A Generalized Avrami Equation for Crystallization Kinetics of Polymers

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A Generalized Avrami Equation for Crystallization Kinetics of Polymers with Concomitant Double Crystallization Processes Rui-Yang Wang, Shu-Fen Zou, Bai-Yu Jiang, Bin Fan, MengFei Hou, Biao Zuo, Xin Ping Wang, Jun-Ting Xu, and Zhiqiang Fan Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b01016 • Publication Date (Web): 02 Oct 2017 Downloaded from http://pubs.acs.org on October 3, 2017

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Crystal Growth & Design

A Generalized Avrami Equation for Crystallization Kinetics of Polymers with Concomitant Double Crystallization Processes

Rui-Yang Wang,† Shu-Fen Zou,† Bai-Yu Jiang,† Bin Fan,† Meng-Fei Hou,† Biao Zuo,‡ Xin-Ping Wang,‡ Jun-Ting Xu,*,† and Zhi-Qiang Fan† †

MOE Key Laboratory of Macromolecular Synthesis and Functionalization, Department of Polymer Science & Engineering, Zhejiang University, Hangzhou 310027, China ‡ Key Laboratory of Advanced Textile Materials and Manufacturing Technology of the Education Ministry, Department of Chemistry, Zhejiang Sci-Tech University, Hangzhou 310018, China

*Corresponding author. E-mail: [email protected]

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ACS Paragon Plus Environment


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ABSTRACT:

Concomitant

double

crystallization

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processes

widely

occur

in

semicrystalline polymers and their blends because of structural and morphological heterogeneity. In this work, a generalized kinetics model was established for the isothermal crystallization in the presence of concomitant double crystallization processes, based on the Avrami theory for single crystallization process. The different situations of the double crystallization processes, including independent, competitive and composite types, are considered in the model by introducing the concept of "domains". This model was applied to crystallization of triblock terpolymer with morphological defects, polymorphic poly(vinylidene fluoride)/nanoclay nanocomposite, isotactic polypropylene with a broad isotacticity distribution and poly(ethylene terephthalate) ultra-thin film. It is found that, besides the information of crystallization kinetics, other information such as morphological defect, nucleation density, co-crystallization and structure of ultra-thin film can also be provided by this model.

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INTRODUCTION Double or multiple crystallization processes frequently occur in ceramics, alloys, colloidal particles, and polymers.1-6 Polymers provide excellent examples to study the concomitant crystallization behavior because of the existence of different nucleation mechanisms, polymorphism, inhomogeneous spatial environment, and non-uniform chain structure.7-20 For instance, fractional crystallization upon cooling from melt can be observed when the crystalline polymers are dispersed in domains with huge differences in size and thus are nucleated via different mechanisms.21-27 Polymorphism means that more than one crystalline form is formed in the solid state of the same substance, which can happen in many polymers, including polypeptides, isotactic polypropylene (iPP), polyesters, syndiotactic polystyrene, and poly(vinylidene fluoride) (PVDF).28-34 Another example is that, when the polymer chain structure (such as comonomer and tacticity) is not uniform, the polymers with different chain structures may crystallize at the same temperature in a similar time scale, forming crystals with different lamellar thicknesses, melting temperatures and fusion enthalpies.35-36 Moreover, stepwise crystallization behavior was observed in the ultra-thin films of poly(ethylene terephthalate) (PET), resulting from the different glass transition relaxations in the bulk and surface layers.37 The overall kinetics for single crystallization process in polymers, which is controlled by nucleation and growth, can be well described by Avrami equation:38

1− xc (t ) = exp(−kt n )

(1)

where n is the Avrami exponent, k is the overall crystallization rate constant, xc is the 3



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relative crystallinity, and t is crystallization time. Attempts have been made to derive the kinetic equation for the double crystallization processes based on Equation (1). For example, Velisaris and Seferis developed two different equations known as "parallel" and "series" models (Equations (2) and (3)):39

xc (t) = w1 (1− exp(−k1t n1 )) + (1− w1)(1− exp(−k2t n2 ))

(2)

1/ xc (t) = w1 / (1− exp(−k1t n1 )) + (1− w1 ) / (1− exp(−k2t n2 ))

(3)

