Operando Studies of the Evolution of Electrode Inhomogeneities and of Solid Electrolyte Interphase Formation Using Spatial Frequency Heterodyne X-ray Imaging by Alexandra Katarina Stephan B.S., University of Chicago, 2014 Advisor: Professor Christoph Rose-Petruck A dissertation submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the Department of Chemistry at Brown University Providence, RI May 2019 © Copyright 2019 by Alexandra Stephan This dissertation by Alexandra K. Stephan is accepted in its present form by the Department of Chemistry as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date ______________ ______________________________________ (Professor Christoph Rose-Petruck), Advisor Recommended to the Graduate Council Date ______________ ______________________________________ (Professor Brian W. Sheldon), Reader Date ______________ ______________________________________ (Professor Brenda Rubenstein), Reader Approved by the Graduate Council Date ______________ ______________________________________ (Andrew G. Campbell), Dean of the Graduate School iii Curriculum Vitae In the fall of 2010, Alexandra Stephan began her studies at the University of Chicago. During her time there she worked in the laboratory of Professor Paul Nealey in the Institute for Molecular Engineering. She completed an honor’s thesis entitled, “An Aqueous Microfluidic Synthesis of Cadmium Selenide Quantum Dots.” In June 2014 she graduated from the University of Chicago with a Bachelor’s of Science in Chemistry. Alexandra entered the chemistry graduate program at Brown University in August 2014. She joined the laboratory of Professor Christoph Rose-Petruck. Her doctoral research on the characterization of Lithium-Ion Batteries using Spatial Frequency Heterodyne X-ray Imaging was done as a collaboration between the laboratories of Professor Christoph Rose-Petruck and Professor Brian W. Sheldon. Alexandra was a recipient of the Brown Chemistry Department Wernig Fellowship for the 2018-2019 school year. In addition to her doctoral research, Alexandra worked as a teaching assistant for the Chemistry 330, general chemistry, and Chemistry 100, introductory chemistry, problem sessions. She also was a teaching assistant for the Chemistry 330 laboratory. During the summer of 2018, Alexandra worked as an adjunct professor at Providence College and taught two semesters of general chemistry. iv Acknowledgements There are many people who I would like to thank for being instrumental in the completion of doctoral work. I would like to begin by thanking my advisor Christoph Rose-Petruck. When I initially joined his lab, neither of us knew at that time where my project would end up leading us. I am so grateful that he allowed me to initiate a collaboration with the Sheldon group and embark on a project focused on the characterization of lithium-ion batteries. Even though lithium-ion batteries were beyond our area of expertise, he gladly joined me in learning about a field, previously unknown to both of us. His encouragement and guidance helped me to work through the many challenges that accompanied the battery imaging project. From the day I first joined his lab, he never gave up on me. He has always believed that I would eventually make sense of my data or get something that seemed impossible to work. Reflecting upon the past five years, Christoph’s unwavering support throughout my five years has shaped me into the scientist that I am today. I would also like to thank Brian Sheldon who has become a second advisor to me. I am forever grateful that he agreed to work with us on applying X-ray imaging to the characterization of lithium-ion batteries. Brian welcomed me into his group and has always been willing to assist me in any way that he can. He has patiently answered all my questions that have come with learning about lithium-ion batteries. v Former group member Francisco taught me everything I practically needed to know about Spatial Frequency Heterodyne X-ray Imaging. Without the work he did to automate the image collecting and image processing, I could not have completed the work presented in his thesis. Ravi Kumar, formerly from the Sheldon group, also is someone I would like to thank. He worked with me during the early stages of the battery imaging project and went through all the difficulties of starting a new project along with me. He taught me so much about batteries and how to cycle them for experiments. Also, on top of his own projects, he made all the cells used in Chapter 2. I would like to thank Elianna Isaac for being more than just a fellow Rose-Petruck group member, but for also being a friend. It truly has been a pleasure mentoring you on the battery imaging project. I also cannot thank you enough for all the help you have given me with coding. The scientific discussions that we have had and the insights that you have provided have been invaluable. Your encouragement and support over the past two years have truly helped me to get to where I am today. I would like to thank Mok Yun Jin. She seamlessly took over for Ravi as my primary collaborator in the Sheldon group once Ravi graduated. She put her heart and soul into the batteries that she has made for our collaboration. Mok Yun has really helped me to understand the electrochemical processes that take place in lithium-ion batteries. I would not have been able to do the work presented in Chapter 3 without her. I cannot thank her enough for all of the help she has given me over the past two years. vi Former and present Rose-Petruck group members that I would also like to thank include Yishou Jiao, Pawel Chmielniak, Michael Michon, Mengjie Lyu, and Ziyue Li (and honorary member Nate Goff). Thank you for always being supportive of me, providing me with so many ideas and inspirations, giving me a hand around lab, and being great friends. I would also like to thank current and former Sheldon group members Wei Zhang, Christos Athanasiou, Juny Cho, Wesley Cai, and Leah Nation. Thank you for welcoming me into your group with open arms. I have learned so much from all of you. I would like to thank everyone in the Brown Chemistry Department for all support they have provided me over the past five years. There are a few people I would like to particularly thank. I would like to thank Professor Brenda Rubenstein for not only serving as a committee member but also being a supportive mentor to me. Professor Rubenstein’s work as the chair of the Diversity and Inclusion Action Committee has truly been making the department a better and more inclusive place. Thank you Professor Sarah Delaney for all the work you have done to get the Graduate Student Leadership Committee off the ground and also for serving as the Director of Graduate Studies. I would like to thank Professor Doll for supporting me and believing in me at a time when I had a hard time believing in myself. I would also like to thank Professor Williard for being his openness to try new ways to make the department better. I also would like to thank my friends in the chemistry department, in particular Vale Cofer-Shabica, Len Sprague, and Jen Ruddock, for always supporting and encouraging me throughout my time in graduate school. I would like to thank Rose Barreira, Sheila Quigley, Elaine Tucci, vii and Kim Keenan for always lending me a hand with whatever I needed help with and for always being friendly faces in the department. Thank you David Blair and Carol DeFeciani for helping me with all the computing problems that I encountered. Outside of Brown, there are also many people who I would also like to thank. First I would like to start with my family. My parents, Russell and Vivian, have always supported me in my dream to pursue a Ph.D. in chemistry. I would not have reached this point without their love and support. My in-laws, Rabie and Gena, have been incredibly supportive of me throughout this journey. There are not enough words to express my gratitude for my husband, Jules. He not only has supported me through all of graduate school, but also completed a Ph.D. in chemistry himself. Jules has helped me push through every time I have wanted to give up. He has been incredibly patient with me throughout these past five years. The advice and wisdom he has shared with me throughout graduate school has truly been invaluable. I honestly would not have reached this point without his love and support. I cannot thank him enough. In addition to my family, my friends have also supported me throughout my graduate school experience. I would like to think Sophia Koinis and Christina Cook for being life-long friends who have always supported my dreams of being a scientist. I would like to thank Jia Guo and Annie Marsden. Their help and encouragement as fellow chemistry majors in undergrad helped to ensure that I reached graduate school in the first place. I would like to thank Laurie Butler for not only introducing me to quantum mechanics in a way that made me actually like quantum mechanics, but also teaching me that only thing that prevents you from getting to where you want to go is your own viii fear of trying. I would like to thank Paul Nealey and Xiaoying Liu for mentoring me in my undergraduate research. Finally I would like to thank Jay Skipper and Deni Ors for teaching me chemistry in high school. Their love of chemistry inspired me in the first place to pursue a Ph.D. in chemistry. ix Table of Contents CHAPTER 1 INTRODUCTION ......................................................................................... 1 1.1 Lithium-Ion Battery Background .................................................................................... 1 1.2 Lithium-Ion Battery Operation ....................................................................................... 3 1.3 High Capacity Alloying Anode: Silicon ............................................................................ 7 1.4 Problems with Current Techniques to Study Lithium-Ion Batteries............................... 11 1.5 Spatial Frequency Heterodyne X-ray Imaging ............................................................... 13 1.6 Aims of Thesis.............................................................................................................. 16 References......................................................................................................................... 18 CHAPTER 2 OPERANDO IMAGING OF THE EVOLUTION OF ELECTRODE INHOMOGENEITIES IN UNMODIFIED LITHIUM ION CELLS........................................... 27 2.1 Introduction................................................................................................................. 27 2.2 Experimental Methods................................................................................................. 28 2.2.1 SFHXI Experimental Setup............................................................................................28 2.2.2 Cell Assembly ..............................................................................................................30 2.2.3 Cell Cycling ...................................................................................................................31 2.2.4 Modeling of Si Nanoparticle Scatter Intensity ............................................................32 2.3 Results ......................................................................................................................... 35 2.3.1 High and Low Capacity Cells.........................................................................................35 2.3.2 Inhomogeneity of the Lithiation Process .....................................................................38 2.3.3 Modeling Nanoparticle Scatter Intensity .....................................................................44 2.4 Discussion.................................................................................................................... 50 2.5 Conclusion ................................................................................................................... 53 References......................................................................................................................... 55 CHAPTER 3 OPERANDO TRACKING OF SOLID ELECTROLYTE INTERPHASE FORMATION AND LITHIATION OF SILICON USING X-RAY IMAGING ...................................................... 62 3.1 Introduction................................................................................................................. 62 3.2 Experimental Methods................................................................................................. 65 3.2.1 SFHXI Experimental Setup............................................................................................65 3.2.2 Cell Assembly ..............................................................................................................67 3.2.3 Cell Cycling ...................................................................................................................69 3.2.4 Modeling of SEI Thickness ...........................................................................................70 3.3 Results and Discussion ................................................................................................ 73 3.3.1 Electrochemical Analysis of First Cycle and Second Lithiation ....................................73 3.3.2 Carbon Black Electrode ...............................................................................................75 3.3.3 20 mV Voltage Hold .....................................................................................................78 3.3.4 High Voltage Holds ......................................................................................................81 3.3.5 SEI Thickness Modeling ...............................................................................................90 3.3.6 Homogeneity of SEI Formation and Si Lithiation .........................................................94 3.4 Conclusion .................................................................................................................. 99 References .......................................................................................................................100 APPENDIX A DETECTOR FIXED PATTERN NOISE CORRECTION ........................................106 A.1 Introduction ...............................................................................................................106 A.2 Development of a Flattener........................................................................................111 A.3 Flatteners and Effective Exposure Time .....................................................................118 x A.3.1 Second Order Polynomial Flatfield Fit ......................................................................118 A.3.2 Third Order Polynomial Flatfield Fit...........................................................................119 A.3.3 Logistic Function Flatfield Fit ....................................................................................121 A.4 Conclusion .................................................................................................................127 References .......................................................................................................................128 APPENDIX B ORIGINS OF AND SOLUTIONS TO NOISE PRESENT IN SCATTER IMAGES 129 B.1 Introduction ...............................................................................................................129 B.2 Artificial Images and Noise .........................................................................................130 B.3 Fourier Transform Region of Interest Sizes..................................................................137 B.4 Fourier Transform Region of Interest Shape................................................................142 B.5 Over Exposing Images ................................................................................................146 B.6 Conclusion ..................................................................................................................150 References........................................................................................................................152 APPENDIX C MATLAB CODING .................................................................................153 C.1 Introduction................................................................................................................153 C.2 Scatter Data Analysis Scripts .......................................................................................153 C.2.1 Scatter Movie Script ..................................................................................................154 C.2.2 Scatter Histogram Script ...........................................................................................155 C.3 Scatter Intensity Modeling Scripts...............................................................................156 C.3.1 Model of Scatter Intensity of Silicon Nanoparticles ..................................................156 C.3.2 Model of Scatter Intensity of Core-Shell Silicon Nanoparticles ................................157 C.3.3 Model of Scatter Intensity of Silicon Nanoparticles with SEI .....................................160 C.4 Fixed Pattern Noise Correction Scripts ........................................................................161 C.5 Script to Create Artificial Images ................................................................................162 References........................................................................................................................162 xi List of Tables Table 2.1: Percent of electrode area with normalized scatter intensity less than 1 for Cells 1-5 at the end of the first lithiation .........................................................................................................41 Table 3.1: Table with the 3 different electrode compositions used in this chapter .........................68 Table B.1: Table showing six region of interest sizes and their resulting average scatters and standard deviations of their average scatter .................................................................................... 139 xii List of Figures Figure 1.1: Comparison of theoretical energy densities of nickel cadmium (Ni/Cd), lead acid (Pb acid), nickel metal hydride (Ni/MH), and lithium ion (Li/C-CoO2) batteries ............................................... 3 Figure 1.2: Charging and discharging LIB with graphite and LiCoO2 electrodes ................................ 4 Figure 1.3: Depiction of SEI formation on electrode surface a) Initial formation of SEI b) Incorporation of organic components into SEI, and c) Incorporation of inorganic components into SEI.......................... 6 Figure 1.4: Gravimetric and volumetric capacities of different group IV elements ........................... 8 Figure 1.5: Plot of specific capacity of Si as a function of the stoichiometric ratio of Li in lithiated Si b) Common lithiated Si phases and the stoichiometric ratio of Li to which they correspond .................... 9 Figure 1.6: Schematic of the a) Pulverization of Si nanoparticles as a result of cycling b) Delamination of Si nanoparticles from electrode surface as a result of cycling ..................................................... 10 Figure 1.7: Diagram showing the formation of SEI during lithiation and then the deformation of SEI upon delithiation ............................................................................................................ 11 Figure 1.8: Diagram of SFHXI technique ............................................................................. 14 Figure 2.1: a) Spatial Frequency Heterodyne X-ray Imaging setup b) Fourier analysis methods ......... 30 Figure 2.2: Lithiation, scatter behavior, and inhomogeneities of high and low capacity cells a) Plot of normalized scatter intensity versus total capacity per gram of Si for the first lithiation of Cells 1-5 b) Plot of voltage as a function of total capacity per gram of Si for Cells 1-5 c) Histograms of scatter intensity distribution for Cells 1-5 at the end of the first lithiation ........................................................... 37 Figure 2.3: 4 cycles of lithiation and delithiation voltage and scatter intensity plotted as a function of elapsed time in thousands seconds for Cell 2 ......................................................................... 38 Figure 2.4: Inhomogeneity of lithiation process a) X-ray scatter image showing regions a, b, c, d, e, and the average over the electrode surface for Cell 2. The average is the same area shown in the movie. b) Plot of normalized scatter intensity versus voltage (V) for the first lithiation of six regions of Cell 2. Each scatter intensity curve is normalized by the initial scatter value for that curves .............................. 39 Figure 2.5: Snapshots from the scatter intensity movie during the first lithiation of Cell 2 and the corresponding voltages. Each pixel is normalized by its initial scatter intensity. Area shown in snapshots corresponds to the average area in Figure 2.4 ....................................................................... 39 Figure 2.6: Histograms of the normalized scatter intensity of Cell 2 during four cycles .................... 42 Figure 2.7: a) Scatter image of Cell 2 showing strips 1, 2, and 3 shown as surfaces. b) Evolution of size and intensity of inhomogeneities in Strip 1 region over 4 cycles. c) Evolution of size and intensity of inhomogeneities in Strip 2 region over 4 cycles. d) Evolution of size and intensity of inhomogeneities in Strip 3 region over 4 cycles ................................................................................................ 43 Figure 2.8: Snapshots from the scatter intensity movie during four cycles of Cell 2 and the corresponding voltages. Each pixel is normalized by its initial scatter intensity. Area shown in snapshots corresponds to the average area in Figure 2.3. ....................................................................... 43 xiii Figure 2.9: a) Plot of calculated scatter intensity as a function of NP diameter for crystalline Si NPs in a vacuum b) Plot of calculated normalized scatter intensity as a function of NP diameter for crystalline Si NPs in a vacuum with diameters in range relevant to those found in the cells ................................ 44 Figure 2.10: Experimental X-ray scatter intensity of crystalline Si NP particle powders with various diameters ..................................................................................................................... 45 Figure 2.11: a) and b) Post mortem SEM of Si composite electrode taken after 13 cycles ............... 45 3 Figure 2.12: Electron density in units of electrons/cm for possible lithiated silicon phases formed during lithiation. Also shown are the electron densities for crystalline Si (cr-Si) and for amorphous Si (a-Si), both indicated with yellow. The phases most likely to form in a Li-ion cell are shown in black. ............ 46 Figure 2.13: Core-Shell particle diagram and modeled scatter intensity of core-Shell NPs in different media a) Diagram of core-shell NP lithiation where the purple shell denotes the LixSiy phase and the yellow core denotes the crystalline Si core where d is the diameter of the core and D is the diameter of the entire particle b) Diagram of Si NP (yellow) and carbon black (grey) on top of copper electrode (copper) all surrounded by electrolyte (green) c) Plot of calculated scatter intensity as a function of the calculated capacity per gram of Si for core-shell NPs in CB medium, electrolyte medium, and a mixture of CB and electrolyte medium. The CB plot is divided by 5. The calculated increase in NP diameter as a function of calculated capacity is shown on the right axis ......................................................... 48 Figure 2.14: a) X-ray diffraction spectrum for a pristine electrode and Si and Cu reference spectra b) X- ray diffraction spectrum for an electrode lithiated to 0.05V and Si and Cu reference spectra .............. 49 Figure 3.1: a) Spatial Frequency Heterodyne X-ray Imaging setup b) Fourier Analysis Methods ........ 67 Figure 3.2: a) Plot of voltage as a function of capacity for a cell for the first lithiation of a cell, b) Plot of voltage as a function of capacity for a cell for the first delithiation of a cell, and c) Plot of voltage as a function of capacity for a cell for the second lithiation of a cell ................................................... 74 Figure 3.3: a) Plot of scatter intensity and voltage for a cell with just CB as the active electrode material and b) Capacity for the three cycles of the cell ....................................................................... 77 Figure 3.4: a) Plot of scatter intensity and voltage for a cell with a 20 mV hold during the first lithiation, b) Magnification of plot a from 0 s to 15500 s, which corresponds to the first cycle, and c) Capacity for the three cycles of the cell ..................................................................................................... 80 Figure 3.5: a) Plot of scatter intensity and voltage for a cell with a 500 mV hold during the first lithiation, b) Magnification of plot a from 3600 s to 53600 s, which corresponds to the first lithiation, and c) Capacity for the six cycles of the cell ............................................................................................... 83 Figure 3.6: a) Plot of scatter intensity and voltage for a cell with a 400 mV hold during the first lithiation, b) Magnification of plot a from 0 s to 180000 s, which corresponds to the first lithiation, and c) Capacity for the four cycles of the cell ............................................................................................. 85 Figure 3.7: a) Plot of scatter intensity and voltage for a cell with a 300 mV hold during the first lithiation, b) Magnification of plot a from 0 s to 12000 s, which corresponds to the first lithiation, c) Magnification of plot a from 11690 s to 14690 s, which corresponds to the second and third cycles, and d) Capacity for the four cycles of the cell ....................................................................................................... 87 Figure 3.8: Nanoparticle diagram and modeled scatter intensity of SEI formation a) Cartoon of Si NP (yellow) and carbon black (grey) on top of copper electrode (copper) all surrounded by electrolyte (light xiv green) and cartoon of SEI growth (dark green) on Si NP (yellow) b) Plot of modeled normalized scatter intensity as a function of SEI thickness in Ångstroms ............................................................... 92 Figure 3.9: Histograms after a high voltage hold where SEI was formed and after the cr-Si was first lithiated a) 500 mV hold cell, b) 400 mV hold cell, and c) 300 mV hold cell .................................... 96 Figure 3.10: Histograms at various points in cycling for the 20 mV voltage hold cell and voltage trace showing the time points of the histograms ........................................................................... 