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: ∂ ∂F ∂F ! ρ u + ∇ u = grad − ∂ ρ ·∇n , 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, := + ∇u = + u · ∇ ∂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, ∇n) = 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) ∈ GL(3)+ := {A ∈ GL(3) | det(A) > 0}.
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 ∈ GL(3) | there exists λ ∈ 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: D ∂Ψ ∂Ψ ! ∂Ψ ! ∂Ψ ρ = ∂ − ∂ ρ a , ρσ = ∂ ρ − ε 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 + (εkprjlp + εlprjkp)νr = 0, Dt Dt D a b a D ∂ γ = ∂ νa + ν γ − γ ∂ ur, := + u · ∇ mat. deriv. Dt l l ab l r l Dt ∂t sum on repeated indices, u ∈ X(D) Eulerian velocity, ρ ∈ F(D) mass 3 density, ν ∈ F(D, R ) microrotation rate, where we use the standard 3 isomorphism between so(3) and R , jkl ∈ F(D, Sym(3)) microinertia tensor (symmetric), σk spin inertia is defined by D D σ := j ν + ε jmnν νn = (j ν ), k klDt l klm l Dt kl l ab 1 γ = (γi ) ∈ Ω (D, so(3)) wryness tensor, related to (η, χ) by −1 a γ = −η∗(∇χ)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 = ∇n×n.
Once we have the Euler-Poincaré formulation, it will be clear that γ = ∇n × 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 := L|g : g → R. For g(t) ∈ G, let ξ(t) = g(t) g˙(t) g˙(t)g(t) ∈ 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. d δl δl (iii) The Euler-Poincaré equations hold: = ±ad∗ . dtδξ ξ δξ (iv) The Euler-Poincaré variational principle Z b δ l(ξ(t))dt = 0 a holds on g, for variations δξ =η ˙ ± [ξ, η], where η(t) is an arbitrary path in g that vanishes at the endpoints, i.e η(a) = η(b) = 0.
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 20 Geometry has led to analytic questions. I am not aware of any serious analysis results for such constrained variational principles. Reconstruction
Solve the Euler-Lagrange equations for a left invariant L : TG → R
• Form l := L|g : g → R d δl ∗ δl • Solve the Euler-Poincaré equations: dt δξ = adξ δξ , ξ(0) = ξ0 • Solve linear equation with time dependent coefficients (quadra- ture): g˙(t) = g(t)ξ(t), g(0) = e
• For any g0 ∈ G the solution of the Euler-Lagrange equations is V (t) = g0g(t)ξ(t) with initial condition V (0) = g0ξ0.
EP reduction: free rigid body, ideal fluids, KdV EP reduction for semidirect products: heavy rigid body, compress- ible fluids, MHD, GFD. Holm, Marsden, Ratiu [1998] Geometry of complex fluids.Holm [2002], Gay-Balmaz, Ratiu [2009]
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 21 AFFINE EULER-POINCARÉ REDUCTION
Right G-representation on V , (v, g) ∈ V × G 7→ vg ∈ V , induces:
• right G-representation on V ∗: (a, g) ∈ V ∗ × G 7→ ag ∈ V ∗
• right g-representation on V : (v, ξ) ∈ V × g 7→ vξ ∈ V
• right g-representation on V ∗: (a, ξ) ∈ V ∗ × g 7→ aξ ∈ V ∗
∗ ∗ Duality pairings: h , ig : g × g → R and h , iV : V × V → R
∗ Affine right representation: θg(a) = ag + c(g), where c ∈ F(G, V ) is a right group one-cocycle, i.e., c(fg) = c(f)g + c(g), ∀f, g ∈ G. This implies that c(e) = 0 and c(g−1) = −c(g)g−1. Note that
d ∗ θexp(tξ)(a) = aξ + dc(ξ), ξ ∈ g, a ∈ V , dt t=0 ∗ where dc : g → V is defined by dc(ξ) := Tec(ξ). Useful to introduce:
