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Low Density Phases in a Uniformly Charged Liquid

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Abstract

This paper is concerned with the macroscopic behavior of global energy minimizers in the three-dimensional sharp interface unscreened Ohta–Kawasaki model of diblock copolymer melts. This model is also referred to as the nuclear liquid drop model in the studies of the structure of highly compressed nuclear matter found in the crust of neutron stars, and, more broadly, is a paradigm for energy-driven pattern forming systems in which spatial order arises as a result of the competition of short-range attractive and long-range repulsive forces. Here we investigate the large volume behavior of minimizers in the low volume fraction regime, in which one expects the formation of a periodic lattice of small droplets of the minority phase in a sea of the majority phase. Under periodic boundary conditions, we prove that the considered energy \({\Gamma}\)-converges to an energy functional of the limit “homogenized” measure associated with the minority phase consisting of a local linear term and a non-local quadratic term mediated by the Coulomb kernel. As a consequence, asymptotically the mass of the minority phase in a minimizer spreads uniformly across the domain. Similarly, the energy spreads uniformly across the domain as well, with the limit energy density minimizing the energy of a single droplet per unit volume. Finally, we prove that in the macroscopic limit the connected components of the minimizers have volumes and diameters that are bounded above and below by universal constants, and that most of them converge to the minimizers of the energy divided by volume for the whole space problem.

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References

  1. Abrikosov A.A.: Some properties of strongly compressed matter. I. Sov. Phys. JETP 12, 1254–1259 (1961)

    Google Scholar 

  2. Acerbi E., Fusco N., Morini M.: Minimality via second variation for a nonlocal isoperimetric problem. Commun. Math. Phys. 322, 515–557 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Alberti G., Choksi R., Otto F.: Uniform energy distribution for an isoperimetric problem with long-range interactions. J. Am. Math. Soc. 22, 569–605 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baym G., Bethe H.A., Pethick C.J.: Neutron star matter. Nucl. Phys. A 175, 225–271 (1971)

    Article  ADS  Google Scholar 

  5. Bohr N.: Neutron capture and nuclear constitution. Nature 137, 344–348 (1936)

    Article  ADS  MATH  Google Scholar 

  6. Bohr N., Wheeler J.A.: The mechanism of nuclear fission. Phys. Rev. 56, 426–450 (1939)

    Article  ADS  MATH  Google Scholar 

  7. Bonacini M., Cristoferi R.: Local and global minimality results for a nonlocal isoperimetric problem on \({\mathbb{R}^{N}}\). SIAM J. Math. Anal. 46, 2310–2349 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  8. Brezis H., Browder F.: A property of Sobolev spaces. Commun. Partial Differ. Equ. 4, 1077–1083 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen X., Oshita Y.: An application of the modular function in nonlocal variational problems. Arch. Ration. Mech. Anal. 186, 109–132 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Choksi R.: Scaling laws in microphase separation of diblock copolymers. J. Nonlinear Sci. 11, 223–236 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Choksi R.: On global minimizers for a variational problem with long-range interactions. Q. Appl. Math. 70, 517–537 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Choksi R., Peletier M.A.: Small volume fraction limit of the diblock copolymer problem: I. Sharp interface functional. SIAM J. Math. Anal. 42, 1334–1370 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Choksi R., Peletier M.A.: Small volume fraction limit of the diblock copolymer problem: II. Diffuse interface functional. SIAM J. Math. Anal. 43, 739–763 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Choksi R., Ren X.: On the derivation of a density functional theory for microphase separation of diblock copolymers. J. Stat. Phys. 113, 151–176 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Choksi R., Sternberg P.: On the first and second variations of a nonlocal isoperimetric problem. J. Reine Angew. Math. 611, 75–108 (2007)

    MATH  MathSciNet  Google Scholar 

  16. Cicalese M., Spadaro E.: Droplet minimizers of an isoperimetric problem with long-range interactions. Commun. Pure Appl. Math. 66, 1298–1333 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  17. Cohen S., Plasil F., Swiatecki W.J.: Equilibrium configurations of rotating charged or gravitating liquid masses with surface tension. II. Ann. Phys. 82, 557–596 (1974)

    Article  ADS  Google Scholar 

  18. Cohen S., Swiatecki W.J.: The deformation energy of a charged drop: IV. Evidence for a discontinuity in the conventional family of saddle point shapes. Ann. Phys. 19, 67–164 (1962)

    Article  ADS  MATH  Google Scholar 

  19. Cook N.D.: Models of the Atomic Nucleus. Springer, Berlin (2006)

    Google Scholar 

  20. Dobrynin A.V., Rubinstein M.: Theory of polyelectrolytes in solutions and at surfaces. Progr. Polym. Sci. 30, 1049–1118 (2005)

