Microscopic foundation of the kinetic chain-folding mechanism in polymer crystallization: stem-formation and attachment-rate parameters, entanglement effects, and lamellar thickening kinetics

Stepanow's kinetic mechanism of chain folding in polymer crystallization (Phys. Rev. E 90, 032601, 2014) selects the lamellar thickness by balancing a phenomenological stem formation time $\tau_s = d_l/(v_0 + c\Delta T)$ against the Rouse time of a chain segment of arc length $d_l$, and derives the growth rate from a phenomenological master equation $dG/dt = -\gamma G t^{-\alpha}$ with the ad hoc choice $\alpha = 1/2$. Every load-bearing ingredient — the constants $v_0$ and $c$ (with the monomer friction $\zeta$), the exponent $\alpha$, the parameter $\gamma$ and its dependence on molecular weight, pressure, and entanglements, the quantitative role of entanglements in stem selection, and the repulsion-driven picture of logarithmic lamellar thickening — is inserted by hand rather than derived from microscopic statistical mechanics. The open problem is to give this mechanism a microscopic, quantitative foundation: derive (or microscopically correct) $\tau_s$, $v_0$, $c$, the attachment-rate law with $\alpha = 1/2$ and $\gamma$'s parameter dependences, incorporate entanglement constraints quantitatively into the two-time-scale framework, ground the repulsion-driven picture of logarithmic lamellar thickening microscopically, and implement the result in coarse-grained analytic/numerical models allowing more quantitative predictions than the illustrative poly(ε-caprolactone) fits. No work deriving these ingredients is known in the literature to date.

Background

When a flexible polymer crystallizes from the melt, it forms lamellae: thin (typically 10–20 nm) plate-like crystals in which each chain is nearly extended along the lamellar normal and folds back on itself at the crystal surface. The straight segments between consecutive folds are called stems, and the emergence of such regular chain folding from a random-coil melt is the defining feature of polymer crystallization. Classical theories of this process, going back to Lauritzen and Hoffman, are nucleation-based and phenomenological: the lamellar thickness is identified with the size of a critical secondary nucleus — a small crystalline embryo forming on the surface of an already-growing crystal — and the theory contains fit parameters rather than quantities computed from the microscopic physics of the melt. Here supercooling $\Delta T$ denotes the difference between the equilibrium melting temperature $T_m^0$ and the crystallization temperature; the trans and gauche states are the two characteristic local conformations of a carbon backbone bond, the extended trans state being energetically favored in the crystal; the Rouse time is the relaxation time of an unconstrained polymer segment in the melt, proportional to $\zeta d_l^2$ for a segment of arc length $d_l$, where $\zeta$ is the monomer friction coefficient; and entanglements are the topological constraints that interpenetrating long chains impose on each other's motion, characterized by the entanglement molecular weight $M_e$. Stepanow (Phys. Rev. E 90, 032601, 2014) proposed an alternative kinetic mechanism in which stems form in the supercooled melt because trans states are preferentially occupied, and their growth competes with the random-coil structure of the chains. Equating the average formation time of a stem of length $d_l$, postulated as $\tau_s = d_l/(v_0 + c\Delta T)$ with constants $v_0$ and $c$ (Eq. 1 of that paper), with the Rouse time $\tau_{\mathrm{Rouse}}(d_l) = \zeta d_l^2/(3\pi^2 k_B T_m^0)$ of a segment of the same arc length, yields a lower bound on the lamellar thickness, $d_l \simeq 3\pi^2 k_B T_m^0/(\zeta c (T_c^0 - T))$, where $T_c^0 = T_m^0 + v_0/c > T_m^0$; the thickness is inversely proportional to supercooling in the primary-crystallization limit $v_0 \simeq 0$, where $T_c^0 = T_m^0$. The growth rate is then derived by treating growth as a dynamic multistage process governed by the phenomenological master-type equation $dG/dt = -\gamma G t^{-\alpha}$ for the attachment rate of a single stem, with $\alpha < 1$ and the choice $\alpha = 1/2$ motivated only by an analogy to multiplicative renormalization in the theory of critical phenomena; integration up to the time $t_l$ at which, on average, one stem attaches to the crystal surface gives the growth velocity $G \equiv G(t_l) = G_0 \exp(-2\gamma\sqrt{t_l})$, with a temperature-dependent prefactor $G_0 = \tilde{G}_0 \exp(-a/T)$ introduced to account for the increase of the relaxation time (viscosity) on cooling, while the growth-rate expression actually uses the larger attachment time $t_g$ obtained by replacing $T_c^0$ with the zero-growth temperature $T_{zg}$. Fits to poly(ε-caprolactone) data fix the growth-rate combination $\gamma/\sqrt{c} \simeq 7.8$ (through the characteristic temperature $T_G = 397\,^{\circ}\mathrm{C}$) and, from the crystallization line, the empirical values $v_0/c = T_c^0 - T_m^0 = 36$ K, $v_0\zeta \simeq 610\,k_B$ and $c\zeta \simeq 17\,k_B$. The author further argues that repulsive (excluded-volume) interactions drive the logarithmic lamellar thickening observed at long times and the increase of lamellar thickness with pressure, and that entanglements should weakly affect the thickness-selection condition while affecting growth more strongly. All of these ingredients — $\tau_s$, $v_0$, $c$, $\alpha$, $\gamma$, the entanglement treatment, and the repulsion-driven thickening picture — are introduced without microscopic derivation, as the paper itself acknowledges. The mechanism should also be seen in the context of the broader, still-unsettled question of lamellar thickness selection: experimental and simulation work since (notably by the Halle group around Saalwächter and Thurn-Albrecht) has established that in so-called crystal-mobile polymers the competition between crystal growth and intracrystalline chain diffusion — the diffusive sliding of chains through their own crystals via the $\alpha_c$-relaxation process, a thermally activated defect dynamics — has a decisive influence on the final lamellar thickness, a picture that competes with, but does not microscopically derive, the kinetic-selection ansatz of Stepanow.

Problem Statement

Provide a microscopic, quantitative foundation for Stepanow's kinetic mechanism of chain folding in polymer crystallization, replacing its phenomenological ingredients by derived or microscopically calibrated quantities. Concretely, from the statistical mechanics of a supercooled polymer melt (or from controlled coarse-grained simulations calibrated against microscopic physics), obtain: (1) the stem formation time law $\tau_s = d_l/(v_0 + c\Delta T)$, where $d_l$ is the stem length (of the order of the final lamellar thickness) and $\Delta T$ is the supercooling — i.e., microscopic values or derivations of the constants $v_0$ and $c$ and of their combinations with the monomer friction coefficient $\zeta$ entering the Rouse balance $\tau_s \simeq \zeta d_l^2/(3\pi^2 k_B T_m^0)$ with $T_m^0$ the equilibrium melting temperature, to be confronted with the poly(ε-caprolactone) values $v_0/c = 36$ K, $v_0\zeta \simeq 610\,k_B$, $c\zeta \simeq 17\,k_B$ already fitted in the source — or a microscopic correction showing precisely how this ansatz must be modified; (2) the functional form of the single-stem attachment master equation $dG/dt = -\gamma G t^{-\alpha}$, including a derivation of the exponent $\alpha$ (the source theory adopts $\alpha = 1/2$) and of the dependence of $\gamma$ on molecular weight, pressure, entanglement, and other external parameters; (3) a quantitative incorporation of entanglement constraints into the two-time-scale framework, deciding whether and how entanglements influence stem (thickness) selection as opposed to growth — the source theory assumes Gaussian-coil statistics make selection weakly entanglement-sensitive, while experiments on poly(L-lactic acid) indicate entanglement induces folding; (4) a microscopic foundation for the source's expectation that repulsive interactions cause the slow (approximately logarithmic) growth of the lamellar thickness $d_l$ at long times, replacing the qualitative rationale by a derived account of the thickening mechanism; and (5) an implementation of the resulting theory in coarse-grained analytic or numerical models that allows more quantitative predictions of lamellar thickness, growth rate, and thickening than the source's illustrative poly(ε-caprolactone) fits. These five deliverables are jointly the ingredients of one objective — converting the kinetic chain-folding mechanism into a predictive microscopic theory. Each ingredient must individually be either microscopically derived within the source framework or replaced by a microscopic correction showing precisely how that framework must be amended; resolving some of the deliverables does not resolve the others.

References

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