The grain boundary (GB) mobility relates GB velocity to the thermodynamic driving forces and is central to our understanding of microstructure evolution in polycrystals. Recent molecular dynamics (MD) and experimental studies have shown that the temperature-dependence of the GB mobility is much more varied than is commonly thought. GB mobility may increase, decrease, remain constant or show multiple peaks with increasing temperature. We propose a mechanistic model for GB migration, based on the formation and migration of line defects (disconnection) within the GB. We implement this model in a kinetic Monte Carlo and statistical mechanics framework; the results capture all of these observed temperature dependencies and are shown to be in quantitative agreement with each other and direct MD simulations of GB migration for a set of specific grain boundaries. Examination of the dependence of GB mobility on disconnection mode and temperature provides new insight into how GBs migrate in polycrystalline materials.
Conventional grain growth is rate-limited by the mobility of grain boundary. To describe similar phenomena limited by the mobility of other grain junctions, we have developed a general theory allowing for size-dependent mobility and its statistical variance. We obtained analytic solutions for the steady-state size distribution and the growth exponent, defined as (grain size)n ~ time, down to n=1, which arises when the mobility of three-grain lines is rate-limiting. When the mobility of four-grain junctions is rate-limiting, exponential growth and a bifurcating size distribution result. These s…
Recent grain growth experiments have revealed that the same type of grain boundary can have very different mobilities depending on its local microstructure. In this work, we use molecular dynamics simulations to quantify uncertainty in the reduced mobility of curved grain boundaries for different types of boundary conditions and over a range of initial velocity seeds. We consider cylindrical island grains in Ni with a [001] tilt axis as a model system. Unexpectedly, we find large variation in the reduced mobility of curved grain boundaries depending on both the imposed constraints and the init…
Faceted grain boundaries, where grain boundary area is increased in the name of producing low-energy segments, can exhibit new and unexpected migration trends. For example, several faceted $Σ$3 boundaries have demonstrated anti-thermal and thermally damped mobility. $Σ$11 <110> tilt boundaries represent another promising but relatively unexplored set of interfaces, with a (113) low-energy plane that can lead to faceting. In this study, molecular dynamics simulations are used to study grain boundary migration of an asymmetric $Σ$11 <110> grain boundary in two face centered cubic metals. Mobilit…
Grain boundary (GB) mobility has been conventionally computed as a single value; however, a recent study has suggested that GB mobility should be expressed as a tensor. In this work, by using atomistic simulations, the concept of GB mobility being applied to the shear direction was re-examined and it is found that it follows the same physical rule as the conventionally defined GB mobility based on the normal direction. The interface random walk method was then used to compute the intrinsic GB mobility tensor at the zero-driving force limit. In order to compute the off-diagonal elements of the …
Stagnation of grain growth is often attributed to impurity segregation. Yttria-stabilized cubic zirconia does not evidence any segregation-induced slowdown, as its grain growth obeys the parabolic law when the grain size increases by more than one order of magnitude. However, lowering the temperature below 1300 oC triggers an abrupt slowdown, constraining the average grains to grow by less than 0.5 $μ$m in 1000 h despite a relatively large driving force imparted in the fine grains of ~0.5 $μ$m. Yet isolated pockets of abnormally large grains, along with pockets of abnormally small grains, emer…
We report field-effect transistors (FETs) with single-crystal molybdenum disulfide (MoS2) channels synthesized by chemical vapor deposition (CVD). For a bilayer MoS2 FET, the mobility is ~17 cm2V-1s-1 and the on/off current ratio is ~108, which are much higher than those of FETs based on CVD polycrystalline MoS2 films. By avoiding the detrimental effects of the grain boundaries and the contamination introduced by the transfer process, the quality of the CVD MoS2 atomic layers deposited directly on SiO2 is comparable to the best exfoliated MoS2 flakes. It shows that CVD is a viable method to sy…
Grain boundary (GB) kinetics is important for many applications in 2d materials and metal thin films. To study how the substrate shape affects GB mobility and kinetics, we develop a kinetic Monte Carlo (kMC) simulation method and an analytical model for GBs on the curved substrate by combining disconnection theory and by Foppl von Karman equations. Using sinusoidal MoS2 as an example, we can increase its GB mobility more than 50 times by changing substrate shape amplitudes and periods. We find that amplitude change GB mobility exponentially while wave vector change GB mobility linearly. The si…
