A Blend of Stretching and Bending in Liquid Crystal Networks Epifanio G. Virga Department of Mathematics University of Pavia Italy eg.virga@unipv.it Summary Nematic Elastomers and Networks Neo-Classical Energy Stretching Energy Bending Energy
Nematic Elastomers and Networks When nematogenic molecules are appended to an elatomeric network at the crosslinking time, essentially two outcomes are possible, depending on the interaction between orientational and elastic degrees of freedom. liquid crystal elastomers White & Broer (2015) The nematic director influences the network deformation.
liquid crystal networks White & Broer (2015) The nematic director is linked to the network deformation. thermal (or optical) stimulus The degree of orientational order can be acted upon by external stimuli. Warner & Terentjev (2003)
Here we shall focus on liquid crystal networks, treated as incompressible ordered elastic materials, mostly two-dimensional. stimulated deformations Kowalski, Mostajeran, Godman, Warner & White(2018)
White & Broer (2015) There is a wealth of fascinating experiments with thermally activated thin sheets that are still in want of an appropriate mathematical explanation. White & Broer (2015)
Neo-Classical Energy Warner & Terentjev (2003) derived an expression for the soft elastic free-energy density of incompressible nematic elastomers: f soft := 1 2 µ tr( F T L − 1 n FL m ) µ > 0 elastic modulus F deformation gradient m blueprinted reference nematic field n conveyed present nematic field L n := a ( I + s n ⊗ n ) present step length tensor L m := a 0 ( I + s 0 m ⊗ m ) reference step length tensor a, a 0 fixed positive constants s, s 0 present and reference scalar order parameters actuation parameter s can be driven away from s 0 , thus serving as an actuation parameter for the deformation of the body. Other energy contributions, such as that connected with semi-soft elasticity are neglected. Nguyen & Selinger (2107)
kinematic prescription n = F m | F m | scaled energy f soft = 1 2 µa 0 a F ( C f ) m · C 2 f m s 0 s F ( C f ) = tr C f + s + 1 m · C f m − s + 1 m · C f m f (tree-dimensional) deformation F = ∇ f deformation gradient C f = F T F right Cauchy-Green tensor
Stretching Energy A naive, but effective approach is treating an elasomer sheet as an inextensible , two-dimensional membrane: y : S → S n = F m F = ∇ y n ⊥ := ν × n m ⊥ := e 3 × m | F m | S planar set e 3 unit normal to S ν unit normal to S
stretching tensor C 2 = (tr C ) C − I C := ( ∇ y ) T ( ∇ y ) det C = 1 s F ( C ) = tr C + s 0 m · C m + m · C m energy minimizer C = λ 2 1 m ⊗ m + λ 2 2 m ⊥ ⊗ m ⊥ principal stretches � s + 1 λ 2 = 1 λ 1 := 4 and s 0 + 1 λ 1 For s > s 0 , which is obtained upon cooling , the polymer network tends to extend along m , whereas for s < s 0 , which is obtained upon heating , the polymer network tends to extend along m ⊥ .
A vast, beautiful literature is concerned with finding the surfaces S that comply with the required principal stretches : Modes, Bhattacharya & Warner (2010) Modes, Bhattacharya & Warner (2011) Cirak, Long, Bhattacharya & Warner (2014) Modes & Warner (2015) Mostajeran (2015) Mostajeran, Warner, Ware & White (2016) Plucinsky, Lemm & Bhattacharya (2016) Mostajeran, Warner & Modes (2017) Kowalski, Mostajeran, Godman, Warner & White(2018) Plucinsky, Lemm & Bhattacharya (2018) Warner & Mostajeran (2018) C can easily be recognized to be the metric tensor of S ; most of the relevant literature belongs to the field of what has recently come to be known as geometric elasticity : Aharoni, Sharon & Kupferman (2014) Aharoni, Xiab, Zhang, Kamien & Yang (2018) Griniasty, Aharoni & Efrati (2019)
Cartesian connectors ◮ On S , ∇ m = m ⊥ ⊗ c ∇ m ⊥ = − m ⊗ c ◮ On S , ∇ n = n ⊥ ⊗ c ∗ + ν ⊗ d ∗ 1 ∇ n ⊥ = − n ⊗ c ∗ + ν ⊗ d ∗ 2 ∇ ν = − n ⊗ d ∗ 1 − n ⊥ ⊗ d ∗ 2 compatibility conditions curl c = 0 c ∗ · m ⊥ = λ 2 λ 1 c · m ⊥ c ∗ · m = λ 1 λ 2 c · m d ∗ 1 · m ⊥ = λ 2 λ 1 d ∗ 2 1 = c ∗ × d ∗ curl d ∗ 2 curl d ∗ 2 = d ∗ 1 × c ∗ curl c ∗ = d ∗ 2 × d ∗ 1
curvature tensor � d 11 � n ⊗ n + d 12 ( n ⊗ n ⊥ + n ⊥ ⊗ n ) + d 22 ∇ s ν = − n ⊥ ⊗ n ⊥ λ 1 λ 2 λ 2 d ∗ d ∗ 1 = d 11 m + d 12 m ⊥ 2 = d 21 m + d 22 m ⊥ � � d 11 λ 1 + d 22 H = − 1 mean curvature 2 λ 2 1 Gaussian curvature K = λ 1 λ 2 ( d 11 d 22 − d 12 d 21 ) Gauss’ Theorema Egregium 1 � λ 1 � − λ 2 [( c · m ⊥ ) 2 − ( c · m ) 2 + m · ( ∇ c ) m ⊥ ] K = λ 1 λ 2 λ 2 λ 1 Since c is determined by m , this acts as a constraint on the surfaces with prescribed principal stretches that are to be found.
