Numerische Simulationen von Kristallen mit multiplen Orientierungen sind im letzten Jahrzehnt zum Gegenstand intensiven Interesses der Indus- trie geworden, da eine große Bandbreite von industriellen Materialien - von Polykristallen bis zu Nanopartikeln - in diese Kategorie fallen. In dieser Ar- beit stelle ich mehrere detaillierte Studien vor, die das Potential von drei verschiedenen Phasenfeldmodellen f¨ ur die Beschreibung solcher Systeme un- tersuchen. Als erstes erweitere ich einen existierenden gekoppeltes Phasen- feld/Monte Carlo-Ansatz [H. Assadi, A **Phase**-**Field** **Model** for Crystalliza- tion into Multiple Grain Structures, in Solidification and Crystallization (2004), ed. von D. Herlach], um Gittereffekte zu eliminieren und verwende diesen Ansatz in Kombination mit dem zuvor entwickelten erweiterten Monte Carlo-Algoritmus, um Kristall-Wachstumswettbewerb in Gesteinskluften zu beschreiben. Anschliessend stelle ich ein neues Modell f¨ ur das Wachstum von metallischen Nanopartikeln in ionischen L¨osungen vor, welches auf einem klassischen Phasenfeldmodell basiert [Wheeler et al., Phys. Rev. A 45 (1992) 7424] und verwenden es f¨ ur erste qualitative Studien. Danach ver- wende ich die neue Phasenfeld-Kristall-Methode [K. R. Elder et al., Phys. Rev. Lett. 88 (2002) 235702-2 ] um die Korellation zwischen thermalem

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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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Polycrystalline materials are widely used in engineering and material science applications, e.g. automobile, aerospace or renewable energy. The macroscopic defects are generally strongly influenced by the fracture be- havior of the polycrystalline materials at meso- and microscopic level. In this paper, the proposed **phase**-**field** **model** for anisotropic fracture, which accounts for the preferential cleavage directions within each randomly ori- ented **crystal**, as well as an anisotropic material behavior with cubic symmetries, has been used to simulate the complex crack pattern in solar-grade polycrystalline silicon. Furthermore, the proposed **phase**-**field** **model** allows to distinguish the loading under tension and compression. The finite element implementation of the **model** has been realized by using a monolithic solution scheme. Three representative numerical examples are carried out, i.e. anisotropic crack propagation (i) in a sole material, (ii) in a bi-material with different crack orientation and (iii) in multi-grains with randomly distributed anisotropy. It is demonstrated that the proposed **phase**-**field** **model** is capable of characterizing fracture propagation in anisotropic solids under static loading.

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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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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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We acknowledge useful discussion with Manuel Brando and Christoph Geibel. We thank Hanoh Lee for advice on crys- tal growth. We thank the EPSRC (Grants No. EP/1031014/1 and No. EP/G03673X/1) and the Max Planck Society for financial support. We also acknowledge the support of the LNCMI-CNRS, member of the European Magnetic **Field** Laboratory (EMFL). E.A.Y. acknowledges support from the Royal Society. P.C.C. was supported, in part, by the U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Sciences and Engineering through the Ames Laboratory. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358. V.F. acknowledges support by the Deutsche Forschungsgemeinschaft through FOR Grant No. 960. H.S. gratefully acknowledges fellowships from the Canon Foundation.

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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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The modelling of **phase** transformation (austenite formation from ferrite pearlite and ferrite formation from austenite) will be performed separately and the results will be combined in one through process simulation. For the simulation of pearlite dissolution a simplified approach is used. Pearlite in this approach is considered as an effective pseudo-**phase** with mixed properties of ferrite and cementite. The thermodynamic driv- ing force for dissolution is calculated from overheating obtained by a linearization of the **phase** diagram. The interactions of pearlite with other phases are defined by assumed slopes which restrict the two **phase** regions [13]. Based on the metallographic observations, nucleation of austenite during heating is assumed to take place only within pearlitic grains. The number of austenite nuclei is set to 100 so that in the large pearlitic grains several austenite nuclei could form, similar to the metallographic finding.

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The modelling of **phase** transformation (austenite formation from ferrite pearlite and ferrite formation from austenite) will be performed separately and the results will be combined in one through process simulation. For the simulation of pearlite dissolution a simplified approach is used. Pearlite in this approach is considered as an effective pseudo-**phase** with mixed properties of ferrite and cementite. The thermodynamic driv- ing force for dissolution is calculated from overheating obtained by a linearization of the **phase** diagram. The interactions of pearlite with other phases are defined by assumed slopes which restrict the two **phase** regions [13]. Based on the metallographic observations, nucleation of austenite during heating is assumed to take place only within pearlitic grains. The number of austenite nuclei is set to 100 so that in the large pearlitic grains several austenite nuclei could form, similar to the metallographic finding.

