Self-assembly Monte Carlo reveals localized entanglement in giant polymer melts

Abstract

Topological entanglements are central to understanding and predicting the properties of polymer melts. Yet, they make equilibrium sampling computationally challenging, as decorrelation times grow rapidly with chain length. Here, we introduce a Monte Carlo scheme that bypasses typical computational bottlenecks by working in a self-assembly ensemble rather than at fixed composition. Strictly local moves efficiently propagate backbone reconnections across scales while conserving the number of linear chains, achieving near-linear scaling of decorrelation time with system size, τ_eq ~ V ^1.0. With this method, formulated for a fully-packed lattice, we equilibrate periodic systems totalling up to  ≃ 1.1 × 10⁹ monomers, accessing a universal melt regime insensitive to lattice details. We analyze intra- and inter-chain entanglements for chains of up to N ≃ 5 × 10⁵ monomers, revealing that they manifest as localized knots and links rather than as global tangles. Finally, we show that the magnitude of the Gauss linking integral between neighbouring chains grows only as N^1/4.

Simulating dense polymer melts remains challenging due to the rapid increase in computational cost with chain length. Fornasa et al. report a Monte Carlo scheme that addresses this bottleneck by employing a self-assembly ensemble, achieving linear scaling with system size and typical chain length.

Introduction

Polymer melts owe many of their physical properties to topological entanglements established by the densely packed chains. Examples range from the structural plasticity of eukaryotic genomes to the viscoelasticity of synthetic materials and the mechanics of interlocked filaments^1–9. The regime most relevant to soft matter, materials science, and biological physics occurs when the chain length, N, largely exceeds the entanglement length, the typical contour separation between entanglement points^10,11. In this regime, equilibration is governed by the slow reptation of the individual filaments within the constraining mesh, with disengagement times growing at least as O(N³) ^12–15. In molecular dynamics simulations with cell or neighbor lists, the additional cost of computing interactions leads to wall-clock decorrelation times scaling as  ≳ O(N⁴)¹⁶. Analogous considerations hold for Monte Carlo methods with topology-preserving local moves. Consequently, characterizing and predicting the properties of dense polymer melts with realistic simulations becomes impractical as N grows well beyond the entanglement length.

To make the problem more tractable, several accelerated sampling strategies have been proposed, including hierarchical coarse-graining, thermodynamic reweighting, and enrichment^17–36. Off-lattice bridging moves that cut and rejoin bonds while conserving melt composition (i.e., polymer lengths, chain count and density), can bypass topological constraints and reduce the effective dynamic exponents to the 2–2.5 range^37–39. On lattices, local reconnection moves for space-filling walks can achieve O(N²) decorrelation scaling^19,40, and efficient hierarchical construction algorithms can be employed too^41,42. Despite these advances, most approaches to equilibrating melts are formulated for the fixed-composition ensemble, where enforcing constraints of chain lengths and connectivity remains a major computational bottleneck.

Here, we introduce an efficient self-assembly Monte Carlo (SAMC) scheme for dense polymers on a cubic lattice at full packing. The system, composed of a periodic ring-linear blend, is evolved with local endpoint bond-swaps that allow the composition to fluctuate while still conserving density, self-avoidance, and the number n of linear chains. By allowing reconnections, SAMC avoids the aforementioned expensive backbone tracing checks that hinder sampling methods at fixed composition. As a result, the global decorrelation time, measured in Monte Carlo steps, scales approximately linearly with system size, τ_eq ~ V ^1.001±0.002. Because the CPU time per Monte Carlo step varies only weakly with V, the near-linear scaling applies also to wall-clock decorrelation time.

This upends the usual hierarchy of computational bottlenecks: while for conventional methods the most onerous step is the collection of uncorrelated states, for SAMC it is instead their analysis, as the computational cost of calculating physical observables often scales quadratically or worse with chain lengths.

In the stationary SAMC ensemble, giant linear components spontaneously emerge against a background of finite and non-extensive rings. In fact, the n linear chains are individually macroscopic, with average lengths 〈N〉 proportional to the system volume, which is almost entirely (~ 90%) occupied by them.

Leveraging the efficiency of SAMC and the emergence of giant components, we equilibrate fully packed systems in which only a few giant chains collectively comprise up to  ≃ 1.1 × 10⁹ monomers. By analyzing chains with millions of monomers, we show that their metric statistics are consistent with those of pure linear melts, indicating that their scaling properties lie in a universal melt regime, insensitive to microscopic lattice details.

We thus push the topological characterization of melts to previously unexplored chain densities and lengths. We profile intra- and inter-chain entanglements across scales and show that they manifest as localized knots and links rather than as global tangles. Finally, we show that the characteristic Gauss linking integral between neighboring chains grows only as N^1/4.

Results

Self-assembly Monte Carlo

We study self-assembled blends of linear chains and rings that are fully packed on a simple-cubic lattice. The system is evolved by a Monte Carlo scheme with strictly local endpoint bond swaps that efficiently propagate backbone reconnections across scales while conserving the number of linear chains. This self-assembly Monte Carlo (SAMC) method has near-linear global decorrelation time with system size and yields macroscopic linear chains with melt-like statistics. This makes it possible to efficiently equilibrate very large systems and thus investigate and establish large-scale properties that are insensitive to microscopic lattice details.

We consider a fully-packed ring-linear blend on a L × L × L periodic lattice, containing n linear chains. Accordingly, all sites have connectivity degree 2 except those hosting the 2n endpoints of the chains, which have degree 1.