97 Figure A.1: a) Background, b) Flatfield, c) Grid, and d) Image .................................................108 Figure A.2: Cross sections of intensity modulations for a) Flatfield and b) Battery image ...............109 Figure A.3: a) Image corrected using FFC, b) Absorption image generated using image corrected using FFC, c) Scatter image generated using image corrected using FFC, and d) Cross section of image corrected using FFC ....................................................................................................................111 Figure A.4: Plot of intensity as a function of exposure time for pixel (300, 300) ...........................112 Figure A.5: Plot of second order polynomial background fit for pixel (300, 300) ..........................114 Figure A.6: Plot of second order polynomial flatfield fit for pixel (300, 300) ................................115 Figure A.7: Using flattener from Equation 3 a) Flattened flatfield, b) Flattened grid, c) Flattened image, and d) Flattened image cross section .................................................................................117 Figure A.8: Flattened using inverse second order polynomial to select flattener on a pixel by pixel basis a) Flattened grid, b) Flattened image, and c) Cross section of flattened image ...............................119 Figure A.9: Plot of logistic function flatfield fit for pixel (300, 300) ............................................122 Figure A.10: a) Flattened grid, b) Flattened image, and c) Cross section of flattened image ............124 Figure A.11: Processed images using logistic flattener a) Absorption image, b) Scatter image, and c) Cross section of the scatter image .....................................................................................126 Figure A.12: Logistic flattener applied to different Image a) Flattened image and b) Cross section of flattened image ............................................................................................................126 Figure B.1: Scatter image generated ................................................................................129 Figure B.2: Artificial images with no noise a) Artificial grid image b) Artificial image produced using Gaussian blur c) Generated (0,1) scatter image d) Generated (1,0) scatter image e) Generated Fourier spectrum ....................................................................................................................131 Figure B.3: a) Fourier spectrum generated from battery image b) Fourier spectrum generated from a non-battery (noncircular) object .......................................................................................132 Figure B.4: Artificial images with 5% salt and pepper noise introduced in artificial image and not in artificial grid a) Artificial image produced using Gaussian blur b) Generated Fourier spectrum c) Generated (0,1) scatter image d) Generated (1,0) scatter image ..............................................................134 Figure B.5: Artificial images with 10% salt and pepper noise introduced in artificial image and not in the artificial grid a) Artificial image produced using Gaussian blur b) Generated Fourier spectrum c) Generated (0,1) scatter image d) Generated (1,0) scatter image ...............................................................135 xv Figure B.6: Artificial images with 5% salt and pepper noise introduced in artificial image and in artificial grid a) Artificial grid image b) Artificial image produced using Gaussian blur c) Generated (0,1) scatter image d) Generated (1,0) scatter image e) Generated Fourier spectrum ......................................136 Figure B.7: Fourier spectra and cross sections of Fourier spectra for the following regions of interest a) 650 x 650 b) 900 x 900 c) 1000 x 1000 ................................................................................140 Figure B.8: Scatter images and cross sections of scatter images for the following regions of interest a) 650 x 650 b) 900 x 900 c) 1000 x 1000 ................................................................................141 Figure B.9: Cross sections of Fourier spectrum for a centered square region of interest a) Horizontal harmonics b) Vertical Harmonics .......................................................................................143 Figure B.10: Cross sections of Fourier spectrum for a horizontally off-centered square region of interest a) Horizontal harmonics b) Vertical Harmonics .....................................................................144 Figure B.11: Cross sections of Fourier spectrum for a centered horizontal rectangle region of interest a) Horizontal harmonics b) Vertical Harmonics .........................................................................145 Figure B.12: Cross sections of Fourier spectrum for a centered horizontal rectangle region of interest that extends across entire image a) Horizontal harmonics b) Vertical Harmonics ...........................146 Figure B.13: Fourier spectra and cross sections of Fourier spectra for the following exposure times a) 20 s with no over exposure b) 25 s with over exposure c) 30 s with over exposure .............................148 Figure B.14: Scatter images and cross sections of scatter images for the following exposure times a) 20 s with no over exposure b) 25 s with over exposure c) 30 s with over exposure ...............................149 xvi CHAPTER 1 INTRODUCTION 1.1 Lithium-Ion Battery Background As consumer demand for portable electronic devices has increased, there has been a parallel effort to develop compact, lightweight, rechargeable batteries to power these devices. Also, as more grid energy is being generated from renewable energy sources i.e. wind, solar, and geothermal, there has been a growing need for batteries to be used for grid storage. Additionally, the introduction of electric vehicles to the consumer market has created another source of demand for compact and lightweight rechargeable batteries. When designing rechargeable batteries, the desirable characteristics of these batteries are high energy densities, high specific energies, low manufacturing and material costs, safe designs, and long cycle lives.1,2,3 Energy density refers to the amount of electrical energy produced per unit volume, and specific energy refers to the amount of electrical energy produced per unit of weight. A battery with both a high 1 energy density and a high specific energy would be compact in size and lightweight and subsequently ideal for use in portable electronic devices and electric vehicles. In the late 1970s, lithium was initially investigated as an electrode material due to its low atomic weight and its electropositivity (-3.04 V versus standard hydrogen electrode). However when Li-metal was used in rechargeable batteries, lithium dendrites would form. These dendrites caused short circuits and consequently presented safety concerns with regards to the implementation of Li-metal batteries. To circumvent the dendrite problem of Li-metal batteries but still take advantage of lithium’s low atomic weight and its electropositivity, Armand proposed a battery where Li-ions moved from being intercalated in an electrode with one potential to being intercalated in another electrode with a different potential.4,5 The Sony Corporation began commercially selling the first lithium-ion battery (LIB) in 1991.4 This first commercial LIB by Sony was modeled after the intercalation battery proposed by Armand where LiCoO2 was the Li-ion source and the Li-ions were intercalated- deintercalated into a petroleum coke carbon electrode.2 The energy density of this initial LIB was 200Wh/l and its specific energy was 80 Wh/kg.4 When compared to other commercially available batteries such as nickel cadmium, lead acid, and nickel metal hydride, the LIB clearly has the highest energy density (Figure 1.1).6 2 Figure 1.1: Comparison of theoretical energy densities of nickel cadmium (Ni/Cd), lead acid (Pb acid), nickel metal hydride (Ni/MH), and lithium ion (Li/C-CoO2) batteries6 1.2 Lithium-Ion Battery Operation Modern commercially available LIBs can have an energy density of 600 Wh/l and a specific energy of 200 Wh/kg.7 Modern LIBs are similar to those first produced by Sony in that they use LiCoO2 as the source of lithium and then intercalate-deintercalate the lithium into graphite in place of petroleum coke carbon (Figure 1.2).7 Graphite eventually replaced coke because graphite has twice the lithium intercalation capacity of coke.4,8 Although graphite has become the standard carbon active material, other carbon active materials have been investigated, particularly those that are nanostructured, in order to further optimize the performance of LIBs.9,10 3 Figure 1.2: Charging and discharging LIB with graphite and LiCoO2 electrodes7 The charging process of a LIB requires the input of electrons so that the lithium ions leave the LiCoO2 cathode and enter the graphite anode. During the discharging process, the lithium ions exit the graphite anode and consequently supply electrons to power the device to which the battery is connected (Figure 1.2). The reaction that the LiCoO2 undergoes during the charging process is LiCoO2 !" Li1-xCoO2 + xLi+ + xe- The reaction that graphite undergoes during the charging process is C6 + xLi+ + xe- !" LixC6 Also as can be seen in Figure 1.2, between the two electrodes there is a liquid electrolyte that allows for the transport of Li+. This liquid electrolyte is usually composed of organic solvents such as diethyl carbonate (DEC), ethylene carbonate (EC), and dimethyl carbonate (DMC) and a lithium salt such as LiPF6, LiBF4, or LiClO4.7 4 During the operation of batteries, an electric potential is applied to the battery, which results in the reduction or oxidation of the electrolyte at the anode or cathode, respectively. The potential at which the decomposition of a particular electrolyte species occurs is variable and is one of the reasons why different electrolyte compositions are explored. Although there is some variability in the potential at which the SEI begins to form, practically 0.8 V is considered the voltage at which the electrolyte becomes unstable.11,12 The decomposition of electrolyte occurs at the surface of the electrode, and the decomposition products deposit onto the surface of the electrode forming the solid electrolyte interphase (SEI). Though the SEI forms on both the cathode and the anode, the anode’s SEI layer is much thicker than that of the cathode.13 The SEI is a permanent insulating passivation layer that has low electronic conductivity but high ionic conductivity.7 The high ionic conductivity allows Li+ transport while its low electronic conductivity ideally prevents further decomposition of the electrolyte.11 The SEI is composed of inorganic components from salt degradation and of organic components from the electrolyte solvent decomposition. The organic components of the SEI start to decompose at higher voltages than the inorganic components.12,14 The thickness of the SEI can range from a few Angstrom to a few thousands of Angstrom.11,12,15,16,17 The exact structure and composition of the SEI is widely debated. Though a two-layered structure consisting of organic and inorganic layers has become somewhat agreed upon (Figure 1.3). 12,14,18 The lack of agreement in SEI structure and in composition stems from the difficulty in measuring and 5 characterizing the SEI and from the many variables such as electrolyte composition, cycling conditions, and electrode material, which affect SEI growth. However, it is agreed upon that there is some degree of porosity in the SEI structure (Figure 1.3). This porosity is what allows the SEI to have a high ionic conductivity but low electronic conductivity. The pore size of the SEI is such that Li+ can pass through the pores but not other electrolyte components. Figure 1.3: Depiction of SEI formation on electrode surface a) Initial formation of SEI b) Incorporation of organic components into SEI, and c) Incorporation of inorganic components into SEI13 The formation of SEI plays an important role in the performance of the battery. The large number of lithium ions that are incorporated into the SEI are considered to be the main contributors to irreversible capacity loss in batteries, particularly during early cycles.7,19,20 Additionally once the SEI layer has grown, the SEI provides a more uniform current distribution across the electrode by reducing concentration polarization and overvoltage, which facilitate Li+ transport on the electrode.7 6 Although the SEI primarily forms during the first cycle, it does continue to evolve in composition and structure with further cycling.21,22 After the first cycle though, this evolution slows because the SEI itself is electrochemically stable.14,15 1.3 High-Capacity Alloying Anode: Silicon Another important metric to consider when assessing batteries is specific capacity, which is the amount of charge per gram of material. Similar to specific energy, batteries with high specific capacities are desirable. LIBs with a graphite anode have a theoretical specific capacity of 372 mAh/g; modern LIBs with graphite anodes have specific capacities of about 360 mAh/g.7 Subsequently, the specific capacity of LIBs with graphite anodes has been maximized. In an effort to increase the specific capacity of LIBs, other electrode materials are being investigated, specifically alloying electrodes in place of intercalating electrodes such as graphite. As early as 1981, before the first commercial LIB was produced, silicon was being investigated as a possible alloying anode material in LIBs.23,24 Alloying electrodes composed of various group IV elements have been explored due to their high capacity Li-rich binary alloys.25,26 Si has great potential as an alloying electrode material because it has a theoretical specific capacity of 4200 mAh/g, the largest specific capacity of any group IV element (Figure 1.4).26 Notably the specific capacity of Si is over an order of magnitude larger than that of graphite. This difference in theoretical specific capacities is due to the larger stoichiometric capacity of Si with regards to lithium ion uptake. Also, Si has an advantage of being an abundant and economical element with low toxicity.26 7 Figure 1.4: Gravimetric and volumetric capacities of different group IV elements 26 Crystalline or amorphous Si can be used as an active material.27,28 If the Si is fully lithiated during its first lithiation, then after the first delithiation, regardless of whether its pristine phase was crystalline or amorphous, the Si will become amorphous.27 It will then remain in the amorphous phase for all of its subsequent delithiated states. The lithiating reaction that Si undergoes during the discharging process is Si + xLi+ + xe- " LixSi The delithiating reaction that Si undergoes during the charging process is LixSi " xLi+ + xe- + Si As seen in Figure 1.5, lithiated silicon can form phases with a stoichiometric ratio of up to Li22Si5. However, in LIBs the highest capacity phase commonly reported is the crystalline phase of Li15Si4.28,29,30,31 This phase forms when the cell potential is below 50 mV vs. Li/Li+.26 8 Figure 1.5: a) Plot of specific capacity of Si as a function of the stoichiometric ratio of Li in lithiated Si b) Common lithiated Si phases and the stoichiometric ratio of Li to which they correspond The large stoichiometric uptake of Li+ by Si results in a large volume increase of the Si.32 It has been shown that the corresponding volume increase of up to 300% is more effectively managed with Si nanoparticles (NPs), since larger particles are more prone to cracking and pulverization during electrochemical cycling (Figure 1.6).1,33,34,35 Even when using Si NPs, repeated volume changes during cycling can crack the NPs and cause capacity loss after a few cycles.34,36 Also, an additional contributor to the loss of 9 capacity of a battery is the debonding of the Si from the current collector, which also can result from the large volume changes that the Si undergoes.1,37 Debonding of the Si means that the Si loses its electrical connection and is subsequently unable to lithiate or delithiate further, freezing its state of charge and reducing its capacity. Overall the rapid capacity loss that is characteristic of Si anodes has impeded the commercialization of this type of Li-ion battery. Figure 1.6: Schematic of the a) Pulverization of Si nanoparticles as a result of cycling b) Delamination of Si nanoparticles from electrode surface as a result of cycling 1 Just as in LIBs with graphite anodes, SEI also forms in LIBs with Si anodes. The composition and possibly structure of the SEI will be somewhat different from that found on the surface of a graphite anode. However, like the SEI on graphite anodes, the SEI on Si anodes begins to form during the first lithiation, and the composition of the electrolyte is dependent on the cell potential.38,22 Even though, the SEI formed during the first lithiation plays an important role in how the cell will function, the SEI does continue to evolve with further cycling. In Si anodes this evolution is in part due to the 10 volume changes that the Si undergoes. During the contraction of the Si during delithiation, the SEI can fracture and lose contact with the Si (Figure 1.7).29 Similar to the delamination of the Si from the current collector, on a smaller scale the loss of contact between the SEI and Si causes electrical isolation of the Si and increases the electrical resistance within the battery.29 This loss of contact leaves a new surface of the Si exposed to the electrolyte which then causes more SEI to form during delithiation.29 This process leads to the depletion of the electrolyte in the battery, and if this occurs with enough frequency, the battery can short circuit due to a lack of electrolyte and thermal runaway can occur. Figure 1.7: Diagram showing the formation of SEI during lithiation and then the deformation of SEI upon delithiation29 1.4 Problems with Current Techniques to Study Lithium-Ion Batteries The study of LIBs has been challenging because they are a closed system that is often encased in a rather impenetrable material such as stainless steel. As a result, the techniques used to study LIBs can often be categorized as requiring post-mortem analysis or the use of modified cells. Post-mortem techniques by definition require the 11 disassembly of the cells. The act of disassembling a cell can crack or delaminate the active material, which leads to incorrect characterizations of the state of the electrode post-cycling. The rinsing of the electrode to remove excess electrolyte from the surface of the electrode often erodes the SEI and can lead to incorrect thickness and compositional analyses.18,15 The exposure of the electrode to air once the cell is disassembled can also lead to changes in chemical composition.15 Additionally once the cell is disassembled and post-mortem analysis has been performed, the practicality of re-assembling the cell and performing additional cycling of the cell is limited. Subsequently post-mortem techniques are an impractical way to characterize electrode changes at multiple points throughout cycling. Modifications to cells that allow for operando or in-situ characterization techniques to be utilized affect the orientation of components within the cell. Subsequently, the pressure that the components experience is altered when compared to an unmodified cell. There is some evidence that these modifications do in fact affect the behavior of the cell.39,40 However, in general the effect of these modifications are not well understood, which is inherently problematic. In summary, these obstacles to studying LIBs lead to conflicting observations and demonstrate the need for in-situ or better yet operando techniques to study unmodified LIBs. The operando and in-situ techniques that are currently being used to study LIBs lack the ability to track the uniformity of the changes that the electrode undergoes during cycling. For instance, X-ray diffraction (XRD), X-ray absorption spectroscopy (XAS), and X-ray photoelectron spectroscopy (XPS) require scanning across a sample and 12 therefore have limited time resolution with regards to providing information about the spatial uniformity of a process.41,33,42,30,43,35,44 Additionally, the high intensity X-rays used in these types of measurements can damage the batteries and affect the cycling behavior of the batteries.45,46 Nanoscale measurements such as Atomic force microscopy (AFM) and transmission electron microscopy (TEM) provide detailed local information; however, they either require scanning across the sample or assuming that one small area of an electrode is representative of the entire electrode.40,41 These techniques lead to an incomplete picture of the changes that LIBs undergo during cycling. The motivation for the work presented in this thesis was to develop a technique to study LIBs that would work on unmodified cells, would be nondestructive, and would provide information about the homogeneity of the processes that occur during cycling. The technique that met these requirements and was subsequently developed for use on LIBs is Spatial Frequency Heterodyne X-ray Imaging. 1.5 Spatial Frequency Heterodyne X-ray Imaging Spatial Frequency Heterodyne X-ray Imaging (SFHXI) uses the small angle scattering of X-rays to produce images of objects.47,48,49,50 For this technique a grid, made from uniformly spaced stainless steel wires is placed in the viewing field between the X-ray source and the detector. An exposure of just the grid is taken. A crisp projection of the grid lines is visible on the detector. Then the object of interest and the grid are placed in series, and an exposure of the grid and the object is taken (Figure 1.8). The presence of the object in the viewing field scatters the X-rays, which blurs the 13 previously sharp projection of the grid wires on the detector. Fourier analysis methods are used to extract a measure of the wire projection blurring and consequently the radiation scattered by the cell (Figure 1.8). Figure 1.8: Diagram of SFHXI technique In the first step of the image processing, a Fourier transform of the acquired image generates a spatial frequency spectrum of the grid and the object in reciprocal space. The grid produces a two-dimensional lattice of peaks at its spatial frequency harmonics. The spatial frequencies of the grid and the object are heterodyned in real space, which results in a convolution of their frequencies in reciprocal space. As a result of this convolution, the spatial frequency of the cell is replicated about each grid harmonic peak. Next, the vicinity of each harmonic is inverse Fourier transformed separately. The central harmonic, (0,0), corresponds to spatial frequencies near a zero scatter angle, and consequently, at this harmonic the scattered radiation is not 14 distinguishable from transmitted radiation. Therefore, the inverse transform of the (0,0) harmonic produces a traditional X-ray absorption image. X-ray scatter information can be obtained by comparing the signals of the central (0,0) harmonic to that of higher harmonics. Specifically, an image, called a scatter image, can be produced where the intensity at each point is proportional to the integral of X-ray scatter intensity over a range of small angles at that location. This image is achieved by calculating the inverse Fourier transforms of the (0,1) or the (1,0) harmonic and normalizing it by the (0,0) image. We have previously found the amount of scattered radiation to be correlated to the size of NPs as well as mesoscopic changes that the system undergoes.51,52,53 Because the anodes are made of Si nanoparticles, the sensitivity of SFHXI to different sized nanoparticles was verified. For 25 nm, 35 nm, 60 nm, and 70 nm crystalline spherical nanoparticles distinct scatter intensities were detected which confirms that this technique would detect the size changes that the nanoparticles would undergo during the lithiation and delithiation processes. SFHXI is an X-ray imaging technique with a large field of view that can image an entire battery during an exposure and that measures the integral of the X-rays scattered at small angles, simultaneously at millions of image points. A synchrotron is not needed to perform SFHXI. This scattering intensity is sensitive to nanoscale and mesoscale changes in structure.51,52,53 The hard X-rays used here can easily penetrate the stainless steel cell casing but are low in intensity so as to not damage the cell. As a result, acquired operando images show changes in NP structure across the entire anode. 15 1.6 Aims of Thesis The fundamental question that initiated the research presented in this thesis was how does the X-ray scatter intensity measured by SFHXI correspond to the material and chemical changes that occur in unmodified Li-ion cells during cycling? As this research was exploratory in nature, the aims of this research became more nuanced as a better understanding of the experimental system developed. Presented below is a brief summary of the topics addressed in this thesis. In Chapter 2 the correlation between scatter intensity and the capacity of cells is explored by investigating the scatter behavior of cells with different capacities. Low capacity cells are identified by their SFHXI scatter intensity during the first lithiation. Additionally, the inhomogeneities of the lithiation and delithiation processes are quantified. High and low capacity regions on individual electrodes are observed and statistically analyzed. A model of the scatter intensity changes that would result from the lithiation of the nanoparticles was developed and used to assist in the interpretation of the experimentally measured scatter behavior. Underperforming electrode regions were found to result from the detachment of Si nanoparticles from the electrode surface. These observations offer new insight into why the actual capacity of Si composite electrodes is lower than their theoretical capacity. Chapter 3 investigates the sensitivity of SFHXI scatter intensity to SEI formation. Scatter intensity is shown via experiment and modeling to be a measure of SEI thickness, with thicker SEI layers resulting in a larger increase in scatter intensity during the first lithiation than thinner layers. Also, SEI formation was found to occur rather 16 homogeneously across the electrode whereas the lithiation of Si was more spatially inhomogeneous than the SEI formation. Thus even though one of the functions of the SEI to provide a more uniform current density on the electrode surface, a homogeneous SEI layer is not sufficient to guarantee the homogeneous lithiation of the active material. Appendix A catalogues various image correction techniques that were tested in order to reduce the fixed pattern noise in the collected transmission images. The flatfield correction that had traditionally been used did not fully correct the fixed pattern noise. This meant that the fixed pattern noise was visible in the scatter images and the absorption images that SFHXI produces. Subsequently, this appendix explains in detail the various image corrections that were investigated in order to reduce the fixed pattern noise. Appendix B describes the work done to understand the noise (other than fixed pattern noise) that is visible in generated scatter images. This work looks at how the introduction of artificial noise into transmission images as well as how various SFHX image processing parameters affect the noise in scatter images. Although this work demonstrates that noise cannot be entirely removed from scatter images, it does present optimal ways to reduce noise such as selecting the appropriate region of interest size. Appendix C presents the MATLAB scripts that were written in order to perform the work included in this thesis. This appendix does not include the code itself, but 17 rather a brief explanation of what each of the scripts does as well as important inputs and parameters that are needed to run the scripts. References (1) Sun, Y.; Liu, N.; Cui, Y. 