Introduction to Liquid Crystalline Elastomers

Transcription

1 Introduction to Liquid Crystalline Elastomers Harald Pleiner Max Planck Institute for Polymer Research, Mainz, Germany "Ferroelectric Phenomena in Liquid Crystals" LCI, Kent State University, Ohio, USA June 18-28, pleiner/ferrofl.html Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

2 Collaborations Elastomers, Liquid Crystals, Rheology Univ. Bayreuth: H.R. Brand, A. Menzel IMFS, Univ. Strasbourg: P. Martinoty, (D. Collin) Univ. Freiburg: H. Finkelmann ALCT: P.E. Cladis Univ. Tübingen: M. Liu, (O. Müller, D. Hahn) Univ. Ottawa: J.L. Harden MPI-P Mainz: S. Bohlius Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

3 Outline 1 LC Elastomers Polymer Liquid Crystals and Crosslinking Single Liquid Crystal Elastomers Early Experiments 2 Linear Elasticity Elastic Strains and their Dynamics Linear Elasticity of Nematic Elastomers Soft Elasticity and Rotational Goldstone Mode 3 Nonlinear Elasticity Nonlinear Elastic Strains and their Dynamics Nonlinear (Relative) Rotations "Semisoftness" and Stripe Patterns Polydomain Plateau Shifted Rotational Goldstone Mode 4 Summary Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

4 LC Elastomers LC Elastomers New class of material combining elasticity of rubbers orientational order of liquid crystals Polymer Liquid Crystals and Crosslinking Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

5 LCE Samples LC Elastomers Side-chain elastomers (Finkelmann) Polymer Liquid Crystals and Crosslinking Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

6 LC Elastomers Polydomain - Monodomain Single Liquid Crystal Elastomers a polycrystalline nematic elastomer becomes a single crystal under elongation (Finkelmann) second crosslinking step in the elongated state liquid single crystal elastomer (LSCE) J. Küpfer, H. Finkelmann, Macromol. Rapid Commun. 12 (1991) 717. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

7 LC Elastomers Polydomain - Monodomain Single Liquid Crystal Elastomers a polycrystalline nematic elastomer becomes a single crystal under elongation (Finkelmann) second crosslinking step in the elongated state liquid single crystal elastomer (LSCE) J. Küpfer, H. Finkelmann, Macromol. Rapid Commun. 12 (1991) 717. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

8 Stretching LC Elastomers stress-strain relation during the polyto monodomain transition; the "plateau shrinks towards T c J. Schätzle, W. Kaufhold, and H. Finkelmann, Makromol. Chem., 190 (1989) Early Experiments stress-strain relation during the reorientation of the director (LSCE) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

9 Stretching LC Elastomers stress-strain relation during the polyto monodomain transition; the "plateau shrinks towards T c J. Schätzle, W. Kaufhold, and H. Finkelmann, Makromol. Chem., 190 (1989) Early Experiments stress-strain relation during the reorientation of the director (LSCE) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

10 Strain Linear Elasticity Elastic Strains and their Dynamics crystals (solids): positional order broken translational symmetry rigid translations do not cost energy Goldstone mode f = f ( u) with u displacement vector decompose displacement gradients (strain and rotation) u 1 2 ( ju i + i u j ) ( ju i i u j ) ɛ ij + Ω ij (1) rigid rotations must not cost energy f = f (ɛ) f el = 1 2 c 11(ɛ 2 xx + ɛ 2 yy) + c 12 ɛ xx ɛ yy + c 13 ɛ zz (ɛ xx + ɛ yy ) c 33ɛ 2 zz + 2c 44 (ɛ 2 yz + ɛ 2 xz) + 2(c 11 c 12 )ɛ 2 xy (2) 5 elastic moduli (3 in the case of incompressibility, tr ɛ = 0); [isotropic case: 2(1)] Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

11 Strain Linear Elasticity Elastic Strains and their Dynamics crystals (solids): positional order broken translational symmetry rigid translations do not cost energy Goldstone mode f = f ( u) with u displacement vector decompose displacement gradients (strain and rotation) u 1 2 ( ju i + i u j ) ( ju i i u j ) ɛ ij + Ω ij (1) rigid rotations must not cost energy f = f (ɛ) f el = 1 2 c 11(ɛ 2 xx + ɛ 2 yy) + c 12 ɛ xx ɛ yy + c 13 ɛ zz (ɛ xx + ɛ yy ) c 33ɛ 2 zz + 2c 44 (ɛ 2 yz + ɛ 2 xz) + 2(c 11 c 12 )ɛ 2 xy (2) 5 elastic moduli (3 in the case of incompressibility, tr ɛ = 0); [isotropic case: 2(1)] Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

12 Strain Linear Elasticity Elastic Strains and their Dynamics crystals (solids): positional order broken translational symmetry rigid translations do not cost energy Goldstone mode f = f ( u) with u displacement vector cf. nematics: broken rotational symmetry f = 1 2 K ijkl( j n i )( l n k ) decompose displacement gradients (strain and rotation) u 1 2 ( ju i + i u j ) ( ju i i u j ) ɛ ij + Ω ij (1) rigid rotations must not cost energy f = f (ɛ) f el = 1 2 c 11(ɛ 2 xx + ɛ 2 yy) + c 12 ɛ xx ɛ yy + c 13 ɛ zz (ɛ xx + ɛ yy ) c 33ɛ 2 zz + 2c 44 (ɛ 2 yz + ɛ 2 xz) + 2(c 11 c 12 )ɛ 2 xy (2) 5 elastic moduli (3 in the case of incompressibility, tr ɛ = 0); [isotropic case: 2(1)] Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

13 Strain Linear Elasticity Elastic Strains and their Dynamics crystals (solids): positional order broken translational symmetry rigid translations do not cost energy Goldstone mode f = f ( u) with u displacement vector decompose displacement gradients (strain and rotation) u 1 2 ( ju i + i u j ) ( ju i i u j ) ɛ ij + Ω ij (1) rigid rotations must not cost energy f = f (ɛ) f el = 1 2 c 11(ɛ 2 xx + ɛ 2 yy) + c 12 ɛ xx ɛ yy + c 13 ɛ zz (ɛ xx + ɛ yy ) c 33ɛ 2 zz + 2c 44 (ɛ 2 yz + ɛ 2 xz) + 2(c 11 c 12 )ɛ 2 xy (2) 5 elastic moduli (3 in the case of incompressibility, tr ɛ = 0); [isotropic case: 2(1)] Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

