A STRUCTURE PRESERVING FEM FOR UNIAXIAL NEMATIC LIQUID CRYSTALS · Liquid Crystals with Variable...

A STRUCTURE PRESERVING FEM FORUNIAXIAL NEMATIC LIQUID CRYSTALS

Ricardo H. Nochetto

Department of Mathematics andInstitute 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 Perspectivesfor 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


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 localarrangement 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 canbe orientable (±1 degree) or non-orientable (±1/2 degree).

• Computation of LCs should allow for defects and yield convergentapproximations of relevant physical quantities.



Half-integer order defects

Figure: Singularities (defects) of degree (b) 1/2 and (c) −1/2. Taken from Sanchez etal., Nature, 2012.



Orientability

• Head-to-tail symmetry: director fields (unit vectors n) introduce anorientational bias into the model that is not physical (Ericksen’s model).

• Defects of degree ±1/2 are not orientable (cannot be described by directorfields but rather by line fields); Landau-DeGennes model.



Ensemble Averaging

• Probability distribution ρ: for x ∈ Ω and p ∈ S2 the unit sphere, ρ satisfies

ρ(x,p) ≥ 0,

∫S2

ρ(x,p)ds(p) = 1, ρ(x,p) = ρ(x,−p)

• First moment:∫S2 p ρ(x,p)ds(p) = −

∫S2 p ρ(x,−p)ds(p) = 0

• Second moment:

M(x) =

∫S2

p⊗ p ρ(x,p)ds(p) ∈ R3×3 ⇒ tr(M) = 1.

• Isotropic uniform distribution:

ρ(x,p) =1

4π⇒ M =

1

3I.

• The Q-tensor: measures deviation from isotropic uniform distribution

Q := M− 1

3I, ⇒ Q = QT , tr(Q) = 0.



The Q-tensor

• Biaxial form of Q: for n1,n2 ∈ S2 director fields and s1, s2 ∈ R scalar orderparameters, Q reads

Q = s1

(n1 ⊗ n2 −

1

3I)

+ s2

(n2 ⊗ n2 −

1

3I)

The signs of n1 and n2 have no effect on Q (head-to-tail symmetry).

• Eigenvalues λi(Q) of Q: probability distribution implies − 13≤ λi(Q) ≤ 2

3

λ1(Q) =2s1 − s2

3, λ2(Q) =

2s2 − s1

3λ3(Q) = −s1 + s2

3

• Uniaxial form of Q: either s1 = 0, s2 = 0 or s1 = s2 and

Q = s(n⊗ n− 1

3I), −1

2≤ s ≤ 1.



The One-Constant Landau - de Gennes Model

• One-constant LdG energy: sum of elastic and potential energies

E[Q] := E1[Q] +1

εE2[Q]

where ε > 0 is small (the nematic correlation length).

• Elastic (Frank) energy:

E1[Q] :=1

2

∫Ω

|∇Q|2dx.

• Double-well potential energy: E2[Q] :=∫

Ωψ(Q)dx where

ψ(Q) = A tr(Q2) +B tr(Q3) + C tr(Q2)2

for A,B,C ∈ R. The minimizer of ψ(Q) is a uniaxial state.



Uniaxial vs Biaxial States

• Biaxial: for thermotropic LCs, the nematic biaxial phase remained elusivefor a long period until

I Acharya, et al, PRL, 2004;I Madsen, et al, PRL, 2004;I 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 inhomogeneous equilibrium states, show any sign of biaxiality.

• LdG-model: does not enforce uniaxiality which, however, is still prevalent inmany situations.

• FEM: we present a finite element method for uniaxial LCs.

• Comparisons: we compare uniaxial and biaxial behavior near a Saturn-ringdefect at the end of this talk.



Q-Model with Uniaxial Constraint

• One-constant energy: E[Q] = E1[Q] + E2[Q] where Q = s(n⊗ n︸︷︷︸=Θ

− 1dI)

E1[Q] =

∫Ω

|∇Q|2, E2[Q] =

∫Ω

(A tr(Q2) +B tr(Q3) + C

(tr(Q2)

)2).