The elementary SAMC move is sketched in Fig. 1a and is articulated over the following stages. First, one of the 2n endpoints, denoted A, is chosen uniformly. Next, a site B is chosen uniformly among the non-bonded nearest neighbors of A. Then, a site C is chosen uniformly among the bonded neighbors of B (one choice if B is an endpoint, two otherwise). If C is an endpoint, the move is rejected, and the current configuration is retained, otherwise bond BC is replaced with AB.

Fig. 1. Endpoint bond-swap move and decorrelation time.

a Sketch of the three stages involved in an endpoint bond-swap move. For clarity, the sketch is shown for a blend with n = 2 linear chains on a two-dimensional (8 × 8) periodic square lattice. First, an endpoint A is randomly chosen among the 2n = 4 degree-1 sites (open circles). Next, a site B is randomly chosen among the non-bonded nearest neighbors of A. Then, a site C is randomly chosen among the bonded neighbors of B. If C is an endpoint, the move is rejected, otherwise bond BC is replaced by AB, obtaining the evolved configuration shown on the right. The move conserves packing, self-avoidance, and the total number of linear chains, while chain lengths and ring counts typically vary. b Global decorrelation time τ_eq for systems with a single linear chain (n = 1) per periodic box as a function of the box volume V = L³. τ_eq was obtained as the integrated autocorrelation of edge-state overlap q, and is measured in Monte Carlo steps. The power law best fit yields a practically linear dependence of τ_eq on V, as indicated, with the prefactor equal to 1.05 ± 0.01. The relative statistical uncertainty on τ_eq is estimated to be smaller than 1% (Supplementary Fig. 3 and Supplementary Note 5).

After each accepted move, every site retains its degree except for sites A, whose degree changes from 1 to 2 after the addition of bond AB, and site C, whose degree changes from 2 to 1 after removing bond BC. As a result, C replaces A as the free end. Rejecting proposals in which C is an endpoint is necessary because the bond swap would disconnect that site, violating the conservation of the number of free ends. With the above move set, such rejections occur with probability O(n/V), where V = L³ is the lattice volume, and hence become negligible as V grows at fixed n (Supplementary Fig. 2). An optional lower-rejection variant, discussed in Supplementary Note 6, reduces this scaling to O(n²/V²).

Thus, the SAMC evolution conserves full packing, self-avoidance, and the number of endpoints (and thus the number of linear chains, n). Instead, the linear chains’ lengths and the number and lengths of rings can fluctuate.

The move proposal is symmetric on admissible endpoint bond swaps, and all such proposals are accepted. The rejected proposals instead contribute only to self-loops. Therefore, the resulting Markov chain satisfies detailed balance with respect to a uniform stationary measure, see Supplementary Note 1. On the considered periodic hypercubic lattices at full packing, the dynamics is free of blocked (immobile) configurations except in the limiting case n = V/2, where the system is made entirely of chains of length 1 (Supplementary Note 2). While a general proof of ergodicity is lacking, comparisons against exhaustively enumerable systems with up to  ≃ 10⁵ states show that a single extensive SAMC run samples uniformly the full configuration space (Supplementary Fig. 1). These results support the effective ergodicity and uniformity of SAMC.

Global decorrelation: near-linear scaling

For a conservative measure of global decorrelation, we take the integrated autocorrelation time of the edge overlap, τ_eq, expressed in Monte Carlo steps. This overlap measures the fraction of lattice edges that have the same state (empty or occupied) in two configurations (“Methods” and Supplementary Note 5).

To characterize the scaling of τ_eq with system size, we computed τ_eq for L ranging from 20 to 800, corresponding to V spanning more than four decades, from 8.0 × 10³ to 5.12 × 10⁸. A power-law fit of the data yields a practically linear scaling of the decorrelation time with system size, τ_eq ∝ V^1.001±0.002 (Fig. 1b). The near-linear decorrelation reflects three features of SAMC. First, working in a self-assembly ensemble circumvents constraints inherent to the conventional fixed-composition ensemble. Second, the endpoint bond-swap move is local, and yet it efficiently propagates connectivity changes across scales. Lastly, the SAMC evolution requires tracking only the 2n endpoints, dispensing with the onerous tracing of entire backbones that fixed-composition accelerated schemes require at each bond swap. With SAMC, backbone tracing is needed only at the analysis stage, and exclusively for the independent samples used to compute observables of interest. In this regard, we collect configurations at time intervals of 3τ_eq to ensure their statistical independence.

In terms of wall-clock runtime, the single-core CPU time per Monte Carlo step on standard workstations is approximately independent of V and equal to  ≃ 0.2 μs as long as RAM capacity is not exceeded (Supplementary Fig. 5). Accordingly, the near-linear scaling of τ_eq on V is carried over to the wall-clock decorrelation time. On the tested workstations, decorrelating configurations of 100 × 100 × 100 systems (V = 10⁶) at 3τ_eq intervals requires  ≃ 0.6 s.

Finally, regarding the analysis, we note that common observables of interest, including those used later to study entanglement, have an O(N²) cost or worse. As a consequence, for the large systems considered here, the dominant computational bottleneck shifts from equilibration to post-processing analysis.

Giant linear components

An emergent property of the SAMC ensemble is that the n linear chains are individually giant or macroscopic, i.e., with length growing proportionally to V. By contrast, the complementary ring background consists of numerous finite loops.

Figure 2 shows a typical SAMC-equilibrated configuration of a ring-linear blend with n = 10 chains fully occupying a 1024 × 1024 × 1024 periodic lattice (V ≃ 1.1 × 10⁹), the largest addressable system with the RAM available to us. We note that this size exceeds the maximum one used for the decorrelation analysis of Fig. 1b because the latter required the simultaneous storage of multiple configurations in memory, while SAMC sampling requires storing only one configuration. Figure 2a shows the blend with coordinates wrapped around the boundaries of the periodic L × L × L box. The linear chains are manifestly the giant components of the system, collectively occupying  ≃ 9.7 × 10⁸ monomers, or 90% of the box volume. Figure 2b shows one of the possible unwrapped representations of the same linear chains in panel a, obtained by selecting a reference periodic box and then tracing each backbone continuously across periodic boundaries, starting from one of its ends in that box.

