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A phase-field approach for phase transformations (PTs) between three different phases at nonequilibrium temperatures is developed. It includes advanced mechanics, thermodynamically consistent interfacial stresses, and interface interactions. A thermodynamic Landau-Ginzburg potential developed in terms of polar order parameters satisfies the desired instability and equilibrium conditions for homogeneous phases. The interfacial stresses were introduced with some terms from large-strain formulation even though the small-strain assumption was utilized. The developed model is applied to study the PTs between two solid phases via highly disordered intermediate phase ($IP$) or intermediate melt (IM) hundreds of degrees below the melting temperature. In particular, the $\beta \leftrightarrow \delta$ PTs in HMX energetic crystal via IM is analyzed. The effects of various parameters (temperature, ratios of widths and energies of solid-solid (SS) to solid-melt (SM) interfaces, elastic energy, and interfacial stresses) on the formation, stability, and structure of the IM within a propagating SS interface are studied. Interfacial and elastic stresses within SS interphase and their relaxation and redistribution with appearance of partial or complete IM are analyzed. Energy and structure of the critical nucleus (CN) of IM are studied as well. In particular, the interfacial stresses increase the aspect-ratio of the CN. Although including elastic energy can drastically reduce the energy of CN of IM, the activation energy of the CN of IM within SS interface increases when interfacial tension is taken into account. The developed thermodynamic potential can also be modified to model other multiphase physical phenomena, such as multi-variant martensitic PTs, grain boundary and surface-induced pre-melting and PTs, as well as developing phase diagrams for IPs.

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... Two different PFAs to the IM were developed using two order parameters: one describing solid-solid PT and another one for melting, see Fig. 20(c) for approach in [277,351,353,354] and Fig. 20(f) for approach in [282], as well as Sections 16.2.3 and 16.2.5 for details. ...

... and 16.2.5 for details. Also, papers [277,352,353] include coupling with elasticity and [354] include interfacial stresses. Internal elastic stresses promote the existence and persistence of the IM . ...

... Such a theory possesses two characteristic nanoscale parameters: widths of the crack surface δ c and the A − M interface width δ p . Then the dimensionless scale parameterδ = δ c /δ p significantly affects PT and fracture, similar to other PFAs with two scale parameters, see Section 11.6 and [268,269], Section 12 and [277,282,351,353,354], as well as review [251]. It was found that the lower surface energy of M than that of A (i.e.,γ = γ M /γ A < 1) promotes nucleation of M at the crack tip, its stabilization at the crack surface as a nanolayer ("wetting" by martensite), as well as nucleation of the pre-martensite or M at the crack surfaces, even in the pseudoelastic regime, when stress release near the crack surface has to lead to the reverse PT (Fig. 28). ...

Review of selected fundamental topics on the interaction between phase transformations, fracture, and other structural changes in inelastic materials is presented. It mostly focuses on the concepts developed in the author's group over last three decades and numerous papers that affected us. It includes a general thermodynamic and kinetic theories with sharp interfaces and within phase field approach. Numerous analytical (even at large strains) and numerical solutions illustrate the main features of the developed theories and their application to the real phenomena. Coherent, semicoherent, and noncoherent interfaces, as well as interfaces with decohesion and with intermediate liquid (disordered) phase are discussed. Importance of the surface-and scale-induced phenomena on interaction between phase transformation with fracture and dislocations as well as inheritance of dislocations and plastic strains is demonstrated. Some nontrivial phenomena, like solid-solid phase transformations via intermediate (virtual) melt, virtual melting as a new mechanism of plastic deformation and stress relaxation under high strain rate loading, and phase transformations and chemical reactions induced by plastic shear under high pressure are discussed and modeled. * Extended version of paper: Levitas V.I. Phase transformations, fracture, and other structural changes in inelastic materials.

... Two different PFAs to the IM were developed using two order parameters: one describing solid-solid PT and another one for melting, see Fig. 20(c) for approach in [277,351,353,354] and Fig. 20(f) for approach in [282], as well as Sections 16.2.3 and 16.2.5 for details. ...

... and 16.2.5 for details. Also, papers [277,352,353] include coupling with elasticity and [354] include interfacial stresses. Internal elastic stresses promote the existence and persistence of the IM . ...

... Such a theory possesses two characteristic nanoscale parameters: widths of the crack surface δ c and the A − M interface width δ p . Then the dimensionless scale parameterδ = δ c /δ p significantly affects PT and fracture, similar to other PFAs with two scale parameters, see Section 11.6 and [268,269], Section 12 and [277,282,351,353,354], as well as review [251]. It was found that the lower surface energy of M than that of A (i.e.,γ = γ M /γ A < 1) promotes nucleation of M at the crack tip, its stabilization at the crack surface as a nanolayer ("wetting" by martensite), as well as nucleation of the pre-martensite or M at the crack surfaces, even in the pseudoelastic regime, when stress release near the crack surface has to lead to the reverse PT (Fig. 28). ...

Review of selected fundamental topics on the interaction between phase transformations, fracture, and other structural changes in inelastic materials is presented. It mostly focuses on the concepts developed in the author's group over last three decades and numerous papers that affected us. It includes a general thermodynamic and kinetic theories with sharp interfaces and within phase field approach. Numerous analytical (even at large strains) and numerical solutions illustrate the main features of the developed theories and their application to the real phenomena. Coherent, semicoherent, and noncoherent interfaces, as well as interfaces with decohesion and with intermediate liquid (disordered) phase are discussed. Importance of the surface- and scale-induced phenomena on interaction between phase transformation with fracture and dislocations as well as inheritance of dislocations and plastic strains is demonstrated. Some nontrivial phenomena, like solid-solid phase transformations via intermediate (virtual) melt, virtual melting as a new mechanism of plastic deformation and stress relaxation under high strain rate loading, and phase transformations and chemical reactions induced by plastic shear under high pressure are discussed and modeled.

... Since for the analytical solution for the variant-variant interfaces was lacking for the most popular MPFA-II, the interfacial stresses were introduced therein at a best guess. The interfacial stresses were correctly introduced for a three-phase model in the polar order parameters (Momeni and Levitas (2016)) and the models with multivariant martensitic transformations in Levitas and Roy (2015); Levitas et al. (2013)) (without detailed derivations). However, these models have drawbacks as discussed above. ...

... If in A gradient of some η i is not zero, this corresponds to a complex M i -A-M j interface, or triple and multiple junction, and the stresses have a more complex structure, which does not have counterparts for a sharp interface. Some examples of interfacial stresses for complex interfaces can be found in Momeni and Levitas (2016). ...

A thermodynamically consistent, novel multiphase phase field approach for stress-and temperature-induced martensitic phase transformations at finite strains and with interfacial stresses has been developed. The model considers a single order parameter to describe the austenite↔martensitic transformations, and another N order parameters describing N variants and constrained to a plane in an N-dimensional order parameter space. In the free energy model coexistence of three or more phases at a single material point (multiphase junction), and deviation of each variant-variant transformation path from a straight line have been penalized. Some shortcomings of the existing models are resolved. Three different kinematic models (KMs) for the transformation deformation gradient tensors are assumed: (i) In KM-I the transformation deformation gradient tensor is a is a linear function of the Bain tensors for the variants. (ii) In KM-II the natural logarithms of the transformation deformation gradient is taken as a linear combination of the natural logarithm of the Bain tensors multiplied with the interpolation functions. (iii) In KM-III it is derived using the twinning equation from the crystallographic theory. The instability criteria for all the phase transformations have been derived for all the kinematic models, and their comparative study is presented. A large strain finite element procedure has been developed and used for studying the evolution of some complex microstructures in nanoscale samples under various loading conditions. Also, the stresses within variant-variant boundaries, the sample size effect, effect of penalizing the triple junctions, and twinned microstructures have been studied. The present approach can be extended for studying grain growth, solidifications, para↔ferro electric transformations, and diffusive phase transformations.

