Anisotropic Misfit Strains Research Articles

3D microstructure evolution in Na[formula omitted]FePO[formula omitted] storage particles for sodium-ion batteries

The cathode material NaxFePO4 of sodium-ion batteries exhibits complex phase segregation thermodynamics with the existence of an intermediate phase, and large volume change during (dis)charging. A virtual multiscale modeling chain is established to construct a 3D anisotropic electro-chemo-mechanical phase-field model based on first-principles calculations for NaxFePO4, which considers phase changes, electrochemical reactions, anisotropic diffusion, anisotropic misfit strain, and anisotropic elasticity, as well as the concentration-dependence of the elasticity tensor. The elastic properties of NaxFePO4 are determined by first-principles for the first time. We investigate how surface reaction kinetics and crystal anisotropy influence the full 3D microstructure evolution, with results that include phase evolution, interface morphology, and stress evolution in NaxFePO4 particles. We find that the existence of 1D Na diffusion channels leads to a kinetically arrested state of single wave propagation along [010]. Furthermore, defect-actuated in-plane diffusion induces low-energy single wave propagation along [100] controlled by the concentration dependent anisotropic elasticity tensor. In addition, the morphology of the double wave propagation along [010] is more prone to particle cracking and mechanical degradation. Beyond NaxFePO4, the findings of this work point towards opportunities to engineer desired phase behavior with better mechanical stability by defect-actuated out-of-1D diffusion of an intercalation electrode material.

The Elastic Effect of Evolving Precipitate Shapes on the Ripening Kinetics of Tetragonal Phases

Coherent tetragonal precipitates, such as the Ni3Nb phase γ″ found in Ni-base superalloys, appear as plate-shaped particles. These shapes are the result of anisotropic elastic misfit strains. We present 3D sharp phase-field simulations that capture this circumstance well due to the inclusion of the elastic effects from the misfit. These simulations reveal that the ripening behavior of γ″ precipitates deviates significantly from the classical LSW theory of Ostwald ripening. A ripening exponent of 2 rather than 3 describes the simulated γ″ size evolution at temperatures between 700 °C and 760 °C best. Employing a quantitative distinction argument, we show that 60 pct of this deviation is attributed to the elastically induced size dependence of the precipitate shapes. With increasing precipitate size, the minimization of elastic energy leads to steadily increasing plate aspect ratios. The precipitate ripening kinetics accelerate with increasing aspect ratio. Fitting the newly received square root time dependence to experimental data yields a physically conclusive activation energy of ripening close to the activation energy of Nb diffusion in the alloy.

Low-symmetry ferroelectric phases stabilized by anisotropic misfit strain

The complete thermodynamic potential of (111) ferroelectric thin films including anisotropic in-plane misfit strain, in-plane shear strain, and external stress is developed based on the Landau-Ginzburg-Devonshire thermodynamic theory. Anisotropic strain-tuned low-symmetry equilibrium phases in (111) single-domain ferroelectric $\mathrm{PbTi}{\mathrm{O}}_{3}$ films were investigated via the developed nonlinear thermodynamic potential and the misfit strain--misfit strain phase diagram was constructed. It was found that the anisotropic epitaxial strain can effectively control the orientations of polarization vectors, which determine the symmetry of equilibrium phases, and their functionalities. In particular, two types of low-symmetry triclinic phases, disappearing under isotropic strain states, can be stabilized by anisotropic strains. Low-symmetry monoclinic and triclinic ferroelectric phases and field-induced phase transitions were also analyzed in depth. It was indicated that the monoclinic (${M}_{A}^{1}$) symmetry rhombohedral-like--tetragonal-like transition and triclinic (${T}_{\mathrm{r}}^{1}$)-monoclinic (${M}_{B}^{1}$) phase transition would give rise to the superior piezoelectric properties as compared to other structure phase transitions. Electric field-enhanced piezoelectric response in triclinic structure originates from the elastic softening that is related to the shear ${u}_{5}$. These results provide guidance for interpreting and understanding more experimental observations as well as the designing of high-performance piezoelectric films.