Fig. 2. Emergent giant linear components.

a Wrapped and (b) unwrapped representations of an SAMC-equilibrated configuration with n = 10 linear chains on a fully-packed 1024 × 1024 × 1024 periodic lattice, totalling  ≃ 1.1 × 10⁹ monomers. The 10 linear chains are distinguished by a rainbow palette, while the ring background is shown in black. Collectively, the linear chains comprise  ≃ 9.7 × 10⁸ monomers, about 90% of the volume.

Indeed, for the considered range of n, the distribution of the volume fraction occupied by the linear chains ϕ_lin converges with increasing V to an asymptotic curve sharply peaked at ${\overline{\varphi }}_{\mathrm{lin}}\simeq 0.9$ (Supplementary Fig. 7 and Supplementary Table 1). The ϕ_lin distributions for the effectively asymptotic 100 × 100 × 100 system are shown in Fig. 3a. The tails beyond the peak have a common rapid decay, while the shoulders at ${\varphi }_{\mathrm{lin}}<{\overline{\varphi }}_{\mathrm{lin}}$ become visibly sharper with increasing n. Thus, the combined linear components are most likely to occupy about 90% of the lattice sites across the explored n.

Fig. 3. Statistics of the macroscopic linear chains and ring background.

a Probability distribution of ϕ_lin, the volume fraction collectively occupied by the indicated number of linear chains, n. Inset: close-up near the modal value, ${\overline{\varphi }}_{\mathrm{lin}}\simeq 0.9$. b Tail probability, P_n(x), that individual linear chains at ${\overline{\varphi }}_{\mathrm{lin}}$ occupy a portion larger than x of the linear volume, $V_{\mathrm{lin}}={\overline{\varphi }}_{\mathrm{lin}}V$. The P_n(x) data are well approximated by the theoretical Dirichlet distribution (solid lines) for random n-partitioning of the linear volume, $P_n^{\mathrm{theor}}\left(x\right)={\left(1-x\right)}^{n-1}$. (c) Average number density of rings of length k, 〈n_k〉/V, for configurations conditioned at ${\overline{\varphi }}_{\mathrm{lin}}$. The solid line is the parameter-free theoretical prediction ${⟨n_k⟩}^{\mathrm{theor}}/V=A/\left[k{\left(k-1\right)}^{3/2}\right]$, with A ≃ 0.1379 (Supplementary Note 10). All plots are for independent SAMC-equilibrated samples of 100 × 100 × 100 systems. For each n, 10⁵ independent configurations were used for panel (a) and 10³ independent conditioned configurations for panels (b, c). Error bars in panels (a) and (b) represent standard errors.

Analysis of how the collectively macroscopic linear volume is partitioned among the n linear chains shows that the latter are individually macroscopic. To establish this, we first conditioned the system to the macroscopic linear regime (${\overline{\varphi }}_{\mathrm{lin}}$) by retaining from the independent configurations only those with ϕ_lin ∈ [0.90, 0.91]. From the individual chain lengths, N_i=1,...,n, we next computed the corresponding linear volume portions, x_i = N_i/V_lin, where $V_{\mathrm{lin}}={\overline{\varphi }}_{\mathrm{lin}}V$.

For n = 1, the conditioning clearly yields a near monodisperse chain length, N₁ ≃ 0.9 V, hence macroscopic. To characterize the n > 1 case, we consider the tail probability P_n(x) that x_i is larger than x and compare it with the Dirichlet expression for uniform random n-partitioning of the unit interval, $P_n^{\mathrm{theor}}\left(x\right)={\left(1-x\right)}^{n-1}$. Figure 3b shows that the observed P_n(x) follows closely the theoretical expression. Notice that the agreement is not the result of a best-fit procedure, since the theoretical expression has no adjustable parameters.

We conclude that, at the modal linear volume fraction ${\overline{\varphi }}_{\mathrm{lin}}$, the n linear chains are statistically equivalent and individually giant or macroscopic, with average lengths scaling as O(V/n), see also Supplementary Fig. 8.

By contrast, the ring background conditioned at ${\overline{\varphi }}_{\mathrm{lin}}$ is instead finite and dominated by short rings. Figure 3c shows that the average number density 〈n_k〉/V of rings of length k collapses on the solid curve, corresponding to the theoretical prediction ${⟨n_k⟩}^{\mathrm{theor}}/V=A/\left[k{\left(k-1\right)}^{3/2}\right]$, with A ≃ 0.1379. This expression is based on the statistics of loops formed by chain closure in melt-like conditions and, notably, has no free parameters, Supplementary Note 10. Consistent with the theoretical scaling, the data for k ≫ 1 decay rapidly as 〈n_k〉/V ~ k^−5/2. As a consequence, the average ring length at ${\overline{\varphi }}_{\mathrm{lin}}$ remains finite across all considered n and V and stays close to the theoretical estimate $⟨l_{\mathrm{ring}}^{\mathrm{theor}}⟩\simeq 8.26$ (Supplementary Note 10 and Supplementary Table 3).

Two observations are in order. First, the regime of giant chains must eventually break down at sufficiently large n, since for n = O(V) the typical chain length is necessarily O(1). Second, our observation of a giant linear component at n = 1 connects to recent mathematical developments for the three-dimensional double-dimer model, for which rigorous proofs have established the existence of a macroscopic open path upon insertion of a single pair of defects⁴³. While the double-dimer model and ours differ for both the configurational averaging and the nature of the ring background, our results provide a direct physical realization of this prediction, and offer a quantitative basis for motivating extensions of the rigorous proof to n > 1.

