Some topics about Nematic and Smectic-A Liquid Crystals

Transcription

1 Some topics about Nematic and Smectic-A Liquid Crystals Chillán, enero de 2010 Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 1/25

2 Table of contents 1 Introduction 2 The Models 3 Statement of the Problems 4 Nematic Case 5 Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 2/25

3 Table of contents 1 Introduction 2 The Models 3 Statement of the Problems 4 Nematic Case 5 Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 2/25

4 Table of contents 1 Introduction 2 The Models 3 Statement of the Problems 4 Nematic Case 5 Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 2/25

5 Table of contents 1 Introduction 2 The Models 3 Statement of the Problems 4 Nematic Case 5 Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 2/25

6 Table of contents 1 Introduction 2 The Models 3 Statement of the Problems 4 Nematic Case 5 Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 2/25

7 Table of contents 1 Introduction 2 The Models 3 Statement of the Problems 4 Nematic Case 5 Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 3/25

8 Introduction Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 4/25

9 Introduction Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 5/25

10 Introduction Natural examples: Soap, soup Biological membranes The protein solution to generate silk of a spider DNA and polypeptides can form LC phases Applications: Liquid Crystal Displays: wrist watches, pocket calculators, flat screens... Liquid Crystal Thermometers: to show a map of temperatures to find tumors, bad connections on a circuit board... Windows that can be changed from clear and opaque with the flip of a switch. To make a stable hydrocarbon foam. Optical Imaging and recording. Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 6/25

11 Introduction Natural examples: Soap, soup Biological membranes The protein solution to generate silk of a spider DNA and polypeptides can form LC phases Applications: Liquid Crystal Displays: wrist watches, pocket calculators, flat screens... Liquid Crystal Thermometers: to show a map of temperatures to find tumors, bad connections on a circuit board... Windows that can be changed from clear and opaque with the flip of a switch. To make a stable hydrocarbon foam. Optical Imaging and recording. Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 6/25

12 Introduction Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 7/25

13 Introduction Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 8/25

14 Table of contents 1 Introduction 2 The Models 3 Statement of the Problems 4 Nematic Case 5 Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 9/25

15 The models d : Orientation of liquid crystal molecules (unit vector). n : Single optical axis perpendicular to the layer. d = 1 f Ginzburg-Landau penalization function f(d) = 1 ε 2 ( d 2 1)d n = 0 n = ϕ ϕ : Layer variable d = n ϕ = 1 Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 10/25

16 The models Penalized Oseen Frank energy: (NC) E e = Ω ( 1 2 d 2 + F(ϕ)) (SAC) E e = Ω ( 1 2 ϕ 2 + F( ϕ)) where f(n) = n F(n). F(n) = 1 4ε 2 ( n 2 1) 2 potential function of f(n) = 1 ε 2 ( n 2 1)n. Minimization problem Euler-Lagrange equation (NC) ω d f(d) = 0, (SAC) ω 2 ϕ f( ϕ) = 0, Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 11/25

17 The models Penalized Oseen Frank energy: (NC) E e = Ω ( 1 2 d 2 + F(ϕ)) (SAC) E e = Ω ( 1 2 ϕ 2 + F( ϕ)) where f(n) = n F(n). F(n) = 1 4ε 2 ( n 2 1) 2 potential function of f(n) = 1 ε 2 ( n 2 1)n. Minimization problem Euler-Lagrange equation (NC) ω d f(d) = 0, (SAC) ω 2 ϕ f( ϕ) = 0, Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 11/25

18 The models Ω IR N (N = 2 or 3), Ω regular, Q = Ω (0, + ) ( Ericksen-Leslie, Lin, E) Angular momentum (NC) t d + u d + γω = 0 (SAC) t ϕ + u ϕ + γω = 0 Linear momentum ρ( t u + (u )u) (σ d + λ σ e ) + p = 0, u = 0 where Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 12/25

19 The models Ω IR N (N = 2 or 3), Ω regular, Q = Ω (0, + ) ( Ericksen-Leslie, Lin, E) Angular momentum (NC) t d + u d + γω = 0 (SAC) t ϕ + u ϕ + γω = 0 Linear momentum ρ( t u + (u )u) (σ d + λ σ e ) + p = 0, u = 0 where Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 12/25

20 The models (NC) (SAC) σ d = µ 4 D(u), σ e = λ ( d d) σ d = µ 1 (n t D(u)n)n n + µ 4 D(u) + µ 5 (D(u)n n + n D(u)n), σ e = f(n) n + ( n) n ( n) n Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 13/25

21 Table of contents 1 Introduction 2 The Models 3 Statement of the Problems 4 Nematic Case 5 Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 14/25

22 Nematic Model The equations t u + (u )u ν u + p = d t d, u = 0, t d + (u )d = ( d f (d)), d 1, in Q + time-dependent (bc) on Σ = (0, ) Ω. + (iv) or (tp) in Ω. Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 15/25

23 Smectic Model The equations t u + (u )u ν u σnl d ( 2 ϕ f( ϕ)) ϕ + q = 0, u = 0, t ϕ + u ϕ + ( 2 ϕ f( ϕ)) = 0, in Q where σ d nl := (n t D(u)n)n n + D(u)n n + n D(u)n + time-dependent (bc) on Σ = (0, ) Ω. + (iv) or (tp) in Ω. Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 16/25

