Nonconserved Order Parameter Research Articles

Crystal plasticity-phase–field based analyses of interfacial microstructural evolution during dynamic recrystallization in a dual phase titanium alloy

In this study, an integrated crystal plasticity finite element–phase–field (CPFE–PF) model is developed to examine dynamic recrystallization (DRX) in a dual phase Ti alloy. The CP framework is coupled with PF by updating the free energy density with energy contributions due to plasticity. The evolution of grain boundaries through evolving non-conserved order parameters in the PF model is tracked using the Allen–Cahn equation. Nucleation is allowed to occur if the dislocation density exceeds a critical value. DRX is studied in various Ti morphologies such as an α−Ti single crystal containing a stiff elastoplastic particle, α-Ti bicrystals with low and high misorientation between grains, an α−β bicrystal and a globular α−β Ti structure with β phase at α−α interfaces. For an α−Ti bicrystal, a high misorientation facilitates the onset of DRX at the α−α interface at a significantly lower strain than the bicrystal with low misorientation. In an α−β bicrystal, DRX is only observed at the α−β interface. For the globular morphology, nucleation is observed at both α−β interfaces and inside α grains, which is consistent with previous experimental observations for a similar morphology. Nucleation inside α grains is explained by the correlation between SSD density and misorientation indicators such as KAM and GROD at the nucleus site. To correlate slip activity with nucleation propensity immediately prior to different nucleation events, the dislocation density, shear rate and Schmid factors on different slip systems are evaluated at nucleation sites.

Non-reciprocal Phase Separations with Non-conserved Order Parameters

Non-reciprocal Phase Separations with Non-conserved Order Parameters

Machine learning based prediction of phase ordering dynamics.

Machine learning has proven exceptionally competent in numerous applications of studying dynamical systems. In this article, we demonstrate the effectiveness of reservoir computing, a famous machine learning architecture, in learning a high-dimensional spatiotemporal pattern. We employ an echo-state network to predict the phase ordering dynamics of 2D binary systems-Ising magnet and binary alloys. Importantly, we emphasize that a single reservoir can be competent enough to process the information from a large number of state variables involved in the specific task at minimal computational training cost. Two significant equations of phase ordering kinetics, the time-dependent Ginzburg-Landau and Cahn-Hilliard-Cook equations, are used to depict the result of numerical simulations. Consideration of systems with both conserved and non-conserved order parameters portrays the scalability of our employed scheme.

Reversible Microscale Assembly of Nanoparticles Driven by the Phase Transition of a Thermotropic Liquid Crystal.

The arrangement of nanoscale building blocks into patterns with microscale periodicity is challenging to achieve via self-assembly processes. Here, we report on the phase-transition-driven collective assembly of gold nanoparticles in a thermotropic liquid crystal. A temperature-induced transition from the isotropic to the nematic phase under anchoring-driven planar alignment leads to the assembly of individual nanometer-sized particles into arrays of micrometer-sized agglomerates, whose size and characteristic spacing can be tuned by varying the cooling rate. Phase field simulations coupling the conserved and nonconserved order parameters exhibit a similar evolution of the morphology as the experimental observations. This fully reversible process offers control over structural order on the microscopic level and is an interesting model system for the programmable and reconfigurable patterning of nanocomposites with access to micrometer-sized periodicities.

Coupling stress fields and vacancy diffusion in phase-field models of voids as pure vacancy phase

High vacancy concentrations in crystals may lead to formation and growth of voids, which is connected to swelling under irradiation and degradation of material properties. Recent experimental work suggests that vacancy condensation may also be involved in nucleation and evolution of porosity in early stages of ductile fracture. Mostly in the realm of irradiation effects, void formation and growth has been simulated with phase-field methods, where voids are treated as pure vacancy phases. Since vacancies induce an eigenstrain field, it is well-known that the evolution of vacancy concentrations is coupled to the elastic stress field. However, the few existing elastically coupled diffuse interface models of void growth seem to face a conceptual problem in the diffuse interface; in the center of which they predict the highest eigenstrains, which results in unrealistically high, fluctuating stresses. In the current work, we present a new model for coupling elastically driven vacancy diffusion with a diffuse interface model of void surfaces, which overcomes the named short-comings and closely reproduces the sharp interface solution. This is achieved by making the eigenstrain a function of the non-conserved order parameter used to distinguish the crystal and void phase. The model is verified for two-dimensional example problems by comparison to the analytical solution of the according sharp interface model. Eventually, the model is used to show the impact of elasto-diffusional coupling on void growth in mechanically loaded systems. We analyze the model with regard to the bi-stable energy landscape and discuss limitations and future prospects of the approach.

