Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations

Multi-objective optimization algorithms aim at finding Pareto-optimal solutions. Recovering Pareto fronts or Pareto sets from a limited number of function evaluations are challenging problems. A popular approach in the case of expensive-to-evaluate functions is to appeal to metamodels. Kriging has been shown efficient as a base for sequential multi-objective optimization, notably through infill sampling criteria balancing exploitation and exploration such as the Expected Hypervolume Improvement. Here we consider Kriging metamodels not only for selecting new points, but as a tool for estimating the whole Pareto front and quantifying how much uncertainty remains on it at any stage of Kriging-based multi-objective optimization algorithms. Our approach relies on the Gaussian process interpretation of Kriging, and bases upon conditional simulations. Using concepts from random set theory, we propose to adapt the Vorob’ev expectation and deviation to capture the variability of the set of non-dominated points. Numerical experiments illustrate the potential of the proposed workflow, and it is shown on examples how Gaussian process simulations and the estimated Vorob’ev deviation can be used to monitor the ability of Kriging-based multi-objective optimization algorithms to accurately learn the Pareto front

Process Characterization and Optimization of Roll-to-Roll Plasma Chemical Vapor Deposition for Graphene Growth

Large-scale production of graphene and other nanostructures remains a hindrance to their adoption in the semiconductor and materials manufacturing industries. The main purpose of this thesis is to develop an efficient and scalable technique for depositing graphene on various flexible substrates. Hence, a custom-built roll-to-roll capacitively coupled plasma chemical vapor system for deposition of graphene on flexible substrates is thoroughly described in this work. Graphene quality on Cu foil has been optimized for a roll-to-roll process using statistical optimization methods. Since graphene quality and uniformity depend on plasma input parameters, such as plasma power, gas pressure, and the gas mixture used, effects of input parameters have been explored to maximize graphene quality, as quantified by Raman spectroscopy using the ID/IG intensity ratio. Furthermore, in situ optical emission spectroscopy (OES) has been developed and utilized to determine the effects of several plasma species on graphene growth and quality. OES results demonstrate that graphene quality on Cu foil increases with CH radical emission; however, O and H atoms, C2 and CN radicals, and Ar+ ion all negatively correlate to graphene quality. Results aid in developing a conceptual model for a graphene growth mechanism that indicates the adverse impact of ion bombardment on graphene quality in the low-frequency capacitively coupled plasma. However, the existence of active carbon species in the plasma, such as CH radical, accelerates the growth process and leads to moderate-quality graphene deposition on Cu foil at web speeds reaching as high as 1 m/min. Nevertheless, graphene quality measured from Raman spectroscopy declines significantly with increased Cu foil velocity (web speed) in the roll-to-roll process, inducing a critical limitation in current production rates for roll-to-roll CVD nonmanufacturing techniques. With the aid of heat transfer modeling of the moving foil, we show that the graphene quality decrease is primarily due to Cu foil temperature decline with increased web speed. The Cu foil temperature distribution is determined both experimentally and numerically during roll-to-roll graphene growth as a function of web speed, plasma power and plasma length. The maximum Cu foil temperature in the plasma rises with increased plasma power due to increased heating from the plasma. However, the maximum Cu foil temperature decreases with increased web speed caused by higher heat advection by the moving foil. In addition, shortening the plasma slit (by decreasing the electrodes length) cools the Cu foil temperature and diminishes its temperature uniformity in the plasma region. Consequently, graphene crystallization, identified using Raman spectroscopy, improves with higher Cu foil temperatures. As a result, an optimum condition is defined by raising the plasma power, lowering the web speed and increasing the plasma region length, which consistently produces high-quality graphene on Cu foil. The throughput of graphene production can be increased by utilizing Ni foil as a substrate since carbon solubility in Ni is higher than in Cu. Thus, the effects of web speed and plasma power on Ni foil temperature distribution are evaluated during graphene deposition in the roll-to-roll process. Furthermore, the Ni foil cooling rate, which strongly affects carbon atom segregation from Ni after the growth