• Property 1: ∇Q = ∇s⊗(Θ− 1

dI)

+ s∇Θ, ∇Θ : Θ = ∇Θ : I = 0

|∇Q|2 = |∇s|2∣∣∣Θ− 1

dI∣∣∣2︸︷︷︸

= d−1d

+s2|∇Θ|2 + 2s∇s ·[∇Θ :

(Θ− 1

dI)]

︸︷︷︸=0

.

• Property 2: A direct calculation gives

s2 = C1tr(Q2), s3 = C2tr(Q3), s4 = C3

(tr(Q2)

)2.

• Equivalent one-constant energy: κ = d−1d

, ψ suitable double-well potential

E1[Q] =

∫Ω

κ|∇s|2 + s2|∇Θ|2, E2[Q] =

∫Ω

ψ(s).



One-Constant Ericksen’s Model

• Model: The equilibrium state minimizes (one-constant Ericksen’s model):

E[s,n] :=

∫Ω

κ|∇s|2 + s2|∇n|2dx︸︷︷︸:=E1[s,n]

+

∫Ω

ψ(s)dx︸︷︷︸:=E2[s]

where κ > 0 and ψ is a double well potential with domain (− 12, 1).

• Director field: |n| = 1 (unit vector).

25.74°

n

nn θ

well-alignedlocaldefect perpendicular

s ≈ 1 s ≈ 0 s ≈ −1/2

• Scalar order parameter: s is the degree of orientation (−1/2 < s < 1).I s = 1: perfect alignment with n.I s = 0: no preferred direction (isotropic). This defines the set of defects:

S = x ∈ Ω, s(x) = 0.I s = −1/2: perpendicular to n.



Ericksen Model vs. Oseen-Frank Model

• When s = s0 > 0, the energy reduces to the Oseen-Frank energy:

E :=

∫Ω

|∇n|2dx.

• One disadvantage of Oseen-Frank model is that the director field with finiteenergy is inconsistent with both line and plane defects.

r

z

C

I Line defects:

n =r

|r|, |∇n| =

2

|r|.

I Compute∫C |∇n|

2dx :∫ 1

0

4

r2rdr =∞.

I Compute∫C κ|∇s|

2 + s2|∇n|2dx :∫ 1

0s2

4

r2rdr <∞.

• Ericksen’s model allows for defects to have finite energy (regularization).



Ericksen’s Model: Point Defect of Degree 1 in 2d

• Dirichlet boundary conditions: s = s∗, n = x|x|

• Director field: κ = 2



Ericksen’s Model: Point Defect of Degree 1 in 2d

• Scalar order parameter: smin = 2.02 · 10−2

• Movie: Director field nh and order parameter sh



Q-Tensor Model: Point Defect of Degree 1 in 2d

• Dirichlet boundary conditions: s = s∗, n = x|x|

• Line field Θ = n⊗ n: κ = 12



Q-Tensor Model: Point Defect of Degree 1 in 2d

• Scalar order parameter sh: smin = 0.02, Eh = 9.18

• Movie: Line field Θh and order parameter sh



Q-Tensor Model: Point Defects of Degree ±1/2 in 2d

• Dirichlet boundary conditions: s = s∗ corresponds to absolute minimum ofdouble well potential ψ and n = n∗ corresponds to defect of degree 3/2 at(0., 0.).

• Scalar field s and line field Θ = n⊗ n:

• 5 defects: 4 defects of degree 12

on the diagonals of Ω = (0, 1)2 and 1defect of degree − 1

2at the origin.



Primary Goals

• Structure preserving FEM: Design discrete energies Eh[Qh] = Eh[Θh, sh] toapproximate the Landau-deGennes energy

E[Θ, s] =

∫Ω

κ|∇s|2 + s2|∇Θ|2︸︷︷︸=E2[Q]

+

∫Ω

ψ(s)dx︸︷︷︸=E2[s]

, Q = s(n⊗ n︸︷︷︸

=Θ

−1

dI)

• Γ-convergence: if Qh are absolute minimizers of Eh

I Qh converge to absolute minimizers Q of E as h→ 0

I Eh[Qh]→ E[Q] as h→ 0.

• Gradient flow: Design a gradient flow to find a minimizer (equilibrium stateor stationary point) Qh of the discrete energy Eh.

• Monotonicity: Energy decrease property of the gradient flow.