... The problem of introducing of the interfacial stresses s st was solved for melting [47,48] and the solid-solid interface for small [3,46,49] and large [50,51] strains, including cases with anisotropic interface energy [51]. The interfacial stresses were also introduced and studied for a complex solid-melt-solid interface [5,52], which appears during solid-solid PT via the intermediate melt. ...

... Elastic interfacial stresses s S e (with resultant force per unit interface length s S e ) appear automatically (i.e., without extra terms in the constitutive equations) as a result of solution of the coupled Ginzburg-Landau and elasticity equations, due to heterogeneity of the transformations strain and the elastic properties within interface. They were found numerically for a solid-melt interface [45,47,48], for the austenite-martensite [3] and martensitemartensite [4,33] interfaces, as well as for a complex solid-meltsolid interface [5,52,53]. In contrast, the theory in Ref. [54] introduces the explicit dependence of the gradient energy on the interfacial strain. ...

The origin of a large elastic stress within an interface between martensitic variants (twins) within a finite strain phase field approach has been determined. Notably, for a sharp interface this stress is absent. Three different constitutive relations for the transformation stretch tensor versus order parameters have been considered: a linear combination of the Bain tensors (kinematic model-I, KM-I), an exponential-logarithmic combination (KM-II) of the Bain tensors, and a stretch tensor corresponding to simple shear (KM-III). An analytical finite-strain solution has been found for an infinite sample for tetragonal martensite under plane stress condition. In particular, explicit expression for the stresses have been obtained. The maximum interfacial stress for KM-II is more than twice that which corresponds to KM-I. Stresses are absent for KM-III, but it is unclear how to generalize this model for multivariant martensitic transformation. An approximate analytical solution for a finite sample has been found as well. It shows good correspondence with numerical results obtained using the finite element method. The obtain results are important for developing phase field approaches for multivariant martensitic transformations coupled to mechanics, especially at the nanoscale.

... Efficient computational tools have been developed to perform simulations within timespans not accessible to experimental studies [22,23]. We may refer to large-scale continuum simulations , phase-field simulations for capturing the microstructure [48][49][50][51][52][53][54][55], MD simulations to capture the atomistic mechanisms [2,47,[56][57][58][59][60][61][62][63][64][65][66][67], and multiscale simulations to capture the broad spectrum of materials and processes response [24,25,46,[68][69][70][71][72][73]. Out of several computational methods, molecular dynamics simulation allows the capturing of materials evolution with atomistic accuracy, including the radiation damage mechanism. ...

Ferritic-martensitic steels, such as T91, are candidate materials for high-temperature applications, including superheaters, heat exchangers, and advanced nuclear reactors. Considering these alloys’ wide applications, an atomistic understanding of the underlying mechanisms responsible for their excellent mechano-chemical properties is crucial. Here, we developed a modified embedded-atom method (MEAM) potential for the Fe-Cr-Si-Mo quaternary alloy system—i.e., four major elements of T91—using a multi-objective optimization approach to fit thermomechanical properties reported using density functional theory (DFT) calculations and experimental measurements. Elastic constants calculated using the proposed potential for binary interactions agreed well with ab initio calculations. Furthermore, the computed thermal expansion and self-diffusion coefficients employing this potential are in good agreement with other studies. This potential will offer insightful atomistic knowledge to design alloys for use in harsh environments.

... Hyperspherical phase-field models for rapid solidification neglecting the surface energy inhomogeneities have recently been developed for diffusionless processes neglecting elasticity [12], with elasticity [13][14][15], and with elasticity and surface tension [16] that satisfy all stability conditions for a three-phase system. Multiphase-field models have been developed and utilized to study the microstructure of printed Inconel 718 alloy [17] and solute trapping behavior during rapid solidification [18]. ...

The integrity of the final printed components is mostly dictated by the adhesion between the particles and phases that form upon solidification, which is a major problem in printing metallic parts using available In-Space Manufacturing (ISM) technologies based on the Fused Deposition Modeling (FDM) methodology. Understanding the melting/solidification process helps increase particle adherence and allows to produce components with greater mechanical integrity. We developed a phase-field model of solidification for binary alloys. The phase-field approach is unique in capturing the microstructure with computationally tractable costs. The developed phase-field model of solidification of binary alloys satisfies the stability conditions at all temperatures. The suggested model is tuned for Ni-Cu alloy feedstocks. We derived the Ginzburg-Landau equations governing the phase transformation kinetics and solved them analytically for the dilute solution. We calculated the concentration profile as a function of interface velocity for a one-dimensional steady-state diffuse interface neglecting elasticity and obtained the partition coefficient, k, as a function of interface velocity. Numerical simulations for the diluted solution are used to study the interface velocity as a function of undercooling for the classic sharp interface model, partitionless solidification, and thin interface.

... This technique avoids applying boundary conditions at an interface that is mathematically difficult and computationally expensive. Instead, it uses additional internal variables, called order parameters, to model the interfaces and microstructure of the material (Ref [11][12][13][14][15][16][17][18]. The method captures intermediate phases and applies to particles with a size comparable to the solid-melt interface width. ...

Aluminum alloys are among the top candidate materials for in-space manufacturing (ISM) due to their lightweight and relatively low melting temperature. A fundamental problem in printing metallic parts using available ISM methods, based on the fused deposition modeling (FDM) technique, is that the integrity of the final printed components is determined mainly by the adhesion between the initial particles. Engineering the surface melt can pave the way to improve the adhesion between the particles and manufacture components with higher mechanical integrity. Here, we developed a phase-field model of surface melting, where the surface energy can directly be implemented from the experimental measurements. The proposed model is adjusted to Al 7075-T6 alloy feedstocks, where the surface energy of these alloys is measured using the sessile drop method. Effect of mechanics has been included using transformation and thermal strains. The effect of elastic energy is compared here with the corresponding cases without mechanics. Two different geometric samples (cylindrical and spherical) are studied, and it is found that cylindrical particles form a more disordered structure upon size reduction compared to the spherical samples.

... However, it is still about an order of magnitude larger than the Inconel/Ni interface width (~3 nm), allowing us to study the atomistic mechanisms and interfacial phases forming due to irradiation at high temperatures. This study benefits development of novel MMLCs for nuclear cladding by elucidating the fundamental mechanisms governing the properties of interlayers and providing the information needed to perform simulations at higher length and temporal scales, e.g., thermodynamically consistent phase-field models of interfaces require a pre-knowledge of the interface thickness [27][28][29][30][31]. This paper provides guidelines for selecting the thickness of each metallic layer considering the lifetime and radiation exposure of associated components. ...

Multimetallic layered composites (MMLCs) have shown an excellent potential for application under extreme environments, e.g., accident-tolerant fuel cladding, because of their low oxidation tendency and high corrosion resistance. Interfacial phases or complexions in nanocrystalline materials accelerate the annihilation of defects and enhance the radiation resistance of materials, making MMLCs with engineered interlayer phases compelling to deploy in extreme conditions. However, implementation of MMLCs in full capacity remained a challenge due to a lack of fundamental understanding of the underlying mechanisms governing the characteristics of the interface between the metallic layers. The precise role of interlayer phases in MMLCs and their interaction with defects, specifically under extreme conditions, is still unexplored. Pursuing atomistic simulations for various Inconel-Ni MMLCs model materials, we revealed accelerated defect mobility in interlayers with larger crystalline misorientation and the inverse relationship between the interface sink strength to the misorientation angle. Furthermore, we found a linear relation between interlayer misorientation angle with the density of radiation-induced defects and radiation enhanced displacements. Finally, our results indicate that radiation-induced material degradation is accelerated by the higher defect formation tendency of MMLCs with a high-angle interlayer interface.
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... The exhaustive trial and error experimentations are prohibited due to significant time and costs. Analytical and computational models provide an alternative approach for designing and optimizing process parameters and alloy compositions [1,[14][15][16][17][18][19][20][42][43][44][45][46][47]. ...

