## Solidification, self-assembly, kinetics of diffusion-controlled transformations, pattern formation, twinned dendrite growth, and stressed epitaxial growth

### April 26, 2009

P L Schaffer, R H Mathiesen and L Arnberg

Studies of Al-based hypermonotectics have so far focused mainly on droplet motion and coagulation dynamics, with limited attention given to the interaction between droplets and the advancing solidification front which is decisive for the final distribution of the second phase within the α-Al matrix. The current work presents results from directional solidification experiments with Al–8 wt.% Bi alloys. It was found that droplets with large radii were frequently pushed and small droplets were engulfed. This is contradictory to the many models that have been proposed to explain pushing/engulfment of solid particles and can in part be ascribed to the fact that while solid-particle models only consider single, non-interacting particles that remain unaffected by solutal gradients ahead of the advancing solidification front, droplet–droplet interaction and local solute gradients have been found to be critical for droplet pushing/engulfment behaviour in hypermonotectic alloys.

P C Millett, and Y U Wang

A novel mesoscale simulation approach to modeling the collective interactions of charged colloidal particles allowing investigation of complex self-assembly processes is presented. Diffuse interface field variables are used to describe the shape, size, location and orientation of each individual particle within the computational domain. In addition, these field variables are used to determine the spatially resolved particle charge density distributions as well as the magnitude and direction of the electric field throughout the medium. Individual particle positions and rotations are updated in time as a result of long-range electrostatic and short-range repulsive forces and torques. Illustrative results of the model’s capability to evolve both monodisperse and binary distributions of charged particles of various shapes and charge characteristics are presented.

H Wang et al

An analytical model has been developed to describe the overall kinetics of diffusion-controlled transformations assuming site saturation or continuous nucleation, in combination with one-dimensional growth. On the basis of linear approximation of the concentration gradient, the method of probability calculation is adopted to model the transformed fraction. First, the so-called geometrical model was re-derived, assuming that the diffusion-controlled growth, according to the parabolic growth law, stops due to geometrical impingement of grains plus their diffusion fields. Then, the transformation kinetics subjected to soft impingement was described, following an analogous approach. The effect of soft impingement, depending on the degree of supersaturation, has been interpreted by evolution of the transformed fraction and the growth exponent. The model was applied to describe the isothermal austenite–ferrite transformation of 0.37C–1.45Mn–0.11 V microalloyed steel, and a good agreement between model predictions and experimental results has been obtained.

I Steinbach

Competing self-organization between the solidification pattern and the convection pattern in a directional solidification environment is investigated theoretically and by phase-field simulations. Melt flow introduces a mode of transport with broken symmetry dependent on the direction of growth relative to the vector of gravity. A stable and an unstable regime can be distinguished. The interaction between spacing selection and convection leads to a new type of scaling that explains results from phase-field simulations and solidification experiments under enhanced gravity.

M A Salgado-Ordorica, J Vallonton and M Rappaz

Under certain thermal conditions (G ≈ 1 × 104 K m−1, νs ≈ 1 × 10−3 m s−1), left angle bracket1 1 0right-pointing angle bracket twinned dendrites appear in aluminum alloys and can overgrow regular columnar dendrites, provided that some convection is also present in the melt. In order to check the stability of such morphologies, directionally solidified twinned samples of Al–Zn were partially remelted in a Bridgman furnace and then resolidified under controlled conditions, with minimal convection. It was found that, although twin planes remain stable during partial remelting, non-twinned dendrites regrow during solidification. They have a crystallographic orientation given by those of the twinned and untwinned “seed” regions, and grow along preferred directions that tend to be those of normal specimens.

N G Rudawski and K S Jones

A dual-timescale model of stressed solid-phase epitaxial growth is developed to provide a basis for the atomistic interpretation of experiments where the macroscopic growth velocity of (0 0 1) Si was studied as a function of uniaxial stress applied in the plane of the growth interface. The model builds upon prior empirical modeling, but is a significant improvement as it provides solid physical bases as to the origin of growth being dual-timescale and more accurately models growth kinetics.

## Atomic structure determination at nanoscale

### April 28, 2007

Here is a review from the recent issue of Science by Simon J L Billinge and Igor Levin on the available experimental and theoretical methods for the determination of atomic structure at the nanoscale. Here is the abstract:

Emerging complex functional materials often have atomic order limited to the nanoscale. Examples include nanoparticles, species encapsulated in mesoporous hosts, and bulk crystals with intrinsic nanoscale order. The powerful methods that we have for solving the atomic structure of bulk crystals fail for such materials. Currently, no broadly applicable, quantitative, and robust methods exist to replace crystallography at the nanoscale. We provide an overview of various classes of nanostructured materials and review the methods that are currently used to study their structure. We suggest that successful solutions to these nanostructure problems will involve interactions among researchers from materials science, physics, chemistry, computer science, and applied mathematics, working within a “complex modeling” paradigm that combines theory and experiment in a self-consistent computational framework.

