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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 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. 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 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. 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 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. 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 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. 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 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. 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 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. 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

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} $
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