1. Introduction

Crystallization is a special type of self-assembly process, which leads to crystals consisted of orderly packed lattices in space. During crystallization, particular configurations and orientations of the motifs (molecules, molecular complex or chain segments) are chosen from enormous random trials and the process is driven by thermal fluctuation and the molecular interactions. Compared to small molecules, polymer chains consisted of covalently linked monomer units show quite different crystallization behaviors [1, 2]. For example, flexible polymer chains usually fold back and forth to form lamellar crystals with thickness ranging from several to tens of nanometers, which results in semi-crystalline polymer materials. These lamellar crystals are metastable compared to the most thermodynamically stable extended chain crystals and may reorganize during heating. Depending on the crystallization conditions, polymer crystals show a rich variety of shapes and appearances, termed as morphologies, though the basic crystal lattice maybe the same. The rich morphologies are reflectance of the microscopic molecular processes during crystallization [3-5].

Polymer crystallization is a rich field spanning crystallography, microscopy, experimental and theoretical modeling, so it is difficult to summarize all the interesting findings in such a short summary. In this review, we choose only some relevant points that we have been engaged so far: Twisting chirality of lamellar crystals due to surface stresses, recognition of chain units during crystallization and the competition of different crystal modifications and different nuclei precursors with various sizes. In common, the three topics are reflectance of two competing factors during the crystallization process: Minimization of free energy in bulk and surface, attaching and detaching of a chain stem, thickening and widening, respectively.

2. What directs the twisting chirality of lamellar crystals of chiral polymers?

Chiral growth of crystals is observed in not only polymers [6-9] but also small molecules [10-13]. In addition, living creatures (e.g., snails) and some twining plants [14, 15] in nature demonstrate chiral features as well. Namely, chirality is expressed in hierarchical levels, as schemed in Fig. 1. However, the mechanism of chiral growth has not been fully understood. Several mechanisms, e.g., Eshelby twist [16], screw dislocations [17, 18], surface stresses [5, 19, 20], heterometry stresses [12, 21-23], enantio-selective adsorption [24], have been proposed to account for chiral growth of crystals. In the field of polymer crystallization, the mechanical origin of lamellar twisting has been attributed to the unbalanced surface stresses arisen from the different degrees of overcrowding on the two fold surfaces of the lamellar crystals [5, 19, 20]. However, what directs the chirality of polymer lamellar twisting has been an open question for decades. It was generally believed that molecular chirality of polymers determined the twisting handedness of lamellae [25], for instance, the lamellar twisting chirality is right-handed in poly(L-lactide) [26], poly(R-epichlorohydrin) [27], and poly(R-propylene oxide) [28], while left-handed in poly(D-lactide), poly(S-epichlorohydrin) (S-PECH), and poly(S-propylene oxide). In contrast, the stereocomplex of chiral polymers does not show lamellar twisting and banded spherulites [29-32]. There are also exceptions. For example, poly (R-3-hydroxybutyrate) and poly(R-3-hydroxyvalerate) have the same configurational chirality (with R chiral center in the main chain) and conformational chirality (both with left-handed helical chain stems in the crystalline core), but the former gives lefthanded twist and the latter right-handed twist [33]. In addition, Li et al. reported that only a methylene difference in a main chain chiral liquid crystalline polyester led to the opposite twist handedness of the lamellar crystals though the two polymers have the same configurational chirality [34]. Based on the above experimental results, it is not the molecular chirality that directly determines the lamellar twisting chirality.

Our real-time AFM observations on poly(R-3-hydroxybutyrateco-17 mol% R-3-hydroxyhexanoate) (PHBHHx) showed that the full lamellae demonstrated right-handed twisting and the half top lamellae exhibited counter-clockwise bending [8]. This suggests that the distribution of surface stresses on a polymer lamella should have a 2-fold rotational symmetry axis, namely, rotation of the lamella by 180° will not change the surface stress distribution. The idea is schemed in Fig. 2.

