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The Lie Theoretical Structure of Liquid Crystal Dynamics

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The Lie Theoretical Structure of Liquid Crystal Dynamics THE LIE THEORETICAL STRUCTURE OF LIQUID CRYSTAL DYNAMICS Tudor S. Ratiu Section de Mathématiques and Bernoulli Center Ecole Polytechnique Fédérale de Lausanne, Switzerland François Gay-Balmaz and Cesare Tronci Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 1 PLAN OF THE PRESENTATION Liquid crystal dynamics: two models Affine Euler-Poincaré reduction Geometry of non-visocous Ericksen-Leslie nematodynamics Nonviscous Eringen micropolar model Eringen implies Ericksen-Leslie In the variational formulation, the terms modeling dissipation have been eliminated. Reason: want to understand the geometric nature of these equations. The dissipative terms are added later. Think: Euler versus Navier-Stokes. Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 2 LIQUID CRYSTALS Tutorial: http://dept.kent.edu/spie/liquidcrystals/ by B. Senyuk Liquid crystal: state of matter between crystal and isotropic liq- uid. Liquid behavior: high fluidity, formation and coalescence of droplets. Crystal behavior: anisotropy in optical, mechanical, elec- trical, magnetic properties. Long-range orientational order in their molecules and sometimes translational or positional order. Many phases; differ by their structure and physical properties. History: Friedrich Reinitzer (1857 Prague – 1927 Graz) botanist and chemist. In 1888 he accidentaly discovered a strange behavior of cholesteryl benzoate that would later be called “liquid crystal". Work continued by physicist Otto Lehmann (1855 Konstanz – 1922 Karlsruhe), the “father" of liquid crystal technology. The discovery received plenty of attention at the time but due to no practical use the interest dropped soon. Lehmann was nominated 10 times (1913-22) for the Nobel Prize; never got it. Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 3 Georges Friedel (1865 Mulhouse – 1933 Strasbourg) mineralogist and crystallographer; in 1922 first classification of liquid crystals. Carl Oseen (1879 Lund – 1944 Uppsala) theoretical physicist; first formulation of elasticity theory of liquid crystals. Since the mid 1960s the entire theoretical and experimental devel- opment in liquid crystals has been influenced by the physicist Pierre- Gilles de Gennes (1932 Paris – 2007 Orsay) who got the Nobel Prize in 1991. Unfortunately, his book is not very useful to mathemati- cians. Better books are by Subrahmanyan Chandrasekhar and es- pecially Epifanio Virga, the most mathematical book I came across. Math. form., engineers: Ericksen-Leslie, Lhuillier-Rey, Eringen. Main phases of liquid crystals are the nematic, smectic, cholesteric (chiral nematic); molecules behave differently. nematic ∼ “nema" (thread); smectic ∼ “smektos"(smeared), or “smechein" (to wash out) + “tikos" (suffix for adjectives of Greek origin); cholesteric ∼ “khole"(bile) + “steros"(solid, stiff) + “tikos"; chiral ∼ “cheir"(hand), introduced by Kelvin. 4 From: DoITPoMS, University of Cambridge From: http://boomeria.org/chemtextbook/cch16.html 5 Photos by Brian Johnstone. Cooling from liquid crystal state to solid crystal state. The circular discs are the solid crystals. Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 6 Nematic polymer; from: DoITPoMS, University of Cambridge Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 7 From B. Senyuk at http://dept.kent.edu/spie/liquidcrystals/ Cholesteric liquid crystal Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 8 M. Mitov/CNRS Photothèque. Cholesteric liquid crystal CNRS-CEMES 2006: Iridescent colors of this rose chafer beetle are due to the organization in cholesteric liquid crystal phase of the chitin molecules of the outer part of the shell 9 LIQUID CRYSTAL DYNAMICS Director theory due to Oseen, Frank, Zöcher, Ericksen and Leslie Micropolar, microstretch, and micropmorphic theories, due to Eringen, which take into account the microinertia of the particles and which is applicable, for example, to liquid crystal polymers Ordered micropolar approach, due to Lhuillier and Rey, which combines the director theory with the micropolar models. We discuss only nematic liquid crystals (no chirality n · curl n). We set all dissipation equal to zero; want to understand the conservative case first. 3 D ⊂ R bounded domain with smooth boundary. All boundary con- ditions are ignored: in all integration by parts the boundary terms vanish. We fix a volume form µ on D. Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 10 ERICKSEN-LESLIE DIRECTOR THEORY For nematic and cholesteric liquid crystals Key assumption: only the direction and not the sense of the molecules matter. The preferred orientation of the molecules around a point is described by a unit vector n : D ! S2, called the director, and n and −n are assumed to be equivalent. Ericksen-Leslie equations (Ericksen [1966], Leslie [1968]) in a domain D, constraint knk = 1, are: 8 @ @F @F ! > ρ u + r u = grad − @ ρ ·rn ; > u −1 j > @t @ρ @n;j > <> D2 @F @F ! ρJ n − 2qn + h = 0; h = ρ − @i ρ > Dt2 @n @n > ;i > > @ D @ @ > ρ + div(ρu) = 0; := + ru = + u · r : @t Dt @t @t 3 u Eulerian velocity, ρ mass density, n : D ! R director (n equiva- lent to -n), J microinertia constant, and F (n; n;i) is the free energy: Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 11 A standard choice for F is the Oseen-Zöcher-Frank free energy: −1 1 2 1 2 ρF (ρ ; n; rn) = K1 (div n) + K2 (n · curl n) 2 | {z } 2 | {z } splay twist 1 2 + K3 kn × curl nk ; 2 | {z } bend associated to the basic type of director distorsions nematics: (a) splay, (b) bend, (c) twist WHAT IS THE VARIATIONAL/HAMILTONIAN STRUCTURE OF THESE EQUATIONS? Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 12 ERINGEN MICROPOLAR THEORY First key assumption: Replace point particles by small deformable bodies: microfluids. Examples: liquid crystals, blood, polymer melts, bubbly fluids, suspensions with deformable particles, biolog- ical fluids. Eringen [1978], [1979], [1981],... A material particle P in a microfluid is characterized by its position X and by a vector Ξ attached to P that denotes the orientation and intrinsic deformation of P . Both X and Ξ have their own motions: X 7! x = η(X; t) and Ξ 7! ξ = χ(X; Ξ; t) called, respectively, the macromotion and micromotion. Second key assumption: Material bodies are very small, so a linear approximation in Ξ is permissible for the micromotion: ξ = χ(X; t)Ξ; where χ(X; t) 2 GL(3)+ := fA 2 GL(3) j det(A) > 0g. Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 13 The classical Eringen theory considers only three possible groups in the description of the micromotion of the particles: GL(3)+(micromorphic) ⊃ K(3)(microstretch) ⊃ SO(3)(micropolar); n + T o K(3) = A 2 GL(3) j there exists λ 2 R such that AA = λI3 is a closed subgroup of GL(3)+; associated to rotations and stretch. The general theory admits other groups describing the micromotion. We will study only micropolar fluids, i.e., the order parameter group is O := SO(3) . Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 14 Eringen’s equations for non-dissipative micropolar liquid crystals: 8 D @Ψ @Ψ ! @Ψ ! @Ψ > ρ = @ − @ ρ a ; ρσ = @ ρ − " ρ a; > ul l 1 k aγl l k l lmn γn > Dt @ρ− @γ @γ @γa > k k m <> D D ρ + ρ div u = 0; jkl + ("kprjlp + "lprjkp)νr = 0; > Dt Dt > > D a b a D @ > γ = @ νa + ν γ − γ @ ur; := + u · r mat. deriv: :> Dt l l ab l r l Dt @t sum on repeated indices, u 2 X(D) Eulerian velocity, ρ 2 F(D) mass 3 density, ν 2 F(D; R ) microrotation rate, where we use the standard 3 isomorphism between so(3) and R , jkl 2 F(D; Sym(3)) microinertia tensor (symmetric), σk spin inertia is defined by D D σ := j ν + " jmnν νn = (j ν ); k klDt l klm l Dt kl l ab 1 γ = (γi ) 2 Ω (D; so(3)) wryness tensor, related to (η; χ) by −1 a γ = −η∗(rχ)χ =: γb = (dγi ); −1 and Ψ = Ψ(ρ ; j; γ): R × Sym(3) × gl(3) ! R is the free energy. Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 15 VARIATIONAL/HAMILTONIAN STRUCTURE? Result: Well posedness of the full EL system with viscosity RELATION BETWEEN ERICKSEN-LESLIE AND ERINGEN? Eringen’s claim: Eringen theory recovers Ericksen-Leslie theory in the rod-like assumption j = J(I−n⊗n) with the choice γ = rn×n. Once we have the Euler-Poincaré formulation, it will be clear that γ = rn × n cannot be considered as a definition! This statement has been controversial due to mistakes: Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 16 e.g. paper by Rymarz [1990] Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 17 This was soon reconsidered in Eringen [1993] more precisely Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 18 Since then, this remains an open problem. We solve this problem using techniques of geometric mechanics: (1) find under what assumptions, Eringen reduces to Ericksen-Leslie (2) find correct relation between γ and n with these assumptions ; Both Eringen and Rymarz are partially right! Expect to use equivalent formulations of viscous Eringen for well posedness. Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 19 EULER-POINCARÉ REDUCTION Poincaré 1901: Left (right) invariant Lagrangian L : TG ! R, −1 −1 l := Ljg : g ! R. For g(t) 2 G, let ξ(t) = g(t) g_(t) g_(t)g(t) 2 g . Then the following are equivalent: (i) g(t) satisfies the Euler-Lagrange equations for L on G. (ii) The variational principle Z b δ L(g(t); g_(t))dt = 0 a holds, for variations with fixed endpoints.
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