Recent advances in the numerical representation of materials opened the way for successful machine learning of grain boundary (GB) energies and the classification of GB mobility and shear coupling. Two representations were needed for these machine learning applications: 1) the ASR representation, based on averaged local environment descriptors; and 2) the LER descriptor, based on fractions of globally unique local environments within the entire GB system. We present a detailed tutorial on how to construct these two representations to learn energy, mobility and shear coupling. Additionally, we …
We explore the effects of surface tension and mobility models in simulations of grain growth using threshold dynamics algorithms that allow performing large scale simulations, while naturally capturing the Herring angle condition at junctions and automatically handling topological transitions. The results indicate that in two dimensions, the different surface tension / mobility models considered do not play a significant role in the stationary grain size distribution. However, in three dimensions, there is a substantial difference between the distributions obtained from the same three models, …
The temperature dependence of experimental charge carrier mobility is commonly used as a predictor of the dominant carrier scattering mechanism in semiconductors, particularly in thermoelectric applications. In this work, we critically evaluate whether this practice is well founded. A review of 47 state-of-the-art mobility calculations reveals no correlation between the major scattering mechanism and the temperature trend of mobility. Instead, we demonstrate that the phonon frequencies are the prevailing driving forces behind the temperature dependence and can cause it to vary between $T^{-1}$…
Faceted grain boundaries exhibit unusual segregation and migration tendencies. To gain a deeper understanding of how solute atoms interact with faceted interfacial structures during migration, this study probes the migration behavior of a faceted $Σ$11 boundary in Cu doped with Ag atoms. The solutes are found to segregate to the facet with more free volume and strongly reduce boundary velocity in one migration direction, but not the other, due to the presence of a directionally-dependent motion mechanism that can escape solute pinning and therefore speed up migration. Hence, a new mechanism of…
Helium diffusion in metals is the basic requirement of nucleation and growth of bubble, which gives rise to adverse degradation effects on mechanical properties of structural materials in reactors under irradiation. Multi-scale modeling scheme has been developed to study effects of helium on the long-term microstructural evolution. However, the implementation of Arrhenius law based on the quasi-equilibrium reaction process is not appropriate to predict the migration behavior of helium in metals due to low-energy barrier. A coarse-grained formula is required to incorporate the non-equilibrium n…
The mobility of charge carriers in a semiconductor nanowire is explored as a function of increasing radius, assuming low temperatures where impurity scattering dominates. The competition between increased cross-section and the concurrent increase in available scattering channels causes strongly non-monotonic dependence of the mobility on the radius. The inter-band scattering causes sharp declines in the mobility at the wire radii at which each new channel becomes available. At intermediate radii with the number of channels unchanged the mobility is seen to maintain an exponential growth even w…
We present a case study of the complex temperature dependence of grain boundary mobility. The same general incoherent twin boundary in different FCC metals is found to display antithermal, thermal, and mixed mobility during molecular dynamics synthetic driving force simulations. A recently developed energy metric known as the generalized interfacial fault energy (GIFE) surface is used to show that twin boundaries moving in an antithermal manner have a lower energetic barrier to motion than twin boundaries moving in a thermally activated manner. Predicting the temperature dependence of grain bo…
Computational experiments are used to show that grain boundary mobility is independent of driving force in a two-dimensional, square-lattice Ising model with Metropolis kinetics. This is established over the entire Monte Carlo temperature range. A calibration methodology is then introduced which endows the Monte Carlo algorithm with time and length scales and allows the Monte Carlo parameters to be expressed directly in terms of experimentally measurable parameters. A comparison of results obtained for a variety of driving forces and temperature conditions indicates that such Calibrated Monte …
