Major categories: - UCI Mathematics



CDI: Development and validation of novel computational tools for modeling the growth and self assembly of crystalline nanostructures

CDI Type II.

Themes: Understanding complexity in natural, built and social systems (primary), From data to knowledge (secondary)

M. Asta (co-PI), Dept. Chem. Eng. Mater. Sci., UC Davis

J. Lowengrub (PI), Dept, Math, UC Irvine

K. Thornton (co-PI), Dept. Mater. Sci., U. Michigan

P. Voorhees (co-PI), Dept. Mater. Sci. Eng., Northwestern U.

1. Introduction

For a wide variety of applications, ranging from energy harvesting and toxin detection in the environment to biomedicine, nanocrystalline materials promise to yield revolutionary new technologies. The realization of such advances has been to date hindered by the challenges inherent in reproducibly synthesizing nanocrystalline materials with controlled morphologies Critical to the practical application of nanotechnology is the ability to to assemble nanostructures into larger, functional device components. The use of self-assembly as a manufacturing strategy is highly desired as it is very efficient. However, this requires a fundamental understanding of natural tendency of nanostructures to form larger structures and patterns. Because of the complexity of the processes and the difficulties associated with obtaining experimental data, computational modeling is essential. Unfortunately, currently available methods are not well suited to modeling nanoscale systems and self-assembly. For example, atomistic models based on molecular dynamics simulations are limited to very short time and space scales, and continuum models fail to capture the atomic processes that govern the evolution of the phase boundaries (i.e., three-phase contact lines and interfaces between phases). What is required is a computational model that accounts for atomic structures of crystalline materials while at the same time is capable of simulating the interaction of these processes over larger scales that control the self-assembly process.

The phase field crystal (PFC) model [1,2], a technique for simulating crystal growth at atomic spatial scales, but on diffusive time scales, promises to revolutionize computational modeling of nanostructure growth by incorporating atomically accurate physics while at the same time allowing for large-scale self-organization to be modeled. The PFC method has its origins in classical density functional theory, and thus models a solid by a periodically varying order parameter representing the time-averaged atomic density. The model naturally incorporates anisotropies resulting from the crystal structure, elastic and plastic deformations, multiple crystal orientations and crystalline defects. The PFC has been used already to simulate a variety of microstructures controlled by the motion of non-faceted solid-liquid interfaces [2]. However, current models are limited in their application beyond simple systems

In this research we propose to develop new multiscale simulation methodologies for the growth and self-assembly of nanostructures. This will involve developing new PFC models applicable to solid-vapor and liquid-vapor surfaces, as well as faceted solid-liquid interfaces. The model development will involve extensive validation efforts employing large-scale atomistic simulations as benchmarks. To apply these models to realistic growth processes, we will develop efficient and accurate state-of-the-art numerical methods for solving PFC equations. Finally, we will develop methods for coarse-graining PFC models that will yield continuum phase field models that capture the atomic-scale physics, while still being capable of simulating the larger length scale self-assembly of these nanostructures that is a route to usable devices. These methodological advances are critical to enabling simulations of the growth and self-assembly of nanocrystals by complex experimental methods in which liquid-solid, solid-vapor, liquid-vapor interfaces coexist.

2. Appropriateness for CDI Program

To achieve the goals of this research, a transformative new computational methodology will be developed for simulating the growth and self-assembly of nanostructures. This requires innovation and advances in computational thinking well beyond the current state-of-the-art. We will develop new, validated atomic-scale and coarse-grained models, new highly efficient numerical algorithms and new tools for knowledge extraction and visualization. The proposed work is highly interdisciplinary and we have assembled a team with expertise in computational and applied mathematics, and atomistic and continuum materials simulations. This team has a demonstrated and successful record of collaborative research, supported under a previous NSF-EU Cooperative Activity in Computational Materials Science, devoted to multiscale simulation of nanoscale magnetic thin films. The proposed work has the potential to create a paradigm shift in understanding how to integrate nanostructures in usable practical devices and thereby realize of the promise of revolutionary new technologies in applications ranging from energy harvesting and toxin detection in the environment to biomedicine.

3. Background

When the size scale of a material is decreased to the scale of nanometers the electronic properties of the material change. This change in properties can be used to design a device that can operate in a regime that is inaccessible with bulk materials. For example, quantum dots, see figure N, hold the promise of unique detectors and lasers since the electronic structure of these (Germanium, Ge) dots (on a silicon (Si) substrate) is different than in bulk Ge. Quantum dots form during heteroepitaxy, e.g. molecular beam epitaxy, to relieve the stress generated by the misfit between the film and substrate crystal lattices. Atoms diffuse along the surface to relieve the stress causing islands to form. Thus the evolution of the dots takes place on diffusional time scales. The structure of the surfaces is much different from that of the bulk. The lines shown in the substrate surface of Figure N (a) are due to the unique surface reconstructions that occur on the Ge-coated Si surface; namely the structure of the crystal lattice of the Ge surface differs from that of bulk Ge. These surface reconstructions alter the diffusivity of the deposited Ge atoms and thus the evolution of the dots.

Since the properties of the device depend on the size and morphology of the dot the challenge is to control their sizes and morphologies and to produce large numbers of them in a well-controlled manner. While there has been much work on understanding the growth processes of a single dot, a much-neglected issue is the integration of these dots into usable device components. This is why there are very few commercial devices available, for example, that are based on quantum dots in Si or Ge systems. Integration requires control of the location and spacing of the dots. Because this involves large numbers of processes and highly nonlinear dynamics, and is very difficult to approach by experimental methods alone, these processes can only be understood using mathematical models and sophisticated computer simulations in addition and complementary to experimental studies. The computational models must accurately account for the atomic-scale motion and structure of the dots shown in Figure N (a) while still capturing the larger scale self-assembly process at the length scales shown in Figure N (b). Thus, the development and application of a comprehensive computational methodology is critical to making the breakthroughs necessary to make quantum-dot–based devices can become a commercial reality

The PFC model provides an ideal basis for a computational methodology as it is designed to simulate crystal growth at atomic spatial scales, but on diffusive time scales. However, there are a number of significant challenges that we will need to overcome in order to develop such a methodology. For example, there are currently no PFC models that can model the surface reconstructions or the diffusion of atoms along a solid-vapor surface, as discussed above. Can one develop a PFC model that accounts for these effects? In addition, as described in section 4.1 below, interface pinning may occur in some PFC models. Can a PFC model be developed that does not exhibit unphysical interface pinning? Further, while large scale simulations of PFC models may be performed it is unlikely that this approach will allow the resolution of the larger length scales necessary to model the long-range, three-dimensional self-assembly process of the islands. Can a coarse-grained PFC model be developed that does justice to the atomic scale motion of atoms while capturing the large length-scale self-assembly process? Moreover, one promising approach to controlling the self-assembly process of quantum dots is to use patterned surfaces or imbedded stressors to direct the locations where quantum dots form as well as their size and structure [Kiravittaya, Eggleston, Wise, Thornton]. Can such a directed self-assembly process be modeled using the PFC?

