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

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22 CHAPTER 2. HAIRPIN CHAIN ELASTOMERSFig. 2.8 shows a comparison between the different approximations to the partitionfunction for the hairpin chain. It shows two different Gaussian approximationsas well as the asymptotic approximation. The asymptotic approximationis not as good as the others because it requires a larger fN valuebefore it becomes accurate. As with most asymptotic results it is difficult topredict at what values of fN the asymptotic approximation will improve. The10.8Z hp0.60.4Actual0.2CurvatureAreaAsymptotic0-1 -0.5 0 0.5 1z/NFigure 2.8: A comparison of the actual partition function ofhairpinned chains at fN = 3 to two Gaussian approximationsand to an asymptotic approximation.approximations are all of remarkable power when one considers that there aredistributionsthatonetruncatedatextents wheretheyarestilllarge. Moreoverthe mean number of hairpins is very small in the example chosen (fN = 3).The strong Gaussian character arises because despite there being few steps inthe random walk they are of variable, random lengths.2.2.4 Comparison of spring constantsA main chain elastomer is made up of polymer chains of different character. Itis important to estimate the moduli of the different types of chains so that theway in which the rubber stretches can be modelled. To this end a comparisonof the spring constants, k, of three different chain is now made. These chainsare; the Gaussian chain, the hairpin chain, and the undulating chain. For theGaussian chainF g = −k B T R22l 2 N(2.56)
2.2. MODEL OF HAIRPIN CHAINS 23f g = − ∂F g∂R = k BTRl 2 N(2.57)⇒ k g = k BTl 2 N . (2.58)For the hairpin chain the asymptotic limit is used. This will provide an overestimateof the spring constant for smaller values of fN.F hp ≈ −k B T R2 fN2N 2 l 2 (2.59)f hp ≈ k B T fNN 2 l2R (2.60)⇒ k hp = k B T fNl 2 N 2 (2.61)This shows that when fN ∼ 5 the Gaussian chains are much stiffer than thehairpin chains. The estimate of the spring constant of the undulating chain isgiven in §2.E. The result obtained isThe ratio of moduli of chains isf z ≈ 4u hk B T⇒ k u = 4u hJk B TLJδR (2.62)L(2.63)≈ 4u hNl 2 (2.64)k u : k g : k hp = 4u hBl 2 : k BTNl 2 : k BTfNN 2 l 2 (2.65)here the combination fN is retained since it is an estimate of the number ofhairpins per chain, n hp . Then u hk B T ≈ ln Nn hp, using the definition of f. In anyevent, one expects u h > k B T because hairpins are well-defined and relativelyinfrequent events. Thus the ratio of the spring constants becomesk u : k g : k hp = 4ln N n hp: 1 : n hpN(2.66)From these estimates it is clear that when the hairpins are removed from achain it becomes significantly harder to stretch. The chains in this state canonly have their end-to-end distance increased by a small amount because theirlongitudinal spatial extent is near to their arc length.2.2.5 Perpendicular directionIn the plane perpendicular to the nematic direction the hairpin chains havetwo sources that can give rise to an end-to-end distance: Gaussian distributed
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Page 17 and 18: Chapter TwoThe elasticity of hairpi
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72 CHAPTER 3. POLARISATION OF CHIRA
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Chapter FourThe elasticity of smect
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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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4.5. 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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Statistical models of elasticity in main chain and smectic liquid ...

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