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Droplet phase in a nonlocal isoperimetric problem under confinement

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  • 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 $ \eta $ to represent the size of the domains of the minority phase, and study the resulting droplet regime as $ \eta\to 0 $. 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 $ M $ which gives the total volume of the droplets at order $ \eta^3 $ and its relation to existence and non-existence of Gamow's Liquid Drop model on $ \mathbb{R}^3 $. For large values of $ M $, the minority phase splits into several droplets at an intermediate scale $ \eta^{1/3} $, while for small $ M $ minimizers form a single droplet converging to the maximum of the confinement density.

    Mathematics Subject Classification: 35Q70, 49Q20, 49S05, 74N15, 82D60.


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  • Figure 1.  The attraction to the origin and scaling at the rate $ \delta = \eta^{1/3} $

    Figure 2.  Minimizing configurations of the second-order energy $ \mathsf{F}_{m^1,\dots,m^n} $ with equal mass $ m^i = 1/100 $ for 100 particles with varying powers $ q $ of degenerate penalization $ \rho(x)-\rho_0\sim |x|^q $. Minimizing configurations are obtained as steady-states of the gradient flow of the energy $ \mathsf{F}_{m^1,\dots,m^n} $

  • [1] Emilio Acerbi, Nicola Fusco and Massimiliano Morini, Minimality via second variation for a nonlocal isoperimetric problem, Comm. Math. Phys., 322 (2013), 515–557. doi: 10.1007/s00220-013-1733-y.
    [2] Stan Alama, Lia Bronsard and Ihsan Topaloglu, Sharp interface limit of an energy modelling nanoparticle-polymer blends, Interfaces Free Bound, 18 (2016), 263–290. doi: 10.4171/IFB/364.
    [3] Frank S. Bates and Glenn H. Fredrickson, Block copolymers–designer soft materials, Physics Today, 52 (1999), 32-38. 
    [4] Marco Bonacini and Riccardo Cristoferi, Local and global minimality results for a nonlocal isoperimetric problem on $\mathbb{R}^ N$, SIAM J. Math. Anal., 46 (2014), 2310–2349. doi: 10.1137/130929898.
    [5] Almut Burchard, Rustum Choksi and Ihsan Topaloglu, Nonlocal shape optimization via interactions of attractive and repulsive potentials, Indiana Univ. Math. J., 67 (2018), 375–395. doi: 10.1512/iumj.2018.67.6234.
    [6] Djalil Chafaï, Nathael Gozlan and Pierre-André Zitt, First-order global asymptotics for confined particles with singular pair repulsion, Ann. Appl. Probab., 24 (2014), 2371–2413. doi: 10.1214/13-AAP980.
    [7] Rustum ChoksiCyrill B. Muratov and Ihsan Topaloglu, An old problem resurfaces nonlocally: Gamow's liquid drops inspire today's research and applications, Notices Amer. Math. Soc., 64 (2017), 1275-1283. 
    [8] Rustum Choksi and Mark A. Peletier, Small volume fraction limit of the diblock copolymer problem: Ⅰ. Sharp-interface functional, SIAM J. Math. Anal., 42 (2010), 1334-1370.  doi: 10.1137/090764888.
    [9] Yao-Li ChuangMaria R. D'OrsognaDaniel MarthalerAndrea L. Bertozzi and Lincoln S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Phys. D, 232 (2007), 33-47.  doi: 10.1016/j.physd.2007.05.007.
    [10] Rupert Frank, Rowan Killip and Phan Thành Nam, Nonexistence of large nuclei in the liquid drop model, Lett. Math. Phys., 106 (2016), 1033–1036. doi: 10.1007/s11005-016-0860-8.
