Leader: Prof. László
Gránásy (DSc)

Members: Tamás
Pusztai (PhD), György Tegze (PhD), Gyula I. Tóth (PhD student)

Previous
members: László Környei (PhD); Tamás Börzsönyi (PhD)

__Latest results:__

**Dendrites
Regularized by Spatially Homogeneous Time-Periodic Forcing**

*T.
Börzsönyi, T. Tóth-Katona,
Á. Buka, and L. Gránásy
*

*Research
Institute for Solid State Physics and Optics, Hungarian Academy of Sciences,
P.O.B. 49, H-1525 Budapest, Hungary*

The effect of spatially homogeneous time-periodic
external forcing on dendritic solidification has been
studied by phase-field modeling and experiments on liquid crystal. It is shown
that the frequency of dendritic sidebranching
can be tuned by oscillating pressure or heating. The main parameters that
influence this phenomenon are identified. [Phys. Rev. Lett. **83**, 2853-2856 (1999)].

**Nucleation
and Bulk Crystallization in Binary Phase Field Theory**

*László** Gránásy, ^{1} Tamás
Börzsönyi,^{1,2} and Tamás Pusztai^{1}*

^{1}*Research Institute for
Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary*

^{2}Groupe de Physique des Solides, CNRS UMR 75-88, Universités
Paris VI at VII, Tour 23, 2 place Jussieu, 75251 Paris Cedex 05, France

We present a phase field theory for binary crystal
nucleation. In the one-component limit, quantitative agreement is achieved with
computer simulations (Lennard-Jones system) and
experiments (ice-water system) using model parameters evaluated from the free
energy and thickness of the interface. The critical undercoolings
predicted for Cu-Ni alloys accord with the measurements, and indicate
homogeneous nucleation. The Kolmogorov exponents
deduced for dendritic solidification and
for soft impingement of particles via diffusion fields are consistent
with experiment. [Phys. Rev. Lett. **88**, 206105
(2002)].

**Crystal
nucleation and growth in binary phase-field theory**

*László** Gránásy, ^{1} Tamás
Börzsönyi,^{1,2} and Tamás Pusztai^{1}*

^{1}*Research Institute for
Solid State Physics and Optics, POB 49, H-1525 Budapest, Hungary*

^{2}Groupe de Physique des Solides, CNRS UMR 75-88, Universités
Paris VI at VII, Tour 23, 2 place Jussieu,75251, Paris Cedex 05, France

Nucleation and growth in unary and binary systems is investigated in the
framework of the phase-field theory. Evaluating the model parameters from the
interfacial free energy and interface thickness, a quantitative agreement is
found with computer simulations and experiments on the ice water system. The
critical undercoolings predicted for a simple binary
system are close to experiment. Phase-field simulations for isotropic and
anisotropic systems show that due to the interacting diffusion fields the Avrami Kolmogorov exponent varies
with transformed fraction and initial concentration. [Journal of Cryst. Growth, **237-239**, 1813 (2002)

**Diffuse
interface analysis of crystal nucleation in hard-sphere liquid**

*László** Gránásy and Tamás Pusztai*

*Research
Institute for Solid State Physics and Optics, H 1525 Budapest, POB 49, Hungary*

We show that the
increase of the interface free energy with deviation from equilibrium seen in
recent Monte Carlo simulations [S. Auer and D. Frenkel,
Nature, London, **413**, 711 (2001)] can be recovered if the molecular
scale diffuseness of the crystal liquid interface is considered. We compare two
models, Gránásy’s phenomenological diffuse interface
theory, and a density functional theory that relies on the type of Ginzburg-Landau expansion for fcc nucleation, that Shih *et al.* introduced
for bcc crystal. It is shown that, in the range of Monte Carlo simulations, the
nucleation rate of the stable fcc
phase is by several orders of magnitude higher than for the metastable
bcc phase, seen to nucleate first in other fcc
systems. The nucleation barrier that the diffuse interface theories predict for
small deviations from equilibrium is in far better agreement with the
simulations than the classical droplet model. The behavior expected at high
densities is model dependent. Gránásy s
phenomenological diffuse interface theory indicates a spinodal
point close to glass transition, while a nonsingular behavior is predicted by
the density functional theory with constant Ginzburg-Landau
coefficients. Remarkably, a minimum of the nucleation barrier, similar to the
one seen in polydisperse systems, occurs if the known
density dependence of the Ginzburg-Landau
coefficients is considered. [J.
Chem. Phys. B, **117**, 11121, (2002)].

**Phase Field
Theory of Nucleation and Growth in Binary Alloys**

*László** Gránásy, ^{1} Tamás
Börzsönyi,^{1,2} and Tamás Pusztai^{1}*

^{1}*Research Institute for
Solid State Physics and Optics, POB 49, H-1525 Budapest, Hungary*

^{2}Groupe de Physique des Solides, CNRS UMR 75-88, Universités Paris
VI at VII, Tour 23, 2 place Jussieu,75251, Paris Cedex 05, France

We present a phase field theory for binary crystal
nucleation. Using the physical interface thickness, we achieve quantitative
agreement with computer simulations and experiments for unary and binary
substances. Large-scale numerical simulations are performed for multi-particle
freezing in alloys. We deduce the Kolmogorov
exponents for dendritic solidification and for the
"soft-impingement" of crystallites interacting via diffusion fields.
[Presented at International
Workshop on "Computational Physics of Transport and Interface
Dynamics" February18-March 8, 2002. MPIPKS Dresden, Germany; Appeared in
Interface and Transport Dynamics, edited by H. Emmerich,
B. Nestler and M. Schreckenberg,
Lecture Notes in Computational Science and Engineering, **32**, Springer,
Berlin, (2003) pp 190-195.]

