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

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32 CHAPTER 2. HAIRPIN CHAIN ELASTOMERSThus for small fN the hairpin part of the rubber will spontaneously elongateas it has a negative nominal stress and hence has no mechanism for competingwith the transverse degrees of freedom. Note that the hairpin degrees offreedom on their own undergo spontaneous expansion here. The transversedegrees of freedom have not yet been included, but will cause the rubber toelongate via volume conservation.2.4.1 Transverse degrees of freedomIn the transverse direction there are contributions from both the small undulationsabout the nematic direction and the transverse extent of the hairpins.As mentioned previously, these processes will be treated together, producingaGaussian distribution of end-to-end distances in the perpendicular plane andhence a Gaussian rubber in the perpendicular direction. As a result the freeenergy in the transverse direction depends on the macroscopic deformationlike a Gaussian rubber) 1F ⊥ = 2(2 k BTΛ 2 ⊥ , (2.98)the factor of 2 coming from the two transverse dimensions. For a volumeconserving uniaxial deformation then1 = Λ ‖ Λ 2 ⊥ . (2.99)For uniaxial deformations writing Λ ‖ = Λ gives the free energy expression asF ⊥ = k BTΛ . (2.100)Thusastheelastomeriselongated, thefreeenergyoftheperpendiculardegreesoffreedomisreducedbecausetheinitial andfinalpointsofstrandsarebroughtinto line and as a result the conformational entropy increases.2.4.2 Free energy curvesTocalculate thetotal freeenergyoftherubberthetransverseGaussiandegreesof freedom and the hairpin degrees of freedom parallel to the nematic fieldare combined. The free energy depends on weakly on the length N of thepolymerchains. TheN dependenceislogarithmic inEq. (2.89) becauseit onlyenters Z hp . Fig. 2.11 shows the free energy as a function of the macroscopicdeformation Λ. For the purposes of numerical illustrations the value N =100 is used (experimentally it is difficult to work with very long main chainsbecause of their slow dynamics). The minimum in the free energy is at a Λgreater than one when there are a few hairpins, and the hairpin degrees offreedom are weak. As the number of hairpins is increased the minima movesslightly below Λ = 1 and then back to Λ = 1 for very large fN. This showsthat on cross-linking the rubber will elongate to a new equilibrium value. As
2.4. HAIRPIN CHAIN NETWORK FREE ENERGY 330.5F/k B T0-0.5-1-1.5-24.5567-2.51 1.5 2 2.5 3 3.5Figure 2.11: The free energy of a hairpin rubber as a functionofthemacroscopicdeformationfortemperaturessuchthatfN = 4.5,5,6,7 andforachain oflength N = 100. Thecurvesare calculated up to the end of the stress plateau, Λ(∞). Afterthis the rubber rapidly attains the modulus of extended wormchains.Λthe rubberis deformed further, the freeenergy gradually becomes linear in thedeformationΛbecausemoreofthechainsarenowinertwiththeresultthatthemicroscopic deformation, λ, is concentrated in very few chains. The numberof chains where energy is being stored reduces with Λ and the deformationbecomes easier subsequently. This linear increase with Λ contrasts with thequadratic Λ 2 response of a Gaussian rubber.2.4.3 Nominal stress curvesThe spontaneous elongation is even clearer on the nominal stress curves.Fig. 2.12 shows the nominal stress dFdΛas a function of the macroscopic deformationΛ. Each of the curves shows the nominal stress beginning to plateauas hairpins are pulled out. After all the hairpins have been pulled out, thenonly the stiff undulating chains remain so the nominal stress will then increasevery rapidly (not modelled here). As the temperature of the rubberis reducedthe average number of hairpins on a chain reduces. This causes the paralleldirection to become weaker and hence the spontaneous elongation on crosslinkingincreases. Eventually the hairpin degrees of freedom will be so weakthat the rubber will immediately elongate to its maximum extent so that itconsists of undulating chains alone. The very large modulus of the undulatingchains will then resist the transverse Gaussian degrees of freedom so that therubber can find its equilibrium. The average number of hairpins on a chain
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
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Page 45 and 46: 2.5. CONCLUSIONS 35to their limit
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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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