A flexible implementation of strong segregation theory for two dimensional ABC star terpolymer morphologies

TL;DR

This work introduces a flexible, polygon-based implementation of strong segregation theory (SST) for two-dimensional ABC star terpolymers, enabling efficient calculation of free energies and construction of phase diagrams across a wide range of compositions and interaction strengths. By tessellating space with Strongly Segregated Polygons (SSPs) around core cylinders and optimizing node positions, the method captures both single-core and multi-core morphologies and supports extension to three-dimensional or quasicrystalline patterns. Key insights include the dominance of single-core structures in many phase diagrams, the appearance of Sigma-phase and other multi-core motifs, and the ability to explore asymmetric interactions with relatively low computational cost. The approach provides a rapid screening tool that complements SCFT and experiments, and its framework could be adapted to other architectures and higher-dimensional systems, including potential quasicrystal approximants.

Abstract

We present a novel computational implementation of strong segregation theory, developed specifically for calculations of phase separated ABC star terpolymers. The method allows calculation of free energies of common two-dimensional morphologies for these polymers and the efficient construction of phase diagrams. The branch points of the ABC star terpolymers are localized in core regions, modeled as cylinders in three dimensions, and our framework is applicable to morphologies with single and multiple core types. Our central idea is that all the structures we wish to model can be assembled from a flexible base motif, which we call Strongly Segregated Polygons. This method is useful for exploring a wide range of complex morphologies, using a range of compositions and interaction strengths. We focus on 2D morphologies of ABC star terpolymers, but our method could be extended into three dimensions and to other molecular architectures, and in principle to large, irregular quasiperiodic two-dimensional structures.

Figures (10)

• Figure 1: A graphical representation of (a) an ABC star terpolymer and (b) phase separation in ABC star terpolymer melts. The three branches (red, blue and yellow) join at their core, and in three dimensions, the different polymer types collect into three domains, with the cores aligned in a straight line, forming columnar structures in one dimension and ordered geometrical patterns in the other two. The cores form a structure that can be taken as a narrow cylinder (gray) to which the polymer branches are grafted. The radius of this cylinder is comparable to the length of a monomer unit. The interfaces between the three domains are indicated by thick black lines. The stretching free energy is calculated by slicing each domain into small wedges, indicated by dashed lines.
• Figure 2: Geometrical structure of a Strongly Segregated Polygon (SSP). In (a), the six-sided SSP has one node (Node 0) in the center and six nodes around the edges. The three colors indicate the regions of three types of polymer. In (b), we show triangle 061 of the SSP, with a coordinate $t$ that runs from $0$ to $1$ along the bottom of the triangle. The dimensionless height of the triangle is $H_{061}$ and the length of its base is $L_{061}$. A wedge (at position $t$) within this triangle is illustrated.
• Figure 3: Geometrical structures of SSPs. In (a) we demonstrate how to place six SSPs to form a periodic patch of the $[6.6.6]$ morphology. The black dots indicate the nodes of the SSPs, and one SSP is marked out with larger dots. Colored lines along edges indicate matching rules which makes the morphology periodic. In (a), the same type of lines indicates joining of edges to make the periodic patch. In (b) we give an illustration of the SSP optimisation framework. The pink dot is the reference node, green dots are the free nodes and light blue dots are the periodic nodes. The black arrows indicates the periodicity, $[v_{xx},v_{xy},v_{yy}]$ of the repeating patch. One SSP is indicated by dashed lines.
• Figure 4: All candidate morphologies considered in this work are illustrated here. The color scheme is A (red), B (blue) and C (yellow). The bold triangles are the SSPs, and each SSP contains six triangles and a single ABC core. In these initial configurations, the SSPs are triangles but there are nodes at the domain boundaries on each edge, so they are in fact six-sided polygons (as in \ref{['fig:sspimplementation']}). The exception is the $L+C$ example, which has an eight-sided SSP containing eight triangles: two in the red and blue regions and four in the yellow. The SSP polygons are different sizes, but the proportion of red, blue and yellow in each SSP is equal to $\phi_A$, $\phi_B$ and $\phi_C$, which in these examples are all equal to one third. These initial configurations are the starting points for minimizing the free energy.
• Figure 5: Phase diagrams for ABC star terpolymer with symmetric interaction strengths and with $c=100$. On the left (a) is the phase diagram created using only six-sided SSPs. The stable morphologies are marked around the phase diagram, with the six permutations of each morphology indicated with shades of color. On the right (b) is the phase diagram where we include eight-sided SSPs, which allow lamellar phases. The data for this figure is available from Joseph2025data.