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 22 T ∗ T • dc : V → g by hdc (v), ξig := hdc(ξ), viV , for ξ ∈ g, v ∈ V
∗ ∗ ∗ • : V × V → g by hv a, ξig := − haξ, viV for ξ ∈ g, v ∈ V , a ∈ V
T • then: haξ + dc(ξ), viV = hdc (v) − v a, ξig
• the semidirect product S = G s V with group multiplication
(g1, v1)(g2, v2) := (g1g2, v2 + v1g2), gi ∈ G, vi ∈ V • its Lie algebra s = g s V with bracket ad (ξ , v ) := [(ξ , v ), (ξ , v )] = ([ξ , ξ ], v ξ − v ξ ) (ξ1,v1) 2 2 1 1 2 2 1 2 1 2 2 1
• then for (ξ, v) ∈ s and (µ, a) ∈ s∗ = g∗ × V ∗ we have ∗ ∗ ad(ξ,v)(µ, a) = (adξ µ + v a, aξ)
In a physical problem (like liquid crystals) we are given:
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 23 ∗ • L : TG × V → R right G-invariant under the action ∗ g ∗ (vh, a) ∈ ThG × V 7−→ (vhg, θg(a)) = (vhg, ag + c(g)) ∈ ThgG × V .
∗ • So, if a0 ∈ V , define La0 : TG → R by La0(vg) := L(vg, a0). Then is right invariant under the lift to of right translation of c La0 TG Ga0 on , where c is the -isotropy group of . G Ga0 θ a0
∗ • Right G-invariance of L permits us to define l : g × V → R by −1 l(vgg , θg−1(a0)) = L(vg, a0). −1 ∗ • Curve g(t) ∈ G, let ξ(t) :=g ˙(t)g(t) ∈ g, a(t) = θg(t)−1(a0) ∈ V Then a(t) as the unique solution of the following affine differential equation with time dependent coefficients
a˙(t) = −a(t)ξ(t) − dc(ξ(t)), ∗ with initial condition a(0) = a0 ∈ V .
The following are equivalent: Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 24 (i) With a0 held fixed, Hamilton’s variational principle
Z t2 δ La0(g(t), g˙(t))dt = 0, t1 holds, for variations δg(t) of g(t) vanishing at the endpoints.
(ii) g(t) satisfies the Euler-Lagrange equations for La0 on G.
(iii) The constrained variational principle
Z t2 δ l(ξ(t), a(t))dt = 0, t1 holds on g × V ∗, upon using variations of the form ∂η δξ = − [ξ, η], δa = −aη − dc(η), ∂t where η(t) ∈ g vanishes at the endpoints.
(iv) The affine Euler-Poincaré equations hold on g × V ∗: ∂ δl δl δl δl = − ad∗ + a − dcT . ∂tδξ ξ δξ δa δa
25 Lagrangian Approach to Continuum Theories of Perfect Complex Fluids
To apply the previous theorem to complex fluids one makes two key observations: 1. Complex fluids have internal degrees of freedom encoded by the order parameter Lie group O D a b a 2. New kind of advection equation: γ = ∂ νa + ν γ − γ ∂ ur Dt l l ab l r l
Geometrically, this means:
1. Enlarge the “particle relabeling group" Diff(D) to the semidirect product G = Diff(D) s F(D, O), F(D, O) := {χ : D → O smooth} 2. The usual advection equations (for the mass density, the entropy, the magnetic field, etc) need to be augmented by a new advected quantity on which the group G acts by an affine representation.
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 26 Algebraic structure of the symmetry group of complex fluids:
Diff(D) acts on F(D, O) via the right action
(η, χ) ∈ Diff(D) × F(D, O) 7→ χ ◦ η ∈ F(D, O). Therefore, the group multiplication is given by
(η, χ)(ϕ, ψ) = (η ◦ ϕ, (χ ◦ ϕ)ψ). Fix a volume form µ on D, so identify densities with functions, one-form densitities with one-forms, etc.