Phase-field simulation of misfit dislocations in two-phase electrode particles: Driving force calculation and stability analysis

Dislocations in lithium-ion battery materials significantly influence the performance of the battery. A chemo-mechanical phase-field model combined with a non-singular continuum theory of dislocations is developed to simulate misfit dislocations at the phase boundary of two-phase LiFePO4 particles. The configurational mechanics of dislocations in the mechanically coupled phase transformation problem is proposed and then used to formulate a dislocation formation criterion. The model is numerically implemented using the finite element method to compute driving forces on misfit dislocations. The driving force is further used to study the stability of the misfit dislocation to predict the critical size of dislocation-free two-phase electrode particles. The orthotropic and non-homogeneous elastic constants in two phases, anisotropic misfit strain, different aspect ratios of the particle, and different positions of the phase boundary in equiaxed particles are considered. The large misfit strain at the phase boundary is found to cause a redistribution of the concentration and thus a relaxation of the stress field at the interface. Stress relaxation is crucial to the driving force on the dislocation and ignoring the stress relaxation leads to an underestimation of the critical particle size. The critical particle size decreases with increasing aspect ratios and shows a saturation after aspect ratios above 2. An optimum aspect ratio of the dislocation-free electrode particle is around 1.5 which features the highest specific surface area.

Effects of anisotropic misfit strains on equilibrium phases and domain structures in (111)-oriented ferroelectric PbTiO3 films

Advances in high-index-oriented ferroelectric films offer the exciting possibility of utilizing strain engineering to tune the desired properties related to the nanostructures. Previous studies often considered ferroelectric films grown on cubic substrates (isotropic strain), while the effects of anisotropic strains introduced by orthorhombic substrates are still not well understood. In our recent work, we grew (111)-oriented PbTiO3 (PTO) films on orthorhombic GdScO3 substrates and studied the domain structure evolution with film thickness. However, the theoretical understanding is lacking. In this work, the physical mechanism of the formation and evolution of domain structures was clearly revealed from the viewpoint of energy competition. Moreover, through the stereographic projection (SP) analysis of polarization vectors, a wide variety of ferroelectric phases for (111)-oriented PTO films under anisotropic strains were identified, and the misfit strain-misfit strain domain stability diagram was then constructed at room temperature. It was indicated that anisotropic misfit strains not only lead to the separation of ferroelectric domain variants that could co-exist under isotropic strain states for tetragonal-like d and monoclinic ma phases, but also stabilize the monoclinic mb phase. Their representative domain morphologies with separated domain variants under different anisotropic strains were analyzed in detail. These results provide guidance for interpreting and understanding more experimental data as well as the designing of (111)-oriented epitaxial ferroelectrics under anisotropic strains.

Temperature dependence of three-dimensional domain wall arrangement in ferroelectric K0.9Na0.1NbO3 epitaxial thin films

The three-dimensional arrangement and orientation of domain walls in ferroelectric K0.9Na0.1NbO3/(110)NdScO3 epitaxial thin films were investigated at different temperatures both experimentally by means of piezoresponse force microscopy and three-dimensional x-ray diffraction and theoretically by three-dimensional phase-field simulations. At room temperature, a well-ordered herringbone-like domain pattern appears in which there is a periodic arrangement of a1a2/MC monoclinic phases. Four different types of domain walls are observed, which can be characterized by out-of-plane tilt angles of ±45° and in-plane twist angles of ±21°. For the orthorhombic high-temperature phase, a periodic a1/a2 stripe domain pattern with exclusive in-plane polarization is formed. Here, two different types of domain walls are observed, both of them having a fixed out-of-plane domain wall angle of 90° but distinguished by different in-plane twist angles of ±45°. The experimental results are fully consistent with three-dimensional phase-field simulations using anisotropic misfit strains. The qualitative agreement between the experiment and the theory applies, in particular, to the wide phase transition range between about 180 °C and 260 °C. In this temperature range, a complex interplay of coexisting monoclinic a1a2/MC and orthorhombic a1/a2 phases takes place.