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Spatial Frequency Heterodyne Imaging of Aqueous Phase Transitions inside Multi-Walled Carbon Nanotubes. Phys. Chem. Chem. Phys. 2015, 17 (46), 31237–31246. https://doi.org/10.1039/C5CP04508H. 26 CHAPTER 2 OPERANDO IMAGING OF THE EVOLUTION OF ELECTRODE INHOMOGENEITIES IN UNMODIFIED LITHIUM ION CELLS 2.1 Introduction Silicon has a theoretical specific capacity an order of magnitude larger than that of the traditional graphite anode.1,2 This is due to the large storage capacity of lithium in silicon. It has been shown that the corresponding volume increase of 300% is more effectively managed with Si nanoparticles (NPs), since larger particles are more prone to cracking and pulverization during electrochemical cycling.3,4,5 The anode undergoes the most significant changes during the first lithiation of the battery. This is when the solid electrolyte interphase (SEI) passivation layer forms and the particles transform from crystalline Si to a LixSiy phase.6,7 During the first delithiation the Si NPs do not return to a crystalline Si phase but rather to an amorphous Si phase and remain amorphous during subsequent cycles.8 During cycling, volume changes can crack the NPs and cause capacity loss after a few cycles.4,9 The rapid capacity loss has impeded the application of this type of Li-ion battery. Here we demonstrate for the first-time operando Spatial Frequency Heterodyne X-ray Imaging (SFHXI) of unmodified Li-ion cells with Si NP anodes. The scattering 27 intensity measured by SFHXI is sensitive to nanoscale and mesoscale changes in structure.17,18,19 The hard X-rays used here can easily penetrate the stainless steel cell casing but are low in intensity so as to not damage the cell. As a result, acquired operando images show changes in NP structure across the entire anode. SFHXI is sensitive to electron density changes of the NPs, i.e. the lithiation of Si, to nanoscale size changes, i.e. the swelling of NPs during lithiation, and to electron density changes of the medium surrounding the NP, i.e. delamination of the NPs from carbon black. With this technique by using cells with different capacities, it is shown that high capacity and low capacity cells can be distinguished early during the first lithiation step, based solely on their scatter behavior. Inhomogeneous lithiation of Si NPs across the electrodes was characterized and provides insight into capacity disparities across individual electrodes. In the low capacity cells and low capacity regions on individual electrodes, debonding of the Si NPs from the carbon black was detected and confirmed by modeling. 2.2 Experimental Methods 2.2.1 SFHXI Experimental Setup The cell was placed 1.2 m below a 12 bit remote RadEye 200 CMOS detector in a vertical imaging arrangement with a 1.6 m source to detector distance shown in Figure 2.1. The X-ray source was a True Focus X-ray tube, model TFX-3110EW with a Tungsten anode and a 10 µm focus size. The tube operated at 80 kV and 0.2 mA. The absorption grid was a two dimensional 150 line per inch stainless steel wire mesh with 0.0026 inch gauge. The grid was placed directly below the cell, between the source and the cell. X- 28 ray images were taken using a 20 s exposure time. For a single image to be produced, a set of 10 images was taken and then averaged together in order to improve the signal- to-noise ratio. A set of ten images was taken every ten minutes throughout the cycling process. The error bars corresponding to a standard deviation for the scatter plots shown in the work are smaller than the width of the line. In the first step of the image processing, a Fourier transform of the acquired image generates a spatial frequency spectrum of the grid and the object in reciprocal space. The grid produces a two-dimensional lattice of peaks at its spatial frequency harmonics. The spatial frequencies of the grid and the object are heterodyned in real space, which results in a convolution of their frequencies in reciprocal space. As a result of this convolution, the spatial frequency of the cell is replicated about each grid harmonic peak. Next, the vicinity of each harmonic is inverse Fourier transformed separately. The central harmonic, (0,0), corresponds to spatial frequencies near a zero scatter angle, and consequently, at this harmonic the scattered radiation is not distinguishable from transmitted radiation. Therefore, the inverse transform of the (0,0) harmonic produces a traditional X-ray absorption image. X-ray scatter information can be obtained by comparing the signals of the central (0,0) harmonic to that of higher harmonics. Specifically, an image, called a scatter image, can be produced where the intensity at each point is proportional to the integral of X-ray scatter intensity over a range of small angles at that location. This image is achieved by calculating the inverse Fourier transforms of the (0,1) or the (1,0) harmonic and normalizing it by the (0,0) image. 29 Figure 2.1: a) Spatial Frequency Heterodyne X-ray Imaging setup b) Fourier analysis methods 2.2.2 Cell Assembly Composite electrodes were prepared with Si NPs (average particle size 100 nm, 99% purity, specific surface area > 80 m2/g, from MTI Corp.) as the active material, Super P carbon black (Timcal) as the conductive additive, and Sodium Alginate (Sigma- Aldrich) as the binder. The slurry was made with 300 mg of Si/carbon black/Na-Alginate in a mass ratio of 60/20/20. The Na-Alginate binder solution was made with 2% by mass of Na-Alginate powder with DI water as the solvent. Additional DI water was added to optimize the viscosity of the slurry. The Si NPs and carbon black were dry mixed using a mortar and pestle for 30 minutes. Then, the binder solution was added and mixed for an additional 15 minutes to create a slurry. The slurry was then transferred to a small beaker and was homogenized using a high-speed stirrer at ~500 rpm for 30 minutes. The 30 slurry was type cast using a doctor blade set to ~200 µm thickness onto a ~25 µm thick copper foil and dried in air at room temperature for 2 hours and then under vacuum at 100ºC for 5 hours. Similar methods for preparing Si composite electrodes using a slurry can also be found in these publications.20–25 The Si NP composite electrodes were used as the working electrode with 0.45 mm thick pure lithium metal foil from MTI Corp. as the counter electrode in CR-2032 coin cells. The coin cells were assembled using the working electrode, a counter electrode, and a 20 µm thick 2320 Celgard separator with 30-50 µL of electrolyte composed of 1 M LiPF6 in ethylene carbonate (EC) and diethyl carbonate (DEC) (1:1 volume ratio). The thickness of the dried electrode coating was ~50 µm. The Si loadings for Cells 1, 2, and 3 is 4.20 mg/cm2, for Cell 4 is 3.62 mg/cm2, and for Cell 5 is 1.365 mg/cm2. Additional information about cell assembly can be found in a recent publication.20 2.2.3 Cell Cycling Cell cycling was performed using a Princeton Applied Research VersaSTAT3 potentiostat. The cell was cycled at room temperature using chronopotentiometry, meaning that a constant current was applied until the cutoff voltage was reached. For lithiation the cutoff voltage was 10 mV and for delithiation the cutoff voltage was 0.9 V. The current that was applied was calculated using the cell capacity C in mAhg-1Si so that the cell would lithiate to a capacity of 1200 mAhg-1Si at a C/20 rate. The second cycle for Cell 2 was performed at a C/5 rate in order to test rate sensitivity. The lithiation or delithiation was time limited by the number of hours dictated by the C rate. We cycled 31 the cells using these conditions to practically limit the total experiment time. The current densities used for cycling were relatively low (0.156 mA/cm2) which we believe is well below the threshold for concern with regards to changes in surface morphology on the Li counter electrode. Also, due to the few number of cycles performed at this current density the Li counter electrode should not undergo significant morphological changes. The X-ray images were collected as current was being applied, not during voltage holds, thus resulting in an operando measurement. No voltage holds were used in the experiments reported in this work. 2.2.4 Modeling of Si Nanoparticle Scatter Intensity The scatter amplitude as a function of angle for a spherical particle can be modeled by Equation 2.1 where q=(4πSin(θ))/λ, d is the diameter of the spherical particle in meters, λ is the average wavelength of the X-rays emitted in meters, ρe is the electron density of the particle in electrons/m3, and θ is the scattering angle in radians.26 If the particles are not in vacuum, ρe becomes ρparticle- ρmed where ρparticle is the electron density of the particle and ρmed is the electron density of the surrounding medium. Equation 2.1: Scatter amplitude as a function of angle and diameter for a spherical particle 32 Our experimental setup measures the integral under the curve, A2, over the angular range from θmin to θmax for which our experimental setup is sensitive, Equation 2.2. A traditional Small Angle X-ray Scattering (SAXS) measurement performed using a monochromatic collimated beam would measure A2. The X-ray scatter intensity, I, increases with the square of the NPs’ electron density and with approximately the square of the NPs’ diameter. Equation 2.2: Scatter intensity as a function of angle and diameter for a spherical particle Using SpekCalc27,28,29 I calculated the X-ray emission spectrum of our X-ray tube and estimated an average X-ray wavelength of 42.7 pm for our system. The minimum scattering angle was calculated by first determining the number of pixels that a gridline spans. For the work done in this chapter, a gridline spanned three pixels, which corresponds to a distance of 288 µm. Then given this distance as well as the distance between the grid and the detector, the minimum scattering angle that the imaging setup can detect was determined to be 7.5x10-4 radians. The maximum detection angle for our system was calculated in a similar way to the minimum scatter angle. The maximum scatter angle corresponds to half the number of pixels on the detector and was found to be 0.042 radians. In order to model the core-shell nature of lithiated Si NPs I used Equation 2.3 where d is the diameter of the inner core of the particle in meters, D is the diameter of 33 the entire particle in meters, ρLi13Si4 is the electron density of the Li13Si4 phase in electrons/m3, ρSi is the electron density of crystalline Si in electrons/m3, and ρmed is the electron density of the medium in electrons/m3. With this equation, I calculate the scatter intensity for an entire NP composed of Li13Si4, subtract out the scattering intensity of the Li13Si4 core, and then add the scattering intensity of the Si core. with Equation 2.3: Scatter intensity as a function of angle and diameter for spherical core shell particles To perform this calculation, I first determined the size of lithiated core-shell NPs at different capacities. To do this, capacities starting at 0 mAh to 6.25 mAh in increments of 0.25 mAh were converted to the number of electrons that flowed in the cell, assuming that each electron produced one Li+. The amount of the Li+ that lithiates the Si NPs was used to find the number of lithiated Si atoms by assuming that the lithiated phase formed is Li13Si4. Given that the average NP size is 100 nm, each Si NP was assumed to have a diameter of 100 nm. The number of Si atoms in a 100 nm NP was calculated using the density of crystalline Si, 6.86 x 1023 electrons/cm3.30 The total number of silicon atoms 34 on an electrode was determined using the mass of Si on the electrode. By dividing the total number of Si atoms on the electrode by the number of Si atoms per NP, the number of Si NPs on the electrode was found. Then by dividing the number of lithiated Si atoms on the electrode by the number of Si NPs, the number of lithiated Si atoms per NP was determined. This value was subtracted from the number of Si atoms per NP to find the number of unlithiated Si atoms per NP. The unlithiated Si atoms comprise the core of the core-shell particle, and the lithiated Si atoms make up the shell of the core- shell particle. Using the density of Li13Si4 and the number of lithiated Si atoms per NP, the volume of the Li13Si4 shell was found. The volume, and subsequently the diameter d, of the unlithiated Si core were calculated using the number of unlithiated Si atoms per NP and the density of crystalline Si. By finding the total volume of the core-shell particle, the unlithiated Si core plus the Li13Si4 shell, the diameter, D, of the entire particle can be determined. 2.3 Results 2.3.1 High and Low Capacity Cells Scatter intensity during the first lithiation was plotted as a function of capacity per gram of Si NPs shown in Figure 2.2a. Cells with a range of different capacities were intentionally used in order to determine the relationship between scatter intensity and capacity. The capacities reported in this section and the subsequent section are the total capacity of the cell divided by the mass of Si and includes the irreversible capacity lost to 35 SEI formation and a small amount of capacity associated with the lithiation of carbon black (CB) (~400 mAh/gCB). The maximum allowed total capacity of the cells investigated was 1366 mAh/gSi. Capacity values similar to 1366 mAh/gSi were observed in three “high capacity” cells, Cells 2, 3, and 5, while much smaller values were observed in two “low capacity” cells, Cells 1 and 4. The terms high and low capacity are used with respect to the cells studied in this paper and are qualitative rather than quantitative. The results and analysis in this paper focus on the first lithiation because SFHXI provides valuable predictive information about the cell by solely looking at the first lithiation. Electrochemical data and scattering data from subsequent cycles are shown in Figure 2.3, and additional electrochemical data can be found in another recent publication.20 The results in Figure 2.2a show that there are three distinct scatter behaviors for the cells; the scatter either increases, is constant, or decreases during lithiation. As lithiation proceeds, increasing scatter intensity is only observed for high capacity cells, Cells 2, 3, and 5. Constant or decreasing scatter intensity is seen for low capacity cells, Cells 1 and 4. Consequently, it can be determined from just the scatter curve shapes that Cells 2, 3, and 5 are high capacity cells and Cells 1 and 4 are low capacity cells. There is an early divergence between the scatter for the high capacity cells and the low capacity cells shown in Figure 2.2a. Specifically this divergence occurs within the first hour or two of C/20 cycling. This indicates that SFHXI can be used to predict the difference between the high or low capacity cells at a point where lithiation of Si is just beginning, well before the cell reaches the cutoff voltage during the first cycle. Because these cells are cycled galvanostatically, the point at which the cutoff voltage is reached 36 reflects both the state of charge of the electrode and the overpotential. In general, the lower capacity cells have a higher overpotential (Figure 2.2b), which reflects higher impedance somewhere in the cell. Figure 2.2: Lithiation, scatter behavior, and inhomogeneities of high and low capacity cells a) Plot of normalized scatter intensity versus total capacity per gram of Si for the first lithiation of Cells 1-5 b) Plot of voltage as a function of total capacity per gram of Si for Cells 1-5 c) Histograms of scatter intensity distribution for Cells 1-5 at the end of the first lithiation 37 Figure 2.3: 4 cycles of lithiation and delithiation voltage and scatter intensity plotted as a function of elapsed time in thousands seconds for Cell 2 2.3.2 Inhomogeneity of the Lithiation Process Because SFHXI captures an image of the entire electrode in a single exposure, the homogeneity of the lithiation and delithiation processes can be assessed. For Cell 2, 1.26 mm by 1.26 mm regions of the electrode were selected, and the scatter signals of these regions were plotted as a function of voltage (regions a-e) in order to quantify the homogeneity of the lithiation process (Figure 2.4). Different regions of the electrode have markedly diverse X-ray scatter intensity curves (Figure 2.4b). The scatter curves diverge almost immediately, which indicate that Si is not lithiating uniformly across the electrode (Figure 2.4b). At optimal lithiation voltages for Si (voltages less than 0.1 V), the scatter curve shapes are markedly different. In this voltage range some of the scatter curves decrease, some increase, and some appear to have a peak at 0.05 V. Region a, whose scatter intensity decreases, is likely a low capacity region because its scatter curve shape is similar to that of low capacity Cell 4. The average final scatter intensity value for the normalized curves is 1.05 with a standard deviation of 0.08. This is 38 remarkable given that 1.06 is the maximum value of the average scatter intensity curve. Consequently, the standard deviation in the final lithiation values for different electrode regions is greater than the change in scatter intensity that the average scatter curve undergoes. Thus, it can be firmly concluded that the different regions of the electrode do not lithiate uniformly. Figure 2.4: Inhomogeneity of lithiation process a) X-ray scatter image showing regions a, b, c, d, e, and the average over the electrode surface for Cell 2. The average is Figure 2.5: Snapshots from the scatter intensity the same area shown in the movie. b) Plot of movie during the first lithiation of Cell 2 and the normalized scatter intensity versus voltage corresponding voltages. Each pixel is normalized (V) for the first lithiation of six regions of Cell by its initial scatter intensity. Area shown in 2. Each scatter intensity curve is normalized snapshots corresponds to the average area in by the initial scatter value for that curves Figure 2.4 39 Further evidence of inhomogeneities is seen in a movie of the electrode’s changes in scatter for Cell 2 Figure 2.5. The region shown is the largest rectangular region of the electrode that could be selected, with dimensions 10.1 mm by 10.1 mm. In each frame, each pixel is normalized by the initial scatter intensity of the pixel. Normalization allows for easy observation of how each pixel changes relative to itself. The snapshots in Figure 2.5 clearly show the inhomogeneities that evolve during cycling. Regions that develop a high scatter intensity at relatively high voltages, such as 0.06 V, maintain an increased scatter intensity throughout the entire first lithiation. Also, regions that decrease in scatter intensity early on in the first lithiation, remain low in scatter intensity throughout the entire lithiation. This indicates that early lithiation behavior determines how the entire lithiation process proceeds. The intensity and location of the inhomogeneities vary from cell to cell; however, scattering inhomogeneities are universal in all cells studied. For the 5 cells, the inhomogeneity of the scatter intensity at the end of the first lithiation was assessed by calculating the percentage of electrode area that decreased in scatter intensity (Table 2.1). Specifically, the lowest capacity cell, Cell 4, has the largest percentage of low scatter intensity regions, 61.8%, where high-capacity cells, Cells 2,3,5, have the smallest percentage of low-scatter regions, on average 30.8%. The other low capacity cell, Cell 1, has a medium percentage of low scattering regions, 53.3%. It is apparent that for high capacity and low capacity cells that there is a wide distribution of scatter intensities across the electrode indicating the widespread inhomogeneities that are present on all electrodes during and after the first lithiation. The evolution of the scattering intensity 40 histrogram for Cell 2 during 4 lithiation and delithiation cycles is shown in Figure 2.6. The distribution quickly broadens early during the first lithiation. Subsequent cycles further broaden this distribution, indicating a diversification of the extent of lithiation and delithiation of individual electrode regions rather than the expected homoginization of the extent of lithiation and delithiation. Cell % of Electrode Area with Normalized Scatter Intensity <1 Cell 1 53.3% Cell 2 28.7% Cell 3 31.4% Cell 4 61.8% Cell 5 32.1% Table 2.1: Percent of electrode area with normalized scatter intensity less than 1 for Cells 1-5 at the end of the first lithiation Further SFHX image analysis of the first four lithiations and delithiations, reveals that the locations of inhomogeneous scatter features do not change after the first lithiation. For example, regions on the electrode that become highly scattering early during the first lithiation remain highly scattering throughout all four cycles (Figure 2.5, Figure 2.7, and Figure 2.8). This illustrates the impact of early lithiation behavior on the later performance of the electrode. Figure 2. 7 shows how the size and locations of inhomogeneities change for three horizontal strips across the electrode as a function of time. The width of the inhomogeneities is on the millimeter scale. Although the widths of the inhomogeneities vary some during the four cycles, the locations of the inhomogeneities are fixed. Also, the scatter intensity changes very little after the first lithiation. Particularly noteworthy is that after the first delithiation the scatter intensity 41 does not return to a value similar to that of the cell prior to lithiation. Further evidence of the stationary position and rather constant intensity of the inhomogeneities is shown in Figure 2.8. Due to the complexity of the changes that the electrode undergoes during cycling, the interpretation of the scatter intensity is still evolving with respect to the behavior in cycles subsequent to the first lithiation. Consequently I have focused on the correlation between the scatter intensity and the state of lithiation for the first cycle. Figure 2.6: Histograms of the normalized scatter intensity of Cell 2 during four cycles 42 Figure 2.7: a) Scatter image of Cell 2 showing strips 1, 2, and 3 shown as surfaces. b) Evolution of size and intensity of inhomogeneities in Strip 1 region over 4 cycles. c) Evolution of size and intensity of inhomogeneities in Strip 2 region over 4 cycles. d) Evolution of size and intensity of inhomogeneities in Strip 3 region over 4 cycles. Figure 2.8: Snapshots from the scatter intensity movie during four cycles of Cell 2 and the corresponding voltages. Each pixel is normalized by its initial scatter intensity. Area shown in snapshots corresponds to the average area in Figure 2.3. 43 2.3.3 Modeling Nanoparticle Scatter Intensity In order to better interpret the X-ray scatter data obtained through SFHXI, I modeled the scatter intensity of Si NPs using the NP diameters, their radial electron densities, and the wavelengths of the detected X-rays (see Equation 2.1). The sensitivity of our imaging setup as a function of scattering angles is incorporated into the model. Figure 2.9: a) Plot of calculated scatter intensity as a function of NP diameter for crystalline Si NPs in a vacuum b) Plot of calculated normalized scatter intensity as a function of NP diameter for crystalline Si NPs in a vacuum with diameters in range relevant to those found in the cells The calculated normalized scatter intensity for different sized crystalline Si NPs is shown in Figure 2.9 (the experimental scatter intensity of Si NPs is shown in Figure 2.10). The general trend is that the scatter intensity increases with increasing NP diameter. There are some small dips in the scatter intensity due to the sinusoidal form of Equation 2.1 and the limited angular range of integration as expressed in Equation 2.2. These calculations assume that the particles are spherical, which is a reasonable approximation, based on SEM images taken of the electrode after 13 cycles, which show that the particles remain roughly spherical (Figure 2.11). 44 Figure 2.10: Experimental X-ray scatter intensity of crystalline Si NP particle powders with various diameters Figure 2.11: a) and b) Post mortem SEM of Si composite electrode taken after 13 cycles As demonstrated in Figure 2.9 and 2.10, the Si NPs’ swelling during the first cycle lithiation should increase the scatter intensity. During cycling, the scatter from the other components of the cell should remain constant because the anode is the only component undergoing changes to which X-ray scattering is sensitive under the experimental conditions. However, during the first lithiation, the NPs not only grow in size but also change in composition. Crystalline Si NPs transform into lithiated Si NPs. The electron density of lithiated Si phases will be lower than that of pure Si because Li has a lower electron density than crystalline or amorphous Si (Figure 2.12). Li13Si4 was considered as a possible lithiated phase reached by our cell because it was observed by Bordes et al. 45 during the first lithiation under similar cycling conditions (current applied, time lithiated, etc.).31,20 For Li13Si4 the electron density was calculated using the density of Li13Si4 given by Braga et al. and was found to be 3.61 x 1023 electrons/cm3.32 This value is 53% of the electron density of crystalline Si, 6.86 x 1023 electrons/cm3.30 Although it is unlikely, if Li15Si4 were formed as suggested by Obrovac et al., the electron density of the Li15Si4 phase is 49% that of the electron density of crystalline Si.11,32 The 4% difference between the electron density ratios of Li15Si4 and Li13Si4 leads to small differences in overall scatter intensity and does not affect the trends reported in this paper. Because the scattering intensity is dependent on the square of the electron density, Equation 2.2, a decrease in electron density would lead to a decrease in scatter intensity. During the first lithiation the NPs are not only decreasing in electron density but are also increasing in diameter, which are competing effects with regards to how the scatter intensity will change. Figure 2.12: Electron density in units of electrons/cm3 for possible lithiated silicon phases formed during lithiation. Also shown are the electron densities for crystalline Si (cr-Si) and for amorphous Si (a- Si), both indicated with yellow. The phases most likely to form in a Li-ion cell are shown in black. 46 Modeling the scattering behavior of the first lithiation is not straightforward because the Si NPs form a core-shell structure in which the outer shell is a lithiated silicon phase and the inner core remains crystalline Si (Figure 2.13a). 