On the macroscopic scale, cellular automata and geometric **models** are most efficient, but lack physical details. Atomistic molecular dynamics simulations have been suc- cessfully applied to simulate solidification and thus offer a route for revealing details of the growth kinetics [18, 19, 81]. Because of the enormous computational effort, however, these **models** are restricted to relatively small system sizes of typically not more than a couple of million atoms. This is why for modelling phenomena on the microscopic scale, **phase** **field** **models** have emerged as a powerful tool for simulating free boundary problems with complex morphological evolution. Since the transport equation for heat and the **phase** **field** are solved simultaneously, the effects of surface tension, nonequilibrium, and anisotropy can be directly included.

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of driving the state y , that is the **phase**-eld in Cahn-Hilliard equation (1.14), as
lose as possible to a desired state y d . The stru
ture of the obje
tive fun
tion J is standard: the rst term in J measure the distan
e between the state y and the de- sired state y d ; the se
ond is a regularization term whi
h guarantees well-posedness of the problem. The
onstant parameter α is usually small (α ∈ [10

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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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In the last chapter a phenomenological two-dimensional Landau-type **phase** **field** representation of the nonpolar matrix material is suggested, which can be easily combined with existing **phase** **field** **models** for polar materials. In contrast to the equivalent circuit model, it is possible to simulate parts of the microstructure in the model to study the influence of the grain boundaries, the grain sizes, the grain shapes, the grain distribution, and the inter- and intra-granular electrical and mechanical interactions on the macroscopic polarization and strain. The **phase** separation energy, i.e., the part of the free energy density that depends on the polarization, is phenomenologically chosen to create the typical double loop polarization hysteresis curve of a nonpolar material. Necessary material parameters were determined from measurements or reports on similar materials. With this model it was possible to compare the behavior of a two layer composite, which is a representation of the equivalent circuit, to a simple grain model of an individual polar grain surrounded by matrix material. It was found that the boundary condi- tions at the grain boundary between both materials have a significant influence on the domain structure inside the polar seed material and a direct influence on the macroscopic strain and polarization of the composite. However the presented studies are only a proof of concept. Future studies have to address the influence of the character and shape of the grain boundary in more detail. The implementation of a parallelized solver will allow the simulation of more complex microstructures and the expansion of the model to three dimensions. This will allow for the study of interactions between different seed grains. An important point that should be addressed in future studies is the accuracy of the model parameters. Moreover, recent in situ transmission electron microscopy measurements revealed that the electric fields required for the nonpolar material to transform into its polar **phase** are actually smaller than the coercive **field** of the polar **phase** [271]. At the **phase** transition **field** the material transforms from the nonergodic relaxor state to a poly-domain polar state similar to that of a ferroelectric material. With further increas- ing electric fields these domains begin switching and continue until the material is fully poled. This is not predicted by the presented **phase** **field** model. Only transitions from nonpolar to a poled polar state are possible, domain walls can only occur at grain boundaries or highly spatially fluctuating electric fields. More advanced approaches are necessary to consider this experimental observation in the **phase** **field** model, for example the introduction of a new order parameter that describes polar or nonpolar **phase** of the material and couples to the electric **field**.

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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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The first **phase**-**field** concept was proposed in an unpublished work by Langer [10] and was first publicly documented by Fix [11] and Caginalp [12]. The simulation of the evolution of complex 3D dendritic structures using **phase**-**field** **models** by Kobayashi [13] initiated extensive use of this methodology in materials sciences. The binary transitions/equilibria between two states were later extended to multi-**phase** equilibria in a multi-**phase**-**field**-model [14]. Higher order derivatives of the order parameter eventually lead to atomic resolution of rigid lattices in so called **phase**-**field** crystal **models** [15]. Currently, **phase**-**field** **models** have reached a high degree of maturity and found applications in describing complex microstructures in technical alloy systems [16]. Reviews on **phase**-**field** modelling are found, for instance, in [17,18].

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Figure 4.4 presents the complete map of a single 8.7-µm-long GaAs nanowire performed by NSOM at a temperature of 10K. The full map of the wire was composed of three subsequent rectangular NSOM scans covering the region on the substrate that contains the nanowire of interest. The top panel (Fig. 4.4a) shows a minor bending of the topographical image of the wire, caused by a slow drift of the sample during the scan processes. The black arrows indicate end facets of the wire used as markers to set an arbitrary beginning of the coordinate axis along the wire. In Figures 4.4b and 4.4c, spatial images of the local PL peak intensity and the PL peak energy are presented. When two or more PL peaks arise in a single pixel of the image, the most intense PL peak of the spectrum of a given pixel is selected and plotted. It is seen that the spatial distribution of the PL peak intensity can be described by a set of clearly separated intensity domains of a various axial extent (Fig. 4.4b). The axial segmentation of optical properties is even more pronounced in the image of the PL peak energy position (Fig. 4.4c, right panel). The sharp borders between some PL peak energy domains, i.e. segments of a constant energy, coincide with abrupt borders of PL peak intensity domains, as emphasised by the dashed lines. Also, it is found that the axial extent of some segments of constant energy is as small as 270 nm. In Figure 4.4c, there is a minor variation of PL peak energy at some positions in the direction that is perpendicular to the nanowire axis. As it can be seen, when the near-**field** probe moves away from the actual wire position, the PL intensity drops drastically (Fig. 4.4b). Therefore, the determination of the PL peak energy is less accurate in these pixels due to the low signal-to-noise ratio and might cause such deviation.

