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Propagating Phase Interface with Intermediate Interfacial Phase: Phase Field Approach

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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. No full-text available ... 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]. ... ... In this study, we develop a phase-field potential for binary alloys that satisfies the stability conditions at all temperatures by capitalizing on our models for diffusionless melting/solidification [12,15,33] and materials growth [34,35]. We analytically solved the governing equations for dilute solution approximation and calculated interface velocity as a function of undercooling. ... ... Now applying chain rule and Putting value from Equations (12) and (14), ... Article Full-text available 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. ... 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. ... ... Despite these drawbacks, the results in [379] make valuable contribution by direct confirmation and visualization of the crystal-crystal PT via IM for − reconstructive PT in colloidal films. They in-situ confirm, specify, and quantify main statements in [249,260,261,277,281,351], including fast growth kinetics for S 1 M S 2 interface (consistent with the absence of the athermal friction) while a coherent crystal-crystal interface is arrested due to the athermal threshold. ... ... 20(c)), without a constraint. In [277,351,353], such a model was developed and applied to PTs between two solid phases and melt, in particular, for solid-solid PT via intermediate (virtual) melt (see also Section 12). ... Preprint Full-text available 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. ... ... Despite these drawbacks, the results in [379] make valuable contribution by direct confirmation and visualization of the crystal-crystal PT via IM for − reconstructive PT in colloidal films. They in-situ confirm, specify, and quantify main statements in [249,260,261,277,281,351], including fast growth kinetics for S 1 M S 2 interface (consistent with the absence of the athermal friction) while a coherent crystal-crystal interface is arrested due to the athermal threshold. ... ... 20(c)), without a constraint. In [277,351,353], such a model was developed and applied to PTs between two solid phases and melt, in particular, for solid-solid PT via intermediate (virtual) melt (see also Section 12). ... Preprint Full-text available 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. ... Following Refs. Levitas et al. (2003) and Momeni and Levitas (2014), the polar order parameters, the radial ! and the angular #, are introduced in a plane (Fig. 1). Geometrically, p#=2, is the angle between the radius vector !, and the axis 1. ... ... Elastic energy has the simplest form corresponding to the linear elasticity rule. All other terms are justified in Ref. Momeni and Levitas (2014). They reduce to the equations in Ref. for any two phases. ... ... The monotonous interpolating functions connecting properties of phases are justified in Ref. Momeni and Levitas (2014): q y; a ð Þ¼ ay 2 À 2ða À 2Þy 3 þ ða À 3Þy 4 ¼ y 2 aðy À 1Þ 2 þ ð4 À 3yÞy ... ... In Javanbakht (2013, 2014) a simplified system of equations for the interaction between a single martensitic variant and dislocation evolution was presented but without any derivations and justification. A number of important problems have been solved in Levitas and Javanbakht (2012, 2014, which include revealing scale-dependent athermal hysteresis for the semicoherent interface motion, pushing and inheriting dislocations by a moving interface, the generation of dislocations by a growing martensitic plate (including its arrest), and an order of magnitude reduction in PT pressure due to dislocations generated by applied shear stresses. ... ... Expression for thermal energy includes the thermal driving force for PT, double-well barriers between all phases and cross terms that allow us to satisfy Condition I and instability conditions (62)-(63). This expression was derived in Levitas and Preston (2002b) and used in all our simulations (Levitas and Javanbakht, 2012, 2014. Here G Δ θ and s Δ are the differences between the thermal part of free energy and entropy of M and A, respectively; A and Ā are the double-well barrier between A and M and between martensitic variants, respectively; parameters B and D control energy away from both the A and M i minima and do not affect the phase equilibrium or PT conditions; functions K ji are related to the jump in elastic moduli during PTs; e θ is the equilibrium temperature for stress-free A and M; A 0 is a parameter, and c θ is the critical temperature at which stress-free A loses its thermodynamic stability. ... ... Complete system of coupled phase field and mechanics equations is presented for a general case and for small strain approximation. Simplified cases of the developed theory have already been used for the finite element solution of various problems on the interaction between PTs and plasticity (Levitas and Javanbakht, 2012, 2014. Application of the current theory to solution of the some physically important problems is presented in the accompanied paper . ... ... 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. ... ... We consider two axisymmetric samples, i.e., a spherical and a cylindrical feedstock particle. The developed Helmholtz free energy with the elastic, thermal, double-well barrier, and gradient energies is 13,16,17). The elastic energy, w e , in terms of total strain, e, elastic strain, e e , transformation strain, e t , and thermal strain, e h , is ... Article 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. ... Implementing these constraints in Eqs. (25)e (26) and (35)e(37), we express them in terms of the single order parameter h i : ... ... Properties of all three phases are collected in Refs. [24,25,30]. Thus, in our simulations we use Ds 10 ¼ À793.79 kJ/ m 3 K, Ds 20 ¼À935.45 ... Article Full-text available The main conditions for the thermodynamic potential for multiphase Ginzburg-Landau theory are formulated for temperature-induced phase transformations (PTs). Theory, which satisfies all these conditions for n- phase material, is developed. The key point is a new penalizing term in the local energy that allows controlling absence or presence and the extent of the presence of the third phase within the interface between two other phases. A finite-element method is applied for studying PT between β and δ phases of HMX energetic crystal via intermediate melting more than 100°C below melting temperature. Depending on material