Molecular origins of polymer crystal growth rates and regime transitions: replacing the phenomenological parameters of the Lauritzen-Hoffman growth equation with a quantitative molecular account

The temperature dependence of the lateral growth rate of lamellar polymer crystals is conventionally fitted by a Lauritzen-Hoffman-type equation whose parameters - the prefactor $G_0$, the short-range diffusion activation energy $U^*$, and the Vogel temperature $T_\infty$ - are phenomenological, and whose predicted regime I-II-III transitions lack a molecular derivation. Dynamic Monte Carlo simulations by Hu and Cai (2008) reproduced regime II-III transitions, tied them to a morphological transition of the growth front, interpreted them through the intramolecular-nucleation model, but could not observe regime I in the melt and left the molecular origins of the growth-rate parameters and of post-nucleation thickness selection unquantified. The open problem is to give a quantitative molecular account of crystal growth rates, their regime transitions, and lamellar thickness selection: deriving $G_0$, $U^*$, and $T_\infty$ (including molecular-weight, entanglement, stiffness, and concentration dependences) from chain mechanisms, characterizing the post-nucleation processes at the growth front, and establishing regime-I kinetics in bulk melts within (or against) the intramolecular-nucleation picture.

Background

When a flexible polymer crystallizes from the melt or from solution, chains fold back and forth and pack into thin plate-like lamellar crystals typically 5-30 nm thick; a straight crystalline segment between two folds is called a stem. Lateral growth of a lamella is controlled by secondary nucleation - the formation of a new crystalline patch on the growth face - followed by lateral spreading of that patch, in contrast to small-molecule crystals where growth units attach one by one without chain connectivity. The classical kinetic description is the Lauritzen-Hoffman (LH) secondary-nucleation theory (Lauritzen and Hoffman, 1960), in which the linear growth rate $G(T)$ follows $G(T)=G_0\,\exp[-U^*/R(T-T_\infty)]\,\exp[-K_g/(f\,\Delta T\,T)]$: the first exponential describes transport of chain segments across the crystal-melt interface (with activation energy $U^*$ and a Vogel-type temperature $T_\infty$ typically fixed near the glass transition $T_g$ minus about 30 K), and the second describes the barrier for secondary nucleation at undercooling $\Delta T=T_m^0-T$ with equilibrium melting temperature $T_m^0$, correction factor $f$, and nucleation constant $K_g$. The relative rates of secondary nucleation and lateral spreading define three regimes: in regime I a single surface nucleus spreads completely across the face before the next forms (smooth front, mononucleation); in regime II many nuclei spread simultaneously (rough front); in regime III nucleation dominates again. Regime transitions appear as breaks in the slope of $\log G$ versus $T_m^0/(T\,\Delta T\,f)$ and are among the most robust experimental signatures of polymer crystallization, yet the LH interpretation of these breaks has been debated for decades. Hu, Frenkel, and Mathot (2003) proposed the intramolecular nucleation model, in which the rate-determining event is the folding of a single chain on itself onto the growth face, and derived a rate equation of the same LH form. Hu and Cai (2008) then performed dynamic Monte Carlo simulations of lattice polymers (a coarse-grained model in which chain conformations evolve by stochastic local moves) of crystal growth induced by a template layer in the melt and in semidilute solutions. They reproduced the regime II-III transition, found that it coincides with a morphological transition from single to multiple lamellar growth, found that dilution makes chain folding more perfect, but could not observe the smooth-front regime I within accessible simulation times, and they reinterpreted the regimes on the intramolecular-nucleation basis. In the LH-type rate equation, however, the parameters $G_0$, $U^*$, and $T_\infty$ remain fitted constants: no molecular theory or simulation has yet derived them from chain parameters, and the elementary post-nucleation processes at the growth front - instant thickening of just-deposited stems, long-distance feeding of stems by chain segments that are stored far away along the chain contour (which for entangled melts is limited by reptation, the tube-like curvilinear diffusion of a chain along its own contour), and subsequent crystal perfection - are not quantitatively characterized, although they plausibly control the final lamellar thickness and defect content of semicrystalline polymers.

Problem Statement

Provide a quantitative molecular-level account of the lateral growth rate $G(T)$ of lamellar polymer crystals and its regime transitions, replacing the phenomenological parameters of the Lauritzen-Hoffman-type rate equation $G(T)=G_0\,\exp[-U^*/R(T-T_\infty)]\,\exp[-K_g/(f\,\Delta T\,T)]$ with derived molecular mechanisms within (or as a correction to) the intramolecular-nucleation picture, as left open by Hu and Cai (2008). The single common objective is a predictive, molecularly grounded theory of growth rates, regime transitions, and lamellar thickness selection; its jointly required, scientifically linked components are: (i) the molecular origin and scaling of the prefactor $G_0$, including its molecular-weight dependence and the contributions of chain connectivity and entanglements; (ii) the role of interfacial diffusion - the short-range diffusion activation energy $U^*$ and the effective Vogel/$T_g$-related temperature $T_\infty$ - and of its molecular-weight and concentration dependences, in setting absolute growth rates; (iii) the influence of chain stiffness on the relative importance of intramolecular versus intermolecular nucleation pathways at the growth front; (iv) the kinetics and mechanisms of the processes following nucleation - instant crystal thickening at the growth front, long-distance feeding of stems along chains (reptation-limited spreading in entangled melts), and subsequent crystal perfection - and how they control final lamellar thickness selection and defect statistics; (v) the pathway from width-restricted nucleated layers to the mature crystal thickness, including whether mesophase transitions intervene during maturation; and (vi) the observation and quantitative description of regime-I growth kinetics in bulk polymer melts, or a molecularly derived demonstration that regime I cannot occur there, completing the link between regime-I kinetics and the intramolecular-nucleation model. A complete answer derives, at a quantitative molecular level, both a mandatory core and the remaining components (iii)-(vi). The core is $G(T)$, with $G_0$, $U^*$, and $T_\infty$ expressed in chain parameters (length, stiffness, entanglement density, concentration), together with the regime boundaries and the selected lamellar thickness, without treating $G_0$, $U^*$, or $T_\infty$ as free fitted constants; component (vi) counts as resolved by either an observation and quantitative description of regime-I growth kinetics in a bulk melt or a molecularly derived demonstration that regime I cannot occur there. No component is satisfied by a qualitative statement alone.

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