Molecular and thermodynamics descriptions of flow-induced crystallization in semi-crystalline polymers Free

Semicrystalline polymers are the most widely used polymeric materials, and polyethylene (PE) and polypropylene (PP) together accounted for 57% of all synthetic polymers.¹ Almost all semicrystalline polymers need to be processed from the initial melt state to the final semicrystalline state, where the processing condition plays a central role. Since the properties of final products are closely related to the final structure, which is significantly influenced by processing, researchers and engineers try to establish the processing–structure–property relationship. Concerning processing conditions, flow and temperature are the most important parameters. The influence of temperature on the crystallization behavior of semicrystalline polymers has been extensively explored ever since the early investigation of polymer crystallization. Most conclusions are obtained in quiescent conditions. Readers interested in such topics are suggested to other reviews or books.^2–10 The polymer crystallization influenced by flow, namely, flow-induced crystallization (FIC), will be discussed in the current work.

In industry, various processing techniques, i.e., melt-spinning, film casting, film blowing, and injection molding, are used to obtain different products meeting different application requirements. During these processes, the external flow is found to significantly enhance the crystallization kinetics and generate oriented crystallites. Both phenomena are closely related to nuclei density and oriented crystallite (point-like¹¹ and oriented¹² ones as shown in Fig. 1). Therefore, most research studies in FIC attempt to couple the external flow parameters (e.g., shearing rate and time) with final crystal morphology. However, different from quiescent crystallization, the FIC is far away from the equilibrium state, and the flow is always complicated in the real application, i.e., multidimensional and non-uniform, resulting in a great challenge in capturing the in situ microstructure evolution under flow.

Despite external flow conditions, the semicrystalline polymer is also intrinsically complicated compared with a small molecule. The long-chain nature of polymer results in much longer relaxation time as compare with normal Newton fluids.¹³ As a result, the processing flow can significantly distort local chain conformation leading to the anisotropic feature of the polymer melt.¹⁴ Later on, the anisotropic melt suffering from cooling starts to transform into the semi-crystalline state, where crystallization and chain relaxation start to compete with each other.¹⁵ Therefore, both the rheological behavior of the polymer melt and microstructure evolution under flow are crucial to understand the FIC phenomenon. However, several molecular characteristics of the polymer, i.e., polydispersity and branched architecture, severely complicate the analysis of experimental data and final establishment of the process–structure–property relationship.

As a tutorial, we aim to present the basic issues of FIC in the semicrystalline polymer system. The detailed historical summary of FIC is far beyond the scope of the current tutorial. The current tutorial is organized as follows. In Sec. II, the general phenomenon of FIC is presented including oriented crystallite and enhanced crystallization kinetics induced by the flow. So readers can get a general idea of FIC. In Sec. III, we present methods for observing FIC. Here, the synchrotron radiation x-ray scattering and non-equilibrium molecular dynamic simulation are discussed in detail. In Sec. IV, the theories accounting for the formation of different crystal morphologies are presented. The coil–stretch transition (CST) theory and later developed stretched network model are discussed in detail. These two theories together with their modifications can well explain the morphology formation under flow from dilute solution to high concentrated and melt states. In Sec. V, the thermodynamics treatment of FIC is presented. Although FIC is a typical non-equilibrium process, the FIC can be still dealt with by conventional thermodynamics concepts, i.e., Gibbs free energy, to present a thermodynamics phenomenological image, especially under a weak flow condition. The thermodynamics treatment can qualitatively or sometimes even quantitatively reproduce the enhanced crystallization kinetics. Finally, the key features together with our perspectives of FIC are presented.

The flow is inevitable in the real processing of polymers. Before further addressing the influence of flow on polymer crystallization, the definition and classification of different flow fields are briefly discussed. Figure 2(a) shows the inner structure of an extruder. The extrusion is commonly the very first step before any further processing of a polymer. The flow field, as shown in Fig. 2(b), is quite complicated. However, such complicated flow can be decomposed into two basic flows: the shear flow [Fig. 2(c)] and extension one [Fig. 2(d)]. An apparent characteristic of the shear flow is the velocity gradient. For the extension flow, the strain rate is the same for different positions along the vertical direction. Since the flow is crucial for FIC, the key parameters, i.e., the strain rate and strain, have been extensively studied. Most of the research studies related to FIC adopts one flow field to investigate the influence of different processing parameters on final crystal morphology and crystallization kinetics.