parameters (ratio of the width and energy of the solid-solid (SS) to solid-melt interface and the magnitude of the penalizing term), there are either two (meta)stable stationary interfacial nanostructures, corresponding to slightly and strongly disordered interfaces (in the limits, pure SS interface or complete melt within SS interface), or these nanostructures coincide. A parametric study of these nanostructures is presented. The developed requirements and approach are applicable to various PTs between multiple solid and liquid phases and can be elaborated for PTs induced by mechanical and electromagnetic fields, diffusive PTs, and the evolution of multi-grain and multi-twin microstructures. ... Such a non-classical nucleation mechanism, originally proposed by Cahn and Hilliard in 1959 (ref. 3 ), was later predicted to happen even during the precipitation of an ordered intermetallic: the composition and order parameter profiles of a nucleus can both be distinctly different from a classic nucleus having uniform composition and order parameter of the product phase [3][4][5][6][7][8][9][10][11][12] . There have been recent reports suggesting non-classical nucleation during the formation of inorganic nanoparticles from solutions [13][14][15] , crystallization of proteins 16 and nucleation of diamond crystals 17 as well as crystallization of amorphous tungsten carbides 18 , but the atomic details of complex reactions in metallurgic systems remain elusive. ... Article Full-text available Two-phase titanium-based alloys are widely used in aerospace and biomedical applications, and they are obtained through phase transformations between a low-temperature hexagonal closed-packed α-phase and a high-temperature body-centred cubic β-phase. Understanding how a new phase evolves from its parent phase is critical to controlling the transforming microstructures and thus material properties. Here, we report time-resolved experimental evidence, at sub-ångström resolution, of a non-classically nucleated metastable phase that bridges the α-phase and the β-phase, in a technologically important titanium–molybdenum alloy. We observed a nanosized and chemically ordered superstructure in the α-phase matrix; its composition, chemical order and crystal structure are all found to be different from both the parent and the product phases, but instigating a vanishingly low energy barrier for the transformation into the β-phase. This latter phase transition can proceed instantly via vibrational switching when the molybdenum concentration in the superstructure exceeds a critical value. We expect that such a non-classical phase evolution mechanism is much more common than previously believed for solid-state transformations. ... No issues have remained only for two variants or three phases, the polar coordinates with a radial and single angular order parameters (Fig. 10(c)), without a constraint. In Momeni and Levitas, 2014;Momeni et al., 2015), such a model was developed and applied to PTs between two solid phases and melt, in particular, for solid-solid PT via intermediate (virtual) melt. ... Article Full-text available 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. ... Phase field (PF) or Ginzburg-Landau (G-L) theory is commonly used to capture various first-order solid-solid phase transitions (PTs) [1,2] such as surface-induced martensitic PT [3][4][5], premelting at the external surfaces [6][7][8], solidsolid PT via intermediate melt [9][10][11][12], dislocation evolution [13][14][15], nanoscale interaction of PTs and dislocations [16][17][18][19][20], martensitic PT in diffrent length scale [21][22][23][24][25][26][27][28][29], crack propagation [30][31][32][33], nanovoids evolution [34,35] etc. In this paper, PTs between austenite to martensitic variants and twinning between martensitic variants are considered. ... Article Full-text available A phase field approach for phase transition between austenite to martensitic variants and twinning between martensitic variants is presented with the main focus on the effects of interfacial stress on phase transformations. In this theory, each variant-variant phase transformations and twinnings within each martensitic variant can be represented by only one-order parameter. Thus, it allows us to get the analytical solution of the martensite-martensite interface profile, energy, and width. Moreover, this model allows us to include interface stress which is consistent with the sharp interface limit. The finite element method is utilized to solve the coupled phase field and elasticity equations for a cubic to tetragonal phase transformation in NiAl shape memory alloy. The stress fields are obtained and the effects of interfacial stress on the stress field at the interfaces are studied for both austenite–martensite and martensite–martensite interfaces in detail. Additionally, the temperature-induced growth of the martensitic phase inside austenite, martensitic phase transformation, and twining are solved. The evolution of the microstructure and stress fields are obtained and the effects of interfacial stress on the morphological evolution of martensitic nanostructures are examined. It is found that the interfacial stress is an important factor, influencing the stress distribution at interfaces and the phase field solution significantly. This theory can be extended for electric, reconstructive, and magnetic phase transformations. ... 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]. ... Article 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. ... For example, it may require quantum mechanical and atomistic reactive force-field calculations to determine the activation energies for atomic migration on a surface 6 and understand the atomistic surface reaction mechanisms 7 , and then a finite element method (FEM) to model the mesoscale mass transport phenomena 8 . Other challenges include incorporating the effects of substrates including the types of substrate defects 9 , the possible wrinkling of 2D films 10 , the effect of van der Waals (vdW) interactions at the mesoscale 11 , and the growth kinetics unique to atomically thin materials 12 . Also reproducing quadratic dispersion for the flexural acoustic modes of 2D materials using classical or reactive potentials may not be straightforward, it has been already formulated 13 . ... Article Full-text available The successful discovery and isolation of graphene in 2004, and the subsequent synthesis of layered semiconductors and heterostructures beyond graphene have led to the exploding field of two-dimensional (2D) materials that explore their growth, new atomic-scale physics, and