ARTICLE pubs.acs.org/IECR
Improved Kinetic Model of Crystallization for Isotactic Polypropylene Induced by Supercritical CO2: Introducing Pressure and Temperature Dependence into the Avrami Equation Ren-Han Zhang,† Xue-Kun Li,† Gui-Ping Cao,*,† Yun-Hai Shi,† Hong-Lai Liu,† Wei-Kang Yuan,† and George W. Roberts‡ † ‡
UNILAB, State Key Lab of Chemical Engineering, East China University of Science and Technology, Shanghai 200237, China Department of Chemical and Biomolecular Engineering, North Carolina State University, Box 7905, Raleigh, North Carolina 27695-7905, United States ABSTRACT: Supercritical carbon dioxide (SC-CO2) provides a highly tunable technique to induce changes in morphology and crystallization kinetics of various polymers. In this study, the effect of SC-CO2 treatment on the crystallinity of isotactic polypropylene (i-PP) was analyzed by differential scanning calorimetry (DSC). The Avrami equation is suitable for describing crystallization kinetics of various solids. However, the assumption that the crystallinity approaches 1.0 as the time approaches infinity is not valid for semicrystalline polymers, especially in the atmosphere of SC-CO2. An improved kinetic model for the CO2-induced crystallization of i-PP was purposed by introducing equilibrium crystallinity, as well as temperature- and pressure-dependent terms into the Avrami equation. The parameters of the crystallization kinetics model were obtained by least-squares fitting of the DSC data. The results show that the improved kinetic model provides a reasonable description of the crystallization behavior of i-PP induced by SC-CO2. The successful application of the improved kinetic model to the CO2-induced crystallization of i-PP suggests that this model may be adopted to other SC-CO2-semicrystalline polymer systems.
’ INTRODUCTION Crystallinity is one of the most important characteristics for semicrystalline polymers. In microcrystalline regions, chains of polymer are essentially held together by dipolardipolar interaction, hydrogen bonding, or van der Waals forces. The crystalline regions have a higher density than the amorphous regions and could be considered as cross-links among the amorphous regions.1 As the crystalline cross-links stiffen and toughen the polymers, by reducing the swelling in solvents, introduction of microcrystallinity changes a rubbery elastomeric polymer into a tough, flexible material on a macroscopic level. In addition, the microcrystalline regions influence the transmittance,2 resistance to degradation, and barrier properties of polymer.3 It is evident, therefore, that the polymer crystallization process is significant to both industry and academy. The traditional method for polymer crystallization is cooling the polymer melt with or without adding nucleating agent, which often causes brittleness because of the stress concentration during the deformation. Chiou,4 Mizoguchi,5 Beckman,6 and Baldwin,7 et al. studied the crystallization of polymers induced by high pressure CO2. Hirota et al.8 investigated the effect of highpressure carbon dioxide on the crystallization behavior of PLLA/ PMMA blends, which could eliminate the brittleness. Asai et al.9 studied the crystallization behavior of poly(ethylene-2,6naphthalate) under supercritical CO2 (SC-CO2). They reported that the nucleation became more dominant with decreasing the temperature of SC-CO2 treatment. Kim et al.10 found that SCCO2 could thicken crystal lamella, increase the crystallinity, and occlude end-group in crystalline regions when they studied the effect of prepolymer molecular weight on solid-state polymerization of r 2011 American Chemical Society
poly(bisphenol A carbonate) with either SC-CO2 or N2 as the sweep fluid. The interesting properties of supercritical carbon dioxide (SCCO2), including an easily accessible critical point (31.4 °C and 7.38 MPa), low toxicity, low cost, chemical inertness, easy availability, and environmentally benign nature, have led to CO2 receiving considerable attention as an alternative to traditional solvents. Supercritical fluids have been considered to be “hybrid solvents” having liquidlike densities and gaslike diffusivities.11 SC-CO2 provides a highly tunable technique to adjust the morphology and/or crystallinity of polymers in a controlled manner. Moreover, CO2 can be easily removed from the polymer matrix by depressurizing the system once the desired morphology is achieved. Because of these advantageous properties compared to conventional solvents, SC-CO2 is being applied in a wide range of polymer processes1122 including modification, deposition, foaming, grafting, and dyeing. Polymer chains are relaxed by CO2 through the interaction of Lewis acidbase or electrostatic attraction/repulsion. SCCO2 can plasticize amorphous regions of semicrystalline polymers, leading to depression in the glass transition temperature, Tg, which permits the movement of polymer chains at a lower temperature. The Avrami equation describes how solids transform from one phase to another at constant temperature, and is especially useful for describing the kinetics of crystallization.2325 Takada et al.2628 Received: August 1, 2010 Accepted: August 9, 2011 Revised: July 14, 2011 Published: August 09, 2011 10509
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Figure 1. Schematic illustration of the experimental setup for the CO2-induced crystallization of i-PP.
