Phase-Field Modeling of Biomineralization in Mollusks and Corals: Microstructure vs Formation Mechanism

László Gránásy1,2, László Rátkai1, Gyula Tóth3, Pupa Gilbert, Igor Zlotnikov4, Tamás Pusztai1

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
3Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, U.K.
4B CUBE - Center for Molecular Bioengineering, Technische Universität Dresden, Germany

While biological crystallization processes have been studied on the microscale extensively, there is a general lack of models addressing the mesoscale aspects of such phenomena. In this work, we investigate whether the phase-field theory developed in materials’ science for describing complex polycrystalline structures on the mesoscale can be meaningfully adapted to model crystallization in biological systems. We demonstrate the abilities of the phase-field technique by modeling a range of microstructures observed in mollusk shells and coral skeletons, including granular, prismatic, sheet/columnar nacre, and sprinkled spherulitic structures. We also compare two possible micromechanisms of calcification: the classical route, via ion-by-ion addition from a fluid state, and a nonclassical route, crystallization of an amorphous precursor deposited at the solidification front. We show that with an appropriate choice of the model parameters, microstructures similar to those found in biomineralized systems can be obtained along both routes, though the time-scale of the nonclassical route appears to be more realistic. The resemblance of the simulated and natural biominerals suggests that, underneath the immense biological complexity observed in living organisms, the underlying design principles for biological structures may be understood with simple math and simulated by phase-field theory.

Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification

Frigyes Podmaniczky1, László Gránásy1,2

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

Two-step nucleation and subsequent growth processes were investigated in the framework of the single mode phase-field crystal model combined with diffusive dynamics (corresponding to colloid suspensions) and hydrodynamical density relaxation (simple liquids). It is found that independently of dynamics, nucleation starts with the formation of solid precursor clusters that consist of domains with noncrystalline ordering (ringlike projections are seen from certain angles), and regions that have amorphous structure. Using the average bond order parameter q¯6, we distinguished amorphous, medium range crystallike order (MRCO), and crystalline local orders. We show that crystallization to the stable body-centered cubic phase is preceded by the formation of a mixture of amorphous and MRCO structures. We have determined the time dependence of the phase composition of the forming solid state. We also investigated the time/size dependence of the growth rate for solidification. The bond order analysis indicates similar structural transitions during solidification in the case of diffusive and hydrodynamic density relaxation.

Topics: Phase field crystal

Phase field benchmark problems for nucleation

W. Wu, D. Montiel, J.E. Guyer, P.W. Voorhees, Warren JA, D. Wheeler, L Gránásy, Pusztai T, O.G. Heinonen

We present nucleation phase field model benchmark problems, expanding on our previous benchmark problems on diffusion, precipitation, dendritic growth, linear elasticity, fluid flow and electrochemistry. Nucleation is the process in which either a new thermodynamic phase or a new structure is created, such as solidification from the melt, or self-assembly of particulates. Based on where the nucleation occurs, it can be divided into two main categories: homogeneous nucleation and heterogeneous nucleation. In the first nucleation benchmark problem, we focus on homogeneous nucleation for both single seed under different initial conditions and multiple seeds. The second nucleation benchmark problem focuses on athermal heterogeneous nucleation and nucleation behavior near the free growth limit with different undercooling driving forces.

Topics: Heterogeneous nucleation

Two- and three-dimensional simulations of Rayleigh–Taylor instabilities using a coupled Cahn–Hilliard/Navier–Stokes model

R. Zanella, G. Tegze, R. Le Tellier, H. Henry

We report on two- and three-dimensional numerical simulations of Rayleigh–Taylor instabilities in immiscible fluids. A diffuse-interface model that combines the Cahn–Hilliard equation, governing the evolution of the volume fraction of one fluid, and the Navier–Stokes equations, governing the bulk velocity and pressure, is used. The study is limited to low Atwood numbers owing to the use of the Boussinesq approximation. The code is based on a pseudo-spectral method. A linear analysis is first performed in a two-dimensional case of Rayleigh–Taylor instability to confirm that the model very well captures this phenomenon in the case of inviscid or viscid fluids. One key aspect of this work is that the influence of the thermodynamic parameters related to the Cahn–Hilliard equation (interface thickness and mobility) is quantitively studied. Three-dimensional results of Rayleigh–Taylor instabilities in viscous fluids are then presented to show the possibilities of this modeling. We observe the effect of the viscosity and the wavelength of an initial single-mode perturbation on the mass transport during the nonlinear regime.

Ultrafine Fe-Fe2Ti eutectics by directed energy deposition: Insights into microstructure formation based on experimental techniques and phase field modelling

G. Requena, K. Bugelnig, F. Sket, S. Milenkovic, G. Rödler, A. Weisheit, J. Gussone, J. Haubrich, P. Barriobero-Vila, Tamás Pusztai1, Gránásy László, A. Theofilatos, J.C. da Silva, U. Hecht

1Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary

We investigated the Fe-Fe2Ti eutectic microstructure obtained by Directed Energy Deposition (DED) with a hypereutectic composition of Fe-17.6 at.% Ti. Ultrafine lamellar spacings as low as 200 nm were achieved, features which otherwise can only be obtained in thin specimens, e.g. by suction casting. However, at interlayer boundaries (ILBs) a globular morphology of the primary Fe2Ti phase is observed with halos of the Fe phase. For the given DED conditions the crystalline structure is thus discontinuous across the ILBs. Both 2D and 3D analysis methods were used to quantify the microstructure, including high resolution synchrotron holographic X-ray computed tomography (HXCT). The generic behaviour of eutectic systems under conditions that qualitatively correspond to those of laser additive manufacturing was explored by phase-field modelling for selected nucleation scenarios and alloy compositions spanning from eutectic to hyper-eutectic. While providing valuable insights into microstructure formation, the simulations point out the need to further deepen our understanding about melting under additive manufacturing conditions in order to implement suitable nucleation and / or free growth models. The simulations also show that globular ILBs can be prevented when using exactly eutectic alloy compositions.

