Phase field approach for stress- and temperature-induced phase transformations that satisfies lattice instability conditions. Part I. general theory

Recently, results of molecular dynamics (MD) simulations were obtained for the crystal lattice instability conditions for the phase transformations (PTs) between semiconducting Si I and metallic Si II under action of all six components of the stress tensor (Levitas et al. (2017a, b)). These conditions are linear in terms of stresses normal to the cubic faces of Si I and are independent of the shear stresses. In the current paper, we (a) formulated the requirements for the thermodynamic potential and transformation deformation gradient tensors and (b) developed a phase field approach (PFA) for the stress-induced martensitic PTs for large strains while allowing for interfacial stresses, which are consistent with the obtained instability conditions. The general system of equations for coupled PFA and nonlinear elasticity is presented. Crystal lattice instability criteria are derived within a PFA, and it is proven that they are independent of the prescribed stress measure. In order to reproduce the lattice instability conditions obtained with MD: (a) one has to use the fifth degree polynomial interpolation functions of the order parameter for all material parameters; (b) each component of the transformation strain tensor should have a different interpolation functions; and (c) the interpolation functions for tensors of the elastic moduli of all ranks should have zero second derivatives for the parent and product phases, so that terms with elastic moduli, which are nonlinear in stresses, do not contribute to the lattice instability conditions. Specific interpolation and double-well functions have been derived for all parts of the Helmholtz free energy and for two models for the transformation deformation gradient. For these models, explicit expressions for the Ginzburg-Landau equations and lattice instability conditions are derived. Material parameters have been calibrated using results of MD simulations. In Part II of this paper, the developed model is further refined and studied, and applied for the finite element simulations of the nanostructure evolution in Si under triaxial loading

Unambiguous Gibbs dividing surface for nonequilibrium finite-width interface: Static equivalence approach

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

High-Pressure Phase Transformations under Severe Plastic Deformation by Torsion in Rotational Anvils

Numerous experiments have documented that combination of severe plastic deformation and high mean pressure during high-pressure torsion in rotational metallic, ceramic, or diamond anvils produces various important mechanochemical effects. We will focus here on four of these: plastic deformation (a) significantly reduces pressure for initiation and completion of phase transformations (PTs), (b) leads to discovery of hidden metastable phases and compounds, (c) reduces PT pressure hysteresis, and (d) substitutes a reversible PT with irreversible PT. The goal of this review is to summarize our current understanding of the underlying phenomena based on multiscale atomistic and continuum theories and computational modeling. Recent atomistic simulations provide conditions for initiation of PTs in a defect-free lattice as a function of the general stress tensor. These conditions (a) allow one to determine stress states that significantly decrease the transformation pressure and (b) determine whether the given phase can, in principle, be preserved at ambient pressure. Nanoscale mechanisms of phase nucleation at plastic-strain-induced defects are studied analytically and by utilizing advanced phase field theory and simulations. It is demonstrated that the concentration of all components of the stress tensor near the tip of the dislocation pileup may decrease nucleation pressure by a factor of ten or more. These results are incorporated into the microscale analytical kinetic equation for strain-induced PTs. This equation is part of a macroscale geometrically-nonlinear model for combined plastic flow and PT. This model is used for finite-element simulations of plastic deformations and PT in a sample under torsion in a rotational anvil device. Numerous experimentally-observed phenomena are reproduced, and new effects are predicted and then confirmed experimentally. Combination of the results on all four scales suggests novel synthetic routes for new or known high-pressure phases (HPPs), experimental characterization of strain-induced PTs under high-pressure during torsion under elevated pressure

High pressure phase transformations revisited

High pressure phase transformations play an important role in the search for new materials and material synthesis, as well as in geophysics. However, they are poorly characterized, and phase transformation pressure and pressure hysteresis vary drastically in experiments of different researchers, with different pressure transmitting media, and with different material suppliers. Here we review the current state, challenges in studying phase transformations under high pressure, and the possible ways in overcoming the challenges. This field is critically compared with fields of phase transformations under normal pressure in steels and shape memory alloys, as well as plastic deformation of materials. The main reason for the above mentioned discrepancy is the lack of understanding that there is a fundamental difference between pressure-induced transformations under hydrostatic conditions, stress-induced transformations under nonhydrostatic conditions below yield, and strain-induced transformations during plastic flow. Each of these types of transformations has different mechanisms and requires a completely different thermodynamic and kinetic description and experimental characterization. In comparison with other fields the following challenges are indicated for high pressure phase transformation: (a) initial and evolving microstructure is not included in characterization of transformations; (b) continuum theory is poorly developed; (c) heterogeneous stress and strain fields in experiments are not determined, which leads to confusing material transformational properties with a system behavior. Some ways to advance the field of high pressure phase transformations are suggested. The key points are: (a) to take into account plastic deformations and microstructure evolution during transformations; (b) to formulate phase transformation criteria and kinetic equations in terms of stress and plastic strain tensors (instead of pressure alone); (c) to develop multiscale continuum theories, and (d) to couple experimental, theoretical, and computational studies of the behavior of a tested sample to extract information about fields of stress and strain tensors and concentration of high pressure phase, transformation criteria and kinetics. The ideal characterization should contain complete information which is required for simulation of the same experiments