**Growth
of “dizzy dendrites” in a random field of foreign particles**

*László** Gránásy, ^{1} Tamás Pusztai^{,1} James A. Warren,^{2}
Jack F. Douglas,^{3} Tamás Börzsönyi,^{1}
and Vincent Ferreiro^{4}*

^{ }

^{1}*Research Institute
for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary*

^{2}Metallurgy and ^{3}Polymers Divisions, National
Institute of Standards and Technology, Gaithersburg, Maryland 20899,USA

^{4}Laboratoire de Structure et Properiétés
de l Etat Solide, CNRS, Batiment C6, 59655 Villeneuve d Ascq, France^{}

Microstructure plays an essential role in determining
the properties of crystalline materials. A widely used method to influence
microstructure is the addition of nucleating agents1. Observations on films
formed from clay polymer blends indicate that particulate additives, in
addition to serving as nucleating agents, may also perturb crystal growth, leading
to the formation of irregular dendritic morphologies.
Here we describe the formation of these dizzy dendrites using a phase-field
theory, in which randomly distributed foreign particle inclusions perturb the
crystallization by deflecting the tips of the growing dendrite arms. This
mechanism of crystallization, which is verified experimentally, leads to a
polycrystalline structure dependent on particle configuration and orientation.
Using computer simulations we demonstrate that additives of controlled crystal
orientation should allow for a substantial manipulation of the crystallization
morphology*. *[Nature
Materials, **2**, 92 (2003)].

**Phase field
theory of crystal nucleation in hard sphere liquid**

*László** Gránásy, ^{1} Tamás
Pusztai,^{1} Gyula Tóth,^{1} Zoltán Jurek,^{1} Massimo Conti,^{2} and Bjřrn Kvamme^{3}*

^{ }

^{1}*Research Institute
for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary*

^{2}Dipartimento di Matematica
e Fisica, Universita di Camerino, and Istituto Nazionale di Fisica della
Materia, Via Madonna delle Carceri, I-62032, Camerino, Italy

^{3}University of Bergen, Department of Physics, Allégaten
55, N-5007 Bergen, Norway

The phase field theory of crystal
nucleation described in L. Gránásy, T. Börzsönyi, and T. Pusztai, Phys.
Rev. Lett.
**88,** 206105 (2002) is applied for nucleation in hard-sphere liquids. The
exact thermodynamics from molecular dynamics is used. The interface thickness
for phase field is evaluated from the cross-interfacial variation of the height
of the singlet density peaks. The model parameters are fixed in equilibrium so
that the free energy and thickness of the (111), (110), and (100) interfaces
from molecular dynamics are recovered. The density profiles predicted without adjustable
parameters are in a good agreement with the filtered densities from the
simulations. Assuming spherical symmetry, we evaluate the height of the
nucleation barrier and the Tolman length without
adjustable parameters. The barrier heights calculated with the properties of
the (111) and (110) interfaces envelope the Monte Carlo results, while those
obtained with the average interface properties fall very close to the exact
values. In contrast, the classical sharp interface model considerably underestimates
the height of the nucleation barrier. We find that the Tolman
length is positive for small clusters and decreases with increasing size, a
trend consistent with computer simulations. [Journal of Chemical Physics, **119**, 10376 (2003)]. ^{}

**Phase-field
models for eutectic solidification**

*Daniel Lewis, ^{1} Tam*

^{ }

^{1}*Metallurgy and Polymers Divisions, National Institute
of Standards and Technology, Gaithersburg, Maryland 20899, USA ^{}*

^{2}*Research Institute for Solid State Physics and Optics,
PO Box 49, H-1525 Budapest, Hungary*

This article discusses two methods for modeling eutectic solidification using the
phase-field approach. First, a multi-phase-field model is used to study the
three-dimensional morphological evolution of binary eutectics. Performing the
calculations in three dimensions allows observation of both lamellar and
rod-like structures as well as transient phenomena such as lamellar fault
motion, rod-branching, and nucleation or elimination of phases as
solidification progresses. The second approach models multiple eutectic grains
where the crystallizing phases have an orientation relationship. This approach
is promising for modeling complex solidification microstructures. [JOM, **56**,
34-39 (2004)].

**Kinetics of
solid hydrate formation by carbon dioxide: Phase field theory of hydrate
nucleation and magnetic resonance imaging**

*B.
Kvamme, ^{1} A. Graue,^{1} E. Aspenes,^{1} T.
Kuynetsova,^{1} L. Gránásy,^{2} G. Tóth,^{2} T.
Pusztai,^{2} and G. Tegze^{2}*

^{1}*University of
Bergen, Department of Physics, Allégaten 55, N-5007
Bergen, Norway*

^{2}Research Institute for Solid State Physics and Optics, PO Box
49, H-1525 Budapest, Hungary

In the course of developing a general kinetic model of
hydrate formation/reaction that can be used to establish/ optimize technologies
for the exploitation of hydrate reservoirs, two aspects of CO2 hydrate
formation have been studied. (i) We developed a
phase field theory for describing the nucleation of CO2 hydrate in aqueous
solutions. The accuracy of the model has been demonstrated on the hard-sphere
model system, for which all information needed to calculate the height of the
nucleation barrier is known accurately. It has been shown that the
phase field theory is considerably more accurate than the sharp-interface
droplet model of the classical nucleation theory. Starting from realistic
estimates for the thermodynamic and interfacial properties, we have shown that
under typical conditions of CO2 formation, the size of the
critical fluctuations (nuclei) is comparable to the interface thickness,
implying that the droplet model should be rather inaccurate. Indeed the phase
field theory predicts considerably smaller height for the nucleation barrier
than the classical approach. (ii) In order to provide accurate transformation
rates to test the kinetic model under development, we applied magnetic
resonance imaging to monitor hydrate phase transitions in porous media under
realistic conditions. The mechanism of natural gas hydrate conversion to
CO2-hydrate implies storage potential for CO2 in natural gas hydrate reservoirs,
with the additional benefit of methane production. We present the
transformation rates for the relevant processes (hydrate formation,
dissociation and recovery). [Phys.
Chem. Chem. Phys., **6**, 2327 (2004)].