Crystal nucleation and growth of spherulites demonstrated by coral skeletons and phase-field simulations

Chang-Yu Sun, László Gránásy1,2, Cayla Stifler, Tal Zaquin, Rajesh Chopdekar, Nobumichi Tamura, James Weaver, Jun Zhang, Stefano Goffredo, Giuseppe Falini, Matthew Marcus, Tamás Pusztai1, Vanessa Schoeppler3, Tali Mass, Pupa Gilbert

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
3B CUBE - Center for Molecular Bioengineering, Technische Universität Dresden, Germany

Spherulites are radial distributions of acicular crystals, common in biogenic, geologic, and synthetic systems, yet exactly how spherulitic crystals nucleate and grow is still poorly understood. To investigate these processes in more detail, we chose scleractinian corals as a model system, because they are well known to form their skeletons from aragonite (CaCO3) spherulites, and because a comparative study of crystal structures across coral species has not been performed previously. We observed that all 12 diverse coral species analyzed here exhibit plumose spherulites in their skeletons, with well-defined centers of calcification (CoCs), and crystalline fibers radiating from them. In 7 of the 12 species, we observed a skeletal structural motif not observed previously: randomly oriented, equant crystals, which we termed “sprinkles”. In Acropora pharaonis, these sprinkles are localized at the CoCs, while in 6 other species, sprinkles are either layered at the growth front (GF) of the spherulites, or randomly distributed. At the nano- and micro-scale, coral skeletons fill space as much as single crystals of aragonite. Based on these observations, we tentatively propose a spherulite formation mechanism in which growth front nucleation (GFN) of randomly oriented sprinkles, competition for space, and coarsening produce spherulites, rather than the previously assumed slightly misoriented nucleations termed “non-crystallographic branching”. Phase-field simulations support this mechanism, and, using a minimal set of thermodynamic parameters, are able to reproduce all of the microstructural variation observed experimentally in all of the investigated coral skeletons. Beyond coral skeletons, other spherulitic systems, from aspirin to semicrystalline polymers and chocolate, may also form according to the mechanism for spherulite formation proposed here.

Orientational order in dense suspensions of elliptical particles in the non-Stokesian regime

György Tegze1, Frigyes Podmaniczky1, Ellák Somfai, Börzsönyi Tamás, László Gránásy1,2

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

Suspensions of neutrally buoyant elliptic particles are modeled in 2D using fully resolved simulations that provide two-way interaction between the particle and the fluid medium. Forces due to particle collisions are represented by a diffuse interface approach that allows the investigation of dense suspensions (up to 47% packing fraction). We focus on the role inertial forces play at low and high particle Reynolds numbers termed low Reynolds number and inertial regimes, respectively. The suspensions are characterized by the orientation distribution function (ODF) that reflects shear induced rotation of the particles at low Reynolds numbers, and nearly stationary (swaying) particles at high Reynolds numbers. In both cases, orientational ordering differs qualitatively from the behavior observed in the Stokesian-regime. The ODF becomes flatter with increasing packing fraction, as opposed to the sharpening previous work predicted in the Stokesian regime. The ODF at low particle concentrations differs significantly for the low Reynolds number and inertial regimes, whereas with increasing packing fraction convergence is observed. For dense suspensions, the particle–particle interactions dominate the particle motion.

Phase-field lattice Boltzmann model for dendrites growing and moving in melt flow

László Rátkai1, Tamás Pusztai1, László Gránásy1,2

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

The phase-field and lattice Boltzmann methods have been combined to simulate the growth of solid particles moving in melt flow. To handle mobile particles, an overlapping multigrid scheme was developed, in which each individual particle has its own moving grid, with local fields attached to it. Using this approach we were able to simulate simultaneous binary solidification, solute diffusion, melt flow, solid motion, the effect of gravity, and collision of the particles. The method has been applied for describing two possible modes of columnar to equiaxed transition in the Al–Ti system.

Crystal growth kinetics as an architectural constraint on the evolution of molluscan shells

Vanessa Schoeppler1, Robert Lemanis, Elke Reich1, Tamás Pusztai2, László Gránásy2,3, 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

Molluscan shells are a classic model system to study formation–structure–function relationships in biological materials and the process of biomineralized tissue morphogenesis. Typically, each shell consists of a number of highly mineralized ultrastructures, each characterized by a specific 3D mineral–organic architecture. Surprisingly, in some cases, despite the lack of a mutual biochemical toolkit for biomineralization or evidence of homology, shells from different independently evolved species contain similar ultrastructural motifs. In the present study, using a recently developed physical framework, which is based on an analogy to the process of directional solidification and simulated by phase-field modeling, we compare the process of ultrastructural morphogenesis of shells from 3 major molluscan classes: A bivalve Unio pictorum, a cephalopod Nautilus pompilius, and a gastropod Haliotis asinina. We demonstrate that the fabrication of these tissues is guided by the organisms by regulating the chemical and physical boundary conditions that control the growth kinetics of the mineral phase. This biomineralization concept is postulated to act as an architectural constraint on the evolution of molluscan shells by defining a morphospace of possible shell ultrastructures that is bounded by the thermodynamics and kinetics of crystal growth.

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