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

More documents

Recommendations

Info

56 CHAPTER 3. POLARISATION OF CHIRAL ELASTOMERSaveraging over formation conditions, is given by∫ (〈V〉 R0 ∝ dR 0 exp − 3 )( 3bN( )2La RT 0 ·l −10 ·R 02aL 2 λ·R 0)×l −1 ·(λ·R 0= b ] [λ T ×l −1 ·λ·l 02aThis resulting binormal can now be summed over all the network strands,n s per unit volume. After weighting each elementary binormal vector by thedipole moment d that it carries, the resulting expression for the polarisationisP i = n s d〈V i 〉 R0 = 1 2 n sd(b/a)ǫ ijk (λ·l 0 ·λ T ·l −1 ) jk , (3.8)where P i is the component of the polarisation in the i direction and n s isthe number of strands per unit volume. To evaluate this expression after therubber has been deformed, the new equilibrium orientation of the directormust be calculated. This is explored in the next section.3.2 Equilibrium orientation of the directorThe free energy density of a liquid crystal elastomer is given byf = 1 2 µλ ijl 0jk λ T kl l−1 li. (3.9)Given the initial director orientation and the applied deformation, the equilibriumorientation of the director can be calculated after the deformation hasbeen applied by minimising the free energy density with respect to the directorn whilst keeping its magnitude fixed via a Lagrange multiplier, χ. Thusthe following quantity should be minimised w.r.t. ng = µ 1 (2 λij l 0jk λ T kl l−1 li+χ[1−n i n i ] ) (3.10)Note that whentheconstraint is satisfied g and f are equal. Ondifferentiationwith respect to n α the result is∂g∂n α= 1 2 µ( 1r −1)( n i λ ij l 0jk λ T kα +λ αjl 0jk λ T kl n l)−µχδiα n iThe stationary points in the free energy density occur when n obeys thecondition( ) 1r −1 M·n = χn (3.11)( ) 1χ =r −1 n i λ ik l 0kj λ T jα n α (3.12)where M iα = λ ij l 0jk λ T kα. This equation provides a useful general way of determiningthe final orientation of the director at equilibrium. From Eq. (3.11),n isaprincipalaxis ofM. Thefreeenergywill bestationary whenthedirectoris aligned with any one of the three principal axes of M.
3.2. EQUILIBRIUM ORIENTATION OF THE DIRECTOR 573.2.1 Nature of the stationary pointsThe free energy density minimum corresponds to putting the director n alongthe principal axis of M with largest principal value when r > 1 and alongthe principal axis with smallest principal value when r < 1. This follows fromwriting the free energy density asf = 1 2 µ[ 1r m 1 +m 2 +m 3], (3.13)where m 1 , m 2 and m 3 are the principal values of the M tensor and 1, 1 and1r are the principal values of the l−1 tensor. For r > 1 then m 1 must be thelargest principal value for the free energy density to be minimal. Similarlyfor r < 1 then m 1 must be the smallest principal value for the free energydensity to be minimal. The stability of aligning the director n with each ofthe principal axes of M can be analysed by looking at the derivatives of thefree energy density∂g∂n = 1 [( ) 1 ( )2 µ r −1 n.M+M.n∂ 2 g∂n 2 = 1 [( ) ] 12 µ r −1 2M−2χ k δ]−2χ k nwhere the label k corresponds to the principal axis under consideration. Theprincipal values of the second derivative matrix will dictate the stability ofthe direction n k . If all the principal values are positive then the point isa minimum, if they are all negative then the point is a maximum, and noconclusion can be drawn if they are neither all positive nor all negative. Inthe principal frame it is clear that for r < 1 all the χ k are positive. In thisframe the matrix ( 1r −1) M has only diagonal entries of χ 1 , χ 2 and χ 3 . Ifthe χ k are non-degenerate then the second derivative matrix has the followingproperties:• Two negative principal values and one zero when k corresponds to thelargest χ value.• One positive, one zero and one negative principal value when k correspondsto the middle χ value.• Two positive principal values and one zero when k corresponds to thesmallest χ value.Note that there is always one zero in the tensor because fluctuations alongthe director do not keep the director of length unity. When r > 1 all the χacquire a minus sign so the behaviour of the stationary points swap. Thus thestable axis swaps from the axis with the smallest m principal value to thatwiththe largest m principal value. Thesystem has onestable axis along which
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 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 117 and 118:
4.3. FINITE DEFORMATION EXAMPLES 10
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 ...

Create successful ePaper yourself

Delete template?

Save as template?