5,31 This model is confirmed by our X-ray diffraction results which still show crystalline Si peaks after the first lithiation (Figure 2.14). Subsequently, I developed Equation 2.3 to model the scatter from a core-shell particle. Figure 2.13a diagrams the formation of the core-shell particles during the first lithiation in which the yellow Si shrinks in size as some of the silicon is consumed to form the purple lithiated silicon shell. During this process, the overall size of the core-shell NP increases. When calculating the scatter intensity of the NP, the scattering of the media surrounding the NP must be accounted for; in this case, the carbon black (CB) and the electrolyte are the media surrounding the NPs in the cell (Figure 2.13b). Using Equation 2.3, the scatter intensity of the core-shell NPs at various states of lithiation were calculated with a medium of purely CB, purely electrolyte, and a 50:50 mixture of CB and electrolyte by electron density. The NPs surrounded entirely by CB or electrolyte are the limiting cases and likely do not exist within the cell. The uptake of Li by the CB is small compared to Si and thus was not included in the model. The formation of the SEI was not directly included in the model because it is difficult to accurately model the SEI due to the complex composition of the SEI and because the electrolyte medium factor should in part account for the SEI. 47 Figure 2.13: Core-Shell Particle Diagram and Modeled Scatter Intensity of Core-Shell NPs in Different Media a) Diagram of Core-Shell NP lithiation where the purple shell denotes the LixSiy phase and the yellow core denotes the crystalline Si core where d is the diameter of the core and D is the diameter of the entire particle b) Diagram of Si NP (yellow) and carbon black (grey) on top of Copper electrode (copper) all surrounded by electrolyte (green) c) Plot of calculated scatter intensity as a function of the calculated capacity per gram of Si for core-shell NPs in CB medium, electrolyte medium, and a mixture of CB and electrolyte medium. The CB plot is divided by 5. The calculated increase in NP diameter as a function of calculated capacity is shown on the right axis. 48 Figure 2.14: a) X-ray diffraction spectrum for a pristine electrode and Si and Cu reference spectra b) X- ray diffraction spectrum for an electrode lithiated to 0.05V and Si and Cu reference spectra 49 The results of the modeling are shown in Figure 2.13c. The plotted calculated capacity only includes the Si contribution. The capacities included in the simulation are higher than those achieved experimentally and were included for completeness. Unlithiated crystalline 100 nm Si NPs are the starting point of the simulation corresponding to 0 mAhg-1. Each curve has been normalized to the initial scatter intensity at 0 mAhg-1. Core-shell NPs in a purely CB medium display a large increase in normalized scatter intensity and then a decay in scatter intensity as lithiation progresses. Core-shell NPs in pure electrolyte show a small decrease followed by a small increase in the scatter intensity. In contrast, core shell NPs in a mixture of electrolyte and CB show a slight increase and subsequent decrease in scatter intensity, which is much smaller than that of core-shell particles in a purely CB medium. 2.4 Discussion The experimental scatter curves for high capacity cells follow the same trends as the calculated curve for core-shell NPs in a mixture of electrolyte and CB, while the experimental scatter of low capacity cells corresponds more closely to the modeled scatter curve for core-shell NPs in a purely electrolyte medium (Figure 2.2a). The modeled scatter for NPs in the mixed medium does not exactly match the experimental high capacity cell results, which is not surprising given the complex nature of the actual electrode structures. Possible causes for discrepancies include the estimation of the ratio of CB to electrolyte surrounding an individual NP, the initial particle size distribution of the NPs, and the state of charge variations of NPs across the electrode. 50 Regardless, the similarity in shape of the curves demonstrates that the calculations are a reasonable model of the experimentally observed scatter behavior. By comparing Figure 2.13c to the end point of Figure 2.2a, the predicted experimental capacity of the Si is ~700 mAhg-1. This is reasonable, since there is additional measured capacity in Figure 2.2a due to CB lithiation and SEI formation. Also, the similarity in the low capacity cells’ scatter and the modeled results for particles in pure electrolyte may provide an explanation for the different scatter behavior seen in the low capacity cells. A reasonable hypothesis is that the NPs experience a change in medium and go from being surrounded by a mixture of CB and electrolyte to being surrounded by mostly electrolyte. This change would result from a loss of contact with the CB during lithiation. An extreme loss of contact scenario, such as total delamination, would result in decreasing scatter behavior while a less extreme loss of contact scenario, such as electrolyte or SEI seeping between the CB and the NP, would result in constant scatter behavior. Because the CB serves as a conductive matrix, loss of contact with the carbon black would prevent the full lithiation of the NPs. The scatter of Cells 1 and 4 appear to flatten out which is consistent with NPs that lose contact with CB, since this debonding would cause the NPs to stop lithiating and effectively “freeze” their state of charge (Figure 2.2a). In order to interpret the scatter intensity exhibited by Cells 1 and 4, many different calculations were performed. The only scenario in which the model resulted in a decrease in scatter intensity is via a change in medium from a 50:50 CB and electrolyte medium to a medium with a higher electrolyte ratio. Based on this interpretation, the regions of the cells with constant or decreasing scatter intensities are 51 likely low capacity regions, where the NPs have lost contact with the carbon black (Figure 2.4b). Given this interpretation even high capacity cells have on average 30% of the electrode that is not performing optimally (Table 2.1). These low performing regions of the electrode may explain the frequently observed difference between the theoretical and actual reported capacity of Si composite electrodes.20, 33 Also the debonding of the nanoparticles explains why the electrode becomes more inhomogeneous with further cycling. The debonded nanoparticles are frozen in their state of charge and remain inhomogeneous and as more nanoparticles debond due to volume changes during cycling, the electrode becomes more inhomogeneous. If the delithiation process were completely reversible, the scatter behavior of the batteries should be the scatter curve of the lithiation process but in reverse. However, this is not the observed scatter behavior (Figure 2.3, Figure 2.5, Figure 2.7, and Figure 2.8). The scatter behavior during delithiation cannot be explained by the transformation of lithiated silicon NPs into amorphous Si NPs. The electron density for amorphous Si is 6.85 x 1023 electrons/cm3.30 Therefore, the difference in scatter intensity between crystalline and amorphous Si NPs of the same size is only 0.3%. The density difference alone does not account for the 7% difference in scatter intensity between the crystalline Si NPs prior to first lithiation and the amorphous Si NPs found at the end of the first delithiation. There are two possible hypotheses for the cause of the observed scatter behavior. The first is that the NPs are not significantly decreasing in size as delithiation is occurring. Instead, the NPs would remain swollen and become porous due to the vacancies left behind by Li. The vacancies in the NPs would fill because there is no 52 observed drop in scatter intensity, which would be indicative of a decrease in electron density. In this scenario the SEI would fill the vacancies, maintain the electron density, and result in a slightly increased scatter intensity. This agrees with the delithiation mechanism proposed by Breitung et al.7 The second hypothesis is that the NPs shrink and that the medium surrounding the NP changes. The medium would change so that the electron density increases and the scatter intensity is maintained. The change in medium could be due to the SEI. Our observed scatter behavior during delithiation could be a result from a combination of the two hypotheses. Chen et al. observed Si shrinking some during the first delithiation, but not returning to its original size prior to lithaition.34 Further experiments are needed to determine which hypothesis results in the observed scatter behavior. 2.5 Conclusion SFHXI is a novel, effective, nondestructive operando method to study the electrode materials of Li-ion cells. This technique has the unique ability to take non- destructive images of the entire electrodes simultaneously at hundreds of thousands of points as the cell is cycling without the need to modify the cell. With this capability, it has been conclusively shown that electrodes do not lithiate uniformly, and this inhomogeneity can be quantified. This result indicates that techniques such as AFM, SEM, and SAXS, which only observe a small area of the electrode during a measurement, provide data that are not representative of the entire electrode. Also, variations in scatter intensity in individual regions during the first lithiation can be correlated to the capacity of those regions. This is particularly valuable for investigating the performance 53 of full electrodes because it is not possible to directly and simultaneously measure the capacity of individual regions of an electrode with other non-synchrotron based techniques while the cell is operating. High capacity electrodes have low capacity regions, which contribute to a lower actual capacity than the predicted theoretical capacity. The detailed statistics about the capacity of individual electrode regions provided by SFHXI can be used to create electrodes that are uniformly high performing. Also, we have found that early lithiation behavior of an electrode region impacts the performance of that region during later cycles. If a region is a low performing during the first lithiation, it does not improve in performance with further cycling. Overall electrodes with Si NPs become more inhomogeneous with more cycling rather than more homogeneous with cycling. By modeling the X-ray scattering of the Si NP lithiation, I have shown that the experimental data is consistent with the formation of core-shell NPs during the first lithiation. Based on the modeling results, SFHXI is capable of detecting the debonding of Si NPs from the surrounding matrix in low capacity regions. Because the cell did not have to be disassembled, the delamination of Si NPs has been shown to occur within operating cells and is not a consequence of disassembling a cell for post-mortem analysis. To our knowledge, this is the first time that an operando X-ray scatter imaging technique has been used to capture images of an entire electrode in an unmodified Li- ion battery coin cell using a single exposure. The data provided by SFHXI about Si NP electrodes is comprehensive. The utility of this technique is not limited to just Li-ion 54 cells with Si NP electrodes; SFHXI can be used to evaluate any type of battery with an electrode material that undergoes nanoscale or mesoscale changes during operation. X- ray scatter imaging provides novel insight into the function of batteries, and this insight can ultimately be used to advance the development of high performance and safe batteries. References (1) Zhang, W.-J. A Review of the Electrochemical Performance of Alloy Anodes for Lithium-Ion Batteries. J. Power Sources 2011, 196 (1), 13–24. https://doi.org/10.1016/j.jpowsour.2010.07.020. (2) Key, B.; Bhattacharyya, R.; Morcrette, M.; Seznéc, V.; Tarascon, J.-M.; Grey, C. P. Real-Time NMR Investigations of Structural Changes in Silicon Electrodes for Lithium-Ion Batteries. J. Am. Chem. Soc. 2009, 131 (26), 9239–9249. https://doi.org/10.1021/ja8086278. (3) Iwamura, S.; Nishihara, H.; Kyotani, T. Fast and Reversible Lithium Storage in a Wrinkled Structure Formed from Si Nanoparticles during Lithiation/Delithiation Cycling. J. Power Sources 2013, 222, 400–409. https://doi.org/10.1016/j.jpowsour.2012.09.003. (4) Liu, X. H.; Zhong, L.; Huang, S.; Mao, S. X.; Zhu, T.; Huang, J. Y. Size-Dependent Fracture of Silicon Nanoparticles During Lithiation. ACS Nano 2012, 6 (2), 1522– 1531. https://doi.org/10.1021/nn204476h. (5) Taiwo, O. O.; Paz-García, J. M.; Hall, S. A.; Heenan, T. M. M.; Finegan, D. P.; Mokso, R.; Villanueva-Pérez, P.; Patera, A.; Brett, D. J. L.; Shearing, P. R. Microstructural 55 Degradation of Silicon Electrodes during Lithiation Observed via Operando X-Ray Tomographic Imaging. J. 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Nanosilicon Electrodes for Lithium-Ion Batteries: Interfacial Mechanisms Studied by Hard and Soft X-Ray Photoelectron Spectroscopy. Chem. Mater. 2012, 24 (6), 1107–1115. https://doi.org/10.1021/cm2034195. (14) Lim, C.; Kang, H.; De Andrade, V.; De Carlo, F.; Zhu, L. Hard X-Ray-Induced Damage on Carbon–Binder Matrix for in Situ Synchrotron Transmission X-Ray Microscopy Tomography of Li-Ion Batteries. J. Synchrotron Radiat. 2017, 24 (3), 695–698. https://doi.org/10.1107/S1600577517003046. (15) Taiwo, O. O.; Finegan, D. P.; Paz-Garcia, J. M.; Eastwood, D. S.; Bodey, A. J.; Rau, C.; Hall, S. A.; Brett, D. J. L.; Lee, P. D.; Shearing, P. R. Investigating the Evolving Microstructure of Lithium Metal Electrodes in 3D Using X-Ray Computed Tomography. Phys. Chem. Chem. Phys. 2017, 19 (33), 22111–22120. https://doi.org/10.1039/C7CP02872E. 57 (16) Harris, S. J.; Timmons, A.; Baker, D. R.; Monroe, C. Direct in Situ Measurements of Li Transport in Li-Ion Battery Negative Electrodes. Chem. Phys. 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Systematic Investigation of Binders for Silicon Anodes: Interactions of Binder with Silicon Particles and Electrolytes and Effects of Binders on Solid Electrolyte Interphase Formation. ACS Appl. Mater. Interfaces 2016, 8 (19), 12211–12220. https://doi.org/10.1021/acsami.6b03357. (25) Kovalenko, I.; Zdyrko, B.; Magasinski, A.; Hertzberg, B.; Milicev, Z.; Burtovyy, R.; Luzinov, I.; Yushin, G. A Major Constituent of Brown Algae for Use in High- Capacity Li-Ion Batteries. Science 2011, 334 (6052), 75–79. https://doi.org/10.1126/science.1209150. (26) Guinier, A. X-Ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies; A Series of books in physics; W.H. Freeman: San Francisco, 1963. (27) Poludniowski, G.; Landry, G.; DeBlois, F.; Evans, P. M.; Verhaegen, F. SpekCalc : A Program to Calculate Photon Spectra from Tungsten Anode x-Ray Tubes. Phys. Med. Biol. 2009, 54 (19), N433. https://doi.org/10.1088/0031-9155/54/19/N01. 59 (28) Poludniowski, G. G.; Evans, P. M. Calculation of X-Ray Spectra Emerging from an x- Ray Tube. Part I. Electron Penetration Characteristics in x-Ray Targets. Med. Phys. 2007, 34 (6Part1), 2164–2174. https://doi.org/10.1118/1.2734725. (29) Poludniowski, G. G. Calculation of X-Ray Spectra Emerging from an x-Ray Tube. Part II. X-Ray Production and Filtration in x-Ray Targets. Med. Phys. 2007, 34 (6Part1), 2175–2186. https://doi.org/10.1118/1.2734726. (30) Custer, J. S.; Thompson, M. O.; Jacobson, D. C.; Poate, J. M.; Roorda, S.; Sinke, W. C.; Spaepen, F. Density of Amorphous Si. Appl. Phys. Lett. 1994, 64 (4), 437–439. https://doi.org/10.1063/1.111121. (31) Bordes, A.; De Vito, E.; Haon, C.; Boulineau, A.; Montani, A.; Marcus, P. Multiscale Investigation of Silicon Anode Li Insertion Mechanisms by Time-of-Flight Secondary Ion Mass Spectrometer Imaging Performed on an In Situ Focused Ion Beam Cross Section. Chem. Mater. 2016, 28 (5), 1566–1573. https://doi.org/10.1021/acs.chemmater.6b00155. (32) Braga, M. H.; Dębski, A.; Gąsior, W. Li–Si Phase Diagram: Enthalpy of Mixing, Thermodynamic Stability, and Coherent Assessment. J. Alloys Compd. 2014, 616, 581–593. https://doi.org/10.1016/j.jallcom.2014.06.212. (33) Beattie, S. D.; Larcher, D.; Morcrette, M.; Simon, B.; Tarascon, J.-M. Si Electrodes for Li-Ion Batteries—A New Way to Look at an Old Problem. J. Electrochem. Soc. 2008, 155 (2), A158–A163. https://doi.org/10.1149/1.2817828. (34) Chen, C.-Y.; Sawamura, A.; Tsuda, T.; Uchida, S.; Ishikawa, M.; Kuwabata, S. Visualization of Si Anode Reactions in Coin-Type Cells via Operando Scanning 60 Electron Microscopy https://pubs.acs.org/doi/abs/10.1021/acsami.7b12340 (accessed Jul 25, 2018). https://doi.org/10.1021/acsami.7b12340. 61 CHAPTER 3 OPERANDO TRACKING OF SOLID ELECTROLYTE INTERPHASE FORMATION AND LITHIATION OF SILICON USING X-RAY IMAGING 3.1 Introduction As the need for batteries to power portable consumer electronics and electric vehicles products has grown, electrode materials with high specific capacities have become increasingly important.1,2 Traditionally graphite has been used as the anode material in Li-ion batteries, but silicon has become a promising anode material because its theoretical specific capacity is an order of magnitude larger than that of graphite.3,4,1 The difference in specific capacity of these two anode materials is due to the larger number of lithium ions that Si can store per atom of Si, resulting in lithium rich phases such as Li15Si4.1,2,5 However, the large capacity of Si causes the Si to undergo volume changes of up to 300% during cycling.1,6 These volume changes lead to rapid capacity loss in batteries with Si anodes.7 One of the components of a Li-ion battery that can suffer from these drastic volume changes that Si undergoes during cycling is the solid electrolyte interphase (SEI) layer.8,9 The solid electrolyte interphase passivation layer begins to form during the first cycle of the battery and results in an initial irreversible capacity loss in Li-ion batteries.8,10,11 The SEI results from the decomposition of electrolyte components as an 62 electric potential is applied during cycling and forms in two stages with an organic SEI forming at higher potentials, ~ 1.2 V, and with an inorganic SEI forming at lower potentials, ~ 0.8 V.12,13 Although the formation of SEI initially sounds detrimental to the operation of a Li-ion battery, the SEI actually plays an important role in the overall function of a battery by being a permanent insulating passivation layer that has low electronic conductivity but high ionic conductivity.14 The high ionic conductivity allows Li+ transport while its low electronic conductivity helps prevent further decomposition of the electrolyte by preventing the passage of electrons.13,15 The SEI provides a more uniform current distribution across the electrode by reducing concentration polarization and overpotential.14 This in turn facilitates uniform Li+ transport on the electrode.14 Throughout the initial cycles, the SEI will continue to change in composition as well as structure.16,17 Any damage to the SEI can result in a shortened cycle life for Li-ion batteries. Despite how critical the SEI is for battery performance, information about the SEI is still limited. The SEI layer can be quite thin, with some measurements reporting thicknesses as small as tens of Ångstroms.11,13,17,18 The thinness of the SEI has created problems in accurately probing the SEI thickness, which can be seen in the large range of thickness values that have resulted from the various methods used to measure the thickness.11,13,17,18 Also, the heavily carbon based composition of the SEI provides a barrier to determining the chemical composition of the SEI.10,13,17 If a post-mortem characterization technique is used, the standard protocol of rinsing the electrode to 63 remove excess electrolyte can also rinse away the SEI itself.10 In summary, gaining a comprehensive and accurate picture of the SEI has been challenging. In previous work using Spatial Frequency Heterodyne X-ray Imaging (SFHXI) to study operating Li-ion cells with Si composite electrodes, high and low capacity cells could be determined solely based off their scatter behavior during the first lithiation, electrode inhomogeneities could be quantified, and low capacity cells were found to result from the debonding of Si nanoparticles from the electrode surface. These observations were possible because the scatter intensity measured by SFHXI is sensitive to changes in electron density, size, and, surrounding medium. Due to the many simultaneous electrochemical and material changes that occur during the cycling of a cell, the factors affecting the scatter intensity could not be further disentangled in our previous work. However, it was observed that the scatter intensity would change early during the lithiation at potentials at which the SEI was forming. Subsequently, our hypothesis is that the scatter intensity measured by SFHXI is predominantly tracking SEI formation. The changes in electron density and nanoscale size that SFHXI is sensitive to not only pertains to the lithiation of the Si nanoparticles themselves, but also to the formation of SEI. In this work, we use carefully designed voltage holds during the cycling of cells to restrict the electrochemical processes that are occurring. This allows the formation of SEI to be isolated from the lithiation of the Si nanoparticles. We demonstrate that the relative thickness of the SEI layer can be determined from the change in scatter intensity during the early stages of the first lithiation of the cell. Also, the SEI layer was found to 64 form rather uniformly across the entire electrode. However, once the lithiation of Si begins, the electrode becomes inhomogeneous. Thus homogeneous SEI formation does not result in homogeneous lithiation of the Si composite electrodes. These observations provide additional insight into the formation of the SEI and its role in determining homogeneous lithiation of the active material in a cell. Thus SFHXI provides a nondestructive way to probe the formation of SEI in Lion-cells without the need for modification to cells or post mortem analysis, which can affect the integrity of the SEI. 3.2 Experimental Methods 3.2.1 SFHXI Experimental Setup The cells were placed 1.2 m below a 12 bit remote RadEye 200 CMOS detector in a vertical imaging arrangement with a 1.6 m source to detector distance shown in Figure 3.1. The X-ray source was a True Focus X-ray tube, model TFX-3110EW with a Tungsten anode and a 10 µm focus size. The tube operated at 80 kV and 0.2 mA. The absorption grid was a two dimensional 150 line per inch stainless steel wire mesh with 0.0026 inch gauge. The grid was placed directly below the cell, between the source and the cell. X- ray images were taken using a 20 s exposure time. For a single image to be produced, a set of 10 images was taken and then averaged together in order to improve the signal- to-noise ratio. A set of ten images was taken every two minutes throughout the cycling process. The error bars corresponding to a standard deviation for the scatter plots shown in the work are smaller than the width of the line. 65 In the first step of the image processing, a Fourier transform of the acquired image generates a spatial frequency spectrum of the grid and the object in reciprocal space. The grid produces a two-dimensional lattice of peaks at its spatial frequency harmonics. The spatial frequencies of the grid and the object are heterodyned in real space, which results in a convolution of their frequencies in reciprocal space. As a result of this convolution, the spatial frequency of the cell is replicated about each grid harmonic peak. Next, the vicinity of each harmonic is inverse Fourier transformed separately. The central harmonic, (0,0), corresponds to spatial frequencies near a zero scatter angle, and consequently, at this harmonic the scattered radiation is not distinguishable from transmitted radiation. Therefore, the inverse transform of the (0,0) harmonic produces a traditional X-ray absorption image. X-ray scatter information can be obtained by comparing the signals of the central (0,0) harmonic to that of higher harmonics. Specifically, an image, called a scatter image, can be produced where the intensity at each point is proportional to the integral of X-ray scatter intensity over a range of small angles at that location. This image is achieved by calculating the inverse Fourier transforms of the (0,1) or the (1,0) harmonic and normalizing it by the (0,0) image. 66 Figure 3.1: a) Spatial Frequency Heterodyne X-ray Imaging setup b) Fourier Analysis Methods 3.2.2 Cell Assembly Silicon nanoparticles (average particle size 30nm, 99% purity, US Research Nanomaterials Inc.) were used as the primary active material in the composite electrodes. Super P carbon black (CB, Timcal) was used as either a conductive additive for the silicon composite electrodes or as a low capacity active material for reference. Sodium Alginate (Sigma-Aldrich) was used as a binder. The slurry was prepared to contain silicon nanoparticles (Si-NPs), CB, and the binder in 60, 20, and 20 weight percent, respectively. In addition, another type of slurry was also prepared using just the CB with the binder to look further into the distinctive effects of CB when cycled. 67 Si-NP (D=30nm) Carbon Black Sodium Alginate Slurry Type Ratio (Weight Percent) 1 60 20 20 2 - 80 20 Table 3.1: Table with the 3 different electrode compositions used in this chapter Sodium Alginate was dissolved in distilled (DI) water to obtain the 2 wt% binder solution. Si-NP and CB powders were first dry-mixed using a mortar and a pestle. Then the binder solution was added into the mortar and mixed further. Additional DI water was introduced (~2ml) to optimize the viscosity of the slurry for adequate coating. The slurry was then transferred into a small beaker (10ml) and homogenized further using a homogenizer (Bio-Gen PRO200, PRO Scientific) at 500 rpm for 30 minutes. The slurry was coated onto a 25um thick copper foil using a doctor blade. The blade clearance was set to 180um for thickness adjustment. The coated foil was first air- dried inside the fume hood at room temperature for an hour and dried further in a vacuum oven at 75⁰C overnight to fully evaporate the remaining solvent of the electrode. For coin cell assembly, 12.5mm electrode disks were cut and the loading of the active materials was about 1mg/cm2. Electrode thicknesses of 35–45 μm (including the copper substrate which is 25um) were measured with a digital micrometer (Mitutoyo) before cycling. The composite electrodes described above was placed and assembled in CR- 2032 coin cells inside an argon-filled glove box. The prepared electrodes were used as working electrodes and lithium foil was used as both the counter and reference electrode. Celgard 2320 (PP/PE/PP, 20um) was used as the separator. The electrolyte 68 was a mixed solution of 1M LiPF6 in ethylene carbonate (EC) and diethyl carbonate (DEC) (1:1 volume ratio). 