Nanowires represent another important nanocrystalline structure. Unlike quantum dots, which are zero-dimensional, nanowires are one-dimensional and thus there is quantum confinement in two directions, but not three as in a quantum dot. Nanowires grown by a vapor-liquid-solid (VLS) process yield approximately cylindrical wires that are 5-100 nm diameter, but 1-10 mm long. The extraordinarily large surface area of these wires allows them to be used, for example, as detectors since the effect of gas adsorption on the electronic properties of the wire scales with the surface area of the wire. Nanowires also have unique electronic properties that make them ideal for use in lasers and as interconnects in advanced electronic circuits and photovoltaic devices. During Si-nanowire growth, Si atoms diffuse through an Au-rich liquid catalyst to a faceted solid-liquid interface, see Figure N. The challenge with making devices using nanowires a reality is not the self-assembly process of many wires, since this is easily controlled by placing the Au catalysts at specific locations prior to growth, but rather in controlling the wire diameter since this alters the electronic properties of the wire. Thus to understand the processes that control the wire diameter, we must extend our computational methodology to accurately simulate VLS nanowire growth.

During Si-nanowire growth, the solid thus grows by the nucleation of a monotomic layer on the interface. The faceted solid-liquid interface joins solid-vapor and solid-liquid interfaces at a trijunction. It is well known that due to the faceted interface the contact angle at the interface is not fixed. Consequently, there is no energy minimization principle that sets the wire diameter [gibbs]. In addition, the nucleation of the monoatomic layer of crystal at the trijunction is thought to control the formation of a twinned microstructure and the crystal structure of III-V and Si nanowires [pic](Koguchi, Kakibayashi et al. 1992; Hiruma, Yazawa et al. 1995; Glas, Harmand et al. 2007; Caroff, Dick et al. 2009). Twinning is a common microstructural defect and corresponds to the formation of a grain boundary with a high degree of symmetry [REF]. The atomic scale nature of the trijuction may be further complicated by the possible existence of an ordered liquid-vapor interface in the Au-Si system [citations in Mark’s section]. Thus, the dynamics of the trijunction are central in setting both the radius of a nanowire and its crystal structure. However, understanding the factors controlling the atomistic scale behavior of these trijunctions requires a method that has atomic-scale resolution and can follow processes on diffusional timescales. Thus, a PFC-based computational methodology is ideal. However, can a PFC model be developed that captures the atomic scale nature of the VLS growth process?

In this proposal, we will address these questions and develop and validate a novel PFC-based computational methodology for simulating solid-vapor and liquid-vapor systems.

4. Proposed Work

The proposed work is roughly divided into five categories: (1) Development of PFC models for solid-vapor and liquid-vapor systems, (2) Validation and refinement of the PFC models using results from atomistic simulations, (3) Development of coarse-grained PFC models (amplitude and phase equations), (4) Development of numerical methods for large-scale simulations and of algorithms for knowledge extraction and visualization, and (5) Capstone simulations of thin film heteroepitaxy and nanowire growth.

4.1 Phase Field Crystal Models: Beyond Swift-Hohenberg (Coordinator: Voorhees)

Interfaces play a critical role during the growth of nanostructures. To predict accurately the evolution of these nanostructures it is thus necessary to develop new PFC models that capture the structure and transport properties of these interfaces. We will focus our attention on the solid-vapor, solid-liquid, and liquid-vapor interfaces that are relevant to quantum dot growth on surfaces and nanowire growth by the vapor-liquid-solid mechanism. Despite this focus, these models will be applicable to broad range of other physical situations and provide new insight into pattern forming systems in general.

Unfortunately, existing PFC models, which are appropriate for certain solid-liquid phase transitions, model the solid as a small perturbation in the density of the liquid. In this approach, currently used PFC models may be derived from a perturbation expansion of classical density functional theory [Elder et al. 2007]. This expansion yields a conserved Swift-Hohenberg-like equation for the density of atoms as a function of position [REF]:

[pic] where [pic]

and ψ is the atomic density of the solid, [pic] is the variational derivative of the PFC energy Fpfc with respect to ψ and a2, l, ko and d, are parameters that fix, for example, the density of the solid and liquid and the lattice parameter of the solid. The term [pic] models the atomic correlation function in the liquid. The PFC equation is a high-order (6th order) nonlinear equation and is highly challenging to solve numerically (see Sec. 4.4).

It is well known that the Swift-Hohenberg equation can yield pinned interfaces when the ordered (crystal) phase grows into a uniform phase (liquid) [pomeau]. This difficulty is thus present in the PFC model. It is particularly acute when epsilon is large, which is the limit necessary to produce faceted interfaces. While pinning can be physically meaningful in the crystal growth case as a manifestation of a finite interfacial mobility, the standard PFC equation requires unphysically large random noise for the interfaces to move. Thus understanding the cause of the pinning and developing a method to control the strength of the pinning will impact the pattern formation community as well as enable physically realistic PFC simulations of nanostructure growth.

Since the PFC equation follows from an assumption that the density difference between solid and liquid is small, it clearly will not be applicable to solid-vapor or liquid-vapor systems where there is a significant density difference between the two phases. The large density difference allows for surface reconstructions of the solid, that is a different arrangement of atoms at the surface compared to that in the bulk. These surface reconstructions result from the manner in which the atoms are bonded and the surface strains that accompany the reconstructions. It is unknown if the physics necessary to yield surface reconstructions can be added to the PFC model. In addition, the liquid-vapor interfaces also may be ordered in the tangent plane to the interface. It is clear that a new PFC model for systems involving a vapor phase needs to be developed.

A new approach to developing PFC models is required so that dynamics of faceted, liquid-vapor and solid-vapor interfaces can be modeled. It is possible that the presence of a very low-density vapor phase can be added to the PFC model by allowing the free energy of the phases to depend on an additional order parameter, h that describes the change in density across the surface (h’0 denotes the vapor phase and h’1 the solid phase). This approach employs a free energy density F,

[pic],

where Fpfc is the usual free energy of the PFC described earlier, g(h) is a smooth interpolating function from the solid or liquid to the vapor,[pic], Fvapor is the free energy of the vapor,

[pic],

and b is a parameter that fixes the curvature of the free energy of a vapor with density ψv. In addition, U is a double-well free energy function, [pic]where Wh is the height of the double well barrier and Wh is a gradient energy coefficient that yields a liquid-vapor interfacial energy. Preliminary results based on this approach are illustrated in Fig. 1(a) and yield the description of the liquid-vapor interface described in the following section. However, an important question is whether this approach captures the correct physics of such interfaces, such as surface reconstructions. This will be assessed and modifications to the PFC model will be introduced as needed.

It is also clearly necessary to develop a PFC model for faceted interfaces that does not exhibit interfacial pinning as a model artifact. One of our approaches will be to use a different description of the atomic correlation function in the liquid, on which the PFC models are based on. This may yield higher order spatial derivatives into the PFC model that while potentially removing the unrealistic pinning would then pose a significant challenge computationally (see section 4.4).

We will explore methods to introduce surface reconstructions in the PFC with the goal of capturing the unique surface reconstructions of diamond cubic materials such as Si or Ge and, possibly, their stress dependence. In developing these models, we will use the atomistic calculations, described below, to provide the required two- and many-body correlation functions, diffusion constants, density relaxation times as well as surface free energy, surface diffusion coefficients and surface structure for both faceted and atomistically rough interfaces. We will compare the step-edge energies computed using the PFC with those computed from the atomistic methods, as well as step dynamics at very high temperatures where the surface diffusion coefficients are large enough to allow for significant step mobility.

