# Prediction of Onsager and Gradient Energy Coefficients from Microstructure Images with Machine Learning

## Additional Funding Sources

The project described was supported by the National Science Foundation via the Research Experience for Undergraduates Site: Materials for Society (Award No. 1950305) and by the Micron School of Materials Science & Engineering at Boise State University.

## Abstract

Some material parameters are very difficult to estimate using experimental methods. As an alternative, machine learning is a valuable tool for estimating material parameters since it obviates the need for expensive and time-consuming experiments. To demonstrate this strategy, this work focuses on the spinodal decomposition of an FeCrCo alloy at 873 K after 100 hours. Spinodal decomposition is modeled using the phase-field method, in which the Cahn-Hilliard (CH) equation is solved for the composition of each element with respect to time and position. The Onsager coefficients (M22, M23, M32, and M33) and the gradient energy coefficient (κ) are parameters in the CH equation that affect the rate of decomposition of the alloy. A database of microstructure images with varying values for these parameters was generated using the Multiphysics Object-Oriented Simulation Environment (MOOSE), which numerically solves the CH equation. A machine learning model was then developed which predicts the Onsager and gradient energy coefficients using the microstructure images as inputs. After optimization, this model predicted M22 with 6.4% difference and κ with 2.5% difference from the actual values in the test dataset. The optimized model was leveraged to predict the parameters for experimental microstructure images obtained with transmission electron microscopy (TEM). Inconsistent predictions were obtained for the TEM images, which illustrates how a multitude of parameter combinations can produce similar microstructure morphologies. Nonetheless, this work highlights how machine learning is a valuable tool for accurately and precisely predicting material parameters from microstructure morphologies.

Prediction of Onsager and Gradient Energy Coefficients from Microstructure Images with Machine Learning

Some material parameters are very difficult to estimate using experimental methods. As an alternative, machine learning is a valuable tool for estimating material parameters since it obviates the need for expensive and time-consuming experiments. To demonstrate this strategy, this work focuses on the spinodal decomposition of an FeCrCo alloy at 873 K after 100 hours. Spinodal decomposition is modeled using the phase-field method, in which the Cahn-Hilliard (CH) equation is solved for the composition of each element with respect to time and position. The Onsager coefficients (M22, M23, M32, and M33) and the gradient energy coefficient (κ) are parameters in the CH equation that affect the rate of decomposition of the alloy. A database of microstructure images with varying values for these parameters was generated using the Multiphysics Object-Oriented Simulation Environment (MOOSE), which numerically solves the CH equation. A machine learning model was then developed which predicts the Onsager and gradient energy coefficients using the microstructure images as inputs. After optimization, this model predicted M22 with 6.4% difference and κ with 2.5% difference from the actual values in the test dataset. The optimized model was leveraged to predict the parameters for experimental microstructure images obtained with transmission electron microscopy (TEM). Inconsistent predictions were obtained for the TEM images, which illustrates how a multitude of parameter combinations can produce similar microstructure morphologies. Nonetheless, this work highlights how machine learning is a valuable tool for accurately and precisely predicting material parameters from microstructure morphologies.