Extension-Induced Nucleation under Near-Equilibrium Conditions

Sep 24, 2014 - (34-36) Other models like phase separation and ghost nucleation were ..... which comes from the stacks of lamellar crystals; while WAXD...
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Extension-Induced Nucleation under Near-Equilibrium Conditions: The Mechanism on the Transition from Point Nucleus to Shish Dong Liu, Nan Tian, Ningdong Huang, Kunpeng Cui, Zhen Wang, Tingting Hu, Haoran Yang, Xiangyang Li, and Liangbin Li* National Synchrotron Radiation Lab and College of Nuclear Science and Technology, CAS Key Laboratory of Soft Matter Chemistry, University of Science and Technology of China, Hefei 230026, China ABSTRACT: Extension-induced crystallization under nearequilibrium condition has been studied in a series of lightly cross-linked high density polyethylene (XL-HDPE) with a combination of extensional rheology and in situ synchrotron radiation small-angle X-ray scattering (SAXS) and wide-angle Xray diffraction (WAXD) measurements. According to crystal morphology and structure, four regions were defined in straintemperature space, namely “orthorhombic lamellar crystal” (OLC), “orthorhombic shish crystal” (OSC), “hexagonal shish crystal” (HSC) and “oriented shish precursor” (OSP), respectively. This indicates that flow not only induces entropic reduction of initial melt, but also modifies the free energies of the final states, which is overlooked in the classical stretched network model (SNM) for flow induced crystallization (FIC). Incorporating the free energies of various final states, a modified SNM is developed and employed to analyze strain-temperature equivalence on nucleation in FIC, which reveals that the critical nucleus thickness l* at different regions leads to a natural transition from lamellar to shish nuclei. The results suggest that classical nucleation theory is still valid for FIC under near-equilibrium condition provided that the free energy changes of initial melt and final states induced by flow are taken into account.

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under extension flow produced by two opposed jets, Keller et al. provided the evidence of the existence of CST in dilute polyethylene (PE) solution, and proposed CST model for shish-kebab formation, where the extend chain is assigned to form shish and random coils is to kebab.21 As the shish-kebab structure or fiber-plateled duality is also widely observed in concentrated solution and melt, Keller extended the CST model to polymer melt. The final morphology of fiber-plateled duality was suggested to be a pointer to the existence of chain extension (CST). Keller’s CST model for shish-kebab formation elegantly established a one-to-one correlation between chain conformations and crystal morphologies and gained great success in FIC, which has been investigated experimentally during the past several decades. Among them the roles of long and short chains were the most used scenarios as the critical strain rate for CST depends on molecular weight.27−33 However, it should be aware that de Gennes’s theory of CST was precisely deduced from polymer dilute solution, while hydrodynamic effect, the essential ingredient for the theory of CST, does not exist in polymer melt. Thus, the argument of CST in polymer melt lacks strict physical basis and Keller’s experiments can only provide convincing evidence

INTRODUCTION Flow-induced crystallization (FIC) of polymers is of vital importance in industry and academy, since flow is inevitably involved in polymer processing, including extrusion, injection, fiber spinning, etc.1−6 The introduction of external flow has a magnificent impact on crystallization of polymer, such as generating different kinds of superstructures by inducing orientated nuclei.7−14 Among those unique superstructures formed in FIC, the most striking feature is shish-kebab, which has been widely studied since it is first found by pioneering work of Pennings et al.15−18 and Keller et al.19,20 Soon afterward great endeavors have been made to unveil the underlying physics of FIC such as the mechanism for the transition from point nucleus to shish. Up to now, two main models are dominant in the community of FIC, namely coil− stretch transition (CST)21 and stretched network model (SNM).22−25 CST model in FIC was proposed to account for the morphology of nucleation (shish-kebab) under flow. The theory of CST was first presented by de Gennes based on the investigation of single chain dynamics under external flow.26 A solute polymer random coil will unwind abruptly when a critical velocity gradient is reached. It generates dual populations of chain conformations, namely random coil and fully extended chains, no stable intermediate chain conformation exists in between. With the birefringent experiments of polymer chain © 2014 American Chemical Society

Received: July 18, 2014 Revised: September 5, 2014 Published: September 24, 2014 6813

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in either models, which may be the essential ingredients in FIC. Flow can induce various morphologies of nucleation, such as point, scaffold and thread-like shish nuclei.42,54−59 The surface free energy σ and the melt temperature Tm of crystal may vary with different morphologies. For example, the free energies of folded-chain surface in point nucleus and the end surface in fringe micellar or extended chain surface are different; meanwhile the latter crystal with larger thickness should also have higher melting point.60,61 On the other hand, new crystal formation can also be induced by flow, like β modification of isotactic polypropylene62,63 and hexagonal phase of polyethylene.64−67 Evidently, different packing lattices of nucleus have different internal bulk free energy or melting enthalpy ΔHf. Either variation of surface free energy due to different morphologies or change of bulk free energy due to different crystal structures affects nucleation barrier ΔG* as ΔG* ∝ σ3/ ΔH2,68 which, however, is still not incorporated in the current theories of FIC. Moreover, the occurrence of intermediate structures like flow-induced preordering or precursors pushes FIC to a more complicated situation.33,60,69−74 As evidenced by the sequential appearance of synchrotron radiation small-angle X-ray scattering (SAXS) and wide-angle X-ray diffraction (WAXD) signal,75 conformational ordering before crystallization detected from spectroscopic methods,76−78 study on lifetime of preordering,33,79 computer simulation,80−82 etc. flow can induce precursor with structural order in between disordered amorphous and periodic packed crystal. The intermediate order breaks the direct pathway from chain conformation to crystal in CST and SNM, which may guide the study of FIC away from classical nucleation theory. Unfortunately, the progress in this aspect is halted as verification of the existence and unveiling the natures of precursors are still an ongoing process. In this work, three lightly cross-linked high density polyethylene (XL-HDPE) samples with different total irradiation dose have been studied with a combination of extensional rheology and in situ SAXS and WAXD measurements, during which the strain-temperature equivalence on nucleation is investigated. The samples are under slow deformation, which is treated as a system under near-equilibrium condition. Different crystal morphologies and structures are observed in straintemperature space, which can be defined into four regions. A modified SNM (mSNM) is proposed and suggests that the transition from lamellar to shish nucleus is determined by critical thickness of nucleus rather than CST at segment level.

