Phase-field Simulations of Coherent Precipitate Morphologies and Coarsening Kinetics
Open Access
- Author:
- Vaithyanathan, Venugopalan
- Graduate Program:
- Materials Science and Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- January 21, 2002
- Committee Members:
- Zi Kui Liu, Committee Member
John Richard Hellmann Jr., Committee Member
Long Qing Chen, Committee Chair/Co-Chair
Darrell G Schlom, Committee Member
Qiang Du, Committee Member - Keywords:
- phase-field model
3D simulation
first-principles phase-field coupling
coarsening of gamma-prime precipitates
multiscale model
anomalous coarsening - Abstract:
- The primary aim of this research is to enhance the fundamental understanding of coherent precipitation reactions in advanced metallic alloys. The emphasis is on a particular class of precipitation reactions which result in ordered intermetallic precipitates embedded in a disordered matrix. These precipitation reactions underlie the development of high-temperature Ni-base superalloys and ultra-light aluminum alloys. Phase-field approach, which has emerged as the method of choice for modeling microstructure evolution, is employed for this research with the focus on factors that control the precipitate morphologies and coarsening kinetics, such as precipitate volume fractions and lattice mismatch between precipitates and matrix. Two types of alloy systems are considered. The first involves L1$_2$ ordered precipitates in a disordered cubic matrix, in an attempt to model the $gamma'$ precipitates in Ni-base superalloys and $delta'$ precipitates in Al-Li alloys. The effect of volume fraction on coarsening kinetics of $gamma'$ precipitates was investigated using two-dimensional (2D) computer simulations. The temporal evolution of precipitate morphologies and coarsening kinetics were characterized by the time-dependence of average aspect ratios of precipitates, average particle size, and precipitate size distributions. With increase in volume fraction, larger fractions of precipitates were found to have smaller aspect ratios in the late stages of coarsening, and the precipitate size distributions became wider and more positively skewed. The most interesting result was associated with the effect of volume fraction on the coarsening rate constant. Coarsening rate constant as a function of volume fraction extracted from the cubic growth law of average half-edge length was found to exhibit three distinct regimes: {em anomalous behavior} or decreasing rate constant with volume fraction at small volume fractions ($lesssim$20\%), {em volume fraction independent} or constant behavior for intermediate volume fractions ($sim$20-50\%), and the {em normal behavior} or increasing rate constant with volume fraction for large volume fractions ($gtrsim$50\%). The simulation results are in agreement with the experimental results of Ardell and coworkers in Ni-Al, Ni-Ti and Ni-Si alloys. Based on non-linear growth law fit to the coarsening data, the growth exponent in the presence of coherency stress was found to be $sim$3 for the average precipitate size, while the exponents were $sim$4 for the average spacing between the aligned precipitate arrays. The effect of simulation dimension (2D and 3D) on the coarsening kinetics was compared for stress-free $delta'$ precipitates in Al-Li alloys and stressed $gamma'$ precipitates in Ni-base alloys, at 20\% volume fraction. The main difference in the kinetics from 2D and 3D simulations was the increase in coarsening rate constant with the simulation dimension. The second alloy system considered was Al-Cu with the focus on understanding precipitation of metastable tetragonal $ heta'$-Al$_2$Cu in a cubic Al solid solution matrix. In collaboration with Chris Wolverton at Ford Motor Company, a {em multiscale model}, which involves {em a novel combination of first-principles atomistic calculations with a mesoscale phase-field microstructure model}, was developed. Reliable energetics in the form of bulk free energy, interfacial energy and parameters for calculating the elastic energy were obtained using accurate first-principles calculations. With the help of anisotropies incorporated in the multiscale model based on first-principles calculations, the equilibrium precipitate morphology of $ heta'$ was found to be governed by the combination of interfacial and elastic energy anisotropy. Quantitative results from the model in the form of length, thickness and aspect ratio of precipitates were comparable with the experimental results. With sufficient large-scale simulations, which will become possible in the coming years, this multiscale model can be used to provide quantitative precipitation kinetics and coarsening information, which are very useful in designing the processing parameters of these alloys.