The crystal morphology in FIC has already shown in Fig. 1, where both point-like and shish–kebab structures could be formed. Since the finding of the “shish–kebab” structure initiated the research of FIC, we will first summarize the state-of-the-art understanding of the formation of the shish–kebab structure.

As shown in Fig. 3, the extended chain formed the shish core, whereas the polymer lamellae or kebab is normal to the extended shish.¹⁶ The lamellae could twist under the low-stress condition as it did in normal polymer spherulite,^16,17 while such twist is largely suppressed under large stress. Also for kebab, there are two different kinds of kebab, namely, macro-kebab and micro-kebab.¹⁸ The macro-kebab can be removed away from the shish through the addition of a solvent or increasing the temperature, while the micro-one remains alive within the shish. Therefore, the micro-kebab is thought to be as stable as the shish core. The micro-kebab is proposed to be inside the macro-shish and perpendicular to the macro-shish [Figs. 3(a)–3(c)]. After the discovery of the shish–kebab structure, researchers proposed various theories and models to explain the formation of this shish–kebab structure. Pioneers such as Penning,^19,20 Hoffman,²¹ and Petermann and Keller^22,23 have contributed to the understanding of the nucleation and growth of the shish core. Detailed summary and comparison of relevant research studies are referred to a recent review.²⁴ 

The flow does not necessarily lead to the formation of a shish structure. As shown in Fig. 4, the crystal morphology and crystallization kinetics (t/t_Q, t_Q is the half-time of crystallization under quiescent condition) are determined by the external flow parameters (shear/strain rate $γ ˙$⁠, strain time t_s, and strain γ) and intrinsic molecular parameters (reptation time τ_rep and Rouse time τ_s).²⁵ The boundary between two neighboring regimes is defined by rheological parameters, Deborah number, D_e, or Weissenberg number, $Wi = γ ˙ τ$⁠. De_rep and De_s represent chain orientation and chain stretch. In regime I (De_rep< 1 and De_s< 1), the polymer chains remain in the coil state. As a result, the crystal morphology keeps spherulite, and the crystallization kinetics is the same as that in the quiescent condition. In regime II (De_rep> 1 and De_s< 1), the whole chain starts to be orientated while the local chain conformation is not stretched. The enhanced crystallization kinetics appears (t/t_Q< 1), but the general crystal morphology is still spherulite. In regimes III and IV (De_rep> 1 and De_s> 1), the chain starts to be stretched. Despite enhanced crystallization kinetics (t/t_Q< 1), the shish structure appears. The prerequisite for the formation of the shish structure, or the classification of regimes III and IV, is the temperature dependence of the strain λ (T). This is because sufficient strain needs to be satisfied to generate shish, which is extensively confirmed by experiments and modeling.^26–28 Admittedly, the criteria of different regimes are not strictly represented by D_e, namely, D_e= 1, as mentioned above. The D_e used here mostly represents the relative difference between the external flow rate and the internal relaxation rate.

Besides the crystal morphology, another apparent change is the flow-enhanced crystallization kinetics compared with that of quiescent crystallization. Taking i-PP as an example, the half crystallization time decreases by the orders of magnitude due to the flow as shown in Fig. 5(a).²⁹ The crystallization kinetics is independent of the shear rate and shearing time.³¹ Such enhanced crystallization kinetics is mainly attributed to the significantly increased nucleus density and growth rate. Figure 5(b) shows the influence of the shear rate on the nucleation rate. Compared with that in quiescent crystallization, the nucleation rate can be increased up to two orders of magnitude.³⁰ Such an increase seems to follow the exponential function. The flow would eventually lead to an increasing nucleation rate once the flow rate exceeds a critical value. For example, for i-PP (M_w= 55 600, M_n= 376 000, and tacticity = 87.6%), the critical shear rate is 0.048 s⁻¹ (T = 140 °C).³² 

