Energy driven systems from Liquid Crystals and Epitaxy Xin Yang Lu - - PowerPoint PPT Presentation

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Energy driven systems from Liquid Crystals and Epitaxy Xin Yang Lu - - PowerPoint PPT Presentation

Feb 02, 2024 541 likes •946 views

ADL regime Nematic Liquid Crystals Energy driven systems from Liquid Crystals and Epitaxy Xin Yang Lu Lakehead University BIRS Workshop Topics in the Calculus of Variations: Recent Advances and New Trends Banff, 2018-05-24 1 / 25 ADL

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ADL regime Nematic Liquid Crystals

Energy driven systems from Liquid Crystals and Epitaxy

Xin Yang Lu

Lakehead University

BIRS Workshop “Topics in the Calculus of Variations: Recent Advances and New Trends” Banff, 2018-05-24

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ADL regime Nematic Liquid Crystals

A nice piece of technology... Lots of silicon Liquid Crystal Display

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ADL regime Nematic Liquid Crystals

Epitaxy models: with elastic forces on vicinal surfaces: the 1 + 1 dimensional case, with elastic forces on vicinal surfaces: the 2 + 1 dimensional case, with wetting, attachment-detachment regime, and many, many others... Nematic Liquid Crystals: Landau-De Gennes model. Q: What do they have in common? A1: All of these are governed by highly irregular PDEs... A2: All these are variational.

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ADL regime Nematic Liquid Crystals

Epitaxy models: with elastic forces on vicinal surfaces: the 1 + 1 dimensional case, with elastic forces on vicinal surfaces: the 2 + 1 dimensional case, with wetting, attachment-detachment regime, and many, many others... Nematic Liquid Crystals: Landau-De Gennes model. Q: What do they have in common? A1: All of these are governed by highly irregular PDEs... A2: All these are variational.

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ADL regime Nematic Liquid Crystals

Epitaxy models: with elastic forces on vicinal surfaces: the 1 + 1 dimensional case, with elastic forces on vicinal surfaces: the 2 + 1 dimensional case, with wetting, attachment-detachment regime, and many, many others... Nematic Liquid Crystals: Landau-De Gennes model. Q: What do they have in common? A1: All of these are governed by highly irregular PDEs... A2: All these are variational.

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ADL regime Nematic Liquid Crystals

Epitaxy models: with elastic forces on vicinal surfaces: the 1 + 1 dimensional case, with elastic forces on vicinal surfaces: the 2 + 1 dimensional case, with wetting, attachment-detachment regime, and many, many others... Nematic Liquid Crystals: Landau-De Gennes model. Q: What do they have in common? A1: All of these are governed by highly irregular PDEs... A2: All these are variational.

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ADL regime Nematic Liquid Crystals

Gradient flows in non reflexive spaces

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ADL regime Nematic Liquid Crystals

Burton-Cabrera-Frank (BCF) type models ˙ xi = D ka2

  • µi+1 − µi

xi+1 − xi + D

k

− µi − µi−1 xi − xi−1 + D

k

  • , for 1 ≤ i ≤ N.

where D is the terrace diffusion constant, k is the hopping rate of an adatom to the upward or downward step, µ is the chemical potential

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ADL regime Nematic Liquid Crystals

Attachment-detachment-limited (ADL) regime the diffusion across the terraces is fast, i.e. D

k ≫ xi+1 − xi, so the

dominated processes are the exchange of atoms at steps edges, i.e., attachment and detachment. The step-flow ODE in ADL regime becomes ˙ xi = 1 a2

  • µi+1 − 2µi + µi−1
  • , for 1 ≤ i ≤ N.

step slope as a new variable is a convenient way to derive the continuum PDE model (Al Hajj Shehadeh, Kohn and Weare, 2011)

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ADL regime Nematic Liquid Crystals

Evolution equation ut = −u2(u3)hhhh, u(0) = u0. If we take whh + c0 = 1/u: wt = (whh + c0)−3

hh ,

with proper, convex, lower semicontinuous energy φ(w) := 1 2 1 (whh + c0)−2dh. So far, so good... except???