• Simulation of defects: Line and plane defects, saturn ring, degree ±1 and±1/2 defects, etc (caused by electric fields and/or colloidal inclusions).



(Incomplete) Literature Review

Modeling and PDEHardt Kinderlehrer Lin (1986), Kinderlehrer (1991), Kinderlehrer, Ou (1992)Brezis (1987, 1989), Ericksen (1991)Virga (1994), DeGennes-Prost (1995)Ambrosio (1990)Lin (1989, 1991)Ball Zarnescu (2011); Ball Majumdar (2010).

Numerical methods for director fields:Cohen Lin Luskin (1989)Alouges (1997)Liu Walkington (2000)Calderer Golovaty Lin Liu (2002)Bartels (2010); Bartels, Dolzmann, Nochetto (2012)Yang Forest Li Liu Shen Wang (2013)

Numerical method for Ericksen’s model and Q-tensor modelBarrett Feng Prohl (2006) (2D-FEM via regularization)James Willman Fernandez (2006) (Q tensor method)Shin Cho Lee Yoon and Won (2008) (Q tensor method)Lin Lin Wang (2010)Bartels Raisch (2014)

Walkington (2011, 2017).



Outline



Γ-Convergence


Simulations





PDE Structure of Ericksen’s Model

• One constant model:

E[s,n] :=

∫Ω

κ|∇s|2 + s2|∇n|2dx︸︷︷︸:=E1[s,n]

+

∫Ω

ψ(s)dx︸︷︷︸:=E2[s]

Euler-Lagrange equation: (degenerate elliptic)

div (s2∇n)− s2|∇n|2n = 0.

−κ∆s− s|∇n|2 +1

2ψ′(s) = 0.

• Energy identity: since |n| = 1, we have ∇|n|2 = 2nT (∇n) = 0.∫Ω

|∇ (sn)︸︷︷︸:=u

|2dx =

∫Ω

|n⊗∇s+ s∇n|2dx =

∫Ω

|∇s|2 + s2|∇n|2dx.

• Equivalent energy: [Ambrosio 90, Lin 91] E1[s,n] = E1[s,u] with

E1[s,u] :=

∫Ω

(κ− 1)|∇s|2 + |∇u|2dx, (u = sn)

i.e. a simple quadratic functional, but with a negative term (κ < 1).



PDE Structure of the Q-Model: Relation to Ericksen’s Model

• One-constant energy: If κ = d−1d

and ψ is a suitable double-well potential,then

E1[Q] =

∫Ω

κ|∇s|2 + s2|∇Θ|2, E2[Q] =

∫Ω

ψ(s).

• Auxiliary variable: If U = sΘ = s(n⊗ n

), then

∇U = ∇s⊗Θ + s∇Θ ⇒ |∇U |2 = |∇s|2 + s2|∇Θ|2

because ∇Θ : Θ = 0. Therefore E1[Q] = E1[s,U ] where

E1[s,U ] =

∫Ω

(κ− 1)|∇s|2 + |∇U |2.

• Structural conditions:

−1

2< s < 1, U = sn⊗ n︸︷︷︸

=Θ

, n ∈ Sd−1.

• Admissible class:

A :=

(s,U) ∈ H1(Ω)×[H1(Ω)]d×d : (s,U) satisfy structural conditions a.e..



Numerical Difficulties

• PDEs: Degenerate nonlinear elliptic PDEs.

• Line field: Construct a vector field Θh = nn ⊗ nh such that nh has unitlength at nodes (locking otherwise).

• Scalar field: Construct sh such that − 12< sh < 1 (at the nodes).

• Auxiliary tensor field: Construct a rank-one tensor field Uh that satisfiesthe structural condition at the nodes Nh

Uh(xi) = sh(xi)(nh(xi)⊗ nh(xi)

)∀xi ∈ Nh.

• Γ-convergence: theory without regularization that allows for defects.

• Computation of minimizers: discrete gradient flow.



Finite Element Spaces

• Meshes: Let Th = T be a conforming, shape-regular triangulation of Ω,with set of nodes (vertices) denoted by Nh.