14 Strain Linear Elasticity Elastic Strains and their Dynamics crystals (solids): positional order broken translational symmetry rigid translations do not cost energy Goldstone mode f = f ( u) with u displacement vector decompose displacement gradients (strain and rotation) u 1 2 ( ju i + i u j ) ( ju i i u j ) ɛ ij + Ω ij (1) rigid rotations must not cost energy f = f (ɛ) f el = 1 2 c 11(ɛ 2 xx + ɛ 2 yy) + c 12 ɛ xx ɛ yy + c 13 ɛ zz (ɛ xx + ɛ yy ) c 33ɛ 2 zz + 2c 44 (ɛ 2 yz + ɛ 2 xz) + 2(c 11 c 12 )ɛ 2 xy (2) 5 elastic moduli (3 in the case of incompressibility, tr ɛ = 0); [isotropic case: 2(1)] Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

15 Elastodynamics Linear Elasticity Elastic Strains and their Dynamics decompose velocity gradients (deformational and rotational flow) v 1 2 ( iv j + j v i ) ( iv j j v i ) A ij + ω ij (3) the time derivative of a displacement is related to the velocity ɛ ij A ij = 0?D 2 ( ) f / ɛ ij A ij is not the strain rate (in Eulerian description, even linearly) ω i 1 2 ɛ ijkω jk = 1 2 (curlv) i is the vorticity f / ɛ ij ψ ij is called the elastic stress Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37 (4)

16 Elastodynamics Linear Elasticity Elastic Strains and their Dynamics decompose velocity gradients (deformational and rotational flow) v 1 2 ( iv j + j v i ) ( iv j j v i ) A ij + ω ij (3) the time derivative of a displacement is related to the velocity ɛ ij A ij = 0?D 2 ( ) f / ɛ ij A ij is not the strain rate (in Eulerian description, even linearly) ω i 1 2 ɛ ijkω jk = 1 2 (curlv) i is the vorticity f / ɛ ij ψ ij is called the elastic stress Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37 (4)

17 Elastodynamics Linear Elasticity Elastic Strains and their Dynamics decompose velocity gradients (deformational and rotational flow) v 1 2 ( iv j + j v i ) ( iv j j v i ) A ij + ω ij (3) the time derivative of a displacement is related to the velocity ɛ ij A ij = D 2 ( ) f / ɛ ij vacancy diffusion (permeation) - neglected in solid state 1 A ij is not the strain rate (in Eulerian description, even linearly) ω i 1 2 ɛ ijkω jk = 1 2 (curlv) i is the vorticity f / ɛ ij ψ ij is called the elastic stress 1 For transient elasticity - viscoelasticity - the diffusion is replaced by a relaxation, α ( f / ɛ ij ), cf. Temmen et al., Phys. Rev. Lett. 84 (2000) 3228, and 86 (2001) 745. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37 (4)

18 Elastodynamics Linear Elasticity Elastic Strains and their Dynamics decompose velocity gradients (deformational and rotational flow) v 1 2 ( iv j + j v i ) ( iv j j v i ) A ij + ω ij (3) the time derivative of a displacement is related to the velocity ɛ ij A ij = D 2 ( ) f / ɛ ij A ij is not the strain rate (in Eulerian description, even linearly) ω i 1 2 ɛ ijkω jk = 1 2 (curlv) i is the vorticity f / ɛ ij ψ ij is called the elastic stress Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37 (4)

19 Stress Linear Elasticity Elastic Strains and their Dynamics crosscoupling of flow with dynamic elasticity ɛ ij A ij = D 2 ψ ij (5) requires a corresponding crosscoupling of elasticity in the stress tensor σ ij, which is the current of the local momentum conservation (g = ρv) equation ( / t)(ρv i ) + j σ ij = 0 with σ ij = pδ ij ψ ij (6) (Onsager, based on 2nd law of thermodynamics 1 ) linear elasticity adds 3 normal modes to the spectrum 1 H.P. and H.R. Brand in in Pattern Formation in Liquid Crystals, eds. A. Buka and L. Kramer, Springer New York, p (1995) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

20 Stress Linear Elasticity Elastic Strains and their Dynamics crosscoupling of flow with dynamic elasticity ɛ ij A ij = D 2 ψ ij (5) requires a corresponding crosscoupling of elasticity in the stress tensor σ ij, which is the current of the local momentum conservation (g = ρv) equation ( / t)(ρv i ) + j σ ij = 0 with σ ij = pδ ij ψ ij (6) (Onsager, based on 2nd law of thermodynamics 1 ) linear elasticity adds 3 normal modes to the spectrum 1 H.P. and H.R. Brand in in Pattern Formation in Liquid Crystals, eds. A. Buka and L. Kramer, Springer New York, p (1995) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

21 Stress Linear Elasticity Elastic Strains and their Dynamics crosscoupling of flow with dynamic elasticity ɛ ij A ij = D 2 ψ ij (5) requires a corresponding crosscoupling of elasticity in the stress tensor σ ij, which is the current of the local momentum conservation (g = ρv) equation ( / t)(ρv i ) + j σ ij = 0 with σ ij = pδ ij ψ ij (6) (Onsager, based on 2nd law of thermodynamics 1 ) linear elasticity adds 3 normal modes to the spectrum 1 H.P. and H.R. Brand in in Pattern Formation in Liquid Crystals, eds. A. Buka and L. Kramer, Springer New York, p (1995) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

22 Statics Linear Elasticity Linear Elasticity of Nematic Elastomers elastic and nematic degrees of freedom, but also relative rotations 2, Ω δn Ω, where δn i (with δn n = 0) describes rotations of the director and Ω i = Ω ij n j describes rotations of the network perpendicular to n. free energy f = f el + f n + f rr with f el the elastic free energy as before, f n the Frank energy, and f rr the energy involving relative rotations f rr = 1 2 D 1( Ω 2 x + Ω 2 y) + D 2 ( Ωy ɛ xz + Ω ) x ɛ yz (7) where D 1 is the stiffness modulus of the relative rotations and D 2 is the coupling constant between relative rotations and elastic deformations. no coupling between relative rotations and gradients of n. 2 P.G. de Gennes, in Liquid Crystals of One- and Two-Dimensional Order, eds. W. Helfrich and G. Heppke, Springer, New York, p. 231 (1980). Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

23 Statics Linear Elasticity Linear Elasticity of Nematic Elastomers elastic and nematic degrees of freedom, but also relative rotations 2, Ω δn Ω, where δn i (with δn n = 0) describes rotations of the director and Ω i = Ω ij n j describes rotations of the network perpendicular to n. free energy f = f el + f n + f rr with f el the elastic free energy as before, f n the Frank energy, and f rr the energy involving relative rotations f rr = 1 2 D 1( Ω 2 x + Ω 2 y) + D 2 ( Ωy ɛ xz + Ω ) x ɛ yz (7) where D 1 is the stiffness modulus of the relative rotations and D 2 is the coupling constant between relative rotations and elastic deformations. no coupling between relative rotations and gradients of n. 2 P.G. de Gennes, in Liquid Crystals of One- and Two-Dimensional Order, eds. W. Helfrich and G. Heppke, Springer, New York, p. 231 (1980). Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

