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

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54 CHAPTER 3. POLARISATION OF CHIRAL ELASTOMERSabvuFigure 3.3: A single monomer of a chiral main chain polymer.The monomer has two arms of different lengths a, and b. Eachmonomer is then assigned a dipole using the right hand rule(in this case out of the page) so that each one is chiral.tions only simple shear defines a direction for the polarisation to point along.The effect of a shear on a rubber consisting of nematic, chiral main chains isillustrated in Fig. 3.4.Figure 3.4: A shear deformation applied to a network of chiralmolecules (a) can cause an overall binormal vector (b) andhence a polarisation to develop (schematic only).The joint probability distribution of end-to-end distance of a long chiralmain chain, R, and the total binormal, V, is now calculated. The two directionsfor the long and the short arms of each monomer can be defined asu and v as shown in Fig. 3.3, so that the end-to-end distance of a monomeris given by: w α = au α +bv α where α labels the monomer. The end-to-enddistance of the chain and the chain binormal are then given byR = ∑ αV = ∑ αw α (3.4)u α ×v α (3.5)Since the u α of each monomer is on average parallel or anti-parallel to thenematic field (Q ≠ 1) and each monomer is independent, then it has thequadrupolar average〈u α i u β j 〉 u = δ αβ q ij , (3.6)
3.1. INTRODUCTION 55where the tensor order parameter is given by: q ij = q ⊥ δ ij + (q ‖ − q ⊥ )n i n j .If θ is the angle that a monomer makes with the nematic direction, n, thenq ‖ = 〈cos 2 θ〉 and q ⊥ = 1 2 〈sin2 θ〉. Since v is perpendicular to u it has theaverage〈v α i vβ j 〉 vu = 1 2 δ αβ(δ ij −q ij ) = 1 2 δ αβM ij (3.7)These averages can be used to evaluate the joint probability distribution byaveraging over delta functions used to count the configurations. Exponentiatingthese delta functions and expanding down from the resulting exponentallows their evaluation. The result is〈 (W(R,V) = δ R− ∑ ) ((au α +bv α ) δ V− ∑ )〉u α ×v ααα×∫ ∫ dηdζ=(2π) 6exp(iη·R+iζ ·V)〈 ( ) 〉∑1−i η ·(au α +bv α )+ζ ·(u α ×v α ) − 1 2 (···)2 +··· .αThe first few averages of this expansion can be evaluated, ignoring termsof order ( b 2,a)and then re-exponentiate and using the method of steepestdescents to evaluate the integral. An iterative method can be used to solvefor the points of steepest descent. The most interesting term comes from thecubic part. The resulting distribution function is( −3W (R,V) ∝ exp2La RT ·l −1 ·R− 1 )N VT ·M −1 ·V(× 1+ 3b [ ])aL 2 R×l −1 ·R ·M −1 ·V+O(R 4 ,R 2 V 2 ) ,where l = δ + (r −1)nn ≈ 3q. This tensor provides information on theanisotropy of the chain shape. The anisotropy of the chains is given by r =l ‖ /l ⊥ . Theimportant correction term is the couplingbetween RandV. Usingthis probability distribution the average binormal, V, can be calculated.∫〈V〉 ∝dVexp= 3bN2aL 2R×l−1 ·R,(− 1 )N VT ·M −1 ·VV(1+ 3b [ ] )aL 2 R×l −1 ·R ·M −1 ·Vwhereclearly in the ∫ dV the first(linear) term gives zero, so theO(V 2 ) termsmust be evaluated. In a rubber the polymer chains will have their ends fixed.It is assumed that deformations move these ends affinely. If the quenched-inend-to-end vector is denoted by R 0 then after a deformation, λ, the end-toenddistance becomes R = λ · R 0 . The quenched-in binormal vector, after
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iStatistical models of elasticity i
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AcknowledgementsDuring the course o
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viiiCONTENTS5 Soft elasticity of sm
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2 CHAPTER 1. INTRODUCTION1.2 Neo-cl
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Page 17 and 18: Chapter TwoThe elasticity of hairpi
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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
Page 47 and 48: 2.A. GEOMETRY OF A HAIRPIN AT ZERO
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Page 55 and 56: 2.D. INTERPRETATION OF FN 452.D Int
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Page 91 and 92: Chapter FourThe elasticity of smect
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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4.3. FINITE DEFORMATION EXAMPLES 10
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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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