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has revealed molecular motions in the high and low temperature phases (Phase I and II, respectively) of 4- chlorobenzyl alcohol (pCBA) [1].. It has been pointed out that the crystal dy[r]

Solid lines are gener- ated from (1) and it's derivative with determined values of the coefficients A 0, B and C. Temperature at which the maximum of permittivity oc- curs as a functio[r]

interest. The interplay between bulk and surface forces gives rise to a complex **phase** behavior. In experiments the confinement is found to induce capillary condensation[2]. Fluids can be studied with the help of interaction potentials, which **model** the po- tential energy between two particles. The purpose of statistical physics is to predict macroscopic properties from such microscopic models. This can be done by computer simulations. The development of computer simulation techniques in statistical physics started in 1953, when Metropolis et al. showed how to obtain the equation of state for a given interaction potential[3]. They used the Monte Carlo (MC) integration scheme – integrating over a random sample of points. An alternative method, called Molecular Dynamics (MD) was presented in 1959[4]. In MD simulations the Newtonian equation of motion is numerically solved.. The success of these methods in predicting thermodynam- ical properties for **model** fluids composed of hard discs[5], Lennard-Jones molecules[6] and for many other fluids and solids in the following years have made computer simu- lations an important tool in theoretical physics. Computer experiments have two main advantages: First, theoretical **model** systems can be developed by testing and com- paring its properties with experimental results. Second, simulations give insight into processes, which are not accessible to the experiment, e.g. the observation of the trace of a single particle. Limitations of molecular simulations are the capacity of computers and the inaccuracies of the theoretical models. During the last decades modifications and combinations of these methods were used – supported by the rapid development of computer hardware. Furthermore, computer simulations are nowadays not only used for physics, but also for chemistry, car traffic, meteorology, climate and forest fire mod- eling, networks of biology, economics, telecommunications, internet and society, and many others. The development of these research areas was highly inspired by physics.

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The **phase** transition **phase**-**field** **model** we present can be derived from variational prin- ciples and converges to the desired sharp interface limit without the problems encountered in the surface diffusion case. Our simulations show that it describes the complicated tip behavior and the elastic far-**field** behavior correctly, also allowing the numerical extrac- tion of quantities like the stress intensity factor (Fig. 5.7). However, despite the benign asymptotic behavior of the **phase**-**field** **model**, we showed clearly that finite-size effects and insufficient separation of the tip radius and the numerical **phase** **field** interface width can strongly influence the numerical results. As a central result, we present an extrapolation technique that enables us to remedy this problem in a systematic manner. In a two step process, we extrapolate the obtained data first to infinite system sizes and then to the limit of vanishing interface widths. The necessary large scale simulations required that the simulation program was parallelized to make efficient use of the supercomputers JUMP, JUBL and JUGENE, which are operated at the Research Center J¨ ulich; we obtained an excellent scaling behavior up to several thousand processors. Our extrapolated values were then compared to the results of a recently developed sharp interface description based on a multipole expansion technique [97] with the same underlying sharp interface equations. The comparison exhibited a very convincing agreement between the completely unrelated methods (Fig. 5.11). We can confirm that the sharp interface **model** has a regime in which the crack reaches a steady state in which it grows with speeds comparable to, but consid- erably slower than the Rayleigh speed, in agreement with experimental observations. This is also in agreement with results from molecular dynamics simulations and experiments. Treating the crack as a fully time dependent free boundary problem, both the propagation velocity and the crack shape are determined in a fully self-consistent manner. We also find the tip velocity to be a weakly decaying function of the external driving force as tip blunting is the preferred mechanism to release the elastic energy (Fig. 5.12).

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( q ) ± |( q )| as the Macauley brackets. Herein, G d represents the material degradation function,
µ S and κ S are the bulk and shear moduli of the porous matrix, respectively, m
θ := −3 κ S α S θ with α S θ as the coefficient
of solid thermal expansion, and θ α is the Kelvin temperature of each constituent. The **phase**-**field** evolution is expressed

