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Some scale-up methodologies for advanced manufacturing

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Some scale-up methodologies for advanced manufacturing

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Content Some Scale-up Methodologies for Advanced Manufacturing by Yanqing Duanmu A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Industrial and Systems Engineering) August 2017 Copyright 2017 Yanqing Duanmu Acknowledgments Joining the Trojan Family is a wonderful decision. I feel very fortunate to have so many people helping me, and sharing with me passion and thoughts. No person deserves my thanks more than Prof. Qiang Huang, my PhD adviser. Without his constant inspirations and help, I could never finish my PhD study. Prof. Huang encouraged me to take advantage of my Physics background and learn more about Statistics. I benefit a lot from the mixed background. Prof. Huang not only offered me careful guidance in research, but also helped me to grow stronger in other aspects. He is strict with us and cares a lot about us. He is always ready to guide us through hard times, and never neglects our progresses, big or small. He expects us to be our personal best and helps us tap our potential. I feel greatly honored to be his PhD student, and I believe the guidance I have obtained from Prof. Huang will be extremely beneficial in my future life and career. I also want to express my most sincere gratitude to my thesis committee members, Prof. Berok Khoshnevis and Prof. Yong Chen. I want to thank them for their generous help in serving as my committee members. My qualifying exam committee members Prof. Sheldon M. Ross, Prof. Noah Malmstadt and Prof. Wei Wu have also given me great supports. Thank you so much for providing valuable comments on my thesis work. My sincere gratitude also goes to other faculty members within Daniel J. Epstein Department of Industrial & Systems Engineering. Their interesting courses and seminars have helped me a lot during my study and research. ii I am also very grateful to the staff in Daniel J. Epstein Department of Industrial & Systems Engineering: Roxanna Carter, Shelly Lewis, Mary Ordaz and Tiffany Ting, who assisted me in teaching, preparing materials, etc. Thank you so much for being so nice and helpful. I also want to thank my friends and colleagues. My PhD study would not be such a pleasant experience without their support and company. In particular, I want to thank Lijuan Xu, one of our previous group members, who has set an outstanding example to us in all aspects as a PhD student. My first research task would never have been accomplished without her previous work and tremendous help. I also benefited from our cooperation with Prof. Noah Malmstadt’s group in Mork Family Department of Chemical Engineering and Materials Science, and particularly learned much from Dr. Carson Riche. I want to thank my lab mates Yuan Jin and He Luan, and my friends in USC, who have brought me much joy during my graduate study. Because of them, I feel myself a member of the Trojan Family. Finally, I would like to give my deepest thanks to my parents for their consistent help and support, and to my husband for loving me and supporting me. Without their love and support, I can never arrive at this stage. iii Contents Acknowledgments ii List Of Tables vi List Of Figures vii Abstract ix 1 Introduction 1 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Importance of Scale-up Study for Advanced Manufacturing . . . . . . . . . 2 1.1.2 Challenges of Scaling up Advanced Manufacturing Processes . . . . . . . . 5 1.2 State of the Art on Scale-up Methodology Research . . . . . . . . . . . . . . . . . . 7 1.2.1 Dimensional Analysis and the Scaling Law . . . . . . . . . . . . . . . . . . . 7 1.2.2 Scale-up Modeling Methodologies for Advanced Manufacturing . . . . . . . 9 1.2.3 Design of Experiments for Scale-up Processes . . . . . . . . . . . . . . . . . 10 1.3 Problems Addressed in Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Physical-Statistical Scale-up Modeling: Analysis and Optimization of III-V Nanowire Synthesis via SA-MOCVD 16 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Our Previous Model and Skirt Area Effect . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 A Growth Model Counting Blocking Effect . . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 Blocking effect modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.2 Choice of concentration function . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.3 Statistical model formulation and estimation . . . . . . . . . . . . . . . . . 26 2.4 Physical Interpretation of Skirt Area Effect . . . . . . . . . . . . . . . . . . . . . . 28 2.5 Optimal Width of Skirt Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3 Scale-up Methodology under Uncertainties: Study of Gas-Solid Separation in a Cyclone Separator 35 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Scale-up Methodology Under Uncertainties . . . . . . . . . . . . . . . . . . . . . . 37 3.2.1 Dimensionality Reduction via Dimensional Analysis . . . . . . . . . . . . . 38 3.2.2 Experimental Setup and Data Collection . . . . . . . . . . . . . . . . . . . . 40 3.2.3 Identification of Random Effects and Mixed Modeling of Cyclone Separation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Robust Parameter Design based on Generalized Linear Mixed Model . . . . . . . . 50 3.4 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 iv 4 Multiple-Domain Scale-up Modeling: Droplet Formation in a Coated Microflu- idic T-junction 53 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Scalable Modeling Methodology for Multiple-domain Manufacturing Process . . . . 56 4.3 Scale-up Modeling of Droplet Formation in a Coated Microfluidic T-junction . . . 61 4.3.1 Dimensional Analysis and Dimension Reduction . . . . . . . . . . . . . . . 61 4.3.2 Experimental Setup and Data Collection . . . . . . . . . . . . . . . . . . . . 63 4.3.3 Model Structures and Basis Functions . . . . . . . . . . . . . . . . . . . . . 64 4.3.4 Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4.1 Physical Insights through High-dimensional Modeling . . . . . . . . . . . . 67 4.4.2 Identification of Physical Domains and Boundaries . . . . . . . . . . . . . . 69 4.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5 Design of Experiments for Domain-Dependent Scale-up Manufacturing Pro- cesses 74 5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 A Sequential Design for Multiple-Domain Scale-up . . . . . . . . . . . . . . . . . . 76 5.2.1 Initial Design to Identify Primary Factor . . . . . . . . . . . . . . . . . . . 76 5.2.2 Optimality Proof of the Separation Scheme in Sequential Design . . . . . . 79 5.2.3 Domain Identification: Sequential DOE for PF . . . . . . . . . . . . . . . . 79 5.2.4 Weight function Identification: Sequential DOE for SF . . . . . . . . . . . 84 5.3 Case Study: Drag Coefficient of a Sphere in Fluid . . . . . . . . . . . . . . . . . . 85 5.4 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6 Discussions and Future Extensions 91 Bibliography 96 v List Of Tables 2.1 Model estimation for various skirt area widths . . . . . . . . . . . . . . . . . . . . . 28 3.1 Relevance list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 GLMM model estimation (RMSE=1.739359%) . . . . . . . . . . . . . . . . . . . . 45 3.3 Reduced GLMM estimation (RMSE=1.717276%) . . . . . . . . . . . . . . . . . . 45 3.4 Comparison of root-mean-square error . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1 Relevant List of Physical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 Experimental settings of physical quantities . . . . . . . . . . . . . . . . . . . . . . 64 4.3 Reduced model estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4 Ca 1 (Q,T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5 Ca 2 (Q,T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.1 Comparison of different experimental designs . . . . . . . . . . . . . . . . . . . . . 90 vi List Of Figures 1.1 Geometric and physical similarity [129] . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Applicability of modeling methodologies [143] . . . . . . . . . . . . . . . . . . . . . 9 1.3 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1 Schematic of SA-MOCVD fabrication [142] . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Lateral diffusion from skirt area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Four sources of precursor contributions in SA-MOCVD [142] . . . . . . . . . . . . 20 2.4 Model prediction in [142]. Growth data are extracted from [21, 20] . . . . . . . . . 22 2.5 Skirt diffusion without “blocking” (skirt width w = 150 μm) . . . . . . . . . . . . . 24 2.6 Model fitting errors with different L d . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7 Predicted models with varying skirt area widths (L d = 43μm). Growth data are extracted from [21, 20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.8 Uniformity for different w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.1 Flowchart of a systematic approach to the study of manufacturing processes under uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 Sample standard deviation of grade-efficiency against Stk and δ dm . . . . . . . . . . 44 3.3 Fitted results of GLMM for grade-efficiency . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Comparison of fitted values of grade-efficiency models . . . . . . . . . . . . . . . . 47 3.5 μ and ∂μ/∂g(μ) as a function of g(μ) . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.1 Methodology to construct the scale-up model for a multivariate physical system . . 58 4.2 Structure of the two-phase microfluidic T-junction [67] . . . . . . . . . . . . . . . . 61 vii 4.3 Change in dimensionless droplet length ¯ L with respect to the capillary numberCa, flow rate ratio Q (from 0.05 to 2) and depth width ratio W h . . . . . . . . . . . . . 68 5.1 Flowchart of overall design strategy for domain-dependent scale-up experiments . . 77 5.2 Flowchart of identifying the primary factor . . . . . . . . . . . . . . . . . . . . . . 78 5.3 Flowchart of sequential design for the primary factor . . . . . . . . . . . . . . . . . 80 5.4 Flowchart of sequential design for the secondary factor . . . . . . . . . . . . . . . . 84 5.5 Initial 10 samples and selection of the 11 th design point . . . . . . . . . . . . . . . 86 5.6 First 11 samples and selection of the 12 th design point . . . . . . . . . . . . . . . . 87 5.7 First 12 samples and selection of the 13 th design point . . . . . . . . . . . . . . . . 88 5.8 First 17 samples and selection of the 18 th design point . . . . . . . . . . . . . . . . 89 viii Abstract Advanced manufacturing that involves cutting edge sciences such as materials, physical and biological science, has great potentials to revolutionize the industry and improve our life. For example, nanomanufacturing has shown great promise in addressing important issues in various fields such as energy, medicine, food, and environmental science. Despite of numerous laboratory successes, we have seen a relatively small number of commercial-scale examples due to difficulties in scaling up nanomanufacturing processes. Early-stage study of advanced manufacturing processes mainly faces two challenges. Firstly, a conclusive understanding of the process is often not available due to limited physical knowledge and process uncertainties, making it difficult to obtain a precise characterization and prediction of the process. For example, although selective area metal organic chemical vapor deposition (SA- MOCVD) has been recognized as a promising process for producing nanowires, existing research has not provided a conclusive understanding of the synthesis process. Influence of controllable variables needs further study to enable accurate prediction of the process, which hinders full-scale nanomanufacturing. Secondly, experiments for understanding advanced manufacturing processes are often costly in time and resources, which restricts extensive experimental research. Various methods have been developed to support scale-up advanced manufacturing, including dimensional analysis (DA), scale-up modeling and design of experiments (DOE). DA serves as an important tool to achieve dimension reduction and ensure the scalability of results. Scale-up modeling methods for advanced manufacturing can be classified into four categories based on the requirement of process knowledge and data, namely physical, statistical, physical-statistical ix and cross domain modeling approach. DOE research has been recognized as an important task due to high cost of experimentation in scale-up manufacturing. However, existing research has not provided methodologies to address all issues that arise in scaling up advanced manufacturing processes. This dissertation not only seeks to deepen the understanding of some advanced manufacturing processes, but also aims to provide some scale-up methodologies for early-stage research on ad- vanced manufacturing processes to bridge the gap from laboratory successes to scale-up advanced manufacturing. We explore four problems, each focusing on a different challenge that arises in scaling up advanced manufacturing processes. In the first scale-up problem of nanowire synthesis via SA-MOCVD, existing studies cannot explain well the distribution of nanowire growth on the substrate, which is critical for commercial- scale fabrication that requires accurate control of nanowire growth. To fill the research gap, we propose a physical-statistical model that interprets well the local dependence of nanowire growth, and allows the optimization of skirt area width of the substrate for uniform growth of nanowires. The second problem investigates uncertainties in scale-up processes, and proposes a mixed model framework to characterize scale-up process uncertainties. Specifically, we propose a Gener- alized Linear Mixed Model (GLMM) to characterize the gas-solid separation process in a cyclone separator, which captures well the scale-up process under uncertainties and enables a robust parameter design to minimize process uncertainties. The third and fourth tasks are motivated by the study of droplet formation process in mi- crofluidic devices, where the process exhibits different characteristics in different feasible domains of settings and is therefore defined as a multiple-domain process. Existing scale-up models that capture the process in a specific physical domain cannot be applied to the prediction under new process settings. To fill the research gap, we explore scale-up modeling and design of experiments for multiple-domain processes in the third and fourth task respectively. In the third task, we establish a novel model framework and interpret a multiple-domain process as the outcome of x coexisting physical mechanisms with different weights. A sequential modeling strategy is also provided to obtain the scale-up model for multiple-domain processes. In the fourth task, we pro- pose a sequential design to detect domains adaptively, where the clustering scheme is introduced to DOE for the first time. Our work in this dissertation will not only further the understanding of specific advanced manufacturing processes, but also provide some generic methodologies for early-stage study of scale-up advanced manufacturing processes to bridge the gap between laboratory successes and scale-up advanced manufacturing. Both modeling and experimental design methodologies are explored to address scale-up challenges in advanced manufacturing due to limited knowledge, process uncertainties and the existence of multiple domains. xi Chapter 1 Introduction Advanced manufacturing involves cutting edge sciences and has shown great promise in solving important global issues in various fields such as energy, security, medicine, food, and environmental science. Despite of extensive research on advanced manufacturing processes, there still exists a gap between laboratory successes and large-scale advanced manufacturing. The early-stage study of advanced manufacturing processes mainly faces two challenges: (i) lack of understanding of the process and (ii) restrictions on experimental resources. To address the first challenge, we study scale-up modeling methodologies of characterizing and predicting advanced manufacturing processes through three representative problems, each focusing on a different type of issues. To address the second challenge, we explore the design of experiments for economizing experimental resources in our last task. Nanowire synthesis via SA-MOCVD, studied in our first task, represents a class of advanced manufacturing processes, where the physical mechanism of the process is unclear and experiments are costly. Our second task focuses on capturing and minimizing nonnegligible uncertainties in scale-up processes, motivated by the study of gas-solid separation process in a cyclone separator, where the prediction accuracy of separation efficiency suffers from nonnegligible process uncertain- ties. Our third and fourth task focus on the study of multiple-domain advanced manufacturing 1 processes, where the process is likely to be dominated by different physical mechanisms in differ- ent domains of settings. Existing scale-up methods that capture the process in a specific physical domain cannot be applied to the prediction under new process settings. Moreover, an efficient experimental design is important for detecting physical domains at an early stage in the study of multiple-domain manufacturing processes. Our dissertation seeks to fill these research gaps and support advanced manufacturing. In the following part of this chapter, we will first discuss the general background and motiva- tion for the study of scale-up methodologies in advanced manufacturing. Following that we will summarize the challenges for early-stage study of advanced manufacturing processes, and review existing literature on scale-up methodologies for advanced manufacturing. We will define the research gaps in more detail and list our tasks and objectives of this dissertation. 1.1 Background and Motivation 1.1.1 Importance of Scale-up Study for Advanced Manufacturing Different from traditional manufacturing, advanced manufacturing involves cutting edge sci- ences, such as materials, physical and biological science. There are two main objectives of advanced manufacturing. The first objective is to develop new ways to manufacture existing products with an expectation of high performance, high precision, high rate and environmental friendliness. The second objective is to manufacture innovative products with high performance. Advanced manufacturing has great potentials to revolutionize the industry and improve our life in various fields such as energy, security, medicine, food, and environmental science. For exam- ple, nanomanufacturing has the great potential to address challenging issues that are not solvable before. One of the most impressive examples is perhaps the usage of nanocarriers such as nanopar- ticles and liposomes to deliver drug to the precise region of the patient’s body where tumors are located. This is widely expected to create novel therapeutics for successful cancer treatment, a 2 dream that humans have strove for in the past few centuries. Nanotechnology-based drug delivery overcomes many biological, biophysical and biomedical barriers, and proves to be more effective than traditional chemotherapy [90, 90, 1]. There are many other possibilities to improve can- cer treatment, and more generally, medicine production and usage through nanomanufacturing [85, 86]. The potential of nanomanufacturing in renewable energy such as solar energy and wind energy, is even more extensively studied [81, 56, 82, 65, 132, 69, 148, 61]. Besides, nanoman- ufacturing also finds its potential applications in environmental protection [123, 64, 149], food production [109], material science and chemical engineering [133, 135, 67], etc. Our first task studies nanowire synthesis via selective area metal organic chemical vapor depo- sition (SA-MOCVD), a promising process for scale-up nanomanufacturing. A nanowire is simply a wire with nanoscale diameter. Due to excellent physical properties, nanowires have shown great potentials in producing electronic and optoelectronic devices especially given the demand for low-cost, more compact, yet more powerful systems nowadays [71, 4, 127]. Among many other potential applications, nanowire sensors have been proved to be superior than traditional sensors for sensing proteins [151] and chemicals [31]. Quality control of nanowire growth is thus critical for scale-up nanomanufacturing [71, 143, 29]. One of the promising nanowire fabrication techniques for scale-up nanomanufacturing is SA-MOCVD. Despite much related work, the growth mecha- nism still remains unclear and the spatial patten of nanowire growth has not been fully captured due to an idealized assumption of uniform precursor distribution in the growth chamber. There- fore, we are motivated to fill this research gap and develop a nanowire growth model to reinforce our understanding of the synthesis mechanism and capture the spatial pattens of nanowire growth for supporting scale-up manufacturing. We investigate the gas-solid separation process in a cyclone separator in our second task. Cy- clone separators are designed to remove particles from a gas stream and have grown in popularity since 1885 due to advantages in various aspects including: lack of moving parts, simple construc- tion, relative ease of maintenance, low energy consumption, high-temperature resistance, large 3 treatment capacity, dust loading adaptability and high separation efficiency. Cyclone separators have been widely applied in virtually every traditional industry where there is a need to achieve gas-solid or gas-liquid separation, for example, in power, mineral and petrochemical industries. Extensive studies [70, 27, 83, 88, 54] have not offered a conclusive understanding of the process, and the prediction of separation efficiency suffers from nonnegligible process uncertainties, which hinders the application of cyclone separators in advanced manufacturing industries that require more precise control of the process. Therefore we are motivated to establish a scale-up model that captures well the influence of controllable variables and process uncertainties to fill the research gap and support scale-up advanced manufacturing. Our third and fourth task stem from the study of droplet formation in microfluidic devices, a promising scale-up process for the synthesis of high-quality nanoparticles. Nanoparticles serve as an important nanostructure. Besides the applications in drug delivery as discussed above, nanoparticles also possess excellent optical properties since they are small enough to confine their electrons and produce quantum effects, leading to promising applications in solar PV and thermal [119, 118]. Nanoparticles also exhibit great potentials for clinical diagnosis [48] and laser [34, 95]. Achieving large-scale fabrication, however, faces the challenge of controlling the size, shape, and variety of nanoparticles. One viable technique for nanoparticle fabrication is through droplet formation in microfluidic devices, which represents a class of advanced manufacturing processes that involve multiple physical domains dominated by different mechanisms. Scale-up models have been developed to characterize the process in specific physical domains [43, 25, 42, 19, 40, 140, 128, 152]. However, there is lack of an overall methodology to scale up the process across all physical domains, hindering large-scale fabrication where physical domains are unknown under new settings. Therefore we are motivated to fill this research gap and support the scale-up of multiple-domain advanced manufacturing processes. Specifically, a scale-up modeling approach is important for obtaining a scale-up model that can be applied to capture the process across all physical domains. In addition to scale-up modeling, design of experiments is also important for 4 scaling up multiple-domain advanced manufacturing processes. Unawareness of physical domains can be dangerous. One reason of the financial crisis in 2008 is due to inaccurate risk modeling under extreme situations. In other words, people were unaware of tiny domains where extreme situations occurred and the model collapsed. Therefore it is important to ensure enough design points in each domain so that all situations are well represented. 