From Fig. 2, we obtained the deduction that the lamella growing along two orthogonal growth axes should show the opposite lamellar twisting chirality. The deduction was validated by the study on poly(R-3-hydroxyvalerate) (PHV) banded spherulites [35]. A PHV banded spherulite consists of eye-like double-banded region surrounded by normal single-banded region. Wide-angle Xray diffraction (WAXD) and in situ heating demonstrated that the two regions were of the same crystal modification. Using an advanced optical microscope, Polscope that decoupled birefringence and orientation of the slow optical axis, we confirmed that the two regions grow along two axes perpendicular to each other. Via home-made tilting sample stage, the lamellar twisting chirality in the eye-like and normal region was determined to be lefthanded and right-handed, respectively, as presented in Fig. 3. Our finding reveals that the lamella of a chiral polymer can exhibit different twisting chirality, depending on the grow axis. This finding explains why the previous attempts to correlate the lamellar twisting chirality to molecular chirality one by one should inevitably fail. The work was highlighted at the cover of the issue of Macromolecules [35].

The origin and distribution of surface stresses arising from the folds are schemed in Fig. 4. Each lamella consists of four sectors. Fig. 4a and b show the different fold encumferences on the neighboring sectors. As a result, the surface stresses are different in the top and bottom surface in each sector and they are different in the neighboring sectors on the top surface, as presented in Fig. 4c. Both the experimental results and the physical model reveal that it is the distribution of surface stresses that directly determines the twisting chirality of polymer lamellar crystals.

By altering the chemical groups on the lamellar surface, we might change the surface stresses so as to invert the lamellar twisting chirality without change of molecular chirality. In poly(R-3-hydroxybutyrate-co-R-3-hydroxyvalerate) (PHBV) random copolymers and poly(R-3-hydroxybutyrate)/poly(R-3-hydroxybutyrate-co-4.6 mol% R-3-hydroxyhexanoate) (PHB/PHBHHx-4.6) blends, the comonomers are enriched in the fold surfaces of the lamellae. With increase of the comonomer content, the twisting chirality changes from left-handed to right-handed, as shown in Fig. 5 [36]. For bulkier comonomer, the inversion happens at a smaller comonomer content. The above results demonstrate unambiguously that it is the surface stresses that determine the lamellar twisting chirality.

The above experimental results demonstrate that the lamellar twisting chirality is determined directly by the distribution of the surface stresses. The latter depends on the molecular chirality, growth axis and surface chemical groups, etc.

Crystallization conditions, such as temperature and film thickness, may have considerable effect on the pitch of lamellar twisting, or the band spacing. For instance, in ultrathin films of polylactide, the lamellae usually adopted flat-on orientation, thus lamellar twisting is hard to observe [37]. In contrast, bending edgeon lamellae and sometimes dendritic curving flat-on lamellae could be observed [38].

Though the mechanical origin of polymer lamellar twisting has been revealed so far, the thermodynamics of lamellar twisting is not clear yet. For the above polymers that can show lamellar twisting when crystallized from melt, the single crystals formed from dilute solution can be flat [39]. This implies that the flat single crystals are more near the equilibrium than the twisted lamellar crystals. The twisted lamellae should possess elastic strain, which will increase the free energy of the crystalline core [40]. These observations suggest that the folds have not obtained the equilibrium conformations when the crystalline lamellar core is initially formed. During the further stabilization of the lamellar core, the folds are constrained so as to produce the repulsive interaction at the lamellar surfaces, which tends to swell the surfaces. However, the surfaces are covalently bound to the lamellar core, so they cannot swell freely. As a result, lamellar twisting makes the sum of the two energies at interior lamellar core and the amorphous surfaces reach the minimum at a certain degree of curvature.