Particularly, Equation (2) is widely applied to describe the kinetics of double crystallization processes. However, there are some defects in these two models. Firstly, two crystallization processes are assumed to start at the same time in both models, which may not be the real situation. Secondly, two crystallization processes are believed to be independent and not correlated in Equation (2). This is true only when two crystallization processes occur in separated domains without any interference, i.e. the two crystallization processes are independent. However, Equation (2) is not applicable when two kinds of crystals are competitively transformed from shared amorphous substance. Sometimes the independent and competitive double crystallization processes may even co-exist, which is the composite type of double crystallization processes. Thirdly, the physical meaning of Equation (3) is not very explicit. Considering that structural heterogeneity and morphological heterogeneity are the characteristics of polymers and their blends, double crystallization processes are frequently observed. Therefore, construction of a generalized kinetics model that can deal with various situations of concomitant double crystallization processes is necessary. In this model the concept of "domains", in which different 4


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crystallization processes occur, is introduced. The two processes are independent when they happen in different domains, while are competitive in the same domain. The independent and competitive crystallization types may also co-exist in a realistic crystallization system, which is defined as composite type. Transformation among different crystallization types is allowed when the domain boundary is broken. EXPERIMENTAL SECTION Materials. Styrene and n-butyl acrylate (nBA) were washed with 10 wt% sodium hydroxide and deionized water, then dried with calcium hydride overnight and distilled under vacuum. ε-Caprolactone (ε-CL) was distilled over calcium hydride under reduced pressure. Toluene was refluxed with sodium flakes and freshly distilled. Anisole was distilled over magnesium sulfate under vacuum. Copper bromide and copper chloride were washed with acetic acid, acetone, and diethyl ether for several times and dried under vacuum. Stannous 2-ethylhexanoate (Sn(Oct)2) (99%) was purchased from Acros and distilled under reduced pressure. The other commercially available chemicals were used as-received. The synthesis of macro-initiator poly(ε-caprolactone) with a Br end group (PCL-Br) was reported in our previous work.40 PVDF (Solef 1015) used in this study was purchased from Solvay (Brussels, Belgium) in powdery form with a weight-average molecular weight (Mw) of 970 000 g/mol. The organically modified montmorillonite (OMMT) nanoclay, Nanomer 1.44P was purchased from Nonocor Inc. (USA). The OMMT was prepared via ion-exchanging Na+MMT with dimethyl, dehydrogenated tallow by the supplier. Metallic magnesium (200 mesh) and idodine (I2) were used as 5



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received. 1-Chlorobutane was purchased from Adamas and dried with 4 Å molecular sieves. n-Heptane and n-octane were purified by refluxing over sodium for 6 h and distilled before use. TiCl4 (Adamas) was used as received. Triethyl aluminum (AlEt3) (Albemarle) and 2-thiophenecarbonyl chloride (TPCC, Alfa Aesar) were diluted with n-heptane to 2 mol/L before use. Propylene gas (polymerization grade, SINOPEC) used in the polymerizations was passed through columns of 4 Å molecular sieves and PEE deoxygenate catalyst (Dalian Samat Chemicals) to remove moisture and oxygen. The MgCl2-supported TiCl4 Ziegler-Natta catalyst was prepared according to the procedure reported in literature.41 Preparation of the Samples. The poly(ε-caprolactone)-b-polystyrene-b-poly(n-butyl acrylate) (PCL-b-PS-b-PnBA) triblock terpolymer was prepared by ring-opening polymerization of ε-caprolactone,42 followed by atom transfer radical polymerization of styrene and n-butyl acrylate. The synthetic route is shown in Figure S1. The PCL-b-PS-b-PnBA triblock terpolymer has a molecular weight of 75.5×103 g/mol and a polydispersity (Mw/Mn) of 1.14 (Table S1). The weight percentages of PCL, PS and PnBA are 25.1%, 58.4% and 16.5%, respectively. The triblock terpolymer is denoted as C170S421B96, in which the subscripts stand for the polymerization degrees of the corresponding blocks. The film of PVDF/OMMT nanocomposite was prepared by mixing the solutions of PVDF and OMMT in N,N-dimethyl formamide (DMF). The weight percentage of OMMT in the PVDF/OMMT nanocomposite is 0.06%. The propylene polymerization was carried out in a 100 mL Schlenck flask at 1 atm and 60 °C with 6


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n-heptane and AlEt3 used as solvent and co-catalyst, respectively. After 10 min, the polymerization was terminated by sequential addition of 2-thiophenecarbonyl chloride and acidic ethanol.