In addition to increased nucleation density, the later growth is also significantly different from that in a quiescent condition. As shown in Fig. 6(a), the multiple-shish, rather than single shish, is observed in PE blends, where 2 wt. % is the ultra-high-molecular-weight PE (UHMWPE) and the remaining is the non-crystalline PE matrix.³³ With the presence of this multiple-shish, the growth rate of kebab, or more precisely, the diameter of the kebab, can be obtained by in situ SAXS measurement as shown in Fig. 6(b). The calculated growth rate as a function of time can be fitted by the diffusion-controlled model [G(t) ∼ t^−1/2]. Such a conclusion is different from that in quiescent conditions, where the growth rate in the secondary nucleation theory is constant. This phenomenon is well consistent with the AFM^34–36 and simulation results.³⁷ More detailed experimental result related to the growth is shown in Fig. 6(c), where increasing shear rate up to 0.3 s⁻¹ results in one order enhanced growth rate.³⁰ 

The polymorphism is a common phenomenon for the semicrystalline polymer. Even for the simplest PE, at least three crystalline phases exist: orthorhombic, monoclinic, and hexagonal phases. Since the macroscopic performance of polymer products is closely related to the crystal structure,³⁹ an apparent question is how the external flow would influence the polymorphism. Compared with crystallization at quiescent conditions, the FIC happens far from the equilibrium state. As a result, the kinetics favored, rather than thermal stable crystalline phases could be obtained. Taking i-PP as an example,³⁸ as shown in Fig. 7(a), the non-equilibrium phase diagram of i-PP is obtained with the assistance of ultra-fast x-ray scattering technique. Since all experiments were conducted well above the melting temperature of both crystals, such results suggest that the strong flow can reverse the thermal stability of crystals and melts. For i-PP, there are numerous crystal structures, such as α, β, γ, and mesomorphic phases, whose helix conformation is 3₁. Despite the same helix conformation, different phases take different packing structures. And among all phases, the α crystal is the most thermally stable. However, with increasing strain rate and temperature, the content of the metastable β crystal also increases. The β crystal is thought to be a kinetically favored phase under a strong flow. A similar phenomenon is also observed in other semicrystalline polymer systems, i.e., poly(1-butene)⁴⁰ and PE.⁴¹ These results suggest that the competition between thermodynamics stability and crystallization kinetics dominates the non-equilibrium phase diagram under flow. This structural information could provide roadmaps for obtaining polymer products with desired properties through tuning flow parameters.

The ex situ studies of the FIC are the mainstream during the last century, and even up to date, this strategy still has great vitality. Experimental evidence, including the crystal structure and morphology, are crucial to the understanding of the FIC by retracing the FIC process. This strategy is applicable in general polymer crystallization research.⁷ However, the in situ observation of FIC is preferred, which can provide direct evidence of micro-structural information during FIC. With the fast development in characterization techniques in the last two decades, the in situ characterization of FIC from the polymer melt to the final crystalline state becomes possible. Techniques, such as rheo-AFM,³⁴ rheo-X-ray,^33,42 rheo-IR,⁴³ and rheo-Optics,^14,44 have been successfully utilized to track the microstructure evolution during FIC, especially the initial part. Also, the development of both hardware and software in molecular dynamics (MD) simulation provides a great opportunity for us to track FIC at the molecular level. Here, the x-ray scattering based on the synchrotron radiation facility and MD simulation is selected as typical examples to show current efforts to elucidate the detailed molecular origins of FIC.

With the fast development of synchrotron radiation facility, the extremely high brilliance ensures the microstructure characterization of FIC with both high spatial (<100 nm) and time resolution (<1 ms). For the collection of most recent achievements in applying synchrotron radiation x-ray scattering (SRXS) in the investigation of FIC, readers are referred to other reviews.^12,45 Here, the selected examples mainly related to the setup of the experiment are shown below.