• Finite element spaces:

Uh := uh ∈ [H1(Ω)]d : uh|T is linear

Sh := sh ∈ H1(Ω) : sh|T is linearNh := nh ∈ Uh : nh|T is linear

|nh(xi)| = 1 at all nodes xi ∈ NhTh := Nh ⊗ Nh.

• Weakly acute mesh: the entries of the stiffness matrix kij satisfy

kij = −∫

Ω

∇φi · ∇φjdx ≥ 0 for i 6= j,

where φi denotes the continuous piecewise linear “hat” basis functionswhich satisfy φi(xj) = δij for all nodes xj ∈ Nh.




• Dirichlet forms: For a piecewise linear function sh, we have∫Ω

|∇sh|2dx =1

2

∑i,j

kij(si − sj)2,

∫Ω

|∇Θh|2dx =1

2

∑i,j

kij(Θi −Θj)2,

where si = sh(xi) and Θi = Θh(xi) for all nodes xi ∈ Nh.

• Discrete energy: approximate E1[s,n] =∫

Ωκ|∇s|2 + s2|∇Θ|2dx by

Eh1 [sh,Θh] :=κ

2

N∑i,j=1

kij (si − sj)2

︸︷︷︸=κ

∫Ω |∇sh|2dx

+1

2

N∑i,j=1

kij

(s2i + s2

j

2

)|Θi −Θj |2︸︷︷︸

≈∫Ω s

2h|∇Θh|2dx

.

• Nodal structural conditions: let

U i = siΘi = si(ni ⊗ ni

)∀xi ∈ Nh.



Stability of the Discrete Energy

• Dirichlet form: Let uh be piecewise linear with nodal values U i = siΘi

2

∫Ω

|∇Uh|2dx =∑i,j

kij(siΘi − sjΘj)2

=∑i,j

kij

(si + sj

2(Θi −Θj) + (si − sj)

Θi + Θj

2

)2

• Orthogonality: Exploiting the relation (Θi −Θj) : (Θi + Θj) = 0 yields

2

∫Ω

|∇Uh|2dx =∑i,j

kij

[(si + sj2

)2

(Θi −Θj)2 + (si − sj)2

(Θi + Θj

2

)2]

=∑i,j

kijs2i + s2

j

2

(Θi −Θj

)2+∑i,j

kij(si − sj)2 −∑i,j

kij(si − sj)2∣∣∣Θi −Θj

2

∣∣∣2• Energy inequalities: U i = siΘi, si = |si|, U i = siΘi (structural conditions)

Eh1 [sh,Θh] ≥ (κ− 1)

∫Ω

|∇sh|2dx+

∫Ω

|∇Uh|2dx =: Eh1 [sh,Uh]

Eh1 [sh,Θh] ≥ (κ− 1)

∫Ω

|∇sh|2dx+

∫Ω

|∇Uh|2dx =: Eh1 [sh, Uh].



Discrete Admissible Class

• Discrete structural conditions:

−1

2< si < 1, U i = si

(ni ⊗ ni

), |ni| = 1 ∀xi ∈ Nh.

• Discrete admissible class:

Ah :=

(sh,Uh) ∈ Sh×Uh : (sh,Uh) satisfies discrete structural conditions.

• Discrete restricted admissible class:

Ah(Ihg, IhR) :=

(sh,Uh) ∈ Ah : sh = Ihg, Uh = IhR on ∂Ω,

where Ih is the Lagrange interpolation operator.



Outline



Γ-Convergence


Simulations





Assumptions for Γ-Convergence

• Meshes: Let Thh≥0 be a sequence of weakly acute meshes.

• Lim-sup inequality (consistency or recovery sequence): Given (s,U) ∈ A,namely s,U = sΘ ∈ H1(Ω), and satisfy Dirichlet boundary conditionss = g,U = R, there exists (sh,Uh) ∈ Ah(Ihg, IhR) such that(sh,Uh)→ (s,U) in H1(Ω) and

lim suph→0

Eh1 [sh,Θh] ≤ E1[s,Θ].

• Lim-inf inequality (stability or lower semi-continuity): Given a sequence

(sh,Uh) ∈ Ah(Ihg, IhR), let sh = Ih|sh|, Uh = Ih[shΘh] be so that

(sh, Uh) converges weakly in H1(Ω) to (s, U). Then we have

E1[s, U ] ≤ lim infh→0

Eh1 [sh, Uh].

