The kinetics with which a single polyelectrolyte chain collapses into, or expands from, a compact globule after a sudden change in solvent quality are summarized by power laws tcol∼Nαt_{\rm col}\sim N^{\alpha} and texp∼Nβt_{\rm exp}\sim N^{\beta} in the chain length NN. Dissipative-particle-dynamics simulations with explicit ions, long-range electrostatics, and hydrodynamic interactions (Yuan and Curk, 2024) established α≈1.4\alpha\approx1.4 and β≈2\beta\approx2 for charged chains, but only within a bounded window: N≤200N\le200, salt concentrations c=10−4c=10^{-4} to 10−210^{-2} M, two discrete ion-to-monomer size ratios (σs/σ=1\sigma_{\rm s}/\sigma=1 across that salt range, with σs/σ=0.5\sigma_{\rm s}/\sigma=0.5 measured only at c=10−2c=10^{-2} M), monovalent ions, and a single instantaneous-quench protocol. The open problem is the parameter-dependent scaling map: whether α\alpha decreases to lower values for N>200N>200 and c>10−2c>10^{-2} M as the source authors speculate, how α\alpha varies continuously with σs/σ\sigma_{\rm s}/\sigma — the authors anticipating that further decreasing the size ratio brings α\alpha closer to the neutral-polymer value α≈1\alpha\approx1 — and how far the expansion scaling β≈2\beta\approx2, reported for all conditions investigated inside that window, extends beyond it.

Background

A polyelectrolyte (PE) is a polymer whose monomers carry ionized charges; in solution each chain is accompanied by neutralizing counterions and, usually, added salt. A flexible polymer in a poor solvent (where monomer–monomer attraction dominates monomer–solvent affinity) collapses from an expanded coil to a compact globule when solvent quality is suddenly reduced — an instantaneous quench — and re-expands when the attraction is switched off. The kinetics are quantified by the time tcolt_{\rm col} for the radius of gyration RgR_{\rm g} (the root-mean-square monomer distance from the chain's center of mass) to reach its collapsed value, and the expansion time texpt_{\rm exp}, both following power laws in chain length, tcol∼Nαt_{\rm col}\sim N^{\alpha} and texp∼Nβt_{\rm exp}\sim N^{\beta}. Operationally, the source defines both characteristic times by the same threshold criterion, Rg(t∗)=Rg,init−fc (Rg,init−Rg,final)R_{\rm g}(t^{*})=R_{\rm g,\rm init}-f_{\rm c}\,(R_{\rm g,\rm init}-R_{\rm g,\rm final}) with fc=0.9f_{\rm c}=0.9, where Rg,initR_{\rm g,\rm init} and Rg,finalR_{\rm g,\rm final} are the initial and final equilibrated values of RgR_{\rm g}; that is, tcolt_{\rm col} (texpt_{\rm exp}) is the time at which RgR_{\rm g} has covered 90% of the change from its expanded (collapsed) initial value toward its collapsed (expanded) final value, a choice that keeps the extracted timescales unaffected by the slow local rearrangements of the collapsed globule at late times. Hydrodynamic interactions (HI) — the coupling of monomer motion through the velocity field of the surrounding solvent — are absent in implicit-solvent Langevin simulations but captured by dissipative particle dynamics (DPD), a mesoscale simulation method whose thermostat conserves momentum so the solvent transmits flow. Electrostatic coupling is measured by the Bjerrum length (the separation at which two unit charges interact with energy kBTk_{\rm B}T), and added salt screens the interactions over the Debye length; strongly charged PEs in poor solvents can form pearl-necklace structures, strings of compact partially neutralized beads connected by stretched segments.

For neutral polymers with HI, DPD simulations give tcol∼N0.98±0.09t_{\rm col}\sim N^{0.98\pm0.09} for collapse from an initially equilibrated coiled state (Guo, Liang, and Wang, 2011), while theory predicts α=4/3\alpha=4/3 for collapse from a fully expanded state with Rg∼NR_{\rm g}\sim N (Kikuchi, Ryder, Pooley, and Yeomans, 2005); HI also select the kinetic pathway of the neutral coil–globule transition (Kamata, Araki, and Tanaka, 2009). For charged polymers, Langevin simulations without HI showed that the collapse rate is strongly sensitive to counterion valence and size (Lee and Thirumalai, 2001; Ghosh and Vemparala, 2021), and a companion study demonstrated that HI substantially accelerate PE collapse without establishing the chain-length scaling (Yuan and Tanaka, 2024). Before Yuan and Curk (2024), no study had measured how tcolt_{\rm col} and, in particular, texpt_{\rm exp} scale with NN for a PE with explicit ions, long-range electrostatics, and HI.