3. Recognition: How do crystals recognize the different type of chain units?

In the above text, we mention that for random copolymers composed of two types of chain units, the minor chain units are usually excluded from the crystalline lamellar cores of the major units (Fig. 6a), leading to depressed melting point, degree of crystallinity and the radial growth rate of spherulite. Since the molecules are blind, how can crystallization separate the two types of chain units? Due to the weak intermolecular interaction between the two types of chain units, among the random trials during the crystallization process, only the trials choosing sufficient sequence lengths of the major chain units are effective to form the thermodynamically stable nuclei. However, there exist exceptions. For instance, in the poly(butylene succinate-cobutylene fumarate) (PBSF) random copolymers, the two types of chain units co-crystallize together to form the similar crystal lattice in the whole composition range (Fig. 6b), which was attributed to the very similar size and configuration of the two types of units so as to be accommodated in the same crystal lattice [41]. The phenomenon that two types of units co-crystallize in the same crystal lattice is termed as isomorphism [42-44]. As a result, the degree of crystallinity of the PBSF copolymers remains almost the constant and the melting point varies linearly with the copolymer composition. Namely, the crystallization process could not distinguish the two types of monomer units due to the almost same configurations and sizes. In contrast, in the poly(butylene succinate-co-butylene malerate) random copolymers, the butylene malerate units are excluded from the crystalline lamellar cores consisted of butylene succinate units. Poly(butylene fumarate) is an effective polymeric nucleating agent for poly(butylene succinate) (PBS) and its copolymers [41, 45], which provides a new guideline for designing nucleating agents for polymers. The mechanism of PBF as nucleating agent was attributed to epitaxial nucleation, which could enhance the rate of both primary and secondary nucleation [46, 47]. Compared to small molecular nucleating agents, the polymeric nucleating agent is of low cost and easy to disperse in the polymer matrix. Isomorphism has also been observed in the polyesters synthesized from other diols, succinic acid and fumaric acid [48].

For crystalline/noncrystalline polymer blends, the selection rule works as well. However, the kinetics will be different. According to the intramolecular chain nucleation [49], only the first stem of a crystalline polymer chain should be randomly selected for the secondary nucleation in polymer blends. In contrast, each chain stems with sufficient sequence lengths of the crystallizable units should be statistically chosen in the random copolymers.

4. Competition during nucleation

It is now generally accepted that polymer crystallization proceeds via the steps of nucleation and growth and the lamellar thickness is chosen during the secondary nucleation process. However, the molecular process of nucleation of polymer lamellae is not clear yet. We just list a few questions in the following:

(1) If a polymer can show two or more types of crystal forms, which one will be chosen?

(2) How do polymer lamellae choose the final lamellar thickness?

4.1. Cross nucleation of polymorphism: Appearing later but growing faster

A homopolymer can show two or more types of crystal modifications, which is termed as polymorphism. For example, poly(butylene adipate) (PBA) can form α and β modification from melt crystallization at high and low temperature range, respectively [50]. More interestingly, α and β modifications of PBA can coexist during isothermal crystallization at a narrow temperature range [51-53]. The nonbanded α form spherulites crystallize first, then banded β form regions appear at the periphery of the α form spherulites, as shown in Fig. 7 [52]. Since the β form has larger radial growth rate than the α form, the later-appeared β form will finally engulf the earlier-formed α form. This phenomenon that one modification nucleates on the surface of another form is termed as cross nucleation, which has been first observed in drug molecules [54, 55].

The competition growth of polymorphism can be attributed to the kinetics of crystallization, especially nucleation [56]. Variation of the energy barriers for primary and secondary nucleation of the two PBA modifications with temperature is schemed in Fig. 8. At the lowest temperature range blow T₁, both the primary and secondary nucleation barriers of β form are smaller than that of α form, leading to only β form. At the highest temperature range above T₂, both the energy barriers of α form are smaller than those of β form, results in appearance of α form. In the middle temperature range, the primary nucleation of α form and the secondary nucleation of β form are preferred, so the former appears in the spherulite center and the latter nucleates at the surface of the former. The mechanism of cross nucleation is not clear yet and deserves further study.

During heating, the β form PBA spherulites will transform into the α form, which indicates that the α form is the thermodynamically stable phase and β form is the metastable phase [57-59]. However, we observed that the α form PBA can transform into β form when a tensile force was applied and the strain exceeded a critical point [60]. Namely, the mechanical stress can alter the stability of the crystal modifications. In addition, nucleating agent, such as boron nitride nanosheets (BNNSs), can widen the formation temperature range for the α form PBA [61]. This can be attributed to the decreased energy barrier for primary nucleation after addition of the nucleating agent.

4.2. Nucleation of secondary nuclei: Thickening or widening?

In the above, we demonstrate that the primary nucleation chooses which type of crystal form crystallizes first and the secondary nucleation decides which form grows faster. In addition, the secondary nucleation process determines the proper lamellar thickness. The classical Hoffman-Lauritzen surface nucleation theory considers only the starting state of the amorphous melt and the final state of lamellar crystals with fixed lamellar thickness (may with small fluctuation) [62, 63]. This assumption has been questioned by the other researchers, e.g., Frank and Tosi [64], Point [65], Sadler and Gilmer [66], etc. They proposed that the stem length in the lamellae or the precursors could vary continuously and suggested that the chosen lamellar thickness corresponded to the fastest radial growth rate or the attracting point enabling steady lamellar growth. In addition, Strobl [67, 68] proposed a mesophase mediated polymer crystallization model, in which the crystallization process might consist of two successive pathways, melt to mesophase and the following mesophase to native crystals.

It is not logically reasonable that the nuclei should initially adopt the final lamellar thickness. In fact, the secondary nucleation of polymer crystallization is probably a competition of evolution of the intermediate states with different stem lengths and cluster widths. During the nucleation stage, both lamellar thickening, widening and melting of the precursors can occur simultaneously. The energy barrier to form extended chain stems is of entropic origin and that to melt the precursors is mainly of enthalpic origin. At the critical point, the two energy barriers reach the same.

We simulated the secondary nucleation of lamellar nuclei on the existing crystal surface via a microscopic kinetics model [69]. Compared to the previous attempts by other researchers, longrange correlation was considered in the attaching and detaching rate constants. The results showed that at first both thickening and widening happened simultaneously. Later, whether thickening or widening is preferred depends on the long-range correlation factor. Stronger inter-stem interaction favors widening of lamella and thus chain folding.

How the lamellae are stabilized is still an open question. It is generally believed that the lamellae are stabilized by thickening. Hoffman-Weeks plot (T_m-T_c line, H-W plot) has long been applied to obtain the equilibrium melting point via linear extrapolation [70]. The reciprocal of the slope of H-W plot, 1/γ, is defined as the thickening ratio, which is the ratio of the lamellar thickness at the melting point T_m to the minimum thickness of the stable lamellae at the crystallization temperature T_c. For polymer lamellae without α_c relaxation, the lamellar thickening during heating can be negligible, thus the thickening ratio, 1/γ, reflects the degree of stabilization during the isothermal crystallization stage.

Our study on melting of PBS revealed that its H-W plot comprised of two lines with different slopes: The line with slope 0.64 below T_c=110 ℃ and that with slope of 1.01 when T_c ×110 ℃, as shown in Fig. 9a [71]. The H-W plot parallel to T_m= T_c line at high temperature range indicates that T_m⁰ could not be obtained by linear extrapolation of the H-W plot. Two questions arise: (1) What is themechanismbehind the slopechangeof H-Wplot?(2) What is the temperature obtained by the usual linear extrapolation of H-W plot at the low temperature range?

Via small-angle X-ray scattering (SAXS) measurements, we obtained both the crystallization and melting line, as presented in Fig. 9b. The results demonstrate that the crystallization line changes slope at T_c=110 ℃, which agrees with turning point on the H-W plot. Via quantitative deductions, we established the correlation between the H-W line, crystallization line and Gibbs-Thomson melting line. The slope of the H-W line equals the ratio of the slope of the melting line and that of the crystallization line. The linearly extrapolated temperature T_x of H-W line at the low temperature range is the temperature where the extrapolated crystallization line intersects with the melting line, as shown in Fig. 9b. For PBS, T_x is much lower than the equilibrium melting point.

We suggest that lamellar widening is the major stabilization mechanism of the lamellar nuclei in the post-nucleation stage, while there is only very limited degree of thickening in the postnucleation stage. The change of slope of the crystallization line with temperature is due to the different stabilization mechanisms. At large super-cooling, the widening happens via coalescence of the neighboring nuclei, as schemed in Fig. 10. At small supercooling, the stabilization effect of widening is limited since the probability of nuclei coalescence is rare. The effect of lamellar widening has seldom be considered before. Besides supercooling, the minimum lamellar thickness depends on the lamellar width as well.

Via step heating differential scanning calorimetry (DSC), we investigated the melting and recrystallization behavior of isothermally crystallized PBS lamellae, as demonstrated in Fig. 11 [72]. The multiple melting-recrystallization peaks were clearly observed via the method. The reorganization is a cooperative process needing partial melting of at least two neighboring preformed secondary nuclei. The kinetics equations of the melting and recrystallization were established quantitatively. The original and the reorganized lamellae formed via melting-recrystallization exhibited quite different melting kinetics, which was attributed to the different stabilization mechanisms. The former was stabilized via coalescence and the latter by limited widening.

5. Summary and perspective

In this review, we briefly discuss several topics of polymer crystallization: Chirality of lamellar twisting, competitive nucleation of different forms of the same polymer, recognition of different chain units in random copolymers and the competition of different intermediates during crystallization.

Crystallization process, not limited to polymer crystallization, is interplay of two counter factors: order and disorder, thermodynamics and kinetics, entropy and enthalpy, surface and bulk, local and global optimization, etc. As a result, the finally formed polymer crystalline materials exhibit various levels of hierarchical order, e.g., chain conformation, lamellar crystals, crystalline textures (such as dendrite and spherulites), etc.

Crystallization leads to minimization of the total free energy of the system. Ideally, a single crystal with infinite size should be the most thermodynamically stable one. However, the formed crystals are not in equilibrium with the surrounding amorphous molecules any more, which makes the minimization operable infinite time (e.g., in the nucleation stage). The short time minimization results in metastable crystals, such as the crystals with small size, metastable crystal forms, etc. On the other hand, the large size of polymer chains often exceeds the size of a critical nuclei, thus the free energy minimization may happen in only a part rather than the whole polymer chain. The local minimization of the free energy results in metastable crystals, such as the folded chain lamellar crystals with finite thickness.

Though intensive efforts have been devoted to understanding polymer crystallization, there are still some open questions remained. For instance, if there are more than one intermediate states, how to describe them in thermodynamics? What is the effect of cooperativity of chain folding and long-range multi-body molecular interactions on polymer crystallization? The researchers in the field of protein folding and those in the field of polymer crystallization can learn from each other. How can the concept of chaperones in the protein folding help to tune the polymorphism of polymer crystals? We expect that new theories based on microscopic kinetics are beneficial to answer the above questions.

Old dogs can be taught to play new tricks. Recently, polymer lamellar crystals have found new applications. For instance, polymer lamellae are used as building blocks of functional materials [73, 74]. Inorganic crystals, such as molecular sieves, have long worked as catalysts. Can polymer crystals work as catalysts? A polymer single crystal can clone many crystals with the same orientation by self-seeding [75]. How to make the polymer crystals mutate so as to be more like life?

Schrödinger [76] proposed that the information carrier of life should be aperiodic crystals, which were later known as DNAs and RNAs. Now the crystallization of homopolymers, random and block copolymers, small molecule-polymer host-guest inclusion compounds has been extensively studied. In the future, more attention should be paid to crystallization of heterogeneous polymer chains with tailored sequences, supramolecular systems, multi-component dynamic systems, etc. It is of particular interest that sleep is a type of phase transition. Consciousness is probably also a type of phase transition. We believe that the ongoing further study on polymer crystallization will deepen our understanding of order, morphogenesis and life.

Acknowledgments

The authors are indebted to the National Natural Science Foundation of China (No. 21374054) and Fund of Key Laboratory of Advanced Materials of Ministry of Education (No. 2017AML07) for financial support.