Figure 8(a) shows the extensional rheometer commonly coupled with synchrotron radiation x-ray scattering for in situ measurement.^46–48The strain rate is controlled by the rotational rate of two rotational clamps, which rotate oppositely. Since the positions of these two clamps are fixed, the sample cavity is fixed, which ensures high flow homogeneity. Figure 8(b) shows a typical 2D SAXS pattern of high-density PE (HDPE) at a strain of 2.5 (strain rate is 15.7 s⁻¹, T = 125 °C). The two streaks appear perpendicular to the extensional direction are assigned to the shish structure. Therefore, the formation of the shish core can be in situ captured by SR-SAXS as summarized in Fig. 8(c). It shows that the shish structure can only appear with a strain larger than 1.57. For the customized extensional rheometer, parameters including extensional temperature, rate, and time can be well controlled, and it can be easily coupled with in situ characterization techniques, i.e., IR and x-ray scattering.

Despite the extensional mode, another common method to apply external flow is shearing. There are numerous modes for shearing, i.e., cone-plate [Fig. 9(a)],⁴⁹ rectilinear parallel plates [Fig. 9(b)],^50,51 rotating parallel plates [Fig. 9(c)],^52,53 and slot flow [Fig. 9(d)].^44,54 Both cone-plate and rectilinear parallel plates provide a uniform shear rate, whereas the other two show the shear rate gradient. Here, one most widely used shear stage—the Linkam shear stage, or more specifically Linkam CSS450—is presented. The Linkam shear stage is based on the rotating parallel plates. Figure 9(e) shows a typical setup of the Linkam shear stage in the synchrotron radiation x-ray scattering beamline.⁵⁵ Such shear rate gradient leads to more convenient detection of FIC under different shear rates. Also, all measurements could be done with a fixed temperature, which eliminates possible uncertainty reproducing the thermal history with repeated measurements for different shear rates.

Despite above shear geometry, another kind of shear is achieved through the introduction of fiber. Figure 10(a) shows the design of a shearing device coupled with micro-focus SRXS, where fiber is used for applying the shear flow. The micro-focus x-ray scattering can provide a high spatial resolution of the whole sample after step shearing as shown in Fig. 10(b). Therefore, the detailed micro-structure evolution after step shearing can be obtained [Fig. 10(C)]. If the shearing is applied well above the melting temperature of the sample (T_m= 165 °C), no crystal is observed [Figs. 10(c)–2(d)]. Furthermore, cooling the temperature to 138 °C leads to the appearance of cylindrical crystallites together with enhanced crystallization kinetics. Also, as shown in Figs. 10(c)–10(c), an induction period is obtained when later crystallized at 138 °C. Such results suggest that the shear-induced precursor is non-crystalline, and sufficient time is required for transformation from non-crystalline state to crystalline one.⁵⁶ 

As discussed in Sec. II A, in the real industry, the multidimensional, rather than simple shear or extensional flow, would be encountered. Here, the biaxial stretching, portable extruder, and film blowing are selected typical examples to show recent efforts attempting to the in situ characterization of microstructural evolution under different flow conditions.

Figure 11(a) shows the design of a portable extruder which can be installed in the x-ray beamline of the synchrotron radiation facility.⁵⁷ The extruder is usually the first step for polymer processing. The flow field is not uniform as already shown in Fig. 2. The flow field condition can be controlled by the mandrel and extrusion piston. With the assistance of synchrotron x-ray scattering, Chang et al. investigated the influence of different extrusion conditions on the microstructure evolution of i-PP. Thus, obtained structural information together with the well-controlled rheological information provides sufficient hint for the correlation between flow parameters and microstructural evolution during extrusion.

Another frequently encountered flow is the biaxial stretching as shown in Fig. 11(b). Limited by the available space in the synchrotron beamline, the biaxial stretching machine needs to be minimized but kept the main feature of biaxial stretching. Chen et al. designed and successfully applied the biaxial stretching machine in BL16B of Shanghai Synchrotron Radiation Facility (SSRF), where the biaxial stretching of natural rubber was conducted.⁵⁸ With the continuous increase of the stretching ratio along with two perpendicular directions, it was found that the strain-induced crystallization (SIC) of natural rubber could be suppressed with λ_x/λ_y< 1.6. Such results challenge the widely accepted self-reinforcement mechanism of natural rubber under deformation.