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ADL regime Nematic Liquid Crystals

Evolution equation ut = −u2(u3)hhhh, u(0) = u0. If we take whh + c0 = 1/u: wt = (whh + c0)−3

hh ,

with proper, convex, lower semicontinuous energy φ(w) := 1 2 1 (whh + c0)−2dh. So far, so good... except???

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ADL regime Nematic Liquid Crystals

the “natural” functional space is W 2,1(0, 1)′′... not H2(0, 1) (or any W 2,p(0, 1) with p ≥ 1)... otherwise lack of coercivity means J + εξ, ξ ∈ ∂φ is not surjective... the “natural” convergence on whh is the weak-* convergence

  • f Radon measures...

Also... Subdifferential of φ(w) = 1 2 1 (whh + c0)−2dh... what does this even mean? φ does not charge very large whh...

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ADL regime Nematic Liquid Crystals

the “natural” functional space is W 2,1(0, 1)′′... not H2(0, 1) (or any W 2,p(0, 1) with p ≥ 1)... otherwise lack of coercivity means J + εξ, ξ ∈ ∂φ is not surjective... the “natural” convergence on whh is the weak-* convergence

  • f Radon measures...

Also... Subdifferential of φ(w) = 1 2 1 (whh + c0)−2dh... what does this even mean? φ does not charge very large whh...

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So φ(w) = 1 2 1 (whh + c0)−2dh is more like φ(w) = 1 2 1 (whh + c0)−2dh... And ∂φ(w) = −(whh + c0)−3 + singular measures

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ADL regime Nematic Liquid Crystals

Set E(w) := 1 2 1 [(whh + c0)−3]2

hhdh =

1 w2

t dh

and note: dE(w) dt = 1 [(whh + c0)−3]hh[(whh + c0)−3]hhtdh = 1 −3[(whh + c0)t]2 (whh + c0)4 dh ≤ 0, and d dt 1 (whh + c0)dh = 1 [(whh + c0)−3]hhhhdh = 0,

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And note there is bound on 1 [whh + c0]dh hence there is an invariant ball of the form {whBV ≤ C}... So consider the evolution equation wt ∈ −∂φ(w) − ∂ψ(w), ψ(w) := χ{whBV ≤C} and we can recover coercivity via ψ...

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ADL regime Nematic Liquid Crystals

And note there is bound on 1 [whh + c0]dh hence there is an invariant ball of the form {whBV ≤ C}... So consider the evolution equation wt ∈ −∂φ(w) − ∂ψ(w), ψ(w) := χ{whBV ≤C} and we can recover coercivity via ψ...

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Gao, Liu, L., Xu, 2018 Given T > 0, initial data w0 ∈ D(B), there exists a strong solution w of wt = (ηhh + c0)−3

hh ,

for a.e. (t, h) ∈ [0, T] × [0, 1]. Besides, we have ((ηhh + c0)−3)hh ∈ L∞([0, T]; L2(0, 1)) and the dissipation inequality E(t) := 1 2 1

  • ((ηhh + c0)−3)hh

2dh ≤ E(0), where ηhh is the absolutely continuous part of whh. However, whh might have singular parts... (Liu and Xu, Ji and Witelski)

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ADL regime Nematic Liquid Crystals

Similarly, the multidimensional model ut = ∆e−∆u can be treated with the same techniques: Gao, Liu, L., 2017 Given T > 0, initial data u0, there exists a strong solution w of ut = ∆e−∆u, for a.e. (t, h) ∈ [0, T] × Ω. Moreover, (∆e−∆u) ∈ L2(0, T; L2(Ω)). However, ∆u might have singular parts... (Ji and Witelski)

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Nematic Liquid Crystals

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Liquid crystals (LC): a state of the matter between crystalline and liquid...

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Different states of LC:

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Landau-De Gennes theory for nematic liquid crystals: the evolution is driven by the energy of the form E[Q] :=

  • F(∇Q(x), Q(x))dx − κQ2

L2(Ω), ,

κ > 0 Q varies in the Q-tensor space S(d) := {symmetric, trace free matrices of Rd×d}. Interesting case d = 3.