Optimal control of a nonconserved phase field model of Caginalp type with thermal memory and double obstacle potential

In this paper, we investigate optimal control problems for a nonlinear state system which constitutes a version of the Caginalp phase field system modeling nonisothermal phase transitions with a nonconserved order parameter that takes thermal memory into account. The state system, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. It consists of two nonlinearly coupled partial differential equations that govern the phase dynamics and the universal balance law for internal energy, written in terms of the phase variable and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. We extend recent results obtained for optimal control problems in which the free energy governing the phase transition was differentiable (i.e., of regular or logarithmic type) to the nonsmooth case of a double obstacle potential. As is well known, in this nondifferentiable case standard methods to establish the existence of appropriate Lagrange multipliers fail. This difficulty is overcome utilizing of the so-called deep quench approach. Namely, the double obstacle potential is approximated by a family of (differentiable) logarithmic ones for which the existence of optimal controls and first-order necessary conditions of optimality in terms of the adjoint state variables and a variational inequality are known. By proving appropriate bounds for the adjoint states of the approximating systems, we can pass to the limit in the corresponding first-order necessary conditions, thereby establishing meaningful first-order necessary optimality conditions also for the case of the double obstacle potential.

Energy-lowering and constant-energy spin flips: Emergence of the percolating cluster in the kinetic Ising model.

After a sudden quench from the disordered high-temperature T_{0}→∞ phase to a final temperature well below the critical point T_{F}≪T_{c}, the nonconserved order parameter dynamics of the two-dimensional ferromagnetic Ising model on a square lattice initially approaches the critical percolation state before entering the coarsening regime. This approach involves two timescales associated with the first appearance (at time t_{p_{1}}>0) and stabilization (at time t_{p}>t_{p_{1}}) of a giant percolation cluster, as previously reported. However, the microscopic mechanisms that control such timescales are not yet fully understood. In this paper, to study their role on each time regime after the quench (T_{F}=0), we distinguish between spin flips that decrease the total energy of the system from those that keep it constant, the latter being parametrized by the probability p. We show that observables such as the cluster size heterogeneity H(t,p) and the typical domain size ℓ(t,p) have no dependence on p in the first time regime up to t_{p_{1}}. Furthermore, when energy-decreasing flips are forbidden while allowing constant-energy flips, the kinetics is essentially frozen after the quench and there is no percolation event whatsoever. Taken together, these results indicate that the emergence of the first percolating cluster at t_{p_{1}} is completely driven by energy decreasing flips. However, the time for stabilizing a percolating cluster is controlled by the acceptance probability of constant-energy flips: t_{p}(p)∼p^{-1} for p≪1 (at p=0, the dynamics gets stuck in a metastable state). These flips are also the relevant ones in the later coarsening regime where dynamical scaling takes place. Because the phenomenology on the approach to the percolation point seems to be shared by many 2D systems with a nonconserved order parameter dynamics (and certain cases of conserved ones as well), our results may suggest a simple and effective way to set, through the dynamics itself, t_{p_{1}} and t_{p} in such systems.

Critical dynamics of the antiferromagnetic O(3) nonlinear sigma model with conserved magnetization.

We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a nonconserved order parameter reversibly coupled to the conserved total magnetization. To calculate response and correlation functions, we set up a description in terms of Langevin stochastic equationsof motion, and their corresponding Janssen-De Dominicis response functional. We find that in equilibrium, the dynamics is well-separated from the statics, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d=2+ɛ expansion about its lower critical dimension d_{lc}=2, whereas the dynamical mode-coupling terms are governed by the upper critical dimension d_{c}=4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an ε=4-d expansion near d_{c}. This yields anomalous scaling features induced by the massless Goldstone modes, namely subdiffusive relaxation for the conserved magnetization density with asymptotic scaling exponent z_{Γ}=d-2, which may be observable in neutron scattering experiments. Intriguingly, if initialized near the critical point, the renormalization group flow for the effective dynamical exponents recovers their universal critical values z_{c}=d/2 in an intermediate crossover region.

