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January  2020, 19(1): 175-202. doi: 10.3934/cpaa.2020010

Droplet phase in a nonlocal isoperimetric problem under confinement

 1 Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada 2 Department of Mathematics and Statistics, McGill University, Montréal, QC, Canada 3 Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, USA

Received  August 2018 Revised  March 2019 Published  July 2019

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.

Citation: Stan Alama, Lia Bronsard, Rustum Choksi, Ihsan Topaloglu. Droplet phase in a nonlocal isoperimetric problem under confinement. Communications on Pure & Applied Analysis, 2020, 19 (1) : 175-202. doi: 10.3934/cpaa.2020010
References:
 [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. Google Scholar [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. Google Scholar [3] Frank S. Bates and Glenn H. Fredrickson, Block copolymers–designer soft materials, Physics Today, 52 (1999), 32-38. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] Rustum Choksi, Cyrill 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. Google Scholar [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. Google Scholar [9] Yao-Li Chuang, Maria R. D'Orsogna, Daniel Marthaler, Andrea 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] Glenn Fredrickson, Equilibrium Theory of Inhomogeneous Polymers, Oxford Science Publications, 2005. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] Hans Knüpfer, Cyrill 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. Google Scholar [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. Google Scholar [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. Google Scholar [22] Jiangfeng Lu and Felix Otto, An isoperimetric problem with Coulomb repulsion and attraction to a background nucleus, arXiv: 1508.07172, 2015.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [26] Takao Ohta and Kyozi Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621–2632.Google Scholar [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. Google Scholar [28] An-Chang Shi and Baohui Li, Self-assembly of diblock copolymers under confinement, Soft Matter, 9 (2013), 1398–1413.Google Scholar [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. Google Scholar

show all references

References:
 [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. Google Scholar [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. Google Scholar [3] Frank S. Bates and Glenn H. Fredrickson, Block copolymers–designer soft materials, Physics Today, 52 (1999), 32-38. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] Rustum Choksi, Cyrill 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. Google Scholar [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. Google Scholar [9] Yao-Li Chuang, Maria R. D'Orsogna, Daniel Marthaler, Andrea 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] Glenn Fredrickson, Equilibrium Theory of Inhomogeneous Polymers, Oxford Science Publications, 2005. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [19] Hans Knüpfer, Cyrill 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. Google Scholar [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. Google Scholar [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. Google Scholar [22] Jiangfeng Lu and Felix Otto, An isoperimetric problem with Coulomb repulsion and attraction to a background nucleus, arXiv: 1508.07172, 2015.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [26] Takao Ohta and Kyozi Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621–2632.Google Scholar [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. Google Scholar [28] An-Chang Shi and Baohui Li, Self-assembly of diblock copolymers under confinement, Soft Matter, 9 (2013), 1398–1413.Google Scholar [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. Google Scholar
The attraction to the origin and scaling at the rate $\delta = \eta^{1/3}$
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 $\mathsf{F}_{m^1,\dots,m^n}$
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