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

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

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

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

Abstract / Introduction Full Text(HTML) Figure(1) / Table(2) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: 82B24, 82D60, 92C15.

    Citation:

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  • Figure 1.  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 $

    Table 1.  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
     | Show Table
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    Table 2.  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
     | Show Table
    DownLoad: CSV
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