Spec. Matrices 2019; 7:1–19 Research Article Open Access Kazumasa Nomura* and Paul Terwilliger e-Polymers, 2019; 19: 470–476 Self-dual Leonard pairs Research Article Open Access https://doi.org/10.1515/spma-2019-0001 Received May 8, 2018; accepted September 22, 2018 Xiaodong Wang1,2, Yuan Hu1,*, Lei Li2,3, Haiyan Fang2,3, Xu Fan2,3 and Shengfei Li2,3 Abstract: Let F denote a eld and let V denote a vector space over F with nite positive dimension. Consider Preparation and performancea pair A, A∗ of diagonalizable ofF-linear polypropylene maps on V, each of which acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which separator modifiedthere existsby anSiO automorphism2/PVA of thelayer endomorphism for algebralithium of V that swaps A and A∗. Such an automorphism is unique, and called the duality A A∗. In the present paper we give a comprehensive description of this ↔ batteries duality. In particular, we display an invertible F-linear map T on V such that the map X TXT− is the duality → A A∗. We express T as a polynomial in A and A∗. We describe how T acts on ags, decompositions, ↔ https://doi.org/10.1515/epoly-2019-0049 and 24 bases for V. Received March 27, 2019; accepted June 13, 2019. 1 Introduction Keywords: Leonard pair, tridiagonal matrix, self-dual Abstract: An aqueous silica (SiO ) dispersion was prepa- 2 Lithium-ion batteries are widely used in the field of red by using silica as ceramic particles, polyvinyl alcohol Classication: 17B37,15A21electronic equipment due to their high energy density (PVA) as a binder and deionized water as a dispersion and long cycle life (1,2). As a key component of lithium medium. A SiO ceramic layer was applied to the surface 2 batteries, battery separators mainly serve to isolate of the polypropylene (PP) separator by dip coating. The positive and negative electrodes, retain electrolyte, and separators before and after modification were charac- Introductionallow lithium ions to pass through (3,4). At present, terized by XRD, SEM, DTA, lyophilic performance test, commercial lithium battery separators are mainly contact angle test and heat resistance test. The separators Let F denote a eldpolyolefin and let Vporousdenote membranes a vector spacesuch as over polyethyleneF with nite (PE) positive dimension. We consider a were assembled into lithium-ion batteries for electroche- pair A, A∗ of diagonalizableand polypropyleneF-linear maps(PP) (5-9). on V Although, each of these which membranes acts on an eigenbasis for the other one in an mical performance test. The results show that after the irreducible tridiagonalhave fashion. the advantages Such a pairof chemical is called inert, a Leonard easy processing pair (see [13, Denition 1.1]). The Leonard pair successful introduction of SiO /PVA coating on the surface 2 A, A∗ is said to be self-dualand low cost, whenever the low there surface exists energy an automorphismand hydrophobicity of the endomorphism algebra of V that of PP separator, the lyophilic and heat resistance and swaps A and A∗. Inof this polyolefin case such materials, an automorphism as well as isthe unique, stretching and calledpore the duality A A∗. electrochemical performance of PP separator have been ↔ forming process, make this kind of membrane have poor improved significantly. The battery rate performanceThe literature and contains many examples of self-dual Leonard pairs. For instance (i) the Leonard pair associ- heat resistance and poor wetting to electrolyte, which cycle performance are significantly improved.ated with Especially an irreducible module for the Terwilliger algebra of the hypercube (see [4, Corollaries 6.8, 8.5]); (ii) a greatly affects the cycle and safety performance of the the capacity retention rate of the originalLeonard separator pair ofwas Krawtchouk type (see [10, Denition 6.1]); (iii) the Leonard pair associated with an irreducible battery (10-12). only 75.79% at 100 charge-discharge times,module and that for theof the Terwilliger algebra of a distance-regular graph that has a spin model in the Bose-Mesner alge- In order to solve the above problems, the simplest and modified separator was as high as 87.18%.bra (see [1, Theorem], [3, Theorems 4.1, 5.5]); (iv) an appropriately normalized totally bipartite Leonard pair most effective method is to introduce a ceramic layer on the (see [11, Lemma 14.8]);surface (v) theof the Leonard membrane pair to consisting prepare a ofceramic any two composite of a modular Leonard triple A, B, C (see [2, Keywords: silicon dioxide; polypropylene separator; Denition 1.4]); (vi)separator. the Leonard After pair the ceramic consisting layer of is a applied pair of to opposite the surface generators for the q-tetrahedron alge- lithium ion battery; thermal resistance; electrochemical bra, acting on an evaluationof the polyolefin module separator, (see [5, Proposition the thermal 9.2]). stability The