Take a look!

## Sparse matrix algorithms for phase field models of grain growth

### April 23, 2007

Phase field models are used, among other things, for a study of grain growth [1,2]. In one type of phase field models used to study grain growth, each grain orientation is represented by an order parameter $\eta_{i}$, where the index $i$ runs from 1 to $N$, where $N$ is the total number of orientations present in the system [3,4]. Though this method is simple to implement and has been used extensively, it is computationally intensive. So, recently there had been at least two attempts to come up with an efficient numerical implementation:

1. Efficient numerical algorithm for multiphase field simulations. Srikanth Vedantam and B S V Patnaik, Phys Rev E 73, 016703, 2006.

Phase-field models have emerged as a successful class of models in a wide variety of applications in computational materials science. Multiphase field theories, as a subclass of phase-field theories, have been especially useful for studying nucleation and growth in polycrystalline materials. In theory, an infinite number of phase-field variables are required to represent grain orientations in a rotationally invariant free energy. However, limitations on available computational time and memory have restricted the number of phase-field variables used in the simulations. We present an approach by which the time and memory requirements are drastically reduced relative to standard algorithms. The proposed algorithm allows us the use of an unlimited number of phase-field variables to perform simulations without the associated burden on computational time or memory. We present the algorithm in the context of coalescence free grain growth.

2. Sparse data structure and algorithm for the phase field method. J Gruber, N Ma, Y Wang, A D Rollett, and G S Rohrer, Modelling and Simulation in Materials Science and Engineering, 14, 1189, 2006.

The concepts of sparse data structures and related algorithms for phase field simulations are discussed. Simulations of polycrystalline grain growth with a conventional phase field method and with sparse data structures are compared. It is shown that memory usage and simulation time scale with the number of nodes but are independent of the number of order parameters when a sparse data structure is used.

The source code for a C++ implementation of the method described in the paper of Gruber et al is available for download here (for non-profit scientific research purposes).

The idea behind these implementations is rather simple. Consider an arbitrary mesh point in a simulation cell. The mesh point either lies in the bulk of a given grain, or it lies in the interface. If it lies in the bulk, all the order parameters except the one corresponding to the grain orientation are zero, and there is nothing much to be done about the calculation at that point. On the other hand, if it lies in the interface, the total number of order parameters which have non-zero values are still a very small number as compared to the total number of orientations present in the entire system. Hence, if there is a database such that for any point (and its neighbours) we have the information of non-zero order parameters, then the calculation can be made more efficient.

Take a look and have fun!

References:

[1] Phase field methods for microstructure evolution. Long-Qing Chen, Annu. Rev. Mater. Res., 32, 113, 2002.

[2] Modeling grain boundaries using a phase-field technique. J A Warren, R Kobayashi, and W Craig Carter, J Cryst Growth, 211, 1, 18, 2000.

[3] A novel computer simulation technique for modeling grain growth. Long-Qing Chen, Scripta Metallurgica et Materialia, 32, 1, 115, 1995.

[4] A phase field concept for multiphase systems. I Steinbach, F Pezzolla, B Nestler, M Seeszelberg, R Prieler, G J Schmitz, and J L L Rezende, Physica D, 94, 3, 135, 1996.

## Wobbling plate, Feynman and computer simulations

### April 1, 2007

Title: Feynman’s wobbling plate

Authors: Slavomir Tuleja, Boris Gazovic, Alexander Tomori and Jozef Hanc

Source: American Journal of Physics, Vol. 75, No. 3, pp. 240–244, March 2007

Abstract: In the book Surely You Are Joking,Mr. Feynman! Richard Feynman tells a story of a Cornell cafeteria plate being tossed into the air. As the plate spun, it wobbled. Feynman noticed a relation between the two motions. He solved the motion of the plate by using the Lagrangian approach. This solution didn’t satisfy him. He wanted to understand the motion of the plate by analyzing the motion of its individual particles and the forces acting on them. He was successful, but he didn’t tell us how he did it. We provide an elementary explanation for the two-to-one ratio of wobble to spin frequencies, based on an analysis of the motion of the particles and the forces acting on them. We also demonstrate the power of numerical simulation and computer animation to provide insight into a physical phenomenon and guidance on how to do the analysis.

Internet resources: Java applets and supplementary information