Collective Behavior at the Interface of Lithium-Ion Batteries under Cyclic Lithiation By Hsiao-Mei Wu B. Sc., National Taiwan University, 2004 M. Sc., University of Illinois at Urbana-Champaign, 2006 M. Sc., Brown University, 2009 Thesis Submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in the School of Engineering at Brown University PROVIDENCE, RHODE ISLAND MAY 2014 © Copyright 2014 by Hsiao-Mei Wu This dissertation by Hsiao-Mei Wu is accepted in its present form by the School of Engineering as satisfying the dissertation requirement for the degree of Doctor of Philosophy.       Date   Kyung-Suk Kim, Advisor         Recommended to the Graduate Council         Date   Pradeep R. Guduru, Reader         Date   Brian W. Sheldon, Reader         Approved by the Graduate Council       Date   Peter M. Weber, Dean of the Graduate School                         iii Curriculum Vitae Hsiao-Mei Wu was born on December 27, 1981, at Taipei, Taiwan, Republic of China. She attended National Taiwan University in 2000 and got her B. Sc. degree in Civil Engineering in 2004. Later she completed her M. Sc. in Civil and Environmental Engineering at University of Illinois at Urbana-Champaign in December 2006. She entered the Mechanics of Solids program at Brown University in 2008 and was awarded an M. Sc. in 2009. Publications Wu, H.-M., Yi, J. W., Moon, M.-W., & Kim, K.-S. (2011). Nanobearings in Nature. Poster, Future Directions in Mechanics Research, NSF Workshop and Symposium in honor of Professor L. B. Freund. Wu, H.-M., Tokranov, A., Xiao, X., Qi, Y., Verbrugge, M. W., Sheldon, B. W., & Kim, K.-S. (2014). Characterization of Interfacial Sliding Properties at Amorphous Si/Cu Interface due to Li-ion Intercalation by Using Self-Adjusting Liquid Linnik Interferometer. Extended Abstract, 17th U.S. National Congress on Theoretical and Applied Mechanics. iv Acknowledgments I would like to express the deepest appreciation to my advisor Professor Kyung-Suk Kim, who inspired me the beauty of experiment, taught me the importance of uniqueness, and cultivated me the persistence of chasing principles. Without his guidance and continually help, this research work would not have been possible. I am grateful to Professor Pradeep Guduru and Professor Brain Sheldon for their insightful comments leading to significant improvement in the quality of this work. Thank all other faculty members and staffs at School of Engineering for providing me a robust and warm academic environment. I would like to thank the research scientists at the General Motors R&D center, especially Dr. Yue Qi, Dr. Xingcheng Xiao, Dr. Mark Verbrugge, and Dr. Qian Lin for giving me the suggestions and materials to complete my studies. I am indebted to the funding support from General Motors through the GM/Brown collaborative research lab. I also thank my fellow lab mates in Nano and Micromechanics Laboratory, Moon-Hyun Cha, Huck Beng Chew, Mazen Diab, Hyun-Gyu Kim, Sang-Pil Kim, Jahn Torres, Shuman Xia, Jin Woo Yi, Cheng Zhang, Ruike Zhao, and my fellow graduate students, Jay Sheth, Maria Stournara, Anton Tokranov, Ravi Kumar, Xin Yi, Teng Zhang, for their friendships and helpful assistances. Moreover, I thank to my friends in Brown Taiwanese Graduate Student Association who made my life at Brown enjoyable. Last but not the least, I would like to express my gratitude to my parents, my brother, and my husband Dr. Chien-Kai Wang for always supporting and encouraging me. v Table of Contents Curriculum Vitae ............................................................................................................... iv  Acknowledgments............................................................................................................... v  List of Tables ..................................................................................................................... ix  List of Figures ..................................................................................................................... x  Chapter 1.  Introduction ....................................................................................................... 1  1.1  Background of the Li-Ion Battery ............................................................................ 1  1.2  Mechanical Issues of the Li-Ion Battery .................................................................. 5  1.3  Outline of Present Work ......................................................................................... 10  Chapter 2.  In Situ Optical Measurement of Displacement Fields of Si Thin Film Anode Using Self-Adjusting Liquid Linnik Interferometer (SALLI) .......................................... 13  2.1  Introduction ............................................................................................................ 13  2.2  Introduction to Interference Microscopes .............................................................. 16  2.2.1  The principle of Michelson interferometer ................................................... 16  2.2.2  Summary of scanning white-light interferometers (SWLIs) ......................... 18  2.2.3  Limitations of existing SWLIs ...................................................................... 20  2.3  Principles of SALLI ............................................................................................... 23  vi 2.4  Algorithms for SALLI ............................................................................................ 25  2.4.1  Detection of the edge of the sample .............................................................. 26  2.4.2  Computation of phase map ............................................................................ 27  2.4.3  Noise reduction technique ............................................................................. 31  2.5  Experimental Procedures ........................................................................................ 33  2.5.1  Optical setup of SALLI ................................................................................. 33  2.5.2  Electrochemical half cell preparation and test .............................................. 38  2.6  Results and Discussions ......................................................................................... 40  2.7  Concluding Remarks .............................................................................................. 47  Chapter 3.  Shear Characteristics of a-SiLix//Cu Interface under Electrochemically Active Segregation of Lithium ..................................................................................................... 48  3.1  Introduction ............................................................................................................ 48  3.2  Formulation of Plate Bending Distribution Sensor (PBDS) .................................. 50  3.2.1  Substrate bending in the presence of interfacial sliding ................................ 51  3.2.2  PBDS for experimental data .......................................................................... 54  3.3  Algorithms and Finite Element Model for PBDS .................................................. 56  3.4  Optimization Scheme ............................................................................................. 60  3.5  Results and Discussions ......................................................................................... 62  Chapter 4.  In Situ Observation and Modeling of the Li-Ion Battery Cell Deformation... 72  4.1  Introduction ............................................................................................................ 72  vii 4.2  In Situ Observation of the Non-homogeneous Deformation in a Free Li-Ion Battery Cell through Electronic Speckle Pattern Interferometer (ESPI) ................................... 