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

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88 CHAPTER 4. THE ELASTICITY OF SMECTIC-A ELASTOMERSa) b)Figure 4.4: a) The experimental results of [66] for a smecticelastomer stretched parallel to the layer normal, b) a photographof the elastomer before the threshold (clear) and after(opaque).the balloon can be altered and the change in radius measured yielding thestress strain characteristics of the film. In these experiments it was observedthat in weakly coupled smectic elastomers there is no signature of the smecticlayer system [68] and [69]. These systems are not considered here.4.1.7 Continuum model of a smectic elastomerThe elastic properties of smectic liquid crystals can be calculated for small deformationsbased on the continuum free energy. The terms that are includedin this free energy are the sum of the contributions for the ordinary nematic,the ordinary smectic, and the elastomer. These different degrees of freedomare then coupled together. The different contributions to the continuum freeenergy will be pointed out here. A full discussion can be found in [46]. Thecontributions to the free energy density will be denoted as: f elastic for the uniaxialelastic energy density, f nem for the Frank elastic terms and the couplingbetween the elastomer and the director, f smA for the Landau-de Gennes smecticorder terms and f smA−el for the terms pertaining to the coupling betweenthe layers and the rubber matrixf elastic = C 1 (n·ǫ·n) 2 +2C 2 Tr[˜ǫ](n·ǫ·n)+C 3(Tr[˜ǫ]+2C 4 [n×ǫ×n] 2 +4C 5 [n×ǫ·n] 2 . (4.30)In this equation ˜ǫ denotes the symmetric part of the deformation tensor λ,and ǫ denotes the same symmetric part after the non-volume preserving partshave been removed (i.e. it has been made traceless). Note C 1 ,C 4 and C 5 areof order µ whereas C 3 is of order the bulk modulus and C 2 is smaller that µ.[f nem = 1 2 D 1 (v xz (a) −δn x) 2 +(v yz (a) −δn y) 2]) 2
4.1. INTRODUCTION 89−D 2[(v (a)xz −δn x )ǫ xz +(v (a)yz −δn y )ǫ yz]+ 1 2 K 1(div δn) 2 + 1 2 K 2(curl δn) 2 + 1 2 K 3(∂ z δn) 2 (4.31)where v (a)ijis the antisymmetric part of the deformation tensor, λ.f smA = 1 2(τ|ψ| 2 + 1 2 β|ψ|4 +g ‖ |∂ z ψ| 2 +g ⊥ |(∇ ⊥ −iq 0 δn)ψ| 2) (4.32)whereψ(r) = |ψ(T)|e −iq 0u(r) isthesmecticorderparameter. Thefirstgradientterm expresses the rigidity of the smectic layers: B = g ‖ q 2 0 |ψ|2 and the secondgradient term expresses the coupling of the director to the layer normal.f smA−el = Λ[V z (r)−u(r)] 2∼ Λ[V z (0)−u(0)] 2 +ΛL 2∑ [∇ j u(0)−v zj ] 2j∫+Λ [ũ(r)−Ṽz(r)] 2 (4.33)where u denotes the layer displacement, V z denotes the displacement of theelastic matrix in the z direction (r → r+V), v zj is defined by the expansionof V: V j (r) = V j (0) + r i v ji + Ṽj and ũ(r) and Ṽz(r) denote the fluctuatingcomponents of the layer displacement and the matrix displacement in thez direction respectively. It is remarkable that the system size L occurs inthe second term. This term is a very rigid constraint which is derived anddiscussed below. This contribution to the free energy density is studied in[70].The free energy terms outlined above are used in the continuum theoryof smectic elastomers. To study the elastic properties of the network one canaverage over the nematic director position, and to study the nematic directorone can average over the elastic degrees of freedom (see [46] for more details).In addition to the above terms the effects of the cross-links have also beenconsidered close to the smectic-nematic transition. In elastomers the volumefractionofcross-linksismuchsmallerthanthatofthebackboneandmesogenicgroups so the cross-links exist in a mesogenic environment. The cross-link’seffect on the smectic order is to enhancesmectic order and locally fix its phase.This has a free energy density contributionf RF = ∑ αγ|ψ(R α )|cos{q 0 [z α −u(R α )]} (4.34)where R α is the position of the αth cross-link point, and γ is the couplingconstant. The effects of this term on the N-SmA transition have been studiedin detail in [71].The interplay of these free energy terms has been studied in [72], whereit was reported that this coupling destroys the conditions required for soft
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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
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Chapter TwoThe elasticity of hairpi
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2.1. INTRODUCTION 9Fig. 2.1. This b
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2.2. MODEL OF HAIRPIN CHAINS 11nFig
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2.2. MODEL OF HAIRPIN CHAINS 13za)
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2.2. MODEL OF HAIRPIN CHAINS 15in
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2.2. MODEL OF HAIRPIN CHAINS 17The
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2.2. MODEL OF HAIRPIN CHAINS 1910.8
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2.2. MODEL OF HAIRPIN CHAINS 21This
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2.2. MODEL OF HAIRPIN CHAINS 23f g
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2.3. NON-AFFINE DEFORMATION MODEL 2
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2.4. HAIRPIN CHAIN NETWORK FREE ENE
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2.4. HAIRPIN CHAIN NETWORK FREE ENE
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2.5. CONCLUSIONS 35to their limit
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Page 91 and 92: Chapter FourThe elasticity of smect
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Page 127: 4.5. CONCLUSIONS 117formation in th
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Page 147 and 148: 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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