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C-NMR spectrum shows that the average content of the pentad

[mmmm] in the obtained iPP is 59 mol% (Figure S16). The details for preparation and characterization of C170S421B96 triblock terpolymer, PVDF/OMMT nanocomposite and iPP are given in Supporting Information. Preparation of the PET ultra-thin films was reported in our previous publication.37 Isothermal Crystallization Kinetics and Data Fitting. The thermal behavior of the samples was characterized using differential scanning calorimetry (DSC) on a TA Q200 instrument. About 3~5 mg of the sample was sealed in an aluminum pan. The samples were first heated from 40 °C to a high temperature (typically 40 °C above the melting temperature), kept for 5 min to eliminate the thermal history, and then cooled to a low temperature (typically 40 °C below the crystallization temperature) to obtain the nonisothermal DSC curves. The heating and cooling rates are 10 °C/min. The crystallization temperatures are determined from the nonisothermal experiments. For isothermal crystallization kinetics, the samples were first heated to melt and kept for 5 min to eliminate the thermal history, and then rapidly cooled to pre-set isothermal crystallization temperature (Tc) and the curves of heat flow versus crystallization time were recorded. We set the zero time moment (t =0) when temperature approaches the pre-set Tc and is barely changed. The heat flow is normalized before fitting with the proposed model. For PET ultra-thin films, the ellipsometric angles (ψ and ∆) were 7



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collected to study the isothermal crystallization. The film thickness changes corresponding to the crystallinity were recorded in our previous work.37 Here, we use the derivative form of the crystallinity (dxc/dt) to reduce the fitting errors (Figure S22). A program written in MATLAB software was used to fit the experiment curves.

RESULTS Model construction. As is well known, the relative crystallinity xc in Equation (1) can be expressed as: t

∞

0

0

xc (t ) = ∫ H (t )dt / ∫ H (t )dt

(4)

where H(t) is the heat flow measured by DSC at crystallization time t. Similarly, the relative crystallinity xci in "crystallization process i" (CPi) (i=1 or 2) can be defined as follows: t

∞

0

0

xci (t) = ∫ Hi (t)dt / ∫ H (t)dt

(5)

where Hi(t) is the heat flow of the CPi. Note that the Hi(t) and xci(t) can be added directly. Since the heat flow curve is more sensitive to the parameters during fitting, the derivative form of the generalized kinetic equation for double crystallization processes is given as follows (see Supporting Information for detailed deduction):

(w1 + w12 )n1k1t n1 −1 exp(−k1t n1 ) (t ≤ t0 ) ∞  H (t ) / ∫ H (t )dt = w1n1k1t n1 −1 exp(−k1t n1 ) + w2 n2 k2 (t − t0 )n2 −1 exp(−k2 (t − t0 )n2 ) 0  n1 −1 n2 −1 n1 n2 (t > t0 ) +w12 (n1k1t + n2 k2 (t − t0 ) ) exp(−k1t − k2 (t − t0 ) )

(6)

The parameter t0 is the time difference between the starting points of CP1 and CP2. The 8


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parameters w1 and w2 stand for the weight fractions of the individual amorphous phases with occurrence of sole CPi and w12 represents the weight fraction of the amorphous phase shared by both processes. In this model, the two crystallization processes can occur either in separated individual amorphous phases ("independent type" in Figure 1b) or in a shared amorphous phase ("competitive type" in Figure 1c). The independent and competitive types of crystallization can also take place simultaneously ("composite type" in Figure 1d). The first and second terms in Equation (6) (t > t0) reflect independent CP1 and CP2, respectively. The third term in Equation (6) for t > t0, which is related to the competitive CP1/CP2 crystallization, can be transformed into following expression:

w12n1k1t n1 −1exp(−k1t n1 ) exp(−k2 (t − t0 )n2 ) + w12n2k2 (t − t0 )n2 −1 exp(−k2 (t − t0 )n2 ) exp(−k1t n1 ) (7) The first and second terms in the above expression correspond to CP1 and CP2 in the competitive mode, respectively. One can see that the expression of CP1 in the competitive mode is different from that in the independent mode. Therefore, consideration of the competitive type of double crystallization processes is the biggest difference between this model and other models. From the viewpoint of kinetic condition, the different expressions of the heat flow in Equation (6) for these two types of crystallization only arise from the difference in the continuity of the amorphous phase. The amorphous phase may be divided into different domains, inside which the substance is continuous. To make this discussion easier but still without loss of generality, we force t0=0 in in Equation (6), and get its integration form (crystallinity): 9



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xc (t) = w1 (1− exp(−k1t n1 )) + w2 (1− exp(−k2t n2 )) + w12 (1− exp(−k1t n1 − k2t n2 )

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(8)

