### Kapitel 1 Einführungsrechnungen verstanden

 ... ... @@ -5,7 +5,7 @@ - *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 - *Landau-de Gennes*: $Q = s(n \otimes n - \tfrac13\operatorname{Id}$ - *Landau-de Gennes*: $Q = s(n \otimes n - \tfrac13\operatorname{Id})$ - Resultat: - Theorien sind gleich für einfach zusammenhängende Urbilder und $W^{1,2}$ - Unterschiede in anderen Fällen sind beschrieben: ... ... @@ -14,21 +14,66 @@ ## 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”) - each point has preferred direction but can have any direction → probability measure $μ(x, ·) : ℒ(S^2) → [0,1]$ ($ℒ$ being all Lebesque sets) - symmetry modeled as $μ(x, A) = μ(x, -A) ⇒ ⟨p⟩ = ∫_{S^2} p dμ(p) = 0$ (first moment or average) - ... (some more steps to reach the $Q$ form from above) - including $-\frac12 \leq s \leq 1$: $s$ says how much the molecules agree, are „in order“ ### 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 $Q$* - usual assumption: $s$ is constant $⇒$ space of $Q$-Tensors "is" $ℝP^2$ ### 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 $L → 0$ ## Oseen-Frank - simpler, but sometimes wrong - (here:) with orientation ($n ∈ S^2$, not $n ∈ ℝP^2$) - problem: *„fake defects”*: „non-orientable line field” - (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)  ## 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 - $P : S^2 → 𝒬$ removes orientation. Orientable = in the image of $P$. Same for $Q$ only defined on $∂Ω$ - $P : 𝕊^2 → 𝒬$ removes orientation. Orientable = in the image of $P$. Same for $Q$ only defined on $∂Ω$ - 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 $p < 2$ ... ... @@ -37,3 +82,8 @@ ## 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 ## 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.