**A general
mechanism for polycrystalline growth**

*László** Gránásy, ^{1} Tamás
Pusztai,^{1} Tamás Börzsönyi,^{1}
James A. Warren^{2} and Jack F. Douglas^{3} *

^{ }

^{1}*Research Institute
for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary*

^{2}Metallurgy and ^{3}Polymers Divisions, National
Institute of Standards and Technology, Gaithersburg, Maryland 20899,USA

Most research into microstructure formation during
solidification has focused on single-crystal growth ranging from faceted
crystals to symmetric dendrites. However, these growth forms can be perturbed
by heterogeneities, yielding a rich variety of polycrystalline growth patterns.
Phase-field simulations show that the presence of particulates (for example,
dirt) or a small rotational translational mobility ratio (characteristic of
high supercooling) in crystallizing fluids give rise
to similar growth patterns, implying a duality in the growth process in these
structurally heterogeneous fluids. Similar crystallization patterns are also
found in thin polymer films with particulate additives and pure films with high
supercooling. This duality between the static and
dynamic heterogeneity explains the ubiquity of polycrystalline growth patterns
in polymeric and other complex fluids. [Nature
Materials, **3**, 645 (2004)].

**Nucleation
and polycrystalline formation in binary phase field theory**

*László** Gránásy, ^{1} Tamás
Pusztai,^{1} Tamás Börzsönyi,^{1}
James A. Warren,^{2 }Bjřrn Kvamme,^{3}
and P.F. James^{4}*

^{1}*Research Institute
for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary*

^{2}Metallurgy Division, National Institute of Standards and
Technology, Gaithersburg, Maryland 20899,USA

^{3}University of Bergen, Department of Physics, Allégaten
55, N-5007 Bergen, Norway

^{4}Glass Research Centre, Department of Engineering Materials, The
University of Sheffield, Sir Robert Hadfield
Building, Mappin Street, Sheffield S1 3JD, UK

We present a phase field theory for the nucleation and
growth of one and two phase crystals solidifying with different
crystallographic orientations in binary alloys. The accuracy of the model is
tested for crystal nucleation in single component systems. It is shown that
without adjustable parameters the height of the nucleation barrier is predicted
with reasonable accuracy. The kinetics of primary solidification is
investigated as a function of model parameters under equiaxial
conditions. Finally, we study the formation of polycrystalline growth
morphologies (disordered dendrites, spherulites and
fractal-like aggregates). [Phys.
Chem. Glass, **45**, 107-115 (2004)].

*We
thank V. Ferreiro and J. F. Douglas for the
experimental images (darker pictures).*

**Modelling**** polycrystalline solidification using
phase field theory**

*László** Gránásy, ^{1} Tamás Pusztai,^{1} and James A. Warren,^{2}*

^{1}*Research Institute for Solid State Physics and Optics,
PO Box 49, H-1525 Budapest, Hungary
^{2}Metallurgy Division, National Institute of Standards and
Technology, Gaithersburg, Maryland 20899,USA
*

We
review recent advances made in the phase field modelling
of polycrystalline solidification. Areas covered include the development of
theory from early approaches that allow for only a few crystal orientations, to
the latest models relying on a continuous orientation field and a free energy
functional that is invariant to the rotation of the laboratory frame. We
discuss a variety of phenomena, including homogeneous nucleation and
competitive growth of crystalline particles having different crystal
orientations, the kinetics of crystallization, grain boundary dynamics, and the
formation of complex polycrystalline growth morphologies including disordered
(dizzy) dendrites, spherulites, fractal-like polycrystalline
aggregates, etc. Finally, we extend the approach by incorporating walls, and
explore phenomena such as heterogeneous nucleation, particle front interaction,
and solidification in confined geometries (in channels or porous media). [J.Phys.
Condens. Matter **16,** R1205 (2004)]

**Multiphase
solidification in multicomponent alloys**

*U. Hecht, ^{1} L.
Gránásy,^{2} T. Pusztai,^{2} B. Böttger,^{1} M. Apel,^{1}
V. Witusiewicz,^{1} L. Ratke,^{3} J. De Wilde,^{4} L.
Froyen,^{4} D. Camel,^{5} B. Drevet,^{5} G. Faivre,^{6}
S.G. Fries,^{1} B. Legendre,^{7} and S. Rex^{1} *

^{1}*ACCESS e.V, Aachen, Germany
^{2}Research Institute for Solid State Physics and Optics of the
Hungarian Academy of Sciences, Budapest, Hungary
^{3}Institute of Space Simulation DLR Köln, Germany
^{4}Departement MTM, Katholieke Universiteit Leuven, Faculteit Toegepaste Wetenschappen, Leuven, Belgium
^{5}CEA-Grenoble, Grenoble, France
^{6}Groupe de Physique des Solides (GPS), Université Paris 6, Paris, France
^{7}Laboratoire de Chimie Physique Minérale et Bioinorganique, EA
401, Faculté de Pharmacie, Chatenay-Malabry, France
*

Multiphase solidification in multicomponent
alloys is pertinent to many commercial materials and industrial processes,
while also raising challenging questions from a fundamental point of view.
Within the past few years, research activities dedicated to multiphase
solidification of ternary and multicomponent alloys
experienced considerable amplification. This paper gives an overview of our
present understanding in this field and the experimental techniques and
theoretical methods research relies on. We start with an introduction to
thermodynamic databases and computations and emphasize the importance of thermophysical property data. Then, we address pattern
formation during coupled growth in ternary alloys and cover microstructure
evolution during successive steps of phase formation in solidifying multicomponent alloys. Subsequently, we review advances
made in phase field modeling of multiphase solidification in binary and multicomponent alloys, including various approaches to
crystal nucleation and growth. Concluding, we address open questions and
outline future prospects on the basis of a close interaction among scientists
investigating the thermodynamic, thermophysical and microstructural properties of these alloys. [Materials Science and
Engineering **R 46**, 1 (2004)]

**Nucleation
and the solid-liquid free energy**

*David
T. Wu, ^{1} László
Gránásy,^{2} and Frans Spaepen^{3}*

This article reviews the current understanding of the
fundamentals of nucleation theory and its use to extract values for the solid liquid
interfacial free energy from experimental and simulation data. [MRS Bulletin, December 2004]

**Growth and
form of spherulites**

*L.
Gránásy, ^{1} T. Pusztai,^{1} G. Tegze,^{1}
J.A. Warren^{2} and J.F. Douglas^{3} *

^{1}*Research Institute
for Solid State Physics and Optics, PO Box 49, H-1525 Budapest, Hungary*

^{2}Metallurgy and ^{3}Polymers Divisions, National
Institute of Standards and Technology, Gaithersburg, Maryland 20899,USA