The Lie algebra g of the semidirect product group is
g = X(D) s F(D, o) 3 (u, ν), and the Lie bracket is computed to be
ad(u,ν)(v, ζ) = (adu v, adν ζ + dν · v − dζ · u), where adu v = −[u, v], adν ζ ∈ F(D, o) is given by adν ζ(x) := adν(x) ζ(x), and dν · v ∈ F(D, o) is given by dν · v(x) := dν(x)(v(x)). Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 27 The dual Lie algebra is identified with ∗ 1 ∗ g = Ω (D) s F(D, o ) 3 (m, κ), through the pairing Z h(m, κ), (u, ν)i = (m · u + κ · ν) µ. D The dual map to ad(u,ν) is ∗ ∗ ad(u,ν)(m, κ) = £um + (div u)m + κ · dν, adν κ + div(uκ) .
Explanation of the symbols:
• κ · dν ∈ Ω1(D) denotes the one-form defined by κ · dν (vx) := κ(x)(dν(vx)) ∗ ∗ ∗ • adν κ ∈ F(D, o ) denotes the o -valued mapping defined by ∗ ∗ adν κ (x) := adν(x)(κ(x)). • uκ is the 1-contravariant tensor field with values in o∗ defined by ∗ (uκ)(αx) := αx(u(x))κ(x) ∈ o . Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 28 So uκ is a generalization of the notion of a vector field. X(D, o∗) denotes the space of all o∗-valued 1-contravariant tensor fields.
• div(u) denotes the divergence of the vector field u with respect to the fixed volume form µ. Recall that it is defined by the condition
(div u)µ = £uµ. This operator can be naturally extended to the space X(D, o∗) as ∗ a a follows. For w ∈ X(D, o ) we write w = waε where (ε ) is a basis ∗ ∗ ∗ of o and wa ∈ X(D). We define div : X(D, o ) → F(D, o ) by a div w := (div wa)ε . Note that if w = uκ we have
div(uκ) = dκ · u + (div u)κ.
Split the space of advected quantities in two: usual ones and new ones that involve affine actions and cocycles.
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 29 GEOMETRY OF THE ERICKSEN-LESLIE EQUATIONS
• Symmetry group: G = Diff(D) s F(D,SO(3)) 3 (η, χ), macromo- tion and micromotion.
∗ 3 • Advected variables: V = F(D) × F(D, R ) 3 (ρ, n), mass density and director field.
• Representation of G on V ∗: (ρ, n) 7→ J(η)(ρ ◦ η), χ−1(n ◦ η) . • Associated infinitesimal actions and diamond operations: T nu = ∇n·u, nν = n×ν, m1n = −∇n ·m and m2n = n×m, 3 where ν, m, n ∈ F(D, R ).
• No cocycle.
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 30 3 • EP equations for Diff(D) s F(D, SO(3)) s F(D) × F(D, R ) : [ ∂ δ` δ` δ` δl δ` T δ` = −£u − div u − ·dν + ρ d − ∇n · , ∂tδu δu δu δν δρ δn ∂ δ` δ` δ` δ` = ν × − div u + n × , ∂tδν δν δν δn • The advection equations are: ∂ ρ + div(ρu) = 0, ∂t ∂ n + ∇n·u + n × ν = 0. ∂t • Reduced Lagrangian for nematic and cholesteric liquid crystals: 1 Z 1 Z Z `(u, ν, ρ, n) := ρkuk2µ + ρJkνk2µ − ρF (ρ−1, n, ∇n)µ. 2 D 2 D D
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 31 • EP equations for this `: yield ! ∂ ∂F ∂F ρ u + ∇uu = grad − ∂i ρ ·∇n , ∂t ∂ρ−1 ∂n (motion) ,i D ρJ ν = h × n, Dt ∂ ρ + div(ρu) = 0, ∂t (advection) D n = ν × n, Dt
• Recovering the Ericksen-Leslie equations: Observation: if ν and n are solutions of the EP equations then:
(i) kn0k = 1 implies knk = 1 for all time.
