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We develop a continuum phase-field model for the simulation of diffusion limited solid-solid phase transformations during lithium insertion in LiFePO4-nano-particles. The solid-solid phase boundary between the LiFePO4 (LFP)-phase and the FePO4 (FP)-phase is modeled as a diffuse interface of finite width. The model-description explicitly resolves a single LiFePO4-particle, which is embedded in an elastically soft electrolyte-phase. Furthermore, we explicitly include anisotropic (orthorhombic) and inhomogeneous elastic effects, resulting from the coherency strain, as well as anisotropic (1D) Li-diffusion inside the nano-particle. In contrast to other related research work, we employ an Allen-Cahn-type phase-field approach for the diffuse interface modeling of the solid-solid phase boundary. The model contains an extra non-conserved order parameter field to distinguish the two different phases. The evolution of this order parameter field is controlled by an extra kinetic parameter independent from the Li-diffusion. Further, the effect of the nano-particle’s size on the kinetics of FP to LFP phase transformations is investigated by means of both model. Both models predict a substantial increase in the steady state transformation velocity as the particle-size decreases down to dimensions that are comparable with the width of the interface between the FP and the LFP-phase. However, the extra kinetic parameter of the Allen-Cahn-type description may be used to reduce the strength of the velocity-increase with the decreasing particle size. Further, we consider the influence of anisotropic and inhomogeneous elasticity on the lithiation-kinetics within a rectangularly shaped LiFePO4-particle embedded in an elastically soft electrolyte. Finally, the simulation of equilibrium shapes of LiFePO4-particles is discussed. Within a respective feasibility study, we demonstrate that also the simulation of strongly anisotropic particles with aspect ratios up to 1/5 is possible.

The mathematical model for intercalation dynamics in phase-separating materials (Singh et al., 2008) is a powerful tool for the investigation of the spinodal decomposition in nanoparticles. By means of this model, we conduct a careful mathematical analysis of the intercalation dynamics in nanoparticles to study the dependence of spinodal gap on the boundary reaction rate and the particle size, which can be used for LiFePO 4 battery material application. Consistent with previous investigations, we found that for some range of the boundary reaction rate and the particle size the concentration spinodal gap is not continuous, but it has stable “islands” where no spinodal decomposition is expected. The new important observation is that the presence of an infinitesimally small boundary reaction rate will destabilize nanoparticles even for infinitesimal length. In particular for nanoparticles having the size of order or less than interphase width λ , the spontaneous charge or discharge will occur at the reaction rate of order 0.1 D / λ . The further raise of the intercalation rate will stabilize the system until some size limit of order two diffusion length. The intercalation effects are proven by means of numerical simulations. We also show that the increasing enthalpy of the spinodal mixture as well as increasing elastic energy due to the lattice misfit can destabilize the particles and increase the spinodal gap.

We develop a continuum phase-field model for the simulation of diffusion limited solid-solid phase transformations during lithium insertion in LiFePO4-nano-particles. The solid-solid phase boundary between the LiFePO4 (LFP)-phase and the FePO4 (FP)-phase is modeled as a diffuse interface of finite width. The model-description explicitly resolves a single LiFePO4-particle, which is embedded in an elastically soft electrolyte-phase. Furthermore, we explicitly include anisotropic (orthorhombic) and inhomogeneous elastic effects, resulting from the coherency strain, as well as anisotropic (1D) Li-diffusion inside the nano-particle. The effect of the nano-particle's size on the kinetics of FP to LFP phase transformations is investigated by means of both model. Both models predict a substantial increase in the steady state transformation velocity as the particle-size decreases down to dimensions that are comparable with the width of the interface between the FP and the LFP-phase. However, the extra kinetic parameter of the Allen-Cahn-type description may be used to reduce the strength of the velocity-increase with the decreasing particle size. Further, we consider the influence of anisotropic and inhomogeneous elasticity on the lithiation-kinetics within a rectangularly shaped LiFePO4-particle embedded in an elastically soft electrolyte. Finally, the simulation of equilibrium shapes of LiFePO4-particles is discussed. Within a respective feasibility study, we demonstrate that also the simulation of strongly anisotropic particles with aspect ratios up to 1/5 is possible.