We report herein that the carrier mobility of the 2%-La-doped BaSnO3 (LBSO) films on (001) SrTiO3 and (001) MgO substrates strongly depends on the thickness whereas it is unrelated to the lattice mismatch (+5.4% for SrTiO3, -2.3% for MgO). Although we observed large differences in the lattice parameters, the lateral grain size (~85 nm for SrTiO3, ~20 nm for MgO), the surface morphology and the density of misfit dislocations, the mobility increased almost simultaneously with the thickness in both cases and saturated at ~100 cm2 V-1 s-1, together with the approaching to the nominal carrier conce…
Two finite element level-set (FE-LS) formulations are compared for the modeling of grain growth of 316L stainless steel in terms of grain size, mean values and histograms. Two kinds of microstructures are considered, some are generated statistically from EBSD maps and the others are generated by immersion of EBSD data in the FE formulation. Grain boundary (GB) mobility is heterogeneously defined as a function of the GB disorientation. On the other hand, GB energy is considered as heterogeneous or anisotropic, respectively defined as a function of the disorientation and both the GB misorientati…
The conductivity $σ$ and carrier density $n$ of the conducting polymer polyaniline were investigated by changing the concentration $x$ of a secondary dopant, meta-cresol. We found that $σ$ changes by four orders of magnitude within the $x$-range of 1-10 %, while $n$, as estimated from the Hall measurements, shows a weak dependence on $x$ in the region of 2% <$x$< 50%. These results suggest that $σ$ can be enhanced by the change in the mobility $μ$. We analyzed the temperature dependence of $μ$ by not only the combination of two different types of scattering mechanism, but also by the polaron h…
We study the dynamics and morphology of grain growth with anisotropic energy and mobility of grain boundaries using a generalized phase field model. In contrast to previous studies, both inclination and misorientation of the boundaries are considered. The model is first validated against exact analytical solutions for the classical problem of an island grain embedded in an infinite matrix. It is found that grain boundary energy anisotropy has a much stronger effect on grain shape than that of mobility anisotropy. In a polycrystalline system with mobility anisotropy, we find that the system evo…
While it is known that alloy components can segregate to grain boundaries (GBs), and that the atomic mobility in GBs greatly exceeds the atomic mobility in the lattice, little is known about the effect of GB segregation on GB diffusion. Atomistic computer simulations offer a means of gaining insights into the segregation-diffusion relationship by computing the GB diffusion coefficients of the alloy components as a function of their segregated amounts. In such simulations, thermodynamically equilibrium GB segregation is prepared by a semi-grand canonical Monte Carlo method, followed by calculat…
We report on the temperature dependence of the mobility, $μ$, of the two-dimensional electron gas in a variable density AlGaN/GaN field effect transistor, with carrier densities ranging from 0.4$\times10^{12}$ cm$^{-2}$ to 3.0$\times10^{12}$ cm$^{-2}$ and a peak mobility of 80,000 cm$^{2}$/Vs. Between 20 K and 50 K we observe a linear dependence $μ_{ac}^{-1} = α$T indicating that acoustic phonon scattering dominates the temperature dependence of the mobility, with $α$ being a monotonically increasing function of decreasing 2D electron density. This behavior is contrary to predictions of scatte…
The growth and the density dependence of the low temperature mobility of a series of two-dimensional electron systems confined to un-intentionally doped, low extended defect density InAs quantum wells with Al$_{1-x}$Ga$_{x}$Sb barriers are reported. The electron mobility limiting scattering mechanisms were determined by utilizing dual-gated devices to study the dependence of mobility on carrier density and electric field independently. Analysis of the possible scattering mechanisms indicate the mobility was limited primarily by rough interfaces in narrow quantum wells and a combination of allo…
Pentacenequinone (PnQ) impurities have been introduced into a pentacene source material at number densities from 0.001 to 0.474 to quantify the relative effects of impurity content and grain boundary structure on transport in pentacene thin-film transistors. Atomic force microscopy (AFM) and electrical measurements of top-contact pentacene thin-film transistors have been employed to directly correlate initial structure and final film structures, with the device mobility as a function of added impurity content. The results reveal a factor four decrease in mobility without significant changes in…
