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Synthesis, Characterization and Properties of Biomass and Carbon Dioxide Derived Spec. Matrices 2019; 7:1–19 Research Article Open Access Kazumasa Nomura* and Paul Terwilliger e-Polymers 2019; 19: 535–544 Self-dual Leonard pairs Research Article Open Access https://doi.org/10.1515/spma-2019-0001 Received May 8, 2018; accepted September 22, 2018 Cheng-Hung Chung*, Wen-Chang Shih and Wei-Ming Chiu Abstract: Let F denote a eld and let V denote a vector space over F with nite positive dimension. Consider Synthesis, characterizationa pair A, A∗ of diagonalizable andF-linear properties maps on V, each ofof 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 biomass and carbonthere exists dioxide an automorphism derived of the endomorphism polyurethane algebra 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 ↔ reactive hot-meltduality. adhesives 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-2019-0057 and 24 bases for V.Keywords: reactive polyurethane; renewable resource; Received June 05, 2019; accepted July 31, 2019. biomass; carbon dioxide; adhesive strength Keywords: Leonard pair, tridiagonal matrix, self-dual Abstract: Polyurethane reactive hot-melt adhesives (PURHs) are frequently employed in industries;Classication: however,17B37,15A21 there is still a need to develop more sustainable and ver- 1 Introduction satile methodologies to expand the functions and fabri- cation of these important materials. Renewable feedstock Polyurethanes reactive hot-melt adhesives (PURHs) can give PURHs with new functions, andIntroduction reduce envi- are 100% solid, isocyanate-terminated urethane ronmental impact. This study focuses on synthesizing prepolymers that are synthesized by reacting polyols PURHs using polyols derived from biomassLet F (plants)denote and a eld with and excess let V isocyanatedenote a (Scheme vector space 1) (1). overPURHsF arewith applied nite positive dimension. We consider a greenhouse gas (CO ) resources. These PURHs were cha- as adhesives in a molten state, cooled to solidify, and 2 pair A, A∗ of diagonalizable F-linear maps on V, each of which acts on an eigenbasis for the other one in an racterized by multiple techniques, includingirreducible solid-state tridiagonal can fashion.be subsequently Such a pair cured is called through a Leonard the reaction pair (see of [13, Denition 1.1]). The Leonard pair 13C nuclear magnetic resonance (NMR), a dynamic mecha- the isocyanate groups and ambient moisture (2). The A, A∗ is said to be self-dual whenever there exists an automorphism of the endomorphism algebra of V that nical analysis (DMA), single-lap adhesive joints strength moisture-cured mechanism is the isocyanate groups to swaps A and A∗. In this case such an automorphism is unique, and called the duality A A∗. of stainless steel, and hydrolytic ageing. The PURH film react with moisture to form an unstable intermediate, ↔ based on biomass poly(tetramethylene ether)The glycol literature (bio- containscarbamic many acid. examplesThe unstable of self-dual carbamic Leonard acid decomposes pairs. For instance (i) the Leonard pair associ- PTMEG) exhibited better water vapor permeability,ated with an tensile irreducible into carbon module dioxide for the and Terwilliger an amine; algebra then the of amine the hypercube reacts (see [4, Corollaries 6.8, 8.5]); (ii) a strength, and adhesive joints propertiesLeonard than pair PURHs of Krawtchouk with isocyanate type (see group [10, Denitionto form a 6.1]);urea (iii)linkage the Leonard(3). In pair associated with an irreducible based on cashew nutshell liquid (CNSL)module polyester for diol the and Terwilliger adhesive algebra joints application, of a distance-regular the formula graphof PURHs that contain has a spin model in the Bose-Mesner alge- poly(propylene carbonate)-poly(propylenebra (seeglycol) [1, Theorem],(PPC- crystalline [3, Theorems and linear 4.1, 5.5]); aliphatic (iv) an polyester appropriately polyols normalized can totally bipartite Leonard pair PPG) copolymer diol. The polyols blend of(see bio-PTMEG [11, Lemma with 14.8]); developed (v) the fast Leonard adhesive pair strength consisting with of increasing any two ofjoined a modular Leonard triple A, B, C (see [2, biomass and CO based polycarbonate diols respectively assemblies productivity (4). In contrast, polyether polyols, 2 Denition 1.4]); (vi) the Leonard pair consisting of a pair of opposite generators for the q-tetrahedron alge- provided PURHs films excellent hydrolysisbra, resistance acting on and an evaluationsuch as poly(propylene module (see glycol) [5, Proposition (PPG), are 9.2]). used Theto produce example (i) is a special case of (ii), and the adhesive strength on single-lap adhesively bonded stain- PURHs with low temperature flexibility and softness (5). examples (iii), (iv) are special cases of (v). less steel specimens. The work herein demonstrates that Furthermore, PURHs made from poly(tetramethylene Let A, A∗ denote a Leonard pair on V. We can determine whether A, A∗ is self-dual in the following way. various renewable polyols can be employed in a sustai- ether) glycol (PTMEG) can exhibit better mechanical d nable fashion to optimize the structuresBy and [13, properties Lemma of 1.3] eachproperties eigenspace and hydrolytic of A, A stability∗ has dimension than PPG (6,7). one. Let θi i= denote an ordering of the eigen- { } d PURHs for important applications. values of A. For ≤ i Most≤ d letof thevi denotepolyester a andθi-eigenvector polyether polyols for A .currently The ordering θi i= is said to be standard d { } d whenever A∗ acts onused the for basis PURHsvi i=arein via an from irreducible petroleum-based tridiagonal resource fashion. If the ordering θi i= is standard d { } { } then the ordering θandd−i hasi= beenis also extensively standard, studied. and no In further contrast, ordering renewable is standard. Similar comments apply to d { } d A∗. Let θ denoteresource a standard makes ordering it possible ofthe to eigenvaluesdevelop the polyols of A. Then withA , A∗ is self-dual if and only if θ { i}i= { i}i= * Corresponding author: Cheng-Hung Chung, Graduateis a standard Institute orderingof new of functions, the eigenvalues which are of Aless∗ (see accessible [7, Proposition via petroleum- 8.7]). Precision Manufacturing, National Chin-Yi University of Technology, based products. These polyols derived from plant source Taichung 41170, Taiwan (R.O.C.), email: [email protected] (e.g., bio-based acids and diols, natural oil polyols, etc.) Wei-Ming Chiu, Graduate Institute of Precision Manufacturing, and carbon dioxide (CO ) can provide PURHs with novel National Chin-Yi University of Technology, Taichung 41170, Taiwan 2 properties and sustainable application (8-10). Earlier (R.O.C.), email: [email protected] *Corresponding Author: Kazumasa Nomura: Tokyo Medical and Dental University, Ichikawa, 272-0827,Japan, Wen-Chang Shih, Department of Chemical and E-mail:Materials [email protected] study has been done on the polyester polyols blend of Engineering, National Chin-Yi University of Technology,Paul Terwilliger: Taichung Departmentbio-based of Mathematics,1,3-propanediol University (1,3-PDO) of Wisconsin, backbone Madison, with WI53706, USA, E-mail: 41170, Taiwan (R.O.C.), email: [email protected]@math.wisc.edupetroleum-based 1,4-butanediol (1,4-BDO) backbone for 4, Open Access. © 2019 Chung 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. 536 C.-H. Chung et al.: Synthesis, characterization and properties of biomass and carbon dioxide derived Scheme 1: Scheme for the sythesis and moisture-curing processes of PURHs. 4’-methylene diphenyl diisocyanate (MDI) based PURHs moisture permeability, tensile strength, adhesive joint (11). The addition of bio-based 1,3-PDO backbone polyol strength, and hydrolysis resistance properties. In this in the PURHs, the molten viscosity could be lowered while study, the bio-based poly(tetramethylene ether) glycol retaining the adhesively bonded strength of stainless (bio-PTMEG) as a standard specimen to contrast cashew steel specimens, and humidity resistance as compared nutshell liquid (CNSL) based polyester diol, the copolymer to petroleum-based 1,4-BDO backbone polyol PURHs. diol of CO2-based PPC and PPG, CO2-based PPC diol and Another application to PURHs, the poly(propylene biomass-based aliphatic polycarbonate diol on the PURHs carbonate) polyols (PPC polyols) prepared via CO2 and properties. epoxides copolymerization have been substituted some traditional transesterification process products as an inexpensive, abundant, nontoxic and renewable resource. The addition of PPC diols in the MDI based PURHs showed 2 Experimental better adhesive strength than the petroleum-based polyols such as poly(1,6-hexamethylene adipate) and poly(1,4- 2.1 Materials butylene adipate) on metal (aluminum and stainless steel), plastic (poly(methyl methacrylate), polycarbonate), and A biomass based poly(tetramethylene ether) glycol leather to halogenated styrene-butadiene rubber (SBR) (bio-PTMEG, bio-based PolyTHF® 1000) was obtained joints of upper-sole bonding (12,13). from BASF Co., Ltd., Taiwan. A cashew nutshell liquid Although many studies have attempted to optimize (CNSL) based polyester diol (Cardolite® NX-9203LP) was and maximize the renewable resource polyols in PURHs as obtained from Cardolite Co., Ltd., China. A CO2-based adhesive joints, little work had been done on expanding the polycarbonate polyol, poly(propylene carbonate) diol functional applications of PURHs, such as their application (PPC diol, Converge® polyol 212-10) and poly(propylene in breathable clothing and biomedical materials (14,15).
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