4.2 Model Parameterization and Validation via Atomistic Simulations (Coordinator: Asta). Atomistic simulations will be used in this research as a tool to guide development of PFC models for solid-vapor and liquid-vapor surfaces, and for faceted solid-liquid interfaces. Our approach will be similar to that undertaken by Wu et al. 1 in the development and validation of Ginzburg-Landau theories for rough solid-liquid interfaces – bulk properties required to parameterize PFC free-energy models will be computed by molecular dynamics (MD) simulations, along with data for interfacial properties derived from the same underlying interatomic potential. The MD-calculated interfacial properties will provide a direct means for assessing the accuracy of the PFC models, and for suggesting strategies by which they can be improved. The MD simulations will focus on model elemental systems, chosen to probe different interfacial processes, motivated by the targeted application of the PFC models to nanoscale crystal growth.

For liquid surfaces atomistic simulations will focus on structural and thermodynamic properties for elemental metal systems. The structure of liquid surfaces will be characterized by MD, focusing specifically on the strength and range of induced density oscillations normal to the surface, as well as the nature of lateral short-range ordering within the surface layers. In addition to structural properties, we will compute density, energy, stress and diffusion profiles, and associated excess thermodynamic properties, to provide a comprehensive set of data that can be used to test the accuracy of PFC models. Due to the relevance to VLS growth, we will focus initially on surfaces of liquid Au, building on previous classical MD simulations by Celestini et al. 2. In this previous work, a close-packed surface layer with a structure corresponding to a defected two-dimensional triangular lattice was observed to form. A sharp growth of orientational correlation length and time is found with decreasing temperature, consistent with a fluid-hexatic transition about 350 K below the melting point, which is pre-empted by surface-induced crystallization. What ingredients are needed in the PFC models to observe such interfacial phase transitions, and the corresponding precursor structural short-range order? To address this question we will begin by computing the bulk properties required to parametrize a PFC model for the Au potential used in Ref. 2, to enable PFC simulations of the liquid-vapor surface for comparisons with the MD data. As a framework for eventually extending the models to alloys, we will also pursue application of ab-initio MD methods (e.g., [pic]3-5) to liquid Au. The ab-initio simulations will be used initially to test the key predictions of the classical potential model used in Ref. 2; in the later stages of the work these simulations will be applied to Au alloys, specifically eutectic Au-Si mixtures, where a variety of ordered structures have been observed to form at the liquid surface [pic]6-8.

For solid surfaces, atomistic simulations will focus on calculations of temperature-dependent surface free energies, surface stresses, and their associated crystalline anisotropies. Specifically, we will pursue the development of a new atomistic simulation methodology to compute anisotropic surface free energies and the properties of steps, based on the formalism of interfacial thermodynamics developed by Cahn 9. In this approach, demonstrated recently by Frolov and Mishin 10 for a Cu(110) surface, temperature-dependent surface free energies are computed through thermodynamic integration, employing MD-calculated excess energies and surface stresses. Here the technique will be extended in the study of surface orientations vicinal to low-energy facets. Calculations of surface free energies as a function of vicinal angle will enable determination of temperature-dependent facet energies, step free energies, and step interaction energies at temperatures below the roughening temperature. Such simulations will be employed initially for surfaces of a classical model of bcc Fe, for which bulk PFC parameterizations already exist 1. This information will be used to determine what ingredients are needed in a PFC model to quantitatively model the thermodynamic properties of highly anisotropic surface thermodynamic properties. As the PFC methods are extended to more complex crystal structures (such as Si) featuring reconstructed surface structures, the MD simulations will be extended accordingly.

For faceted solid-liquid interfaces, our previous MD simulations of Si (111) solid-liquid interfaces [pic]11, 12 will provide an extensive set of data against which the PFC models can be tested. Our previous work for this faceted interface has involved detailed analysis of interface structure 12, calculations of step kinetic coefficients 11, step free energies and terrace nucleation kinetics 13. Additional simulations for this project will focus on calculations of the bulk static and dynamic correlation functions required to parameterize the PFC models for this diamond-cubic structured material. The atomistic work will thus aim to guide the development of PFC methods capable of modeling the kinetics of faceted interfaces.

Finally, MD simulations will be used as a framework to perform “numerical experiments” of nanoscale crystal growth, building on our previous NSF-supported research. An example is shown in Fig. xx, taken from MD simulations of nanowire growth from a model of elemental Si 14, which establish a capillarity-induced size dependence for the nucleation barrier underlying the formation of new crystal terraces in a layer-by-layer growth mode. Such simulations will be extended to consider the kinetic processes at the solid-liquid interface underlying the growth of twinned nanowires, and the formation of stacking faults during growth. In these simulations the focus will be on understanding the thermodynamic and kinetic processes underlying crystal growth from a liquid with nanometer-scale dimensions, including potential size effects. Such information will be used to guide the formulation of quantitative PFC models for nanoscale crystal growth.

4.3 Density-Amplitude Formulations for Solid-Vapor and Liquid-Vapor Models (Coordinator: Thornton)

Although we will develop new, highly efficient algorithms for performing direct numerical simulations of the atomistic PFC models (see Sec. 4.4), it is still unlikely that this will be sufficient to achieve large enough length scales—e.g. micron scales in three-dimensions—needed to accurately capture the self-assembly process and thus provide a route to the development of usable devices. Thus, in addition to using high performance numerical algorithms, we propose to also take a complementary approach toward enabling large-scale simulations faithful to atomic-scale processes.

The PFC model can be extended to larger spatial scales by coarse-graining to obtain equations for the amplitude of the atomic-scale periodic variation and the local average of the atomic density field. The density-amplitude coarse-grained model will not only be computationally efficient; we also expect that it will provide a foundation for building a new phase field model that incorporates atomic phenomena accounted for in the PFC models. This would allow, for example, crystalline effects such as anisotropy, strain, and defects such as dislocations and grain boundaries, to be naturally captured in a continuum phase field model. The development of a new phase field model that includes phenomena stemming from the atomic scale without resolving all of the atoms, could transform the field of phase field modeling.

In preliminary work as part of prior NSF support we have performed a heuristic multiple-scale analysis [8,9] of the PFC equations for solid/liquid systems where density variations are small [7]. This yields a conserved equation for the average density field [pic] and a non-conserved equation for the amplitudes of the atomic-scale periodic variation of the density [pic], where [pic] specifies the corresponding wave vector [pic]. One of the ways to obtain the equations involves applying a single-mode approximation to the free energy expressions of the PFC model. The density-amplitude equations are capable of simulating length scales relevant to nanostructures and beyond. The PFC atomic density can then be reconstructed by [pic] to describe the evolution at the atomic scale [8,9]. A result for 3D crystallite growth in a liquid is shown in Fig. XXX1. More than 65,000 atoms are simulated using this coarse-grained model. In Fig. XXX2, the reconstructed atomic density from the coarse-grained model is compared to the full PFC solution. There is very good agreement between the two. The main differences between the coarse-grained and full PFC results stem from the fact that the relaxation of the lattice near the solid-liquid interface cannot be exactly represented by a single-mode approximation. Interestingly, it was found that Gibbs-Thomson condition, which is important in determining the dynamics of the interface, is accurately represented by the coarse-grained model [7].

Although these preliminary results are very promising, there are significant mathematical challenges associated with justifying the heuristic multiple-scale analysis. For example, the analysis involves performing expansions where uncontrolled approximations are made and certain terms thought to be small are dropped even though formally they occur at the same order in the expansion as terms that are kept. A mathematical theory that validates this approach is needed.