The flow conditions become even more complicated during film blowing. As shown in Fig. 12, to mimic the film blowing process and meet the space requirement in synchrotron radiation beamline, the film blowing is minimized, whereas the key functions are the same as those of apparatus in the real industry.^59–65 The flow field during the film blowing process is intrinsically biaxial, where the horizontal extension of the bubble induced by blowing air and the vertical one induced by the nip rolls. Here, the flow is characterized by the particle tracking method where the particle is captured by CCD, and the temperature is detected by the infrared (IR) thermometer. The in situ synchrotron radiation x-ray scattering, WAXS/SAXS, was applied to capture the microstructure evolution right above the die. It was found that the crystallinity at the frost line, where the bubble diameter reaches the maximum, is determined by both external flow field and internal molecular characteristics of polymers. And the non-deformed crystal-based scaffold is formed at the frost line. Despite FIC, the temperature gradient exists during the film blowing process, namely, the temperature induced crystallization (TIC) competes with FIC through the whole process.⁶⁰ 

The flow in real processing is always complicated. But the knowledge obtained by the above simple extension and shear rheometers is useful to understand the FIC in such non-uniform flow and temperature fields. Moreover, the final crystal morphology, either point-like or shish structure, can be obtained through proper selection of a polymer material and processing parameters.⁶⁰ The development of such in situ equipment provides great opportunities for researchers in both industry and academics to obtain the desired properties of polymer products.

One great advantage of simulation is the capturing of the detailed molecular feature under flow. Utilizing non-equilibrium molecular dynamics (NEMD) simulations, the researchers can track the trajectory of every single monomer upon crystallization under flow, and the information at the molecular level is, therefore, accessible. The local/global order parameters are introduced in a simulation study to extract evolutions of many properties such as local structure, conformation, orientation, density, and crystallinity. In Fig. 13, the snapshots of systems clearly show the crystallization process of the polymer melt under extension flow, where the crystalline domains are colored in white.^66–68 Meanwhile, the structures within amorphous domains are identified, the distributions of folds (yellow), tie chains (red) and cilia (green) tell how the networks forming during FIC.

On the other hand, the thermodynamic quantities of polymer crystallization can be calculated by MD simulations even though the FIC is non-equilibrium in nature. Recently, Nicholson et al. developed a method based on the mean first passage time (MFPT) to quantitatively characterize the kinetics and physics of FIC.⁶⁹ In Fig. 14, how the critical free energy barrier (ΔG*), the critical size of the nucleus (n*), and the monomer attachment rate (f₁) change with shear rate (or Weissenberg number W_i) are given for both shear and extension flows. That information extracted from MD simulations shed some light on understanding polymer crystallization as the nucleation events at temporal and spatial resolutions that present particular challenges to the experiment.

In Secs. I–III, the general phenomena of FIC, i.e., crystal morphology and crystallization kinetics, and techniques used for tracking FIC are presented. In Sec. IV, the molecular origin of FIC will be addressed first, which could provide a theoretical explanation for the formation of different crystal morphologies.

Following Peterlin's work,⁷¹ De Gennes theoretically predicted that an abrupt transition from a random coil to extended chain conformation, namely, coil–stretch transition (CST), when the strain rate $ε ˙$ exceeds a critical value $( ε ˙ c )$ [Fig. 15(a)].⁷² In CST, no stable intermediate chain conformation exists.²⁴ One parameter, namely, Deborah number $D e = ε ˙ ∗ τ c$⁠, can be used to quantify such phenomenon: $ε ˙$ is the macroscopic strain rate, and τ_c is the terminal relaxation time of the polymer chain, which is closely related to the molecular weight.

Later on, Keller and Kolnaar first validated this prediction through the elegant design of the extensional flow device as shown in Fig. 15(b).⁷⁰ The birefringence of PE dilute solution starts to change abruptly when the CST happens [Fig. 15(c)]. The results also show that the critical strain rate $ε ˙ c$ is molecular weight dependence: $ε ˙ c$∝M^−1.5. Therefore, the longer chain is more easy to be stretched. This provides a hint for the mechanism for the formation of the shish–kebab structure, where the long-chain mostly participates in the shish formation.

