The Tough Journey of Polymer Crystallization: Battling with Chain

Perspective Cite This: Macromolecules XXXX, XXX, XXX−XXX

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The Tough Journey of Polymer Crystallization: Battling with Chain Flexibility and Connectivity Xiaoliang Tang, Wei Chen, and Liangbin Li*

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National Synchrotron Radiation Lab, Anhui Provincial Engineering Laboratory of Advanced Functional Polymer Film, CAS Key Laboratory of Soft Matter Chemistry, University of Science and Technology of China, Hefei 230026, China ABSTRACT: Theoretical approaches addressing the mechanism of polymer crystallization remain the great challenge in polymer science. Numerous different, or even conflicting, models/theories have been proposed during the past several decades. However, none of them can fully satisfy the whole community. In this Perspective, we first trace the roots of these models/ theories back to the classical and nonclassical nucleation theories. The correlation between these theories and milestone theoretical works in polymer crystallization is elucidated together with their intrinsic drawbacks. Then the newly proposed two-step nucleation scenarios, with either bondorientational order or density fluctuation as precursors, are introduced, which, in our view, may stimulate the development of polymer crystallization theory. Afterward, the peculiarities of polymer crystallization due to chain flexibility and connectivity are discussed. A personal outlook on the ultimate polymer crystallization theory is given at last, which is suggested to address the following three questions: (i) How do flexible chains transform into rigid conformational ordered segments? (ii) How do interlamellar amorphous layers form? (iii) Are polymer chains dragged by force to the growth front? Answering these questions may eventually end the tough journey for the establishment of polymer crystallization theory, though polymer crystallization never completes fully. the review of Armistead and Goldbeck-Wood in 1992.13 Starting in the early 1990s, new experimental observations, mainly coming from scattering techniques, suggest that preordering or precursors emerge before the onset of crystallization.14−18 A spinodal decomposition-assisted nucleation model and a multistage crystallization model were proposed by Olmsted et al.19 and Strobl,20,21 respectively, to challenge the HL model. These new observations and ideas trigger a vivid discussion for over two decades around 2000,22−25 but no consensus has ever been reached in the community of polymer crystallization. In celebrating the 50th anniversary of Macromolecules, Prof. Timothy P. Lodge listed the theory of polymer crystallization as the top one challenge remaining in polymer physics.26 This Perspective does not intend to review the current models of polymer crystallization, as several comprehensive reviews have been published before, in which the basic assumptions, physical pictures, and predictions are discussed in detail.13,27−38 The purpose of this Perspective is to raise questions rather than give answers, which we wish to stimulate discussions and new ideas for the development of polymer crystallization theory. The Perspective is organized as follows. In section 2, we first take one step back to briefly introduce the classical nucleation theory (CNT)39−41 and nonclassical

1. INTRODUCTION The worldwide polymer industry produces about 400 million metric tons of polymeric materials with a total sale of about $500 billion annually, among which 70% are semicrystalline.1,2 Tuning crystallization behavior with either primary molecular structures like chain length and distribution of branches or processing parameters like flow and temperature, the mechanical performances of polymer materials have been greatly improved over the past decades. Polyethylene (PE) may serve as one of the best examples, which continuously exceeds our expectations with super performance to replace traditional materials in various fields such as oil tanks, pipes, and bulletproof jackets.3−6 Therefore, understanding the mechanism of polymer crystallization is of critical importance due to the significant impact of the crystallization process on the properties of materials. The Hoffmann and Lauritzen (HL) theory of polymer crystallization was proposed in 1960,7 just 3 years after Keller’s discovery of PE single crystals.8 After many extensions and modifications in the past 50 years,9 HL theory becomes the most widely used model as the theoretical expressions on lamellar thickness and growth rate are ready to be applicable in data evaluation. Nonetheless, HL theory has also been challenged by various new observations and ideas.10,11 The 1979 Faraday Discussion in Cambridge12 is one important cornerstone in polymer crystallization, after which the HL model gains more attention over others. The theories of polymer crystallization were comprehensively discussed in © XXXX American Chemical Society

Received: December 24, 2018 Revised: April 11, 2019

A

DOI: 10.1021/acs.macromol.8b02725 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Classical Nucleation Theory. One critical assumption of CNT is the capillary approximation where the macroscopic characteristics of the nucleus, i.e., density/composition/ structure, are the same as the bulk crystal and a sharp density boundary exists between the nucleus and liquid (Figure 1a).

nucleation theories (non-CNT) including diffuse interface theory (DIT),42−45 as polymer crystallization models are essentially rooted in the general nucleation theories. Section 3 is dedicated to polymer crystallization models, where we briefly introduce basic ideas of the HL model9 and Strobl’s multistage model. The HL model is a secondary nucleation and growth theory describing polymer crystal growth and shares similar physical features as those of CNT,13,34 while Strobl’s multistage model is a typical nonclassical approach widely discussed in the polymer community in recent years. Strobl’s model highlights the existence of the mesomorphic phase as a structural intermediate but without a detailed molecular description on nucleation and growth as the general nucleation theories and HL model do. Therefore, the arrangement here is a compromise to group polymer crystallization models in one section. In section 4, the recent “two-step” nucleation scenarios, assisted by either density fluctuation46,47 or bondorientational order (BOO),48,49 are discussed. Although the two-step nucleation scenarios are proposed as general nucleation models and mainly contributed by the colloid and biomacromolecules communities, they are expected to stimulate the development of synthetic polymer crystallization. In section 5, we focus on the roles of the two peculiar features of long chains in polymer crystallization, namely, flexibility and connectivity. With regard to chain connectivity, the discussion is mainly restricted on crystallization of bulk or highly entangled polymer solution. Three questions will be discussed. (i) How do flexible chains transform into rigid conformational ordered segments (COS) during polymer crystallization? (ii) How do interlamellar amorphous layers form? (iii) Are polymer chains dragged by force to the growth front? This Perspective summarizes some suggested future research directions in section 6. Note that this work mainly focuses on the early stage of polymer crystallization; thus, many interesting and important phenomena at the perfection process including crystal−crystal transformation, lamellar branching or lamellar twisting are not addressed here. This topic has been comprehensively discussed in a recent Perspective by Lotz, Miyoshi, and Cheng,35 which we recommend the reader to refer to.

Figure 1. (a) Schematic illustration of CNT, where the nucleus owns the same properties as bulk crystal and a sharp boundary exists between nucleus and liquid, and the particles deposit onto or detach from nucleus within one step. (b) Free energy profile according to eq 1, and the Gibbs free energy is the summation of surface and volume terms.

With respect to the formation of the spherical nucleus with a radius of r during the homogeneous nucleation process, the Gibbs free energy change ΔG after the formation of the spherical nucleus is given by ΔG = −

4πr 3 Δgsl + 4πr 2σ 3

(1)

where Δgsl and σ are the bulk free energy density difference and the surface free energy between the nucleus and surrounding liquid, respectively. The surface and volumetric terms as well as ΔG versus radius of the nucleus are schematically illustrated in Figure 1b, where the volumetric and surface terms in eq 1 show the opposite effect. The radius r* of the critical nucleus can be obtained when dΔG/dr = 0:

r* =

2σ Δgsl

ΔG* =

2. CLASSICAL AND NONCLASSICAL NUCLEATION THEORIES A description of the classical nucleation theory can be generally classified into two categories: thermodynamics and kinetics.50 The thermodynamic description of the nucleation can be dated back to the 1870s when Gibbs proposed the theory of thermodynamic fluctuation.39,40 The thermodynamic driving force for nucleation is the free energy difference between liquid and solid. Later on, the kinetics description of the nucleation was formulated by Volmer and Weber,51 which is further developed by Szilard/Farkas,52 Volmer,53 Becker/Döring,54 Frenkel,55 Turnbull and Fisher,41 and other pioneers. The CNT provides a simple framework to interpret the crystallization phenomena. CNT is flexible and has successfully predicted the nucleation rate in many systems due to its reasonable assumptions; however, many complicating effects also stem from the simple treatment resulting in discrepancies between theoretical predictions and experimental results.56,57 In this section, we will briefly introduce the CNT and other non-CNT, and hopefully, the basic physical picture, as well as the difference between them, could be clear.