Giant linear melts

Consistent with the fact that the linear chains are the sole macroscopic components at ${\overline{\varphi }}_{\mathrm{lin}}$, we observe that their metric properties follow closely the scaling laws expected for melts over the investigated range.

Figure 4a shows the intra-chain contact statistics at ${\overline{\varphi }}_{\mathrm{lin}}$ for selected values of n and V; additional combinations are in Supplementary Fig. 9. The data are for the probability, P_c(s), that pairs of monomers at contour distance s are in contact, i.e., on nearest neighboring sites. To connect with known scaling properties for melts, the analysis was performed on near monodisperse configurations. Specifically, the configurations were selected a posteriori such that each of the n chain lengths fell within 10% of the equisubdivision value, ${\overline{\varphi }}_{\mathrm{lin}}V/n$. The P_c(s) data follow the  ~ s^−3/2 scaling typical of melts and ideal chains in three dimensions, the effective exponent from a power-law best fit to the data for 150 × 150 × 150 systems being  − 1.505 ± 0.003 (solid line). Also, the mean square internal distance (inset) exhibits the expected scaling for melts, 〈R²(s)〉 ∝ s, the best fit exponent being 1.00 ± 0.01.

Fig. 4. Melt-like metric scaling of giant linear chains.

a Contact probability P_c and mean square internal distance 〈R²〉 (inset) of giant linear chains as a function of contour separation s. Power-law best fits (solid lines) to P_c and 〈R²〉 data (L = 150, n = 1, s < 10⁴) yield respectively the exponents  − 1.505 ± 0.003 and 1.00 ± 0.01, consistent with melt behavior. b Average total number of inter-chain monomeric contacts and average number of distinct overlapping chains as a function of chain length, N. Again, the indicated exponents of power-law best fits (solid lines) are consistent with melt behavior. Data are for various system sizes and numbers of linear chains n; additional combinations are in Supplementary Fig. 9. For each combination of size and n, data are from 10² independent configurations conditioned at ${\overline{\varphi }}_{\mathrm{lin}}$ and with near-monodisperse lengths, $N\simeq {\overline{\varphi }}_{\mathrm{lin}}V/n$. Error bars in panel (b) represent standard errors.

For the inter-chain contact statistics, shown in Fig. 4b, we computed two distinct observables for the same near monodisperse samples. First, we addressed the length dependence of the average number of inter-chain monomeric contacts, n_contacts, made by a given linear chain. Next, we computed the number of overlapping chains, n_overlap, i.e., chains making at least one contact with a given one. Contributions from periodic images were included in both cases. We observed that n_contacts grows as $N^{{\beta }_1}$, with β₁ = 0.994 ± 0.006, while n_overlap scales as N^a, with a = 0.48 ± 0.01. We recall that the expected theoretical exponents for melts are ${\beta }_1^{\mathrm{theor}}=1+d\left(2\nu -1\right)$ and a^theor = ν d − 1, where d is the dimensionality and ν is the metric exponent⁴². Our observed β₁ and a are compatible with these theoretical expressions specialized to three-dimensional melts (d = 3 and ν = 1/2), ${\beta }_1^{\mathrm{theor}}=1$ and a^theor = 1/2.

Notice that all considered observables collapse onto the same master curves across various n and V, even though the density of periodic chain images varies substantially across these combinations. This indicates that the metric scaling of the giant linear chains at ${\overline{\varphi }}_{\mathrm{lin}}$ effectively lies in a universal melt regime, insensitive to microscopic lattice details, and interactions with both periodic images and the finite ring background.

Capitalizing on this, we use the giant chains equilibrated by SAMC as a gateway to access properties of linear melts at previously unexplored chain lengths and densities. Given the centrality of topological constraints for rationalizing and predicting the physical properties of melts, we next direct our analysis to intra- and inter-chain entanglement across scales.

Knotting in giant linear melts

For the intra-chain topological entanglements, we study how the abundance and complexity of physical knots vary with the length of the macroscopic linear chains. The most straightforward setting for this analysis is to consider the n = 1 case conditioned at ${\overline{\varphi }}_{\mathrm{lin}}$, which yields giant chains with near monodisperse lengths, N ≃ 0.9V. The physical knotted state of the (unwrapped) macroscopic linear chains was established by joining the ends with minimally invasive auxiliary arcs and then evaluating topological invariants, see Methods section. Because this topological profiling is more costly than SAMC sampling, we performed it up to N ≃ 1.1 × 10⁵, corresponding to systems of size 50 × 50 × 50.

Figure 5 shows the (un-)knotting probability of the giant chains as a function of N. The data indicate an effective exponential decay, $P_{\mathrm{unknot}}\left(N\right)~\exp \left(-N/N_0\right)$, with characteristic knotting length N₀ = (4.8 ± 0.1) × 10³. This value lies between those of two well-studied cases. It is an order of magnitude smaller than for isolated self-avoiding chains (N₀ ≃ 10⁵ for both open and closed chains on- and off-lattice) but an order of magnitude larger than for ideal chains (N₀ ≃ 300 Kuhn lengths^44,45 about 450 steps in our model, see “Methods”). The intermediate value of N₀ reflects the crossover of effective interactions in dense melts, where excluded volume is screened at large scales while it persists locally. For completeness, we note that the observed N₀ for the giant melts is also much larger than for self-avoiding polymers below the θ-point. In this regime, corresponding to bad solvent conditions, polymer chains are individually collapsed and the reported characteristic knotting lengths of these dense and hence highly entangled globules, which have been studied for circular chains^46–48, are of the order of 10² − 10³. Consequently, melt chains are substantially underknotted relative to pure random walks⁴⁹ and collapsed polymers, while also being overknotted compared to isolated self-avoiding chains.