Phase transformations, fracture, and other structural changes in inelastic materials

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.Comment: 132 pages, 29 figures, 3 tables. Extended version of paper: Levitas V.I. Phase transformations, fracture, and other structural changes in inelastic materials. International Journal of Plasticity, 2021, Vol. 140, 102914, 51 pp., invited review. DOI 10.1016/j.ijplas.2020.10291

Phase field approach to martensitic phase transformations with large strains and interface stresses

Thermodynamically consistent phase field theory for multivariant martensitic transformations, which includes large strains and interface stresses, is developed. Theory is formulated in a way that some geometrically nonlinear terms do not disappear in the geometrically linear limit, which in particular allowed us to introduce the expression for the interface stresses consistent with the sharp interface approach. Namely, for the propagating nonequilibrium interface, a structural part of the interface Cauchy stresses reduces to a biaxial tension with the magnitude equal to the temperature-dependent interface energy. Additional elastic and viscous contributions to the interface stresses do not require separate constitutive equations and are determined by solution of the coupled system of phase field and mechanics equations. Ginzburg-Landau equations are derived for the evolution of the order parameters and temperature evolution equation. Boundary conditions for the order parameters include variation of the surface energy during phase transformation. Because elastic energy is defined per unit volume of unloaded (intermediate) configuration, additional contributions to the Ginzburg-Landau equations and the expression for entropy appear, which are important even for small strains. A complete system of equations for fifth- and sixth-degree polynomials in terms of the order parameters is presented in the reference and actual configurations. An analytical solution for the propagating interface and critical martensitic nucleus which includes distribution of components of interface stresses has been found for the sixth-degree polynomial. This required resolving a fundamental problem in the interface and surface science: how to define the Gibbsian dividing surface, i.e., the sharp interface equivalent to the finite-width interface. An unexpected, simple solution was found utilizing the principle of static equivalence. In fact, even two equations for determination of the dividing surface follow from the equivalence of the resultant force and zero-moment condition. For the obtained analytical solution for the propagating interface, both conditions determine the same dividing surface, i.e., the theory is noncontradictory. A similar formalism can be developed for the phase field approach to diffusive phase transformations described by the Cahn-Hilliard equation, twinning, dislocations, fracture, and their interaction

Sublimation via virtual melting inside an elastoplastic material

Thermodynamic, kinetic, and mechanical approaches for sublimation inside elastoplastic material via intermediate (virtual) melting under tensile pressure are developed for a spherical nucleus. Virtual melting represents the appearance of subcritical liquid drop that immediately transforms to gas bubble. The variety of mechanisms and transformation paths are revealed in different pressure ranges. The radius of the critical gas nucleus differs from the classical one because elastic energy of melt is size dependent due to surface tension. Our developed approach can be extended for various structural changes in nanoparticles within a void inside elastoplastic material and two-stage and multistage nucleation processes. Universal mechanical gas bubble instability is revealed

Coherent solid/liquid interface with stress relaxation in a phase-field approach to the melting/solidification 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

Size and mechanics effects in surface-induced melting of nanoparticles

Various melting-related phenomena (like surface melting, size dependence of melting temperature, melting of few nm-size particles and overheating at a very fast heating rate) are of great fundamental and applied interest, although the corresponding theory is still lacking. Here we develop an advanced phase-field theory of melting coupled to mechanics, which resolves numerous existing contradictions and allowed us to reveal exciting features of melting problems. The necessity of introducing an unexpected concept, namely, coherent solid-melt interface with uniaxial transformation strain, is demonstrated. A crossover in temperature dependence of interface energy for radii below 20 nm is found. Surface-induced premelting and barrierless melt nucleation for nanoparticles down to 1 nm radius is studied, and the importance of advanced mechanics is demonstrated. Our model describes well experimental data on the width of the molten layer versus temperature for the Al plane surface and on melting temperature versus particle radius

Anisotropic Compositional Expansion and Chemical Potential for Amorphous Lithiated Silicon under Stress Tensor

Si is a promising anode material for Li-ion batteries, since it absorbs large amounts of Li. However, insertion of Li leads to 334% of volumetric expansion, huge stresses, and fracture; it can be suppressed by utilizing nanoscale anode structures. Continuum approaches to stress relaxation in LixSi, based on plasticity theory, are unrealistic, because the yield strength of LixSi is much higher than the generated stresses. Here, we suggest that stress relaxation is due to anisotropic (tensorial) compositional straining that occurs during insertion-extraction at any deviatoric stresses. Developed theory describes known experimental and atomistic simulation data. A method to reduce stresses is predicted and confirmed by known experiments. Chemical potential has an additional contribution due to deviatoric stresses, which leads to increases in the driving force both for insertion and extraction. The results have conceptual and general character and are applicable to any material systems