restricted to the situation of dilute or semidilute solutions. Polymer melt contains large amount of entanglement points to force different molecular chains to connect together. It seems unlikely to reach full extension at the whole chain level. Experimental evidence on the undiscriminated role of long and short chains and the critical strain for shish formation clearly revealed that CST and full chain extension is not a precondition to induce shish.34−36 Other models like phase separation and ghost nucleation were also proposed to account for shish formation in different systems.37,38 With some experiments on entanglement network and cross-linked network, CST model at segment level, instead of the whole chain, was proposed.36,39 Though the compromised CST model already loses the original physics of de Gennes’s theory of CST, it is widely adopted to explain shish-kebab formation in polymer melt in the FIC community. Unfortunately, the idea of chain extension at the segment level in the compromised CST model is still lacking of direct evidence.36,37,40−42 The SNM in FIC mainly concerns the nucleation kinetic. The essential idea of the SNM is that the entropic reduction due to flow-induced orientation and stretch leads to decrease of nucleation barrier and enhancement of nucleation rate. In early 1940s, Flory developed the pioneer theory of SNM through the application of statistical mechanics similar to those employed in rubber elasticity theory.22−25 When the polymer network is stretched, chains between network junctions are deformed from their equilibrium configurations. Then the corresponding configurational entropies of these chains decrease and crystallization occurs when TΔSf > ΔHf. Focusing on the entropic change caused by stretch, Yeh et al. had endeavored great effort on the morphology development during strain induced crystallization.43−46 They proposed that flow decreased the entropies of chain segments, and increased the thermodynamic driving force of crystallization ΔG. On the basis of the expression of nucleation rate from classical nucleation theory N = Nc exp(−ED/kT) exp(−ΔG*/kT), the nucleation rate accelerated by flow field can be obtained through a modification of ΔG*, which is the energy change for the formation of critical nucleus. Similar SNM for FIC was also developed by other authors, where the differences were mainly related to the expression of entropic reduction.46−48 All these SNMs developed in early time focus mainly on covalent crosslinked network, where strain, instead of strain rate, plays the critical role on the acceleration of nucleation rate. Dealing with entangled polymer melt, Coppola et al.49 proposed a microrheological model to describe FIC based on Doi− Edwards polymer dynamics theory.50−52 Here the role of strain rate was emphasized to act with chain relaxation in steady flow and the strain rate dependent entropic reduction was coupled into classical nucleation theory. This model was extended to step-strain case and guides the understanding of the evolution of long period after step extension.53 Nevertheless, all theories of SNM only consider the entropic reduction of the initial melt, while flow-induced change of free energy of the final nucleus are not taken into account, which, however, may be of vital importance on controlling the nucleation kinetics and final morphology and structure of crystal. Though CST and SNM concern different aspects of FIC, the approach from the conformation transition of polymer chains to crystallization is in common. Nevertheless, the variation of nucleus morphology and structure as well as possible intermediate state like the so-called precursor are not specified

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EXPERIMENT

The raw high density polyethylene (HDPE) granules used in this study were supplied by Sinopec Qilu Co. Ltd.. The number-average (Mn), weight-average (Mw), and Z-average molecular weight (Mz) are about 42 kg/mol, 823 kg/mol, and 4395 kg/mol, respectively. The granules were first molded to plates with a thickness of 1 mm by a vulcanizing press at 180 °C and then cooled down to room temperature. The plates were annealed under vacuum at 90 °C for 24 h in order to eliminate residual stress. After that the plates were exposed to a 60Co γ-ray radiation source (located in USTC, Hefei, China) at room temperature. The dose rate is 35 Gy/min, and the total absorbed doses of three samples used in this work are 15, 30, and 50 kGy, respectively. For the sake of briefness, the three samples are hereafter referred to S15, S30, and S50, respectively. In order to reduce the formation of peroxide radicals, oxygen was isolated from the sample during the radiation process. The trapped free radicals were further eliminated through annealing at 90 °C for 24 h under vacuum. The cross-linking degree of the lightly cross-linked high density 6814

