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# Closed Loop Recycling of Electric Vehicle Batteries to Enable Ultra-high Quality Cathode Powder

## Abstract

The lithium-ion battery (LIB) recycling market is becoming increasingly important because of the widespread use of LIBs in every aspect of our lives. Mobile devices and electric cars represent the largest application areas for LIBs. Vigorous innovation in these sectors is spurring continuous deployment of LIB powered devices, and consequently more and more LIBs will become waste as they approach end of life. Considering the significant economic and environmental impacts, recycling is not only necessary, but also urgent. The WPI group has successfully developed a closed-loop recycling process, and has previously demonstrated it on a relatively small scale 1 kg spent batteries per experiment. Here, we show that the closed-loop recycling process can be successfully scaled up to 30 kg of spent LIBs from electric vehicle recycling streams, and the recovered cathode powder shows similar (or better) performance to equivalent commercial powder when evaluated in both coin cells and single layer pouch cells. All of these results demonstrate the closed-loop recycling process has great adaptability and can be further developed into industrial scale.

## Introduction

With the development of mobile devices and electric cars, the demand of lithium-ion batteries (LIBs) keeps increasing. The market value of global lithium-ion battery was $29.86 billion in 2017 and estimated to reach$139.36 billion in 20261. Because of the decreasing cost and increasing efficiency of LIBs, the rechargeable battery market is facing a major transformation. Bernatein estimates that LIBs will occupy 70% of the rechargeable battery market by 20252. Accordingly, the amount of end-of-life LIBs will rise significantly, lagging only in time. It is known that some countries use unsustainable ways to deal with battery waste such as incinerating or landfilling. The materials’ value is lost if no suitable recycling process is applied, and thus valuable resources are lost. Considering both the economical and environmental implications, LIBs entering the waste stream require efficient and environmentally friendly recycling processes3,4,5,6. Favorable economics would encourage collection, and follow the successful effective recycling precedent set by the lead acid industry.

Currently, recycling approaches can be divided into three main types: pyrometallurgical, hydrometallurgical and direct recycling7. Pyrometallurgy uses high temperature to smelt valuable metals in spent LIBs, a temperature above 1000 °C is used to form alloys8. High use of energy restrains its lab-scale research, however, pyrometallurgy is widely used in industry because of its simplicity and high productivity. Hydrometallurgy employs chemical process to recycle, multi-step treatments including acid–base leaching, solvent extraction, precipitation and ion exchange and electrolysis are involved due to the chemical complexity of LIB itself  9,10,11,12,13,14,15,16,17. Direct recycling recover different materials by physical processes. With minimal destruction, the recovered material retains its crystal structure and has a good electrochemical performance18. Pyrometallurgy, hydrometallurgy and direct recycling processes can be combined together to accommodate different incoming chemistry and expected outcome materials.

Over the past few years, many different recycling approaches and methods have been proposed and studied although much of the research is still in the lab scale phase. Ren et al. employed a novel slag system FeO-SiO2-Al2O3 to recover spent batteries8. In situ recycling was developed by Li et al., they used oxygen-free roasting and wet magnetic separation technique to recover spent LiCoO2/graphite batteries19. Tanong et al. tested several leaching reagents – inorganic acids, organic acids, chelating agents and alkaine agents, and found sulfuric acid was the most efficient solution for solubilizing metals from spent batteries10. They further optimize the best leaching condition using a three level Box-Behnken design10. Zhan et al. used froth flotation technique and separated fine battery electrode materials efficiently20. Lien concentrated valuable metals and graphite using membrane technologies21. Sonoc et al. firstly employed Donnan dialysis with cation exchange membranes and recovered lithium, transition metals16. Meng et al. proposed an electrochemical cathode-reduction method to leach LiCoO2 from spent LIBs and mechanism was revealed by kinetic analysis17. Shi et al. developed a simple process to regenerate spent LiCoO2 cathode, and the resulting cathode had a high electrochemical performance18. In addition, a number of research development specifically related to hydrometallurgical technologies in recent years are listed in Table 1. Hydrometallurgical recycling mainly involves leaching, solvent extraction and chemical precipitation. Leaching steps can be divided into alkali leaching and acid leaching, and acid leaching is more favorable because of its higher efficiency. Acid leaching includes inorganic acid and organic acid leaching, and inorganic leaching involves strong acid and can produce secondary pollution, while organic leaching can reach similar efficiency under a milder environment. Another leaching process is bioleaching, and it utilizes the acids generated during microorganisms’ metabolism processes. Inorganic acid leaching has the advantages of low cost while organic acid leaching and bioleaching are more environmentally friendly. Solvent extraction is the process that follows leaching and to separate metal ions or to remove impurities, and it is accomplished because of the various distribution of metal ions between organic solvent and aqueous solution. Due to the high purity of products, solvent extraction is adopted in industry. However, there is still room for improvements to eliminate the complex procedures and high cost of solvent. Chemical precipitation is widely used for separating metals from complex systems due to the varied solubilities at a certain pH. Common precipitants are NaOH, H2C2O4, C4H8N2O2, H3PO4, and Na2CO3, which can react with transition metal ions or Li+ and forms insoluble precipitates. Ni, Mn and Co have similar properties and thus can be co-precipitated as hydroxides, which can be further fabricated into cathode. As such, complex separation steps are avoided and all the values can be recovered. In addition to the primary chemical processes discussed above, other recycling techniques including electrolysis, ion exchange and sol-gel processes are also studied for recycling. However, most of these processes only use single stream of spent batteries for recycling experiments. The produced materials are normally evaluated in the university lab.

Our group has developed a lab-scale and highly efficient closed-loop recycling process previously, which combines hydrometallurgical and direct recycling processes22,23,24,25,26. The inorganic acid leaching and co-precipitation reaction in our closed-loop recycling process are the typical hydrometallurgical processes. While different from industrial available hydrometallurgical recycling processes, in which the recovered materials are meal oxides or raw metal alloys, our closed-loop recycling process produced industrial-grade cathode material directly from recycling stream. However, not restricted to the success of recycling in lab scale, this closed-loop recycling process can be transferred into industrial scale and it paved a path to commercially recycle LIBs in a more economic and environmental friendly way. The main and exclusive advantages of our recycling process are as follow: (1) it can accommodate a wide range of incoming LIB chemistry feed. EV batteries from General Motors, Ford and Fiat Chrysler Automobiles were used to demonstrate this flexibility. (2) the recovered cathode material has equal or, in some instances, better electrochemical performance compared with commercial cathode material. (3) the electrochemical results haven been proved by both WPI and A123 Systems independently. (4) the recycling process has been scaled up to 30 kg spent batteries per experiment. Battery Resourcers is further scaling up the recycling work to 0.5ton spent lithium ion batteries per day.

Our previous work has focused primarily on the recycling of electronics battery waste and experiments were conducted up to a maximum scale of 1 kg of battery feed. The properties of recovered cathode materials show good performance; however, they are not as good as the industrial material. Here, with the utilization of the closed-loop recycling process, we report the successful scaling-up of the process with different EV recycling streams and produce very high quality cathode materials-LiNi1/3Mn1/3Co1/3O2 (NMC 111). The large scale experiment (7-day) also reveals how the particles are evolved, which has not been reported before. In this work, the end-of-life LIBs come from General Motor Chevrolet Volt (GM), Ford Focus (Ford) and Fiat Chrysler Automobiles 500 e (FCA) vehicles. The batteries used in both GM and Ford’s EV batteries use a cathode which consists of LiNi1−x−yMnxCoy-LiMn2O4 (NMC-LMO), while FCA’s battery supplier employs a cathode comprised of LiNi1−x−yMnxCoy-LiMn2O4-LiNixCoyAlzO2 (NMC-LMO-NCA)27,28,29,30,31,32. The successful scaling-up of the process was confirmed by our own analyses and independent electrochemical test results from A123 Systems. Compared with the commercially sourced cathode material that was used as our control, our recycled cathode shows comparable electrochemical performances and, notably, superior rate capability.

## Results and Discussions

Parameters optimization of the co-precipitation reaction was accomplished using experiments whose reaction time was 30 hours and in which ~1 kg spent lithium ion batteries were processed. Then parameters were translated to the larger scale experiments, in which 30 kg batteries were used and the reaction time were 168 hours. The ultimate success of the scaling-up of the closed-loop recycling process was verified both by our results and independent electrochemical tests from A123 Systems.

### Small-scale Experiment

In the co-precipitation reaction, parameters that need to be optimized include pH number, residence time, reaction time, the feed rates of the ammonia solution and metal sulfate, etc. Here, metal sulfate solution is added in a range of 2.4–3.5 mL/min, and ammonia water is 0.4–0.9 mL/min. NaOH solution is fed automatically and the pH is controlled in a range of 10–11. In the small-scale experiment with a reaction time of 30 hours, samples are taken every five hours to monitor the process. Every sample is filtered and dried for SEM observation and tap density tests. While the as-filtered sample is pink, it turns into black when dried. It is believed that the black powder is a mixture of hydroxide and oxyhydroxide33.