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In this work, the parallel scalability of the finite element program for the sim- ulation of fracture in elastic-plastic solids is analyzed. The in house finite element program, from here on referred to as PLEANv1.0, is written as a user element and a user material model for the research code Finite Element Analysis Program (FEAP) developed by Prof. R. L. Taylor from the University of California Berke- ley. After introducing the fundamentals of the background theory of the finite element method, the **phase** **field** method of fracture and the underlying constitu- tive relations of elastic-plastic solids, the scalability analysis is performed by solv- ing a mechanical problem with 3D unit cube geometry and uniaxial tensile load. The scalability and performance of the PLEANv1.0 is obtained for computations performed over a high performance computing (HPC) cluster using 576 CPUs. A comparison of the scalability and performance of PLEANv1.0 with the research code FEAP for different solution options is also discussed. The promising scala- bility of PLEANv1.0 allows solution of very large problems in a computationally efficient manner by using the HPC cluster.

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For the treatments I, II and IV signal periodicity is as- sumed and thus the ideal **phase** shift filter via DFT (Sec. 2.2) was applied. To create treatments V and III, no signal periodicity was assumed for the whole musical piece as well as for the generated pink noise raw material of 6 minutes duration. Thus FIR filtering according to Sec. 2.1 was realized. Considering the audio contents as rectangularly windowed signals of infinite duration, the filter order of 3 963 530 (≈ 90 s!) ensures that linear convolution of the chosen excerpt of Hotel California is complete. The resulting magnitude ripple of the Blackman windowed FIR is negligible for the relevant reproduction bandwidth. Since the pink noise length can be arbitrarily set, the same FIR filter was utilized for consistence.

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Finally, there is one more aspect of strong external magnetic fields, that is central to the in- vestigations presented in this chapter. Besides their presence in experimental situations and the extension of the ’standard’ QCD **phase** diagram, they have a more technical application in researching confinement. This connection was introduced in form of a dual condensate in [156]. The dual condensate with respect to the magnetic fields, the dressed Wilson loop, is an observable that is sensitive to confinement while at the same time derived from the chiral condensate. It is equal to the conventional Wilson loop in the case of infinitely heavy quarks and thus allows to probe the string tension between two static quarks. The conventional Wil- son loop is in the first place defined on the lattice, and not directly accessible from functional methods. In consequence, the dressed Wilson loop could be a very useful observable for functional methods, due to its derivation from the chiral condensate. Nevertheless, there is no work reporting on dressed Wilson loops from other approaches than from lattice gauge theory. Therefore, we strive to investigate the possibility to recover the dressed Wilson loop with the help of external magnetic fields in a Dyson–Schwinger framework.

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To tackle the moving boundary problem of dynamic crack propagation, we properly extended the above mentioned fully dynamic **phase** **field** model for solid-state transfor- mations to a **phase** **field** description of crack propagation by first order **phase** transforma- tion processes. Apart from that, also efficient steady state sharp interface methods based on the expansion of the elastic state in eigenfunctions of the straight mathematical cut were used. In contrast to the **phase** **field** technique this method easily could cover both the **phase** transformation kinetics as well as surface diffusion. However, it provides only accurate results in either the limit of vanishing viscosity or the limit of static elasticity (slow crack motion). Both quite complementary methods together then offer a profound insight into the phenomenon of fracture within the frame work of the present continuous description. Considering mode I cracks our theory predicts three generic features of frac- ture: The saturation of the steady state velocity appreciably below the Rayleigh speed; parameter regimes of normal crack behavior where the steady state velocity increases with increasing driving force; and the tip splitting instability for high applied tensions. However, concerning mixtures of mode I and mode III loadings the situation changed quite significantly. In contrast to pure mode I loadings, the cracks undergo now the tip splitting instability even without the inclusion of dynamic effects due to a finite mode III loading contribution. Furthermore, it turns out that the propagation via **phase** transforma- tions does not lead to a valid description of fracture due to a weak logarithmic increase of the fracture opening, which is revealed by an asymptotic analysis of the chemical poten- tial far behind the tip. However, the model of crack propagation by surface diffusion does not suffer from this problem, since in this case the material conservation condition leads to a suppression of this effect.