potential device applications. This review aims to provide an overview of theoretical, computational, and machine learning methods and tools at multiple length and time scales, and discuss how they can be utilized to assist/guide the design and synthesis of 2D materials beyond graphene. We focus on three methods at different length and time scales as follows: (i) nanoscale atomistic simulations including density functional theory (DFT) calculations and molecular dynamics simulations employing empirical and reactive interatomic potentials; (ii) mesoscale methods such as phase-field method; and (iii) macroscale continuum approaches by coupling thermal and chemical transport equations. We discuss how machine learning can be combined with computation and experiments to understand the correlations between structures and properties of 2D materials, and to guide the discovery of new 2D materials. We will also provide an outlook for the applications of computational approaches to 2D materials synthesis and growth in general. ... Recently, a much more detailed phase field theory of virtual melting has been developed and simulations were performed for crystal-crystal phase transitions. 19,29,30 This more sophisticated model takes into account both the reduction in the interfacial energy and mechanics. Numerous nontrivial scale and kinetic effects have been revealed through this work. ... Article 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. ... For our computations with two variants, we consider the first two Bain tensors without loss of generality. In all the simulations, we will consider a θ = a b = a β = 3, a 0 = 10 −4 (see Momeni and Levitas (2014)), ρ 0 A 0M = 1744.7 MPa, ρ 0Ā = 5320 MPa, β 0M = 0.97 × 10 −10 N, and β 12 = 2.96 × 10 −10 N. Thus Eqs. (140) and (141) (135) and (136) along with all other constitutive relations have been solved simultaneously using the finite element method (Zienkiewicz and Taylor (2000a,b)). ... Article Full-text available 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. ... It may lead to reshaping and faceting of nanowires (Levitas et al., 2012) and other nanoobjects. It also can be included for melting within grain boundaries (Lobkovsky and Warren, 2002) and at the interfaces between two solid phases (Levitas, 2005;Levitas et al., 2012;Levitas and Momeni, 2014;Luo and Chiang, 2008;Momeni and Levitas, 2014;Momeni et al., 2015). 2. Micropolar theory. ... ... It may lead to reshaping and faceting of nanowires [75] and other nanoobjects. It also can be included for melting within grain boundaries [76] and at the interfaces between two solid phases [75,[77][78][79][80][81]. Note that reorientation of an interface may occur due to applied stresses and corresponding thermodynamic driving force is found in Refs. ... Article Full-text available 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. ... A new expression for surface stresses and finite strains can be introduced in the phase field approach for an external surface ( Levitas and Javanbakht, 2011b), for single and multiple martensitic variants, and for melting ( Levitas and Samani, 2011a). This would allow for more precise study of surface-stress-induced PTs in nanowires ( Diao et al., 2003) and the contribution of surface stresses to formation of nanometer size third phases at the interfaces between phases ( Luo and Chiang, 2008;Levitas et al., 2012;Levitas and Momeni, 2014;Momeni and Levitas, 2014) and within grain boundaries (Lobkovsky and Warren, 2002). ... Article 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. Data availability All data that was obtained during this project is available from the authors. Article 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. Article 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. Article Full-text available Surface tension plays an integral part in our discussions of formation of new phases from melt, although a quantitative understanding of the surface tension is almost completely lacking, especially in the synthesis of polymorphic solids. This lacuna can be appreciated if we realize that even a semi-quantitative understanding of the celebrated Ostwald’s step rule does not exist. A dramatic lowering of the surface energy of a growing nucleus happens when the stable phase grows from melt in the presence of several amorphous metastable phases. The latter have character intermediate between the initial and the final phases. This reduction in surface tension may dictate rate of both nucleation and growth in a manner hitherto unexplored. In this work a new order parameter based approach is developed and applied to quantify the effects of such metastable phases. Interestingly, the total surface energy between melt and stable solid phase display a fractional dependence on the number of metastable phases (NMS). At higher temperature this effect is observed to be much stronger. This fractional dependence of surface energy is related to the curvature of free energy surface (that varies with temperature, pressure etc.) and assumed plausible ordering of metastable phases in the interface, and have consequence in nanomaterial synthesis. Article Full-text available 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. Article Full-text available 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. Article 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. Article Full-text available 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. Article Full-text available 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. Article Full-text available 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. Article Full-text available 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. Article Full-text available 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. Article Full-text available 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. Article Full-text available 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. Article Full-text available 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. Article Full-text available Recent experimental measurements of Ag impurity diffusion in the Σ5(310) grain boundary (GB) in Cu revealed an unusual non-Arrhenius behavior suggestive of a possible structural transformation Divinski et al., [Phys. Rev. B 85, 144104 (2012)]. On the other hand, atomistic computer simulations have recently discovered phase transformations in high-angle GBs in metals Frolov et al., [Nat. Commun. 