24 Statics Linear Elasticity Linear Elasticity of Nematic Elastomers elastic and nematic degrees of freedom, but also relative rotations 2, Ω δn Ω, where δn i (with δn n = 0) describes rotations of the director and Ω i = Ω ij n j describes rotations of the network perpendicular to n. free energy f = f el + f n + f rr with f el the elastic free energy as before, f n the Frank energy, and f rr the energy involving relative rotations f rr = 1 2 D 1( Ω 2 x + Ω 2 y) + D 2 ( Ωy ɛ xz + Ω ) x ɛ yz (7) where D 1 is the stiffness modulus of the relative rotations and D 2 is the coupling constant between relative rotations and elastic deformations. no coupling between relative rotations and gradients of n. 2 P.G. de Gennes, in Liquid Crystals of One- and Two-Dimensional Order, eds. W. Helfrich and G. Heppke, Springer, New York, p. 231 (1980). Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

25 Statics Linear Elasticity Linear Elasticity of Nematic Elastomers elastic and nematic degrees of freedom, but also relative rotations 2, Ω δn Ω, where δn i (with δn n = 0) describes rotations of the director and Ω i = Ω ij n j describes rotations of the network perpendicular to n. free energy f = f el + f n + f rr with f el the elastic free energy as before, f n the Frank energy, and f rr the energy involving relative rotations f rr = 1 2 D 1( Ω 2 x + Ω 2 y) + D 2 ( Ωy ɛ xz + Ω ) x ɛ yz (7) where D 1 is the stiffness modulus of the relative rotations and D 2 is the coupling constant between relative rotations and elastic deformations. no coupling between relative rotations and gradients of n. 2 P.G. de Gennes, in Liquid Crystals of One- and Two-Dimensional Order, eds. W. Helfrich and G. Heppke, Springer, New York, p. 231 (1980). Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

26 Dynamics Linear Elasticity Linear Elasticity of Nematic Elastomers in the (linear) dynamics relative rotations couple 3 reversibly to the stress tensor - similar to back flow in ordinary nematics influences flow alignment behavior irreversibly to the director dynamics changes the usual director relaxation irreversibly to an electric current consequences not obvious 3 H.R. Brand and H.P., Physica A, 208 (1994) 359 Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

27 Shear Elasticity Linear Elasticity the shear elastic free energy rewritten Soft Elasticity and Rotational Goldstone Mode f shear = 2c 44 (ɛ 2 yz + ɛ 2 xz) + 2(c 11 c 12 )ɛ 2 xy ) D 1( Ω 2 x + Ω 2 y) + D 2 ( Ω y ɛ xz + Ω x ɛ yz = 2 c 44 (ɛ 2 xz + ɛ 2 yz) + 2(c 11 c 12 )ɛ [ 2 xy D 1 ( Ω y + D 2 ɛ xz ) 2 + ( Ω x + D ] 2 ɛ yz ) 2 2D 1 2D 1 with the renormalized shear modulus c 44 = c 44 D 2 2 /4D 1. under shear the system minimizes the free energy by choosing Ω y = D 2 2D 1 ɛ xz and Ω x = D 2 2D 1 ɛ yz leading to (8) (9) f shear = 2 c 44 (ɛ 2 xz + ɛ 2 yz) + 2(c 11 c 12 )ɛ 2 xy (10) soft elasticity means c 44 = 0 (C R 5 = 0) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

28 Shear Elasticity Linear Elasticity the shear elastic free energy rewritten Soft Elasticity and Rotational Goldstone Mode f shear = 2c 44 (ɛ 2 yz + ɛ 2 xz) + 2(c 11 c 12 )ɛ 2 xy ) D 1( Ω 2 x + Ω 2 y) + D 2 ( Ω y ɛ xz + Ω x ɛ yz = 2 c 44 (ɛ 2 xz + ɛ 2 yz) + 2(c 11 c 12 )ɛ [ 2 xy D 1 ( Ω y + D 2 ɛ xz ) 2 + ( Ω x + D ] 2 ɛ yz ) 2 2D 1 2D 1 with the renormalized shear modulus c 44 = c 44 D 2 2 /4D 1. under shear the system minimizes the free energy by choosing Ω y = D 2 2D 1 ɛ xz and Ω x = D 2 2D 1 ɛ yz leading to (8) (9) f shear = 2 c 44 (ɛ 2 xz + ɛ 2 yz) + 2(c 11 c 12 )ɛ 2 xy (10) soft elasticity means c 44 = 0 (C R 5 = 0) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

29 Shear Elasticity Linear Elasticity the shear elastic free energy rewritten Soft Elasticity and Rotational Goldstone Mode f shear = 2c 44 (ɛ 2 yz + ɛ 2 xz) + 2(c 11 c 12 )ɛ 2 xy ) D 1( Ω 2 x + Ω 2 y) + D 2 ( Ω y ɛ xz + Ω x ɛ yz = 2 c 44 (ɛ 2 xz + ɛ 2 yz) + 2(c 11 c 12 )ɛ [ 2 xy D 1 ( Ω y + D 2 ɛ xz ) 2 + ( Ω x + D ] 2 ɛ yz ) 2 2D 1 2D 1 with the renormalized shear modulus c 44 = c 44 D 2 2 /4D 1. under shear the system minimizes the free energy by choosing Ω y = D 2 2D 1 ɛ xz and Ω x = D 2 2D 1 ɛ yz leading to (8) (9) f shear = 2 c 44 (ɛ 2 xz + ɛ 2 yz) + 2(c 11 c 12 )ɛ 2 xy (10) soft elasticity means c 44 = 0 (C R 5 = 0) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

30 Shear Elasticity Linear Elasticity the shear elastic free energy rewritten Soft Elasticity and Rotational Goldstone Mode f shear = 2c 44 (ɛ 2 yz + ɛ 2 xz) + 2(c 11 c 12 )ɛ 2 xy ) D 1( Ω 2 x + Ω 2 y) + D 2 ( Ω y ɛ xz + Ω x ɛ yz = 2 c 44 (ɛ 2 xz + ɛ 2 yz) + 2(c 11 c 12 )ɛ [ 2 xy D 1 ( Ω y + D 2 ɛ xz ) 2 + ( Ω x + D ] 2 ɛ yz ) 2 2D 1 2D 1 with the renormalized shear modulus c 44 = c 44 D 2 2 /4D 1. under shear the system minimizes the free energy by choosing Ω y = D 2 2D 1 ɛ xz and Ω x = D 2 2D 1 ɛ yz leading to (8) (9) f shear = 2 c 44 (ɛ 2 xz + ɛ 2 yz) + 2(c 11 c 12 )ɛ 2 xy (10) soft elasticity means c 44 = 0 (C R 5 = 0) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