1.1.2 Challenges of Scaling up Advanced Manufacturing Processes The early-stage study of advanced manufacturing processes faces two main challenges: (i) A conclusive understanding of the process is often not available. (ii) Extensive experimental research is restricted due to high costs. Due to lack of knowledge and process uncertainties, a conclusive understanding of an advanced manufacturing process is often not available. Unlike traditional manufacturing processes, most advanced manufacturing processes involve physical or chemical changes, which complicates the scale-up of a process. Moreover, the relationship between inputs and outputs of an advanced manufacturing process can be very complex due to the large number of controllable variables. Some advanced manufacturing processes involve multiple physical domains dominated by different physical mechanisms, which further complicates the scale-up of these processes. In our first task, the growth mechanism of nanowires remains to be clarified for nanowire synthesis via SA-MOCVD. There have been debates on whether the process is governed by self- catalyzed VLS/VS growth [14, 76, 80] or diffusion based selective area epitaxy [23, 87, 58, 52, 21, 44]. Although diffusion based selective area epitaxy is more widely accepted as the growth mechanism of nanowires synthesized via SA-MOCVD, the diffusion sources and their contribu- tions still remain unclear. Empirical models [122, 59] and physical models [21, 87, 103] even produce conflicting results. Existing studies based on idealized assumptions cannot capture well the distribution of nanowire growth on the substrate, which is critical for scaling up the process. 5 A conclusive understanding of the process is also not available for gas-solid separation in a cyclone separator. Existing studies include computational models based on idealized assumptions of physics behind the process [70, 27, 83], and empirical models that only count the influence of a few selected controllable variables [88, 54]. However, none of the existing models interpret the nonnegligible process uncertainties, which leads to inaccurate prediction of separation effi- ciency and hinders the application of cyclone separators in advanced manufacturing industries that require more precise control of the process. Droplet formation process in microfluidic devices represents a class of multiple-domain ad- vanced manufacturing processes. Extensive work has been presented to characterize influence of some variables on the process and partially explain the mechanism in specific physical domains [5, 19, 46, 39, 40, 41, 42, 43, 47, 24, 25, 128, 125]. There are still active debates on domain identification and the influence of some controllable variables. It follows that scale-up models developed to capture the process in a specific domain cannot be applied to the prediction under new process settings. Scaling up advanced manufacturing processes requires accurate process models based on ex- tensive experimental research. However, experiments for the scale-up study of advanced man- ufacturing processes are often costly. For example, noble metals, such as gold and platinum, are often used in nanomanufacturing. When the materials are not expensive, experimentation is still expected to be environmentally friendly. It is therefore important to establish efficient de- signs of experiments to economize on experimental resources and extract maximum information from limited runs of experiments. Design of experiments (DOE) for multiple-domain advanced manufacturing processes is tough. Note that physical domains and the process model are often unknown at the early stage of study. It follows that an efficient optimal design that relies on model cannot be directly applied. Traditional model-independent methods tend to evenly dis- tribute design points over the entire range of settings, which is not applicable for multiple-domain processes. To illustrate this, suppose a system involves two domains, with one much larger than 6 the other in area. If assigned evenly, design points are much less likely to lie in the tiny domain. In an extreme situation, we might have all design points assigned in the larger physical domain and fail to discover the tiny physical domain within limited experiments. Therefore a novel de- sign of experiments is required to economize experimental resources for multiple-domain advanced manufacturing processes. Note that the two challenges aforementioned are closely related. On one hand, a conclusive understanding of an advanced manufacturing process normally relies on extensive experimental research; on the other hand, design of experiments seeks guidance from understanding of the process. 1.2 State of the Art on Scale-up Methodology Research Various methodologies have been proposed to address the aforementioned challenges and sup- port the scale up of advanced manufacturing processes, including dimensional analysis, scale-up modeling methodologies, and design of experiments. 1.2.1 Dimensional Analysis and the Scaling Law In order to scale up manufacturing processes, we need to first address the so-called scale-up issue: how to translate the understanding of a process from lab scale to commercial scale such that the optimal properties can be determined priorly for future operations. The science base of achieving the scalability of engineering models is the scaling law of engineering systems [7, 156], which signifies the property of physical similarity as a generalization of the concept geometric similarity. As shown in Figure 1.1 (from Wang and Huang’s technical report [129]), two triangles are similar if their length ratios of three sides, i.e., π = l 1 : l 2 : l 3 , are the same, such that the side lengths are scaled up or down proportionally. Physical similarity is similar to geometric similarity but more complex, which requires the equivalence of a set of dimensionless numbers 7 between the laboratory (with superscript L) and the full-scale process (with superscript F), i.e. π L =π F , π L 1 =π F 1 , ..., π L p−k =π F p−k , where the dimensionless numbers are proper combinations of physical quantities. These “scale-invariant” dimensionless numbers capture all characteristics of the system except scales. One system is predictable from the other if they are characterized by the same group of dimensionless numbers, i.e. they have physical similarity. l 1 l 3 l 2 l 1 ' l 3 ' l 2 ' Geometric similarity Physical similarity Laboratory Process x 1 L x p L :" y L Full-scale Process x 1 F x p F :" y F Scale(up" Scale(up" y L = f(x L 1 ,...,x L p ) y F = f 0 (x F 1 ,...,x F p ) ⇡ L =⇡ F ⇡ L = l 1 : l 2 : l 3 ⇡ F = l 0 1 : l 0 2 : l 0 3 ⇡ = (⇡ 1 ,...,⇡ p k ) ⇡ L =⇡ F ⇡ L 1 =⇡ F 1 . . . ⇡ L p k =⇡ M p k ⇡ F = y x a1 1 ···x ak k ⇡ L = y x a1 1 ···x ak k ⇡ F j = x k+j x aj1 1 ···x ajk k ,j=1,··· ,p k ⇡ L j = x k+j x aj1 1 ···x ajk k ,j=1,··· ,p k Figure 1.1: Geometric and physical similarity [129] The dimensionless numbers π i ’s are derived through dimensional analysis (DA) [12, 11, 117]. The essential idea of DA originated from the principle of dimensional homogeneity, i.e. only quantities of the same dimension may be compared, equated, added, or subtracted. Dimensional analysis enables (i) reduction in the number of quantities to formulate the problem and model the process, (ii) a flexible selection of parameters that result in the same group of dimensionless numbers, and (iii) an insight into physical mechanisms of the process. Popular methods of DA include Rayleigh’s method [99] and Buckingham Π-theorem [10]. Note the form of dimensionless numbers is not stipulated by the pi-theorem while the quantity of them does. There exist con- ventional dimensionless numbers in some particular fields, such as the Reynolds number (Re) and the capillary number (Ca) in fluid dynamics. 8 Dimensional analysis (DA) has been applied to various engineering fields, yet the scale-up prob- lem is only partially solved after performing dimensional analysis [60]. More precisely, DA enables an equivalent formulation of the original problem in a dimensionless Π-space, which achieves di- mension reduction and ensures scalability. To completely solve a scale-up problem, methodologies, such as modeling and design of experiments, need to be established in the Π-space. 1.2.2 Scale-up Modeling Methodologies for Advanced Manufacturing Scale-up modeling methodologies for advanced manufacturing can be classified into four cate- gories based on the requirement of domain knowledge and data as shown in Figure 1.2 [143]. Recall that dimensional analysis (DA) can be applied to achieve dimension reduction before modeling, which ensures the scalability of results. Figure 1.2: Applicability of modeling methodologies [143] Physical models, based on recognized principles and analysis of mechanisms, provide reliable understandings of the system and process. Though widely applied in traditional manufactur- ing, physical modeling approach is used less frequently in the study of advanced manufacturing processes, where systems are usually too complex to be characterized precisely by physical mod- els. Moreover, mechanisms of many potential scale-up processes are still not clear. Approximate 9 physical models are sometimes built to capture main characteristics of the process. For exam- ple, approximate physical models are built to characterize nanowire synthesis via SA-MOCVD [21, 87, 103]. Statistical models are empirical and exclusively rely on experimental data. Although physical knowledge is not required, purely statistical modeling is normally not preferred in the scale-up research of advanced manufacturing due to multiple considerations including the cost, sensitive responses, inestimable deviation and poor interpretation power. For example, empirical models are established to characterize gas-solid separation in cyclone separators [88, 54]. Physical-statistical modeling has been recognized as a promising approach in scale-up method- ology research for advanced manufacturing, consolidating the advantages and disadvantages of both purely physical and statistical modeling. For example, this strategy has been applied to modeling nanowire synthesis via SA-MOCVD [144] in the parameter space and modeling droplet formation in microfluidic devices [19, 42, 43, 40, 140, 128, 152] in the dimensionless Π-space. Cross-domain modeling was first devised by [130] in the parameter space when physical knowledge and data are both limited in nanomanufacturing for developing credible models us- ing physical-statistical modeling. The model was given in the form of a weighted combination of a physical model and a statistical model. 1.2.3 Design of Experiments for Scale-up Processes Experimentation is essential for investigating scale-up processes. Design of experiments (DOE), allowing the researcher to economize on experimental resources and extract maximum information from experimentation, has become an extremely important task due to high cost of experimenta- tion when innovative technologies are exploited. Experimental designs for scale-up processes can be divided into two categories based on the design space: either the dimensional parameter space or the dimensionless Π-space. 10 Design of experiments in the parameter space Most experiments in advanced manufacturing are designed and conducted in the parameter space. Various DOE methods have been employed. Traditional DOE methods have been applied to some nanomanufacturing processes. For in- stance, a factorial design followed by response surface analysis was applied to identify significant control factors in the preparation of colloidal nanoparticles [9, 126] for producing drug delivery systems, which demands accuracy and monodispersity in the size of nanoparticles. However, these designs are not applicable for complex experimental systems involving nonlinearity, high dimensionality and multiple domains. Space-filling designs are commonly used when the unknown response surface is possibly highly complex and nonlinear. Latin hypercubes [79, 115, 116, 15, 94, 57], uniform design [33] and other regular space-filling designs are appropriate for experiments with known feasible region, but may result in a waste of runs and resources when there exists infeasible region [154]. Unlike non-optimal design, optimal designs rely on process models and maximize the informa- tion matrix under a certain criterion. A smaller number of runs is required for optimal designs than non-optimal designs to achieve the same precision for parameter estimation, providing the possibility of a reduction in experimental costs. For instance, Zhu et al. present D-optimal de- signs for parameter estimation exponential-linear models proposed in [55] to characterize silica nanowire growth [155]. However, since proper models are often unknown priorly due to lack of physical knowledge, we often cannot directly apply an optimal design for scale-up experiments. Sequential design of experiments was first developed as a tool for quality control during World War II and has been recognized as an important strategy ever since then [16, 17, 137]. Design points are added sequentially according to information obtained from previous experiments. Most of the existing work in sequential designs are based on knowledge of proper scale-up models. For example, Zhu proposed a sequential Bayesian D-optimal design for a nanostructure growth model 11 [154]. Like regular optimal designs, the application of sequential optimal designs is hindered by lack of prior knowledge on the scale-up process. Robust parameter designs are particularly used to identify conditions that minimize response variation from uncontrollable factors by investigating the interactions between control and un- controllable noise variables. For example, in the study of ZnO nanoparticle synthesis via sol-gel process, molar concentration ratio of [LiOH]/[Zn(Ac) 2 ] proved to have significant effect on the size distribution [28] and a robust parameter design was employed to identify the optimal exper- imental settings. A combined use of DA and DOE Since advanced manufacturing often involves physical or chemical processes, which are likely to behave differently at different scales, models obtained in the parameter space at lab-scale may not be applicable for full-scale production. Recall that dimensional analysis enables an equivalent formulation of the original problem in a dimensionless Π-space, which achieves dimension reduc- tion and ensures scalability. A combined use of dimensional analysis and DOE methodologies has been proposed to address the scaling issue [2, 73, 92, 107]. Dimensional analysis (DA) was systematically introduced to statisticians by Albrecht et al. in [2]. The paper concerns the use of DA in DOE and its potential drawbacks from a statistician’s perspective. The authors point out the possibility of model misspecification when a key explana- tory variable is omissive in dimensional analysis. A robust-DA approach is proposed to produce designs that are capable of estimating both the DA and the empirical model simultaneously with high efficiency. Piepel [92] admitted the ground-breaking importance of the paper mentioned above, but also pointed out potential challenges in combining DA with DOE due to (i) potential correlations among the dimensionless variables (π numbers) introduced by shared base quantities, and (ii) the irregularity (e.g., non-convexity) of the feasible region of the dimensionless variables. The paper concludes with encouragement of more research on handling these challenges. Lin 12 and Shen also pointed out these two challenges in [73] together with another issue regarding the existence of dimensional physical constants that are not scalable. Although the combined use of DA and DOE methods has been recognized as a promising approach for scale-up research [2, 73, 92], fundamental challenges arise in two aspects: (i) High dimensionality may lead to complex response surfaces in the dimensionless Π-space, (ii) An ad- vanced manufacturing process may involve multiple unknown physical domains driven by differ- ent physical mechanisms, for example, the droplet formation process in microfluidic devices are dominated by different physical mechanisms in different domains while the domain identification remains unclear [42, 43, 19, 125, 5]. 1.3 Problems Addressed in Thesis Although there have been extensive research on advanced manufacturing processes, there still exists a gap between laboratory successes and large-scale advanced manufacturing. The early- stage study of advanced manufacturing processes mainly faces two challenges: (i) lack of under- standing of the process and (ii) restrictions on experimental resources. These two challenges are closely related. On one hand, a conclusive understanding of an advanced manufacturing process normally relies on extensive experimental research; on the other hand, design of experiments seeks guidance from understanding of the process. Therefore, scale-up modeling for the understanding of the process and design of experiments for economizing experimental resources are both essential to translating laboratory successes to large-scale advanced manufacturing. As shown in Fig. 1.3, we study scale-up modeling methodologies of characterizing, predicting and optimizing advanced manufacturing processes through three representative problems, each focusing on a different type of issues. The main issue in each task is due to (i) lack of physical knowledge and extensive data, (ii) nonnegligible process uncertainties, and (iii) the possibility of existing multiple physical mechanisms that dominate a manufacturing process in different feasible 13 Scale-up research in advanced manufacturing Process characterization, prediction & optimization via scale-up modeling Process observation via experimental research Design of experiments Issue I: Lack knowledge & extensive data guide support Issue II: Nonnegligible uncertainties Issue III: Multiple domains Physical-statistical model for nanowire growth Mixed model for a cyclone separator Microfluidic droplet formation model Experimental design for multiple-domain processes Figure 1.3: Thesis structure domains. As an extension of Task III, the last task investigates the design of experiments for the scale-up of multiple-domain manufacturing processes. The rest of thesis is organized as follows. In Chapter 2, we propose a physical-statistical model to utilize limited process knowledge and experimental resources in the study of nanowire synthesis. This work aims to analyze and optimize edge effect for III-V nanowire synthesis via SA-MOCVD process, which has been recognized as a promising scale-up process for the synthesis of nanowires. Existing studies based on idealized assumptions cannot capture well the distribution of nanowire growth on the substrate [59, 142, 21], which is critical for commercial-scale application that requires accurate control of nanowire growth. Therefore we propose a physical-statistical model that interprets well the local dependence of nanowire growth and enables the optimization of skirt area width on the substrate for uniform growth of nanowires. Chapter 3 investigates advanced manufacturing processes under nonnegligible uncertainties. We interpret the process uncertainties with random effects of some controllable variables and propose using mixed scalable models to capture manufacturing processes under uncertainties for the first time, which enables a robust parameter design to minimize process uncertainties. Specifically, we propose a Generalized Linear Mixed Model (GLMM) to characterize the gas-solid 14 separation process in a cyclone separator, and provide a robust parameter design to minimize uncertainties during the separation process. Our third and fourth task are motivated from the study of droplet formation in microfluidic devices, a promising process for large-scale production of high-quality metal nanoparticles. In order to realize industrial application, microfluidic reactors need to be scaled up with the foremost control over nanoparticle size. However, the droplet formation process is found to be characterized by different scale-up models in different physical domains of settings. Existing scale-up models that capture the process in a specific physical domain cannot be applied to the prediction under new process settings. Therefore we are motivated to investigate scale-up methodologies for multiple- domain processes. In Chapter 4, we address a class of scale-up modeling problems for multiple-domain processes. A multiple-domain process is dominated by different physical mechanisms in different feasible domains of settings. This work is motivated by the study of droplet formation in microfluidic devices, which is a multiple-domain manufacturing process. We establish a novel model framework and interpret a multiple-domain process as the outcome of coexisting physical mechanisms with different weights. Motivated by the third modeling problem, Chapter 5 explores the experimental design problem for multiple-domain processes. The objective is to identify physical domains for a multiple-domain process so that the modeling methodology discussed in Chapter 4 can be applied to obtain a comprehensive model. We propose a sequential design to detect domains adaptively, where the clustering scheme is introduced to DOE for the first time. To summarize, our work in this dissertation not only seeks to deepen our understanding of advanced manufacturing processes that have shown great promise in improving and even revo- lutionizing related industries, but also aims to provide some generic scale-up methodologies to bridge the gap between laboratory successes and large-scale advanced manufacturing. Possible extensions are discussed in Chapter 6. 