4, 1899 (2013)]. In this Letter we report on atomistic simulations of Ag diffusion and segregation in two different structural phases of the Cu Σ5(310) GB which transform to each other with temperature. The obtained excellent agreement with the experimental data validates the hypothesis that the unusual diffusion behavior seen in the experiment was caused by a phase transformation. The simulations also predict that the low-temperature GB phase exhibits a monolayer segregation pattern while the high-temperature phase features a bilayer segregation. Together, the simulations and experiment provide the first convincing evidence for the existence of structural phase transformations in high-angle metallic GBs and demonstrate the possibility of their detection by GB diffusion measurements and atomistic simulations. Article Full-text available 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. Article Full-text available The surface of ice exhibits the swath of phase-transition phenomena common to all materials and as such it acts as an ideal test bed of both theory and experiment. It is readily available, transparent, optically birefringent, and probing it in the laboratory does not require cryogenics or ultrahigh vacuum apparatus. Systematic study reveals the range of critical phenomena, equilibrium and nonequilibrium phase-transitions, and, most relevant to this review, premelting, that are traditionally studied in more simply bound solids. While this makes investigation of ice as a material appealing from the perspective of the physicist, its ubiquity and importance in the natural environment also make ice compelling to a broad range of disciplines in the Earth and planetary sciences. In this review we describe the physics of the premelting of ice and its relationship with the behavior of other materials more familiar to the condensed-matter community. A number of the many tendrils of the basic phenomena as they play out on land, in the oceans, and throughout the atmosphere and biosphere are developed. Article Full-text available Several mechanisms can extend the equilibrium domain of a liquid phase into the solid region of the normal phase diagram. The causes of premelting, which include surface melting, interface curvature and substrate disorder, occur in all types of substances, including H2O. In the case of H2O, premelting can have important environmental consequences, among which are the heaving of frozen ground, breakdown of rock and concrete, sintering of snow, flow of glaciers, scavenging of atmospheric trace gases by snow and ice, and the electrification of thunderclouds. The article reviews the basic mechanisms of premelting and discusses their roles in the environmental phenomena. The principal results of numerous studies are reviewed, and trends in current research are outlined. Article Full-text available 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. Article Full-text available 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. Article Full-text available Structural transformations at interfaces are of profound fundamental interest as complex examples of phase transitions in low-dimensional systems. Despite decades of extensive research, no compelling evidence exists for structural transformations in high-angle grain boundaries in elemental systems. Here we show that the critical impediment to observations of such phase transformations in atomistic modelling has been rooted in inadequate simulation methodology. The proposed new methodology allows variations in atomic density inside the grain boundary and reveals multiple grain boundary phases with different atomic structures. Reversible first-order transformations between such phases are observed by varying temperature or injecting point defects into the boundary region. Owing to the presence of multiple metastable phases, grain boundaries can absorb significant amounts of point defects created inside the material by processes such as irradiation. We propose a novel mechanism of radiation damage healing in metals, which may guide further improvements in radiation resistance of metallic materials through grain boundary engineering. Article Full-text available A mechanism for crystal-crystal phase transformations (PTs) via surface-induced virtual premelting is justified thermodynamically and confirmed experimentally for the PTs in PbTiO3 nanofibers. When the thickness of the surface melt (which appears much below the melting temperature, especially for nano-objects) exceeds the size of the critical product nucleus, nucleation and growth of the product crystal occur. For nanowires, premelting starts near the smallest size, and hydrodynamic flow driven by reduction in the external surface leads to a large change in shape and further promotion of crystal-crystal PT. During the product crystal-growth stage, virtual melting is observed experimentally within the crystal-crystal interface. Article Full-text available 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. Article Full-text available We use the phase-field-crystal (PFC) method to investigate the equilibrium premelting and nonequilibrium shearing behaviors of$[001]$symmetric tilt grain boundaries (GBs) at high homologous temperature over the complete range of misorientation$0<\theta<90^\circ$in classical models of bcc Fe. We characterize the dependence of the premelted layer width$W$as a function of temperature and misorientation and compute the thermodynamic disjoining potential whose derivative with respect to$W$represents the structural force between crystal-melt interfaces due to the spatial overlap of density waves. The disjoining potential is also computed by molecular dynamics (MD) simulations, for quantitative comparison with PFC simulations, and coarse-grained amplitude equations (AE) derived from PFC that provide additional analytical insights. We find that, for GBs over an intermediate range of misorientation ($\theta_{\rm min}<\theta<\theta_{\rm max}$),$W$diverges as the melting temperature is approached from below, corresponding to a purely repulsive disjoining potential, while for GBs outside this range ($\theta<\theta_{\rm min}$or$\theta_{\rm max}<\theta<90^\circ$),$W$remains finite at the melting point, with its value corresponding to a shallow attractive minimum of the disjoining potential. In response to a shear stress parallel to the GB plane, GBs in PFC simulations exhibit coupled motion normal to this plane, with a discontinuous change of the coupling factor as a function of$\theta\$ that reflects a transition between two coupling modes, and/or sliding (shearing of the two grains). The coupling factor for the two coupling modes is in excellent quantitative agreement with previous theoretical predictions [J. W. Cahn, Y. Mishin, and A. Suzuki, Acta Mater. 54, 4953 (2006)].