3.2.3 Cell Cycling All cell cycling was performed using a Princeton Applied Research VersaSTAT3 potentiostat at room temperature. Current densities used for cycling were relatively low (0.156 mA/cm2) which we believe is well below the threshold for concern with regards to changes in surface morphology on the Li counter electrode. Also, due to the few number of cycles performed at this current density the Li counter electrode should not undergo significant morphological changes. The cell with the 20 mV hold was cycled at a C/20 rate for each of the lithiations until a voltage of 10 mV hold was reached. Then the cell was held at a voltage of 10 mV for five hours to ensure a deep lithiation of the Si. The delithiations were performed at a C/20 rate until a voltage cutoff of 0.9 V was reached. The CB cell was cycled at a C/20 rate with respect to the CB capacity with voltage cutoffs of 10 mV and 0.9 V for all three cycles. The cell with the 500 mV voltage hold was first lithiated at a C/20 rate until 500 mV was reached. Then the cell was held at 500 mV until the current decayed to -2 µA, which corresponded to a voltage hold for 150000 s. The first lithiation was continued at a C/20 rate until a voltage of 10 mV was reached. The cell was then cycled for the remaining cycles at a C/10 rate and with voltage cutoffs of 0.9 V and 10 mV. For the 400 mV voltage hold cell during the first lithiation, the cell was cycled at a C/20 rate until a voltage of 400 mV was reached. Then a 400 mV voltage hold was 69 applied until the current decayed to -5 µA, which corresponded to a voltage hold for 4600 s. A higher current cutoff was used to have a shorter voltage hold and test the effect of a shorter voltage hold on SEI formation. After the voltage hold the cell was cycled at a C/10 rate until a voltage of 20 mV was reached. Another voltage hold of 20 mV was applied until the current decayed to -5 µA. Then the cell was delithiated at a C/20 rate until the voltage reached 0.9 V. The remaining cycles were then performed at a C/10 rate with voltage cutoffs of 10 mV and 0.9 V. For the cell with the 300 mV voltage hold during the first lithiation, the cell was first cycled at a C/20 rate during the first lithiation until a voltage of 300 mV was reached. Then the cell was held at a voltage of 300 mV until the current decayed to -5 µA, which corresponded to a voltage hold for 8450 s. The cell was then delithiated at a C/20 rate and then lithiated again at a C/20 rate. For the second delithiation through the fourth delithiation, the cell was cycled at a C/10 rate. The voltage cutoffs used after the first lithiation are 10 mV and 900 mV. 3.2.4 Modeling of SEI Thickness In order to analyze the effect of SEI thickness on scatter intensity, a model of the scatter intensity generated by a 30 nm Si NP with SEI was developed. The scatter amplitude as a function of angle for a spherical particle can be modeled by Equation 3.1 where q=(4πSin(θ))/λ, d is the diameter of the spherical particle in meters, λ is the average wavelength of the X-rays emitted in meters, ρe is the electron density of the particle in electrons/m3, and θ is the scattering angle in radians.19 If the particles are not 70 in vacuum, ρe becomes ρparticle- ρmed where ρparticle is the electron density of the particle and ρmed is the electron density of the surrounding medium. Equation 3.1: Scatter amplitude as a function of angle and diameter for a spherical particle Our experimental setup measures the integral under the curve, A2, over the angular range from θmin to θmax for which our experimental setup is sensitive, Equation 3.2. A traditional Small Angle X-ray Scattering (SAXS) measurement performed using a monochromatic collimated beam would measure A2. The X-ray scatter intensity, I, increases with the square of the NPs’ electron density and with approximately the square of the NPs’ diameter. Equation 3.2: Scatter intensity as a function of angle and diameter for a spherical particle Using SpekCalc20,21,22 we calculated the X-ray emission spectrum of our X-ray tube and estimated an average X-ray wavelength of 42.7 pm for our system. The minimum scattering angle that our imaging system can detect was determined to be 7.5x10-4 radians, which corresponds to the number of pixels on the detector that a grid line pair occupies. The maximum detection angle for our system was calculated to be 0.042 radians, which corresponds to half the number of pixels on our detector. 71 In order to model the formation of SEI on the spherical nanoparticles we used Equation 3.3 where d is the diameter of cr-Si NPs in meters, D is the diameter of the Si NP and the SEI in meters, ρSEI is the electron density of the SEI in electrons/m3, ρSi is the electron density of crystalline Si in electrons/m3, and ρmed is the electron density of the medium in electrons/m3. With this equation, we calculate the scatter intensity for a spherical particle NP composed of SEI, subtract out the scattering intensity of the SEI core, and then add the scattering intensity of the cr-Si core. Equation 3.3: Scatter intensity as a function of angle and diameter for spherical core shell particles To perform this calculation, the diameter of the crystalline Si core was set to 30 nm because the average size of the nanoparticles used for the electrodes in this work was 30 nm. Then starting with an SEI thickness of 0 Å, which would signify a crystalline- Si NP prior to any SEI formation, the scattering intensity was iteratively calculated for an SEI thickness of up to 1000 Å. The value used of the electron density of the cr-Si was the 72 density of crystalline Si, 6.86 x 1023 electrons/cm3 and the value used for the electron density of the SEI was 6.00 x 1023 electrons/cm3.23,11 The electron density of the media surrounding the NP is a 50:50 mixture of CB and electrolyte by electron density. 3.3 Results and Discussion 3.3.1 Electrochemical Analysis of First Cycle and Second Lithiation In order to interpret the SFHXI scatter intensities presented in this section, it is first necessary to understand the various electrochemical changes that occur during the first lithiation (Figure 3.2a). The representative cell shown in Figure 3.2 was cycled at a C/20 rate with 20 mV voltage holds at the end of each lithiation. These voltage holds ensured a deep lithiation of the Si NPs. The first lithiation of the cells presented in this paper is unique from all subsequent lithiations in that SEI will form for the first time, carbon black (CB) will lithiate, and the crystalline Si (cr-Si) will form a lithiated Si phase.10,24,25,26 During the first lithiation the cell’s potential starts high and then decreases; once a potential of around 1.2 V is reached, an organic SEI layer begins to form.12,13 The organic SEI in the cells in this paper consist of compounds such as poly(ethylene oxide) oligomers, which are the decomposition products of ethylene carbonate (EC) and diethylene carbonate (DEC), the organic components of the electrolyte.10 Once a potential of 0.8 V is reached, the organic SEI formation continues and the lithiation of CB begins.10,25 Then at potentials around 300 mV the inorganic SEI begins to form.12 The inorganic SEI consists of compounds such as Li2O, LiF, and Li2CO3.12 At a potential of 170 mV the lithiation of cr-Si begins.26 Even though the lithiation of Si 73 begins, the SEI continues to form and evolve.10,11 The voltage inflection around 70 mV is indicative of the formation of amorphous lithiated silicon phases.10,26 At voltages below 50 mV the crystalline Li15Si4 phase forms.5 Figure 3.2: a) Plot of voltage as a function of capacity for a cell for the first lithiation of a cell, b) Plot of voltage as a function of capacity for a cell for the first delithiation of a cell, and c) Plot of voltage as a function of capacity for a cell for the second lithiation of a cell 74 During the first delithiation, the lithiated silicon will delithiate and amorphous Si (a-Si) will form26,8 (Figure 3.2b). Also during the delithiation, the SEI layer can contract as the Si nanoparticles undergo a volume decrease.8,17 Then during the subsequent lithiation, a-Si rather than cr-Si is being lithiated, which results in a different voltage trace than that of the first lithiation.26 There are two voltage plateaus, 0.25 V and 90 mV that are characteristic of the lithiation of a-Si (Figure 3.2c).26,8 Once again at voltages below 50 mV the crystalline Li15Si4 phase forms.5,26 Additional SEI forms during the second lithiation; however, it is much less than during the first lithiation.12 3.3.2 Carbon Black Electrode During cycling, especially the first lithiation, there are many simultaneous processes occurring. In order to disentangle which electrochemical events are affecting measured scatter intensities, control cells were run that just used carbon black (CB) as the active material. This control eliminates the lithiation of Si from affecting the scatter behavior. During the first lithiation of the CB cell, the processes that will occur are the lithiation of CB and the formation of SEI.25,27 The SEI that forms on this electrode will be slightly different in chemical composition due to the lack of Si as well as in morphology due to the lack of distinct nano-sized features on the electrode’s surface.28 In order to keep the SEI that forms on the Si composite electrodes as similar as possible to the SEI that forms on the CB electrodes, the same electrolyte composition as well as the same voltage cutoffs, 0.9 V and 0.01 V, were used for both types of cells. The scatter behavior and the electrochemical data for a representative CB cell are shown in Figure 3.3. The plateau in the voltage trace at approximately 0.9 V 75 corresponds to the lithiation of the CB.29 Prior to this voltage, just SEI forms. The scatter intensity for the CB cell sharply increases during the potentials at which the only electrochemical event is the formation of SEI (Figure 3.3a). As the CB begins to lithiate at the voltage plateau of 0.9V, the scatter intensity continues to sharply increase. The scatter intensity levels off after this voltage plateau. The first cycle does have an irreversible capacity loss of approximately 300 mAh/gCB due in part to the irreversible lithiation of CB but most significantly to the formation of SEI (Figure 3.3b). The irreversible capacity loss of the first cycle in conjunction with the sharp rise in scatter intensity prior to the lithiation of the CB support that the scatter intensity is tracking the formation of SEI in the CB cell. For the second and third cycles, the scatter intensity is quite constant. Also during these two cycles the irreversible capacity loss becomes nearly nonexistent, which indicates that significant SEI formation is no longer occurring. During the second and third cycles, the reversible capacity of 160 mAh/gCB is from the reversible lithiation of the CB. The stabilization of the scatter intensity during the second and third cycles, which have minimal SEI formation and some reversible lithiation of the CB, demonstrates that the scatter intensity is tracking the formation of SEI and not the lithiation and delithiation of CB. 76 Figure 3.3: a) Plot of scatter intensity and voltage for a cell with just CB as the active electrode material and b) Capacity for the three cycles of the cell The lithiation of CB does not result in the large volume changes that accompany the lithiation of Si.30 Thus the contribution of large volume changes to the scatter intensity is minimized. Additionally, the lithiation of CB should not correspond to a significant change in electron density because the electron densities of Li and C are closer than that of Li and Si. Also, stoichiometrically C incorporates far fewer Li ions; this also minimizes the change in electron density from the lithiation of the active material 77 from contributing to the scatter intensity. Thus the only factors that can be significantly contributing to scatter intensity are the changing electron density and nanoscale size due to SEI formation. This in turn means that the scatter intensity change seen in the CB cell is due to the formation of SEI. 3.3.3 20 mV Voltage Hold For the cell shown in Figure 3.2, the scatter intensity and additional electrochemical data for three cycles are shown in Figure 3.4. As soon as the lithiation starts and up to the lithiation of cr-Si begins, 170 mV, there is a sharp increase in scatter intensity. At these potentials the increase in scatter intensity could only be due to the formation of SEI or the lithiation of carbon black. However, the lithiation of CB has been ruled out as causing the scatter intensity increase at these potentials. The amount of SEI formed during this first lithiation is quite substantial as indicated by the large irreversible capacity loss of 700 mAh/gSi during the first cycle (Figure 3.4c). Thus there was significant SEI formation occurring during the first lithiation to which the scatter intensity could be sensitive. During the initial lithiation of the cr-Si NPs, voltages greater than 170 mV, the sharp increase in scatter intensity continues. Even though the Si itself is lithiating, it is important to note that the SEI is still forming at these voltages and that both of these electrochemical events will affect the scatter intensity because they are leading to changes in electron density and also in nanoscale volumes. Midway into the formation of the amorphous lithiated Si phase, indicated by the voltage inflection during the first lithiation at approximately 70 mV, the sharp increase in scatter intensity stops, and the 78 scatter intensity levels out during the remainder of the first lithiation. As discussed in our previous work, the electrochemical and material changes that the electrode undergoes during the first lithiation result in competing factors that affect scatter intensity. These competing factors, which include SEI formation, particle size increases, and electron density decreases, have likely canceled each other to result in a steady scatter intensity. Thus nearly constant scatter intensity does not imply that the electrode is no longer undergoing electrochemical and physical change, but rather that the many varying factors that contribute to the scatter intensity have reached a point in their evolution where they no longer result in a net change in scatter. There are distinct modulations in the scatter intensity during the subsequent lithiations and delithiation. These are likely due to the deep lithiation of the Si NPs, which is reflected in the cell capacity of 3000 mAh/gSi. Also, from the voltage traces, it can be seen that the Li15Si4 phase is likely formed during the second and third cycles. The voltage plateau near 0.4 V during the second and third delithiations is characteristic of the delithiation of the Li15Si4 phase (Figure 3.4a).10,5 Interestingly, the scatter intensity does not undergo significant changes during the deep lithiation of the Si during these cycles which indicates that if Li15Si4 is formed, it does not affect the scatter intensity. The lithiated state of the Si, particularly the deep lithiation and subsequent delithiation of the Si that occurs in this cell, will affect the scatter intensity because it results in significant size and electron density changes of the NPs. However, the smaller intensity of these modulations when compared to the initial rapid increase in scatter intensity further suggests that the scatter intensity is predominantly sensitive to SEI formation. 79 Figure 3.4: a) Plot of scatter intensity and voltage for a cell with a 20 mV hold during the first lithiation, b) Magnification of plot a from 0 s to 15500 s, which corresponds to the first cycle, and c) Capacity for the three cycles of the cell 80 Especially given that the SEI stabilizes during the second and third cycles, which is supported by the practically nonexistent irreversible capacity loss during the second and third cycles, the modulations in scatter intensity during these cycles must be due to the lithiation and delithiation of the Si NPs (Figure 3.4c). 3.3.4 High Voltage Holds To further confirm the sensitivity of X-ray scatter intensity to SEI formation, cells were cycled with voltage holds during the first lithiation at potentials above the potential at which the lithiation of cr-Si occurs. First a cell was cycled with a 500 mV voltage hold during the first lithiation, which is well above the potential at which cr-Si lithiation occurs.26 Thus at this voltage SEI, primarily organic in composition, is forming and the lithiation of CB is occurring. It is important to note that the amount of CB present on this electrode is 20% by mass of the composite electrode. Therefore, the lithiation of CB would only contribute to a small amount capacity during this first lithiation. The corresponding electrochemical data and X-ray scatter data for a representative cell with a 500 mV voltage hold are shown in Figure 3.5. Leading up to and during the 500 mV hold, there is a sharp increase in scatter intensity as was also seen in the 20 mV hold and the CB cell. By the end of the 500 mV hold, the scatter intensity increased by 8% (Figure 3.5b). At this potential this scatter intensity increase could only be due to the formation of SEI. During the first cycle, the cell had an irreversible capacity loss of 500 mAh/gSi, which confirms the formation of SEI (Figure 81 3.5c). This amount of SEI formation could account for the change in scatter intensity during the first lithiation. When the cell is allowed to cycle at a C/20 rate until it reaches the voltage cutoff of 10 mV, the scatter intensity continues to increase, but the rate of increase is much less than during the 500 mV hold (Figure 3.5b). During this period of C/20 cycling, the SEI continues to form and evolve; the Si NPs transform from cr-Si to an amorphous lithiated Si phase. Even though the cutoff potential of 10 mV which would imply the formation of the crystalline phase Li15Si4, the reversible capacity of 1500 mAh/gSi and the lack of a delithiation plateau at 0.4 V indicate that the Si NPs have not reached the Li15Si4 phase. Thus during this portion of cycling the scatter intensity is tracking the continuation of SEI formation and the formation of amorphous lithiated Si phases. The slower rate of scatter intensity change suggests that, similar to the 20 mV hold cell, these simultaneous processes are resulting in slower net rate of scatter intensity change. This is due to the competing effects of increasing particle size and decreasing electron density on the net scatter intensity. Although the most substantial SEI formation occurred during the first cycle, there is some SEI formation during the second and third cycles as seen by the irreversible capacity loss during these cycles Figure (3.5c). However, by the fourth cycle SEI formation stabilizes as seen by the negligible irreversible capacity loss. Thus the scatter intensity modulations during the fourth through sixth cycles are caused by the lithiation and delithiation of Si NPs rather than additional SEI formation (Figure 3.5a). 82 Figure 3.5: a) Plot of scatter intensity and voltage for a cell with a 500 mV hold during the first lithiation, b) Magnification of plot a from 3600 s to 53600 s, which corresponds to the first lithiation, and c) Capacity for the six cycles of the cell 83 These scatter intensity modulations closely resemble the scatter intensity modulations during the second and third cycles of the 20 mV voltage hold cell (Figure 3.4a). Thus the initial sharp increase in scatter in this 500 mV hold cell is due to SEI formation while the variations in the scatter intensity after the initial SEI formation are caused by the lithiation and delithiation of Si. Because the composition and thickness of the SEI varies at different voltages, a 400 mV voltage hold was also performed. The cell also had a 20 mV voltage hold at the end of the first lithiation to ensure the deep lithiation of the Si NPs. The 400 mV hold is still above the potential at which the lithiation of cr-Si occurs, so scatter intensity changes during this voltage hold will reflect the formation of SEI. The potential of 400 mV is still at a point when the composition of the SEI will be primarily organic. The corresponding scatter intensity and electrochemical data for a representative cell with a 400 mV voltage hold are shown in Figure 3.6. During the first lithiation, the scatter intensity increases sharply resulting in a 3% increase in scatter by the end of the 400 mV voltage hold (Figure 3.6b). This initial scatter increase is much lower than that of the 500 mV hold cell. However, this discrepancy is due to the shorter duration of the 400 mV hold which corresponds to less time for SEI formation during the 400 mV hold than during the 500 mV hold. 84 Figure 3.6: a) Plot of scatter intensity and voltage for a cell with a 400 mV hold during the first lithiation, b) Magnification of plot a from 0 s to 180000 s, which corresponds to the first lithiation, and c) Capacity for the four cycles of the cell 85 The irreversible capacity loss of ~700 mAh/gSi during the first cycle indicates that significant SEI was formed during the first lithiation (Figure 3.6c). The shortness of the 400 mV voltage hold indicates that this amount of SEI was not formed solely during the 400 mV hold. Subsequently, a continuation of SEI growth must have occurred during the 20 mV voltage hold, which is supported by the continued sharp increase in scatter during the 20 mV hold. Given the scatter intensity behavior for the 500 mV hold cell as well as the 20 mV hold cell, it is likely that the point at which the scatter intensity rate starts to decrease is the point at which the lithiation of the Si NPs begins to replace SEI formation as the dominate contributor to scatter intensity signal. During the 20 mV voltage hold, the crystalline Li15Si4 phase was reached because the signature 0.4 V voltage plateau, indicating the delithiation of this phase, is visible during the first delithiation (Figure 3.6b). During the end of the 20 mV hold when the Li15Si4 phase is forming, the scatter intensity does become more constant. As with the previous cells the more constant net scatter intensity is due to the competing factors of electron density, particle size, and SEI formation. After the first lithiation, the scatter intensity levels off with some small intensity modulations present during the second cycle. These small modulations once again reflect the lithiation and delithiation of the Si NPs because the irreversible capacity loss during the second cycle is quite small (Figure 3.6c), which indicates that SEI formation was minimal during the second cycle. Thus the electrochemical event that the scatter intensity modulations are tracking during the second cycle is the lithiation and delithiation of Si and not the formation of SEI. However, the absence of the 0.4 V voltage plateau during the second and subsequent 86 delithiations indicates that the Li15Si4 is not reached during the later cycles, which is also reflected in the lower reversible capacity of approximately 1700 mAh/gSi compared to the reversible capacity of 2500 mAh/gSi during the first cycle (Figure 3.6c). The final high voltage hold cell had a 300 mV hold during the first lithiation (Figure 3.7). The cycling for this cell is distinct from that of the 500 mV hold cell and the 400 mV hold cell in that after the 300 mV hold was completed, instead of lithiating the cr-Si, the cell was allowed to delithiate and then cycle at a C/10 rate. By not lithiating the cr-Si, the scatter behavior during the first lithiation is entirely due to the changes occurring in the SEI. Also, at the potential of 300 mV, the SEI will be organic as well as inorganic in composition. Figure 3.7: a) Plot of scatter intensity and voltage for a cell with a 300 mV hold during the first lithiation, b) Magnification of plot a from 0 s to 12000 s, which corresponds to the first lithiation, c) Magnification of plot a from 11690 s to 14690 s, which corresponds to the second and third cycles, and d) Capacity for the four cycles of the cell 87 The voltage trace for the cell clearly shows that the lithiation of cr-Si did not occur during the 300 mV voltage hold during the first lithiation as it is missing all the signature features of the lithiation of cr-Si (Figure 3.7b). During the fourth lithiation the voltage trace has a clear voltage plateau at 70 mV, which is characteristic of the formation of an amorphous lithiated silicon phase from cr-Si (Figure 3.7a). This means that the Si remained crystalline (pristine) prior to the fourth lithiation. Also, the low capacity during the first lithiation as well as the irreversible capacity loss during the first cycle is indicative of SEI formation and is further confirmation that the Si NPs are not lithiated (Figure 3.7d). The scatter intensity during the 300 mV hold sharply increases, irrefutably confirming that the early sharp increase in scatter intensity results from the formation of SEI. As with the 500 mV hold cell, the normalized scatter intensity increases by approximately 8% during this first voltage hold (Figure 3.7b). This indicates that the 300 mV and 500 mV hold cells likely grew similar thicknesses of SEI. The reversible capacity for cycles 2 and 3 is below 100 mAh/gSi, which indicates that minimal electrochemistry, SEI formation and lithiation of Si, is occurring during these two cycles (Figure 3.7d). The scatter intensity for these two cycles is indicative of this lack of electrochemical change because, as can be seen in Figure 3.7c, the scatter intensity does not vary during these two cycles. During the fourth lithiation, the scatter intensity increases again. This increase begins at voltages above 170 mV, which would be the potentials at which SEI would form but cr-Si would not lithiate. Evidence of continued SEI formation during the fourth 88 lithiation can be seen in the irreversible capacity of 280 mAh/gSi loss during the fourth cycle. Thus the continued increase in scatter intensity during the fourth cycle is resulting from SEI formation. The scatter intensity becomes more constant during the latter portion of the fourth lithiation when amorphous lithiated silicon phases are forming, 70 mV plateau (Figure 3.7a). Given the capacity reached during the fourth cycle, 1580 mAh/gSi, and the lack of a 0.4 V plateau during the fourth delithiation, it can be concluded that the Li15Si4 phase is not formed during this cycle. The scatter behavior during the fourth cycle, confirms that the scatter intensity changes are sensitive predominantly to SEI formation but also to some extent to the lithiation of cr-Si. Although, the amount of CB present on the Si composite electrodes is small, 20% by mass, the question may arise as to how exactly the irreversible capacity loss due to CB corresponds to the irreversible capacity loss in the Si composite cells. The irreversible capacity loss for the CB cell during the first cycle is comprised of the irreversible CB lithiation and SEI formation. The capacity loss of 290 mAh/gCB in the CB cell this would correspond to a capacity loss of approximately 55 mAh/gSi in the Si composite cells. For the cycles after the first cycle, the reversible capacity of the CB cell is 160 mAh/gCB; given the small amount of CB present in the Si composite cells, the reversible lithiation of CB in the Si composite cells accounts for approximately 30 mAh/gSi of the total reversible capacity of the Si composite cells. The irreversible capacity loss during the first lithiation was 170 mAh/gSi for the 300 mV hold cell, 720 mAh/gSi for the 400 mV hold cell, 450 mAh/gSi for the 500 mV hold cell, and 700 mAh/gSi for the 20 mV hold cell. The capacity loss for these Si composite cells is much larger than the contribution of 55 89 mAh/gSi by the CB, which indicates that the capacity loss due to CB is minimal in these Si composite cells. Subsequently the irreversible capacity loss in the Si composite cells is due largely to SEI formation and potentially some incomplete delithiation of Si.8 Thus the sharp increase in scatter intensity during the first lithiation at potentials above 170 mV indicates that the scatter intensity is tracking SEI formation not lithiation of CB. 3.3.5 SEI Thickness Modeling In order to further verify that SFHXI is sensitive to SEI formation, the effect of SEI thickness on scatter intensity was modeled. The model that was used is based on the core-shell model used in our previous work. However, instead of having a lithiated core- shell particle, in this model the particle has a SEI shell and a cr-Si core (Figure 3.8a). The model is discussed in detail in section 3.2.4. Because of the complex electrochemical processes that occur during the first lithiation, a few assumptions had to be made in order to create this model. The model assumes that the cell is at a potential above 170 mV and consequently that the cr-Si core is not lithiating. Also, the SEI electron density value used in this model is 0.6 electrons/ Å3; this is the SEI electron density value that Cao and coworkers found using X-ray reflectivity (XRR) for SEI formed on cr-Si that resulted from an electrolyte with a 1:1 ratio by mass of ethylene carbonate: dimethyl carbonate with 1 molar LiPF6.11 The system used in by Cao and coworkers is quite similar to that of the cells used in this work with the main difference being that diethyl carbonate was used in this work instead of dimethyl carbonate. The difference between two molecules is two methyl (CH3) groups. Given this small difference, the electron 90 density found by Cao and coworkers is a reasonable value to use in a model of the SEI formed by the electrolyte used in this work. The modeling results for the effect of SEI thickness on the scatter intensity are shown in Figure 3.8b. The range of thickness values selected for the model was from 0 Å, when no SEI has formed, to 1000 Å. A range of values was chosen because the SEI is growing in thickness as the first lithiation progresses and because the scatter intensity is an operando measurement it is tracking this change. The exact thickness of the SEI on the electrode is not known for the experimental cells but given the extensive work done to measure the SEI thickness on Si electrodes, we are confident that the actual SEI thickness falls within the range that was modeled.11,13,17,18 The general scatter intensity trend seen in Figure 3.8b is that as the SEI thickness increases the scatter intensity will also increase. Within this general trend, there are intensity fluctuations. As discussed in previous work, these fluctuations result from the sinusoidal form of Equation 3.1, the limited angular range of integration as expressed in Equation 3.2, as well as the competing factors of particle size and electron density. The agreement between the modeled scatter intensity trend as SEI thickness increases and the experimental scatter intensity’s sharp increase at potentials where only SEI formation is occurring demonstrates that the experimental scatter intensity is in fact tracking the thickness of the SEI layer in addition to the lithiated state of the nanoparticles. 