phase-field crystal method

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Efficient phase field simulations of multiple crystal orientations

Efficient phase field simulations of multiple crystal orientations

Numerical simulations of crystals with multiple crystal orientations have become the subject of intense interest in the last decade, as a vast range of industrial materials - from polycrystals to nanoparticles - falls into that category. In this work, I present several detailed studies which examine the potential of three different phase-field models for such systems. First, I ex- pand an existing coupled phase-field/Monte Carlo approach [H. Assadi, A Phase-Field Model for Crystallization into Multiple Grain Structures, in So- lidification and Crystallization (2004), ed. by D. Herlach] to eliminate lattice effects and use it to simulate crystal growth competition in veins. Secondly, I propose a new model for the growth of metal nanoparticles in ionic liquids based on a classical phase-field model [Wheeler et al., Phys. Rev. A 45 (1992) 7424] and use it in combination with the extended Monte Carlo algorithm developed earlier for some first qualitative studies. Next, I use the newly introduced phase-field crystal method [K. R. Elder et al., Phys. Rev. Lett. 88 (2002) 235702-2 ] to investigate the correlation between thermal noise and nucleation rates, which will be of use in future nucleation studies - including the nanoparticle growth described earlier. Finally, I discuss the parallels be- tween the three different models and how to combine the knowledge gained
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Field-temperature phase diagram and entropy landscape of CeAuSb2

Field-temperature phase diagram and entropy landscape of CeAuSb2

Single CeAuSb 2 crystals were grown by a self-flux method, similar to that described in Refs. [ 24 , 25 ]. High-purity ingots of Ce (99.99%, Ames Laboratory), Au (99.999%, Alfa Aesar), and Sb (99.999%, Alfa Aesar) were placed in an alumina crucible with a Ce:Au:Sb atomic ratio of 1:6:12. The crucible was then sealed in an evacuated quartz ampoule and heated to 1100 ◦ C for 10 h, followed by cooling to 700 ◦ over a period of 100 h. The excess flux was decanted with a centrifuge at 700 ◦ C. Measurement by energy-dispersive x-ray spectroscopy (EDX) confirmed that the crystals are stoichiometric to within the 5% measurement precision. The residual resistivity ratios of the crystals used in this work were between 6 and 9. A photograph of an as-grown CeAuSb 2 crystal is shown in Fig. 2 .
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Phase field modeling of ferroelectrics with
point defects

Phase field modeling of ferroelectrics with point defects

Bassiouny et al. first considered the multi-axial loading case. They introduced the Helmholtz free energy and used the yield surface in plasticity and work hardening as a source of reference. The theory described the initial polarization and the hysteresis loop under cyclic loading, but did not present the butterfly-loop [28, 29]. Based on the motion of domain walls, Huber et al. developed a micromechanical constitutive law, which is similar to crystal sliding [30]. Chen et al. develpoed a model which introduced volume fraction of domains as an internal variable, considering the interaction between crystals by a mean field method and obtained the behaviour of polycrystalline ferroelectrics [31]. Lu et al. conducted a study based on micromechanics and established a criterion taking into account the difference between the 90 ◦ switching and the 180 ◦ switching by a thermodynamic approach [32]. Shaikh et al. proposed a domain switching crite- rion for a generalized electromechanical loading based on an estimation of the existing domain switching criteria for ferroelectrics [33]. Kamlah and Tsakmakis constructed a phenomenolog- ical model of ferroelectricity for general loading histories [34]. Kamlah et al. also presented a complete phennomenological theory. They introduced several nonlinear functions to describe the switching behaviour and defined the domain switching driving forces for different loading cases [35]. McMeeking et al. presented a phenomenological theory. They defined the domain switching criterion similar to the yield surface in plasticity theory [36, 37]. Based on domain- switching mechanisms, Zhang et al. proposed a new domain-switching criterion [38].
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Near-field spectral imaging of crystal-phase heterostructures in GaAs nanowires

