Elastic energies for nematic elastomers Antonio DeSimone SISSA International School for Advanced Studies Trieste, Italy http://people.sissa.it/~desimone/ Drawing from collaborations with: S. Conti, Bonn; G. Dolzmann, Regensburg; K. Urayama, Kyoto; A. Di Carlo and L. Teresi, Roma Tre, P. Cesana, Trieste. 5 th ILCEC: Kent, 25.09.2009
Outline Part 1 Three key experiments: Kuepfer, Finkelman Martinoty et al. Urayama et al. Part 2 Energy-based models for nematic elastomers vs. benchmark experiments Part 3 Numerical simulation of the key experiments. (for complex materials simple mechanical experiments give rise to complex BVPs) Part 4 Conclusions 2
1: Uniaxial stretching Kuepfer, Finkelmann, 1991,1994. 3
2: Simple Shear Simple shear parallel to N Simple shear perpendicular to N Rogez, Francius, Finkelmann, Martinoty : Shear mechanical anisotropy of SCLCE: influence of sample peparation, Eur. Phys. J. E, vol. 20, p. 369 (2006). 4
3: E-field applied to free-standing (!) film equil. opitcal birefringence cos 2 (θ) n e θ equil. horizontal strain A. Fukunaga, K. Urayama, T. Takigawa, A. DeSimone, L. Teresi: Dynamics of electro-opto-mechanical effects in swollen nematic elastomers, Macromolecules, vol.41, p. 9389 (2008). 5
Warner-Terentjev model energy, corrections M. Warner, E. Terentjev, Liquid crystal elastomers, Clarendon Press, Oxford 2003. W (F, n ) = ½ μ (F F T ) (L n ) -1 L n = F n2, F n = a 1/3 n n + a -1/6 (I - n n) transformation strain W is minimized i i iff current direction of maximal stretch t aligned with n n and principal stretches = a 1/3, a -1/6, a -1/6. This energy density is isotropic i (no memory of cross-linking state): t Min W (F,n) = ½ μ (λ 2 min + λ 2 int + λ 2 max / a) λ principal stretches n F n is an isotropic function of n: F Rn = R F n R T Golubovic-Lubensky applies (c 44 vanishes) Anisotropic corrections: W α, W β,... most general exp with correct invariance Also: small strain versions, Frank elasticity, electrostatic energy, dynamics... 6
Anisotropic corrections to W W (F, n ) = ½ μ (F F T ) (L n ) -1 n r n* (F) isotropic energy (Bladon, Terentjev, Warner 94) Introduce memory of n r, director orientation at crosslinking W -1 β (F, n ) = W(F,n) + ½ β μ (F T F) (L nr ) anisotropic through residual stress (Verwey, Terentjev, Warner 96) W α (F, n ) = W(F,n) + ½ α μ (1 (n*(f) n ) 2 ) anisotropic through angle between current n and n*(f) = current n r (DeSimone, Teresi 09, in a spirit similar to Pleiner et al. see: Fukunaga, Urayama, Takigawa, DeSimone, Teresi, 08) n 7
The W-T trace formula revisited A. DeSimone, L. Teresi: Elastic energies for nematic elastomers, Eur. Phys. J. E, vol. 29, p. 191 (2009). 8
shear δ Energy landscape through homogeneous stretch and shear Stretch by λ Shear by δ Min W (F,n) (,) (isotropic) Bifurcation=shear band instability n stretch λ energy at λ=1 shear δ n=e 3 n=e 2 P.-G. de Gennes: Weak nematic gels, in: Helfrich and Heppke (eds.), LC of one- and two-dimensional order, p. 231 (1980). A. DeSimone, L. Teresi: Elastic energies for nematic elastomers, Eur. Phys. J. E, vol. 29, p. 191 (2009). 9
At a mesoscopic scale, the effective energy density is... n=e n=e 3 2 Effective mesoscopic energy (quasi-convex envelope W qc ) Can turn this into a computational strategy: avoid resolving fine scales explicitly, though they are accounted for in the energetics (homogenization, relaxation...) 10
shear δ Bifurcation is simply shifted by anisotropy Stretch by λ Min W (F,n) (,) (isotropic) n n=e 3 n=e 2 energy at λ=1 shear δ Shear by δ Min W β (F,n),) (anisotropic) n energy at λ=1 stretch λ shear δ stretch λ shear δ A. DeSimone, L. Teresi: Elastic energies for nematic elastomers, Eur. Phys. J. E, vol. 29, p. 191 (2009). M. Warner, E. Terentjev, Liquid crystal elastomers, Clarendon Press, Oxford 2003. 11
Initial shear moduli parallel and perp to N Shear by δ (parallel to N) Shear by ε (perpendicular to N) As β ranges from 0 to + G par goes from 0 to G perp /a < G perp λ crit goes from a -1/3 to a 1/12 Same qualitative picture from W α (F,n),) shear δ Min W β (F,n) (anisotropic) i n stretch λ n r energy at λ=1 shear δ DeSimone, Teresi: Elastic energies for nematic elastomers, Eur. Phys. J. E, vol. 29, p. 191 (2009). Rogez, Francius, Finkelmann, Martinoty : Shear mechanical anisotropy of SCLCE: influence of sample peparation, Eur. Phys. J. E, vol. 20, p. 369 (2006). 12
