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 and Continuous Dynamical Systems, 2020, 40 (6) : 3957-3979. doi: 10.3934/dcds.2020048
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]

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

[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.

[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.

[5]

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.

[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.

[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.

[8]

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

[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.

[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.

[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.

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]

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

[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.

[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.

[5]

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.

[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.

[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.

[8]

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

[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.

[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.

[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.

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
$ 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
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
$ 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
[1]

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

[2]

Jianghong Bao. Complex dynamics in the segmented disc dynamo. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3301-3314. doi: 10.3934/dcdsb.2016098

[3]

Matthias Ngwa, Ephraim Agyingi. A mathematical model of the compression of a spinal disc. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1061-1083. doi: 10.3934/mbe.2011.8.1061

[4]

Jinzhi Lei, Frederic Y. M. Wan, Arthur D. Lander, Qing Nie. Robustness of signaling gradient in drosophila wing imaginal disc. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 835-866. doi: 10.3934/dcdsb.2011.16.835

[5]

Jaume Llibre, Arefeh Nabavi. Phase portraits of the Selkov model in the Poincaré disc. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022056

[6]

Zhisong Chen, Shong-Iee Ivan Su. Assembly system with omnichannel coordination. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1863-1889. doi: 10.3934/jimo.2021047

[7]

Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121

[8]

Florian Dorsch, Hermann Schulz-Baldes. Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 945-976. doi: 10.3934/dcdsb.2021076

[9]

Song Wang, Xia Lou. An optimization approach to the estimation of effective drug diffusivity: From a planar disc into a finite external volume. Journal of Industrial and Management Optimization, 2009, 5 (1) : 127-140. doi: 10.3934/jimo.2009.5.127

[10]

Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (I) --- Basic state solutions. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1091-1116. doi: 10.3934/dcdsb.2004.4.1091

[11]

Jianghong Bao, Dandan Chen, Yongjian Liu, Hongbo Deng. Coexisting hidden attractors in a 5D segmented disc dynamo with three types of equilibria. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6053-6069. doi: 10.3934/dcdsb.2019130

[12]

Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (II): interfacial instability and pattern formation at early stage of growth. Communications on Pure and Applied Analysis, 2004, 3 (3) : 527-543. doi: 10.3934/cpaa.2004.3.527

[13]

Zhouchao Wei, Fanrui Wang, Huijuan Li, Wei Zhang. Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021263

[14]

Liping Wang, Juncheng Wei. Solutions with interior bubble and boundary layer for an elliptic problem. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 333-351. doi: 10.3934/dcds.2008.21.333

[15]

Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228

[16]

Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271

[17]

Genni Fragnelli, Gisèle Ruiz Goldstein, Jerome Goldstein, Rosa Maria Mininni, Silvia Romanelli. Generalized Wentzell boundary conditions for second order operators with interior degeneracy. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 697-715. doi: 10.3934/dcdss.2016023

[18]

Yuebin Hao. Electromagnetic interior transmission eigenvalue problem for an inhomogeneous medium with a conductive boundary. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1387-1397. doi: 10.3934/cpaa.2020068

[19]

Jérôme Coville, Juan Dávila. Existence of radial stationary solutions for a system in combustion theory. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 739-766. doi: 10.3934/dcdsb.2011.16.739

[20]

Stéphanie Portet, Julien Arino. An in vivo intermediate filament assembly model. Mathematical Biosciences & Engineering, 2009, 6 (1) : 117-134. doi: 10.3934/mbe.2009.6.117

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (191)
  • HTML views (458)
  • Cited by (0)

Other articles
by authors

[Back to Top]