15 Chapter 2 Physical-Statistical Scale-up Modeling: Analysis and Optimization of III-V Nanowire Synthesis via SA-MOCVD Selective Area Metal-Organic Chemical Vapor Deposition (SA-MOCVD) is a promising tech- nique for scale-up nanowire fabrication. Our previous study investigated the growth mechanism of SA-MOCVD processes by quantifying contributions from various diffusion sources. Yet the edge effect on nanostructure uniformity captured by skirt area diffusion has not been quantitatively analyzed. This work further improves the process understanding by considering the edge effect as a superposition of skirt area diffusion and “blocking effect” and optimizing the edge effect for uniformity control of nanowire growth. We directly model the blocking effect of nanowires in the process of precursor diffusion from the skirt area to the center of a substrate. The obtained physical-statistical model captures well the nanowire length distribution across the substrate and enables optimization of the skirt area width to improve the predicted structure uniformity of SA-MOCVD growth. 2.1 Introduction SA-MOCVD (Selective Area Metal-Organic Chemical Vapor Deposition) is a promising fabri- cation process for III-V nanowires such as GaAs, InP and InGaAs nanowires. Being economical, 16 flexible, and robust, SA-MOCVD preserves the strength of top-down lithography approach and produces high purity structures in high growth rates [110, Chapter 1]. As illustrated in Fig. 2.1 [142], SA-MOCVD synthesizes GaAs nanowires through well-patterned openings in the dielec- tric mask with precise position and shape control. Compared to conventional Au-catalyzed VLS (Vapor Liquid Solid) synthesis, the catalyst-free property of SA-MOCVD also avoids any metal contamination. Figure 2.1: Schematic of SA-MOCVD fabrication [142] Despite of its great potentials, controlled SA-MOCVD synthesis is limited by the understand- ing of its growth mechanisms. There have been debates on growth mechanisms of whether the process is governed by self-catalyzed VLS/VS growth [14, 76, 80] or diffusion based selective area epitaxy [23, 87, 58, 52, 21, 44]. Although selective area epitaxy is more widely accepted as the dominant mechanism, the diffusion sources and their relative contributions are still controversial. Empirical models [122, 59] or physical models [21, 87, 103] often produce conflicting results. For instance, some argue that nanowire height h should be proportional to the inverse of diameter d i.e. h∝ 1/d [49, 89, 59] while others believe h∝ 1/d 2 [122]. Side facet diffusion introduced in [59] is assumed to be proportional to the height of nanowires. However, side facet adsorption cannot be simplified as if the distribution of precursors in the space is uniform. Edge effect has to be considered when discussing the uniformity of nanowire growth in a selected area, as we could imagine intuitively, nanowires close to the boundaries tend to grow higher than those near the center. Normalized “precursor concentration” is then suggested [21] 17 to depict the side facet adsorption at different locations. But diffusion from skirt area is still neglected in nanowires near the boundaries. Our previous work attempted to develop a nanowire growth model for SA-MOCVD synthe- sis based on selective area epitaxy mechanism [142]. By quantifying contributions from various diffusion sources, we established a functional relation between nanowire height and process pa- rameters, and examined the relation based on real experimental data. The model explained both spatial and temporal growth patterns observed in various experiments and provided a more com- prehensive description of nanostructure growth variations in a general view. However, the work in [142] did not give a pertinent explanation to the variations near the center of the substrate, and no discussion or method was provided to optimize the skirt area setting in the substrate for uniformity control. This deviation originates from the omission of the influence of obstruction from surrounding nanowires in the process of precursor diffusion from the skirt area to the center of a substrate as shown in Figure 2.2. We call this influence on nanowire growth the “blocking effect” . Diffusion from skirt area and the blocking effect comprise the edge effect. Following this introduction, we will briefly review in section II our previously developed model and point out the missing blocking effect in the model. Section III establishes the SA-MOCVD growth model with consideration of edge effect as a superposition of skirt diffusion and blocking effect. Section IV attempts to provide a physical interpretation of edge effect. Section V develops an optimization method to find the optimal skirt area width so as to improve the nanostructure length uniformity. Summary is given in Section VI. 2.2 Our Previous Model and Skirt Area Effect As shown in Fig. 2.3, our previous study of III-V nanowire synthesis via SA-MOCVD process considered four sources of precursors: (i) adsorption on top surface, (ii) diffusion absorbed on facets, (iii) pervasion from the surface of the patterned mask, and (iv) diffusion from the skirt 18 Figure 2.2: Lateral diffusion from skirt area area [142]. The location of a nanowire is characterized by the coordinates of the center of its bottom surface on the mask. By modeling the volume increment of a nanowire at location (x,y) and time t, the proposed model considered skirt diffusion: √ 3 2 d 2 Δh(x,y,t) =C 1 √ 3 2 d 2 Δt +C 2 (2 √ 3d)h(x,y,t)Δt +C 3 √ 3 2 (p 2 −d 2 )Δt +C 4 S(x,y)Δt (2.1) with four terms at the right-hand side corresponding to the four sources of precursor diffusion respectively. In Equation 2.1, d refers to the side length of the hexagonal top surface, p refers to the distance between the center of two adjacent openings, and S(x,y) refers to the skirt area diffusion absorbed by the nanowire located at (x,y). With the implied assumption that the precursors from the top are provided at a constant speed and all the patterned openings are congruent hexagons distributed at equal distance, the 19 Figure 2.3: Four sources of precursor contributions in SA-MOCVD [142] adsorption on top surface and the pervasion from the surface of the mask are independent of the locations of nanowires. Since the growth is assumed to be well controlled before lateral overgrowth occurs, the growth of the height is proportional to the volume increment, and could thus be written as following [142]: Δh(x,y,t) Δt =C 0 +C 1 h(x,y,t) +C 2 S(x,y). (2.2) Here we abuse the notation C i in Equation 2.1 by merging some terms, where C 0 denotes the location-independent growth components, consisting of the adsorption from the top surface and the pervasion from the patterned mask surface; the second term on the right hand side corresponds to the side facet diffusion and the third refers to the skirt area diffusion. All the C 0 i s are constant coefficients when the growth conditions are determined, but may vary under different growth conditions such as varying rates of gas flow. 20 Taking the limit Δt→ 0, we could obtain a differential equation from the model in Equation 2.2. An observation of the growth length distribution at a fixed time can be simplified as a linear model: h(x,y) =a +bS(x,y). (2.3) As a main term of modeling skirt area diffusion, S(x,y) in [142] represents the ideal case of precursor diffusion from the skirt area to any nanowire at location (x,y) by neglecting edge effects. With the probability for an atom to travel a distance r being 1/(2πrL d )exp(−r/L d ) and L d being diffusion length [53], skirt area diffusion S(x,y) is defined in [142] as S(x,y)≈ Z (u,v)∈skirt area 1 2πL d p (x−u) 2 + (y−v) 2 exp ( − p (x−u) 2 + (y−v) 2 L d ) dvdu (2.4) One-dimensional growth distribution under skirt width 350μm, 250μm, and 150μm is observed along a straight line at a fixed time (see Figure 2.4) with one definite coordinate, which is sufficient to describe a symmetric two-dimensional scenario. The origin of the coordinate plane is selected to be the location of the nanowire at one corner of the growth area. Although the model in [142] could give an overall good explanation for the rapid change near the boundaries, there seems a lack of fit in the middle region. Authors in [142] pointed out that the lack of fit in the middle region might be due to the negligence of the “blocking effect”, i.e., surrounding nanowires hinder the diffusion from skirt area to the center region of a substrate. This study therefore intends to address this issue for improved control of nanowire growth by optimizing the skirt area width. 21 0 200 400 600 800 1000 1000 1500 2000 2500 3000 Position in the patterned region (um) Height (nm) ● ● ● ● ● ● ● ● ● ● ● ● ● ● Skirt 350 um Skirt 250 um Skirt 150 um Figure 2.4: Model prediction in [142]. Growth data are extracted from [21, 20] 2.3 A Growth Model Counting Blocking Effect 2.3.1 Blocking effect modeling An intuitive attempt to take the blocking effect into consideration is to modify the skirt diffusion term in the growth model. Since nanowires around the boundary of a substrate will block the precursor diffusion from the skirt area to nanowires in the middle region, it is expected that the blocking effect has a location-dependent property. One approach is to introduce a weight function W (x,y) into the growth model: h(x,y,t) = 1 C 1 {exp(C 1 t)− 1}{C 0 +C 2 W (x,y)S(x,y)} (2.5) 22 If we observe the growth at a certain fixed time, the revised model can be simplified into the same form as Equation 2.3. However, as shown in Figure 2.4, the skirt area mainly contributes to a narrow region near the boundaries due to limited diffusion length of precursor atoms. The modification via weight function W (x,y) therefore has limited impact on regions away from the skirt area. The limitation of using “blocked” skirt area diffusion to capture edge effect is reflected directly in the model coefficients where b proves to be much smaller than a in Equation 2.3. The ineffec- tiveness of such modification indicates that,a in Equation 2.3 actually includes some latent terms which are location-dependent. In other words, the blocking effect might impact some of the other sources of precursor transport: top surface absorption, side facet diffusion, and the pervasion from patterned mask area. Since the first and the third source of precursor transport are both height-independent, the side facet diffusion is the only source among the three that depends on the location of nanowires and thus is the only term that might be influenced by the blocking effect. It is also reasonable from physical point of view that the surrounding nanowires would block the diffusion of precursors in the space and thus block facet adsorption. Diffusion absorbed from side facets of a particular nanowire is then no longer proportional to its height, but is modified by its location in a more complicated way taking the effect of its neighbors into account. It is reasonable to assume that the side facet diffusion is positively correlated with the height of the nanowire, but with a location-dependent coefficient to weight the effect at different loca- tions. We introduce f(x,y) to describe the location dependence, which can be interpreted as the “concentration distribution” of precursors diffused from the skirt area to location (x,y). Here we assume that the concentration distribution f(x,y) is time-independent (this will be explained in Section 2.3.2), and the differential equation of the growth model is revised in Eq. 2.6 with the solution in Eq. 2.7. 23 0 200 400 600 800 1000 0 20000 40000 60000 80000 100000 120000 x-Position(um) S(x,y) Figure 2.5: Skirt diffusion without “blocking” (skirt width w = 150 μm) ∂h(x,y,t) ∂t = C 0 +C 1 f(x,y)h(x,y,t) +C 2 S(x,y). (2.6) h(x,y,t) = 1 C 1 {exp[C 1 f(x,y)t]− 1}{C 0 +C 2 S(x,y)} (2.7) As can be seen from the solution, nanowire height at a fixed time t 0 , i.e., h(x,y,t 0 ), could no longer be simplified in a linear form of Equation 2.3. h(x,y) turns out to be a product of two factors instead, both of which are location-dependent. S(x,y) is dominated by the number of precursor atoms that directly reach the nanowires and therefore its effect is limited to a narrow region near the boundaries (see Figure 2.5) . The side facet diffusion absorbed by nanowires near the boundaries is also dominated by the precursor atoms that arrive directly without obstruction, while the effect on nanowires farther from the boundaries is more likely to be dominated by detoured precursors. 24 2.3.2 Choice of concentration function In order to choose a proper function to capture the side facet diffusion, we focus our attention on the functions that are coincident with the following basic intuitions. 1. Time-independence We consider the concentration function to be independent of time. At the steady state, time-independent function is sufficient to describe the blocking effect. 2. Continuity A continuous concentration function is preferred over a discrete one because (i) the atom concentration in a connected space has to be continuous when the system is under control, (ii) although the diameter of nanowires and the patterning pitch are restricted by fabrication limitations, shapes and size of patterning can vary, therefore a discrete form lacks flexibility, (iii) the superposition of blocking effect from the surrounding nanowires has to be included with a discrete function, which can make the case unnecessarily complicated. 3. Trend The concentration function f(x,y) should be decreasing with the distance from the skirt area, since the atom concentration is higher near the boundaries and lower in the middle region due to the blocking effect. For the data observed in one dimension (Fig. 2.4), we choose the concentration function to be the following: f(x) = 1−exp − (x−μ) 2 2σ 2 (2.8) where μ = ¯ x, σ 2 =Var(x). 25 To make a comparison between the outcome of this model and the one in [142], we use the same series of x 0 i s for an interval [0, 1000μm] in Fig. 2.4, {x i |x i = 2i, i = 0, 1,..., 500}. (2.9) The mean value and the variance of the series are μ = 500 μm, σ 2 = 83666.7 μm 2 . 2.3.3 Statistical model formulation and estimation Considering the process and modeling uncertainties, we introduce Gaussian noise into Equa- tion 2.7. Furthermore, we eliminate time variable t as we observe the height at a fixed time. By merging constants we obtain the following statistical model h(x,y) =exp [a +bf(x,y)] [c +S(x,y)] +. (2.10) Here a, b, c and L d inS(x,y) (see Equation 2.4) are parameters to be estimated with the experi- mental data, some of which may vary under different experiment conditions. Figure 2.6: Model fitting errors with different L d 26 The estimation of model parameters can be accomplished through nonlinear least squares (nls) estimation using a form of Gauss-Newton iteration that employs derivatives approximated numerically [37, Appendix]. Since the unknown diffusion lengthL d is embedded inS(x,y) without analytical expression, we propose the following iterative estimation procedure: S.1 For an initial value of L d , estimate model (Equation 2.10) using nls based on the observed experimental data with skirt width w (e.g., w = 350μm, 250μm, and 150μm in our case). LetRSS(L d ,w) denote the residual sum squares (RSS) of the estimated model for given L d and w. S.2 LetT (L d ) = Σ w RSS(L d ,w) denote the sum ofRSS(L d ,w) over different skirt widths (call it “total error”) for the value of L d given in Step 1. S.3 Compute T (L d ) for all possible values of L d following Step 1 and Step 2. S.4 Find the real diffusion length L ∗ d that satisfiesdT (L d )/dL d = 0 for any skirt widthw. Here we assume the experimental condition remains the same except varying skirt area widths. Therefore, the diffusion length L d is identical for different w. To obtain appropriate starting values of model parameters in Equation 2.10, we use the log- transformation of Equation (2.10) to obtain the initial values of a, b and c. The total errorT (L d ) as a function of diffusion lengthL d is shown in Figure 2.6. Each possible value ofL d from 1μm to 100μm by an increment of 1μm is explored to determine the diffusion length. For every fixed value of L d , we compute a and b and the corresponding error. As shown in Figure 2.6, the minimum value of the residual sum squares arises when L ∗ d = 43 μm in this case. Given the diffusion length L ∗ d , the model coefficients of Equation 2.10 for different width w of the skirt area are listed in Table 2.1. As can be observed from the table, the width of the skirt area not only determines the effect of skirt diffusion, but also influences the extent of adsorption 27 on lateral surfaces through the varying coefficients a and b. The model apparently improves the fitting (see Figure 2.7) comparing with the fitting results in [142] (see Fig. 2.4). Table 2.1: Model estimation for various skirt area widths Skirt width Parameter Estimate Std. Error P-value a -5.181283 0.05524 4.64e-16 w = 350 μm b 0.4588492 0.03665 1.96e-07 c 161481.2 7315 8.16e-10 a -4.987778 0.05359 5.02e-16 w = 250 μm b 0.145533 0.03691 2.76e-03 c 117211.2 5408 9.77e-10 a -5.406677 0.05813 < 2e-16 w = 150 μm b -0.0976809 0.02970 7.21e-03 c 173848.3 9237 1.02e-09 Remark: A very interesting phenomenon discovered by the new model from Figure 2.7 is that the nanowire length shows a slight increase in the middle region when the width of the skirt area is 150μm, which suggests that the“blocking effect” is not limited to inhibit the growth of nanowires. The “blocking effect” may also result in a local reinforcement in the middle region under certain conditions, which cannot be explained under the previous assumption that the precursor atoms are distributed uniformly in the space. 2.4 Physical Interpretation of Skirt Area Effect The assumption of uniform precursor concentration in the growth chamber has been widely used in the literature of nanowire growth. Our study suggests that this assumption might not work well, at least for the case where edge effects cannot be ignored. Even if we may assume that the absorption speed on top of the nanowires is under perfect control and the speed of the pervasion from the mask surface maintains constant, the assumption of the uniform distribution of atoms in the growth chamber may not be appropriate for a finite region with boundaries and a surrounding skirt area. We attempt to use “diffraction” theory to explain the lateral diffusion of precursor atoms shown in Figure 2.7. In physics, diffraction occurs when a wave encounters an obstacle or emerges 28 0 200 400 600 800 1000 1000 1500 2000 2500 3000 Position in the patterned region (um) Height (nm) ● ● ● ● ● ● ● ● ● ● ● ● ● ● Skirt 350 um Skirt 250 um Skirt 150 um Figure 2.7: Predicted models with varying skirt area widths (L d = 43μm). Growth data are extracted from [21, 20] from an opening. Similar phenomenon may also exist when a lateral flow of precursor atoms encounters the nanowires, in other words, the blocking effect might be explained analogously with wave behavior in a general scenario. LetN direct (x,y) denote the number of precursor atoms that arrive directly at the side facet of a certain nanowire located at (x,y) without obstruction, and letN blocked (x,y) denote the number of precursor atoms that encounter “blocking” nanowires and finally reach the nanowire located at (x,y) after making a detour. As can been seen in Figure 2.2, the “wave fronts” depicted by semicircles represent the periodic lateral diffusion, where the periodicity is attributed to the precursor atoms periodically introduced in each reaction cycle. The solid directed lines describe the trajectories of precursor atoms that arrive directly at the facet of a certain nanowire without obstruction, while the directed dashed lines depict the trajectories of precursor atoms that detour. Under this framework, we conclude from observing Figure 2.7: 29 1. If the skirt area is large enough to provide strong lateral diffusion, for example, when the width of the skirt area equals 350 μm, N direct (x,y) dominates the side facet diffusion in the whole region. The wave periodicity is then not explicit in this case since N direct (x,y) is positively correlated with the number of possible “blocking” nanowires and thus the distance from the center of the region. The closer a nanowire is located to the center of the region, the larger the number of possible “blocking” nanowires is and thus the weaker the side facet diffusion it can receive. 2. If the skirt area is not so large, the periodicity starts to emerge, for example, when the width of the skirt area equals 150 μm. In this case scenario, N blocked (x,y) can only be neglected near the boundaries, and even dominates the distribution of lateral diffusion in a region near the center. Although we haven’t realize the theoretical analysis, the lateral flow of the precursor atoms seemingly behaves like a wave. The nonlinear property of the superposition of wave in amplitude and the intensity distribution’s dependence in symmetry may also be applied to lateral diffusion and thus provide a reasonable explanation for the local reinforcement in Figure 2.7 with the width of the skirt area being 150 μm. 3. We may predict that if the skirt area becomes even narrower, the periodicity of the lateral diffusion will be more explicit. Besides a rapid decrease in the growth near the boundaries, there might be a small-range fluctuation in the growth of the rest nanowires. 2.5 Optimal Width of Skirt Area The uniformity of the height of nanowires is an important performance metric in fabrication. The delicate relationship between the skirt diffusion and the blocking effect provides a possibility to further improve the predicted growth uniformity by properly setting the width for skirt area. 30 Specifically, the optimal width of skirt area will minimize the fluctuation of nanowire lengths in a certain region. We are now in position to define the uniformity of nanowire growthU as following: U(w) = Var[h(w)] E[h(w)] 2 , (2.11) whereVar[h(w)] andE[h(w)] represent the variance and expected mean value of height in obser- vational region given skirt width w. Smaller U indicates better uniformity. The variance itself is not sufficient to reflect the uniformity since it depends heavily on the mean value of the height. A square in the denominator is also necessary to make U(w) a pure number with no units. Noting that Equation 2.11 does not explicitly include time since uniformity is defined for a given growth time. The choice of growth time is mainly decided by other critical growth variables such as temperature and precursor flow rate. It can be true that at different time, the optimal skirt area width can take different values to maximize the uniformity. But for a given growth time T, there is one optimal choice of skirt area width to maximizeU(w), which is the case demonstrated in this case study. Our developed method shows that once the growth time is determined, we could use skirt area width as an adjustment factor to achieve better uniformity. Letw ∗ denote the optimal width such thatU(w) is minimized atw =w ∗ . An iteration method of estimating w ∗ is proposed as follows: S.1 Choose a search interval of skirt area width such that|w ∗ −w 0 | < Δ. The interval can be determined by experimental observation. Pick an initial value w 0 from the interval as a guess of w ∗ . S.2 Compute the uniformity U(w) in Equation 2.11 for all possible values of w 0 s within the interval [w 0 − Δ,w 0 + Δ], using the model parameters (a 0 ,b 0 ,c 0 ) of h(w) for the skirt area width of w 0 . 