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

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84 CHAPTER 4. THE ELASTICITY OF SMECTIC-A ELASTOMERSof the director from the layer displacement. The layer normal is parallel to∇φ where φ is defined in Eq. (4.3). The free energy cost of bending the layerscan be calculated from the Frank elastic term associated with splay, but withrenormalised elastic constants compared with a nematic because of the layerformation. Assuming that ∇u is small then the following results are obtainedn x = − ∂u∂xn y = − ∂u∂yf = 1 2 B ( ∂u∂z) 2+ 1 2 K 1(4.14)(4.15)( ∂ 2 u∂x 2 + ∂2 u∂y 2 ) 2. (4.16)Note that the associated length scale ξ = (K 1 /B) 1/2 is of the order of the layerthickness, and that there can be no twist deformation because n·∇×n = 0by the commutative property of the partial derivatives for sufficiently smoothfunctions. Typically, the Frank elastic constant, K 1 , is larger than that ofnematics because of the higher degree of order in smectics.The Helfrich-Hurault effect requires the calculation of the contribution ofthe magnetic field to the free energy. The susceptibility tensor χ is uniaxialwith its principal axis along the director n, so the magnetisation (induced bythe external magnetic field) is given by: M = χ·H = χ ⊥ H+(χ ‖ −χ ⊥ )(H·n)n.The part of the magnetic free energy density that depends on the director isthus given byf mag = − 1 2 µ 0χ a (n·H) 2 , (4.17)where χ a = χ ‖ −χ ⊥ is the anisotropy in the susceptibility of the rods.4.1.4 The Helfrich-Hurault effect in smectic liquid crystalsThe Helfrich-Hurault effect has been analysed in [61] as follows. Consider anSmA liquid crystal between two plates, with the layers parallel to the walls.Applying a magnetic field to the system (Fig. 4.2) in the plane of the layersgenerates a torque that acts to turn the layers. The clamping at the walls willprevent the rotation of layers occurring because as soon as the layer rotationstarts there is an infinite energy cost as the layers pile up at the walls, andtheir spacing there collapses.An alternative to this bulk rotation is local rotation of different parts oflayers in opposite directions, so that the layers lower their energy with respectto the magnetic field whilst avoiding the cost associated with reducing thelayer spacing, as well as any global layer movement. Above a certain thresholdmagnetic field, H, the layers start to rotate. This can represented in terms ofthe layer displacement variableu(x,z) = u 0 (z)coskx. (4.18)
4.1. INTRODUCTION 85HzyxGlobal rotation of layersHigh energy cost at boundaryLocal rotation of layersFlat at boundaryFigure 4.2: An illustration of the magnetic field configurationrequiredtoseetheHelfrich-Huraulteffect. Afterthefieldisappliedin the direction shown we can either rotate all the layersor locally rotate different parts of layers in opposite directions.This is subject to the constraint that the amplitude of the displacement is zeroat the edges of the sample, z = 0,L. Only the first harmonic is consideredhere as an illustrationu 0 (z) = u 0 sink z z, (4.19)where k z = π/L. The director can be calculated from the layer displacementbecause it remains normal to the layersn x ≈ − ∂u∂x = ǫsin(k zz)sin(kx) (4.20)n y = 0 (4.21)n z ≈ 1, (4.22)whereǫ = u 0 k. Substitutingthis expressionfor thelayer displacement into theFrank elastic and magnetic energy densities of the liquid crystal the followingis obtained[]〈f el 〉 = 1 2 ǫ2 B k2 zk 2〈cos2 k z z〉〈cos 2 kx〉+K 1 k 2 〈sin 2 k z z〉〈sin 2 kx〉 (4.23)〈f mag 〉 = − 1 2 χ aµ 0 H 2 〈n 2 x 〉 = −1 2 χ aµ 0 H 2 ǫ 2 〈sin 2 k z z〉〈sin 2 kx〉. (4.24)The minimum in the total free energy occurs when k 2 = π/(ξL), whereξ 2 = K B. This is the optimal wavelength of the distortion that reaches acompromise between bending the layers and keeping their spacing fixed. Thecritical magnetic field can be calculated from the condition that the elasticand the magnetic terms exactly cancelχ a µ 0 H 2 c = 2πBξL . (4.25)The strength of the field required to see this transition is extremely strong,and the required sample must be extremely uniform. However, this transitioncan be seen in a simpler, mechanical context.
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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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Page 45 and 46: 2.5. CONCLUSIONS 35to their limit
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Page 127: 4.5. CONCLUSIONS 117formation in th
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134 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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