Droplet phase in a nonlocal isoperimetric problem under confinement
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Bibliographic record
Abstract
<p style='text-indent:20px;'>We address small volume-fraction asymptotic properties of a nonlocal isoperimetric functional with a confinement term, derived as the sharp interface limit of a variational model for self-assembly of diblock copolymers under confinement by nanoparticle inclusion. We introduce a small parameter <inline-formula><tex-math id="M1">\begin{document}$ \eta $\end{document}</tex-math></inline-formula> to represent the size of the domains of the minority phase, and study the resulting droplet regime as <inline-formula><tex-math id="M2">\begin{document}$ \eta\to 0 $\end{document}</tex-math></inline-formula>. By considering confinement densities which are spatially variable and attain a unique nondegenerate maximum, we present a two-scale asymptotic analysis wherein a separation of length scales is captured due to competition between the nonlocal repulsive and confining attractive effects in the energy. A key role is played by a parameter <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> which gives the total volume of the droplets at order <inline-formula><tex-math id="M4">\begin{document}$ \eta^3 $\end{document}</tex-math></inline-formula> and its relation to existence and non-existence of Gamow's Liquid Drop model on <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{R}^3 $\end{document}</tex-math></inline-formula>. For large values of <inline-formula><tex-math id="M6">\begin{document}$ M $\end{document}</tex-math></inline-formula>, the minority phase splits into several droplets at an intermediate scale <inline-formula><tex-math id="M7">\begin{document}$ \eta^{1/3} $\end{document}</tex-math></inline-formula>, while for small <inline-formula><tex-math id="M8">\begin{document}$ M $\end{document}</tex-math></inline-formula> minimizers form a single droplet converging to the maximum of the confinement density.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.005 | 0.004 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it