Position-dependent numerical prefactors for the rate constant between reactive groups at arbitrary positions n and m on two Rouse chains in a nonentangled polymer melt

In his 1982 analysis of diffusion-controlled processes in dense polymer systems, P. G. de Gennes derived, at scaling level, the rate constant for a reaction between groups A and B attached to two long flexible chains in a nonentangled melt: at times below the Rouse time the exploration is compact and $k(t)\propto t^{-1/4}$; at later times $k\cong D_R R_0\propto N^{-1/2}$. The numerical prefactors were constructed only for a representative interior placement (central monomers); the paper states that the prefactors of the short-time displacement law depend on the positions $n$ and $m$ of the reactive monomers along their respective chains, are obtainable in principle for all $n, m$, but require a rather heavy calculation, and that the short-time analysis is otherwise mainly qualitative. Subsequent work — generalized melt kinetics for end-functionalized chains, exact single-chain first-passage solutions, cyclization simulations, and sticker-association scaling theories — has refined the exponents and resolved the single-chain analog at arbitrary positions, and a formal Markovian (Wilemski–Fixman-type) framework for interchain reactions between groups at arbitrary positions on two chains exists in the literature; but that framework is left unevaluated, and no published work constructs the position-dependent two-chain prefactors themselves. The problem is to construct them: the short-time displacement law $x^2(t; n)$ and its interior-to-end crossover, the full numerical coefficient of $k(t; n, m)$, and the position dependence and finite-$N$ corrections of the long-time rate constant.

Background

Diffusion-controlled reactions — reactions whose rate is limited by how fast the reactants find each other by transport rather than by intrinsic chemistry — are classically described by the Smoluchowski theory for small molecules: a time-independent second-order rate constant of order $k\sim 4\pi D b$, where $D$ is the relative diffusivity of the two reactants and $b$ their capture radius, i.e. the separation below which reaction is instantaneous. In a dense polymer system (a melt or a concentrated solution) the reacting groups A and B are not free particles: each is attached to a long flexible macromolecule, and its transport reflects the collective motion of the whole chain.

P. G. de Gennes (1982) analyzed this problem for two reacting groups carried by two different long chains in a melt, in the regime where the chains are shorter than the entanglement threshold $N_e$ (the chain length above which topological constraints from overlapping chains dominate the dynamics; the entangled regime is treated separately in the companion Part II via reptation, the model in which an entangled chain diffuses back-and-forth along a tube-like region defined by its neighbors). In this regime the chain dynamics follow the Rouse model — the standard model in which a Gaussian chain (a chain of $N$ beads of size $a$, connected by ideal springs, whose equilibrium conformation is a random walk) moves through independent normal diffusion modes of the bead sequence, with $W$ the microscopic monomer jump frequency. The slowest mode relaxes at the Rouse time $T_R \approx W^{-1}N^2$. A key feature of Rouse dynamics is that a tagged monomer moves subdiffusively at short times: during $t < T_R$ displacement propagates along the chemical sequence by curvilinear diffusion, so a cooperative segment of $\Delta n \sim (Wt)^{1/2}$ monomers shares the motion, giving a mean-square displacement $x^2(t) \propto a^2(Wt)^{1/2}$, i.e. $x(t)\propto t^{1/4}$, before crossover at $t \gtrsim T_R$ to ordinary diffusion of the whole coil with the Rouse diffusivity $D_R = Wa^2/(3N)$ and coil size $R_0 = N^{1/2}a$.

De Gennes's central distinction is compact versus noncompact exploration. If $x^3(t)/t$ decreases with time (compact exploration), the explorer repeatedly revisits the same volume, so as soon as the volumes explored by A and B overlap the reaction fires, and the rate constant is time dependent, $k(t)\propto x^3(t)/t$. If $x^3(t)/t$ increases (noncompact exploration), exploration is effectively irreversible and a time-independent rate constant exists. For nonentangled chains he concluded: (a) at $t < T_R$ exploration is compact, $k(t)\propto t^{-1/4}$, independent of $N$ and — for capture radius $b < x(t)$ — of $b$; (b) at $t \gtrsim T_R$, $k \cong D_R R_0 \propto N^{-1/2}$. The quantitative vehicle is a memory-function formulation: with the one-particle propagator $G_t(\mathbf{r})$ (the probability density for the group's displacement $\mathbf{r}$ after time $t$), the memory function $S(t) = \int [G_t(\mathbf{r})]^2\, d^3\mathbf{r}$ measures the self-overlap of exploration; for a Gaussian propagator $S(t) = \sigma x^{-3}(t)$ with $\sigma = 2^{-3}\pi^{-3/2}$, and $k^{-1} = \int_0^\infty S(t)\,dt$ in the noncompact case while $k(t) = x^3(t)/(\sqrt{2\pi}\,\sigma\,t)$ in the compact case.