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ADL regime Nematic Liquid Crystals

Energy F = Fel + FBM, where Fel(∇Q) :=

d

  • i,j,k=1
  • L1|Qij

xk|2 + L2Qik xj Qij xk + L3Qij xjQik xk

  • , L1 ≫ L2, L3

FBM(Q) := inf

ρ∈AQ

  • S2 ρ(p) log ρ(p)dp

(Ball & Majumdar, 2009) AQ :=

  • ρ : S2 −

→ R : ρ ≥ 0,

  • S2 ρdx = 1,
  • S2 ρ(x)
  • x ⊗ x − id/3
  • dx = Q
  • .

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ADL regime Nematic Liquid Crystals

Energy F = Fel + FBM, where Fel(∇Q) :=

d

  • i,j,k=1
  • L1|Qij

xk|2 + L2Qik xj Qij xk + L3Qij xjQik xk

  • , L1 ≫ L2, L3

FBM(Q) := inf

ρ∈AQ

  • S2 ρ(p) log ρ(p)dp

(Ball & Majumdar, 2009) AQ :=

  • ρ : S2 −

→ R : ρ ≥ 0,

  • S2 ρdx = 1,
  • S2 ρ(x)
  • x ⊗ x − id/3
  • dx = Q
  • .

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About FBM(Q): well defined convex and isotropic log speed asymptote if any eigenvalue of Q approaches −1/3, 2/3 smooth in its effective domain

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Energy E[Q] =

  • Fel(∇Q) + FBM(Q)dx − κQ2

L2(Ω),

κ > 0 satisfies Boundedness from below: inf E > −∞ since convex functions are bounded from below, and Q2

L2(Ω) is also bounded due to

requirement that all eigenvalues of Q are in (−1/3, 2/3). Lower semicontinuity: consider a sequence Qn → Q strongly: we have then lim inf

n→+∞ E(Qn) ≥ E(Q).

λ-convexity along segments, with λ = −2κ: we have indeed E((1 − t)Q + tP) ≤ (1 − t)E(Q) + tE(P) + κt(1 − t)Q − P2

L2(Ω).

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Existence and regularity For any initial datum Q0 ∈ D(E) there exists a unique function Q such that: Regularizing effect: Q is locally Lipschitz regular, and Q(t) ∈ D(|∂E|) ⊆ D(E) for all t > 0. In particular, all eigenvalues of Q stay in (−1/3, 2/3) for a.e. x and all t > 0. Variational inequality: Q is the unique solution of the evolution variational inequality 1 2 d dt Q(t) − P2

L2(Ω) − κQ(t) − P2 L2(Ω) + E(Q(t)) ≤ E(P)

among all the locally absolutely continuous curves such that Q(t) → Q0 as t ↓ 0+.

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More properties: Exponential semigroup formula: Q(t) = limn→+∞ Jn

t/n(Q0),

with J denoting the resolvent X ∈ Jτ(Y ) ⇐ ⇒ X ∈ argmin

  • E(·) + 1

2τ Y − ·2

L2(Ω)

  • .

Contraction semigroup: for initial data Q0, P0 ∈ D(E), the corresponding solutions Q, P satisfy Q(t) − P(t)L2(Ω) ≤ e2κtQ0 − P0L2(Ω). This is not enough... Physicality fails if eigenvalues reach −1/3, 2/3 anywhere...

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More properties: Exponential semigroup formula: Q(t) = limn→+∞ Jn

t/n(Q0),

with J denoting the resolvent X ∈ Jτ(Y ) ⇐ ⇒ X ∈ argmin

  • E(·) + 1

2τ Y − ·2

L2(Ω)

  • .

Contraction semigroup: for initial data Q0, P0 ∈ D(E), the corresponding solutions Q, P satisfy Q(t) − P(t)L2(Ω) ≤ e2κtQ0 − P0L2(Ω). This is not enough... Physicality fails if eigenvalues reach −1/3, 2/3 anywhere...