31 S.3 Let w ∗ denote the skirt width with minimum U(w) among all the possible skirt widths studied in Step 2. Stop if w 0 =w ∗ . Else, go to Step 4. S.4 Withw ∗ obtained in Step 3, search around the neighborhood of model parameters (a 0 ,b 0 ,c 0 ) and compute the uniformities for all possible groups of model parameters in that neighbor- hood. Select the group with minimum U(w) to replace/update (a 0 ,b 0 ,c 0 ). Go to Step 2 with w 0 updated by w ∗ . We demonstrate this optimization procedure using the example in Fig. 2.7. We may neglect nanowires near the boundaries due to the dramatic change of heights in those regions; the ob- servational region is determined by the producer’s tolerance of the waste of nanowires. Suppose an 80% output is required by a producer, we could then reduce our observational region from [0, 1000μm] to [100μm, 900μm]. 0 200 400 600 800 1000 0 500 1000 1500 2000 2500 Position in the patterned region (um) Height (nm) Skirt 350 um Skirt 250 um Skirt 247.8 um Figure 2.8: Uniformity for different w 32 Our experimental data provides a potential appropriate initial value of skirt width, w 0 = 250μm. As shown in Figure 2.7 and predicted by our physical interpretation, “blocking effect” is still weak when w = 350μm, however, it gradually becomes stronger as to the decrease in skirt width and starts to balance the direct diffusion from the skirt area when w = 250μm; further decrease in skirt width results in the dominant position of “blocking effect” in the distribution of lateral diffusion in the observational area. Therefore it is reasonable to assume that w ∗ could be obtained in the region [247.5μm, 252.5μm], with Δ = 2.5μm. For example, we choose{w i |w i = (247.5 + 0.1i)μm, i = 0, 1,..., 50} in our case. The iteration turns out to be convergent at w ∗ = 247.8μm, with corresponding parameters a =−5.034887, b = 0.113087, c = 109091.3. The neighborhood of each parameter is constructed as 95% confidence interval using t-distribution from nls fitting (see results in Table 1). Therefore w ∗ = 247.8μm is a reasonable and convincing estimation of the optimal width of skirt area in our case, giving the best predicted uniformity of nanowire growth, which provides a promising selection of skirt width for industrial production with a high uniformity demand (see Figure 2.8 where E[h(w)] 0 s are marked in dashed lines). 2.6 Summary and Conclusion In this chapter, we investigate edge effect, which is common in nanostructure growth where the nanostructures around the boundary of growth region vary significantly from those in the center. In the Selective Area Metal-Organic Chemical Vapor Deposition (SA-MOCVD), the width of skirt area is a critical factor of edge effect. This study identifies that side facet diffusion is influenced by the skirt area and the diffusion of precursors from skirt area will be blocked by nanowires. We further introduce an analytical model to quantify the effect of skirt area and its impact on side facet diffusion through a location-dependent concentration function. The nonlinear least square model fitting not only shows much improved prediction of growth distribution compared to the 33 previous model in [142], but also discovers that the blocking effect of nanowires to skirt area diffusion can either inhibit or facilitate the growth of nanowires, depending on the width of the skirt area. The edge effect is therefore a superposition of the skirt area diffusion and the blocking effect. This phenomenon cannot be explained by the prevailing assumption of uniform precursor distribution in literature. We attempt to interpret this skirt area effect using the diffraction theory. The established model of skirt area effect and physical interpretation leads to a procedure to optimize the skirt area width for structure uniformity of SA-MOCVD growth. 34 Chapter 3 Scale-up Methodology under Uncertainties: Study of Gas-Solid Separation in a Cyclone Separator Cyclone separators, designed for gas-solid and gas-liquid separation, have been widely utilized in industry due to various advantages. The performance of a cyclone separator is often measured by grade-efficiency, which captures the efficiency of separating particles of a given feed size from the dusty gas stream. Accurate prediction of grade-efficiency has huge implications on design, safety, energy consumption, and environment. However, the grade-efficiency of a cyclone separator is not well explored under uncertainties. This study presents a systematic approach to scale up a manufacturing process under uncertainties, involving dimensionality reduction via dimensional analysis, data collection, modeling under uncertainties and a robust parameter design. Specifically, in the study of gas-solid separation via cyclone separators, we interpret the process uncertainties with random effects and identify the Generalized Linear Mixed Model (GLMM) to be the best possible model in characterizing grade-efficiency, which achieves dramatic reduction in prediction variability. The improved understanding also leads to a robust parameter design to minimize process uncertainties. 35 3.1 Introduction Cyclone separators are designed to remove particles from a gas stream and have been widely applied in virtually every industry where there is a need to achieve gas-solid or gas-liquid sepa- ration, for example, in power, mineral and petrochemical industries. The first cyclone separator patent (No. US 325521) was granted to John M. Finch back in 1885, and cyclones have grown in popularity due to advantages in various aspects including: lack of moving parts, simple construc- tion, relative ease of maintenance, low energy consumption, high-temperature resistance, large treatment capacity, dust loading adaptability and high separation efficiency. However, an individual cyclone separator may exhibit a low efficiency for particles with size below some “cut size”, and the grade-efficiency, which captures the separation efficiency for a given feed particle size, is impacted by numerous physical parameters. Variations in those parameters can lead to poor prediction even if the grade-efficiency law holds in general. A predominant strategy of addressing this issue is to enhance models based on the first princi- ples. In the case of cyclone separators, separation efficiency has been modeled as a function of flow and structure parameters based on the computational analysis of gas-solid flow [70, 27, 83]. How- ever, these simplified computational models are subject to loss of precision due to the complexity in physical modeling of multi-phase flow. An alternative strategy is purely data-driven approach by deriving empirical models from data. In the case of cyclone separators, linear regression has often be applied in practice to obtain empirical grade-efficiency models. These models, however, are not scalable from lab environments to industrial settings since not all control factors are con- sidered. For instance, the separation efficiency is related to the Reynolds and Stokes number [88, 54]. Furthermore, little engineering insights can be obtained for improving system design. Despite its practical significance and extensive studies on the mechanism and grade-efficiency of cyclone separators, there is still a gap between the predicted behavior of a cyclone separator based 36 on existing models and experimental results, largely due to the complexity of flow distribution and uncertainties of measurement. This study aims to obtain a systematic strategy to address the aforementioned issue of uncer- tainties in scaling up manufacturing processes. A comprehensive scale-up methodology is estab- lished in Section 3.2 followed by a robust parameter design is proposed in Section 3.3. Analytical and experimental analysis is demonstrated in cyclone separation processes. Summary is given in 3.4. 3.2 Scale-up Methodology Under Uncertainties The performance of a cyclone separator is often measured by the grade-efficiency η δ , or the separation efficiency associated with particle diameter δ. A widely accepted approach to study the performance of a cyclone separator is to model the grade-efficiency empirically in the form of Eq. 3.1 [54], where α and β are functions of different settings regarding operation conditions, cyclone separator structures and particle properties. η δ = 1− exp −αδ β (3.1) Existing empirical models adopt linear regression based on strong assumptions, which suffer from prediction bias of expected grade-efficiency. To advance understanding of the separation process in a cyclone separator, we propose a systematic approach in Fig. 3.1. Our systematic approach starts with identifying initial set of factors through dimensional anal- ysis (DA) in the Π-space (spanned by dimensionless numbers) to achieve dimensionality reduction and enable model scalability. Extensive experiments are then conducted for model building and validation. To obtain a predictive model, we first analyze possible source of uncertainties in the 37 Study of manufacturing process under uncertainty Dimensionality reduction via dimensional analysis Experimental setup and data collection Yes Random effects Measurement uncertainties No Mixed Model Robust parameter design Fixed effects model Figure 3.1: Flowchart of a systematic approach to the study of manufacturing processes under uncertainty manufacturing process. If replicated experiments indicate the existence of random effects or mea- surement uncertainties, we establish a mixed model to capture both fixed and random effects on the process, which enables robust parameter design of the manufacturing system. Otherwise, we construct fixed effects models, such as Generalized Linear Models (GLM). 3.2.1 Dimensionality Reduction via Dimensional Analysis Instead of investigating the joint effect of all related physical quantities in the parameter space, we formulate the grade-efficiency modeling problem in the Π-space. Dimensional analysis is applied to generate dimensionless numbers that form the Π-space [12, 11, 117, 156], which enables dimension reduction and ensures the scalability of models. Note the selection of dimensionless numbers is not unique, yet the Π-spaces spanned by different selections of dimensionless numbers are equivalent and transferrable. In the study of gas-solid separation through a cyclone separator, 38 dimensionless numbers can be divided into three types, related to operation, cyclone separator structure and particles respectively. Table 3.1: Relevance list Variable Description Dimension Gas ρ gas density ML −3 μ gas viscosity ML −1 T −1 V i inlet velocity LT −1 V tm maximum tangential LT −1 velocity in separator Particle δ particle diameter L d m mean diameter at inlet L σ ξ mean deviation of L diameter at inlet ρ p particle density ML −3 C i gas-solid mass ratio 1 at inlet Cyclone D body diameter L H body height L a,b inlet heights L Gravity g gravitational constant LT −2 1. Operation-related dimensionless numbers include the Reynolds number represented byRe = ρViD μ and the Froude number denoted by Fr = gH V 2 i . 2. The dimensionless number purely related to the cyclone separator structure is k a = 4ab πD 2 3. Particle-related dimensionless numbers are derived as S tk = ρpδ 2 Vtm 18μD (Stokes number), δ dm , σ ξ , dm D and C i . With these dimensionless numbers, the grade-efficiency modeling problem is naturally formu- lated in the form of Eq. 3.2, where η δ denotes the grade-efficiency of the cyclone separator with respect to particles of diameter δ. η δ =F (S tk , δ d m ,σ ξ , d m D ,C i ,Re,Fr,k a ). (3.2) 39 The rest of the paper aims to figure out the explicit form of F (S tk , δ dm ,σ ξ , dm D ,C i ,Re,Fr,k a ), which captures the joint effect of the dimensionless numbers on the grade-efficiency. 3.2.2 Experimental Setup and Data Collection The cyclone separator achieves gas-solids separation as follows. A dusty gas stream is initially introduced tangentially into the cyclone separator utilizing pressure difference, and then passes through the scroll-type inlet, where the linear motion is transformed to high-speed rotation. When dense particles fail to follow the tight curve of the rotating stream due to high inertia, they strike the outside wall of the cyclone, and fall to the bottom of the cyclone separator where they can be removed. As the rotating flow moves towards the narrow end of the cyclone, smaller particles are separated. It follows that smaller particles are removed with a lower efficiency. The grade- efficiency is thus adopted to capture the efficiency for a given feed particle size, which is critical to industrial applications. In order to systematically understand the joint effect of controllable parameters on the grade- efficiency, we conducted 693 groups of experiments and measured the grade-efficiency under dif- ferent treatments of all the influential dimensionless numbers in Eq. 3.2. Experiments can be divided into two parts. 1. Measurement of the gas flow field utilizes the Laser Doppler velocimetry (LDV). The dis- tribution of tangential and axial velocity of gas in the cyclone separator is measured to determine the maximum tangential velocity of gas in the cyclone separator V tm and calcu- late the Stokes number S tk . This measurement can bring random effects. η δ = 1− (1−η T ) f e (δ) f f (δ) . η T = 1− C e C i . (3.3) 40 2. Measurement of the gradient-efficiency is realized through the use of WELAS2000 Aerosol Spectrometer in the outlet of cyclones. Particles following a priorly known size distribution are first introduced into the inlet of cyclones, with given gas-solid mass ratioC i and velocity V i . The aerosol spectrometer then detects online the gas-solid mass ratio in the outlet of cyclones C e . The grade-efficiency η δ with respect to particle diameter δ can be calculated using Eq. 3.3, where f f (δ) and f e (δ) refer to the fraction of feed and escaped solids with diameter δ respectively, and η T refers to the total efficiency. Remark: We exploit Monte Carlo cross validation (MCCV) [146] as a measure of fitting level. The data collected from experiments is randomly split into training data of 493 groups and validation data of 200 groups. For each of the 10 random splits, we fit a model based on the training data and apply the model to the validation data. The predictive accuracy of the model is assessed by the squared root of the average mean squared errors of 10 splits, namely the root-mean-square error (RMSE) of cross-validation. 3.2.3 Identification of Random Effects and Mixed Modeling of Cyclone Separation Process We are motivated to establish a mixed model in the study of cyclone separators since both hard-to-control factors and uncertainties of measurement arise in the cyclone separation process. For example, particle sizes and velocities are expected to follow certain distributions, as opposed to being constant values. The uncertainties in particle size and velocity may lead to uncertainties in grade-efficiency predictions. We therefore introduce the framework of mixed-effects modeling, where hard-to-control factors are treated as random effects. Experiments are replicated under selected settings to observe the variation of grade-efficiency with respect to dimensionless numbers S tk , δ dm , Re and Fr, where either the particle size or the velocity is involved. The mixed-effects models are established to further identify the sources of 41 grade-efficiency for variation reduction purpose. Mixed effects models [66, 74, 136] have been widely applied to interpret repeated measurements of the same statistical units, where effects on the response can be decomposed into two parts, namely, the fixed and random effects. As shown in previous sections, the expected value of grade-efficiency can be well captured by fixed effects of relevantπ numbers, while the local variation can only be interpreted by random effects. In order to accommodate hard-to-control factors which bring random effects, we introduce the general framework of Generalized Linear Mixed Models (GLMM) proposed by Wolfinger and O’connell [136] in Eq. 3.4, where ∈ R n denotes the vector of observed grade-efficiency and e is a noise vector with E(e|) = 0. = +e, g() = X +Z. (3.4) The monotonic link functiong() defines a mapping from (0, 1) toR. The vectors∈R p and ∈ R q characterize the fixed and random effects respectively, associated with model matrices X∈R n×p andZ∈R n×q . Each row ofX corresponds to a set of (S tk , δ dm ,σ ξ , dm D ,C i ,Re,Fr,k a ). Log-transformed values ofπ numbers are used to avoid skewness. The random effects are assumed to be normally distributed with mean 0 due to the symmetric distribution of observed values with respect to the expected value of grade-efficiency, i.e. ∼ N(0,D), where D is diagonal with nonzero entries D ii = i for some vector ν∈R q . We consider a grade-efficiency model under the GLMM framework that relates the local vari- ation toπ numbers directly in the form of Eq. 3.5, where g(μ) is an element of g(), (π 1 ... π 8 ) corresponds to a setting of (S tk , δ dm ,σ ξ , dm D ,C i ,Re,Fr,k a ), β 0 i s are fixed effects, and φ 0 i s charac- terize the random effects that follow a normal distribution N(0,σ 2 i ), i = 0,..., 8. Both β 0 i s and σ 0 i s are unknown parameters to be estimated. 42 g(μ) = β 0 + 8 X i=1 (β i +φ i ) ln π i , φ i |π i ∼ N (0,σ 2 i ), i = 1,..., 8. (3.5) In the experiments of gas-solid separation via cyclone separators, the variation of grade- efficiency is mainly introduced by the control of particle size and velocity. Therefore we only need to consider the random effects of relevant dimensionless numbers, S tk , δ/d m , Re and Fr. As shown in Fig. 3.2, the grade-efficiency is measured repeatedly under certain settings of π numbers in our experiments, and the variation exhibits dependence on π numbers. We exclude the random effects of Fr when investigating the variation of grade-efficiency with respect to two coupled settings of Re and Fr. It follows that the GLMM can be reduced to Eq. 3.6. g(μ) = β 0 + (β 1 +φ 1 ) ln(S tk ) + (β 2 +φ 2 ) ln (δ/d m ) + β 3 ln(σ ξ ) +β 4 ln(d m /D) +β 5 ln(C i ) + (β 6 +φ 6 ) ln(Re) +β 7 ln(Fr) +β 8 ln(k a ), φ i |π i ∼ N (0,σ 2 i ), i = 1, 2, 6. (3.6) Remark In the GLMM of grade-efficiency, we assume that the variation is linear as a function of the absolute value of log-transformed dimensionless numbers. This assumption captures well the trend of variation with respect to S tk and δ/d m . The exponential decrease in the variation of grade-efficiency at low values of S tk andδ/d m is characterized by the decrease in|ln(S tk )| and |ln(δ/d m )| whenln(S tk ) andln(δ/d m ) are negative. A slight increase is observed at larger values ofδ/d m , whereln(δ/d m )> 0 and|ln(δ/d m )| is increasing at a decreasing rate. While the notation of ln(Re) is used in GLMM, the random effects φ 6 is actually treated as a categorical variable 43 Figure 3.2: Sample standard deviation of grade-efficiency against Stk and δ dm with two levels. We may consider a more general nonlinear mixed-effects model to include the random effects of Re and Fr. We exploit the Matlab function ‘fitglme’ to achieve maximum likelihood estimation through Laplace approximations [78, 98]. The ‘log-log’ link function is selected as a result of comparison with other link functions to attain the maximum likelihood approximations and minimum RMSE. Estimation results of the model in Eq. 3.6 are listed in Table 3.2, including the 95% confidence interval of random effects. We further reduce the GLMM to Eq. 3.7 by excluding insignificant effects via model selection. Estimation results of the reduced GLMM are listed in Table 3.3, including the 95% confidence interval of random effects. g(μ) = β 0 + (β 1 +φ 1 )ln(S tk ) + (β 2 +φ 2 )ln (δ/d m ) + β 3 ln(σ ξ ) +β 4 ln(d m /D) + (β 6 +φ 6 )ln(Re) + β 7 ln(Fr), φ i |π i ∼ N (0,σ 2 i ), i = 1, 2, 6. (3.7) 44 Table 3.2: GLMM model estimation (RMSE=1.739359%) Estimate Std. Error P-value β 0 61.635 6.1895 6.6148e-22 β 1 -6.7138 0.73396 6.5979e-19 β 2 14.426 1.4674 1.9987e-21 β 3 27.059 5.2852 3.9814e-07 β 4 14.144 1.5315 3.181e-19 β 5 -0.043003 0.11178 0.70056 β 6 1.556 0.45066 0.00058912 β 7 -2.1894 0.38867 2.5856e-08 β 8 -0.79902 1.0722 0.45638 Estimate Lower Upper σ 1 0.0011361 0.00034872 0.003701 σ 2 0.16096 0.13412 0.19317 σ 6 0.037954 0.02775 0.051911 Table 3.3: Reduced GLMM estimation (RMSE=1.717276%) Estimate Std. Error P-value β 0 64.952 4.7019 1.7896e-38 β 1 -6.8608 0.64062 7.4357e-25 β 2 14.721 1.2805 4.303e-28 β 3 23.673 3.2352 7.0839e-13 β 4 14.794 0.98611 3.2965e-44 β 6 1.6486 0.40073 4.3607e-05 β 7 -2.2869 0.36419 6.0287e-10 Estimate Lower Upper σ 1 0.0011382 0.00035134 0.0036874 σ 2 0.16095 0.13411 0.19315 σ 6 0.038589 0.028213 0.052781 As shown in Fig. 3.3, the residual not captured by the GLMM model is identically distributed across the range of observations. To further illustrate the existence of random effects in the cyclone separation process, we compare the performance of GLMM with that of Generalized Linear Models (GLM). In the study of grade-efficiency for a cyclone separator, we set up the GLM framework in the form of Eq. 3.8, where T (·) defines a transformation of the response with parameter λ. d(·) refers to the distribution of (transformed) responses with mean μ and variance φ. Both Gaussian 45 Figure 3.3: Fitted results of GLMM for grade-efficiency and beta regression model are examined, with “complementary log-log” and “loglog” link chosen respectively to maximize likelihood and minimize RMSE based on cross-validation results. T(η|x;λ) ∼ d(μ,φ) μ = A 0 +A 1 ln(S tk )+A 2 ln(δ/dm) + A 3 ln(σ ξ )+A 4 ln(dm/D)+A 5 ln(C i ) + A 6 ln(Re)+A 7 ln(Fr)+A 8 ln(ka). (3.8) As shown in Table 3.4, GLMM achieves the lowest RMSE of grade-efficiency among four models, which is approximately 72% lower than that of the ordinary linear model and 63% lower than that of the best possible Generalized Linear Model. This indicates that random effects are important in predicting the behavior of a cyclone separator. The existence of random effects in the cyclone separation process is further illustrated in Fig. 3.4. Although the beta regression model captures well the expected value of the grade-efficiency 46 Table 3.4: Comparison of root-mean-square error Model Linear Gaussian Beta GLMM RMSE(%) 6.15548 4.91621 4.63503 1.71728 (both the linear and the Gaussian model exhibit asymmetric distribution of observations along the fitted curve), the amplitude of variation is not identical across the range of grade-efficiency, which illustrates the existence of random effects that are not captured in GLMs. Figure 3.4: Comparison of fitted values of grade-efficiency models The generalized linear mixed model not only achieves a high accuracy of characterizing the grade-efficiency, but also provides further insights into the engineering design of a cyclone sepa- rator to achieve variation reduction, which is critical to industrial implementations. Insights of engineering design can be obtained through analysis on both fixed and random effects. 