Modelling the statics and the dynamics of fluctuations in the ordering alloy AuAgZn2

The ordering alloy AuAgZn2 has a Heusser second-order transition at Tc ≃ 336.4°C. Static measurements of the critical scattering were carried out at the BM02 beamline of the European Synchrotron Radiation Facility (ESRF). These results are compared with Monte-Carlo simulations of the Ising model and show that the model with a simple interaction between two neighbouring atoms of the simple cubic Au/Ag lattice fully explains the X-ray diffuse scattering. Dynamic measurements obtained from X-ray scattering below Tc and the observation of X-ray photon correlations at the ESRF ID10 beamline are compared with dynamic simulations. It is shown that this system follows the predictions of “model A” [P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49, 436 (1977)] for a transition with non-conserved order parameter. The dynamics of ordering with nearest neighbour exchange of atoms in the simple cubic lattice is shown to be equivalent to the usual Ising spin flip model, but with a different time scale. A comparison between the kinetics of ordering and the dynamics of the observed speckles arising from critical fluctuations shows some discrepancy suggesting the need for further experiments.

Phase Field vs Gradient Enhanced Damage Models: A Comparative Study

A comparison of the two different approaches for modelling damage in material in an infinitesimal strain setting is studied. The first approach is the nonlocal gradient enhanced damage model where the damage variable is taken as an independent variable which will be determined based on the local strain measure. Here, the nonlocal integral form is approximated to an implicit or explicit differential form using the Taylor’s series expansion for simpler numerical implementation. The second approach is the phase field damage model where a Helmholtz free energy density function is considered that includes a new energy degradation function along with a phase field non-conserved order parameter. The first variational principle on this energy density functional with respect to the corresponding order parameter variable will reach a stationarity value resulting in the non-conserved Allen-Cahn equation. The relationship of the order parameter with the damage variable gives the Allen-Cahn evolution equation for damage. A 1D bar example is considered for commenting on the similarities and differences of the two approaches to damage based on the mesh convergence studies based on the results obtained for various meshing profiles, changing length scale parameter values and the obtained damage profiles.

Analysis and optimal control theory for a phase field model of Caginalp type with thermal memory

A nonlinear extension of the Caginalp phase field system is considered that takes thermal memory into account. The resulting model, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. Two equations, resulting from phase dynamics and the universal balance law for internal energy, are written in terms of the phase variable (representing a non-conserved order parameter) and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. Existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem. Then, an optimal control problem is investigated for a suitable cost functional, in which two data act as controls, namely, the distributed heat source and the initial temperature. Fr\'echet differentiability between suitable Banach spaces is shown for the control-to-state operator, and meaningful first-order necessary optimality conditions are derived in terms of variational inequalities involving the adjoint variables. Eventually, characterizations of the optimal controls are given.

Cell dynamics simulations of coupled charge and magnetic phase transformation in correlated oxides.

We present a comprehensive numerical study on the kinetics of phase transition that is characterized by two nonconserved scalar order parameters coupled by a special linear-quadratic interaction. This particular Ginzburg-Landau theory has been proposed to describe the coupled charge and magnetic transition in nickelates and the collinear stripe phase in cuprates. The inhomogeneous state of such systems at low temperatures consists of magnetic domains separated by quasimetallic domain walls where the charge order is reduced. By performing large-scale cell dynamics simulations, we find a two-stage phase-ordering process in which a short period of independent evolution of the two order parameters is followed by a correlated coarsening process. The long-time growth and coarsening of magnetic domains is shown to follow the Allen-Cahn power law. We further show that the nucleation-and-growth dynamics during phase transformation to the ordered states is well described by the Kolmogorov-Johnson-Mehl-Avrami theory in two dimensions. On the other hand, the presence of quasimetallic magnetic domain walls in the ordered states gives rise to a very different kinetics for transformation to the high-temperature paramagnetic phase. In this scenario, the phase transformation is initiated by the decay of magnetic domain walls into two insulator-metal boundaries, which subsequently move away from each other. Implications of our findings to recent nano-imaging experiments on nickelates are also discussed.