Evolution of the precursor particle morphology is observed in the SEM images in Fig. 1, and the tap densities detailed in Supporting Information (Fig. S1). The precursor particles are secondary particles that are spherical in shape, and are aggregates of plate-shaped primary particles. As the reaction progresses, the primary particles keep filling the internal void space available within the secondary particle. Via this mechanism, the precursor particles grow progressively larger and denser. Tap densities in Supporting Information (Fig. S1) supports this trend. At some point, there is no more space for thickening and filling, and particle size achieves steady state and remains constant, beyond which tap density stops increasing significantly. This signals that the co-precipitation reaction has reached equilibrium. In the small-scale experiment, this occurs at 25 hours. Samples collected after 25 hours can be regarded as the same, which laid the foundation for further scale-up and eventual industrial application. In the later scale-up experiments, it was more evident that the reaction reaches an equilibrium after a transient non-steady state period. It is worth mentioning that particle size and morphology play important roles in electrochemical performances. Uniform, spherical particles improve tap density and further boost the capacity/energy density of the electrodes34. Smaller particles are beneficial for cell power performance because of shorter ionic diffusion distance. However, the total higher surface area also adversely impacts performance due to increased contact resistance35. Optimized particle size distribution benefits both rate tests and cycle performance, and it is believed 10 ± 2 µm of D50 is an ideal range for NMC111 based on the size of commercial NMC 111 powder. Here, the 30-hour precursor sample has a tap density of 2.05 g/mL. The high tap density, uniform morphology and suitable size of the precursor particles make them good candidates for sintering into cathode powder.

Cathode material is fabricated by sintering the final precursor and lithium carbonate. SEM images are included in Supporting Information (Fig. S2), and it can be observed that the cathode powder maintains good morphology and the size of the precursor, which highlights the importance of synthesizing high quality precursor. The tap density of the recycled cathode power is as high as 2.75 g/mL, which is beneficial for specific capacity and electrochemical performance, as will be reported later. The crystalline property and purity of the synthesized cathode are confirmed by XRD and ICP-OES, respectively. The XRD pattern and refinement are presented in Supporting Information (Fig. S2 and Table S1). The peaks of synthesized cathode are sharp and it is classified as α-NaFeO2 structure (R$$\bar{3}\,$$m). Distinct separations of (006)/(012) and (018)/(110) are due to the highly ordered layered structure. The c/3a has a value of 1.6587, which is also an indicator of good layered structure34. The XRD pattern difference of synthesized cathode and simulation is marginal, not only from observation but also from the Rwp number (6.50%). The magnitude of ICP-OES tests is in ppm, and both precursor and cathode powder are dissolved and diluted first. Results are shown in Supporting Information (Table S2), and ratios of metal ions are exactly 1:1:1:3 (Ni:Mn:Co:Li) with less than 3% measurement error. Moreover, no impurities are detected in ppm order.

Parameters optimization in small-scale experiments was not only successful in generating high quality cathode material, but offers a valuable tool from which the parameters can be translated to inform larger scale experiments. Cathode material synthesized using our closed-loop recycling process exhibits excellent performances, including tap density, particle morphology and size distribution, crystallization and purity tests (XRD and ICP-OES). It not only demonstrates the viability of our recycling method, but also provides parameters for the following scale-up experiments.

### Scale-up

Herein, the reaction time is increased to seven days compared with thirty hours in the small experiment. For the adjustments in larger scale experiments, we transferred all the parameters into larger scale experiments in their initial status (168 hrs reaction time) after finding the optimized parameters in smaller scale experiments (30 hrs reaction time). However, in order to accommodate to requirements of particle size distribution, tap density, et al., parameters including pH number, flow rate, etc would be adjusted during the larger scale experiments while monitoring the experiment closely. Therefore, the parameters in larger scale experiments are not exactly the same with those in smaller experiments.

Samples are taken every 24 hours and the experiment is monitored to ensure scaled-up experiments are suited to eventual industrial application. It is well known that industrial-scale co-precipitation processes employ continuous reactions. For example, production runs for 30 days are common, with material being collected daily over the course of the run. The key to successful industrialization is to guarantee every batch of power can be used, and thus every day’s precursor is collected for further testing. Tap densities are displayed in Supporting Information (Fig. S3), and they remain constant from day to day, and as high as 2.00 g/mL. Particle size distribution seems to show some differences over the first three days, as observed in the SEM images in Fig. 2. Firstly, the particles have uniform size. With the progress of the experiment, the particles grow larger and new particles are formed. After 3~4 days, the reaction reaches to steady state and at this time the particle size and distribution will not change much with further progress of the experiment. The particles collected after steady state have bimodal size distribution. Although it takes longer to attain equilibrium in large-scale experiment, considering industrialization in the future, three days has marginal influence for a one-month production run. Factors including particle morphology, size distribution, and tap density are monitored and once stable, samples from Day 3 to Day 7 are combined together and sintered into cathode powder. It can be seen by the SEM images presented in Fig. 2, that the cathode particles are densely packed and quite spherical, and a tap density of 2.52 g/mL was achieved. Particle size distribution of the synthesized cathode has been characterized by A123 systems and results are shown in Supporting Information (Fig. S4). Material has a D50 of a value of 11.7 µm and the electrochemical performance is presented in later sections. Although the large-scale experiment in our lab was only carried out for seven consecutive days, it is believed that with a larger reactor and constant supply of reactants, commercialization of our closed-loop recycling process is very promising.

These results demonstrate that our recycling process was successfully scaled up from a spent battery feed of 1 kg to one of 30 kg. The synthesized powder characterization results, including particle size, particle morphology, tap density, purity and crystalline state, and their proximity to results from reference commercial cathode material, suggest that the recycled cathode will deliver comparable electrochemical performance in coin cells and pouch cells. This will be addressed in the next section.

### Electrochemical performance

Ultimately the success of our scale-up must be validated by evaluating the electrochemical performance of the recycled cathode material in cells. Accordingly, both half coin cell and full coin cell were assembled in our lab to evaluate rate capability and cycle performance. Results are shown in Fig. 3. Specific capacity is 158 mAh/g, 155 mAh/g, 149 mAh/g, 140 mAh/g, 133 mAh/g, 125 mAh/g, 113 mAh/g, 79 mAh/g for 0.1C, 0.2C, 0.5C, 1C, 2C, 3C, 5C, 10C, respectively. The capacity of 158 mAh/g at 0.1C is comparable with commercial NMC 111 capacity. The high rate performance is quite impressive (113 mAh/g and 79 mAh/g for 5C and 10C, respectively). Besides the excellent rate capability results, the cycle life tests, conducted using 0.5C/0.5C cycling, are also promising. The coulombic efficiency stays above 99% for all of the 100 cycles. After 100 cycles, the capacity retention is nearly 100%. There is a slightly specific capacity increase during the cycling. In general, the slight increase of the specific capacity is due to the following reasons: (1) the increased conductivity of electrode materials. (2) the increase of surface area caused by electrode materials’ minimal breakage. (3) the continuous activation of electrode materials due to better infiltration of electrolyte. (4) the slight increase of room temperature. Here, the increase of specific capacity was very small (1.55 mAh/g, ~1% of the specific capacity of the material) and we believed the slight increase of room temperature maybe the main reason. The coin cells were tested at room temperature, and there was a slight variation of temperature between daytime and nighttime. Also, the continuous stabilization of interface in coin cells and activation of electrode materials may help the slight capacity increase. Compared with commercial NMC 111 powder, our recycled product is comparable, with better rate capability. Considering the aggregate results including SEM image, tap density, size distribution, purity and crystal structure, one can conclude that the closed-loop recycling process described herein has very promising commercial potential.

### Independent Testing at A123 Systems

Synthesized cathode powders from the recycling process were sent to A123 Systems for independent electrochemical performance tests. A123 Systems selected a commercially available NMC111 cathode as a control powder, and compared the control powder with WPI synthesized cathode powder (Fig. 4) in coin cells and single layer pouch cells (SLPs) that were otherwise identical. Detailed comparison of physical properties between WPI synthesized cathode powder with control cathode powder tested by A123 Systems is depicted in Supporting Information (Table S3). It was observed that WPI synthesized powder performed better than the control reference at all rate tested up to 10C (Fig. 4a). A comparison between WPI synthesized cathode powder and commercial control cathode powder is shown in Fig. 5. WPI synthesized cathode is more porous than control cathode, which is especially clear in small particles. Porous cathode particles have more electrolyte uptake and diffusion is easier because ionic diffusion in liquid (diffusion coefficien t ~10−6 cm2/s)36 is faster than that in solid (diffusion coefficient ~10−10 cm2/s)37. It explains that at higher rate, WPI synthesized cathode outperforms control cathode. Specific capacity of WPI synthesized cathode powder has a value 153 mAh/g, 145 mAh/g, 139 mAh/g, 132 mAh/g, 111 mAh/g, 38 mAh/g for 0.2C, 0.5C, 1C, 2C, 5C, 10C, respectively, which is similar to the results in the previous section. Based on those results, the synthesized cathode material from our closed-loop recycling process is well positioned to compete with non-recycled counterparts for Li-ion battery market. Besides coin cell, A123 systems also fabricated single layer pouch (SLP) cell to further study the electrochemical performance of WPI recovered cathode powder.