We developed a combined finite element and CALPHAD based model of the Laser Powder Bed Fusion (LPBF) process for AA7075 alloy that considers the effect of feedstock composition and print parameters. A single-pass of a laser on a layer of AA7075 alloy powder has been considered. Sensitivity of temperature evolution and melt pool geometry to variation in the stoichiometry of the feedstock powder and laser source characteristics have been studied. Our results indicate that deviation (up to 10%) of the feedstock composition from the AA7075 raises the maximum temperature and increases melt pool size. Excess Cu content shows the largest melt pool width and depth among all the cases. The peak temperature is higher than the standard feedstock composition in all cases, except when the Cu concentration is reduced. Increasing the scan power also results in a higher peak temperature and a larger melt pool size. Furthermore, the temperature's rise time increases by lowering the scan speed.

... Further, the properties 20 of a crystalline solid are necessarily anisotropic [1], and, hence, like the interfacial free energy and interface stress is also known to be anisotropic: see, for example [15,16,17]. The effect of such interface stress on solid-melt equilibrium is well known; see [14,18,19] for example. ...

The interface stresses at of the solid-melt interface are, in general, anisotropic. The anisotropy in the interfacial stress can be evaluated using molecular dynamics (MD) and phase field crystal (PFC) models. In this paper, we report our results on the evaluation of the anisotropy in interface stress in a BCC solid with its melt. Specifically, we study Fe using both MD and PFC models. We show that while both MD and PFC can be used for the evaluation, and the PFC and the amplitude equations based on PFC give quantitatively consistent results, the MD and PFC results are qualitatively the same but do not match quantitatively. We also find that even though the interfacial free energy is only weakly anisotropic in BCC interfacial stress anisotropy is strong. This strong anisotropy has implications for the equilibrium shapes, growth morphologies and other properties at nano-scale in these materials.

... Such a theory possesses two characteristic nanoscale parameters: widths of the crack surface δ c and the A − M interface width δ p . Then the dimensionless scale parameterδ = δ c /δ p significantly affects PT and fracture, similar to other PFAs with two scale parameters, see Section 10 and [208,209], and [217,222,275,277], as well as review [193]. It was found that the lower surface energy of M than that of A (i.e., γ = γ M /γ A < 1) promotes nucleation of M at the crack tip, its stabilization at the crack surface as a nanolayer ("wetting" by martensite), as well as nucleation of the pre-martensite or M at the crack surfaces, even in the pseudoelastic regime, when stress release near the crack surface has to lead to the reverse PT (Fig. 15). ...

Review of selected fundamental topics on the interaction between phase transformations, fracture, and other structural changes in inelastic materials is presented. It mostly focuses on the concepts developed in the author's group over last three decades and numerous papers that affected us. It includes a general thermodynamic and kinetic theories with sharp interfaces and within phase field approach. Numerous analytical (even at large strains) and numerical solutions illustrate the main features of the developed theories and their application to the real phenomena.
Coherent, semicoherent, and noncoherent interfaces, as well as interfaces with decohesion and with intermediate liquid (disordered) phase are discussed. Importance of the surface- and scale-induced phenomena on interaction between
phase transformation with fracture and dislocations as well as inheritance of dislocations and plastic strains is demonstrated. Some nontrivial phenomena are discussed and modeled.

... The solid-solid phase transformation, including the graphene→diamond transformation, is a complicated procedure. It depends on various parameters like defect quantity in the parent solid-phase, the direction of the compression with respect to the basal plane, local stress state, intermediate amorphous states, and surface energy [25][26][27][28][29]. ...

Diamond is the hardest superhard material with excellent optoelectronic, thermomechanical, and electronic properties. Here, we have investigated the possibility of a new synthesis technique for diamane and diamond thin films from multilayer graphene at pressures far below the graphite → diamond transformation pressure. We have used the Molecular Dynamics technique with reactive force fields. Our results demonstrate a significant reduction (by a factor of two) in the multilayer graphene → diamond transformation stress upon using a combined shear and axial compression. The shear deformation in the multilayer graphene lowers the phase transformation energy barrier and plays the role of thermal fluctuations, which itself promotes the formation of diamond. We revealed a relatively weak temperature dependence of the transformation strain and stresses. The transformation stress vs. strain curve for the bulk graphite drops exponentially for finite temperatures.

... 25 A more complete theory of virtual melting in these systems can be found here. 26 (5) In aluminum and copper, plastic deformation and stress relaxation under high strain rates have also been predicted thermodynamically and by MD simulations to occur by virtual melting. 27 In this case, melting occurred 4000 K below the thermodynamic equilibrium temperature at equilibrium pressure and was caused by the relation of deviatoric stresses during melting. ...

The solid, secondary explosive nitramine-octahydro-1,3,5,7-tetranitro-1,3,5,7 or HMX has four different stable polymorphs which have different molecular conformations, crystalline structures, and densities, making structural phase transitions between these nontrivial. Previous studies of the kinetics of the β-δ HMX structural transition found this to happen by a nucleation and growth mechanism, where growth was governed by the heat of fusion, or melting, even though the phase transition temperature is more than 100 K below the melting point. A theory known as virtual melting could easily justify this since the large volume difference in the two phases creates a strain at their interface that can lower the melting point to the phase transition temperature through a relaxation of the elastic energy. To learn more about structural phase transitions in organic crystalline solids and virtual melting, here we use time-resolved X-ray diffraction to study another structural phase transition in HMX, γ-δ. Again, second order kinetics are observed which fit to the same nucleation and growth model associated with growth by melting even though the volume change in this transition is too small to lower the melting point by interfacial strain. To account for this, we present a more general model illustrating that melting over a very thin layer at the interface between the two phases reduces the total interfacial energy and is therefore thermodynamically favorable and can drive the structural phase transition in the absence of large volume changes. Our work supports the idea that virtual melting may be a more generally applicable mechanism for structural phase transitions in organic crystalline solids.

... To accurately model the crack initiation and shortcrack growth behavior, it is critical to model both the microstructure morphology evolution of the material including the micro-crack itself, and the microscale deformation rules and mechanisms that governs the initiation and growth of the short cracks under the influence of microstructures. For the microstructure modeling, the phase-field (PF) approach (Provatas and Elder 2011;Chen 2002), has been widely applied to study the microstructure evolution process including phase transformations (Levitas 2013;Levitas and Momeni 2014;Levitas 1841, 2015;Momeni and Levitas 2016), grain boundary migration and recrystallization (Jafari et al. 2017;Chen et al. 2015) and crack modeling (Arriaga and Waisman 2017; Spatschek et al. 2011;Bleyer et al. 2016;Tanné et al. 2018). The PF method can well-describe the complex microstructure morphologies using order parameters, which make it an appropriate simulation technique for modeling the heterogeneous polycrystal microstructures. ...

This paper presents a physics-based prediction of crack initiation at the microstructure level using the phase field (PF) model without finite element discretization, coupled with an efficient and accurate modeling of crack propagation at macro-scale based on extended finite element method (XFEM). Although the macro-scale model assumes linear elastic material behavior, at micro-scale the behavior of plastically deforming heterogeneous polycrystals is taken into account by coupling the PF model and a crystal plasticity model in the fast Fourier transform computational framework. A sequential coupling has been established for the multiscale modeling where the macro-scale finite element (FE) model determines the hot spots at each cyclic loading increment and passes the associated stress/strain values to the unit-cell phase-field model for accurate physics-based microstructure characterization and prediction of plasticity induced crack initiation. The PF model predicts the number of cycles for the crack initiation and the phenomenological crack growth models are employed to propagate the initiated crack by the appropriate length to be inserted in the FE mesh. Finally, the XFEM solution module is activated to perform mesh independent crack propagation from its initial crack size to the final size for the total life prediction. The effectiveness of the proposed multiscale method is demonstrated through numerical examples.

... Also, the intermediate β phase may appear at the interface between α and ω phases and it can be stabilized by reduction in the total interfacial energy. Similar PTs via intermediate phases were discussed in [53,54] and described by phase field approach [55]. ...