The direct visualization of the CST of the single polymer chain is also achieved in the fluorescently labeled DNA system.^73–78 The experimentally direct visualization of the CST is merely valid in the dilute solution of the macromolecules with long Kuhn lengths, such as DNA or protein. The direct visualizations of common commercial polymers are not accessing in the experiments. The molecular dynamics (MD) simulation provides a good supplement for the study of CST in the polymer systems.^77–82 The MD simulations extend the study of CST into the situation of stronger external fields and variable polymer systems including solutions of linear polymers, branched polymers, ring polymers, and tethered polymers.

Although CST originally proposed to describe the chain dynamics of the polymer chain under flow in dilute solution, the extension of CST to the concentrated solution and even the melt state is also attempted. Different from the polymer chain in the dilute solution, the polymer chains in the concentrated solution and melt are highly entangled with each other. In other words, the polymer network, rather than a single chain, starts to be considered in dealing with the mechanism for the formation of flow-induced morphology.⁷⁰ As shown in Fig. 16 by Keller, further increasing the strain rate to $ε ˙ n$ leads the whole polymer chains deform as a network. The two lines of $ε ˙ c and ε ˙ n$ get close to each other with increasing the concentration. If there is no crossover between these two lines, the CST holds for both the solution and the melt state as shown in Fig. 16(a). Otherwise, CST holds for concentration below C*, and for the melt state, the molecular origin needs to be revised. However, the prerequisite for the foundation of CST is the hydrodynamic effect, which exists in a dilute solution case. In other words, for the homopolymer, the theoretical foundation of CST in the melt or highly entangled states does not exist anymore.¹² The sudden transition from a random coil state to a stretched one would be replaced by transient transition with multiple intermediate states.

In addition to CST, there exist other models accounting for the formation of shish–kebab structure formation. Among them, the stretched network model stands out due to its success in explaining the formation of the shish–kebab structure in a highly entangled state. As early as 1984, Smook and Pennings proposed that the stretched network during flow is responsible for the formation of the shish–kebab structure during gel spinning as shown in Fig. 17.^83,84 For the ultra-high molecular weight PE (UHMWPE), highly entangled network is formed in the gel state before spinning, and the elastic flow instability is also encountered at a low flow rate. During extrusion, the inhomogeneous network consisting of entanglements of different lifetime deforms inhomogeneously: the entanglement with short lifetime migrates or relaxes easily, while that with long lifetime bears the external flow leading to the formation of oriented chains. During the continuous flow, the oriented chains form the shish while the unoriented ones form the kebab.

For the polymer melt under flow, Janeschitz-Kriegl et al. introduced the dormant nuclei, which is intrinsically a precursor with a shape of fringed micelle.^86–88 Once the flow strength exceeds the critical value, those dormant nuclei would transform into point-like nuclei and align along the flow direction. And further increasing the flow strength leads to the merge of these nuclei to form the shish-like nuclei. Later on, some modified models are also proposed. Figure 18 shows one model proposed by Seki et al. for depicting the molecular picture of the formation of nucleation by the flow.⁸⁵ The flow could lead to the formation of precursors close to the initial nucleus, and these precursors would be further transformed into more thermal stable nucleus along the flow field. This eventually leads to the trace of precursor clusters or, namely, shish nuclei. Later, Cui et al. proposed a ghost nucleation model, which highlights the importance of the movement of these precursors for the formation of shish nuclei and enhanced crystallization kinetics.⁸⁹ Despite existing different models, the key feature induced by the flow is the significant reduction of the nucleation barrier. As a result, a more point-like nucleus or shish-nucleus could be formed under flow.

In all, for the highly entangled polymer, concentrated polymer solution, or melt states, the entanglement between physically crosslinking points, rather than the single-chain, is responsible for the shish–kebab structure formation.

