Modern Polymer Spect..

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3.5 Towards Lnrger Molecules: From Oligoiiiers to <strong>Polymer</strong>s 101 fact that y is the phase shift of two equivalent oscillators belonging to two adjacent chemical units within the crystallographic repeat unit (see above). Following the same algebraic procedures as for the finite molecule the final secular equation to be solved is [50] where GR(~) = (GR)’ + c [(GR)S eisq + (GR’)S ePisV‘] (3-44) S FR(~) = (FR)’ + c [(FR)~ eisv + (FR’)~ eCiSv] S (3-45) Equation (3-43) is of 3pth degree in A. There are 3p characteristic roots (i = 1 . . .3p) for each value of the phase y. The r = 3p functions i~,(v,) can be interpreted as branches of a multiple valued function which is the dispersion relation for a onedimensional polymer chain. The solution reached in Eq. (3-43) is analogous to that previously attained by many authors in solid-state physics for simple tridimensional lattices [51]. The important digerelice is that Eq. (3-43) treats the problem in internal chemical coordinates, while, generally, simple lattices are treated in Cartesian coordinates (Eq. (3-6)). The fhction is periodic with period ~ Tand C calculations are limited to values of q~ within the first Brillouin zone [51, 521 (BZ) (-TC I y i TC). In most cases, the results for only half of the BZ are reported, the second half being symmetrical through y z 0. In the spectroscopy of polymers as one-dimensional ID lattices it is generally easier to deal with the phase shift y at odds with solid-state physics, dealing with three-dimensional 3D crystal, which treats the whole dynamics in terms of the vector k. For ID lattices we can describe the wave-motion (phonon) propagating along the 1D chain either by the phase shift v, or by the vector Ik/ = y/d where k has only one component along the chain axis k,. In contrast, in the spectroscopy and dynamics of 3D lattices (where intermolecular forces are active in all directions) phonons are labeled with k vectors with three components (kk, k,, k,) [51]. When k is used in polymer dynamics Eq. (3-43) can be easily rewritten as It should be remembered that in a single and isolated polymer, chain atoms are allowed to move in the tridimensional space even if phonons are considered to propagate in one direction along the chain axis. This means that no neighboring intermolecular interactions are taken into account, i.e., the dynamics is that of a chain ‘in uacuo’.
102 3 l,i'brationcil <strong>Spect</strong>ra as n Probe of Structural Order 300G i - 2Got 3000 r 7 250{ - 1500 l 5 O 0 L = loook Figure 3-1. Dispersion curves of single-chain trans planar polyethylene. (a) v/p; (b) i$/k. As an example, in Figure 3-1 we report the dispersion curves of a single chain of polyethylene in v, and k space. The chain of trims-planar polyethylene -(CW?),,- is generated by application to a single CH2 group of the operator $(n. d/3). It follows that the crystallographic repeat unit contains two CH? groups. The physics and the numbers are not changed. The dispersion curves in Figure 3-lb are obtained by just halving Figure 3-la at .n/2 [53]. As an example of a more complex polymer chain we quote the case of polytetrafluoi-oethylene -(CFZ)n- for which one can identify the rototranslational operator $(2n/15, d/15) which describes a polymer chain made up by z = 15 CF2 units coiled in a helix with t = 7 turns. In cases such as polytetrafluoroetliylene the
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Modern Polymer Spectroscopy Edited
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Modern Polymer Spectroscopy Edited
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Preface For unfortunate reasons the
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However, theory and calculations yi
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Contents 1 Two-Dimensional Infrared
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Con ten t~ xi Index 5.2 Force Field
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Contributors M. Del Zoppo Dipartime
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1 Two-Dimensional Infrared (2D IR)
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1.2 Brick~qrorrnrl 3 . Figure 1-3.