Ferroelectric phase transitions in multi-domain K0.9Na0.1NbO3 epitaxial thin films

A high-temperature phase transition in strained ferroelectric K0.9Na0.1NbO3 thin films epitaxially grown on orthorhombic (110) NdScO3 substrates is identified and investigated by in situ x-ray diffraction and piezoresponse force microscopy. At room temperature, the thin films exhibit a highly anisotropic misfit strain, inducing the occurrence of monoclinic a1a2/MC phases and manifesting itself in the formation of a highly regular, herringbone-like domain arrangement. With increasing temperature, a ferroelectric-to-ferroelectric phase transition to an orthorhombic a1/a2 phase with exclusive lateral electrical polarization takes place. Within a wide temperature range from 180 °C to about 260 °C, a coexistence of the monoclinic a1a2/MC room temperature phases and the orthorhombic a1/a2 high temperature phase is observed. Finally, at higher temperatures only the orthorhombic a1/a2 phase, which is arranged in a regular stripe domain pattern, is present. Corresponding simulations of the scattered x-ray intensity patterns show that the orthorhombic unit cells undergo a small in-plane rotation. This leads to four different in-plane orientations of the orthorhombic unit cells and four corresponding variants of superdomains.

Stress-Controlled Three-Dimensional Phase Evolution in Lithium Iron Phosphate

Many battery compounds undergo first-order phase transformations induced by ion (de)intercalation during battery operation, and their rate capability is often controlled by the kinetics of two-phase boundary movement. LiFePO4, a commercial cathode materials for Li-ion batteries, is a model system for studying the unique aspects of phase transitions in battery intercalation compounds. In a recent in-situ scanning transmission X-ray microscopy (STXM) study [Ohmer et al. Nature Communication 6, 6045 (2015)], Li deintercalation and phase evolution in microsized LiFePO4 single crystal upon delithiation are found to be highly non-uniform. FePO4 phase was observed to form [010]-aligned filaments growing into the LiFePO4 crystal, which is postulated to be dominated by elastic effects. In this work, we performed three-dimensional phase-field simulation to examine in the details phase evolution in LiFePO4 and the effect of misfit strain energy arising from the lattice match between FePO4 and LiFePO4. Our simulation reproduces the filamentary growth morphology of FePO4 phase (Figure 1) as observed in experiment, and confirms the dominating role of misfit stress in causing such behavior. We show that phase evolution in LiFePO4 goes through two steps during delithiation. In the first step, phase separation occurs in LiFePO4 via surface-mode coherent spinodal decomposition and results in the formation of individual FePO4 domains on particle surface. Next, FePO4 grows into the LiFePO4 matrix as wedge shaped domains that coarsen at the same time. The filamentary morphology of FP phase within (001) plane is mainly a result of the anisotropic misfit strain between LiFePO4 and FePO4, which favors the elongation of FP domains along the [010] axis to reduce elastic strain energy. Simulation reveals that Li diffusion anisotropy also significantly influences the phase growth morphology. The existence of three-dimensional Li diffusion in LiFePO4, which results from the presence of antisite defects, is found to be essential for the development of filamentary FePO4. The excellent agreement between the phase-field simulation and in-situ STXM experiment suggests that misfit stress plays an important role in triggering non-uniform Li (de)intercalation especially in large electrode particles, which is detrimental to battery performance and life and deserves close attention. Figure 1