74  4.2.1  Experimental setup ........................................................................................ 75  4.2.2  Thickness variation during charge and discharge cycles .............................. 78  4.3  In Situ Stress Variation of Confined Li-Ion Battery Cells via Amplified Pressure Distribution Sensor (APDS) ......................................................................................... 85  4.3.1  Experimental setup ........................................................................................ 86  4.3.2  Confinement-pressure variations during charge and discharge cycles ......... 88  4.4  Micromechanical Internal State Model of the Li-Ion Battery Cell ........................ 92  4.4.1  Formation of the model ................................................................................. 92  4.4.2  Finite element analysis of the model ............................................................. 97  4.4.3  Verification of the internal state model with experimental data ................... 99  4.5  Concluding Remarks ............................................................................................ 103  Appendix A: The LG Pouch Li-Ion Battery Cells ...................................................... 105  Appendix B: The Details of Finite Element Model .................................................... 106  Chapter 5.  Conclusions and Future Works ..................................................................... 109  5.1  Concluding Remarks ............................................................................................ 109  5.2  Future Works ........................................................................................................ 112  References ....................................................................................................................... 114  viii List of Tables Table 4.1 A list of the cells tested in the confinement experiment. (*The cell (c) is continuously prestressed and measured after cell (b).) ......................................... 87  ix List of Figures Figure 1.1 Micrographs of Si films on Cu substrate before (first row) and after (2nd and 3rd row) first cycle. The features near and away from the edge of Si films shown in 2nd and 3rd row respectively. (a) Uniaxial and biaxial delamination buckles near and away from the film edge respectively in a 89.1 nm thick Si film. Circular blister buckling shapes are observed at a region distant from the edge after cycling. (b) The notch at the edge became sharper and the film delamination happened at a region away from the edge of a 95.2 nm thick film. (c) Grid fracture patterns along the edge of a 108.7 nm film. Note that the grid patterns could be induced by the photoresist processes while making island markers. ........................................ 8  Figure 1.2 Plan-views of a 52.6 nm thin Si film on a Cu substrate (a) before and (b) after the first cycle. Note that the dotted line and solid line represent the edge of the film for initial and final configurations respectively. The figures show large-scale interfacial sliding of the film on the substrate and no fracture features observed after cycling. ........................................................................................................... 9  Figure 2.1 (a) A schematic of a basic Michelson interferometer. M2’ – the image of M2 in the beam splitter. The interference is considered as superposition of the real reflecting surface M1 and the virtual reflecting surface M2’. (b) Plane parallel mirrors: illustrating the distance between two mirrors, M1 and M2’ (M2) in Michelson interferometer. ..................................................................................... 17  x Figure 2.2 Schematics of three different types of SWLIs: (a) Michelson interferometer (b) Mirau interferometer (c) Linnik interferometer. Parts of the interferometers: MO – microscopic objective lens, BS – beam splitter, O – object, R – reference mirror. ................................................................................................................... 19  Figure 2.3 The scenario with same image path lengths (IPLs) in both lines: (a) optical and image paths in RL, (b) physical (not phase) optical (solid line) and image paths (dashed line) in OL, (c) the optical path lengths (OPLs) for both RL and OL; analogous figures for the case with same OPLs in both lines: (d)-(f). ................. 22  Figure 2.4 The schematic of the principal of self-adjusting liquid Linnik interferometer (SALII). Parts of the interferometer: BS – beam splitter, MO1 & MO2 – microscopic objective lenses, RPS – relative phase shifter, O – object, R – reference mirror. The optical paths inside RPS can be detoured and RPS can be placed between BS and MOs or MOs and CH. ................................................... 24  Figure 2.5 (a) An optical image obtained from SALLI. (b) The intensity distribution of the image. (c) The intensity profile of the AA’ row. Black dots are experimental data, red and blue lines are the best fitting function. Blue circle represents the position of the edge. (d) The optical image with a cyan line representing the edge of the sample. ........................................................................................................ 27  Figure 2.6 (a) A typical interferogram acquired from SALLI after background intensity removal (b) relative phase map (c) distribution of fringe numbers (d) surface profile. Dotted lines in the figures denoted the edge of the sample. The object O is in left region and the reference R is in right region. ............................................. 28  xi Figure 2.7 (a) An interferogram obtained from SALLI after background removal (b) relative phase map (c) surface profile before adjustment (d) surface profile after adjustment. Dotted lines in the figures denoted the edge of the sample. The transition region for adjusting fringe connections is enclosed by two solid lines (white and black)................................................................................................... 30  Figure 2.8 (a) A schematic of the sample design. (b) The corresponding interferogram, a real experimental image acquired from SALLI with 50% transparency. (c) A schematic of the flattened surface profile along A-A’ line. .................................. 32  Figure 2.9 The optical setup of the self-adjusting liquid Linnik interferometer (SALLI).34  Figure 2.10 Expanded side view of the key elements, relative phase shift (RPS), of the SALLI. Only illumination light is shown in the figure; reflected light is too complicated to draw in the same figure. Note that the angles of M4 and M5 were fixed and they were mounted on one stage. .......................................................... 35  Figure 2.11 Optical interference fringes for validation of SALLI((a)-(c)) and for testing the function of RPS ((d)-(f)). The same object was measured by a commercial (air) Linnik-interferometer microscope, (air) SALLI, and (water) SALLI shown in (a)- (c) respectively. ..................................................................................................... 37  Figure 2.12 Schematic illustration of the half cell battery (HCB). Insets are the detailed dimensions of the Si electrode and top view of the sample. ................................. 39  Figure 2.13 The time-dependent in situ SALLI results on a 50 nm Si film during the first cycle. The edge of the film is denoted as dotted line. Note that in the image domain a-Si is on the left and Cu on the right side. Distinct movement of the electrode edge, up to 200 µm, was observed from the fringe images (first column). xii 2D and 3D contours illustrate the out-of-plane deformation field. The color bar is shared for all 2D and 3D contours. ....................................................................... 