Fig. 5. Intra-chain entanglement: physical knotting of giant linear chains.

a Knotting probability of giant chains versus their length N. The complementary unknotting probability is shown in the inset along with the exponential best fit to the data. b Stacked histogram illustrating the contributions of various prime and composite knots to the overall knotting probability. c Typical configuration (unwrapped representation) of a giant chain of N ≃ 5.8 × 10⁴ monomers before (top) and after (bottom) a topology-preserving geometrical simplification. In the simplified configuration, spatially separated localized prime components are clearly visible, see also close-ups in the callouts. The corresponding knotted regions are highlighted with matching colors in the original configuration. Data are based on 10⁴ independent configurations conditioned at ${\overline{\varphi }}_{\mathrm{lin}}$ for various system sizes and n = 1 linear chains. d Average trefoil knot length, 〈l_tref〉 as a function of N/n_tref, the chain contour length rescaled by the number of trefoil components. Only (L, n_tref) combinations with at least 50 measurements of trefoil knot length were considered. Data for these combinations, distinguished by different colors and symbols, collapse onto a master curve that is well described by the power law best fit $⟨l_{\mathrm{tref}}⟩\propto {\left(N/n_{\mathrm{tref}}\right)}^{0.36\pm 0.05}$ (dashed line). The outlier data point for (L = 10, n_tref = 1) was excluded from the fit. e Probability density of l_tref for various (L, n_tref) combinations. The corresponding cumulative distributions for n_tref = 3 and various L are shown in panel (f). Error bars in panels (a) and (d) represent standard errors.

The knot spectrum, which we could resolve near exhaustively for lengths up to N ≃ 2.4 × 10⁴, is dominated by composite knots, which are connected sums of prime components (Fig. 5 and Supplementary Table 2). A sliding-window scan along the chain contour reveals that for N ≫ N₀ these composites typically manifest as multiple localized simple prime knots separated by long unknotted segments (Fig. 5c). The observed spatial factorization is nontrivial, since the multiple prime components could instead coalesce in a single delocalized tangle.

Motivated by the abundance of trefoils (3₁) among prime knots and prime components (Supplementary Table 2), we analyzed their average contour length, 〈l_tref〉. To this end, we isolated individual trefoil-knotted components with a bottom-up scan of the chains (Supplementary Note 8). Figure 5d shows that 〈l_tref〉 depends on both the overall chain length, N, and the number of trefoil-knotted components, n_tref, accommodated by the chain. Except for the smallest system (L = 10), the 〈l_tref(N, n_tref)〉 data points fall on the same master curve when plotted versus the reduced chain length N/n_tref. The master curve is well described by the power law (dashed line in Fig. 5d)

$$⟨l_{\mathrm{tref}}⟩\propto {\left(\frac{N}{n_{\mathrm{tref}}}\right)}^{0.36\pm 0.05}.$$ 1

Several observations are in order. First, the result is qualitatively consistent with previous studies of single self-avoiding open and closed chains, which found that the average contour length of single prime knots scaled sublinearly with chain length, indicating weak knot localization^47,50–53.

Second, it is interesting to compare the observed exponent with previous estimates for trefoil-knotted polymers in the absence of attractive interactions. Reported values span a broad range, possibly also due to the limited length of the studied chains, as noted by Mansfield and Douglas⁵². Specifically, previous estimates include 0.74 ± 0.14 and 0.54 (⁵¹ and⁵², respectively) for lattice rings, 0.63 for a polyethylene model⁴⁷, 0.4 ± 0.1 as inferred from the mechanical response of chains stretched between two plates⁵⁰, and 0.44 ± 0.08 for fully-flexible off-lattice chains of up to 1.5 × 10⁴ monomers⁵³. The value found here for the giant chains in a melt, 0.36 ± 0.05, is at the lower end of the above range, and is closest to the last two estimates.

Third, our results show that this scaling extends to the individual localized prime components of composite knots, with N/n_tref emerging as the appropriate scaling variable to account for the competition of the multiple components that share the same chain contour.

Finally, further insight into the characteristic knot length emerges from the probability distributions of the individual trefoil-knot lengths, l_tref, shown in Fig. 5e,f for various (N, n_tref) combinations. In qualitative accord with ref. ⁵³, the data show that: (i) the distributions are unimodal, (ii) they are peaked at about ${\overline{l}}_{\mathrm{tref}}~100$ almost independently of N and n_tref, and (iii) have a slowly decaying tail at large knot lengths. At the same time, the tails of the distributions decay slowly enough that a non-negligible portion of the probability density extends to the largest observed knot lengths (Fig. 5f). Therefore, while the modal trefoil-knot length is independent of N and n_tref, the average one depends on N/n_tref because increasing this rescaled length extends the cutoff of the slowly decaying probability tail.

As a complement to the above considerations on the length and localization of trefoil knots, we discuss a minimal stochastic model for their multiplicity in composite knots. To this end, we consider a Poisson insertion process in which localized trefoils are placed independently along the chain contour. Inspired by theoretical arguments for isolated chains^46,54–56, we take the insertion rate as ${\overline{N}}_{\mathrm{tref}}^{-1}$, where ${\overline{N}}_{\mathrm{tref}}\simeq \left(5.0\pm 0.2\right)\times 10^3$ is the chain length corresponding to the peak abundance of trefoils (Fig. 5 and Supplementary Fig. 10). This process, despite underestimating total knotting by neglecting the insertion of other prime types, yields $P_{\mathrm{unknot}}\left(N\right)~\exp \left(-N/{\overline{N}}_{\mathrm{tref}}\right)$, capturing both the observed exponential decay and the correct order of magnitude for N₀. Thus, the model ties both results to the observed spatial localization of the simple prime components.