1. Analyze Fixed Effects to Maximize Grade-Efficiency Fixed effects of the GLMM model enable prior control of expected grade-efficiency μ. We have g(μ) = −ln(−ln(μ)) by using the ‘log-log’ link function. It follows that μ = exp[−exp(−g(μ))] and ∂μ/∂g(μ) = exp[−exp(−g(μ))−g(μ)] as illustrated in Fig. 3.5, 47 Figure 3.5: μ and ∂μ/∂g(μ) as a function of g(μ) where g(μ) is a function of the π numbers given in Eq. 3.7. The grade-efficiency mono- tonically increases with respect to the function of π numbers g(μ), with the increase rate decreasing exponentially before approaching zero. Combination of these characteristics and the GLMM model gives conditions of settings for a desired range of grade-efficiency. For example, the GLMM model suggests g(μ)> 2.97 for grade-efficiency to be higher than 95% and g(μ)> 4.60 to let the expected grade-efficiency μ> 99%, which gives conditions of π numbers for an acceptable grade-efficiency. The effect of most physical parameters can be observed directly from model coefficients. Note that the “log-log” link corresponds to a monotonically increasing function. For example, to investigate the effect of inlet velocity, we need to look at the effect of related dimensionless numbers Re =ρV i D/μ andFr =gH/V 2 i . Since the model coefficient with respect to Re is positive while that ofFr is negative, it follows that the grade-efficiency can be enhanced by increasing the inlet velocityV i . This is consistent with the mechanism of a cyclone separator since particles are easier to strike the wall and fall down to be removed in a stream rotating at a higher speed. However, the effect of some physical parameters cannot be directly observed from the model coefficients. For example, the body diameter of the cyclone separator, denoted byD, appears in Re = ρV i D/μ, k a = 4ab/(πD 2 ), S tk = ρ p δ 2 V tm /(18μD) and d m /D. On one hand, the 48 sign of model coefficients with respect to Re, S tk , k a suggests a larger D to enhance the grade-efficiency, while that associated withd m /D suggests a smallerD. Therefore the effect of D on the grade-efficiency need to be analyzed in the physical parameter space. In our experiments, ∂T (η δ )/∂D = (A 6 −A 1 −A 4 − 2A 8 )/D < 0, indicating that a smaller body diameter results in higher grade-efficiency. This can be explained by the fact that particles are easier to be removed when the gas stream rotates with a smaller radius. We may need to further consider the trade-off between a high grade-efficiency and a high separation throughput in engineering design. 2. Analyze Random Effects to Minimize Variance of Grade-Efficiency Random effects of the GLMM model in Eq. 3.7 capture the variation of grade-efficiency with respect to dimensionless numbers that involve hard-to-control variables, the particle size and velocity. Besides distinct performance at two levels of Re and Fr, the variation of grade- efficiency decreases with respect to the decrease in absolute values of log-transformed di- mensionless numbersS tk andδ/d m . Smaller absolute values are thus suggested for|ln(S tk )| and|ln(δ/d m )| for variance reduction, i.e. values closer to 1 are suggested for both S tk and δ/d m . Specifically, a smaller value ofS tk ∈ (0, 1) is suggested to reduce variation associated with random effects characterized by φ 1 . So far, we have analyzed fixed and random effects independently. However, suggestions on settings of cyclone separators obtained from two perspectives may not consist with each other, and there might exist a “trade-off” between large value and small variation of grade-efficiency. For instance, on one hand, the negative sign of fixed effect β 1 suggests a smaller value of S tk for higher grade-efficiency; on the other hand, a smaller value ofS tk ∈ (0, 1) leads to a larger absolute value of ln(S tk ) and results in amplification of variation characterized by the random effect φ 1 . To achieve an engineering design of cyclone separators for both high and stable grade-efficiency, we are motivated to forward a robust parameter design based on generalized linear mixed models. 49 3.3 Robust Parameter Design based on Generalized Linear Mixed Model Robust parameter design has been recognized as an important methodology of reducing the performance variation of an engineering system [111, 113, 112, 84, 91]. Factors are divided into two categories based on whether the values remain fixed once they are chosen, namely control factors and noise (hard-to-control) factors. In the case of gas-solid separation using cyclone separators, the particle size and velocity are hard-to-control factors among all adjustable factors. Robust parameter design achieves variation reduction by selecting the setting of control factors to make the system less sensitive to noise variation. We apply the responding modeling approach [134, 108] to obtain a robust parameter design of the gas-solid separation using cyclone separators. The generalized linear mixed model in Eq. 3.7 is regarded as the response model, which involves both control and hard-to-control factors. Note g(μ) is used as an equivalent response of μ for computation simplification, which monotonically increases with respect to μ via a link function. Suppose a larger value is a priority compared to less variation for the grade-efficiency of a cyclone separator, which is normally the case. It follows that the robust parameter design becomes a larger-the-better problem. We may consider a three-step procedure to obtain a robust parameter design based on the generalized linear mixed model as a generalization of the two-step procedure suggested in [138]. 1. Recommend on Fixed Effects Fixed effects β 3 and β 4 are independent of the hard-to-control factors, particle size and velocity. The positive sign ofβ 3 andβ 4 indicates that the value of grade-efficiency increases with respect to dimensionless numbers σ and d m /D. 2. Analysis of Control-by-Noise Interaction Effects 50 According to Fig. 3.2, the couple settings Re = 27668.93, Fr = 40.30048 results in smaller variance of the grade-efficiency and is thus recommended. Also, an observation of the response model in Eq. 3.7 suggests values closer to 1 for both S tk and δ/d m , since the variation is reduced with respect to|ln(S tk )| and|ln(δ/d m )|. 3. Minimize Predicted Variance based on GLMM We only include variance due to particle sizeδ in the transmitted variance model Eq. 3.9 since the choice between two coupled levels of Re and Fr is obvious from Fig. 3.2. Quantitative analysis on the influence of velocity requires subsequent experiments. Var[g(μ)] = ∂g(μ) ∂δ 2 Var(δ) = (2β 1 + 2φ 1 +β 2 +φ 2 ) 2 Var(δ) δ 2 . (3.9) Although there is no adjustment variable in the transmitted variance model apart from par- ticle sizeδ, the third step of the procedure still provides valuable insights into the engineering design of the gas-solid separation process through a cyclone separator. Since particle size is normally harder to control for smaller particles, it is normal that Var(δ)/δ 2 increases expo- nentially with respect to the decrease in particle size δ. The grade-efficiency of separation is more robust for larger particles. 3.4 Summary and Conclusion In this paper, we proposed a systematic approach to scale up a manufacturing process un- der uncertainties. Our approach starts with dimensionality reduction via dimensional analysis (DA), and formulates the modeling and design problem in the dimensionless Π-space. Extensive 51 experiments are then conducted to support model building and validation. We introduced ran- dom effects to capture process uncertainties, and established the framework of Generalized Linear Models (GLM) to examine the expected grade-efficiency of cyclone separators, which achieves dramatic reduction in prediction variability and provides insights into the engineering design of the cyclone separator system through robust parameter design. 52 Chapter 4 Multiple-Domain Scale-up Modeling: Droplet Formation in a Coated Microfluidic T-junction Nanoparticles has great potentials to revolutionize the industry and improve our life in var- ious fields such as energy, security, medicine, food, and environmental science. Droplet-based microfluidic reactors serve as an important tool to facilitate monodisperse nanoparticles with a high yield. Depending on process settings, droplet formation in a typical microfluidic T-junction is explained by different mechanisms, squeezing, dripping or squeezing-to-dripping. Therefore the manufacturing process can potentially operate under multiple physical domains due to uncer- tainties. Although mechanistic models have been developed for individual domains, a modeling approach for the scale-up manufacturing of droplet formation across multiple domains does not exist. Establishing an integrated and scalable droplet formation model, which is vital for scaling up microfluidic reactors for large-scale production, faces two critical challenges: high dimensional- ity of modeling space and ambiguity among boundaries of physical domains. This work establishes a novel and generic formulation for the scale-up of multiple-domain manufacturing processes and provides a scalable modeling approach for the quality control of products, which enables and supports the scale-up of manufacturing processes that can potentially operate under multiple physical domains due to uncertainties. Specifically, we reduce dimensionality of modeling space by identifying low-dimensional model structures through dimensional analysis and adoption of 53 primary factor. A set of basis functions are chosen to characterize the process across multiple physical domains. The obtained model for droplet formation in a coated microfluidic T-junction not only explains the experimental results well but also provides new physical insights for scale-up production. 4.1 Introduction Synthesis of solid particles, liquid droplets and gas bubbles is critical for pharmaceutical and chemical engineering applications [26, 131, 145]. Droplet-based microfluidic devices manipulate immiscible fluids in channels of micrometer size [3, 75, 18, 153, 39, 114, 121, 42, 19]. The high surface-area-to-volume ratio within microchannels guarantees uniform temperature throughout the reaction volume, and convective mixing within droplets ensures rapid homogenization. These properties make droplet microfluidic reactors a viable technology for the scalable synthesis of high-quality metal nanoparticles. In order to produce particles on an industrial scale with low cost, microfluidic reactors need to be scaled up for high throughput and high yield with the foremost control over droplet size [19, 120, 67, 100]. Scale-up modeling, which refers to the process modeling approaches that enable and support economical production at commercial scale, is thus crucial for the quality control of nanoparticles[143]. Scale-up modeling of the droplet formation in the microfluidic channels face several key challenges: • High dimensionality. Description of droplet formation in microfluidic channel involves a large number of physical parameters or quantities. Even by conducting dimensional analysis using scaling law [43, 25, 42], the number of obtained dimensionless numbers can be large, which gives rise to a high-dimensional problem. For instance, droplet size after scaling is proposed to be a function of more than five dimensionless numbers in squeezing-to-dripping domain with each being a combination of multiple physical parameters [47, 43]. This poses 54 both experimental and modeling challenges to understand the response surface in a high- dimensional space. Currently, the droplet formation experiment in practice uses the one- factor-at-a-time approach, i.e., testing one dimensionless number at a time and fixing the rest [43, 25, 42, 19, 40, 140, 128, 152], which only guarantees the understanding in a projected low-dimensional space. • Multiple physical domains. The droplet formation in a microfluidic channel is multiple- domain [19, 40, 140, 128, 152], and a microfluidic T-junction can produce droplets either in squeezing, dripping or squeezing-to-dripping domain (Due to our focus on producing monodisperse droplets, jetting domain is not herein discussed). Since different domains are dominated by different mechanisms, model structures depicting droplet formation vary with physical domains, which significantly increases the complexity in experimentation and modeling. Current practice is to discover individual domains through experimentation and then establish individual models within each domain. For instance, the domains are classified based on the capillary number Ca: squeezing with Ca< 0.002, squeezing-to-dripping with 0.002 < Ca < 0.01, dripping with 0.01 < Ca < 0.3, and jetting domain with Ca > 0.3 [141]. In the squeezing domain, the droplet formation process is explained by the pressure drop across the droplet during formation; while in the dripping domain, the process is interpreted by the balance between shear force and interfacial force. However, there are inconsistencies and ambiguity in defining the boundaries of physical domains [25], which hinders the application of these models in full-scale manufacturing. To address these challenges for full scale production, this work aims at establishing a unified scale-up model across multiple domains under uncertainties. In the rest of the paper, Section 4.2 introduces our model formulation and methodology for the multiple-domain scale-up modeling problem. Section 4.3 applies and demonstrates the proposed generic formulation and methodology 55 to predict droplet formation in the coated microfluidic T-junction, with discussions in Section 4.4. We conclude in Section 4.5. 4.2 Scalable Modeling Methodology for Multiple-domain Manufacturing Process In order to scale up manufacturing processes, we need to first address the so-called scale-up issue: how to translate the understanding of a process from lab scale to commercial scale such that the optimal properties can be determined a priori for future operations. The science base of achieving the scalability of engineering models is the scaling law, which captures the scale-invariant characteristics of an engineering system [7, 156]. Dimensional analysis serves as an important tool to transform the parameter space to the so-called Π-space spanned of dimensionless numbers, which achieves dimension reduction and ensures the scalability of models. Note although there is no unique selection of dimensionless numbers, the dimension of Π-space is uniquely determined by dimensional analysis. Droplet formation in microfluidic channels represents a class of “multiple-domain” scale-up modeling problems, which involve different physical phenomena and mechanisms [55, 130]. Since the boundaries between different physical domains can sometimes be ambiguous, it is often not known a priori which physical domain the process belongs to. Our objective is to establish a unified scale-up model across multiple domains under uncertainties to facilitate full-scale production. Our methodology is illustrated in Fig. 4.1. We first conduct dimensional analysis to formulate the problem in the Π-space. Then, we identify the primary factor and secondary factor for further dimension reduction. Definition 1. The primary factor consists of dimensionless numbers that characterize the physical domains of engineering systems; the rest of dimensionless numbers form the secondary 56 factor. Each physical mechanism, which is likely to dominate a certain domain, is then charac- terized by a model structure that only depends on the primary factor, termed as a basis function. Basis functions are assumed to be linearly independent. The rationale of this definition is based on the observation that physical domains of engineering systems are often be classified by one or only a very few dimensionless numbers, which form the primary factor henceforth. For example, the Reynolds number is widely used to predict flow patterns in the case of a bounding surface. Laminar flow occurs at low Reynolds numbers where viscous forces dominate, and turbulent flow occurs at high Reynolds numbers where the flow is dominated by inertial forces [104]. In the case of microfluidic droplet formation, four domains are differentiated by the capillary number defined in Eq. (4.5). Since the multiple-domain property of the process is fulled captured by the primary factor, the original scale-up modeling problem can be divided into several subproblems in lower-dimensional subspaces. Each physical domain is dominated by one mechanism, and the secondary factor contributes to the weights of each mechanism. For instance, the effect of flow rate ratio on the dimensionless size of droplets appears to be linear across different domains [42, 40, 140, 128, 152], and can thus be treated as a weighting factor of domains. Based on this rationale, we are now in the position to formulate the high-dimensional multiple- domain scale-up modeling problem. Let z denote the response of interest, and let x ∗ , x ◦ denote the primary factor (PF) and the secondary factor (SF) respectively. In the high-dimensional Π-space, there exists the functional relation in Eq. (4.1), where and denote the random noise and model parameters respectively. z = Φ(x ∗ , x ◦ ) + (4.1) 57 Objective: unified scale-‐up model Dimension Reduction: identify low-‐dimensional structure Dimensional Analysis: transform parameter space to the pi-‐space Unified Scale-‐up Model: by integrating basis functions across domains Identify Model Structure in Each Domain Figure 4.1: Methodology to construct the scale-up model for a multivariate physical system Projecting the response surface Φ(x ∗ , x ◦ ) onto the lower-dimensional space spanned by the primary factor x ∗ , we obtain Φ(x ∗ |x ◦ ), i.e., the response conditioning on given settings of the secondary factor x ◦ . Since the model structure of Φ(x ∗ |x ◦ ) in each domain is dictated by x ∗ only, we adopt a set of linearly independent basis functions S ={f i (x ∗ )} K i=1 to represent conditional response function Φ(x ∗ |x ◦ ) in Eq. (4.2), where f i (x ∗ ) is the ith basis function used to charac- terize a model structure, and its coefficient β i characterizes the effect of secondary factor, i.e., a scaling factor for modeling structure f i (x ∗ ). Note the definition of primary and secondary factor guarantees the existence of model decomposition in Eq. 4.2. Φ(x ∗ |x ◦ ) = K X i=1 β i f i (x ∗ ) (4.2) wheref i (x ∗ ) is theith basis function used to characterize a model structure, and its coefficient β i characterizes the effect of secondary factor, i.e., a scaling factor for modeling structure f i (x ∗ ). In dripping domain with high capillary number, for instance, experimental studies show that 58 droplet size is proportional to Ca −0.25 [125, 40], i.e., a candidate basis function in the dripping domain is f i (Ca) = Ca −0.25 . It is important to note that the basis functions may vary across domains. Based on the conditional response model in Eq. (4.2), we deduce the full model in Eq. (4.3). Since the measured response is finite, there exists a reference frame such that the response is always non-negative, and Tonelli’s theorem holds. The exchange between sum and integration is thus valid. Φ(x ∗ , x ◦ ;) = Z x ◦ Φ(x ∗ |x ◦ )dx ◦ = Z x ◦ K X i=1 β i (x ◦ )f i (x ∗ )dx ◦ = K X i=1 Z x ◦ β i (x ◦ )dx ◦ f i (x ∗ ) = K X i=1 g i (x ◦ )f i (x ∗ ) (4.3) Proposition 2. Weight functions g(x ◦ ; ◦ i ) 0 s satisfy K X i=1 α i g(x ◦ ; ◦ i ) =g(x ◦ ; ◦ ) (4.4) for all constants α 0 i s∈R whereg i (x ◦ ) can be interpreted as a weight function forf i (x ∗ ) given the settings of secondary factor x ◦ , noting that g i (x ◦ ) 0 s share the function form with different parameters. Remark: The model formulation in (4.3) essentially suggests the statistical additive model frame- work [51, 13, 38] for the high-dimensional multiple-domain scale-up modeling problem. The formulation enables the application of modeling techniques in statistics for model building and estimation. 59 Proposition 2 is obtained from Definition 1, since, otherwise,∃j≤K such that g(x ◦ ; ◦ j ) and K P i=1,i6=j α i g(x ◦ ; ◦ i ) are linearly independent, which results in basis functions and contradicts with the definition of secondary factor. Although experimental literature assist to identify candidate basis functions, there exist dis- crepancies due to incomplete physical understanding and variations in experimental conditions and facilities. Furthermore, disagreement and ambiguity in defining the boundaries of physical domains will increase complexity of model building as well. To accommodate these uncertainties, we will investigate the following framework. Assumption Let S d denote the subset of basis functions characterizing the model structure of Φ(x ∗ | x ◦ ) in the dth domain, d = 1, 2,...,D. We assume S d =S for all d. Under this framework, the equality f i (x ∗ ) = 0 holds only for countable settings of x ∗ . So the basis functions of Φ(x ∗ | x ◦ ) do not degenerate in any continuous domain. Physically, the assumption means that all physical mechanisms, which are characterized by the complete set of basis functions, co-exist in all physical domains with different weights. Remark Note that the setup of this framework is able to avoid the issue of defining the transition points or boundaries between different physical domains upfront. However, implicitly the model Φ(x ∗ , x ◦ ) selects proper basis function(s) to characterize different physical domains by varying the weight g i (x ◦ ) for each basis function f i (x ∗ ). The influence of a particular mechanism is described by some subset of{g i (x ◦ )f i (x ∗ ),i = 1,...,K}. It follows that both domains and domain transitions can be obtained from the model Φ(x ∗ , x ◦ ;): (i) A domain is identified to be dominated by a certain mechanism if this mechanism contributes to the majority of response in this domain; (ii) If none of the mechanisms contributes to the majority of response in certain domain, this domain is a transitional domain. 60 4.3 Scale-up Modeling of Droplet Formation in a Coated Microfluidic T-junction This section presents the detailed solution procedure for the high-dimensional scale-up mod- eling problem formulated in Section 4.2. 