Here we propose to develop a coarse-grained model (density-amplitude equations) for the new solid-vapor and liquid-vapor systems by performing multiscale expansions and/or variational analyses of the new PFC model described in Sec. 4.1. A preliminary analysis of a possible model described in Section XXXPWV has already been performed to examine whether the structure of liquid surfaces are captured in multiscale analysis of the three-phase PFC model. By applying the single-mode approximation and multiscale analysis to the equations presented in Sec. 3.1, we have

[pic]

and obtain the governing equation for the slowly varying amplitude,

[pic],

for the equilibrium profile of the planar liquid-vapor interface. Here, [pic]is the average density in the liquid, [pic], [pic]and [pic]are rapidly and slowly varying spatial coordinates, respectively, and other variables have been defined in Sec. 3.1. The solution can be obtained analytically, yielding

[pic].

This equilibrium solution predicts density oscillations normal to the surface at liquid-vapor interface, as shown in Fig. XXX3. As mentioned in Section 4.2, such a structure has been observed in MD simulations [Celestini] and can affect the properties of the interface. The rate of the decay of oscillations with depth was determined to be a function of parameters that enters into the free energy functional, indicating that the emergence of this phenomenon is clearly dependent on the material and the condition. Does this prediction in fact reflect the physics underlying liquid-vapor interface? If so, does the prediction quantitative and how do material properties alter the prediction? Can the model provide insights into phenomena such as prefreezing of nanoscaled liquid droplets and how freezing proceed in such systems? We will address these questions.

In addition, the density-amplitude formulation will allow studies of atomic structures that are different from the body centered cubic structure that is stable as formulated in the PFC model [1]. By choosing Fourier bases for expansion of the PFC atomic density such that structures like the face centered cubic are represented, we will develop a model for a wider variety of crystal structures. These models will be just as efficient as the body-centered-cubic model because the approach does not require introduction of higher-order derivatives. Such models will be compared to the corresponding full PFC models to quantify the changes in the equilibrium and dynamics predicted by the new models.

Finally, we will consider further improvements of the density-amplitude formulation by including higher harmonics. This is likely to be important in accurately representing faceted structures (at very low temperatures), and atomic structures near surfaces or interfaces, or when large strains exist.

[pic]

[pic]

Fig. XXX2: Comparisons between the order parameter calculated by the original PFC model (PFCM, solid line) and the order parameter reconstructed from the average density and amplitude (MPFC-RG, dotted line). The difference is mainly due to the single-model approximation, in which only one periodicity is considered.

[pic]

Fig. XXX3: The equilibrium profile of the two order parameters describing vapor and liquid (η ) and the atomic probability density (φ ) at a liquid surface. The atomic probability density clearly exhibits oscillation, showing a structure forming in the liquid due to the presence of surface. The result indicates that the new vapor/solid/liquid PFC model capture an important, but subtle aspect of liquid surface. (FIGURE NEEDS TO BE REMADE TO MAKE THE FONT BIGGER.)

4.4 Numerical Methods for Large-Scale Simulations, Data Reduction and Visualization (Coordinator: Lowengrub): We propose to develop new adaptive, parallel algorithms to solve the phase-field crystal (PFC) equations. Because the equations are high-order in space, this is a formidable computational challenge. The standard PFC models are 6th order in space and the use of new atomic correlation functions in the liquid, or taking into account different crystal lattice symmetries (e.g. FCC instead of BCC), may introduce even higher derivatives. Thus, explicit methods have severe time step constraints for stability [1, Kim, Hirouchi, Achim, Xu] and are problematic for large-scale simulations, particularly in three dimensions. Semi-implicit algorithms have recently been developed for solid-liquid systems using pseudo-spectral [11, Ohnogi, Mellenthin, Tegze], finite-element [12] and finite-difference [13,14] methods. In semi-implicit algorithms, however, care needs to be taken to handle the discretization of nonlinear and backwards-diffusive terms as these can introduce limiting time step restrictions in spite of the implicit discretization. For example, the incorporation of backwards-diffusive terms in the integrating factor approach in [Mellenthin] may significantly amplify low-to-moderate wavenumbers at large time steps and introduce accuracy and possibly stability related time step restrictions. The implicit treatment of backwards-diffusive terms in [12] is expected to lead to time step restrictions for solveability of the implicit system. The treatment of nonlinear terms in the operator-splitting algorithm [Tegze], recently developed for a binary version of the PFC equation where the components have similar densities [Elder07], is found to limit the time steps required for stability. In addition, filtering is used at high wavenumbers to control numerical instability.

In recent work, we have developed energy-stable semi-implicit finite difference methods [13,14] that ensure that the discrete form of the energy is non-increasing for any time step while at the same time ensuring that the (nonlinear) equation at the implicit time level is uniquely solveable. The algorithms rely on splitting the PFC energy Fpfc =Ec-Ee where Ec and Ee are both convex functionals. For example, if a2>1/12, we may take [pic] and [pic]. We may use the scheme [14]:

[pic],

where dψ denotes the variational derivative with respect to y and centered finite difference approximations are used for the spatial derivatives with finite sums replacing the continuous energy integrals. This scheme satisfies [pic]and is uniquely solveable for yn+1. The above splitting is not unique and other splittings could be used (e.g., [Xu]) although the above approach has the advantage that the coefficients in the splitting scheme are independent of the solution unlike the scheme described in [Xu]. The above algorithm is related to schemes developed earlier [Eyre] although we have developed a more direct and general approach. Energy-stability guarantees the unconditional stability and convergence of the numerical schemes [14]. In addition, the well-posedness of the PFC equation may also be established by obtaining a priori estimates for solutions of an analogous spatially-continuous, temporally-discrete scheme [14] and taking the limit Dt to 0. While the scheme above is first-order accurate in time, we have also developed a second-order unconditionally stable and solveable version [13] that is weakly energy stable such that [pic].

Here, we propose to develop second-order accurate, finite-difference methods that are energy-stable for solid-vapor and liquid-vapor PFC systems where there are large differences in density. The challenge is to develop a more general framework by which discrete versions of the binary PFC energy, which is more complicated than the simple version given above, can be analyzed and split into differences of convex components. The system of discrete equations for the two order parameters will involve gradients of convex functions. In addition, the nonlinear equations at the implicit time level should represent the gradient of a strictly convex function so that they are uniquely, and unconditionally solvable [13,14]. Another advantage of this approach is that it not limited to periodic boundary conditions. We will prove the stability and convergence of the numerical algorithms. In addition, the existence of solutions to the continuous time and space binary PFC equations can also be addressed using this framework [14].

To perform large-scale simulations, we will use adaptive, block-structured Cartesian spatial mesh refinement [15] (see Fig. 1d) where adaptivity is expected to be useful in allowing a coarse mesh description of the vapor layer. In addition, we will also implement an adaptive time stepping algorithm. To solve the nonlinear equations at the implicit time level, we will use an optimal-order nonlinear multilevel multigrid method [Trottenberg,15]. We note that this approach involves only local linearizations of the implicit equations and is typically much more efficient at solving nonlinear systems than algorithms involving global (Newton-type) linearization methods. In addition, block-structured Cartesian mesh refinement is an ideal framework for applying the nonlinear multigrid method in a locally-refined, multi-level way (e.g. [Brandt, 15]). This yields efficient nonlinear solvers which is critical as the computational cost of the solvers dominates the algorithm.