It is easy to be aware that the kinetic equations for the independent type (the first two terms of Equation (8)) and competitive type (the third term of Equation (8)) are different, which originates from the different boundary conditions when solving the mathematics. Two independent crystallization processes happen in their individual domains (Figure 1b), while there only exists one domain for competitive double crystallization processes (Figure 1c). Inside the domain, all of the substance is continuous. Among the domains there exist boundaries and the continuous condition is broken. The appearance of domains is accompanied by the interruption of continuous condition and occurrence of boundaries, which determines the final crystallinity form when solving the kinetic equation (see Supporting Information). Interestingly, if we force n1=n2=n, and w1=w2=0, the Equation (8) transforms into the following form:

xc (t) = 1− exp((−k1 − k2 )t n )

(9)

Equation (9) is similar to the Avrami equation for single crystallization process. However, if we force n1=n2=n, and w12=0, we get a different form:

xc (t) = w1(1− exp(−k1t n )) + w2 (1− exp(−k2t n ))

(10)

Equation (10) is quite different from the Avrami equation. This is a natural result of the domains, because there is only one domain in the single crystallization process and the competitive type of double crystallization processes, whereas there exist two domains for the independent type. One may ask why the continuous condition is broken at the domain boundaries or 10


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why different domains are formed in the independent type. In fact the boundaries can be viewed as energy barriers. The domains are not produced arbitrarily but determined by energy barriers, which may originate from a variety of factors, including spatial separation, structural dissimilarity, crystal thermal stability et al. In the independent type, the different crystallization processes occur in their individual domains, thus the corresponding amorphous phase can only be transformed via a single crystallization process. By contrast, in the competitive type, the amorphous phase can be reached by both crystallization processes freely in a single domain. However, the transition between different crystallization types is not forbidden. When the energy barrier is overcome, the boundaries between the domains are broken. Since the transition among different crystallization type is related to the change of domains, this model can provide more details than the crystallization kinetics parameters when fitting the experimental data. The overall crystallization enthalpy of CPi, ∆Hi, can be obtained by integration of the fitting data, which is the contribution of CPi in both independent and competitive crystallization types (Equations (S28), (S29) in Supporting Information). The parameter r reflecting the fraction of CP1 in these two crystallization processes is defined as:

r = ∆H1 / (∆H1 +∆H2 )

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(11)


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Figure 1. Schematic descriptions for the single crystallization process (a), independent type of double crystallization processes (b), competitive type of double crystallization processes (c), and composite type of double crystallization processes (d). The solid blue circles and red pentagrams represent the nuclei of CP1 and the CP2, respectively. Different domains are separated by the black solid lines. The independent, competitive and composite types of double crystallization processes have two, one and three domains, respectively. The parameters w1, w2 and w12 are the weight fractions of the domains, in which independent CP1, CP2 and competitive CP1/CP2 take place, respectively (w1 + w2 + w12 =1). 12


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Application to Bi-phased C170S421B96 Triblock Terpolymer with Morphological Defects. We first test this model in a PCL-b-PS-b-PnBA triblock terpolymer, named as C170S421B96, with cylindrical crystalline PCL microdomains. Among these three blocks, the C and S blocks are the most incompatible and the B block is compatible with both S and C blocks. Since the B block is located at the chain end, CSB terpolymers form frustrated bi-phase structures with more morphological defects, as compared with CBS ones.40 It is found that the higher order scattering peaks are absent in the small angle X-ray scattering (SAXS) profile of C170S421B96, while the higher order scattering peaks can be clearly seen in the SAXS profile of C170B95S409 (Figure 2a). Distorted and dislocated hexagonal arrangements of PCL cylindrical microdomains can be observed in the atomic force microscopy (AFM) phase image of C170S421B96 (Figure 2c). The SAXS and AFM results confirm that the microphase structure in C170S421B96 is less ordered and there exist more morphological defects. Similar phenomena were observed in other triblock terpolymers.43-45 Figure 3 shows the representative experimental and fitting heat flow curves and relative crystallinity curves of C170S421B96 isothermally crystallized at Tc = 7 °C. As one can see, C170S421B96 exhibits two overlapped exothermic peaks in the heat flow curve upon isothermal crystallization, showing the existence of double crystallization processes. If we use the Avrami equation for single crystallization process to fit the heat flow curves of C170S421B96, upturning can be observed at the late stage of crystallization in the Avrami plots (Figure S5), which is different from the frequently observed leveling-off caused by secondary crystallization. This shows that the Avrami 13



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equation for single crystallization process cannot describe the crystallization kinetics of C170S421B96. When Equation (6) for double crystallization processes is used to fit the experimental data, the goodness of fit (R2) is 0.9958 and the mean square error (MSE) is 1.81×10-6. The large R2 and small MSE show that crystallization kinetics of C170S421B96 can be well described using Equation (6) for double crystallization processes. Similar results are obtained for C170S421B96 at other crystallization temperatures (Figure S6, S7 and Table S3).