Using DPD simulations of a bead-spring PE with explicit monovalent counterions and salt, Yuan and Curk (2024) found α=0.94±0.01\alpha=0.94\pm0.01 and β=1.97±0.10\beta=1.97\pm0.10 for neutral chains, in agreement with the earlier DPD result for collapse from an equilibrated conformation. For charged chains at salt concentrations c=10−4c=10^{-4} to 10−210^{-2} M, they found α≈1.4±0.1\alpha\approx1.4\pm0.1, close to the 4/34/3 prediction for collapse from a fully extended state, which they attribute to electrostatic repulsion swelling the initial coil. Reducing the ion-to-monomer size ratio to σs/σ=0.5\sigma_{\rm s}/\sigma=0.5 at c=10−2c=10^{-2} M — ions small enough to be incorporated into the globule, rendering it charge-neutral — lowers α\alpha to 1.15±0.041.15\pm0.04 (with β=1.94±0.09\beta=1.94\pm0.09 unaffected), whereas with σs=σ\sigma_{\rm s}=\sigma the excluded ions force a more expanded, pearl-necklace-like equilibrium. The expansion exponent stays at β≈2\beta\approx2 for all conditions tested, so β>α\beta>\alpha throughout: collapse proceeds by pearl formation and coarsening assisted by directional solvent flow, while expansion is isotropic and diffusion-limited. All of these conclusions were established for N≤200N\le200 (simulation boxes up to L=180σL=180\sigma), c≤10−2c\le10^{-2} M, two discrete ion sizes (the smaller one only at c=10−2c=10^{-2} M), monovalent ions, one solvent parametrization, and instantaneous quenches. Extrapolating to a real PE with N≈104N\approx10^{4} at c=10−2c=10^{-2} M, the authors estimate tcol≈13 μt_{\rm col}\approx13\,\mus and texp≈1350 μt_{\rm exp}\approx1350\,\mus, so the problem also bears on the timescales accessible to fast single-molecule experiments.

Problem Statement

For a single flexible polyelectrolyte chain in dilute solution with explicit counterions and salt, long-range electrostatics, and hydrodynamic interactions — the setting of Yuan and Curk, J. Chem. Phys. 160, 161103 (2024) — determine how the kinetic scaling exponents α\alpha and β\beta in tcol∼Nαt_{\rm col}\sim N^{\alpha} and texp∼Nβt_{\rm exp}\sim N^{\beta} (collapse and expansion times after an instantaneous solvent-quality quench, chain length NN, defined by the source's threshold criterion Rg(t∗)=Rg,init−fc(Rg,init−Rg,final)R_{\rm g}(t^{*})=R_{\rm g,\rm init}-f_{\rm c}(R_{\rm g,\rm init}-R_{\rm g,\rm final}) with fc=0.9f_{\rm c}=0.9) depend on the physical parameters NN, cc, and σs/σ\sigma_{\rm s}/\sigma outside the already-simulated window (N≤200N\le200; c=10−4c=10^{-4}–10−210^{-2} M with σs/σ=1\sigma_{\rm s}/\sigma=1; σs/σ=0.5\sigma_{\rm s}/\sigma=0.5 at c=10−2c=10^{-2} M only; monovalent ions). Concretely: (i) does the charged-chain collapse exponent α≈1.4\alpha\approx1.4 persist for substantially longer chains (N>200N>200) and higher salt (c>10−2c>10^{-2} M), or does it change to lower values, as the source authors explicitly speculate; (ii) how does α\alpha vary continuously with the ion-to-monomer size ratio σs/σ\sigma_{\rm s}/\sigma between and beyond the two tested values, and does it approach the neutral-polymer value α≈1\alpha\approx1 as σs/σ\sigma_{\rm s}/\sigma decreases (strengthened screening, small ions incorporated into a charge-neutral globule), as the authors anticipate; and (iii) how far does the expansion scaling β≈2\beta\approx2 — reported for all conditions investigated inside that window and therefore window-bound — continue to hold outside it, in particular for substantially longer chains? The single scientific objective uniting these questions is the parameter-dependent scaling map of single-chain PE collapse and expansion kinetics with hydrodynamic interactions: an answer establishes where α≈1.4\alpha\approx1.4 and β≈2\beta\approx2 hold, and how α\alpha crosses over to the neutral-chain regime. Sub-questions (i)–(iii) are jointly required to establish this map; each can be answered independently, and a result on any single one is a partial answer that leaves the remaining directions open.

References

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