February  2017, 37(2): 983-1012. doi: 10.3934/dcds.2017041

A stationary core-shell assembly in a ternary inhibitory system

Department of Mathematics, The George Washington University, Washington, DC 20052, USA

* Corresponding author: Xiaofeng Ren

Received  March 2015 Revised  January 2016 Published  November 2016

A ternary inhibitory system motivated by the triblock copolymer theoryis studied as a nonlocal geometric variational problem. The free energyof the system is the sum of two terms: the total size of the interfacesseparating the three constituents, and a longer ranging interaction energythat inhibits micro-domains from unlimited growth. In a particular parameterrange there is an assembly of many core-shells that exists as a stationaryset of the free energy functional. The cores form regions occupied by thefirst constituent of the ternary system, the shells form regionsoccupied by the second constituent, and the background is taken by thethird constituent. The constructive proof of the existence theorem revealsmuch information about the core-shell stationary assembly: asymptoticallyone can determine the sizes and locations of all the core-shells in theassembly. The proof also implies a kind of stability for the stationaryassembly.

Citation: Xiaofeng Ren, Chong Wang. A stationary core-shell assembly in a ternary inhibitory system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 983-1012. doi: 10.3934/dcds.2017041
References:
[1]

E. AcerbiN. Fusco and M. 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

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R. Choksi and M.A. Peletier, Small volume fraction limit of the diblock copolymer problem: I. sharp inteface functional, SIAM J. Math. Anal., 42 (2010), 1334-1370.  doi: 10.1137/090764888.  Google Scholar

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R. Choksi and X. Ren, Diblock copolymer -homopolymer blends: Derivation of a density functional theory, Physica D, 203 (2005), 100-119.  doi: 10.1016/j.physd.2005.03.006.  Google Scholar

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R. Choksi and P. Sternberg, On the first and second variations of a nonlocal isoperimetric problem, J. Reine Angew. Math., 611 (2007), 75-108.  doi: 10.1515/CRELLE.2007.074.  Google Scholar

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P.C. Fife and D. Hilhorst, The Nishiura-Ohnishi free boundary problem in the 1D case, SIAM J. Math. Anal., 33 (2001), 589-606.  doi: 10.1137/S0036141000372507.  Google Scholar

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X. Kang and X. Ren, The pattern of multiple rings from morphogenesis in development, J. Nonlinear Sci, 20 (2010), 747-779.  doi: 10.1007/s00332-010-9072-z.  Google Scholar

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M. Morini and P. Sternberg, Cascade of minimizers for a nonlocal isoperimetric problem in thin domains, SIAM J. Math. Anal., 46 (2014), 2033-2051.  doi: 10.1137/130932594.  Google Scholar

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C.B. Muratov, Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions, Comm. Math. Phys., 299 (2010), 45-87.  doi: 10.1007/s00220-010-1094-8.  Google Scholar

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H. Nakazawa and T. Ohta, Microphase separation of ABC-type triblock copolymers, Macromolecules, 26 (1993), 5503-5511.  doi: 10.1021/ma00072a031.  Google Scholar

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Y. Nishiura and I. Ohnishi, Some mathematical aspects of the microphase separation in diblock copolymers, Physica D, 84 (1995), 31-39.  doi: 10.1016/0167-2789(95)00005-O.  Google Scholar

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T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.  doi: 10.1021/ma00164a028.  Google Scholar

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Y. Oshita, Singular limit problem for some elliptic systems, SIAM J. Math. Anal., 38 (2007), 1886-1911.  doi: 10.1137/060656632.  Google Scholar

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X. Ren and J. Wei, On the multiplicity of solutions of two nonlocal variational problems, SIAM J. Math. Anal., 31 (2000), 909-924.  doi: 10.1137/S0036141098348176.  Google Scholar

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X. Ren and J. Wei, Triblock copolymer theory: Ordered ABC lamellar phase, J. Nonlinear Sci., 13 (2003), 175-208.  doi: 10.1007/s00332-002-0521-1.  Google Scholar

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X. Ren and J. Wei, Many droplet pattern in the cylindrical phase of diblock copolymer morphology, Rev. Math. Phys., 19 (2007), 879-921.  doi: 10.1142/S0129055X07003139.  Google Scholar

[22]

X. Ren and J. Wei, Single droplet pattern in the cylindrical phase of diblock copolymer morphology, J. Nonlinear Sci., 17 (2007), 471-503.  doi: 10.1007/s00332-007-9005-7.  Google Scholar

[23]

