# American Institute of Mathematical Sciences

June  2020, 40(6): 3957-3979. doi: 10.3934/dcds.2020048

## The impact of the domain boundary on an inhibitory system: Interior discs and boundary half discs

 1 The George Washington University, Washington, DC 20052, USA 2 Alvernia University, Reading, PA 19607, USA

* Corresponding author. Phone: 1 202 994-6791; Fax: 1 202 994-6760

Received  March 2019 Revised  August 2019 Published  October 2019

Fund Project: Xiaofeng Ren is supported in part by NSF grant DMS-1714371

When the Ohta-Kawasaki theory for diblock copolymers is applied to a bounded domain with the Neumann boundary condition, one faces the possibility of micro-domain interfaces intersecting the system boundary. In a particular parameter range, there exist stationary assemblies, stable in some sense, that consist of both perturbed discs in the interior of the system and perturbed half discs attached to the boundary of the system. The circular arcs of the half discs meet the system boundary perpendicularly. The number of the interior discs and the number of the boundary half discs are arbitrarily prescribed and their radii are asymptotically the same. The locations of these discs and half discs are determined by the minimization of a function related to the Green's function of the Laplace operator with the Neumann boundary condition. Numerical calculations based on the theoretical findings show that boundary half discs help lower the energy of stationary assemblies.

Citation: Xiaofeng Ren, David Shoup. The impact of the domain boundary on an inhibitory system: Interior discs and boundary half discs. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3957-3979. doi: 10.3934/dcds.2020048
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##### References:
 [1] E. Acerbi, N. 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] 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 [3] 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 [4] 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 [5] D. Goldman, C. 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 [6] 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 [7] 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 [8] T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.  doi: 10.1021/ma00164a028.  Google Scholar [9] X. F. Ren and D. Shoup, The impact of the domain boundary on an inhibitory system: Existence and location of a stationary half disc, Comm. Math. Phys., 340 (2015), 355-412.  doi: 10.1007/s00220-015-2451-4.  Google Scholar [10] X. F. Ren and J. C. 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 [11] X. F. Ren and J. C. 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
From the left of the first row with $n_i = 10$ and $n_b = 0$ to the right of the last row with $n_i = 0$ and $n_b = 20$, these assemblies, of $n_i + \frac{n_b}{2} = 10$, minimize $F$. Among them, the right one on the first row has the least $F$ value. Here $\omega = 0.2$
Stationary assemblies with $n_i+\frac{n_b}{2}$ less than or equal to 4
 $n_i + \frac{n_b}{2}$ $n_i$ $n_b$ Minimum $F$ 1 1 0 -0.0796 1 0 2 -0.0307 1.5 1 1 -0.1365 1.5 0 3 -0.1131 2 2 0 -0.2221 2 1 2 -0.2333 2 0 4 -0.2025 2.5 2 1 -0.3440 2.5 1 3 -0.3374 2.5 0 5 -0.2922 3 3 0 -0.4619 3 2 2 -0.4706 3 1 4 -0.4421 3 0 6 -0.3780 3.5 3 1 -0.5955 3.5 2 3 -0.5890 3.5 1 5 -0.5707 3.5 0 7 -0.4573 4 4 0 -0.7301 4 3 2 -0.7287 4 2 4 -0.6783 4 1 6 -0.6963 4 0 8 -0.5280
 $n_i + \frac{n_b}{2}$ $n_i$ $n_b$ Minimum $F$ 1 1 0 -0.0796 1 0 2 -0.0307 1.5 1 1 -0.1365 1.5 0 3 -0.1131 2 2 0 -0.2221 2 1 2 -0.2333 2 0 4 -0.2025 2.5 2 1 -0.3440 2.5 1 3 -0.3374 2.5 0 5 -0.2922 3 3 0 -0.4619 3 2 2 -0.4706 3 1 4 -0.4421 3 0 6 -0.3780 3.5 3 1 -0.5955 3.5 2 3 -0.5890 3.5 1 5 -0.5707 3.5 0 7 -0.4573 4 4 0 -0.7301 4 3 2 -0.7287 4 2 4 -0.6783 4 1 6 -0.6963 4 0 8 -0.5280
Stationary assemblies with $n_i+\frac{n_b}{2} = 10$
 $n_i + \frac{n_b}{2}$ $n_i$ $n_b$ Minimum $F$ 10 10 0 -2.5781 10 9 2 -2.5819 10 8 4 -2.5885 10 7 6 -2.5793 10 6 8 -2.5644 10 5 10 -2.5433 10 4 12 -2.4791 10 3 14 -2.2864 10 2 16 -1.9222 10 1 18 -1.3549 10 0 20 -0.3911
 $n_i + \frac{n_b}{2}$ $n_i$ $n_b$ Minimum $F$ 10 10 0 -2.5781 10 9 2 -2.5819 10 8 4 -2.5885 10 7 6 -2.5793 10 6 8 -2.5644 10 5 10 -2.5433 10 4 12 -2.4791 10 3 14 -2.2864 10 2 16 -1.9222 10 1 18 -1.3549 10 0 20 -0.3911
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