• Equi-coercivity: There exists a constant C > 0 such that

‖sh‖H1(Ω), ‖Uh‖H1(Ω) ≤ CEh1 [sh,Θh].



Theorem: Γ-Convergence for the Q-Model

Let Thh≥0 be a sequence of weakly acute meshes. Then the discrete energyEh1 [sh,Θh] satisfies the lim-sup, lim-inf, and equi-coercivity properties.Moreover,

• If Eh[sh,Θh] ≤ Λ uniformly, then there exist subsequences (not relabeled)(sh,Uh), (sh, Uh), and Θh such that

I (sh,Uh) converges to (s,U) weakly in H1 and strongly in L2;

I (sh, Uh) converges to (s, U) weakly in H1 and strongly in L2;

I The limits satisfy s = |s| = |U | = |U |;I There exists a line field Θ defined in the complement of the singular setS = s = 0 such that Θh converges to Θ in L2(Ω \ S) and U = sΘ and

U = sΘ;

I Θ admits a Lebesgue gradient ∇Θ and |∇U |2 = |∇s|2 + s2|∇Θ|2 a.e. inΩ \ S.

• If (sh,Θh) is a sequence of global minimizers of Eh, then every cluster point(s,Q) is a global minimizer of E.



Consistency or Recovery Sequence: Regularization of Functions in A(g,R)

• Regularization: Given ε > 0 and (s,u) ∈ A(g,R), i.e. satisfying s = g,U = R on ∂Ω, there exists (sε,U ε) ∈ A(g,R) ∩ [W 1

∞(Ω)]d+1 such that

‖(s,U)− (sε,U ε)‖H1(Ω) ≤ ε.

• Difficulties: pair (sε,U ε) must satisfy

I Structural conditions

−1

2< sε < 1, rank(Uε) ≤ 1, tr(Uε) = sε a.e. in Ω;

I Dirichlet boundary conditions

(sε,Uε) = (g,R) on ∂Ω;

I Approximate the pair (s,U) in H1(Ω).

• Lim-sup equality: For any ε > 0, we obtain (Ihsε, IhU ε) ∈ Ah(Ihg, IhR) bynodewise interpolation of (sε,Uε) and

E1[sε,Θε] = limh→0

Eh1 [Ihsε, IhΘε] = limh→0

Eh1 [Ihsε, IhU ε] = E1[sε,U ε].



Weak Lower Semi-continuity: Lim-Inf Inequality

• Energy inequality: Recall that

Eh1 [sh,Θh] ≥ Eh1 [sh, Uh] = −1

d

∫Ω

|∇sh|2dx+

∫Ω

|∇Uh|2dx,

where sh = Ih|sh| and uh = Ih[shΘh].

• Difficulties: is the right-hand side convex in ∇Uh? Note negative first term!

I True if |sh| = |Uh| in Ω

I But this relation |sh(xi)| = |Uh(xi)| is only valid at the nodes xi ∈ Nh.

• Weak lower semi-continuity: Let W h in Uh converge weakly to W in theH1-norm. Then

lim infh→0

∫Ω

−1

d

∣∣∇Ih|W h|∣∣2 +

∣∣∇W h

∣∣2 ≥ ∫Ω

−1

d

∣∣∇|W |∣∣2 +∣∣∇W ∣∣2.



Equi-Coercivity

For any (sh,Uh) ∈ Ah(Ihg, IhR) we have Uh = Ih[shΘh], Uh = Ih[shΘh]and

• Estimate for (sh,Uh):

Eh1 [sh,Θh] ≥ d− 1

dmax

∫Ω

|∇Uh|2,∫

Ω

|∇sh|2

• Estimate for (sh, Uh):

Eh1 [sh,Θh] ≥ d− 1

dmax

∫Ω

|∇Uh|2,∫

Ω

|∇sh|2.



Outline



Γ-Convergence


Simulations





Discrete Gradient Flow for the Q-Model

• Three-step algorithm: Given s = sh and Θ = Θh = Ih[nh ⊗ nh], iterate

I Weighted gradient flow for Θ: find T k = nk ⊗ tk + tk ⊗ nk such that tk istangent to nk and for all v tangent to nk (Bartels, Raisch (2014))

1

τ

∫Ωs2k∇tk : ∇v + tk · v+δΘE

h1 [sk,Θk+T k;V ] = 0 ∀V = nk⊗v+v⊗nk.

