Tamás Pusztai

Phase Field Theory of Heterogeneous Crystal Nucleation

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.

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

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 (2007) 064107] 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.

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

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.

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

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).

Interfacial properties deduced from nucleation experiments: A Cahn–Hilliard analysis

We apply a single-order-parameter Cahn–Hilliard theory to deduce properties of the fluid–crystal interface from nucleation experiments: The two Cahn–Hilliard parameters (the free energy scale and the coefficient of the square-gradient term) are chosen so that the experimentally determined interfacial free energy of nuclei is recovered. The theory is then used to predict the thickness and free energy of the equilibrium planar interface, and other quantities such as the Tolman length and characteristic thickness, which describe the curvature dependence of the interfacial free energy. The accuracy of the method is demonstrated on systems (Lennard-Jones and ice-water) for which these properties are known. Experimental data available for five stoichiometric oxide glasses are then analyzed. The reduced interfacial free energy (Turnbull’s α) and the interface thickness, we obtained, cover the α=0.28–0.51 and the d=0.8–1.6 nm ranges. For oxide glasses we find that α scales with n−1/3, where n is the number of mol...

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

A phase field theory with model parameters evaluated from atomistic simulations/experiments is applied to predict the nucleation and growth rates of solid CO(2) 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 CO(2) 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.

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

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 (Granasy et al 2007 Phys. Rev. Lett. 98 035703), 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 2005 Europhys. Lett. 71 131) 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-field model.

Amorphous Nucleation Precursor in Highly Nonequilibrium Fluids

Dynamical density-functional simulations reveal structural aspects of crystal nucleation in undercooled liquids: The first appearing solid is amorphous, which promotes the nucleation of bcc crystals but suppresses the appearance of the fcc and hcp phases. These findings are associated with features of the effective interaction potential deduced from the amorphous structure.

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

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.

Phase field theory of crystal nucleation in hard sphere liquid

The phase field theory of crystal nucleation described in L. Granasy, T. Borzsonyi, 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.