41  Figure 2.14 The time-dependent in situ SALLI results on a 50 nm Si film during 6.39 − 8.51 hour of the first cycle. Continue from the previous figure. The movement of the electrode edge was approximately 50 µm. The thickness variation is inhomogeneous especially during delithiation cycle. ........................................... 42  Figure 2.15 The time-dependent in situ SALLI results on a 50 nm Si film during the second cycle. The movement of the electrode edge was within 50 µm. It expanded during lithiation (8.5 − 11.73 hour) and shrank back to the bulk during delithiation (11.73 – 14.07 hour). The thickness distribution is non-uniform. ........................ 43  Figure 2.16 The area-average thickness from different average domains during the first two cycles.............................................................................................................. 44  Figure 2.17 The sliding distances in situ measured by SALLI along with the potential history.  es in positive represents the film shrinks in in-plane direction; otherwise, the film expands. ................................................................................................... 45  Figure 3.1 (a) Initial configuration of a thin film on substrate. (b) The configurations of the initial (dashed line) and current (solid line) Si film configurations in the top view. (c) The free body diagram of half of the sliding film. (d) Corresponding shear diagram. (e) The free body diagram of bending moment and thin film force diagram (f). ........................................................................................................... 52  Figure 3.2 Free body diagrams (first column) and the diagrams of the corresponding shear force (second column) for each stage: (a) initial configuration, (b) stage A, (c) stage B, (d) stage C1, and (e) stage D1. A thin film is plotted in gray solid line xiii and its direction of motion is shown by a gray arrow. Notation P and Q represent the position of the film edge and that of the slip front. The character in the superscript denotes the parameter formed in which stage, and is indicates the final size of the slip zone. ..................................................................................... 55  Figure 3.3 (a) Schematic figure of the finite element analysis model. (b) Out-of-plane deformation field from ABAQUS analysis while the sliding distance   250 μm . ............................................................................................................................... 57  Figure 3.4 The polynomial coefficients ci   for expressing displacement field u3 ( s1 , s2 ,  ) obtained from finite element analysis. .............................................. 58  Figure 3.5 Contour plots in the 1st and 3rd rows are the thickness variation obtained from experiment and those in the 2nd and 4th rows are calculated from simulation. Selected figures are from (a) stage A, (b) stage B, (c) stage C1, (d) stage D1, (e) stage C2, and (f) stage D2. .................................................................................... 61  Figure 3.6 The evolution of the zone sizes for interfacial sliding between a 50 nm thin a- Si film and a 200 nm Cu current collector for the first two cycles. ...................... 63  Figure 3.7 The relative thickness variations of the a-Si film and the SEI layer on top of Cu. Inset: A schematic shows the measuring spots and indicates that the thickness measured from SALLI is the relative thickness on top of the Cu film. ................ 65  Figure 3.8 The in situ stress data of a 50 nm Si thin film along with potential history for the first two cycles. ............................................................................................... 67  Figure 3.9 The critical energy release rate for debonding between a-SiLix//Cu interface. Inset: a free body diagram of a thin film with the concentrated stress at the edge area. ....................................................................................................................... 68  xiv Figure 3.10 The interfacial shear strength at a-SiLix//Cu during the first two cycles. Inset: a schematic of the interface segregated by electro-chemically active lithium-ion liquid. .................................................................................................................... 70  Figure 4.1 (a) The optical setup of ESPI system for out-of-plane displacement field measurement. (b) The experimental setup for in situ deformation measurement of a free Li-ion battery cell and the typical interferograms acquired from ESPI. (c) The out-of-plane displacement field post-processed from the interferometer figures. .................................................................................................................. 77  Figure 4.2 The normalized variation of the thickness, area, and volume of the cell during the first cycle (charged in 1C rate and discharge in 2C rate). ............................... 80  Figure 4.3 The contours of the accumulative thickness variation reference to a fresh, free standing cell at the 1st and 3rd charge-discharge cycles. ....................................... 81  Figure 4.4 The contours of the accumulative thickness variation reference to a fresh, free standing cell at the 6th and 12th charge-discharge cycles. ..................................... 82  Figure 4.5 The area-average thickness variation with extreme values for the first charge- discharge cycle (charged in 1C rate and discharge in 2C rate). ............................ 84  Figure 4.6 The experimental setups for in situ stress distribution measurements: (a) under higher constrained pressures. (b) under lower constrained pressures. .................. 87  Figure 4.7 (a) The history and (b) the contours of area-average stress distributions during C-D cycles under the displacement constraint with -13 psi prestressed pressure. 89  Figure 4.8 (a) The area-average stress variation for a brand-new cell (cell (a) in Table 4.1) under high prestressed pressures during C-D cycles. (b) The area-average stress xv for one new cell (cell (a)) and four aged cells (cell (b) − (e) listed in Table 4.1) under low constrained pressures. .......................................................................... 90  Figure 4.9 The schematic of the mechanical internal state model. ................................... 93  Figure 4.10 (a) The schematic of finite element models. In the auxiliary field (sys. B), the undeformed model is free at the top and the cavity between separators is vacant. In the real field (sys. A), the solid is deformed from the undefromed configuration due to initial prestressed and then fixed at the top. The cavity is filled with gas and electrolyte. (b) The stress contour  22 of the deformed solid with -36 psi prestressed pressure. ............................................................................................. 98  Figure 4.11 The simulation results and experimental data of the area-average pressure under (a) high prestressed pressures. (b) low initial prestressed pressures for cell (a) − (d) in Table 4.1. Note that the modeling results without the contact pressure term under prestressed pressure -30psi is shown in (a) for comparison. ............ 100  Figure 4.12 The gas generation rates estimated from internal state model for cell (a) − (d) under low initial prestressed pressures. .............................................................. 101  Figure 4.13 Average capacities for the cells (a), (d), and (e) under different initial prestressed pressures. Inset: A schematic of interface bubble-gas generation (IB- GG) by confinement control. .............................................................................. 