Near-field spectral imaging of crystal-phase heterostructures in GaAs nanowires

In III–V semiconductor nanowires, many exciting physical phenomena are determined by the spatial inhomogeneity of the crystal structure and material composition at the nanometer length scale. As it was previously discussed, the current understanding of the relationship be- tween morphology and optoelectronic properties of individual crystal-phase heterostructures in single nanowires is still incomplete. Structural parameters of nanowires are highly inho- mogeneous within the same growth batch. The investigation of nanowire ensembles provides at most the information about predominant optical and structural properties. Therefore, an accurate determination of structure-property relationships requires the application of single- nanowire measurements. The main method widely used to correlate optical and structural properties is a combination of micro-photoluminescence (µ-PL) and transmission electron mi- croscopy (TEM) measurements performed on the same single nanowires [22, 60, 111, 112]. In previous studies, this method was successfully applied to probe local photoluminescence spec- tra and the crystal structure information of individual nanowires [22, 60, 111, 112]. However, the diffraction-limited spatial resolution of the µ-PL spectral imaging (the smallest laser spot diameter is ∼0.8 µm [22]) reduces the overall resolution of the structure-property correlation technique. Such limiting factor induces uncertainties on the attribution of optical transi- tions to a particular crystal structure and hence on the interpretation of experimental data. This also hinders the use of experimental data for modelling of the electronic band structure and quantum confinement effects in crystal-phase heterostructures. Alternatively, spectral imaging by cathodoluminescence (CL) spectroscopy can provide the sub-wavelength spatial resolution of different optical properties at different locations in the wire [25, 64, 113, 114]. Recently, an attempt to directly correlate the spatially resolved spectral characteristics with the crystal structure of nanowires by applying the CL-TEM strategy was made [114]. So far, results of correlated studies were mostly based on the demonstration of the correspondence of the average emission energy shift of PL spectra to the predominate crystal structure in nanowires [64]. However, the assignment of individual emission lines to the local crystal struc- ture is required. In addition, the variation of experimental parameters, model structures, or substrates for single-nanowire measurements may cause the inconsistency in experimental results of different studies. Therefore, an advanced procedure providing a direct nanoscale correlation of high spectrally and spatially resolved local optical properties with particular
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Quantum phase transitions in magnetic systems: application of coupled cluster method

Quantum phase transitions in magnetic systems: application of coupled cluster method

Quantum phase transitions between semiclassical magnetically ordered phases and magnetically disordered quantum phases which are driven by frustration attract much interest, see, e.g., Ref. [2]. In particular, frustration may lead to the breakdown of semi- classical N´ eel LRO in 2D quantum antiferromagnets. There are a variety of models which are known to exhibit the so-called frustration effects. A typical model is the Shastry-Sutherland antiferromagnet introduced in the eighties [74], which has special arrangement of frustrating next-nearest-neighbor J 2 bonds on the square lattice, cf. Fig. 4.2. We note that for bonds of equal strength, i.e., J 1 = J 2 , the Shastry-Sutherland model is equivalent to a Heisenberg model on one of the eleven uniform Archimedean lattices [1]. Although the initial motivation to study this special frustrated square- lattice antiferromagnet is related to the existence of a simple singlet-product eigenstate (which becomes the ground state (GS) for strong frustration), the renewed interest in the last years was stimulated by the discovering of the new quantum phase in SrCu(BO 3 ) 2 [14,75] which can be understood in terms of the Shastry-Sutherland model (see Fig. 4.1). Although the GS of this model in the limit of small frustration J 2 and large J 2 is well understood, the GS phase at moderate J 2 is still a matter of discussion.
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Free-boundary problem of crack dynamics: phase field modeling

Free-boundary problem of crack dynamics: phase field modeling

There are many phase-field models for fracture based on a non-conserved order parameter that encompass much of the expected behavior of cracks [62, 52, 2, 30]. But as long as the scale of the growing patterns is set by the phase-field interface width ξ, these models do not possess a quantitatively valid sharp interface limit. While phase-field models are often used to model physical processes associated with microstructural evolution, we want to emphasize that this is not our goal here. The diffuse interface model we present in this section is instead used primarily as a numerical tool to solve the sharp interface equations Eqs. (5.3)-(5.6). This means that the phase-field model permits a strict physical interpretation only in the limit of vanishing interface thickness, ξ → 0, requiring that ξ is much smaller than any characteristic lengthscale in the problem. In order to succeed in this undertaking, our phase-field model has to fulfill two fundamental requirements: it has to have a valid sharp interface limit for the case ξ → 0, and the results must not depend on the phase-field width ξ. Both requirements are directly connected to the fact that ξ is a purely numerical parameter and not directly connected to physical properties. Alternative descriptions, which are intended to investigate the influence of elastic stresses on the morphological deformation of surfaces due to phase transition processes, are also based on macroscopic equations of motion. But they suffer from inherent finite time singularities which do not allow steady state crack growth unless the tip radius is again limited by the phase-field interface width [69].
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Modeling Crystal Growth and Multi-Phase Flow in Metal-Air Batteries

Modeling Crystal Growth and Multi-Phase Flow in Metal-Air Batteries

electrochemical dynamics within a GDE including the effects of pressure-driven convection and multi- phase coexistence with continuum models and Lattice-Boltzmann theory [1,2]. The lithium hydroxide concentration in alkaline lithium-air batteries is accumulating during discharge until it precipitates. We rationalize that this precipitation is inhomogeneous due to fundamental transport effects in alkaline electrolytes and discuss adjusted cell designs [1]. On a microscopic level, we study the elementary kinetics of the oxygen reduction reaction on the active surfaces [3].