applied the Avrami equation to the semicrystalline polymer CO2 system to describe the isothermal crystallization of polypropylene, poly(ethylene terephthalate), and poly(L-lactide). In these papers, the parameters of the Avrami equation, n and k, were determined over a range of pressures (below 3.0 MPa), and temperatures (378.2483.2 K). They found that n and k decreased with the increment of pressure and temperature. Li et al.29 used the Avrami equation to describe the polypropylene crystallization with SC-CO2 assisted dispersion of a nucleating agent of sodium benzoate. They found that n decreased with pressure and mass percentage of nucleating agent. Yu et al.30 described polylactide crystallization with SC-CO2-assisted dispersion of talc nucleating agent using the Avrami equation. Varma-Nair et al.31 studied crystallization kinetics of polypropylene compressed by CO2 using a high pressure DSC. It is evident from those papers that the temperature and pressure of SC-CO2 had a remarkable influence on the behavior of polymer crystallization. On the other hand, in previous studies the ranges of temperature were somewhat narrow and the pressures were relatively low, some even below the supercritical region. Furthermore, previous studies did not describe the action of CO2 clearly, since the Avrami eqution was used directly as the kinetic model for CO2-induced polymer crystallization. The Avrami equation was deduced without consideration of the pressure and temperature of the CO2. Therefore, the parameters of n and k estimated in those studies can hardly be used for engineering processes. The Avrami equation has been playing an important role in describing crystallization kinetics behavior of polymers. The crystallinity in Avrami equation can be defined as follows3 ϕc, A ¼
Vt V0 V∞ V0
ð1Þ
where V0 is the specific volume of a sample at t = 0, V∞ is the volume at t = ∞, and Vt is the volume at any intermediate time. So ϕc,A = 0 at t = 0, and ϕc,A = 1 at t = ∞. In the practical application, for a semicrystalline polymer, neither its initial crystallinity could be exactly zero nor can it crystallize completely when it held for an infinite time. So we will
only consider here the crystallization process of a semicrystalline polymer in the practical application. In this paper, we establish an improved kinetic model for crystallization of the i-PP-SC-CO2 system. Here, the Avrami equation is modified by introducing the Arrhenius equation directly into the Avrami equation, and then expressing the activation energy and pre-exponential factor in the Arrhenius equation in terms of pressure. These modifications are based on qualitative analysis of the effect of SC-CO2 on Gibbs free energy. Then, integral and differential methods were used to introduce the equilibrium crystallinity32 and initial crystallinity to modify the Avrami equation, which could overcome the two drawbacks of the Avrami equation. The parameters of the improved crystallization kinetics model were obtained by least-squares estimation of DSC data.
’ EXPERIMENTAL SECTION Materials. Commercially available i-PP (Y1600, weight-average molecular weight = 1.90 106 g/mol, polydisersity index = 5.1, melt flow index = 16 g/10 min at 230 °C, Tg = 130 °C, and isotacticity g96%) used in this study was presented by Sinopec Shanghai Petrochemical Ltd. Co., China. The polymer was free of additives and in pellet form. The i-PP pellets were first put into a 1-mm thick mold, and then melted in an oven at 200 °C in an atmosphere of high-purity nitrogen. The molten polymer was then pressed using a tablet compression machine under 10 MPa to eliminate air bubbles. Then, i-PP was melted again in the oven at 200 °C in high-purity nitrogen. Finally, the molten polymer and its mold were quickly cooled in liquid nitrogen. The prepared i-PP samples were sealed in zip-lock bags and stored at 18 °C, which was lower than the glass transition temperature of i-PP. The CO2, which was purchased from Shanghai Jifu Gas Ltd. Co., was used as received with the purity more than 99.9%. Apparatus and Operation. A schematic illustration of the experimental apparatus for the CO2-induced crystallization of i-PP is shown in Figure 1. The sample chamber was constructed of stainless steel and was entirely within a temperature-controlled oil bath ((0.5 K) with forced agitation. Two tube connectors 10510
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Figure 2. Crystallinity of i-PP treated by SC-CO2 for 6 h at different temperature and SC-CO2 pressure. (Initial crystallinity = 0.365.)