Pouch cells are widely used in consumer, military and automotive applications because of their simple and lightweight design38. The SLP cell is a simple yet representative form factor which largely mimics the multi-layered prismatic cells. Screening battery materials at the SLP level offers a useful research tool for early stage technology development where material quantity may limit the choice of form factors between coin cell and large format pouch cells. Therefore, the electrochemical performance of SLP cells fabricated by A123 Systems has great values and offers insight to inform subsequent work. Results are presented in Fig. 4(b), where control and WPI synthesized cathode show similar trends in both discharge ΔSOC tests and direct current resistance (DCR) tests. After 1000 cycles, 90% of the discharge capacity is retained, and after 1700 cycles, 80% of its discharge capacity remains. Impedance is also comparable for the control and WPI synthesized cathode powders. Independent electrochemical performance tests again yield encouraging results. WPI synthesized cathode powder has comparable or better (for high rates) performance than the commercially sourced control powder. The results suggest that our recycling process is capable of being scaled up. The critical next phase of technology development and large scale validation is presently being conducted by Battery Resourcers, Inc.

## Conclusions

In this study, it is confirmed that our closed-loop recycling process can handle large format spent batteries that come from different commercial electrified vehicles (GM, Ford and FCA), having different cathode chemistries. Moreover, recycled cathode powder synthesized by WPI has similar or better performance when compared to equivalent stoichiometry commercial powder. This is validated by electrochemical testing of WPI assembled coin cells, and independent electrochemical testing of coin cells and SLP cells made and tested by A123 Systems. In addition, this recycling process is scalable (30 kg spent batteries are recycled in each experiment), with efforts towards further scale-up underway. The very promising results obtained to date of this timely and important work suggest a viable path towards commercialization exists. Toward this end, scale-up and development work continues at Battery Resourcers and further results can be expected in the near future.

## Experimental Section

### Overall Recycling Process

In general, the closed-loop recycling stream can be summarized in Supporting Information (Fig. S5). End-of-life batteries are first cut, shredded and sieved. Batteries of various form factors, package design (pouch or metal can) and multiple chemistries can be combined in a single feed lot. After removal of the casing, aluminum etc., what remains is the graphite, carbon and cathode powders. The different cathode powders are dissolved together in a leaching solution of sulfuric acid (H2SO4) and hydrogen peroxide (H2O2). In this step, some impurities are also dissolved in the leaching solution. In order to synthesize NMC111, impurities such as Cu, Fe, Al are removed by strategically controlling the pH. Inductively coupled plasma optical emission spectrometry (Perkin Elmer Optima 8000 ICP-OES) is used to determine the concentrations of various metal ions, and nickel sulfate hexahydrate, manganese sulfate monohydrate and cobalt sulfate are added to reach the desired ratio, which is 1:1:1 for Ni, Mn, Co in this study. The co-precipitation reaction will be discussed in detail later. After filtering and drying, precursors (Ni1/3Mn1/3Co1/3(OH)2) mix with Li2CO3. Then the mixture is sintered at 450 °C for 5 hours and 900 °C for 14 hours.

### The Co-precipitation Reaction – Precursor Synthesis

After cutting, shredding and sieving to remove foils and cell case or pouch materials, the remaining powders, which are a mixture of carbon, graphite and cathode powder, are leached in acid. After removing the impurities in the leachate, nickel sulfate hexahydrate (GFS Chemicals), manganese sulfate monohydrate (GFS Chemicals) and cobalt sulfate (GFS Chemicals) are added to adjust the ratio of Ni, Mn, Co. This ratio is tested again using ICP, after which the metal sulfate solution undergoes a co-precipitation reaction. Chemical reagents fed into the co-precipitation reactor are (Ni/Mn/Co) metal sulfate solution, ammonia solution (32%, EMD Millipore) and sodium hydroxide (VWR). A 5 L jacketed glass cylinder with customized feed ports is used for the co-precipitation reaction, which is conducted under nitrogen. Process parameters including pH, flow rate and temperature are controlled throughout the reaction. After a certain transient period, the co-precipitation reaction reaches equilibrium, or steady state, and particle size, morphology and tap density remain constant39.

At the end of experiment, the suspension is filtered and washed thoroughly to remove residual or absorbed salts. Then, the particles need to be dried in the oven for about 12 hrs at 130 °C. Tap densities are measured manually, whereby a graduated cylinder is tapped constantly until the level stops changing. JEOL JSM 7000 F is used for obtaining SEM images of the particles. X-ray diffraction (XRD) patterns are obtained using PANalytical Empyrean, with highScore software employed to obtain Rietveld refinement.

### Cathode Sintering

To synthesize cathode active material, 1 mol of precursor is mixed with 1.05 mol of Li2CO3 (VWR). An excess of 5% Li2CO3 compensates for the Li loss during firing40. Before sintering, the mixture must be uniform in color. Otherwise, it needs to be mixed again. Sintering conditions are 450 °C for 5 hours and 900 °C for 14 hours. Both heating and cooling rates are 2 °C/min. After cooling down to room temperature, the cathode powder must undergo grinding to ensure the material does not contain any agglomerates.

### Electrochemical Testing

Cathode powder, conductive carbon (super C65), and Polyvinylidene fluoride (PVDF) dissolved in N-Methyl-2-pyrrolidone (NMP) are mixed uniformly. Then the slurry is cast onto an aluminum foil and dried at 60 °C for 8 hours. 14 mm diameter electrode disks are punched and pressed to reach a desired porosity prior to assembling into coin cells. After being dried in vacuum oven for 8 hours, any excess solvent is removed, and the cathode electrode is ready for coin cell assembly. In the WPI lab, we assemble half coin cells to test the rate performance and full coin cells to test the cycle performance. Half coin cells use a lithium metal anode while full coin cells use a graphite anode. They are both assembled in glove box. The electrolyte used is 1M LiPF6 in ethylene carbonate (EC), diethyl carbonate (DEC), and dimethyl carbonate (DMC) (1:1:1). The separator used is a 25um trilayer polypropylene-polyethylene-polypropylene membrane (MTI Corporation). Electrochemical performance tests are conducted with an Arbin instrument (Model BT2043). Both coin cells and single layer pouch cells are also assembled at A123 Systems for independent evaluation following similar cell assembly and testing protocols.

## Data Availability

The dataset generated during the current study are not publicly available due to intellectual property purposes but are available from the corresponding author on reasonable request.

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28. 28.

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29. 29.

SB LiMotive to Provide Lithium-ion Battery Packs for Fiat 500EV, http://detroit.cbslocal.com/2010/11/08/sb-limotive-to-provide-lithium-ion-battery-packs-for-fiat-500ev/ (2010).

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Blomgren, G. E. The Development and Future of Lithium Ion Batteries. Journal of The Electrochemical Society 164, A5019–A5025, https://doi.org/10.1149/2.0251701jes (2017).

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Warner, J. T. The Handbook of Lithium-Ion Battery Pack Design: Chemistry, Components, Types and Terminology. (Elsevier Science, 2015).

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Xiao, Q., Li, B., Dai, F., Yang, L. & Cai, M. Application of Lithium Ion Battery for Vehicle Electrification. Electrochemical Energy: Advanced Materials and Technologies (2015).

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Van Bommel, A. & Dahn, J. Synthesis of Spherical and Dense Particles of the Pure Hydroxide Phase Ni1∕3Mn1∕3Co1∕3 (OH) 2. Journal of The Electrochemical Society 156, A362–A365 (2009).

34. 34.

Nam, K.-M., Kim, H.-J., Kang, D.-H., Kim, Y.-S. & Song, S.-W. Ammonia-free coprecipitation synthesis of a Ni–Co–Mn hydroxide precursor for high-performance battery cathode materials. Green Chemistry 17, 1127–1135, https://doi.org/10.1039/C4GC01898B (2015).

35. 35.

Pohjalainen, E., Rauhala, T., Valkeapää, M., Kallioinen, J. & Kallio, T. Effect of Li4Ti5O12 Particle Size on the Performance of Lithium Ion Battery Electrodes at High C-Rates and Low Temperatures. The Journal of Physical Chemistry C 119, 2277–2283, https://doi.org/10.1021/jp509428c (2015).

36. 36.

Ong, M. T. et al. Lithium Ion Solvation and Diffusion in Bulk Organic Electrolytes from First-Principles and Classical Reactive Molecular Dynamics. The Journal of Physical Chemistry B 119, 1535–1545, https://doi.org/10.1021/jp508184f (2015).

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Capron, O. et al. On the Ageing of High Energy Lithium-Ion Batteries—Comprehensive Electrochemical Diffusivity Studies of Harvested Nickel Manganese Cobalt Electrodes. Materials 11, 176 (2018).

38. 38.

BU-301a: Types of Battery Cells, http://batteryuniversity.com/learn/article/types_of_battery_cells (2017).

39. 39.

van Bommel, A. & Dahn, J. R. Analysis of the Growth Mechanism of Coprecipitated Spherical and Dense Nickel, Manganese, and Cobalt-Containing Hydroxides in the Presence of Aqueous Ammonia. Chemistry of Materials 21, 1500–1503, https://doi.org/10.1021/cm803144d (2009).

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Cho, J. LiNi0.74Co0.26-xMgxO2 Cathode Material for a Li-Ion Cell. Chemistry of Materials 12, 3089–3094, https://doi.org/10.1021/cm000153l (2000).