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Signal interpolation method for quadrature phase-shifted Fabry-Perot interferometer

Signal interpolation method for quadrature phase-shifted Fabry-Perot interferometer

Quadrature phase-shifted Fabry-Perot interferometer (QFPI) is a kind of Fabry-Perot interferometer (FPI) which is applicable to displacement measurement. QFPI differs from the conventional FPI. It has an orthogonal interference signal. So the quantity and direction of the measured displacement can be obtained simultaneously.

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A Method of Calculating Bound States in a Unified Field Model 

A Method of Calculating Bound States in a Unified Field Model 

In a model in which the usual elementary particles (leptons, quarks, photons, weak bosons, gluons, and so on) are bound states of truly elementary fermions we present a method for the [r]

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Color current induced by gluons in background field method of QCD

Color current induced by gluons in background field method of QCD

terms [1]. This formalism needs to be extended by incorporating classical chromofield. The evolution of partons can be obtained by solving the Schwinger-Dyson equation defined on a closed-time-path. This is formidable task because it involves a non-local non-linear integrodifferential equation and because of its quantum nature. There are two scales in the system: the quantum (microscopic) scale and the statistical-kinetic (macroscopic) one. When the statistical-kinetic scale is much larger than the quantum one, the Schwinger-Dyson equation may be recasted into a much simpler form of the kinetic Boltzmann equation by a gradient expansion. The Boltzmann equation describes evolution of the particle distribution function in the phase space of momentum-coordinate, and can be solved numerically for practical purpose. For a non-equilibrium system of quarks and gluons in a chromofield, the distribution function also depends on the classical color charge of the parton, since the color is exchanged between the chromofield and partons and among partons themselves. In this case, the Boltzmann equation also describes the evolution of the parton distribution function in the color space [2,3]. While the Boltzmann equation describes the kinetic and color evolution of the hard parton system, soft partons are normally treated as a coherent classical field whose evolution may be described by a equation which is similar to Yang- Mills equation. Therefore, one should study the transport problem for hard partons in the presence of a classical background field. For example, in high energy heavy-ion collisions, minijets (which are hard partons) are initially produced and then propagate in a classical chromofield created by the soft partons [4–6]. In this situation it is necessary to derive the equation of the gluon and the classical chromofield to study the formation and equilibration of quark-gluon plasma. Hence we see that the issues we want to address in this paper have practical implication in ultra-relativistic heavy ion collisions.
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Phase-Field Modeling of Relaxor Ferroelectrics and Related Composites

Phase-Field Modeling of Relaxor Ferroelectrics and Related Composites

Although extensive efforts have been made to explore the physical origin of relaxors, there are limited works on the discussion of the electromechanical couplings in relaxors, which are more important in applications. Phase-field model has been proved to be an efficient tool to study the evolution of domain structure in ferroelectrics, [ 13 , 14 , 15 , 16 ] and the order parameter can be fully coupled with electrical and mechanical quantities. Recently, the phase-field methods have also been employed in the field of relaxors, see Refs. [ 17 , 18 ]. Rather than generic re- laxor models, these works regards the relaxor features are the results of either point defects or localized nanoscale polar volumes inside a different phase. Moreover, the piezoresponse cannot be directly simulated from these models. Therefore, a generic fully coupled phase-field relaxor model which can reproduce and predict relaxor characteristics found in experiments is required. There are many important issues which could be investigated with the phase-field relaxor model. Here more attention is paid to the following two topics. (1) The role of relaxors in the relaxor/ferroelectric composite structure. Experimentally, the relaxor/ferroelectric composites show excellent large-signal piezoresponse. Understanding the role of relaxors in such compos- ites is important for the designing of relaxor-based piezoelectric devices. (2) The role of relaxors in the core-shell structure. The core-shell structures have been found in some relaxor systems, for instance, 0.75Bi 1/2 Na 1/2 TiO 3 -0.25SrTiO 3 (BNT-25ST). The phase-field relaxor model will help in understanding the coupling effect at the core-shell interface and the abnormal macro- scopic electromechanical behaviors in such core-shell structured relaxors. In the following, these two questions are elaborated in detail.
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A Field-Matching Method for Sound Field Synthesis for Large Scale Sound Reinforcement Systems