91 a) b) Figure 3.8: Nanoparticle diagram and modeled scatter intensity of SEI formation a) Cartoon of Si NP (yellow) and carbon black (grey) on top of copper electrode (copper) all surrounded by electrolyte (light green) and cartoon of SEI growth (dark green) on Si NP (yellow) b) Plot of modeled normalized scatter intensity as a function of SEI thickness in Ångstroms Because the scatter intensity is dependent on the thickness of the SEI, the scatter intensity can provide information about the thickness of the SEI while cells are operating. At the end of the 500 mV voltage hold and the 300 mV voltage hold the scatter intensity had increased by 8% and at the end of the 400 mV voltage hold the scatter intensity had increased by 3%. Given the trend between SEI thickness and scatter intensity, this indicates that a thicker SEI layer was formed during the 500 mV and 300 mV voltage holds than during the 400 mV voltage. The correlation between scatter intensity increase and SEI thickness is supported by the brevity of the 400 mV hold. Additionally, at the end of the 500 mV hold the capacity was 112 mAh/gSi, at the end of the 400 mV hold the capacity was 11 mAh/gSi, and at the end of the 300 mV hold the capacity was 200 mAh/gSi. The order of magnitude difference in capacity between the 400 mV hold cell and the 500 mV and 300 mV hold cells, supports that the difference in 92 scatter intensity values at the end of these holds does in fact reflect the thickness of the SEI at these points. Thus cells with a small increase in scatter intensity during the initial part of the first lithiation when SEI is primarily forming, have a thin SEI layer where as cells with a large increase in scatter intensity during the initial part of the first lithiation have a thick SEI layer. Subsequently, SFHXI is a method to probe the thickness of SEI without post-mortem analysis, which could easily disturb the SEI and therefore not provide an accurate measure of the thickness of the SEI. The sensitivity of scatter intensity to SEI thickness explains results in our previous work where the scatter intensity behavior during delithiation was not simply the reverse of the scatter intensity behavior during lithiation. The SEI, though it may shrink and lose contact with the Si NPs during the delithiation process, does remain present during cycling.8 The scatter intensity tracks the initial formation of the SEI and because the SEI does not disappear during the subsequent delithiation and additional cycles, the scatter intensity does not undergo as drastic of a change as it does during the first lithiation. With any model it is important to discuss the assumptions that go into the model and how they affect the results produced by the model. This model of the SEI does assume uniform SEI composition through the use of a uniform SEI electron density on the NPs. However, the composition of the SEI itself is not uniform. The assumption that the SEI is uniform is a valid assumption when calculating the scatter intensity because the electron density differences that result from the differences in composition are trivial. This is because the compositional differences in SEI primarily involve carbon atoms, which have a small electron density. Additionally this model used 30 nm Si NPs. 93 The nanoparticles on the electrode are a range of sizes. With a distribution of nanoparticle sizes, the modeled scatter intensity would still increase as the SEI grows in thickness. However, the modeled scatter intensity for a distribution of particle sizes would be smoother than the curve shown in Figure 3.8b because it would be an average over many particle sizes. In fact it would likely more closely match the experimentally measured scatter intensity during SEI growth. Thus the assumptions that went into the SEI model were reasonable. 3.3.6 Homogeneity of SEI formation and Si Lithiation Because SFHXI allows for an image of the entire cell to be captured in a single exposure, the scatter images from SFHXI can be used to quantify the homogeneity of the electrodes. For the 500 mV, 400 mV, and 300 mV hold cells, histograms of their scatter intensities after the voltage holds and after the lithiation of Si are shown in Figure 3.9. To create the histograms, each scatter image for a cell is normalized by the first scatter image so that the histogram tracks the changes in homogeneity. Subsequently a broader histogram indicates a greater range of scatter intensity values on the electrode, which means that the electrode has become more inhomogeneous. Though it is important to note that due to the pixel size, which is hundreds of micrometers, and the nanoparticle size, which is tens of nanometers, the scatter intensity measured at each pixel is the scatter intensities contributed by many nanoparticles. After the high voltage holds where SEI formed but cr-Si did not lithiate, the full width half maximums of the histograms are quite narrow; however, after the lithiation 94 of cr-Si occurs, the full width half maximums of the histograms become much broader (Figure 3.9). Subsequently, the SEI forms more homogeneously across the electrode, but the lithiation of the Si itself is inhomogeneous. It should be noted that the time of the post-lithiation of cr-Si histograms was at the end of the first lithiation for the 500 mV hold cell, at the end of the first lithiation for the 400 mV hold cell, and at the end of the fourth lithiation for the 300 mV hold cell. These times were selected because these times were all after the voltage plateau at 70 mV signifying the formation of amorphous lithiated silicon phases from cr-Si. Thus the cr- Si was in fact lithiated at the time corresponding to the second histograms. 95 Figure 3.9: Histograms after a high voltage hold where SEI was formed and after the cr-Si was first lithiated a) 500 mV hold cell, b) 400 mV hold cell, and c) 300 mV hold cell 96 Figure 3.10: Histograms at various points in cycling for the 20 mV voltage hold cell and voltage trace showing the time points of the histograms 97 Histograms for the 20 mV hold cell at various time points during cycling are shown in Figure 3.10. Like the 500 mV, 400 mV, and 300 mV hold cells, the 20 mV hold cell has a rather narrow scatter intensity distribution prior to the lithiation of cr-Si. During the formation of amorphous lithiated Si phases, the histograms dramatically broaden in width, reflecting the inhomogeneous lithiation of cr-Si. The most likely reason for the development of inhomogeneities during the lithiation of cr-Si is that the particles are lithiating to different extents. These particle to particle variations in the extent of lithiation would lead to a variety of scatter intensities, which would result in a broader distribution of scatter intensity values as seen in Figure 3.10. Once deep lithiation of the Si occurs, the inhomogeneities on the electrode are set. Even though small fluctuations in the widths of the histograms occur during the subsequent delithiations and lithiation, the histograms remain broad which indicates that the electrode does not become more uniform with subsequent cycling. What these results show is that homogeneous SEI formation does not result in homogeneous lithiation across the electrode. Given that one of the roles of the SEI is to provide a more uniform current distribution across the electrode, it is unexpected that a uniform current distribution itself is not adequate to ensure a homogeneous lithiation of Si NPs. The inhomogeneous lithiation of the Si could result from uneven distribution of the Si NPs on the electrode surface or the poor electrical contact of some Si NPs. These hypotheses are supported by the inhomogeneity during the cycles subsequent to the first lithiation in the 20 mV hold cell. Once the inhomogeneity of the electrode develops during the lithiation of Cr-Si it remains inhomogeneous. 98 3.4 Conclusion By strategically utilizing voltage holds that isolate the formation of SEI from the lithiation of cr-Si and by carefully correlating scatter intensity to phase transformation voltage signatures, it has been shown that X-ray scatter intensity is sensitive to SEI formation and also to the lithiation of Si. The relative thickness of the SEI can be determined solely by the change in scatter intensity during the early part of the first lithiation. Larger increases in scatter intensity correlate to greater SEI thickness, while smaller increases in scatter intensity correspond to thinner SEI layers. SFHXI provides feedback about the thickness of SEI formation without the inaccuracies that can arise when SEI thickness is determined via post-mortem analysis and without the need to wait until the end of the first cycle to estimate the amount of SEI formation from the irreversible capacity loss. Because an image of the entire cell is captured in a single exposure, SFHXI also allows for the characterization of the homogeneity of SEI formation and lithiation of Si. In all the cells studied, the SEI layer formed homogeneously across the electrode surface. However, uniform SEI formation did not lead to the spatially uniform lithiation of the Si. All of the cells imaged experienced a significant increase in inhomogeneity once the lithiation of cr-Si began. Furthermore, electrode inhomogeneities that develop during the lithiation of Si remain present during subsequent cycling. This result provides additional insight into the causes of inhomogeneous lithiation of Si and shows that uniform SEI formation alone is not a determinant of the uniformity of the lithiation of the active material on an electrode. Thus, efforts to improve the uniformity of lithiation 99 and delithiation on the entire electrode surface and subsequently increase the working capacity of these cells cannot focus solely on controlling SEI formation. Other solutions such as improving the electrical contact of the Si with the rest of the electrode and improving the uniform distribution of Si on the electrode surface should be investigated. The work presented demonstrates that SFHXI is a unique tool that not only tracks the lithiation of Si but also the formation of SEI in unmodified operating cells. Thus SFHXI does provide comprehensive insight into the processes that dictate the performance of cells. Although SFHXI was applied to primarily Si composite cells in this work, its utility with carbon based cells was also demonstrated with the CB cell. The ability of the scatter intensity to track SEI formation means that SFHXI is a valuable SEI probing technique in a variety of different battery types such as Li metal and even Na and K batteries. Though there is still work to be done in disentangling the electrochemical processes that result in the scatter signal, these obstacles have been surmountable obstacles in the other techniques that have been used to study batteries. Subsequently, the unique characterization ability of unmodified operating batteries that SFHXI provides will progress the understanding of the processes that occur in batteries and subsequently lead to the development of batteries with improved cycle lives and safety. 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Solid State Ion. 2001, 143 (2), 173–180. https://doi.org/10.1016/S0167- 2738(01)00852-9. (30) Whitehead, A. H.; Edström, K.; Rao, N.; Owen, J. R. In Situ X-Ray Diffraction Studies of a Graphite-Based Li-Ion Battery Negative Electrode. J. Power Sources 1996, 63 (1), 41–45. https://doi.org/10.1016/S0378-7753(96)02440-8. 105 APPENDIX A DETECTOR FIXED PATTERN NOISE CORRECTION A.1 Introduction SFHXI relies on the capture of four different types of X-ray images, background, flatfield, grid, and the image, in order to generate two scatter images and an absorption image. For more information on the production of scatter images see section 1.5 of Chapter 1. The background is taken by reading out the noise from the detector for a given exposure time without the detector being exposed to X-rays (Figure 1a). The background should provide a measure of the offset of every pixel as well as any fixed pattern noise (FPN). The flatfield is generated by exposing the detector to X-rays with nothing present in the viewing field (Figure A.1b); the flatfield is a measure of the pixel response to X-rays for a given exposure time, gain. The grid image is produced by exposing the detector to X-rays while a grid made of evenly spaced stainless steel wires is present in the viewing field (Figure A.1c). The image is produced in the same way that the grid image is produced, but, in addition to a grid, the object of interest is also present in the viewing field. For the work presented in this thesis a battery is the object of interest. The Complementary Metal Oxide Semiconductor (CMOS) detector used in our experimental setup has always had defects that cause the fixed pattern noise (FPN) to 106 appear in the images produced by the detector. FPN can be comprised of a gain component and an offset component.1 The FPN appears in the background, flatfield, grid, and object images (Figure A.1). The most obvious FPN is the checkerboard like pattern that is predominantly seen on the left half of the background, flatfield, grid, and image. Some of the checkerboard pattern can be seen more faintly on the right half of the detector, and some bright diagonal stripes appear on both halves of the detector. The appearance of the checkerboard pattern in all four types of images, especially the background, indicates that the FPN is in large part due to an offset and can thus be classified as dark signal non-uniformity.1 The difficulty with correcting the FPN stems from the fact that although the location of the FPN remains the same, the intensity of the noise does vary with exposure time, becoming significantly worse with shorter exposure times. Also, not readily apparent in Figure A.1 are the rogue pixels. These pixels were named rogue pixels because they are completely defective and show no response when exposed to X-rays. Because the rogue pixels are constant in intensity and location throughout all the images, Rogue pixels are properly addressed by the median filter used in the scatter image processing. 107 a) b) c) d) Figure A.1: a) Background, b) Flatfield, c) Grid, and d) Image Although the defects in the detector have been present since the detector was purchased, they have not presented as much of a challenge for previous projects using SFHXI because the other projects did not focus on looking at the scatter images themselves but rather the measured changes in scatter signal. When the scatter images were used, the object of interest often occupied a smaller portion of the viewing field and thus the regions with large FPN could be avoided. A key component of the battery imaging work presented in this thesis is being able to look at a scatter image of a battery as it is evolving during the cycling process. As seen in Figure 1d, the battery occupies the both halves of the detector and thus is affected by the FPN. Having significant FPN that 108 is visible in the battery image is not desirable. Figure A.2 shows a cross section across the detector for a flatfield and an image of a battery. The checkerboard pattern on the left side of the detector is clearly visible in the pixel range of 200 to 500 pixels. The FPN shows up as intensity modulations. a) b ) Figure A.2: Cross sections of intensity modulations for a) Flatfield and b) Battery image FPN is a rather common problem in the field of X-ray imaging. There are standard preprocessing techniques for correcting FPN, particularly for Charge Coupled Device (CCD) detectors. The most common CCD preprocessing correction is called the flatfield correction (FFC) and this correction is given by Equation A.1.2 The FFC is considered to be a quite effective and low computational cost method for correcting fixed pattern noise.3 Although the FFC was developed for X-ray imaging using CCD detectors, the FFC was applied to our CMOS detector. Prior to the CMOS camera that is used in the current experimental setup, a CCD detector was used. The FFC had been part of the preprocessing for images taken with the CCD, so when the CCD was replaced by the current CMOS we chose to use the FFC correction. We had assumed that the fixed pattern noise on a CMOS detector would behave similarly to a CCD detector and thus be corrected in the same way. However, as seen in Figure A.3, the FFC is not 109 effective in eliminating the noise from the CMOS detector. Though in hindsight this is not surprising given that CMOS detectors are prone to more FPN than CCD’s given that there are more sources of FPN in CMOS detectors such as pixel transistors and column amplifiers.1 What is most concerning about the FFC not adequately correcting the FPN from the CMOS detector is that the defects are still visible in the absorption and scatter images. Consequently, it was concluded that FFC was probably not the best preprocessing image correction, and an investigation into a better way to correct images was launched. Image Corrected= Image – Background Flatfield - Background Equation A.1: FFC The fixed pattern corrections that will be mentioned in the following sections were programmed as MATLAB scripts. Further information regarding those scripts can be found in Appendix D. 110 a) b) c) d) Figure A.3: a) Image corrected using FFC, b) Absorption image generated using image corrected using FFC, c) Scatter image generated using image corrected using FFC, and d) Cross section of image corrected using FFC A.2 Development of a Flattener The aim of the work presented in this section was to design a preprocessing noise correction algorithm that would better correct the FPN than the FFC. Ideally, this algorithm would be universal and could be used for all CMOS detectors. What seemed like the most straightforward way to correct the noise would be to develop a matrix of values corresponding to every pixel. This matrix could then be multiplied with the raw Images in order to correct the defects (Equation A.2). This matrix will subsequently be 111 referred to as the flattener. It is so named the flattener because it should flatten all the pixels to the same value by reducing pixel to pixel variations. Image Corrected=Flattener*Image with Defects Equation A.2: Image correction using flattener In order to develop the flattener, it was necessary to characterize how the pixels behave when exposed to X-rays. The initial hypothesis was that the pixels would behave linearly with exposure time, meaning that the intensity values of the pixels would be a linear function of exposure time. However, by collecting flatfields and by plotting the intensity values of the example pixel (300, 300) with respect to different exposure times it became rapidly apparent that the correlation was non-linear as seen in Figure A.4. Although the plot shown in Figure A.4 is for the pixel (300, 300), the shape of the plot is representative of the response of all pixels on the detector to different exposure times. Intensity (Counts) Exposure Time (s) Figure A.4: Plot of intensity as a function of exposure time for pixel (300, 300) 112 The shape of the plot in Figure A.4 appears to be closer to that of a second order polynomial rather than a straight line. The flattening out of the plot at high exposure times, 25 s and 30 s corresponds to the saturation of the detector. Subsequently we determined that the maximum exposure that could be used with our CMOS detector was 20 s. Any longer exposure time than 20 s would result in the saturation of the detector’s pixels and would subsequently not be an accurate measure of the number of photons hitting the detector. Because intensity was not linear as a function of exposure time as initially expected, we decided to take a step back and look at a more straightforward case, the background. There are no X-rays involved in the generation of the background, so the background should be a measure of the detector offset plus the noise generated when collecting an image. In a similar manner to determining the relationship between exposure time and intensity for flatfields, the same was done for backgrounds. The backgrounds with varying exposure times were fit to a function that would allow us to determine how noise and offset varies with exposure time. This was done for every pixel meaning that each pixel had its own unique fit. It was done for every pixel, because as evidenced by the detector defects, not all the pixels behave in the same way. At an exposure time of zero, the only signal from the detector should be the offset of the detector; no noise should be present. Determining the offset for every pixel is important information to incorporate into a flattener. The fit was done pixel by pixel for backgrounds with exposure times of 5 s, 7 s, 10 s, 12 s, 15 s, 20 s, and 25 s. These were the same exposure 113 times that were also used for the flatfields. The fit used was a second order polynomial. The y-intercept of the fit, the third term of the polynomial, is the offset of the detector. It should be noted that the fit for the background pixels was almost linear. The curvature is very slight, and is concave down as seen in Figure A.5. Figure A.5 uses example pixel (300, 300); however, the other pixels had the same general relationship between exposure time and intensity. The main pixel to pixel variation in the curves was in their offset values. Intensity (Counts) Exposure Time (s) Figure A.5: Plot of second order polynomial background fit for pixel (300, 300) Next a similar fit was done using flatfields. The first function that was tested was a second order polynomial. This was chosen as a reasonable test fit because as the detector was saturated by longer exposure times, the slope between points of higher exposure times approached zero. The plot of a second order polynomial fit for pixel (300, 300) is shown in Figure A.6. Comparing Figure A.4 to Figure A.6, it can be seen that the curvature of the detector response was modeled well by a second order polynomial. Through studying the behavior of the flatfields and backgrounds with different exposure times, the shortcomings of the FFC became apparent. One of the reasons why 114 the FFC does not adequately compensate for the detector defects is because the background and images, particularly with long exposure times, have large differences in magnitudes (Figure A.4 and Figure A.5). Subtracting the Background from the image and the flatfield respectively in the FFC resulted in subtracting a number on the order of hundreds of counts from a number on the order of thousands of counts. Also, dividing the image by the flatfield was not an adequate correction factor because when there is an object placed in the viewing field the subsequent intensities of the defects decrease (Figure A.1). This is because the object decreases the effective exposure time that the detector sees. The flatfield used in the FFC has the same exposure time as the image and does not take the effective exposure time into account. As previously mentioned, for shorter exposure times, which is what the effective exposure time corresponds to, the fixed pattern noise becomes more prevalent because of its dark signal non- uniformity component. This means that the flatfield in the FFC underestimates the fixed pattern noise and therefore cannot properly correct it. Intensity (Counts) Exposure Time (s) Figure A.6: Plot of second order polynomial flatfield fit for pixel (300, 300) 115 The flattener that was then developed is given by Equation A.3. The third coefficient of the second order polynomial fit of the flatfields, the y-intercept, is used as the flatfield offset in this flattener. The background offset is the third coefficient of the second order polynomial fit of the backgrounds, the y-intercept. The idea behind this flattener is to “walk” the flatfield and background closer to each other in terms of magnitude in order to maximize the ability to correct the FPN. Flattener=FlatfieldExposure Time -Background Offset BackgroundExposure Time-Flatfield Offset- Background Offset Equation A.3: Flattener using second order polynomial flatfield fit In order to test the effectiveness of the flattener, the flattener from Equation A.3 was multiplied with the image, grid, and flatfield, and the exposure times of the image, grid, flatfield, and background correspond to the exposure time used to generate the flattener. As seen in Figure A.7, this flattener improved the FPN, but the FPN is still visible. In the flattened flatfield the FPN on the left half of the detector where the checkerboard noise was initially present is now brighter than the rest of the image, which indicates that the FPN has been overcorrected. The flattener correction appears to be most effective for the grid image. The reason for this effectiveness is likely because the effective exposure time for the grid is approximately the same as the actual exposure time because there is not a highly absorbing object present in the viewing field. Though the FPN is still quite visible in the image, which is probably due to the object in the viewing field altering the effective exposure time that the detector sees. 116 The design of the flattener in Equation A.3 does not take into account effective exposure time when it is significantly different than the actual exposure time. The flattener assumes that the effective exposure of the image is the same as the actual exposure of the image. By using a flattener generated for the new effective exposure time caused by the object rather than the original flatfield exposure time, the flattener that would be more effective at flattening the image. a) b) c) d) Figure A.7: Using flattener from Equation 3 a) Flattened flatfield, b) Flattened grid, c) Flattened image, and d) Flattened image cross section 117 A.3 Flatteners and Effective Exposure Time A.3.1 Second Order Polynomial Flatfield Fit To solve the problem of effective exposure time versus actual exposure time, the effective exposure time of each pixel needs to be calculated prior to the generation of the flattener. The flatfield fit polynomials were inverted so that the effective exposure time was a function of intensity rather than the intensity being a function of exposure time. Then each image was fed through these inverted polynomials to