4.3.1 Dimensional Analysis and Dimension Reduction Before conducting dimensional analysis to obtain the transformed Π-space, we first introduce the droplet formation process in a microfluidic T-junction to identify relevant physical quantities in the parameter space. As shown in Fig. 4.2, the carrier oil (continuous phase) is injected via inlet 1, while the reagent streams are introduced via inlets 2 and 4. A stream injected via inlet 3 is used to prevent diffusive mixing between reagent streams before droplet formation. The immiscible fluid of droplets is called “dispersed phase” . Figure 4.2: Structure of the two-phase microfluidic T-junction [67] Notations of physical quantities used in the rest of the paper are listed below. The parameter space consists of parameters that characterize the geometric structure, proper- ties of the materials and mechanical control variables, which are listed in the relevance list (Table 4.1). Dimensional analysis is then conducted to generate dimensionlessπ numbers (Equation 4.5) that span the transformed Π-space [43, 25, 42]. 61 Notation Physical quantity w c (w d ) width of channel into which continuous (dispersed) phase flows h depth of channel μ c (μ d ) dynamic viscosity of continuous (dispersed) phase ρ c (ρ d ) density of continuous (dispersed) phase Q c (Q d ) volumetric flow rate of continuous (dispersed) phase σ interfacial tension Table 4.1: Relevant List of Physical Quantities Quantity Dimension Geometry w c , w d L h L Material μ c , μ d ML −1 T −1 ρ c , ρ d ML −3 Mechanics Q c , Q d L 3 T −1 σ MT −2 π 0 = ¯ L = L w c , π 1 =Ca = μ c Q c σw c h , π 2 =λ = μ d μ c , π 3 =Q = Q d Q c , π 4 =W w = w d w c , π 5 =W h = h w c , π 6 =ρ = ρ d ρ c , π 7 =Re = ρ c Q c μ c w c . (4.5) Corresponding to the formulation in Section 4.2, we choose the response to be the dimen- sionless droplet length z = π 0 = ¯ L due to our interest in droplet size. The scale-up model- ing problem in the transformed Π-space is then formulated by the π numbers in the form π 0 = Φ(π 1 ,π 2 ,π 3 ,π 4 ,π 5 ,π 6 ,π 7 ), i.e. ¯ L = Φ(Ca,λ,Q,W w ,W h ,Re,ρ), with (x ∗ , x ◦ ) = (Ca,λ,Q,W w ,W h ,Re,ρ). To reduce dimensionality, the capillary number is selected as the primary factor, i.e. x ∗ =Ca, whereas the remaining dimensionless numbers are identified as secondary factor, which will be explained in Section 4.3.3. For typical microchannel flows, the Reynolds numberRe is very small. Once the microfluidic T-junction design and fluidic materials are determined, the only remaining controllable dimensionless number other than Ca is the flow rate ratio, the effect of which has been investigated in each domain. We also qualitatively investigate the effect ofW h by comparing 62 droplet formation in a microfluidic T-junction with W h = 1 and W h = 2. Therefore Q and W h form the two-dimensional secondary factor x ◦ = (Q,W h ). Despite the demonstration in the study of droplet formation in a coated microfluidic T-junction, the strategy to reduce the dimension of a high-dimensional scale-up problem by identifying low-dimensional structures can be applied to a generic high-dimensional scale-up problem. 4.3.2 Experimental Setup and Data Collection In order to systematically characterize droplet formation across multiple domains, we first select a reference geometry and keep the fluid pair fixed. We used two geometries that maintained a w d : w c ratio of 1 : 4, with w c = 200 or 400 μm. Microfluidic devices were coated with a low-surface-energy fluoropolymer coating using initated chemical vapor deposition (iCVD) as described previously [67, 100]. The continuous phase was a polychlorotrifluoroethene (PCTFE) oil (trade name/vendor: Halocarbon oil), with μ c = 100mPa·s and the dispersed phase was de-ionized water. Dimensional parameters for the reference system are given in Table 4.2. The width ratio W w is set to be 0.25, while the depth width ratio W h = 1, 2. Droplet length was measured as a function of capillary number Ca and flow rate ratio Q for the reference system by selecting five different flow rate ratios Q = 0.05, 0.25, 0.5, 1, 2. For each fixed flow rate ratio, the capillary number is varied from Ca = 0.00193 to 0.15432. To keep the flow rate ratio fixed, both Q c and Q d must vary as Ca varies. As reported in [100], for Ca < 0.05 with w c = 200μm or Ca < 0.01 with w c = 400μm respectively, droplet size increased with increasing flow rate ratio at each fixed capillary number explored in the experiment and decreased with increasing Ca at each fixed flow rate ratio (see Figure 4.3). Unlike droplet formation in uncoated microfluidic channels, droplet formation in the coated device remained in the dripping regime above the threshold of Q = 0.05, and the droplet size appeared to either plateau or increase according to different flow rate ratios. 63 Table 4.2: Experimental settings of physical quantities Fluid System Viscosity [mPa·s] Continuous phase Halocarbon oil 100 Dispersed phase De-ionized water 1 Device geometry Dimension [μm] Channel width w c 200− 400 w d 50− 100 Channel depth h 400 Flow rate control Dimension μL/h Continuous phase Q c 250− 200000 Dispersed phase Q d 12.5− 200000 4.3.3 Model Structures and Basis Functions Candidates of basis functions that characterize model structures for droplet formation in un- coated microfluidic T-junction in each domain can be acquired from experimental literature. In squeezing domain (Ca . 0.01 according to [25], Ca < 0.002 in [141]), the scaling law is given in the form ¯ L = σ +ωQ [46], where σ and ω are parameters determined by channel geome- try. In dripping domain with high Ca where droplets are unconfined, the dimensionless droplet length is proportional to Ca −0.25 approximately [40, 124]. In the squeezing-to-dripping domain, according to the correlations discussed in existing studies [19, 40, 140, 128, 152], the scale-up model regarding the capillary number with all the other dimensionless numbers fixed is either given in the general form ¯ L(Ca)∝ Ca −α with 0 < α < 0.25 or by the dimensionless droplet volume ¯ V (Ca) being linear combination of Ca −α1 and Ca −α2 based on approximation models. According to the assumption in Section 4.2, droplet formation in the squeezing-to-dripping do- main exhibits intermediate phenomenon. This explains why ¯ L(Ca)∝Ca −α with 0<α< 0.25 in squeezing-to-dripping domain. We therefore choose the basis functions f 1 (Ca) = 1, f 2 (Ca) = Ca −α to characterize the decrease of droplet size in squeezing and dripping domain respectively at relatively low capillary numbers, where α is a positive parameter to be determined for coated devices. 64 In addition to the decrease in droplet size as the capillary number increases at relatively low Ca, we observed either a plateau or an increase of droplet size at higher capillary numbers before jetting occurred. We attribute this to the difference between our system and most in the literature using PDMS channels. The water contact angle of PDMS channels is 112 ◦ while the low surface energy coating applied to our system renders the channels more hydrophobic, resulting a water contact angle larger than 120 ◦ [100]. Wall effects by coating become significant when the droplet length is larger than the channel geometry. Therefore an additional basis function f 3 (Ca) =Ca γ is proposed to characterize the increase in droplet size at higher capillary numbers with γ > 0. The total number of basis functions is K = 3. Besides the primary factor x ∗ = Ca, two secondary factor x = (Q,W h ) are considered to explore the form of g i (x) in Equation 4.3. From observation and existing literature, it is known that (i) The droplet size is linear to flow rate ratio with remainingπ numbers fixed at high viscosity contrast [42, 40, 140, 128, 152]; (ii) Droplet formation with regard to Ca does not depend on the geometry of microfluidic T-junction in squeezing [46], squeezing-to-dripping [19] and in dripping domain [40], indicating the independence of α and γ in W h . The weight function g i (Q,W h ) is proposed to be g i (Q,W h ) =β i,0 (W h ) +β i,1 (W h )Q. (4.6) where the intercept and slope of the linear model β i,0 , β i,1 are functions of the other secondary factor W h . We analyze the effect of W h qualitatively by regarding it as a treatment factor due to the following reasons: (i) The effect of geometry has not been well investigated across domains, i.e., concrete forms of β i0 and β i1 are not available for some i; (ii) W h is a two-level factor in our experiments, which can be represented by j = W h with j = 1, 2. Similar to [101], an equivalent 65 expression of Equation 4.6 can be given in Equation 4.7, using a dummy variable T defined such that T = 0 for j = 1 and T = 1 for j = 2. g i (T,Q) =η i +τ i T +β i Q +w i TQ, i = 1,...,K. (4.7) Remark This is essentially an ANCOVA (Analysis of Covariance) model setup in statistical design of experiments, which is a generalized linear model blending ANOVA (Analysis of Variance) with regression. g i (T,Q) is the response for a given value of covariateQ and a selection of treatmentT , τ i refers to the difference inβ i,0 (W h ) due to the treatment effect,β i is the regression coefficient for the covariate Q, and w i is the contrast in β i,1 (W h ) due to an interaction between the treatment factor T and the covariate Q. Based on the assumption formulated in Section 4.2 and discussions above, the basis functions of Φ(Ca|Q) are obtained asS ={1,Ca −α ,Ca γ }. According to Equation 4.7, the unified scale-up model under the assumption is given in the form of Equation 4.8, with K = 3 and T = 0, 1 denoting W h = 1, 2 respectively. Φ(Ca,Q,T ) = K X i=1 {ηi +τiT +βiQ +wiTQ}fi(Ca) (4.8) 4.3.4 Model Estimation The nls() function in R language was applied for Nonlinear Least Squares (NLS) estimation [37, Appendix], using a form of GaussâĂŞNewton iteration that employs numerically approximated derivatives. The full model in Equation 4.8 is further reduced to the model in Equation 4.9 by eliminating redundant terms to minimize the residual standard error, estimation given in Table 4.3. 66 Φ(Ca,Q,T ) = (η2 +τ2T )Ca −α + [η3 +β3Q +w3TQ]Ca γ (4.9) Table 4.3: Reduced model estimation Parameter Estimate Std.error P-value α 0.39953 0.04079 3.12e-15 γ 0.24503 0.01758 < 2e-16 η 2 0.09094 0.02596 0.000764 τ 2 0.05991 0.01521 0.000177 η 3 0.83829 0.11187 8.94e-11 β 3 1.70621 0.11252 < 2e-16 w 3 0.84682 0.08789 6.36e-15 4.4 Discussions 4.4.1 Physical Insights through High-dimensional Modeling Compared to one-factor-at-a-time approach, our generic methodology allows investigation into the overall model structure in the high dimensional space, providing additional insights into the droplet formation process in coated microfluidic devices. 1. Flexibility for full-scale production As observed from Figure 4.3, the concise model in Equation 4.9 captures well the influence ofCa,Q andW h simultaneously, providing an opportunity to optimize multiple parameters simultanenously for full-scale production. 2. Interpretation of multiple physical domains Domains are identified by dominant physical mechanisms characterized by corresponding basis functions. As observed from Equation (4.9), dripping mechanism characterized by 67 1.0 2.0 W h =1 Ca L 0.00193 0.07716 0.1929 2 1 0.5 0.25 0.05 1.0 2.0 3.0 4.0 W h =2 Ca L 0.00193 0.0463 0.15432 2 1 0.5 0.25 0.05 2 1 0.5 0.05 0.25 2 1 0.5 0.05 0.25 Figure 4.3: Change in dimensionless droplet length ¯ L with respect to the capillary number Ca, flow rate ratio Q (from 0.05 to 2) and depth width ratio W h . 68 f 2 (Ca) dominates the decrease in droplet size beforef 3 (Ca) starts to dominate the increase in droplet size. The transition between domains depends on the secondary factor. 3. Effect of coating Coating enables a wider range of producing stable droplets, allowing higher flow rate ratios and capillary numbers. Although physical domains can still be classified according to dif- ferent dominant mechanisms characterized by corresponding basis functions of the capillary numbers, the range of each physical domain could be very different from that of droplet for- mation in uncoated microfluidic devices, e.g. squeezing mechanism characterized by f 1 (Ca) is not significant within our explored range, whereas in a coated T-junction, squeezing dom- inates droplet formation for Ca < 0.002 and is competitive with dripping mechanism for 0.002<Ca< 0.01. 4. Effect of secondary factor Transitions between physical domains are not independent of secondary factor (Q, W h dis- cussed in our case). This will be further explained in the discussion of identifying physical domains. 4.4.2 Identification of Physical Domains and Boundaries The scale-up model in Equation 4.9 not only provides some insights into the droplet formation process in a coated microfluidic T-junction, but also demonstrates the possibilities of detecting physical domains dominated by different mechanisms without ambiguity in boundaries that are characterized by values of primary factor in literature. Physical domains in this scale-up droplet formation process are detected by identifying dominant mechanisms. 1. Identification of relevant mechanisms 69 As observed in model reduction from Equation 4.8 to 4.9, f 1 (Ca) is eliminated from the final model since it is not significant across investigated domains. This implies that squeez- ing mechanism characterized by f 1 (Ca) in our coated devices is not significant within the explored range, i.e. coating leads to a decrease in the lower bound of dripping domain with respect to the primary factor Ca as a benefit of low surface energy. Remark As observed from Figure 4.3, there exists a slight lack of fit under W h = 2. This can be attributed to the unconspicuous postponement of dripping domain compared to the case when W h = 1. In other words, when W h = 2, droplet formation near the lower bound of explored Ca may actually fall in the squeezing-to-dripping domain, although the basis functionf 1 (Ca) is still recognized as insignificant due to the narrow range of the squeezing- to-dripping domain even if there exists. 2. Identification of dominant mechanisms Dripping mechanism characterized by f 2 (Ca) dominates the decrease of droplet size at low capillary numbers within our explored range, while droplet size starts to increase and then tends to reach a plateau as Ca increases to a certain level. The domain in which the size of stable droplets increases is identified as ”dripping-to-jetting” domain due to the coexistence of partial characteristics: (i) Stable droplets are produced in this domain with no observations of jets; (ii) Droplet size tends to reach a plateau near the explored upper bound of Ca, consistent with the scale-up model in jetting domain [41] which merely depends on Ca. 3. Identification of boundaries 70 Noting that f 2 (Ca) = Ca α ) and f 3 (Ca) = Ca γ are both monotonic with respect to Ca, a transition point between dripping and dripping-to-jetting domain can be defined as the solution to ∂Φ(Ca,Q,T )/∂Ca = 0 given in Equation 4.10. Ca 1 (Q,T ) = α(η 2 +τ 2 T ) γ[η 3 + (β 3 +w 3 T )Q] 1 α+γ . (4.10) When Ca<Ca 1 (Q,T ), droplet size decreases, i.e., dripping mechanism depicted by Ca −α explains a larger portion of change in droplet size; whenCa>Ca 1 (Q,T ), droplet size starts to increase, i.e., dripping-to-jetting mechanism depicted by Ca β contributes to a larger percentage of size change. Table 4.4: Ca 1 (Q,T ) Q 0.05 0.25 0.5 1 2 T = 0 0.0055 0.012 0.023 0.036 0.059 T = 1 0.0071 0.017 0.036 0.062 0.12 As shown in Table 4.4 and Figure 4.3 (dashed lines), the value of Ca 1 (Q,T ) is smaller at higher flow rate ratio Q, and the transition is postponed due to extra confinement with W h = 2 (T = 1) compared to the case with W h = 1 (T = 0). 4. Effect of secondary factor To further demonstrate the effect of secondary factor on boundaries between physical do- mains, we define Ca 2 (Q,T ) as the solution to ∂ 2 Φ(Ca,Q,T )/∂Ca 2 = 0 given in Equation 4.11, noting that Ca 2 (Q,T )>Ca 1 (Q,T ). Ca 2 (Q,T ) = α(1 +α)(η 2 +τ 2 T ) γ(1−γ)[η 3 + (β 3 +w 3 T )Q] 1 α+γ . (4.11) 71 Table 4.5: Ca 2 (Q,T ) Q 0.05 0.25 0.5 1 2 T = 0 0.014 0.032 0.060 0.094 0.15 T = 1 0.019 0.044 0.093 0.16 0.31 In dripping domain with Ca < Ca 1 (Q,T ) < Ca 2 (Q,T ), ∂ 2 Φ(Ca,Q,T )/∂Ca 2 is positive with Ca 2 (Q,T ) decreasing in Q, which explains the sharper decrease at lower Q when Ca is close to the lower bound of explored range. In dripping-to-jetting domain with Ca > Ca 1 (Q,T ), the increase in droplet size with respect to Ca first accelerates in the range of Ca<Ca 2 (Q,T ) and then decelerates to reach a plateau at higher Ca>Ca 2 (Q,T ). 4.5 Summary and Conclusion We formulated a generic multiple-domain scale-up problem and proposed a scalable modeling approach to predict manufacturing processes that can potentially operate under multiple physical domains due to uncertainties. The approach addresses two critical challenges in scale-up modeling for multiple-domain manufacturing processes: high dimensionality and multiple physical domains. The challenge of high dimensionality is addressed by identifying low-dimensional model structures through dimensional analysis and adoption of primary factor. The ambiguity in boundaries be- tween physical domains is addressed by interpreting multiple-domain manufacturing processes as an outcome of coexisting mechanisms. Physical domains are identified by identifying dominant mechanisms. The proposed formulation and approach have been applied and demonstrated to investigate the scale-up droplet formation process in the coated microfluidic T-junction. The unified scale-up model across multiple domains not only captures well the joint effect of capillary number, flow rate ratio and depth width ratio, but also leads to some physical insights into the process: (i) Droplet formation in the coated device can be explained by coexisting mechanisms with weights varying 72 across domains. (ii) Variables besides the capillary number influence the transition from dripping to dripping-to-jetting domain, which is postponed at lower flow rate ratio and higher depth width ratio. (iii) The low-surface-energy fluoropolymer coating has proved to highly extend the range of domain for producing stable droplets either in dripping or dripping-to-jetting domain. 73 Chapter 5 Design of Experiments for Domain-Dependent Scale-up Manufacturing Processes In Chapter 4, a modeling methodology has been proposed to study domain-dependent scale-up manufacturing processes. Since domains are often a priori unknown, an efficient design of exper- iments is essential for detecting all domains. Although we have established an additive model to characterize domain-dependent scale-up manufacturing processes in Section 4.2, an optimal de- sign cannot be directly applied due to lack of knowledge in concrete model structures. Traditional methods tend to evenly distribute design points over the entire range of settings, which is not applicable for domain-dependent processes. Suppose a system involves two domains, with one much larger than the other in area. If design points are assigned evenly, the tiny domain might not be discovered within a limited number of experiments, which could result in a risky model that collapses in the tiny domain. One reason of the financial crisis in 2008 is due to inaccurate risk modeling under extreme situations. In other words, extreme situations that occur in a tiny domain are poorly represented. Therefore it is important to ensure enough design points in each domain so that all situations are well represented. A sequential design is proposed in this Chapter to detect domains adaptively. 74 5.1 Problem Formulation Let x = (x 1 ,...,x n ), S ∗ ={x j : x 0 j s form the primary factor x ∗ }, S ◦ ={x j : x j / ∈ S ∗ }. Define x c j such that x = (x j , x c j ). Initial designs are designed for identifying whether x j ∈S ∗ or x j ∈ S ◦ . It follows from Proposition 2 that, if variable x j is a component of secondary factor, then Φ(x j |x c j ) = K P i=1 c i h(x j ; j i ) =h(x j ; j ) for some functionh(·). Noting the following discussion holds for all possible j, we use u instead of x j for short and omit superscript j in components of parameter j and j i . Let j = (θ 1 ,...,θ d )∈R d , and j i = (θ i,1 ,...,θ i,d ). Comparing the Taylor series expansion of K P i=1 c i h(u; j ) and h(u; j i ) at = 0, we obtain Eq. 5.1 and 5.2. h(u; j ) = ∞ X n1=0 . . . ∞ X n d =0 (θ 1 ) n1 . . . (θ d ) n d n 1 !. . .n d ! ∂ n1+···+n d h(u;) (∂θ 1 ) n1 . . . (∂θ d ) n d (0) = ∞ X n1=0 . . . ∞ X n d =0 (θ 1 ) n1 . . . (θ d ) n d α n1,...,n d (u) (5.1) K X i=1 c i h(u; j i ) =h(u; j )⇔ K X i=1 c i (θ i,1 ) n1 . . . (θ i,d ) n d = (θ 1 ) n1 . . . (θ d ) n d for all n 1 ,...,n d such that α n1,...,n d (u)6= 0. (5.2) Since j ∈ R d , we must have exactly d groups of n 1 ,...,n d such that α n1,...,n d (u)6= 0 to obtain a unique solution of j . Assumption 3. Suppose x ∗ ∈R n ∗ with n ∗ ≤ (n ∗ ) max , and assume (n ∗ ) max is known. For each i,∃ a domainD i 0 such that f i (x ∗ ) is the only dominant basis function withinD i 0. Remark: Assumption 3 can always be satisfied by choosing the functions that dominate some physical domain to form the set of basis functions. 75 5.2 A Sequential Design for Multiple-Domain Scale-up We proposed the sequential design in Figure 5.1 for domain-dependent scale-up experiments. Dimensional analysis is first applied to transfer the parameter space to the Π-space for both priori dimension reduction and scalability of results [2]. The primary factor can normally be identified by prior knowledge of the engineering process. For example, nanoparticle synthesis via microfluidic channels exhibits different characteristics as the capillary number Ca varies across the range of feasible values [42, 40, 140, 128, 152, 67, 100]. In classical fluid dynamics, for instance, domains of creeping, laminate, and turbulent flow are classified by Reynolds number Re [104]. If prior knowledge is not sufficient to identify components of the primary factor, initial experiments are necessary for the identification as explained in Section 5.2.1. With the primary factor identified, we propose a sequential design to obtain the model structure in Eq. 4.3, followed by an optimal design in the feasible space. A separation scheme is introduced to the sequential design, where the basis functionsf i (x ∗ ) 0 s and the weight functiong(x ◦ ) are explored separately. An optimality proof of this separation scheme is given in Section 5.2.2. Sequential experimental designs for the primary and secondary factor are discussed in Section 5.2.3 and 5.2.4 respectively. An integral scale-up model is expected to be obtained hierarchically from previous steps, resulting in the feasibility of exploiting an optimal design in the entire feasible space. We will end the process only if all the domains are well captured by the integral model, otherwise, we will go back to the step in Section 5.2.3 under another setting of secondary factor to identify underlying domains. 5.2.1 Initial Design to Identify Primary Factor Let (x j ) 1 , (x j ) 2 ,... denote arbitrary settings of x j , and x c j 1 , x c j 2 ,... arbitrary settings of x c j . For x j ∈S ∗ , define x ∗ j such that x ∗ = (x j , x ∗ j ). Similarly, for x j ∈S ◦ , define x ◦ j such that x ◦ = (x j , x ◦ j ). 