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Magnetic field processing is promising for directing and enhancing self-assembly of diamagnetic block copolymers (BCPs) via domain alignment, but is typically limited to high field strengths and few polymer chemistries. Herein, a novel magnetic field-induced ordering mechanism distinct from domain alignment is demonstrated in aqueous, spherical BCP micelles. Here, low-intensity magnetic fields (B< 0.5 T) induce an anomalous disorder-to-order transition, accompanied by a several order-of-magnitude increase in shear modulus-- effectively transforming a low viscosity liquid into an ordered soft solid. The induced moduli are orders of magnitude larger than those resulting from thermally-induced ordering. Further magnetization induces cubic-to-cylinder order-to-order transitions. Comprehensive characterization via magnetorheology, small- and wide-angle X-ray scattering, differential scanning calorimetry, and vibrational spectroscopy reveals a significant reduction in micelle size and aggregation number relative to zero-field temperature- or concentration-induced ordering, suggesting that B-fields strongly alter polymer-solvent interactions. This extraordinary BCP ordering strategy enables discovery of structures and d-spacings inaccessible via traditional processing routes, thus providing a new platform for developing advanced materials with precisely-controlled features.
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This content will become publicly available on February 5, 2026
Ray class groups and ray class fields for orders of number fields
We contribute to the theory of orders of number fields. We define a notion of ray class group associated to an arbitrary order in a number field together with an arbitrary ray class modulus for that order (including Archimedean data), constructed using invertible fractional ideals of the order. We show existence of ray class fields corresponding to the class groups. These ray class groups (resp., ray class fields) specialize to classical ray class groups (resp., fields) of a number field in the case of the maximal order, and they specialize to ring class groups (resp., fields) of orders in the case of trivial modulus. We give exact sequences for simultaneous change of order and change of modulus. As a consequence, we identify the ray class field of an order with a given modulus as a specific subfield of a ray class field of the maximal order with a larger modulus. We also uniquely describe each ray class field of an order in terms of the splitting behavior of primes.
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- Award ID(s):
- 2302514
- PAR ID:
- 10618633
- Publisher / Repository:
- MSP
- Date Published:
- Journal Name:
- Essential Number Theory
- Volume:
- 4
- Issue:
- 1
- ISSN:
- 2834-4626
- Page Range / eLocation ID:
- 1 to 65
- Subject(s) / Keyword(s):
- class field theory orders of number fields ray class fields ring class fields
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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