Many structural materials (metal alloys, polymers,
minerals, etc.) are formed by quenching liquids into crystalline solids.
This highly nonequilibrium process often leads to
polycrystalline growth patterns that are broadly termed "spherulites" because of their large-scale average
spherical shape. Despite the prevalence and practical importance of spherulite formation, only rather qualitative concepts of
this phenomenon exist. It is established that phase field methods naturally
account for diffusional instabilities that are
responsible for dendritic single-crystal growth. However,
a generalization of this model is required to describe spherulitic
growth patterns, and in the present paper we propose a minimal model of this
fundamental crystal growth process. Our calculations indicate that the
diversity of spherulitic growth morphologies arises
from a competition between the ordering effect of discrete local
crystallographic symmetries and the randomization of the local crystallographic
orientation that accompanies crystal grain nucleation at the growth front growth front nucleation (GFN). This
randomization in the orientation accounts for the isotropy of spherulitic growth at large length scales and long times.
In practice, many mechanisms can give rise to GFN, and the present work
describes and explores three physically prevalent sources of disorder that lead
to this kind of growth. While previous phase field modeling elucidated two of
these mechanisms - disorder created by particulate impurities or other static
disorder or by the dynamic heterogeneities that spontaneously form in supercooled liquids (even pure ones) - the
present paper considers an additional mechanism, crystalline branching induced
by a misorientation-dependent grain boundary energy,
which can significantly affect spherulite morphology.
We find the entire range of observed spherulite
morphologies can be reproduced by this generalized phase field model of
polycrystalline growth. [Phys.
Rev. E **72**, 011605 (2005)]

**Phase
field theory of polycrystalline solidification in three dimensions**

*T.
Pusztai, G. Bortel and L. Gránásy*

*Research
Institute for Solid State Physics and Optics H-1525 Budapest, POB 49, Hungary*

A phase field theory of polycrystalline solidification
is presented that describes the nucleation and growth of anisotropic particles
with different crystallographic orientation in 3D dimensions. As opposed with
the two-dimensional case, where a single orientation field suffices, in three
dimensions, minimum three fields are needed. The free energy of grain
boundaries is assumed to be proportional to the angular difference between the
adjacent crystals expressed here in terms of the differences of the four
symmetric Euler parameters. The equations of motion for these fields are
obtained from variational principles. Illustrative
calculations are performed for polycrystalline solidification with dendritic, needle and spherulitic
growth morphologies. [Europhys. Lett. **71**, 131 (2005)]

**Phase field
modeling of polycrystalline freezing**

*T.
Pusztai, G. Bortel and
L. Gránásy*

*Research
Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest,
Hungary*

The formation of two and three-dimensional
polycrystalline structures are addressed within the framework of the phase
field theory. While in two dimensions a single orientation angle suffices to
describe crystallographic orientation in the laboratory frame, in three
dimensions, we use the four symmetric Euler parameters to define
crystallographic orientation. Illustrative simulations are performed for
various polycrystalline structures including simultaneous growth of randomly oriented
dendritic particles, the formation of spherulites and crystal sheaves. [Materials Science and
Engineering A **413–414**, 412–417 (2005)]

**Phase field
simulation of liquid phase separation with fluid flow**

*G.
Tegze, T. Pusztai and L. Gránásy*

*Research
Institute for Solid State Physics and Optics, PO Box 49, H-1525 Budapest,
Hungary*

A phase-field theory of binary liquid phase separation
coupled to fluid flow is presented. The respective Cahn–Hilliard-type
and Navier–Stokes equations are solved numerically.
We incorporate composition and temperature dependent capillary forces. The free
energies of the bulk liquid phases are taken from the regular solution model.
In the simulations, we observe Marangoni motion, and
direct and indirect hydrodynamic interactions between the droplets. We find
that coagulation is dramatically accelerated by flow effects. Possible
extension of the model to solidification is discussed. [Materials Science and
Engineering A **413–414**, 418–422 (2005)]

**Phase field
theory of crystal nucleation and polycrystalline growth: A review
**

*L. Gránásy, ^{1} T. Pusztai,^{1} T. Börzsönyi,^{1}
G. Tóth,^{1} G. Tegze,^{1} J.A. Warren,^{2} and J.F.
Douglas^{2}*

^{1}*Research Institute
for Solid State Physics and Optics, H-1525 Budapest, Hungary*

^{2}*National Institute of Standards and Technology,
Gaithersburg, Maryland 20899*

We
briefly review our recent modeling of crystal nucleation and polycrystalline
growth using a phase field theory. First, we consider the applicability of
phase field theory for describing crystal nucleation in a model hard sphere
fluid. It is shown that the phase field theory accurately predicts the nucleation
barrier height for this liquid when the model parameters are fixed by
independent molecular dynamics calculations. We then address various aspects of
polycrystalline solidification and associated crystal pattern formation at
relatively long timescales. This late stage growth regime, which is not
accessible by molecular dynamics, involves nucleation at the growth front to
create new crystal grains in addition to the effects of primary nucleation.
Finally, we consider the limit of extreme polycrystalline growth, where the
disordering effect due to prolific grain formation leads to isotropic growth
patterns at long times, i.e., spherulite formation. Our model of spherulite growth exhibits
branching at fixed grain misorientations, induced by
the inclusion of a metastable minimum in the orientational free energy. It is demonstrated that a
broad variety of spherulitic patterns can be
recovered by changing only a few model parameters. [J. Mater. Res., **21**, 309 (2006)]

**Multiscale**** approach to CO2 hydrate formation in
aqueous solution: Phase field theory and molecular dynamics. Nucleation and
growth
**

*György** Tegze, ^{1} Tamás
Pusztai,^{1} Gyula Tóth,^{1} László Gránásy,^{1} Atle
Svandal,^{2} Trygve Buanes,^{2} Tatyana Kuznetsova,^{2} and Bjřrn
Kvamme^{2}*

^{1}*Research Institute for Solid State Physics and Optics,
P.O. Box 49, H-1525 Budapest, Hungary*

^{2}Institute of Physics and Technology, University of Bergen, Allégaten 55, N-5007 Bergen, Norway