Linking in giant linear melts

For the inter-chain topological entanglement, we investigate the pairwise physical linking of contacting giant chains, studying its localization and dependence on chains’ length. To achieve, again, near monodisperse conditions, we considered the case n = 2 conditioned at ${\overline{\varphi }}_{\mathrm{lin}}$, further restricting the dataset to configurations with a  < 10% difference between the two chain lengths, hence equal to N₁ ≃ N₂ ≃ 0.45 V. To quantify the topological interlocking of the two chains, we computed the Gauss linking integral L_G (Supplementary Note 9) of their closest periodic images, defined as the unwrapped images with the smallest center-of-mass separation (see “Methods”).

Because the ensemble average 〈L_G〉 vanishes by symmetry, we characterized the entanglement magnitude using the root-mean-square and mean absolute values, $\sqrt{⟨L_{\mathrm{G}}^2⟩}$ and 〈∣L_G∣〉. Note that for linear chains, L_G is generally not an integer. The analysis was performed up to N ~ 4.5 × 10⁵ monomers, corresponding to systems of size 100 × 100 × 100.

Figure 6 a illustrates the linking magnitude as a function of N. Intuitively, one might expect that tightly contacting chains in densely packed systems should form tangles of complexity rapidly growing with N. In stark contrast, the linking magnitude increases only slowly with chain length,

$$\sqrt{⟨L_{\mathrm{G}}^2⟩}~N^{\gamma },\gamma \simeq 0.25\pm 0.01,$$ 2

and remains modest even for our longest chains. Indeed, the most probable value of ∣L_G∣ remains of order unity throughout the entire explored length range (Fig. 6b).

Fig. 6. Inter-chain entanglement: physical linking of giant linear chains.

a Root-mean-square Gauss linking integral $\sqrt{⟨L_{\mathrm{G}}^2⟩}$ versus chain length N. The solid line is a power-law best fit to the data; the exponent is indicated. b Histogram of the ∣L_G∣ probability distribution, with bins centered at integer values. Data are for 10³ independent configurations conditioned at ${\overline{\varphi }}_{\mathrm{lin}}$ for various system sizes and n = 2 near-monodisperse linear chains. c Typical heatmap $M_{L_{\mathrm{G}}}$ of local segment-segment contributions to the Gauss linking integral of two contacting (unwrapped) giant chains in a 100 × 100 × 100 system. The heatmap is computed at the resolution of 10⁴ monomers per segment. Finer resolution close-ups are shown in panels (d, e). Error bars in panels (a) and (b) represent standard errors; those in (a) are smaller than the symbols.

To resolve the spatial organization of the overall weak interlockings, we decomposed L_G into segment-segment contributions (Methods and Supplementary Note 9). The heatmaps $M_{L_{\mathrm{G}}}$ constructed through this double-contour scan are remarkably sparse even for the longest chains, as illustrated in Fig. 6c–e for a representative $M_{L_{\mathrm{G}}}$. At the segment resolution of Δ = 10000 monomers, only about 20 segment pairs contribute significantly to the overall linking integral (L_G ≃ 1.19). Importantly, these contributions appear with both signs. Using finer segment subdivisions reveals that the significant entries correspond to localized clusters.

We conclude that, despite the overall tight packing, nearby giant chains have sparse and localized entanglements. Even at the largest lengths, pairwise chain interlockings occur over only a few and widely separated distinct regions.

Both this linking factorization and the observed exponent γ ≃ 0.25 ± 0.01 can be rationalized and connected by a simple scaling argument for monodisperse melts at fixed density. A given chain overlaps with $n_{\mathrm{overlap}}~\sqrt{N}$ neighboring ones (see also Fig. 4b), with the total number of interlockings scaling as  ~ N/L_e, where L_e is the entanglement length. Thus, the typical number of distinct interlockings between two neighboring chains scales as $n_{\mathrm{links}}^{\mathrm{pair}}~N/\left(L_e{n}_{\mathrm{overlap}}\right)\propto N^{1/2}$. Since these topological constraints are sparse and independent, they contribute to L_G with random (unbiased) signs. Resorting to the central limit theorem, the argument gives,

$$\sqrt{{⟨L_{\mathrm{G}}^2⟩}^{\mathrm{theor}}}~\sqrt{n_{\mathrm{links}}^{\mathrm{pair}}}\propto N^{1/4},$$ 3

thus accounting for both the observed scaling exponent and the sparsity of the $M_{L_{\mathrm{G}}}$ heatmaps.

As a complement, we note that the L_G distributions have fatter than Gaussian tails (Supplementary Fig. 12), suggesting long-range correlations between interlockings, a feature warranting future investigation.

Discussion

We introduced a self-assembly Monte Carlo (SAMC) scheme for equilibrating fully packed ring-linear blends on a lattice via strictly local endpoint bond swaps. The method samples an ensemble in which the number of linear chains, n, is conserved while chain lengths and ring statistics fluctuate.

This allows SAMC to bypass the bottlenecks of conventional fixed-composition schemes, achieving a linear scaling of the global decorrelation time with system size, τ_eq ~ V ^1.001±0.002. This scaling inverts the usual hierarchy of computational costs, shifting it from the collection of independent states to their analysis. Accordingly, we equilibrated fully packed blends in which a few linear chains collectively comprise  ~ 9.7 × 10⁸ monomers, and computed complex observables for individual chains of length up to N ~ 10⁶.