X. Ren and J. Wei, Spherical solutions to a nonlocal free boundary problem from diblock copolymer morphology, SIAM J. Math. Anal., 39 (2008), 1497-1535.  doi: 10.1137/070690286.  Google Scholar

[24]

X. Ren and J. Wei, Oval shaped droplet solutions in the saturation process of some pattern formation problems, SIAM J. Appl. Math., 70 (2009), 1120-1138.  doi: 10.1137/080742361.  Google Scholar

[25]

X. Ren and J. Wei, A toroidal tube solution to a problem involving mean curvature and Newtonian potential, Interfaces Free Bound., 13 (2011), 127-154.  doi: 10.4171/IFB/251.  Google Scholar

[26]

X. Ren and J. Wei, A double bubble in a ternary system with inhibitory long range interaction, Arch. Rat. Mech. Anal., 208 (2013), 201-253.  doi: 10.1007/s00205-012-0593-5.  Google Scholar

[27]

X. Ren and J. Wei, Asymmetric and symmetric double bubbles in a ternary inhibitory system, SIAM J. Math. Anal., 46 (2014), 2798-2852.  doi: 10.1137/140955720.  Google Scholar

[28]

X. Ren and J. Wei, Double tori solution to an equation of mean curvature and Newtonian potential, Calc. Var. Partial Differential Equations, 49 (2014), 987-1018.  doi: 10.1007/s00526-013-0608-6.  Google Scholar

[29]

X. Ren and J. Wei, A double bubble assembly as a new phase of a ternary inhibitory system, Arch. Rat. Mech. Anal., 215 (2015), 967-1034.  doi: 10.1007/s00205-014-0798-x.  Google Scholar

[30]

P. Sternberg and I. Topaloglu, A note on the global minimizers of the nonlocal isoperimetric problem in two dimensions, Interfaces Free Bound., 13 (2011), 155-169.  doi: 10.4171/IFB/252.  Google Scholar

[31]

I. Topaloglu, On a nonlocal isoperimetric problem on the two-sphere, Comm. Pure Appl. Anal., 12 (2013), 597-620.  doi: 10.3934/cpaa.2013.12.597.  Google Scholar

[32]

L. Xie, Analysis of the Long Range Interation in the Ternary System, PhD Thesis, The George Washington University. Google Scholar

show all references

References:
[1]

E. AcerbiN. Fusco and M. 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]

G. AlbertiR. Choksi and F. Otto, Uniform energy distribution for an isoperimetric problem with long-range interactions, J. Amer. Math. Soc., 22 (2009), 569-605.  doi: 10.1090/S0894-0347-08-00622-X.  Google Scholar

[3]

F.S. Bates and G.H. Fredrickson, Block copolymers -designer soft materials, Phys. Today, 52 (1999), 32-38.  doi: 10.1063/1.882522.  Google Scholar

[4]

R. Choksi and M.A. Peletier, Small volume fraction limit of the diblock copolymer problem: I. sharp inteface functional, SIAM J. Math. Anal., 42 (2010), 1334-1370.  doi: 10.1137/090764888.  Google Scholar

[5]

R. Choksi and X. Ren, Diblock copolymer -homopolymer blends: Derivation of a density functional theory, Physica D, 203 (2005), 100-119.  doi: 10.1016/j.physd.2005.03.006.  Google Scholar

[6]

R. Choksi and P. Sternberg, On the first and second variations of a nonlocal isoperimetric problem, J. Reine Angew. Math., 611 (2007), 75-108.  doi: 10.1515/CRELLE.2007.074.  Google Scholar

[7] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.   Google Scholar
[8]

P.C. Fife and D. Hilhorst, The Nishiura-Ohnishi free boundary problem in the 1D case, SIAM J. Math. Anal., 33 (2001), 589-606.  doi: 10.1137/S0036141000372507.  Google Scholar

[9]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, Basel, Stuttgart, 1984. Google Scholar

[10]

D. GoldmanC.B. Muratov and S. Serfaty, The Gamma-limit of the two-dimensional Ohta-Kawasaki energy. I. droplet density, Arch. Rat. Mech. Anal., 210 (2013), 581-613.  doi: 10.1007/s00205-013-0657-1.  Google Scholar

[11]

X. Kang and X. Ren, Ring pattern solutions of a free boundary problem in diblock copolymer morphology, Physica D, 238 (2009), 645-665.  doi: 10.1016/j.physd.2008.12.009.  Google Scholar

[12]

X. Kang and X. Ren, The pattern of multiple rings from morphogenesis in development, J. Nonlinear Sci, 20 (2010), 747-779.  doi: 10.1007/s00332-010-9072-z.  Google Scholar