D (ii) Dt(n·ν) = 0. Therefore, n0·ν0 = 0 implies n·ν = 0 for all time.
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 32 (iii) Suppose that n0·ν0 = 0 and kn0k = 1. Then D D n = ν × n becomes ν = n × n Dt Dt and D D2 ρJ ν = h × n becomes ρJ n − 2qn + h = 0. Dt Dt2 Therefore: If (u, ν, ρ, n) is a solution of the Euler-Poincaré equations with initial conditions n0 and ν0 satisfying kn0k = 1 and n0 · ν0 = 0, then (u, ρ, n) is a solution of the Ericksen-Leslie equations. Conversely: if (u, ρ, n) is a solution of the Ericksen-Leslie equations, define
D 3 ν := n × n ∈ F(D, R ). Dt Then, (u, ν, ρ, n) is a solution of the Euler-Poincaré equations.
Use these equations plus add dissipation, get well posedness.
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 33 GEOMETRY OF THE ERINGEN EQUATIONS
• Symmetry group: same group as before G = Diff(D) s F(D, O).
• Advected variables: V ∗ = F(D) × F(D, Sym(3)) × Ω1(D, so(3)) 3 (ρ, j, γ), mass density, microinertia tensor, strain.
• Representation: (η, χ) ∈ Diff(D) s F(D, SO(3)) acts linearly on the advected quantities (ρ, j) ∈ F(D) × F(D, Sym(3)), by (ρ, j) 7→ J(η)(ρ ◦ η), χT(j ◦ η)χ , χT = χ−1.
• Affine representation: (η, χ) ∈ Diff(D) s F(D, SO(3)) acts on γ ∈ Ω1(D, so(3)) by an affine representation
γ 7→ χ−1(η∗γ)χ + χ−1∇χ. Note that γ transforms as a connection.
So, Eringen’s wryness tensor is a connection one-form.
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 34 • The reduced Lagrangian of Eringen’s theory: h 3 i h 1 i ` : X(D) s F(D, R ) s F(D) ⊕ F(D, Sym(3)) ⊕ Ω (D, so(3)) → R 1 Z 1 Z Z `(u, ν, ρ, j, γ) = ρkuk2µ + ρ (jν ·ν) µ − ρΨ(ρ−1, j, γ)µ. 2 D 2 D D • The affine Euler-Poincaré equations for ` are: ! ∂ ∂Ψ ∂Ψ ρ u + ∇ u = grad − ∂ ρ γa , u −1 k a ∂t ∂ρ ∂γ k ! D 1 ∂Ψ a ∂Ψ j ν − (jν) × ν = − div ρ + γ × , Dt ρ ∂γ ∂γa ∂ D ρ + div(ρu) = 0, j + [j, ν] = 0, ∂t Dt ∂ γ γ + £uγ + d ν = 0, νˆ = ν ∈ F(D, so(3)), ∂t where dγ is the covariant γ-derivative defined by dγν(v) := dν(v) + [γ(v), ν].
This system is identical Eringen’s equations. Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 35 The general affine Euler-Poincaré theory applied to many other complex fluids: spin chain, Yang-Mills MHD (classical and super- fluid), Hall MHD, multivelocity superfluids (classical and super- fluid), HBVK dynamics for superfluid 4He, Volovik-Dotsenko spin glasses, microfluids, Lhuillier-Rey equations (see Gay-Balmaz & Ratiu [2009]).
