Nonlinear Stress - Strain Behavior of Nematic Elastomers using - PowerPoint PPT Presentation

Nonlinear Stress - Strain Behavior of Nematic Elastomers using Relative Rotations Andreas Menzel, 1 Harald Pleiner, 2 and Helmut R. Brand 1 , 2 1 Theoretische Physik III, Universität Bayreuth, Germany 2 Max Planck Institute for Polymer Research, Mainz, Germany ISSP/SOFT 2010 ISSP International Workshop on Soft Matter Physics, August 9 - 13, 2010, Tokyo University, Kashiwa Campus, Tokyo, Japan H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 1 / 27

Outline Introduction 1 Elasticity Including Nonlinear Relative Rotations 2 Energetics Perpendicular Stretching Linear Response under Pre-Strain 3 Effective Linear Shear Modulus Director Reorientability Conclusions 4 H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 2 / 27

Introduction Monodomain Side-Chain Nematic Elastomers Experiment: linear anisotropic elasticity nonlinear stress-strain plateau for perpendicular stretching accompanied by a complete director reorientation Description and Interpretation: effective linear modulus and director relaxation under pre-strain? H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 3 / 27

Introduction Monodomain Side-Chain Nematic Elastomers Experiment: linear anisotropic elasticity nonlinear stress-strain plateau for perpendicular stretching accompanied by a complete director reorientation Description and Interpretation: effective linear modulus and director relaxation under pre-strain? H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 3 / 27

Introduction Results Stretching a mono-domain nematic elastomer perpendicularly the resulting elastic plateau at finite strains comes with a vanishing effective linear modulus and a divergent director reorientability at its beginning and end (soft mode) this bifurcation-type behavior is a genuine manifestation of the role of nonlinear relative rotations it requires two independent preferred directions and discriminates nematic LSCEs from simple anisotropic solids and this soft mode behavior is not related to the proposed Nambu-Goldstone mode ("soft-elasticity"), nor is any closeness to an ideal soft-elastic behavior ("semi-soft elasticity") required: the described scenario is found also for cases, where the plateau starts at very large applied strains H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 4 / 27

Introduction Results Stretching a mono-domain nematic elastomer perpendicularly the resulting elastic plateau at finite strains comes with a vanishing effective linear modulus and a divergent director reorientability at its beginning and end (soft mode) this bifurcation-type behavior is a genuine manifestation of the role of nonlinear relative rotations it requires two independent preferred directions and discriminates nematic LSCEs from simple anisotropic solids and this soft mode behavior is not related to the proposed Nambu-Goldstone mode ("soft-elasticity"), nor is any closeness to an ideal soft-elastic behavior ("semi-soft elasticity") required: the described scenario is found also for cases, where the plateau starts at very large applied strains H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 4 / 27

Elasticity Including Nonlinear Relative Rotations Energetics Elastic and Orientational Degrees of Freedom Network: da α = R α j Ξ jk dr k Eulerian strain tensor 1 ε ik = 2 [ δ ik − Ξ ij Ξ ik ] 1 = 2 [ δ ik − ( ∂ a α /∂ r k )( ∂ a α /∂ r i )] 1 = 2 [ ∂ u i /∂ r k + ∂ u k /∂ r i − ( ∂ u j /∂ r i )( ∂ u j /∂ r k )] Nematic: Director n = S · ˆ ˆ n 0 and textures ( ∇ j n i ) H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 5 / 27

Elasticity Including Nonlinear Relative Rotations Energetics Relative Rotations Coupling: n nw = R − 1 · ˆ n nw ˆ rotations of the anisotropic network 0 (there is no closed expression for R − 1 in terms of ∂ u j /∂ r i ) ˆ n = S · ˆ rotations of the nematic director n 0 relative rotations (projections) 1 ˜ n nw n − γ ˆ ˆ Ω ≡ n nw + γ ˆ ˜ Ω nw − ˆ ≡ n n nw = 0 = ˜ Ω nw · ˆ n nw resulting in ˜ with γ ≡ ˆ n · ˆ Ω · ˆ n 1A. M. Menzel, H. Pleiner and H. R. Brand, J. Chem. Phys. 126 (2007) 234901. H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 6 / 27