31 Linear Elasticity Classical Rubber Elasticity Soft Elasticity and Rotational Goldstone Mode Conformation of a long polymer chain has an entropic maximum deformation costs energy by the affine deformation approximation this elasticity is scaled up to the macroscopic network Mooney-Rivlin f = 1 2 µ(λ2 + 2/λ 3) with µ = n s k B T and λ the linear expansion based on several (unrealistic) assumptions: Gaussian phantom chains (interpenetration) crosslinking neglected affine deformation result is not convincing Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

32 Linear Elasticity Classical Rubber Elasticity Soft Elasticity and Rotational Goldstone Mode Conformation of a long polymer chain has an entropic maximum deformation costs energy by the affine deformation approximation this elasticity is scaled up to the macroscopic network Mooney-Rivlin f = 1 2 µ(λ2 + 2/λ 3) with µ = n s k B T and λ the linear expansion based on several (unrealistic) assumptions: Gaussian phantom chains (interpenetration) crosslinking neglected affine deformation result is not convincing Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

33 Linear Elasticity Classical Rubber Elasticity Soft Elasticity and Rotational Goldstone Mode Conformation of a long polymer chain has an entropic maximum deformation costs energy by the affine deformation approximation this elasticity is scaled up to the macroscopic network Mooney-Rivlin f = 1 2 µ(λ2 + 2/λ 3) with µ = n s k B T and λ the linear expansion based on several (unrealistic) assumptions: Gaussian phantom chains (interpenetration) crosslinking neglected affine deformation result is not convincing Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

34 Linear Elasticity Classical Rubber Elasticity Soft Elasticity and Rotational Goldstone Mode Conformation of a long polymer chain has an entropic maximum deformation costs energy by the affine deformation approximation this elasticity is scaled up to the macroscopic network Mooney-Rivlin f = 1 2 µ(λ2 + 2/λ 3) with µ = n s k B T and λ the linear expansion based on several (unrealistic) assumptions: Gaussian phantom chains (interpenetration) crosslinking neglected affine deformation result is not convincing Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

35 Linear Elasticity Soft Elasticity and Rotational Goldstone Mode Neo-Classical (Nematic) Rubber Elasticity classical rubber elasticity plus anisotropy (of the single chains!) due to the anisotropic environment of the nematic side-chains by the affine deformation approximation this elasticity is scaled up to the macroscopic network Warner - Terentjev s neo-classical free energy (nonlinear) when linearized, the shear elastic modulus ( c 44 ) turns out to be zero, exactly 4, soft elasticity. based on the same (unrealistic) assumptions as classical rubber theory "spectacular" type of material: conventional rubber above T c, that below T c turns into a liquid for shears containing the preferred direction, but remains an elastic body perpendicular! however, there must be a hidden reason for c 44 0! 4 P.D. Olmsted, J. Physique II, 4 (1994) 2215 Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

36 Linear Elasticity Soft Elasticity and Rotational Goldstone Mode Neo-Classical (Nematic) Rubber Elasticity classical rubber elasticity plus anisotropy (of the single chains!) due to the anisotropic environment of the nematic side-chains by the affine deformation approximation this elasticity is scaled up to the macroscopic network Warner - Terentjev s neo-classical free energy (nonlinear) when linearized, the shear elastic modulus ( c 44 ) turns out to be zero, exactly 4, soft elasticity. based on the same (unrealistic) assumptions as classical rubber theory "spectacular" type of material: conventional rubber above T c, that below T c turns into a liquid for shears containing the preferred direction, but remains an elastic body perpendicular! however, there must be a hidden reason for c 44 0! 4 P.D. Olmsted, J. Physique II, 4 (1994) 2215 Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

37 Linear Elasticity Soft Elasticity and Rotational Goldstone Mode Neo-Classical (Nematic) Rubber Elasticity classical rubber elasticity plus anisotropy (of the single chains!) due to the anisotropic environment of the nematic side-chains by the affine deformation approximation this elasticity is scaled up to the macroscopic network Warner - Terentjev s neo-classical free energy (nonlinear) when linearized, the shear elastic modulus ( c 44 ) turns out to be zero, exactly 4, soft elasticity. based on the same (unrealistic) assumptions as classical rubber theory "spectacular" type of material: conventional rubber above T c, that below T c turns into a liquid for shears containing the preferred direction, but remains an elastic body perpendicular! however, there must be a hidden reason for c 44 0! 4 P.D. Olmsted, J. Physique II, 4 (1994) 2215 Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

38 Linear Elasticity Soft Elasticity and Rotational Goldstone Mode Neo-Classical (Nematic) Rubber Elasticity classical rubber elasticity plus anisotropy (of the single chains!) due to the anisotropic environment of the nematic side-chains by the affine deformation approximation this elasticity is scaled up to the macroscopic network Warner - Terentjev s neo-classical free energy (nonlinear) when linearized, the shear elastic modulus ( c 44 ) turns out to be zero, exactly 4, soft elasticity. based on the same (unrealistic) assumptions as classical rubber theory "spectacular" type of material: conventional rubber above T c, that below T c turns into a liquid for shears containing the preferred direction, but remains an elastic body perpendicular! however, there must be a hidden reason for c 44 0! 4 P.D. Olmsted, J. Physique II, 4 (1994) 2215 Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

39 Linear Elasticity Shape Orientational Goldstone Mode Soft Elasticity and Rotational Goldstone Mode if an elastic body undergoes a spontaneous phase transition from an isotropic shape to an anisotropic one, then it must have c 44, due to the Goldstone theorem 5 a shear in a plane containing the preferred axis (z), e.g. z u x = ɛ zx + Ω zx, contains a rotation however, since the orientation of the shape anisotropy is undetermined energetically, such a rotation must not cost energy the shear elastic modulus c 44 has to be zero! remarks argument is independent of liquid crystal nature does it apply to LSCE? 5 L. Golubovic and T.C. Lubensky, Phys. Rev. Lett., 1989, 63, 1082 and T.C. Lubensky, R. Mukhopadhyay, L. Radzihovsky, and X. Xing, Phys. Rev. E, 2002, 66, Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

