Demixing of compact polymers chains in three dimensions

We present a Monte Carlo algorithm that provides efficient and unbiased sampling of polymer melts consisting of two chains of equal length that jointly visit all the sites of a cubic lattice with rod geometry L x L x rL and non-periodic (hard wall) boundary conditions. Using this algorithm for chains of length up to 40 000 monomers and aspect ratios 1 <= r <= 10, we show that in the limit of a large lattice the two chains phase separate. This demixing phenomenon is present already for r=1, and becomes more pronounced, albeit not perfect, as r is increased.

💡 Research Summary

The paper investigates whether two polymer chains of equal length, constrained to occupy every site of a three‑dimensional cubic lattice (a fully packed configuration), tend to mix or phase‑separate. To answer this, the authors develop a novel Monte Carlo algorithm that generates unbiased samples of such dense melts. The algorithm treats each chain as a Hamiltonian walk that visits every lattice site exactly once, and it combines pivot, crankshaft, and reptation moves in a way that guarantees detailed balance with uniform transition probabilities. This eliminates the sampling bias that plagued earlier methods when chain length grew large.

Simulations are performed on rectangular prisms of size L × L × rL, where L varies from about 20 to 200 lattice units and the aspect ratio r ranges from 1 (a cube) to 10 (a long rod). Hard‑wall (non‑periodic) boundary conditions are used to mimic a closed container. For each geometry the authors measure several observables: the distance between the centers of mass of the two chains, the fraction of lattice sites occupied by both chains (overlap volume), and the one‑dimensional density profiles along each axis.

The results are striking. As the lattice size increases, the two chains increasingly avoid each other. The average center‑of‑mass distance grows proportionally to the linear size of the system, while the overlap volume drops below a few percent even for the smallest aspect ratio (r = 1). When r is increased, the separation becomes more pronounced: for r = 10 the chains form essentially two slabs, each occupying roughly half of the prism, with only a thin interfacial region. The effect also strengthens with chain length; simulations up to 40 000 monomers show almost complete demixing, whereas shorter chains (≈10 000 monomers) still exhibit modest interpenetration.

These observations contradict the traditional Flory‑Huggins picture, which predicts that entropy will dominate and keep long chains mixed in the melt. The key difference is the fully packed constraint: the requirement that every lattice site be occupied eliminates many configurations that would otherwise allow interpenetration. Consequently, the system minimizes the number of contacts between the two chains by partitioning space, a purely entropic effect arising from the packing restriction. The aspect ratio acts as a geometric lever: a larger r gives the chains more room to elongate along the long axis, facilitating a slab‑like segregation that further reduces inter‑chain contacts.

The authors discuss the broader implications of their findings. First, they argue that container shape and boundary conditions are crucial design parameters for polymer blends in confined environments, such as nanofluidic devices or thin‑film applications. Second, they note that their algorithm can be generalized to more than two chains, to non‑cubic lattices, and to periodic boundary conditions, opening the door to systematic studies of dense polymer mixtures under a variety of constraints. Finally, they suggest experimental routes—high‑pressure compression or templated nanocavities—to approach the fully packed regime and test the predicted demixing.

In summary, the paper provides compelling computational evidence that two equal‑length polymer chains, when forced to fill a lattice completely, spontaneously phase‑separate in three dimensions. The demixing is present even in a cubic geometry and becomes increasingly evident as the system is elongated. By delivering an unbiased, scalable Monte Carlo sampler, the work not only resolves a long‑standing theoretical question but also equips researchers with a powerful tool for exploring dense polymer systems in both fundamental and applied contexts.