A STRUCTURE PRESERVING FEM FOR UNIAXIAL NEMATIC LIQUID CRYSTALS - PowerPoint PPT Presentation

A STRUCTURE PRESERVING FEM FOR UNIAXIAL NEMATIC LIQUID CRYSTALS Ricardo H. Nochetto Department of Mathematics and Institute for Physical Science and Technology University of Maryland Joint work with Juan Pablo Borthagaray , University of Salto, Uruguay Shawn Walker , Louisiana State University Wujun Zhang , Rutgers University Numerical Methods and New Perspectives for Extended Liquid Crystalline Systems ICERM, December 9-14, 2019

LdG Model Structure Preserving FEM Γ -Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems Outline The Landau - de Gennes Model Structure Preserving FEM Γ -Convergence Discrete Gradient Flow Simulations Electric and Colloidal Effects Conclusions and Open Problems A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ -Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems Outline The Landau - de Gennes Model Structure Preserving FEM Γ -Convergence Discrete Gradient Flow Simulations Electric and Colloidal Effects Conclusions and Open Problems A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ -Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems Liquid Crystals with Variable Degree of Orientation • Nematic liquid crystal (LC) molecules are often idealized as elongated rods. Modeling is further simplified by an averaging procedure to replace local arrangement of rods by a few order parameters. Left: Thermotropic LC; Right: Schlieren texture of liquid crystal nematic phase with surface point defects (boojums). Picture taken under a polarization microscope. • Defects are inherent to LC modeling, analysis and computation. They can be orientable ( ± 1 degree) or non-orientable ( ± 1 / 2 degree). • Computation of LCs should allow for defects and yield convergent approximations of relevant physical quantities. A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ -Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems Half-integer order defects Figure: Singularities (defects) of degree (b) 1 / 2 and (c) − 1 / 2 . Taken from S´ anchez et al., Nature , 2012. A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ -Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems Orientability • Head-to-tail symmetry: director fields (unit vectors n ) introduce an orientational bias into the model that is not physical (Ericksen’s model). • Defects of degree ± 1 / 2 are not orientable (cannot be described by director fields but rather by line fields); Landau-DeGennes model. A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ -Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems Ensemble Averaging • Probability distribution ρ : for x ∈ Ω and p ∈ S 2 the unit sphere, ρ satisfies � ρ ( x , p ) ≥ 0 , S 2 ρ ( x , p ) ds ( p ) = 1 , ρ ( x , p ) = ρ ( x , − p ) � � • First moment: S 2 p ρ ( x , p ) ds ( p ) = − S 2 p ρ ( x , − p ) ds ( p ) = 0 • Second moment: � S 2 p ⊗ p ρ ( x , p ) ds ( p ) ∈ R 3 × 3 M ( x ) = ⇒ tr( M ) = 1 . • Isotropic uniform distribution: ρ ( x , p ) = 1 M = 1 ⇒ 3 I . 4 π • The Q -tensor: measures deviation from isotropic uniform distribution Q := M − 1 Q = Q T , 3 I , ⇒ tr( Q ) = 0 . A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ -Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems The Q -tensor • Biaxial form of Q : for n 1 , n 2 ∈ S 2 director fields and s 1 , s 2 ∈ R scalar order parameters, Q reads � � � � n 1 ⊗ n 2 − 1 n 2 ⊗ n 2 − 1 Q = s 1 + s 2 3 I 3 I The signs of n 1 and n 2 have no effect on Q (head-to-tail symmetry). • Eigenvalues λ i ( Q ) of Q : probability distribution implies − 1 3 ≤ λ i ( Q ) ≤ 2 3 λ 1 ( Q ) = 2 s 1 − s 2 λ 2 ( Q ) = 2 s 2 − s 1 λ 3 ( Q ) = − s 1 + s 2 , 3 3 3 • Uniaxial form of Q : either s 1 = 0 , s 2 = 0 or s 1 = s 2 and � � n ⊗ n − 1 − 1 2 ≤ s ≤ 1 . Q = s 3 I , A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ -Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems The One-Constant Landau - de Gennes Model • One-constant LdG energy: sum of elastic and potential energies E [ Q ] := E 1 [ Q ] + 1 ǫ E 2 [ Q ] where ǫ > 0 is small (the nematic correlation length). • Elastic (Frank) energy: � E 1 [ Q ] := 1 |∇ Q | 2 d x . 