An Experimental Design Approach

Transcription

1 An Experimental Design Approach to Process Design Martha Grover Gallivan School of Chemical & Biomolecular Engineering Georgia Institute of echnology ebruary 11, 2008

2 actors Influencing Material Properties When chemical composition alone does not determine the material properties, we need to simultaneously consider the process, the material structure, and the resulting properties. Perfect Crystal Non-Equilibrium Structures: t Glass Polycrystalline Disordered Liquidid kinetics = dynamics

3 Process Systems Engineering Design, simulation, optimization, and control actorial experimental design Chemistry Properties Group contribution models Optimization Molecular Dynamics Control Processing Monte Carlo Structure Historically, systems engineering has been applied to systems in which physics and chemistry are well understood. he new challenge is to design systems with only a partial model. e.g. materials, nanotechnology, synthetic biology

4 Approach Combine general methodology and specific applications Barriers to systems engineering g in material structure design 1. Models are not accurate enough 2. Computational demands of models are too high 3. In-situ sensing is difficult Methodology development 1. Experimental design 2. Model reduction 3. Real-time estimation Develop and demonstrate in several specific experimental applications

5 Current practice in materials development Design of materials and processes is largely empirical Macroscopic models are used in process design, but molecular/microscopic models are not Materials properties (advanced materials) require consideration of molecular structure

6 Coupling of models and experiments With partial models, cannot decouple the experiments Partial models Parameter estimation Candidate mechanisms and models Multiscale models: models at each scale Need experiments at each scale Need statistical analysis to build models: which model is best to make decisions about experiments Exploration Design objectives Alternative is system identification: best for linear systems

7 Chemical vapor deposition Commonly used process for depositing thin films hermal CVD Case study Volatile precursor Heated substrate Chemical reactions ormation of solid film Yttrium oxide (Y 2 O 3 ) stabilized thin films Silicon substrate Polycrystalline Application hermal barrier coatings Solid oxide fuel cells Microelectronics gate dielectric Oxygen sensors Microstructure High grain density, strong Graded microstructure Amorphous Nanocrystalline

8 Materials Design via Process Design hree main steps in the design of a material and process Should simultaneous consider the entire problem, but can decompose into: 1. Design of hardware (geometry) 2. Design of process settings (open loop control) 3. Correct for disturbances (closed loop control) C t Plant K t

9 Experimental testbed Enables case study and demonstration of methods reflectometer CVD reactor Schematic of CVD testbed

10 wo Challenges in CVD Both require formal consideration of dynamics and statistics In-situ sensing Approach: estimation theory Process and sensor modeling and validation Provide theoretical underpinning and improved performance for existing fitting methods Design of process settings Approach: sequential experimental design Bayesian estimation Model discrimination Get the most information out of each experiment

11 Designing g with partial models Much design work goes on without physics-based models Ideal case Experiments Models exist at various resolutions: constant trend fit continuum lumped models molecular Build models empirical bulk mechanistic high spatial resolution Design system More understanding of process

12 Previous approaches: empirical No mechanistic model is required in this approach actorial experimental design Use when experiments are costly and there is little understanding of the process Empirical modeling Response surface models Design criterion Nominal performance y Robustness (aguchi) Not explicitly sequential. Analogy to gradient based optimization Resulting model is not adaptable x 2.. x 1 x D. C. Montgomery, Design and Analysis of Experiments, John Wiley (2005).

13 Previous approaches: mechanistic Approaches for mechanistic models not based on design criteria Parameter estimation e.g. D-optimal, Bayesian, Maximum Likelihood Objective: estimate unknown parameters accurately Model discrimination Design experiment at settings where model predictions disagree most / 2 ˆ ν / 2 j j e P ( M Y, S, ν ) P ( M ) 2 S j D m, n e e j ( y ( x) y ( x) ) p j ( m n x) = 2 2 σ + σ 2 m ( x) + σ ( x ) Methods may or may not be sequential Box on Quality and Discovery: with Design, Control, and Robustness, Box on Quality and Discovery: with Design, Control, and Robustness, ed. G. C. iao,wiley (2000). Buzzi-erraris, A new sequential design procedure for discriminating among rival models, Chemical Engineering Science, 38, (1983). n 2

14 Desired features in new approach Want to retain the best features of both approaches 1. Mathematical ti and statistical ti ti underpinnings i 2. ractable computation e.g. 1 day 3. Logic consistent with the empirical design approach e.g. include design objectives 4. ransition from low to high resolution models as knowledge is gained 5. radeoff between exploration and refinement Global versus local minima

15 Challenges and open questions Commonly used process for depositing thin films Experimental design criterion Multiple and competing objectives Local minima Parameter estimation Selection of experiments Need to avoid repeating the same experiment Kriging / spatial statistics e.g. batch-wise design: multiple local minima Effect of initial experiments Rate of convergence Steady state

16 Approach for process design Need close coupling between experiments and models for design Experiments are costly: need to get the most information possible from them. Empirical fits may be most probable/useful, especially with limited data. Need rigorous statistics for decision-making. Run experiments Start Probability of models Performance at various settings Design of experiments Stop? Model 1 (empirical) inal process recipe Model 2 (mechanistic) Parameter estimation New discrimination function based on performance criteria D x) = ( P ( x) f ( yˆ ( x) + P ( x) f ( yˆ ( x) ) ( yˆ ( x) yˆ ( x) ) m 2 2σ ( x) + σ 2 n ( x) + σ ( x) n mn ( m m n n 2 2 m

17 Case study In simulation quickly assess tradeoffs and options Process inputs temperature ( K) flux ( s -1 ) Mechanistic model Nucleation of yttria clusters on silicon surface Parameters: E i, E d, σ dn 1 (1 θ ) ( i + 1) Knuc ( σ,, Ei, Ed ) Kagg ( σ,, Ei, E dt dnisl = Knuc ( σ,, Ei, N 1 ) dt Empirical model N Parameters: A, B, C isl = A + B + C Goal: maximize grain density N isl Initial experiments: 2 2 factorial d )

18 Iteration 1 Experimental design function Design portion Discrimination portion

19 Iteration 2 Experimental design function Design portion Discrimination portion

20 Iteration 3 Experimental design function Design portion Discrimination portion

21 0.06 Iteration 4 Experimental design function Design portion Discrimination portion

22 Conclusions: experimental design Experimental repetitions reduce model discrepancy, and thus bias the experiments toward other points his may be an inefficient way to explore the space Spatial statistics (kriging) provides a more direct way to incorporate Multiple local minima in the discrimination function should be expected near the boundaries and corners, especially for crude empirical models Discriminate between the two most probable models Batchwise sequential experimental design may be preferable to sample multiple local minima

23 Acknowledgments Paul Wissmann Rentian Xiong Cihan Oguz National Science oundation CAREER: A systems approach to materials processing US Air orce Research lab and Air orce Office of Scientific Research