24 Lifting functions Boundary condition depending on the time (NC) d = d(t) stationary (for weak norms) or non-stationary (for regular norms) lifting function d(t) = d(t) d(t), d = 0 on Ω Unknows: u, p, d (SAC) non stationary lifting function ϕ = ϕ(t) ϕ(t) = ϕ(t) ϕ(t), Unknows: ϕ = 0 on Ω u, p, ϕ Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 17/25

25 Table of contents 1 Introduction 2 The Models 3 Statement of the Problems 4 Nematic Case 5 Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 18/25

26 Nematic Case Weak solution: u L (0, + ; L 2 ) L 2 w(0, + ; H 1 ), d L (0, + ; H 1 ) L 2 w(0, + ; H 2 ) Regular solution: u L (0, + ; H 1 ) L 2 w(0, + ; H 2 ), d L (0, + ; H 2 ) L 2 w(0, + ; H 3 ) Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 19/25

27 Nematic Case Asymptotic Stability: E(t) = 1 2 u(t) 2 + E e (t) E, u(t) 0 in H 1 0(Ω), ω(t) = ( ϕ f (ϕ))(t) 0 in L 2 (Ω) when t +, where E = E e,d = 1 2 d 2 + F(d) and d is a critical point of E e. Moreover, d d for subsequences in H 2 (Ω)-weak. Ω Stability: If initial data are small, u, ω, and E(t) are small for each t 0 Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 20/25

28 Nematic Case Previous result: (iv)-problem, boundary condition independent of time. Existence of a weak global solution. Existence and uniqueness of a regular solution for larger viscosity. [Lin,Liu 95] Goal Time-dependent (bc) case [Climent,Guillén,Rojas 06] (tp)-problem. Existence of a weak periodic solution. Regularity N = 2 [Climent,Guillén,Moreno 08] (iv)-problem. Existence of a weak global solution. Existence of a global strong solution, ν big enough. Uniqueness of strong/weak solutions. (tp)-problem. Existence of a regular solution, ν big enough. [Climent,Guillén,Rodríguez] Stability and asymptotic stability (tp)-problem, time-independent (bc) case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 21/25

29 Nematic Case Previous result: (iv)-problem, boundary condition independent of time. Existence of a weak global solution. Existence and uniqueness of a regular solution for larger viscosity. [Lin,Liu 95] Goal Time-dependent (bc) case [Climent,Guillén,Rojas 06] (tp)-problem. Existence of a weak periodic solution. Regularity N = 2 [Climent,Guillén,Moreno 08] (iv)-problem. Existence of a weak global solution. Existence of a global strong solution, ν big enough. Uniqueness of strong/weak solutions. (tp)-problem. Existence of a regular solution, ν big enough. [Climent,Guillén,Rodríguez] Stability and asymptotic stability (tp)-problem, time-independent (bc) case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 21/25

30 Table of contents 1 Introduction 2 The Models 3 Statement of the Problems 4 Nematic Case 5 Smectic-A Case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 22/25

31 Smectic-A Case Weak solution: u L (0, + ; H 1 ) L 2 w(0, + ; H 2 ), ϕ L (0, + ; H 4 ) L 2 w(0, + ; H 6 ) Regular solution: u L (0, + ; H 1 ) L 2 w(0, + ; H 2 ), ϕ L (0, + ; H 2 ) L 2 w(0, + ; H 3 ) Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 23/25

32 Smectic-A Case Asymptotic Stability: u(t) 0 in H 1 0 (Ω) ( 2 ϕ f ( ϕ))(t) 0 in L 2 (Ω) when t +. Moreover, ϕ ϕ for sequences in H 4 (Ω)-weak, where ϕ is a solution of a Euler-Lagrange problem. Stability: If initial data are small, u, ϕ, and ω are small for each t 0 Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 24/25

33 Smectic-A Case Previous result: (iv)-problem, time-independent boundary conditions. Existence of weak solutions in [0, T ], global regularity of weak solutions (for big enough viscosity). [Liu 00] Goal Time-dependent (bc) case [Climent,Guillén] 1 Uniqueness weak/strong solutions (iv)-problem, 2 Existence of global weak solutions (iv)-problem, bounded up to infinity time, 3 Existence of weak time-periodic solutions, 4 Existence of regular solutions for both previous cases (dominant viscosity coefficient). 5 Stability and asymptotic stability Time-independent (bc) case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 25/25

34 Smectic-A Case Previous result: (iv)-problem, time-independent boundary conditions. Existence of weak solutions in [0, T ], global regularity of weak solutions (for big enough viscosity). [Liu 00] Goal Time-dependent (bc) case [Climent,Guillén] 1 Uniqueness weak/strong solutions (iv)-problem, 2 Existence of global weak solutions (iv)-problem, bounded up to infinity time, 3 Existence of weak time-periodic solutions, 4 Existence of regular solutions for both previous cases (dominant viscosity coefficient). 5 Stability and asymptotic stability Time-independent (bc) case Blanca Climent Ezquerra. Universidad de Sevilla. III WIMA Nematics and Smectic-A Liquid Crystals 25/25