examples m = cos α e r + sin α e ϑ α ( r ) = πr 2 R K = ( λ 2 1 − λ 2 2 ) κ ( r ) � 2 � R � π � πr sin πr R + cos πr κ ( r ) = − 2 R R 2 R m = cos ω e x + sin ω e y ω ( y ) = πy 2 L L K = ( λ 2 1 − λ 2 2 ) κ ( y ) � π � 2 cos πy κ ( y ) = − 2 L L a
Plentitude of Surfaces Perhaps, the easiest case arises when S is a disk and m = e r , that is, for α ≡ 0, which, since c = 1 r e ϑ , enforces K ≡ 0. cones and anticones Modes, Bhattacharya & Warner (2010) problem There is no stretching-energy minimizer of class C 1 .
folded cones Pedrini & Virga (2019)
folded cones Pedrini & Virga (2019)
folded cones Pedrini & Virga (2019)
folded cones Pedrini & Virga (2019)
folded cones Pedrini & Virga (2019)
folded cones Pedrini & Virga (2019)
Bending Energy All folds cost no stretching energy, but there the mean curvature is indefinite . Modified Kirchhoff-Love Hypothesis S is the mid-surface of a planar slab of thickness 2 h f ( x + x 3 e 3 ) = y ( x ) + φ ( x , x 3 ) ν ( x ) x ∈ S − h ≦ x 3 ≦ h φ ( x , 0) ≡ 0 |∇ φ | ≪ 1
A general representation for ∇ y is the following, ∇ y = a ⊗ m + b ⊗ m ⊥ a = ( ∇ y ) m b = ( ∇ y ) m ⊥ a , b present Cosserat directors C = ( ∇ y T )( ∇ y ) = a 2 m ⊗ m + ( a · b )( m ⊗ m ⊥ + m ⊥ ⊗ m ) + b 2 m ⊥ ⊗ m ⊥ det C = a 2 b 2 − ( a · b ) 2 = 1 ν = a × b C f := ( ∇ f ) T ( ∇ f ) = C φ + φ ′ 2 e 3 ⊗ e 3 C φ := C + φ C 1 + φ 2 C 2 C 1 = 2( ∇ y ) T ( ∇ C 2 = ( ∇ y ) T ( ∇ s ν ) 2 ( ∇ y ) s ν )( ∇ y ) det C f = φ ′ 2 det C φ = 1 3 + 1 φ = x 3 − H ( x ) x 2 3[6 H 2 ( x ) − K ( x )] x 3 3 + O ( x 4 3 )
remarks ◮ The classical Kirchhoff-Love hypothesis, which assumes that φ ≡ x 3 so that the thickness of the sheet is preserved by deformation, has proven incorrect. Friesecke, James & M¨ uller (2002) ◮ Here the thickness of the deformed sheet is not uniform: � + h φ ′ d x 3 = 2 h + 2 2 h ′ = 3 h 3 (6 H 2 − K ) + O ( h 5 ) − h neo-classical energy 1 1 � tr C φ + s 0 m · C φ m + s det C φ � F ( C f ) = + det C φ s + 1 m · C φ m Integrating F ( C f ) over the sheet thickness f ( ∇ y , ∇ 2 y ) = f s ( C ) + f b ( ∇ y , ∇ 2 y ) + O ( h 5 ) energy density per unit area of S f stretching energy density f s bending energy density f b
energy densities 1 s � � �� f s = 2 h 1 + tr C + s 0 m · C m + s + 1 m · C m f b = 2 1 � 2 s s 0 + 1 + 3 s 2(8 H 2 − K ) + 3 h 3 � � � a 2 K − 2 H a · ( ∇ s ν ) a s + 1 a 4 1 + s 0 − s � � s ν ) 2 a − 2 H b · ( ∇ s ν ) b + a · ( ∇ a 4 s ν ) 2 b + 4 s � 2 �� � + b · ( ∇ a · ( ∇ s ν ) a a 6 two-step minimization 1. Assuming h 2 H 2 ≪ 1 and h 2 K ≪ 1, we make f s the prevailing energy; it is minimized for C = λ 2 1 m ⊗ m + λ 2 2 m ⊥ ⊗ m ⊥ � s + 1 λ 2 = 1 λ 1 = 4 s 0 + 1 λ 1 a = λ 1 n b = λ 2 n ⊥ Then K is prescribed .
2. Representing ∇ s ν in the frame ( n , n ⊥ ) as ∇ s ν = κ 11 n ⊗ n + κ 12 ( n ⊗ n ⊥ + n ⊥ ⊗ n ) + κ 22 n ⊥ ⊗ n ⊥ we minimize f b subject to κ 11 κ 22 − κ 2 12 = K target curvature √ 1. K ≧ 0 ∇ s ν = ± K ( n ⊗ n + n ⊥ ⊗ n ⊥ ) √ 2. K ≦ 0 ∇ s ν = ± − K ( n ⊥ ⊗ n + n ⊗ n ⊥ ) compatibility problem In the language of geometric elasticity, a target metric C is supplemented by a target curvature ∇ s ν . The compatibility problem becomes more complicated, and possibly with less solutions.
double hit Perfect compatibility is ensured by special blueprinted m fields, 1 div m + 2 K ∇ K · m = ± c 2 1 curl m · e 3 − 2 K ∇ K · m ⊥ = ± c 1 � λ 1 � 1 − λ 2 ( c 2 2 − c 2 K = 1 + c 12 ) λ 1 λ 2 λ 2 λ 1 to be continued . . .
Acknowledgements Collaboration O. Ozenda A. Pedrini A.M. Sonnet Discussion P. Palffy-Muhoray J.V. Selinger Soft Matter Mathematical Modelling
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