Bismuth selenide Bi$_2$Se$_3$ was grown by molecular beam epitaxy while carrier density and mobility were measured directly \emph{in situ} as a function of film thickness. Carrier density shows high interface n-doping (1.5 x 10$^{13}$ cm$^{-2}$) at the onset of film conduction, and bulk dopant density of $\sim$5 x 10$^{18}$ cm$^{-3}$, roughly independent of growth temperature profile. Mobility depends more strongly on the growth temperature and is related to the crystalline quality of the samples quantified by \emph{ex-situ} AFM measurements. These results indicate that Bi$_2$Se$_3$ as prepare…
Thermal stability of nanocrystalline multilayer thin film is of paramount importance as the applications often involve high temperature. Here we report on the layer instability phenomenon in binary polycrystalline thin film initiating from the grain boundary migrations at higher temperatures using phase-field simulations. Effect of layer thickness, bilayer spacing and the absence of grain boundary are also investigated along with the grain boundary mobility of individual phases on the layer stability. Layer instability in the polycrystalline film is shown to arise from the grain boundary groov…
We investigate the carrier mobility in mono- and bi-layer graphene with a top HfO2 dielectric, as a function of the HfO2 film thickness and temperature. The results show that the carrier mobility decreases during the deposition of the first 2-4 nm of top dielectric and remains constant for thicker layers. The carrier mobility shows a relatively weak dependence on temperature indicating that phonon scattering does not play a dominant role in controlling the carrier mobility. The data strongly suggest that fixed charged impurities located in close proximity to the graphene are responsible for th…
Coupled grain boundary (GB) motion has been studied in a two-dimensional tricrystal where a cylindrical grain is embedded at the center of the planar GB of a large bicrystal. Kinetic relations for GB dynamics, grain orientations, and junctions have been derived within the framework of Gibbs thermodynamics. These are solved numerically to investigate the shrinkage of the embedded grain while emphasizing the role of coupled motion as well as junction mobility in the shape evolution.
Experiments, theory and atomistic simulations show that finite triple junction mobility results in non-equilibrium triple junction angles in evolving polycrystalline systems. These angles have been predicted and verified for cases where grain boundary migration is steady-state. Yet, steady-state never occurs during the evolution of polycrystalline microstructures as a result of changing grain size and topological events (e.g., grain face/edge switching - "$T_1$" process, or grain disappearance "$T_2$" or "$T_3$" processes). We examine the non-steady evolution of the triple junction angle in th…
Faceted grain boundaries can migrate in interesting and unexpected ways. For example, faceted Σ11 <110> tilt grain boundaries were observed to exhibit mobility values that could be strongly dependent on the direction of migration. In order to understand whether this directionally-anisotropic mobility is a general phenomenon and to isolate mechanistic explanations for this behavior, molecular dynamics simulations of bicrystals evolved under an artificial driving force are used to study interface migration for a range of boundary plane inclination angles and temperatures in multiple face centere…
A continuum theory based on thermodynamics has been developed for modeling diffusional creep of polycrystalline solids. It consists of a coupled problem of vacancy diffusion and mechanics where the vacancy generation/absorption at grain boundaries is driven by grain boundary dislocations climb. The model is stated in terms of general balance laws and completed by the choice of constitutive equations consistent with classical non-equilibrium thermodynamics. The kinetics of diffusional creep is derived from physically-based mechanisms of climb of dislocations at grain boundaries, thus introducin…
The high-frequency mobility in disordered systems is governed by transport properties on mesoscopic length scales, which makes it a sensitive probe for the amount of local order. Here we present a method to measure the energy dependence of the high frequency mobility by combining an electrochemically gated transistor with in-situ quasi-optical measurements in the sub-terahertz domain. We apply this method to poly([2-methoxy-5-(3',7'-dimethylocyloxy)]-p-phenylene vinylene) (OC_1C_10-PPV) and find a mobility at least as high as 0.1 cm^2V^-1s^-1.
The generalized eguation that describes the yield stress dependence upon the grain size in the wide range of grain sizes has been obtained. There are two critical grain sizes that corresponds to the changes in strengthening mechanism. The equation includes Hall - Petch relation in the range of d>dcr1. For dcr1>d>dcr2, the power in the Hall - Petch relation varies from -1/2 to -1. In a nanometer (nm) range (d<dcr2) there are posibileties of softerning (in case of "weak" boundaries) as well as significant strengthening (in case of "srtong" boundaries).