For a reactive monomer in the middle of its chain, de Gennes constructed the full numerical prefactors: $x^2(t) = \frac{2}{3}a^2(Wt/\pi)^{1/2}$, hence $k(t)$ with its complete coefficient, and the long-time $k \cong D_R R_0$. But the coefficients depend on where along the chain the reactive group sits: the displacement statistics of a monomer near a free chain end differ from those of an interior monomer because the Rouse-mode amplitudes carry position-dependent weights and boundary corrections at the free ends. The paper states that these position-dependent coefficients can be obtained in principle for all positions $n$ and $m$ of the two reactive monomers along their respective chains, but that their explicit construction requires a rather heavy (lengthy and detailed) calculation, and that the short-time analysis is otherwise mainly qualitative.

Later work sharpens what such a construction entails. For a single Rouse chain, the first-passage problem for reactive groups at arbitrary positions has been solved: an exact analytical solution exists for the end reaction, and explicit rate formulas for reactants at arbitrary positions along the chain were derived within the Wilemski–Fixman closure — the standard self-consistent Markovian approximation for diffusion-influenced reactions, in which the reacting pair separation is treated as a Markov process governed by an effective renormalized propagator. Moreover, a non-Markovian analysis of reactive conformations showed that such Markovian closures can misestimate numerical prefactors, because the chain conformation at the reaction instant is biased relative to its equilibrium statistics. For the two-chain melt problem itself, a formal apparatus does exist: the Seoul school of Sung, Park, and Lee set up hierarchical reaction–diffusion equations for interchain reactions in which the reactive groups sit at arbitrary positions — the $m$th bead of chain A and the $n$th bead of chain B — with the rate constant expressed in the Laplace domain through a Green's function (the re-encounter probability of the two reactive groups) built from position-dependent single-chain propagators. However, this apparatus is explicitly left unevaluated (the review presents it as enabling future investigation), its evaluated applications concern energy transfer and excitation migration rather than the de Gennes melt rate constant, and its Markovian closure is exactly the kind of approximation known to shift numerical prefactors. The explicit position-resolved prefactor construction for the original two-chain melt problem therefore remains to be done.

Problem Statement

Within the Rouse-model framework of de Gennes's 1982 Part I analysis, construct the explicit numerical prefactors — position-resolved, not just scaling — for the diffusion-controlled reaction between groups A and B attached at positions $n$ and $m$ along two different long flexible chains in a nonentangled polymer melt (degree of polymerization below the entanglement threshold). The required results are: (i) the short-time ($t < T_R$) mean-square displacement law $x^2(t; n)$ of a reactive monomer at position $n$ along its chain, with its full numerical coefficient (de Gennes's $x^2(t) = \frac{2}{3}a^2(Wt/\pi)^{1/2}$ being the representative interior/central case) and the crossover of this coefficient as $n$ moves from the chain interior toward a free end; (ii) the resulting time-dependent two-chain rate constant $k(t; n, m)$ between groups at positions $n$ and $m$ on their two chains — the full numerical coefficient of the $t^{-1/4}$ compact-exploration law, with its domain of validity in $t$, $n$, $m$, and capture radius $b$ (instant reaction inside the capture sphere and $b < x(t)$ may be assumed, as in the source); and (iii) the long-time ($t \gtrsim T_R$) rate constant $k(n, m) \cong D_R R_0 f(n/N, m/N)$ with the explicit position-dependence function $f$ and finite-$N$ corrections, generalizing the central-monomer result $k \cong D_R R_0$. The coefficients depend on the chain discretization, on the boundary conditions imposed on the Rouse modes at the free ends (equal or unequal chain lengths may be treated), and on the closure adopted for the two-chain relative-displacement propagator; at $n = m = N/2$ they reduce to de Gennes's published central-monomer prefactors, and the crossover of the coefficient as $n$ moves from the chain interior toward a free end is part of the target. The target is the evaluated, position-resolved coefficients of items (i)-(iii); the formal Wilemski–Fixman-type Green's-function or Laplace-domain expressions for the two-chain problem — which already exist in the literature unevaluated — are the starting point rather than the target.