Once we have developed an efficient serial solver, we plan to develop a parallel implementation to gain further efficiency. In general, the development of a parallel adaptive mesh refinement (AMR) method requires a significant software infrastructure because each processor computes only within a partition of the full domain and thus substantially affects the set-up, solution and communication among neighboring meshes and processors. However, to begin we propose to use an adaptive full-domain covering approach to develop an efficient parallel version of an existing sequential code. An advantage of this approach is that it provides a straightforward way to parallelize serial codes without incurring the overhead needed by traditional parallelization approaches. There are many ways of doing this. One approach, developed by Bank and Holst [Bank] and later refined by Mitchell [Mitchell], and Vey and Voigt [Vey], is to use domain decomposition to divide the domain into regions of approximately equal load balance. Each processor solves the full problem on the whole domain and adaptively coarsens subdomains corresponding to other processors but refines its own subdomain as well as some overlap region with neighboring domains. During each iteration, or time step, the load balance quality and repartitioning overhead can assessed to determine whether the domain should be repartitioned after the refinement of the subregions. Finally, the global solution is constructed on the composite mesh that consists of all the refined partitions from all the processors. This can be done either by using a parallel multigrid solver or a partition of unity method. Up to now, this algorithm has been applied to finite element methods. We will adapt the algorithm to the finite difference context.

While the adaptive full domain covering approach is expected to give good speed-ups, it may prove necessary to develop even more efficient parallel algorithms. If this is the case, we will also explore the use of other more traditional parallel approaches, such as recent algorithm developed by Henshaw & Schwendeman [Henshaw] using domain decomposition and overlapping grids with AMR, combined with existing infrastructure codes such as CHOMBO [Colella], which uses the block-structured Cartesian mesh adaption approach that we adopt here. Another interesting approach we will consider is the development parallel implementations specialized to shared memory architectures such as multi-core/multi-thread CPUs. To gain further experience, the postdoctoral research and students will attend parallel computing workshops held regularly at the San Diego Supercomputer Center.

In addition, the adaptive, parallel algorithms described above for the PFC will be modified to provide efficient simulation tools for the coarse-grained amplitude and phase equations that describe evolution on a larger scale. Here, adaptivity will play a very significant role in reducing computational cost [9]. To provide benchmark simulations, we will also develop parallel implementations of pseudo-spectral and explicit finite-difference algorithms currently used [7].

Because a large amount of data will be generated, new methods will be developed to visualize and interpret the results. In particular, using image-processing techniques (e.g., [18]) we will automatically extract the solid-vapor and liquid-vapor interfaces to quantify their dynamics as well as those of steps and kinks. We will also extract step-edge energies, grain boundaries, atomic scale twinning, dislocations, and the dynamic stress state. We will accomplish this by extending and generalizing the approach in [18] where grain boundaries and the stress state were determined from 2D simulation results of the PFC equation at a fixed time, as well as TEM experimental data, by minimizing a Mumford-Shah [Mumford,Chan] type functional for grain orientations and the deformation. In particular, we will extend the functional to incorporate the dynamics of the processes, as well as defects, variable lattice spacing and type, as well as the large density differences between the phases, which will enable the simultaneous extraction of the solid-vapor and liquid-vapor interfaces. We will also consider extensions of the approach to three dimensions.

4.5 Applications to Thin-Film and Nanowire Growth (Coordinator: Lowengrub): NEEDS TO BE UPDATED. Multiphase systems are commonly used during the processing of nanomaterials. Solid-vapor systems arise during thin film growth wherein atoms of one material are evaporated onto a substrate of another material. During thin film growth dots, or islands, form to reduce the strain energy that is generated by the difference in lattice parameters between the film and substrate. The objective is to control the size, shape and location of the (quantum) dots or islands. The atomic scale nature of the process is essential in understanding the dynamics of the island evolution. For example, the evolution of the islands can be altered by the arrangement of the atoms at the surface. Furthermore this arrangement can be a function of the stress at the surface. In addition, the stress can be sufficiently large to nucleate dislocations. All of these phenomena involve processes at the atomic scale, yet are linked to the much longer length scale elastic and diffusion fields. Thus following the evolution of quantum dots on surfaces demands a method that can both capture atomic scale events while still enabling the long-range fields to be computed that evolve on diffusive time-scales. Finally much of these phenomena are inherently three-dimensional, such as capturing the correct dislocation structure.

Another nanoscale materials processing method is the Vapor-Liquid-Solid (VLS) process that is used to grow nanowires. In this method a nanowire grows by atoms descending from the vapor onto a liquid droplet that sits atop a solid nanowire. The atoms then diffuse through the liquid to the solid-liquid interface causing the solid to grow. The result is a wire that is 1-10 μm long and 5-100 nm in diameter. Thus, unlike the quantum dot growth process, it is necessary to consider three phases instead of two. A further complication is that the interface between the liquid droplet and solid is usually faceted. The facetted solid-liquid interface moves by the diffusive motion of atomic-height steps. Steps may nucleate at the liquid-vapor-solid trijunction, thus it is only through a model of the atomic scale nature of this trijunction that an accurate model of the nucleation process can be developed. Finally, there is evidence that the atoms in the liquid droplet may be ordered near the liquid-vapor interface. This atomic scale ordering, which should be describable using a PFC model, could affect the diffusion process of atoms through the liquid-vapor interface.

Describing either VLS growth or stress-driven island formation requires predictive multiscale models that capture both atomic scale processes and the long-range fields that evolve on diffusive time scales. This will require a combination of new PFC models, coarse-grained models, state-of-the-art algorithms, and parallel computations.

5. Broader impacts, education and outreach activities

Broader Impacts to Science and Engineering: The development of phase field models that incorporate atomistic features and crystal defects, and yet are able to simulate processes on the diffusive time scales, will improve our ability to predictively model crystal growth from liquid and vapor phases, and provide a framework to dramatically enhance our fundamental understanding of how elasticity, plasticity, and crystal defects control growth morphologies at the nanoscale. This will have a significant impact on the development of novel devices for diverse applications, such as advanced optoelectronic and magnetic storage units, photovoltaics and catalysts. The proposed numerical methods will transform the use of high performance computing in materials science and will impact the development of numerical methods for studying other systems such as emulsions and biological tissues. The results will be widely disseminated through presentations and publications, as well as the proposed workshop on phase field crystal models.

Education and Outreach: The grant will yield broader impacts by educating the future scientists and engineers. Courses on crystal growth for high school students are proposed as part of the California State Summer School for Mathematics and Science (COSMOS) at UC Irvine and Davis. Outreach activities are planned at the Science Safari at the Ann Arbor Hands-On Museum. This will help to recruit new math and science majors and enhance the participation of high school students in research. Graduate students and a postdoctoral researcher will also receive interdisciplinary training. These researchers will present their findings at conferences, thus enhancing their participation in professional societies. The PIs will make every effort to recruit undergraduate and graduate students with diverse backgrounds. The proposed work also broadens the participation of underrepresented groups as one of the PIs and several of the students involved in the project will be women.

[pic]

Fig. XXX. Outreach activities enabled by the prior NSF support. Under the NSF-EC funding, the team held several workshops at the Ann Arbor Hands-On Museum and two summer schools on crystal growth for high school students, as well as developing modules for these activities.

