Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0901.0898

Asymptotically periodic L^2 minimizers in strongly segregating diblock copolymers

Adam Chmaj
View original source ↗Read source document ↗Export metadata ↓

Abstract

Using the delta correction to the standard free energy \cite{bc} in the elastic setting with a quadratic foundation term and some parameters, we introduce a one dimension only model for strong segregation in diblock copolymers, whose sharp interface periodic microstructure is consistent with experiment in low temperatures. The Green's function pattern forming nonlocality is the same as in the Ohta-Kawasaki model. Thus we complete the statement in [31,p.349]: ``The detailed analysis of this model will be given elsewhere. Our preliminary results indicate that the new model exhibits periodic minimizers with sharp interfaces.'' We stress that the result is unexpected, as the functional is not well posed, moreover the instabilities in $L^2$ typically occur only along continuous nondifferentiable ``hairs''. We also improve the derivation done by van der Waals and use it and the above to show the existence of a phase transition with Maxwell's equal area rule. However, this model does not predict the universal critical surface tension exponent, conjectured to be 11/9. Actually, the range $(1.2,1.36)$ has been reported in experiments [21,p. 360]. By simply taking a constant kernel, this exponent is 2. This is the experimentally ($ \pm 0.1$) verified tricritical exponent, found e.g., at the consolute $0.9$ K point in mixtures of ${}^3$He and ${}^4$He. Thus there is a third unseen phase at the phase transition point.

Explore related subjects

math.GM

Keep this discovery

BibTeXRIS

Adam Chmaj. 2009-01-07. Asymptotically periodic L^2 minimizers in strongly segregating diblock copolymers. https://arxiv.org/abs/0901.0898

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