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

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34 CHAPTER 2. HAIRPIN CHAIN ELASTOMERS2Nominal stress σ/(n s k B T)1.510.57654.501 1.5 2 2.5 3 3.5ΛFigure 2.12: The nominal stress as a function of the macroscopicdeformation of a hairpin rubber for chains of lengthN = 100.when the rubber first elongates to its maximum extent can be calculated asfollows. If the rubber just reaches its maximum extent then as λ → ∞ thenominal stress will be zero at this point. Using Eq. (2.89) and replacing theintegral using the Gaussian approximation thendFdΛλ→∞ = 2k BTΛ(∞)(lnZ hp (N)+ 2N2 α3)− k BTΛ(∞) 2 (2.101)Using the simplest approximation for N 2 α = fN 2from the asymptotic expansionyields a value of approximately fN = 4.3 when N = 100. This correspondsto the lowest average number of hairpins possible on the chain beforethe chain spontaneously extends to its maximum extent on cross-linking. Inpractise it will not elongate to its maximum extent because there will bemore and more percolating paths of inert chains which will gradually makethe rubber harder. However, before this situation is reached the above modelapplies.The spontaneous elongation on cross-linking the polymer chains has tobe taken into account in subsequent measurements. The initial spontaneouselongation of Λ s occurs on cross-linking. After this elongation any furtherdeformation carried out will be in addition to this deformation. Fig. 2.13shows the nominal stress curves plotted as a function of the experimentallymeasured deformation, Λ ′ Λ ′ Λ s = Λ. (2.102)Thus networks with hairpins should have a longer stress-strain plateau, terminatedby the final removal of hairpins. Networks that spontaneously deform
2.5. CONCLUSIONS 35to their limit Λ(∞) will be instantly strong in extension but rubbery in compressionalong the director — most unusual materials.2Nominal stress σ/(n s k B T)1.510.576501 1.5 2 2.5 3 3.5Λ ’ 4.5Figure 2.13: The nominal stress as a function of the measuredmacroscopic deformation of a hairpin rubber for chainsof length N = 100 .The spontaneous extension shown in this model may be a consequence ofthe way in which the chains are split up into non-hairpinned and hairpinnedchains in the deformation mechanism. Since the non-hairpinned chains areseparatedoutfromthehairpinnedchainsthentheremaininghairpinnedchainswill try to achieve their equilibrium distribution. Consequently the sampleextends. In order to explore this effect in more detail the splitting up of thedeformation between the chains should be done in a less singular way, suchthat the non-hairpinned chains experience some small elongation dependingon their stiffness. The limit of infinite chain stiffness should then be taken.2.5 ConclusionsAn elastomer composed of main chain liquid crystalline polymers was considered.The strong nematic field results in creation of hairpin defects. Thestatistics of these hairpin chains can be calculated and can be approximatedwell by a truncated Gaussian distribution. The transverse degrees of freedomthat accompany the hairpins are undulations of the chains about thenematic director. These undulations produce a Gaussian distribution for thetransverse extent. The difference in the spring constants of undulating andhairpinned chains motivated a non-affine stretching mechanism of the rubber.Cross-linking the chains results in a rubber that spontaneously extends atlow temperatures. As the rubber is deformed along the director it exhibits a
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
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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 47 and 48: 2.A. GEOMETRY OF A HAIRPIN AT ZERO
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Page 57: 2.E. SPRING CONSTANT OF AN EXTENDED
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Page 91 and 92: Chapter FourThe elasticity of smect
Page 93 and 94: 4.1. INTRODUCTION 83wherer 12 is th
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4.1. INTRODUCTION 85HzyxGlobal rota
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4.1. INTRODUCTION 87Figure 4.3: The
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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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