... Formation of different interfacial phases, driven by the relaxation of excess interfacial energy and stresses, is also reported. [4][5][6][7][8][9] Formation of nanotubes by rolling up thin solid films due to misfit strains [10][11][12] or imbalance of surface stresses, 13 as well as transformation of 1D nanowires to 0D nanoparticles driven by Rayleigh-instability 14 is also reported. Throughout the structural transformation, relaxation of surface energy and stresses as well as the change in material properties, such as anisotropic surface strains, 15 result in the formation of a set of new structures and kinetics. ...

Reducing the dimensions of materials to atomic scales results in a large portion of atoms being at or near the surface, with lower bond order and thus higher energy. At such scales, reduction of the surface energy and surface stresses can be the driving force for the formation of new low-dimensional nanostructures, and may be exhibited through surface relaxation and/or surface reconstruction, which can be utilized for tailoring the properties and phase transformation of nanomaterials without applying any external load. Here we used atomistic simulations and revealed an intrinsic structural transformation in monolayer materials that lowers their dimension from 2D nanosheets to 1D nanostructures to reduce their surface and elastic energies. Experimental evidence of such transformation has also been revealed for one of the predicted nanostructures. Such transformation plays an important role in bi-/multi-layer 2D materials.

In this work, molecular dynamics simulations have been used to undertake a computational study of the equilibrium crystal-melt interface stresses in face-centered-cubic (FCC) Ni and body-centered-cubic (BCC) Fe, BCC Nb, and a model BCC soft-sphere elemental system, for three different interface orientations, i.e., (100), (110), and (111). The sign, magnitude, and anisotropy of the excess interface stresses and their relationships with the corresponding interfacial free energies have been examined. The universality of a few trends regarding the interfacial stresses observed in FCC crystal-melt interfaces has been assessed for the BCC crystal-melt interfaces. The role of the interatomic bonding that affects the shape of the interfacial stress profiles, thus modulating the magnitude or sign of the excess interface stress, has been discussed through inspecting a particular type of crystal-melt interface over different materials. Besides, for the first time, we have demonstrated that the Irving-Kirkwood fine-grained algorithm for depicting microscopic pressure components and stresses in the vicinity of the crystal-melt interface is superior to the previously used per-particle virial stress algorithm. The reported data and new knowledgqe could enrich the accumulation for theory breakthroughs in predicting interface stresses and motivate future studies on the interfacial stresses for more types of solid-liquid interfaces.

We developed a coupled CALPHAD and finite element-based computational model of the Laser Powder Bed Fusion (LPBF) process for HAYNES230, considering the feedstock composition and packing density. We further used this model to investigate the effect of variation in feedstock composition and print parameters on the quality of the final printed part. Sensitivity of the maximum reached temperature to variations in characteristics of the laser source is also studied considering a single-track laser scan on a layer of metal powder. We analyzed temperature evolution in the powder bed and melt pool geometry along the path of the laser. Our results indicate that the LPBF process of HAYNES230 alloy requires a powder layer thickness of ∼20μm and laser spot size ∼30μm radius compared to other alloys. It is essential to achieve sufficient melt pool depth necessary for cohesion with the substrate while avoiding large melt pool width that adversely affects the formation of cracks and residual stresses. We also revealed that reducing the laser power or increasing scan speed drastically reduces peak temperature while less susceptible to solute composition.

The interface stresses at of the solid–melt interface are, in general, anisotropic. The anisotropy in the interfacial stress can be evaluated using molecular dynamics (MD) and phase field crystal (PFC) models. In this paper, we report our results on the evaluation of the anisotropy in interface stress in a bcc solid with its melt. Specifically, we study Fe using both MD and PFC models. We show that while both MD and PFC can be used for the evaluation, and the PFC and the amplitude equations based on PFC give quantitatively consistent results, the MD and PFC results are qualitatively the same but do not match quantitatively. We also find that even though the interfacial free energy is only weakly anisotropic in bcc-Fe, the interfacial stress anisotropy is strong. This strong anisotropy has implications for the equilibrium shapes, growth morphologies and other properties at nano-scale in these materials.

Grain boundary-induced transformations between solid, premelt, and melt are studied using a phase field approach. The effect of grain boundary width and energy and triple junction energy is studied.

In this work, TiC ceramic particle reinforced TiAl-based composites were synthesized by selective laser melting (SLM) by using Ti, Al, and TiC multi-component powder. The results showed that a plenty of TiC dendritic crystals formed during the solidifying process. It was found that TiC dendrites exhibited three different modes of nucleation and growth, corresponding to the dissolution-precipitation of fully melted fine TiC particles, the epitaxial growth along the margin of partly melted TiC particle, and the recrystallization growth based on stress-induced nonequilibrium melting. Subsequently, the influence of laser energy density on microstructure evolution and high-temperature oxidation behaviour of SLM fabricated TiC/TiAl composites were investigated. At a high laser energy density of 189 J/mm³, the relatively full dense SLM-fabricated part was obtained, accompanying the formation of a variety of TiC dendrites with the coarsening structure, which made a contribution to the elevated high-temperature oxidation resistance with a low oxidation kinetics constant of 1.32 × 10⁻⁵ mg ⁶ cm⁻¹² h⁻¹.

We review findings obtained within an advanced Ginzburg-Landau theory that the ratio of two nanoscale parameters (e.g., width of two different interfaces or width of the interface and the Burgers vector of interfacial dislocations) drastically affects transformation nano and macroscale behavior. The ratio of two nanoscale lengths induces new phenomena, changes transformation parameters and mechanisms, and should be considered as a new dimension in a "phase diagram." Examples include surface-induced melting of nanoparticles and martensitic transformations, solid-solid transformation via an intermediate phase, and interaction between phase interface and dislocations.

When two materials interact, the processes between the phases determine the functional properties of the compound. Pivotal interface phenomena are diffusion and redistribution of atoms (molecules). This is especially of interest in Lithium ion batteries where the interfacial kinetics determines the battery performance and impact cycling stability. A new phase field model, which links the atomistic processes at the interface to the mesoscale transport by a redistribution flux controlled by the so called ‘interface permeability’ was developed. The model was validated with experimental data from diffusion couples. Calculations of the concentration profiles of the species at the electrode–electrolyte interface are reported. Active particle size, morphology and spatial arrangement were put in correlation with diffusion behavior for use in reverse engineering.

The main focus of this paper is to introduce, in a thermodynamically consistent manner, an anisotropic interface energy into a phase field theory for phase transformations. Here we use a small strain formulation for simplicity, but we retain some geometric nonlinearities, which are necessary for introducing correct interface stresses. Previous theories have assumed the free energy density (i.e., gradient energy) is an anisotropic function of the gradient of the order parameters in the current (deformed) state, which yields a nonsymmetric Cauchy stress tensor. This violates two fundamental principles: the angular momentum equation and the principle of material objectivity. Here, it is justified that for a noncontradictory theory the gradient energy must be an isotropic function of the gradient of the order parameters in the current state, which also depends anisotropically on the direction of the gradient of the order parameters in the reference state. A complete system of thermodynamically consistent equations is presented. We find that the main contribution to the Ginzburg-Landau equation resulting from small strains arises from the anisotropy of the interface energy, which was neglected before. The explicit expression for the free energy is justified. An analytical solution for the nonequilibrium interface and critical nucleus has been found and a parametric study is performed for orientation dependence of the interface energy and width as well as the distribution of interface stresses.

Thermodynamic Ginzburg-Landau potential for temperature-and stress-induced phase transformations (PTs) between n phases is developed. It describes each of the PTs with a single order parameter without an explicit constraint equation, which allows one to use an analytical solution to calibrate each interface energy, width, and mobility; reproduces the desired PT criteria via instability conditions; introduces interface stresses; and allows for a controlling presence of the third phase at the interface between the two other phases. A finite-element approach is developed and utilized to solve the problem of nanostructure formation for multivariant martensitic PTs. Results are in a quantitative agreement with the experiment. The developed approach is applicable to various PTs between multiple solid and liquid phases and grain evolution and can be extended for diffusive, electric, and magnetic PTs.

