Precursor location and structure for secondary nuclei in lamellar polymer crystal growth

The kinetic sizing method of Zhang et al. (Macromolecules, 2019) infers the number of crystallizable repeat units, stems, and distinct chains in a critical secondary nucleus on a lamellar polymer growth front from how spherulitic growth rates fall when crystallizable units are diluted by noncrystallizable comonomers or a miscible amorphous blend partner. The method, however, assumes but does not establish that the precursors of secondary nuclei form at the growth front by stochastic selection of units from a homogeneous melt; its validity window in supercooling is bounded at both ends; and its stated extension to primary nucleation is unimplemented. This problem asks for a direct determination of where the precursors of critical secondary nuclei form during lamellar crystal growth of flexible polymers — at the lateral growth front or in the bulk melt — and of their molecular structure, accompanied by two related open items noted in the same sizing program: establishing the method's applicability window at large supercooling (where restricted diffusion may break its equal-mobility assumption) and at low supercooling (where random-copolymer growth rates become extremely small), and implementing the probability-based method for primary nucleation. Later work by the same group extended the method to further materials, to two-component stereocomplex crystals (co-crystals of stereochemically complementary chains), and to solution crystallization, and a 2022 study sized a critical primary nucleus in metastable solution via the nucleation theorem (a general relation between the supersaturation dependence of the nucleation rate and the number of units in the critical nucleus), but no direct determination of precursor location or structure exists.

Background

In semicrystalline polymers, crystallization proceeds by chain folding: flexible chains deposit on a growth face and fold back and forth, forming lamellar crystals — platelets typically about 10 nm thick — which in the bulk aggregate into spherulites, radially symmetric polycrystalline objects whose radial growth rate $G$ is a standard kinetic observable. Lateral growth of a lamella is controlled by secondary nucleation: the formation of a first crystallized chain segment, or stem (a straight segment about one lamellar thickness long), on the growth face, after which further stems attach by lateral spreading. The classical Lauritzen–Hoffman theory describes this process in three regimes: in regimes I and III the growth rate is proportional to the secondary-nucleation rate $i$, while in regime II nucleation and lateral spreading (rate $g$) alternate, so that $G\propto(i g)^{1/2}$. A critical nucleus is a cluster sitting at the top of the free-energy barrier: smaller clusters tend to dissolve, larger ones grow. Its size — the number of crystallized repeat units it contains — is the central mechanistic quantity, because it discriminates between competing pictures of growth, such as the single marginally stable stem of the classical theory of Lauritzen and Hoffman versus the multi-stem nuclei suggested by rate-theory analyses like that of Sadler and Gilmer and by simulations. Critical secondary nuclei are too small, too rare, and too short-lived for direct imaging: real-time observations generally see objects already larger than critical, and small-angle X-ray scattering (SAXS) measures the mature lamellar thickness, not nuclei. The classical treatment of random copolymers by Sanchez and Eby showed that noncrystallizable comonomer units are excluded from the crystal, which makes random copolymerization a controlled dilution of crystallizable units.

Zhang et al. (2019) turned this dilution into a quantitative sizing method. If a critical secondary nucleus contains $m$ crystallizable repeat units selected at random from a homogeneous melt, then in a random copolymer with fraction $p_A$ of crystallizable units the secondary-nucleation rate is suppressed to $i'=i\,p_A^{m}$; in growth regimes where $G\propto i$, the slope of $\ln G$ versus $\ln p_A$ gives $m$ directly, while in regime II the slope equals $(x+y)/2$ with $x$ and $y$ the unit counts controlling secondary nucleation and lateral spreading, so that $m\le x\le 2m$. Analogously, diluting a crystallizable homopolymer with a miscible amorphous blend partner at volume fraction $\phi_A$ gives $i_{blend}=\phi_A^{n}\,i_{hom}$, whose exponent counts the number of distinct chains $n$ contributing stems to the nucleus. For poly(butylene succinate) (PBS) and its random copolymers crystallized from the melt at 70–95 °C, this yielded $m=15$ to $27$ butylene-succinate units — combined with SAXS lamellar core thicknesses, roughly 5 to 8 stems — and $n\approx2$ contributing chains at lower temperatures, tending toward $n\approx1$ (intramolecular nucleation) at higher temperatures. The method rests on mechanistic assumptions — homogeneous random selection of units, equal chain mobility between homopolymer and copolymer, and a composition-independent nucleus size — that are plausible but not independently verified. The authors were explicit about what remained unresolved. First, the evidence locating nucleus precursors is only indirect: homogeneous SAXS of the melt and the strong growth-rate depression in random copolymers argue against precursor formation by microphase separation in the bulk melt, but do not directly determine where precursors form or what their structure is. Second, the method's window in supercooling (the temperature gap below the melting point, $\Delta T$) is bounded: at large $\Delta T$, restricted diffusion may invalidate the assumptions connecting composition and dilution exponents to nucleation probabilities, and at small $\Delta T$ the radial growth rates of random copolymers are too small for precise measurement. Third, although the approach is stated to be extendable to primary nucleation (nucleation in the bulk rather than on a growth face), this extension remains to be implemented and validated.

Problem Statement

The problem centers on one mechanistic core — the direct determination of where the precursors of critical secondary nuclei form and of their molecular structure — accompanied by two related open items of the same research program noted by the source: the bounded supercooling window of the kinetic sizing method, and its stated but unimplemented extension to primary nucleation.

(1) Mechanistic core. Directly determine where the precursors of critical secondary nuclei form during lamellar crystal growth of flexible polymers — at the lateral growth front itself or in the surrounding bulk melt — and determine their molecular structure (for example, whether a precursor is a single adsorbed stem, a multi-stem cluster, or a mesomorphic, partially ordered domain). 'Directly' means by an experimental determination (or an experimentally validated simulation) that does not merely infer the location indirectly from melt SAXS homogeneity or from growth-rate depression in random copolymers. Any such determination must fix and apply an operational definition of 'precursor' that (i) picks out a sub-critical entity on the pathway to the critical secondary nucleus — one that is on the nucleation pathway but is not yet the critical nucleus — distinct from equilibrium conformational fluctuations and from a quiescent adsorbed chain layer; (ii) is resolvable by the chosen experimental or simulated method independently of the location and structure outcome; and (iii) is stated without presupposing the answer, in particular without defining the precursor by its location (growth front versus bulk melt) or by its number of stems. A determination may be delivered as an executed measurement or simulation, or argued from a fully specified experiment or simulation, provided the operational definition above is fixed in advance and the observable it predicts is falsifiable; for a simulation, 'experimentally validated' means that the simulation reproduces, without being fitted to the target quantity, at least one independently measured observable of the same crystallization process before it is used to locate or characterize the precursor. The sizing method and the mechanistic question were established by the source on melt-crystallized poly(butylene succinate) and its random copolymers and blends at 70–95 °C, with the approach stated by the source to be generally applicable to the lamellar crystals of other crystallizable flexible polymers; a direct determination of precursor location and structure in a representative chain-folded polymer system addresses the mechanistic core as posed.

(2) Validity window. Establish the applicability window of the kinetic sizing method in supercooling: extend it into the large-supercooling regime, where restricted diffusion may invalidate the equal-mobility assumption connecting composition and dilution exponents to nucleation probabilities, and into the low-supercooling regime, where the radial growth rates of random copolymers are too small for precise measurement. Both limits are already documented as standing limitations by the source and by later work of the same group, so resolving this item requires new measurements or analysis establishing where the dilution-exponent method remains quantitatively valid; a restatement of the noted limitations does not settle it.

(3) Primary nucleation. Implement and validate the stated extension of the probability-based sizing approach to primary nucleation: the source states that the approach is "not limited to secondary nucleation and can be adopted for primary nucleation as well", and this extension remains unimplemented.

The mechanistic core (1) is the deliverable whose discharge resolves the task: a direct determination of precursor location and structure as specified in (1) answers the problem as posed. Items (2) and (3) are related open limitations and directions noted by the source within the same research program, not conjunctive acceptance requirements; an answer that also discharges (2) or (3) adds to the program but is not needed to resolve the task.

References

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