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rheometer, strain rate as well as Hencky strain ε can be varied independently. Figure 1 also depicts the experimental procedure in this work. Each sample was first heated up to 200 °C and held for 10 min in order to erase thermal and mechanical histories. Then it was cooled to a certain temperature with a rate of 2 °C/min for extension. Hereafter the extension temperature is referred to Tc. A nitrogen gas flow was used to homogenize temperature and prevent samples from degradation. The temperature fluctuation was within ±0.5 °C. Immediately after reaching Tc, step extensions with constant strain rate of 0.02 s−1 and a total strain of 2.8 were imposed on the molten samples. Torque was recorded continuously during and after step extensions with the torque sensor. Simultaneously, the structure evolution processes were monitored by in situ two-dimensional (2D) SAXS and WAXD measurement at the beamline BL16B of Shanghai Synchrotron Radiation Facility (SSRF), separately as two sets of experiments. Subsequent isothermal evolution processes on Tc were also monitored. However, it is not the focus of this work. The X-ray wavelength was 0.124 nm and a Mar165 CCD detector (2048 × 2048 pixels with pixel size of 80 μm) was employed to collect time-resolved 2D SAXS and WAXD patterns. The exposure time was 1.5 s with an additional 2.5 s for reading and cleaning (i.e., patterns were acquired at a rate of 4 s/ frame). The sample-to-detector distances were calibrated to be 5357 mm by beef tendon for SAXS, while 218 mm by yttrium sesquioxide (Y2O3) for WAXD, respectively. SAXS and WAXD in current work can cover a q range of 0.056−0.76 nm−1 and 2.9−17.9 nm−1, respectively. Fit2D software from European Synchrotron Radiation Facility was used to analyze the SAXS and WAXD data, which were corrected for background scattering through subtracting contributions from the extensional rheometer and air. The 2D SAXS patterns were integrated to obtain one-dimensional (1D) azimuthal intensity distribution, as well as 1D scattering profile as a function of 2d(sin θ) = nλ. The 2D WAXD patterns were integrated to obtain 1D diffraction profile as a function of q = 4π(sin θ)/λ and 2d(sin θ) = nλ, where q is the module of scattering vector, d the interplanar spacing, n the positive integer, 2θ the scattering angle, and λ the X-ray wavelength.

polyethylene (XL-HDPE) sample is expressed as gel fraction. The samples were extracted in boiling xylene for 48 h with Soxhlet extractor, and then dried in a vacuum oven at 70 °C for 24 h, until the weight of residual no longer decreased. The gel fraction was calculated gravimetrically from the weight of the sample before and after extraction. The gel fraction of the original HDPE sample is 0 wt %, while the values of S15, S30, and S50 are 0, 17.1, and 23.3 wt %, respectively. The XL-HDPE samples for extensional flow test were cut into rectangular shapes with length, width and thickness of 30, 18, and 1 mm, respectively. A homemade two-drum extensional rheometer used in this work can apply well-defined thermal history and impose extensional flow field. The details of this apparatus have already been described elsewhere.53 Its design is similar to the commercial Sentmanat extensional rheometer, as schematically shown in Figure 1. The ends of samples are secured to geared drums by means of

Figure 1. Experimental procedure and the schematic drawing of the homemade two-drum extensional rheometer device used in this in situ experiment.

clamps. The length of samples (L0) subjected to stretching equals the distance between the axes of two drums, where L0 is kept at a constant value of 20 mm during the test. With a constant angular velocity ω of drums, the extensional strain rate is constant as ε̇ = ΔL/tL0 = ωd/L0, where d is the diameter of drums. With this uniaxial extensional

Figure 2. Selected engineering stress−Hencky strain curves of the extension processes. (a) Same XL-HDPE sample in different temperature, using S50 as a representative. (b) Different samples in the same temperature of 145 °C. The blue dash lines indicate strain of 2.8 where the extension ceased. Parts a′ and b′ are corresponding true stress−Hancky strain curves. 6815

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Figure 3. Selected in situ 2D SAXS (a) and WAXD (b) pattern of XL-HDPE samples, using S50 during extension in 136 °C, as the representatives. The frames with sign of incipient strain (εi) in the upper right corner are the ones where the signals first came out, the signals are indicated by the red arrows and labels. The numbers in the bottom right corner of each pattern represent experimental strain. The extensional flow direction (FD) is horizontal, as indicated by the double-headed arrow.

Figure 4. Overall incipient strains for the initial signals to be induced as a function of temperature, detected by both SAXS (a) and WAXD (b), respectively.

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extension at 136 °C. Combining SAXS or WAXD patterns with stress−strain curves, the incipient strain (εi) is obtained, which is the strain corresponding to the first SAXS or WAXD frame with ordered structures. In this work, the “ordered structures” can be lamellar crystal and shish detected with SAXS or orthorhombic and hexagonal phases measured with WAXD. Analogue to homogeneous nucleation temperature at quiescent condition, εi can be denoted as homogeneous nucleation strain in strain-temperature space, which represents a strain-temperature equivalence on nucleation. As highlighted with the sign εi at the upper right corner of the patterns in Figure 3, εi obtained by SAXS measurements is 1.20 for S50 stretched at 136 °C, which is indicated by two streaks, characteristic signal of shish, in the SAXS pattern. The incipient stain εi obtained from WAXD is 1.60 for the same experiment, indicated by the appearance of diffractions from orthorhombic crystal. The crystalline diffractions emerge at larger strain than εi observed with SAXS. This phenomenon occurs at high temperatures for all samples, while εi obtained from SAXS and WAXD reach the same value at low temperatures. Following the same approach described above, we extracted εi at different temperatures for different samples, which are summarized in Figure 4, parts a and b, from SAXS and WAXD, respectively. Figure 4 shows that εi increases with the increase of temperature for each sample, while at the same temperature εi decreases with the increase of irradiation dose or cross-link density. The curves in Figure 4 are boundary lines for homogeneous nucleation in strain-temperature space, where either cooling temperature or increasing strain across the lines will induce homogeneous nucleation. As mentioned above, when the temperature is relatively low, εi acquired from SAXS and WAXD are similar. For example, at 129 °C, εi obtained