Phase-field modeling of θ′ precipitation kinetics in 319 aluminum alloys

Understanding the morphological evolution of precipitates is critical for evaluating their hardening effects and therefore improving the yield strength of an alloy during aging. Here we present a three-dimensional phase-field model for capturing both the nucleation and the growth kinetics of the precipitates and apply it to modeling θ′ precipitates in 319 aluminum alloys. The model incorporates the relevant thermodynamic data, diffusion coefficients, and the anisotropic misfit strain from literature, together with the anisotropic interfacial energy from first-principles calculations. The modified classical nucleation theory is implemented to capture the nucleation kinetics. The model parameters are optimized by comparing the simulation results to the experimentally measured peak number density, average diameters, average thicknesses and volume fractions of precipitates during isothermal aging at 463 K (190 °C), 503 K (230 °C) and 533 K (260 °C). Further model improvements in terms of prediction accuracy of the precipitate kinetics in 319 alloys and the remaining challenges are discussed.

Anisotropic strain relaxation and high quality AlGaN/GaN heterostructures on Si (110) substrates

We have investigated the growth and relaxation mechanisms of anisotropic lattice misfit strain in AlN and GaN layers on Si (110) substrates. A qualitative model is proposed to explain the relaxation process. It is revealed that the anisotropic misfit strain is quickly relaxed in the low temperature AlN layer by the formation of interface misfit dislocations, small misoriented grains, and lattice distortion. As a result, isotropic properties and atomically smooth surface are observed in the high temperature AlN layer. Based on this isotropic AlN layer, a high quality GaN layer and AlGaN/GaN heterostructures with a high electron mobility of 2160 cm2/(V · s) have been obtained. This work will have important impacts on the understanding of the epitaxy of isotropic semiconductor films on anisotropic substrates.

Single-domain (110)PbTiO3thin films: Thermodynamic theory and experiments

We report the thermodynamic potential for single-domain (110) thin films epitaxially grown on dissimilar cubic substrates. Using different sets of paraelectric phase elastic compliance coefficients of ${\mathrm{PbTiO}}_{3}$ single crystal, calculated from the experimental room-temperature values, we predict rotational phases similar to those observed in (001) thin films under anisotropic biaxial misfit strain. The new sets of elastic compliance coefficients also predict a triclinic phase that could potentially lead to the enhancement of both dielectric and piezoelectric properties. We also conducted experimental studies on highly tetragonal monocrystalline ${\mathrm{PbZr}}_{0.05}{\mathrm{Ti}}_{0.95}{\mathrm{O}}_{3}$ thin films of different thicknesses, epitaxially grown on (110) ${\mathrm{SrTiO}}_{3}$ substrate by pulsed laser deposition technique. Piezoresponse force microscopy measurements showed that the as-grown films were single domain with the ${a}_{2}c$ phase, which corroborates with the prediction of the theory. Moreover, the ${T}_{c}$ values of both thin and thick films (17--90 nm) also fell within the predicted range $(540--600\phantom{\rule{0.16em}{0ex}}{}^{\ensuremath{\circ}}\mathrm{C})$. The measured remanet polarization of $57\ensuremath{\mu}\mathrm{C}/{\mathrm{cm}}^{2}$ was in good agreement with the theoretical values of $55--58\ensuremath{\mu}\mathrm{C}/{\mathrm{cm}}^{2}$. Small-signal piezoelectric response measurements gave a piezoelectric coefficient of 40 pm/V, which is also in good agreement with the numerically calculated values of 38--42 pm/V.

Out-of-plane three-stable-state ferroelectric switching: Finding the missing middle states

By realizing a nonvolatile third intermediate ferroelectric state through anisotropic misfit strain, we demonstrate electrical switching among three stable out-of-plane polarizations in bismuth ferrite thin films grown on ${(110)}_{\mathrm{pc}}$-oriented gadolinium scandate substrates (where pc stands for pseudocubic) by the use of an asymmetric external electric field at the step edge of a bottom electrode. We employ phenomenological Landau theory, in conjunction with electrical poling experiments using piezoresponse force microscopy, to understand the role of anisotropic misfit strain and an in-plane electric field in stabilization of multiple ferroelectric states and their competition. Our finding provides a useful insight into multistep ferroelectric switching in rhombohedral ferroelectrics.