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Zheng, X. et al. Spent lithium-ion battery recycling – Reductive ammonia leaching of metals from cathode scrap by sodium sulphite. Waste Management 60, 680–688, https://doi.org/10.1016/j.wasman.2016.12.007 (2017).

42. 42.

Barik, S. P., Prabaharan, G. & Kumar, L. Leaching and separation of Co and Mn from electrode materials of spent lithium-ion batteries using hydrochloric acid: Laboratory and pilot scale study. Journal of Cleaner Production 147, 37–43, https://doi.org/10.1016/j.jclepro.2017.01.095 (2017).

43. 43.

Li, L. et al. Sustainable Recovery of Cathode Materials from Spent Lithium-Ion Batteries Using Lactic Acid Leaching System. ACS Sustainable Chemistry & Engineering 5, 5224–5233, https://doi.org/10.1021/acssuschemeng.7b00571 (2017).

44. 44.

Bahaloo-Horeh, N. & Mousavi, S. M. Enhanced recovery of valuable metals from spent lithium-ion batteries through optimization of organic acids produced by Aspergillus niger. Waste Management 60, 666–679, https://doi.org/10.1016/j.wasman.2016.10.034 (2017).

45. 45.

Virolainen, S., Fallah Fini, M., Laitinen, A. & Sainio, T. Solvent extraction fractionation of Li-ion battery leachate containing Li, Ni, and Co. Separation and Purification Technology 179, 274–282, https://doi.org/10.1016/j.seppur.2017.02.010 (2017).

46. 46.

Yang, Y., Xu, S. & He, Y. Lithium recycling and cathode material regeneration from acid leach liquor of spent lithium-ion battery via facile co-extraction and co-precipitation processes. Waste Management 64, 219–227, https://doi.org/10.1016/j.wasman.2017.03.018 (2017).

47. 47.

Chen, X., Fan, B., Xu, L., Zhou, T. & Kong, J. An atom-economic process for the recovery of high value-added metals from spent lithium-ion batteries. Journal of Cleaner Production 112, 3562–3570, https://doi.org/10.1016/j.jclepro.2015.10.132 (2016).

48. 48.

Yang, Y., Huang, G., Xie, M., Xu, S. & He, Y. Synthesis and performance of spherical LiNixCoyMn1-x-yO2 regenerated from nickel and cobalt scraps. Hydrometallurgy 165, 358–369, https://doi.org/10.1016/j.hydromet.2015.11.015 (2016).

## Acknowledgements

This material is based upon work supported by the Department of Energy, National Energy Technology Laboratory under Award Number DE-EE0006250) with the United States Advanced Battery Consortium LLC (USABC LLC)”. M.C. and Z.Z. contributed to the paper equally. This report was prepared as an account of work sponsored by an agency of the United States Government and USABC LLC. Neither the USABC LLC, the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, or those of USABC LLC.

## Author information

Authors

### Contributions

M.C., Z.Z. and Y.W. designed the experiments, M.C., Z.Z., Q.W., Y.Z., X.M., C.S., D.X., J.L., Y.L., E.G. conducted the experiments, P.G., I.O., L.P., J.W. helped in evaluating cathode powder, R.A. gave suggestions for the experiments design. M.C. and Y.W. wrote the manuscript and all authors reviewed the manuscript.

### Corresponding author

Correspondence to Yan Wang.

## Ethics declarations

### Competing Interests

Worcester Polytechnic Institute has filed a patent that uses this recycling process.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Chen, M., Zheng, Z., Wang, Q. et al. Closed Loop Recycling of Electric Vehicle Batteries to Enable Ultra-high Quality Cathode Powder. Sci Rep 9, 1654 (2019). https://doi.org/10.1038/s41598-018-38238-3

• Accepted:

• Published:

• ### Exploring a green route for recycling spent lithium-ion batteries: Revealing and solving deep screening problem

• , Quanyin Tan
•  & Jinhui Li

Journal of Cleaner Production (2020)

• ### Systematic Study of Al Impurity for NCM622 Cathode Materials

• Ruihan Zhang
• , Zeyi Yao
• , Panawan Vanaphuti
• , Xiaotu Ma
• , Sungyool Bong
• , Mengyuan Chen
• , Yangtao Liu
• , Feng Cheng
• , Zhenzhen Yang
•  & Yan Wang

ACS Sustainable Chemistry & Engineering (2020)

• ### Recycling of mixed cathode lithium‐ion batteries for electric vehicles: Current status and future outlook

• Tyler Or
• , Storm W. D. Gourley
• , Karthikeyan Kaliyappan
• , Aiping Yu
•  & Zhongwei Chen

Carbon Energy (2020)

• ### Recycling End-of-Life Electric Vehicle Lithium-Ion Batteries

• Mengyuan Chen
• , Xiaotu Ma
• , Bin Chen
• , Renata Arsenault
• , Peter Karlson
• , Nakia Simon
•  & Yan Wang

Joule (2019)

• ### High-Performance Graphite Recovered from Spent Lithium-Ion Batteries

• Xiaotu Ma
• , Mengyuan Chen
• , Bin Chen
• , Zifei Meng
•  & Yan Wang

ACS Sustainable Chemistry & Engineering (2019)


text:
# Category Archives: MTBF

Mean Time Between Failures or MTBF is a common metric for reliability and is often misused or misunderstood.

# Sample Size and Duration and MTBF

If you have been a reliability engineer for a week or more, or worked with a reliability engineer for a day or more, someone asked about testing planning. The conversation may have started with “how many samples and how long will the test take?”

You have heard the sample size question.

# Learn to Notice MTBF Everyday

Did you notice the speed limit signs in your neighborhood today?

If like me, you went about your commute or regular travels relatively blind. You watched for the neighbor’s dog that jumped into the road last week, yet didn’t register seeing the speed limit sign.

It’s a cognitive burden to notice the mundane or known. Continue reading Learn to Notice MTBF Everyday

# The 3 Best Reasons to Use MTBF

This may seem an odd article for the NoMTBF site. Stay with me for a moment longer.

Over the years of speaking out on the perils of MTBF, there has been some push back. A few defend using MTBF. Here are three of the most common (maybe not exactly the best, per se) reasons to use MTBF. Continue reading The 3 Best Reasons to Use MTBF

# Illuminating MTBF’s Lack of Information

Here’s a simple illustration of how MTBF oversimplifies data concealing essential information.

By convention, we tend to use MTBF for repairable data. That is fine.

You may also be aware of my dislike for the use of MTBF, for many different reasons. If you find yourself suggesting your organization, customer, industry or whomever to stop using MTBF, you may want to use this simple example to illustrate the ‘value’ of MTBF. Continue reading Illuminating MTBF’s Lack of Information

# A Quick MTBF Search Reveals Distressing Results

I was preparing to write this article and wondered how many search hits would appear for MTBF? So, opened Google and did an MTBF search. It is a common if misunderstood, acronym.

Beyond the 5,200,000 Google search results, it was the first page results that got me thinking. Keep in mind that Google often serves up a combination of what it thinks you are seeking and which sites have been useful for others.

Let’s break down what you find when you do an MTBF search. Continue reading MTBF Search Result Sadness

# 4 Questions to Ask When Confronted with MTBF

MTBF comes up a bit too often. When it does I have found rolling my eyes and arguing against using MTBF is not very effective.

So, what should a knowing reliability professional do instead?

Let’s explore four questions that you can ask that may help others find the value in no longer talking about MTBF. Continue reading 4 Questions to Ask When Confronted with MTBF

# Replace After MTTF Time To Avoid Failures – Right?

If we replace an item after a duration equal to the MTTF value, we would avoid failures, right?

Well, no, most likely not, was my response. What is your response? How would you answer this question? Continue reading Replace After MTTF Time To Avoid Failures – Right?

# Yet Another Confused MTBF Definition

Just when I thought we had experienced every possible MTBF definition confusion, here’s another.

This one is courtesy the thread concerning the impact to reliability when adding redundancy to a system. Continue reading Yet Another Confused MTBF Definition

# Yet Another Way to Misunderstand MTBF

In a Q&A forum, the response to a question concerning failure rate and repair times for a redundant system demonstrated yet another person confusing MTBF with something it is not.

The responder to the question mentioned the reference to repair time implied the need for MTBF as a metric. Then went on to describe MTBF as the duration of repair time, which should not change given a redundant system over a non-redundant system. Continue reading Another Way to Spot Someone Confusing MTBF

# A Few Simple Ideas to Improve Your Reliability Program

Spending too much on reliability and not getting the results you expect? Just getting started and not sure where to focus your reliability  program? Or, just looking for ways to improve your program?

There is not one way to build an effective reliability program. The variations in industries, expectations, technology, and the many constraints, shape each program. Here are three suggestions you can apply to any program at any time. These are not quick fix solutions, nor will you see immediate results, yet each will significantly improve your reliability program and help you achieve the results you and your customers expect. Continue reading 3 Ways to Improve your Reliability Program

# 5 Clues Using MTBF is Not Helping

Have you ever heard the claim that “We use MTBF, as it’s working just fine”?

They may be profitable and successful in the marketplace. Is MTBF serving them well?

