Density-Selected Topological Pathways in the Melting of Single-Particle-Thick Stripes

Technical Summary: Density-Selected Topological Pathways in the Melting of Single-Particle-Thick Stripes

Problem Statement
The melting of microphase-separated systems, such as those forming stripes, involves complex changes in connectivity that are not fully captured by conventional structural (positional) and orientational descriptors. While the Kosterlitz–Thouless–Halperin–Nelson–Young framework describes melting in ordinary 2D crystals via defect unbinding, stripe-forming systems possess emergent connectivity at the domain scale. It remains unclear whether the melting of an ordered, one-particle-thick stripe phase necessarily destroys its system-spanning topology or if the resulting fluid topology is uniquely determined by the initial state. Furthermore, it is unknown whether the same ordered stripe phase can melt into topologically distinct fluids depending on density, specifically distinguishing between finite polymer-like clusters and a percolated, system-spanning network.

Methodology
The study employs molecular dynamics simulations of a two-dimensional monodisperse system interacting via a Lennard-Jones plus Gaussian (LJG) pair potential. This potential features a repulsive core-softened shoulder arising from competing particle separations, which stabilizes single-particle-thick stripes.

• Simulation Protocol: Seven isochoric heating paths were analyzed across a density range ($\rho = 0.395$ to $0.455$) where the stripe morphology is stable. Simulations utilized $N=4096$ particles (with system-size checks up to $N=16384$) and were heated from $T=0.04$ to $T=0.20$.
• Observables: The study combines standard thermodynamic and orientational metrics with graph-theoretic and topological analyses.
  • Thermodynamic/Orientational: Specific heat ($c_V$), nematic order parameter ($S_2$), local twofold order ($\langle|\psi_2|\rangle$), and orientational susceptibility ($\chi_2$).
  • Topological/Graph: Instantaneous particle configurations were mapped to undirected graphs (cutoff $r_c = 1.5\sigma$). Key metrics included the first Betti number ($\beta_1$), decomposed into contributions from components with and without periodic winding ($\beta_1^{wind}$ and $\beta_1^{nonwind}$), the fraction of particles in the graph 2-core, cluster-size distributions, and fundamental cycle lengths.
  • Dynamical: Long-time self-diffusion coefficients ($D_{MSD}$) and topology-conditioned mobilities (comparing endpoints, branch points, and backbone particles).

Key Results

1. Decoupling of Orientational and Topological Melting: The loss of global stripe alignment (marked by the peak in $R_2 = -\partial S_2/\partial T$ and $c_V$) occurs over a specific temperature interval that is largely independent of density. However, the topological reconstruction of the fluid occurs over a broader, density-dependent range. The orientational melting does not automatically imply the destruction of system-spanning connectivity.
2. Density-Selected Pathways: The final state of the melted fluid is selected by density:
   • Low Density ($\rho \lesssim 0.425$): Upon heating, the winding stripe backbones open and fragment. The system transitions into a fluid of finite, polymer-like clusters. The first Betti number ($\beta_1$) exhibits a non-monotonic behavior: an initial decrease (loss of winding) followed by an increase due to the proliferation of short, non-winding local loops within finite clusters.
   • High Density ($\rho \gtrsim 0.435$): The system retains a system-spanning, percolated network even after the stripes lose their orientational order. Here, $\beta_1$ increases monotonically with temperature. This growth is driven by the formation of additional independent cycles (branching and reconnection) within the winding component, rather than the creation of finite clusters. The fluid becomes "locally reticulated" but "globally connected."
3. Dynamical Fluidity: Both the finite-cluster fluid (low density) and the percolated network fluid (high density) remain dynamically fluid. The long-time diffusion coefficient increases smoothly through the melting transition. The persistent connectivity at high density is not a sign of gelation or arrest; rather, it represents a continuously restructuring fluid where bonds are constantly renewed while maintaining global percolation.
4. Mechanism of Reconstruction: The transition is mediated by local graph operations. At low densities, the dominant process is the opening of contours to create mobile endpoints. At high densities, the dominant process is the contact and reconnection of bent filaments to form junctions (branch points) and loops, which suppresses endpoint formation and maintains the cyclic backbone.
5. Non-Monotonic Stability: The temperature of maximum orientational response is non-monotonic with density, peaking near $\rho \approx 0.417$. This corresponds to a geometric matching point where the imposed stripe spacing aligns with the two characteristic length scales of the interaction potential.

Significance and Claims
The paper claims that the melting of single-particle-thick stripes is not a single transformation where all order parameters vanish simultaneously. Instead, it is a multistage process where orientational disordering and topological reconstruction are distinct events.

• Topological Distinction: The study demonstrates that the same ordered microphase can melt into topologically distinct fluids (finite clusters vs. percolated networks) solely based on density. This outcome cannot be resolved by thermodynamic or orientational descriptors alone; graph-based observables are required to distinguish these states.
• Beyond Defect-Mediated Melting: The results extend defect-based descriptions of melting by distinguishing between the opening of winding backbones, the creation of local cycles, and the reticulation of a spanning component.
• Fluid vs. Gel: The work clarifies that a percolated network in competing-interaction fluids does not necessarily imply an arrested gel state; the high-density network remains a diffusive, reconfiguring fluid.
• Methodological Framework: The paper establishes a framework combining orientational, thermodynamic, and topological observables to describe the melting of narrow filamentary microphases, highlighting that global connectivity is a distinct degree of freedom from local packing and orientation.