Asymmetric and Symmetric Double Bubbles in a Ternary Inhibitory System

A ternary inhibitory system contains two terms in its free energy: the interface energy that favors microdomain growth and the longer ranging confinement energy that prevents unlimited spreading. In a parameter regime where two constituents are small in size compared to the third constituent and the longer ranging energy does not dominate, there is a double-bubble-like stable stationary point of the energy functional. The two minority constituents occupy the two bubbles of the double bubble, respectively, and the majority constituent fills the background. A special way of perturbing an exact double bubble leads to a restricted class of perturbed double bubbles that can be described by internal variables which are elements in a Hilbert space. The exact double bubble is nondegenerate in this class and nearby there is a perturbed double bubble that locally minimizes the free energy within the restricted class. This perturbed double bubble satisfies three of the four equations for stationary points of the fre...

[1]  H. A. Schwarz.  Beweis des Satzes, dass die Kugel kleinere Oberfläche besitzt, als jeder andere Körper gleichen Volumens , 1884 .

[2]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[3]  Xiaofeng Ren,et al.  The Pattern of Multiple Rings from Morphogenesis in Development , 2010, J. Nonlinear Sci..

[4]  Mark A. Peletier,et al.  Small Volume Fraction Limit of the Diblock Copolymer Problem: I. Sharp-Interface Functional , 2009, SIAM J. Math. Anal..

[5]  J. Zimba,et al.  The standard double soap bubble in R2 uniquely minimizes perimeter , 1993 .

[6]  Peter Sternberg,et al.  Cascade of Minimizers for a Nonlocal Isoperimetric Problem in Thin Domains , 2013, SIAM J. Math. Anal..

[7]  X. Ren,et al.  A Double Bubble in a Ternary System with Inhibitory Long Range Interaction , 2013 .

[8]  X. Ren,et al.  Ring pattern solutions of a free boundary problem in diblock copolymer morphology , 2009 .

[9]  Peter Sternberg,et al.  On the first and second variations of a nonlocal isoperimetric problem , 2007 .

[10]  Peter Sternberg,et al.  On the global minimizers of a nonlocal isoperimetric problem in two dimensions , 2011 .

[11]  Juncheng Wei,et al.  Single Droplet Pattern in the Cylindrical Phase of Diblock Copolymer Morphology , 2007, J. Nonlinear Sci..

[12]  Xiaofeng Ren,et al.  Diblock copolymer/homopolymer blends: Derivation of a density functional theory , 2005 .

[13]  Frank Morgan,et al.  Proof of the Double Bubble Conjecture , 2000, Am. Math. Mon..

[14]  I. Topaloglu.  On a nonlocal isoperimetric problem on the two-sphere , 2012 .

[15]  N. Fusco,et al.  Communications in Mathematical Physics Minimality via Second Variation for a Nonlocal Isoperimetric Problem , 2013 .

[16]  J. Hass,et al.  Double bubbles minimize , 2000, math/0003157.

[17]  Xiaofeng Ren,et al.  Triblock copolymer theory: free energy, disordered phase and weak segregation , 2003 .

[18]  J. Taylor,et al.  The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces , 1976 .

[19]  F. Almgren,et al.  Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints , 1975 .

[20]  S. Serfaty,et al.  The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. I. Droplet Density , 2011, 1201.0222.

[21]  Xiaofeng Ren,et al.  Oval Shaped Droplet Solutions in the Saturation Process of Some Pattern Formation Problems , 2009, SIAM J. Appl. Math..

[22]  Xiaofeng Ren,et al.  Triblock Copolymer Theory: Ordered ABC Lamellar Phase , 2003, J. Nonlinear Sci..

[23]  Cyrill B. Muratov,et al.  Communications in Mathematical Physics Droplet Phases in Non-local Ginzburg-Landau Models with Coulomb Repulsion in Two Dimensions , 2010 .

[24]  Yasumasa Nishiura,et al.  Some mathematical aspects of the micro-phase separation in diblock copolymers , 1995 .

[25]  G. Fredrickson,et al.  Block Copolymers—Designer Soft Materials , 1999 .

[26]  Xiaofeng Ren,et al.  MANY DROPLET PATTERN IN THE CYLINDRICAL PHASE OF DIBLOCK COPOLYMER MORPHOLOGY , 2007 .

[27]  Xiaofeng Ren,et al.  On the Multiplicity of Solutions of Two Nonlocal Variational Problems , 2000, SIAM J. Math. Anal..

[28]  Ohta,et al.  Dynamics of phase separation in copolymer-homopolymer mixtures. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[29]  Yoshihito Oshita.  Singular Limit Problem for Some Elliptic Systems , 2007, SIAM J. Math. Anal..

[30]  Juncheng Wei,et al.  A toroidal tube solution to a problem involving mean curvature and Newtonian potential , 2011 .

[31]  Giovanni Alberti,et al.  Uniform energy distribution for an isoperimetric problem with long-range interactions , 2008 .

[32]  Takao Ohta,et al.  Microphase separation of ABC-type triblock copolymers , 1993 .

[33]  B. Reichardt.  Proof of the Double Bubble Conjecture in Rn , 2007, 0705.1601.

[34]  Xiaofeng Ren,et al.  Spherical Solutions to a Nonlocal Free Boundary Problem from Diblock Copolymer Morphology , 2008, SIAM J. Math. Anal..

[36]  Juncheng Wei,et al.  Double tori solution to an equation of mean curvature and Newtonian potential , 2014 .

[37]  H. Meinhardt,et al.  A theory of biological pattern formation , 1972, Kybernetik.

[38]  Vesa Julin,et al.  Minimality via second variation for microphase separation of diblock copolymer melts , 2013, 1301.7213.

[39]  Cyrill B. Muratov,et al.  The Gamma-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement at the sharp interface level via the renormalized energy , 2012, 1210.5098.

[40]  K. Kawasaki,et al.  Equilibrium morphology of block copolymer melts , 1986 .

[41]  C. Muratov,et al.  On an Isoperimetric Problem with a Competing Nonlocal Term II: The General Case , 2012, 1206.7078.