16πσ 3 3(Δgsl )2

(2)

(3)

where ΔG* is free energy of the critical nucleus, which can be obtained by the statistical−mechanical treatment as proposed by ten Wolde and Frenkel.58 With energy barrier ΔG* and the assumption of reversible reactions in nucleation, CNT gave the mathematic expression of nucleation rate I, which can be experimentally accessible. I is first semiquantitatively proposed by Becker and Döring54 and later derived by Turnbull and Fisher41 on the basis of the theory of absolute reaction rates: I≅

ji ΔG* + ΔGη zyz NkT zz expjjjj− zz j h kT k {

(4)

where ΔGη is the activation energy for the diffusion of a particle with an atomic distance at the growth front and N is the total number of particles in the system. k is the Boltzmann constant, T is the temperature, and h is the Planck constant. Qualitative elucidation of the nucleation process can be achieved with the assistance of the above equations deduced from CNT; however, the quantitative description often fails. Such failure mainly results from the drawbacks of CNT: (i) An B


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Figure 2. (a) Capillary approximation of CNT that clusters own the same density as bulk crystal and a sharp boundary exists between solid/liquid phases. The periodic oscillation of density ρ represents the ordered structure of the crystalline phase, whereas the liquid phase shows a constant value. The lower plot is the corresponding free energy density of solid state gs and liquid state gl. (b) Possible density profile and corresponding free energy density landscape in DIT, and the free energy density of interface is a function of radius. (c) An alternative approach addressing the interface in DIT in forms of the step function of free energy and density. gi is the free energy density of the interface; Δgsi, Δgil, and Δgsl are the solid/ interface, interface/liquid, and solid/liquid free energy density difference, respectively. The thickness of the interface is equal to d.

including oxide glasses and the metallic glasses.62,63 However, DIT is a phenomenological thermodynamic model, and the diffuse interface layer is an ad hoc unproven assumption without detailed molecular information. The classical density functional theory (cDFT) can describe the density and free energy profiles of the diffuse interface layer in DIT, which has been implemented to study the nucleation phenomena in gas−liquid and liquid−solid transitions. This method is mathematically rigorous, and the only required input is the interaction potentials; it has been proved to have the potential to calculate properties including critical cluster size, density profile, and critical grand potential. The study of polymer crystallization using cDFT is approaching; here the references are listed for the reader to refer to.64−70

energy barrier always exists even if the systems approach the spinodal decomposition. (ii) The nucleus always has bulk properties even if it is small and there is a sharp interface between nucleus and mother phase. (iii) The CNT represents the curved surface of the nucleus by an infinite flat planner surface.57,59 To address these problems, numerous theories, namely, nonclassical nucleation theories, have been proposed to challenge the CNT. Among them, the diffuse interface theory and the density functional theory are two worthy of note. Diffuse Interface Theory (DIT). Figure 2a illustrates the distribution of density ρ, and the free energy density profile across the interface between the nucleus and the surrounding liquid according to CNT, where a sharp boundary is assumed. Such a sharp boundary between the two phases of CNT results in a large discrepancy of nucleation rate between theoretical predictions and experimental results with a deviation of up to several orders of magnitude.60,61 To address this problem, Gránásy42−44 and Spaepen45 developed the DIT in the early 1990s, in which an interface layer with a finite thickness of d, instead of a sharp interface with 0 thickness in CNT, is proposed to sit between solid crystal nucleus and liquid as shown in Figure 2b. Similar to CNT, the free energy change of forming a nucleus in DIT is contributed from the solid core with radius rs and the interface layer with thickness d = rl − rs, which can be written as ΔG = 4π

∫0

rs

r 2(gs − gl ) dr + 4π

∫r

rs + d

3. POLYMER CRYSTALLIZATION MODELS The CNT and non-CNT are mostly developed for understanding the primary nucleation of small molecules; they also largely influence the development of polymer crystallization. In the following section, polymer crystallization models, including Hoffman−Lauritzen theory and Strobl’s multistage model, will be briefly summarized. Hoffman and Lauritzen Theory. Inspired by the general concepts in CNT, Hoffman and Lauritzen7,9 proposed the secondary nucleation theory to account polymer crystallization with chain folding.8 Admittedly, CNT deals with the primary nucleation, whereas HL theory describes the crystal growth of polymer with the secondary or surface nucleation. But similar to CNT, HL theory describes polymer crystallization as a kinetically reversible reaction process with segments attaching onto and detaching from the crystal growth front. Although the polymer community is familiar with HL theory, we still want to highlight the key characteristics of this theory to be compared with the above theories. HL theory describes the crystal growth with surface nucleation rate Isn and substrate completion rate Isg (see Figure 3). During surface nucleation, the first chain stem, with the length l equaling the thickness of crystal substrate, deposits onto an atomically smooth growth front with a bulk energy gain of a0b0lΔg and a penalty of 2b0lσ due to creating two lateral surfaces, where a0 and b0 are the width and thickness of the stem, respectively, Δg is the free energy density difference between crystal and polymer melt, and σ is the free energy density of the lateral surface. The further deposition of other stems requires chains to fold back and forth with the same energy gain of a0b0lΔg

r 2[gi (r ) − gl ] dr

s

(5)

where gs, gl, and gi(r) are free energy densities of crystal, liquid, and the interface layer, respectively. As shown in Figure 2b, gi(r) is not constant and varies with r, which, unfortunately, cannot be easily obtained. As an alternative approach, a step function is introduced as shown in Figure 2c to simplify the mathematical treatment. Then the free energy of nucleus can be expressed as ΔG = −

4πrs 3 4π Δgsl + [(rs + d)3 − rs 3]Δgil 3 3

(6)

where Δgsl and Δgil are the solid−liquid and interface−liquid free energy density differences, respectively. Following a similar approach as CNT, the radius rs* of the critical nucleus and the nucleation barrier ΔG* can be obtained. DIT has proven its ability to well reproduce the nucleation rate of various systems, C


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I, II, and III, are predicted by Hoffman−Lauritzen theory, which are expressed as b0IsnL, b0(2IsnIsg)1/2, and b0Isn(nIIIa0), respectively.9 nIII is a number on the order of 2.0−2.5. The detailed form of IG in different regimes can be found in the latest version of the HL model,9 which will not be discussed here. After several decades of investigation, the microscopic parameters shown in eqs 7−11 are obtained, which are summarized by Armistead and Hoffman.71 Definitely, the HL model has constructed a good framework to understand the polymer crystallization, which not only describes how chain folding occurs but also provides the expressions of lamellar thickness l* and growth rate IG that can be ready to compare with experimental observations. However, the HL model has been challenged by many recent experimental and simulation investigations.19,21 Other comprehensive approaches have been continuously proposed to understand the polymer crystallization, such as the fine-grained model,10 Point’s multipath model,11,72 Sadler−Gilmer’s model,73−75 Hikosaka’s sliding diffusion model,76,77 Wunderlich’s molecular nucleation model,78 Hu’s intramolecular nucleation model,79 and Muthukumar’s continuum model,27 just to name a few. These remarkable theoretical approaches proposed solutions for the crystallization of some specific cases, i.e., extended-chain crystal and single-chain nucleation, or focused on the effect of system parameters, i.e., molecular weight and concentration dependence. Admittedly, these theories are self-consistent in their own cases and provide possible insight into the polymer crystallization; it would be better to consider them as compensations or extensions of the HL model rather than inevitable approaches. As the HL model is essentially based on CNT, it naturally inherits all the limitations of CNT such as capillary assumption and the sharp interface between crystal and liquid as discussed above. Additionally, because of the long chain nature of polymer, the following comments are necessarily emphasized here: (i) The HL model assumes that rigid conformational ordered segments, with the length equaling the thickness of crystal substrate l, attach onto or detach from the growth front, but it does not mention when and how conformational ordering takes place. (ii) Different from atoms or small molecules, polymer crystallization requires to consider the chain connectivity. Although the concept of reptation was introduced in the HL model,9,80 theoretical treatment of the nucleation process considering chain connectivity is not yet fully resolved. Multistage Crystallization. The proposal of DIT overcame the drawbacks of CNT by eliminating the capillary approximation, and similarly, people start to consider there may be no sharp boundary between polymer melt/solution and crystal. The multistage crystallization model proposed by Strobl21 is the most widely discussed one in recent two decades (Figure 4). In the multistage crystallization model, a mesomorphic phase, composed with orientation ordered stems, is proposed to sit between polymer liquid and crystal growth front, and crystallization is a transition from the liquid to mesomorphic phase and further to the crystal rather than a one-step process directly from the liquid to crystal. Accompanied by this liquid− mesomorphic phase−crystal transition, a continuous thickening is also suggested, which reduces the mobility of stems, resulting in the formation of metastable crystals with limited thickness rather than extended-chain crystals. The multistage nucleation model supposes the mesomorphic layer is a distinct