RESULTS In Situ SAXS and WAXD. Extension with a constant strain rate of 0.02 s−1 and a total strain of 2.8 were imposed on three XL-HDPE samples (S15, S30, and S50) at series Tc. Figure 2 presents the selected representative engineering stress−Hencky strain curves, where Figure 2a depicts the same XL-HDPE sample (S50) at different temperatures, and Figure 2b different samples at the same temperature of 145 °C. The engineering stress−Hencky strain curve can be roughly defined into three zones, namely linear elastic, stress plateau and strain hardening zones. Figure 2a shows that the moduli slightly decrease with the increase of temperature for the same sample. In Figure 2b, at the same temperature, different samples show some differences in moduli and the stress plateau zones. Increasing the irradiation dose or cross-link density leads to a reduction of the stress plateau zone and an increase of the modulus. For comparison, corresponding true stress−Hencky strain curves are calculated with σtrue = σeng exp(ε), as depicted in Figure 2, parts a′ and b′, respectively. Strain hardening points in true stress−Hencky strain curves start at nearly the same strain of about 2.0 as observed in engineering stress−Hencky strain curves. The strain hardening points are acquired from the cross point of two tangent lines of the strain−stress curves. During the whole extension process, in situ 2D SAXS and WAXD measurements were conducted continuously. With the aforementioned experimental conditions, each extension experiment took 140 s. With an acquisition rate of 4 s/frame for SAXS and WAXD, 35 frames were collected for each extension experiment, corresponding to a resolution of 0.08 in strain space. Figure 3 presents several representative 2D SAXS (a) and WAXD (b) patterns, which were taken from S50 during 6816

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Table 1. Four Temperature Regions of the Three Samples, Divided According to the Different Crystal Morphologies and Structures Induced by Extension Revealed by SAXS and WAXD temperature (°C) sample

gel fraction (%)

region I

region II

region III

region IV

S15 S30 S50

0.0 17.1 23.3

126−132 126−132 126−132

134−149 134−142 134−149

− 149−154 154−184

154−157 157−160 193−200

Figure 5. Different temperature regions divided according to the initial structures revealed by the SAXS (a) and WAXD (b) results of S50. The first column depicts the patterns of strain less than εi as a reference. The relative q range of SAXS is indicated by the red scale plates. The red arrows and labels indicate the relative structure signals. The extensional flow direction (FD) is horizontal.

Figure 6. Corresponding 1D SAXS scattering profiles of 2D SAXS pattern of S50 in both meridian and equator of 129 °C (a) and 136 °C (b), respectively. (c) Azimuthal intensity distribution of the SAXS pattern of S50 in a narrow q range. 2D SAXS pattern in upper right corner depicts the q range with black concentric circles, and shows the start of the azimuthal angle and direction as indicated by the red horizontal line and arrow. (d) Integrated 1D WAXD curves of S50. Scattering vector q (top) as well as 2θ (bottom) are both presented. 2D WAXD pattern in upper left corner shows the mask protocol used for integration.

149, 154, and 157 °C for S15, S30, and S50, respectively, while SAXS signal still arises up to 157, 160, and 200 °C for S15, S30, and S50, respectively. The above incipient strains in Figure 4 do not specify the detailed structures, which in fact vary at different temperatures. With the combination of morphologies and crystal structures

from SAXS and WAXD are of the same values around 1.20, 0.96, and 0.96 for S15, S30, and S50, respectively. However, when the temperature exceeds 132 °C, εi in WAXD was always delayed compared to that of SAXS. Moreover, with the strain of 2.8 studied in this work, no crystalline diffraction appears in WAXD during the extension when the temperature exceeds 6817

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elongation ratio (α) follows the function of α = exp(ε). The entropic decrease of extension deformation (ΔSelg) from undeformed (α = 1) to deformed states (α = αi) is related to the tensile stress (σ(α)) by,87