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In this paper, the three-dimensional Landau model of austenite-martensite transformations constructed in Part I is generalized to include transformations between an arbitrary number of martensitic variants. The model can incorporate all temperature-dependent thermomechanical properties of both phases for arbitrary crystal symmetries, including higher-order elastic constants, and it correctly describes the characteristic features of stress-strain curves for shape-memory alloys and steels, namely, constant transformation strain tensors, con-stant or weakly temperature dependent stress hysteresis, and transformation at nonzero tangent moduli. Geo-metric representations of the conditions for phase equilibrium and phase transformations in six-dimensional stress space are developed. For the cubic-tetragonal phase transformation, equilibrium and transformation surfaces in three-dimensional stress space and the corresponding lines in the deviatoric-stress plane are found at various temperatures, and transformation processes are analyzed. All model parameters are obtained for the NiAl cubic-tetragonal phase transformation using the results of molecular dynamics simulations available in the literature.
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Based on the phase-field total free energy functional and free-end nudged elastic band (NEB) algorithm, a new methodology is developed for finding the saddle-point nucleus in solid-state transformations. Using cubic → tetragonal transformations in both two and three dimensions as examples, we show that the activation energy and critical nucleus configuration along the minimum energy path (MEP) can be determined accurately and efficiently using this new approach. When the elastic energy contribution is dominant, the nucleation process is found to be collective with the critical nucleus consisting of two twin-related variants. When the elastic energy contribution is relatively weak, the critical nucleus consists of a single variant, and the polytwinned structure develops during growth through a stress-induced autocatalytic process. A nontrivial two-variant critical nucleus configuration is observed at an intermediate level of the elastic energy contribution. This general method is applicable to any thermally activated process in anisotropic media, including nucleation of stacking faults and dislocation loops, voids and microcracks, and ferroelectric and ferromagnetic domains. It is able to treat nucleation events involving simultaneously displacive and diffusional components, and heterogeneous nucleation near pre-existing lattice defects.
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A new model for homogeneous nucleation of structural phase transformations, which can span the range of nucleation from classical to nonclassical, is presented. This model is extended from the classical nucleation theory by introducing driving-force dependencies into the interfacial free energy, the misfit strain energy, and the nucleus chemical free-energy change in order to capture the nonclassical nucleation phenomena. The driving-force dependencies are determined by matching the asymptotic solutions of the new model for the nucleus size and the nucleation energy barrier to the corresponding asymptotic solutions of the Landau-Ginzburg model for nucleation of solid-state phase transformations in the vicinity of lattice instability. Thus, no additional material parameters other than those of the classical nucleation theory and the Landau-Ginzburg model are required, and nonclassical nucleation behavior can be easily predicted based on the well-developed analytical solutions of the classical nucleation model. A comparison of the new model to the Landau-Ginzburg model for homogeneous nucleation of a dilatational transformation is presented as a benchmark example. An application to homogeneous nucleation of a cubic-to-tegragonal transformation is presented to illustrate the capability of this model. The nonclassical homogeneous nucleation behavior of the experimentally studied fcc → bcc transformation in the Fe-Co system is examined by the new model, which predicts a 20 pct reduction in the critical driving force for homogeneous nucleation.
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An improved way of estimating the local tangent in the nudged elastic band method for finding minimum energy paths is presented. In systems where the force along the minimum energy path is large compared to the restoring force perpendicular to the path and when many images of the system are included in the elastic band, kinks can develop and prevent the band from converging to the minimum energy path. We show how the kinks arise and present an improved way of estimating the local tangent which solves the problem. The task of finding an accurate energy and configuration for the saddle point is also discussed and examples given where a complementary method, the dimer method, is used to efficiently converge to the saddle point. Both methods only require the first derivative of the energy and can, therefore, easily be applied in plane wave based density-functional theory calculations. Examples are given from studies of the exchange diffusion mechanism in a Si crystal, Al addimer formation on the Al(100) surface, and dissociative adsorption of CH4 on an Ir(111) surface. (C) 2000 American Institute of Physics. [S0021-9606(00)70546-0].