The effect of elastic energy on nucleation and disappearance of a nanometer size intermediate melt (IM) region at a solid-solid (S1S2) phase interface at temperatures 120 K below the melting temperature is studied using a phase-field approach. Results are obtained for broad range of the ratios of S1S2 to solid-melt interface energies, kE, and widths, kδ. It is found that internal stresses only slightly promote barrierless IM nucleation but qualitatively alter the system behavior, allowing for the appearance of the IM when kE < 2 (thermodynamically impossible without mechanics) and elimination of what we termed the IM-free gap. Remarkably, when mechanics is included within this framework, there is a drastic (16 times for HMX energetic crystals) reduction in the activation energy of IM critical nucleus. After this inclusion, a kinetic nucleation criterion is met, and thermally activated melting occurs under conditions consistent with experiments for HMX, elucidating what had been to date mysterious behavior. Similar effects are expected to occur for other material systems where S1S2 phase transformations via IM take place, including electronic, geological, pharmaceutical, ferroelectric, colloidal, and superhard materials.

In this Letter, continuum thermodynamic and phase field approaches (PFAs) predicted internal stress-induced reduction in melting temperature for laser-irradiated heating of a nanolayer. Internal stresses appear due to thermal strain under constrained conditions and completely relax during melting, producing an additional thermodynamic driving force for melting. Thermodynamic melting temperature for Al reduces from 933.67 K for a stress-free condition down to 898.1 K for uniaxial strain and to 920.8 K for plane strain. Our PFA simulations demonstrated barrierless surface-induced melt nucleation below these temperatures and propagation of two solid-melt interfaces toward each other at the temperatures very close to the corresponding predicted thermodynamic equilibrium temperatures for the heating rate Q≤1.51×1010K/s. At higher heating rates, kinetic superheating competes with a reduction in melting temperature and melting under uniaxial strain occurs at 902.1 K for Q = 1.51 × 1011 K/s and 936.9 K for Q = 1.46 × 1012 K/s.

The definition of all properties of the nonequilibrium interface depends on the choice of the position of the dividing surface. However, the definition of its position has been an unsolved problem for more than a century. A missing principle to unambiguously determine the position of the Gibbs' dividing surface is found: the principle of static equivalence. A sharp interface (dividing surface) is statically equivalent to a nonequilibrium finite-width interface with distributed tensile stresses if it possesses (a) the same resultant force, equal to the interface energy, and (b) the same moment, which is zero about the interface position. Each of these conditions determines the position of a sharp interface, which may be contradictory. This principle is applied to resolve another basic problem: the development of a phase field approach to an interface motion that includes an expression for interface stresses, which are thermodynamically consistent, and consistent with a sharp-interface limit. Using an analytical solution for a curved propagating interface, it is shown that both conditions determine the same dividing surface, i.e., the theory is self-consistent. The expression for the interface energy is also consistent with the expression for the velocity of the curved sharp interface. Applications to more complex interfaces that support elastic stresses are discussed.

A generalization of the phase-field theory for multivariant martensitic phase transformations is suggested that allows one to vary martensite-martensite interface energy independent of energy for austenite-martensite interfaces. The finite element method is utilized to solve the coupled phase-field and elasticity equations. Width and energy of the austenite-martensite interfaces are determined. Splitting of the martensite-martensite interface into two austenite-martensite interfaces, leading to barrierless austenite nucleation, is obtained. The effect of the martensite-martensite interface energy and grain size on the stationary and non-stationary nanostructure inside the transforming grain embedded in the austenitic matrix is determined. Some nano-structures differ essentially from the prediction of crystallographic theory. Relationships between the number of twins in grain vs. grain size, and width of twin vs. its length are found. Two unexpected stress-relaxation mechanisms at the boundary of transforming grain are revealed.

General phase-field theory for multivariant martensitic phase transformations and explicit models are formulated at large strains. Each order parameter is unambiguously related to the transformation strain of the corresponding variant. Thermodynamic potential includes energy related to the gradient of the order parameters that mimics the interface energy. Application of the global form of the second law of thermodynamics resulted in the determination of the driving force for change of the order parameters and the boundary conditions for the order parameters. Kinetic relationships between the rate of change of the order parameters and the conjugate driving force lead to the Ginzburg-Landau equations. For homogeneous fields, conditions for instabilities of the equilibrium states (which represent criteria for the phase transformation between austenite and martensitic variants and between martensitic variants) are found for the prescribed Piola-Kirchoff stress tensor. It was proved that these criteria are invariant with respect to change in the prescribed stresses. The expression for the rigid-body rotation tensor is derived for the prescribed Piola-Kirchoff stress. The explicit expressions for the Helmholtz free energy and for transformation strain in terms of order parameters are derived for the most general case of large elastic and transformational strains, rotations, as well as nonlinear, anisotropic, and different elastic properties of phases. For negligible elastic strains, explicit expression for the Gibbs potential is formulated. Results are obtained for fifth- and sixth-degree potentials in Cartesian order parameters and for similar potentials in hyperspherical order parameters. Geometric interpretation of transformation conditions in the stress space and similarity with plasticity theory are discussed. All material parameters are obtained for cubic to tetragonal transformation in NiAl. Phase transformations in NiAl, boron nitride, and graphite to diamond under uniaxial loading are described explicitly, and the importance of geometrically nonlinear terms is demonstrated. A similar approach can be applied for twinning, dislocations, reconstructive transformations, and fracture. (c) 2013 Elsevier Ltd. All rights reserved.

Previously unknown phenomena, scale, and kinetic effects are revealed by introducing the finite width Δξ of the particle-exterior interface as the additional scale parameter and thermally activated melting in the phase field approach. In addition to traditional continuous barrierless premelting and melting for Δξ= 0, barrierless hysteretic jumplike premelting (melting) and thermally activated premelting (melting) via critical nucleus are revealed. A very rich temperature θ-Δξ transformation diagram is found, which includes various barrierless and thermally activated transformations between solid, melt, and surface melt, and complex hysteretic behavior under various temperature and Δξ trajectories. Bistable states (i.e., spontaneous thermally activated switching between two states) between solid and melt or surface melt are found for Al particles. Strong dependence of the melting temperature (which, in contrast to previous approaches, is defined for thermally activated premelting and melting) for nanoparticles of various radii on Δξ is found. Results are in good agreement with experiments for Al for Δξ=0.8-1.2nm. They open an unexplored direction of controlling surface melting and melting or solidification by controlling the width of the external surface and utilizing predicted phenomena. They also can be expanded for other phase transformations (e.g., amorphization, solid-solid diffusionless, diffusive, and electromagnetic transformations) and phenomena, imbedded particles, and mechanical effects.

Numerical simulations of the heating with constant rate of a PBX (plastic-bonded explosive) 9501 formulation consisting of the energetic crystal HMX embedded in a polymeric binder inside of a rigid cylinder is performed. The continuum thermo-mechanochemical model of the behavior of a PBX 9501 developed in the preceding paper [V. I. Levitas, B. F. Henson, L. B. Smilowitz, D. K. Zerkle, and B. W. Asay, J. Appl. Phys. 102, 113502 (2007)] is applied. The model describes the β↔δ phase transformations in crystalline HMX, chemical decomposition of the HMX and binder leading to gas formation, gas leaking from the cylinder, elastic, thermal, and transformational straining, as well as straining due to mass loss. We study the kinetics of the β↔δ phase transformations and pressure buildup, as well as how they are affected by the heating rate, initial porosity and prestrain, HMX and binder decomposition, and gas leaking rule.

