Biomineralization as a Paradigm of Directional Solidification: A Physical Model for Molluscan Shell Ultrastructural Morphogenesis

Vanessa Schoeppler1, László Gránásy2,3, Elke Reich1, Nicole Poulsen1, René de Kloe4, Phil Cook5, Alexander Rack, Tamás Pusztai2, Igor Zlotnikov1

1B CUBE - Center for Molecular Bioengineering, Technische Universität Dresden, Germany
2Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
3BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom
4EDAX, Tilburg, The Netherlands
5ESRF – The European Synchrotron, Grenoble, France

Molluscan shells are a model system to understand the fundamental principles of mineral formation by living organisms. The diversity of unconventional mineral morphologies and 3D mineral-organic architectures that comprise these tissues, in combination with their exceptional mechanical efficiency, offers a unique platform to study the formation-structure-function relationship in a biomineralized system. However, so far, morphogenesis of these ultrastructures is poorly understood. Here, a comprehensive physical model, based on the concept of directional solidification, is developed to describe molluscan shell biomineralization. The capacity of the model to define the forces and thermodynamic constraints that guide the morphogenesis of the entire shell construct-the prismatic and nacreous ultrastructures and their transitions-and govern the evolution of the constituent mineralized assemblies on the ultrastructural and nanostructural levels is demonstrated using the shell of the bivalve Unio pictorum. Thereby, explicit tools for novel bioinspired and biomimetic bottom-up materials design are provided.

Topics: Polycrystalline solidification

Topological defects in two-dimensional orientation-field models for grain growth

Bálint Korbuly1, Mathis Plapp2, Hervé Henry2, James A. Warren3, László Gránásy1,4, Tamás Pusztai1

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2Laboratoire Physique de la Matière Condensée, École Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France
3Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
4BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

Standard two-dimensional orientation-field-based phase-field models rely on a continuous scalar field to represent crystallographic orientation. The corresponding order parameter space is the unit circle, which is not simply connected. This topological property has important consequences for the resulting multigrain structures: (i) trijunctions may be singular; (ii) for each pair of grains there exist two different grain boundary solutions that cannot continuously transform to one another; (iii) if both solutions appear along a grain boundary, a topologically stable, singular point defect must exist between them. While (i) can be interpreted in the classical picture of grain boundaries, (ii) and therefore (iii) cannot. In addition, singularities cause difficulties, such as lattice pinning in numerical simulations. To overcome these problems, we propose two formulations of the model. The first is based on a three-component unit vector field, while in the second we utilize a two-component vector field with an additional potential. In both cases, the additional degree of freedom introduced makes the order parameter space simply connected, which removes the topological stability of these defects.

Topics: Orientation field models, Polycrystalline solidification

Hydrodynamic theory of freezing: Nucleation and polycrystalline growth

Frigyes Podmaniczky1, Gyula Tóth2,1, György Tegze1, László Gránásy1,3

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
3BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

Structural aspects of crystal nucleation in undercooled liquids are explored using a nonlinear hydrodynamic theory of crystallization proposed recently [G. I. Tóth et al., J. Phys.: Condens. Matter 26, 055001 (2014)], which is based on combining fluctuating hydrodynamics with the phase- field crystal theory. We show that in this hydrodynamic approach not only homogeneous and heterogeneous nucleation processes are accessible, but also growth front nucleation, which leads to the formation of new (differently oriented) grains at the solid-liquid front in highly undercooled systems. Formation of dislocations at the solid-liquid interface and interference of density waves ahead of the crystallization front are responsible for the appearance of the new orientations at the growth front that lead to spherulite-like nanostructures.

Videos of growth front nucleation

Topics: Polycrystalline solidification

Grain coarsening in two-dimensional phase-field models with an orientation field

Bálint Korbuly1, Tamás Pusztai1, Hervé Henry2, Mathis Plapp2, Markus Apel3, László Gránásy1,4

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2Laboratoire Physique de la Matière Condensée, École Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France
3Access e.V., Intzestr. 5, 52072 Aachen, Germany
4BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

In the literature, contradictory results have been published regarding the form of the limiting (long-time) grain size distribution (LGSD) that characterizes the late stage grain coarsening in two-dimensional and quasi-two-dimensional polycrystalline systems. While experiments and the phase-field crystal (PFC) model (a simple dynamical density functional theory) indicate a lognormal distribution, other works including theoretical studies based on conventional phase-field simulations that rely on coarse grained fields, like the multi-phase-field (MPF) and orientation field (OF) models, yield significantly different distributions. In a recent work, we have shown that the coarse grained phase-field models (whether MPF or OF) yield very similar limiting size distributions that seem to differ from the theoretical predictions. Herein, we revisit this problem, and demonstrate in the case of OF models [by R. Kobayashi et al., Physica D 140, 141 (2000) and H. Henry et al. Phys. Rev. B 86, 054117 (2012)] that an insufficient resolution of the small angle grain boundaries leads to a lognormal distribution close to those seen in the experiments and the molecular scale PFC simulations. Our work indicates, furthermore, that the LGSD is critically sensitive to the details of the evaluation process, and raises the possibility that the differences among the LGSD results from different sources may originate from differences in the detection of small angle grain boundaries.

