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

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 regime Nematic Liquid Crystals A nice piece of technology... Lots of silicon Liquid Crystal Display 2 / 25

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 . 3 / 25

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 . 3 / 25

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 . 3 / 25

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 . 3 / 25

ADL regime Nematic Liquid Crystals Gradient flows in non reflexive spaces 4 / 25

ADL regime Nematic Liquid Crystals Burton-Cabrera-Frank (BCF) type models � µ i +1 − µ i µ i − µ i − 1 � x i = D ˙ − , for 1 ≤ i ≤ N . ka 2 x i +1 − x i + D x i − x i − 1 + D k k 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 5 / 25

ADL regime Nematic Liquid Crystals Attachment-detachment-limited (ADL) regime the diffusion across the terraces is fast, i.e. D k ≫ x i +1 − x i , 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 x i = 1 � � ˙ µ i +1 − 2 µ i + µ i − 1 , for 1 ≤ i ≤ N . a 2 step slope as a new variable is a convenient way to derive the continuum PDE model (Al Hajj Shehadeh, Kohn and Weare, 2011) 6 / 25

ADL regime Nematic Liquid Crystals Evolution equation u t = − u 2 ( u 3 ) hhhh , u (0) = u 0 . If we take w hh + c 0 = 1 / u : w t = ( w hh + c 0 ) − 3 hh , with proper, convex, lower semicontinuous energy � 1 φ ( w ) := 1 ( w hh + c 0 ) − 2 dh . 2 0 So far, so good... except??? 7 / 25

ADL regime Nematic Liquid Crystals Evolution equation u t = − u 2 ( u 3 ) hhhh , u (0) = u 0 . If we take w hh + c 0 = 1 / u : w t = ( w hh + c 0 ) − 3 hh , with proper, convex, lower semicontinuous energy � 1 φ ( w ) := 1 ( w hh + c 0 ) − 2 dh . 2 0 So far, so good... except??? 7 / 25

ADL regime Nematic Liquid Crystals the “natural” functional space is W 2 , 1 (0 , 1) ′′ ... not H 2 (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 w hh is the weak-* convergence of Radon measures... Also... Subdifferential of � 1 φ ( w ) = 1 ( w hh + c 0 ) − 2 dh ... 2 0 what does this even mean? φ does not charge very large w hh ... 8 / 25

ADL regime Nematic Liquid Crystals the “natural” functional space is W 2 , 1 (0 , 1) ′′ ... not H 2 (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 w hh is the weak-* convergence of Radon measures... Also... Subdifferential of � 1 φ ( w ) = 1 ( w hh + c 0 ) − 2 dh ... 2 0 what does this even mean? φ does not charge very large w hh ... 8 / 25

ADL regime Nematic Liquid Crystals So � 1 φ ( w ) = 1 ( w hh + c 0 ) − 2 dh 2 0 is more like � 1 φ ( w ) = 1 ( w hh � + c 0 ) − 2 dh ... 2 0 And ∂φ ( w ) = − ( w hh � + c 0 ) − 3 + singular measures 9 / 25

ADL regime Nematic Liquid Crystals Set � 1 � 1 E ( w ) := 1 [( w hh + c 0 ) − 3 ] 2 w 2 hh dh = t dh 2 0 0 and note: � 1 dE ( w ) [( w hh + c 0 ) − 3 ] hh [( w hh + c 0 ) − 3 ] hht dh = dt 0 � 1 − 3[( w hh + c 0 ) t ] 2 = ( w hh + c 0 ) 4 dh ≤ 0 , 0 and � 1 � 1 d [( w hh + c 0 ) − 3 ] hhhh dh = 0 , ( w hh + c 0 ) dh = dt 0 0 10 / 25

ADL regime Nematic Liquid Crystals And note there is bound on � 1 [ w hh + c 0 ] dh 0 hence there is an invariant ball of the form {� w h � BV ≤ C } ... So consider the evolution equation w t ∈ − ∂φ ( w ) − ∂ψ ( w ) , ψ ( w ) := χ {� w h � BV ≤ C } and we can recover coercivity via ψ ... 11 / 25

ADL regime Nematic Liquid Crystals And note there is bound on � 1 [ w hh + c 0 ] dh 0 hence there is an invariant ball of the form {� w h � BV ≤ C } ... So consider the evolution equation w t ∈ − ∂φ ( w ) − ∂ψ ( w ) , ψ ( w ) := χ {� w h � BV ≤ C } and we can recover coercivity via ψ ... 11 / 25