6. Mentoring Plan.

One postdoctoral researcher and four graduate students will be funded by this project. Their development will be enhanced through a program of structured mentoring activities. The goal of the mentoring program will be to provide the skills, knowledge and experience to prepare the postdoctoral researcher and students to excel in their career paths. To accomplish this goal, our mentoring plan will follow the guidance of the National Academies of Science and Engineering on how to enhance their experiences by providing a structured mentoring plan, career planning assistance, and opportunities to learn a number of career skills such as writing grant proposals, teaching students, writing articles for publication and communication skills [NAS]. Specific elements of the mentoring plan will include:

• Working with the postdoc and students to establish and implement an Individual Development Plan that identifies both professional development needs and career objectives.

• Providing opportunities to network with visiting scholars who are leaders in the field by having lunch or dinner with them when they participate in the PI department’s seminar series.

• Encouraging participation in a journal club (for graduate students and postdocs), in which participants meet weekly, along with a faculty facilitator, to discuss and critique recent journal articles in the field and to discuss how to write and submit journal articles.

• Enabling travel to at least one conference each year (travel funds are included in the budget), with the goal of presenting a poster or paper at the conference.

• Enabling travel to each of the PI institutions to interact with other students and PIs to forge long-lasting collaborations, gain experience in different research environments and to better integrate the research components proposed in the project. The postdoc will spend month-long visits at Northwestern (years 1 and 3) and Michigan (years 2 and 4).

• Participating in each PI’s weekly research group meetings and in periodic tele- and video- conferences involving all the PIs, in which members will be expected to present their research regularly, and feedback and coaching will be given to help all members to develop their communication and presentation skills.

• Attending career planning workshops and seminars offered regularly by each of the PI institutions.

In addition, the postdoc-specific mentoring activities will include

• Attending seminars, workshops and individual consultations on how to identify research funding opportunities and write competitive proposals. Such programs are offered regularly in the Graduate Division at the University of California, Irvine.

• Participating in seminars and workshops on teaching and learning, as well as access to a teaching mentoring program, conducted by the Teaching, Learning & Technology Center at the University of California, Irvine.

Success of this mentoring plan will be assessed by tracking the progress of the postdoctoral researcher and graduate students through their Individual Development Plans, interviews to assess satisfaction with the mentoring program, and tracking of the postdoc and students’ progress toward their career goals after finishing the project.

7. Results from Prior Support

NSF-EC Cooperative Activity in Computational Materials Research: Bridging Atomistic to Continuum - Multiscale Investigation of Self-Assembling Magnetic Dots During Epitaxial Growth 8/1/05-7/31/10 (in no-cost extension), Total amount funded: $1,218,000. PI: Katsuyo Thornton, co-PIs: Mark Asta, John Lowengrub, Peter Voorhees (with EC counterparts: Axel Voigt, Tapio Ala-Nissila, Miroslav Kotrla, Olivier Fruchart)

Funded as a part of the NSF-EC Cooperative Activity in Computational Materials Research program, this grant aims to advance multiscale computational materials science through development of a suite of tools for modeling directed self-assembly of nanoscale magnetic dots. The researchers have developed mathematical and computational tools that integrates multiple length/time scales and examined issues associated with controlled growth of metallic thin films and dots. Examples of results relevant to the proposed work were presented throughout this proposal, including Figs. XXX. The results are disseminated through presentations and publications [REFS]. Broader impact of the project includes the summer school for talented high school students and activities at Ann Arbor Hands-On Museum (see Fig. XXX).

NIRT: Multiscale Modeling of Nanowire Growth: From Atoms to Wires 9/1/05-8/31/09, Total amount funded: $1,100,000. PI: Peter Voorhees, coPIs: Mark Asta, Stephen Davis, Alexander Golovin, Lincoln Lauhon

The objective of this research is to develop and test theories describing the vapor-liquid-solid (VLS) approach to nanowire growth. Comprehensive investigations of the mechanisms and processes that govern nanowire growth by the VLS method are underway within the framework of a multiscale-modeling approach, closely coupled with experimental synthesis and characterization efforts providing the framework for validation of the theoretical models. The research addresses outstanding issues in VLS nanowire growth that are relevant to the realization of nanowire device technology, such as: (i) the factors controlling growth rate and growth direction of the wire [1-4] (ii) the mechanisms underlying the formation of complex branched nanowire morphologies [5-7], and (iii) factors controlling composition of alloy nanowires [8-13]. An example of the work is presented in Fig. X.

8. Management Plan (NEEDS EXPANDING-move to the end?): Each task will be performed in active collaboration of multiple PIs. Lowengrub, the directing PI, will oversee the overall project, as well as task 4 described above, while each PI will manage one task. Voorhees, an expert on phase-field modeling of solid-vapor systems, will oversee task 1. Asta, who is an expert in atomistic modeling, will manage the parametrization- and validation-related task 2. Thornton, who developed a new amplitude formulation for solid-liquid systems, will coordinate the amplitude formulation effort, task 3. All of the PIs will collaborate on the applications described in task 5, which will be overseen by Lowengrub. The team has a track record of collaboration, demonstrated by over 15 co-authored publications ADD REFs. The PIs and their students will gather annually at a selected institution, which will be rotated. We will continue to take advantage of collaborative tools such as video conferencing, as well as major professional society meetings for additional face-to-face interactions. Finally, a workshop will be held in conjunction with SIAM Conference on Mathematical Aspects of Materials Science in 2012, which will also facilitate dissemination of the results from the grant.

Make role of students and postdocs more precise.

References:

[1] K.R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystal, Phys. Rev. E 90:051605, 2004.

[2] N. Provatas, J.A. Dantzig, B. Athreya, P. Chan, P. Stefanovic, N. Goldenfeld and K.R. Elder, Using the phase-field crystal method in the multiscale modeling of microstructure evolution, JOM, 59:83, 2007.

[3] K.-A. Wu, A. Karma, J. J. Hoyt and M. Asta, “Ginzburg-Landau theory of crystalline anisotropy for bcc-liquid interfaces,” Phys. Rev. B 73, 094101 (2006).

[4] T. Frolov and Y. Mishin, “Temperature dependence of the surface free energy and surface stress: an atomistic calculation for (110) Cu,” Phys. Rev. B (submitted).

[5] D. Buta, M. Asta and J. J. Hoyt, “Kinetic Coefficient of Steps at the Si(111) Crystal-Melt Interface from Molecular Dynamics Simulations,” J. Chem. Phys. 127, 074703 (2007). D. Buta, M. Asta and J. J. Hoyt, “Structure and Dynamics of a Faceted Crystal-Melt Interface,” Phys. Rev. E 78, e031605 (2008)

[6] T. Haxhimali, D. Buta, J. J. Hoyt, P. W. Voorhees and M. Asta, in preparation.

[7] Yeon, D. H., Elder, K. R., Huang, Z. F. & Thornton, K. Amplitude-Based Phase-Field Crystal Model for Liquid-Solid Systems with Miscibility Gap. Submitted to Philosophical Magazine.

[8] Goldenfeld, N., Athreya, B. P. & Dantzig, J. A. Renormalization group approach to multiscale modelling in materials science. J. Stat. Phys., 125, 1019-1027 (2006).

[9] Athreya, B. P., Goldenfeld, N., Dantzig, J. A., Greenwood, M. & Provatas, N. Adaptive mesh computation of polycrystalline pattern formation using a renormalization-group reduction of the phase-field crystal model. Phys. Rev. E 76, 056706 (2007).

[10] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys., 49:435, 1977.

[11] M. Cheng and J.A. Warren, An efficient algorithm for solving the phase-field crystal model, J. Comput. Phys. 227:6241, 2008.

