Finite volume method for self-consistent field theory of polymers: Material conservation and application

For the purpose of checking material conservation of various numerical algorithms used in the self-consistent-field theory (SCFT) of polymeric systems, we develop an algebraic method using matrix and bra-ket notation, which traces the Hermiticity of the product of the volume and evolution matrices. Algebraic tests for material conservation reveal that the popular pseudospectral method in the Cartesian grid conserves material perfectly, while the finite-volume method (FVM) is the proper tool when real-space SCFT with the Crank-Nicolson method is adopted in orthogonal coordinate systems. We also find that alternating direction implicit methods combined with the FVM exhibit small mass errors in the SCFT calculation. By introducing fractional cells in the FVM formulation, accurate SCFT calculations are performed for systems with irregular geometries and the results are consistent with previous experimental and theoretical works.

Figure 1

Pictorial representation of the discretized spherical coordinate system in two dimensions for the case of $I=3$, $J=6$, $r_1=5\mathrm{\Delta }r/2$, ${\theta }_1=\mathrm{\Delta }\theta /2$, and $\mathrm{\Delta }\theta =\pi /6$. The system boundary is at $r_{1/2}=2\mathrm{\Delta }r$ and $r_{3+1/2}=5\mathrm{\Delta }r$ in the $r$ direction and at ${\theta }_{1/2}=0$ and ${\theta }_{6+1/2}=\pi$ in the $\theta$ direction. Each green dot represents a grid point $\mathbf{i}=(i,j)$ and the neighboring region surrounded by dotted lines indicates the grid cell $C_{\mathbf{i}}$. The cell $C_{2,2}$ is highlighted with red lines as an example.

Figure 2

The $\mathcal{O}$ matrix for a 12-point ($3\times 4$) discretized domain in the two-dimensional spherical coordinate system, with $i\in [1,3]$ and $j\in [1,4]$. Only part of the $12\times 12$ matrix is shown. If Neumann boundary conditions are chosen in the $r$ direction, $O_{1,j}^{r-}$ and $O_{3,j}^{r+}$ become zeros.

Figure 3

The ${\mathcal{O}}_r$ matrix for a 12-point ($3\times 4$) discretized domain in the two-dimensional spherical coordinate system, with $i\in [1,3]$ and $j\in [1,4]$. Depending on the boundary conditions, a few of the elements shown may become zeros.

Figure 4

The ${\mathcal{O}}_{\theta }$ matrix for a 12-point ($3\times 4$) discretized domain in the two-dimensional spherical coordinate system, with $i\in [1,3]$ and $j\in [1,4]$. Depending on the boundary conditions, a few of the elements shown may become zeros.

Figure 5

Pictorial representation of orthogonal coordinates for polymers in irregular geometries. (a) Cartesian coordinate system with curved boundary. Cell $C_{3,j,2}$ is highlighted with red lines as an example. In the $y$ direction, the two-dimensional slice at the $j\mathrm{th}$ cells is shown. (b) Cylindrical coordinate system modeling a thin film with thickness variation. The slope of the upper surface is $\alpha$. In the $z$ direction, the two-dimensional slice is shown. Green dotted cells have the whole volumes and yellow dotted cells have only fractional volumes. Red dotted cells are skipped in the calculation because no flux flows in or out of the cells.

Figure 6

Possible AB diblock copolymer morphologies in a thickness-modulated geometry. The segment concentrations of the A block (${\varphi }_A$) and the B block (${\varphi }_B$) are shown in red and blue, respectively. The lamellae are aligned in the (a) $x$, (b) $y$, and (c) $z$ directions.

Figure 7

Simulation of AB block copolymers in a large thickness-modulated film with surface interaction (a) ${\eta }_A=0$, (b) ${\eta }_A=0.2$, and (c) ${\eta }_A=-0.3$. For clarification, only the regions with ${\varphi }_A\ge 0.5$ are shown with colors.

Figure 8

Convergence and error analysis for the cylindrical grid in Fig. 5. (a) Distorted bcc morphology observed for block copolymers with $f=0.25$ and $\chi N=18$, inside a box with varying thickness. (b) The FVM free energies at various $I=J=K$ values plotted as functions of $1/\mathrm{\Delta }s$. The FVM(OS) (solid lines) and FVM(non-OS) (dashed lines) are shown along with our best estimation for the free energy drawn with a dotted line [FVM(OS), with $I=250$ and $1/\mathrm{\Delta }s=700$].

Figure 9

Pictorial representation of the discretized cylindrical coordinate system, for the case of $I=3$, $J=12$, ${\rho }_{1/2}=2\mathrm{\Delta }\rho$, and $\mathrm{\Delta }\phi =\pi /6$. In the $z$ direction, the two-dimensional slice at the $k\mathrm{th}$ cells is shown. The system boundary is at ${\rho }_{\frac{1}{2}}=2\mathrm{\Delta }\rho$ and ${\rho }_{3+1/2}=5\mathrm{\Delta }\rho$ in the $\rho$ direction and there is a periodic boundary in the $\phi$ direction so that ${\phi }_{1/2}=0$ is equivalent to ${\phi }_{12+1/2}=2\pi$. Each green dot represents a grid point $(i,j,k)$ and the neighboring region surrounded by dotted lines indicates the grid cell $C_{i,j,k}$. The cell $C_{2,2,k}$ is highlighted with red lines as an example. Each blue dot denotes the midpoint of a surface where the scale factor for the gradient calculation must be selected.