Active grain growth control with distributed heating

Abstract

Microstructure affects the physical properties and behavior of materials. While metallurgists have long studied microstructure characterization and evolution, thermo-mechanical material processing to achieve a desired microstructure remains largely experience-based. This paper presents a distributed thermal control methodology for the microstructure evolution. We consider the problem of achieving a uniform microstructure, starting from a non-uniform initial distribution. This is a common goal in material processing, as uniform microstructure implies consistent macroscopic properties. To illustrate the approach, we consider an example process with a multi-zone micro-heater array controlling the grain growth of a copper thin film. Cascaded temperature and grain-growth models characterize the process dynamics – finite element method (FEM) models the temperature field in response to the heater input, which in turn drives the microstructure evolution through a biased Monte-Carlo (MC) model. The high order combined FEM/MC model is used as the validation “truth” model. For the control design and analysis, a simplified model is developed to only capture the essential trend in the full model. Using the simplified model and dividing the copper thin film into multiple spatial zones with measurable grain statistics in each zone, we obtain a nonlinear multi-input/multi-output control design model. Using the simplified model, this paper presents the development and comparison of three control methods: 1. Direct output feedback from the measured mean local grain sizes to the heater current. 2. Model predictive control (MPC) using a finite horizon optimization to compute the required heat input at each control step. 3. Inner-outer loop control with temperature as the surrogate input for the outer loop and using the heater current to achieve the required temperature in the inner loop. All three methods achieve uniform microstructure in grain growth in the higher order FEM/MC simulation. Direct output feedback is the simplest to implement, but has the slowest convergence. MPC shows fast convergence but requires model-dependent on-line optimization. Inner-outer loop demonstrates good compromise between model-dependence and rate of convergence.