Small Volume-Fraction Limit of the Diblock Copolymer Problem: II. Diffuse-Interface Functional
Bibliographic record
Abstract
We present the second of two articles on the small volume-fraction limit of a nonlocal Cahn–Hilliard functional introduced to model microphase separation of diblock copolymers. After having established the results for the sharp-interface version of the functional [SIAM J. Math. Anal., 42 (2010), pp. 1334–1370], we consider here the full diffuse-interface functional and address the limit in which $\varepsilon$ and the volume fraction tend to zero but the number of regions (called particles) associated with the minority phase remains $O(1)$. Using the language of $\Gamma$-convergence, we focus on two levels of this convergence, and derive first- and second-order effective energies, whose energy landscapes are simpler and more transparent. These limiting energies are finite only on weighted sums of delta functions, corresponding to the concentration of mass into “point particles.” At the highest level, the effective energy is entirely local and contains information about the size of each particle but no information about its spatial distribution. At the next level we encounter a Coulomb-like interaction between the particles, which is responsible for the pattern formation. We present the results in three dimensions and comment on their two-dimensional analogues.
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How this classification was reachedexpand
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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".