Probably not. Continue reading 5 Clues Using MTBF is Not Helping

# The Magic Math of Meeting MTBF Requirements

Recently heard from a reader of NoMTBF. She wondered about a supplier’s argument that they meet the reliability or MTBF requirements. She was right to wonder.

Estimating reliability performance a new design is difficult.

There are good and better practice to justify claims about future reliability performance. Likewise, there are just plain poor approaches, too. Plus there are approaches that should never be used.

## The Vendor Calculation to Support Claim They Meet Reliability Objective

Let’s say we contract with a vendor to create a navigation system for our vehicle. The specification includes functional requirements. Also it includes form factor and a long list of other requirements. It also clearly states the reliability specification. Let’s say the unit should last with 95% probability over 10 years of use within our vehicle. We provide environmental and function requirements in detail.

The vendor first converts the 95% probability of success over 10 years into MTBF. Claiming they are ‘more familiar’ with MTBF. The ignore the requirements for probability of first month of operation success. Likewise they ignore the 5 year targeted reliability, or as they would convert, MTBF requirements.

[Note: if you were tempted to calculate the equivalent MTBF, please don’t. It’s not useful, nor relevant, and a poor practice. Suffice it to say it would be a large and meaningless number]

RED FLAG By converting the requirement into MTBF it suggests they may be making simplifying assumptions. This may permit easier use of estimation, modeling, and testing approaches.

## The Vendor’s Approach to ‘Prove’ The Meet the MTBF Requirement

The vendor reported they met the reliability requirement using the following logic:

Of the 1,000 (more actually) components we selected 6 at random for accelerated life testing. We estimated the lower 60% confidence of the probability of surviving 10 years given the ALT results. Then converted the ALT results to MTBF for the part.

We then added the Mil Hdbk 217 failure rate estimate to the ALT result for each of the 6 parts.

RED FLAG This one has me wondering the rationale for adding failure rates of an ALT and a parts count prediction. It would make the failure rate higher. Maybe it was a means to add a bit of margin to cover the uncertainty? I’m not sure, do you have any idea why someone would do this? Are they assuming the ALT did not actually measure anything relevant or any specific failure mechanisms, or they used a benign stress? ALT details were not provided.

## The Approach Gets Weird Here

Then we use a 217 parts count prediction along with the modified 6 component failure rates to estimate the system failure rate, and with a simple inversion estimated the MTBF. They then claimed the system design will meet the field reliability performance requirements.

RED FLAG Mil HDBK 217 F in section 3.3 states

Hence, a reliability prediction should never be assumed to represent the expected field reliability …

If you are going to use a standard, any standard, one should read it. Read to  understand when and why it is useful or not useful.

## What Should the Vendor Have Done Instead?

There are a lot of ways to create a new design and meet reliability requirements.

• The build, test, fix approach or reliability growth approach works well in many circumstances.
• Using similar actually fielded systems failure data. It may provide a reasonable bound for an estimate of a new system. It may also limit the focus on the accelerated testing to only the novel or new or high risk areas of the new design — given much of the design is (or may be) similar to past products.
• Using a simple reliability block diagram or fault tree analysis model to assembly the estimates, test results, engineering stress/strength analysis (all better estimation tools then parts count, in my opinion) and calculate a system reliability estimate.
• Using a risk of failure approach with FMEA and HALT to identify the likely failure mechanisms then characterize those mechanisms to determine their time to failure distributions. If there is one or a few dominant failure mechanisms, that work would provide a reasonable estimate of the system reliability.

In all cases focus on failure mechanisms and how the time to failure distribution changes given changes in stress / environment / use conditions. Monte Carlo may provide a suitable means to analysis a great mixture of data to determine an estimate. Use reliability, probability of success over a duration.

In short, do the work to understand the design, it’s weaknesses, the time to failure behavior under different use/condition scenarios, and make justifiable assumptions only when necessary.

## Summary

We engage vendors to supply custom subsystems given their expertise and ability to deliver the units we need for our vehicle. We expect them to justify they meet reliability requirements in a rationale and defendable manner. While we do not want to dictate the approach tot he design or the estimate of reliability performance, we certainly have to judge the acceptability of the claims they meet the requirements.

What do you report when a customer asks if your product will meet the reliability requirements? Add to the list of possible approaches in the comments section below.

Related

How to Calculate MTBF

MTBF: According to a Component Supplier

# How to Properly Calculate System Availability

Recently received a request for my opinion concerning the calculation of system availability using the classic formula

$\displaystyle A=\frac{MTBF}{MTBF+MTTR}$

The work is to create a set of goals for various suppliers and contractors to achieve. The calculation values derive from vendor data sheets and available information concerning MTBF and MTTR. The project is in the design phase thus they do not have working system’s available to measure actual availability.

How would you go about improving on this approach? Continue reading Calculating System Availability

# Considerations When You Calculate MTBF

It is deceptively easy to calculate MTBF given a count of failure and an estimate of operating hours. Just tally up the total hours the various systems operate and divide by the number of failures. Easy.

This simple calculation is the unbiased estimator for the inverse of the parameter lambda for the exponential distribution, or directly to estimate theta (MTBF). We use theta to represent the 1 / lambda.

What could go wrong with such a simple calculation?

## What is a failure?

Let’s start with what we count or do not count as a failure. This directly changes the resulting MTBF value. If we only count confirmed hardware failures, and do not count intermittent or unreproducible or software failures, are we under counting what the customer experiences as a failure?

Over what duration do we count the failures? Should we focus only on the first month of operation, the first year, the warranty or service contract period or the entire operating life of the system? How do you calculate MTBF?

Some organizations only count failures they expect to occur. The unexpected ones are ‘special’ causes and require further study before counting as failure officially.

Another organization only counted failures that completely shut down the system. A partial loss of functionality, a degradation of capability or the failure of a redundant element all did not count a system failure.

In my opinion if the customer calls it a failure, it’s a failure. If a failure, by any definition, costs your organization time and money to address, acknowledge, resolve or repair, it’s a failure.

## What is operating time?

This one is tricky. If the system does include the appropriate sensors and tracking mechanisms (hour meter) and a way to gather that operating time of units both failed and still operating, then we have a pretty good way to track total operating hours. Some situations and systems make this easy.

Most do not.

Let’s say we ship 100 systems a month for 10 months. At the end of ten months the first shipments have accumulated 10 months of operating time. IF….

… They are all placed into service immediately

… They are all operated full time for the full 10 months

… They are have each failure reported including down time

In general, we do have to make a few assumptions to determine the operating time for shipped systems. We tend to be conservative and err on the side that would make the MTBF value a little smaller than if we had the full set of carefully tracked data. Or do we?

• Some organization count from date/time of shipment ignoring shipping and installation time.
• Some organization assume all systems are installed and operated 24/7.
• Some organization assume no news is good news and the systems with no information are still operating.

And a few organization assume systems run indefinitely, even systems 20 years old, unless notified that it is decommissioned, assume it is still running full tilt. i.e. No retirement or replacement policy.

## How about when you calculate MTBF?

By convention when there are no failures we assume in the next instant there will be one failure. This avoid dividing by zero which causes fits for calculators and spreadsheets and mathematicians.

Another issue is how often are the calculations made? Do we gather data hourly, daily, weekly, monthly, annually? Some use a rolling set of data, for example only units shipped in the last year count for both operating time and failures. This result will ignore or discount the longer term wear out failures as the bulk of the units are young.

Some organization do the calculations weekly in order to detect trends. If there are trends you probably should not be using MTBF…. If it’s changing, if there are early life or wear out failure mechanisms, you should not be using MTBF.

Even though you can calculate MTBF easily, the complexities of getting it right still do not provide a useful metric. Instead focus on getting better data including time to failure information so you can explore and report the data with other tools and methods. Treat the data appropriately and make better decisions

Sure, better data will improve the ability to calculate the MTBF value, if you’d like to be like some organizations, that is fine.

How have you seen MTBF calculated poorly? Share your thoughts and stories in the comments below.

Related:

How to calculate MTTF

Perils of using MTBF

# Fred talks about the NoMTBF blog and movement

Did you know I was interviewed for the Dare to Know podcast? The interview was fun, check it out here.

## The Dare to Know podcast Interview

Tim Rodgers interviews Fred Schenkelberg concerning his blog, No MTBF and his mission to eradicate the common mis-use of MTBF.

Fred Schenkelberg is a reliability engineering and management consultant at FMS Reliability. He’s a lecturer at the University of Maryland, and he’s been an active contributor to both the IEEE and the ASQ Reliability Divisions.

Fred re-established Hewlett Packard’s corporate reliability program in the late 1990s, and also worked as a reliability consultant at Microsoft and a manufacturing engineer at Raychem.

In this episode, Fred Schenkelberg discusses:

• Why MTBF is a poor reliability metric
• Common objections to eliminating MTBF
• Alternatives to MTBF

format: plain


text:
# Let the section of the part recount [duplicate]

I want every part have a new counter.