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1.0 , Figure 1-6. The in-phtrse and
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1.2 Bcrckgroinizd 7 where AA (i~) i
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1.2 Background 9 1.2.5 Two-Dimensio
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1.3 Bnsic Properties of 20 IR Corre
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2D IR spectrometer coupled with a d
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1.5 Applirtrtions 15 Unlike a dispe
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\ '\ Melliyleiie 1'7- -3000 Posiliv
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11 Amorphous Crystalline ,...." I F
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1.5 Applications 21 / I 1430 Figure
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1.5 Appliccitioiis 23 Methylene Fig
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1.5 Ajqdicrrtions 25 3024 Figure 1-
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1.5 Applicutions 27 Furthermore. th
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1.7 Coizclirsions 29 derived in a s
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1.9 References 31 [9] Colthup, N. B
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2 Segmental Mobility of Liquid Crys
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SRMPLE/DETECTOR e3 ‘retardation
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2.4 Srvuc me-Dependenr Alignment 37
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2.5 Electric Field-Innductid Orirnt
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2.5 Electric Field-nclticetl Orient
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a -@ COWER SRHPLE ._ 2.5 Elec tric
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2.5 Electric Field-Itzduced Orienta
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2.5 Electric Field-IwrElrced Orient
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2.5 Electric Field-Im/irced Orientu
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Page 72 and 73: 2.5 Elrc tric Field-Iiidircwl Orim
Page 74 and 75: 2500 2008 I s00 WRVtNdMRtR CM-I Fig
Page 76 and 77: -3.6 Aligmient OJ' Side- Clinin Liy
Page 78 and 79: ~ polyester 2.6 Alignnient qf Side-
Page 80 and 81: I." 0.9 0.8 Figure 2-34. FTlR 0.7 p
Page 82 and 83: induced alignment of the investigat
Page 84 and 85: 2.7 Orientation oj Liquid Ci:i,stal
Page 86 and 87: 2.7 Orieritation of Liquid Ciysttrl
Page 88 and 89: 0 8 18 16 1 a W u z a m a 0 Ln m Q
Page 90 and 91: 2.7 Orientation oj Liquid Crystals
Page 92 and 93: 2.7 Orientation of Liquid Crjvtals
Page 94 and 95: 2.7 Orientatioiz of Liquid Crystals
Page 96 and 97: 2.8 Conclusions 81 strain. As A0 is
Page 98 and 99: 3. I0 Refcreiice.r 83 [I 71 Hoffina
Page 100 and 101: 2.10 References 85 [92] Wiesner, U.
Page 102 and 103: t Order/Disorder in Chain Molecules
Page 104 and 105: onds which hold atoms together thro
Page 106 and 107: 3.2 The Dyriamical Case qf Simll ar
Page 108 and 109: 3.2 The Dyrinniical Case of Sinnll
Page 110 and 111: 3.4 S11or.f- and Loizg-Rcrizye Vibr
Page 112 and 113: 3.4 Slzort- and Loiig-Range Vibrati
Page 114 and 115: eyularity), e.g., 2. During the pol
Page 118 and 119: 3.5 TOIIYW~ Laiyer Molertiles: Fvoi
Page 120 and 121: 3.5 Towmu" Larger Molecules: F~om O
Page 122 and 123: 1700 - CIO --- ca c 22 _- c 12 l l
Page 124 and 125: 3.6 From Dynamics to Vibrational Sp
Page 126 and 127: 3.6 Froin Dynaiiiics to T’ihratio
Page 128 and 129: 3.7 The Case of Isotactic Polypropy
Page 130 and 131: 3.7 The Cuse of Isotactic Polypvop.
Page 132 and 133: 3.8 Density of Vibrational States a
Page 134 and 135: 3.8 Dtvz.sit,v of L’ihrntioizal S
Page 136 and 137: 3.8 Driisity of Vibratioiid StLitrs
Page 138 and 139: 3 9 Mouiiiq ToIvmds Rerrlitv: From
Page 140 and 141: 3.9 Moving Towards Reality: From Or
Page 142 and 143: 3.9 Moving Towards Reality: Froin O
Page 144 and 145: 3.10 Wlmt Do We Learn from Cnlcailc
Page 146 and 147: + 3.11 A Very Siriiple Ccrse: Latti
Page 148 and 149: 3 I1 A Yey) Siiiiple Case: LLittice
Page 150 and 151: 3.11 A Vq> Siiiiplc Crrw: Luttice D
Page 152 and 153: Figure 3-22. Sample eigenvectors in
Page 154 and 155: 3.12 CIS-trans Opening @the Double
Page 156 and 157: 3.13 Defect Modes cis Structtrr.al
Page 158 and 159: 3.13 Defect Mo&s cis Structurcil Pr
Page 160 and 161: 0 3.14 Case Studies N 0 d - Y N o m
Page 162 and 163: 3.14 Case Studies 147 OC 60 58 57.