2 Ω � • Double-well potential energy: E 2 [ Q ] := Ω ψ ( Q ) d x where ψ ( Q ) = A tr( Q 2 ) + B tr( Q 3 ) + C tr( Q 2 ) 2 for A, B, C ∈ R . The minimizer of ψ ( Q ) is a uniaxial state. A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ -Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems Uniaxial vs Biaxial States • Biaxial: for thermotropic LCs, the nematic biaxial phase remained elusive for a long period until ◮ Acharya, et al, PRL, 2004; ◮ Madsen, et al, PRL, 2004; ◮ Prasad, et al, J. Amer. Chem. Soc., 2005. • Uniaxial: 2012 book by Sonnet and Virga (sec. 4.1) says The vast majority of nematic liquid crystals do not, at least in homogeneous equilibrium states, show any sign of biaxiality. • LdG-model: does not enforce uniaxiality which, however, is still prevalent in many situations. • FEM: we present a finite element method for uniaxial LCs. • Comparisons: we compare uniaxial and biaxial behavior near a Saturn-ring defect at the end of this talk. A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ -Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems Q -Model with Uniaxial Constraint − 1 • One-constant energy: E [ Q ] = E 1 [ Q ] + E 2 [ Q ] where Q = s ( n ⊗ n d I ) � �� � =Θ � � � � 2 � � |∇ Q | 2 , A tr( Q 2 ) + B tr( Q 3 ) + C tr( Q 2 ) E 1 [ Q ] = E 2 [ Q ] = . Ω Ω � � Θ − 1 • Property 1: ∇ Q = ∇ s ⊗ d I + s ∇ Θ , ∇ Θ : Θ = ∇ Θ : I = 0 |∇ Q | 2 = |∇ s | 2 � � � � �� � Θ − 1 2 Θ − 1 � � + s 2 |∇ Θ | 2 + 2 s ∇ s · d I ∇ Θ : d I . � � �� � � �� � = d − 1 =0 d • Property 2: A direct calculation gives � � 2 . s 2 = C 1 tr( Q 2 ) , s 3 = C 2 tr( Q 3 ) , s 4 = C 3 tr( Q 2 ) • Equivalent one-constant energy: κ = d − 1 d , ψ suitable double-well potential � � κ |∇ s | 2 + s 2 |∇ Θ | 2 , E 1 [ Q ] = E 2 [ Q ] = ψ ( s ) . Ω Ω A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ -Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems Q -Model with Uniaxial Constraint − 1 • One-constant energy: E [ Q ] = E 1 [ Q ] + E 2 [ Q ] where Q = s ( n ⊗ n d I ) � �� � =Θ � � � � 2 � � |∇ Q | 2 , A tr( Q 2 ) + B tr( Q 3 ) + C tr( Q 2 ) E 1 [ Q ] = E 2 [ Q ] = . Ω Ω � � Θ − 1 • Property 1: ∇ Q = ∇ s ⊗ d I + s ∇ Θ , ∇ Θ : Θ = ∇ Θ : I = 0 |∇ Q | 2 = |∇ s | 2 � � � � �� � Θ − 1 2 Θ − 1 � � + s 2 |∇ Θ | 2 + 2 s ∇ s · d I ∇ Θ : d I . � � �� � � �� � = d − 1 =0 d • Property 2: A direct calculation gives � � 2 . s 2 = C 1 tr( Q 2 ) , s 3 = C 2 tr( Q 3 ) , s 4 = C 3 tr( Q 2 ) • Equivalent one-constant energy: κ = d − 1 d , ψ suitable double-well potential � � κ |∇ s | 2 + s 2 |∇ Θ | 2 , E 1 [ Q ] = E 2 [ Q ] = ψ ( s ) . Ω Ω A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto

LdG Model Structure Preserving FEM Γ -Convergence Gradient Flow Simulations Electric and Colloidal Effects Open Problems One-Constant Ericksen’s Model • Model: The equilibrium state minimizes (one-constant Ericksen’s model): � � κ |∇ s | 2 + s 2 |∇ n | 2 dx E [ s, n ] := + ψ ( s ) dx Ω Ω � �� � � �� � := E 1 [ s, n ] := E 2 [ s ] where κ > 0 and ψ is a double well potential with domain ( − 1 2 , 1) . • Director field: | n | = 1 (unit vector). n n θ 25.74° n local well-aligned defect perpendicular s ≈ 1 s ≈ 0 s ≈ − 1 / 2 • Scalar order parameter: s is the degree of orientation ( − 1 / 2 < s < 1 ). ◮ s = 1 : perfect alignment with n . ◮ s = 0 : no preferred direction (isotropic). This defines the set of defects : S = { x ∈ Ω , s ( x ) = 0 } . ◮ s = − 1 / 2 : perpendicular to n . A structure preserving FEM for uniaxial nematic liquid crystals Ricardo H. Nochetto