process, is derived from the heat transfer model. Plasma power has negligible effects on the cooling rate, whereas the web speed has a significant impact on the cooling rate. Consequently, graphene has comparable quality at different plasma powers, whereas web speed controls graphene quality, particularly with regards to uniformity and thickness. Our work highlights the benefits of using Ni foil in a roll-to-roll process for graphene deposition at higher web speeds and lower substrate temperatures, rather than using Cu foil, which requires significantly more substrate heating. Plasma plays a crucial role in heating the foil for graphene deposition in the roll-to-roll process, without the need of a supplemental heating source. Thus, accurate measurement of the translational gas temperature in the plasma is vital, since gas temperature strongly influences the foil temperature distribution, which, in turn, affects graphene growth kinetics. Optical emission spectroscopy (OES) is used to measure the rotational temperatures of N2 + (B-X), CN (B-X) and H2 (d3Πu → a3Σg +), and to determine accurate translational gas temperatures. Power dissipation in the plasma is also measured to understand gas temperature variation for the experimental input conditions. Thus, the effects of plasma power, gas pressure and the addition of nitrogen (N2), oxygen (O2) and methane (CH4) gases on power dissipation and gas temperature in a hydrogen (H2) plasma are assessed. The rotational temperatures measured from the gas species have different values due to the non-equilibrium nature of the plasma. Of the gases measured, the rotational temperature of N2 + is most accurate in representing the translational gas temperature. These results improve the understanding and control of the thermochemical environment for carbon nanostructure growth in the plasma chemical vapor deposition processes. Graphene quality significantly depends on gas pressure since our plasma roll-to-roll system is sustained by a capacitively coupled plasma that operates in two modes, depending on the gas pressure and discharge gap. The modes are identified as alpha and gamma modes, and are sustained by volume ionization and secondary electron emission processes, respectively. Up to our knowledge, the presence of both modes at 80 kHz plasma frequency has not previously been reported. Thus, a detailed characterization of argon plasma is attempted to determine the underlying plasma physics of the low-frequency plasma. Due to strong ion bombardment on the electrodes, the gamma mode coexists with the alpha mode, resulting in a hybrid mode. The voltage square waveform is found to play an important role in sustaining this hybrid mode. The hybrid mode exists at low gas pressures of 5.5 and 9.5 mbar in the plasma set power ranges from 300 to 1100 W. However, the plasma at 13.8 mbar gas pressure transforms from hybrid to gamma mode when the plasma set power is beyond 750 W due to increased secondary electron emission processes. The emission spectra measured from optical emission spectroscopy reveal the presence of non-Ar species in the gamma mode, such as H, CH, and C2. These species are sputtered from the graphite electrodes by ion bombardment to produce secondary electrons that sustain the gamma discharge. Results show the possibility of sustaining the hybrid mode at a low plasma frequency using a tailored waveform. As a results of these plasma characterization tools, we report a continuous and rapid rollto- roll deposition of thin graphite film on Cu foil. The composition of the Ar/H2/CH4/N2/O2 plasma plays significant role in the successful direct growth of the thin graphite film on copper foil. Optical emission spectroscopy is used to characterize the plasma during graphite synthesis and show that the addition of N2 enhances the plasma reactivity, and O2 was found to increase the deposition rate of the graphite film. The film was characterized by Raman spectroscopy, scanning electron microscopy (SEM), transmission electron microscopy (TEM) and X-ray photoelectron spectroscopy (XPS). The described large-scale graphite production can produce a graphite-Cugraphite structure or uniform thin graphite films for thermal management applications in electronics devices. Graphene growth optimization, substrate thermal analysis, and plasma characterizations are used to control graphene mass-production in a custom-built roll-to-roll plasma CVD system. These techniques are addressed to provide a route for nanomanufacturing of graphene and graphite on Cu and Ni foils. These methods aid in understanding the correlations between process conditions and graphene quality, as well as the interactions between the plasma and the substrate, to yield high-throughput production of high-quality graphene. The procedure outlined here can be applied to efficiently scale-up the production of other micro- and nanomaterials