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A continuum thermomechanochemical model of the behavior of a plastic-bonded explosive (PBX) 9501 formulation consisting of the energetic crystal octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) embedded in a polymeric binder is developed. Our main focus is on the study of the β↔δ phase transformations (PTs) in crystalline HMX under a complex pressure-temperature path. To reproduce the pressure-temperature path, in particular during heating of PBX inside of a rigid cylinder, the β↔δ PTs in HMX are coupled to 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. A fully physically based thermodynamic and kinetic model of the β↔δ PT in HMX crystal is developed. It is based on a suggested nucleation mechanism via melt mediated nanocluster transformation and the recently revealed growth mechanism via internal stress-induced virtual melting. During the nucleation, nanosize clusters of the β phase dissolve in a molten binder and transform diffusionally into δ phase clusters. During the interface propagation, internal stresses induced by transformation strain cause the melting of the stressed δ phase much below (120 K) the melting temperature and its immediate resolidification into the unstressed δ phase. These mechanisms explain numerous puzzles of HMX polymorphism and result in overall transformation kinetics that is in good agreement with experiments. Simple phenomenological equations for kinetics of chemical decomposition of the HMX and the binder are in good correspondence with experiments as well. A continuum deformation model is developed in two steps. The geometrically linear (small strain) theory is use- d to prove that the internal stresses and macroscopic shear stresses are negligible. Then a large strain theory is developed under hydrostatic loading. The developed continuum thermomechanochemical model is applied in the accompanying paper [V. I. Levitas, B. F. Henson, L. B. Smilowitz, D. K. Zerkle, and B. W. Asay, J. Appl. Phys. (submitted)] to modeling the heating of PBX inside of a rigid cylinder.
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It is shown that in any two-phase mixture of fluids near their critical point, contact angles against any third phase become zero in that one of the critical phases completely wets the third phase and excludes contact with the other critical phase. A surface layer of the wetting phase continues to exist under a range of conditions when this phase is no longer stable as a bulk. At some temperature below the critical, this perfect wetting terminates in what is described as a first-order transition of the surface. This surface first-order transition may exhibit its own critical point. The theory is qualitatively in agreement with observations.
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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.
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To investigate the effect of dislocations on grain growth in polycrystals, two sets of experiments were performed using SrTiO3 single crystals and SrTiO3 powder compacts. In the first set, with single crystals embedded in 2.0-mol%-TiO2-exess powder compacts, the growth of the single crystal was not affected by dislocations at 1300 °C, below the eutectic temperature, while it was enhanced by dislocations at 1470 °C, above the eutectic temperature, where the grain boundaries are wetted by an amorphous phase. In the second set, with 0.5-mol%-Nb2O5-doped powder compacts, the single crystal grew considerably into the SrTiO3 matrix grains in the presence of an amorphous film between the grains at 1470 °C both in 95N2-5H2 and in air, similar to the case of the TiO2-exess samples. However, when annealed at 1470 °C in 95N2-5H2 after pre-annealing at 1250 °C in 95N2-5H2, the amorphous phase remained at triple junctions and did not penetrate the grain boundaries, implying that this boundary configuration also is thermodynamically stable above the eutectic temperature. In this case, growth of the embedded single crystal was insignificant in spite of the presence of dislocations. These experimental observations indicate that the growth of SrTiO3 is promoted by dislocations only when an intergranular amorphous film is present at grain boundaries. The apparent ineffectiveness of dislocations on grain growth promotion without an intergranular amorphous film is discussed in terms of a low fraction of facetted grain boundaries and grain boundary drag by triple junctions.
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Plastic-bonded explosives are heterogeneous materials. Initiation of a PBX is dominated by hot spots which are subgrain in size. Conse- quently, simulations of hot spots require resolving individual explosive grains. Computations on the grain scale are called meso-scale simu- lations. At the grain level an explosive is crystalline and, by its very nature, anisotropic. This has an eect on the dissipative mechanisms leading to the formation and evolution of hot spots. Here we focus on the explosive HMX. Properties of HMX needed for meso-scale simu- lations are discussed and the available data are reviewed.
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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.