Kelvin-Noether circulation theorem for micropolar liquid crystals d I I ∂Ψ ∂Ψ 1 ∂Ψ! u[ = dj + i dγ − div ρ γ. dt Ct Ct ∂j ∂γ ρ ∂γ 3 The γ-circulation formulated in R d I I γ = ν × γ dt Ct Ct
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 36 ERINGEN IMPLIES ERICKSEN-LESLIE
Physically, the Eringen equations should imply the Ericksen-Leslie equations. Eringen [1993] proposes
j := J(I3 − n ⊗ n), γ := ∇n × n to pass from his equations to the Ericksen-Leslie equations. This is FALSE! Two arguments: brute force computation and symmetry considerations. So, one needs to do something else. However, not all is wrong: 1. it is true that there is Ψ(j, γ) such that
Ψ (J(I3 − n ⊗ n), ∇n × n) = F (n, ∇n).
2. the definition j := J(I3 − n ⊗ n) is geometrically consistent.
WE SHALL USE THE TOOLS OF GEOMETRIC MECHANICS TO GIVE A DEFINITIVE ANSWER.
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 37 Note: For simplicity, we consider motionless nematics. The present approach easily generalizes to the flowing case.
STEP I: γ-formulation of Ericksen-Leslie The material Lagrangian for nematic motionless liquid crystals L : 3 T F(D, SO(3)) → R, D ⊂ R , is thus given by
1 Z 2 Z L(χ, χ˙) = J kχ˙n0k µ − F (χn0, ∇(χn0))µ, 2 D D where, usually n0 = ˆz, J is the microinertia constant, and F is the Oseen-Frank free energy:
1 2 1 2 F (n, ∇n) = K2 (n · curl n) + K11 (div n) + K22 (n · curl n) | {z } 2 | {z } 2 | {z } chirality splay twist 1 2 + K33 kn × curl nk . 2 | {z } bend
IDEA: Apply two different EP reductions to this Lagrangian.
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 38 FIRST EULER-POINCARÉ REDUCTION FOR NEMATICS
Write L(χ, χ˙) = Ln0(χ, χ˙), where the Lagrangian
Ln0 : T F(D,SO(3)) → R is invariant under the right action
−1 (χ, n0) 7→ χψ, ψ n0 of ψ ∈ F(D, SO(3))n0 (the Ga0 of the general theory). So get the reduced Euler-Poincaré Lagrangian
1 Z 2 Z `1(ν, n) = J kν × nk µ − F (n, ∇n)µ, 2 D D −1 νb =χχ ˙ , n = χn0. The Euler-Poinaré equations are d δ`1 δ`1 δ`1 = ν × + n × dt δν δν δn ∂tn + n × ν = 0
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 39 More explicitly, upon denoting h = −δ`1/δn, one has ( J∂tν = h × n ∂tn + n × ν = 0, which are the Ericksen-Leslie equations of nematodynamics if kn0k = 1 and ν0 · n0 = 0: d2n d2n! J − 2 n · h + J n · n + h = 0. dt2 dt2 | ={zq } d D Since there is no macromotion, dt = Dt.
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 40 SECOND EULER-POINCARÉ REDUCTION FOR NEMATICS
Start with the same Lagrangian. If n0 is constant, we can write
1 Z 2 Z L(χ, χ˙) = J kχ˙n0k µ − F (χn0, ∇(χn0))µ 2 D D Z Z 1 2 −1 = J kχ˙n0k µ − F (χn0, (∇χ) χ · χn0))µ, 2 D D and we view L as
L(χ, χ˙) = L (χ, χ˙). (n0,γ0=0) This Lagrangian is invariant under the right action
−1 −1 −1 (χ, n0, γ0) 7→ χψ, ψ n0, ψ γ0ψ + ψ ∇ψ of the isotropy subgroup F(D,SO(3)) = F(D,S1)∩SO(3) = S1 (n0,0) (the c of the general theory). Ga0
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 41 So we get the reduced affine Euler-Poincaré Lagrangian
1 Z 2 Z `2(ν, n, γ) = J kν × nk µ − F (n, −γ × n)µ. 2 D D −1 −1 1 νb =χχ ˙ , n = χn0, γ = −(∇χ) χ ∈ Ω (D, so(3)).
; The correct relation bewteen γ and n is ∇n = n × γ and not γ := ∇n × n. Note: γ is NOT determined by n! It does not matter.
