On the Ternary Ohta-Kawasaki Free Energy and Its One Dimensional Global Minimizers

We study the ternary Ohta–Kawasaki free energy that has been used to model triblock copolymer systems. Its one-dimensional global minimizers are conjectured to have cyclic patterns. However, some physical experiments and computer simulations found triblock copolymers forming noncyclic lamellar patterns. In this work, by comparing the free energies of the cyclic pattern and some noncyclic candidates, we show that the conjecture does not hold for some choices of parameters. Our results suggest that even in one dimension, the global minimizers may take on very different patterns in different parameter regimes. To unify the existing choices of the long range coefficient matrix, we present a reformulation of the long range term using a generalized charge interpretation, and thereby propose conditions on the matrix in order for the global minimizers to reproduce physically relevant nanostructures of block copolymers.

[1]  Bang Wong,et al.  Points of view: Color blindness , 2011, Nature Methods.

[2]  E Zaccarelli,et al.  Ground-state clusters for short-range attractive and long-range repulsive potentials. , 2004, Langmuir : the ACS journal of surfaces and colloids.

[3]  M. Novaga,et al.  Low Density Phases in a Uniformly Charged Liquid , 2015, 1504.05600.

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

[6]  K. Birdi.  Introduction to Surface and Colloid Chemistry , 2015 .

[7]  Frank S. Bates,et al.  Model ABC triblock copolymers and blends near the order-disorder transition , 2002 .

[8]  Emanuele Spadaro,et al.  Uniform energy and density distribution: diblock copolymers' functional , 2009 .

[9]  Yuliang Yang,et al.  Morphology and phase diagram of ABC linear triblock copolymers: parallel real-space self-consistent-field-theory simulation. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  M. Schick,et al.  Self-assembly of block copolymers , 1996 .

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

[12]  Kun Zhou,et al.  Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints , 2012, SIAM Rev..

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

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

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

[16]  Sara Daneri,et al.  Pattern Formation for a Local/Nonlocal Interaction Functional Arising in Colloidal Systems , 2018, SIAM J. Math. Anal..

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

[18]  X. Ren,et al.  A Double Bubble Assembly as a New Phase of a Ternary Inhibitory System , 2015 .

[19]  G. Lawlor.  Double Bubbles for Immiscible Fluids in ℝn , 2014 .

[20]  Juncheng Wei,et al.  Lamellar phase solutions for diblock copolymers with nonlocal diffusions , 2019, Physica D: Nonlinear Phenomena.

[21]  L. Bronsard,et al.  Droplet phase in a nonlocal isoperimetric problem under confinement , 2016, 1609.03589.

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

[23]  Y. Mogi,et al.  Superlattice Structures in Morphologies of the ABC Triblock Copolymers , 1994 .

[24]  Qiang Du,et al.  Nonlocal Modeling, Analysis, and Computation , 2019 .

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

[26]  A. Alexander-Katz,et al.  Dissipative particle dynamics for directed self-assembly of block copolymers. , 2019, The Journal of chemical physics.

[27]  Yuliang Yang,et al.  Microphase Ordering Mechanisms in Linear ABC Triblock Copolymers. A Dynamic Density Functional Study , 2005 .

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

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

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

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

[32]  A. Jayaraman,et al.  Molecular Dynamics Simulation and PRISM Theory Study of Assembly in Solutions of Amphiphilic Bottlebrush Block Copolymers , 2018, Macromolecules.

[33]  G. Gamow.  Mass Defect Curve and Nuclear Constitution , 1930 .

[34]  S. Serfaty,et al.  The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. Droplet Arrangement via the Renormalized Energy , 2014 .

[35]  M. Peletier,et al.  Stability of monolayers and bilayers in a copolymer-homopolymer blend model , 2007, 0710.3298.

[36]  M. Peletier,et al.  Copolymer–homopolymer blends: global energy minimisation and global energy bounds , 2007, 0704.3222.

[37]  Glenn H. Fredrickson,et al.  Parallel algorithm for numerical self-consistent field theory simulations of block copolymer structure , 2003 .