A Field-Matching Method for Sound Field Synthesis for Large Scale Sound Reinforcement Systems

5.1 Setup Listening tests were conducted to evaluate the performance of the method. The setup shown in Figure 3 consisted of 16 loudspeakers located with spacing of 53cm, same as the simulation in 4. The loudspeakers were hidden behind an acoustically transparent curtain to exclude visual cues. Four listeners positions were defined to represent the entire listening area. Virtual sources are located in the “stage” area behind the loudspeakers; one virtual source is placed far off the stage on the side. The test signals were three pulses of broadband pink noise. Each listener position was provided with a disk marked in degrees on the floor. Participants were asked to write down the perceived direction of the source which was later compared with the geometrical direction.
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Phase-Field Modelling of Crack Propagation in Anisotropic Polycrystalline Materials

Phase-Field Modelling of Crack Propagation in Anisotropic Polycrystalline Materials

In this contribution, we firstly provided a brief overview of the phase-field model for crack propagation in isotropic and anisotropic materials, respectively. Next, we presented the phase-field model for anisotropic fracture which can distinguish the loading under tension and compression. The fully coupled monolithic solution scheme within the finite element framework was formulated. Comprehensive parameter studies for the proposed phase-field model have been done and the results have been analyzed. Simulations of the anisotropic fracture within the lower value of the crack orientation θ are also considered, though the resulting crack path is no longer in alignment with the predefined crack orientation, and the opposite direction with fluctuating crack paths can be observed in the results. Furthermore, it is necessary to account for such phenomena that the widely used anisotropic materials (e.g. woods) consist of the lower value of the crack orientation in the structure and it can play an important role in material design processes. This topics will be addressed elsewhere. Representative numerical examples of the crack propagation in solar-grade polycrystalline silicon are carried out which can validate its capability of modelling of inter- and transgranular fracture process. Last but not least, the damage and failure analysis of solar-grade polycrystalline silicon using phase-field method will also be compared with experimental results in future.
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Image Space Tensor Field Visualization Using a LIC-like Method

Image Space Tensor Field Visualization Using a LIC-like Method

In the 1980’s several method where introduced to avoid triangulation of clouds, dusty surfaces, or other density based volumes. These methods, namely [Bli82, KVH84], adopted the idea of ray-tracing to visualize volumetric data and therefore build the foundation of direct volume rendering. It is assumed, that the volume to be rendered is filled with a media, whose optical properties are defined by the scalar values inside the volume. To reach maximum flexibility, the mapping from a value at a given point to a color can be done using transfer functions, similar to those described in Section 2.3.1. This allows emphasizing different parts of the volume through varying the emission and absorption prop- erties of a certain structure. For diffusion tensor images, anisotropy measures have proven to be appropriate [KWH00] as scalar property to be used in transfer function definition and, therefore, for color and opacity.
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Solid state transformations and crack propagation : a phase field study

Solid state transformations and crack propagation : a phase field study

Cracks concentrate the macroscopic elastic energy of a stressed solid to release it spon- taneously on atomic scales. It is particularly this multiscale nature that makes fracture being a quite challenging subject, and that eventually requires a detailed understanding of crack motion on atomistic as well as on macroscopic scales. However, discussing fracture on the level of elastically driven pattern formation processes, we rather address aspects of the latter: like velocity selection and scaling; stability of motion and energy release mech- anisms. Here, we aim to find minimal models for fracture in such a way that not only the crack speed, but also the crack shape can be determined self-consistently. Those mini- mal models are designed such that all the microscopic details which occur at the crack tip are modeled effectively by a single kinetic mass transport coefficient. Apart from that we only take into account well established macroscopic bulk theories. In particular we consider the dynamic theory of linear elasticity, and viscous bulk friction, with which we go beyond the usual small scale yielding of brittle fracture. In this respect, both the ki- netics of phase transformations and fracture have in common, that they are both moving boundary problems.
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Detection of Constant Phase Shifts in Filters for Sound Field Synthesis