Notations: i 1 3 1 3 γ = γb. For γ = γidx ∈ Ω (D; R ), define γ × n ∈ Ω (D, R ) by i γ × n = (γi × n) dx , or
(γ × n)(vx) = γ(vx) × n, vx ∈ TxD.
Important: L(χ, χ,˙ n0, γ0) may not be defined when γ0 6= 0. `2 is −1 only defined on the orbit of γ0 = 0, i.e., if γ = −(∇χ)χ . However, this does not affect reduction, as long as the expression L(χ, χ,˙ n0, 0) is invariant under the isotropy group of γ0 = 0. This occurs in the reduction for molecular strand dynamics with nonlocal interactions (Ellis, Gay-Balmaz, Holm, Putkaradze, Ratiu [2010]). Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 42 The affine Euler-Poincaré equations are d δ` δ` δ` δ` ! δ` 2 2 2 2 2 = ν × + div + Tr γ × + n × dt δν δν δγ δγ δn ∂tn + n × ν = 0 ∂tγ + γ × ν + ∇ν = 0, γ0 = 0.
If γ0 6= 0, these reduced equations still make sense, and they are an extension of EL dynamics to account for disclination dynamics. Note that these equations consistently preserve the relation ∇n = n × γ, since ∂ − ν× (∇n − n × γ) = 0 . ∂t
EP equations for `1 and AEP equations `2 are equivalent since they are induced by the SAME Euler-Lagrange equations for L(χ, χ˙) on T F(D, SO(3)). Moreover, the AEP equations allow for a generalization of Ericksen- Leslie to the case with disclinations. Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 43 STEP II: Eringen micropolar theory contains Ericksen-Leslie direc- tor theory as a particular case
Recall: 1. Eringen’s Lagrangian (motionless case = no macro motion) Z Z 1 −1 T −1 −1 −1 −1 L(χ, χ˙) = Tr (i0χ χ˙) χ χ˙ µ− Ψ(χj0χ , χ∇χ +χγ0χ )µ, 2 D D was interpreted as L = L , where i := 1 Tr(j )I − j . This (j0,γ0) 0 2 0 3 0 Lagrangian is invariant under the right affine action −1 −1 −1 (χ, j0, γ0) 7→ χψ, ψ j0ψ, ψ γ0ψ + ψ ∇ψ of the isotropy subgroup F(D,SO(3)) . (j0,γ0) 2. Reduced Lagrangian 1 Z Z `2(ν, j, γ) = (jν) · νµ − Ψ(j, γ)µ. 2 D D 3. Eringen’s equation are the affine Euler-Poincaré equations for: G = F(D,SO(3)) V ∗ = F(D, Sym(3)) × Ω1(D, so(3)). Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 44 II.1 Rod-like assumption
Take as initial condition j0 = J(I − n0 ⊗ n0). This definition is F(D,SO(3))-equivariant, so that j = J(I − n ⊗ n) for all time.
Consider L(χ, χ˙) = L (χ, χ˙) := L (χ, χ˙). This (n0,γ0) (j0=J(I−n0⊗n0),γ0) Lagrangian is invariant under the right action −1 −1 −1 (χ, n0, γ0) 7→ χψ, ψ n0, ψ γ0ψ + ψ ∇ψ of the isotropy subgroup F(D,SO(3)) . (n0,γ0) Reduced Lagrangian 0 `2(ν, n, γ) : = `2(ν,J(I − n ⊗ n), γ) J Z Z = kν × nk2µ − Ψ(J(I − n ⊗ n), γ)µ, 2 D D 0 Affine Euler-Poincaré equations for `2 are equivalent to Eringen’s equations in which the rod-like assumption has been assumed.