In the SAMC stationary ensemble, macroscopic linear components emerge spontaneously, collectively occupying  ≃ 90% of the volume. At the modal linear volume fraction, ${\overline{\varphi }}_{\mathrm{lin}}\simeq 0.9$, the n linear chains are statistically equivalent and individually extensive, while the complementary ring background remains non-extensive and dominated by short loops. As a result, the giant chains reproduce the metric statistics of dense linear melts, placing the large-scale behavior accessed by SAMC in the melt universality class.

We harnessed SAMC to characterize topological entanglement in melts at packing density and chain lengths up to N ~ 10⁶, a regime typically inaccessible to equilibrium sampling. For intra-chain entanglement, i.e., physical knotting, the unknotting probability is well described by an exponential decay with characteristic length N₀ ≃ 4.8 × 10³, substantially smaller than in dilute self-avoiding conditions. In addition, the growth of topological complexity with N is driven by the spatial factorization into localized prime knots separated by extended unknotted segments, similar to a Poisson insertion process.

For inter-chain entanglement, i.e., physical linking, we observed an analogous spatial factorization of the interlocking between contacting chain pairs. The Gauss linking integral, L_G, remains modest and of order unity even at the longest analyzed lengths, N ~ 4.5 × 10⁵, and its magnitude $\sqrt{⟨L_{\mathrm{G}}^2⟩}$ grows only as N^1/4. Analysis of segment-segment contributions to L_G shows that this weak growth arises from sparse and spatially localized interlockings. A simple scaling argument based on the number of such topological constraints and their unbiased random-sign contribution to L_G rationalizes both the exponent and the observed sparsity.

The performance of SAMC, which is also straightforward to implement, makes it well-suited to tackle strongly entangled polymer systems at lengths where conventional equilibration is typically unfeasible. A standing open problem addressable with SAMC is the characterization of multi-chain entanglements^57,58, which remain poorly understood and, e.g., may account for the interlocking correlations suggested by the non-Gaussian tails of the measured ∣L_G∣ distributions. Spatial confinements relevant to experimental setups (channels, slits and cavities) can be accessed with suitable embedding geometries and open boundaries (Supplementary Note 2). The microscopic details of the model can be refined too. The resolution can be enhanced with higher-coordination lattices, and bending rigidity can be treated with the same local updates supplemented by a Metropolis acceptance criterion. More generally, SAMC-generated configurations can be used, after off-lattice relaxation, as equilibrated initial states for molecular dynamics simulations of large dense polymer systems. This would enable studying the manifestations and implications of topological constraints across scales, a central problem from designed materials, such as slide-ring polymer networks, to biological soft matter, such as eukaryotic chromosomes.

In principle, SAMC could be extended to control the dispersity of the linear chains. For instance, the original SAMC could be used to sample equilibrated configurations with a substantial number of distinct chains, e.g., $n=O\left(\sqrt{V}\right)$, thus ensuring a non-giant average chain length, $O\left(\sqrt{V}\right)$. To achieve near-monodispersity, these configurations could then be separately evolved with a modified SAMC in which moves are accepted or rejected with a Metropolis-like criterion favoring small chain-length dispersity. This additional evolution, requiring backbone tracing at each step, would steer the original configurations towards near-monodispersity. Among other applications, this would enable more direct comparisons with standard ring-linear blends and linear melts.

A further interesting future direction is to extend SAMC beyond the fully packed regime. To this end, one could equip SAMC with additional moves, e.g., attaching bonds to empty lattice sites while deleting terminal bonds to conserve density. Though the sampling efficiency would expectedly degrade as the packing fraction is lowered, with such an extension it might become possible to study how knotting and linking statistics cross over from dense melts to dilute self-avoiding polymers.

SAMC could also be generalized to simulate blends of living polymers⁵⁹ by supplementing the present dynamics with scission and recombination moves that create or anneal nearby pairs of free ends. More specifically, the number of linear chains, n, could be allowed to fluctuate by removing non-terminal bonds through split moves that increment n by 1, and by bridging nearest-neighbor endpoints through merge moves that decrease n by 1. This would allow for sampling living polymer blends in which the linear chains fluctuate in both number and length, as the circular chains in the background already do. Though the efficiency of such extensions would need to be determined, it would be interesting to use the approach to study how reversible breakage and recombination modify the equilibrium properties of dense ring-linear blends.

Methods

Model and initialization

Our approach is inspired by recent work that recasts the sampling of dense polymer systems as a binary optimization (QUBO) problem amenable to quantum annealing machines^6,60–62. The QUBO representation is based on binary variables, corresponding to occupation states of lattice sites, edges, and corner turns, that interact via local bonding rules but without global connectivity constraints. This motivated our modeling of ring-linear polymer blends in the self-assembly ensemble rather than at fixed composition.

The initial space-filling SAMC state for n linear chains and the ring background on L × L × L periodic simple-cubic lattice can be conveniently prepared with the following two-step procedure. First, one generates a deterministic Hamiltonian path that, starting from a corner of the L × L × L box, fills successive layers in a serpentine fashion. Removing n − 1 non-consecutive bonds from the obtained path yields the desired initial space-filling configuration with n open chains, with an initial ring count of zero. The procedure can be used with all admissible values of n, from 1 to $n_{\max }=\lfloor \left(V-1\right)/2\rfloor$. Excluding the n = V/2 case, when admissible by lattice parity constraints, is necessary because it corresponds to configurations entirely filled by chains of length 1, which are blocked under the SAMC evolution (Supplementary Note 2). Under the subsequent Monte Carlo dynamics, the number of linear chains, n, is conserved while chain lengths and ring counts vary. The tests of uniformity and ergodicity of the self-assembly Monte Carlo, including a statistical analysis after ref. ⁴⁰, are presented in Supplementary Fig. 1.

