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

More documents

Recommendations

Info

122 CHAPTER 5. SOFT ELASTICITY OF SMECTIC ELASTOMERSwavelength deformations arise because of the broken-symmetry of the nematicelastomer. This expression also describes the soft modes for biaxial nematicelastomers [88].For the case of a uniaxial nematic elastomer Eq. (5.6) can be rearrangedinto the formλ = W ·l 1/2n·l −1/2′ 0 (5.7)where n ′ is a unit vector on the surface of a sphere and n = W · n ′ . In thisform it is clear that the soft modes available to a nematic elastomer can befound by selecting n ′ from the surface of a sphere, and then carrying out ageneral rotation, W (i.e. n ′ defines the shape of the body which can then berotated).For the case of a biaxial nematic elastomer the same rearrangement can becarried out, but this time after the choice of n ′ has been made, the position ofthe secondary axis can be selected from any point on a specified great circle.Soft modes in biaxial nematicsAs pointed out in §5.1.1 biaxial liquid crystalline phases are rare. Howeverbiaxial nematic phases [89] biaxial smectic A phases [90] have been found inliquid crystalline polymers and in principle it should be possible to make abiaxial nematic or biaxial SmA liquid crystal elastomer. The properties of abiaxial phase must necessarily be described by biaxial tensors, and the case ofthe shape tensor of a biaxial polymer is no different. Suppose that the shapetensor has principal axes l, n and m which are orthogonal and normalised.The shape tensor can be expressed asl = l ‖ nn T +l 1 mm T +l 2 ll T (5.8)= δ +(r −1)nn T + p 2 mmT − p 2 llT , (5.9)where p = (l 1 − l 2 )/l ⊥ and l ⊥ = 1 2 (l 1 + l 2 ). In biaxial nematics rotationsof the shape tensor about the principal director, n are significant, whereas ina uniaxial nematic they are not. An explicit example of a soft mode arisingfrom this sort of rotation is presented in [88]. This mode is given here as itwill prove useful in our discussion of soft modes in smectic A elastomers. Fora soft mode with angle of rotation φ about n 0 ≡ n = z of the form l 1/2φ ·l−1/2 0the following deformation tensor is obtained⎛λ = ⎝1−(1−1/ √ r ⊥ )sin 2 φ (1− √ r ⊥ )sinφcosφ 0(1/ √ r ⊥ −1)sinφcosφ 1+( √ r ⊥ −1)sin 2 φ 00 0 1⎞⎠ (5.10)where r ⊥ = l 1 /l 2 is a material parameter expressing the degree of biaxiality.
5.1. INTRODUCTION 123InSmAelastomers thelayers break therotational symmetryof thestate sothat only rotations of the matrix points about the layer normal leaves the freeenergy unchanged. It will be shown that since the director has no componentin the layers there can be no soft modes in SmA elastomers, except for thecase of a biaxial order parameter.5.1.4 Experimental literature on smectic C elastomersSeveral investigations on the preparation of monodomains from SmC elastomershave been performed. The principles of aligning the SmC elastomerare the same as those of a nematic elastomer, i.e. cross-linking under a load.However, aligning smectic elastomers, and particularly SmC elastomers, ismore involved because both the director and the layers must be aligned.An investigation by Semmler and Finkelmann [91] into aligning SmC elastomersby found some success by adopting a two stage process of non-collinearstretches. On straining a polydomain chiral SmC sample by λ = 1.6 the sampleremains opaque, indicating that there is no director alignment. X-rayscattering shows a small degree of alignment, which does not improve on annealing.A new approach was adopted whereby the SmC sample was firstloaded a small amount and then swollen with toluene, until it reached anisotropic state. It was then gradually deswollen and reformed into the smecticstate whilst under load. The sample shows good director alignment, but thelayers can have any orientation on the surface of a cone. The resulting samplewas centrosymmetric. A second stretch is then applied at an angle of 90 ◦ −θwhereθ isthetilt angle. Asaresultthelayers rotatearoundsothatthestretchdirection is contained within the layer. The result is a non-centrosymmetricmonodomain. This alignment can be locked in by another cross-linking stage.An alternative approach of Hiraoka and Finkelmann [92, 93] was to usea mechanical shear field. The same procedure as described above was usedto obtain a uniaxially aligned director field. The sample was then shearedthrough an angle equal to the tilt angle of the director w.r.t. the layersin the SmC phase. Again a well aligned monodomain, with aligned layersis obtained. This technique of making monodomains has recently made itpossible to observe spontaneous deformations in the SmC phase [94].SmC ⋆ elastomers may also prove to be of great technological significance,for example they possess the correct phase symmetry to exhibit second harmonicgeneration, that is generate light at a frequency of twice that of theincident light. This has been the subject of experimental research [95]. Theirpiezoelectric properties have also been studied experimentally [96, 97]. Thismay prove to be significant because whilst conventional piezoelectric materialscan only achieve small strains, rubbery materials could achieve muchlarger strains. The ferroelectric properties of these materials have also beenintensively studied [98–102]. As a result of these ferroelectric properties SmCelastomers show giant electristriction effects (analogous to piezoelectricity but
Page 3:
iStatistical models of elasticity i
Page 7:
AcknowledgementsDuring the course o
Page 10 and 11:
viiiCONTENTS5 Soft elasticity of sm
Page 12 and 13:
2 CHAPTER 1. INTRODUCTION1.2 Neo-cl
Page 14 and 15:
4 CHAPTER 1. INTRODUCTIONticular in
Page 17 and 18:
Chapter TwoThe elasticity of hairpi
Page 19 and 20:
2.1. INTRODUCTION 9Fig. 2.1. This b
Page 21 and 22:
2.2. MODEL OF HAIRPIN CHAINS 11nFig
Page 23 and 24:
2.2. MODEL OF HAIRPIN CHAINS 13za)
Page 25 and 26:
2.2. MODEL OF HAIRPIN CHAINS 15in
Page 27 and 28:
2.2. MODEL OF HAIRPIN CHAINS 17The
Page 29 and 30:
2.2. MODEL OF HAIRPIN CHAINS 1910.8
Page 31 and 32:
2.2. MODEL OF HAIRPIN CHAINS 21This
Page 33 and 34:
2.2. MODEL OF HAIRPIN CHAINS 23f g
Page 35 and 36:
2.3. NON-AFFINE DEFORMATION MODEL 2
Page 41 and 42:
2.4. HAIRPIN CHAIN NETWORK FREE ENE
Page 43 and 44:
2.4. HAIRPIN CHAIN NETWORK FREE ENE
Page 45 and 46:
2.5. CONCLUSIONS 35to their limit
Page 47 and 48:
2.A. GEOMETRY OF A HAIRPIN AT ZERO
Page 49 and 50:
2.B. UNDULATING CHAIN STATISTICS 39
Page 51 and 52:
2.C. COUNTING CONFIGURATIONS BY IND
Page 53 and 54:
2.C. COUNTING CONFIGURATIONS BY IND
Page 55 and 56:
2.D. INTERPRETATION OF FN 452.D Int
Page 57:
2.E. SPRING CONSTANT OF AN EXTENDED
Page 60 and 61:
50 CHAPTER 3. POLARISATION OF CHIRA
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83: 72 CHAPTER 3. POLARISATION OF CHIRA
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
Page 101 and 102: 4.2. MICROSCOPIC, FINITE DEFORMATIO
Page 107 and 108: 4.3. FINITE DEFORMATION EXAMPLES 97
Page 123 and 124: 4.4. COMPARISON WITH EXPERIMENT 113
Page 125 and 126: 4.4. COMPARISON WITH EXPERIMENT 115
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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?