I Projection: update Θ by

Θk+1 :=nk + tk

|nk + tk|⊗

nk + tk

|nk + tk|.

I Gradient flow for s: find sk+1 such that

1

2τ

∫Ω|sk+1 − sk|2 + δsE

h1 [sk+1,Θk+1; z] + δsE

h2 [sk+1; z]︸︷︷︸

convex splitting

= 0 ∀z,

where δSE2[s; z] =∫Ω ψ′(s)z and ψ is a double-well potential.

• Relaxation: the effect of weight s2k is to allow for larger variations of tk.



Energy Decrease Property for the Q-Model

• Variations in line field: avoid nonlinear term t⊗ t (Bartels & Raisch 2014).

• Projection step: is energy decreasing (Alouges 1997, Bartels 2010).

• Convex splitting: to treat the double-well potential term in the gradientflow for sk.

• Theorem (energy decrease property) If the meshes Th are weakly acuteand the time step τ ≤ C0h

d/2, then there holds

Eh1 [sN ,ΘN ] +2

τ

N−1∑k=0

(‖sk∇tk‖2L2(Ω) + ‖tk‖2L2(Ω) + ‖sk+1 − sk‖2

)≤ Eh1 [s0,Θ0] ∀N ≥ 1.

Thus, the algorithm stops in a finite number of steps for any tolerance ε.



Why do we have a CFL condition?

• Tangential variations: In Step 1 we update Θk = Ih(nk ⊗ nk) according to

Θk + Tk, T k = nk ⊗ tk + tk ⊗ nk

with tk(xi) · nk(xi) = 0 (nodal tangential update).

• Rank-one update: In Step 2 we create the rank-1 update(nk + tk

)⊗(tk + nk

)= Θk + Tk + tk ⊗ tk

and next normalize it (projection into discrete line fields update).

• Correction term: We must thus accound for tk ⊗ tk∑i,j

kij(sk(xi)

2 + sk(xj)2)∣∣δij(tk ⊗ tk)

∣∣2 ≈ h−d(∫Ω

s2k|∇tk|2 + |tk|2

)2

.

An induction argument together with τ ≤ C0hd/2, with C0 proportional to

Eh[s0,Θ0], allows for control of this term.



Outline



Γ-Convergence


Simulations





Ericksen’s Model: “Propeller” Defect κ = 0.1

• Director field at 4 slices: z = 0.2, 0.4, 0.6, 0.8

0 0.5 10

0.5

1Z = 0.2 Slice

X

Y

0 0.5 10

0.5

1Z = 0.4 Slice

X

Y

0 0.5 10

0.5

1Z = 0.6 Slice

X

Y

0 0.5 10

0.5

1Z = 0.8 Slice

X

Y

• Movie: Director field at z = 0 Movie: 3d defect is marked in red.



Q-Model: 3-D Line Defect of Degree 1

• Line field at 4 slices: z = 0.1, 0.35, 0.65, 0.9; smin = 1.0× 10−2.

• Movie: Line field at z = 0.5. Gradient flow starts with line defect of degree1 located at (0.25, 025).



Q-Model: 3-D Line Defect of Degree 1/2

• Line field at 4 slices: z = 0.1, 0.35, 0.65, 0.9; smin = 6.8× 10−2.

• Movie: Line field at z = 0.5. Gradient flow starts with line defect of degree1 at (0.25, 025).



Twisting Defect: Non-Orientable Line Defect of Degree + 12

in a Cube

• Computation of line field and line defect: Horizontal slices of the +1/2degree line defect at levels z = 0.0, 0.5, 1.0 colored by scalar field s. Thenon-straight defect is the s = 0.05 iso-surface.

• Movie: Display of line field at levels z = 0.0, 0.5, 1.0 colored by scalar field.



Outline



Γ-Convergence


Simulations





Electric Field Effect with Q-Model: Freedericksz Transition

• Electric energy: let E be a given (fixed) external field,

Eext[s,Θ] = −Kext

2

(ε

∫Ω

(1− sγa)|E|2 + εa

∫Ω

s (ΘE ·E)

).