The nominal Kuhn length of polymers on simple-cubic lattices can be estimated by considering non-backtracking random walks, for which the mean scalar product of two consecutive unit-length steps is 〈c〉 = 1/5. Using the Kuhn length expression for freely-rotating chains, (1 + c)/(1 − c), and replacing c with 〈c〉 yields a nominal value of 3/2.

Global decorrelation time

To quantify global decorrelation, we use the normalized edge-state overlap q between two configurations Γ₀ and Γ₁:

$$q\left({\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1\right)=\frac{n_{\mathrm{eq}}\left({\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1\right)}{E},$$ 4

where n_eq counts the lattice edges with identical state (empty or occupied) in both configurations and the normalizing factor E = 3V is the total number of edges.

The normalized autocorrelation function for q is:

$$C\left(t\right)=\left(⟨q\left(t\right)⟩-\overline{q}\right)/\left(1-\overline{q}\right),$$ 5

where 〈q(t)〉 is the average overlap of configurations at time lag t, measured in Monte Carlo steps, and $\overline{q}$ is the average overlap of independent configurations, estimated numerically (Supplementary Fig. 3). The global decorrelation time, τ_eq, is obtained by integrating C(t) from t = 0 up to the first time lag when C(t) < 10⁻².

Statistical analysis of observables

Power-law best fits were performed via linear regression of the log-log data. The reported errors of the exponents correspond to the 3σ confidence interval. For each combination of system size and n, data in Fig. 4a involves a double average. First, for each considered giant chain, we compute the contact probability of all pairs of monomers at separation s. Next, we compute P_c(s) as the mean of these individual contact probabilities at fixed s. In the plots of Fig. 4b, 5, 6, where data for near-monodisperse chains are presented as a function of chain length, the reported value of N corresponds to ${\overline{\varphi }}_{\mathrm{lin}}V/n$, i.e., the reference length around which 10% length fluctuations are allowed. Statistical errors on the knotting probabilities were obtained from block analysis, using m = 5 or m = 10 blocks per dataset, depending on data numerosity.

Detection of physical knots

The knotting probability and knot spectrum of the giant linear chains were established by computing topological invariants on their circularized versions, obtained by applying the minimally invasive closure procedure of ref. ⁶³. For a given open chain, the latter involves comparing the termini separation, D_TT = d_t1,t2, with the sum of their distances to the chain convex hull, D_TCH = d_t1,ch + d_t2,ch. If D_TT≤D_TCH, the termini are bridged directly; otherwise, each terminus is prolonged along its distance vector to the convex hull by a length ten times the chain gyration radius and the prolonged ends are then connected. Choosing the smaller of D_TT and D_TCH is a practical way to limit spurious entanglement from closure arcs by minimizing the contour length of the portion inside the convex hull, which can topologically interfere with the open chain. To speed up the calculation of topological invariants for the giant chains, we apply topology-preserving geometrical simplifications, first on-lattice using BFACF-type moves^64,65 and then off-lattice with KMT-type moves^66,67. After simplification, we compute Alexander determinants and Dowker codes. After algebraic simplification and factorization, the latter are compared against lookup tables of prime knots with up to 16 crossings, allowing near complete (> 95%) determination of the knot spectrum for giant chains with up to N ≃ 2.4 × 10⁴ monomers. For longer chains, we conservatively discriminate between knotted and unknotted states using the determinants of the Alexander polynomial. Applying the same knot-profiling procedure to systematic bottom-up scans of chain segments of increasing length enables localizing the knotted components. Details of the scanning procedure and knot spectrum, including a best-fit analysis of the trefoil-knot probability after ref. ⁶⁸, are provided in Supplementary Note 8, Supplementary Table 2 and Supplementary Fig. 10.

Detection of physical links

To quantify the degree of entanglement of two giant linear chains, we compute the Gauss linking integral L_G of their closest unwrapped periodic images, i.e., the pair of images with the minimal center-of-mass separation. Except for accidental degeneracies, the distance-minimizing pair is unique up to common translations by integer linear combinations of the simulation-box basis vectors, under which L_G is invariant. The latter was evaluated using the method detailed in Supplementary Note 9. The direct evaluation of L_G involves O(N²) elementary operations. Applying on- and off-lattice topology-preserving simplifications afforded an order-of-magnitude speed-up while leaving L_G unchanged to within 1% (Supplementary Fig. 11). To establish the degree of localization of physical links, we subdivide each chain into equally long segments and compute the segment-segment contributions to L_G, yielding matrices $M_{L_{\mathrm{G}}}$ capturing the degree of physical interlocking between the various regions of the two chains.

Author contributions

E.F., F.S., and C.M. designed research, analyzed the data and wrote the paper; E.F. implemented the model and carried out simulations.

Peer review

Peer review information

Nature Communications thanks Gerard Barkema and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

This study was funded in part by the European Union - NextGenerationEU, in the framework of the PRIN Project “The Physics of Chromosome Folding" (code: 2022R8YXMR, CUP: G53D23000820006) and by PNRR Mission 4, Component 2, Investment 1.4_CN_00000013_CN-HPC: National Center for HPC, Big Data and Quantum Computing - spoke 7 (CUP: G93C22000600001). The views and opinions expressed are solely those of the authors and do not necessarily reflect those of the European Union, nor can the European Union be held responsible for them.

Data availability

The data generated in this study and presented in the main text and Supplementary Information plots have been deposited together with the corresponding Jupyter notebooks in the Zenodo repository under accession code 10.5281/zenodo.20039444⁶⁹.

Code availability

The source code of SAMC has been deposited in the GitHub repository indicated in ref. ⁷⁰. Compilation requires users to provide their own licensed copy of the specified random number generator.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-74480-4.