generate the effective exposure time seen by each pixel. These exposure times were then used to select the flattener corresponding to the exposure time on a pixel by pixel basis. The flatteners for each pixel were multiplied by the corresponding pixels in the original images in order to flatten the images. Figure A.8 shows the resulting images from using the inverse second order polynomials to select the flattener. These resulting images look almost the same as the resulting images from the flattener that did not take into account the effective exposure time (Figure A.7 and Figure A.8). Considering that the inverse polynomial approach was much more computationally expensive and yielded little to no improvement, this flattener was also not deemed an acceptable flattener. When designing another flattener, it was important to look at what components of the previous flattener did not seem correct. The first aspect of the previous flattener that did not seem logical was that the y-intercept for the second order polynomial flatfield fit is negative, which implies that for an exposure time of zero that there are a negative number of photons hitting the detective. This problematic part of the fit 118 probably arises from the fact that the shortest exposure time that the detector can reliably measure is 5 s. Thus the extrapolation to an exposure time of zero is not that accurate. Thus in the next iteration of the flattener, either the flatfield offset term needs to be either redefined or removed from the flattener. a) b) c) Figure A.8: Flattened using inverse second order polynomial to select flattener on a pixel by pixel basis a) Flattened grid, b) Flattened image, and c) Cross section of flattened image A.3.2 Third Order Polynomial Flatfield Fit A value that makes more physical sense for the zero exposure time of the flatfield is the extrapolated detector offset generated from the second order polynomial fit of the background. Consequently, the exposure time of zero for the flatfield was set to the extrapolated y-intercept of the background fit. However, once this was done a second order polynomial was no longer a good fit for the flatfield data. Thus a third 119 order polynomial was selected as a trial fit. The s-shape of a third order polynomial better matched the shape of the data with the set y-intercept. As the detector is reaching saturation, the count intensity should flatten out. Also, as the detector approaches zero exposure time, the count intensity should approach the detector offset. With this approach, the flatfield offset term in the previous flattener becomes irrelevant. Now that the flatfield offset term was redefined to a value that made physical sense, it was necessary to develop a new flattener. This new flattener is given by Equation A.4. It has a few differences when compared to the previous iteration of the flattener given by Equation A.3. These differences result from keeping the flatfield and background close to each other in terms of magnitude in order to maximize its ability to correct the FPN and from accounting for the now irrelevant flatfield offset term. Flattener=FlatfieldExposure Time-BackgroundExposure Time-Background Offset BackgroundExposure Time-Background Offset Equation A.4: Flattener for third order polynomial fit for other exposure times Flattener5s=Flatfield5s -Background Offset Background5s-Background Offset Equation A.5: 5 s Flattener for third order polynomial fit Unfortunately using the process described, it was not possible to develop a flattener that worked universally for all exposure times. A separate flattener had to be developed specifically for the 5 s exposure time (Equation A.5). This is probably due to the fact that the 5 s flatfield has a significantly lower intensity than all the other 120 flatfields (Figure A.4). If the additional background term is subtracted from the 5 s flattener, the result is an image that is too dark, and the FPN is not adequately corrected. In order to select the appropriate flattener for each pixel, the flatteners for each exposure time were fit to a second order polynomial. The inverse of the third order polynomials for the flatfield fit were used to find the effective exposure times for the image and flatfield. These effective exposure times were then fed through the second order polynomial fit of the flatteners in order to determine the appropriate flattener values. Then the flattener values are multiplied with their respective flatfields and images. The flatfields are effectively flattened by the flattener; however, the image was made significantly worse by the application of the flattener. This likely results from the difficulty of solving for a unique inverse function for a third order polynomial, which this approach required. The lack of a unique inverse function makes the effective exposure times inaccurate, which subsequently limits the effectiveness of the flattener. The resulting images were not included because they provided no useful information other than that they were not acceptable. A.3.3 Logistic Function Flatfield Fit Consequently, a new fit function for the flatfields was needed that has a similar shape to the third ordered polynomial, is easily solvable, and has a unique inverse. The function that best matched these criteria was the logistic function (Equation A.6). The four coefficients act as four fit parameters for the logistic function fit, and this is the same number of fit parameters present in a third order polynomial fit. For the logistic 121 function there is an analytical inverse function, and as a result, it is straightforward to find the effective exposure time of each pixel in an image (Equation A.7). Intensity=D+ A-D (1+ e-B*(t-C ) Equation A.6: Logistic function where A, B, C, and D are fit coefficients t=log((A-Intensity)/(Intensity-D)) (B-C) Equation A.7: Inverse logistic function As was done with the third order polynomial fit, the zero exposure time value was set to the background y-intercept value. It is important to note that the background fit was kept as a second order polynomial. A plot of the logistic function flatfield fit is shown for pixel (300, 300) (Figure A.9). The shape of the logistic function fit provides the s-shaped fit that is necessary to fit the flatfield data with a set y-intercept. Intensity (Counts) Exposure Time (s) Figure A.9: Plot of logistic function flatfield fit for pixel (300, 300) The use of the logistic function fit required another iteration of the flattener. The equation of this new flattener is given by Equation A.8. The n coefficient is used to 122 account for the difference in the intensity of the defect that results from different exposure times. The n is the same for all pixels. However, there is no systematic way to calculate n. It had to be found by iteratively adjusting the value of n and then determining whether the flattened image improved or worsened. The inability to rigorously calculate n was a significant drawback to this flattener because n becomes a bit of a “fudge factor” that is subjectively determined. Flattener=FlatfieldExposure Time-Flatfield Intercept Background +n*Flatfield Equation A.8: Flattener for the logistic fit where n is a coefficient that is set for each exposure time The flatteners are then fit with third order polynomials. The effective exposure times are calculated using the inverse logistic function fit for the flatfield data. Then these effective exposure times are fed through the polynomial fit of the flatteners in order to select the corresponding flattener values. Once the flattener values are selected, they are then multiplied with their respective images and grids. The flattened grid and flattened image are shown in Figure A.9. The FPN is improved is the most corrected out of all the iterations of flatteners. The cross section of the flattened image shows significantly fewer modulations than the FFC Image (Figure A.3 and Figure A.10). 123 a) b) c) Figure A.10: a) Flattened grid, b) Flattened image, and c) Cross section of flattened image Because there was such a great improvement in the FPN in the flattened image and the grid, the grid and image were fed through a modified scatter processing script in order to generate scatter images. The modification in the processing script was the removal of the FFC algorithm. Also because the correction to the images was performed in a separate script, the modified scatter processing script did not need to use flatfields and backgrounds; only the flattened grid and image were necessary to generate scatter images. The generated absorption image and scatter image are shown in Figure A.11. Upon visual inspection, the absorption and scatter images do not appear more uniform than the original scatter image produced using the FFC method. In fact the regions where the FPN was present appear to have become exaggerated, particularly on the left 124 side of the detector. What is even more troubling is that the cross section of the scatter image indicates regions of negative scatter. Such a result is not physically possible and points to the flattener as not being an effective way to correct the FPN. When the standard deviations of pixels in regions of the detector that should be homogenous were examined, it was found that the standard deviations were larger for the scatter images generated with the logistic flattener than with the original FFC method. Prior to completely abandoning the logistic flattener, the logistic Flattener that was used on the images in Figure A.10 and A.11 was tested on a different battery image. This means that the n value was kept constant between the two sets of data. Once again the image was passed through the inverse logistic function to generate effective exposure times for the image. Then these exposure times were fed into the third order polynomial to which the flatteners had been fit. Then the selected flattener was multiplied by the image. The resulting flattened image is shown below in Figure A12. As can be seen in the figure below, the flattener was not at all effective in removing the FPN from this different set of battery images. 125 a) b) c) Figure A.11: Processed images using logistic flattener a) Absorption image, b) Scatter image, and c) Cross section of the scatter image a) b) Figure A.12: Logistic flattener applied to different Image a) Flattened image and b) Cross section of flattened image 126 This means that this iteration of the flattener is not universal and cannot be applied to all images, which significantly reduces the viability of the flattener. A new flattener cannot be generated for every set of images that needs to be processed. Firstly, that is extremely tedious and completely inefficient for large sets of data. Secondly, the purpose of a flattener is to be a universal correction. It should be applicable without modification to all images generated by that detector. A.4 Conclusion Although much effort was put into correcting the FPN caused by defects in the detector, the work did not lead to significant improvements in the reduction of the FPN in corrected images. The use of the flattener, especially with inverse functions to calculate effective exposure times, substantially increased the computational expense of the image processing without yielding results that merited the additional computational time. Our original conclusion still stands that the FFC is not the optimal way to correct fixed pattern noise on a CMOS detector especially because FPN level are so much higher on CMOS detectors than on CCD detectors. The FFC does not account for the effective exposure time that the detector sees when an object is placed in the viewing field and that the FPN intensity does vary with effective exposure time. However, the inability to generate a better correction has led to the continued use of the FFC in our image processing. Although the work presented in this section did not yield the result hoped for, it should serve as a guide for future investigations into FPN correction techniques. It is 127 possible that the best solution to this problem may be to return to using a CCD detector with fewer native defects and subsequently less fixed pattern noise. References (1) Lee, H.; Ham, D.; Westervelt, R. M. CMOS Biotechnology; Springer Science & Business Media, 2007. (2) Stock, S. R. MicroComputed Tomography: Methodology and Applications; CRC Press, 2008. (3) Nieuwenhove, V. V.; Beenhouwer, J. D.; Carlo, F. D.; Mancini, L.; Marone, F.; Sijbers, J. Dynamic Intensity Normalization Using Eigen Flat Fields in X-Ray Imaging. Opt. Express 2015, 23 (21), 27975–27989. https://doi.org/10.1364/OE.23.027975. 128 APPENDIX B ORIGINS OF AND SOLUTIONS TO NOISE PRESENT IN SCATTER IMAGES B.1 Introduction Because SFHXI relies on extensive image processing, which utilizes Fourier transforms and inverse Fourier transforms, noise can be easily introduced and amplified during the image processing. This noise is visible in final scatter images and results in the characteristic speckled appearance of the scatter images (Figure B.1). However, the exact causes of the speckles was not entirely understood, which subsequently presented a difficulty in reducing the noise present in scatter images. Figure B.1: Scatter image generated 129 The work presented in this appendix is an investigation into the origin of the noise in scatter images and into various solutions to reduce the amount of noise. Although this appendix like Appendix A deals with noise in images, it is distinct from Appendix A in that this appendix focuses on noise that is pervasive across all scatter images rather than the fixed pattern noise arises from the detector. In this appendix, artificial transmission images are utilized to control the introduction of noise into scatter images and thus lead to an understanding of how different amounts of noise affect the generated scatter images. Also, the effect of adjusting the region of interest parameters on noise in generated Fourier spectra and scatter images is explored. The results of increasing exposure time in order to increase signal to noise ratios is detailed. Overall this work provides a comprehensive examination of noise in scatter images as well as noise-reduction solutions. B.2 Artificial Images and Noise The first step in producing artificial transmission images is to create an artificial grid. This was done using a MATLAB script, which utilizes two sinusoidal functions; see Appendix C Section C.4.1 for more details regarding the script. The artificial grid image produced had perfectly straight and evenly spaced grid lines and no noise (Figure B.2a). The next step was to produce an artificial object image. The way in which this was done was to take the artificial grid image and apply a circular Gaussian blur with a radius of 1 using ImageJ (Figure B.2B). For the ImageJ Gaussian blur function, the radius is the radius of decay to e-0.5.1 130 a) b) c) d) e) Figure B.2: Artificial images with no noise a) Artificial grid image b) Artificial image produced using Gaussian blur c) Generated (0,1) scatter image d) Generated (1,0) scatter image e) Generated Fourier spectrum Generating a scatter image of an object using SFHXI works because the object blurs the previously crisp gridlines. The Gaussian blur is a noiseless blurring of the artificial gridlines and thus results in a perfectly noiseless object image that can be used 131 in SFHX image processing. A circular Gaussian blur was selected to mimic an object because the batteries utilized in this thesis are circular. Previous observations have shown that the circularity versus non-circularity of the object affects the generated Fourier spectrum in the SFHX image processing (Figure B.3). a) b) Figure B.3: a) Fourier spectrum generated from battery image b) Fourier spectrum generated from a non-battery (noncircular) object The artificial grid and artificial object image were then processed to produce scatter images. No image correction such as the flatfield correction was used in the processing because there was no noise to correct. The resulting (0,1) and (1,0) scatter images are shown in Figure B.2c-d. Clear concentric circles can be seen in the object. The outermost ring is highest in intensity due to edge effects. When there is a sudden change in material, such as at an edge, this causes a higher scatter intensity. Visible in the (0,1) scatter image are horizontal lines and visible in the (1,0) scatter image are vertical lines. These are due to the sensitivity of the different harmonics to horizontal features and vertical features respectively. What is most important to note in these scatter images produced from noiseless artificial images are the clear rings and lines. 132 The Fourier spectrum generated by the artificial images is also shown (Figure B.2e). When comparing the Fourier spectrum from the artificial noise free images to the Fourier spectra from the experimental images, the artificial Fourier spectrum is quite distinct. The artificial Fourier spectra has many sharp higher order harmonics that have perfectly even spacing between them (Figure B.3). The question is whether the uniqueness of the artificial Fourier spectrum results is from the perfect symmetry of the Gaussian blur used or from the lack of noise. With the noiseless artificial images as a control, the logical next step was to introduce some noise into the artificial images and determine how the scatter images and Fourier spectrum are affected as a result. Initially 5% noise was introduced to the artificial image not the artificial grid using ImageJ. With all noise introduced in this section using ImageJ, the noise introduced is randomly distributed and is Gaussian in value with the average noise value at zero and the standard deviation being the specified percentage.1 The resulting artificial images and generated scatter images are shown in Figure B.4. Notably with the introduction of just 5% noise in only the object image and not the grid, there is already a dramatic increase in noise in the scatter images as well as the loss of higher order harmonic peaks in the Fourier spectrum. The scatter images have lost their concentric rings and the horizontal and vertical lines are no longer as clearly visible; instead, the scatter images have the speckles that are characteristic of the experimentally generated scatter images (Figure B.1, Figure B.2, and Figure B.4). This result demonstrates that the speckles can never fully be eliminated from scatter images 133 because the speckles are generated when the noise level in the image is only ±5%. A 5% noise level could easily come from detector noise (see Appendix A) or even from material variations in the object being imaged. a) b) c) d) Figure B.4: Artificial images with 5% salt and pepper noise introduced in artificial image and not in artificial grid a) Artificial image produced using Gaussian blur b) Generated Fourier spectrum c) Generated (0,1) scatter image d) Generated (1,0) scatter image Also, the disappearance of the higher order harmonics in the Fourier spectrum with added noise seems to indicate that the unique Fourier spectrum in Figure B.2e is most likely due to the completely noiseless images used to generate it rather than the perfect symmetry of the Gaussian blur. Subsequently if harmonics higher than the first order harmonics were to be used in SFHX image processing, then the noise level of 134 object and grid images should be reduced in order to increase the intensity and sharpness of these higher order harmonics. a) b) c) d) Figure B.5: Artificial images with 10% salt and pepper noise introduced in artificial image and not in the artificial grid a) Artificial image produced using Gaussian blur b) Generated Fourier spectrum c) Generated (0,1) scatter image d) Generated (1,0) scatter image In order to test how higher noise levels affect scatter images and the generated Fourier spectrum, the noise level in the artificial images was then increased to 10% in just the image and not the grid. The resulting artificial image, Fourier spectrum, and scatter images are shown in Figure B.5. Additionally, another set of artificial grid, object, Fourier spectrum, and scatter images was generated where 5% noise was added to both the artificial grid and artificial image (Figure B.6). 135 a) b) c) d) e) Figure B.6: Artificial images with 5% salt and pepper noise introduced in artificial image and in artificial grid a) Artificial grid image b) Artificial image produced using Gaussian blur c) Generated (0,1) scatter image d) Generated (1,0) scatter image e) Generated Fourier spectrum Increasing the level of noise and adding noise to both the artificial grid and artificial image does not significantly affect the Fourier spectra nor the scatter images 136 when compared to the Fourier spectrum and scatter images for 5% noise in just the object image (Figure B.4, Figure B.5, and Figure B.6). The Fourier spectra with produced from artificial images with noise are all missing the higher order harmonics. The scatter images are also speckled though the speckle pattern, location, and intensities are varied for all the different levels of noise. The creation of artificial images provided insight into how noise affects scatter images by allowing for the first time to generate a perfectly noiseless scatter image. Then by adding in noise the difference between noiseless scatter images and Fourier spectra and scatter images and Fourier spectra with noise could be characterized. In conclusion the most important determinant of how the scatter images will look is the addition of noise rather than the level of noise itself. Going from a noiseless image to an image with noise causes the greatest change in the scatter images and Fourier spectra whereas increasing the level of noise has less of an impact on the appearance of the scatter images and Fourier spectra. Thus noise reduction attempts can only minimally improve the appearance of scatter images. B.3 Fourier Transform Region of Interest Sizes In the Fourier analysis process, a region of interest is selected. This is the region of the image that is Fourier transformed and then subsequently inversely Fourier transformed. Outside this region of interest, the image is padded with zeros. This padding was done to reduce the amount of image information that needed to be Fourier transformed and inversely Fourier transformed in an attempt to reduce the amount of noise in the generated images. As a consequence, the question arose as to 137 how the size of the region of interest actually affects the generated images and whether having as small of a region of interest as possible would reduce the noise in generated images. In order to test the effect of changing the region of interest size, a set of battery images were Fourier transformed and inverse Fourier transformed with different sized region of interests. For the results shown in Table B.1, for each image a region was selected so that the average scatter intensity and standard deviation of the average scatter intensity could be calculated. The size and location of the average scatter intensity region was the same for all the tested region of interest sizes. The average scatter intensity does not significantly vary when the size of the region is varied which is a good control because the average scatter intensity should not change given that the same region is used for all the average scatter calculations. Also, if the size of the region of interest changed the average scatter intensity value, then the appropriateness of utilizing a region of interest would have to be investigated. Although the average scatter intensity does not change with region of interest size, the standard deviation of the average scatter intensity does increase as the region of interest size becomes larger. This demonstrates that the region of interest size does in fact affect the noise present in the generated images. 138 Region of Interest Size (pixels) Average Scatter Intensity Standard Deviation 650 x 650 1.679 0.267 890 x 890 1.680 0.310 900 x 900 1.680 0.310 910 x 910 1.680 0.313 1000 x 1000 1.680 0.327 Table B.1: Table showing six region of interest sizes and their resulting average scatters and standard deviations of their average scatter In order to understand the role of the region of interest size on the Fourier transform itself, the Fourier spectrum must be examined. The Fourier spectra and cross sections of the Fourier spectra are shown in Figure B.7 for the following regions of interest: 650 x 650, 900 x 900, and 1000 x 1000. These regions of interest sizes were selected from all the regions of interest shown in Table B.1 because they included the smallest and the largest regions tested as well as the size that has most commonly been used in SFHX image processing, 900 x 900. There is a distinct difference between the Fourier spectrum for the 650 x 650 region and the other two regions of interest sizes. The 650 x 650 region Fourier spectrum lacks the higher order harmonics that are present in the other spectra. Another surprising result is that the Fourier spectra for the 900 x 900 and 1000 x 1000 are less noisy than the 650 x 650 Fourier spectrum; this can be most easily seen in the cross sections of the Fourier spectra. The presence of the higher order harmonics as well as the less noisy Fourier spectra seem to indicate that larger regions of interest are 139 preferable at least with regards to the Fourier transform and inverse Fourier transform processing steps. a) b) c) Figure B.7: Fourier spectra and cross sections of Fourier spectra for the following regions of interest a) 650 x 650 b) 900 x 900 c) 1000 x 1000 Initially, the noise in the Fourier spectra was thought to correspond to the noise in the generated scatter images. As this is not the case, it is of higher priority to reduce the noise present in the scatter images rather than in the Fourier spectrum. Figure B.8 shows the generated scatter images and corresponding cross sections for the 650 x 650, 140 900 x 900, and 1000 x 1000 regions of interest. Although the amount of noise present in the three images is quite similar, the noise in the scatter image from the 650 x 650 region of interest is less than that of the other two regions of interest. a) b) c) Figure B.8: Scatter images and cross sections of scatter images for the following regions of interest a) 650 x 650 b) 900 x 900 c) 1000 x 1000 141 The cross sections of the scatter images indicate that a smaller region of interest size is favorable, which supports the initial hypothesis that instigated this investigation (Figure B.8). However, the reduction of noise in the Fourier spectra for the larger region of interest sizes would suggest that a larger region of interest size is advantageous (Figure B.7). Subsequently this investigation has not provided a decisive result as to what is the optimal region of interest size. Given the mixed results, the region of interest size of 900 x 900 pixels has continued to be used as the region of interest size for image processing particularly because it allows for a scatter image of an entire cell to be generated. The question that remains from this region of interest work is why a larger of region of interest results in a less noisy Fourier spectrum but a noisier scatter image. B.4 Fourier Transform Region of Interest Shape In addition to the size of the region of interest, the shape and location of the region of interest and its impact on the generated Fourier spectra were investigated. Square and rectangular regions of interests were tested because these shapes could easily be generated using MATLAB, which is the programming language currently used to process images. Circular regions of interest could be created in ImageJ, but this would require an additional manual step in image processing. Not only were the shapes of the regions of interest varied, but also their location in the image itself was altered. For determining the optimal shape and location of the region of interest, the cross sections of the generated Fourier spectra were compared in both the horizontal and vertical directions. 