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\part{part $2$}
\section{  how to make this section label become 1}
%(* now  it is 1 *)
\end{document}

• article does not provide chapters. Are you looking for report class? – Johannes_B Dec 3 '16 at 7:45
• To the actual problem: \usepackage{chngcntr} \counterwithin{section}{part}  – Johannes_B Dec 3 '16 at 7:50
• this is useful, but have a problem. section name and content is partial overlap – tiankonghewo Dec 3 '16 at 7:54

With chngcntr it's easy:

\documentclass[10pt,a4paper]{article}
\usepackage{chngcntr}

\counterwithin*{section}{part}

\author{wsy}
\title{test}
\begin{document}
\maketitle

\tableofcontents

\part{part 1}
\section{part 1, section 1}

\part{part 2}
\section{how to make this section label become 1}

\end{document}

Note \counterwithin*, so the definition of \thesection is not changed.

If you want arabic numbers for parts and sections in the style part.section, you can do like

\documentclass[10pt,a4paper]{article}
\usepackage{chngcntr}
\usepackage{xpatch}

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\makeatletter
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text:
## An adaptive algorithm for the Crank-Nicolson scheme applied to a time-dependent convection-diffusion problem

An a posteriori upper bound is derived for the nonstationary convection-diffusion problem using the Crank-Nicolson scheme and continuous, piecewise linear stabilized finite elements with large aspect ratio. Following Lozinski et al. (2009) [13], a quadratic time reconstruction is used.

Published in:
Journal Of Computational And Applied Mathematics, 233, 1139-1154
Year:
2009
Publisher:
Elsevier
ISSN:
0377-0427
Keywords:
Laboratories:


text:
## Thinking Mathematically (6th Edition)

{$\frac{7}{5}$}
9(5x -2) = 45 Step 1 : Use distributive property 9.(5x) - 9.2 =45 Step 2: Multiply 9.(5x) =45x , 9.2 = 1 45x - 18 = 45 Step 1: Add 18 on both the sides 45x - 18 + 18 = 45 + 18 Step 2: Simplify 45x = 63 Step 3: Divide both the sides by 45 $\frac{45x}{45}$ = $\frac{63}{45}$ Step 4: Simplify x = $\frac{7}{5}$ Now we check the proposed solution, $\frac{7}{5}$, by replacing x with $\frac{7}{5}$ in the original equation. Step 1: the original equation 9(5x-2) = 45 Step2: Substitute $\frac{7}{5}$ for x 9(5($\frac{7}{5}$) -2) = 45 Step 3: Multiply 5($\frac{7}{5}$) = 7 9(7 -2) = 45 Step 4: Solve parenthesis 7-2 = 5 9(5) =54 Step 5: Multiply 9(5) = 45 45 = 45 Since the check results in true statement, we conclude that the solution set of the given equation is {$\frac{7}{5}$}


text:
## University Calculus: Early Transcendentals (3rd Edition)

a. Domain : The domain is set of all $(x,y)$ in the xy -plane. b. Range: A set of all Real numbers c. Level Curves: These are Hyperbolas with the x- axis and y-axis as asymptotes when $f(x,y) \ne 0$ also, the x-axis and y-axis hen $f(x,y)=0$ d. No boundary points e. Both closed and open. f. Unbounded


text:
Faculty of Physics
M.V.Lomonosov Moscow State University
Regular Article

# The effect of constant magnetic and light-wave fields on the luminescence polarization degree in an aqueous glycyltryptophan solution

## B.D. Ryzhikov, R.E. Shikhlinskaya

### Moscow University Physics Bulletin 1991. 46. N 6. P. 53

• Article
Annotation

The effect of external fields on the luminescence of the (0.1-1) $mg$ $ml^{-1}$ aqueous glycyltryptophan solution was studied. It was found that the solution "memorizes" the prolonged external action of both a constant magnetic field with induction of 0.1 T and a linearly polarized quasimonochromatic light-wave field with intensity of about 1 $W m^{-2}$. On termination of the exposure, the relaxation processes had a nonmonotonic behavior of a cyclic character with characteristic times of the order of a few tens of minutes. The results obtained were interpreted in terms of the concept of an interrelation between structural rearrangements in water and conformational variations of solute molecules.

Rissian citation:
Б.Д. Рыжиков, Р.Э. Шихлинская.
Вестн. Моск. ун-та. Сер. 3. Физ. Астрон. 1991. № 6. С. 57.
Authors
B.D. Ryzhikov, R.E. Shikhlinskaya
Department of General Physics for the Physics Faculty, Faculty of Physics, Moscow State University, Leninskie Gory, Moscow, 119992, Russia

### Science News of the Faculty of Physics, Lomonosov Moscow State University

This new information publication, which is intended to convey to the staff, students and graduate students, faculty colleagues and partners of the main achievements of scientists and scientific information on the events in the life of university physicists.


text:
On Hamiltonicity Of Claw- And Net-Free Graphs

Author: Alexander Kelmans

ABSTRACT
The results of this paper include necessary and sufficient conditions for a claw- and net-free graph $G$ with $s,t \in V(G)$ and $e \in E(G)$ to have $(a1)$ a Hamiltonian path and Hamiltonian $s$--path, $(a2)$ a Hamiltonian $st$--path and Hamiltonian $s$- and $st$--paths containing $e$ if $G$ has connectivity one, and $(a3)$ a Hamiltonian cycle containing $e$ if $G$ is 2--connected. For example, we show that {\em if $G$ is a 2--connected claw- and net-free graph and $e = xy \in E(G)$ then $e$ belongs to a Hamiltonian cycle of $G$ if and only if $G - \{x,y\}$ is connected.} These results imply that a connected claw- and net-free graph has a Hamiltonian path and a 2--connected claw- and net-free graph has a Hamiltonian cycle [D. Duffus, R.J. Gould, M.S. Jacobson, Forbidden Subgraphs and the Hamiltonian Theme, in The Theory and Application of Graphs $($Kalamazoo, Mich., 1980$)$, Wiley, New York (1981) 297--316], and are used in [A. Kelmans, On the Claw- and Net-Free Graphs, a manuscript (1999)] to obtain more general Hamiltonicity results for $k$--connected graphs. Our proofs of $(a1)-(a3)$ are shorter, than the proofs of their corollaries in [D. Duffus, R.J. Gould, M.S. Jacobson, Forbidden Subgraphs and the Hamiltonian Theme, in The Theory and Application of Graphs $($Kalamazoo, Mich., 1980$)$, Wiley, New York (1981) 297--316], and provide polynomial time algorithms for solving the corresponding Hamiltonicity problems.

Keywords: claw, net, graph, Hamiltonian path, Hamiltonian cycle, polynomial time algorithm.

Paper Available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2004/2004-34.ps.gz


text:
# Mods that will rack your brain

1. Jul 3, 2004

### 1+1=1

show that a^13 is congruent to a (modulo 2730)

if a is an integer such that a is not divisidble by 3, or such that a IS divisible by 9, then a^7 is congruent to a (mod 63)

with these, could i just divide the mod numbers by the powers? that is what i am doing, but it seems like as though it ISN'T right... anyone help ?

2. Jul 4, 2004

### Gokul43201

Staff Emeritus
Have you come across Fermat's Little Theorem ?

a^p == a (mod p) for prime p

3. Jul 4, 2004

### 1+1=1

gokul, could i say that b/c of fermat's thm, i can say that it IS == to a, because of the fact that 13 is prime? would i need to prove any further? also, could that work for the second one? anyone else think so?

4. Jul 4, 2004

### AKG

What exactly are you asking? According to that theorem

a^13 = a (mod 13), but you need to show a^13 = a (mod 2730), so of course you need to prove more! I've never taken any courses on number theory, but you might know a theorem that takes advantage of the fact that 2730 = 0 (mod 13).

5. Jul 4, 2004

### 1+1=1

ok so i could set it up, then say, "according to fermat's theorem,... and apply the theorem to this probloem?

6. Jul 4, 2004

### AKG

Uhh.. are you asking if you're allowed to use Fermat's Little Theorem? I suppose you should ask your teacher/prof/t.a.

7. Jul 4, 2004

### Gokul43201

Staff Emeritus
I haven't taken any courses either...but this is what I would do. The folks that really know some number theory probably have a far better approach.

2730 = 13*7*5*3*2
So we are done if we can show that a^13 == a (mod n), for n=2,3,5,7, and 13.
Use both forms of the Little Theorem...
a^p == a (mod p), for all prime p and a^(p-1) == 1 (mod p) for prime p, when p does not divide a.

Clearly a^13 == a (mod 13) ---- (1)

Also a^7 == a (mod 7) and
a^6 == 1 (mod 7), so
(a^7)*(a^6) = a^13 == a*1 (mod 7) ---(2)

Again, a^5 == a (mod 5)
and a^4 == 1 (mod 5) => a^8 == 1 (mod 5)
Hence a^13 == a (mod 5) ---(3)

Likewise, show this for n=3 and n=2..and you are done.

This proof lacks some rigor (in that, for each result, the exceptions need to be covered. This is not hard at all), but I'm going to leave that to you to find and fill in.

Last edited: Jul 5, 2004
8. Jul 4, 2004

### AKG

Gokul43201

I don't know why it's sufficient to show that $a^{13} \equiv a\ (\mbox{mod }n)$ for those n, but unless a = 1 or a = 0, a cannot be congruent to a (mod 2).