[12] R. Backofen, A. Ratz and A. Voigt, Nucleation and growth by a phase field crystal model, Phil. Mag. Lett., 87:813, 2007.

[13] Z. Hu, S.M. Wise, C. Wang, and J.S. Lowengrub, Stable and efficient finite difference nonlinear multigrid schemes for the phase field crystal equation, J. Comput. Phys., in review.

[14] S.M. Wise, C. Wang, and J.S. Lowengrub, An energy and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., in review.

[15] S.M. Wise, J.S. Kim and J.S. Lowengrub, Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method, J. Comput. Phys., 226:414, 2007.

[16] R.E. Bank and M. Holst, A new paradigm for parallel adaptive meshing algorithms, SIAM Rev., 45:291, 2003.

[17] W.D. Henshaw and D.W. Schwendeman, Parallel computation of three-dimensional flows using overlapping grids with adaptive mesh refinement, J. Comp. Phys., 227:7469, 2008.

[18] B. Berkels, A. Ratz, M. Rumpf, and A. Voigt, Extracting grain boundaries and macroscopic deformations from images on atomic scale, J. Sci. Comput., 35:1, 2008.

Additional References (from Peter)

Gibbs: J.W. Gibbs, The Scientific Papers of J. Williard Gibbs, Dover publications, 1961.

Pomeau: Y. Pomeau, Physica D, V. 23 735 (1986).

Perea – 90: Daniel E. Perea, Eric R. Hemesath, Edwin J. Schwalbach, Jessica L. Lensch-Falk, Peter W. Voorhees and Lincoln J. Lauhon, Nature Nanotechnology, online March 2009

Caroff, P., K. A. Dick, et al. (2009). "Controlled polytypic and twin-plane superlattices in III-V nanowires." Nature Nanotechnology 4(1): 50-55.

Semiconductor nanowires show promise for use in nanoelectronics, fundamental electron transport studies, quantum optics and biological sensing. Such applications require a high degree of nanowire growth control, right down to the atomic level. However, many binary semiconductor nanowires exhibit a high density of randomly distributed twin defects and stacking faults, which results in an uncontrolled, or polytypic, crystal structure. Here, we demonstrate full control of the crystal structure of InAs nanowires by varying nanowire diameter and growth temperature. By selectively tuning the crystal structure, we fabricate highly reproducible polytypic and twin-plane superlattices within single nanowires. In addition to reducing defect densities, this level of control could lead to bandgap engineering and novel electronic behaviour.

Glas, F., J. C. Harmand, et al. (2007). "Why does wurtzite form in nanowires of III-V zinc blende semiconductors?" Physical Review Letters 99(14): 4.

We develop a nucleation-based model to explain the formation of the wurtzite phase during the catalyzed growth of freestanding nanowires of zinc blende semiconductors. We show that in vapor-liquid-solid nanowire growth, nucleation generally occurs preferentially at the triple phase line. This entails major differences between zinc blende and wurtzite nuclei. Depending on the pertinent interface energies, wurtzite nucleation is favored at high liquid supersaturation. This explains our systematic observation of zinc blende during early growth of gold-catalyzed GaAs nanowires.

Hiruma, K., M. Yazawa, et al. (1995). "GROWTH AND OPTICAL-PROPERTIES OF NANOMETER-SCALE GAAS AND INAS WHISKERS." Journal of Applied Physics 77(2): 447-462.

Koguchi, M., H. Kakibayashi, et al. (1992). "CRYSTAL-STRUCTURE CHANGE OF GAAS AND INAS WHISKERS FROM ZINCBLENDE TO WURTZITE TYPE." Japanese Journal of Applied Physics Part 1-Regular Papers Short Notes & Review Papers 31(7): 2061-2065.

Crystal structures of GaAs and InAs whiskers grown by metalorganic vapor phase epitaxy are evaluated by means of a transmission electron microscope. The whiskers are grown epitaxially on GaAs substrates with diameters of 20-100 nm and lengths of 1-5 mum. They have the following characteristics. 1) GaAs whiskers have layered structures with 2-30 nm period, that are the 111 rotating twins of the zinc-blende type. 2) InAs whiskers also have layered structures which consist of wurtzite and zinc-blende type crystals. The wurtzite type InAs is observed for the first time in this study. The volume ratio of these two types strongly depends on the growth conditions, such as substrate temperature and material gas pressure. This suggests that defect-free whiskers with a single phase that are useful for quantum wire devices can be grown by controlling the growth conditions.

Additional refs from Mark:

1. Wu, K. A.; Karma, A.; Hoyt, J. J.; Asta, M., Ginzburg-Landau theory of crystalline anisotropy for bcc-liquid interfaces. Phys. Rev. B 2006, 73 (9), 7.

2. Celestini, F.; Ercolessi, F.; Tosatti, E., Can liquid metal surfaces have hexatic order? Phys. Rev. Lett. 1997, 78 (16), 3153-3156.

3. Fabricius, G.; Artacho, E.; Sanchez-Portal, D.; Ordejon, P.; Drabold, D. A.; Soler, J. M., Atomic layering at the liquid silicon surface: A first-principles simulation. Phys. Rev. B 1999, 60 (24), 16283-16286.

4. Walker, B. G.; Marzari, N.; Molteni, C., Layering at liquid metal surfaces: Friedel oscillations and confinement effects. J. Phys.-Condes. Matter 2006, 18 (19), L269-L275.

5. Walker, B. G.; Marzari, N.; Molteni, C., In-plane structure and ordering at liquid sodium surfaces and interfaces from ab initio molecular dynamics. J. Chem. Phys. 2007, 127 (13), 8.

6. Shpyrko, O. G.; Streitel, R.; Balagurusamy, V. S. K.; Grigoriev, A. Y.; Deutsch, M.; Ocko, B. M.; Meron, M.; Lin, B. H.; Pershan, P. S., Surface crystallization in a liquid AuSi alloy. Science 2006, 313 (5783), 77-80.

7. Shpyrko, O. G.; Streitel, R.; Balagurusamy, V. S. K.; Grigoriev, A. Y.; Deutsch, M.; Ocko, B. M.; Meron, M.; Lin, B. H.; Pershan, P. S., Crystalline surface phases of the liquid Au-Si eutectic alloy. Phys. Rev. B 2007, 76 (24), 9.

8. Halka, V.; Streitel, R.; Freyland, W., Is surface crystallization in liquid eutectic AuSi surface-induced? J. Phys.-Condes. Matter 2008, 20 (35), 4.

9. Cahn, J., In Interface Segregation, Johnson, W.; Blakely, J., Eds. American Society for Metals: Metals Park, OH, 1979.

10. Frolov, T.; Mishin, Y., Temperature dependence of the surface free energy and surface stress: An atomistic calculation for Cu(110). Phys. Rev. B 2009, 79 (4), 10.

11. Buta, D.; Asta, M.; Hoyt, J. J., Kinetic coefficient of steps at the Si(111) crystal-melt interface from molecular dynamics simulations. J. Chem. Phys. 2007, 127 (7), 10.

12. Buta, D.; Asta, M.; Hoyt, J. J., Atomistic simulation study of the structure and dynamics of a faceted crystal-melt interface. Phys. Rev. E 2008, 78 (3), 11.

13. Buta, D.; Asta, M., In preparation. 2009.

14. Haxhimali, T.; Buta, D.; Asta, M.; Hoyt, J. J.; Voorhees, P. W., Size-dependent nucleation kinetics at non-planar nanowire growth interfaces. In Physical Review Letters (submitted). 2009.