    Article  Google Scholar 

  21. Evans L.C., Gariepy R.L.: Measure Theory and Fine Properties of Functions. CRC, Boca Raton (1992)

    MATH  Google Scholar 

  22. Feenberg E.: On the shape and stability of heavy nuclei. Phys. Rev. 55, 504–505 (1939)

    Article  ADS  MATH  Google Scholar 

  23. Figalli A., Fusco N., Maggi F., Millot V., Morini M.: Isoperimetry and stability properties of balls with respect to nonlocal energies. Commun. Math. Phys. 336, 441–507 (2015)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Foldy L.L.: Phase transition in a Wigner lattice. Phys. Rev. B 3, 3472–3479 (1971)

    Article  ADS  Google Scholar 

  25. Förster S., Abetz V., Müller A.H.E.: Polyelectrolyte block copolymer micelles. Adv. Polym. Sci. 166, 173–210 (2004)

    Article  Google Scholar 

  26. Frank R.L., Lieb E.H.: A compactness lemma and its application to the existence of minimizers for the liquid drop model. SIAM J. Math. Anal. 47, 4436–4450 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  27. Frenkel J.: On the splitting of heavy nuclei by slow neutrons. Phys. Rev. 55, 987 (1939)

    Article  ADS  MATH  Google Scholar 

  28. Fuchs K.: A quantum mechanical investigation of the cohesive forces of metallic copper. Proc. R. Soc. Lond. A 151, 585–602 (1935)

    Article  ADS  MATH  Google Scholar 

  29. Fusco N., Maggi F., Pratelli A.: The sharp quantitative isoperimetric inequality. Ann. Math. 168, 941–980 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Gamow G.: Mass defect curve and nuclear constitution. Proc. R. Soc. Lond. A 126, 632–644 (1930)

    Article  ADS  MATH  Google Scholar 

  31. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  32. Goldman D., Muratov C.B., Serfaty S.: The \({\Gamma }\)-limit of the two-dimensional Ohta–Kawasaki energy. I. Droplet density. Arch. Ration. Mech. Anal. 210, 581–613 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  33. Goldman D., Muratov C.B., Serfaty S.: The \({\Gamma }\)-limit of the two-dimensional Ohta–Kawasaki energy. Droplet arrangement via the renormalized energy. Arch. Ration. Mech. Anal. 212, 445–501 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  34. Goldman M., Novaga M.: Volume-constrained minimizers for the prescribed curvature problem in periodic media. Calc. Var. PDE 44, 297–318 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  35. Hashimoto M., Seki H., Yamada M.: Shape of nuclei in the crust of neutron star. Prog. Theor. Phys. 71, 320–326 (1984)

    Article  ADS  Google Scholar 

  36. Julin V.: Isoperimetric problem with a Coulombic repulsive term. Indiana Univ. Math. J. 63, 77–89 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  37. Julin, V., Pisante, G.: Minimality via second variation for microphase separation of diblock copolymer melts. J. Reine Angew. Math. (2015) (published online)

  38. Kirzhnits D.A.: Internal structure of super-dense stars. Sov. Phys. JETP 11, 365–368 (1960)

    Google Scholar 

  39. Knüpfer H., Muratov C.B.: On an isoperimetric problem with a competing non-local term. I. The planar case. Commun. Pure Appl. Math. 66, 1129–1162 (2013)

    Article  MATH  Google Scholar 

  40. Knüpfer H., Muratov C.B.: On an isoperimetric problem with a competing non-local term. II. The general case. Commun. Pure Appl. Math. 67, 1974–1994 (2014)

    Article  MATH  Google Scholar 

  41. Koester D., Chanmugam G.: Physics of white dwarf stars. Rep. Prog. Phys. 53, 837–915 (1990)

    Article  ADS  Google Scholar 

  42. Landkof N.S.: Foundations of Modern Potential Theory. Springer, New York (1972)

    Book  MATH  Google Scholar 

  43. Lattimer J.M., Pethick C.J., Ravenhall D.G., Lamb D.Q.: Physical properties of hot, dense matter: the general case. Nucl. Phys. A 432, 646–742 (1985)

    Article  ADS  Google Scholar 

  44. Lorenz C.P., Ravenhall D.G., Pethick C.J.: Neutron star crusts. Phys. Rev. Lett. 70, 379–382 (1993)

    Article  ADS  Google Scholar 

  45. Lu J., Otto F.: Nonexistence of a minimizer for Thomas–Fermi–Dirac–von Weizsäcker model. Commun. Pure Appl. Math. 67, 1605–1617 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  46. Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems. Cambridge Studies in Advanced Mathematics, vol. 135. Cambridge University Press, Cambridge (2012)

  47. Meitner L., Frisch O.R.: Disintegration of uranium by neutrons: a new type of nuclear reaction. Nature 143, 239–240 (1939)