Influence of anisotropic strain relaxation on the magnetoresistance properties of epitaxial Fe3O4 (110) films

We studied Fe3O4 (110) films grown epitaxially on MgO (110) substrates using oxygen plasma assisted molecular beam epitaxy. The films with thickness of 30–200 nm showed anisotropic in-plane partial strain relaxation. Magneto resistance (MR) measurements with current and magnetic field along ⟨001⟩ direction showed higher MR compared to ⟨1¯10⟩ direction. Maximum value of MR was measured at Verwey transition temperature for both directions. We explain the observed anisotropy in the MR on the basis of the effects of anisotropic misfit strain, and the difference between the density of antiferromagnetically coupled antiphase boundaries formed along ⟨001⟩ and ⟨1¯10⟩ crystallographic directions, suggesting the dependence of spin polarisation on the anisotropic strain relaxation along the said crystallographic directions.

Elastic consideration of the precipitation in model alloys of maraging steels: theory and experimental validation

The unique combination of high strength and high fracture toughness in maraging steels is attributed to the precipitation of mainly two phases, Ni3Mo and Ni3Ti, in the soft bcc martensitic matrix. Experimentally, it is difficult to distinguish between these phases; moreover, the effect of each precipitating phase on the properties of the steel is ambiguous. In the present work, we tackle these questions by analyzing the elastic fields and elastic energy associated with each precipitating phase and their role in the determination of the shape and relative arrangement. Using a semi-analytic approach based on the solution of the equations of elasticity by Fourier transform, the elastic fields and elastic energy associated with each precipitate were calculated. The calculations show that the minimum self-energy of both precipitates is accomplished when the smallest crystallographic misfit lies parallel to the longest geometric axis. The combination of elastic energy and surface energy provides an explanation to the observed morphologies of the precipitates, each type having an elongated shape but with a different aspect ratio. According to the calculations, the minimum self-elastic energy associated with Ni3Ti coherent precipitate is 2.8-fold larger than the energy of Ni3Mo. This implies for a larger strengthening effect of the former, as was confirmed experimentally in model alloys, one without Mo and the other without Ti. The elastic model also predicts that elongated shape and the anisotropic misfit strains favor several vertical arrangements of both precipitates, as was also verified experimentally by transmission electron microscopy of aged model alloys.

Shape transition and multi-stability of helical ribbons: a finite element method study

Helical structures are among the most universal building blocks in nature and engineering. In this work, I performed three-dimensional finite element simulations to study the transitions of shapes and multi-stability in the mechanically self-assembled helical structures driven by anisotropic misfit strains. The shape transition between a purely twisted ribbon, or a helicoid, and a general helical ribbon can be achieved by tuning a few relevant geometric and mechanical parameters, including the misfit strains, the geometric misorientation angle, the dimensions, and the mechanical properties of the composite layers. The results of our work show good agreement with the recent theoretical works and will serve as a powerful tool to facilitate on-demand designs of spontaneously curved structures at both macroscopic and microscopic scales, for a number of engineering applications including nanoelecromechanical systems, drug delivery, sensors, drug delivery, active materials, optoelectronics, and microrobotics.

Giant electrocaloric effect of PbTiO3 thin film tuned in a wide temperature range by the anisotropic misfit strain

The influence of anisotropic in-plane strains on the electrocaloric effect (ECE) in PbTiO3 (PT) epitaxial ferroelectric thin films is investigated by using a Landau-Devonshire thermodynamic theory. The calculation results show that the anisotropic strain can tune the ECE of PT ferroelectric thin films to obtain a large adiabatic temperature change in a wide temperature range which is attributed to the shift of c-phase boundary of PT thin films under the anisotropic strains with an external electric field. These results indicate that the anisotropic strain can provide an efficient way to adjust the ECE of ferroelectric thin films to refrigerate in a wide temperature range.