protruded about 50 mm from the outside of the sample chamber wall. The first connected to the buffer vessel via valve A was for gas inlet, while the second connected to a vacuum pump via valve B for gas outlet. The pressures of the sample chamber and the buffer vessel were measured by the pressure transducers (NS-I1, 0.1% F.S; purchased from Shanghai TM Automation Instruments Co., Ltd.). The pressures were recorded automatically by a computer. The i-PP samples were treated with CO2 at different pressures (921 MPa) and temperatures (323413 K) for different times (6160 h) to investigate the effects of temperature, pressure, and treating time on the CO2-induced crystallization of i-PP. Each of the experiments was operated at isothermal and isobaric conditions. After the treatment with SC-CO2, the i-PP samples were sealed in zip-lock bags and stored at 18 °C. Crystallization Analysis. The crystallization measurements were carried out with a NETZSCH SAT 409PC/PG system under nitrogen between room temperature and 250 °C. The DSC data were analyzed by using NETZSCH Proteus Analysis software. In order to ensure the reproducibility of the DSC method used in this work, a series of repeated measurements of the crystallinity of the Y1600 raw material were carried out by DSC with the heating rate of 10 °C/min, and the heating rate of 10 °C/min of DSC was adopted for all further DSC experiments.
’ RESULTS AND DISCUSSION Crystallinity. For each experiment, the crystallinity was measured at a constant temperature and pressure. The crystallinity, ϕc, was calculated from eq 233,34
ϕc ¼
ΔH ΔH0
ð2Þ
where ΔH is the enthalpy of polymer crystallization per unit mass as measured by DSC, J/g, and ΔH0 is the value of the enthalpy change between 100% crystallization and 0% crystallization. For i-PP, the value of ΔH0 is 209 J/g.33 The crystallinity of the Y1600 raw material for five repeated measurements were 0.542, 0.546, 0.535, 0.545, and 0.536, respectively. The average of crystallinity and standard deviation were 0.541 and 0.5%, respectively. By the same method, the crystallinity of the starting material with cooling operation for all of the crystallization experiments was 0.365. The crystallinity of i-PP treated by CO2 for 6 h at different temperatures and pressures is shown in Figure 2. The crystallinity
Figure 3. Variation of the crystallinity with treatment time at 50 °C, 12 MPa.