A total negative field dependence of hole mobility down to low temperature was observed in N,N'-diphenyl-N,N'-bis(3-methylphenyl)-(1,1'-biphenyl)-4,4'diamine (TPD) doped in Polystyrene. The observed field dependence of mobility is explained on the basis of low values of energetic and positional disorder present in the sample. The low value of disorder is attributed to different morphology of the sample due to aggregation/crystallization of TPD. Monte Carlo simulations were also performed to understand the influence of aggregates on charge transport in disordered medium with correlated site ene…
The temperature dependence of the charge carrier density, mobility and Seebeck coefficient of melt-grown, bulk ZnGa2O4 single crystals was measured between 10 K and 310 K. The electrical conductivity at room temperature is about s = 286 S/cm due to a high electron concentration of n = 3.26*10^(19) cm^(-3), caused by unintenional doping. The mobility at room temperature is mu = 55 cm^2/Vs, whereas the scattering on ionized impurities limits the mobility to mu =62 cm^2/Vs for temperatures lower than 180 K. The Seebeck coefficient relative to aluminum at room temperature is S_(ZnGa2O4-Al) = (-125…
We have investigated the low-frequency 1/f noise of both suspended and on-substrate graphene field-effect transistors and its dependence on gate voltage, in the temperature range between 300K and 30K. We have found that the noise amplitude away from the Dirac point can be described by a generalized Hooge's relation in which the Hooge parameter αH is not constant but decreases monotonically with the device's mobility, with a universal dependence that is sample and temperature independent. The value of αH is also affected by the dynamics of disorder, which is not reflected in the DC transport ch…
We examine the effect of changing the linear polarization angle $θ$ of incident microwaves with respect to the dc current on radiation-induced magnetoresistance oscillations in a two-dimensional (2D) system within the balance-equation formulation of the photon-assisted magnetotransport model, considering the radiative decay as the sole damping mechanism. At an extremum the amplitude of oscillatory magnetoresistance $R_{xx}$ exhibits a sinusoidal, up to a factor of 5, magnitude variation with rotating the polarization angle $θ$. The maximal amplitude shows up generally at a nonzero $θ$, which i…
Dislocation motion in body centered cubic (bcc) metals displays a number of specific features that result in a strong temperature dependence of the flow stress, and in shear deformation asymmetries relative to the loading direction as well as crystal orientation. Here we develop a generalized dislocation mobility law in bcc metals, and demonstrate its use in discrete Dislocation Dynamics (DD) simulations of plastic flow in tungsten (W) micro pillars. We present the theoretical background for dislocation mobility as a motivating basis for the developed law. Analytical theory, molecular dynamics…
We develop a broadly applicable transport-based technique, GAte Modulated activation Energy Spectroscopy (GAMEaS), for determining the density of states (DOS) in the energy gap. GAMEaS is applied to field effect transistors made from different single crystal oligomer semiconductors to extract the free-carrier mobility, u_0, from the field effect mobility, u_eff. Samples with a lower DOS exhibit higher u_eff. Values of u_0 up to 100 +/- 40 cm2/Vs at 300K are observed, showing that performance can be greatly enhanced by improving sample purity and crystal quality.
Optimizing high-temperature electron mobility in single-crystal Bi$_2$O$_2$Se based on its unconventional dependence on concentration
University of Pardubice, Faculty of Chemical Technology（帕尔杜比采大学化学技术学院）；Institute of Physics of the Czech Academy of Sciences（捷克科学院物理研究所）；Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University（查理大学数学与物理学院凝聚态物理系）；Masaryk University, Department of Condensed Matter Physics（马萨里克大学凝聚态物理系）
Instead of normal non-Arrhenius relationship, the carrier mobility $ln(μ)$ v.s. $1/T^2$ showed abnormal dependence in an MEH-PPV / InP nanocrystal composite system that a critical temperature $(T_c)$ behavior is prominent in temperature range of 233 K to 333 K. Here, in the model of variable range hopping theory, an analytical model is developed within a Gaussian trap distribution, which is successfully implemented on that phenomenon. The results show that Tc becomes the transition temperature as long as trap-filling factor (FF) ~1, which means a transition point from Boltzmann to Fermi distri…
We report low temperature scanning tunneling microscopy characterization of MoSe2 crystals, and the fabrication and electrical characterization of MoSe2 field-effect transistors on both SiO2 and parylene-C substrates. We find that the multilayer MoSe2 devices on parylene-C show a room temperature mobility close to the mobility of bulk MoSe2 (100 cm2V-1s-1 - 160 cm2V-1s-1), which is significantly higher than that on SiO2 substrate (~50 cm2V-1s-1). The room temperature mobility on both types of substrates are nearly thickness independent. Our variable temperature transport measurements reveal a …