Figure 2. (a) SAXS profiles of C170S421B96 and C170B95S409 at 110 °C. (b) AFM phase image of C170S421B96 prepared by spin-coating from 10.0 mg/mL toluene solutions. The red, green and blue frames in Figure 2(b) indicate normally coordinated hexagons, distorted hexagons and dislocated hexagons, respectively.

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Figure 3. Experimental and fitting curves of heat flow (a) and relative crystallinity (b) of C170S421B96 isothermally crystallized at Tc = 7 °C. experimental data; CP2;



CP1;



 fitting curve. Some of the experimental points are skipped to draw the picture.

The value of R2 is 0.9958 and the MSE is 1.81×10-6 for the fitting of heat flow curve in Figure 3a.

Application to Polymorphic PVDF/OMMT Nanocomposite. The kinetics model is also tested in polymorphic PVDF. A small amount of organically modified montmorillonite (OMMT) nanoclay is added into PVDF, which usually forms non-polar

α crystals under common crystallization conditions. It is reported that, OMMT may induce conformational change of nearby PVDF chains, which is favorable to the formation of polar β and γ phases.32,46 The weight percentage of OMMT in the PVDF/OMMT nanocomposite is 0.06%. The results of wide angle X-ray scattering (WAXS), Fourier-transform infrared spectroscopy (FT-IR), polarized optical microscopy (POM), and DSC show that both α and γ crystals are formed in the PVDF/OMMT nanocomposite after isothermal crystallization (Figure 4, Figure S9 and S10), confirming 15



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the occurrence of double crystallization processes, although no obvious double peaks are observed in the heat flow curves upon isothermal crystallization. The representative heat flow curve and evolution of relative crystallinity of the PVDF/OMMT nanocomposite at Tc=156 °C are presented in Figure 5. The results at other Tcs are shown in Figure S12, S13 and Table S5. Again large values of R2 and small values of MSE are obtained when fitting the heat flow curves using Equation (6), showing the applicability of double crystallization processes to the crystallization kinetics of PVDF/OMMT nanocomposite.

Figure 4. FTIR absorbance spectrum (a) and WAXS profile (b) of PVDF/OMMT nanocomposite after isothermal crystallization at Tc = 153 °C.

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Figure 5. Experimental and fitting curves of heat flow (a) and relative crystallinity (b) of PVDF/OMMT nanocomposite isothermally crystallized at Tc = 156 °C. experimental data;  CP1;  CP2;  fitting curve. Some of the experimental points are skipped to draw the picture. The values of R2 and MSE are 0.9994 and 3.49×10-6, respectively, for the fitting of heat flow curve in Figure 5(a).

Application to iPP with a Broad Isotacticity Distribution. Due to the nature of plural active sites or alteration of polymerization condition during reaction, the obtained polymers usually exhibit a heterogeneous distribution of chain structure. Herein we synthesized an iPP using a MgCl2-supported TiCl4 Ziegler-Natta catalyst containing multiple types of active centers.47-49 The prepared iPP can be fractionated into three fractions: the fraction soluble in n-octane at room temperature (C8-SOL), the fraction soluble in boiling n-heptane (C7-SOL) and the fraction insoluble in boiling n-heptane (C7-INSOL) (Table S6).

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C-nuclear magnetic resonance (NMR) spectra reveal that the

isotaciticty of C8-SOL, C7-SOL and C7-INSOL fractions is low, moderate and high, respectively (Figure 6a). DSC result shows that the C8-SOL fraction is un-crystallizable, 17



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but the C7-SOL and C7-INSOL fractions are crystallizable (Figure 6b). However, the melting temperature of C7-INSOL is higher than that of C7-SOL, and multiple melting peaks are observed for the whole sample iPP (Figure S15). The

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C-NMR and DSC

characterizations confirm that the prepared iPP is a mixture of fractions with different crystallizability, thus double crystallization processes may be applicable to isothermal crystallization of iPP. However, WAXS shows that only α crystals are formed after isothermal crystallization (Figure S17), indicating that the double crystallization processes are not caused by polymorphism. The representative heat flow curves and the evloution of relative crystallinity for iPP at Tc=128 °C are presented in Figure 7, and the results at other Tcs are given in Figure S20, S21 and Table S8. As we can see from Figure 7, the fitting curves using Equation (6) agree well with the experimental data.

