Phase-field crystal modeling of heteroepitaxy and exotic modes of crystal nucleation

Frigyes Podmaniczky1, Gyula Tóth2,1, György Tegze1, 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

We review recent advances made in modeling heteroepitaxy, two-step nucleation, and nucleation at the growth front within the framework of a simple dynamical density functional theory, the Phase-Field Crystal (PFC) model. The crystalline substrate is represented by spatially confined periodic potentials. We investigate the misfit dependence of the critical thickness in the StranskiKrastanov growth mode in isothermal studies. Apparently, the simulation results for stress release via the misfit dislocations fit better to the PeopleBean model than to the one by Matthews and Blakeslee. Next, we investigate structural aspects of two-step crystal nucleation at high undercoolings, where an amorphous precursor forms in the first stage. Finally, we present results for the formation of new grains at the solid-liquid interface at high supersaturations/supercoolings, a phenomenon termed Growth Front Nucleation (GFN). Results obtained with diffusive dynamics (applicable to colloids) and with a hydrodynamic extension of the PFC theory (HPFC, developed for simple liquids) will be compared. The HPFC simulations indicate two possible mechanisms for GFN.

Topics: Phase field crystal

Consistent multiphase-field theory for interface driven multidomain dynamics

Gyula Tóth1,2, Tamás Pusztai2, 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

We present a new multiphase-field theory for describing pattern formation in multi-domain and/or multi-component systems. The construction of the free energy functional and the dynamic equations is based on criteria that ensure mathematical and physical consistency. We first analyze previous multiphase-field theories, and identify their advantageous and disadvantageous features. On the basis of this analysis, we introduce a new way of constructing the free energy surface, and derive a generalized multiphase description for arbitrary number of phases (or domains). The presented approach retains the variational formalism; reduces (or extends) naturally to lower (or higher) number of fields on the level of both the free energy functional and the dynamic equations; enables the use of arbitrary pairwise equilibrium interfacial properties; penalizes multiple junctions increasingly with the number of phases; ensures non-negative entropy production, and the convergence of the dynamic solutions to the equilibrium solutions; and avoids the appearance of spurious phases on binary interfaces. The new approach is tested for multi-component phase separation and grain coarsening.

Topics: Emulsion

Phase-field theory of multicomponent incompressible Cahn-Hilliard liquids

Gyula Tóth1,2, Mojdeh Zarifi, Bjørn Kvamme1

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

In this paper, a generalization of the Cahn-Hilliard theory of binary liquids is presented for multicomponent incompressible liquid mixtures. First, a thermodynamically consistent convection-diffusion-type dynamics is derived on the basis of the Lagrange multiplier formalism. Next, a generalization of the binary Cahn-Hilliard free-energy functional is presented for an arbitrary number of components, offering the utilization of independent pairwise equilibrium interfacial properties. We show that the equilibrium two-component interfaces minimize the functional, and we demonstrate that the energy penalization for multicomponent states increases strictly monotonously as a function of the number of components being present. We validate the model via equilibrium contact angle calculations in ternary and quaternary (four-component) systems. Simulations addressing liquid-flow-assisted spinodal decomposition in these systems are also presented.

Topics: Emulsion

Growth control of peptide-nanotube spherulitic films: Experiments and simulations

Netta Hendler1,2, Elad Mentovich1,2, Bálint Korbuly3, Tamás Pusztai3, László Gránásy3,4, Shachar Richter1,2

1Department of Materials Science and Engineering, Faculty of Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
2Center for Nanoscience and Nanotechnology, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
3Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, Budapest H-1525, Hungary
4BCAST, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom

Multi-hierarchical self-assembly (MHSA) is a key process responsible for the spontaneous formation of many complex structures. However, because of the complexity of the process, the underlying mechanism remains largely unclear. Thus, a deeper understanding of MHSA is required, especially for the preparation of MHSA systems via bottom-up methodologies. We show here, experimentally and theoretically, that the complex-formation MHSA of peptide nanotube films can be controlled solely by manipulating the experimental parameter of humidity. Furthermore, we identify growth-front nucleation (GFN; the formation of new grains at the perimeter) as the physical background for the observed morphological transitions by correlating experimental observations with phase-field modeling of the morphological evolution. Our findings indicate a simple way to control multi-hierarchical morphologies, crucial for the employment of bottom-up techniques in constructing complex structures for practical applications.

Recent Developments in Modeling Heteroepitaxy/Heterogeneous Nucleation by Dynamical Density Functional Theory

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

Crystallization of supersaturated liquids usually starts by epitaxial growth or by heterogeneous nucleation on foreign surfaces. Herein, we review recent advances made in modeling heteroepitaxy and heterogeneous nucleation on flat/modulated surfaces and nanoparticles within the framework of a simple dynamical density functional theory, known as the phase-field crystal model. It will be shown that the contact angle and the nucleation barrier are nonmonotonous functions of the lattice mismatch between the substrate and the crystalline phase. In continuous cooling studies for substrates with lattice mismatch, we recover qualitatively the Matthews–Blakeslee mechanism of stress release via the misfit dislocations. The simulations performed for particle-induced freezing will be confronted with recent analytical results, exploring thus the validity range of the latter. It will be demonstrated that time-dependent studies are essential, as investigations based on equilibrium properties often cannot identify the preferred nucleation pathways. Modeling of these phenomena is essential for designing materials on the basis of controlled nucleation and/or nano-patterning.