Solid–solid (SS)(SS) phase transformations via nanometer-size intermediate melts (IMs)(IMs) within the SS interface, hundreds of degrees below melting temperature, were predicted thermodynamically and are consistent with experiments for various materials. A necessary condition for the appearance of IMs, using a sharp interface approach, was that the ratio of the energies of SS and solid–melt (SM)(SM) interfaces, kEkE, were >2. Here, an advanced phase-field approach coupled with mechanics is developed that reveals various new scale and interaction effects and phenomena. Various types of IM are found: (i) continuous and reversible premelting and melting; (ii) jump-like barrierless transformation to IMs, which can be kept at much lower temperature even for kE<2kE<2; (iii) unstable IMs, i.e. a critical nucleus between the SS interface and the IM. A surprising scale effect related to the ratio of widths of SS and SM interfaces is found: it suppresses barrierless IMs but allows IMs to be kept at much lower temperatures even for kE<2kE<2. Relaxation of elastic stresses strongly promotes IMs, which can appear even at kE<2kE<2 and be retained at kE=1kE=1. The theory developed here can be tailored for diffusive phase transformations, formation of intergranular and interfacial phases, and surface-induced phase transformations.

A phase-field theory of transformations between martensitic variants and multiple twinning within martensitic variants is developed for large strains and lattice rotations. It resolves numerous existing problems. The model, which involves just one order parameter for the description of each variant-variant transformation and multiple twinnings within each martensitic variant, allows one to prescribe the twin interface energy and width, and to introduce interface stresses consistent with the sharp interface limit. A finite-element approach is developed and applied to the solution of a number of examples of twinning and combined austenite-martensite and martensite-martensite phase transformations (PTs) and nanostructure evolution. A similar approach can be developed for reconstructive, electric, and magnetic PTs.

An exact expression for the temperature-dependent interface stress tensor (tension) and energy is derived within a phase field approach. The key problem, of which part of the thermal energy should contribute to the surface tension, is resolved with the help of an analytical solution for a nonequilibrium interface. Thus, for a propagating interface at any temperature, the interface stress tensor represents biaxial tension with magnitude equal to the temperature-dependent interface energy. Explicit expressions for the distributions of interface stresses are obtained for a nonequilibrium interface and a critical nucleus. The results obtained are applicable for various phase transformations (solid-solid, melting-solidification, sublimation, etc.) and structural changes (twinning, grain evolution), and can be generalized for anisotropic interface energy, for dislocations, fracture, and diffusive phase transformations described by Cahn-Hilliard theory.

Thermodynamically consistent phase field theory for multivariant martensitic transformations is developed with the main focus on introducing correct interface stresses (tension). The nontrivial point is that the interface tension (physical phenomenon) is introduced with the help of some geometric nonlinearities, even when strains are infinitesimal. Total stress at the diffuse interface consists of elastic and dissipative parts which are determined by the solution of the coupled system of phase field and viscoelasticity equations and the introduced interface stresses. An explicit expression for the free energy is derived that results in the desired expression for the interface stresses consistent with the sharp interface for the propagating nonequilibrium interface. Analytical expressions for nonequilibrium interface energy, width, entropy excess, as well as distribution of the interface tension are derived and parametrically studied. Interface stress tensor distribution is also obtained and analyzed for a critical martensitic nucleus. The possibility of extending the developed approach to other phenomena and more general models is discussed.

Capillarity: Unconstrained Interfaces / Capillarity and Gravity / Hysteresis and Elasticity of Triple Lines / Wetting and Long-Range Forces b/ Hydrodynamics of Interfaces -- Thin Films, Waves, and Ripples / Dynamics of the Triple Line / Dewetting / Surfactants / Special Interfaces / Transport Phenomena

A three-dimensional Landau theory of stress-induced martensitic phase transformations is presented. It describes transformations between austenite and martensitic variants and transformations between martensitic variants. The Landau free energy incorporates all temperature-dependent thermomechanical properties of both phases. The theory accounts for the principal features of martensitic transformations in shape memory alloys and steels, namely, stress-strain curves with constant transformation strain and constant, or weakly temperature dependent, stress hysteresis, as well as nonzero tangent elastic moduli at the phase transformation point. In part I, the austenitemartensite phase transformation is treated, while transformations between martensitic variants are considered in part II.

Structural and functional materials inherit their macroscopic properties from the complex microstructures they develop at mesoscale. We discuss here the ability of the phase field method to capture the physical mechanisms at the origin of these complex morphologies in two different situations. First, we analyze the polytwinned microstructures observed in martensitic alloys, and show that, due to the large rotations involved in the accommodation mechanism, a correct modeling of the microstructures requires the use of a geometrically nonlinear model. Second, we present an elasto-viscoplastic phase field model and show its application to the understanding of the rafting phenomena observed in superalloys under creep.

Semi-infinite systems which undergo a first-order bulk transition are considered. A new type of surface phase transition is predicted which has two unexpected features: (1) It exhibits some universal properties since a variety of surface exponents can be defined although there are no bulk exponents; (2) a layer of the disordered phase appears between the free surface and the ordered bulk. The interface between the disordered and the ordered phases becomes delocalized as in the wetting and in the pinning transition.

An advanced Ginzburg-Landau (GL) approach to melting and solidification coupled with mechanics is developed. It is based on the concept of a coherent solid-liquid interface with a transformation strain tensor, the deviatoric part of which is described by a thermodynamically consistent kinetic equation. Due to the relaxation of the elastic energy, a promoting contribution to the driving force for phase transformation in the GL equation appears, both for melting and solidification. Good agreement with known experiments is obtained for Al nanoparticles for the size-dependent melting temperature and temperature-dependent thickness of the surface molten layer. All types of interface stress distributions from known molecular dynamics simulations are obtained and interpreted. A similar approach can be applied for sublimation and condensation, amorphization and vitrification, diffusive transformations, and chemical reactions.

In part III of this paper, alternative Landau potentials for the description of stress-and temperature-induced martensitic phase transformations under arbitrary three-dimensional loading are obtained. These alternative potentials include a sixth-degree (2-4-6) polynomial in Cartesian order parameters and a potential in hyperspherical order parameters. Each satisfies all conditions for the correct description of experiments. The unique features of the potentials are pointed out and a detailed comparison of the potentials is made for NiAl alloy. Analytic solutions of the one-dimensional time-independent Ginzburg-Landau equations for the 2-3-4 and 2-4-6 potentials for a constant-stress tensor and invariant-plane strain are obtained and compared. Solutions include martensitic and austenitic critical nuclei and diffuse martensite-austenite and martensite-martensite interfaces. The widths and energies of the nuclei and interfaces are functions of the thermodynamic driving force, the gradient energy coefficient, and a parameter that characterizes the stability of austenite. The splitting of a martensite-martensite interface into two austenite-martensite interfaces is interpreted as a potentially new mechanism—namely, barrierless austenite nucleation—which might be observed experimentally at the interface between two invariant-plane-strain variants. The widths, energies, and gradient energy coefficients of the martensite-martensite and austenite-martensite interfaces are estimated for NiAl. Finally, we outline a version of phase field theory for dislocations based on our theoretical framework for phase transformations.

Certificate of reviewing from Elsevier
Physica A: Statistical Mechanics and its Applications

The various models which have been proposed to explain the stability of the icosahedral phases1–6 start from the isotropic symmetry of the liquid phase, assumed to be the parent phase. This assumption stems from the fact that initially, the icosahedral structure could only be obtained in certain cooling conditions, below the range of stability of the liquid phase. Further experiments revealed however, that one could get the icosahedral phase under as equilibrium conditions7 , and that it could coexist8 with, and recrystallize into9 , other crystalline structures.

Abstract The phase-field method has recently emerged as a powerful computational approach to modeling and predicting mesoscale morphological and microstructure evolution in materials. It describes a microstructure using a set of conserved and nonconserved field variables that are continuous across the interfacial regions. The temporal and spatial evolution of the field variables is governed by the Cahn-Hilliard nonlinear diffusion equation and the Allen-Cahn relaxation equation. With the fundamental thermodynamic and kinetic information as the input, the phase-field method is able to predict the evolution of arbitrary morphologies and complex microstructures without explicitly tracking the positions of interfaces. This paper briefly reviews the recent advances in developing phase-field models for various materials processes including solidification, solid-state structural phase transformations, grain growth and coarsening, domain evolution in thin films, pattern formation on surfaces, dislocation microstructures, crack propagation, and electromigration.

