Ultraspherical Spectral Method for Block Copolymer Systems on Unit Disk

TL;DR

This work develops an ultraspherical spectral method-based framework to simulate the Ohta–Kawasaki and Nakazawa–Ohta block copolymer models on the unit disk, incorporating long‑range interactions. A stabilized second‑order BDF time discretization is paired with a Chebyshev–Fourier spatial discretization to form semi‑ and fully‑discrete energy‑stable schemes, with proofs and demonstrations of second‑order temporal convergence. Numerically, the approach reveals rich coarsening dynamics and pattern formation in the disk, including interior and boundary bubble assemblies for the binary OK model and diverse head‑to‑tail bubble patterns for the ternary NO model. The method provides efficient, spectrally accurate simulations in polar geometries and offers insight into how long‑range interactions shape pattern formation on disk domains, with potential implications for theoretical analysis and materials design.

Abstract

In this paper, we investigate ultraspherical spectral method for the Ohta-Kawasaki (OK) and Nakazawa-Ohta (NO) models in the disk domain, representing diblock and triblock copolymer systems, respectively. We employ ultraspherical spectral discretization for spatial variables in the disk domain and apply the second-order backward differentiation formula (BDF) method for temporal discretization. To our best knowledge, this is the first study to develop a numerical method for diblock and triblock copolymer systems with long-range interactions in disk domains. We show the energy stability of the numerical method in both semi-discrete and fully-discrete discretizations. In our numerical experiments, we verify the second-order temporal convergence rate and the energy stability of the proposed methods. Our numerical results show that the coarsening dynamics in diblock copolymers lead to bubble assemblies both inside and on the boundary of the disk. Additionally, in the triblock copolymer system, we observe several novel pattern formations, including single and double bubble assemblies in the unit disk. These findings are detailed through extensive numerical experiments.

Figures (4)

• Figure 4.1: Coarsening dynamics in binary system on a unit disk with $\gamma = 2500$(top), $\gamma = 3500$(middle), and $\gamma = 4000$(bottom).
• Figure 4.2: Coarsening dynamics in ternary system on a unit disk with $\gamma_{11} = \gamma_{22}=6000, \gamma_{12} = \gamma_{21}=0$ (top), and $\gamma_{11} = \gamma_{22} = 15000,\gamma_{12} = \gamma_{21}=0$ (bottom).
• Figure 4.3: Coarsening dynamics in ternary system on a unit disk with $\gamma_{11} = \gamma_{22}=6000, \gamma_{12} = \gamma_{21}=3000$ (top), $\gamma_{11} = \gamma_{22} = 6000,\gamma_{12} = \gamma_{21}=6000$ (middle), and $\gamma_{11} = \gamma_{22} = 6000, \gamma_{12} = \gamma_{21}=8000$ (bottom).
• Figure 4.4: different equilibrium in the ternary system on a unit disk with $\gamma_{11} = \gamma_{22}=6000, \gamma_{12} = \gamma_{21}=0$, and the equilibrium energy: $E_{\text{(a)}}<E_{\text{(b)}}$.

Theorems & Definitions (8)

• Definition 2.1: BMC-II function Wilber_Townsend_Wright2017
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.2