Topics: Heterogeneous nucleation

Phase field modelling of spinodal decomposition in the oil/water/asphaltene system

Gyula Tóth1,2, Bjørn Kvamme1

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

In this paper the quantitative applicability of van der Sman/van der Graaf type Ginzburg–Landau theories of surfactant assisted phase separation [van der Sman et al., Rheol. Acta, 2006, 46, 3] is studied for real systems displaying high surfactant concentrations at the liquid–liquid interface. The model is applied for the water/heptane/asphaltene system (a model of heavy crude oil), for which recent molecular dynamics (MD) simulations provide microscopic data needed to calibrate the theory. A list of general requirements is set up first, which is then followed by analytical calculations of the equilibrium properties of the system, such as the equilibrium liquid densities, the adsorption isotherm and the interfacial tension. Based on the results of these calculations, the model parameters are then determined numerically, yielding a reasonable reproduction of the MD density profiles. The results of time-dependent simulations addressing the dynamical behaviour of the system will also be presented. It will be shown that the competition between the diffusion and hydrodynamic time scales can lead to the formation of an emulsion. We also address the main difficulties and limitations of the theory regarding quantitative modelling of surfactant assisted liquid phase separation.

Topics: Emulsion

Ternary eutectic dendrites: Pattern formation and scaling properties

László Rátkai1, Attila Szállás1, Tamás Pusztai1, Tetsuo Mohri, 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

Extending previous work [Pusztai et al., Phys. Rev. E 87, 032401 (2013)], we have studied the formation of eutectic dendrites in a model ternary system within the framework of the phase-field theory. We have mapped out the domain in which two-phase dendritic structures grow. With increasing pulling velocity, the following sequence of growth morphologies is observed: flat front lamellae → eutectic colonies → eutectic dendrites → dendrites with target pattern → partitionless dendrites → partitionless flat front. We confirm that the two-phase and one-phase dendrites have similar forms and display a similar scaling of the dendrite tip radius with the interface free energy. It is also found that the possible eutectic patterns include the target pattern, and single- and multiarm spirals, of which the thermal fluctuations choose. The most probable number of spiral arms increases with increasing tip radius and with decreasing kinetic anisotropy. Our numerical simulations confirm that in agreement with the assumptions of a recent analysis of two-phase dendrites [Akamatsu et al., Phys. Rev. Lett. 112, 105502 (2014)], the Jackson-Hunt scaling of the eutectic wavelength with pulling velocity is obeyed in the parameter domain explored, and that the natural eutectic wavelength is proportional to the tip radius of the two-phase dendrites. Finally, we find that it is very difficult/virtually impossible to form spiraling two-phase dendrites without anisotropy, an observation that seems to contradict the expectations of Akamatsu et al. Yet, it cannot be excluded that in isotropic systems, two-phase dendrites are rare events difficult to observe in simulations.

Topics: Spiral eutectic dendrites

Analysis of Ginzburg-Landau-type models of surfactant-assisted liquid phase separation

Gyula Tóth1,2, Bjørn Kvamme1

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

In this paper diffuse interface models of surfactant-assisted liquid-liquid phase separation are addressed. We start from the generalized version of the Ginzburg-Landau free-energy-functional-based model of van der Sman and van der Graaf. First, we analyze the model in the constant surfactant approximation and show the presence of a critical point at which the interfacial tension vanishes. Then we determine the adsorption isotherms and investigate the validity range of previous results. As a key point of the work, we propose a new model of the van der Sman/van der Graaf type designed for avoiding both unwanted unphysical effects and numerical difficulties present in previous models. In order to make the model suitable for describing real systems, we determine the interfacial tension analytically more precisely and analyze it over the entire accessible surfactant load range. Emerging formulas are then validated by calculating the interfacial tension from the numerical solution of the Euler-Lagrange equations. Time-dependent simulations are also performed to illustrate the slowdown of the phase separation near the critical point and to prove that the dynamics of the phase separation is driven by the interfacial tension.

Topics: Emulsion

Phase-Field Modeling of Solidification in Light-Metal Matrix Nanocomposites

Tamás Pusztai1, László Rátkai1, Attila Szállás1, 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 quantitative phase-field approach has been adapted to model solidification in the presence of Metal Matrix Nanocomposites (MMNCs) in a single-component liquid. Nanoparticles of fixedsize and shape are represented by additional fields. The corresponding equations of motion are assumed to ensure relaxation dynamics, and can be supplemented by random forces (realizing Brownian motion) or external fields. The nanoparticles are characterized by two model parameters: their mobility and the contact angle they realize with the solid-liquid interface. We investigate the question how grain size distribution can be influenced by heterogeneous nucleation on the nanoparticles and by the front-particle interaction. We explore, furthermore, how materials and process parameters, such as temperature, density and size/shape distribution of the nanoparticles, influence microstructure evolution.

Topics: Heterogeneous nucleation

Heterogeneous nucleation of/on nanoparticles: a density functional study using the phase-field crystal model

László Gránásy1,2, Frigyes Podmaniczky1, Gyula Tóth3,1, György Tegze1, 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
3Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway

Crystallization of supersaturated liquids usually starts by heterogeneous nucleation. Mounting evidence shows that even homogeneous nucleation in simple liquids takes place in two steps; first a dense amorphous precursor forms, and the crystalline phase appears via heterogeneous nucleation in/on the precursor cluster. Herein, we review recent results by a simple dynamical density functional theory, the phase-field crystal model, for (precursor-mediated) homogeneous and heterogeneous nucleation of nanocrystals. It will be shown that the mismatch between the lattice constants of the nucleating crystal and the substrate plays a decisive role in determining the contact angle and nucleation barrier, which were found to be non-monotonic functions of the lattice mismatch. Time dependent studies are essential as investigations based on equilibrium properties often cannot identify the preferred nucleation pathways. Modeling of these phenomena is essential for designing materials on the basis of controlled nucleation and/or nano-patterning.

Topics: Heterogeneous nucleation, Phase field crystal

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