76 Design of domain-dependent scale-up experiments Dimensional analysis: parameter spaceàΠ-space Identification of primary factor Domain identification: DOE for primary factor Weight identification: DOE for secondary factor DOE for integral model in entire feasible space All domains captured No Model obtained for scale-up manufacturing Yes Figure 5.1: Flowchart of overall design strategy for domain-dependent scale-up experiments Definition 4. For the jth component of x = (x 1 ,...,x n ), namely x j , we define d j such that Φ x j | x c j 1 ,..., Φ x j | x c j k are linearly dependent if and only if k≥d j + 1, j = 1,...,n. Assumption 5. max xj∈S ◦d j < min xj∈S ∗d j . The assumption can be satisfied by restricting the number of unknown parameters in the univariate functionh ◦ (x j ; j ) for eachx j ∈S ◦ , which reduces the risk of overfitting. An approach of identifying primary factor is proposed based on this assumption. As shown in Fig. 5.2, we first identify d j for each independent variable x j . Based on Assumption 3 and 5, if max j d j > 1, then the upper bound ofn ∗ can be updated ton ∗ u = min (n ∗ ) max , max i :d (n−i) <d (n−i+1) . DefineS ∗ u = x (n−n ∗ u +1) ,...,x (n) , thenS ∗ ⊆S ∗ u . Ifd j = 1 for allj = 1,...,n, then Φ(x ∗ , x ◦ ) = θ h(x ◦ ) K P i=1 f i (x ∗ ) =θ h(x ◦ )F (x ∗ ), we define S ∗ u ={x 1 ,...,x n } in this case. Note S ∗ u consists of candidate components of the primary factor. Then we apply the sequential experimental design proposed in Section 5.2.3 to each variable in S ∗ u . 77 Initial design to identify primary factor Identify " for each variable " Identify domains via DOE for each ∈ ∗ Multiple domainsà ∈ ∗ Identify * ∗ , set of candidate PF components Figure 5.2: Flowchart of identifying the primary factor To identify d j for each j = 1,...,n, a sequential design of experiments is proposed. Let S j (1),S j (2),... denote a series of sets, such that (i) S j (m) = (x j ) 1 ,..., (x j ) m is a set of m different values of x j , and (ii) S j (m)⊂S j (m + 1) for any m≥ 1. We define ˆ Φ m x j |x c j to be a discrete function such that ˆ Φ m (x j ) k |x c j = Φ (x j ) k |x c j for k = 1,...,m and 0 otherwise. The sequential design is given as follows. Step 1. Initialization: d← 1, m←m 0 . Step 2. Conduct experiments at n (x j ) 1 , x c j 1 , . . ., (x j ) m , x c j 1 o to obtain ˆ Φ m x j | x c j 1 . Step 3. Conduct m experiments to obtain ˆ Φ m x j | x c j d+1 . If d > 1, conduct d additional experiments to obtain n ˆ Φ m x j | x c j 1 ,..., ˆ Φ x j | x c j d o . Step 4. Stop if n ˆ Φ m x j | x c j 1 ,..., ˆ Φ m x j | x c j d+1 o are linearly dependent, set d j =d; oth- erwise, update d←d + 1, m←m + 1, and go to Step 2. Remark: The sequential design achieves identification of d j within (d j + 1)(m 0 +d j − 1) experiments, j = 1,...,n. The linear dependence of n ˆ Φ x j | x c j 1 ,..., ˆ Φ x j | x c j dj+1 o can be easily checked via linear regression. 78 5.2.2 Optimality Proof of the Separation Scheme in Sequential Design In this section, we prove the optimality of the separation scheme in the sequential design procedure (Fig. 5.1). Note the sufficient number of experiments can be specified according to any precision requirement. For example, the number of experiments can be considered as sufficient when the minimum of pairwise absolute difference in response is smaller than some small number that characterizes the precision. Assumption 6. At least n f samples fromD i are needed to identify f i (x ∗ ), i = 1,...,K, and at least n g samples of x ◦ are needed to identify g(x ◦ ). Proposition 7. At leastn g +Kn f −1 different samples are needed to obtain the explicit function form of the model in Eq. 4.3, i.e. identify f i (x ∗ ) 0 s and g(x ◦ ). Moreover, the sequential design procedure achieves the lower bound number of samples, i.e., it is optimal. Proof. By Assumption 6, to learn f i (x ∗ ) we need n f samples with the x ∗ component fromD i . Moreover, since domains do not overlap with each other due to Assumption 3, theseKn f samples must be different from each other. To learn g(x ◦ ), we need n g samples with fixed x ∗ and different x ◦ . These n g samples can share at most one sample with previous Kn f samples, which have different x ∗ components. So at least n g − 1 new samples are needed. To sum up, in total, n g +Kn f − 1 samples are needed. It is easy to see that the sequential design procedure achieves this lower bounded number of samples, thus is optimal to learn the form of Φ(x ∗ , x ◦ ;). 5.2.3 Domain Identification: Sequential DOE for PF With the secondary factor (SF) x ◦ fixed, we formulate a clustering problem and partition the feasible region into different physical domains (clusters). The following sequential design procedure is proposed for the identification of domains and candidate basis functions, S. 79 Step 1. Repeat the two-stage procedure to identify physical domains based on clustering. (i) Estimation of response surface and the gradient. (ii) Optimization of some criterion function to select a new design point. Step 2. Identify the dominant basis function in each physical domain. A flowchart of the overall sequential design for primary factor is given in Fig. 5.3, where n max and n ∗ denote the maximum number of experiments allowed, and the sufficient number of experiments for each cluster respectively. Domain identification: sequential design for primary factor Initial design of size " (< $%& ), initialize = " Estimation of the response surface and gradient Clustering: partition design points into clusters, count number of design points in each cluster, ( = $%& or ( > * for all No Stop and identify dominant basis function in each domain Yes Selection of next design point in feasible space, =+1 Figure 5.3: Flowchart of sequential design for the primary factor 80 Estimation of Response Surface and Gradient We will use the k-nearest neighbor inverse distance weighting (k-NN IDW) interpolator [63, 154] in Eq. (5.3) to estimate the response surface, where d 2 (u, u j ) represents the Euclidean distance between u and u j , andI j (u m , u) is an indicator function which equals 1 if u j falls in the k-nearest neighborhood of u. ˆ h(u) = n X m=1 v m (u)h(u m ), v m (u) = I k (u m , u)w m (u) P n j=1 I k (u j , u)w j (u) , w j (u) = 1 d 2 (u, u j ) . (5.3) Let u = x ∗ and h(u) = Φ(x ∗ |x ◦ ) in Eq. 5.3. Since the dominant basis function of the conditional response Φ(x ∗ |x ◦ ) varies across different physical domains, we include the gradient of Φ(x ∗ |x ◦ ) as an additional feature to the primary factor x ∗ . An approximation of the gradient, ∇Φ(x ∗ |x ◦ ), can be obtained by using finite difference to estimate partial derivatives. Selection of Next Design Point Assume the feasible region for design points is given by Φ(x ∗ |x ◦ ) > c, for example, the droplet size is always positive, i.e. Φ(x ∗ |x ◦ )> 0. We defineI(y ∗ |x ◦ , x ∗ 1 ,..., x ∗ n ) in some proper transformed feature space to quantify the amount of extra information gained from sample (x ∗ , x ◦ ) with fixed x ◦ and existing samples (x ∗ 1 , x ◦ ),..., (x ∗ n , x ◦ ). A new design point x ∗ n+1 is selected to maximize the amount of extra information. As shown in Eq. 5.4, the transformed feature space is constructed to capture information from both the independent variable x ∗ and the response z. 81 x ∗ n+1 =argmax Φ(x ∗ |x ◦ ) V >c I(x ∗ |x ◦ , (x ∗ 1 ,z 1 ),..., (x ∗ n ,z n )) (5.4) One measurement of extra information is defined in Eq. 5.5, where V (x ∗ ) denotes a neighbor- hood of x ∗ . I(x ∗ |x ◦ , (x ∗ 1 ,z 1 ),..., (x ∗ n ,z n )) = X x ∗ j ∈V(x ∗ ) |d(y, y j )| 2 , y =∇ x ∗Φ(x ∗ |x ◦ ) V , y j =∇ x ∗ j Φ(x ∗ j |x ◦ ) V (5.5) Clustering and Stopping Criteria of Step 1 Clustering aims to partition points into clusters such that the similarity is relatively high within each cluster and low between clusters. Four types of clustering algorithms are most commonly used. Centroid-based clustering [50] characterizes each cluster by a central vector, and minimizes the sum of distance functions of each point to the assigned central vector. We will not adopt this scheme since it does not allow non-convex clusters. Algorithms based on distribution [147] are not applied due to lack of prior knowledge, and density-based clustering [32] is also not appropriate for similar reason. We prefer to use connectivity-based clustering (hierarchical clustering) scheme [62], which forms clusters based on distances or similarity. There are two ways to evaluate clustering results, internal evaluation and external evaluation. External evaluation can be applied if some pre-classified experimental results are available for validation based on existing studies. External criteria includes Rand measure [97], F-measure [93], FowlkesâĂŞMallows index [36], etc. However, additional information are not always available, especially at the initial stage of experimental study. It is therefore more practical to access the 82 quality of clustering based on some internal validation measures. Since we are concerned about both the compactness and separation of the clustering, indices that involve only one aspect are not considered. Candidate validity criteria for connectivity-based clustering includes the well-known Dunn index [30] and the Silhouette index [102]. The Dunn Index (DI) is given in the form of the ratio between separation and compactness, characterized by the minimum intercluster distance and the maximum intra-cluster distance respectively. The Silhouette index uses the difference of between- and within-cluster distances to quantify how appropriately each point lies in its assigned cluster. Assuming that the number of clusters, K, is not known apriori, we choose K such that DI is maximized in each iteration of Step 1. We stop iteration in Step 1 if and only if at least one of the following criteria is satisfied: 1. The maximum number of experiments allowed, n max , is achieved. 2. The number of experiments is sufficient for each cluster, sayn i ≥n f for some predetermined n f and all i, where n i denotes the number of samples in Cluster i. Discussion on Step 2 For Step 2, we will make use of the experimental results in Step 1. Note the basis functions are normally selected among specific families known apriori based on domain-dependent approxi- mation models. For example, the size of droplets synthesized in microfluidic channels is proposed to be a distinct power function of the capillary number (Ca) in each physical domain. Basis func- tionsf i (x ∗ ) 0 s are identified in this step based on observations in each physical domain (cluster in Step 1). 83 5.2.4 Weight function Identification: Sequential DOE for SF We are now in the position of applying sequential design to explore the general form of weight functionsg i (x ◦ ) 0 s, with the primary factor fixed. The IDW-MM algorithm in [154] can be directly applied as shown in the flowchart (Fig. 5.4), where the selection ofn g will be discussed in Section 5.2.2. The conditional response surface is approximated using Eq. 5.3, with u = x ◦ and h(u) = Φ(x ◦ |x ∗ ). In each iteration, we select a new design point x ◦ n+1 such that: x ◦ n+1 =argmax Φ(x ◦ |x ∗ ) V >c min j=1,...,n d(x ◦ , x ◦ j ). (5.6) Weight function identification: sequential design for secondary factor Initial design of size " (< % ), initialize = " Estimation of the response surface Selection of next design point in feasible space, =+1 = % Stop and move on to optimal design for the integral model Yes No Figure 5.4: Flowchart of sequential design for the secondary factor 84 5.3 Case Study: Drag Coefficient of a Sphere in Fluid The study of drag force in uniform flow around a sphere is critical to the design of airplanes, missiles and cars. The drag force is captured by Eq. 5.7 according to fluid dynamics [8], where ρ,A,v refer to the mass density of the fluid, flow velocity relative to the object and reference area respectively. The Drag coefficient C D is found to be a function of the Reynolds number Re, an empirical formula of which is given in Eq. 5.8 [8]. F D = 1 2 ρAv 2 C D , C D =f(Re) (5.7) C D = 24 Re + 2.6 Re 5.0 1 + Re 5.0 1.52 + 0.411 Re 2.63×10 5 −7.94 1 + Re 2.63×10 5 −8 + 0.25 Re 10 6 1 + Re 10 6 (5.8) We can use the model in Eq. 4.3 to interpret the drag force as a function of independent variables ρ,A,v,Re. The response, primary factor and secondary factor are given by z = F D , x ∗ = Re, x ◦ = (ρ,A,v) respectively. Since the primary factor and Eq. 5.7 are given by prior knowledge, it suffices to identify physical domains based on the relationship between the drag coefficient C D and the primary factor Re. We use Eq. 5.8 to generate the true drag coefficient for given Re and apply the sequential design for primary factor in Section 5.2.3. Assume both Eq. 5.7 and Eq. 5.8 are unknown. We formulate the following experimental design problem to learn drag coefficientC D as a function ofRe: If we start withn 0 initial samples uniformly distributed over the range of Re, how should we choose the next design point such that we have at least n f design points in each domain? For example, let n 0 = 10, n f = 4. The sequential design in Section 5.2.3 is applied to identify potential domains adaptively. We will use the measure in Eq. 5.5 by choosing V (x ∗ ) = [x ∗ κ(x ∗ ) , x ∗ κ(x ∗ )+1 ]. Hierarchical clustering is applied on the 1 st order central difference. The processes of choosing the 11 th , 12 th , 13 th and 18 th design point are depicted in Fig. 5.5 - 5.8. 85 Figure 5.5: Initial 10 samples and selection of the 11 th design point The response surface is estimated via k-NN IDW, followed by the estimation of 1 st -order central difference. The “distance” quantifies extra information brought by candidate design points as defined in Eq. 5.5. A candidate that brings maximum extra information is selected. 86 Figure 5.6: First 11 samples and selection of the 12 th design point 87 Figure 5.7: First 12 samples and selection of the 13 th design point 88 Figure 5.8: First 17 samples and selection of the 18 th design point 89 As shown in Table 5.1, our sequential design for the primary factor detects all domains adap- tively with 8 additional design points, while a traditional uniform design in the feasible region requires 49 additional design points. Table 5.1: Comparison of different experimental designs Design 1 st Domain 2 nd Domain 3 rd Domain Total Initial 8 0 2 10 Existing 44 4 11 59 New 9 5 4 18 5.4 Summary and Conclusion In this Chapter, we have considered design of experiments for domain-dependent scale-up manufacturing processes. The objective of this design is to detect all physical domains and obtain a scale-up model that captures the process across all domains. In order to solve this problem, I proposed a novel sequential design, which detects domains adaptively and ensures enough design points in each domain. To achieve this, the clustering scheme is introduced to experimental design for the first time. Simulation studies have been used to evaluate the algorithm. The sequential design highly reduces the number of experiments required to detect all domains for modeling. 90 Chapter 6 Discussions and Future Extensions Although innovative technologies developed in laboratories have huge potential in revolution- izing many industries, we have seen a relatively small number of commercial scale applications in advanced manufacturing. The study of scale-up methodologies is thus essential for the migration from lab-scale success to commercial scale application. The early-stage study of advanced manu- facturing processes mainly faces two closely-related challenges: (i) lack of process understanding and (ii) restrictions on experimental resources. To address the first challenge, we study scale-up modeling methodologies of characterizing and predicting advanced manufacturing processes through three representative problems, each focusing on a different type of issues. In the first task, to address the issue of lacking physical knowledge and extensive data, we make full use of available information and propose a physical- statistical modeling approach to interpret nanowire growth. In the second task, to accommodate nonnegligible process uncertainties, we introduce random effects to capture the influence of process uncertainties on the output, and propose a mixed scalable model to capture a manufacturing process under uncertainties for the first time. In the third task, to address multiple-domain problems, we establish a novel model framework and interpret a multiple-domain process as the outcome of coexisting physical mechanisms with different weights. To address the second challenge for multiple-domain manufacturing processes, we propose a sequential design of experiments to 91 detect physical domains adaptively, where the clustering scheme is introduced to DOE for the first time. To summarize, our work in this dissertation not only deepens our understanding of some advanced manufacturing processes that have great potentials to revolutionize the industry and improve our life, but also contributes to the scale-up methodology research for advanced manu- facturing, especially in scale-up modeling and design of experiments. In the future, we may extend the dissertation work to further support advanced manufacturing scale-up. Possible directions include but are not limited to the following. · A more generic scale-up modeling methodology for multiple-domain processes Recall that Task III is motivated by the study of droplet formation process in coated microfluidic devices. We establish a novel model framework and interpret a multiple-domain process as the outcome of coexisting physical mechanisms with different weights. A more generic framework would allow the absence of some physical mechanisms in certain feasible domains. In other words, explicit boundaries would appear between two domains in the form of “change points” when some physical mechanism appears in one domain but disappears in the other. It is expected that we will have a more generic scale-up modeling approach to accommodate both “change points” and weighted combination of physical mechanisms. The change-point approach has been applied to multiple-domain modeling problems in various fields. In [55], Huang et al. compared the abilities of several change-point models in modeling silica Nanowire growth, where the models are two-domain and one-dimensional. Very little work in the study of advanced manufacturing processes has considered multiple-domain change-point models even in one dimension. Such one-dimensional multiple-domain change-point models can be found in other disciplines like biomedicine [106, 6], neuroscience [72], ecosystem [96] and finance [45, 150]. Research on change-point modeling with multivariate settings are still at the 92 early stage. Most of the proposed strategies are relatively simple, e.g., decomposing the model to several single-dimensional problems by assuming independence among these dimensions [139]. The method of weighted combination in modeling is commonly seen in machine learning litera- ture, particularly, for semantic models of multiple domains [22, 35, 77, 105]. In fact, the recently most popular machine learning model, namely the deep neural network model, can be viewed as a linear combination of a set of submodels where the weights are learned from data. One of the key advantages of the deep neural network model is to combine, with proper weights, the advantages of submodels that perform well in different domains so as to achieve good prediction globally. For more details, see the tutorial in [68] and various references therein. However, these machine learning models suffer from poor physical interpretations. Moreover, they require a large num- ber of training data, thus are not suitable for advanced industrial manufacturing, since data in industrial manufacturing is usually limited (compared with those needed for learning a model) and expensive to obtain. As a result, physical-statistical modeling (or cross-domain modeling when both physical knowledge and data are very limited [130]), which utilizes both available physical knowledge and data, is desired for scale-up modeling in advanced manufacturing. So far, the weighted combination models employed in industrial engineering mostly assume the weights to be either constant or a function of the single variate (in the one-dimensional model) with some parameters to be estimated [55]. Our work in Chapter 4 proposes a more general weighted combination model where the weights are modeled as functions of the secondary factor composed of variables that do not contribute to the domain-dependence of response surfaces. A modeling approach that accommodates both the change-point and the weight-combination situation suffices to address an arbitrary multiple-domain scale-up problem. · Design of experiments for high-dimensional multiple-domain scale-up processes In Task IV, we propose a sequential design to detect domains adaptively for multiple-domain scale-up processes, where the clustering scheme is introduced to DOE for the first time. Design 93 points are added one by one after some initial experiments, and we keep updating our knowl- edge of the scale-up process. Each design point is selected to maximize the amount of extra information additional to knowledge obtained from previous experiments. Our sequential design dramatically reduces the cost of identifying physical domains, and the possibility of omitting domains with small area. Let n denote the number of dimensionless numbers. The sequential design is proposed within the scope of commonly-seen scale-up processes in advanced manufacturing, where n additional experiments are not a concern. In the future, an innovative technology adopted in a scale- up process may involve a large number of controllable variables, leading to a large number of dimensionless numbers. Considering the high costs of experimentation,n additional experiments could become a concern. In this case, it is not efficient to identify whether each dimensionless number is a component of the primary factor. Moreover, computing finite differences to obtain the gradient space would be very costly. In order to address the DOE problem for a high-dimensional multiple-domain scale-up process, we may consider the use of spectral clustering techniques, which performs dimensionality reduc- tion before clustering in fewer dimensions. The intuition is that the primary factor, compared to the secondary factor, captures more information across all physical domains, since the multiple- domain property is fully characterized by the primary factor. Let (x,z) T denote the vector of dimensionless numbers, including the independent variable x∈ R n and the response z∈ R. Note the response surface exhibits different characteristics in different physical domains, while (x does not contain any information regarding the response surface. 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Abstract (if available)