It remains to show that these equations contain as particular case, the Ericksen-Leslie equations. Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 45 II.2 No disclination assumption γ0 = 0
Same step as earlier: suppose that n0 is constant and take γ0 = 0. So the evolution of γ is given by −1 γ = θχ−1(0) = −(∇χ)χ . Since n = χn0, we get ∇n = n × γ. II.3 Recovering the Oseen-Frank free energy
Recall that Ψ = Ψ(j, γ), rod-like assumption j = J(I − n ⊗ n), and
1 2 1 2 F (n, ∇n) = K2 (n · curl n) + K11 (div n) + K22 (n · curl n) | {z } 2 | {z } 2 | {z } chirality splay twist 1 2 + K33 kn × curl nk ; 2 | {z } bend K2 6= 0 for cholesterics, K2 = 0 for nematics. So we need to show that there exists Ψ = Ψ(j, γ) such that
Ψ(j, γ) = Ψ(J(I − n ⊗ n), γ) = F (n, n × γ) = F (n, ∇n). Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 46 Lemma The Oseen-Frank free energy can be expressed in terms of Ψ = Ψ(j, γ) as
K K Ψ(j, γ) = 2 Tr(jγ) + 11 Tr (γA)2 (Tr(j) − J) − 2 Tr j(γA)2 J J 1K K 2 + 22 Tr2(jγ) − 33 Tr (γj)A − JγA . 2 J2 J So we can rewrite the reduced Eringen Lagrangian in the rod-like assumption
0 `2(ν, n, γ) = `2(ν,J(I − n ⊗ n), γ) J Z Z = kν × nk2µ − Ψ(J(I − n ⊗ n), γ)µ 2 D D as Z Z 0 J 2 `2(ν, n, γ) = kν × nk µ − F (n, n × γ)µ, 2 D D Same substitution in the unreduced Eringen Lagrangian in the rod- like assumption yields L(χ, χ˙) = L (χ, χ˙). (n0,γ0=0)
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 47 II.4 Recovering Ericksen-Leslie theory
1. Interpret now this L(χ, χ˙) as L (χ, χ˙) instead of L (χ, χ˙). n0 (n0,γ=0)
2. Check that this Lagrangian is F(D,SO(3))n0-invariant under the −1 action (χ, n0) 7→ χψ, ψ n0 .
3. Implement Euler-Poincaré reduction associated to the action −1 (χ, n0) 7→ χψ, ψ n0 and obtain the reduced Lagrangian Z Z 0 J 2 `1(ν, n) = kν × nk µ − F (n, ∇n)µ 2 D D (Previously we considered affine Euler-Poincaré reduction associ- −1 −1 −1 ated to the action (χ, n0, γ0) 7→ χψ, ψ n0, ψ γ0ψ + ψ ∇ψ , with 0 reduced Lagrangian `2).
0 By general reduction theory: EP equations for `1 and AEP equa- 0 tions `2 are equivalent since they are induced by the SAME Euler- Lagrange equations for L(χ, χ˙) on T F(D, SO(3)).
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 48 0 It remains to show that the EP equations for `1 are the Ericksen- Leslie equations. True, by direct verification.
We have thus proved:
THEOREM: The Eringen micropolar theory of liquid crystals con- tains as a particular case the Ericksen-Leslie director theory. More precisely, the Ericksen-Leslie theory is recovered by assuming rod- like molecules: j = J(I − n ⊗ n) and absence of disclinations γ0 = 0.
Summary of method:
- This is shown by considering two distinct Euler-Poincaré reduc- tions associated with distinct advected quantities.
- This allows us to replace the non-consistent definition γ := ∇n×n by the relation ∇n = n × γ and to solve the inconsistencies in Erin- gen’s approach.
