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

More documents

Recommendations

Info

2 CHAPTER 1. INTRODUCTION1.2 Neo-classical rubber elasticityIn 1969 de Gennes proposed that if an elastomer was cross-linked in the presenceof a liquid crystalline solvent then the resulting polymer network shouldshow anisotropic properties as a consequence [4]. For example, if the networkwas cross-linked in a nematic solvent then the polymer chains should elongatealong the director. Polymer liquid crystals (PLCs) offer a more direct way ofcombining the anisotropic properties of liquid crystals with a polymer backbone.The liquid crystalline molecules can be coupled in several different waysto the polymer as illustrated in Fig. 1.1. If these PLCs are cross-linked intoa)b)c)nFigure1.1: Threeexamples of polymerliquidcrystals: a) mainchain , b) prolate back boneside chain and c) oblate back boneside chain [5].a network, then the coupling between the mesogenic units and the polymerbackbone will result in an elastomer that exhibits the anisotropic propertiesreferred to by de Gennes. An important advance was the synthesis of polymerliquid crystal networks using polysiloxane backbones [6]. This enabledexploration of a several parameters in the construction of the network, forexample the spacer length and the influence of the mesogenic units. Initiallythe samples made were not globally aligned: they were polydomains. Two approacheshave been successfully employed to align the separate domains, andhence prepare monodomain elastomers. The first is to pre-align the polymersin a magnetic field and then cross-link them [7]. On heating these samples tothe isotropic state and returning them to the nematic state, they show completerecovery of the globally aligned nematic phase. The second method is tocarry out a light cross-linking stage of an unaligned elastomer, and then loadthe sample before carrying out a second cross-linking stage [8]. The elasticproperties of the resulting monodomains can then be investigated without thecomplication of the polydomain structure. The anisotropic properties of thenematic monodomain described above can be modelled using the phantom
1.3. APPLICATIONS FOR LIQUID CRYSTAL ELASTOMERS 3network model for anisotropic chains (see e.g. [9]). The resulting free energydensity is)f = 1 2(l µTr 0 ·λ T ·l −1 ·λ ,where l 0 and l are the matrices of effective step lengths before and after thedeformation by λ, respectively. This theory has proved remarkably successfulin describing, amongst other things, the spontaneous elongation of a nematicelastomer on cooling into the nematic state. Some main chain polymers (seeFig. 1.1 a)) show particularly large spontaneous elongation due to the strongnematic state that they form. The unusual elastic properties of main chainliquid crystalline elastomers will be discussed in part I.Cholesteric liquid crystals have a helical structure in their director fields.A cholesteric liquid crystal elastomer also has a twisted director field that iscross-linked into the elastomer. However, by the use of a cholesteric solventand a nematic elastomer, a twist can be cross-linked into the elastomer andcan be adjusted by the concentration of the cholesteric solvent present duringcross-linking. The more concentrated the cholesteric solvent, the more tightlytwisted is the director field on cross-linking. This director field is imprintedon the network by the cross-linking process and remains after the solventhas been removed. These chirally imprinted elastomers show a fascinatingtwisted-untwisted transition as the solvent concentration is varied [10, 11].Cholesteric elastomers show a remarkable ability to separate chiral isomers bypreferentially absorbing one handedness over the other [12, 13].The smectic phase was also discussed by de Gennes and was expectedto have extremely anisotropic mechanical properties. This is because thelayered structure of smectic liquid crystals was expected to force the polymerchains toconcentrate inbetween layers, andas aresulttheelastomer structureis strongly anisotropic. The elastic properties of smectic elastomers will bediscussed in part II.An interesting attempt to connect together the classical theory of rubberelasticity and the neo-classical theory of rubber elasticity has been made byXinget al. [14]. Theyfollow thereplica-based approach of Deam andEdwards[15] and construct a Landau theory of the vulcanization transition. Theirwork also promises to shed light on properties of polydomain liquid crystalelastomers [16–19].1.3 Applications for liquid crystal elastomersLiquid crystal elastomers are an exciting system academically because of theirmany unusual elastic and optical properties for example [20]. Their practicalusefulness is currently limited because of the small quantities available. However,there are many interesting prospective applications including the siftingof chiral molecules mentioned above. One application that has attracted par-
Page 3: iStatistical models of elasticity i
Page 7: AcknowledgementsDuring the course o
Page 10 and 11: viiiCONTENTS5 Soft elasticity of sm
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:
52 CHAPTER 3. POLARISATION OF CHIRA
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:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
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 103 and 104:
Page 105 and 106:
Page 107 and 108:
4.3. FINITE DEFORMATION EXAMPLES 97
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
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 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
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 ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?