102  Figure 4.14 (a) A schematic of the dimensions and components of a LG 1.4 pouch Li-ion battery cell. One mini-cell composes of one anode, cathode, corresponding current collector sheets, and two separators. (b) The typical charge and discharge protocol used in the experiment for a LG pouch battery cell. ............................ 106  Figure 4.15 The flow chart of analyzing the confinement pressure at time  t  t  . .... 107  xvi Chapter 1. INTRODUCTION This thesis presents experimental measurements and modeling of multi-scale collective behaviors characteristics of hierarchical interfaces in lithium-ion batteries (LIBs) during cycling. Two interfacial mechanisms are introduced: one is in-plane sliding between lithiated electrodes and current collectors; the other is normal contact between the internal interfaces of pouch battery cells. Through bridging in situ measurements and numerical models, the multi-scale collective behaviors of LIBs are elaborated. It is hoped that the work carried out in this thesis can contribute to optimal design of new battery cells and maximize cell capacity and life of LIBs. The following sections of this chapter will first review backgrounds of LIBs. Then, briefly introduce the mechanical issues and challenges of current battery technology. Finally, an outline of this thesis will be provided. 1.1 Background of the Li-Ion Battery In modern society, more and more portable electronic devices are invented, so the demands for lighter and miniature energy sources increase. Since Li is the lightest metal 1 (the density is 0.53 g·cm-3) with high positive standard reduction potential (-3.04 V relative to a standard hydrogen electrode), using Li metal as an electrode material has drawn first attention in advanced battery technology (Schalkwijk & Scrosati, 2002). The historical development of lithium metal batteries started in 1970s. Whittingham (1976) made the first lithium battery by using TiS2 as the cathode material and Li metal as the anode electrode. It was widely used in military applications and common electronic devices due to its high capacity. However, the uneven dendritic Li growth on the surface of electrodes during charge-discharge (C-D) cycles was the main drawback of lithium batteries. This dendritic problem resulted in electrical shortage of batteries and further induced explosion hazards (Tarascon & Armand, 2001). In the meanwhile, Steele & Armand (1973) proposed the concept of electrochemical intercalation and suggested insertion compounds as battery electrodes. The major improvement was made by Basu who developed the first lithium intercalated graphite electrode at Bell Labs in 1977. It was found that Li-ion can intercalate into and deintercalate from graphite rapidly and reversibly. Later in 1983, Thackeray et al. identified manganese spinel as a positive electrode with low-cost, good structural stability and electric conductivity. These two discoveries led to significant development of Li-ion batteries technology and provided a substitute to the lithium mental battery. Nowadays, LIBs are considered to be safer batteries than Li-metal cells, since Li presents in ionic state instead of metallic state and the dendrite problem no longer occurs. In 1991, Sony Corporation created the first commercial LIB cell by using graphite as the anode electrode and LiCoO2 as the cathode material (Nagaura & Tozawa, 1990). Till now, LiCoO2 is still used as a common cathode material in modern LIBs. 2 In summary, LIBs with high energy density (225 – 400 W·h·L-1), long cycle life, low self-discharge rate, lightweight design (with specific energy 100 – 175 W·h·kg-1), no memory effect, and no pollution, have been widely used as power supplies for portable electronic devices (Tarascon & Armand, 2001). For several decades, the rapid development of LIB has ensured its dominance of the secondary battery market by replacing traditional lead-acid, nickel cadmium, and nickel-metal hydride batteries (Gomadam et al., 2002). Moreover, in the near future, LIBs have been considered as potential energy sources for electric vehicles (EV) due to environmental pollution and energy shortage problem in the world (Wakihara, 2001). This concern has drawn more and more attentions to study both fundamentals and applications of lithium-ion rechargeable batteries. It can be predicted that LIBs will become the most promising and important chemical energy sources in the 21st century. The basic components of a LIB cell are composed of a cathode and an anode separated by a separator. Both electrodes are soaked in an electrolyte solution containing dissociated salts, which allows ion transfer between them. Here, anode and cathode materials serve as hosts for lithium. While chemical reactions proceed at both electrodes, the materials are not consumed and ideally do not change their structures as lithium ions and electrons are exchanged. During discharge cycles, useful energy comes from electrons moving to lower electrical potential while maintaining charge neutrality (Tarascon & Armand, 2001). Typical electrode redox reactions in the battery are in the following, Cathode oxidation (e.g., lithium cobalt oxide): LiCoO2 ↔ Li1-xCoO2 + xLi+ + xe- Anode reduction (e.g., graphite): xLi+ + xe- + C6 ↔ LiC6 3 As an ideal cathode material, a lithium intercalation compound should react with lithium in a reversible manner (rapid insertion and removal) and with a high free energy of reaction to provide high capacity as well as high voltage. It also needs to be a good electron conductor and a stable material. In a practical point of view, the compound should be cheaper and will not induce environmental pollution. The common cathode materials are layered lithium cobalt oxide (LiCoO2), layered lithium nickel manganese cobalt oxide (LiNi1/3Mn1/3Co1/3O2), spinel lithium manganese oxide (LiMn2O4), and olivine lithium iron phosphate (LiFePO4). In general, layered compounds offer the highest energy density, followed by spinels (slightly less than layered) and olivines (approximately half) (Etacheri et al., 2011; Whittingham, 2004). With the development of LIBs technology, most studies have been focused on graphite, silicon-based, and tin-based materials as anodes. Recently, silicon-carbon composites and other alloys are also considered as potential candidates. Typically, the electrode redox potential of an ideal anode should be as low as possible to provide higher output voltage of a battery. The anode should react with lithium in a reversible manner (rapid insertion and removal) and provide higher capacity density due to more insertion ions. The change of its structure during lithium intercalation and deintercalation needs to be minimal to result in good cycle performance. Besides, the host material must have proper surface configuration for forming a passive layer to protect the electrode. Similar to cathode materials, an ideal anode has to have good conductivity and retain good chemical stability over the entire voltage range (Ji et al., 2011). In a practical point of view, the material should be cheap and environmentally benign. 4 For more than two decades, most of commercial LIB cells have used graphite as an anode-active material. It serves as a reliable host structure for lithium to be easily intercalated and deintercalated. Recently, due to the development of EVs, more and more attentions have been drawn to improve the efficiency of large format pouch cells. The other rising interest is to enhance the performance of LIBs by using Si as an anode material. It is because Si has high theoretical capacity 4200 mAg·g-1 (more than ten times larger than that of graphite) and is the second most abundant element on earth (Kasavajjula et al., 2007). Therefore, the major efforts in this thesis are focused on the collective behaviors of anode materials, particularly emphasizing coated graphite (C) in commercial pouch cells and on nanostructured amorphous silicon (a-Si) anodes in half cells. 