Above, Kext is a weight parameter, and ε, γa, εa are material constants.

• Simulation: Freedericksz transition with Dirichlet boundary conditions onthe left and right of Ω and zero Neumann condition on top and bottom of Ω.The electric field is E = (1, 0)>.



Colloidal Inclusion Effect with Q-Model

• Immersed boundary method: to avoid mesh-acuteness issues (conformingmeshes are hard to construct).

• Phase-field approach: φ ≈ 0 inside the colloid and φ ≈ 1 in the LC region.n

defectring

• Weak anchoring energy: penalization of (s,Θ) = (s∗,ν ⊗ ν) on colloidalsurface with unit normal ν

Ea[s,Θ] =Ka

2C0ε

(∫Ω

s2 (|Θ|2|∇φ|2 −Θ∇φ · ∇φ)

+

∫Ω

|∇φ|2(s− s∗)).

• Movie: 2d simulation. Spherical colloid with center (0.5, 0.7)> and radius0.2. Dirichlet boundary conditions: s = s∗, Θ = n⊗n with n normal to thecolloid on its surface and n = n∗ = (1, 0) on ∂Ω.

• Two defects of degree −1/2: located on top and bottom of colloid.



3d Saturn Ring Defect with Q-Model

• Setting: Dirichlet boundary conditions Θ = (0, 0, 1)⊗ (0, 0, 1) and s = s∗ inΩ = [0, 1]3. A colloid of radius 0.25 is located at the center of the domain.

Left: isosurface s = 0.2. Center: line field slice at plane y = 0.5.Right: degree of orientation slice at plane y = 0.5.

• Defect of degree −1/2: analysis by S. Alama, L. Bronsard, X. Lamy (2016).

• Movie: Evolution of order parameter s



Saturn Ring Defect: Degree of Biaxiality (Majumdar-Zarnescu, S. Walker)

0 ≤ β(Q) = 1− 6

(trQ3

)2(trQ2

)3 ≤ 1 ⇒

β(Q) = 0 uniaxial

β(Q) = 1 biaxial

Figure (a): Measure of biaxiality β(Q) (blue (uniaxial), red (biaxial));Figure (b): plot of the function |QLdG−Quni| (the error can be up to 15-20%).



Outline



Γ-Convergence


Simulations





Conclusions

• Structure preserving FEMs for the one constant Ericksen’s model

E[s,n] :=

∫Ω

κ|∇s|2 + s2|∇n|2dx+

∫Ω

ψ(s)dx, κ > 0.

and the Landau-DeGennes Q-tensor model (Q = sn⊗ n)

E[Q] =

∫Ω

κ|∇s|2 + s2|∇Θ|2 +

∫Ω

ψ(s), κ =d− 1

d< 1.

• Γ-convergence of the FEM for any dimension d: convergence of globaldiscrete minimizers to global minimizers.

• Monotone energy decrease of the quasi-gradient and weighted gradient flows.

• High dimensional defects can be captured by our FEMs for both models.

• 2d and 3d electric and colloidal effects: comparisons between Ericksen andLandau-deGennes models. Computations done within Matlab softwareFELICITY (S. Walker (2017)).



Open Problems

• Explore alternatives to the gradient flow such as direct minimization.

• Γ-convergence to local minimizers.

• Error estimates.

• Comparison of uniaxial Q-model with the full (biaxial) Q-model.

• Coupling of the Q-model with incompressible fluids.

Publications:

• N, S.W. Walker, and W. Zhang, A finite element method for nematicliquid crystals with variable degree of orientation, SIAM J. Numer. Anal. 55,3 (2017), 1357-1386.

• N, S.W. Walker, and W. Zhang, The Ericksen model of liquid crystalswith colloidal and electric effects, J. Comp. Phys. 352 (2018), 568–601.

• J.P. Borthagaray, N, and S.W. Walker, A structure preserving FEMfor the Q-model of uniaxial nematic liquid crystals, (submitted).


A STRUCTURE PRESERVING FEM FOR UNIAXIAL NEMATIC LIQUID CRYSTALS · Liquid Crystals with Variable...

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