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In Section 2.4, we will give a precise derivation of the main model studied in this thesis, after we have introduced some necessary notation and definitions at the beginning of Chapter 2. Nevertheless, we give a first description of the mathematical setting of the main model for introductory purpose. The model involves three variables: the mechanical displacement u, the electric **field** φ and the polarization P . The former two variables are given by an elliptic piezo-system, with coefficients which are functions in variable P (see (2.14) below). Thus roughly speaking, once P is given, the variable (u, φ) can be uniquely determined by certain elliptic existence theory. Hence it suffices to find a solution P which fulfills the evolutionary law

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Foams are complex and challenging materials. The damage process of the foam materials takes place on multiple scales changing several physical and structural properties of the material. In this study, the topology-based vari- able describing the connectivity state of a cell is introduced to formulate a non-variational **phase**-**field** model for the damage evolution in an open-cell foam. The material is considered consisting of the damaged and unimpaired **phase** with the proposed **phase**-**field** variable describing the separation of phases. The performance of the compu- tational model is examined by means of the standard benchmarks such as tensile and simple shear test. The results show a qualitative correspondence with the two-dimensional artificial foam model used as a reference. Further- more, the influence of the directional data extracted from the microstructure is investigated. The utilisation of the connectivity-based damage variable turns out to be a suitable choice for the simulation of the damage evolution in open-cell foam materials.

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for each model are completely new and we obtain optimal regularity results for each problem under consideration. The second part in the analysis of (0.1) and (0.2) is devoted to the study of the long- time behavior of the solutions. To be precise, we will show that each solution converges to a steady state as time tends to infinity, without a restriction on the initial value. In particular we show that for any initial value in an appropriate energy space, there exists a solution of the stationary problem such that the corresponding orbit converges to this steady state. Moreover, we are able to remove the smallness restrictions on a, c and B − I in the Cahn-Hilliard-Gurtin equations, which were assumed by Miranville & Rougirel [31]. A convergence result for the conserved Penrose- Fife equations is not known to the author. The same holds for the non-isothermal Cahn-Hilliard equation. To prove convergence, we need to know that for each of the above **models**, there exists a strict Lyapunov functional E : V → R, defined on a suitable energy space V , which satisfies the Lojasiewicz-Simon inequality near some point ϕ in the ω-limit set of the solution. That is, there exist constants δ, C > 0, s ∈ (0, 1/2] such that for all v ∈ V with |v − ϕ| V ≤ δ there holds

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S := K F (p) 0 S with K F as the fluid compressibility parameter and p as the fluid pressure [4]. Following the kinematics of multiphase materials, a Lagrangian description of the solid matrix via the solid displacement u S and velocity v S is considered. The pore-fluid flow is expressed either in an Eulerian description using the fluid velocity v F or by modified Eulerian settings via the seepage velocity w F := v F − v S . Within a geometrically-linear framework, the solid small strain tensor is defined as ε S := 1 2 (grad u S + grad T u S ). The onset and propagation of brittle fractures are modeled based on the diffusive interface **phase**-**field** modeling (PFM), which uses a scalar **phase**-**field** variable d S to determine the material state, i.e. d S = 1 for the cracked state and d S = 0 for the intact state. Within brittle fracture mechanics, the total potential energy is expressed as the

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The numerical simulations of the eutectoid transformation in binary Fe-C steels is performed using the grand-chemical potential formulation Choudhury and Nestler [2] of the multiphase-ﬁe[r]

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The underlying concept of **phase**-resolving numerical wave **models** is based on the consideration of time and space scales of single waves, which implies that the overall wave **field** is the result from the superposition of multiple individual waves. With the distribution of sea surface elevation around the mean water level (zero-crossing), the free surface evolution is directly affected by the local variations in bathymetry, wave-wave interactions, and wave breaking. These processes require high spatial resolution; therefore **phase**-resolving **models** typically use a much finer mesh than **phase**-averaged **models**. Due to the associated computational constraints, the model domains usually cover areas on the order of several square kilometers in contrast to square degrees. Of course, this is different in the case of tsunami computations where the spatial scale of individual waves spans over several kilometers, therein allowing for the use of much coarser numerical meshes.

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In contrast to these systems and to gas targets, the sub-cycle quiver motion and the cycle-averaged drift motion can not be separated in our experiment, as the ponderomotive shift becomes non-adiabatic due to the small decay length of the near-**field**. Due to the short pulse length and the long wavelength, the pondero- motive shift is clearly visible in our experiment. At low intensities the obtained energy is limited by the drift motion. This leads to a strong linear increase of the peak position with increasing intensity. At intermediate intensities the peak is reached and the maximal ponderomotive shift is acquired. The sub-cycle propaga- tion becomes dominant and in the following, the additionally acquired energy is only a result of the stronger electric **field** strength and thereby comparably small. The appearance of the secondary peaks at lower energies can be attributed to different half cycles of the laser-**field** as indicated by theoretical simulations. Not all features can be explained by the simple considerations above. In the spectra the peak significantly broadens, which is not reproduced by the simple model and has to be analyzed in detail with further experiments.

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As to nonlinear evolutionary equations with memory term there has been only some progress concerning convergence to steady state. The reason for this lies essentially in the fact that these problems do not generate in general a semi-ﬂow in the natural **phase** space. Another diﬃculty consists in ﬁnding Lyapunov functions for such problems which are appropriate to investigate the asymptotic behaviour of global bounded solutions.

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