An atomistically-informed phase-field model for quantifying the effect of hydrogen on the evolution of dislocations in FCC metals

A new atomistically-informed phase-field model is developed to quantify the effect of hydrogen on the evolution of dislocations. In this model, the long-range interaction between dislocation and hydrogen is considered through the elastic interaction energy due to their eigenstrain fields, and the short-range interaction is taken into account by the hydrogen-affected dislocation core energy. The non-conserved phase-field order parameters describing the evolution of dislocations are controlled by the time-dependent Ginzburg-Landau equations, while the conserved order parameter representing the distribution of hydrogen atoms is governed by the Cahn-Hilliard diffusion equation. To explore the capability of this newly developed phase-field model, it is employed to simulate the hydrogen effect on both the static dissociation of dislocations and the dynamics shrinkage of a dislocation glide loop at zero applied stress in nickel (Ni). The simulation results show that, on the one hand, hydrogen can enhance the equilibrium spacing between two partial dislocations mainly due to the short-range interaction. On the other hand, hydrogen can impede the shrinkage of the glide loop by reducing its line energy. These results are well consistent with the atomistic calculations and theoretical predictions. It means that this new model can well capture both the hydrogen-affected dislocation dynamics influenced by long-range interaction and the hydrogen-affected equilibrium configurations of dislocations dominated by short-range interaction. Finally, with minor modifications, this new model can be further employed to study the effect of hydrogen on the evolution of various microstructures beyond dislocation networks at the mesoscale, which is crucial for a thorough understanding of the hydrogen-induced premature failure mechanisms.

Phase-field model for a weakly compressible soft layered material: morphological transitions on smectic-isotropic interfaces.

A coupled phase-field and hydrodynamic model is introduced to describe a two-phase, weakly compressible smectic (layered phase) in contact with an isotropic fluid of different density. A non-conserved smectic order parameter is coupled to a conserved mass density in order to accommodate non-solenoidal flows near the smectic-isotropic boundary arising from density contrast between the two phases. The model aims to describe morphological transitions in smectic thin films under heat treatment, in which arrays of focal conic defects evolve into conical pyramids and concentric rings through curvature dependent evaporation of smectic layers. The model leads to an extended thermodynamic relation at a curved surface that includes its Gaussian curvature, non-classical stresses at the boundary and flows arising from density gradients. The temporal evolution given by the model conserves the overall mass of the liquid crystal while still allowing for the modulated smectic structure to grow or shrink. A numerical solution of the governing equations reveals that pyramidal domains are sculpted at the center of focal conics upon a temperature increase, which display tangential flows at their surface. Other cases investigated include the possible coalescence of two cylindrical stacks of smectic layers, formation of droplets, and the interactions between focal conic domains through flow.

Aging exponents for nonequilibrium dynamics following quenches from critical points.

Via Monte Carlo simulations we study nonequilibrium dynamics in the nearest-neighbor Ising model, following quenches to points inside the ordered region of the phase diagram. With the broad objective of quantifying the nonequilibrium universality classes corresponding to spatially correlated and uncorrelated initial configurations, in this paper we present results for the decay of the order-parameter autocorrelation function for quenches from the critical point. This autocorrelation is an important probe for the aging dynamics in far-from-equilibrium systems and typically exhibits power-law scaling. From the state-of-the-art analysis of the simulation results, we quantify the corresponding exponents (λ) for both conserved and nonconserved (order-parameter) dynamics of the model in space dimension d=3. Via structural analysis we demonstrate that the exponents satisfy a bound. We also revisit the d=2 case to obtain more accurate results. It appears that irrespective of the dimension, λ is approximately the same for both conserved and nonconserved dynamics.

Bifurcations of front motion in passive and active Allen-Cahn-type equations.

The well-known cubic Allen-Cahn (AC) equation is a simple gradient dynamics (or variational) model for a nonconserved order parameter field. After revising main literature results for the occurrence of different types of moving fronts, we employ path continuation to determine their bifurcation diagram in dependence of the external field strength or chemical potential. We then employ the same methodology to systematically analyze fronts for more involved AC-type models. In particular, we consider a cubic-quintic variational AC model and two different nonvariational generalizations. We determine and compare the bifurcation diagrams of front solutions in the four considered models.

