Amplitude representations of a binary phase field crystal model are developed for a two dimensional triangular lattice and three dimensional BCC and FCC crystal structures. The relationship between these amplitude equations and the standard phase field models for binary alloy solidification with elasticity are derived, providing an explicit connection between phase field crystal and phase field models. Sample simulations of solute migration at grain boundaries, eutectic solidification and quantum dot formation on nano-membranes are also presented.
Using small-angle X-ray diffraction and polarizing microscopy, we have investigated the phase and structural behavior of inverse bicontinuous cubic phases in a homologous series of 2:1 (mol:mol) saturated fatty acid/phosphatidylcholine mixtures in water. For the C12 fatty acid/phosphatidylcholine mixtures, we find three inverse bicontinuous cubic phases, which our analyses indicate are based upon the P, D, and G minimal surfaces. Between 35 and 49 °C these phases are observed to occur in the sequence G → D → P as the water content is increased. The excess water P cubic disappears above 49 °C when it is replaced by the D cubic in coexistence with the inverse hexagonal phase. For C14 fatty acid/phosphatidylcholine mixtures, we find only the P cubic in coexistence with the inverse hexagonal phase. For longer chain length fatty acid/phosphatidylcholine mixtures, the inverse bicontinuous cubics are entirely absent, their place being taken by the inverse hexagonal phase. Most of our evidence for the structure of the cubic phases being based on the P, D, and G minimal surfaces comes from our modeling of the isothermal swelling data in the C12 system. This modeling has also been used to fit the swelling behavior in the inverse hexagonal phase and has allowed us to determine the location of the pivotal surface for the 2:1 fatty acid/phosphatidylcholine mixtures for chain lengths between C12 and C18. With little variation as a function of chain length or hydration, the pivotal surface appears to be located between the second and third CH2 groups of the hydrocarbon chains with an average molecular area per 2:1 (mol:mol) saturated fatty acid/phosphatidylcholine complex of 109.5 ± 1.5 Å2. As far as we are aware this is the first self-consistent determination of pivotal surface geometry spanning four phases and four different chain lengths.
The world of two-dimensional crystals is of great significance for the design and study of structural and functional materials with novel properties. Here we examine the mechanisms governing the formation and dynamics of these crystalline or polycrystalline states and their elastic and plastic properties by constructing a generic multi-mode phase field crystal model. Our results demonstrate that a system with three competing length scales can order into all five Bravais lattices, and other more complex structures including honeycomb, kagome and other hybrid phases. In addition, nonequilibrium phase transitions are examined to illustrate the complex phase behavior described by the model. This model provides a systematic path to predict the influence of lattice symmetry on both structure and dynamics of crystalline and defected systems. . On larger length scales much progress has been made in the self-assembly of 2D crystals using particles of nano or micron size that are easier to tailor for specific functionalities and to observe. Colloidal crystals, for example, play a vital role in the study of structural properties of crystalline systems and the development of engineered, functional materials [7][8][9]. In addition, another novel technique for artificial lattice ordering is built on the trapping of ultracold atoms (e.g., 87 Rb) in optical superlattices produced by overlaying laser beams [10,11], as utilized for the study of many-body quantum physics. [7,11]. Thus it is of fundamental importance to identify the universal mechanisms underlying these distinct modes of crystallization, based on the general principle of symmetry [12]. It is also important to understand the nature of topological defects which occur frequently in such systems and are known to determine the electronic and mechanical properties of the sample [2]. Unfortunately it is very difficult to model and predict the nature of such defected states, due to multiple length and time scales involved in the non-equilibrium crystallization processes.In this work we develop a dynamic model that can be applied to the study of crystallization with a variety of ordered and defected structures. We adopt the phase field crystal (PFC) formalism [13][14][15][16], in the spirit of the Alexander-McTague analysis of crystallization based on Landau theory [12]. The advantage of this PFC approach is that one can study polycrystal formation in terms of the atomic number density on diffusive time scales that are many orders of magnitude larger than that of classical microscopic models such as molecular dynamics. One can also apply renormalization techniques [17][18][19] on the PFC equation to study problems that involve both micro and meso scales such as epitaxial growth [20] and surface patterning in ultra-thin films [21].Recently a great deal of progress has been made on generalizing the PFC formulation to include more crystal symmetries [22][23][24][25], although in 2D current PFC studies are restricted to triangular and square states. The basic idea is to incorp...
The instability of strained films for island formation is examined through an approach incorporating both discrete microscopic details and continuum mechanics. A linear relationship between the island wave number and misfit strain is found for large strains, while only in the small strain limit is a crossover to the continuum elasticity result obtained. A universal scaling relation accommodating all range of misfit strains is identified. Our results indicate that continuum mechanics may break down even at relatively small misfit stress due to the discrete nature of crystalline surfaces.
The dynamics of phase field crystal (PFC) modeling is derived from dynamical density functional theory (DDFT), for both single-component and binary systems. The derivation is based on a truncation up to the three-point direct correlation functions in DDFT, and the lowest order approximation using scale analysis. The complete amplitude equation formalism for binary PFC is developed to describe the coupled dynamics of slowly varying complex amplitudes of structural profile, zerothmode average atomic density, and system concentration field. Effects of noise (corresponding to stochastic amplitude equations) and species-dependent atomic mobilities are also incorporated in this formalism. Results of a sample application to the study of surface segregation and interface intermixing in alloy heterostructures and strained layer growth are presented, showing the effects of different atomic sizes and mobilities of alloy components. A phenomenon of composition overshooting at the interface is found, which can be connected to the surface segregation and enrichment of one of the atomic components observed in recent experiments of alloying heterostructures.
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