A
phase field theory with model parameters evaluated from atomistic
simulations/experiments is applied to predict the nucleation and growth rates
of solid CO2 hydrate in aqueous solutions under conditions typical to
underwater natural gas hydrate reservoirs. It is shown that under practical
conditions a homogeneous nucleation of the hydrate phase can be ruled out. The
growth rate of CO2 hydrate dendrites has been determined from phase field
simulations as a function of composition while using a physical interface
thickness 0.85±0.07 nm evaluated from molecular dynamics
simulations. The growth rate extrapolated to realistic supersaturations
is about three orders of magnitude larger than the respective experimental
observation. A possible origin of the discrepancy is discussed. It is suggested
that a kinetic barrier reflecting the difficulties in building the complex
crystal structure is the most probable source of the deviations. [J. Chem. Phys. **124**,
234710 (2006)]

**Phase field
theory of polycrystalline freezing in three dimensions**

*Tamás**
Pusztai, Gábor Bortel and László Gránásy
*

*Research
Institute for Solid State Physics and Optics; H-1525 Budapest, POB 49, Hungary*

A phase field theory, we proposed recently to describe
nucleation and growth in three dimensions (3D), has been used to study the
formation of polycrystalline patterns in the alloy systems Al-Ti and Cu-Ni. In
our model, the free energy of grain boundaries is assumed proportional to the
angular difference between the adjacent crystals expressed in terms of the
differences of the four symmetric Euler parameters called quaternions.
The equations of motion for these fields have been obtained from variational principles. In the simulations cubic crystal
symmetries are considered. We investigate the evolution of polydendritic
morphology, present simulated analogies of the metallographic images, and
explore the possibility of modeling solidification in thin layers.
Transformation kinetics in the bulk and in thin films is discussed in terms of
the Johnson-Mehl-Avrami-Kolmogorov approach. [Modeling of Casting,
Welding and Advanced Solidification Processes- XI, TMS 409 (2006)]

**Phase field
theory of liquid phase separation and solidification with melt flow
**

*György** Tegze and
László Gránásy
*

*Research Institute for
Solid State Physics and Optics; H-1525 Budapest, POB 49, Hungary*

A
phase-field theory of binary liquid phase separation and solidification coupled
to fluid flow is presented. The respective equations of motion and Navier-Stokes equations are solved numerically. We
incorporate composition and temperature dependent capillary forces. The free
energies of the bulk liquid phases are taken from the regular solution model.
In the simulations, we observe Marangoni motion of
the droplets, and direct and indirect hydrodynamic interactions between the
droplets. We observe that capillary effects dramatically accelerate droplet
coagulation and that solidification interacts with liquid phase
separation. [Modeling
of Casting, Welding and Advanced Solidification Processes- XI, TMS 513 (2006)]

**Polycrystalline
patterns in far-from-equilibrium freezing: a phase field study
**

*L. Gránásy, ^{1} T.
Pusztai,^{1} T. Börzsönyi,^{1} G. Tóth,^{1} G. Tegze,^{1}
J.A. Warren,^{2} and J.F. Douglas^{2}*

^{1}Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary

^{2}National Institute of Standards and Technology, Gaithersburg, Maryland 20899

We discuss the formation of polycrystalline
microstructures within the framework of phase field theory. First, the model is
tested for crystal nucleation in a hard sphere system. It is shown that, when
evaluating the model parameters from molecular dynamics simulations, the phase
field theory predicts the nucleation barrier for hard spheres accurately. The
formation of spherulites is described by an extension
of the model that incorporates branching with a definite orientational
mismatch. This effect is induced by a metastable
minimum in the orientational free energy. Spherulites are an extreme example of polycrystalline
growth, a phenomenon that results from the quenching of orientational
defects (grain boundaries) into the solid as the ratio of the rotational to the
translational diffusion coefficient is reduced, as is found at high undercoolings. It is demonstrated that a broad variety of spherulitic patterns can be recovered by changing only a
few model parameters.* *[Philos.
Mag. **86** 3757 (2006)]

**Phase field
theory of nucleation and polycrystalline pattern formation
**

*L. Gránásy,
T. Pusztai and T. Börzsönyi*

*Research Institute for
Solid State Physics and Optics, H-1525 Budapest, Hungary*

We
review our recent modeling of crystal nucleation and polycrystalline growth
using a phase field theory. First, we consider the applicability of phase field
theory for describing crystal nucleation in a model hard sphere fluid. It is
shown that the phase field theory accurately predicts the nucleation barrier
height for this liquid when the model parameters are fixed by independent molecular
dynamics calculations. We then address various aspects of polycrystalline
solidification and associated crystal pattern formation at relatively long
timescales. This late stage growth regime, which is not accessible by molecular
dynamics, involves nucleation at the growth front to create new crystal grains
in addition to the effects of primary nucleation. Finally, we consider the
limit of extreme polycrystalline growth, where the disordering effect due to
prolific grain formation leads to isotropic growth patterns at long times,
i.e., spherulite formation. Our
model of spherulite growth exhibits branching at
fixed grain misorientations, induced by the inclusion
of a metastable minimum in the orientational
free energy. It is demonstrated that a broad variety of spherulitic patterns can be recovered by changing only a
few model parameters. [Handbook
of Theoretical and Computational Nanotechnology, Edited by Michael Rieth and Wolfram Schommers
American Scientific Publishers, Stevenson Ranch, CAL, 2006, Volume **9**:
Pages (525-572)]

**Phase field
theory of heterogeneous crystal nucleation**

*L. Gránásy, ^{1} T. Pusztai,^{1} D. Saylor,^{2}
and J.A. Warren^{3} *

^{1}Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary

^{2}*Food and Drug Administration, Rockville, Maryland
20852, USA
^{3}National Institute of Standards and Technology, Gaithersburg,
Maryland 20899, USA*

The phase field approach is used to model
heterogeneous crystal nucleation in an undercooled
pure liquid in contact with a foreign wall. We discuss various choices for the
boundary condition at the wall and determine the properties of critical nuclei,
including their free energy of formation and the contact angle as a function of
undercooling. For particular choices of boundary
conditions, we may realize either an analog of the classical spherical cap
model or decidedly nonclassical behavior, where the
contact angle decreases from its value taken at the melting point towards
complete wetting at a critical undercooling, an
analogue of the surface spinodal of liquid-wall
interfaces. [Phys. Rev. Lett. **98**, 035703 (2007) ]