[13]

M. Morini and P. Sternberg, Cascade of minimizers for a nonlocal isoperimetric problem in thin domains, SIAM J. Math. Anal., 46 (2014), 2033-2051.  doi: 10.1137/130932594.  Google Scholar

[14]

C.B. Muratov, Droplet phases in non-local Ginzburg-Landau models with Coulomb repulsion in two dimensions, Comm. Math. Phys., 299 (2010), 45-87.  doi: 10.1007/s00220-010-1094-8.  Google Scholar

[15]

H. Nakazawa and T. Ohta, Microphase separation of ABC-type triblock copolymers, Macromolecules, 26 (1993), 5503-5511.  doi: 10.1021/ma00072a031.  Google Scholar

[16]

Y. Nishiura and I. Ohnishi, Some mathematical aspects of the microphase separation in diblock copolymers, Physica D, 84 (1995), 31-39.  doi: 10.1016/0167-2789(95)00005-O.  Google Scholar

[17]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.  doi: 10.1021/ma00164a028.  Google Scholar

[18]

Y. Oshita, Singular limit problem for some elliptic systems, SIAM J. Math. Anal., 38 (2007), 1886-1911.  doi: 10.1137/060656632.  Google Scholar

[19]

X. Ren and J. Wei, On the multiplicity of solutions of two nonlocal variational problems, SIAM J. Math. Anal., 31 (2000), 909-924.  doi: 10.1137/S0036141098348176.  Google Scholar

[20]

X. Ren and J. Wei, Triblock copolymer theory: Ordered ABC lamellar phase, J. Nonlinear Sci., 13 (2003), 175-208.  doi: 10.1007/s00332-002-0521-1.  Google Scholar

[21]

X. Ren and J. Wei, Many droplet pattern in the cylindrical phase of diblock copolymer morphology, Rev. Math. Phys., 19 (2007), 879-921.  doi: 10.1142/S0129055X07003139.  Google Scholar

[22]

X. Ren and J. Wei, Single droplet pattern in the cylindrical phase of diblock copolymer morphology, J. Nonlinear Sci., 17 (2007), 471-503.  doi: 10.1007/s00332-007-9005-7.  Google Scholar

[23]

X. Ren and J. Wei, Spherical solutions to a nonlocal free boundary problem from diblock copolymer morphology, SIAM J. Math. Anal., 39 (2008), 1497-1535.  doi: 10.1137/070690286.  Google Scholar

[24]

X. Ren and J. Wei, Oval shaped droplet solutions in the saturation process of some pattern formation problems, SIAM J. Appl. Math., 70 (2009), 1120-1138.  doi: 10.1137/080742361.  Google Scholar

[25]

X. Ren and J. Wei, A toroidal tube solution to a problem involving mean curvature and Newtonian potential, Interfaces Free Bound., 13 (2011), 127-154.  doi: 10.4171/IFB/251.  Google Scholar

[26]

X. Ren and J. Wei, A double bubble in a ternary system with inhibitory long range interaction, Arch. Rat. Mech. Anal., 208 (2013), 201-253.  doi: 10.1007/s00205-012-0593-5.  Google Scholar

[27]

X. Ren and J. Wei, Asymmetric and symmetric double bubbles in a ternary inhibitory system, SIAM J. Math. Anal., 46 (2014), 2798-2852.  doi: 10.1137/140955720.  Google Scholar

[28]

X. Ren and J. Wei, Double tori solution to an equation of mean curvature and Newtonian potential, Calc. Var. Partial Differential Equations, 49 (2014), 987-1018.  doi: 10.1007/s00526-013-0608-6.  Google Scholar

[29]

X. Ren and J. Wei, A double bubble assembly as a new phase of a ternary inhibitory system, Arch. Rat. Mech. Anal., 215 (2015), 967-1034.  doi: 10.1007/s00205-014-0798-x.  Google Scholar

[30]

P. Sternberg and I. Topaloglu, A note on the global minimizers of the nonlocal isoperimetric problem in two dimensions, Interfaces Free Bound., 13 (2011), 155-169.  doi: 10.4171/IFB/252.  Google Scholar

[31]

I. Topaloglu, On a nonlocal isoperimetric problem on the two-sphere, Comm. Pure Appl. Anal., 12 (2013), 597-620.  doi: 10.3934/cpaa.2013.12.597.  Google Scholar

[32]

L. Xie, Analysis of the Long Range Interation in the Ternary System, PhD Thesis, The George Washington University. Google Scholar

Figure 1.  A double bubble assembly on the left and a core-shell assembly on the right
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