The deposition of a high-$κ$ oxide overlayer is known to significantly enhance the room-temperature electron mobility in single-layer MoS$_{2}$ (SLM) but not in single-layer graphene (SLG). We give a quantitative account of how this mobility enhancement is due to the non-degeneracy of the two-dimensional electron gas system in SLM at accessible temperatures. Using our charged impurity scattering model [Ong and Fischetti, Phys. Rev. B 86, 121409 (2012)] and temperature-dependent polarizability, we calculate the charged impurity-limited mobility ($μ_{\textrm{imp}}$) in SLM with and without a hig…
We study the temperature dependence ($77$ K - $475$ K) of the longitudinal resistivity and Hall coefficient of thin films (thickness $20$ nm) of three dimensional topological Dirac semimetal Na$_3$Bi grown via molecular beam epitaxy (MBE). The temperature-dependent Hall coefficient is electron-like at low temperature, but transitions to hole-like transport around $200$ K. We develop a model of a Dirac band with electron-hole asymmetry in Fermi velocity and mobility (assumed proportional to the square of Fermi velocity) which explains well the magnitude and temperature dependence of the Hall re…
The many-body Monte Carlo method is used to evaluate the frequency dependent conductivity and the average mobility of a system of hopping charges, electronic or ionic on a one-dimensional chain or channel of finite length. Two cases are considered: the chain is connected to electrodes and in the other case the chain is confined giving zero dc conduction. The concentration of charge is varied using a gate electrode. At low temperatures and with the presence of an injection barrier, the mobility is an oscillatory function of density. This is due to the phenomenon of charge density pinning. Mobil…
A discrete model and the regular solution approximation are applied to describe the effect of grain boundary motion on grain boundary phase transformations in a binary alloy. The model predicts all thermodynamic properties of the grain boundary and the solute drag force, and permits a broad exploration of the parameter space and different dynamic regimes. The grain boundary phases continue to exist in the moving grain boundary and show a dynamic hysteresis loop, a dynamic critical line, and other features that are similar to those for equilibrium phases. Grain boundary motion strongly affects …
Mg grain boundary (GB) segregation and GB diffusion can impact the processing and properties of Al-Mg alloys. Yet, Mg GB diffusion in Al has not been measured experimentally or predicted by simulations. We apply atomistic computer simulations to predict the amount and the free energy of Mg GB segregation, and the impact of segregation on GB diffusion of both alloy components. At low temperatures, Mg atoms segregated to a tilt GB form clusters with highly anisotropic shapes. Mg diffuses in Al GBs slower than Al itself, and both components diffuse slowly in comparison with Al GB self-diffusion. …
We calculate the classic Hall conductivity and mobility of the undoped and doped (or in the gate voltage) graphene as a function of temperature, magnetic field, and carrier concentration. Carrier collisions with defects and acoustic phonons are taken into account. The Hall resistivity varies almost linearly with temperature. The magnetic field dependence of resistivity and mobility is anomalous in weak magnetic fields. There is the square root contribution from the field in the resistivity. The Hall mobility diverges logarithmically with the field for low doping.
In order to study the dependence of the coercive force of sintered magnets on temperature, nucleation and domain wall propagation at the grain boundary are studied as rate-determining processes of the magnetization reversal phenomena in magnets consisting of bulk hard magnetic grains contacting via grain boundaries of a soft magnetic material. These systems have been studied analytically for a continuum model at zero temperature (A. Sakuma, et al. J. Mag. Mag. Mat. {\bf 84} 52 (1990)). In the present study, the temperature dependence is studied by making use of the stochastic Landau-Lifshitz-G…
A phase field model with arbitrary misorientation dependence of grain boundary energy
University of Michigan（密歇根大学）；George Mason University（乔治梅森大学）；California Institute of Technology（加州理工学院）；Northwestern University（西北大学）；National Institute of Standards and Technology（美国国家标准与技术研究院）
The phase behavior of grain boundaries can have a strong influence on interfacial properties. Little is known about the emergence of grain boundary phases in elemental metal systems and how they transform. Here, we observe the nanoscale patterning of a grain boundary by two alternating grain boundary phases with distinct atomic structures in elemental copper by atomic resolution imaging. The same grain boundary phases are found by grain boundary structure search indicating a first-order transformation. Finite temperature atomistic simulations reveal a congruent, diffusionless transition betwee…