    Article  ADS  MATH  Google Scholar 

  48. Muratov, C.B.: Theory of domain patterns in systems with long-range interactions of Coulomb type. Phys. Rev. E 66, 066108 (2002)

  49. Muratov C.B.: Droplet phases in non-local Ginzburg–Landau models with Coulomb repulsion in two dimensions. Commun. Math. Phys. 299, 45–87 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  50. Muratov C.B., Zaleski A.: On an isoperimetric problem with a competing non-local term: quantitative results. Ann. Glob. Anal. Geom. 47, 63–80 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  51. Myers W.D., Swiatecki W.J.: Nuclear masses and deformations. Nucl. Phys. 81, 1–60 (1966)

    Article  Google Scholar 

  52. Myers W.D., Swiatecki W.J.: Nuclear properties according to the Thomas–Fermi model. Nucl. Phys. A 601, 141–167 (1996)

    Article  ADS  Google Scholar 

  53. Nagai T., Fukuyama H.: Ground state of a Wigner crystal in a magnetic field. II. Hexagonal close-packed structure. J. Phys. Soc. Jpn. 52, 44–53 (1983)

    Article  ADS  Google Scholar 

  54. Ohta T., Kawasaki K.: Equilibrium morphologies of block copolymer melts. Macromolecules 19, 2621–2632 (1986)

    Article  ADS  Google Scholar 

  55. Okamoto M., Maruyama T., Yabana K., Tatsumi T.: Nuclear “pasta” structures in low-density nuclear matter and properties of the neutron-star crust. Phys. Rev. C 88, 025801 (2013)

    Article  ADS  Google Scholar 

  56. Oyamatsu K., Hashimoto M., Yamada M.: Further study of the nuclear shape in high-density matter. Prog. Theor. Phys. 72, 373–375 (1984)

    Article  ADS  Google Scholar 

  57. Pelekasis N.A., Tsamopoulos J.A., Manolis G.D.: Equilibrium shapes and stability of charged and conducting drops. Phys. Fluids A: Fluid Dyn. 2, 1328–1340 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  58. Pethick C.J., Ravenhall D.G.: Matter at large neutron excess and the physics of neutron-star crusts. Ann. Rev. Nucl. Part. Sci. 45, 429–484 (1995)

    Article  ADS  Google Scholar 

  59. Ravenhall D.G., Pethick C.J., Wilson J.R.: Structure of matter below nuclear saturation density. Phys. Rev. Lett. 50, 2066–2069 (1983)

    Article  ADS  Google Scholar 

  60. Ren X.F., Wei J.C.: On the multiplicity of solutions of two nonlocal variational problems. SIAM J. Math. Anal. 31, 909–924 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  61. Rigot S.: Ensembles quasi-minimaux avec contrainte de volume et rectifiabilité uniforme. Mémoires de la SMF, 2 série 82, 1–104 (2000)

    MATH  MathSciNet  Google Scholar 

  62. Rougerie N., Serfaty S.: Higher dimensional Coulomb gases and renormalized energy functionals. Commun. Pure Appl. Math. 69, 0519–0605 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  63. Salpeter E.E.: Energy and pressure of a zero-temperature plasma. Astrophys. J. 134, 669–682 (1961)

    Article  ADS  MathSciNet  Google Scholar 

  64. Sandier E., Serfaty S.: From the Ginzburg–Landau model to vortex lattice problems. Commun. Math. Phys. 313, 635–743 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  65. Schneider A., Horowitz C., Hughto J., Berry D.: Nuclear “pasta” formation. Phys. Rev. C 88, 065807 (2013)

    Article  ADS  Google Scholar 

  66. Sternberg P., Topaloglu I.: On the global minimizers of the nonlocal isoperimetric problem in two dimensions. Interfaces Free Bound. 13, 155–169 (2010)

    MATH  MathSciNet  Google Scholar 

  67. Tinkham M.: Introduction to Superconductivity, 2nd edn. McGraw-Hill, New York (1996)

    Google Scholar 

  68. von Weizsäcker C.F.: Zur Theorie der Kernmassen. Zeitschrift für Physik A 96, 431–458 (1935)

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Heidelberg, INF 294, 69120, Heidelberg, Germany

    Hans Knüpfer

  2. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 07102, USA

    Cyrill B. Muratov

  3. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Matteo Novaga

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  1. Hans Knüpfer
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  2. Cyrill B. Muratov
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  3. Matteo Novaga
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Correspondence to Cyrill B. Muratov.

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Communicated by H.-T. Yau

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Knüpfer, H., Muratov, C.B. & Novaga, M. Low Density Phases in a Uniformly Charged Liquid. Commun. Math. Phys. 345, 141–183 (2016). https://doi.org/10.1007/s00220-016-2654-3

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