exampleof the (i) is a special case of (ii), and the performance examples (iii), (iv) arecomposite special separator cases of (v).is remarkably improved due to the Let A, A∗ denoteexcellent a Leonard high pair temperature on V. We canresistance determine of the whether ceramicA, A∗ is self-dual in the following way. d By [13, Lemma 1.3] eachparticles. eigenspace At the same of A time,, A∗ sincehas dimensionthe surface of one. the Letceramicθ denote an ordering of the eigen- { i}i= particles contains a hydroxyl group, the wettability of the d values of A. For ≤ i ≤ d let vi denote a θi-eigenvector for A. The ordering θi i= is said to be standard electrolyte to thed separator can be remarkably improved. { } d whenever A∗ acts on the basis vi i= in an irreducible tridiagonal fashion. If the ordering θi i= is standard In addition,d { since} the coating layer can additionally { } then the ordering θd−i i= is also standard, and no further ordering is standard. Similar comments apply to d { retain} a part of the electrolyte, the liquid absorption of d * Corresponding author: Yuan Hu, State Key LaboratoryA∗. Let ofθ Fire denote a standard ordering of the eigenvalues of A. Then A, A∗ is self-dual if and only if θ { i}i= the composite separator is remarkably improved. Juang { i}i= Science, University of Science and Technology of China, Hefei, is a standard ordering of the eigenvalues of A∗ (see [7, Proposition 8.7]). 230026, P. R. China, e-mail: [email protected] et al. used microwave method to deposit titanium dioxide Xiaodong Wang, State Key Laboratory of Fire Science, University of on polymer film to prepare lithium ion battery composite Science and Technology of China, Hefei, 230026, P. R. China film, which improved thermal and dimensional stability Xiaodong Wang, Lei Li, Haiyan Fang, Xu Fan and Shengfei Li, School (13). Xu et al. prepared ultra-thin multifunctional films on of Chemistry & Material Engineering, Chaohu University, Chaohu, *Corresponding Author:polyethylene Kazumasa (PE) Nomura: filmsTokyo by using Medical polyacrylic and Dental acid University, (PAA) Ichikawa, 272-0827,Japan, 238000, P. R. China E-mail: [email protected] and zirconia (ZrO ) through simple self-assembly process Lei Li, Haiyan Fang, Xu Fan and Shengfei Li, KeyPaul Laboratory Terwilliger: of Department of Mathematics,2 University of Wisconsin, Madison, WI53706, USA, E-mail: without destroying the special porous structure of the Functional Material Processing and Application,[email protected] Chaohu University, Chaohu, 238000, P. R. China films (14). Compared with the original PE separator, the Open Access. © 2019 Wang et al., published byOpen De Gruyter. Access. ©This2019 work Kazumasa is licensed Nomura under and the Paul Creative Terwilliger, Commons published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Attribution alone 4.0 License. X. Wang et al.: Preparation and performance of polypropylene separator 471 composite membrane has good electrolyte wettability and continue stirring at 90°C until PVA was completely and absorption, high ionic conductivity, which makes dissolved. 0.5 g of nano-SiO2 (average particle 20 nm, lithium ion batteries have excellent cycle performance. Aladdin) powder was added in the PVA solution system. Shi et al. prepared a separator modified by nanometer After stirring magnetically for 2 h, the PVA/nano-SiO2 ceramic powders for lithium-ion battery, which showed slurry dropped to room temperature. The coating slurry good high temperature stability, high absorption and was prepared after standing for 3 h. wettability of electrolytes (15). Wang et al. applied the The suitable PP separator was cut off, and a suitable Al2O3/phenolphthalein polyetherketone (PEK-C) layer amount of coating solution was applied to the separator by on the surface of the PP film to improve the thermal dipping coating method. The modified coating separator stability of the separator and the assembly safety was dried at room temperature until no obvious liquid performance of the battery, but did not significantly existed. Then the coating film was dried in a vacuum oven improve the hydrophilic performance of the membrane at 60°C for 24 h. (16). A composite membrane as a separator for lithium-ion batteries was studied by Zhang et al. (17). The composite membrane is mainly composed of calcium carbonate 2.2 Performance test powder and polymer binder. It has high porosity and good capillary properties, and has good wettability with The native separator and modified separator were liquid electrolyte. Among the above methods, composite studied by XRD (D/max-2500ps, Regaku). Test coatings can improve the performance of separator to a conditions: room temperature, Ka diffraction of copper certain extent, but the preparation process of composite target, test voltage 40 Kv, current intensity 200 mA, coatings is complex and the cost is high.