Page 164 and 165: 3.14 Case Studies 149 GI A V E N U
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3.14 Case Studies 15 1 correspond t
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3.14 Case Studies 153 (I10 + 200) +
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3.14 Case Studies 155 1500 1480 146
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3.14 Case Studies 157 the molecular
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17E (ir 1, R): (iii) z = 4, t = 1,
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3.16 Stnictural Znlioi~zogeneity an
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3.1 7 Fermi Resonancrs 163 stants.
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3.1 7 Femi Resorinnces 165 2 L 1 29
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lowing. The CHl-bending mode 6 [cer
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3.17 Ferini Re.sonrriire.s 169 a Fi
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3. I7 Fernii Resoiiances 17 1 terin
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3.18 Band Broadening and Confonnati
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3.18 Bmd Brourleiziriy md Cmforrnat
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3.18 Band Bvoaderiing arid Conjonna
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3.18 Band Broadening and Confornmti
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3.19 A Worked E.utinple 181 pre-mel
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3.19 A Wovh-ecl E.uanzple 183
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3.19 A Worked Example 185 19 5°C 2
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3.19 A Workrd Example 187 Figure 3-
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3.19 A Worked E.rcnniplr 189 LAM I
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3.19 A Worked E.xuniple 191 Figure
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3.19 A Worked E.vnn~yle 193 Figure
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3.19 A Worked E.vnrqde 195 009.1 00
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3.19 A Worked Example 197 existence
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3.1 9 A Worked Example 199 The soli
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3.20 References 201 very important,
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3.20 References 203 [41] M. Gussoni
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3.20 Refeeveiicrs 205 [I 171 P. Jon
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4 Vibrational Spectroscopy of Intac
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4.3 Georrietvy oj Intuct Po1vniev.r
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1'45 4.4 Geometric Chunges I?zditce
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4.4 Geometric Chmges Iiid~lrcecl bj
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4 5 Mer1iodolog.v of Raiii~11 Studi
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4.7 Poly(p-pheiq.lene) 217 with an
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4.7 Pol)~(p-pheiz~~lene) 219 ENERGY
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is worthwhile pointing out that the
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~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 801 Table 4-3
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4.7 Poly(y-phenylene) 225 Figure 4-
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Table 4-4. Observed Raman frequenci
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4.8 Other Polwieys 229 Table 4-5. T
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under various models. According to
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4. I0 Mechnriism oj Charge. Transpo
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4.12 Reftrences 235 [ 131 Kuzmany,
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4.12 References 237 [98j Matsunuma,
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5 Vibrational Spectroscopy of Polyp
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5.2 FOYCC Fields 241 such calculati
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5.2 Force Fields 243 accounts for o
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5.2 Force Fields 245 centers of the
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5.2 Force Fields 247 ture with conf
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5.3 Amide Modes 249 to the nh initi
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5.3 Ainide Modes 251 Table 5-1. Ami
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Table 5-2. Some amide modes of four
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5.3 Amidc Modes 255 Figure 5-1. Opt
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5.4 Polypeptides 257 nonhydrogen-bo
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5.4 Polvpeptida 259 Figure 5-2. Ant
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I 5.4 Polypeptides 261 )I .- + $ 80
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1226M 1222s (1 1165W 1167s (1 1120V
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135s 9 1 Ms1i 122Wbr 147 NH ob(37)
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5.4 Polypeptides 267 glycine residu
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5.4 Polypeptides 269 Raman bands in
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5.4 Polypeptides 271 Figure 5-7. St
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Frequency (cm-'1 100% b 700 500 300
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5.4 Polypptidt~s 215 Table 5-9. Obs
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~~ ~~ ~~~ ~ ~ ~~ 5.4 Polypeptides 2
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5.4 Poliyeptides 219 Table 5-11. Co
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Table 5-12. Aniide mode frequencies
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1351 Lifson, S., Stern, P. S., J. C
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5.6 Refeverices 285 [130] Tiffany,
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Index Amide modes 249 Amorphous pol
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Step-scan FTIR spectroscopy 14,34 -
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