Statistical models of elasticity in main chain and smectic liquid ...

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44 CHAPTER 2. HAIRPIN CHAIN ELASTOMERSThe odd result then follows by combining the intermediate result when onlyone hairpin had been added to an even number of hairpinsG n+ (z,N) =1(n−1)!! 2(N2 −z 2 ) n−12G n− (z,N) =1(n−1)!! 2(N2 −z 2 ) n−12G n (z,N) = G n− (z,N)+G n+ (z,N)=2(n−1)!! 2(N2 −z 2 ) n−12 (2.133)This method agrees with the calculation via the use of the exponentiation ofthe delta function/Laplace transform.
2.D. INTERPRETATION OF FN 452.D Interpretation of fNThe average number of hairpins on a chain at zero end-to-end distance can becalculated as follows.〈n〉(0) =1Z hp (0)∞∑nG n (0,N)e −βu hnn=0= − ∂lnZ hp∂(βu h )(2.134)(2.135)The number of hairpins on a chain with zero end-to-end distance must alwaysbe greater than or equal to one because a straightened chain cannot havezero end-to-end distance. The mean number of hairpins on the chain can becalculated usingZ hp (0) = f(I 0 (fN)+I 1 (fN)) (2.136)Using the chain rule results in the following〈n〉(0) = (fN) ∂lnZ hp∂(fN)I 0 (fN)= fN +I 0 (fN)+I 1 (fN)(2.137)(2.138)Since the modified Bessel function is asymptotic to x −1/2 e x then for moderatelylarge fN it follows that〈n〉(z = 0) ≈ fN + 1 2(2.139)Thus fN can be interpreted as the average number of hairpins on a polymerchain with span of z = 0.
Page 3: iStatistical models of elasticity i
Page 7: AcknowledgementsDuring the course o
Page 10 and 11: viiiCONTENTS5 Soft elasticity of sm
Page 12 and 13: 2 CHAPTER 1. INTRODUCTION1.2 Neo-cl
Page 14 and 15: 4 CHAPTER 1. INTRODUCTIONticular in
Page 17 and 18: Chapter TwoThe elasticity of hairpi
Page 19 and 20: 2.1. INTRODUCTION 9Fig. 2.1. This b
Page 21 and 22: 2.2. MODEL OF HAIRPIN CHAINS 11nFig
Page 23 and 24: 2.2. MODEL OF HAIRPIN CHAINS 13za)
Page 25 and 26: 2.2. MODEL OF HAIRPIN CHAINS 15in
Page 27 and 28: 2.2. MODEL OF HAIRPIN CHAINS 17The
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Page 31 and 32: 2.2. MODEL OF HAIRPIN CHAINS 21This
Page 33 and 34: 2.2. MODEL OF HAIRPIN CHAINS 23f g
Page 35 and 36: 2.3. NON-AFFINE DEFORMATION MODEL 2
Page 41 and 42: 2.4. HAIRPIN CHAIN NETWORK FREE ENE
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Page 45 and 46: 2.5. CONCLUSIONS 35to their limit
Page 47 and 48: 2.A. GEOMETRY OF A HAIRPIN AT ZERO
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Page 93 and 94: 4.1. INTRODUCTION 83wherer 12 is th
Page 95 and 96: 4.1. INTRODUCTION 85HzyxGlobal rota
Page 97 and 98: 4.1. INTRODUCTION 87Figure 4.3: The
Page 99 and 100: 4.1. INTRODUCTION 89−D 2[(v (a)xz
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4.2. MICROSCOPIC, FINITE DEFORMATIO
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4.3. FINITE DEFORMATION EXAMPLES 97
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4.4. COMPARISON WITH EXPERIMENT 113
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4.4. COMPARISON WITH EXPERIMENT 115
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4.5. CONCLUSIONS 117formation in th
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120 CHAPTER 5. SOFT ELASTICITY OF S
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Bibliography[1] P. J. Flory. Statis
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139[27] J-.C. Meiners and S. R. Qua
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141[58] W. L. McMillan. Measurement
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143[86] L. Golubovic and T. C. Lube
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