[38]  Thomas H. Epps,et al.  A Noncubic Triply Periodic Network Morphology in Poly(isoprene-b-styrene-b-ethylene oxide) Triblock Copolymers , 2002 .

[39]  Weihua Li,et al.  Theoretical Study of Phase Behavior of Frustrated ABC Linear Triblock Copolymers , 2012 .

[40]  Chong Wang,et al.  Stationary disk assemblies in a ternary system with long range interaction , 2019, Communications in Contemporary Mathematics.

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

[42]  Marco Cicalese,et al.  Ground States of a Two Phase Model with Cross and Self Attractive Interactions , 2015, SIAM J. Math. Anal..

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

[44]  L. Chew,et al.  Self-assembly of complex structures in a two-dimensional system with competing interaction forces. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

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

[47]  X. Ren,et al.  On the Derivation of a Density Functional Theory for Microphase Separation of Diblock Copolymers , 2003 .

[48]  Eris Runa,et al.  Exact Periodic Stripes for Minimizers of a Local/Nonlocal Interaction Functional in General Dimension , 2017, Archive for Rational Mechanics and Analysis.

[49]  G. Fredrickson,et al.  Block copolymer thermodynamics: theory and experiment. , 1990, Annual review of physical chemistry.

[50]  Wei Zheng,et al.  Morphology of ABC Triblock Copolymers , 1995 .

[51]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[52]  Thilo M. Simon,et al.  A Nonlocal Isoperimetric Problem with Dipolar Repulsion , 2018, Communications in Mathematical Physics.

[53]  C. Muratov,et al.  An Old Problem Resurfaces Nonlocally: Gamow's Liquid Drops Inspire Today's Research and Applications , 2017 .

[54]  Pattern formation in systems with competing interactions , 2008, 0811.3078.

[55]  R. Choksi,et al.  Nonlocal Shape Optimization via Interactions of Attractive and Repulsive Potentials , 2015, 1512.07282.

[56]  Y. Mogi,et al.  Preparation and morphology of triblock copolymers of the ABC type , 1992 .

[57]  Xiaofeng Ren,et al.  Asymmetric and Symmetric Double Bubbles in a Ternary Inhibitory System , 2014, SIAM J. Math. Anal..

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

[59]  R. Frank.  Non-spherical equilibrium shapes in the liquid drop model , 2019, Journal of Mathematical Physics.

[60]  HighWire Press.  Proceedings of the Royal Society of London. Series A, Containing papers of a mathematical and physical character , 1934 .

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

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

[63]  Eris Runa,et al.  On the optimality of stripes in a variational model with non-local interactions , 2016, Calculus of Variations and Partial Differential Equations.

[64]  Yuliang Yang,et al.  Morphology and phase diagram of complex block copolymers: ABC linear triblock copolymers. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[65]  C B Muratov.  Theory of domain patterns in systems with long-range interactions of Coulomb type. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[66]  Weiyu Wang,et al.  Block Copolymers: Synthesis, Self-Assembly, and Applications , 2017, Polymers.

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

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

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

[70]  Haojun Liang,et al.  Effect of polydispersity on the phase diagrams of linear ABC triblock copolymers in two dimensions. , 2005, The journal of physical chemistry. B.

[71]  M. Matsen.  Gyroid versus double-diamond in ABC triblock copolymer melts , 1998 .

[72]  X. Ren,et al.  Bubble assemblies in ternary systems with long range interaction , 2017, Communications in Mathematical Sciences.

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

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

[75]  Stan Alama,et al.  Periodic minimizers of a ternary non-local isoperimetric problem , 2019, Indiana University Mathematics Journal.

[76]  F. Bates,et al.  Morphological Behavior Bridging the Symmetric AB and ABC States in the Poly(styrene-b-isoprene-b-ethylene oxide) Triblock Copolymer System , 2001 .

[77]  Alicja Kerschbaum,et al.  Striped patterns for generalized antiferromagnetic functionals with power law kernels of exponent smaller than d+2 , 2021, Nonlinear Analysis.