observed by SAXS and WAXD, respectively, four temperature regions can be defined in strain-temperature space. The temperature ranges of each region for three samples are presented in Table 1, which is explained as following. Taking S50 as the representative, typical features in different temperature regions provided by SAXS and WAXD are presented in Figure 5, parts a and b, respectively. The crystal phases are checked carefully and no crystal−crystal transition occurs during the early stage of extension. For a better view of the peaks of 2D SAXS and WAXD patterns, the corresponding one-dimensional (1D) SAXS scattering profiles of 129 and 136 °C in both equator and meridian are depicted in Figure 6, parts a and b, respectively, the azimuthal intensity distribution of the SAXS pattern in Figure 6c and the 1D WAXD curves in Figure 6d. Region I locates at relatively low temperature range from 126 to 132 °C (strain from 0.88 to 1.02). Represented by patterns at 129 °C in Figure 5, the SAXS signal first appears as a pair of blobs in meridian (also see Figure 6, parts a and c), which comes from the stacks of lamellar crystals; while WAXD signal displays pairs of equatorial (110)o and (200)o diffraction arcs of orthorhombic phase. According to the above structural feature, we name this region as orthorhombic lamellar crystal (OLC) region. In region I, the incipient signals in SAXS and WAXD appear at the same strain. This is different from regions II and III, where the WAXD signals are always delayed compared to the SAXS ones. Region II covers the temperature range from 134 to 149 °C for sample S50 (strain from 1.12 to 1.6). With 136 °C as a representative in Figure 5, the signal first appearing in SAXS is a pair of equatorial streaks from shish (also see Figure 6, parts b and c), while in WAXD it is still the orthorhombic (110)o and (200)o diffractions but with a higher orientation than that in region I. Region II is named as orthorhombic shish crystal (OSC) region. Temperature from 154 to 184 °C is defined as region III (strain from 1.76 to 2.24). In this region, SAXS first gives the same pair of streaks from shish; while instead of forming orthorhombic phase the WAXD pattern shows (100)h diffraction of hexagonal crystal, which is normally observed at high pressure (on the order of kbar) and under stretching of ultrahigh molecular weight PE.83−86 The 1D diffraction positions are presented in Figure 6d for a better view, in regions I and II, (110)o and (200)o reflections of orthorhombic phase locate at q of 14.95 nm−1 and 16.19 nm−1 or 2θ of 16.97° and 18.39°, respectively; while the (100)h reflection of hexagonal crystal in region III is at a q of 14.45 nm−1 or 2θ of 16.39°. Region III is named as the hexagonal shish crystal (HSC) region. Further increasing temperature to region IV (from 193 to 200 °C, strain at 2.48), the pair of streaks from shish in SAXS still persists though weakens, which is not accompanied by any crystalline diffraction. Only amorphous halo is observed in this region as Figures 5b and 6d depict. This region is named as oriented shish precursor (OSP) region. Similar to S50, SAXS and WAXD results from S15 and S30 lead to the same region dividing in temperature space, though the temperature range for each region is varied with samples as listed in Table 1. Note that no region III exists for S15 with low cross-link density. Analysis with Modified SNM. In this section, based on morphology and structure in strain-temperature space of the four regions under near-equilibrium condition, we employ the SNM for FIC to analyze the relation between temperature and the incipient strain. Here we use incipient elongation ratio (αi) instead of strain εi for the convenience of comparison with the SNM. The relationship between the Hencky strain and

ΔSelg ≈ −

1 T

∫1

αi

σ (α ) d α

(1)

,which depends on the description of the strain-energy function. For uniaxial extension as in current work, some direct quantitative relations between ΔS and elongation ratio α are proposed. According to the statistical theory of rubber elasticity, under the assumptions of affine deformation of Gaussian chains, Krigbaum and Roe proposed the expression as48 ⎞ 1 ⎛ 2 ΔSelg = − vk ⎜α 2 + − 3⎟ ⎠ 2 ⎝ α

(2)

,where v is the network-chain density and k the Boltzmann constant. In current work, chains as well as chain segments in between two cross-linked points of XL-HDPE can be assumed to be Gaussian, also assumption of affine deformation of the melt should valid, until incipient strain where induced structures appear. Moreover, v shows a temperature independent until the incipient strain as the cross-linked network-chain density part plays the dominant role under the near-equilibrium condition, which can be seen from Figure 2, parts a and a′. The melting temperature Tm is equivalently elevated, it can be expressed as47 ΔSelg 1 1 = + Tm(αi) Tm(1) ΔH

(3)

,where Tm(αi) and Tm(1) are melt temperatures at elongation ratio of 1 and αi, respectively, ΔH is the melting enthalpy of crystal. Flory defined Tm in his formulation as “incipient crystallization temperature”,23 which might be essentially the melt temperature. In this work, we measured the correlation between elongation ratio αi and homogeneous nucleation temperature under near-equilibrium condition rather than melting temperature. According to the expression of critical nucleus thickness l*88 l* =

4σeTm ΔH(Tm − Tc)

(4)

,we obtained

Tm =

Tc 1−

4σe ΔHl *

(5)

,where (Tm − Tc) is the suppercooling, Tc the crystallization temperature, σe the specific free energy of end surface, ΔH is the melting enthalpy of crystals, these variable parameters together determine the size of the nucleus in different regions. Combining eqs 2, 3, and 5 yields

(

2

)

vk αi 2 + α − 3 1 1 i = − 4σ Tc(αi) Tc(1) 2ΔH 1 − ΔHle *

(

)

(6)

,where Tc(1) and Tc(αi) are the crystallization temperature at elongation ratios of 1 and αi, respectively. In this way, eq 6 presents a modified SNM (mSNM), where we incorporate both entropic reduction of initial melt and free energy change 6818