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Figure 6. (a) 13C-NMR spectra of C7-INSOL, C7-SOL, and C8-SOL fractions with high, moderate, and low isotacticity. The isotacticity is characterized by the pentad [mmmm] content. (b) DSC heating curves of C7-INSOL, C7-SOL, and C8-SOL fractions after isothermal crystallization at 132 °C. The heating rate is 10 °C/min.

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Figure 7. Experimental and fitting curves of heat flow (a) and relative crystallinity (b) of iPP isothermally crystallized at Tc = 128 °C. experimental data;  CP1;  CP2;  fitting curve. Some of the experimental points are skipped to draw the picture. The values of R2 and MSE are 0.9996 and 6.15×10-7, respectively, for the fitting of heat flow curve in Figure 7(a).

Application to PET Ultra-thin Films. Equation (6) can also be applied to crystallization of PET ultra-thin films on SiOx-Si surface. We observed stepwise change of film thickness upon isothermal crystallization, which corresponds to double crystallization processes originating from the different Tgs of the surface and bulk layers.37 Previously we used two independent crystallization processes to fit the data.37 In this work the crystallization kinetics data of PET ultra-thin films were re-fitted with our generalized model and the fitting results are shown in Figure 5a and Table S9. The experimental and fitting curves for dxc/dt and evolution of relative crystallinity for PET ultra-thin film with a thickness of 33 nm are presented in Figure 8, and the results for PET ultra-thin films with different thicknesses are shown in Figure S22, S23 and Table 20


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S9. Similarly, the fitting results are pretty good, as indicated by the large values of R2 and small values of MSE.

Figure 8. Experimental and fitting curves for dxc/dt (a) and relative crystallinity (b) for PET ultra-thin film with a thickness of 33 nm and Tc=98 °C. experimental data;



CP1;  CP2;  fitting curve. Some of the experimental points are skipped to draw the picture. The values of R2 and MSE are 0.9999 and 6.82×10-9, respectively, for the fitting of dxc/dt curve in Figure 8(a)

DISCUSSION Above results reveal that Equation (6) for double crystallization processes can be applicable to describe the crystallization kinetics of different polymer systems. Furthermore, the crystallization mechanism can be inferred from the fitting results. The values of wi and r at various Tcs for C170S421B96 obtained by fitting with Equation (6) are shown in Figure 9a. One can see that, at Tc < 6 °C the value of w2 is nearly 0, implying that the independent CP2 is absent at low Tcs. When Tc is between 6 °C and 8 °C, there is 21



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an abrupt decrease for w12, but a rapid increase for w1, and the value of w2 also becomes larger. At Tc > 8 °C, the values of wi are almost invariant. It is found that the Avrami exponents for CP1 (n1) are around 1.0 and the Avrami exponents for CP2 (n2) range from 2.50 to 2.15 (Table S3). Based on Avrami theory, the Avrami exponent n is determined by nucleation mechanism and growth dimension of crystals. For heterogeneous nucleation, the Avrami exponent is equal to the growth dimension of crystals, while for sporadic homogeneous nucleation, the Avrami exponent is equal to the growth dimension of crystals plus one. Since homogeneous nucleation usually occurs under a huge supercooling,50-52 we believe that heterogeneous nucleation happens in both crystallization processes at the Tcs conducted for C170S421B96 in the present work. The formation of large-sized crystals supports such a conclusion (Figure S8).53-55 Therefore, n1 ≈ 1.0 corresponds to one-dimensional growth of crystals, and CP1 is ascribed to the crystallization in the poorly connected cylindrical PCL microdomains with fewer structure defects (Figure 9b). By contrast, CP2 with n2 = 2.0 ∼ 3.0 is attributed to the crystallization in the highly connected PCL microdomains due to more structure defects. Because of the connectivity of the PCL microdomains, the crystal growth dimension in CP2 is larger, leading to larger values of n. In the moderately connected PCL microdomains, both CP1 and CP2 can competitively take place. Since the values of r are quite large (~ 0.85) in C170S421B96, the fraction of highly connected microdomains is minority. In C170S421B96, the domains for different types of crystallization are simply spatially separated. One can see from Figure 9a that, only independent CP1 and 22


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competitive CP1/CP2 happened at Tc < 6 °C. This is because both CP1 and CP2 proceed quickly at lower Tcs, thus CP1 can compete with CP2 in the moderately and highly connected microdomains, while CP2 cannot occur in the poorly connected microdomains. When Tc is higher than 8 °C, the crystallization rates of both CP1 and CP2 are slow and CP2 starts much behind CP1 (as revealed by the large t0 in Table S3), thus CP1 and CP2 are well-separated and take place independently. In this case, independent CP1 is the main crystallization process in the moderately connected microdomains, evidenced by the sharp increase of w1 and abrupt decrease of w12 when Tc exceeds 6 °C.