However, what the heat pad not directly reveals is how the **phase** transformation dy- namics produce order in a less ordered environment 4 . Although sometimes emerging from rather simple homogeneous initial states, the appearing structures can show a pat- terned ordering of unexpected variety and fascinating beauty. Probably the most famous example, in this context, is the beautifully shaped snowflake, which grows out of homo- geneous undercooled water vapor. However, also in the case of processing Damascus steel ordering of the microstructure is gained through each process step. It is worth to mention that ordering or pattern selection further implies the presence of a stabilizing force, which somehow competes with the undercooling that drives the process. Then, the driving and the stabilizing force counterbalance each other, which results in an optimum that is reflected by the actual selection of length and/or velocity scales. An important stabilizing force is capillarity, meaning that the existence of interfaces is »energetically expensive« for the system. As known from the literature, the selection of solidification patterns like for instance dendritic structures 5 requires the additional stabilising influence from anisotropic capillarity, which results from an orientation dependent interfacial en- ergy. In turn, this kind of selection is not possible without the orientation dependence, i.e. for isotropic capillarity as we will consider here. Also the wootz ingots had been solidified from a homogeneous undercooled liquid, and during this process step dendritic structures were selected [123]. The growth of these dendrites led to a physical effect called microsegregation which finally resulted in the very important initial inhomogene- ity of the solid wootz steel. In later process stages the ribbons grew out preferentially in the so-called interdendritic regions. In this sense it can be said that it is the dendritic selection which sets the characteristic ribbon spacing.

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Table 2 shows the parameters used in the simulations under directional solidiﬁcation conditions for the Fe–Mn
system. Different simulations were performed with different values of ␦–␥ interfacial tension in the range presented in this table in order to check: (1) how the ␦–␥ interfacial ten- sion affects the ␥-**phase** thickness during the steady-state growth on ␦-**phase** (during the peritectic reaction) in the sim- ulation; and, (2) how the simulation results can be compared with the analytical theory presented brieﬂy in Section 3 . Thus, the ␥-**phase** thicknesses were measured during the steady- state growth on the ␦-**phase** at a distance of 3.1 m from the triple-point position. The position of the measurements was deﬁned arbitrarily but considering the interferences of mea- surements excessively close to the triple-point or extremely far way of this point (the inﬂuence of the peritectic transfor- mation – the growth of ␥-**phase** into the liquid and ␦-**phase**). In addition to that, the liquid concentration next to the ␥–L interface was determined in the simulations and these deter- mined values were assumed suitable for being utilized as x L/ B in the analytical **model**. The other composition terms of the analytical **model** were determined in accordance with the **phase** diagram data considering x L/ B . Another assumption is that the speed of the ␥-**phase** growth is equal to the pulling speed of the unidirectional solidiﬁcation on the steady-state. Thus, the analytical values and the simulation results were obtained for directional solidiﬁcation with thermal gradient equal to 100 K/cm and pulling speed equal to 5.0 and 10.0 m/s. The numerical domain for this investigation was set equal to 60 m × 32 m.

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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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separation is readily observed and the assembly of chains is energetically unfavourable. The charge-to-charge separation, d, is systematically decreased, keeping the dumbbell dipole moment, respectively the dumbbell charges, constant. Using this procedure we can approach the dipolar soft sphere limit in such a way that difficulties like the reversible association of particles develop “slowly”, which allows the extrapolation to the desired limit. Even at the smallest d (= 10 −4 ), we do observe a transition terminating in a critical point, which suggests a gas-liquid critical point in the dipolar soft sphere limit. This result is in accordance with the previous simulation study by Ganzenm¨ uller and Camp [31] for dipolar hard spheres. To explain our simulation results, we apply different simple models. However, neither the extension of Flory’s lattice theory to reversibly aggregating polymers [38, 39], nor the defect **model** put forward by Tlusty and Safran [37], yields a consistent description of the simulation results. Only the developed van der Waals mean **field** theory provides a close to quantitative description of the critical parameters obtained from the simulation. The theory combines the Onsager approach to dipolar liquids [40] with the idea that the basic unit is not the single dipole, but rather a small reversible aggregate. To support the results for charged soft dumbbell in the dipole limit, i.e. d → 0, we also conduct simulations for the dipolar soft sphere **model** in its pure form and with an additional parameter which controls the soft sphere repulsion. We see that the soft repulsive interaction can be used as an effective means for limiting the aggregation in the dipole limit. The resulting short reversible chains, which are easy to equilibrate, do exhibit gas-liquid **phase** separation. Subsequently to the investigation at the dipole limit we study the gas-liquid **phase** transition in the dumbbell system for d 1 and observe critical parameters in the whole

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The effect of the **model** parameters α and β on the damage variable evolution is evaluated in a standard tensile test performed with different parameter values. The values of the damage variable ξ are computed in a point close to the notch tip. From Figure (7(a)), one can clearly recognise that both parameters have significant influence upon the evolution of the damage zone. While the increasing values of the parameter β are speeding up the damage progression, performing the same with the values of the second parameter α forces the damage zone propagation to slow down.

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].