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of the final states (i.e., different nuclei) induced by flow. In eq 6, the correlation between Tc and αi are constructed, using l* as a bridge (eq 4) where the end surface free energy and enthalpy of crystal are taken into account. Following eq 6, Figure 7 plots the data from S15 (a), S30 (b), and S50 (c) in Figure 4a with α2 + 2/α − 3 and 1000/Tc as

morphologies and crystal structures in the four regions. Thus, independent linear fitting is applied. The red lines are linear fitting result of experimental data in each region. It generates the slope −1000 vk/2ΔH(1 − ((4σe)/(ΔHl*))) and the intercept 1000/Tc(1), respectively. The linear fitting results on the values of Tc(1) and slopes of S15, S30, and S50 are summarized in Table 2, respectively. The extrapolated crystallization temperatures Tc(1) are almost the same values for all three samples in the first three regions, namely, about 120.0, 128.0, and 141.5 °C for regions I, II, and III respectively. The value of TcIII(1) is close to the reported equilibrium melting point of PE (141 °C).89 Different from the situation of the first three regions, TcIV(1) depends on cross-link densities of samples, which varies from 140 to 160 °C. Table 2 also shows the slopes of different samples in each region. The slope contains the information on surface free energy, fusion enthalpy, thickness of critical nuclei and crosslink density. The absolute value of slope increases with the increase of cross-link density of the samples, and decreases from region I to region IV. The fitting procedure is also performed with the WAXD data in regions II and III, as depicted in Figure 8, parts a and b, respectively. The fitting results are given in Table 3, which shows a similar trend as the SAXS data. TcII(1) from WAXD seems to be same as that obtained from SAXS while TcIII(1) is slightly lower.

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DISCUSSION On the basis of in situ SAXS and WAXD measurements during stretch of XL-HDPE network at different temperatures, some interesting findings can be extracted. (i) According to morphology from SAXS and structure from WAXD, four different regions can be defined in strain-temperature space, namely OLC, OSC, HSC, and OSP for regions from I to IV, as schematically described in Figure 9. (ii) The incipient strain (εi) for nucleation follows a one-to-one correlation with temperature, which indicates strain-temperature equivalence holds for strain-induced nucleation. (iii) Analysis with the mSNM shows that Tc(1) keeps constant for samples with different cross-link densities provided that the final state is crystal, while Tc(1) varies with samples when the final state is noncrystalline. On the basis of these findings, several debating issues on FIC can be clarified: (i) What determines the transition from point nucleus to shish? (ii) Is coil−stretch transition (CST) at segment level a necessary condition for shish formation? (iii) Does flow-induced precursor exist or not and how does it influence crystallization? We will discuss these issues in the following sections. What determine the transition from point nucleus to shish? As the transition from regions I to II exactly corresponds to the transition from point nucleus to shish, analyzing the nucleation mechanism in different regions may help to answer this

Figure 7. Different temperature regions divided according to the entropy change of extension. Blue open rectangles, cycles and triangles are the experimental SAXS data of S15 (a), S30 (b), and S50 (c), respectively. α is the incipient elongation ratio, and T is the absolute temperature. The black dashes denote the temperature boundaries. The red lines are the linear fitting results in each region of each sample, according to the modified SNM. Profiles share a common xaxis on a top−bottom presentation.

x and y axes, respectively. A simple linear relation is expected if only one single type of structure forms in the whole straintemperature space, which, however, is not observed in Figure 7. Interestingly, the data points can also be divided into four linear regions (separated with the horizontal black dash lines in Figure 7), which coincides with the four regions defined above with the combination of morphology and crystal structure obtained from SAXS and WAXD, respectively. This indicates that the nonlinearity is mainly due to the formations of different

Table 2. Fitted Parameters in Four Regions Based on SAXS Data.a region I

region II

region III

region IV

sample

TcI(1)

slopeI

TcII(1)

slopeII

TcIII(1)

slopeIII

TcIV(1)

slopeIV

S15 S30 S50

119.8 120.6 115.8

−0.006 17 −0.010 22 −0.017 89

128.9 127.8 127.3

−0.001 69 −0.004 55 −0.005 90

− 141.5 141.5

− −0.001 77 −0.002 71

140.0 146.1 160.7

−0.000 81 −0.001 25 −0.001 36

Slope equals −1000 vk/2ΔH(1 − ((4σe)/(ΔHl*))), denoting the slopes of the fitting lines; the unit of Tc(1) is °C. Superscripts I, II, III, and IV denote each region.

a

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Figure 8. Correlation between the incipient elongation ratio of the WAXD data and temperature in region II (a) and region III (b), respectively, fitting with the mSNM.

Table 3. Fitted TcII(1) and TcIII(1) Based on WAXD Data WAXD sample

TcII(1) (°C)

TcIII(1) (°C)

S15 S30 S50

126.9 129.1 127.2

− 135.2 134.3

Figure 10. Schematic illustration of the free energy of different morphologies and structures induced by flow as a function of degree of order. (a) Classical SNM only involves the effect of the flow on entropic reduction of melt. (b) mSNM considers the different crystal morphologies (ΔH) and structures (σe) in the final stage of nucleation as well. G is the free energy of certain state, where the subscript L and C represent liquid (melt) and crystal, respectively. ΔGf is free energy change of melt state induced by flow; ΔGq is the fundamental thermodynamic driving force in quiescent condition; ΔGq* is the nucleation barrier of critical nuclei.

ΔG* = Figure 9. Schematic illustration of the morphologies and structures of four regions in the strain-temperature space, based on the information deduced from both WAXD (a) and SAXS (b). The extensional flow direction (FD) is perpendicular to the plane of the paper (the same direction with the c axis, as indicated by the blue dot) for WAXD and is horizontal (as indicated by the blue double-headed arrow) for SAXS deducing, respectively. The red dash open polygons are the crystal units.

