Quantifying microstructural evolution via time-dependent reduced-dimension metrics based on hierarchical $n$-point polytope functions

We devise reduced-dimension metrics for effectively measuring the distance between two points (i.e., microstructures) in the microstructure space and quantifying the pathway associated with microstructural evolution, based on a recently introduced set of hierarchical $n$-point polytope functions $P_n$. The $P_n$ functions provide the probability of finding particular $n$-point configurations associated with regular $n$ polytopes in the material system, and are a special subset of the standard $n$-point correlation functions $S_n$ that effectively decompose the structural features in the system into regular polyhedral basis with different symmetries. The $n\mathrm{th}$ order metric ${\mathrm{\Omega }}_n$ is defined as the ${\mathbb{L}}_1$ norm associated with the $P_n$ functions of two distinct microstructures. By choosing a reference initial state (i.e., a microstructure associated with $t_0=0$), the ${\mathrm{\Omega }}_n\left(t\right)$ metrics quantify the evolution of distinct polyhedral symmetries and can in principle capture emerging polyhedral symmetries that are not apparent in the initial state. To demonstrate their utility, we apply the ${\mathrm{\Omega }}_n$ metrics to a two-dimensional binary system undergoing spinodal decomposition to extract the phase separation dynamics via the temporal scaling behavior of the corresponding ${\mathrm{\Omega }}_n\left(t\right)$, which reveals mechanisms governing the evolution. Moreover, we employ ${\mathrm{\Omega }}_n\left(t\right)$ to analyze pattern evolution during vapor deposition of phase-separating alloy films with different surface contact angles, which exhibit rich evolution dynamics including both unstable and oscillating patterns. The ${\mathrm{\Omega }}_n$ metrics have potential applications in establishing quantitative processing-structure-property relationships, as well as real-time processing control and optimization of complex heterogeneous material systems.

Figure 1

Schematic illustration of the $n$-point polytope functions $P_n$.

Figure 2

Upper panels: Snapshots of the evolving 2D binary system with the double-well potential at selected time points. Lower panels: The associated $n$-point polytope functions $P_n$ at the corresponding time points (indicated by the same color code). The unit of distance $r$ is in pixels.

Figure 3

${\mathrm{\Omega }}_n$ metrics for the 2D binary system in both linear scale (a) and logarithmic scale (b) with least-square linear fits indicated by dashed lines. $\delta {\mathrm{\Omega }}_n\left(t\right)$ for the system is shown in (c).

Figure 4

Representative snapshots of the growing film containing hillocks with contact angle $\theta ={32}^{\circ }$ at film-vapor interface (see Ref. [47] for details).

Figure 5

Upper panels: Representative snapshots of the top surface patterns associated with the blue phase at different time points during the film growth with contact angle $\theta ={32}^{\circ }$. Lower panels: The associated $n$-point polytope functions $P_n$ at the corresponding time points (indicated by the same color code). The unit of distance $r$ is in pixels.

Figure 6

${\mathrm{\Omega }}_n$ metrics for the evolving surface patterns during the film growth with contact angle $\theta ={32}^{\circ }$ in both linear scale (a) and logarithmic scale (b) with least-square linear fit shown as a dashed line. $\delta {\mathrm{\Omega }}_n\left(t\right)$ for the system is shown in (c).

Figure 7

The ${\mathrm{\Omega }}_n^{*}\left(t\right)$ metrics for the the evolving surface patterns during the film growth with contact angle $\theta ={32}^{\circ }$ with a different reference state. (a) respectively shows the original reference (upper panel) and the new reference (lower panel). (b) shows ${\mathrm{\Omega }}_n^{*}\left(t\right)$ in linear scale and (c) shows ${\mathrm{\Omega }}_n^{*}\left(t\right)$ in logarithmic scale with linear fitting shown as a dashed line.

Figure 8

Upper panels: Representative snapshots of the top surface patterns associated with the blue phase at different time points during the film growth with contact angle $\theta ={51}^{\circ }$. Lower panels: The associated $n$-point polytope functions $P_n$ at the corresponding time points (indicated by the same color code). The unit of distance $r$ is in pixels.

Figure 9

${\mathrm{\Omega }}_n$ metrics for the evolving surface patterns during the film growth with contact angle $\theta ={51}^{\circ }$ in both linear scale (a) and logarithmic scale (b) with least-square linear fitting shown as a dashed line. $\delta {\mathrm{\Omega }}_n\left(t\right)$ for the system is shown in (c).