ADL regime Nematic Liquid Crystals Gao, Liu, L., Xu, 2018 Given T > 0, initial data w 0 ∈ D ( B ), there exists a strong solution w of w t = ( η hh + c 0 ) − 3 hh , for a.e. ( t , h ) ∈ [0 , T ] × [0 , 1]. Besides, we have (( η hh + c 0 ) − 3 ) hh ∈ L ∞ ([0 , T ]; L 2 (0 , 1)) and the dissipation inequality � 1 E ( t ) := 1 � 2 dh ≤ E (0) , (( η hh + c 0 ) − 3 ) hh � 2 0 where η hh is the absolutely continuous part of w hh . However, w hh might have singular parts... (Liu and Xu, Ji and Witelski) 12 / 25

ADL regime Nematic Liquid Crystals Similarly, the multidimensional model u t = ∆ e − ∆ u can be treated with the same techniques: Gao, Liu, L., 2017 Given T > 0, initial data u 0 , there exists a strong solution w of u t = ∆ e − ∆ u , for a.e. ( t , h ) ∈ [0 , T ] × Ω. Moreover, (∆ e − ∆ u ) � ∈ L 2 (0 , T ; L 2 (Ω)) . However, ∆ u might have singular parts... (Ji and Witelski) 13 / 25

ADL regime Nematic Liquid Crystals Nematic Liquid Crystals 14 / 25

ADL regime Nematic Liquid Crystals Liquid crystals (LC): a state of the matter between crystalline and liquid... 15 / 25

ADL regime Nematic Liquid Crystals Different states of LC: 16 / 25

ADL regime Nematic Liquid Crystals Landau-De Gennes theory for nematic liquid crystals: the evolution is driven by the energy of the form � F ( ∇ Q ( x ) , Q ( x )) dx − κ � Q � 2 E [ Q ] := L 2 (Ω) , , κ > 0 Q varies in the Q -tensor space S ( d ) := { symmetric, trace free matrices of R d × d } . Interesting case d = 3. 17 / 25

ADL regime Nematic Liquid Crystals Energy F = F el + F BM , where d � x k | 2 + L 2 Q ik � � L 1 | Q ij x j Q ij x k + L 3 Q ij x j Q ik F el ( ∇ Q ) := , L 1 ≫ L 2 , L 3 x k i , j , k =1 � F BM ( Q ) := inf S 2 ρ ( p ) log ρ ( p ) dp (Ball & Majumdar, 2009) ρ ∈ A Q � � ρ : S 2 − A Q := → R : ρ ≥ 0 , S 2 ρ dx = 1 , � � � � S 2 ρ ( x ) x ⊗ x − id / 3 dx = Q . 18 / 25

ADL regime Nematic Liquid Crystals Energy F = F el + F BM , where d � x k | 2 + L 2 Q ik � � L 1 | Q ij x j Q ij x k + L 3 Q ij x j Q ik F el ( ∇ Q ) := , L 1 ≫ L 2 , L 3 x k i , j , k =1 � F BM ( Q ) := inf S 2 ρ ( p ) log ρ ( p ) dp (Ball & Majumdar, 2009) ρ ∈ A Q � � ρ : S 2 − A Q := → R : ρ ≥ 0 , S 2 ρ dx = 1 , � � � � S 2 ρ ( x ) x ⊗ x − id / 3 dx = Q . 18 / 25

ADL regime Nematic Liquid Crystals About F BM ( Q ): well defined convex and isotropic log speed asymptote if any eigenvalue of Q approaches − 1 / 3 , 2 / 3 smooth in its effective domain 19 / 25

ADL regime Nematic Liquid Crystals Energy � F el ( ∇ Q ) + F BM ( Q ) dx − κ � Q � 2 E [ Q ] = L 2 (Ω) , κ > 0 satisfies Boundedness from below: inf E > −∞ since convex functions are bounded from below, and � Q � 2 L 2 (Ω) is also bounded due to requirement that all eigenvalues of Q are in ( − 1 / 3 , 2 / 3). Lower semicontinuity: consider a sequence Q n → Q strongly: we have then lim inf n → + ∞ E ( Q n ) ≥ E ( Q ) . λ -convexity along segments, with λ = − 2 κ : we have indeed E ((1 − t ) Q + tP ) ≤ (1 − t ) E ( Q ) + tE ( P ) + κ t (1 − t ) � Q − P � 2 L 2 (Ω) . 20 / 25