40 Linear Elasticity Shape Orientational Goldstone Mode Soft Elasticity and Rotational Goldstone Mode if an elastic body undergoes a spontaneous phase transition from an isotropic shape to an anisotropic one, then it must have c 44, due to the Goldstone theorem 5 a shear in a plane containing the preferred axis (z), e.g. z u x = ɛ zx + Ω zx, contains a rotation however, since the orientation of the shape anisotropy is undetermined energetically, such a rotation must not cost energy the shear elastic modulus c 44 has to be zero! remarks argument is independent of liquid crystal nature does it apply to LSCE? 5 L. Golubovic and T.C. Lubensky, Phys. Rev. Lett., 1989, 63, 1082 and T.C. Lubensky, R. Mukhopadhyay, L. Radzihovsky, and X. Xing, Phys. Rev. E, 2002, 66, Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

41 Linear Elasticity Shape Orientational Goldstone Mode Soft Elasticity and Rotational Goldstone Mode if an elastic body undergoes a spontaneous phase transition from an isotropic shape to an anisotropic one, then it must have c 44, due to the Goldstone theorem 5 a shear in a plane containing the preferred axis (z), e.g. z u x = ɛ zx + Ω zx, contains a rotation however, since the orientation of the shape anisotropy is undetermined energetically, such a rotation must not cost energy the shear elastic modulus c 44 has to be zero! remarks argument is independent of liquid crystal nature does it apply to LSCE? 5 L. Golubovic and T.C. Lubensky, Phys. Rev. Lett., 1989, 63, 1082 and T.C. Lubensky, R. Mukhopadhyay, L. Radzihovsky, and X. Xing, Phys. Rev. E, 2002, 66, Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

42 No Soft Elasticity early experiments Linear Elasticity finite slope for small strains (linear regime) Soft Elasticity and Rotational Goldstone Mode finite shear modulus c 44 WT coined the notion of "semisoftness" (linear): c 44 is in principle zero, but due to imperfections it acquires a small, but finite value ("imperfections" is explained in their book) is there semisoftness in LSCE? Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

43 No Soft Elasticity early experiments Linear Elasticity finite slope for small strains (linear regime) Soft Elasticity and Rotational Goldstone Mode finite shear modulus c 44 WT coined the notion of "semisoftness" (linear): c 44 is in principle zero, but due to imperfections it acquires a small, but finite value ("imperfections" is explained in their book) is there semisoftness in LSCE? Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

44 No Soft Elasticity early experiments Linear Elasticity finite slope for small strains (linear regime) Soft Elasticity and Rotational Goldstone Mode finite shear modulus c 44 WT coined the notion of "semisoftness" (linear): c 44 is in principle zero, but due to imperfections it acquires a small, but finite value ("imperfections" is explained in their book) is there semisoftness in LSCE? Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

45 No Soft Elasticity early experiments Linear Elasticity finite slope for small strains (linear regime) Soft Elasticity and Rotational Goldstone Mode finite shear modulus c 44 WT coined the notion of "semisoftness" (linear): c 44 is in principle zero, but due to imperfections it acquires a small, but finite value ("imperfections" is explained in their book) is there semisoftness in LSCE? Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

46 No Semisoft Elasticity Linear Elasticity Soft Elasticity and Rotational Goldstone Mode G c 44 as a function of temperature and frequency T G + 1Hz G 10 Hz ( 100 Hz Hz T NI in the static limit there is no anomaly c 44 < c 44, but of the same order of magnitude ( 10 kpa as is a common value for polymeric networks) c 44 is neither zero, nor small 6 P. Martinoty, P. Stein, H. Finkelmann, H. P., and H.R. Brand, Eur. Phys. J. E, 14 (2004) 311. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

47 No Semisoft Elasticity Linear Elasticity Soft Elasticity and Rotational Goldstone Mode G c 44 as a function of temperature and frequency T G + 1Hz G 10 Hz ( 100 Hz Hz T NI in the static limit there is no anomaly c 44 < c 44, but of the same order of magnitude ( 10 kpa as is a common value for polymeric networks) c 44 is neither zero, nor small 6 P. Martinoty, P. Stein, H. Finkelmann, H. P., and H.R. Brand, Eur. Phys. J. E, 14 (2004) 311. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

48 Linear Elasticity Why Are LSCEs Non-Soft? I LSCE are above the critical point Soft Elasticity and Rotational Goldstone Mode in an external field the isotropic to nematic phase transition gets continuous above the critical point in LSCE, internal stresses are fixed by the 2nd crosslinking and act as external field 7 a residual nematic order is already present in the isotropic phase 8 (no spontaneous symmetry breaking) 7 even if the crosslinking (under elongation) is done in the isotropic state, random disorder and internal stresses are generated, cf. H.R. Brand, K. Kawasaki, Makromol. Chem. Rapid Commun., 15 (1994) A. Lebar, et al., Phys. Rev. Lett. 94 (2005) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

49 Linear Elasticity Why Are LSCEs Non-Soft? I LSCE are above the critical point Soft Elasticity and Rotational Goldstone Mode in an external field the isotropic to nematic phase transition gets continuous above the critical point in LSCE, internal stresses are fixed by the 2nd crosslinking and act as external field 7 a residual nematic order is already present in the isotropic phase 8 (no spontaneous symmetry breaking) 7 even if the crosslinking (under elongation) is done in the isotropic state, random disorder and internal stresses are generated, cf. H.R. Brand, K. Kawasaki, Makromol. Chem. Rapid Commun., 15 (1994) A. Lebar, et al., Phys. Rev. Lett. 94 (2005) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

50 Linear Elasticity Why Are LSCEs Non-Soft? II recall the physical picture Soft Elasticity and Rotational Goldstone Mode even in the isotropic phase the shape of a chain is not spherical; a random walk (even a self-avoiding random walk) is highly anisotropic (principal axes are ca. 12:3:1) in this molecular picture the isotropic to nematic transition corresponds to a disorder - order transition (like in lmw nematics) however there is a fundamental difference: the small particles of lmw nematics can easily reorient without distorting their environment (this is still true for the side-chain mesogens due to the spacer) in elastomers the chains are highly entangled and even crosslinked and cannot simply rotate without any energy penalty instead of a macroscopic shape change, there are polydomains Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

51 Linear Elasticity Why Are LSCEs Non-Soft? II recall the physical picture Soft Elasticity and Rotational Goldstone Mode even in the isotropic phase the shape of a chain is not spherical; a random walk (even a self-avoiding random walk) is highly anisotropic (principal axes are ca. 12:3:1) in this molecular picture the isotropic to nematic transition corresponds to a disorder - order transition (like in lmw nematics) however there is a fundamental difference: the small particles of lmw nematics can easily reorient without distorting their environment (this is still true for the side-chain mesogens due to the spacer) in elastomers the chains are highly entangled and even crosslinked and cannot simply rotate without any energy penalty instead of a macroscopic shape change, there are polydomains Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