Abstract Advanced manufacturing that involves cutting edge sciences such as materials, physical and biological science, has great potentials to revolutionize the industry and improve our life. For example, nanomanufacturing has shown great promise in addressing important issues in various fields such as energy, medicine, food, and environmental science. Despite of numerous laboratory successes, we have seen a relatively small number of commercial-scale examples due to difficulties in scaling up nanomanufacturing processes. ❧ Early-stage study of advanced manufacturing processes mainly faces two challenges. Firstly, a conclusive understanding of the process is often not available due to limited physical knowledge and process uncertainties, making it difficult to obtain a precise characterization and prediction of the process. For example, although selective area metal organic chemical vapor deposition (SA-MOCVD) has been recognized as a promising process for producing nanowires, existing research has not provided a conclusive understanding of the synthesis process. Influence of controllable variables needs further study to enable accurate prediction of the process, which hinders full-scale nanomanufacturing. Secondly, experiments for understanding advanced manufacturing processes are often costly in time and resources, which restricts extensive experimental research. ❧ Various methods have been developed to support scale-up advanced manufacturing, including dimensional analysis (DA), scale-up modeling and design of experiments (DOE). DA serves as an important tool to achieve dimension reduction and ensure the scalability of results. Scale-up modeling methods for advanced manufacturing can be classified into four categories based on the requirement of process knowledge and data, namely physical, statistical, physical-statistical and cross domain modeling approach. DOE research has been recognized as an important task due to high cost of experimentation in scale-up manufacturing. However, existing research has not provided methodologies to address all issues that arise in scaling up advanced manufacturing processes. ❧ This dissertation not only seeks to deepen the understanding of some advanced manufacturing processes, but also aims to provide some scale-up methodologies for early-stage research on advanced manufacturing processes to bridge the gap from laboratory successes to scale-up advanced manufacturing. We explore four problems, each focusing on a different challenge that arises in scaling up advanced manufacturing processes. ❧ In the first scale-up problem of nanowire synthesis via SA-MOCVD, existing studies cannot explain well the distribution of nanowire growth on the substrate, which is critical for commercial-scale fabrication that requires accurate control of nanowire growth. To fill the research gap, we propose a physical-statistical model that interprets well the local dependence of nanowire growth, and allows the optimization of skirt area width of the substrate for uniform growth of nanowires. ❧ The second problem investigates uncertainties in scale-up processes, and proposes a mixed model framework to characterize scale-up process uncertainties. Specifically, we propose a Generalized Linear Mixed Model (GLMM) to characterize the gas-solid separation process in a cyclone separator, which captures well the scale-up process under uncertainties and enables a robust parameter design to minimize process uncertainties. ❧ The third and fourth tasks are motivated by the study of droplet formation process in microfluidic devices, where the process exhibits different characteristics in different feasible domains of settings and is therefore defined as a multiple-domain process. Existing scale-up models that capture the process in a specific physical domain cannot be applied to the prediction under new process settings. To fill the research gap, we explore scale-up modeling and design of experiments for multiple-domain processes in the third and fourth task respectively. In the third task, we establish a novel model framework and interpret a multiple-domain process as the outcome of coexisting physical mechanisms with different weights. A sequential modeling strategy is also provided to obtain the scale-up model for multiple-domain processes. In the fourth task, we propose a sequential design to detect domains adaptively, where the clustering scheme is introduced to DOE for the first time. ❧ Our work in this dissertation will not only further the understanding of specific advanced manufacturing processes, but also provide some generic methodologies for early-stage study of scale-up advanced manufacturing processes to bridge the gap between laboratory successes and scale-up advanced manufacturing. Both modeling and experimental design methodologies are explored to address scale-up challenges in advanced manufacturing due to limited knowledge, process uncertainties and the existence of multiple domains.

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Asset Metadata

Creator Duanmu, Yanqing (author)

Core Title Some scale-up methodologies for advanced manufacturing

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Publication Date 06/28/2017

Defense Date 05/10/2017

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag advanced manufacturing,clustering,design of experiments,Modeling,multiple-domain,OAI-PMH Harvest,scale-up

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Huang, Qiang (committee chair)

Creator Email duanmu@usc.edu,graceduanmu@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-391147

Unique identifier UC11264052

Identifier etd-DuanmuYanq-5460.pdf (filename),usctheses-c40-391147 (legacy record id)

Legacy Identifier etd-DuanmuYanq-5460.pdf

Dmrecord 391147

Document Type Dissertation

Rights Duanmu, Yanqing

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA