Molecular-Weight Dependence of Polymer Crystallization: Decomposing the Nucleation-Rate Scaling and Identifying the Active Heterogeneous Nucleation Sites

How chain length (molecular weight, $M$) controls the crystallization kinetics of polymer melts is codified empirically but not mechanistically understood. Okui, Umemoto, Kawano and Mamun (2007) established the maximum-rate framework: the primary nucleation rate $I(T)$ and the crystal growth rate $G(T)$ are bell-shaped in temperature, and their maxima follow power laws in $M$, with $G_{\max}\propto M^{-0.5}$ across many chain-folding polymers, but $I_{\max}\propto M^{1.5}$ established only for fractionated poly(ethylene succinate) and decomposed only speculatively as $I\propto N_i D_i\exp(-\Delta F_i/RT)$, where the sequence-availability exponent $b$ in $N_i\propto M^b$ is undetermined. The mechanistic core remains open: no determination of $b$ is known to date; the universality of $I_{\max}\propto M^{1.5}$ is untested on other polymers; the molar-mass dependence of the nucleation and transport parameters ($\Delta E_g$, $\Delta E_i$, $K_g$, $K_i$, $T_m^0(M)$) is unquantified; and the physical identity, statistics, and thermal stability of the active heterogeneous nucleation sites, whose number and activation rate control the nucleation prefactor and its strong history dependence (self-nucleation, melt memory), remain unidentified. The problem asks for a quantitative, predictive, mechanistic account of how chain length controls bulk crystallization kinetics, including the overall-rate exponent, anchored in a microscopic identification of the active nucleation sites.

Background

When a quiescent polymer melt is cooled below its melting temperature it crystallizes through two separable rate processes: primary nucleation, the rare formation of the first stable ordered domains, and linear crystal growth, the advance of crystal fronts fed by secondary (surface) nucleation of chain segments. For flexible chain-folding polymers the two are measured independently: the primary nucleation rate $I$ (nuclei per unit volume per unit time, counted by polarized optical microscopy or inferred from the induction time $t_0$ preceding any calorimetric signal) and the radial growth rate $G$ of spherulites (the spherical polycrystalline aggregates seen in polarized microscopy). Both $I(T)$ and $G(T)$ are bell-shaped in temperature: at small supercooling $\Delta T = T_m^0 - T$ (where $T_m^0$ is the equilibrium melting temperature) the crystal-nucleation barrier suppresses the rate, while at large $\Delta T$ the slowing of segmental transport suppresses it. The transport factor is described either by an Arrhenius form $\exp(-\Delta E/RT)$ or by a Williams-Landel-Ferry (WLF) type form, the standard empirical description of relaxation kinetics in supercooled liquids; classical fitting forms due to Turnbull-Fisher for nucleation and Lauritzen-Hoffman for growth combine the two factors. The Lauritzen-Hoffman (LH) secondary-nucleation theory also partitions growth into kinetic regimes characterized by a substrate terrace length $H$ and a step-propagation rate $g$. The maxima $I_{\max}$ (at $T_{imax}$) and $G_{\max}$ (at $T_{cmax}$) are experimentally accessible intrinsic reference values largely insensitive to the choice of transport form (Okui, Umemoto, Kawano and Mamun, 2007). In molecular weight, Okui and co-workers compiled maximum-rate data on narrow (fractionated) samples. The maximum growth rate follows $G_{\max}\propto M^{-0.5}$ across roughly ten chain-folding polymers, rationalized by a prefactor $G_o\propto M^{a+d}$ with $a=0.5$ for loop-train adsorption probability and $d=-1$ for reptation-like surface diffusion (reptation is the snake-like tube motion of entangled chains, whose self-diffusion coefficient scales as $M^{-2}$). For primary nucleation, fractionated poly(ethylene succinate) (PES) instead yields $I_{\max}\propto M^{1.5}$: longer chains nucleate faster. The proposed microscopic picture writes $I\propto N_i D_i\exp(-\Delta F_i/RT)$, where $N_i\propto M^b$ counts intramolecular sequence availability, the number of chain segments capable of assembling a nucleus, and $D_i$ captures diffusion slowing; but $b$ has never been determined, so the empirical exponent 1.5 has never been decomposed into its competing contributions (sequence statistics versus diffusion-limited slowing), and its generality beyond PES is untested. Whether the remaining kinetic parameters, the transport activation energies $\Delta E_g$, $\Delta E_i$, the nucleation constants $K_g$, $K_i$, and the equilibrium melting temperature $T_m^0(M)$, can be treated as molar-mass-independent is likewise unresolved. Heterogeneous nucleation introduces history dependence: in bulk samples primary nucleation is almost always heterogeneous, catalyzed by a limited population of active sites of unknown nature (impurity particles, substrate surfaces, density-fluctuation domains, or melt-structure memory). Its kinetics follow the site-limited form $dN/dt = J_o(N_s - N)$, with $N_s$ the number of active sites and $J_o$ a per-site activation rate; $N_s$ and $J_o$ change by orders of magnitude with prior melt temperature, cooling program, aging below the glass transition temperature $T_g$, and sample volume, whereas $G$ is insensitive to all of these. The related phenomena of self-nucleation (seeds produced by the polymer's own previous crystals) and melt memory (accelerated recrystallization after incomplete erasure of previous crystalline order) remain mechanistically unexplained; Muthukumar (2016) calls melt memory a five-decade-old puzzle and models it through an intermediate inhomogeneous melt state, so far only qualitatively. For overall kinetics, the macroscopic Avrami rate constant $k$ of bulk crystallization (from the standard phenomenological conversion law $X(t)=1-\exp(-kt^n)$) mixes both steps, $k\propto I^c G^m$, where $m$ is the growth dimensionality and $c\in[0,1]$ weights the nucleation contribution ($c=1$ sporadic homogeneous nucleation, $c=0$ instantaneous heterogeneous nucleation). With the two maximum-rate scalings this predicts $k_{\max}\propto M^{\gamma}$ with $\gamma = 1.5c - 0.5m$, spanning $-1.5\le\gamma\le 0$ for $m=3$, a mapping that has been drawn but never validated because its nucleation input is unverified.