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ADL regime Nematic Liquid Crystals

Main result (Liu, L. and Xu, 2018) There exists some time T0 > 0 such that all eigenvalues of Q(t) are uniformly bounded away from −1/3, 2/3 for all t > T0. AGS gives more or less −∆Q(t) + F ′

BM(Q(t)) + ξ(Q(t)) ∈ L2(0, T; L2(Ω)).

Issues: ∆Q(t) + F ′

BM(Q(t)) ∈ L2(Ω) does not give ∆Q(t) ∈ L2(Ω)...

(Not enough regularity) ξ(Q(t)) perturbation of Laplacian destroys any comparison/maximum principle... (So no way to follow L1 → L∞ arguments from [Constantin, Kiselev, Ryzhik, Zlatoˇ s, 2008])

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ADL regime Nematic Liquid Crystals

Main result (Liu, L. and Xu, 2018) There exists some time T0 > 0 such that all eigenvalues of Q(t) are uniformly bounded away from −1/3, 2/3 for all t > T0. AGS gives more or less −∆Q(t) + F ′

BM(Q(t)) + ξ(Q(t)) ∈ L2(0, T; L2(Ω)).

Issues: ∆Q(t) + F ′

BM(Q(t)) ∈ L2(Ω) does not give ∆Q(t) ∈ L2(Ω)...

(Not enough regularity) ξ(Q(t)) perturbation of Laplacian destroys any comparison/maximum principle... (So no way to follow L1 → L∞ arguments from [Constantin, Kiselev, Ryzhik, Zlatoˇ s, 2008])

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ADL regime Nematic Liquid Crystals

Main result (Liu, L. and Xu, 2018) There exists some time T0 > 0 such that all eigenvalues of Q(t) are uniformly bounded away from −1/3, 2/3 for all t > T0. AGS gives more or less −∆Q(t) + F ′

BM(Q(t)) + ξ(Q(t)) ∈ L2(0, T; L2(Ω)).

Issues: ∆Q(t) + F ′

BM(Q(t)) ∈ L2(Ω) does not give ∆Q(t) ∈ L2(Ω)...

(Not enough regularity) ξ(Q(t)) perturbation of Laplacian destroys any comparison/maximum principle... (So no way to follow L1 → L∞ arguments from [Constantin, Kiselev, Ryzhik, Zlatoˇ s, 2008])

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Main arguments:

1 Approximate FBM with Fn, and analyze the gradient flow of

En := Fel + Fn.

2 Use the Γ-convergence to infer convergence of gradient flows. 3 Achieve ∆Q(t) ∈ L2(0, T; L2(Ω)). 4 Upgrade to ∆Q(t) ∈ L∞(t0, T; L2(Ω)). 24 / 25

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Main arguments:

1 Approximate FBM with Fn, and analyze the gradient flow of

En := Fel + Fn.

2 Use the Γ-convergence to infer convergence of gradient flows. 3 Achieve ∆Q(t) ∈ L2(0, T; L2(Ω)). 4 Upgrade to ∆Q(t) ∈ L∞(t0, T; L2(Ω)). 24 / 25

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ADL regime Nematic Liquid Crystals

Main arguments:

1 Approximate FBM with Fn, and analyze the gradient flow of

En := Fel + Fn.

2 Use the Γ-convergence to infer convergence of gradient flows. 3 Achieve ∆Q(t) ∈ L2(0, T; L2(Ω)). 4 Upgrade to ∆Q(t) ∈ L∞(t0, T; L2(Ω)). 24 / 25

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Main arguments:

1 Approximate FBM with Fn, and analyze the gradient flow of

En := Fel + Fn.

2 Use the Γ-convergence to infer convergence of gradient flows. 3 Achieve ∆Q(t) ∈ L2(0, T; L2(Ω)). 4 Upgrade to ∆Q(t) ∈ L∞(t0, T; L2(Ω)). 24 / 25

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Thank you for your attention!

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