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.

[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.

[3]

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

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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.

[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.

[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.

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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.

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

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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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

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.

[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.

[3]

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

[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.

[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.

[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.

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

[9]

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

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

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