Anisotropic resistivities in anisotropic-strain-controlled phase-separated La0.67Ca0.33MnO3/NdGaO3(100) films

The anisotropic resistivities (AR) in La0.67Ca0.33MnO3 films grown on orthorhombic NdGaO3(100) substrates were investigated. In this epitaxial system, the large anisotropic misfit strain was demonstrated to induce an antiferromagnetic insulator (AFI) phase transition near ∼250 K, leading to phase separation (PS) with the coexistence of AFI and ferromagnetic-metal (FM) phases at the temperature below. In the PS regime, the resistivity measured along the highly strained b-axis is greater than that along c-axis, giving rise to a huge AR. It can reach ∼12 500% and shows strong dependence on the amplitude and orientation of the magnetic field. We ascribed this unusual AR to the anisotropic-strain-controlled MnO6 octahedral deformations which can organize the competing AFI and FM phases into orientation-preferred PS pattern, thus resulting in the anisotropic percolative transport.

Effects of thickness on the polarization states in epitaxial ferroelectric thin films

The effect of thickness and depolarization field on the polarization states of epitaxial ferroelectric single-domain thin films with anisotropic in-plane misfit strains is studied by using a nonlinear thermodynamic theory with the thickness-dependent Curie temperature. The “thickness-misfit strain” phase diagrams for single-domain PbTiO3 thin films grown on tetragonal substrates are developed with and without depolarization field through minimization of the Helmholtz free energy. The results show that the depolarization field always reduces the region of the c-phase (P3 ≠ 0, P1 = P2 = 0) in the phase diagrams. The critical thickness, at which the ferroelectricity disappears, decreases when the tensile or compressive misfit strain increases.

Determination of heteroepitaxial layer relaxation at growth temperature from room temperature X-ray reciprocal space maps

Lattice-mismatched heteroepitaxial growth in compound semiconductor layer structures, e.g. metamorphic buffers, is frequently used to combine materials with favorable properties. A detailed understanding of the strain relaxation during growth is important to optimize the buffer layers and to achieve high material quality. We present a method to determine the epilayer relaxation at growth temperature from room temperature X-ray diffraction measurements. For this the lattice parameters are measured with (004) and (224) reciprocal space maps along two orthogonal <110> directions to account for anisotropic misfit strain relaxation. Dissimilar thermal expansion coefficients of substrate and epilayer are taken into consideration when calculating the lattice parameters and epilayer relaxation parameters at growth temperature. The application of this method is discussed for GaAsxP1−x metamorphic buffer structure on Si with eight lattice-mismatched layers showing a relaxation parameter of over 130% at room temperature. All layers were found to be nearly 100% relaxed at growth temperature. The use of this method for other compound semiconductors is discussed.

Coherency Strain and the Kinetics of Phase Separation in LiFePO4 Nanoparticles

A theoretical investigation of the effects of elastic coherency strain on the thermodynamics, kinetics, and morphology of intercalation in single LiFePO(4) nanoparticles yields new insights into this important battery material. Anisotropic elastic stiffness and misfit strains lead to the unexpected prediction that low-energy phase boundaries occur along {101} planes, while conflicting reports of phase boundary orientations are resolved by a partial loss of coherency in the [001] direction. Elastic relaxation near surfaces leads to the formation of a striped morphology with a characteristic length scale predicted by the model, yielding an estimate of the interfacial energy. The effects of coherency strain on solubility and galvanostatic discharge are studied with a reaction-limited phase-field model that quantitatively captures the influence of misfit strain, particle size, and temperature on solubility seen in experiments. Coherency strain strongly suppresses phase separation during discharge, which enhances rate capability and extends cycle life. The effects of elevated temperature and the feasibility of nucleation are considered in the context of multiparticle cathodes.