**Phase field
theory of interfaces and crystal nucleation in a eutectic system of fcc structure: I. Transitions in
the one-phase liquid region**

*Gy**.**
I. Tóth and L. Gránásy, *

*Research Institute for
Solid State Physics and Optics, H-1525 Budapest, Hungary*

The
phase field theory PFT has been applied to predict equilibrium interfacial
properties and nucleation barrier in the binary eutectic system Ag–Cu using
double well and interpolation functions deduced from a Ginzburg-Landau
expansion that considers fcc
face centered cubic crystal symmetries. The temperature and composition
dependent free energies of the liquid and solid phases are taken from
Calculation of Phase Diagrams-type calculations. The model parameters of PFT
are fixed so as to recover an interface thickness of 1 nm from molecular
dynamics simulations and the interfacial free energies from the experimental
dihedral angles available for the pure components. A nontrivial temperature and
composition dependence for the equilibrium interfacial free energy is observed.
Mapping the possible nucleation pathways, we find that the Ag and Cu rich
critical fluctuations compete against each other in the neighborhood of the
eutectic composition. The Tolman length is positive
and shows a maximum as a function of undercooling.
The PFT predictions for the critical undercooling are
found to be consistent with experimental results. These results support the
view that heterogeneous nucleation took place in the undercooling
experiments available at present. We also present calculations using the
classical droplet model classical nucleation theory CNT and a phenomenological
diffuse interface theory DIT. While the predictions of the CNT with a purely
entropic interfacial free energy underestimate the critical undercooling,
the DIT results appear to be in a reasonable agreement with the PFT
predictions. [J.
Chem. Phys. **127**, 074709 (2007)]

**Phase field
theory of interfaces and crystal nucleation in a eutectic system of fcc structure: II. Nucleation in
the metastable liquid immiscibility region**

*G. I. Tóth
and L. Gránásy, *

*Research Institute for
Solid State Physics and Optics, H-1525 Budapest, Hungary
*

In
the second part of our paper, we address crystal nucleation in the metastable liquid miscibility region of eutectic systems
that is always present, though experimentally often inaccessible. While this
situation resembles the one seen in single component crystal nucleation in the
presence of a metastable vapor-liquid critical point
addressed in previous works, it is more complex because of the fact that here
two crystal phases of significantly different compositions may nucleate.
Accordingly, at a fixed temperature below the critical point, six different
types of nuclei may form: two liquid-liquid nuclei: two solid-liquid nuclei;
and two types of composite nuclei, in which the crystalline core has a liquid
“skirt,” whose composition falls in between the compositions of the solid and
the initial liquid phases, in addition to nuclei with concentric alternating
composition shells of prohibitively high free energy. We discuss crystalline
phase selection via exploring/identifying the possible pathways for crystal
nucleation. [J.
Chem. Phys. 127, 074710 (2007)]

**Phase-field approach to polycrystalline solidification including
heterogeneous and homogeneous nucleation.**

*T. Pusztai, ^{1} G.
Tegze,^{2} G. I. Tóth,^{1} L. Környei,^{1} G. Bansel,^{2}
Z. Fan,^{2} and L. Gránásy^{2} *

^{1}Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary;

^{2}Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK

Advanced
phase-field techniques have been applied to address various aspects of
polycrystalline solidification including different modes of crystal nucleation.
The height of the nucleation barrier has been determined by solving the
appropriate Euler-Lagrange equations. The examples shown include the comparison
of various models of homogeneous crystal nucleation with atomistic simulations
for the single-component hard sphere fluid. Extending previous work for pure
systems [Gránásy *et al.*, Phys. Rev. Lett. **98**, 035703 (2007)], heterogeneous nucleation
in unary and binary systems is described via introducing boundary conditions
that realize the desired contact angle. A quaternion
representation of crystallographic orientation of the individual particles
[outlined in Pusztai *et al*., Europhys. Lett. **71**, 131 (2005)] has been applied for modeling a
broad variety of polycrystalline structures including crystal sheaves, spherulites and those built of crystals with dendritic, cubic, rhombo-dodecahedral
and truncated octahedral growth morphologies. Finally, we present illustrative
results for dendritic polycrystalline solidification
obtained using an atomistic phase-feld model. [*J. Phys.: Condens. Matter* **20**, 404205 (2008)]

**Advanced operator-splitting-based semi-implicit spectral method to
solve the binary phase-field crystal equation with variable coefficients.**

*G. Tegze, ^{1}
G. Bansel,^{1} G. I. Tóth,^{2} T. Pusztai,^{2} Z. Fan,^{1}
and L. Gránásy^{1} *

^{1}*Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK
^{2}Research Institute for Solid State Physics and Optics, H-1525
Budapest, Hungary
*

We
present an efficient method to solve numerically the equations of dissipative
dynamics of the binary phase-field crystal model proposed by Elder et al. [K.R.
Elder, M. Katakowski, M. Haataja,
M. Grant, Phys. Rev. B 75, 064107 (2007)] characterized by variable
coefficients. Using the operator splitting method, the problem has been
decomposed into sub-problems that can be solved more efficiently. A combination
of non-trivial splitting with spectral semi-implicit solution leads to sets of
algebraic equations of diagonal matrix form. Extensive testing of the method
has been carried out to find the optimum balance among errors associated with
time integration, spatial discretization, and
splitting. We show that our method speeds up the computations by orders of
magnitude relative to the conventional explicit finite difference scheme, while
the costs of the pointwise implicit solution per timestep remains low. Also we show that due to its
numerical dissipation, finite differencing can not compete with spectral
differencing in terms of accuracy. In addition, we demonstrate that our method
can efficiently be parallelized for distributed memory systems, where an
excellent scalability with the number of CPUs is observed. [*J. Comput. Phys.* **228**, 1612 (2009)]

