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

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58 CHAPTER 3. POLARISATION OF CHIRAL ELASTOMERSthe director lies, one saddle point, and one unstable axis. When consideringa soft mode, then two of the principal values are degenerate. In this caseit still follows that the axis along which the director lies is stable, but theother two axes have degenerate values of χ and so will have two zeros in thematrix determining their stability, which cannot be classified as a maximumor a minimum.3.3 The polarisation of a pure liquid crystalelastomersOnce the equilibrium orientation of the director is known, the polarisation ofthe elastomer can be calculated. The tensor M commutes with the tensorl −1 because the director lies along one of the principal axes of M. The objectM·l −1 is thus symmetric and as a result the polarisation of the elastomer iszero since it is the contraction of this symmetric tensor with the antisymmetrictensor ǫ ijk . Two more arguments that produce the same result are nowpresented.3.3.1 Linearization of the polarisation expressionIt is useful to look at small, symmetric deformations and rotations of therubber matrix in which the rods are held. The deformation tensor can bebroken up into the sum of the Kronecker delta, a symmetric tensor, ǫ ij , andan antisymmetric tensor W ij . Theantisymmetric tensor has the property thatW ·a = Ω×a. (3.14)This matrix performs rotations about the axis ˆΩ of magnitude Ω for smallrotations. It is convenient to rewrite the antisymmetric tensor in terms of itsaxial vector, ΩW ij = ǫ ikj Ω k . (3.15)The strain tensor then becomesλ ij = δ ij +ǫ ij +ǫ ikj Ω k , (3.16)where ǫ is a symmetric second rank tensor and Ω is a vector parallel to therotation axis and has a magnitude equal to the angle of rotation, Ω. Thereshould be no confusion between the Levi-Civita symbol, ǫ ijk , and the symmetrictensor ǫ ij here because of the number of indices. This decompositioncan be used in the expressions for free energy and polarisation (dropping theprefactors in both cases for simplicity)[f = Tr l 0 ·l −1 +l 0 ·ǫ·l −1 +ǫ·l 0 ·l −1 +ǫ·l 0 ·ǫ·l −1
3.3. THE POLARISATION OF A PURE LIQUID CRYSTALELASTOMERS 59− l 0 ·W ·l −1 +W ·l 0 ·l −1 −W ·l 0 ·W ·l −1+ W ·l 0 ·ǫ·l −1 −ǫ·l 0 ·W ·l −1] (3.17)[P i = ǫ ijk l 0 ·l −1 +l 0 ·ǫ·l −1 +ǫ·l 0 ·l −1 +ǫ·l 0 ·ǫ·l −1− l 0 ·W ·l −1 +W ·l 0 ·l −1 −W ·l 0 ·W ·l −1+ W ·l 0 ·ǫ·l −1 −ǫ·l 0 ·W ·l −1] jk . (3.18)[Note that the linear terms in ǫ cancel out due to the identity: (r − 1) +( 1r −1) +(r −1) ( 1r −1) = 0]. Suppose now that Ω and ǫ are small so thatsecond order effects in these quantities may be neglected. The resulting freeenergy density and polarisation are[f ≈ Tr l 0 ·l −1 +l 0 ·ǫ·l −1 +ǫ·l 0 ·l −1− l 0 ·W ·l −1 +W ·l 0 ·l −1] (3.19)[P i ≈ ǫ ijk l 0 ·l −1 +l 0 ·ǫ·l −1 +ǫ·l 0 ·l −1− l 0 ·W ·l −1 +W ·l 0 ·l −1] jk . (3.20)To show that the first order shift in the director cancels out the polarisation,consider minimisation of the free energy density with respect to variations inthe order parameter that maintain the normalisation of the director n·n = 1.Differentiation of the free energy density expression above, Eq. (3.19), w.r.t.n and using a Lagrange multiplier, χ, to maintain the constraint results inl 0 ·n+(l 0 ·ǫ+ǫ·l 0 )·n+(W ·l 0 −l 0 ·W)·n = χn (3.21)Perturbation theory can beused to calculate the shift in the principal axis (i.e.the director n) to first order. If ǫ and W are small then the terms containingthem on the left had side of Eq. (3.21) can be written as a perturbationl ′ = (l 0·ǫ+ǫ·l 0 )+(W·l 0 −l 0·W) with a parameter ξ to control the“strength”of the perturbation and keep track of its order. Using this substitution andwriting χ = r +ξr (1) +... and similarly for n results in(l 0 +ξl ′ )(n 0 +ξδn (1) +...) = (r +ξr (1) +...)(n 0 +ξδn (1) +...).(3.22)Collecting powers of ξ produces the following for the 0 th and the 1 st ordershiftsl 0 ·n 0 = rn 0 (3.23)l 0 ·δn (1) +l ′ ·n 0 = rδn (1) +r (1) n 0 (3.24)
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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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4 CHAPTER 1. INTRODUCTIONticular in
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 33 and 34: 2.2. MODEL OF HAIRPIN CHAINS 23f g
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Page 45 and 46: 2.5. CONCLUSIONS 35to their limit
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Page 55 and 56: 2.D. INTERPRETATION OF FN 452.D Int
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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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4.3. FINITE DEFORMATION EXAMPLES 10
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4.3. FINITE DEFORMATION EXAMPLES 11
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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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