52 Linear Elasticity Why Are LSCEs Non-Soft? II recall the physical picture Soft Elasticity and Rotational Goldstone Mode even in the isotropic phase the shape of a chain is not spherical; a random walk (even a self-avoiding random walk) is highly anisotropic (principal axes are ca. 12:3:1) in this molecular picture the isotropic to nematic transition corresponds to a disorder - order transition (like in lmw nematics) however there is a fundamental difference: the small particles of lmw nematics can easily reorient without distorting their environment (this is still true for the side-chain mesogens due to the spacer) in elastomers the chains are highly entangled and even crosslinked and cannot simply rotate without any energy penalty instead of a macroscopic shape change, there are polydomains Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

53 Nonlinear Strain Nonlinear Elasticity Nonlinear Elastic Strains and their Dynamics finite distortions of a solid body can be described in two different ways: by the original frame, where the body was undeformed (material frame, a) and by the lab frame, r, where the final state of the body is measured either the original state is described as a function of the final one, a(r) (Eulerian), or vice versa, r(a) (Lagrange) the two frames may have different origins and different orientations a "displacement" r a does not make sense instead, the infinitesimal distance between two neighboring points is monitored (dr) 2 (da(r)) 2 2 dr i dr j ɛ ij (r) (Eulerian strain tensor) with (11) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

54 Nonlinear Strain Nonlinear Elasticity Nonlinear Elastic Strains and their Dynamics finite distortions of a solid body can be described in two different ways: by the original frame, where the body was undeformed (material frame, a) and by the lab frame, r, where the final state of the body is measured either the original state is described as a function of the final one, a(r) (Eulerian), or vice versa, r(a) (Lagrange) the two frames may have different origins and different orientations a "displacement" r a does not make sense instead, the infinitesimal distance between two neighboring points is monitored (dr) 2 (da(r)) 2 2 dr i dr j ɛ ij (r) (Eulerian strain tensor) with (11) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

55 Nonlinear Strain Nonlinear Elasticity Nonlinear Elastic Strains and their Dynamics finite distortions of a solid body can be described in two different ways: by the original frame, where the body was undeformed (material frame, a) and by the lab frame, r, where the final state of the body is measured either the original state is described as a function of the final one, a(r) (Eulerian), or vice versa, r(a) (Lagrange) the two frames may have different origins and different orientations a "displacement" r a does not make sense instead, the infinitesimal distance between two neighboring points is monitored (dr) 2 (da(r)) 2 2 dr i dr j ɛ ij (r) (Eulerian strain tensor) with (11) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

56 Nonlinear Strain Nonlinear Elasticity Nonlinear Elastic Strains and their Dynamics finite distortions of a solid body can be described in two different ways: by the original frame, where the body was undeformed (material frame, a) and by the lab frame, r, where the final state of the body is measured either the original state is described as a function of the final one, a(r) (Eulerian), or vice versa, r(a) (Lagrange) the two frames may have different origins and different orientations a "displacement" r a does not make sense instead, the infinitesimal distance between two neighboring points is monitored (dr) 2 (da(r)) 2 2 dr i dr j ɛ ij (r) (Eulerian strain tensor) with (11) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

57 Nonlinear Strain Nonlinear Elasticity Nonlinear Elastic Strains and their Dynamics finite distortions of a solid body can be described in two different ways: by the original frame, where the body was undeformed (material frame, a) and by the lab frame, r, where the final state of the body is measured either the original state is described as a function of the final one, a(r) (Eulerian), or vice versa, r(a) (Lagrange) the two frames may have different origins and different orientations a "displacement" r a does not make sense instead, the infinitesimal distance between two neighboring points is monitored (dr) 2 (da(r)) 2 2 dr i dr j ɛ ij (r) (Eulerian strain tensor) with da i = ( j a i )dr j ɛ ij = 1 2 [δ ij ( i a k )( j a k )] = 1 2 [ iu j + j u i ( i u k )( j u k )] (11) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

58 Nonlinear Strain Nonlinear Elasticity Nonlinear Elastic Strains and their Dynamics finite distortions of a solid body can be described in two different ways: by the original frame, where the body was undeformed (material frame, a) and by the lab frame, r, where the final state of the body is measured either the original state is described as a function of the final one, a(r) (Eulerian), or vice versa, r(a) (Lagrange) the two frames may have different origins and different orientations a "displacement" r a does not make sense instead, the infinitesimal distance between two neighboring points is monitored (dr(a)) 2 (da) 2 2 da i da j Uij L (a) (Lagrange strain tensor) with dr i = ( u i / a j )da j U L ij = 1 2 [ u i/ a j + u j / a i + ( u k / a i )( u k / a j )] (11) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

59 Nonlinear Elasticity Nonlinear Elastic Energy Nonlinear Elastic Strains and their Dynamics generalized Hooke s law ψ ij = C ijkl ɛ kl, but with the elastic modulus tensor being a function of all state variables C ijkl = 1 2 C 1(δ ik δ jl C 2(ɛ ik δ jl + ɛ jk δ il + ɛ il δ jk + ɛ jl δ ik ) + δ il δ jk ) +C 3 δ ij δ kl + C 4 [δ ij ɛ kl + δ kl ɛ ij (δ ikδ jl + δ il δ jk )ɛ pp ] +C 5 δ ij δ kl ɛ pp + O(2) (12) generalized Mooney energy 9 f = C ( (κ (i) 1) 2 + C ) 2 (13) i=1 i=1 where κ (i) are the three eigenvalues of ɛ ij. 9 O. Müller, PhD thesis, Univ. Tübingen Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37 κ (i)

60 Nonlinear Elasticity Nonlinear Elastic Energy Nonlinear Elastic Strains and their Dynamics generalized Hooke s law ψ ij = C ijkl ɛ kl, but with the elastic modulus tensor being a function of all state variables C ijkl = 1 2 C 1(δ ik δ jl C 2(ɛ ik δ jl + ɛ jk δ il + ɛ il δ jk + ɛ jl δ ik ) + δ il δ jk ) +C 3 δ ij δ kl + C 4 [δ ij ɛ kl + δ kl ɛ ij (δ ikδ jl + δ il δ jk )ɛ pp ] +C 5 δ ij δ kl ɛ pp + O(2) (12) generalized Mooney energy 9 f = C ( (κ (i) 1) 2 + C ) 2 (13) i=1 i=1 where κ (i) are the three eigenvalues of ɛ ij. 9 O. Müller, PhD thesis, Univ. Tübingen Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37 κ (i)