With the fast development of characterization techniques as mentioned above, a more detailed transient structure, especially the initial flow induce nucleation (FIN), can be obtained. The initial microstructural evolution, or the FIC precursor structure, has been extensively investigated by numerous research groups, i.e., Li and de Jeu,^42,90 Hsiao et al.,^33,91–94 Winter et al.,^95–98 Kornfield et al.,^{44,85,99–101} Peters et al.,^25–28 and our group^{46,56,102–105} (just a few are listed here limited by the length of the current article). Numerous discoveries are found. For instance, the structure of shish can be either amorphous or crystalline, and not all chains within shish are fully extended but share the same molecular feature: chains in shish are highly oriented along the flow direction.²⁴ Here, the multi-step ordering of FIN is presented as a general view of the current understanding of the formation of nucleation under flow.

As shown in Fig. 19, after applying the flow on the polymer melt, the random coil immediately experiences transition from the random coil to the helix.¹⁰⁶ Later, these rigid helixes start the isotropic–nematic transition to form a locally anisotropic structure.^107,108 Then, the helix bundles are formed as the precursors for nucleation and growth. Concerning the direct experimental evidence of the existence of these intermediate states, the IR⁴³ or Raman¹⁰⁹ spectroscopy is extensively used to track the formation of the helix as these techniques are sensitive to local chain conformation change. For i-PP case, different helix lengths result in different and well-resolved IR absorption lines.^110–112 Numerous rheo-IR experiments were conducted to in situ observe the coil–helix transition.^43,113,114 The non-equilibrium MD simulation is also used to investigate the conformational transition of i-PP under extension,¹¹⁵ where more detailed microstructure information including the number and the length of the helix can be quantitatively calculated from the trajectories. These spectroscopy and simulation results together confirm the multi-step ordering of FIN.

Many research groups have made a great contribution to this issue using computational techniques, i.e., Muthukumar et.al.,^126–128 Rutledge et al.,^129,130 Sommer et al.,^131–133 Hu et al.,^134–136 and Yamamoto^137–139 (just a few are listed here due to the length limitation). The microscopic evidence can be a compensation of experimental results and also shed some light on the fundamental understanding of this non-equilibrium crystallization process.

The molecular origin of FIC as discussed above mainly accounts for different crystal morphologies induced by flow, whereas the theoretical treatment of the enhanced crystallization kinetics during FIC is still missing. Herein, the thermodynamics description of FIC will be presented. Although the FIC is an intrinsically non-equilibrium state, the FIC can still be treated by classical thermodynamics concepts, especially for that in weak flow conditions.

The first experimental evidence of FIC is believed to be attributed to Penning's work in the 1960s, where the later well-known shish–kebab structure is found.^19,20 However, almost two decades ago, Flory proposed a model, namely, entropy reduction model (ERM), accounting for the strain-induced crystallization (SIC) in natural rubber.¹⁴⁰ The ERM is extensively used in understanding FIC and owns great success in interpreting the enhanced nucleation rate.

For polymer, due to the connectivity of the segment, the initial melt is no longer isotropic under flow. As a result, the polymer chain in the deformed melt already deviates from the initial random coil state or the Gaussian chain. This is the origin of the name “strain-induced crystallization,” where strain here represents the deviation of the polymer chain from the initial Gaussian state. As shown in Fig. 21(b), such deviation results in the increase in the free energy of the polymer chain in the melt state and finally reduces the nucleation barrier ΔG_f* for the formation of the nucleus. One great success of ERM is the estimation of the reduction of ΔG* in Eq. (2). Rooted in the rubber elasticity theory, the free energy reduction of the oriented polymer melt is attributed to conformation entropy reduction ΔS_f, which can be estimated by the statistical mechanics. Based on ERM, $Δ S f = ( k B N ¯ 2 ) [ ( 24 m π ) 0.5 λ − ( λ 2 + 2 λ ) ]$⁠, where λ is the stretch ratio, and m and $N ¯$ are the numbers of Kuhn segment per network chain and network chain per unit volume, respectively.^140,142 Therefore, the reduced ΔG_f* can be quantitatively calculated.