In this paper diffuse interface models of surfactant-assisted liquid-liquid phase separation are addressed. We
start from the generalized version of the Ginzburg-Landau free-energy-functional-based model of van der Sman
and van der Graaf. First, we analyze the model in the constant surfactant approximation and show the presence
of a critical point at which the interfacial tension vanishes. Then we determine the adsorption isotherms and
investigate the validity range of previous results. As a key point of the work, we propose a new model of the van
der Sman/van der Graaf type designed for avoiding both unwanted unphysical effects and numerical difficulties
present in previous models. In order to make the model suitable for describing real systems, we determine the
interfacial tension analytically more precisely and analyze it over the entire accessible surfactant load range.
Emerging formulas are then validated by calculating the interfacial tension from the numerical solution of the
Euler-Lagrange equations. Time-dependent simulations are also performed to illustrate the slowdown of the phase separation near the critical point and to prove that the dynamics of the phase separation is driven by the interfacial
tension.

This article critically assesses the current status and future directions for the development of interfacial phase diagrams for applications in activated sintering and other fields. The origin of solid‐state activated sintering is attributed to the enhanced mass transport in sintering‐aid‐based, nanoscale, quasi‐liquid, interfacial films that are stabilized below the bulk solidus line. Interfacial thermodynamic models have been developed via extending a phenomenological premelting theory and incorporating the computational thermodynamic (CalPhaD) methods. A primitive type of interfacial phase diagrams, λ‐diagrams, have been computed, and these diagrams have been validated by experiments and proven useful. More rigorous interfacial phase diagrams with well‐defined transition lines and critical points may also be constructed. A long‐range scientific goal is proposed to develop interfacial phase diagrams as a new materials science tool. Future studies should be conducted in several areas to achieve this goal, and special efforts should be made to predict the complex interfacial phase behaviors in multicomponent ceramic materials. Potential broad applications are envisaged.

An advanced three-phase phase-�eld approach (PFA) is suggested for a non-equilibrium phase interface
which contains an intermediate phase, in particular, a solid-solid interface with a nanometersized
intermediate melt (IM). Thermodynamic potential in the polar order parameters is developed,
which satis�es all thermodynamic equilibrium and stability conditions. Special form of the gradient
energy allowed us to include the interaction of two solid-melt interfaces via intermediate melt and
obtain a well-posed problem and mesh-independent solutions. It is proved that for stationary 1D
solutions to two Ginzburg-Landau equations for three phases, the local energy at each point is equal
to the gradient energy. Simulations are performed for � $ � phase transformations (PTs) via IM
in HMX energetic material. Obtained energy - IM width dependence is described by generalized
force-balance models for short- and long-range interaction forces between interfaces but not far
from the melting temperature. New force-balance model is developed, which describes phase �eld
results even 100K below the melting temperature. The e�ects of the ratios of width and energies
of solid-solid and solid-melt interfaces, temperature, and the parameter characterizing interaction
of two solid-melt interfaces, on the structure, width, energy of the IM and interface velocity are
determined by �nite element method. Depending on parameters, the IM may appear by continuous
or discontinuous barrierless disordering or via critical nucleus due to thermal
uctuations.
The IM may appear during heating and persist during cooling at temperatures well below than it
follows from sharp-interface approach. On the other hand, for some parameters when IM is expected,
it does not form, producing an IM-free gap. The developed PFA represents a quite general
three-phase model and can be extended to other physical phenomena, such as martensitic PTs,
surface-induced premelting and PTs, premelting/disordering at grain boundaries, and developing
corresponding interfacial phase diagrams.

The transformation kinetics of the β-γ solid state phase transition in the organic nitramine molecule octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) was discussed using second harmonic generation. The quantitative measurement of the γ phase mole fraction in ensembles of free HMX crystals and crystals embedded in a visco-elastic polymer matrix was discussed. The analysis showed difference in nucleation kinetics between samples of free crystals and crystals embedded in a visco-elastic polymer matrix.

A continuum phase field theory and corresponding numerical solution methods are developed to describe deformation twinning in crystalline solids. An order parameter is associated with the magnitude of twinning shear, i.e., the lattice transformation associated with twinning. The general theory addresses the following physics: large deformations, nonlinear anisotropic elastic behavior, and anisotropic phase boundary energy. The theory is applied towards prediction of equilibrium phenomena in the athermal and non-dissipative limit, whereby equilibrium configurations of an externally stressed crystal are obtained via incremental minimization of a free energy functional. Outcomes of such calculations are elastic fields (e.g., displacement, strain, stress, and strain energy density) and the order parameter field that describes the size and shape of energetically stable twin(s). Numerical simulations of homogeneous twin nucleation in magnesium single crystals demonstrate fair agreement between phase field solutions and available analytical elasticity solutions. Results suggest that critical far-field displacement gradients associated with nucleation of a twin embryo of minimum realistic size are 4.5%–5.0%, with particular values of applied shear strain and equilibrium shapes of the twin somewhat sensitive to far-field boundary conditions and anisotropy of twin boundary surface energy.

It has already been asserted, by Mr. Monge and others, that the phenomena of capillary tubes are referable to the cohesive attraction of the superficial particles only of the fluids employed, and that the surfaces must consequently be formed into curves of the nature of lintearias, which are supposed to be the results of a uniform tension of a surface, resisting the pressure of a fluid, either uniform, or varying according to a given law. Segner, who appears to have been the first that maintained a similar opinion, has shown in what manner the principle may be deduced from the doctrine of attraction, but his demonstration is complicated, and not perfectly satisfactory; and in applying the law to the forms of drops, he has neglected to consider the very material effects of the double curvature, which is evidently the cause of the want of a perfect coincidence of some of his experiments with his theory. Since the time of Segner, little has been done in investigating accurately and in detail the various consequences of the principle. It will perhaps be most agreeable to the experimental philosopher, although less consistent with the strict course of logical argument, to proceed in the first place to the comparison of this theory with the phenomena, and to inquire afterwards for its foundation in the ultimate properties of matter. But it is necessary to premise one observation, which appears to be new, and which is equally consistent with theory and with experiment; that is, that for each combination of a solid and a fluid, there is an appropriate angle of contact between the surfaces of the fluid, exposed to the air, and to the solid. This angle, for glass and water, and in all cases where a solid is perfectly wetted by a fluid, is evanescent: for glass and mercury, it is about 140°, in common temperatures, and when the mercury is moderately clean.

In this paper we develop a free energy function for a Cu–Al–Ni alloy undergoing cubic–orthorhombic phase transitions. We use the irreducible Lagrangian strain polynomial invariants of the cubic austenite parent phase to construct a polynomial expansion for the Helmholtz free energy. Our expansion retains quartic terms of the strain components in order to describe the two-phase material—the cubic austenite phase and six variants of the orthorhombic martensite phase. The coefficients of the free energy polynomial function are temperature dependent and are fitted to appropriate experimental data for austenite and martensite phases from literature. The resulting Helmholtz free energy function is given by (3.20), (4.11). We examine the response predicted by the model for shear in the twinning directions.

Interfacial region and three-phase line region are considered as two-dimensional and one-dimensional Cosserat continua with kinematics described by two independent vectors—a displacement vector ν and a rotation vector ω; in such media a couple-stress tensor μ appears simultaneously with a stress tensor σ. Equations of the linear momentum balance and the moment-of-momentum balance generalize the Laplace equation for surfaces and the Young equation for lines.