Role of hydrodynamics in liquid–liquid transition of a single-component substance

Liquid-liquid transition (LLT) is an unconventional transition between two liquid states in a single-component system. This phenomenon has recently attracted considerable attention not only because of its counterintuitive nature but also since it is crucial for our fundamental understanding of the liquid state. However, its physical understanding has remained elusive, particularly of the critical dynamics and phase-ordering kinetics. So far, the hydrodynamic degree of freedom, which is the most intrinsic kinetic feature of liquids, has been neglected in its theoretical description. Here we develop a Ginzburg-Landau-type kinetic theory of LLT taking it into account, based on a two-order parameter model. We examine slow critical fluctuations of the nonconserved order parameter coupled to the hydrodynamic degree of freedom in equilibrium. We also study the nonequilibrium process of LLT. We show both analytically and numerically that domain growth becomes faster (slower), depending upon the density decrease (increase) upon the transition, as a consequence of hydrodynamic flow induced by the density change. The coupling between nonconserved order parameter and hydrodynamic interaction results in anomalous domain growth in both nucleation-growth-type and spinodal-decomposition-type LLT. Our study highlights the characteristic features of hydrodynamic fluctuations and phase ordering during LLT under complex interplay among conserved and nonconserved order parameters and the hydrodynamic transport intrinsic to the liquid state.

Enhanced Orientational Ordering Induced by an Active yet Isotropic Bath.

Can a bath of isotropic but active particles promote ordering of anisotropic but passive particles? In this Letter, we uncover a fluctuation-driven mechanism by which this is possible. Somewhat counterintuitively, we show that the passive particles tend to be more ordered upon increasing the noise strength of the active isotropic bath. We first demonstrate this in a general dynamical model for a nonconserved order parameter (model A) coupled to an active isotropic field and then concentrate on two examples: (i)a collection of polar rods on a substrate in an active isotropic bath and (ii)a passive apolar suspension in a momentum conserved, actively forced but isotropic fluid, which are relevant for current research in active systems. Our theory, which is relevant for understanding ordering transitions in out-of-equilibrium systems can be tested in experiments, for instance, by introducing a low concentration of passive rodlike objects in active isotropic fluids and, since it is applicable to any nonconserved dynamical field, may have applications far beyond active matter.

Transport and spectral signatures of transient fluctuating superfluids in the absence of long-range order

Results are presented for the quench dynamics of a clean and interacting electron system, where the quench involves varying the strength of the attractive interaction along arbitrary quench trajectories. The initial state before the quench is assumed to be a normal electron gas, and the dynamics is studied in a regime where long-range order is absent, but nonequilibrium superconducting fluctuations need to be accounted for. A quantum kinetic equation using a two-particle irreducible formalism is derived. Conservation of energy, particle-number, and momentum emerge naturally, with the conserved currents depending on both the electron Green's functions and the Green's functions for the superconducting fluctuations. The quantum kinetic equation is employed to derive a kinetic equation for the current, and the transient optical conductivity relevant to pump-probe spectroscopy is studied. The general structure of the kinetic equation for the current is also justified by a phenomenological approach that assumes model-F in the Halperin-Hohenberg classification, corresponding to a non-conserved order-parameter coupled to a conserved density. Linear response conductivity and the diffusion coefficients in thermal equilibrium are also derived, and connections with Aslamazov-Larkin fluctuation corrections are highlighted. Results are also presented for the time-evolution of the local density of states. It is shown that Andreev scattering processes result in an enhanced density of states at low frequencies. For a quench trajectory corresponding to a sudden quench to the critical point, the density of states is shown to grow in a manner where the time after the quench plays the role of the inverse detuning from the critical point.

3D non-isothermal phase-field simulation of microstructure evolution during selective laser sintering

During selective laser sintering (SLS), the microstructure evolution and local temperature variation interact mutually. Application of conventional isothermal sintering model is thereby insufficient to describe SLS. In this work, we construct our model from entropy level, and derive the non-isothermal kinetics for order parameters along with the heat transfer equation coupled with microstructure evolution. Influences from partial melting and laser-powder interaction are also addressed. We then perform 3D finite element non-isothermal phase-field simulations of the SLS single scan. To confront the high computation cost, we propose a novel algorithm analogy to minimum coloring problem and manage to simulate a system of 200 grains with grain tracking algorithm using as low as 8 non-conserved order parameters. Specifically, applying the model to SLS of the stainless steel 316L powder, we identify the influences of laser power and scan speed on microstructural features, including the porosity, surface morphology, temperature profile, grain geometry, and densification. We further validate the first-order kinetics of the transient porosity during densification, and demonstrate the applicability of the developed model in predicting the linkage of densification factor to the specific energy input during SLS.