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 49 ∇j = [j, γb] new - Eringen 6 6 j = J(I − n ⊗ n) j = J(I − n ⊗ n)
∇n = n × γ Ericksen-Leslie - new j = J(I − n ⊗ n) j = J(I − n ⊗ n)
? ∇n = n × γ ? Lhuillier-Rey - new The Lagrangians for different theories. Lagrangians on the center line are for the material descriptions of the models. Slanted arrows are Euler-Poincaré re- duction; vertical arrows show how the theories are embedded in each other. By Euler-Poincaré reduction theory, all the Lagrangians related by a dashed arrow are equivalent. Consequence: any question in a given model can be treated, equivalently, with any of the three Lagrangians in a given triangle. Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 50 −1 bν :=χχ ˙
−1 −1 L(χ, χ,˙ χj0χ , ∇(χj0χ )) 6 −1 j = 0χ γ χj χj = 0 = − χ −1 j (∇ χ) χ −1 - ∇j = j, γ b - `1(ν, j, ∇j) `2(ν, j, γ) 6 6
j0 = J(I − n0 ⊗ n0)
j = J(I − n ⊗ n) L(χ, χ,˙ χn0, ∇(χn0)) j = J(I − n ⊗ n)
n 0 = χn γ = χn = − 0 n (∇ χ) χ −1 - ∇n = n × γ - `1(ν, n, ∇n) `2(ν, n, γ)
j0 = J(I − n0 ⊗ n0)
? −1 j = J(I − n ⊗ n) L(χ, χ,˙ χj0χ , χn0, ∇(χn0)) j = J(I − n ⊗ n) n = χn χn0 γ 0 j, = −1 = = n χ − 0 (∇ χj χj χ 0 χ − = ) − 1 j χ 1 ? - ? ∇n = n × γ - `1(ν, j, n, ∇n) `2(ν, j, n, γ) 51 Final remarks: 1.) All the discussion here can be easily extended to moving liquid crystals. One applies EP, respectively affine EP, theory, as discussed earlier. Then the same considerations as above show that Eringen micropolar theory contains Ericksen-Leslie ne- matodynamics. 2.) Other inconsistencies in the micropolar description: Eringen 1 2 3 defines a smectic liquid crystal by Tr(γ) = γ1 + γ2 + γ3 = 0. This −1 −1 is not preserved by the evolution γ = η∗ χγ0χ + χ∇χ . Consistent with the statement: the equation ∂γ + £uγ + dν + γ × ν = 0 ∂t does not imply that if Tr(γ0) = 0 then Tr(γ) = 0 for all time. Is Eringen’s definition of smectic incorrect? Instead of the trace need an F(D, SO(3))-invariant function (of γ) under the action −1 −1 3 v 7→ χ v + χ ∇χ, v ∈ F(D, R ), χ ∈ F(D, SO(3)). We do not know how to choose a physically reasonable function of this type. 3.) Other difficulties in liquid crystals dynamics may be solved by using the tools of geometric mechanics (disclinations, defects,...) Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 52 References
Gay-Balmaz, F. and Ratiu T.S. [2009] The geometric structure of complex fluids, Advances in Applied Math., 42, 176–275.
Gay-Balmaz, F., Ratiu, T.S., and Tronci, C. [2013] Equivalent the- ories of liquid crystal dynamics, Arch. Rational Mechanics Analysis, 210, 773–811, doi: 10.1007/s00205-013-0673-1.
Chechkin, G.A., Ratiu, T.S., Romanov, M.S., and Samokhin, V.N. [2014] Existence and uniqueness theorems for the two-dimensional Ericksen-Leslie system, submitted.
Chechkin, G.A., Ratiu, T.S., Romanov, M.S., and Samokhin, V.N. [2014] Existence and uniqueness theorems for the full three- dimen- sional Ericksen-Leslie system, submitted.
Lie Theory and Mathematical Physics, CRM Montreal May 19-23, 2014 53