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Carbon bers combined with higher-performance matrix materials such as polyimides or polybenzimidazole offer the potential for further improvements Civilian applications include sporting goods golf-club shafts, tennis racquets, shing rods, and boat masts The lighter weight of the carbon ber composite (compared to the metal component it replaces) allows a redesign of the item for improved performance The lighter golf-club shaft allows a heavier head design Other applications include high-quality loudspeakers and various medical applications, including arti cial limbs and surgical implants Other polymers undergo cyclization, but there are no commercial applications Poly (methacrylic acid) cyclizes by anhydride formation and poly(methyl vinyl ketone) by condensation (with dehydration) between methyl and carbonyl groups 9-8 OTHER REACTIONS Hydrogenation of double bonds is practiced with some elastomers such as SBR and NBR to increase high temperature and oxidative resistance [Hsieh, 1998; Hsieh and Quirk, 1996, Wrana et al, 2001].



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NET code. ' Convert Excel (XLSX) to PDF. ExcelFile.Load("Workbook.xlsx").Save(​"Workbook.pdf"). For a more detailed example, see Convert Excel to PDF in C# ...

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A very powerful rule of thumb for calculations involving the Levy-Civita tensor relates the product of two such tensors to a determinant of Kronecker deltas. We can derive this rule of thumb in the following way. Let Xi, Xj, Xk and Xl. Xm , Xn be two rearrangements of x I, X2, X3 as defined in Section E.2. Now suppose that we try to expand each of the vectors Xi, Xj, Xk in terms of Xl, Xm , Xn:





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Only partial hydrogenation is performed since some double bonds are required to achieve the subsequent crosslinking required to achieve elastic behavior Epoxidation of double bonds in poly(1,3-dienes) yield epoxy resins that are subsequently crosslinked (Sec 2-12d) 9-9 GRAFT COPOLYMERS Graft and block copolymers contain long sequences of each of two different monomers in the copolymer chain A block copolymer contains the sequences of the two monomers in a continuous arrangement along the copolymer chain while a graft copolymer contains a long sequence of one monomer (often referred to as the backbone polymer) with one or more branches (grafts) of long sequences of a second monomer (Secs 2-13a, 6-1b) A variety of techniques for synthesizing such structures have been studied.

+ (Xi' Xm)Xm + (Xi' Xn)Xn , == (Xj . Xz)Xl + (Xj' Xm)Xm + (Xj . Xn)Xn , == (Xk . Xz)X[ + (Xk . Xm)Xm + (Xk xn)x n .

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Most methods of synthesizing graft copolymers involve radical polymerization, but ionic graft polymerizations are also possible [Battaerd and Tregear, 1967; Ceresa, 1973, 1976; Dreyfuss and Quirk, 1986; Gnanou, 1996; Gnanou and Taton, 2002; Rempp and Lutz, 1989] Graft polymerizations are carried out in heterogeneous as well as homogeneous systems Three methods exist for synthesizing graft copolymers Grafting onto involves the reaction between functional groups on two different polymers:.

(E. 13)

9-43

Grafting from involves a polymer with functional groups that initiate polymerization of monomer:

(E. 14)

9-44

(E.IS)

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Grafting through involves the polymerization (or copolymerization) of a macromonomer, usually a vinyl macromonomer:

These expansions will coincide with Xi, Xj, Xk only if the set Xl, Xm , Xn contains all the unit vectors Xl> X2, X3. If any index is repeated in i, m, n, one ofthe unit vectors Xl, X2, X3 will be missing, so that one of x~, xj, X~ will vanish. This means that x~ . (xj A X~) will be equal to eijk, as in (E.II), if i, m, n contain all three indices 1,2,3; otherwise, x~ . (xj A xU will vanish. We can write this statement as (E. 16) where the square of the Levy-Civita is I if l, m, n is any permutation of 1, 2, 3, and it vanishes if any index is repeated. Now, using (E. 13)-(E. 15), we find that

9-45

x~ . (xj A X~)

Grafting from is useful since many polymers can be used together with a range of reaction conditions (radical and ionic). Grafting through is also convenient but requires the synthesis of a macromonomer. Grafting onto is the least useful method since reactions between polymers are slow and there are relatively limited numbers of polymers with functional groups capable of reacting with each other. Graft copolymers can be synthesized with different densities of graft chains attached to the backbone polymer. Higher graft densities are achieved in the grafting-onto and graftingfrom methods by using a backbone polymer with a higher density of A groups. The grafting density in the grafting-through method is controlled by the ratio of macromonomer to comonomer. The graft copolymer is essentially the equivalent of a star polymer when there are only a few graft chains. The terms comb polymer and brush polymer are used to describe a graft copolymer with a high density of graft chains. The term brush polymer is the more correct description because of the conformational freedom allowed the graft chains as well as the backbone polymer. The graft polymer resembles a brush used in the laboratory to clean the inside of a test tube or bottle.

(xm A Xn)](Xj . Xm)(Xk . xn)

9-9a 9-9a-1

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