61 Nonlinear Elasticity Nonlinear Elastodynamics Nonlinear Elastic Strains and their Dynamics since a(r, t) describes the time evolution of a material point, its co-moving time derivative is zero, [( / t) + v j j ]a i = 0 in the absence of any dissipative process this leads immediately to the nonlinear elastodynamic equation ( t + v )ɛ ij A ij + ɛ kj i v k + ɛ ki j v k = [ j (D il k ψ kl ) + i j] where the r.h.s. describes general diffusion (or relaxation) processes; the nonlinear flow couplings constitute a lower convected derivative. the corresponding stress tensor has the form (14) σ ij = pδ ij ψ ij + ψ ki ɛ jk + ψ kj ɛ ik ν ijkl A kl (15) where the material tensor D ij and ν ijkl are a function of the state variables, in particular of ɛ ij. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

62 Nonlinear Elasticity Nonlinear Elastodynamics Nonlinear Elastic Strains and their Dynamics since a(r, t) describes the time evolution of a material point, its co-moving time derivative is zero, [( / t) + v j j ]a i = 0 in the absence of any dissipative process this leads immediately to the nonlinear elastodynamic equation ( t + v )ɛ ij A ij + ɛ kj i v k + ɛ ki j v k = [ j (D il k ψ kl ) + i j] where the r.h.s. describes general diffusion (or relaxation) processes; the nonlinear flow couplings constitute a lower convected derivative. the corresponding stress tensor has the form (14) σ ij = pδ ij ψ ij + ψ ki ɛ jk + ψ kj ɛ ik ν ijkl A kl (15) where the material tensor D ij and ν ijkl are a function of the state variables, in particular of ɛ ij. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

63 Nonlinear Elasticity Nonlinear Elastodynamics Nonlinear Elastic Strains and their Dynamics since a(r, t) describes the time evolution of a material point, its co-moving time derivative is zero, [( / t) + v j j ]a i = 0 in the absence of any dissipative process this leads immediately to the nonlinear elastodynamic equation ( t + v )ɛ ij A ij + ɛ kj i v k + ɛ ki j v k = [ j (D il k ψ kl ) + i j] where the r.h.s. describes general diffusion (or relaxation) processes; the nonlinear flow couplings constitute a lower convected derivative. the corresponding stress tensor has the form (14) σ ij = pδ ij ψ ij + ψ ki ɛ jk + ψ kj ɛ ik ν ijkl A kl (15) where the material tensor D ij and ν ijkl are a function of the state variables, in particular of ɛ ij. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

64 Nonlinear Elasticity Nonlinear Elastodynamics Nonlinear Elastic Strains and their Dynamics since a(r, t) describes the time evolution of a material point, its co-moving time derivative is zero, [( / t) + v j j ]a i = 0 in the absence of any dissipative process this leads immediately to the nonlinear elastodynamic equation ( t + v )ɛ ij A ij + ɛ kj i v k + ɛ ki j v k = [ j (D il k ψ kl ) + i j] where the r.h.s. describes general diffusion (or relaxation) processes; the nonlinear flow couplings constitute a lower convected derivative. the corresponding stress tensor has the form (14) σ ij = pδ ij ψ ij + ψ ki ɛ jk + ψ kj ɛ ik ν ijkl A kl (15) where the material tensor D ij and ν ijkl are a function of the state variables, in particular of ɛ ij. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

65 Nonlinear Rotations Nonlinear Elasticity Nonlinear (Relative) Rotations linear rotations are described by an angle of rotation about a fixed axis nonlinear rotations by a rotation matrix, R ij (R 1 ) ji, which in linear approximation is R ij = δ ij ( iu j j u i ) nonlinear: j a i R ik Ξ kj (Noll s decomposition theorem) Ξ ij is symmetric, contains only deformations, and can be expanded into powers of ɛ ij using Ξ ik Ξ kj = δ ij 2ɛ ij ; Ξ ik Ξ kj is the Cauchy tensor the rotation matrix R can be expressed in powers 10 of j u i and ɛ ij R ij = δ ij ( iu j j u i ) ɛ ikɛ jk ɛ jk k u i + O(3) nonlinear deformations contribute to rotations! 10 up to fourth order cf. A. Menzel, H.P., and H.R. Brand, J. Chem. Phys. 126 (2007) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

66 Nonlinear Rotations Nonlinear Elasticity Nonlinear (Relative) Rotations linear rotations are described by an angle of rotation about a fixed axis nonlinear rotations by a rotation matrix, R ij (R 1 ) ji, which in linear approximation is R ij = δ ij ( iu j j u i ) nonlinear: j a i R ik Ξ kj (Noll s decomposition theorem) Ξ ij is symmetric, contains only deformations, and can be expanded into powers of ɛ ij using Ξ ik Ξ kj = δ ij 2ɛ ij ; Ξ ik Ξ kj is the Cauchy tensor the rotation matrix R can be expressed in powers 10 of j u i and ɛ ij R ij = δ ij ( iu j j u i ) ɛ ikɛ jk ɛ jk k u i + O(3) nonlinear deformations contribute to rotations! 10 up to fourth order cf. A. Menzel, H.P., and H.R. Brand, J. Chem. Phys. 126 (2007) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

67 Nonlinear Rotations Nonlinear Elasticity Nonlinear (Relative) Rotations linear rotations are described by an angle of rotation about a fixed axis nonlinear rotations by a rotation matrix, R ij (R 1 ) ji, which in linear approximation is R ij = δ ij ( iu j j u i ) nonlinear: j a i R ik Ξ kj (Noll s decomposition theorem) Ξ ij is symmetric, contains only deformations, and can be expanded into powers of ɛ ij using Ξ ik Ξ kj = δ ij 2ɛ ij ; Ξ ik Ξ kj is the Cauchy tensor the rotation matrix R can be expressed in powers 10 of j u i and ɛ ij R ij = δ ij ( iu j j u i ) ɛ ikɛ jk ɛ jk k u i + O(3) nonlinear deformations contribute to rotations! 10 up to fourth order cf. A. Menzel, H.P., and H.R. Brand, J. Chem. Phys. 126 (2007) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

68 Nonlinear Rotations Nonlinear Elasticity Nonlinear (Relative) Rotations linear rotations are described by an angle of rotation about a fixed axis nonlinear rotations by a rotation matrix, R ij (R 1 ) ji, which in linear approximation is R ij = δ ij ( iu j j u i ) nonlinear: j a i R ik Ξ kj (Noll s decomposition theorem) Ξ ij is symmetric, contains only deformations, and can be expanded into powers of ɛ ij using Ξ ik Ξ kj = δ ij 2ɛ ij ; Ξ ik Ξ kj is the Cauchy tensor the rotation matrix R can be expressed in powers 10 of j u i and ɛ ij R ij = δ ij ( iu j j u i ) ɛ ikɛ jk ɛ jk k u i + O(3) nonlinear deformations contribute to rotations! 10 up to fourth order cf. A. Menzel, H.P., and H.R. Brand, J. Chem. Phys. 126 (2007) Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