In ERM, the simple affine deformation works quite well for the natural rubber system as the system is highly crosslinked. Concerning the polymer melt, however, only physical entanglements exist in most real applications. Therefore, the chain dynamics is required to be considered. Later on, based on the Doi–Edwards model and independent alignment approximation,^13,143 a microrheological model, namely, (DE-IAA), was proposed.¹⁴⁴ The key feature of this model is the modification of the free energy of the polymer melt under steady flow. The free energy is calculated based on the strain rate and chain relaxation (DE memory function), which is, thus, strain rate dependent. Besides the chain relaxation under steady flow, such relaxation occurs also after the flow. Tian et al. designed a step-strain experiment to check the influence of the strain rate on the nucleation rate.¹⁴⁵ A significant reduction of the entropy change is found after the flow, which is attributed to chain relaxation. In all, despite numerous modifications of calculation of the free energy of the polymer melt under flow, the general scenario as shown in Fig. 21(b) does not change.

Despite the free energy change of the initial polymer melt induced by flow, that of the formed crystal may also be changed. Liu et al. found four different crystal morphologies of lightly cross-linked PE after flow, namely, orthorhombic lamellar crystal (OLC), orthorhombic shish crystal (OSC), hexagonal shish crystal (HSC), and non-crystalline oriented shish precursor (OSP) (Fig. 22).¹⁰³ This finding shows that the invariant crystal structure assumed in classical ERM is not right anymore. The local packing structure (crystal phase) and crystal morphology changes, i.e., from point-like to shish, clearly showing the final free energy should be changed as shown in Fig. 21(c).

In summary, to interpret the enhanced nucleation rate induced by flow, various models, including ERM, DE-IAA, and ER–EC, are proposed based on the classical nucleation theory. However, this classical theory has been challenged during the last two decades, especially in the crystallization of small molecules.¹⁴⁶ Various new proposed models, i.e., crystallization by particle attachment (CPA),¹⁴⁶ are proposed, which mostly thanks to the fast development of new characterization techniques (i.e., Cryo-TEM). Systems, such as inorganic salt^147–149 and bio-systems,^150,151 have shown the multi-step crystallization mechanism. Concerning the polymer system, such a multi-step feature has also been discovered. Readers are referred to a recent review.¹² 

In this tutorial, the general description of the flow-induced crystallization (FIC) was achieved generally through two lines: one is the detailed molecular picture for the formation of different crystal morphologies induced by flow, and the other is the general thermodynamics consideration for enhanced crystallization kinetics induced by the flow. The important models and theories, including the coil–stretch transition theory, the stretched network model, and the entropy reduction model are briefly discussed. Herein, the basic foundations of FIC together with recent achievements are summarized.

Considering the publication of Flory's work in 1947,¹⁴⁰ people have continued working on FIC for more than 70 years. Yet, there are still numerous debates existing. This reflects the huge difficulty in fully understanding FIC. Since the FIC is closely related to industrial application, the continuous effort in understanding FIC and implementing new characterizations to in situ track the microstructure evolution of FIC is required. Since quite few structure features exist in the early stage of FIC, namely, the direct detection of multiple intermediates, different techniques validating various intermediate structures of different polymers are required. The fast development of molecular dynamics simulation and advanced characterization techniques, i.e., synchrotron radiation facility, presents a greatly promising future to investigate this non-equilibrium FIC. For instance, the x-ray photon correlation spectroscopy (XPCS) has been successfully coupled with 3D printing technology to in situ track both structure and dynamics evolution during injecting.¹⁵² The further deep investigation of the “old” FIC will continuously help us to better understand different crystal morphology formations during various processing, and optimization of both external processing parameters, i.e., strain rate and time, and internal molecular characteristics of polymers, i.e., molecular weight and polydispersity.

We appreciate the funding support from the National Natural Science Foundation of China (NNSFC; Nos. 51633009 and 51703217), the National Key R&D Program of China (No. 2016YFB0302500), and the Fundamental Research Funds for the Central Universities (No. WK2310000079). We would like to acknowledge the kind help of co-workers in the National Synchrotron Radiation Laboratory (NSRL) and Shanghai Synchrotron Radiation Facility (SSRF) for beamtime and assistance in conducting experiments during last 15 years. Also, we would like to thank Professor Zhe Ma, Professor Tian Nan, Dr. Kunpeng Cui, and Professor Zhen Wang for their helpful comments on the manuscript.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.