1.2 Mechanical Issues of the Li-Ion Battery Although lithium-ion batteries are regarded as the best potential energy sources for electrical vehicles, there are still some remaining difficulties which need to be resolved. The major design considerations for LIBs involve electrochemistry, thermal management and mechanical integrity. The electrochemistry has been widely studied as it directly determines battery performance (e.g., cell potential, capacity, or energy density) and its life cycle (Etacheri et al., 2011; Lai et al., 2014; Wakihara, 2001). Several efforts have been devoted to predict battery performance as a function of its temperature, since battery efficiency strongly depends on its thermal response (Seong Kim et al., 2011). However, it is not the purpose of this thesis to review the enormous amount of previous research progresses in electrochemistry and thermal management. The main goal of this thesis is 5 to understand the collective behaviors of anode materials (especially graphite and a-Si electrodes) and further enhance the performance of LIBs via improving mechanical integrity. For carbonaceous anode materials (graphite, hard carbon, hybrid carbon blends) and alloy anodes (i.e., Si, Sn, Sb, Al, Mg, and Bi) (W.-J. Zhang, 2011), the operation potential of negative electrodes is below the reduction potential of an electrolyte. Therefore, the electrolyte decomposes and reacts with the new surface of an anode to form a thin (several nanometer) film called solid-electrolyte interphase (SEI). The SEI layer is electrically resistive. It allows lithium-ion transports and behaves as a passivating film on the anode surface (Verma et al., 2010). This formation process results in irreversible capacity loss and generates a certain amount of gases as by-products (depending on the compounds of the electrolyte) (Goers et al., 2004; Kim et al., 2011). Although SEI grows mostly during the first cycle, gases buildup on subsequent cycles due to formation of new SEI layers on freshly exposed anode owing to particle fracture. Evolution and migration of these gases could be critical to cell life especially for large format pouch cells used in EV, since they cause the cells tend to swell leading to highly inhomogeneous current distribution. This inhomogeneity can lead to various failure mechanisms such as local overcharging or lithium deposition (Arora et al., 1999; Kostecki et al., 2006). Another main concern for pouch cells is that electrode particles can easily lose contact (particles to particles, particles to current collectors), as there is no strong adhesive between layers of electrode/separator assembly. This electrical disconnection becomes more severe and brings in significant capacity fading and internal resistance increasing (Vetter et al., 2005), while gas bubbles block Li-ion diffusion. 6 A few research groups have found that a cell under compressive stack pressure limited the porosity of the anode in the prismatic cell and enlarged the internal contact areas resulting in lower resistance and higher capacity (Rubino et al., 2001). However, the influence of gases on porosity evolution in the electrodes leading to disintegration of the cells was not considered in these experiments, since the amount of gases accumulated in these tested cells was negligible unlike large format pouch cells for electrical vehicles. The other major challenge for graphite and alloy anodes is large volume expansion and contraction during lithium insertion and extraction. This can cause serious capacity degradation of cells. For instance, the swelling and shrinking of anodes can lead to delamination of electrodes from current collectors, failure of binders, and movement of conductive carbon, all of which reduce connectivity of active materials and cause poor battery mechanical integrity finally resulting in loss of capacity (Vetter et al., 2005). These lithiation and delithiation induced volume expansion and contraction are especially severe in Si anode system. Along with its ultrahigh theoretical capacity, 4200 mAh·g-1 (Boukamp et al., 1981), Si electrodes expand up to nearly 400% of original volume (Beaulieu et al., 2001) during lithiation. This large strain generates enormous stresses and leads to serious mechanical degradation, such as fracture of electrodes, pulverization of Si particles, and isolation of electrode particles from the adjacent materials (Kasavajjula et al., 2007; Maranchi et al., 2003; Xiao et al., 2011). Many studies have been attempted to alleviate these stresses towards optimal design of Si electrodes through both structural and material optimization. However, the capacity fading due to fracture and delamination are still observed in different Si nanostructures, e.g., thin films (Bourderau et al., 1999; Maranchi et al., 2006), nanowires (Liu et al., 2011; Ryu et al., 2011), nanoparticles (Liu 7 Figure 1.1 Micrographs of Si films on Cu substrate before (first row) and after (2nd and 3rd row) first cycle. The features near and away from the edge of Si films shown in 2nd and 3rd row respectively. (a) Uniaxial and biaxial delamination buckles near and away from the film edge respectively in a 89.1 nm thick Si film. Circular blister buckling shapes are observed at a region distant from the edge after cycling. (b) The notch at the edge became sharper and the film delamination happened at a region away from the edge of a 95.2 nm thick film. (c) Grid fracture patterns along the edge of a 108.7 nm film. Note that the grid patterns could be induced by the photoresist processes while making island markers. et al., 2012; McDowell et al., 2013), and Si-C composites (Guo et al., 2005; L. Q. Zhang et al., 2011). Among these studies, an interesting thickness size effect has been reported for the a-Si thin film electrodes. It has been found that 50 nm a-Si thin film anodes on top of 30 µm thick Ni foils gave excellent electrochemical performance (over 3500 mAh·g-1) for 8 Figure 1.2 Plan-views of a 52.6 nm thin Si film on a Cu substrate (a) before and (b) after the first cycle. Note that the dotted line and solid line represent the edge of the film for initial and final configurations respectively. The figures show large-scale interfacial sliding of the film on the substrate and no fracture features observed after cycling. 200 cycles (Ohara et al., 2004). On the other hand, thicker (300 nm) film had very poor available capacity (Takamura et al., 2004). Moreover, a lot of microcracks have been observed from SEM morphology images of a 100 nm a-Si film on Cu after 12 cycles (Xiao et al., 2011) and a 250 nm a-Si film even after first cycle (Maranchi et al., 2006). Similar results were also observed in our experiments reported in this thesis. The micrographs of a-Si films with different thickness on Cu current collectors were obtained from optical microscope after first cycle. The development of telephone cord instability and circular blister buckling shapes (Argon et al., 1989; Gille & Rau, 1984) were observed on the edge and at the bulk of a 89.1 nm Si film (Figure 1.1(a)). A 95.2 nm thin film delaiminated at the regions far from the edge and the notch at the edge became sharper indicating the film was still under compression (Figure 1.1(b)). For a thicker film (108.7 nm), fracture patterns at the regions far from and near the edge of the Si film were shown in Figure 1.1(c). However, no fracture or buckling features of a 52.6 nm thin film 9 was found after first cycle (Figure 1.2) due to the presence of interfacial sliding between the Si electrode and the Cu current collector. Recently, the experiments and theoretical studies by Soni et al. (2011) and Haftbaradaran & Gao (2012) demonstrated that these large stresses can be mitigated due to interfacial sliding between electrodes and nearby current collectors. Although these results indicate that Si films with appropriate size can effectively prevent formation of microcracks and films delamination, the material properties (i.e., interfacial shear strength, energy release rate) at electrode/current collector interfaces during cyclic intercalation are still waiting to be determined. 