Figure 3. Model of secondary nucleation and growth proposed in the HL theory. The parameters including substrate length L, nucleation rate Isn, spreading rate Isg, free energy density of the fold surface σe, and the lateral surface σ as well as the stem length l. a0 and b0 are the width of the stem and thickness of the layer, respectively. Reproduced with permission from ref 9. Copyright 1997 Elsevier.

and a penalty of 2a0b0σe due to chain folding, and σe is the free energy density of the folded-chain surface. This process eventually leads to the formation of secondary nuclei on the surface with a nucleation rate of Isn. Afterward, the rest of the growth front is covered by following deposited polymer stems until fully covered with a substrate completion rate Isg. Repeating this process leads to the growth of crystal layer by layer with a growth rate of IG. Similar to Turnbull and Fisher’s treatment41 in the kinetic approach of CNT, the forward and backward reactions (attaching and detaching) are employed in the HL model to study the secondary nucleation process.9 According to the HL model, the attaching and detaching rates of the first stem (ν = 1) are given by A 0 = K exp( −2b0σl /kT )

(7)

B = K exp[−a0b0l(Δg )/kT ]

(8)

where the factor K is the frequency trying to attaching or detaching associated with transport of chain segments. The penalty for depositing of other stems (ν ≥ 2) is the work of chain folding 2a0b0σe; therefore, the forward rate is given by A = K exp( −2a0b0σe/kT )

(9)

With these forward and backward reaction rates, an expression for the steady-state flux which depends on l can be given by S(l) = N0A 0(1 − B /A)

(10)

where N0 is the number of reacting species. S(l) can also be interpreted as the probability distribution for a crystal with thickness l.13 Therefore, the average thickness of an crystal could be given by ∞

⟨l⟩ = l* =

∫2σ / Δg lS(l) dl e

∞

∫2σ / Δg S(l) dl e

=

2σe + δl Δg

(11)

where δl accounts for the fluctuation of the lamellar thickness, and the lower bound of integration 2σe/Δg is the minimal stem length required as derived in the thermodynamics approach. The experimental observed lamellar thickness is kinetics controlled in the HL model, during which lamellae with thickness l* have the quickest growth rate overwhelming shorter and longer ones. If the stems are shorter or longer than l*, the nucleation rate would be quite small. With respect to the growth rate, IG, depending on the supercooling condition, three growth regimes, namely, regimes D


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(structure), or vice versa.83−85 This process requires at least two order parametersdensity and structureto differentiate the old and the new phases, which is interpreted with “twostep” nucleation models. Note that “two-step” here is not a necessary limit as multistep could also be possible, and the authors just borrow the name to emphasize the importance of intermediate states. The reported liquid−precursor−crystal sequential ordering process in polymer crystallization can be grouped in the two-step nucleation models,19,86 in which either conformational ordering, orientation ordering, or density fluctuation is suggested involving in precursors. Meanwhile, recent work based on the combination of isotope labeling and 13 C double quantum NMR has revealed that polymer chain tends to form nanoclusters before further depositing onto the growth front.87−89 The Oswald stage rule states that phase transition takes the kinetically favored process via structural intermediates with the lowest energy barrier rather than jumps from the initial state to the final stable one directly.90−94 The scenarios of “two-step nucleation” reported in recent decades demonstrate the phase transition via intermediate states is a preferable pathway.46,48,93 As shown in Figure 5a, the CNT states that all order

Figure 4. Sketch of the multistage model that the crystallization goes through the melt, mesomorphic phase, granular crystal phase, and to the final lamellar crystal. Reproduced with permission from ref 21. Copyright 2009 American Physical Society.

phase which owns an identifiable free energy density, gm. Compared with the free energy densities of isotropic liquid, ga, and bulk crystal, gc, the relation is given by ga > gm > gc (12) This is to say, the mesomorphic phase is also stable against the isotropic melt. The temperature-dependent lamellar thickness and growth rates are key measurable parameters to check the validity of different theories. With respect to the lamellar thickness, both HL and multistage models take the Gibbs−Thomson equation, which shows the relation between melting points and lamellar thickness: T (l) = Tf∞ −

2σeTf∞ 1 Δhf l

(13)

T∞ f

where is the equilibrium melting temperature and Δhf denotes the heat of fusion. But T∞ f is replaced with Tc in the multistage model, where Tc > T∞ f . For the growth rate, the supercooling ΔT = T∞ f − T(l) determines the growth rates as predicted by the HL model. This is also challenged by Strobl, which is proposed to be revised by the introduction of the zero growth temperature Tzg. A detailed description of the difference between these two models can be found elsewhere.20 The supporting experimental evidence of the multistate model mainly comes from different slopes of the crystallization and melting lines of various polymers, such as syndiotactic polypropylene and poly(ε-caprolactone). And recently, combining in situ infrared microscopy and micro-X-ray diffraction, we observed conformational ordered layer exists at the growth front of spherulite.81 The memory effect on the growth of form I in iPB-1 also demonstrated an ordered melt existed at the growth front.82 Whether these ordered melt layers are the mesomorphic phase is not verified yet, but they are at least in line with the DIT. The multistage model introduced the intermediate states, which results in a kinetics pathway significantly different from the one-step HL model. Even though the definition of “mesomorphic phase” is still questionable,23 it could be a possible way to further understand the polymer crystallization.

Figure 5. (a) Schematic diagram of one-step and two-step nucleation scenarios, where ρ is the density and Q is the order parameter for bond orientation or other local structure order. The blue arrow indicates the one-step process that density and structure transform simultaneously. The yellow and red arrows represent two kinds of two-step scenarios with the dense liquid or bond-orientational order as precursor, respectively. (b) Schematic illustration of the phase diagram consisting of solid, liquid, and gas in temperature (T) and pressure (P) space. The green dashed lines indicate the possible boundaries of liquid and bond-orientational order.

parameters transform simultaneously in crystallization. While in the two-step nucleation models, order parameters, i.e., density and structure, transform in sequence. Two different two-step nucleation scenarios have been discussed recently, in which either a dense-liquid precursor or bond-orientational order (BOO) precursor is present as the structural intermediate.83,95,96 Bond-Orientational Order Assisted Nucleation. Bondorientational order (BOO) is proposed by Nelson and coworkers97,98 to identify local ordered structures without longrange order. The Steinhardt order parameter97,99 of BOO, later popularized by Frenkel and co-workers,100 is often used to classify the particles as solid-like or liquid-like in simulation and experimental works:

4. “TWO-STEP” NUCLEATION MODELS In CNT, the cluster formation and reorganization into new phases are assumed to happen simultaneously. As a result, only one order parameter, i.e., density or structure, is sufficient to describe the phase transition process. However, more and more experimental evidence show that the clusters emerge first (density), followed by reorganization to form crystals

ij 4π Q n(i) = jjjj j 2n + 1 k where

E

y ∑ |Q nm(i)| zzz m =−n { +n

2z zz

1/2

(14)


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Figure 6. (a) Schematic temperature−pressure phase diagram of phosphorus. A first-order phase transition between two distinct liquid forms exists, and the red line demonstrates the coexistence. Reproduced with permission from ref 131. Copyright 2004 AAAS. (b) Schematic free energy landscape for the crystallization process starting either from the homogeneous amorphous state (a) or the inhomogeneous melt state (m), and the c is the final crystalline state. G1* and G2* indicate the free energy barriers. Reproduced with permission from ref 137. Copyright 2016 AIP.

