict the tensile strengths of the Ti6Al4V, we take into account the L-PBF process parameters and the as-built microstructures in a comprehensive manner, extracting the pertinent microstructural features. A machine learning (ML)-based model is built to facilitate accurate predictions. Accurate and reliable predictions are demonstrated by this model when applied to Ti6Al4V. This data-driven approach establishes a novel avenue for AM material property prediction and process parameter optimization.

Anomalous Mitigation in Phase Evolution Impacting Thermal Stability in a Rapidly Solidified AA5182 Al–Mg Alloy via Continuous Thin‐Strip Casting

This study, for the first time, reports an anomalous phase evolution and its impact on the thermal and oxidation behavior of rapidly solidified AA5182 Al–Mg alloy strip fabricated using a novel thin‐strip (TS) continuous casting technique. The microstructural analysis reveals a through‐thickness gradient microstructure of distinct types, morphology, and fractions of nonequilibrium Al–Mn–Fe intermetallic and Al–Mg eutectic phases. The rapid solidification experienced in TS casting effectively mitigates the formation of phases, particularly with the Al–Mg (β‐Al3Mg2) eutectic, being nearly absent from the near‐surface regions. The total fraction of formed phases is considerably lower than that in the slowly cooled direct‐chill counterpart. The solute macrosegregation also shows an inverse Mg segregation profile toward the strip surface primarily due to a higher degree of matrix supersaturation closer to the strip surface. Modeling of solidification successfully predicts the influence of cooling rate on the fractions of the nonequilibrium eutectic phase, agreeing well with the experimental data obtained from image analysis. Heat treating the samples over a broad temperature range unveiled unexpected improvements in oxidation resistance and excellent thermal stability, attributed to the absence of the eutectic β‐phase in subsurface regions. The research findings have practical implications for improving the properties of sheet Al–Mg alloys.

Minimum-Action Method for Nonequilibrium Phase Transitions

Minimum-Action Method for Nonequilibrium Phase Transitions

SIR model on one dimensional small world networks

We study the nonequilibrium phase transition for the model of epidemic spreading, Susceptible–Infected–Refractory (SIR), on one dimensional small world networks. This model belongs to the universality class of dynamical percolation class (DyP) and the upper critical dimension corresponding to this class is dc=6. One dimensional case is special case of this class in which the percolation threshold goes to one (boundary value) in thermodynamic limit. This behavior resembles slightly the behavior of equilibrium phase transition in a one dimensional Ising and XY models where the critical thresholds for both models go to zero temperature (boundary value) in thermodynamic limit. By analytical arguments and numerical simulations we demonstrate that, increasing the connectivity (2k) of this model on regular one dimensional lattice does not alter the criticality of the model. However the phase transition study shows that, this model crosses from a one dimensional structure to mean field like for any finite value of the rewiring probability (p). This behavior is similar to what happened in the equilibrium phase transition for the Ising and XY models on small world networks. Thus, this model is a one of nonequilibrium models which behaves similarly to the equilibrium systems on small world networks. Unlike of many nonequilibrium systems on small world networks which have been found to display a mean field like behavior only at finite values of p or even show critical exponents depend on p. Furthermore, we calculate the critical exponents and the full critical phase space of this model on small world network. We also introduce the crossover scaling function of this model from one dimensional behavior to mean field behavior and reveal the similarity between this model and the equilibrium models on the small world networks.

Nonequilibrium phase transition of a one dimensional system reaches the absorbing state by two different ways

We study the nonequilibrium phase transitions from the absorbing phase to the active phase for the model of diseases spreading (Susceptible-Infected-Refractory-Susceptible (SIRS)) on a regular one-dimensional lattice. In this model, particles of three species (S, I, and R) on a lattice react as follows: S+I→2I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S+I\\rightarrow 2I$$\\end{document} with probability λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}, I→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$I\\rightarrow R$$\\end{document} after infection time τI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _I$$\\end{document} and R→I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R\\rightarrow I$$\\end{document} after recovery time τR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _R$$\\end{document}. In the case of τR>τI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au _R>\ au _I$$\\end{document}, this model has been found to have two critical thresholds separating the active phase from absorbing phases. The first critical threshold λc1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _{c1}$$\\end{document} corresponds to a low infection probability and the second critical threshold λc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _{c2}$$\\end{document} corresponds to a high infection probability. At the first critical threshold λc1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _{c1}$$\\end{document}, our Monte Carlo simulations of this model suggest the phase transition to be of directed percolation class (DP). However, at the second critical threshold λc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _{c2}$$\\end{document} we observe that the system becomes so sensitive to initial values conditions which suggest the phase transition to be a discontinuous transition. We confirm this result using order parameter quasistationary probability distribution and finite-size analysis for this model at λc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _{c2}$$\\end{document}.

Thermal-activated escape of the bistable magnetic states in 2D Fe3GeTe2 near the critical point

Great effort has been made recently to investigate the phase transitions in two-dimensional (2D) magnets while leaving subtle quantification unsolved. Here, we demonstrate the thermal-activated escape in 2D Fe3GeTe2 ferromagnets near the critical point with a quantum magnetometry based on nitrogen-vacancy centers. We observe random switching between the two spin states with auto-correlation time described by the Arrhenius law, where a change of temperature by 0.8 K induces a change of lifetime by three orders of magnitude. Moreover, a large energy difference between the two spin states about 51.3 meV is achieved by a weak out-of-plane magnetic field of 1 G, yielding occupation probability described by Boltzmann’s law. Using these data, we identify all the parameters in the Ginzburg-Landau model. This work provides quantitative description of the phase transition in 2D magnets, which paves the way for investigating the critical fluctuation and even non-equilibrium phase transitions in these 2D materials.