Recent observations of three classes of nanometer-thick, disordered, interfacial films in multicomponent inorganic materials are reviewed and critically assessed. The three classes of films are equilibrium-thickness intergranular films (IGFs) in ceramics, their free-surface counterparts, that is, surficial amorphous films (SAFs), and their metallic counterparts. Also briefly reviewed are several related wetting and adsorption phenomena in simpler systems, including premelting in unary systems, prewetting in binary liquids or vapor adsorption on inert walls, and frustrated-complete wetting. Analogous diffuse-interface and force-balance models are discussed with the goal of exploring a unifying thermodynamic framework. In general, the stability of these nanometer-thick interfacial films does not follow bulk phase diagrams. Stabilization of quasi-liquid interfacial films at subeutectic or undersaturation conditions in multicomponent materials can be understood from coupled interfacial premelting and prewetting transitions. More realistic models should include additional interfacial interactions, for example, dispersion and electrostatic forces, and consider the possibility for metastable equilibration. It is suggested that quasi-liquid grain boundary films in binary metallic systems can be used to validate a basic thermodynamic model. These nanoscale interfacial films are technologically important. For example, the short-circuit diffusion that occurs in interface-stabilized, subeutectic, quasi-liquid films explains the long-standing mystery of the solid-state activated sintering mechanism in ceramics, refractory metals, and ice.

This chapter reviews Gibbs' formulation of equilibrium thermodynamics of material systems and discusses modern attempts to extend his method to multi-component solids. The chapter also describes Gibbs' thermodynamics of surfaces and his dividing surface construction. This method allows for a general and comprehensive treatment of non-diffuse fluid surfaces. Gibbs' extension of his method to the case of a surface between a single component solid and multi-component fluid is then examined and the difficulties of generalizing his approach when the solid can have multiple substitutional components are outlined. It is proposed that these difficulties can be addressed if the concept of thermodynamic availability is employed to describe the equilibrium state of a surface. Attempts to extend Gibbs' dividing surface analysis to solid–solid interfaces are discussed. Finally, the chapter presents examples of how Gibbsian surface thermodynamics can be applied to characterize systems for which capillary effects play a central role in the thermodynamic behavior.

A review of the interface stress is carried out to develop a clearer understanding of the origin of the two terms that appear in the most common expression for this quantity: σ = γ + ∂γ ∂ε. The first term in this expression is shown to arise from the choice of the fixed, Eulerian coordinate system used in the description of interface stress and strain. When the interface stress and strain are described using the embedded, Lagrangian coordinate system, the first term does not appear and the interface stress is given purely in terms of the elastic strain dependence of the interfacial free energy. Furthermore, by using a very simple microscopic model, it is illustrated that the interface stress does give the work effect associated with elastic straining of the interface.

The Phase Field Microelasticity theory is developed for proper multivariant martensitic transformations. The model is based on the exact solution of the elasticity problem in the homogeneous modulus approximation. The model takes into account the transformation-induced coherency strain and provides for the strain compatibility throughout the system. Computer simulations are performed for a dilatationless cubic→tetragonal martensitic transformation and for the transformation with parameters corresponding to a martensitic transformation Fe–31%Ni alloy. The development of the martensitic transformation through nucleation, growth and coarsening of orientation variants is simulated at different levels of undercooling. The simulated martensitic structure has a complex polytwinned morphology. Simulation demonstrates that the presence of a non-zero volumetric component in the transformation strain in the Fe–31%Ni system significantly affects the martensitic transformation.

This paper, in line with the previous works (Javili and Steinmann, 2009, 2010), is concerned with the thermomechanically consistent theory and formulation of boundary potential energies and the study of their impact on the deformations of solids. Thereby, the main thrust in this contribution is the extension to thermomechanical effects. Although boundary effects can play a dominant role in the material behavior, the common modelling in continuum mechanics takes exclusively the bulk into account, nevertheless, neglecting possible contributions from the boundary. In this approach the boundary is equipped with its own thermodynamic life, i.e. we assume separate boundary energy, entropy and the like. Afterwards, the derivations of generalized balance equations, including boundary potentials, completely based on a tensorial representation is carried out. The formulation is exemplified for the example of thermohyperelasticity.

We consider a phase field model that includes a stress field during nonisothermal phase transformation of a single-component system. The model is applied to the solidification and melting of confined spherical volumes, where sharp interface solutions can be obtained and compared with the results of the phase field simulations. Numerical solutions to the phase field model for a spherically symmetric geometry have been obtained, with particular emphasis on the computation of surface energy, surface stress, and surface strain. The analysis of the equilibrium states for the phase field model allows us to obtain the value of the surface energy in the presence of stress, which can then be compared to the analogous calculation of the energy of a planar liquid-solid interface. It is also demonstrated that modeling the liquid as a solid with zero shear modulus is realistic by comparing the long-range stress fields in phase field calculations with those calculated using sharp interface models of either a coherent or a relaxed liquid-solid interface.

In this paper we bring together and extend two recent developments in phase-field models, namely, a phase-field model of a multiphase system [I. Steinbach et al., Physica D 94, 135 (1996)] and the extension of the Cahn-Hoffman ξ-vector theory of anisotropic sharp interfaces to phase-field models [A. A. Wheeler and G. B. McFadden, Eur. J. Appl. Math. 7, 369 (1996); Proc. R. Soc. London, Ser. A 453, 1611 (1997)]. We develop the phase-field model of a multiphase system proposed by Steinbach et al. to include both surface energy and interfacial kinetic anisotropy. We show that this model may be compactly expressed in terms of generalized Cahn-Hoffman ξ vectors. This generalized Cahn-Hoffman ξ-vector formalism is subsequently developed to include the notion of a stress tensor, which is used to succinctly derive the leading-order conditions at both moving interfaces and stationary multijunctions in the sharp interface limit.

In this and the accompanying paper [L. Smilowitz et al., J. Chem. Phys. 117, 3789, 2002] we present a theoretical treatment and experimental study, respectively, of the β–δ solid state phase transition in the organic nitramine molecule octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX). The transition is thermodynamically first order with a measured latent heat, occurs via nucleation and growth, and exhibits a thermally activated rate of transformation. We construct a two state kinetic model of the system consisting of equilibrium terms first order in the β or δ mole fraction simulating nucleation, and second order in β and δ simulating growth. The model has four rate constants, the temperature dependence of which is described by eight parameters. We use the transition state formulation of the rate constants and apply a thermodynamic model of the activated state that associates the difference in activated state free energy in forward and reverse directions with the equilibrium transition free energy, and identifies the activated state of the growth process with a metastable melt. By associating components of the activated state free energy with independently measured thermodynamic energies we reduce the degrees of freedom to three, which we fix initially by comparison with previously published kinetic data. We apply the model to both the β–δ and δ–β transformations over a temperature range from 300 to 700 K in order to assess the theoretical validity of the model. The model reproduces the half time of the transition over this entire range, spanning conversion times from 106 to 10−4 s. In the accompanying paper we present an experimental study of the kinetics and mechanism of the phase transition based on second harmonic generation spectroscopy. We use second harmonic generation to verify the nucleation and growth mechanism of the transition and measure the mole fraction change with time over a wide range of temperatures. We use the set of parameters established by theoretical considerations in this paper as an initial parameter set and determine an optimized set by comparison with these data. © 2002 American Institute of Physics.

A recent formulation of a multiphase-field model is presented. The approach is employed to numerically simulate phase transitions in multiphase systems and to describe the evolution of the microstructure during solidification processes in alloy systems. A new method for modelling solute diffusion in a binary alloy within N different phases with varying solubilities and different diffusion coefficients is integrated in the multiphase-field model. The phase-field/diffusion model derived is compared with the previous Wheeler, Boettinger and McFadden (WBM) model in a limiting case. The set of coupled evolution equations, the phase-field model equations and the concentration field equation is solved using control volume techniques on a uniform mesh. With the input of the specific phase diagram, thermophysical and materials data of the chosen real Fe-C alloy system, the multiphase-field method is successfully applied to compute the peritectic solidification process of steel. The numerical calculations of the peritectic reaction and transformation are presented.