Fig 1 - uploaded by Valery I. Levitas

Content may be subject to copyright.

# Gibbs potential G ˜ (r,y,U,W) for HMX at y = y 21 e and principal stresses of r = {1, À2, À4} MPa for different y md c = y 10 c and y mb c = y 20 c values, in the polar system of the order parameters U and W (a-d, f). The 3D plot of the potential surface is shown in (e) along with its contour plot, for the same critical temperatures as plotted in (a).

Source publication

A phase-field approach for phase transformations (PTs) between three different phases at nonequilibrium temperatures is developed. It includes advanced mechanics, thermodynamically consistent interfacial stresses, and interface interactions. A thermodynamic Landau-Ginzburg potential developed in terms of polar order parameters satisfies the desired...

## Contexts in source publication

**Context 1**

... describes the PT between phase 0 and any of phases 1 or 2 along the corresponding basis vectors. The angular order parameter W, where pW/2 is the angle between the radius ! and the positive first basis vector, describes the PT between phases 1 and 2. Examples of the thermodynamic potentials in terms of the polar order parameters are presented in Fig. ...

**Context 2**

... the only solution with the IM exists and U min weakly depends on k d . Simulations are performed in a wide range of temperatures (significantly below the melting tempera- ture) for two small interfacial interactions (a 0 = 0.01 and 0.1), without and with interfacial tension, and disordering, energy, width, and velocity of the IM are reported in Fig. ...

**Context 3**

... tension promotes the formation of the IM, espe- cially at larger interface interactions, in terms of increasing disordering, width, and velocity of the SMS interface. At higher temperatures, increasing interface interactions promotes the formation of more disordered IM, while at low temperatures it results in less disordered IM (Fig. 10a). The critical temperature of the intersection point decreases as interfacial tension is included. The formation of almost complete IM (U min B 0.07) with a width of B1.4 nm is captured at B0.8y 21 e , which is 185 (Kelvin) below the melting ...

**Context 4**

... DG = (DG y 10 + DG y 20 )/2 is proportional to temperature. Solid-solid interface velocity for models with and without mechanics are also presented in Fig. 10d for different tempera- tures. Simulation results for the model without mechanics match the analytical solution, eqn (A.1) 1 , ESI, † which indicates the validity of the developed numerical model. The SS interface temperature, indicating that the interfacial tension slightly increases the IM energy. (c) Normalized IM steady-state width, ...

**Context 5**

... low-temperature phase S 2 has larger volumetric transformation strain with respect to melt than high-temperature phase S 1 and possesses higher stresses s z (Fig. 9) than phase S 1 (due to plane strain conditions), elastic energy promotes its transformation to S 1 and reduces phase equilibrium temperature between them. That is why all lines in Fig. 10d in which mechanics is taken into account do not intersect the zero-velocity line in the temperature range under study. The presence of the IM reduces both the magnitude of the interface velocity and its temperature dependence, as well as introduces some nonlinearity in the temperature depen- dence. Larger interface interaction ...

**Context 6**

... cylindrical sample is considered with a radius of 20 nm and infinite size in the z-direction, which is modeled using COMSOL's ''perfectly matched'' layers at both ends. Two different solutions are found, which are designated CN 1 and CN 2 , that correspond to the CN at the center of the sample and the CN at its surface ( Fig. 11 and 12). The values of U min for the CN 1 for models without and with interfacial tension are 0.30 and 0.19, respectively, which means that the interfacial stresses suppress nucleation and require more-complete melt to be a CN 1 . For both critical nuclei, the ellipsoidal CN 1 and toroidal CN 2 , the aspect ratio of the CN is larger for the ...

**Context 7**

... results presented in Table 3 indicate that all activation energies for the model with interfacial tension are larger than those with neglected interface stresses, i.e., interfacial stresses suppress both the appearance and disappearance of the IM via the thermally activated process. For the nucleation of the IM, Fig. 12 Effects of the interfacial tension on the structure and morphology of CN 2 . The level-plots of solutions U CN 2 (r,z) and W CN 2 (r,z) for a model without (a and b) and with interfacial tension (c and d) are presented. The results are obtained for a 0 = 0.01, k d = 0.7, and k E = 2.6, at y = y 21 e = 432 K. The meaning of the insets ...

**Context 8**

... of the interfacial tension on the structure and morphology of CN 2 . The level-plots of solutions U CN 2 (r,z) and W CN 2 (r,z) for a model without (a and b) and with interfacial tension (c and d) are presented. The results are obtained for a 0 = 0.01, k d = 0.7, and k E = 2.6, at y = y 21 e = 432 K. The meaning of the insets is the same as in Fig. 11. Table 1 Energies per unit area of the ground states for the stationary S 1 S 2 (i.e., E 21 = C SS /A int ) and S 1 MS 2 (i.e., E SMS = C SMS /A int ) interfaces and interface areas for k d = 0.7, k E = 2.6, and y = y e = 432 (Kelvin). Contributions to each energy due to the thermal part C y , gradient energy C r , and elastic energy C ...

## Similar publications

The temperature dependences of the ratios of the average surface tension grain boundaries in a- and b-phase to the average surface tension of a/b interphase boundaries are determined in a wide temperature range in a + b region of Ti– 6Al–4V alloy. The temperature dependences of the average surface tension of grain boundaries in a- and b-phase and a...

## Citations

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

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

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

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

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

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

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

Multimetallic layered composites (MMLCs) have shown an excellent potential for application under extreme environments, e.g., accident-tolerant fuel cladding, because of their low oxidation tendency and high corrosion resistance. Interfacial phases or complexions in nanocrystalline materials accelerate the annihilation of defects and enhance the radiation resistance of materials, making MMLCs with engineered interlayer phases compelling to deploy in extreme conditions. However, implementation of MMLCs in full capacity remained a challenge due to a lack of fundamental understanding of the underlying mechanisms governing the characteristics of the interface between the metallic layers. The precise role of interlayer phases in MMLCs and their interaction with defects, specifically under extreme conditions, is still unexplored. Pursuing atomistic simulations for various Inconel-Ni MMLCs model materials, we revealed accelerated defect mobility in interlayers with larger crystalline misorientation and the inverse relationship between the interface sink strength to the misorientation angle. Furthermore, we found a linear relation between interlayer misorientation angle with the density of radiation-induced defects and radiation enhanced displacements. Finally, our results indicate that radiation-induced material degradation is accelerated by the higher defect formation tendency of MMLCs with a high-angle interlayer interface.
Data availability
All data that was obtained during this project is available from the authors.

... The exhaustive trial and error experimentations are prohibited due to significant time and costs. Analytical and computational models provide an alternative approach for designing and optimizing process parameters and alloy compositions [1,[14][15][16][17][18][19][20][42][43][44][45][46][47]. ...

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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