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

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90 CHAPTER 4. THE ELASTICITY OF SMECTIC-A ELASTOMERSnematic elasticity. It was also found that the smectic layer modulus is responsiblefor the large anisotropy found by Nishikawa and Finkelmann [66].Theundulationinstability inSmAliquidsinglecrystal elastomers hasbeenstudied using this continuum theory [73]. A technique very similar to thatoutlined in §4.1.4 was used to calculate the threshold strain and the criticalwavenumber to compare with the experiments of [66]. The results obtainedwere 10 −3 cm< λ c < 10 −2 cm (here λ c refers to the length scale of the stripesnot to the imposed strain) together with a threshold strain of ǫ c ≈ 1 C 12 B ≈ 1%.4.2 Microscopic, finite deformation model of thesmectic elastomerA microscopic model of a nematic elastomer can be formulated from analysingthe probability distribution of the occurrence of particular end-to-end spans ofa polymer. This average end-to-end distribution gives the average propertiesof all chains, and is used as a representative of the elastomer as a whole. Thisapproach works particularly well for nematic elastomers.For a smectic elastomer both the span of a polymer chain and its positionrelative to the smectic layers are significant. This interaction between thepolymer chains and the smectic order is modelled here as follows. The smecticorder presents a series of potential wells, forming a corrugated potential, inwhich the cross-link points sit as illustrated in Fig. 4.5. Deviation of the crosslinkpoints from these wells is penalised because it disrupts the smectic orderof the layers, and because of the steric repulsion between the cross-link pointand the mesogens.The probability distribution as a function of the positions of the ends ofthe chain, R 1 and R 2 is given by{P 0 (R 1 ,R 2 ) ∝ exp − 32L RT 12 ·l−1 0 ·R 12 +2βcos(α−q T 0 ·R 1)}+2βcos(α−q T 0 ·R 2 )≈ ∑ n,mexp{− 32L RT 12 ·l −10 ·R 12 −β(2πn−q T 0 ·R 1 +α) 2(4.35)−β(2πm−q T 0 ·R 2 +α) 2} (4.36)= ∑ m,nP 0(m,n) (R 1 ,R 2 ), (4.37)where R 12 = R 1 −R 2 , L is the arc length of a polymer, q 0 is the wave vectorof the smectic layers, β defines thestrength of the potential in which thelayersare sitting and we use the definitionl 0 = l ⊥ δ +(l ‖ −l ⊥ )n 0 n T 0 (4.38)
4.2. MICROSCOPIC, FINITE DEFORMATION MODEL OF THESMECTIC ELASTOMER 91R 2R 1OVFigure 4.5: The figure shows an illustration of the microscopicmodel of a smectic elastomer, in which the cross-link pointssit in a periodic potential as a result of the smectic ordering.Note the smectogens are not shown for clarity.Note that this probability distribution does not penalise the polymer chainfor moving across layers (see for example [74]). This effect could be partiallytakenintoaccount byusingadifferentvalueoftheanisotropy, r. Itisassumed,without loss of generality, that the first layer in the system sits at the originα = 0, i.e. there is no displacement w.r.t. the background. In Eq. (4.36)the limit of β ≫ 1 has been taken and the probability distribution writtenas a sum over all the layers labelled by n and m in which the two differentends can sit. The cosine functions have been written as a power series and,since β ≫ 1, only the first term taken. The summation sign can then bebrought down from the exponent because since, β is so large, all the wells ofthe potential are effectively decoupled. This expression is useful in calculatingthe quenched average since each cross-link point has to be quenched into aparticular layer. A one dimensional version of this probability distribution isshown in Fig. 4.6, for P 0(m,n) (0,R 2 ) where R 2 is parallel to q 0 .It is useful to convert to centre of mass and span coordinates as followsP = 1 2 (R 1 +R 2 ) (4.39)
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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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Page 51 and 52: 2.C. COUNTING CONFIGURATIONS BY IND
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Page 55 and 56: 2.D. INTERPRETATION OF FN 452.D Int
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Page 60 and 61: 50 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: 4.1. INTRODUCTION 89−D 2[(v (a)xz
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Page 107 and 108: 4.3. FINITE DEFORMATION EXAMPLES 97
Page 123 and 124: 4.4. COMPARISON WITH EXPERIMENT 113
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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
Page 149 and 150: 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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