Article
Anisotropic formation of equilibrium-thickness Bi2O3-enriched surficial amorphous films (SAFs) on ZnO has been documented [Luo J, Chiang Y-M. Acta Mater 2000;48:4501]. This study further explores anisotropic wetting of ZnO single crystals by Bi2O3-rich liquid with and without SAFs. For Bi2O3 on the ZnO {112¯0} surfaces wherein nanometer-thick SAFs are present in equilibrium with partial wetting drops, the measured (advancing) contact angle decreases with increasing temperature, but it stabilizes at ∼6° above ∼860°C. In contrast, the contact angle is virtually a constant on the {11¯00} surfaces where SAFs are not present. This observation suggests that wetting in the presence of nanoscale SAFs follows a generalized Cahn wetting model. Faceted ridge formation at the triple lines and associated pinning effects are observed. Observation and analysis of unique two-stage isothermal drop receding kinetics show that complete wetting does not occur up to 1050°C and that the receding contact angle is estimated to be ∼4° via two methods. Quantitative evaluation of a thermodynamic model shows that the observed “residual” contact angle of ∼4–6° and the extended SAF stability can be explained on the basis of a significant attractive London dispersion force.
Article
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.
Article
It is shown that the free energy of a volume V of an isotropic system of nonuniform composition or density is given by : NV∫V [f0(c)+κ(▽c)2]dV, where NV is the number of molecules per unit volume, ▽c the composition or density gradient, f0 the free energy per molecule of a homogeneous system, and κ a parameter which, in general, may be dependent on c and temperature, but for a regular solution is a constant which can be evaluated. This expression is used to determine the properties of a flat interface between two coexisting phases. In particular, we find that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc, and that at a temperature T just below Tc the interfacial free energy σ is proportional to (Tc−T.
Article
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.
Article
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.
Article
The stabilization of nanoscale surficial amorphous films (SAFs) for Bi2O3 on ZnO, VOx on TiO2, SiOx on Si, and several other oxide systems provides evidence for the existence of prewetting phenomena with analogies in water and other simple systems, as well as the stabilization of intergranular amorphous films in ceramics. Experimental results show that in the subeutectic regime, the equilibrium film thickness decreases monotonically with decreasing temperature until it vanishes at a dewetting (prewetting) temperature. With increasing temperatures, nanometer-thick SAFs persist into a solid-liquid coexistence regime, in equilibrium with partial-wetting drops, with a gradual decrease in the macroscopic contact angle upon heating. The presence of an attractive dispersion force can significantly delay or inhibit the (otherwise expected) occurrence of complete wetting at higher temperatures. The equilibrium thickness of SAFs is explained from a balance between several interfacial interactions, including dispers...
Article
The London dispersion forces, along with the Debye and Keesom forces, constitute the long-range van der Waals forces. London's and Hamaker's work on the point-to-point dispersion interaction and Lifshitz's development of the continuum theory of dispersion are the foundations of our understanding of dispersion forces. Dispersion forces are present for all materials and are intrinsically related to the optical properties and the underlying interband electronic structures of materials. The force law scaling constant of the dispersion force, known as the Hamaker constant, can be determined from spectral or parametric optical properties of materials, combined with knowledge of the configuration of the materials. With recent access to new experimental and ab initio tools for determination of optical properties of materials, dispersion force research has new opportunities for detailed studies. Opportunities include development of improved index approximations and parametric representations of the optical properties for estimation of Hamaker constants. Expanded databases of London dispersion spectra of materials will permit accurate estimation of both nonretarded and retarded dispersion forces in complex configurations. Development of solutions for generalized multilayer configurations of materials are needed for the treatment of more-complex problems, such as graded interfaces. Dispersion forces can play a critical role in materials applications. Typically, they are a component with other forces in a force balance, and it is this balance that dictates the resulting behavior. The ubiquitous nature of the London dispersion forces makes them a factor in a wide spectrum of problems; they have been in evidence since the pioneering work of Young and Laplace on wetting, contact angles, and surface energies. Additional applications include the interparticle forces that can be measured by direct techniques, such as atomic force microscopy. London dispersion forces are important in both adhesion and in sintering, where the detailed shape at the crack tip and at the sintering neck can be controlled by the dispersion forces. Dispersion forces have an important role in the properties of numerous ceramics that contain intergranular films, and here the opportunity exists for the development of an integrated understanding of intergranular films that encompasses dispersion forces, segregation, multilayer adsorption, and structure. The intrinsic length scale at which there is a transition from the continuum perspective (dispersion forces) to the atomistic perspective (encompassing interatomic bonds) is critical in many materials problems, and the relationship of dispersion forces and intergranular films may represent an important opportunity to probe this topic. The London dispersion force is retarded at large separations, where the transit time of the electromagnetic interaction must be considered explicitly. Novel phenomena, such as equilibrium surficial films and bimodal wetting/dewetting, can result in materials systems when the characteristic wavelengths of the interatomic bonds and the physical interlayer thicknesses lead to a change in the sign of the dispersion force. Use of these novel phenomena in future materials applications provides interesting opportunities in materials design.