1.3 Outline of Present Work This thesis presents the experimental observations and simulation models of multi-scale collective behaviors of lithium-ion batteries (LIBs) caused by deformation characteristics of hierarchical interfaces during cycling. This work particularly focuses on the sliding mechanism between anodes (a-Si) and copper current collectors (Cu) and the contact mechanism between internal interfaces of large format pouch cells. As mentioned in the previous section, although the interfacial properties during lithiation and delithiation cycles have been estimated from DFT calculations (Stournara et al., 2013) and a continuum model (Haftbaradaran et al., 2012), direct experimental measurements are still needed to validate the results. Therefore, an experiment for measuring the interfacial properties between a-Si thin film electrodes and Cu current collector is introduced in Chapter 2. Here, a new apparatus, called “self-adjusting liquid Linnik interferometer (SALLI)”, is invented. SALLI overcomes existing limitations of 10 current interference microscopes with a novel optical configuration that naturally self- compensates for any refractive-index change in the liquid medium. Through direct imaging of the specimen, the lateral deformation can be monitored in situ with few micrometers resolution. At the same time, SALLI also permits high precision real-time measurements of thickness variation (resolution of 1 nm). This time-dependent full-field deformation measurement provides critical information about the Li distribution in the Si, and the corresponding deformation. In Chapter 3, a mechanical model system named as “plate bending distribution sensor (PBDS)” which incorporates substrate bending and interfacial sliding in its calibration, is developed to further extract the sliding characteristics of the interface between a film and a substrate from the SALLI experimental data. By bridging the deformation estimated from the model and those measured from SALLI experiment, the interfacial properties between the electrode film and current collector can be extracted quantitatively. In Chapter 4, we perform two sets of in situ experimental tests and develop a mechanical model to explain the internal contact mechanism and its relationship with gas evolution. Through these experiments and the model, the performance of large format pouch cells under different prestressed pressures are elaborated. Non-uniform thickness variations across the whole surface area of a traction-free LG battery is first measured in situ during charge and discharge (C-D) cycles via employing an electronic speckle pattern interferometry (ESPI). Second, in situ confinement-pressure variations across the whole surface area of constrained battery cells are performed using a high resolution amplified pressure distribution sensor (APDS). These measurements are particularly useful for characterizing the cyclic performance of battery cells, understanding of which 11 must be bridged to that of microstructural behavior to improve structural design of battery packing. Finally, the conclusions of the thesis and some comments for the future study on the interfacial behaviors of LIBs are included in Chapter 5. 12 Chapter 2. IN SITU OPTICAL MEASUREMENT OF DISPLACEMENT FIELDS OF SI THIN FILM ANODE USING SELF-ADJUSTING LIQUID LINNIK INTERFEROMETER (SALLI) 2.1 Introduction Rechargeable lithium-ion batteries (LIBs) are used as the most promising power supply of portable devices due to their superior energy density. The demand of LIBs is increasing due to the miniaturization of electronic appliances and the desire to enhance stretchability and flexibility of electronic devices (Bruce et al., 2008; Schalkwijk & Scrosati, 2002; Xu et al., 2013). Recently, LIBs are considered to be the energy sources for electrical vehicles. However, there are still some remaining difficulties which need to be resolved, e.g., electrode failure induced by large volume expansion during cycling and capacity degradation caused by SEI formation. Therefore, a large variety of in situ techniques have been used for studying electrode deformation and SEI growth ranging from spectroscopy to X-ray diffractometry and nanometer-scale microscopy (Verma et al., 2010). For examples, vibration spectroscopes like Fourier transform infrared spectroscopy (FTIR) (Aurbach et al., 1996) and Raman spectroscopy (Novák et al., 2000) provide valuable surface information regarding the functionality. Synchrotron X-Ray 13 diffraction (XRD) (Poizot et al., 2000) and synchrotron X-ray absorption spectroscopy (XAS) (Balasubramanian et al., 2001) give crystal structure information and electronic structure charges respectively. Besides, a fundamental understanding of the morphologies of electrode materials observed from in situ microscopes, e.g., scanning electron microscopy (SEM) (Orsini et al., 1998), transmission electron microscope (TEM) (Huang et al., 2010), and atomic force microscope (AFM) (Aurbach & Cohen, 1996), is also important to mitigate the mechanical failure during cycling. In situ SEM and TEM can provide useful information on morphological evolution in nm resolution, e.g. micro-cracking of particles (Chen et al., 2011) and the evolution of the phase boundaries of electrodes (J. W. Wang et al., 2013). But both of them have to be operated in ultra-high vacuum, and specially designed cells need to be implemented, i.e., only ionic liquid or solid electrolytes (Li2O) can be used as electrolytes. Therefore, complicated sample preparation is required. Moreover, ionic liquid electrolytes (ILE) are highly sensitive to the electron beam (e-beam). It was observed that strong e-beam dose induced degradation of the ILE (Liu & Huang, 2011). This is a challenging issue especially in employing in situ TEM measurement that requires longer exposure time. Overall, the fields of view of both microscopes are small when we observe the morphological evolution with nm resolution, and both techniques can only provide local nanometrology information instead of the whole deformation fields. In situ AFM techniques have also been used to monitor nanometrology in Li-ion battery cells (BalkeN et al., 2010; Becker et al., 2013). Although AFM can provide the whole out-of-plane deformation fields within a 100 µm observing window, this technique still has a few difficulties. Because the probes have to be immersed in liquid electrolytes 14 during scanning, the diffusion to electrode surface may be influenced by the geometry of AFM probes (Burt et al., 2008) and SEI formation on the tip creates significant challenges with the measurements (Sri Devi Kumari et al., 2013). Also, the scanning rate is the other key issue for this technique. At the slow scan rates needed for high resolution, there will be substantial deformation of the material between the first and last line-scans. Increasing the scan rate to capture these changes will limit resolution. Furthermore, soft materials (i.e., SEI) may be scratched by the tip (Domi et al., 2011; Jeong et al., 2001). To observe the sliding phenomena of a thin film electrode (Figure 1.2), it requires an apparatus that has hundreds of micrometers field of view and nanometer resolution in the out-of-plane direction. Thus nondestructive real-time measurements with optical interference microscopes are ideally suited for in situ study of Li-ion battery materials. However, existing optical microscope interferometers have serious limitations for measuring objects in liquid media directly, largely because of mismatch between optical (phase) path length and image path length in the system. In this chapter, we present the development of a new apparatus, named “self-adjusting liquid Linnik interferometer (SALLI)”. SALLI overcomes these difficulties with a novel optical configuration that naturally self-compensates for any refractive-index change in the liquid medium and provides real-time whole field deformation measurement which is essential information for better understanding the morphologies of LIBs during cycling. 