32σ 2σeTm 2 (ΔT ΔH )2

(7)

,the effect of the equivalent increase of ΔT due to entropic reduction may be partially counteracted by the increase of σe, leading to weaker variation of ΔG* induced by flow. On the other hand, the transition from regions II to III is a transition from orthorhombic to hexagonal lattices, where the latter has a smaller ΔH or higher GC, which may result in a larger ΔG* assuming other parameters in eq 7 remaining the same. Clearly, the changes of the final states play a critical role in FIC. Taking into account flow-induced free energy changes of initial melt and the final states, the transition from point nucleus to shish can be simply attributed to the critical nucleus thickness l*, which can be deduced from the slopes of linear fitting as listed in Table 2. We denote the ratio between slopes from two different regions as A and as assume that the crosslink density of each sample in all regions remains unchanged. Thus, the following relation can be obtained,

question, which can be explained with the mSNM for FIC. The classical SNM for FIC considers strain-induced entropic reduction of polymer melt, while the final state is assumed to be the same. Thus, the effect of strain is only allocated on free energy of initial polymer melt GL, which is elevated by ΔGf = TΔSelg, as schematically illustrated in Figure 10a, where the solid blue line represents the energy landscape under quiescent condition and the dash red line under flow. Consequently, the nucleation barrier ΔG* is also reduced with flow, leading to an enhancement of nucleation rate. The observation of different crystal morphologies and structures in current work indicates that the assumption of the same final state induced by flow does not hold anymore. In another word, FIC in the SNM should take into account not only the change of the initial state but also the final state induced by flow. Considering all these factors, a new schematic free energy landscape is presented in Figure 10b. For the transition from folded-chain lamellae to shish morphology (from regions I to II), the end surface is changed from fold-chain to fringe micellar structures, which corresponds to an increase of specific surface free energy σe. As nucleation barrier ΔG* can be expressed as88

ΔH j − slopei = slope j ΔH i −

4σej l*j 4σei l*i

=A (8)

,where i and j represent two different regions. Combining eqs 4 and 8, the ratio of l* of the two regions is ⎞ Aσei σei Tmi ⎛ ΔH j l*i = + − A⎟ ⎜ j j i i j ⎠ σe σe ΔT ⎝ ΔH l*

(9)

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and strain rate are generally overstated in some studies on FIC. For example, the idea of CST stresses only on chain conformation, while how the oriented or stretched chain form shish is not described. In fact it is rather hard to obtain fully extended chain or chain segment by flow in polymer melt, while extended chains or segments commonly exist in crystal. In another word, the stretched chain conformation in shish nucleus may be generated mainly by crystallization itself rather than by external flow. Flow indeed provides external work and modifies the free energy landscape as illustrated in Figure 10. However, at current near-equilibrium condition, external work more likely plays a role of perturbation; the intrinsic free energies of melt and crystal still take the leading role in FIC. This deduction suggests CST even in segment level is not a necessary condition for shish formation. The one-to-one correlation between strain and temperature or strain−temperature equivalence for nucleation also supports that CST at segment level is not a necessary condition for shish formation. As mentioned in the Introduction, CST in polymer melt does not exist due to lacking of the essential ingredient of hydrodynamic interaction. Even without the initial physic basis, Keller et al. proposed extended chain segment or CST at segment level is the origin for shish formation in cross-linked rubber materials. Without direct verification, this model has also been adopted to explain shish formation in polymer melt. If CST of chain segments between cross-link points is a necessary condition for shish formation, one would expect that a constant critical strain is required to induce the formation of shish, which makes chain segments between adjacent cross-link points to be extended. Oppositely, the strain-temperature equivalence presented in Figure 4 shows that the strain for shish formation obeys a one-to-one correlation with temperature for a sample with a given cross-link density. This one-toone correlation excludes that a constant critical strain is the precondition for shish formation. Moreover, in the three samples the transition from lamellar to shish nuclei occurs at strain smaller than the strain hardening points, which is the starting point for chain segments to be stretched. For example, εi for shish formation is about 1.12 for S50 at 134 °C, while the strain hardening point locates at strain of about 2.0. The formation of shish stems from a coupled effect between temperature and strain rather than CST at chain segment level, where the former determines the critical nucleus thickness l* and leads to the transition from point nucleus to shish. The above discussion is restricted within two preconditions: (i) direct phase transition from melt to crystal; (ii) under nearequilibrium condition. Flow-induced preordering or noncrystalline precursor is a debating issue in polymer community. If precursor appears before crystallization, the direct pathway from melt to crystal is broken and the free energy landscape in Figure 10b needs to be further modified. The occurrence of SAXS signal in regions II and III before WAXD signal and only SAXS but without WAXD signals in region IV support the existence of precursor. Additionally, the variation of Tc(1) in region IV for samples with different cross-link densities indicates the absence of a fixed final structure in this region. This suggests that the noncrystalline precursor is a preordered state rather than a thermodynamic phase; otherwise, a constant Tc(1) should also exist in region IV. For this reason, it is still hard for us to incorporate the preordering into the mSNM. Current experimental result is based on the cross-linked network under slow deformation, which is regarded as a system under near-equilibrium condition. In this case, all the laws of