1+1=1

For your second question, part of it is easy:

To prove $a^7 \equiv a\ (\mbox{mod }63)$ if a is divisible by 9, we can start by saying:

$$a^7 \equiv a\ (\mbox{mod }7)$$

Therefore, for some integer x:

$$a^7 = 7x + a$$

Now, since a is a multiple of 9, $a^7$ must also be a multiple of 9, and thus 7x must also be a multiple of 9. So, for some integer y, x = 9y. Now, we have:

$$a^7 = 63y + a$$

I think this is sufficient to prove $a^7 \equiv a\ (\mbox{mod }63)$ if a is divisible by 9.

Last edited: Jul 4, 2004
9. Jul 4, 2004

### robert Ihnot

No dividing by 0

Much of the work on the problem is good, but remember that we can not divide by 0. For that reason forget about a^p ==a implies a^(p-1) ==1. That is not true in general. For example in the case of a^13 = =a Mod 7, we have a^13 =a^7 x a^6 ==a x a^6 == a^7 ==a Mod 7, this is true even for a=7. The arithmetic is 7^7 ==7==0 Mod 7, but 7^6 ==0 Mod7 also, not 1.

Last edited: Jul 4, 2004
10. Jul 4, 2004

### Gokul43201

Staff Emeritus
If a==b (mod n) and a==b (mod m), then a==b (mod mn), if gcd(m,n)=1.

AND

If n divides a, ie. if a==0 (mod n), then clearly a^k == 0 (mod n), which implies that a^k==a (mod n).

I think I've said too much already.

Last edited: Jul 5, 2004
11. Jul 5, 2004

### Gokul43201

Staff Emeritus
I'm not sure if this is directed at me...

I insist that nothing I have said is wrong, and that I do not carelessly divide congruence relations. Please re-read what I have written.

12. Jul 5, 2004

### Gokul43201

Staff Emeritus
Post#10, line 1.

What ? Any number is always congruent to itself ! Isn't 'congruence' a reflexive relation ? Isn't 0 divisible by all numbers ?

This looks okay. But there's a shorter way...see post#10.

Last edited: Jul 5, 2004
13. Jul 5, 2004

### Muzza

No, it's not true in general, but it IS true when p is prime and p doesn't divide a (i.e a != 0 (mod p)). Besides, Gokul43201 was talking about modulo p, so your counterexample doesn't hold.

Last edited: Jul 5, 2004
14. Jul 5, 2004

### 1+1=1

first of all, thank you to all who helped with the input! i too figured some of it out on my own, but with this it reinforced what i had! isn't that what learning is about, doing some on your own then seeing what others have? i wish i could meet you all and thank yall in person!

15. Jul 5, 2004

### AKG

Oh, you're right. I was thinking that a (mod 2) didn't work if a > 2, I thought 17 (mod 4) would be wrong, and 1 (mod 4) would be the right way to do it, but I suppose they're both right, I've just never seen "17 (mod 4)" or "a (mod 2)" [for a > 2] before.

16. Jul 5, 2004

### robert Ihnot

I thought you said, I took it from what you wrote:

This what you wrote:
Also a^7 == a (mod 7) and
a^6 == 1 (mod 7), so
(a^7)*(a^6) = a^13 == a*1 (mod 7) ---(2)

17. Jul 5, 2004

### robert Ihnot

the fact of the problem

You said, Also a^7 == a (mod 7) and
a^6 == 1 (mod 7), so
(a^7)*(a^6) = a^13 == a*1 (mod 7) ---(2)

Here we really want X^13 ==X Mod 2730, so I thought from how I read you that you were not considering X =7,14,21...etc. That is how I thought about it.


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# Math Help - Poisson

1. ## Poisson

2. (a) ${e^{-4}4^6\over 6!}$.

(b) $P(X\ge 10) =1-P(X\le 9)$ where $\lambda =8$ since this is a two week period and you expect twice as many parts used.

So, $P(X\ge 10) =1-\sum_{k=0}^9 {e^{-8}8^k\over k!}$.

(c) Here you want three 'successes' in three weeks (binomial) where p is from (a)... $\biggl({e^{-4}4^6\over 6!}\biggr)^3$.

Sorry, I do not understand (d).


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# Computability Part 5: Elementary Cellular Automata

In the previous post we talked about Conway’s Game of Life as a well-known cellular automaton. In this post we will cover even simpler automata - the elementary cellular automata. Stephen Wolfram covers them extensively in his book, A New Kind of Science.

To recap, we defined a cellular automaton as a discrete n-dimensional lattice of cells, a set of states (for each cell), a notion of neighborhood for each cell, and a transition function mapping the neighborhood of each cell to a new cell state.

An elementary cellular automaton is 1-dimensional - an array of cells. A cell can be either on or off (just like in Conway’s Game of Life). The neighborhood of a cell, meaning the cells that we take into account when we determine the next state of the next generation, consists of the cell itself and its left and right neighbors.

For example, we can define an elementary cellular automaton with the following rules:

[ on,  on,  on] -> off
[ on,  on, off] -> off
[ on, off,  on] -> off
[ on, off, off] ->  on
[off,  on,  on] -> off
[off,  on, off] ->  on
[off, off,  on] ->  on
[off, off, off] -> off

If we start with a single on cell and produce 10 generations, we get (using # to mean on):

         #
###
#   #
### ###
#       #
###     ###
#   #   #   #
### ### ### ###
#               #
###             ###

## Rule encoding

The elementary cellular automata can easily be enumerated exhaustively: the neighborhood of a cell can be in only one of 8 states, as we saw above: [on, on, on], [on, on, off], … [off, off, off]. The transition function maps each of these possible states to either on or off. If we think of the on/off as a bit, we need 8 bits to represent the transition function.

[ on,  on,  on] -> off
[ on,  on, off] -> off
[ on, off,  on] -> off
[ on, off, off] ->  on
[off,  on,  on] -> off
[off,  on, off] ->  on
[off, off,  on] ->  on
[off, off, off] -> off

can be represented as the binary number 00010110, which, in decimal, is 22 (where [off, off, off] is the least significant bit). We can represent numbers from 0 to 255 in 8 bits, so there are exactly 256 elementary cellular automata. This encoding is referred to as Rule as in “transition rule”. The elementary cellular automata in our above example is called Rule 22 .

## Elementary cellular automata behavior

A common way to plot the evolution of an elementary cellular automata over multiple generation is to render each generation below the previous one, like our above example using # for on. A more condensed version with 1 pixel per cell of running rule 22 for 301 generations looks like this:

At this level, we can clearly see patterns emerging in the automaton. We get an even more interesting view if, instead of starting with just a single on cell, we start with a random state - an array of random on and off cells. Here is rule 22 starting with 301 random cells and running for 301 generations:

We can also easily see some of the automatons are complements of other automatons: if we simply flip each bit, we end up with a complementary version. Rule 22’s complement is Rule 151:

We can also reflect a rule by swapping the transitions for [on, off, off] with [off, off, on] and [on, on, off] with [off, on, on]. This doesn’t work for rule 22, since its reflection is still 22, but, for example, rules 3 and 17 are reflections of each other.

Rule 3:

[ on,  on,  on] -> off
[ on,  on, off] -> off
[ on, off,  on] -> off
[ on, off, off] -> off
[off,  on,  on] -> off
[off,  on, off] -> off
[off, off,  on] ->  on
[off, off, off] ->  on

Renders as:

Rule 17:

[ on,  on,  on] -> off
[ on,  on, off] -> off
[ on, off,  on] -> off
[ on, off, off] ->  on
[off,  on,  on] -> off
[off,  on, off] -> off
[off, off,  on] -> off
[off, off, off] ->  on

Renders as:

That means that, even though there are 256 possible automata, from behavioral perspective, some are complements or reflections of others thus exhibit the same behavior. In fact, there are only 88 uniquely behaving automata, all others being complements and/or reflections of these.

## Implementation

Let’s look at a Python implementation. We will represent the state of an automaton as a list of Boolean cells. We can encode the state of a neighborhood as a 3 bit number: left neighbor * 4 + cell * 2 + right neighbor. Given a list of cells and the index of a cell, we have:

def neighbors(cells, i):
return (cells[i - 1] if i > 0 else False) * 4 + \
cells[i] * 2 + \
(cells[i + 1] if i < len(cells) - 1 else False)

If we run off the ends of the list, we assume the state of that cell is off. In Python, False becomes 0 and True becomes 1 if we do arithmetic with them, so this function will return a number between 0 and 7.

We can derive the transitions from the rule number by taking a rule number and expanding it into a dictionary that maps each value from 0 to 7 to the corresponding bit in the rule number value:

def transition(rule):
return {i: rule & (1 << i) != 0 for i in range(8)}

This might be a bit hard to understand, so let’s work through an example. Let’s take Rule 22. The binary representation of Rule 22 is 00010110. We’re iterating over the range 0…7 (i) and for each of these values, we shift 1 exactly i bits left. Then we check if the rule logic AND this shifted bit is different than 0.

For i == 0: 00010110 & (1 << 0), which is 00010110 & 00000001, we get False, so transitions[0] = False.

For i == 1: 00010110 & (1 << 1), which is 00010110 & 00000010, we get True, so transitions[1] = True.