Topics: Orientation field models, Polycrystalline solidification

Phase-field modeling of eutectic structures on the nanoscale: the effect of anisotropy

László Rátkai1, Gyula Tóth2,1, László Környei3, Tamás Pusztai1, László Gránásy1,4

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
3Department of Mathematics and Computational Sciences, Széchenyi István University, Győr 9026, Hungary
4BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

A simple phase-field model is used to address anisotropic eutectic freezing on the nanoscale in two (2D) and three dimensions (3D). Comparing parameter-free simulations with experiments, it is demonstrated that the employed model can be made quantitative for Ag-Cu. Next, we explore the effect of material properties, and the conditions of freezing on the eutectic pattern. We find that the anisotropies of kinetic coefficient and the interfacial free energies (solid-liquid and solid-solid), the crystal misorientation relative to pulling, the lateral temperature gradient, play essential roles in determining the eutectic pattern. Finally, we explore eutectic morphologies, which form when one of the solid phases are faceted, and investigate cases, in which the kinetic anisotropy for the two solid phases are drastically different.

Investigating Nucleation Using the Phase-Field Method

Frigyes Podmaniczky1, Gyula Tóth2,1, Tamás Pusztai1, László Gránásy1,3

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
3BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

The first order phase transitions, like freezing of liquids, melting of solids, phase separation in alloys, vapor condensation, etc., start with nucleation, a process in which internal fluctuations of the parent phase lead to formation of small seeds of the new phase. Owing to different size dependence of (negative) volumetric and (positive) interfacial contributions to work of formation of such seeds, there is a critical size, at which the work of formation shows a maximum. Seeds that are smaller than the critical one decay with a high probability, while the larger ones have a good chance to grow further and reach a macroscopic size. Putting it in another way, to form the bulk new phase, the system needs to pass a thermodynamic barrier via thermal fluctuations. When the fluctuations of the parent phase alone lead to transition, the process is called homogeneous nucleation. Such a homogeneous process is, however, scarcely seen and requires very specific conditions in nature or in the laboratory. Usually, the parent phase resides in a container and/or it incorporates floating heterogeneities (solid particles, droplets, etc.). The respective foreign surfaces lead to ordering of the adjacent liquid layers, which in turn may assist the formation of the seeds, a process termed heterogeneous nucleation. Herein, we review how the phase-field techniques contributed to the understanding of various aspects of crystal nucleation in undercooled melts, and its role in microstructure evolution. We recall results achieved using both conventional phase-field techniques that rely on spatially averaged (coarse grained) order parameters in capturing the phase transition, as well as molecular scale phase-field approaches that employ time averaged fields, as happens in the classical density functional theories, including the recently developed phase-field crystal models.

Topics: Heterogeneous nucleation

A Physically Consistent Multiphase-Field Theory of First Order Phase Transitions

Gyula Tóth1,2, Tamás Pusztai2, Bjørn Kvamme1, László Gránásy2,3

1Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
2Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
3BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

A multiphase-field theory is presented that describes interface driven multi-domain dynamics. The free energy functional and the dynamic equations are constructed on the basis of criteria of mathematical and physical consistency. First, it is demonstrated that the most widely used multiphase theories are physically inconsistent, therefore, a new theory has to be developed. Combining elements of the investigated models with a new multivariate generalization of the free energy surface results in a general multiphase / multi-component theory, which keeps the variational formalism, reduces / extends naturally on the level of both the free energy functional and the dynamic equations, utilizes arbitrary pairwise equilibrium interfacial properties, features equilibrium = stationary equivalency, and avoids the appearance of spurious phases.

Phase-field crystal modeling of nucleation including homogeneous and heterogeneous processes, and growth front nucleation

László Gránásy1,2, Frigyes Podmaniczky1, Gyula Tóth3,1

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom
3Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway

Structural aspects of crystal nucleation in undercooled liquids are explored using a nonlinear hydrodynamic theory of freezing proposed recently, which is based on combining fluctuating hydrodynamics with the phase-field crystal (PFC) theory. It will be shown that unlike the usual PFC models of diffusive dynamics, within the hydrodynamic approach not only the homogeneous and heterogeneous nucleation processes are accessible, but also growth front nucleation, which leads to the formation of differently oriented grains at the front in highly undercooled systems. Formation of dislocations at the solid-liquid interface and the interference of density waves ahead of the crystallization front are responsible for the appearance of new orientations.

Topics: Phase field crystal

Orientation-field models for polycrystalline solidification: grain coarsening and complex growth forms

Bálint Korbuly1, Tamás Pusztai1, Gyula Tóth2,1, Hervé Henry3, Mathis Plapp3, László Gránásy1,4

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
2Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway
3Laboratoire Physique de la Matière Condensée, École Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France
4BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

We compare two versions of the phase-field theory for polycrystalline solidification, both relying on the concept of orientation fields: one by Kobayashi et al. [Physica D 140 (2000) 141] and the other by Henry et al. [Phys. Rev. B 86 (2012) 054117]. Setting the model parameters so that the grain boundary energies and the time scale of grain growth are comparable in the two models, we first study the grain coarsening process including the limiting grain size distribution, and compare the results to those from experiments on thin films, to the models of Hillert, and Mullins, and to predictions by multiphase-field theories. Next, following earlier work by Gránásy et al. [Phys. Rev. Lett. 88 (2002) 206105; Phys. Rev. E 72 (2005) 011605], we extend the orientation field to the liquid state, where the orientation field is made to fluctuate in time and space, and employ the model for describing of multi-dendritic solidification, and polycrystalline growth, including the formation of “dizzy” dendrites disordered via the interaction with foreign particles.

Topics: Orientation field models, Polycrystalline solidification

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