Additional refs from John

T. Hirouchi, T. Takaki and Y. Tomita, Development of numerical scheme for phase field crystal deformation simulation, Comp. Mater. Sci. 44 (2009) 1192-1197.

C.V. Achim, A.P. Ramos, M. Karttunen, K.R. Elder, E. Granato, T. Ala-Nissila and S.C. Ying, Nonlinear driven response of a phase-field crystal in a periodic pinning potential, Phys. Rev. E 79 (2009) 011606.

G. Tegze, G. Bansel, G.I. Toth, T. Pusztai, Z. Fan and L. Granasy, Advanced operator splitting-based semi-implicit spectral method to solve the binary phase-field crystal equations with variable coefficients, J. Comput. Phys. 228 (2009) 1612-1623.

Y.-C. Xu and B.-G. Liu, Phase field crystal modeling of the (2x1)-(1x1) phase transitions of the Si(001) and Ge(001) surfaces, J. Phys. D: Appl. Phys. 42 (2009) 034202.

H. Ohnogi, and Y. Shiwa, Instability of spatially periodic patterns due to a zero mode in the phase field crystal equation, Physica D 237 (2008) 3046-3052.

J. Mellenthin, A. Karma and M. Plapp, Phase-field crystal study of grain boundary pre-melting, Phys. Rev. B 78 (2008) 184110.

J.H. Kim and S.H. Garofalini, Modeling microstructural evolution using atomic density function and effective pair potentials, Phys. Rev. B 78 (2008) 144109.

K.R. Elder, N. Provatas, J. Berry, P. Stefanovic and M. Grant, Phase field crystal modeling and classical density functional theory of freezing, Phys. Rev. B 75 (2007) 064207.

C. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal. 44 (2006) 1759=1779.

D. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation. In J.W. Bullard, R. Kalia, M. Stoneham and L.Q. Chen, editors, Computational and Mathematical models of Microstructural Evolution, volume 53, pp. 1686-1712, Warrendale, PA, USA, 1998, Materials Research Society.

U. Trottenberg, C. Oosterlee and A. Schuller, Multigrid, Academic Press, New York, 2001.

Achi Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 (1997) 333-390.

S. Vey and A. Voigt, Adaptive full domain covering meshes for parallel finite element computations, Computing 81 (2007) 53-75.

F.W. Mitchell, Parallel adaptive multilevel methods with full domain partitions, Appl. Num. Anal. Comput. Math. 1 (2004) 36-48.

W.D. Henshaw and D.W. Schwendeman, Parallel computation of three-dimensional flows using overlapping grids with adaptive mesh refinement, J. Comput. Phys. 227 (2008) 7469-7502.

P. Colella, D.T. Graves, T.J. Ligocki, D.F. Martin, D.B. Serafin, and B. Van Straalen, CHOMBO software package for AMR applications design document. Technical Report, Lawrence Berkeley National Laboratory.

D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math 42 (1989) 577-586.

T.F. Chan and L.A. Vese, Active contours without edges, IEEE Trans. Image Process. 10 (2001) 266-277.

References.(Peter prior support)

1. Roper, S.M., et al., Steady growth of nanowires via the vapor-liquid-solid method. Journal of Applied Physics, 2007. 102(3): p. 7.

2. Golovin, A.A., S.H. Davis, and P.W. Voorhees, Step-flow growth of a nanowire in the vapor-liquid-solid and vapor-solid-solid processes. Journal of Applied Physics, 2008. 104(7): p. 11.

3. Golovin, A.A., Davis, S.H., Voorhees, P.W., Step-Bunching in Nanowire Growth, in Submitted.

4. Haxhimali, T., et al., Size-Dependent Nucleation Kinetics at Non-planar Nanowire Growth Interfaces, in Submitted.

5. Lensch-Falk, J.L., et al., Vapor-solid-solid synthesis of ge nanowires from vapor-phase-deposited manganese germanide seeds. Journal of the American Chemical Society, 2007. 129(35): p. 10670-+.

6. Adhikari, H., et al., Metastability of Au-Ge liquid nanocatalysts: Ge vapor-liquid-solid nanowire growth far below the bulk eutectic temperature. Acs Nano, 2007. 1(5): p. 415-422.

7. Lensch-Falk, J.L., et al., Vapor-solid-solid synthesis of Ge nanowires from vapor-phase-deposited manganese germanide seeds (vol 129, pg 10671, 2007). Journal of the American Chemical Society, 2008. 130(3): p. 1109-1109.

8. Schwalbach, E.J. and P.W. Voorhees, Phase Equilibrium and Nucleation in VLS-Grown Nanowires. Nano Letters, 2008. 8(11): p. 3739-3745.

9. Perea, D.E., et al., Composition analysis of single semiconductor nanowires using pulsed-laser atom probe tomography. Applied Physics a-Materials Science & Processing, 2006. 85(3): p. 271-275.

10. Allen, J.E., et al., High-resolution detection of Au catalyst atoms in Si nanowires. Nature Nanotechnology, 2008. 3(3): p. 168-173.

11. Nichol, J.M., et al., Displacement detection of silicon nanowires by polarization-enhanced fiber-optic interferometry. Applied Physics Letters, 2008. 93(19): p. 3.

12. Perea, D.E., et al., Tomographic analysis of dilute impurities in semiconductor nanostructures. Journal of Solid State Chemistry, 2008. 181(7): p. 1642-1649.

13. Perea, D.E., et al., Direct Measurement of Dopant Distribution in an Individual Vapour-Liquid-Solid Nanowire. Nature Nanotechnology, 2009.

[NAS] National Academy of Science, National Academy of Engineering, Institute of Medicine, “Enhancing the Postdoctoral Experience for Scientists and Engineers: A Guide for Postdoctoral Scholars, Advisers, Institutions, Funding Organizations, and Disciplinary Societies,” National Academies Press, 2000.

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[pic]

Figure 1: (a) PFC-simulated strain distribution around a step at a solid surface, (b) MD simulation of nanoscale crystal growth from a liquid, (c) Simulation of nanocrystal growth employing amplitude formulation of PFC, (d) Snapshot of adaptive mesh in a phase field simulation of an advancing interface.

[pic] [pic]

Figure N. Ge quantum dots on the surface of Si. (a) a 70 nm X 70 nm scan of the surface showing the arrangement of the atoms on the surface of the dots and the Ge-coated Si substrate [tomitori] (b) a larger field of view showing multiple faceted islands and larger dome-like islands [rastelli]. The challenge is to capture the atomic details shown in (a) while modeling the self-assembly process over the larger length scales shown in (b).

[pic]Fig. N. A Ge nanowire showing the locations of atoms of Ge (blue), Au (yellow) produced using atom-probe tomography [perea-09]. The Au was liquid at the growth temperature, arrow points to the liquid-vapor-solid trijunction. The nanowire is shown slightly off-axis and hence the solid-liquid interface is not perfectly flat. The nanowire is growth vertically in this case.

[pic]

Fig. xx: Snapshot of a MD simulation of a VLS nanowire. Solid and liquid atoms are colored blue and red, respectively.

[pic][pic][pic]

Fig. XXX1: Phase-field crystal simulations of crystal (green) - liquid (transparent) system at t=200, 500, and 3000 (dimensionless units), using our density-amplitude formulation at 90% solid fraction. The simulation took only 18 hours on a single CPU [7].

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