References

1. P. G. de Gennes, Kinetics of diffusion-controlled processes in dense polymer systems. I. Nonentangled regimes, The Journal of Chemical Physics 76, 3316-3321 (1982). DOI 10.1063/1.443328, https://doi.org/10.1063/1.443328
2. P. G. de Gennes, Kinetics of diffusion-controlled processes in dense polymer systems. II. Effects of entanglements, The Journal of Chemical Physics 76, 3322-3326 (1982). DOI 10.1063/1.443329, https://doi.org/10.1063/1.443329
3. B. O'Shaughnessy, Diffusion-controlled reactions in entangled polymer systems, The Journal of Chemical Physics 94, 4042-4054 (1991). DOI 10.1063/1.460680, https://doi.org/10.1063/1.460680
4. B. O'Shaughnessy and D. Vavylonis, Reaction kinetics in polymer melts, The European Physical Journal B 6, 363-372 (1998). DOI 10.1007/s100510050562, https://doi.org/10.1007/s100510050562
5. P. J. Park and S. Lee, Diffusion-influenced reversible energy transfer reactions between polymers, The Journal of Chemical Physics 115, 9594-9600 (2001). DOI 10.1063/1.1413968, https://doi.org/10.1063/1.1413968
6. J. Sung, P. J. Park, J. Lee, W. Lee, J.-H. Kim, and S. Lee, Diffusion-Influenced Kinetics of Reactions Involving Polymers, Bulletin of the Korean Chemical Society 24, 843-852 (2003). DOI 10.5012/bkcs.2003.24.6.843, https://doi.org/10.5012/bkcs.2003.24.6.843
7. I. M. Sokolov, Cyclization of a Polymer: First-Passage Problem for a Non-Markovian Process, Physical Review Letters 90, 080601 (2003). DOI 10.1103/PhysRevLett.90.080601, https://doi.org/10.1103/PhysRevLett.90.080601
8. A. D. Litmanovich, N. A. Platé, and Y. V. Kudryavtsev, Reactions in polymer blends: interchain effects and theoretical problems, Progress in Polymer Science 27, 915-970 (2002). DOI 10.1016/S0079-6700(02)00003-5, https://doi.org/10.1016/S0079-6700(02)00003-5
9. C. Yeung and B. Friedman, Cyclization of Rouse chains at long- and short-time scales, The Journal of Chemical Physics 122, 214909 (2005). DOI 10.1063/1.1924412, https://doi.org/10.1063/1.1924412
10. A. E. Likhtman and C. M. Marques, First-passage problem for the Rouse polymer chain: An exact solution, Europhysics Letters 75, 971-977 (2006). DOI 10.1209/epl/i2006-10206-6, https://doi.org/10.1209/epl/i2006-10206-6
11. R. Everaers and A. Rosa, Multi-scale modeling of diffusion-controlled reactions in polymers: Renormalisation of reactivity parameters, The Journal of Chemical Physics 136, 014902 (2012). DOI 10.1063/1.3673444, https://doi.org/10.1063/1.3673444
12. T. Guérin, O. Bénichou, and R. Voituriez, Non-Markovian polymer reaction kinetics, Nature Chemistry 4, 568-573 (2012). DOI 10.1038/nchem.1378, https://doi.org/10.1038/nchem.1378
13. T. Guérin, O. Bénichou, and R. Voituriez, Reactive conformations and non-Markovian cyclization kinetics of a Rouse polymer, The Journal of Chemical Physics 138, 094908 (2013). DOI 10.1063/1.4792437, https://doi.org/10.1063/1.4792437
14. E. B. Stukalin, L.-H. Cai, N. A. Kumar, L. Leibler, and M. Rubinstein, Self-Healing of Unentangled Polymer Networks with Reversible Bonds, Macromolecules 46, 7525-7541 (2013). DOI 10.1021/ma401111n, https://doi.org/10.1021/ma401111n