We have investigated the microstructure evolution during the isothermal and non-isothermal solidification of ternary Al–Cu–Ni alloys by means of a general multi-phase-field model for an arbitrary number of phases. The stability requirements for the model functions on every dual interface guarantee the absence of “ghost” phases. The aim was to generate a realistic microstructure by coupling the thermodynamic parameters of the phases and the thermodynamically consistent phase-field evolution equations. It is shown that the specially constructed thermal noise terms disturb the stability on the dual interfaces and can produce heterogeneous nucleation of product phases at energetically favorable points. Similar behavior can be observed in triple junctions where the heterogeneous nucleation of a fourth phase is more favorable. Finally, the model predicts the growth of a combined eutectic-like and peritectic-like structure that is comparable to the observed experimental microstructure in various alloys.

Spinodal decomposition leads to spontaneous fluctuations of the local concentration. In the early stage, the resulting pattern provides explicit information about the material properties of the mixture. In the case of two isotropic fluids, the static structure factor shows the characteristic ring shape. If one component is a liquid crystal, the pattern is typically anisotropic and the structure factor is more complex. Using numerical methods, we investigate how structure factors can be used to extract information about material properties like the diffusion constant or the isotropic and the anisotropic contributions to the interfacial tension. The method is based on momenta taken from structure factors in the early stage of the spinodal demixing. We perform phase field calculations for an isotropic and an anisotropic spinodal decomposition. A comparison of the extracted results with analytic values is made. The calculations show that linear modes dominate in the beginning of the growth process, while non-linear modes grow monotonously in the region of the k-space for which damping is predicted by the linearized theory. As long as non-linear modes are small enough, linearized theory can be applied to extract material properties from the structure factor.

We report very simple and accurate algebraic expressions for the van der Waals (VDW) potentials and the forces between two parallel and crossed carbon nanotubes. The Lennard-Jones potential for two carbon atoms and the method of the smeared-out approximation suggested by Girifalco were used. It is found that the interaction between parallel and crossed tubes is described by two universal curves for parallel and crossed configurations that do not depend on the van der Waals constants, the angle between tubes, and the surface density of atoms and their nature but only on the dimensionless distance. The explicit functions for equilibrium VDW distances, well depths, and maximal attractive forces have been given. These results may be used as a guide for the analysis of experimental data to investigate the interaction between nanotubes of various natures.

Phase-field simulations of the martensitic transformation (MT) in the austenitic matrix, which has already undergone the plastic deformation, are carried out. For this purpose the elasto-plastic phase-field approach of incoherent MT developed in the previous work [Kundin et. al. J. Mech. and Phys. Solids 59 (2011) 2012] is used. The evolution equation for the dislocation density field is extended by taking into account the thermal and athermal annihilation of the dislocations in the austenitic matrix and the athermal annihilation at the transformation front. It is shown that the plastic deformation in the austenite caused by the MT interacts with the pre-deformed plastic strain that leads to the inhomogeneous increasing of the total dislocation density. During the phase transformation one part of the dislocations in the pre-deformed austenite is inherited by the martensitic phase and this inheritance depends on the crystallography of MT. An other part of dislocations annihilates at the transformation front and decreases the dislocation density in the growing martensite. Based on the simulation results a phenomenological dependency of the inherited dislocations on the martensitic fraction and the plastic deformation in the martensite and austenitic matrix is proposed., 27 pages, 9 figures

Excellent electron-field emission properties of carbon nanotubes (CNTs) attract essential scientific and practical interests. The small gap between top of the emitter and the anode essentially increases the field enhancement facto. The screening effect reduces the field emission current of a CNT placed in an array of CNTs. This paper presents an analytical expression of the field enhancement factor for the "floating sphere at emitter-plane potential" and the "hemi-ellipsoid on plane" model of CNT array. The field enhancement factor for the "floating sphere at emitter-plane potential" model of CNT array in diode configuration is shown. The explicit formulas for optimum distance between tubes, the areal density of emitters, and the maximum of anode are suggested.

The analytical expressions for the van der Waals potential energy and force between two crossed carbon nanotubes are presented. The Lennard-Jones potential between pairs of carbon atoms and the smeared-out approximation suggested by L. A. Girifalco (J. Phys. Chem. 1992, 96, 858) were used. The exact formula is expressed in terms of rational and elliptical functions. The potential and force for carbon nanotubes were calculated. The uniform potential curves for single- and multiwall nanotubes were plotted. The equilibrium distance, maximal attractive force, and potential energy have been evaluated.