For i == 7: 00010110 & (1 << 7), which is 00010110 & 10000000, we get False, so transitions[7] = False.

Remember the keys of the dictionary are neighborhood states.

Now we just need a function that takes a rule, an initial state, and the number of steps we want to run. The function will start with the initial state, then at each step, update the list of cells using the transition function:

def run(rule, initial_state, steps):
t, cells = transition(rule), initial_state

for _ in range(steps):
yield cells
cells = [t[neighbors(cells, i)] for i in range(len(cells))]

We talked about two ways to look at cellular automata: starting with a single on cell, or starting with a random initial state.

Let’s implement an initial_state function which takes a cell count as input and returns a list of cells, all of which are off except the middle one:

def initial_state(cell_count):
result = [False] * cell_count
result[cell_count // 2] = True
return result

We’ll also want a random_initial_state which takes a cell count and returns a random cell list. We’ll take advantage of the fact that Python supports arbitrarily large integers natively, so we’ll just generate a random number with cell_count bits, then derive the cell list from that (if a bit is 1, the corresponding cell is on):

import random

def random_initial_state(cell_count):
seed = random.randint(0, 2 ** cell_count - 1)
return [seed & (1 << i) != 0 for i in range(cell_count)]

Here is all the code in one listing:

def neighbors(cells, i):
return (cells[i - 1] if i > 0 else False) * 4 + \
cells[i] * 2 + \
(cells[i + 1] if i < len(cells) - 1 else False)

def transition(rule):
return {i: rule & (1 << i) != 0 for i in range(8)}

def run(rule, initial_state, steps):
t, cells = transition(rule), initial_state

for _ in range(steps):
yield cells
cells = [t[neighbors(cells, i)] for i in range(len(cells))]

def initial_state(cell_count):
result = [False] * cell_count
result[cell_count // 2] = True
return result

import random

def random_initial_state(cell_count):
seed = random.randint(0, 2 ** cell_count - 1)
return [seed & (1 << i) != 0 for i in range(cell_count)]

Here is how we can use this to print the first 30 steps of Rule 22:

for state in run(22, initial_state(61), 30):
print(''.join(['#' if e else ' ' for e in state]))

## Wolfram classification

Wolfram analyzed the behavior of cellular automata and classified them in 4 classes (called Wolfram classes). These go beyond elementary cellular automata to cover other cellular automata like, for example, ones where the next generation of a cell is not determined only by the cell and the two cells next to it, rather the neighborhood includes next-next cells. In this post we’ll stick to elementary cellular automata though.

### Class 1

Class 1 automata converge quickly to a uniform state. For example rule 0 becomes all off in one generation:

It’s complement, rule 255, becomes all on in one generation:

### Class 2

Class 2 automata converge quickly to a repetitive state. For example rule 4:

### Class 3

Class 3 automata appear to remain in a random state, without converging. Rule 22, which we started with above, exhibits this type of behavior:

### Class 4

The most interesting class of cellular automata, class 4, has a quite remarkable behavior: areas of cells end up in static or repetitive state, while some cells end up forming structures that interact with each other. Rule 110 is the only elementary cellular automaton that exhibits this behavior:

## Turing completeness

The fact that Rule 110 has areas of cells that are static or repetitive while some other cells form structures should remind you of the Conway’s Game of Life spaceships we discussed in the previous post. In the previous post, we saw that the Game of Life is Turing complete, and how a Turing machine was “implemented” using spaceships as signals “processed” by other patterns.

Turns out Rule 110 is also Turing complete. Stephen Wolfram conjectured this in 1985, and the conjecture was proved in 2004 by Matthew Cook[1]. Cook uses Rule 110 gliders (interacting structures) to emulate a cyclic tag system. We saw in Computability Part 3: Tag Systems that cyclic tag systems can emulate tag systems, and an m-tag system with $$m \gt 1$$ is Turing complete.

Rule 110, an elementary cellular automaton, is also capable of universal computation. And while this all might seem very abstract, cellular automata are so simple they show up in nature:


text:
# [Namazu-users-en] Re: namazu stopped working

Mon Nov 28 20:02:39 JST 2005

IEM - network operating center wrote:
>
> >>i guess it is a problem with some multi-byte characters.
> >
> >
> > The cause might be another one.
> > If the document file can be gotten by specifying the document
> > that makes trouble, it is
> > likely to be able to pinpoint the cause.
>
> i am not sure what you mean here.
> should i try to find the document (or one of the documents) that causes
> the trouble?

It is not understood whether the cause of destroying the cause
(8bit character) to which mknmz puts out Warning and the index
is the same.

Therefore, it is necessary to get the document that destroys the
index and to verify it.

> i was able to track the problem down to following line:
> <html>
> sorry, whenever did you think iĺl put windows code in mine?
> </html>

# Are referred to to the numeric realities?

Does the error occur by nmzchkw.pl when the index is made only with
this document in mknmz?

The index seems not to be broken in the environment in hand.

> if you need the entire document, i can send it of course.

Please send the document that destroys the index.

> > By the way,
> > I think that warning is improved by the following corrections.
> > (no guarantee)
> >
> > -    \$\$contref =~ tr/\x00-\x08\x0b-\x0c\x0e-\x1a/ /;
> > +    \$\$contref =~ tr/\x00-\x08\x0b-\x0c\x0e-\x1a\x80-\xff/ /;
> >
>
> unfortunately this did not help.

How did it become it?
- With Warning generated
- With the index breaks
- The both

> the locale is set to "en_GB.ISO-8859-15": i have no idea how this
> happened, since i am in austria (so there is no need to have a british
> locale); could this be related to the problem? should i choose one with
> ISO-8859-1?

It is not ISO-8859-1, and either C will have to be used.
Because Namazu supports only English (US-ASCII).
--
=====================================================================


text:
# Quark Matter 2018

13-19 May 2018
Venice, Italy
Europe/Zurich timezone
The organisers warmly thank all participants for such a lively QM2018! See you in China in 2019!

## The sPHENIX HF-jet physics program

15 May 2018, 17:00
2h 40m
First floor and third floor (Palazzo del Casinò)

### First floor and third floor

#### Palazzo del Casinò

Poster Future facilities, upgrades and instrumentation

### Speaker

Dr Jin Huang (Brookhaven National Lab)

### Description

Jets initiating from heavy flavor quarks (HF-jets) are sensitive to the collisional energy loss of high energy partons when traversing through Quark Gluon Plasma. Using sPHENIX, a state-of-the-art jet detector, we will perform the first HF-jet measurements at RHIC, which will include the nuclear modification and flow of $b$-jets, and the momentum balance in di-$b$-jet pairs. A variety of $b$-Jet tagging algorithms have been developed, which select the HF-jet sample rich in tracks displaced from the primary collision point as measured by the high precision sPHENIX MAPS vertex tracker. The detection method, physics projection and possible impact to the field of our understanding of parton energy loss will be discussed in this poster.

Content type Experiment sPHENIX Presenter name already specified


text:
# Two-Way ANOVA using SPSS

Nikki Kamouneh

Posted on
ANOVA SPSS Sum of Squares Two-Way

This tutorial is going to take the theory learned in our Two-Way ANOVA tutorial and walk through how to apply it using SPSS. We will use the Moore dataset which can be downloaded using this link.

The data is from an experimental study which consists of subjects in a “social-psychological experiment who were faced with manipulated disagreement from a partner of either of low or high status. The subjects could either conform to the partner’s judgment or stick with their own judgment.” (John Fox, Sanford Weisberg and Brad Price (2018). carData: Companion to Applied Regression Data Sets. R package version 3.0-2. https://CRAN.R-project.org/package=carData).

We are interested in exploring an individuals conformity based on their partner’s status:

• Low
• High

We are also interested in exploring whether their F-score category (a measure of authoritarianism) affects outcomes or interacts with partner status. The three factor levels are:

• Low
• Medium
• High

Our sample size is $$N = 45$$. First, let’s inspect the data for outliers or funky distributions by creating a boxplot of conformity scores within each combination of partner.status and fcategory. To create the boxplot in SPSS, go to Graphs $$\rightarrow$$ Chart Builder…


text:
# What is 0.375 as a fraction ?

#### Risposta:

0.375 = 3/8

#### Spiegazione:

Here's a general method for converting terminating decimal expansions into fractions...

• If the last non-zero decimal digit is 2,4,6 or 8, then write down color(red)(5) and multiply by it.

• If the last non-zero decimal digit is 5, then write down color(red)(2) and multiply by it.

• If the last non-zero digit is something else (i.e. 1, 3, 7 or 9) then write down color(red)(10) and multiply by it.

• Repeat until you end up with an integer.

• Multiply all the color(red)(5)'S, color(red)(2)'s or color(red)(10)'s you found together to form the denominator of your fraction and use the integer you ended up with as the numerator.

Quindi nel nostro esempio:

• 0.375 termina con 5, so write down color(red)(2) and double 0.375 ottenere 0.75

• 0.75 termina con 5, so write down color(red)(2) and double 0.75 ottenere 1.5

• 1.5 termina con 5, so write down color(red)(2) and double 1.5 ottenere 3.

• Having reached an integer 3, we find the required fraction is:

3/(color(red)(2) * color(red)(2) * color(red)(2)) = 3/8