×

zbMATH — the first resource for mathematics

Pattern formation for a local/nonlocal interaction functional arising in colloidal systems. (English) Zbl 07245265
The ability of matter to arrange itself in periodic structures is often referred to as spontaneous pattern formation. This phenomenon is of fundamental importance in science, technology, engineering, and mathematics, and it is often caused by the interaction between local attractive and nonlocal repulsive forces. In this paper, the authors study pattern formation for a physical local/nonlocal interaction functional where the local attractive term is given by the 1-perimeter and the nonlocal repulsive term is the Yukawa (or screened Coulomb) potential [B. Derjaguin and L. Landau, “Theory of the stability of strongly charged lyophobic sols and of the adhesion of strongly charged particles in solutions of electrolytes”, Acta Phys. Chem. USSR, 14, 633–662 (1941)]. This model is physically interesting as it is the \(\Gamma\)-limit of a double Yukawa model used to explain and simulate pattern formation in colloidal systems. The authors prove that in a suitable regime minimizers are periodic stripes in any space dimension.
MSC:
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
49N20 Periodic optimal control problems
49S05 Variational principles of physics (should also be assigned at least one other classification number in Section 49-XX)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. Bomont, J. Bretonnet, D. Costa, and J. Hansen, Communication: Thermodynamic signatures of cluster formation in fluids with competing interactions, J. Chem. Phys., 137 (2012), 011101.
[2] C. Bores, E. Lomba, A. Perera, and N. G. Almarza, Pattern formation in binary fluid mixtures induced by short-range competing interactions, J. Chem. Phys., 143 (2015), 084501.
[3] G. Bouchitté, I. Fonseca, and L. Mascarenhas, A global method for relaxation, Arch. Ration. Mech. Anal., 145 (1998), pp. 51-98. · Zbl 0921.49004
[4] M. Burger, J. A. Carrillo, J.-F. Pietschmann, M. Schmidtchen, Segregation and Gap Formation in Cross-Diffusion Models, e-print, arXiv:1906.03712, 2019. · Zbl 1445.35048
[5] M. Burger, B. Düring, L. M. Kreusser, P. A. Markowich, and C.-B. Schönlieb, Pattern formation of a nonlocal, anisotropic interaction model, Math. Models Methods Appl. Sci., 28 (2018), pp. 409-451. · Zbl 1383.35024
[6] J. A. Carrillo, M. Di Francesco, A. Figalli, T. Laurent, and D. Slep \(\check{{c}}\) ev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J., 156 (2011), pp. 229-271.
[7] J. A. Carrillo, B. Düring, L. M. Kreusser, and C.-B. Schönlieb, Stability analysis of line patterns of an anisotropic interaction model, SIAM J. Appl. Dyn. Syst., 18 (2019), pp. 1798-1845. · Zbl 1430.35026
[8] B. Chacko, C. Chalmers, and A. J. Archer, Two-dimensional colloidal fluids exhibiting pattern formation, J. Chem. Phys., 143 (2015), 244904.
[9] S. Daneri and E. Runa, Exact periodic stripes for a minimizers of a local/non-local interaction functional in general dimension, Arch. Ration. Mech. Anal., 231 (2019), pp. 519-589. · Zbl 1410.82005
[10] B. Derjaguin and L. Landau, Theory of the stability of strongly charged lyophobic sols and of the adhesion of strongly charged particles in solutions of electrolytes, Acta Phys. Chem. USSR, 14 (1941), pp. 633-662.
[11] J. Fröhlich, R. Israel, E. H. Lieb, and B. Simon, Phase transitions and reflection positivity. I. General theory and long range lattice models, Comm. Math. Phys., 62 (1978), pp. 1-34.
[12] J. Fröhlich, R. Israel, E. H. Lieb, and B. Simon, Phase transitions and reflection positivity. II. Lattice systems with short range and Coulomb interactions, J. Stat. Phys., 22 (1980), pp. 297-347.
[13] A. Giuliani, J. L. Lebowitz, and E. H. Lieb, Ising models with long-range dipolar and short range ferromagnetic interactions, Phys. Rev. B, 74 (2006), 064420.
[14] A. Giuliani, J. L. Lebowitz, and E. H. Lieb, Striped phases in two dimensional dipole systems, Phys. Rev. B, 76 (2007), 184426.
[15] A. Giuliani, J. L. Lebowitz, and E. H. Lieb, Periodic minimizers in 1D local mean field theory, Comm. Math. Phys., 286 (2009), pp. 163-177. · Zbl 1173.82008
[16] A. Giuliani, J. L. Lebowitz, and E. H. Lieb, Modulated phases of a one-dimensional sharp interface model in a magnetic field, Phys. Rev. B, 80 (2009), 134420.
[17] A. Giuliani and R. Seiringer, Periodic striped ground states in Ising models with competing interactions, Comm. Math. Phys., 347 (2016), pp. 983-1007. · Zbl 1351.82019
[18] P. Godfrin, R. Castan \(\tilde{n}\) eda-Priego, Y. Liu, and N. Wagner, Intermediate range order and structure in colloidal dispersions with competing interactions, J. Chem. Phys. 139 (2013), 154904.
[19] D. Goldman, C. B. Muratov, and S. Serfaty, The \(\gamma \)-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density, Arch. Ration. Mech. Anal., 210 (2013), pp. 581-613. · Zbl 1296.82018
[20] D. Goldman, C. B. Muratov, and S. Serfaty, The Gamma-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement via the renormalized energy, Arch. Ration. Mech. Anal. 212 (2014), pp. 445-501. · Zbl 1305.35134
[21] M. Goldman and E. Runa, On the optimality of stripes in a variational model with nonlocal interactions, Calc. Var. Partial Differential Equations, 58 (2019), 103. · Zbl 1415.49033
[22] A. Imperio and L. Reatto, Microphase separation in two-dimensional systems with competing interactions, J. Chem. Phys., 124 (2006), 164712.
[23] C. B. Muratov, Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions, Comm. Math. Phys., 299 (2010), pp. 45-87. · Zbl 1205.82107
[24] S. Müller, Singular perturbations as a selection criterion for periodic minimizing sequences, Calc. Var. Partial Differential Equations, 1 (1993), pp. 169-204. · Zbl 0821.49015
[25] K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions, Comm. Math. Phys., 31 (1973), pp. 83-112. · Zbl 0274.46047
[26] W. C. K. Poon, Colloidal suspensions, in The Oxford Handbook of Soft Condensed Matter, E. M. Terentjer and D. A. Weitz, eds., Oxford University Press, Oxford, UK, 2015, pp. 1-50.
[27] A. Stradner, H. Sedgwick, F. Cardinaux, W. C. Poon, S. U. Egelhaaf, and P. Schurtenberger, Equilibrium cluster formation in concentrated protein solutions and colloids, Nature, 432 (2004), pp. 492-495.
[28] E. Verwey and J. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948.
[29] H. Yukawa, On the interaction of elementary particles, I, Proc. Phys. Math. Soc. Jpn., 17 (1935), pp. 48-57. · JFM 61.1592.09
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.