Sep 27, 2011 - Energies for nematic elastomers: small strain theories. Antonio DeSimone. SISSA. International School for Advanced Studies. Trieste, Italy.
Energies for nematic elastomers: small strain theories.
Antonio DeSimone SISSA International School for Advanced Studies Trieste, Italy
http://people.sissa.it/~desimone/
6th International Conference on Liquid Crystal Elastomers, Lisbon 7.9.2011
Elasto-nematic coupling induces spontaneaous deformations Cross-linked networks of polymeric chains containing nematic mesogens: alignment of mesogens along average direction n induces spontaneous distortion of chains Spontaneous distortion (Bain strain): Fn = a1/3 n⊗n + a-1/6(I - n⊗n) (Fn)ij = a1/3 ninj + a-1/6(δij - ninj) a volume preserving uniaxial extension along n of magnitude a 1/3 ≥ 1 (a > 1)
n e2 e1 (H. Finkelmann)
nematic director, a −1/ 6 Fe2 = 0 0
0 a1/ 3 0
|n|=1 0 0 a −1/ 6
Nematic Elastomers, Antonio DeSimone, SISSA (Trieste, ITALY)
n 3
Warner-Terentjev energy, and corrections M. Warner, E. Terentjev, Liquid crystal elastomers, Clarendon Press, Oxford 2003.
W (F, n ) = ½ µ tr ( (F FT) (Fn FnT )-1 ) , det F=1 Fn = a1/3 n⊗n + a-1/6(I - n⊗n) = ½ µ tr (
(B)
( Ln )-1 ) ,
Bain strain
B= F FT, Ln = Fn FnT
n
tr (B Ln-1 ) is minimized iff B= Ln i.e. singular values of F must be a1/3, a-1/6, a-1/6, and n = max stretch direction. Will focus on:
Min W (F,n) n
(isotropic)
(n “slaved” to F)
Anisotropic corrections: Wβ (favoring special “Finkelmann” director no) Also, but not today: Frank elasticity, electrostatic energy, viscous dynamics, ..... Nematic Elastomers, Antonio DeSimone, SISSA (Trieste, ITALY)
4
Energy landscape through homogeneous stretch and shear: non-convexity and instabilities Shear by δ
Stretch by λ
Min W (F,n) n
(isotropic)
shear δ
energy at λ=1
n=e3
shear δ stretch λ
n=e2
(stripe-domain instability explained by nonconvexity of energy landscape, see van der Waals/Maxwell on LV phase transition)
Nematic Elastomers, Antonio DeSimone, SISSA (Trieste, ITALY)
5
Small strain (but large director rotations) versions of Warner-Terentjev ‘93 and ‘96 models a1/3 = 1 + γ , γ
