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

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106 CHAPTER 4. THE ELASTICITY OF SMECTIC-A ELASTOMERSenergy solution are given by{ } 2f = 1 2 µ λ +λ2 +b(λ−1) 2 (4.109)σ nom = µ(λ− 1 )λ 2 +B(λ−1). (4.110)When layer rotation starts at λ cr , numerically it is clear that λ yy = √ 1λcrisa constant. Starting from this assumption avoids having to minimise over λ yyand thereby√simplifies the problem. After the transition occurs it follows thatλ xx = λcrλso that the material now has Poisson ratios (0,1). Minimising thefree energy density w.r.t. λ 2 zx yields, after some simplification[0 = 1+ (r −1)λ4 λ cr(λ cr +λ 2 zxλ 2 ) 2 +b λ 3√ ]λ cr(λ cr +λ 2 zxλ 2 ) 3/2 − λ cr λ 4(λ cr +λ 2 zxλ 2 ) 2 . (4.111)Solution of this equation for λ zx at first sight is complicated, however λ zx onlyappears in the following combinationp 2 = λ 2 zx + λ crλ 2 . (4.112)The combination p is a function only of r − 1, b and λ cr , that is f(r,b,λ cr ),and obeys the equation0 = p 4 +(r −1)λ cr +b(p √ )λ cr −λ cr . (4.113)This is because Eq. (4.111) is a polynomial in p. At the critical value, λ cr ,using the fact that λ zx = 0 it follows that p = √ 1λcr. From Eq. (4.113) thefollowing equation for the critical value of λ cr can be obtainedλ 3 cr (r −b−1)+bλ2 cr +1 = 0. (4.114)This equation can be solved analytically by the general formula for cubicequations. However it is more useful to analyse its limits. The first few termsin the expansion for large b yieldλ cr = 1+ r b +r(r−3) 1 ( ) 1b 2 +O b 3 . (4.115)To obtain a threshold at all, the condition b > r −1 must be obeyed. Belowthis layer modulus there is no instability. From p 2 = 1λ crin Eq. (4.112) theinduced shear is √1λ zx = ± − λ crλ cr λ 2 . (4.116)
4.3. FINITE DEFORMATION EXAMPLES 107Note that there are two solutions here corresponding to the two directionsthat the director can rotate. On substituting the form of λ zx back into thefree energy density the following expression is obtained{ }2f = 1 2 µ +λ 2 cr +r(λ 2 −λ 2λcr)+b(λ cr −1) 2 . (4.117)crFrom Eq. (4.109) and Eq. (4.117) the nominal stress can be calculatedusing the equation: σ nom = ∂f∂λ. The results for the nominal stress are thus{ ( )µ λ−1σ nom = λ +B(λ−1) λ < 2 λcr(4.118)µrλ λ > λ crNote that the continuity of the nominal stress with λ can be used to deriveEq. (4.114). From this result it is clear that the ratio of the two slopes isrelated toλ cr , whichprovidesaconvenient experimental check of thethresholdstretch. Thus for large B the ratio of the two slopes can be calculated, andtheir ratio taken to obtainrµB ≈ λ cr −1. (4.119)Experimentally µ can be obtained from stretching the rubber in the layers,yielding a modulus of E ⊥ = 4µ, and thus obtain the anisotropy of the polymers,r by combining these two results. An illustration of Eq. (4.118) is shownin Fig. 4.14, again for the same small value of b.43σ nom/µ2101.0 1.5 2.0λ zzFigure 4.14: The figure illustrates the nominal stress for asmectic elastomer stretched parallel to the layer normal, withr = 2 and b = 5.It is interesting to calculate the layer spacing of the system as a functionof the imposed stretch. The layer spacing is given bydd 0=1λ yy√λ 2 xx +λ 2 zx=λλ√ xx. (4.120)λ 2 xx +λ 2 zx
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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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2.A. GEOMETRY OF A HAIRPIN AT ZERO
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2.B. UNDULATING CHAIN STATISTICS 39
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2.C. COUNTING CONFIGURATIONS BY IND
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2.C. COUNTING CONFIGURATIONS BY IND
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2.D. INTERPRETATION OF FN 452.D Int
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2.E. SPRING CONSTANT OF AN EXTENDED
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50 CHAPTER 3. POLARISATION OF CHIRA
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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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Page 127: 4.5. CONCLUSIONS 117formation in th
Page 130 and 131: 120 CHAPTER 5. SOFT ELASTICITY OF S
Page 147 and 148: Bibliography[1] P. J. Flory. Statis
Page 149 and 150: 139[27] J-.C. Meiners and S. R. Qua
Page 151 and 152: 141[58] W. L. McMillan. Measurement
Page 153 and 154: 143[86] L. Golubovic and T. C. Lube
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