142 For a region of interest that is a square centered in the image, the cross sections of the Fourier spectrum are shown in Figure B.9. The most notable feature is that the cross sections of the Fourier spectrum are identical for the horizontal and vertical harmonics indicating that a centered square region of interest results in a perfectly symmetric Fourier spectrum. a) b) Figure B.9: Cross sections of Fourier spectrum for a centered square region of interest a) Horizontal harmonics b) Vertical Harmonics Next a horizontally off-centered square region of interest was tested and the resulting cross sections are shown in Figure B.10. The cross sections for the horizontal and vertical harmonics are once again practically identical. Also, a vertically off-centered square region of interest was also tested, and the horizontal and vertical harmonics were also identical. This indicates that when using a square region of interest, the location of the region of interest within the image itself does not affect the generation of the Fourier spectrum. 143 a) b) Figure B.10: Cross sections of Fourier spectrum for a horizontally off-centered square region of interest a) Horizontal harmonics b) Vertical Harmonics Although square regions of interest are often used in SFHX image processing, on occasion a rectangular region of interest is favorable due to the shape of the object being imaged. Consequently, a centered horizontal rectangular region of interest was tested and the resulting cross sections are displayed in Figure B.11. There is a significant difference between the cross section of the horizontal and vertical harmonics when a rectangle is used. Because the rectangle was horizontal, the vertical harmonics are most sensitive the horizontality of the rectangle and thus exhibit a different cross section than the horizontal harmonic. A centered vertical rectangle region of interest was also tested and the horizontal harmonic was affected, confirming the result that the directionality of the rectangle does in fact impact the symmetry of the Fourier spectrum. This result is important because SFHXI can be used to test for the anisotropy of the object being imaged by comparing the scattering signal from the vertical harmonic to that from the horizontal harmonic. The effect that a rectangular region of interest has on the Fourier spectrum demonstrates that if anisotropic probing is being done then a square region of interest needs to be utilized so that a shape of the region of interest does not artificially impact the resulting scatter signal. 144 Additionally, a region of interest consisting of a not centered rectangle was tested. Not centering the rectangle did not significantly change the cross section of the Fourier spectrum when compared to the Fourier spectrum resulting from centering the rectangle. Thus centering a square region of interest has a greater impact on the generation of the Fourier spectrum than does centering a rectangle. a) b) Figure B.11: Cross sections of Fourier spectrum for a centered horizontal rectangle region of interest a) Horizontal harmonics b) Vertical Harmonics The rectangle used for the region of interest in Figure B.11 did not extend horizontally to the edge of the image. If the rectangular region of interest is extended to the edge of the image, then the Fourier spectrum becomes symmetric (Figure B.12). Although the reason for this result is not entirely understood, it suggests that boundary conditions do play an important role in the generation of the Fourier spectrum when an asymmetrical shape, such as a rectangle, is being used for the region of interest. When the region of interest extends to the edge of the image what is changed is the boundary conditions for the Fourier transform and subsequent inverse Fourier transform. As a result of the work presented in the sections B.3 and B.4 of this appendix, it has been concluded that the optimal region of interest for generating scatter images is a rather large centered square. 145 a) b) Figure B.12: Cross sections of Fourier spectrum for a centered horizontal rectangle region of interest that extends across entire image a) Horizontal harmonics b) Vertical Harmonics B.5 Over Exposing Images When imaging batteries, the batteries absorb a large amount of X-rays due to their stainless steel casing. This large absorbance consequently reduces the amount of photons transmitted through the cells. Because only a small subset of transmitted photons are scattered, the increased absorption of photons results in a reduced scatter intensity. When a flatfield with an exposure time of 20 s is taken, the average intensity is 3610 counts, but when a cell image with an exposure time of 20 s is taken, the average intensity where the cell is located is 957 counts. The transmitted intensity is less than a third of the initial intensity of the flatfield image, which results in a significant decrease in the scatter signal. However, as discussed in Appendix A, the detector saturates when an exposure time beyond 20 s is used. The presence of the cell in the viewing field does reduce the effective exposure time that the detector “sees” and subsequently would prevent the region of the detector where the cell is located from being saturated for exposure times beyond 20 s. However, the scatter processing requires the acquisition of grid and 146 flatfield images with the same exposure time as the object image. Thus even though an exposure time beyond 20 s could be used on an image with a cell in the viewing field, a problem arises in the generation of scatter images. In order to increase the scatter signal and circumvent saturating the detector, an over exposure of just the object image was tested. The over exposure times of 22 s, 25 s, 27 s, and 30 s were tested. In this appendix the representative over exposure times of 25 s and 30 s are shown. A 20 s exposure time was still used for the background, flatfield, and grid images to avoid saturating the detector. Also, as discussed in Appendix A, an exposure time of 20 s is closer to the effective exposure time that the detector “sees” when the cell is in the viewing field and should subsequently provide an adequate correction of the fixed pattern noise. The flatfield correction was used when generating the scatter images resulting from over exposed object images. The Fourier spectra resulting from a standard 20 s object image and over exposed 25 s and 30 s object images are shown in Figure B.13. The 20 s Fourier spectrum shows clear first order harmonic peaks. However, the first order harmonic peaks for the over exposed 25 s and 30 s object images are much smaller in magnitude. Their signal to noise ratio is much smaller than that of the 20 s spectrum. Thus over exposing may not be an appropriate method to increase scatter signal. To confirm the ineffectiveness of over exposing, the generated scatter images are shown below in Figure B.14. Ironically, the over exposure actually resulted in a decrease in the average scatter intensity. The average scatter intensity resulting from a 147 20 s exposure was 2.22, but the average scatter intensities resulting from the 25 s over exposure and 30 s over exposure were 1.87 and 2.15 respectively. The decreased scatter intensities show conclusively that over exposing, at least while using the flatfield correction, are ineffective at increasing the scatter intensity. a) b) c) Figure B.13: Fourier spectra and cross sections of Fourier spectra for the following exposure times a) 20 s with no over exposure b) 25 s with over exposure c) 30 s with over exposure Also of note are the differences in the noise levels in the scatter images for the three different exposure times. Given the previous discussion of the sources of noise in 148 scatter images in this appendix and fixed pattern noise correction in Appendix A, these differences in noise are most likely due to the varying effectiveness of the flatfield correction in correcting each of these different exposure times. The different exposure times result in different fixed pattern noise. If the fixed pattern noise is not properly corrected, then the resulting images from the different exposure times will all have different somewhat corrected fixed pattern noise, which would likely account for the noise variations among the scatter images (Figure B.14.). a) b) c) Figure B.14: Scatter images and cross sections of scatter images for the following exposure times a) 20 s with no over exposure b) 25 s with over exposure c) 30 s with over exposure 149 Initially, over exposing the object image seemed like a potential solution to the problem of high absorption of X-rays by batteries. However, the lowered signal-to-noise ratio in the Fourier spectra, the decreased scatter intensities, and inconsistencies in scatter intensity noise demonstrate that over exposing object images is detrimental to improving scatter signal intensity. Thus with the X-ray tube being used in the current experimental setup, the greatest scatter intensity signal that can be generated is with a 20 s exposure. These findings also indicate that materials that are any more absorbing than Li-ion cells cannot be used for SFHXI using the current experimental setup. B.6 Conclusion Due to the complexities of SFHX image processing, there is not a straightforward solution to reduce noise in the generated scattered images. This is especially evident in that the introduction of just 5% noise in an image creates the speckled scatter images that have become synonymous with SFHXI. Fixed pattern noise from the detector that has not been sufficiently corrected or inhomogeneities in the material being imaged could alone produce variations in the image intensity that could be greater than this 5% noise threshold. These are just two examples of where noise could be introduced into an image, but given the overall complexity of the experimental setup small amounts of noise could easily be introduced throughout the entire process. Thus it is impossible to reduce the noise level in captured images to a point where the speckles that are present in the scatter images would be eliminated. Even though the speckles in scatter images cannot practically be eliminated as with any experimental technique, overall noise reduction is still a priority. The size, 150 shape, and location of the region of interest were found to impact the generation of the Fourier spectrum and subsequently the scatter images. Square shaped regions of interest that are centered in the image and that are not too small in size were found to be optimal. This type of region of interest results in Fourier spectra with first order harmonic peaks that were symmetric in the horizontal and vertical direction and have good signal to noise ratios. Also, although heavily absorbing objects such as Li-ion cells reduce the scatter intensity signal, over exposing the object image only results in a reduced scatter intensity signal and subsequently does not improve the signal to noise ratio. Using the current experimental setup, the only viable way to use over exposed object images is to improve the fixed pattern correction process. Although this appendix does not offer a panacea to the noise present in scatter images, it does provide an insight into how noise affects the appearance of scatter images and the optimal experimental conditions to utilize in order to reduce noise in scatter images. It also points to several questions that warrant further investigation. Why does noise in Fourier spectrum and noise in scatter images not appear to be correlated? How do boundary conditions affect the Fourier transforms performed in image processing? Is there a better way to perform a type of over exposure to increase scatter signal? Though with any future work done to reduce noise in scatter images, it should be kept in mind that the only way to entirely eliminating speckles in scatter images is to have no noise in collected images. 151 References (1) ImageJ User Guide - IJ 1.46r https://imagej.nih.gov/ij/docs/guide/146.html (accessed Mar 15, 2019). 152 APPENDIX C MATLAB CODING C.1 Introduction In order to practically handle the large data sets that were generated by SFHXI, scripts are needed to expedite various aspects of the data analysis process. This appendix is a catalogue of some of the various scripts that were written and utilized for the work presented in thesis. A brief summary of what each of the scripts does is included as well as parameters and inputs that are needed for each of the scripts. C.2 Scatter Data Analysis Scripts Several MATLAB scripts were written in order to provide new ways to analyze the scatter images. The handling of large image data sets has uncovered some of the quirks of MATLAB that had to be addressed in the scripts in this section. If future MATLAB scripts are written for data analysis, then these quirks need to also be addressed in those scripts as well. The first quirk is that given the size of images generated from the current imaging setup, 1000 x 1024 pixels, MATLAB cannot store more than 1500 of these images in a single matrix. If more than 1500 images need to be analyzed, then they must be split in to separate matrices. Even with this precaution, it is 153 still necessary to clear variables pertaining to these image matrices as frequently as possible to avoid MATLAB running out of memory and crashing. The second is quirk is that when MATLAB converts a matrix of pixel values into a surface the values get reflected over the x-axis (resulting in the surface being upside down with respect to the image). The flipping seems to originate from matrices being read from the upper left corner and from the xy-coordinate system being read from bottom left corner. This means that the surface is oriented properly from left to right but not top to bottom. The flipping becomes a problem when an image, which is read in as a matrix in MATLAB, is converted into a surface. Subsequently a correction has been added to the scripts that generate surfaces in order to orient the images correctly with respect to the original scatter images. C.2.1 Scatter Movie Script One of SFHXI’s unique capabilities is being able to image the entire electrode of a cell in a single exposure and being able to produce a large number of images during the cycling of the cells. Both of these features lend themselves to the production of scatter movies, which show the evolution of the scatter intensity of the electrodes as the cells are cycling. In order to generate such movies, a MATLAB script was written. Several versions of the scatter movie script were written depending on the size of the data set and whether the user would like to average over several images. The most straightforward version of the script is “Make_video_no_average_8_21_18.m”. This script can be used for small data sets (data sets with less than 1500 Scatter Images) and when averaging over several images is not desirable. As a note whether image 154 averaging should occur depends on the speed of cycling, which subsequently affects the frequency of data points. The decision of whether to average must be made on an experiment-by-experiment basis. The histogram script requires as inputs a text file with the scatter image names and the scatter images, both of which are produced by the SFHXI processing script, as well as a CSV file with the times and voltages corresponding to the scatter images. The script reads in the scatter images and then normalizes the scatter images by the first scatter image. Next the script generates frames of the movie that show the scatter image for a particular frame and the voltage that corresponds with the scatter image. Then the frames are compiled into a GIF. A file format of GIF was selected because this was the easiest file format to use for movie generation in MATLAB. C.2.2 Scatter Histogram Script In order to quantify the inhomogeneity of the scatter intensity on the electrode surface, a MATLAB script was written in order to produce histograms of the scatter intensity. Like the scatter movie script, there are several versions of the histogram script that vary in the number of scatter images that will be read into the script. The script “Histogram_of_Scatter_Intensity_8_22_18.m” can be used for data sets with more than 1000 initial scatter images. This script requires as inputs a text file with the scatter image names and the scatter images, both of which are produced by the SFHXI processing script, as well as a CSV file with the times and voltages corresponding to the scatter images. The starting image, the number of images that will be averaged over, and the size of the region of interest can all be adjusted. 155 The histogram creates scatter images normalized by the initial scatter image prior to creating histograms. Then either histogram frames can be generated of a movie of the histogram evolution with the corresponding voltages. The histogram movie will be in a GIF format. C.3 Scatter Intensity Modeling Scripts C.3.1 Model of Scatter Intensity of Silicon Nanoparticles In order to do more complex modeling of the lithiation of silicon nanoparticles inside of cells, it was important to first develop a model of the scatter intensity of pristine Si nanoparticles. The scatter amplitude as a function of angle for a spherical particle can be modeled by Equation C.1 where q=(4πSin(θ))/λ, d is the diameter of the spherical particle in meters, λ is the average wavelength of the X-rays emitted in meters, ρe is the electron density of the particle in electrons/m3, and θ is the scattering angle in radians.1 Equation C.1: Scatter amplitude as a function of angle and diameter for a spherical particle Our experimental setup measures the integral under the curve, A2, over the angular range from θmin to θmax for which our experimental setup is sensitive, Equation C.2. The X-ray scatter intensity, I, increases with the square of the NPs’ electron density and with approximately the square of the NPs’ diameter. 156 Equation C.2: Scatter intensity as a function of angle and diameter for a spherical particle Using SpekCalc2,3,4 we calculated the X-ray emission spectrum of our X-ray tube and estimated an average X-ray wavelength of 42.7 pm for our system. The minimum scattering angle that our imaging system can detect was determined to be v, which corresponds to the number of pixels on the detector that a grid line pair occupies. The maximum detection angle for our system was calculated to be 0.042 radians, which corresponds to half the number of pixels on our detector. The electron density of crystalline Si is 6.86 x 1023 electrons/cm3.5 This model and these parameters were written into a script, “Calculated_Scatter_Intensity_of_Si_NPs.m” in order to calculate the theoretical scatter intensity that would be produced by different sized Si nanoparticles. The variables that this script utilizes are the radius of the nanoparticles r, the wavelength of the X-ray lambda, the minimum scatter angle theta_min, the maximum scatter angle theta_max, and the electron density of crystalline Si ed_Si. All these variables can be changed to account for different sizes of nanoparticles, different wavelengths of X-rays, different scatter angles, and the electron density of different materials. C.3.2 Model of Scatter Intensity of Core-Shell Silicon Nanoparticles A script entitled “Core_Shell_Nanoparticle_Scatter_Intensity.m” was written to model the scatter intensity of the core-shell nanoparticles that form during the lithiation 157 of the Si nanoparticles. For a further discussion of core-shell particles see Chapter 2. This script builds off the modeling performed in the script “Calculated_Scatter_Intensity_of_Si_NPs.m”. However, the core-shell script is unique in that it accounts for the change in density of the outer shell of the Si nanoparticles during the lithiation process as well as the inner core remaining pristine. Also, this script accounts for three different possible media surrounding the core-shell particles, which include pure carbon black, pure electrolyte, and a mixture of carbon black and electrolyte. The model that the script utilizes is given below in Equation C.3 where d is the diameter of the inner core of the particle in meters, D is the diameter of the entire particle in meters, ρLi13Si4 is the electron density of the Li13Si4 phase in electrons/m3, ρSi is the electron density of crystalline Si in electrons/m3, and ρmed is the electron density of the medium in electrons/m3. with Equation C.3: Scatter intensity as a function of angle and diameter for spherical core shell particles 158 The capacity is the variable that should be changed each time the script is run in order to indicate how far the lithiation has progressed. This input capacity is the entire capacity of the cell. Also, as a note, be careful when inputting the capacity value as to not exceed the capacity of the nanoparticles. The lithiated phase of the shell is coded to be Li13Si4. When inputting the capacity, it is important to not over-lithiate the nanoparticles beyond a capacity corresponding to a fully lithiated particle nanoparticles of Li13Si4 otherwise an error will result. The lithiated phase can be adjusted; however, the number of Li+ that correspond to each Si atom in the lithiated phase would also need to changed along with the electron density of the lithiated phase. Although this script was initially written for Si nanoparticles with a 50 nm radius, the radius of the particle can be modified for initial particles of a different size. Though by adjusting the initial size of the particle this will change the maximum number of lithium ions that the nanoparticles can uptake to reach the Li13Si4 phase without over- lithiating. The electrolyte mixture is defined as being a 1:1 ratio by mass of ethylene carbonate and diethyl carbonate. A different electrolyte mixture could be utilized; however, a new electron density value for the electrolyte would need to be calculated prior to running the script. The mixed surrounding medium was defined as an average of the electron density of the carbon black and the electrolyte. The mixed surrounding medium can be adjusted to be a weighted average of the electron density of carbon black and the electrolyte. 159 The current angular range that the integral in Equation C.3 is integrated over is from 7.5x10-4 radians to 0.042 radians. If the magnification of the imaging system is in any way adjusted or if the stainless steel grid is swapped for a grid with a different gauge and spacing of grid lines then the new angular range of the imaging systems needs to be calculated. C.3.3 Model of Scatter Intensity of Silicon Nanoparticles with SEI The script “Nanoparticle_with_SEI_Scatter_Intensity.m” is based off the script “Core_Shell_Nanoparticle_Scatter_Intensity.m”, and was written in order to model the changes in scatter intensity that would result from different thicknesses of SEI. This script is much simpler than the previous core-shell script because it assumes that the cell is at a voltage above the potential at which crystalline Si lithiates. Thus the only change the Si nanoparticles undergo is the formation of SEI. Subsequently, the script requires an input for the range of SEI thicknesses and then calculates the scatter intensity for each thickness value in that range. The script currently utilizes an electron density value for the SEI from Cao and coworkers paper which found the electron density value of the SEI for a cell with a 1:1 ratio by mass of ethylene carbonate and dimethyl carbonate with 1 Molar LiPF6.6 Although this SEI electron density value is hard coded into the script, it could be modified if a different electrolyte system was utilized. However, at the time that this appendix is written there are very few electron density values for SEI that can be found in literature. In this script the particle encased in SEI is surrounded by a medium consisting of an average of the electron density of CB and the electrolyte. 160 C.4 Fixed Pattern Noise Correction Scripts For the motivation behind and a detailed explanation of the fixed pattern noise correction work see Appendix A. For the generation of the Flattener with a second order polynomial flatfield fit the following scripts were used. The “Background_y_intercept.m” script fits background images of different exposure times to a second order polynomial. This script requires 20 background images of each of the following exposure times 5 s, 7 s, 10 s, 15 s, 20 s, 25 s, and 30 s. Then the third fit term, the y-intercept of the polynomial, is deemed the Background Offset; it is saved as a MATLAB object, “OffsetCorrection”, so that it can be used by the “Flatfield_Minus_Offset.m” script. The “FlatfieldOffset.m” script fits flatfields of different exposure times to a second order polynomial. The script requires twenty flatfield images with each of the following exposure times 10 s, 15 s, and 20 s. The third fit term of the fit, the y- intercept of the polynomial, is deemed the Flatfield Offset; it is saved as a MATLAB object, “OffsetCorrectionFlatfield”, so that it can be used by the “Flatfield_Minus_Offset.m” script. The “Flatfield_Minus_Offset.m” script then reads in the “OffsetCorrectionFlatfield” and the “Offset Correction” to utilize them in the Flattener given by Equation A.3 in Appendix A. This script also requires as inputs twenty background images and flatfield images with the exposure time of 20 s in order to generate the Flattener correction. 161 C.5 Script to Create Artificial Images In order to investigate the origin of noise in the generated scatter images, it was necessary to generate scatter images from entirely noiseless images. The only way this could be achieved was through the creation of artificial image. A MATLAB script was written entitled “Artificial_Grid_Image.m” in order to produce a noiseless grid image with a perfectly spaced grid lines. This script utilizes multiplies the absolute value of two sinusoidal functions in order to create a grid image. The periodicity of the grid can be adjusted using the parameter m. This script produces a 1024 by 1024 pixel artificial grid Tiff. This image can then be used as an artificial grid image or modified to be an artificial object image. References (1) Guinier, A. Théorie et Technique de La Radiocristallographie; W.H. Freeman, 1963. (2) Poludniowski, G.; Landry, G.; DeBlois, F.; Evans, P. M.; Verhaegen, F. SpekCalc : A Program to Calculate Photon Spectra from Tungsten Anode x-Ray Tubes. Phys. Med. Biol. 2009, 54 (19), N433. https://doi.org/10.1088/0031-9155/54/19/N01. (3) Poludniowski, G. G.; Evans, P. M. Calculation of X-Ray Spectra Emerging from an x- Ray Tube. Part I. Electron Penetration Characteristics in x-Ray Targets. Med. Phys. 2007, 34 (6Part1), 2164–2174. https://doi.org/10.1118/1.2734725. (4) Poludniowski, G. G. Calculation of X-Ray Spectra Emerging from an x-Ray Tube. Part II. X-Ray Production and Filtration in x-Ray Targets. Med. Phys. 2007, 34 (6Part1), 2175–2186. https://doi.org/10.1118/1.2734726. 162 (5) Braga, M. H.; Dębski, A.; Gąsior, W. Li–Si Phase Diagram: Enthalpy of Mixing, Thermodynamic Stability, and Coherent Assessment. J. Alloys Compd. 2014, 616, 581–593. https://doi.org/10.1016/j.jallcom.2014.06.212. (6) Cao, C.; Steinrück, H.-G.; Shyam, B.; Toney, M. F. The Atomic Scale Electrochemical Lithiation and Delithiation Process of Silicon. Adv. Mater. Interfaces 2017, 4 (22), 1700771. https://doi.org/10.1002/admi.201700771. 163