**Phase
field approach to heterogeneous nucleation in alloys.**

*J. A. Warren, ^{1}
T. Pusztai,^{2} L. Környei,^{2} and L. Gránásy^{3}*

^{1}Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

^{2}Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary

^{3 }Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK

We
extend the phase field model of heterogeneous crystal nucleation developed
recently [L. Gránásy *et al.*, Phys. Rev. Lett. **98**, 035703 (2007)] to binary alloys. Three
approaches are considered to incorporate foreign walls of tunable wetting
properties into phase field simulations: a continuum realization of the
classical spherical cap model (called model A herein), a nonclassical
approach (model B) that leads to ordering of the liquid at the wall and to the
appearance of a surface spinodal, and a nonclassical model (model C) that allows for the appearance
of local states at the wall that are accessible in the bulk phases only via
thermal fluctuations. We illustrate the potential of the presented phase field
methods for describing complex polycrystalline solidification morphologies
including the shish-kebab structure, columnar to equiaxed
transition, and front-particle interaction in binary alloys. [*Phys. Rev. B*
**79**, 014204 (2009)]

**Crystal
nucleation in the hard-sphere system revisited: A critical test of theoretical
approaches **

*G. I. Tóth ^{1} and
L. Gránásy^{2}*

^{1}*Research Institute for Solid State Physics and Optics,
H-1525 Budapest, Hungary
^{2}Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK
*

The
hard-sphere system is the best known fluid that crystallizes: the solid-liquid
interfacial free energy, the equations of state, and the height of the
nucleation barrier are known accurately, offering a unique possibility for a
quantitative validation of nucleation theories. A recent significant downward
revision of the interfacial free energy from 0.61*kT*/*s*^{2} to 0.56 *kT*/*s*^{2} [Davidchack, R.; Morris, J. R.; Laird, B. B. *J. Chem.
Phys.* **125**, 094710 (2006)] necessitates a re-evaluation of
theoretical approaches to crystal nucleation. This has been carried out for the
droplet model of the classical nucleation theory (CNT), the self-consistent
classical theory (SCCT), a phenomenological diffuse interface theory (DIT), and
single- and two-field variants of the phase field theory that rely on either
the usual double-well and interpolation functions (PFT/S1 and PFT/S2,
respectively) or on a Ginzburg-Landau expanded free
energy that reflects the crystal symmetries (PFT/GL1 and PFT/GL2). We find that
the PFT/GL1, PFT/GL2, and DIT models predict fairly accurately the height of
the nucleation barrier known from Monte Carlo simulations in the volume
fraction range of 0.52 < *f* < 0.54, whereas the CNT, SCCT, PFT/S1, and PFT/S2
models underestimate it significantly. [*J. Phys. Chem.
B* **113**, 5141 (2009)]

**Diffusion-controlled
anisotropic growth of stable and metastable crystal
polymorphs in the phase-field crystal model **

*G. Tegze, ^{1} L.
Gránásy,^{1} G. I. Tóth,^{2} F. Podmaniczky,^{2} A.
Jaatinen,^{3} T. Ala-Nissila,^{3} and T. Pusztai^{2} *

^{1}Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK

^{2}Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary

^{3}Department of Applied Physics, Helsinki University of Technology, Post Office Box 1100, FI-02015 TKK, Finland

We
use a simple density functional approach on a diffusional
time scale, to address freezing to the body-centered cubic (bcc), hexagonal
close-packed (hcp), and face-centered cubic (fcc) structures. We observe
faceted equilibrium shapes and diffusion-controlled layerwise
crystal growth consistent with two- dimensional nucleation. The predicted
growth anisotropies are discussed in relation with results from experiment and
atomistic simulations. We also demonstrate that varying the lattice constant of
a simple cubic substrate, one can tune the epitaxially growing body-centered tetragonal structure
between bcc and fcc, and observe a Mullins-Sekerka/Asaro-Tiller-Grinfeld-type instability. [*Phys. Rev. Lett.* **103**, 035702 (2009)]

**Classical
density functional theory methods in soft and hard matter **

*M. Haataja, ^{1} L.
Gránásy,^{2,3} and H. Löwen^{4} *

^{1}Department of Mechanical and Aerospace Engineering, Institute for the Science and Technology of Materials (PRISM) and Program in Applied and Computational Mathematics (PACM), Princeton University,Princeton NJ 08544, USA

^{2}Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary

^{3}BCAST, Brunel University, Uxbridge UB8 3PH, UK

^{4}Department of Theoretical Physics, Heinrich-Heine-Universität Düsseldorf, D-40225 D

Herein
we provide a brief summary of the background, events and results/outcome of the
CECAM workshop ‘Classical density functional theory methods in soft and hard
matter’ held in Lausanne between October 21 and October 23 2009, which brought
together two largely separately working communities, both of whom employ
classical density functional techniques: the soft-matter community and the
theoretical materials science community with interests in phase transformations
and evolving microstructures in engineering materials. After outlining the
motivation for the workshop, we first provide a brief overview of the articles
submitted by the invited speakers for this special issue of *Journal of
Physics: Condensed Matter*, followed by a collection of outstanding problems
identified and discussed during the workshop. [ *J**. Phys.: Condens.
Matter ***22**, 360301** **(2010)]

**Polymorphism,
crystal nucleation and growth in the phase-field crystal model in 2d and 3d**

G. I. Tóth,^{1} G. Tegze,^{1} T.
Pusztai,^{1} G. I. Tóth,^{2} _{ }and L. Gránásy^{1,3}

^{1}Research Institute for Solid State Physics and Optics, H-1525
Budapest, Hungary

^{2}Institute of Chemistry, Eötv*ö**s University, PO Box 32, H-1518 Budapest, Hungary*

^{3}*Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK
*

We
apply a simple dynamical density functional theory, the phase-field crystal
(PFC) model of overdamped conservative dynamics, to
address polymorphism, crystal nucleation, and crystal growth in the
diffusion-controlled limit. We refine the phase diagram for 3D, and determine
the line free energy in 2D and the height of the nucleation barrier in 2D and
3D for homogeneous and heterogeneous nucleation by solving the respective
Euler–Lagrange (EL) equations. We demonstrate that, in the PFC model, the
body-centered cubic (bcc), the face-centered cubic (fcc),
and the hexagonal close-packed structures (hcp)
compete, while the simple cubic structure is unstable, and that phase
preference can be tuned by changing the model parameters: close to the critical
point the bcc structure is stable, while far from the critical point the fcc prevails, with an hcp
stability domain in between. We note that with increasing distance from the
critical point the equilibrium shapes vary from the sphere to specific faceted
shapes: rhombic dodecahedron (bcc), truncated octahedron (fcc), and hexagonal prism (hcp).
Solving the equation of motion of the PFC model supplied with conserved noise,
solidification starts with the nucleation of an amorphous precursor phase, into
which the stable crystalline phase nucleates. The growth rate is found to be
time dependent and anisotropic; this anisotropy depends on the driving force.
We show that due to the diffusion-controlled growth mechanism, which is
especially relevant for crystal aggregation in colloidal systems, dendritic growth structures evolve in large-scale
isothermal single-component PFC simulations. An oscillatory effective pair
potential resembling those for model glass formers has been evaluated from
structural data of the amorphous phase obtained by instantaneous quenching.
Finally, we present results for eutectic solidification in a binary PFC model. [ *J**. Phys.: Condens.
Matter 22, 364101 (2010).]*