Problem Statement

Establish a quantitative, mechanistic, predictive account of how molecular weight (molar mass, $M$) controls the bulk crystallization kinetics of chain-folding polymers, in place of the current purely empirical maximum-rate scalings. Concretely, building on the framework of Okui, Umemoto, Kawano and Mamun (2007): (1) determine the microscopic decomposition of the primary nucleation rate's molar-mass scaling, that is, the sequence-availability exponent $b$ in $N_i\propto M^b$ together with the associated transport (diffusion) contribution, such that the observed $I_{\max}\propto M^{1.5}$ for fractionated poly(ethylene succinate) is derived rather than fitted, and establish whether this exponent, or its corrected value, is universal across chain-folding polymers or polymer-specific; (2) determine which of the kinetic parameters, the transport activation energies $\Delta E_g$ and $\Delta E_i$, the nucleation constants $K_g$ and $K_i$, and the equilibrium melting temperature $T_m^0(M)$, genuinely depend on molar mass, and over which ranges they may be treated as constant; (3) identify the physical nature, statistical distribution and thermal stability of the active heterogeneous nucleation sites $N_s$, and of the related self-nucleation and melt-memory precursors, whose number and activation rate $J_o$ control the nucleation prefactor and its strong dependence on prior melt temperature, cooling and quench rate, sub-$T_g$ aging and sample volume; and (4) on this basis, provide a predictive theory of the overall crystallization rate's molar-mass dependence ($k_{\max}\propto M^{\gamma}$ with $\gamma = 1.5c - 0.5m$, or its corrected equivalent, as the limiting power-law form, and the deviations from it that produce the observed non-monotonic $k(M)$ behavior), coupling morphological evolution, temperature-dependent chain transport, and surface-nucleation kinetics, including clarification of which apparent regime transitions in growth-rate plots are genuine and which are artifacts of a mis-specified transport term. Items (1)-(3) supply the microscopic inputs, the functional molar-mass dependences of the rate constants and the physical origin of the prefactor and its history dependence, that item (4) integrates into overall-kinetics prediction; they jointly serve one objective. Each of the four items is a required component of this one objective: a result on any single item is a partial advance, and the overall objective is met only when all four are addressed. Standardized measurement and reporting protocols are not part of the scientific core stated here.

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