1 Q nm(i) = N(i)

N(i)

crystal are composed of rotational symmetries (or point group symmetry). If their symmetries are the same or close, local adjustment of particle position can lead to the transformation from BOO to crystal, during which BOO promotes crystallization, while if their symmetries mismatch with each other largely, frustration on crystallization may be imposed by BOO. 106,107 The second is interfacial wetting-induced crystallization.48,108 As BOO is structurally intermediate between the liquid and solid crystal, if crystal nucleation occurs inside BOO domains, BOO is present as the interfacial layer between liquid and crystal, which can reduce surface tension or penalty and consequently enhance nucleation. The interfacial wetting-induced crystallization is rather similar to the DIT or Strobl’s multistage models. In polymer crystallization, BOO as a local favored structure is rarely mentioned yet. Instead, conformational order like gauche−trans and coil−helix transitions (CHT)109−111 is a basic concept in polymer science, which is also local in nature. CHT is intrachain order and results in a one-dimensional crystal-like structure, which may not be treated as BOO. As will be discussed later, similar to BOO local structure order, both intrachain conformational order and interchain interaction may appear in the polymer. Density Fluctuation Assisted Nucleation. Another twostep crystallization scenario is density fluctuation assisted nucleation. As depicted by this model, density or concentration fluctuation occurs first, resulting in two-phase separated liquids, namely high and low density (concentration) liquids, while crystal nucleation appears later inside of the high-density one.46,112−115 As phase separation and crystallization present intrinsically in the polymer solution, blends, and copolymers with multiple components, it is ready to accept that coupling and competition between phase separation and crystallization can take place,116−119 which indeed has been widely observed in experiments,120−123 while it seems harder to reach the consensus for one-component system to separate into highand low-density liquids. Starting from the early 1990s, Kaji and co-workers86,124 reported that phase separation in a series of polymers at temperatures slight above Tg, while density fluctuation and phase separation were suggested to exist in PE and iPP at low supercooling by Ryan and co-workers.16,125,126 Olmsted et al.19 made one step further and proposed the theory of spinodal decomposition assisted crystallization. They consider two order parameters, density

∑ Ynm(θij , ϕij) j=1

(15)

Ynm(θij,ϕij) are the spherical harmonics functions; θij and ϕij are the polar angles between center particle i and N(i) neighboring particles within a certain cutoff distance. As the crystal possesses both translational order (density ρ as order parameter) and BOO (rotational symmetry, Q as order parameter), the one-step CNT states that crystallization occurs with ρ and Q coupled together, while in the two-step nucleation models, BOO and translational order can decouple and occur at temperatures of TQ and Tρ, respectively, as shown in Figure 5b. Determined by the relative values of TQ and Tρ,101 three different crystallization processes can be observed. (i) If TQ > Tρ, BOO takes place when the system is cooled into the region Tρ < T < TQ (the black arrow in Figure 5b indicates the pathway). Here the emergence of BOO is not necessarily coupled with ρ. Further reducing temperature to T < Tρ, crystallization can start, during which BOO can either promote or suppress crystallization determined by the symmetric match between BOO and crystal. (ii) If TQ < Tρ, cooling the system crosses Tρ can trigger crystallization, during which translation order and BOO couple with each other. This process falls in the case of the one-step CNT. (iii) If TQ < Tρ, BOO may still decouple with density to emerge first. As crystallization is a strong first-order phase transition, the high nucleation barrier may suppress its emergence to high supercooling with crystallization temperature Tc < TQ.49 In contrast, BOO is local in nature without a large barrier to prevent it from occurring. In this case, BOO can appear prior to the onset of crystallization, which can either assist or frustrate crystallization. The (iii) situation may be rather common in polymers, as the homogeneous nucleation of polymer generally takes place at large supercooling ΔT = T0m − Tc, where T0m and Tc are equilibrium melting temperature and crystallization temperature, respectively. ΔT for homogeneous nucleation is about 107, 102, and >140 °C for isotactic polypropylene (iPP),102 isotactic polystyrene (iPS),103 and isotactic poly(1butene) (iPB-1),104 respectively. Even for PE with the fastest crystallization rate among polymers, ΔT for homogeneous nucleation is still more than 55 °C.105 The assisting role of BOO on crystallization may stem from two factors: structural and interfacial matching. The first is structural similarity between BOO and crystal. Both BOO and F


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5. PECULIARITIES OF POLYMER CRYSTALLIZATION In the above sections, we briefly discussed some milestones in the development of crystallization theories. Although the discrepancies still remain and the newly proposed two-step nucleation models may require further modifications and confirmations, these new ideas can certainly benefit the development of polymer crystallization theory. In addition to borrowing the physics from general crystallization models, polymer crystallization has its own features different from that of spherical atoms and small molecules. Because of the long chain nature, the polymer has two peculiar features, namely flexibility and connectivity. In this section, we are going to discuss the roles of flexibility and connectivity on polymer crystallization. Flexibility. The classic polymers like PE and iPP are long chains connected by repeated monomeric units. In the liquid state, chains are flexible with randomly distributed trans and gauche conformations while polymer crystals are packed with rigid COS. Polymer crystallization always involves with conformational ordering like trans−gauche transition or CHT. How do flexible chains transform into rigid COS in polymer crystallization? Different from spherical atoms and most small molecules, conformational ordering is one peculiar feature and the rate-limiting step in polymer crystallization,138 which, however, has not gained the attention deserved in current crystallization models. The HL model states that COS, with the same length as the thickness l of substance, deposits one-by-one onto the crystal growth front,9 while Strobl’s mesomorphic layer composes of oriented COS segments.21,22 In the spinodal decomposition model of Olmsted et al.,19 conformational order is coupled with density provided the concentration of COS reaches the critical value. To some extent, all these models take COS as a precondition or attribute the formation of COS as the results of intra- or intermolecular interactions but do not discuss how conformational order takes place from the perspective of kinetics. In addition to the energy gap between trans and gauche conformations, entropy is the dominant penalty in conformational ordering. Without external fields like stretch and flow or strong intrachain interaction like hydrogen bond in α helix of protein, single COS, as one-dimensional crystal, is not stable according to Landau−Peierls instability.139 For most synthetic polymers, conformational ordering has to rely on the assistance from either interchain interactions or external field. Evidently, the theory of polymer crystallization has to find which kinds of interchain interactions assist intrachain conformational order at the quiescent condition. Referring to the symmetries of polymer crystals, four order parameters, including density (ρ), orientation (P), BOO (Q), and conformational order (w), may be involved in polymer crystallization. Among these order parameters, density and orientation are global while BOO and conformational order are local in nature. Considering the same “local” nature, BOO, instead of density and orientation order, has the advantage of assisting conformational ordering in polymer crystallization. To verify whether BOO assists conformational order in polymer crystallization, recently we conducted full-atom molecular dynamics simulation on the nucleation process of PE.140 As the original BOO is defined for spherical atoms and cannot be directly extended to polymers with irregular shape, we introduced a new local structure order parameter (OCB) defined as

and conformation, involved in crystallization, and the free energy density is given as g = g0(ρ ̅ ) + g (ρ ̅ , ρ*) + gη(η , ρ ̅ , ρ ) * *

(16)

where g0(ρ ̅ ) is the free energy density of a melt with random chain conformations, ρ̅ is the average mass density, g (ρ ̅ , ρ*) * is the Landau free energy of crystallization, ρ* is the coefficient in the Fourier expansion of the crystal density, gη(η , ρ ̅ , ρ ) * describes the contribution of chain conformations, and η indicates the conformation states varying smoothly from 0 (random) to 1 (helix). The theory predicts that the conformational order couples with density, which results in a liquid−liquid phase separation (LLPS) first and crystallization takes place in the dense liquid later. The physical picture of LLPS here takes conformational order as one order parameter, which is similar to BOO coupled with density fluctuation inducing phase separation. However, as experimental evidence from scattering techniques are questioned by other groups,127,128 the spinodal decomposition assisted crystallization model is not well appreciated in the polymer community. Can a one-component system separate into two liquids with different structures or densities? Does a one-component system exist as polymorphism of liquid or glass? To answer these questions, we turn our eyes to atomic and small molecular materials, and the answer is “yes”. There are abundant materials with a single component showing polymorphism of liquid or glass in pressure−temperature space.129 A well-known example is water,130 which has high- and lowdensity liquid as well as their corresponding glasses. Even for atomic materials like phosphorus, polymorphisms of liquid have also been reported.131 Figure 6a shows a phase diagram of phosphorus with two liquids. If atomic and small molecule materials have polymorphism of liquid and glass, it seems hard to exclude the existence of LLPS in one-component polymers. The existence of two amorphous phases has been directly demonstrated by Rastogi et al. during pressing poly(4-methyl1-pentene),132,133 although LLPS has not been observed between these two liquids. Studies on memory effects of various polymers also support the existence of different liquids, although their structures are unknown yet.134−136 For instance, heating β and α forms of trans-1,4-polyisoprene (TPI) above their T0m, the melt of β crystals shows a strong memory effect to trigger nucleation of α (not β) crystal after cooling while melting the α crystal loses memory completely.136 Different memory effects are also observed in melting iPP mesophase and α form.134,135 Different crystal forms melt into different liquids resembling the correlation between the two liquids and two glasses of water. To account the memory effect, Muthukumar137 proposed that an inhomogeneous melt exists with intermediate order between fully disorder melt and crystal and presented a theoretical framework to obtain the free energy landscape as shown in Figure 6b, where a, m, and c denote the initial amorphous melt, the metastable inhomogeneous melt (MIM) state, and the final crystalline state, respectively. He suggested that the MIM may originate from orientation fluctuation of COS or local clustering. These efforts strongly suggest the existence of polymorphism of polymer melt and provide a possible way for LLPS. G


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Macromolecules

Figure 7. (a, b) Snapshots of nucleation process of PE with different types of atoms colored differently. Orange and cyan (red and yellow) correspond to the atoms of H−OCB (O−OCB) clusters, respectively. (c) Evolutions of the number of OCB structures at 375 K. Reproduced with permission from ref 140. Copyright 2017 APS. n

Q nm =

∑ |Ynm(θij , ϕij)|2

(17)

m=0

OCB

1 = N(i)

i 2π y Q nmzzz ∑ jjj + n 1 k { j=1 N(i)

1/2

(18)

Qnm in eq 17 is summation of spherical harmonic function Ynm with n = 4 and m ∈ [0, n], θij and ϕij are the polar and azimuthal angles, respectively. N(i) is the number of neighboring atoms j of the center atom i within a certain cutoff distance. Coupling conformational order and OCB as the shape descriptor, we can differentiate hexagonal (H−OCB) and orthorhombic (O−OCB) clusters, which take the hexagonal and orthorhombic crystal lattices of PE as the references. At the quiescent condition, we observed that the hexagonal clusters emerge first and grow with time, which are dynamic in nature. Figure 7a shows a representative snapshot of hexagonal clusters in the simulation. After the hexagonal clusters reach a certain size, the stable orthorhombic crystal nuclei form inside the hexagonal clusters (see Figure 7b). Figure 7c depicts the evolution of hexagonal (H−OCB) and orthorhombic (O−OCB) clusters in atomic numbers during crystallization, which shows H−OCB clusters emerge immediately after the temperature reaches 375 K while it takes about 15 ns of induction time for O−OCB crystal nuclei to appear. These observations demonstrate that the crystal nucleation of PE is a two-step process with hexagonal clusters as the structural intermediate. Note crystal growth is observed to proceed via the coalescence of H−OCB dynamic clusters at the growth front, which is similar to DIT and Strobl’s model but different from the stem-by-stem deposition process in the HL model. More interestingly, the simulation results provide one possible answer for the question of how flexible chains transform into rigid conformational ordered segments. As shown in Figure 8a, both hexagonal LOS atoms i and the neighboring atoms j were displayed to visualize the local conformation, and these COS tilt with each other without the requirement of orientation order. Compared to the matrix melt, the hexagonal cluster has comparable density but lower structural entropy,140 suggesting that the hexagonal cluster is indeed a local ordered structure (LOS) without involving density as the order parameter. After excluding the roles of density and orientation order in assisting conformational order, we turn to the LOS with the similar definition as BOO. As the hexagonal clusters are extracted out with the criterion coupling OCB and conformational order, it is safe to draw the conclusion that conformational order is assisted by LOS here.

Figure 8. (a) Snapshot of the local ordered cluster. (b) Schematic illustration of possible topological constraint among neighboring COS.

The hexagonal cluster observed in PE may be a special LOS, and its definition with eqs 17 and 18 may not be ready to extend to other polymers. Moreover, the simulation only tells us that LOS can assist conformational order but does not say how this works or which kind of physical interaction is involved. Inspired by the snapshot of the hexagonal cluster as shown in Figure 8a, we propose that topological constraint among neighboring segments in the cluster may be a generalized physical mechanism to assist conformational order, which does not require a specific symmetry like hexagonal order in PE and can be ready to extend to all polymers. As illustrated in Figure 8b, the topological constraint not only restricts rotation of these COS but also prevents them from conformational disordering (or bending of the rigid segments). By introduction of barriers for rotation and bending of these COS, it may be possible to construct a general mechanism to resolve the question of how flexible chains transform into rigid conformational ordered segments for all polymers. With respect to experimental verification, the occurrence of intrachain conformational order in advance of crystallization has been reported with Raman and infrared spectroscopy measurements,141−143 while as LOS or the topological constraint is a dynamic with weak interactions and low order it would be a grand challenge to detect experimentally. Imposing flow or stretch can directly transform flexible chain into rigid COS,144−146 which makes flow-induced crystallization (FIC) of polymer to be fundamentally different from quiescent crystallization as the assistance of interchain interactions may not be necessary anymore. With the same full-atom molecular dynamics simulation on FIC of PE, we found that flow-induced conformational ordering occurs first, which does not couple with LOS as that at quiescent condition.147 When the concentration and the length of COS increase to certain values, coupling among conformaH


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Macromolecules

Figure 9. (a) TEM images of the contiguous lamellar structure in a melt-crystallized sample of PE crystallized at low undercooling (128 °C). Reproduced with permission from ref 152. Copyright 1980 Wiley. (b) Schematic illustration of the stacked lamellae.

materials combining high modulus and toughness.159−164 For instance, the recent advance of PE in the application of oil tank and pipe is largely attributed to the increased tie chain fraction in the amorphous layers.3 Based on the probability distribution of chain end-to-end distance, models (like the Huang−Brown model) have been developed to estimate the fraction of tie chain, which seems correlate well with the slow crack resistance and strain hardening modulus.165−167 Nevertheless, these ideas are still immature as we do not have an efficient approach to see and also do not know how to precisely tune the amorphous structure, which relies on new polymer crystallization theory incorporating both crystalline and amorphous layers. The chain connectivity has been considered since the beginning of polymer crystallization study. To justify the folded-chain crystal against Flory’s random switchboard model, Frank and others argued that the interlamellar amorphous segments resemble as polymer brush and would cost too much entropy due to surface crowding if the remaining amorphous segments do not fold back into the crystal.12 A Monte Carlo simulation, performed by Zachmann168,169 in the later 1960s, showed that the entropic loss of the remaining amorphous segments in the fringed-micelle nucleation model can be nearly one order larger than the folded-chain surface free energy σe. Nevertheless, the entropic loss seems to be much less than the value estimated by Zachmann if a transient phase or structural intermediate assists nucleation, suggesting fringed-micelle nucleation may still occur in polymer crystallization.91,92,170 The approaches discussed above on the coupling between conformational order and OCB or topological constraint may solve out the issues in primary nucleation although chain connectivity certainly plays a role in this process.140 Here we will focus on the role of chain connectivity in crystal growth and search possible directions for answering the question of how the accompanied interlamellar amorphous layers are constructed during crystallization. Figure 10a illustrates a polymer crystallization process with the highlighted growth front. The red chain represents a case that one end is attached on the growth front while the rest remains in the melt entangled with other chains. At the same time, plenty of chain segments from other chains sit at the growth front. Taking PE lamellar crystal with a thickness l of 14 nm as an example, a chain with radius gyration of 10 nm (C3000H6002) would entangle with about 130 other chains with the same length.171

tional order, orientation order, and density takes place, which leads to an LLPS first and then eventually nucleation of crystal. This multistage FIC process through conformational order− orientation order/density fluctuation−crystallization has been verified by experiments combining in situ infrared as well as small- and wide-angle X-ray scattering measurements.148−151 The physical picture of FIC is somehow close to Olmsted’s spinodal decomposition model, but here conformational order is promoted by flow first. The difference between quiescent crystallization and FIC is intimately related to the question of how flexible chain transforms into rigid COS. At the quiescent condition, conformational ordering has to be assisted by interchain interactions, where LOS or topological constraint may be the right approach to compensate the entropic penalty. While as flow can stretch polymer chains into rigid COS directly even without specific interchain interactions, during which the entropic penalty is cracked by external work, FIC follows a completely different kinetics pathway from that at the quiescent condition. Therefore, in the development of the theories of polymer crystallization, different models should be considered for quiescent and flow conditions. Connectivity. The connectivity of long-chain polymer brings many peculiar behaviors, which require special care in the development of polymer crystallization theory. Polymer crystallizing from bulk or highly entangled solution always forms lamellar stacks with alternatively arranged amorphous and crystalline nanolayers, which is essentially related to the chain connectivity.153 For instance, Figures 9a and 9b show the TEM image of stacked PE crystals and its schematic model, respectively. These stacked crystal possess strong correlations in mechanical performance,154,155 orientation, and even crystal registry,156−158 suggesting these separated crystals communicate with each other intimately via the interlamellar amorphous chains. Nevertheless, current polymer crystallization models only consider the crystal but somehow ignore the amorphous layer. In our view, the ultimate theory of polymer crystallization should not only describe how the crystals form but also explain how the interlamellar amorphous layers are constructed simultaneously. It is commonly accepted that the interlamellar amorphous layer composes of tie chain/bridge, loop, cilium, and entanglement. From the aspect of the practical application, the amorphous layers share equal importance on the mechanical performance of semicrystalline I


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Macromolecules

To reach such a disentangled state with reduced entropy, the red chain has to sustain the entropic elastic force. In this sense, the growth of polymer crystal at quiescent condition is a stretch-induced crystallization process at the single molecular scale (denoted as SIC@SM), which is fundamentally different from the crystallization of atoms and small molecules. The estimated reeling force is about 5 pN on a single chain of PE according to the HL model,80 which is quite large but has not been measured with experimental techniques directly. To visualize crystallization induces dragging force, we studied crystallization of lightly cross-linked PE after extended to different strains. For the sample and experimental information, please refer to our early work.172,173 This approach has two advantages. First, with the cross-linked network, chain diffusion and relaxation are restricted, which provides a possibility to build up forces along polymer chain during crystallization. Second, under uniaxial stretch polymer chains are oriented in the tensile axis direction, the forces sustained by chains are aligned in the same direction and obey the simple add-up rule, which may be amplified to be detectable macroscopically. Figures 11a and 11b show the evolutions of crystallinity and stress during crystallization at 132 °C after extended to the Hencky strain ε of 0.5 and 1.6, respectively. At ε = 1.6, stress decreases monotonically with the increase of crystallinity, which is in line with current models of FIC172,174 and early experimental observations (Figure 11b).175,176 While at ε = 0.5, stress decreases first due to relaxation, which, however, increases up after crystallization starts (Figure 11a). Here the increased stress certainly stems from crystallization, which may serve as a demonstration on the existence of the dragging force in polymer crystallization. The macroscopic flow observed at quiescent crystallization by Wang and co-workers177 may be another indirect evidence for [email protected] Nevertheless, these results are still preliminary and indirect. Therefore, direct measurements of the dragging force of crystallization state are highly desirable. Apart from depositing the remaining amorphous segments on the growth front of the old crystal, another choice for the red chain is to initiate new nuclei elsewhere (see Figure 10c). Analogous to macroscopic FIC, stretch reduces the entropy of the red chain, which consequently leads to a reduction of the nucleation barrier and enhances the nucleation rate. Considering the surface crowding effect, it would be hard for the remaining amorphous segments of the red chain to move to the old crystal front, especially at high supercooling, while an alternative way is to trigger new nuclei, which is precisely a stretch-induced nucleation at a single molecule (SIN@SM).179 The SIN@SM has been demonstrated on a double crystalline poly(L-lactide)-b-poly(ethylene oxide) (PLLA-b-PEO) block

Figure 10. (a) Schematic illustration of the deposition of the polymer chain at the growth front. The red line represents the depositing chain with one end fixed at the growth front, and the white one indicates the surrounding chains. Further crystallization leads to two different pathways: (b) continuous growth at the same growth front and (c) formation of a new nucleus.

How do chain segments move to the growth front of polymer crystals? In CNT and other nucleation models, moving particles from a certain distance to near the crystal growth front is generally thought to be driven by density or concentration gradient with the assistance of thermal fluctuation. Both density and concentration gradients stem from interparticle interactions, while in polymer melt and highly entangled solution stress could transfer along the chain. Do polymer chains diffuse to the crystal front simply as spherical atoms and small molecules do? Or are polymer chains pulled to the crystal front by a dragging force? Although full disentanglement may only occur in crystallization of short chain polymers at low supercooling, the entanglement state is expected to change in all polymer crystallization as entanglement points are excluded in amorphous layers, during which polymer chain may sustain certain force. In the HL model, polymer chains are thought to be pulled out from the tube with a “reeling force”, and the curvilinear diffusion is described as forced reptation, disturbed reptation, or reptation of slack at different regimes. The last one, reptation of slack, works in regime III at low crystallization temperature, which considers chain segments rather than the whole chain are pulled out from the tube. In Figure 10a, if the remaining amorphous part of the red chain wants to deposit onto crystal front to form a new crystalline stem (Figure 10b), it must move out the tube and disentangle from surrounding chains.

Figure 11. Evolutions of crystallinity (black open rectangle) and stress (blue solid circle) during isothermal crystallization of a lightly cross-linked PE after extension with Hencky strain ε of 0.5 (a) and 1.6 (b), respectively. J


Perspective

Macromolecules copolymer.179 Increasing the crystallization temperature of PLLA block leads to an increase of stretch of PEO block, which enhances the crystallization kinetics of the PEO block during the cooling and isothermal process. The SIN@SM process can serve as a mechanism for understanding lamellar branching in polymer crystallization. In-situ atomic force microscopy shows that newly formed lamella locate closely (∼10 nm) to the old lamella but without direct contact,180−182 implying that long-range interactions in the chain length scale rather than short-range interaction like epitaxial growth play a role in promoting these new nuclei. This supports that SIN@ SM is one possible mechanism to trigger the lamellar branching and the formation of lamellar stacks. Whether the remaining amorphous segments deposit on the old crystal front or initiate new crystal nuclei is determined by molecular parameters (such as molecular weight, distribution, and chain branch) and crystallization temperature. Thermodynamically, it is a competition between secondary (growth) and primary nucleation, which is strongly affected by the retardation term. Increasing molecular weight or short branches increases the difficulty to pull chains to the growth front and favors the formation of new nuclei, while lower crystallization temperature not only enhances dragging force but also reduces the energy barrier difference between primary and secondary nucleation, in which both factors tend to promote the formation of new nuclei. These general trends agree well with experimental observations but require more direct and quantitative experiments and simulations to confirm. If stretch-induced secondary nucleation (SIC@SM) and primary nucleation (SIN@SM) occur simultaneously with competition, polymer crystallization follows a lamellar-stack growth model rather than a single-lamellar growth model. The lamellar stack growth model incorporates both crystalline and amorphous layers, which may explain how interlamellar amorphous layers are constructed in crystallization. HL theory, as well as other models, is a single-lamellar crystal growth model, which solves almost all thermodynamic and kinetic issues constrained within the surface and bulk of one lamella. The dragging force is considered by the HL model in the retardation term, but it is not incorporated in the thermodynamic nucleation barrier. Dragging force leads to chain stretch and reduces its entropy, which contributes an excess surface free energy much larger than σe as estimated by Zachmann and others.168,169 In this sense, chain folding or σe may not be the critical penalty term determining the thermodynamics in polymer crystallization, such as nucleation barrier and lamellar thickness l*. If the dragging force plays a more important role than σe in crystallization, and sliding diffusion76,183 or LOS140 assisted nucleation presents in growth, we speculate the tight fold forms in the later perfection stage rather than in surface nucleation as stated in the HL model. Following this line, σe or folded-chain surface plays a critical role in melting as the Gibbs−Thomson equation describes,78 while crystallization is more influenced by the dragging force. This may give a possible explanation for the different interceptions of melting and crystallization lines observed by Strobl.21,22

the newly proposed two-step nucleation scenarios in the future. Additionally, the peculiarities of polymer crystallization should deserve more dedicated efforts. Polymer crystallization is a tough journey battling with chain flexibility and connectivity, which has never and ever achieved full completion and consequently results in semicrystalline polymeric materials with a combination of high strength and high toughness. On the basis of the above discussions, we finalize this Perspective with the following questions, which we would like to recommend for future research on the development of polymer crystallization theory. How Do Flexible Chains Transform into Rigid COS? At the quiescent condition, we propose that LOS or topological constraint is one choice to assist conformational ordering, which may solve the question of how flexible chains transform into rigid COS. The local order parameter we defined is still limited for PE, while the “topological constraint” used here is vague without detailed definition. A universal local order parameter or descriptor for polymers is required for understanding their crystallization behavior. Molecular dynamics simulation is an effective approach to visualize the structural evolution in the early stage of crystallization,184−186 which, however, is still hard to be extended to other polymers. Crystal nucleation and growth may be only observed in PE with fullatom molecular dynamic simulation, which even does not work for iPP with a slight increase of structural complexity. Thus, either a new simulation method or a more generalized polymer model is highly desirable in this direction. The ultimate verification on LOS assisted conformational ordering needs experimental evidence. Although conformational order has been reported in diverse polymer systems with spectroscopic methods, LOS has not been observed with any experimental techniques before. As shown in Figure 8a, neither density nor orientation order is present in LOS clusters, which somehow excludes structural detecting techniques sensitive to density fluctuation (like X-ray scattering) or orientation order (like birefringence, depolarized light scattering, polarized infrared, and Raman spectroscopies). NMR is thought to be one possible technique to capture the structure and dynamics of LOS. In the solid state, the chemical shift of spin is sensitive to the local magnetic environment. The change of molecular conformation could result in the shift of the NMR signal. For instance, for PE, the two distinct conformations, i.e., trans and gauche, lead to two well-resolved 13C NMR signals, namely ∼30 and ∼33 ppm, which correspond to the trans/gauche fast transition (amorphous) and all-trans (crystalline) conditions.187 Especially, the recently developed diamond sensorbased NMR has the capability to detect the structure and dynamics of a single protein,188,189 which may be a potential technique to follow the structure and dynamic of LOS clusters. How Do Interlamellar Amorphous Layers Form? HL and other current models of polymer crystallization are a single lamellar crystal growth model and have been developed with the thermodynamic parameters of crystal alone, while the rest of the amorphous chain segments are generally ignored, which does not describe how the interlamellar amorphous layers form during crystallization. The ultimate polymer crystallization theory should be able to account both crystalline and amorphous layers, which may be described with the lamellar stack growth model as suggested here. The micelle cluster model190 in the homopolymer melt and the equilibrium folded-chain crystal model191 in the block copolymer consider the free energies contributed from both crystal and amorphous

6. FUTURE RESEARCH DIRECTIONS The development of polymer crystallization theory is an unaccomplished tough journey, during which the polymer community has been (or will be) nourished from general nucleation theories like CNT and non-CNT in the past and K


Perspective

Macromolecules layers, which can be taken as the first trial in this direction. Meanwhile, the crystallinity has never been predicted as the current theories only focus on single lamellae, and this problem might be solved after taking the amorphous parts and lamellar stacks into account. To answer how amorphous layers form, we first should know their structure. Because of disorder and dynamic nature, the structural information about amorphous layers is still scarce. Tie-chain, cilia, and loop plus physical entanglement are generally considered to construct amorphous layers,192 while interfacial rigid amorphous is also thought to be present in some polymers.193−195 A statistical model like the Huang− Brown model167 with molecular weight and branching distribution as input parameters has been proposed to estimate tie-chain concentration, but to visualize tie-chain in interlamellar amorphous layers is still a challenge. Recently, we studied the stretch-induced structure evolution of highly oriented iPP and PE with in situ small- and wide-angle X-ray scattering techniques and observed a stress-induced phase separation between tie-chain and cilia/loop.196,197 This provides a possible method to study the distribution of tie chain and cilia/loop. Meanwhile, the entanglement in the amorphous region can be characterized by the time domain multiple-quantum (MQ) NMR, based on the obtained residual dipole−dipole couplings among protons.198 Recently, this NMR technique has been utilized in combination with SAXS to investigate the crystal-fixed polymer poly(ϵ-caprolactone) with variable molecular weights and crystallization temperatures. Results show that the thickness of the amorphous layer is determined by the entanglement density.199 Moreover, chemical labeling branch or chain segments with the different chemical structure or deuterium is certainly another important approach, which may be visualized with single molecular spectroscopy as well as X-ray and neutron scattering. Are Polymer Chains Dragged by Force to the Crystal Growth Front? As stress can be mainly transferred along the polymer chain, dragging chain segments to the crystal growth front is highly possible to occur, especially at large supercooling with high crystal growth rate and slow chain diffusion rate. Although indirect evidence seems to support this idea, no direct measurements on the dragging force induced by crystallization have been reported yet. With the development of single chain force measurement techniques, we anticipate that direct measurements on the dragging force during crystallization may come soon. Zhang and co-workers200 recently already realized to pull a single chain out from polymer single crystal by AFM fishing, and their technique could be extended to study crystallization. If the dragging force can be measured quantitatively, we may answer the question of how chain connectivity influences polymer crystallization.

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Biographies

Xiaoliang Tang received his B.S. from the College of Nuclear Science and Engineering, Sichuan University, in 2015. He is currently pursuing his doctorate at the University of Science and Technology of China under the guidance of Professor Liangbin Li. Currently, he is using molecular simulations to study the nucleation process of longchain polymers.

Wei Chen received his B.S. (2011) in Polymer Science from the University of Science and Technology of China (USTC). Later, he got his Ph.D. (2016) in Polymer Science from the University of Akron under the supervision of Prof. Toshikazu Miyoshi. After the postdoctoral work in Tongji University (2016−2018), he joined the National Synchrotron Radiation Lab as an associate professor (2018). His research focuses on understanding the polymer crystallization process based on the hierarchical structure and molecular dynamics information. The main characterization techniques he used are synchrotron radiation X-ray scattering and solid-state NMR spectroscopy.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Wei Chen: 0000-0001-8334-0024 Liangbin Li: 0000-0002-1887-9856 Author Contributions

X.T. and W.C. contributed equally. Notes

The authors declare no competing financial interest. L


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Macromolecules

(11) Point, J. J. Reconsideration of kinetic theories of polymer crystal growth with chain folding. Faraday Discuss. Chem. Soc. 1979, 68, 167−176. (12) Point, J. J.; Rault, J.; Hoffman, J. D.; Kovacs, A. J.; Mandelkern, L.; Wunderlich, B.; Dimarzio, E. A.; De Gennes, P. G.; Klein, J.; Ball, R. C. General discussion. Faraday Discuss. Chem. Soc. 1979, 68, 365− 490. (13) Armitstead, K.; Goldbeck-Wood, G.; Keller, A. Polymer crystallization theories In Macromolecules: Synthesis, Order and Advanced Properties; Springer: 1992; pp 219−312. (14) Imai, M.; Mori, K.; Mizukami, T.; Kaji, K.; Kanaya, T. Structural formation of poly (ethylene terephthalate) during the induction period of crystallization: 2. Kinetic analysis based on the theories of phase separation. Polymer 1992, 33 (21), 4457−4462. (15) Okada, T.; Saito, H.; Inoue, T. Time-resolved light scattering studies on the early stage of crystallization in isotactic polypropylene. Macromolecules 1992, 25 (7), 1908−1911. (16) Heeley, E.; Maidens, A.; Olmsted, P.; Bras, W.; Dolbnya, I.; Fairclough, J.; Terrill, N.; Ryan, A. Early stages of crystallization in isotactic polypropylene. Macromolecules 2003, 36 (10), 3656−3665. (17) Heeley, E. L.; Poh, C. K.; Li, W.; Maidens, A.; Bras, W.; Dolbnya, I. P.; Gleeson, A. J.; Terrill, N. J.; Fairclough, J. P. A.; Olmsted, P. D.; et al. Are metastable, precrystallisation, densityfluctuations a universal phenomena? Faraday Discuss. 2003, 122, 343−361. (18) Kaji, K.; Nishida, K.; Kanaya, T.; Matsuba, G.; Konishi, T.; Imai, M. Spinodal crystallization of polymers: Crystallization from the unstable melt In Interphases and Mesophases in Polymer Crystallization III; Springer: 2005; pp 187−240. (19) Olmsted, P. D.; Poon, W. C.; McLeish, T.; Terrill, N.; Ryan, A. Spinodal-assisted crystallization in polymer melts. Phys. Rev. Lett. 1998, 81 (2), 373−376. (20) Strobl, G. Crystallization and melting of bulk polymers: New observations, conclusions and a thermodynamic scheme. Prog. Polym. Sci. 2006, 31 (4), 398−442. (21) Strobl, G. Colloquium: Laws controlling crystallization and melting in bulk polymers. Rev. Mod. Phys. 2009, 81 (3), 1287−1300. (22) Strobl, G. From the melt via mesomorphic and granular crystalline layers to lamellar crystallites: A major route followed in polymer crystallization? Eur. Phys. J. E: Soft Matter Biol. Phys. 2000, 3 (2), 165−183. (23) Lotz, B. What can polymer crystal structure tell about polymer crystallization processes? Eur. Phys. J. E: Soft Matter Biol. Phys. 2000, 3 (2), 185−194. (24) Cheng, S. Z. D.; Li, C. Y.; Zhu, L. Commentary on polymer crystallization: Selection rules in different length scales of a nucleation process. Eur. Phys. J. E: Soft Matter Biol. Phys. 2000, 3 (2), 195−197. (25) Muthukumar, M. Commentary on theories of polymer crystallization. Eur. Phys. J. E: Soft Matter Biol. Phys. 2000, 3 (2), 199−202. (26) Lodge, T. P. Celebrating 50 years of macromolecules. Macromolecules 2017, 50, 9525−9527. (27) Muthukumar, M. Nucleation in polymer crystallization. Adv. Chem. Phys. 2004, 128, 1−64. (28) Muthukumar, M. Modeling polymer crystallization In Interphases and Mesophases in Polymer Crystallization III; Springer: 2005; pp 241−274. (29) Bassett, D. On the Role of the Hexagonal Phase in the Crystallization of Polyethylene In Interphases and Mesophases in Polymer Crystallization I; Springer: 2005; pp 1−16. (30) Lotz, B. Analysis and observation of polymer crystal structures at the individual stem level In Interphases and Mesophases in Polymer Crystallization I; Springer: 2005; pp 17−44. (31) Hu, W.; Frenkel, D. Polymer crystallization driven by anisotropic interactions In Interphases and Mesophases in Polymer Crystallization III; Springer: 2005; pp 1−35. (32) Li, L.; de Jeu, W. H. Flow-induced mesophases in crystallizable polymers In Interphases and Mesophases in Polymer Crystallization II; Springer: 2005; pp 75−120.

Liangbin Li received his Ph.D. (2000) in Polymer Material Processing from Sichuan University. From 2000 to 2004, he was a postdoctoral fellow in FOM-Institute for Atomic and Molecular Physics and Technology University of Delft, The Netherlands. From 2004 to 2006, he worked as a material scientist at the Unilever Food and Health Research Institute. Since 2006, taking the “one-hundred talent program” of the Chinese Academy of Sciences, Dr. Li joined the National Synchrotron Radiation Lab at the University of Science and Technology of China as a professor and started the Soft Matter Group. His primary research interests are polymer physics relevant to processing such as flow-induced crystallization and stress-induced deformation and phase transition of crystalline polymers. He has served as an Associate Editor for Macromolecules since 2018.

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ACKNOWLEDGMENTS We thank Prof. Daan Frenkel (Cambridge) for the stimulating discussions. The authors thank all the reviewers for their constructive and valuable comments and criticism, which help us improving this work a lot. The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (51633009), Royal SocietyNewton Mobility Grant MBAG/240 RG82754, and Chinese Scholarship Council. The X-ray experiments were performed at the Shanghai Synchrotron Radiation Facility (SSRF). L.L also thanks all the group members for their contributions, especially Dr. Dong Liu for Figure 11 and his daughter Annie Li for inspiring discussions on the title and TOC.

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NOMENCLATURE BOO, bond-orientational order; CHT, coil−helix transitions; CNT, classical nucleation theory; DFT, density functional theory; DIT, diffuse interface theory; FIC, flow-induced crystallization; LLPS, liquid−liquid phase separation; LOS, local ordered structure; non-CNT, nonclassical nucleation theory; MIM, metastable inhomogeneous melt; SIC@SM, stretch-induced crystallization at single molecule; SIN@SM, stretch-induced nucleation at single molecule.

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REFERENCES

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The Tough Journey of Polymer Crystallization: Battling with Chain

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