Stretching exp.s: from qualitative to quantitative via FEM numerical simulations AR = min W (F,n) n min W β (F,n) n isotropic: no memory of n at cross-linking β strength of anisotropy S. Conti, A. DeSimone, G. Dolzmann: Soft elastic response of stretched sheets of nematic elastomers: a numerical study, J. Mech. Phys. Solids, vol.50, p. 1431 (2002); Semi-soft elasticity and director reorientation in stretched sheets of nematic elastomers, Phys. Rev. E, vol. 66, p. 061710 (2002). β 13
Numerical stretching experiment using W qc S. Conti, A. DeSimone, G. Dolzmann: J. Mech. Phys. Solids, vol.50, p. 1431 (2002). 16
Theory vs. Experiment Multiscale numerics (Conti, DeSimone, Dolzmann) X-ray scattering (Zubarev et al.) Direct observation (Terentjev) 17
Can use small strain theory as well P. Cesana, A. DeSimone, L. Teresi: Numerical stretching t experiments for nematic elastomers: dynamic attainability of low energy states, t forthcoming (2009). 18
Static and dynamic response to e-field in free-standing swollen films steady states: A. Fukunaga, K. Urayama, T. Takigawa, A. DeSimone, L. Teresi: Dynamics of electro-opto-mechanical effects in swollen nematic elastomers, Macromolecules, vol.41, p. 9389 (2008). 21
Conclusions The big picture: peculiar elastic response ( plateau ) due to director re-orientation. W (trace formula of W&T) is a very helpful conceptual tool to understand shear band instability explaining correlation between plateau and director re-orientation. Anisotropic correction imparts to the material finite shear modulus in planes containing n r, and a hard response to stretching up to critical stretch λ c. At λ= λ c the shear modulus vanishes and response to further stretching is softer due to shear band instability (just as in Ye et al., Biggins et al.) The same qualitative picture (the shear band bifurcation of W is latent at low stretches and it is activated at sufficiently high stretches) emerges from both W α (Pleiner-like) lik and W β (Verwey-Warner-Terentjev). Simple shear experiments by Martinoty et al. are not in contradiction with either W α or W β and (at least at a qualitative level) cannot discriminate between these two models. 22
References A. DeSimone, Energetics of fine domain structures, Ferroelectrics, vol. 222, p. 275 (1999). A. DeSimone and G. Dolzmann, Material instabilities in nematic elastomers, Physica D, vol. 136, p. 175 (2000). A. DeSimone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant i energies, Archive Rat. Mech. Analysis, vol. 161, p. 181 (2002). S. Conti, A. DeSimone, and G. Dolzmann, Soft elastic response of stretched sheets of nematic elastomers: a numerical study, J. Mech. Phys. Solids, vol. 50, p. 1431 (2002). S. Conti, A. DeSimone, and G. Dolzmann, Semi-soft elasticity and director reorientation in stretched sheets of nematic elastomer, Phys. Rev. E, vol. 60, p. 61710 (2002). A.DeSimone and G. Dolzmann, Stripe-domains in nematic elastomers:old and new, in: IMA Volumes in Mathematics and its Applications, Springer Verlag, 2005. J. Adams, S. Conti, and A. DeSimone, Soft elasticity and microstructure in smectic C elastomers, Cont. Mech. and Thermodynamics, vol.18, p.319 (2007). J. Adams, S. Conti, A. DeSimone, and G. Dolzmann, Relaxation of some transversally isotropic energies and applications to smectic A elastomers, Math Mod. Meth. Appl. Sci., vol. 18, p. 1 (2008). 23
References continued A. DeSimone, A. DiCarlo, and L.Teresi, Critical voltages and blocking stresses in nematic gels, Eur. Phys. J. E, vol. 24, p. 303 (2007). A. Fukunaga, K. Urayama, T. Takigawa, A. DeSimone, and L.Teresi, Dynamics of electro-optomechanical effects in swollen nematic elastomers, Macromolecules, vol.,41, p. 9389 (2008). A. DeSimone and L.Teresi, Elastic energies for nematic elastomers, Eur. Phys. J. E, vol. 29, p. 191 (2009). P. Cesana and A. DeSimone, Strain-order coupling in nematic elastomers: equilibrium configurations, Math. Meth. Mod. Appl. Sci., vol. 19, p. 601 (2009). P. Cesana, Relaxation of multiwell energies in linearized elasticity and application to nematic elastomers, Archive Rat. Mech. Analysis, in press (2009). P. Cesana, A. DeSimone, and L. Teresi, Numerical stretching experiments for nematic elastomers: dynamic attainability of low energy states, forthcoming (2009). Many contributions by many other authors... Monograph: M. Warner and E. Terentjev, Liquid Crystal Elastomers, Oxford University Press, 2003. 24