The model proposed by Ahmad and Tripathi (2006 Nanotechnology 17 3798) demonstrates that the field enhancement factor of carbon nanotubes (CNTs) reaches a maximum at a certain length. Here, we show that this behavior should not occur and suggest our correction to this model.

Application of various one-dimensional nanostructure materials as field emission sources has attracted extensive scientific efforts. Elongated structures are suitable for achieving high field-emission-current density at a low electric field because of their high aspect ratio. Area of its application includes a wide range of field-emission-based devices such as flat-panel displays, electron microscopes, vacuum microwave amplifiers, X-ray tube sources, cathode-ray lamps, nanolithography systems, gas detectors, mass spectrometers etc. Since the discovery of carbon nanotubes (CNTs) (Iijima, 1991; Iijima & Ichihashi, 1993; Bethune et al., 1993) and experimental observations of their remarkable field emission characteristics (Rinzler et al., 1995; de Heer et al., 1995; Chernazatonskii et al., 1995), significant efforts have been devoted to the application of using CNTs for electron sources. One of the main problems for design such field emission emitter is the difficulties in estimation of the electric field on the apex of nanotubes. Only a few works considered forces acting on nanoemitters under electric field. Thus far, there is no analytical formula which provides a good approximation to the total current generated by the nanoscale field emitter. In this chapter, we theoretically consider the electric field strength, field enhancement factor, ponderomotive forces, and total current of a metallic elliptical needle in the form of hemi- ellipsoid in the presence of a flat anode. Also we shortly review the history CNT cold emitters and technology of their fabrication. Furthermore we consider the application areas of CNT electron sources.

We present an exact solution for the electrostatic field between a metallic hemi-ellipsoidal needle on a plate (as a cathode) and a flat anode. The basic idea is to replace the cathode by a linearly charged thread in a uniform electric field and to use a set of "image" charges to reproduce the anode. We calculate the field enhancement factor on the needle surface and ponderomotive force acting on the needle. Using the Fowler-Nordheim theory we obtain an exact analytical formula for the total current.

The screened field enhancement factor for a carbon nanotube (CNT) placed in a CNT array (which is reduced due to the screening effect) is derived based on the “floating sphere” model. We obtain an expression for the field enhancement factor for a CNT in the array as γ=3+2(1+η)/{(2+η)[2πα(2+η)δ2+η]}, where ρ is the radius of sphere, h is the distance from cathode to the center of sphere, and D is the distance between the nearest spheres, η=ρ/h, δ=ρ/D, and α=1 for square or 2/3 for hexagonal lattice made of CNTs. Explicit algebraic formulas for optimizing the distance between tubes, areal density of emitters, and the anode current are also obtained.

We have corrected the field enhancement factor for the “floating sphere at emitter-plane potential” model with the finite anode-cathode distance. If ρ is the radius of sphere, h is the distance from cathode to the center of sphere, and l is the distance from the center to the anode, then the field enhancement factor is given as the following expression βsph=(2+7η−η2)(λ2−2λ+2)/[2η(1−λ)(2−λ)], where η=ρ/h, λ=ρ/l. This expression demonstrates reasonable behavior for three limiting cases: if h→ρ, if l→∞, and if l→ρ. We have compared our factor βsph with the field enhancement factor βtube for the “hemisphere on a post” model and the factor βell for the “hemiellipsoid on plane” model. We have shown realization of the approximate evaluation βtube≈(βsph+βell)/2.

We consider field emission from carbon nanotubes and other elongated nanostructures. An exact solution for the electrostatic field between a metallic hemi-ellipsoidal needle on a plate (as a cathode) and a flat anode are presented. The basic idea is to replace the cathode by a linearly charged thread in a uniform electric field and to use a set of "image" charges to reproduce the anode. Exact analytical formulas of the electrical field, field enhancement factor, and electrostatic force are found. Using the Fowler-Nordheim theory we obtain an exact analytical formula for the total current. The field enhancement factor, total force and emission current, as well as their distributions on the top of the needle for a wide range of parameters, have been calculated and analyzed.