    [11] Rupert L. Frank and Elliot H. Lieb, A "liquid-solid" phase transition in a simple model for swarming, based on the "no flat-spots" theorem for subharmonic functions, Indiana Univ. Math. J., 67 (2018), 1547-1569.  doi: 10.1512/iumj.2018.67.7398.
    [12] Rupert L. Frank and Elliott H. Lieb, A compactness lemma and its application to the existence of minimizers for the liquid drop model, SIAM J. Math. Anal., 47 (2015), 4436-4450.  doi: 10.1137/15M1010658.
    [13] Glenn Fredrickson, Equilibrium Theory of Inhomogeneous Polymers, Oxford Science Publications, 2005.
    [14] Valeriy V. Ginzburg, Feng Qiu, Marco Paniconi, Gongwen Peng, David Jasnow and Anna C Balazs, Simulation of hard particles in a phase-separating binary mixture, Phys. Rev. Lett., 82 (1999), 4026-4029.
    [15] Shay Gueron and Itai Shafrir, On a discrete variational problem involving interacting particles, SIAM J. Appl. Math., 60 (2000), 1–17 (electronic). doi: 10.1137/S0036139997315258.
    [16] Vesa Julin,, Isoperimetric problem with a Coulomb repulsive term, Indiana Univ. Math. J., 63 (2014), 77–89. doi: 10.1512/iumj.2014.63.5185.
    [17] Hans Knüpfer and Cyrill B. Muratov, On an isoperimetric problem with a competing nonlocal term Ⅰ: The planar case, Comm. Pure Appl. Math., 66 (2013), 1129-1162.  doi: 10.1002/cpa.21451.
    [18] Hans Knüpfer and Cyrill B. Muratov, On an isoperimetric problem with a competing nonlocal term Ⅱ: The general case, Comm. Pure Appl. Math., 67 (2014), 1974-1994.  doi: 10.1002/cpa.21479.
    [19] Hans KnüpferCyrill B. Muratov and Matteo Novaga, Low density phases in a uniformly charged liquid, Comm. Math. Phys., 345 (2016), 141-183.  doi: 10.1007/s00220-016-2654-3.
    [20] Pierre-Louis Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145.
    [21] Jiangfeng Lu and Felix Otto, Nonexistence of a minimizer for Thomas-Fermi-Dirac-von Weizsäcker model, Comm. Pure Appl. Math., 67 (2014), 1605–1617. doi: 10.1002/cpa.21477.
    [22] Jiangfeng Lu and Felix Otto, An isoperimetric problem with Coulomb repulsion and attraction to a background nucleus, arXiv: 1508.07172, 2015.
    [23] Francesco Maggi, Sets of Finite Perimeter and Geometric Variational Problems, volume 135 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, first edition, 2012. doi: 10.1017/CBO9781139108133.
    [24] Daniela Morale, Vincenzo Capasso and Karl Oelschläger, An interacting particle system modeling aggregation behavior: from individuals to populations, J. Math. Biol., 50 (2005), 49–66. doi: 10.1007/s00285-004-0279-1.
    [25] Phan Thành Nam and Hanne van den Bosch, Nonexistence in Thomas–Fermi–Dirac–von Weizsäcker theory with small nuclear charges, Math. Phys. Anal. Geom., 20 (2017), Art. 6, 1-32. doi: 10.1007/s11040-017-9238-0.
    [26] Takao Ohta and Kyozi Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621–2632.
    [27] Etienne Sandier and Sylvia Serfaty, Vortices in The Magnetic GInzburg-LAndau Model, Progress in Nonlinear Differential Equations and their Applications, 70. Birkhäuser Boston, Inc., Boston, MA, 2007.
    [28] An-Chang Shi and Baohui Li, Self-assembly of diblock copolymers under confinement, Soft Matter, 9 (2013), 1398–1413.
    [29] James H. von Brecht, David Uminsky, L. Bertozzi, Theodore Kolokolnikov and Andrea L. Predicting pattern formation in particle interactions, Math. Models Methods Appl. Sci., 22(suppl. 1) (2012), 1140002, 1-31. doi: 10.1142/S0218202511400021.
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