In the case of regions I and II, the OSC nuclei in region II is fringe micellar structure, where the specific free energy of end surface σIIe is larger than that of OLC (σIe) in region I, as the later is folded-chain surface. According to the estimation of Zachmann et al.,90,91 the chain crowding on the end surface leads to extra surface free energy and σIIe is about 2−6 times of the folded-chain surface free energy σIe. We take σIIe ≈ 3σIe for a rough estimation. As regions I and II form the same orthorhombic crystals, ΔHII = ΔHI = 290 J/g. From Table 2, the average slopes ratio between regions II and I is AII/I av = 0.35. In eq 9, Tm/ΔT is relatively difficult to estimate. Here we take the value at quiescent condition as reference. The homogeneous nucleation temperature and equilibrium melting temperature of PE are about 86 and 141 °C, respectively,92 which leads to TIIm/ΔTII ≈ 7.5. Inserting all the above estimated values into eq 9 we obtain the ratio of l*II/l*I ≈ 15.6. With the thickness of lamellar crystal in the order of 10 nm, we obtain the thickness of OSC critical nuclei l*II in the order of 100 nm, which corresponds to shish structure. The increase of critical nucleus thickness l* in one order gives a straightforward explanation for the transition from point nucleus to shish. Note that the extra surface free energy estimated by Zachmann et al. originates from entropic reduction due to surface crowding may be smaller under strain as the entropic reduction is partially taken by stretch. On the other hand, TIIm/ΔTII ≈ 7.5 may be underestimated as the supercooling is expected to be smaller than that at quiescent condition if the equivalent melting temperature Tm(αi) under strain is used as reference point. Nevertheless, though the above estimation is still rough, the general trend is still valid. The important insight from the above estimation is that the formation of shish is essentially determined by the critical nucleus thickness l* itself rather than by either strain or temperature alone. The ratio l*III/l*II of regions III and II is evaluated in the same way. The enthalpy of fusion for hexagonal crystal of PE varies between 40 and 80 J/g, smaller than that of orthorhombic phase (290 J/g).86,93 Here we take ΔHIII ≈ (1/5) ΔHII. Hexagonal phase of PE is a conformational disorder crystal, where chain mobility inside crystal is high. The entropic reduction due to chain crowding on end surface of nuclei is estimated to be much weaker than that of orthorhombic phase. Following the estimation of Sirota and Milner, the specific free energy of end surface of hexagonal phase is less than that of the folded chain surface of II 94,95 orthorhombic phase. Here we assume σIII e ≈ (1/5)σe . From Table 2, the average slopes ratio between regions III and III = 0.42. TIII II is estimated to be AIII/II av m /ΔT ≈ 7.5 is also used as above. With eq 9 we estimate the ratio l*III/l*II ≈ 6.9, giving rise to the thickness of critical nuclei l*III about the order of 500 nm for HSC, which also corresponds to shish structure. Evidently, under current near-equilibrium condition the formation of shish can be simply explained with classical nucleation theory provided flow-induced free energy changes of the initial melt and the final state are incorporated in. The extrapolated temperature Tc(1) does not vary with samples with different cross-link densities in regions with crystalline final state also suggests that classical nucleation mechanism determines the morphology and structure of nucleus. In regions I, II, and III, the final states are either orthorhombic or hexagonal crystals, i.e., fixed structure with specific free energy minimum in each region. In this case, crystal itself determines where and how the structure evolves during FIC. We emphasize this point because the roles of strain 6821

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equilibrium thermodynamics seem to be valid and classical nucleation theory works rather well with external work treated as perturbation. Both the extrapolation Tc(1) and the critical nucleus thickness l* at different regions supports that crystal itself determines phase transition kinetics as well as the morphology and structure of nuclei. However, under fast deformation FIC should be treated as a system under far-fromequilibrium condition, where classical nucleation theory may not apply anymore. Under fast deformation, chain conformation does not have sufficient time to re-equilibrate and free energy may not distribute homogeneous in one chain, where the classical statistics of polymer chain is not applicable. For this concern, FIC under far-from-equilibrium condition is still a long way to go.

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CONCLUSION Current work is aiming to investigate extensional-induced crystallization in strain-temperature space under a nearequilibrium system. On the basis of in situ SAXS and WAXD measurements during slow stretch of cross-linked PE network at different temperatures, several interesting findings are extracted. First, four regions are defined in strain-temperature space, namely “orthorhombic lamellar crystal” (OLC), “orthorhombic shish crystal” (OSC), “hexagonal shish crystal” (HSC), and “oriented shish precursor” (OSP). With a modified SNM (mSNM), the estimated critical nucleus thickness l* of OLC, OSC, and HSC nuclei are in orders of 10, 100, and 500 nm, respectively. Second, εi for nucleation follows a one-to-one correlation with temperature, which indicates strain-temperature equivalence holds for strain-induced nucleation. Third, the three different samples with different cross-link densities give almost identical Tc(1) for OLC, OSC, and HSC regions, respectively, while in OSP region Tc(1) varies with different samples. These results support that (i) l* determines the transition from lamellar to shish nuclei, (ii) CST even in segment level is not a necessary condition for shish formation, (iii) and that the noncrystalline precursor is a preordered state rather than a thermodynamic phase. Current work suggests that classical nucleation theory is still valid for FIC under nearequilibrium condition provided that the free energy changes of initial melt and final states induced by flow are taken into account.

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AUTHOR INFORMATION

Corresponding Author

*(L.L.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank Prof. Xuewu Ge’s group (USTC, Hefei) for their kindly help on the sample irradiation. This work is supported by the National Natural Science Foundation of China (51325301, 51033004, 51120135002, 51227801), 973 program of MOST (2010CB934504), “the Fundamental Research Funds for the Central Universities” and the Project 2013BB05 supported by NPL, CAEP. The experiment is partially carried out in National Synchrotron Radiation Lab (NSRL) and Shanghai Synchrotron Radiation Facility (SSRF). 6822

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