q4-Orientiation-Equivalence.md 6.09 KB
 Felix Hilsky committed Jan 13, 2022 1 2 3 4 5 6 7 # Orientability and Energy Minimization in Liquid Crystal Models ## Overview - *uniaxial nematic liquid crystals* werden modelliert - *Oseen-Frank*: unit vector field $n$ → nicht als $ℝP^2$ gesehen, sondern mit $n ≠ -n$ (ignoriert Symmetrie) - wird als Standardsichtweise dargestellt  Felix Hilsky committed Jan 15, 2022 8 - *Landau-de Gennes*: $Q = s(n \otimes n - \tfrac13\operatorname{Id})$  Felix Hilsky committed Jan 13, 2022 9 10 11 12 13 14 15 16 - Resultat: - Theorien sind gleich für einfach zusammenhängende Urbilder und $W^{1,2}$ - Unterschiede in anderen Fällen sind beschrieben: - for simple energy functional, holes, various boundary conditions, difference cases are characterised (i.e. $⇔$) ## de Gennes - closer to physics reality - $Q$-tensors are generally more [complex](https://arxiv.org/pdf/1409.3542.pdf), in our case just of the simple form („constrained”)  Felix Hilsky committed Jan 15, 2022 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48  ### Derivation of $Q$-Tensors (from chapter 1 Introduction) - each point has preferred direction but can have any direction → probability measure $μ(x, ·) : ℒ(𝕊^2) → [0,1]$ ($ℒ$ being all Lebesque sets) modelling very small area around $x$ - symmetry modeled as $μ(x, A) = μ(x, -A) ⇒ ⟨p⟩ = ∫_{𝕊^2} p dμ(p) = 0$ (first moment or average) - Tensor of second moments: $M_{ij} = ∫_{𝕊^2} p_ip_j dμ(p)$ ($i, j = 1, 2, 3$) - $M = M^T$, $\operatorname{tr} M = Σ_{i=1}^3 ∫_{𝕊^2} p_i^2 dμ(p) = ∫_{𝕊^2} 1 dμ(p) = 1$ ($μ$ is probability measure) - $e · M e = ∫_{𝕊^2} (e · p)^2 dμ(p) = ⟨\cos^2(θ)⟩$ ($θ$ = angle between $p$ and $e$) for $|e| =1$) - if $μ$ is isotropic (no preferred direction): $μ_0$ with $dμ_0(p) = (4π)^{-1}dA$ with $M_0 = (4π)^{-1} ∫_{𝕊^2} p \otimes p dA = \frac 13 \operatorname{Id}$ - $∫_{𝕊^2} p_1 p_2 dμ(p) = 0$ since what happens on one half-sphere is countered by the other half-sphere ($p_2$, $p_3$ are same but $p_1$ is of opposite sign) - $∫_{𝕊^2} p_i^2 dμ(p)$ are equal ($i=1,2,3$) and in sum $= \operatorname{tr} M_0 = 1$ - Def. *de Gennes order-parameter tensor $Q$*: difference of second moment tensor to isotropic case $Q = M - M_0 = ∫_{𝕊^2}(p \otimes p - \frac13 \operatorname{Id}) dμ(p)$ - $Q$ symmetric, $\operatorname{Q} = 0$ $⇒$ (spectral theorem) $Q = λ_1 ê_1 \otimes ê_2 + λ_2 ê_2 \otimes ê_2 - (λ_1 + λ_2)ê_3 \otimes ê_3$ ($ê_{1,2,3}$ orthonormal eigenvector basis, $λ_1, λ_2$ eigenvectors) - Eigenvectors for different non-zero eigenvalues are orthogonal: $(λ_1e_1) · (λ_2^{-1} e_2) = (Q e_1)^T (Q^{-1}e_2) = e_1 · Q^T Q^{-1} e_2 = e_1 · e_2 ⇒ e_1 ⊥ e_2$ - two eigenvalues equal and non-zero: *uniaxial* (considered here) - otherwise *biaxial* (more complicated) - Constraint on $s$: - $s$ defined by limiting process $L → 0$ and the constants $a, b, c$ - but also $Q$ has an original definition via $μ$. Together: math Qn · n = n^T s (n n^T - \frac 13 \operatorname{Id}) n = s (n^T n n^T n - \frac13 n^T \operatorname{Id}n) = s(1 · 1 - \frac13 · 1) \frac23 s = ∫_{𝕊^2} n^T p p^T n - \frac13 n^T n dμ(p) = ∫_{𝕊^2} (n · p)^2 - \frac 13 dμ(p) = ⟨\cos^2 θ - \frac13⟩ ⇒ s = \frac32 ⟨\cos^2 θ - \frac13⟩  - $⇒ -\frac12 \leq s \leq 1$: $s$ says how much the molecules agree, are „in order“ (max: $s = 1$: perfectly ordered $p∥n$; $s = -\frac12$: $p ⊥ n$; $s=0$: $Q = 0$, isotropic) - $⇒$ s is *scalar order parameter associated to the tensor $Q$*  Felix Hilsky committed Jan 13, 2022 49 50  - usual assumption: $s$ is constant $⇒$ space of $Q$-Tensors "is" $ℝP^2$  Felix Hilsky committed Jan 15, 2022 51 52 53 54 55 56 57 ### Simpliest energy functional (p. 495 = 3) math ℱ_{\text{LG}}[Q] = ∫_{Ω} \left(\sum_{i,j,k = 1}^3 \frac L2 Q_{ij,k} Q_{ij,k} - \frac a2 \operatorname{tr} Q^2 - \frac b3 \operatorname{tr} Q^3 + \frac c4 (\operatorname{tr} Q^2)^2 \right) dx  $a, b, c$ constants, $L$ „*elastic constant*”. Physics is interested in $L → 0$  Felix Hilsky committed Jan 13, 2022 58 59 ## Oseen-Frank - simpler, but sometimes wrong  Felix Hilsky committed Jan 15, 2022 60 61 62 63 64 65 66 67 68 69 70 71 - (here:) with orientation ($n ∈ 𝕊^2$, not $n ∈ ℝP^2$) - problem: *„fake defects”*: „non-orientable line field” (places where vector field has to be rough but would be OK if we took away the orientation ### Simpliest energy functional minimizers of $ℱ_{\text{LG}}$ for $L → 0$ are "suitably approximated" by minimizers of math ℱ_{\text{OF}}[Q] = ∫_{Ω} \sum_{i, j, k = 1}^3 Q_{ij,k} Q_{ij,k} dx  *if* $Q ∈ W^{1,2}$ with $Q$ uniaxial almost everywhere, i.e. math Q = s(n \otimes n - \tfrac13 \operatorname{Id}) \quad (s = -3 λ_1 = -3 λ_2, n = ê_3)   Felix Hilsky committed Jan 13, 2022 72 73 74 75  ## Compatibility - Oseen-Frank and de Gennes are compatible if the oriented line field (= unit vector field) can be oriented (without changing regularity) - otherwise Oseen-Frank might miss a global minimizer because it is not orientable  Felix Hilsky committed Jan 15, 2022 76 - $P : 𝕊^2 → 𝒬$ removes orientation. Orientable = in the image of $P$. Same for $Q$ only defined on $∂Ω$  Felix Hilsky committed Jan 14, 2022 77 78 79 80 81 82 83 84 - in chapter 3.1 firstly for continous $Q$ since that's standard from topology - *Theorem 1*: $𝒬_2$ (one-dim), orientable on boundary of holes, $Q$ continous $⇒$ orientable (long proof with a lot of fiddling!) - chapter 3.2 Theorem 2: $Ω$ simply connected, $Q ∈ W^{1, p}$, $p \geq 2$ (!) $⇒$ orientable with Sobolev-Seminorm estimate. Counterexample for $p < 2$ - chapter 4: different cases of $Q$ orientable on boundary $⇒$/$⇔$/$⇐$ on $Ω$. Sometimes $Q: → 𝒬_2$, sometimes $Q$ continous (on boundary) ## Energy functionals - specific energy functionals are regarded (p. 11/503) because they were looked at before and Oseen-Frank was successful for them - conversion between energy functionals possible  Felix Hilsky committed Jan 15, 2022 85 86 87 88 89  ## Questions - What is the third index $k$ for $Q$ on page 3 (= 495)? - And why is it written as a product instead of a square$^2$? - To make spectral representation of $Q$ fit to $Q = s(n \otimes n - \tfrac13 \operatorname{Id})$, we need $s = -3 λ_1 = -3 λ_2, n = ê_3$. How can $s$ then depend on $a, b, c$ (see formula (3) on page 3 (=495))? If that's the case, $Q$ is further reduced to the choice of $n$ but that's what we assumed as our model at the beginning. So apparently that's fine.