Figure 9. (a) Values of wi and r as a function of Tc for C170S421B96. Crystallization types are denoted by black Arabic numerals with the plus sign and different crystallization types are separated by the vertical dash lines. (b) Schematic morphological defects in bi-phased C170S421B96.

For PVDF/OMMT nanocomposites, the double crystallization processes arise from polymorphism. Due to the nucleation effect of OMMT, CP1 with a faster crystallization rate corresponds to the formation of γ crystals, and CP2 leads to α crystals. This 23



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assignment is also in accordance with the larger r value and POM result (Figure S10). Here, the energy barrier for the domains of different types of crystallization may be caused by the spatial distance and crystal thermal stability. The simulation results for PVDF/OMMT are shown in Figure 10a and Table S5. One can see that, when Tc is lower than 156 °C, only independent CP1 and competitive CP1/CP2 are observed, while independent CP2 is absent. By contrast, independent CP1 also appears at Tc > 157 °C, but w12 becomes quite small. The variations of crystallization behavior in PVDF/OMMT nanocomposite with Tc can be interpreted in terms of nucleation density. At lower Tcs, both the densities of α and γ crystal nuclei are high, thus the α and γ crystal nuclei are very close and the fraction of the interfacial region shared by two types of nuclei that can be transformed into both α and γ crystals is quite large (Figure 10b). Under such a situation, due to the strong nucleation effect of OMMT and accordingly the rapid formation of the γ crystals, the independent CP1 and competitive CP1/CP2 happen but independent CP2 cannot occur. By contrast, at higher Tcs, the density of α crystal nuclei becomes smaller and the crystallization rates of both α and γ crystals are slower, the independent CP2 can occur as well.

24


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Figure 10. (a) Values of wi and r for PVDF/OMMT nanocomposite with an OMMT weight percentage of 0.06% at different Tcs. Crystallization types are denoted by black Arabic numerals with the plus sign and different crystallization types are separated by the vertical dashed lines. (b) Scheme for crystallization mechanism.

The simulation results for isothermal crystallization kinetics of iPP are presented in Figure 11a and Table S8. It is found that independent CP1 and CP2 and competitive CP1/CP2 take place at lower Tcs, but only independent CP1 and CP2 occur at higher Tcs. This result agrees well with the number of melting peak (Figure 11b). Here, the structural dissimilarity leads to occurrence of different types of crystallization, though the amorphous phases are not physically divided into different domains in space. Usually the polymer chains with different chain structures (such as isotacticity) crystallize separately,56 leading to independent CP1 of high isotacticity polymer chains (C7-INSOL fraction) and CP2 of intermediate isotacticity polymer chains (C7-SOL fraction). The occurrence of competitive CP1/CP2 implies that the PP chains with moderate and high isotacticity can co-crystallize to form co-crystals (Figure 11c). The co-crystallization 25



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phenomenon is frequently observed in the blends of ethylene copolymers with different comonomer contents.57 Note that there exists a broad isotacticity distribution for the chain structure of iPP. Since the stability of the formed crystals changes with Tc, the distribution of polymer chains with different isotacticity in different crystals changes accordingly to be commensurate with the stability of crystals. With increasing Tc, more iPP chains of high isotacticity migrate into the co-crystals, but some chains with moderate isotacticity also move out from the co-crystals and enter into the crystals formed by independent CP2. This is the reason why we observe that the highest melting peak diminishes gradually and the lowest melting peak becomes stronger as Tc increases (Figure 11b). When Tc is high enough, there is a large difference in the stability between the crystals formed by the polymer chains of moderate and high isotacticity, thus formation of co-crystals is forbidden and only independent CP1 and CP2 happen.

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Figure 11. (a) Values of wi and r for iPP as a function of Tc. Crystallization types are denoted by black Arabic numerals with the plus sign and different crystallization types are separated by the vertical dash lines. (b) DSC melting traces of iPP after isothermal crystallization at various Tcs. The dash lines are drawn to show the change of melting temperature. (c) The possible crystallization mechanism.