**Phase-field
crystal modelling of crystal nucleation, heteroepitaxy and patterning**

*L.
Gránásy, ^{1,2} G. Tegze,^{1} G. I.
Tóth,^{1} and T. Pusztai^{1}*

^{1}Research Institute for Solid State Physics and Optics, H-1525
Budapest, Hungary

^{2}Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK

A
simple dynamical density functional theory, the phase-field crystal (PFC)
model, was used to describe homogeneous and heterogeneous crystal nucleation in
two-dimensional (2D) monodisperse colloidal systems
and crystal nucleation in highly compressed Fe liquid. External periodic
potentials were used to approximate inert crystalline substrates in addressing
heterogeneous nucleation. In agreement with experiments in 2D colloids, the PFC
model predicts that in 2D supersaturated liquids, crystalline freezing starts
with homogeneous crystal nucleation without the occurrence of the hexatic phase. At extreme supersaturations,
crystal nucleation happens after the appearance of an amorphous precursor both
in two and three dimensions. Contrary to expectations based on the classical
nucleation theory, it is shown that corners are not necessarily favourable places for crystal nucleation. Finally, it is
shown that by adding external potential terms to the free energy, the PFC
theory can be used to model colloid patterning experiments. [ *Philos**. Mag.* **91**, 123-149** **(2011).]

**Tuning the structure
of non-equilibrium soft materials by varying the thermodynamic driving force
for crystal ordering**

*G.
Tegze, ^{1} L. Gránásy,^{1,2} G. I.
Tóth,^{1} J. F. Douglas,^{3} and T. Pusztai^{1}*

^{1}Research Institute for Solid State Physics and Optics, H-1525
Budapest, Hungary

^{2}Brunel Centre for Advanced Solidification Technology, Brunel University, Uxbridge UB8 3PH, UK

^{3}*Polymers Division, National Institute of Standards and
Technology,Gaithersburg, MD,
20899, USA.
*

The
present work explores the ubiquitous morphological changes in crystallizing
systems with increasing thermodynamic driving force based on a novel dynamic
density functional theory. A colloidal ‘soft’ material is chosen as a model
system for our investigation since there are careful colloidal crystallization
observations at a particle scale resolution for comparison, which allows for a
direct verification of our simulation predictions. We particularly focus on a
theoretically unanticipated, and generic, morphological transition leading to
progressively irregular-shaped single crystals in both colloidal and polymeric
materials with an increasing thermodynamic driving force. Our simulation method
significantly extends previous ‘phase field’ simulations by incorporating a
minimal description of the ‘atomic’ structure of the material, while allowing
simultaneously for a description of large scale crystal growth. We discover a
‘fast’ mode of crystal growth at high driving force, suggested before in
experimental colloidal crystallization studies, and find that the coupling of
this crystal mode to the well-understood ‘diffusive’ or ‘slow’ crystal growth
mode (giving rise to symmetric crystal growth mode and dendritic
crystallization as in snowflakes by the Mullins–Sekerka
instability) can greatly affect the crystal morphology at high thermodynamic
driving force. In particular, an understanding of this interplay between these
fast and slow crystal growth modes allows us to describe basic crystallization
morphologies seen in both colloidal suspensions with increasing particle concentration
and crystallizing polymer films with decreasing temperature: compact symmetric
crystals, dendritic crystals, fractal-like
structures, and then a return to compact symmetric single crystal growth again.
[ *Soft** Matter ***7**, 1789-1799 (2011).]

**Ginzburg****-Landau-type multiphase field model for competing fcc and bcc nucleation**

*G.
I. Tóth, ^{1} J. R. Morris,^{2} and L.
Gránásy^{1,3}*

^{1}Research Institute for Solid State Physics and Optics, H-1525 Budapest,
Hungary

^{2}*Oak Ridge National Laboratory, Oak Ridge, Tennessee
37830, USA
^{3}BCAST, Brunel University, Uxbridge,
Middlesex, UB8 3PH, United Kingdom*

We
address crystal nucleation and fcc-bcc phase
selection in alloys using a multiphase field model that relies on Ginzburg-Landau free energies of the liquid-fcc, liquid-bcc, and fcc-bcc
subsystems, and determine the properties of the nuclei as a function of
composition, temperature, and structure. With a realistic choice for the free
energy of the fcc-bcc interface, the model predicts
well the fcc-bcc phase-selection boundary in the
Fe-Ni system. [ *Phys**. Rev. Lett.
***105**, 045701 (2011).]

** Faceting and branching in 2D crystal growth**

*G.
Tegze, ^{1} G. I. Tóth,^{1} and L.
Gránásy^{1,2}*

^{1}Research Institute for Solid State Physics and Optics, H-1525
Budapest, Hungary

^{2}*BCAST, Brunel University,
Uxbridge, Middlesex, UB8 3PH, United Kingdom*

*
*Using atomic scale time-dependent
density functional calculations we confirm that both diffusion-controlled and diffusionless crystallization modes exist in simple 2D
systems. We provide theoretical evidence that a faceted to nonfaceted
transition is coupled to these crystallization modes, and faceting is governed
by the local supersaturation at the fluid-crystalline
interface. We also show that competing modes of crystallization have a major
influence on mesopattern formation. Irregularly
branched and porous structures are emerging at the crossover of the
crystallization modes. The proposed branching mechanism differs essentially
from dendritic fingering driven by diffusive
instability. [