Elasticity Including Nonlinear Relative Rotations Energetics Free Energy Power series expansion in ε ij , ˜ Ω i , ˜ Ω nw , and n i and all its couplings up j to some order here: simplified model (analytical treatment) - elastic nonlinearities neglected c 1 ε ij ε ij + 1 F = 2 c 2 ε ii ε jj + . . . + 1 Ω i ) 2 + D ( 3 ) Ω i + D ( 2 ) 2 D 1 ˜ Ω i ˜ (˜ Ω i ˜ (˜ Ω i ˜ Ω i ) 3 1 1 + D 2 n i ε ij ˜ Ω j + D nw n nw ε ij ˜ Ω nw 2 i j + D ( 2 ) Ω k + D nw , ( 2 ) 2 n i ε ij ε jk ˜ ε ij ε jk ˜ n nw Ω nw i k 2 2 ǫ a ( n i E i ) 2 − 1 reduces in linear order to de Gennes’ expression H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 7 / 27

Elasticity Including Nonlinear Relative Rotations Perpendicular Stretching Plateau for Perpendicular Stretch - Eulerian 250 50 200 dA [kJ dA [kJ dF dF m 3 ] m 3 ] 40 150 30 100 20 50 10 0 0 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 A A Fig.2: Same stress-strain data as in Fig.1 Fig.1: Stress-strain data measured by Urayama et al. a transferred to the with nonlinear purely elastic contributions by the network of polymer backbones representation in terms of the stretch subtracted. The line is the result of the amplitude A = ∂ u z /∂ z and dF / dA . theoretical model a a A. Menzel, H.P ., and H.R. Brand, J. Appl. Phys. a K. Urayama, R. Mashita, I. Kobayashi, and 105 (2009) 013503. T. Takigawa, Macromol. 40 (2007) 7665. H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 8 / 27

Elasticity Including Nonlinear Relative Rotations Perpendicular Stretching Plateau for Perpendicular Stretch - Lagrangian Fig.3: The same stress-strain data points of Urayama et al. and the theoretical line obtained by the present model (with the nonlinear elastic experimental contributions added) – now in the representation of the nominal stress as a function of the true strain. H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 9 / 27

Elasticity Including Nonlinear Relative Rotations Perpendicular Stretching Director Reorientation ϑ [ ◦ ] A Fig.4: Angle ϑ between the director orientation and the x axis under the influence of an externally imposed strain A for various initial director orientations ϑ 0 = ϑ ( A = 0 ) , e.g. 0 ◦ , 0 . 1 ◦ , 2 ◦ , 10 ◦ , . . . 80 ◦ , and 89 . 9 ◦ , respectively. For ϑ 0 = 0 ◦ (perpendicular stretch) a singular threshold behavior is found. H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 10 / 27

Elasticity Including Nonlinear Relative Rotations Perpendicular Stretching Forward bifurcation Fig.4a: ϑ = ϑ ( A ) ; same as Fig.4 with the area around A c enlarged In the vicinity of A c an amplitude equation can be derived analytically for the case ϑ 0 = 0 a ( A c − A ) + g ϑ 2 � + O ( ϑ 5 ) . � 0 = ϑ − → forward bifurcation with exchange of stability between ϑ = 0 for A < A c and ϑ ∼ √ A − A c for A > A c for ϑ 0 > 0 an imperfect bifurcation is obtained H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 11 / 27

Elasticity Including Nonlinear Relative Rotations Perpendicular Stretching Forward bifurcation Fig.4a: ϑ = ϑ ( A ) ; same as Fig.4 with the area around A c enlarged In the vicinity of A c an amplitude equation can be derived analytically for the case ϑ 0 = 0 a ( A c − A ) + g ϑ 2 � + O ( ϑ 5 ) . � 0 = ϑ − → forward bifurcation with exchange of stability between ϑ = 0 for A < A c and ϑ ∼ √ A − A c for A > A c for ϑ 0 > 0 an imperfect bifurcation is obtained H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 11 / 27

Elasticity Including Nonlinear Relative Rotations Perpendicular Stretching Soft mode a forward bifurcation is similar to a second order phase transition an (effective) susceptibility vanishes at the phase transition (at onset) giving rise to diverging fluctuations (soft mode) in contrast to Nambu-Goldstone modes, where a susceptibility is identically zero throughout the whole phase due to symmetry reasons example: director rotations in a smectic C phase: azimuthal (on the cone) Nambu-Goldstone mode tilt angle: soft only at the smectic A to C transition for imperfect bifurcations no diverging fluctuations H.R. Brand (Universität Bayreuth) Nonlinear Stress - Strain Behavior Tokyo, August 9, 2010 12 / 27