For polymer ultra-thin films, the structure and dynamics at interfaces and bulk may be different. In literature, a two-layer model (surface and bulk layers) is usually to describe the structure of polymer ultra-thin films having no strong interaction with the substrate.58-59 The polymer chains in the surface layer exhibit a lower Tg and thus stronger mobility.60 By contrast, polymer chains in the bulk layer usually have a normal Tg, which is a little higher than that of the surface layer. Figure 12a shows the values of wi and r for PET ultra-thin films with different thicknesses. Here CP1 and CP2 represent the fast crystallization in the surface layer due to the stronger mobility and the slower crystallization in the bulk layer with weaker mobility, respectively. The energy barrier for division of the amorphous phases into different domains originates from the different 27



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chain mobility. One can see that, when the film thickness (l) is about 10 nm, only independent CP1 occurs, which means that there exist only surface layer in the thin film. At an intermediate film thickness (10 nm < l < ~140 nm), both independent CP1 and CP2 and competitive CP1/CP2 are observed. We ascribe the competitive CP1/CP2 process to the crystallization in the transition layer between the surface and bulk layer. When the film thickness is very large (l >140 nm), the value of w2 approaches 1.0 and the values of w1 and w12 are quite small. This implies that the thicknesses of the surface and transition layers are negligible, as compared with that of bulk layer, thus independent CP2 is dominant. Our work shows that, the thin film structure does not switch abruptly from the surface layer into bulk layer, and there exists a transition layer between the surface and bulk layers (Figure 12b).

Figure 12. (a) Values of wi and r for PET ultra-thin films at Tc = 98 °C. Crystallization types are denoted by black Arabic numerals with the plus sign. Different crystallization types are separated by the vertical dash lines. (b) The hypothetical structure of the ultra-thin film. 28


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From Figures 9-12, we find that the occurrence of different crystallization types may change with Tc or ultra-film thickness. Competitive double crystallization processes (indicated by the values of w12) tend to occur at lower Tcs or ultra-film of intermediate thicknesses, while this type of crystallization may vanishes at elevated Tcs or become less important in thicker films. As we have pointed out, the transition of crystallization type means the overcomeing of the energy barrier between different crystallization types and accompanied with the change of the domains. Since a lower Tc is favorable to nucleation (if Tc is not too close to the glass transition temperature of polymers), the effect of energy barrier between different crystallization types is insignificant and the energy barrier can be easily overcome. This is reflected by the rapid nucleation of both CP1 and CP2. By contrast, the effect of energy barrier is dominant at higher Tcs and is difficult to overcome, thus independent CP1 and/or CP2 tend to take place.

CONCLUSION In summary, based on the nucleation-growth controlled Avrami equation, we constructed a generalized kinetics equation for different types of double crystallization processes, including independent, competitive and composite types. This equation can be applied to the crystallization of bi-phased triblock terpolymers with morphological defects, polymorphic polymers, polymers with heterogeneous chain structures, and polymer ultra-thin films. We find that the double crystallization processes may change with crystallization temperature. Particularly, this generalized model can supply not only the 29



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information about the crystallization kinetics but also additional information such as morphological defect, nucleation density, co-crystallization and structure of ultra-thin film. It is hopeful that this generalized model can also be applied to the concomitant double crystallization processes in ceramics, alloys, and colloidal particles.

ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: ….. The deduction of the isothermal crystallization kinetics equation, details for synthesis and characterizations of the samples, raw data and fitting results for crystallization kinetics in different systems.

AUTHOR INFORMATION Corresponding Author *(J.-T.X.) E-mail: [email protected]. Telephone: +86-571-87953164. Fax: +86-571-87952400. ORCID Jun-Ting Xu: 0000-0002-7788-9026 Notes The authors declare no competing financial interest. 30


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ACKNOWLEDGMENTS This work was supported by National Natural Science Foundation of China (21574116 and 21774111). The authors would also like to thank beamline BL16B1 at SSRF for providing the beam time.

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A Generalized Avrami Equation for Crystallization Kinetics of Polymers with Concomitant Double Crystallization Processes

Rui-Yang Wang,† Shu-Fen Zou,† Bai-Yu Jiang,† Bin Fan,† Meng-Fei Hou,† Biao Zuo,‡ Xin-Ping Wang,‡ Jun-Ting Xu,*,† and Zhi-Qiang Fan†

TOC Graphic

Synopsis The kenetic equations for three types of double crystallization processes are derived and compared with that of single crystallization process. The concept of energy barrier-related "domains" is introduced, which determines the crystallization type. This model can be applied to crystallization kinetics in different polymer systems with concomitant double crystallization processes.

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A Generalized Avrami Equation for Crystallization Kinetics of Polymers

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