Table 1. Values of the Coefficients in Equations 3 and 4
in eq 3 in eq 4
b0
bT
bP
bt
bTP
STDEV
0.548 83 0.548 00
0.028 15 0.028 32
0.038 35 0.038 26
0.069 76 0.069 03
0.002 43
0.012 101 0.012 042
of i-PP increased with both temperature and pressure. Polymer crystallization is the process by which polymer chains undergo regularly packing, with three levels of structure: unit cells, lamellae, and spherulites. Indeed, the process is just the movement and rearrangement of polymer chains. Energy is required to cause polymer chains to move from an amorphous state into a regular state. Thus, higher temperature is more conducive for polymer chains to move and rearrange into a regular conformation. The dissolution of CO2 into i-PP can plasticize polymer, allowing its chains to relax.3034 In other words, the dissolution of CO2 into i-PP should decrease the activation energy for crystallization of the polymer. Therefore, high CO2 pressure reduces the energy required for polymer chains to rearrange and crystallize. The effect of CO2-treatment time on the crystallinity of i-PP was also studied; a typical result is shown in Figure 3. The crystallinity of i-PP rises with the increased treating time. For the purpose of determining the influences of temperature, pressure, and time on CO2-induced crystallization of i-PP, significant level analysis was applied.35,36 The independent variables (T, P, and t) were transformed into nondimensional variables with values between 1 and 1. Stepwise multiple regression equations, eqs 3 and 4, were used to clarify the relationship among pressure, temperature, and time in the crystallization kinetics model, and the regression coefficients in eqs 3 and 4 were determined by using stepwise multiple regression and are shown in Table 1. ϕc ¼ b0 þ bT xT þ bP xP þ bt xt
ð3Þ
ϕc ¼ b0 þ bT xT þ bP xP þ bt xt þ bTP xT xP
ð4Þ
The data were not adequately fitted if the interaction terms, bTtxTxt between T and t, or bPtxPxt between P and t, were put in eq 3 or 4 during the regression. The three variables, T, P, and t, affect the crystallinity by approximately the same order of magnitude. The effect of time is slightly stronger than that of pressure, and the effect of pressure is slightly stronger than that of 10511
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temperature. There is a very weak interaction between temperature and pressure,bTP, which is 1 order of magnitude smaller than the effects of T, P, and t. There is no interaction between temperature and time, or between pressure and time. The results of regression analysis demonstrate that the effects of temperature, pressure, and time are independent with each other in the crystallization kinetics model. Crystallization Kinetics Model of CO2Polymer System. The length of time required for a macroscopic sample to crystallize is of concern in many areas of polymer processing. For predicting the required time and the crystallinity at a given condition, a kinetic equation is essential. In the earlier studies,2325 the Avrami equation was used to describe the crystalline fraction as a function of time. The Avrami equation was deduced from an analysis of crystal nucleus growth for solids2325 which can be written in the following general form, known as the Avrami equation ϕc ¼ 1 expðkt n Þ
ð5Þ
where k is the rate constant, and n is the Avrami exponent. As the mechanisms and the driving forces between induced crystallization and thermal crystallization are different, the values of n and k in induced crystallization may be quite different than those in thermal crystallization. The Avrami equation has been used directly to describe polymer crystallization in SC-CO2 by determining n and k at different temperatures and pressures.2631 However, the Avrami equation itself does not directly include temperature and gas pressure. The temperature of the system and the pressure of the gas are two of the most important factors which must be considered in the kinetic model of polymer crystallization. As shown in Figure 2, the crystallinity of i-PP increased with temperature. In order to describe the effect of temperature (T), the Arrhenius equation was introduced directly into the Avrami equation by expressing the rate constant, k, as shown in eq 6 E ð6Þ kðTÞ ¼ k0 exp RT where k0 is the pre-exponential factor which describes the probability of effective regular arrangement among polymer chains during their movement. E is the apparent activation energy for the movement of polymer chains. R is the gas constant, and T is the temperature in units of Kelvin. In previous work,37,38 the driving force for nucleation was identified as the difference of Gibbs free energy, ΔG, between the liquid and the crystal. In this study, ΔG is the difference of Gibbs free energy between the amorphous and crystalline phases. Gas pressure affects various characteristics of the polymer, the free volume of polymer and the Gibbs free energy of polymer chains are increased by introducing CO2 into the amorphous regions of polymer. In the view of this analysis, the presence of CO2 decreases ΔG of the polymer system. Since the sign of ΔG is negative, the absolute value of ΔG as well as ΔG2 are increased by the action of CO2; i.e., ΔG2 is proportionate to the gas pressure, and thus, ΔG2 can be written as a function of gas pressure. Considering the equation of activation energy for homogeneous nucleation,1 the activation energy of crystallization is of the form
EðPÞ ¼
16πσ 3 1 1 ¼ a þ bP þ cP2 3 ΔG2
ð7Þ
Table 2. Estimated Parameters no.
