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

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124 CHAPTER 5. SOFT ELASTICITY OF SMECTIC ELASTOMERSwith strain proportional to the square of the applied electric field) [103]. PolydomainSmC elastomers have also been shown to exhibit shape memory whichmay also prove to be important technologically [104].5.2 Soft modes of biaxial smectic A elastomers5.2.1 Model biaxial smectic A elastomerThe model of a biaxial SmA used here is an extension of the model presentedin chapter 4. It is assumed that there is no interaction between the formationof the layers and the secondary alignment axis of the mesogens. As a resultthe required free energy density is given byf = 1 2 µTr [λ·l 0 ·λ T ·l −1] + 1 2 B(d/d 0 −1) 2 (5.11)where both l 0 and l are now biaxial, with principal axes n, m and l. Theprimary alignment axes of the mesogens, n is identified with the smectic layernormal, but the secondary alignment axis is free to rotate in the plane of thelayer to rotate in the plane of the layers. Consequently, the soft modes of thisbiaxial SmA arise because of the freedom of the secondary alignment axis. Ifwe assume that the layers move affinely with the matrix, then it follows thatthe layer spacing is given bydd 0=1|λ −T ·n 0 | , (5.12)where n 0 is the initial direction of the primary axis of alignment.5.2.2 General form of soft modes in biaxial SmA elastomersSince the model of biaxial SmA elastomer considered here is based on that ofa nematic elastomer, the starting point used is the general form of soft modesin nematic elastomersλ = l 1/2 ·W ·l −1/20 , (5.13)where W is a general rotation matrix, l 0 is the initial biaxial anisotropy tensorand l is the current anisotropy tensor of the polymer. A general deformationcould in principle change the layer spacing. However for soft modes the layerspacing must remain fixed. This constraint can be expressed via Eq. (5.12) asn T 0 ·λ −1 ·λ −T ·n 0 = 1. (5.14)Inserting the general form of a soft mode Eq. (5.13) into the layer spacingconstraint Eq. (5.14) yields the following equation1r = nT 0 ·W T ·l −1 ·W ·n 0 . (5.15)
5.3. SOFT MODES OF SMECTIC C ELASTOMERS 125For uniaxial SmA, the only solution to this equation is n = W·n 0 because thequadric surface associated with l −1 only has the correct width at one point.Thus there is no freedom for the director and no soft modes in uniaxial SmAelastomers except for pure rotations. This is because if the layer normal n ismoved by any deformation other than a rotation then the layer spacing will bechanged and the resulting state will be higher in energy. However, in biaxialSmA elastomers the secondary alignment axis means that there is still enoughfreedom for a soft mode to exist.The soft modes can be decomposed into their component rotations togain a better understanding of them as follows. First we write the generaldeformation matrix asW = W R ·W n0 , (5.16)where W n0 is a rotation about n 0 and n = W R ·n 0 . Using this in Eq. (5.13)results in the following expression for the soft mode:λ = l 1/2 ·W R ·W n0 ·l −1/20[ ](5.17)= W R ·W n0 · l 1/2n 0 ,m ′ ,l·l −1/2′ 0 . (5.18)The factor in square brackets here is a familiar soft mode from the examplegiven in §5.1.3. Thus all soft modes in biaxial SmA elastomers can be decomposedinto a rotation of the soft mode that has the primary alignment axis(i.e. n) fixed and a secondary rotation axis displaced from its initial position.5.2.3 Particular example of a soft mode in a SmA elastomerFor later comparison with the soft modes of an SmC elastomer, the soft modeassociated with an imposed λ xx component is now presented. This mode willhave a fixed primary alignment direction, n 0 , but will have a mobile secondaryalignment direction, m. Fig. 5.2 shows an illustration of this mode. In thecentre of the diagram the secondary alignment axis, m is depicted. Along theoutside of the diagram the shape of the biaxial SmA elastomer is illustratedas viewed from above.5.3 Soft modes of smectic C elastomers5.3.1 Model smectic C elastomer free energyThe model of an SmC elastomer described here is again an extension of theSmA elastomer model presented in chapter 4. It is based on a constrainedversion of the nematic elastomer. Whilst it is a specific microscopic model ofthe SmC elastomer phase, its features are more general and the soft modesexplored here are also present in other, continuum, approaches [105].
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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 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
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Page 147 and 148: Bibliography[1] P. J. Flory. Statis
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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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