Detection of Constant Phase Shifts in Filters for Sound Field Synthesis

The audibility of constant phase shifts can be regarded as special issue of the audibility of phase distortion and group delay distortion, cf. [13–18], often evaluated with allpass filters. From these works it is known, that audibility is strongly dependent of the signal’s waveform and spectrum and the amount of the group delay in the critical bands. Generally, sensitivity for phase/group delay distortions decreases with increasing frequency. For low frequency content a different pitch and for high frequency content ringing and different lateralization is reported for group delay distortions. The polarity of highly transient signals plays a role for the audi- bility. It was often shown, that training on phase/group delay distorted audio content increases the sensitivity to detect them. To the authors’ knowledge to date, the perceptual impact of the constant phase shift has not been studied yet. It is of great interest whether the existence or absence of such a phase shift is audible, and in the special context of sound field synthesis, if this affects the authenticity of the synthesized sound fields. The paper discusses the signal processing fundamentals of discrete-time constant phase shift in Sec. II. In Sec. III a listening test is presented for selected audio content and phase shifts to initially evaluate the audibility of constant phase shifts. Sec. IV concludes the paper.
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Scalability analysis of the phase field fracture finite element program

Scalability analysis of the phase field fracture finite element program

The reference is the scalability of ParFEAP 5 as it is given in [ 28 ] and has been used for validation in section 4.1.1 . There a linear elastic problem was solved (simulation A). Subsequently the scalability of ParFEAP with the pc jacobi becomes the perfor- mance reference (simulation B). The serial implementation of the PLEANv1.0 can be compared with ParFEAP with a linear elastic problem, without invoking the fracture problem. Hence the same problem is solved by a serial algorithm on the one hand and on the other hand by a parallel routine (simulation C). Yet there is a slight difference because the phase field means an additional nodal degree of freedom. But because there is no crack initiation the additional serial computation of the phase field can be assumed to be of a negligible amount. If this assumption could not be done the number of nodes or elements would need to be reduced proportionally. Because of the serial computation of the element arrays it is expected that the simulation C will scale less than the simulation of type B.
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Thermomechanical phase‐field fracture modeling of fluid‐saturated porous media

Thermomechanical phase‐field fracture modeling of fluid‐saturated porous media

Cam-Lai Nguyen 1 , Abdel Hassan Sweidan 1 , Yousef Heider 1,∗ , and Bernd Markert 1 1 Institute of General Mechanics, RWTH Aachen University. Eilfschornsteinstr. 18, 52062, Aachen, Germany In this work, the problem of brittle fracture in a fluid-saturated porous material is extended by considering the non-isothermal states of the sample. The temperature field will affect the problem in two aspects: 1) Temperature-dependent material param- eters, such as elasticity modulus (E) and critical energy release rate (Gc). 2) Thermal expansion due to thermo-mechanical volume coupling. In hydraulic fracturing, we further study the effect of the temperature difference between the injected fluid and the surrounding porous media ambient on the crack behavior. The modeling of the porous media domain is based on the macroscopic theory of porous media (TPM), whereas the phase-field method (PFM) is applied to approximate the sharp crack edges by diffusive ones. In the numerical implementation, the coupled system of partial differential equations will be solved using the FEM in order to simulate the heat transition in the crack and non-crack regions.
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f-f transitions of Nd+ in the crystal field / Jan Michael Senekowitsch

f-f transitions of Nd+ in the crystal field / Jan Michael Senekowitsch

The first part being kinetic and potential energies of nuclei and electrons , the electron- nuclei-, nuclei-nuclei- and electron-electron interaction. Hso, Hss, Hhfs, Hext denote spin-orbit, spin-spin, hyperfine structure and extern field couplings. Although we can state the problem obtaining solutions proves technically difficult as the pure amount of necessary wave function and energy eigenvalues scales exponentially with increase of the size of a given molecule. We have therefore to choose an aproximative approach in order to simplify the problem. In assuming that a single electron moves in a mean field of all other electrons we can reduce the necessary equations. Further we can use the Born-Oppenheimer approximation in supposing e- move nearly instantaneously in comparison to the much heavier nuclei decoupling their motion from the electrons position. An altogether different approach is to apply group theoretical considerations to the problem.
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A Phase-Field Approach to Damage Modelling in Open-Cell Foams

A Phase-Field Approach to Damage Modelling in Open-Cell Foams

inspection of the evolution Equation (5) and the Cahn-Hillard or Allen-Cahn equations often used in the phase- field formulations (Qin and Bhadeshia (2010), Moelans et al. (2008) ) reveals these three equations as a reaction- diffusion equations, described in terms of an order-parameter and its gradients. This categorisation and also the non-conserving character of the connectivity-based phase-field variable emphasise the similarity of the proposed formulation with the equations used in variational phase-field formulations.

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