parameter
value
1
ϕc,lm
0.5998
2
n
0.7300
3
α
23.66
4
β
0.7710
5
a
7.400 105
6
b
2.614 106
7
c
1.932 107
where the value of activation energy of crystallization is equal to the apparent activation energy for the movement of polymer chains. Because the CO2 pressure can affect the movement ability of polymer chains, it will then affect the value of k0. Here we assume that k0 can be described by eq 8 k0 ðPÞ ¼ α þ βP
ð8Þ
where a and β are constant model parameters. Thus, eq 6 can be modified by eq 9. EðPÞ kðT, PÞ ¼ k0 ðPÞ exp ð9Þ RT Owing to the semicrystalline polymers not being crystallized to a crystallinity of 1.0, they remain in an enduring nonequilibrium state, even if the polymer is held for an infinite time. In this study, there is a limit between the crystalline and amorphous regions, this limit is close to the equilibrium state, and the crystallinity reaches some “limiting” value, ϕc,lm. Thus, in the practical application, the Avrami equation can be modified straightforwardly ϕc ¼ ϕc, lm ½1 expðkðT, PÞt n Þ
ð10Þ
where ϕc,lm is the fraction of the polymer that is crystalline, once the limit has been reached, i.e., at infinite time. Differentiating eq 10, we get dðϕc ϕc, lm Þ ¼ kðT, PÞnt n 1 dt ðϕc ϕc, lm Þ By integrating eq 11 Z ϕc Z t dðϕc ϕc, lm Þ ¼ kðT, PÞn t n 1 dt 0 ϕc ð0Þ ðϕc ϕc, lm Þ
ð11Þ
ð12Þ
we obtain ϕc ¼ ϕc, lm þ ðϕc ð0Þ ϕc, lm ÞexpðkðT, PÞt n Þ
ð13Þ
Thus, the kinetics of crystallization for the polymer-CO2 system can be described by eqs 13, 9, 8, and 7. There are seven parameters in this model that must be estimated by fitting the crystallinity data, i.e., ϕc,lm, n, a, β, a, b, and c. The data used for fitting are over the ranges of 921 MPa, 323413 K, and 680 h, and the data of 120 h and 160 h are used for examining the accuracy of the model and limiting crystallinity. A nonlinear least-squares method, based on the LevenbergMarquardt algorithm, was used to estimate these parameters. The results are shown in Table 2. The activation energy for CO2-induced crystallization of i-PP can be calculated, as shown in Figure 4, by using the parameters 10512
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Figure 6. Crystallinity after 6 h versus temperature (points, experimental data; line, model data). Figure 4. Model value of activation energy (E) of CO2-induced crystallization and pre-exponential factor (k0) versus pressure.
Figure 5. Crystallinity versus time at 12 MPa and 323.15 K (line, the model values; stars, the data used for fitting; circles, the data used for examination).
a, b, and c in Table 2. The activation energy always decreases with pressure. These results fitted the analysis of activation energy above very well. The pre-exponential factor can be calculated, as shown in Figure 4, by using the parameters of a and β in Table 2. The preexponential factor decreases with pressure. As discussed above, the value of the pre-exponential factor is determined by the two effects of SC-CO2 on the capability of chain movement. The model values and the experimental values of crystallinity, at different times and at 12 MPa and 323.15 K, are shown in Figure 5. The data used for fitting were at the ranges of 921 MPa, 323413 K, and 680 h, and the data of 120 and 160 h were used for examining the accuracy of the model and limiting crystallinity. (The line and the dot line are the model values; the stars are the data used for fitting; and the circles are the data used for examination.) The experimental values are evenly distributed on both sides of the model curve, i.e., deviations between experiment and model appear to be random. As time extends, the data of different treating time are approaching the model value of limiting crystallinity. The model values agree well with the experimental values at 12 MPa and 323.15 K. The effect of temperature on the crystallinity at different pressures and at t = 6 h is shown in Figure 6, where the points are the experimental crystallinity and the line is the crystallinity calculated from the model. The experimental values are evenly
Figure 7. Crystallinity after 6 h versus pressure (points, experimental data; line, model data).