15 2.2 Introduction to Interference Microscopes In nanometrology, an optical interferometry is a main tool for observing minute details of the surface structure. Especially, a scanning white-light interferometer (SWLI), a variation of Michelson interferometer, can perform non-contact measurement of surfaces topography or displacements with nanometer resolution within a fraction of a second. It is a suitable technique for in situ measurement of thin film electrode deformation. In this section, the principle of Michelson interferometer is introduced first and three common types of SWLIs are briefly summarized. Finally, the limitations, i.e., the difficulties of measurements in liquid media, with existing SWLIs are elaborated. 2.2.1 The principle of Michelson interferometer Michelson interferometer was invented by Albert A. Michelson in 1887 (Michelson & Morley, 1887). The setup configuration for Michelson interferometer is shown in Figure 2.1(a). The main optical parts consist of two plane mirrors M1 and M2 and one beam splitter BS. Light from an extended source S is divided by BS into two beams at right angles. These are reflected at M1, M2, and return to BS, where they are re-combined to enter the observing screen. In this setup, M1 is mounted on a translation stage which can move it toward or away from BS. Once the two light waves are united, interference pattern can be observed. A general equation for interference between two superimposed waves with the same frequency, but each with arbitrary amplitude and phase gives i  1   t  i  2   t  ET  E1 e  E2 e . (2.1) 16 Figure 2.1 (a) A schematic of a basic Michelson interferometer. M2’ – the image of M2 in the beam splitter. The interference is considered as superposition of the real reflecting surface M1 and the virtual reflecting surface M2’. (b) Plane parallel mirrors: illustrating the distance between two mirrors, M1 and M2’ (M2) in Michelson interferometer. Note that E1 and E 2 are the magnitudes of their respective fields,  the phase,  the angular frequency, and t time. The intensity of the superposed wave is I  ET* ET  E12  E22  2 E1E2 cos  1   2  (2.2)  I1  I 2  2 I1 I 2 cos  1   2  , where  indicates time averaging and I1 , I 2 are the intensities of the individual fields. Here, we assume I1  I 2 , as we use a 50:50 beam splitter. Then, the intensity can be derived as  I  2I1 1  cos    4I1 cos2 . (2.3) 2 17 If    1   2  2m where m  0, 1, 2, ... , then the waves are in phase. It is called constructive interference with maximum intensity I  4 I1 . Likewise, if    2m  1  , then the two waves are out of phase. The intensity of this destructive interference has minimum intensity ( I  0 ). In the Michelson interferometer, optical path difference (OPD) between the two superimposed waves, reference light (RL) and the objective light (OL), is illustrated in Figure 2.1(b) and estimated as   OPD  n AB  BC  n AD  2nd cos  , (2.4) where n is refractive index of the medium, d is the distance between M1 and M2, and  is the angle of incidence which approaches to 0 in the setup. The relative phase difference according to this OPD is     2nd   2  , where  is the wavelength of the light source. While the two light waves with the same frequency and amplitude are in phase, the constructive fringes are observed. Then, the phase difference    2m , and it yields m d . (2.5) 2n Finally, the distance d between M1 and M2 in the medium with refractive index n can be measured by counting m numbers of fringes. 2.2.2 Summary of scanning white-light interferometers (SWLIs) Following three types of SWLIs, i.e., Michelson, Mirau, and Linnik imaging interferometers illustrated in Figure 2.2, are variations of the basic Michelson interferometer in principle. Here, an object O replaces the mirror M1 in Figure 2.1(a). 18 Figure 2.2 Schematics of three different types of SWLIs: (a) Michelson interferometer (b) Mirau interferometer (c) Linnik interferometer. Parts of the interferometers: MO – microscopic objective lens, BS – beam splitter, O – object, R – reference mirror. Then, the measuring quantity d in equation (2.5) represents the relative distance from the detector to the object O and the reference mirror R (original M2 in Figure 2.1(a)). In any case, the accuracy of length or wavelength measurement depends on how accurate one can determine the fringe structure and position. Therefore, to obtain a high lateral resolution, microscope objectives are used. Besides, to minimize the optical paths in the interferometer arms, a miniaturized interferometer is generally assembled inside the microscope objective (Malacara, 2007). The microscopic interferometer with Michelson setup (Figure 2.2(a)) is designed for the measurement of surfaces with large fields of views. The magnification used in this interferometer is rather low, from 1X to 5X. In this setup, a beam splitter BS is placed below the microscope objective lens MO. The reference mirror R is located outside the image path. Note that the working distance is limited by BS. 19 Mirau interferometer (Mirau, 1952), shown in Figure 2.2(b), uses intermediate magnifications 10X-50X and two parallel plane glass plates are placed in front of MO. Since this setup is most compact, most SWLIs are equipped with a Mirau objective. However, the disadvantage of Mirau interferometer is the small usable numerical aperture. Besides, the position of R may cause central obscuration during measurement (Lehmann, 2010). Figure 2.2(c) is Linnik interferometer (Linnik, 1933) which is suitable for high magnifications (100X-200X) and gives high lateral resolution. In this setup, no components in front of MO are needed. Hence, it provides the longest working distance and highest numerical aperture. In order to achieve equal optical path lengths in both reference and object lights, both MOs are required to have identical wave fronts and chromatic aberrations. On the other hand, this type of SWLI is more sensitivity to mechanical noise and thermal vibration due to its longer working distance (Niehues et al., 2012). These three types of interferometers are performed using white light instead of a laser as the light source due to its short coherence length. Although it is more challenging to match interferometer path lengths, the short coherence length easily eliminates spurious interference fringes caused by any stray reflections from the optical components (Schwider, 1999). Therefore, the accuracy of the interferometers is highly enhanced, up to nanometer range. 