69 Nonlinear Elasticity Nonlinear Relative Rotations Nonlinear (Relative) Rotations starting state: undeformed network and uniform director n 0 i final state: deformed (and rotated) network and director n i the actual state of the director, obtained by an appropriate rotation, is n i = S ji nj 0 the state the director would have obtained, if he had been rotated rigidly with the network, is ni net = R ji nj 0 relative rotations that cost energy are n i (n k R lk nl 0)R jinj 0, where only those rotations perpendicular to n net contribute for infinitesimal rotations it reduces to the linear expression; it has the required n i n i and ni 0 ni 0 (and n net n net ) symmetry i i Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37 i

70 Nonlinear Elasticity Nonlinear Relative Rotations Nonlinear (Relative) Rotations starting state: undeformed network and uniform director n 0 i final state: deformed (and rotated) network and director n i the actual state of the director, obtained by an appropriate rotation, is n i = S ji nj 0 the state the director would have obtained, if he had been rotated rigidly with the network, is ni net = R ji nj 0 relative rotations that cost energy are n i (n k R lk nl 0)R jinj 0, where only those rotations perpendicular to n net contribute for infinitesimal rotations it reduces to the linear expression; it has the required n i n i and ni 0 ni 0 (and n net n net ) symmetry i i Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37 i

71 Nonlinear Elasticity Nonlinear Relative Rotations Nonlinear (Relative) Rotations starting state: undeformed network and uniform director n 0 i final state: deformed (and rotated) network and director n i the actual state of the director, obtained by an appropriate rotation, is n i = S ji nj 0 the state the director would have obtained, if he had been rotated rigidly with the network, is ni net = R ji nj 0 relative rotations that cost energy are n i (n k R lk nl 0)R jinj 0, where only those rotations perpendicular to n net contribute for infinitesimal rotations it reduces to the linear expression; it has the required n i n i and ni 0 ni 0 (and n net n net ) symmetry i i Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37 i

72 Nonlinear Elasticity Nonlinear Relative Rotations Nonlinear (Relative) Rotations starting state: undeformed network and uniform director n 0 i final state: deformed (and rotated) network and director n i the actual state of the director, obtained by an appropriate rotation, is n i = S ji nj 0 the state the director would have obtained, if he had been rotated rigidly with the network, is ni net = R ji nj 0 relative rotations that cost energy are n i (n k R lk nl 0)R jinj 0, where only those rotations perpendicular to n net contribute for infinitesimal rotations it reduces to the linear expression; it has the required n i n i and ni 0 ni 0 (and n net n net ) symmetry i i Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37 i

73 Nonlinear Description Nonlinear Elasticity Nonlinear (Relative) Rotations for simple shear the relative rotational energy contribution leads to director reorientation and to compressional and elongational strains, which are not present in an ordinary anisotropic elastic medium 11 ˆn ẑ nonlinear relative rotations, together with the (uniaxial) nonlinear elasticity and the coupling between them, give the proper macroscopic description for the nonlinear elasticity of LSCE. this is a non-soft description (neither soft- nor semisoftness is implied) 11 A. Menzel et al. cit.op. ˆx Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

74 Nonlinear Description Nonlinear Elasticity Nonlinear (Relative) Rotations for simple shear the relative rotational energy contribution leads to director reorientation and to compressional and elongational strains, which are not present in an ordinary anisotropic elastic medium 11 ˆn ẑ nonlinear relative rotations, together with the (uniaxial) nonlinear elasticity and the coupling between them, give the proper macroscopic description for the nonlinear elasticity of LSCE. this is a non-soft description (neither soft- nor semisoftness is implied) 11 A. Menzel et al. cit.op. ˆx Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

75 Nonlinear Elasticity "Semisoftness" (nonlinear) "Semisoftness" and Stripe Patterns the assumed smallness of the shear elastic modulus c 44 in the semisoft description of WT (which is a linear property) has additional consequences in the nonlinear elastic domain as has been shown by DeSimone and coworkers 12, the semisoft free energy of WT et al. leads to a quasi-plateau in the stress/strain relation (left) resembling experiments (middle ) S. Conti, A. DeSimone, and G. Dolzmann, Phys. Rev. E 66 (2002) Küpfer et al., cit.op. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

76 Nonlinear Elasticity "Semisoftness" (nonlinear) "Semisoftness" and Stripe Patterns the assumed smallness of the shear elastic modulus c 44 in the semisoft description of WT (which is a linear property) has additional consequences in the nonlinear elastic domain as has been shown by DeSimone and coworkers 12, the semisoft free energy of WT et al. leads to a quasi-plateau in the stress/strain relation (left) resembling experiments (middle and right) S. Conti, A. DeSimone, and G. Dolzmann, Phys. Rev. E 66 (2002) Küpfer et al., cit.op. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

77 Stripe pattern as a second consequence of the semisoft free energy of WT et al., again DeSimone and coworkers Nonlinear Elasticity showed that the quasi-plateau in the stress/strain relation is accompanied by a stripe pattern of the director (left) resembling experiments (right) "Semisoftness" and Stripe Patterns however, there are descriptions of these stripe patterns without using semisoftness, e.g. Fried and Sellers 14 using a more general free energy; the onset of an heterogeneous orientation pattern under strain was described by Brand and Weilepp using the concept of relative rotations 15 stripes are no proof for semisoftness 14 E. Fried and S. Sellers, J. Appl. Phys. 100 (2006) J. Weilepp and H.R. Brand, Europhys. Lett. 34 (1996) and 37 (1997) 499. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37

78 Stripe pattern as a second consequence of the semisoft free energy of WT et al., again DeSimone and coworkers Nonlinear Elasticity showed that the quasi-plateau in the stress/strain relation is accompanied by a stripe pattern of the director (left) resembling experiments (right) "Semisoftness" and Stripe Patterns however, there are descriptions of these stripe patterns without using semisoftness, e.g. Fried and Sellers 14 using a more general free energy; the onset of an heterogeneous orientation pattern under strain was described by Brand and Weilepp using the concept of relative rotations 15 stripes are no proof for semisoftness 14 E. Fried and S. Sellers, J. Appl. Phys. 100 (2006) J. Weilepp and H.R. Brand, Europhys. Lett. 34 (1996) and 37 (1997) 499. Pleiner (MPI-P Mainz) Liquid Crystalline Elastomers Kent, June 26th, / 37