Article
Disordered films of 1–2 nm thickness have been observed on the surfaces of binary oxide systems in liquid–solid and solid–state equilibrium. The systems Bi 2 O 3 –ZnO, Bi 2 O 3 –Fe 2 O 3 , WO 3 –TiO 2 and MoO 3 – Al 2 O 3 have been examined. Sufficient data now exist to interpret the observed films as surface phases of thermodynamically determined thickness and composition, coexisting at equilibrium with one or more bulk phases. Phenomenological comparisons to the formation of equilibrium-thickness intergranular films, multi-layer adsorption, prewetting and surface melting are made. A thermodynamic model that allows the prediction of systems likely to form stable films is proposed, in which key variables are the relative surface energies, the sign and strength of the dispersion interaction, the extent of ordering (epitaxy) at the crystal–film interface, and the free energy of amorphization and mixing. It is suggested that the relative interfacial energies and volume thermodynamic terms are always important contributions, while the dispersion interaction is relatively unimportant well below the solidus temperature, but may be critical close to and above the solidus tempera-ture. © 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
Article
The conditions for structural transitions at the core of a grain boundary separating two crystals was investigated with a diffuse interface model that incorporates disorder and crystal orientation [Kobayashi , Physica D 140, 141 (2000)]. The model predicts that limited structural disorder near the grain boundary core can be favorable below the melting point. This disordered material is a precursor to a liquid phase and therefore the model represents grain boundary premelting. This model is shown to be isomorphic to Cahn's critical point wetting theory [J.W. Cahn, J. Chem. Phys. 66, 3667 (1977)] and predicts first- and higher-order structural grain boundary transitions. A graphical construction predicts the equilibrium grain boundary core disorder, the grain boundary energy density, and the relative stability of multiple grain boundary "complexions." The graphical construction permits qualitative inference of the effect of model properties, such as empirical homogeneous free energy density and assumed gradient energy coefficients, on properties. A quantitative criterion is derived which determines whether a first-order grain boundary transition will occur. In those systems where first-order transition does occur, they are limited to intermediate grain-boundary misorientations and to a limited range of temperatures below the melting point. Larger misorientations lead to continuously increasing disorder up to the melting point at which the disorder matches a liquid state. Smaller misorientation continuously disorder but are not completely disordered at the melting point. Characteristic grain boundary widths and energies are calculated as is the width's divergence behavior at the melting point. Grain boundary phase diagrams are produced. The relations between the model's predictions and atomistic simulations and with experimental observations are examined.
Article
The glass-melt penetration of dense TiO2 polycrystals (where the grains are initially in contact) and the sintering of SiO2-coated TiO2 powders (where the grains are initially separated) have been used to investigate the influence of initial particle separation on final microstructures. Grains that were initially in sintered contact resisted penetration by the liquid and retained crystalline boundaries. However, grains that were initially separated by liquid had a tendency to form a final state where they were separated by a glass film ∼1 nm thick. The results imply that crystalline grain boundaries and those that contain a thin amorphous film represent two local thermodynamic minima that are separated by an energy barrier. These observations are in agreement with a thermodynamic model that predicted such a barrier for this system, and these observations show that the stable phase distribution in liquid-phase-sintered ceramics can be dependent on the path.
Article
In this paper the well-known modified (underrelaxed, damped) Newton method is extended in such a way as to apply to the solution of ill-conditioned systems of nonlinear equations, i.e. systems having a nearly singular Jacobian at some iterate. A special technique also derived herein may be useful, if only bad initial guesses of the solution point are available. Difficulties that arose previously in the numerical solution of nonlinear two-point boundary value problems by multiple shooting techniques can be removed by means of the results presented below.
Article
A one-dimensional static Ginzburg-Landau theory for the martensitic phase transitions in shape-memory alloys is developed. From the equilibrium conditions the structure of static domain walls of martensite-martensite as well as of martensite-austenite type is calculated. In the finite crystal a discrete spectrum of domain structures results whereas in an unbounded crystal there are four types of domain walls. For each type of walls the energy is calculated.
Article
In this work the structure and chemistry of intergranular films at metal–ceramic interfaces was investigated via detailed microstructural characterization of model metal–Al2O3 nanocomposites. We report here experimental results indicating the formation and stability of equilibrium nanometer-thick films at metal–ceramic interfaces. Thin ∼1 nm interface films were observed for two different metal–alumina systems (Ni and Cu) doped with glass-forming additives. High spatial resolution energy dispersive spectroscopy showed a difference in the chemical composition of the films at Ni–alumina and Cu–alumina interfaces. These results may have immediate ramifications on structural and functional properties of metal–ceramic interfaces.
Article
A two-dimensional frame-invariant phase field model of grain boundaries is developed. One-dimensional analytical solutions for a stable grain boundary in a bicrystal are obtained, and equilibrium energies are computed. With an appropriate choice of functional dependencies, the grain boundary energy takes the same analytic form as the microscopic (dislocation) model of Read and Shockley [W.T. Read, W. Shockley, Phys. Rev. 78 (1950) 275]. In addition, dynamic (one-dimensional) solutions are presented, showing rotation of a small grain between two pinned grains and the shrinkage and rotation of a circular grains embedded in a larger crystal.