2.2.3 Limitations of existing SWLIs Although SWLIs can easily perform non-destructive full-field height measurement with nanometer resolution, present microscopic interferometers still have difficulties to 20 measure objects in liquid media directly. The major challenge is due to the contradictory phenomena of the optical path length (OPL) and the image path length (IPL) when the light wave goes into a liquid medium. The former becomes n times, but the latter approximately 1 n times the geometric path length, if the refractive index of the medium is n . One scenario is while the microscopic objective (MO) focuses on the surface of the object (O) in the medium, the optical paths (solid line) and image paths (dashed line) are shown in Figure 2.3(b). The physical lengths of optical and image paths in liquid medium with refractive index n are n and  respectively. In the medium, the wave oscillates with the same frequency but travels at a slower speed. This implies that factor of the wavelength shrinkage is 1 n . Hence, the OPL for a path of physical length n equals n 2  in air (Figure 2.3(c)). In the meanwhile, the identical lens MO in RL also focuses on the surface of R (Figure 2.3(a)), and both OPL and IPL are  . In this case, although IPLs in both lines are the same, OPL in OL is n 2 of that in RL. These two waves cannot interfere with each other. On the other hand, while OPLs in both RL and OL are the same (Figure 2.3(d)-(f)), the physical length of optical path is  n in the liquid medium. The lens MO in OL is unable to focus the rays on the surface of O. In this instance, although two waves interfere with each other, the interference image is not observable. Currently, there are two ways to resolve the disentangling phenomena of OPLs and IPLs between the RL and OL. One way is to put a compensating wedge between the lens MO and the surface of R. It is almost infeasible, since the wedge requires having the same thickness and effective refractive index as those of the medium. The other way is 21 Figure 2.3 The scenario with same image path lengths (IPLs) in both lines: (a) optical and image paths in RL, (b) physical (not phase) optical (solid line) and image paths (dashed line) in OL, (c) the optical path lengths (OPLs) for both RL and OL; analogous figures for the case with same OPLs in both lines: (d)-(f). immersing both MOs into the identical media, i.e., immersion Mirau interferometer (Dubois & Boccara, 2008; Lyulko et al., 2010) or directing both RL and OL to pass through indistinguishable liquid chambers, i.e., liquid cell interferometer (Reed et al., 2008). Although the morphology of O in the medium can be measured through the second method, it requires a precise and real-time feedback system to match OPLs and IPLs between RL and OL during measurement by adjusting the position of the lens MO 22 in the reference length RL or that of R in the compensated chamber. Otherwise, the interference fringes disappear. Besides, the immersion approach destroys the major advantage of optical experiment – non-contact measurement of the surface profile of O without influencing the environment and conditions of the specimen. These two significant drawbacks are the major difficulties to have in situ measurement of the morphology of O in liquid media via existing technique of SWLIs. 2.3 Principles of SALLI Self-adjusting liquid Linnik interferometer (SALLI) introduced in this thesis overcomes the above difficulties with a novel optical configuration that naturally self-compensates for any refractive-index change in the liquid medium. Through direct imaging of the specimen, the lateral deformation can be monitored in situ with few-micrometers resolution. At the same time, SALLI also permits high precision real-time measurements of changes in the vertical surface position (resolution of 1 nm). The principle of SALLI is shown in Figure 2.4. A cubic beam splitter (BS), the key element in the interferometer, has to be rotated 45 degrees. The collimated light beam is then split into two parallel lights (RL and OL). In this particular setup, RL and OL are parallel to each other with a small distance and impinge onto the surface of R and O through microscopic objectives (MO1, MO2) simultaneously. Since both R and O are in the same liquid chamber (CH), the OPLs and IPLs of both lights are identical to each other all the time. In this setup, MO1 and MO2 do not require immersing into the liquid medium. Accordingly, the environment of the chamber CH will not be disturbed. Furthermore, the optical system has self-adjusting function. Even though the amount of 23 Figure 2.4 The schematic of the principal of self-adjusting liquid Linnik interferometer (SALII). Parts of the interferometer: BS – beam splitter, MO1 & MO2 – microscopic objective lenses, RPS – relative phase shifter, O – object, R – reference mirror. The optical paths inside RPS can be detoured and RPS can be placed between BS and MOs or MOs and CH. the liquid may change during measurement, the interference fringe patterns still remain with no need of any feedback system. Therefore, SALLI is suitable for in situ measurement of the morphological variation of the live cells, i.e., real-time studying the dynamic motion of electrodes. In this optical setup, unlike traditional Linnik interferometer, the motions of R and O are coupled since they are in the same liquid chamber CH. Consequently, it is difficult to control the tilts of R and O independently. In practice word, this implies that the direction and number of the fringes cannot be appropriately adjusted. This is a significant limitation in enhancing the resolution of the measurement. In order to resolve this problem, a novel optical system, the relative phase shifter RPS, is introduced in this thesis. 24 The system RPS employs odd number of mirrors in RL and even numbers of mirrors in OL together with a piezoelectric phase controller in one of the optical legs. In this way, the relative tilt between the light wave fronts from R and O can be controlled by simply tilting the chamber CH, while the overall relative phase shift is adjusted by the phase controller. Accordingly, the optical phase difference between R and O and its gradient can be properly controlled. The optical system RPS may be placed between BS and MOs as shown in Figure 2.4 or between MOs and CH. Although the gradient can be also controlled by tilting a mirror in one of the optical legs, the mirror tilting cause significant displacement of reflection spots on R or O; therefore, mirror tilting is not a good option of controlling the fringe gradient. The physical experimental setup of SALLI will be further described in section 0. 2.4 Algorithms for SALLI The algorithms to post-process the optical images are elaborated in this section. First, detect the edge of the sample from the intensity distribution of the images. Convert the interference images to the relative phase maps. Further, unwrap the phase maps to calculate the height profile. In time-sequence measurements, the connection of the fringes may need to be adjusted by shifting the fringe-number jump by an appropriate integer, based on known constraints of the physical phenomena. Finally, an innovative technique and useful schemes are introduced to reduce the noise level of the measurement. 25 2.4.1 Detection of the edge of the sample To obtain interfacial properties (discussed in next chapter), it is important to record and detect the edge of an electrode during lithiation and delithiation cycles. Here, a scheme is developed to identify the edge of a sample from the intensity distribution of an image. Figure 2.5(a) is a typical optical image obtained from SALLI. Clearly, the image has been divided into two regions of different intensities. In this case, the reference region R has higher reflectivity than O as shown in Figure 2.5(b). To properly determine the edge of O, we analyze the intensity profiles line by line. Taking AA’ row as an example, the black dots in Figure 2.5(c) are the intensity data. Despite the corner effects in the image, a distinct intensity jump can be observed and estimated as  IO , x1  x1O .   1 I  x1    (n)   I R  I O  x1  x1R I R  x1O I O  , x1O  x1  x1R . (2.6)  x1O  x1R  I R , x1R