distributed around the model curve. The model values agree well with the experimental values at different temperatures, except perhaps at 12 MPa, where all of the data lies above the model line. The effect of pressure on the crystallinity under different temperatures at t = 6 h is shown in Figure 7, where the points are the experimental crystallinity and the line is the crystallinity calculated from the model. The experimental values are evenly distributed on both sides of the model curve. The model values agree well with the experimental values at different pressures. Residual analysis and root-mean-square error analysis of the model values and experimental values were carried out to confirm the conformance, effectiveness, and suitability of the improved kinetic model. The results are shown in Table 3. The standard deviation, σϕc , for ϕcwas calculated from sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ ϕc ¼ ¼
N
∑1 δ2ϕ =ðN 1Þ c
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2:14 103 =ð32 1Þ ¼ 8:32 103
ð14Þ
The standard deviation of the crystallinity ranges from 1.4% to 2.0% of the experimental values. Therefore, the improved kinetic model provides an excellent description of the CO2-induced crystallization of i-PP. The form of the model indicates that the interactions among T, P, and t can be ignored. In other words, the results of the regression analysis demonstrate that the effects of temperature, 10513
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Table 3. Crystallization Degree of of i-PP at Different Temperatures, Pressures, and Times P, MPa
t, h
ϕc
ϕc,moda
δϕc ,b 103
323.15
9
6
0.408
0.412
3.25
The improved model contains seven parameters that were estimated by fitting the experimental crystallinity data for i-PP. The standard error between model and experimental values is 7.10 103. Therefore, the improved model provides an excellent description of the CO2-induced crystallization of i-PP.
323.15 323.15
12 15
6 6
0.415 0.437
0.420 0.440
5.65 2.50
’ AUTHOR INFORMATION
4
323.15
18
6
0.469
0.466
2.59
Corresponding Author
5
323.15
21
6
0.477
0.492
15.15
*Tel.: +86-21-64253934. E-mail:
[email protected].
6
353.15
9
6
0.432
0.436
3.63
no.
T, K
1 2 3
7
353.15
12
6
0.456
0.446
9.77
8
353.15
15
6
0.469
0.468
1.01
9
353.15
18
6
0.512
0.495
17.13
10 11
353.15 373.15
21 9
6 6
0.520 0.456
0.518 0.453
1.64 2.57
12
373.15
12
6
0.461
0.464
3.37
13
373.15
15
6
0.491
0.486
5.04
14
373.15
18
6
0.521
0.512
8.73
15
373.15
21
6
0.529
0.533
4.21
16
393.15
9
6
0.474
0.471
2.72
17
393.15
12
6
0.483
0.482
1.04
18 19
393.15 393.15
15 18
6 6
0.501 0.532
0.504 0.528
3.17 3.67
20
393.15
21
6
0.539
0.546
6.60
21
413.15
9
6
0.500
0.490
9.95
22
413.15
12
6
0.494
0.500
5.60
23
413.15
15
6
0.526
0.520
6.00
24
413.15
18
6
0.545
0.542
3.06
25
413.15
21
6
0.553
0.557
4.02
26 27
323.15 323.15
12 12
12 22
0.440 0.486
0.450 0.483
9.53 3.29
28
323.15
12
31
0.507
0.504
3.22
29
323.15
12
48
0.524
0.531
7.82
30
323.15
12
80
0.550
0.561
11.17
31
323.15
12
120
0.552
0.579
27.03
32
323.15
12
180
0.583
0.588
4.53
ϕc,mod is the modeled values of crystallinity. b δϕc is the difference between the experimental and calculated crystallinity. All of these values retain three significant digits. a
pressure, and time are independent. Furthermore, the parameter n in this kinetic model is smaller than positive integers of 1, 2, and 3, which suggests that the induced crystallization kinetics behave quite differently from the isothermal or nonisothermal crystallization kinetics. The process of polymer induced crystallization is not an ordinary process of one-dimensional, two-dimensional, or three-dimensional nuclei growth.
’ CONCLUSIONS The effects of temperature, pressure, and treating time were studied with the CO2-ploymer system. The crystallinities of i-PP increased with temperature, pressure, and treating time. On the basis of significance testing, plasticization analysis, and interaction analysis between i-PP and CO2 molecules, temperature and pressure terms were introduced into the Avrami equation and the Arrhenius equation. This leads to an improved kinetic model for crystallization in the CO2-polymer system that contains the three variables, T, P, and t.
’ ACKNOWLEDGMENT We are grateful to Hui-Ming Zhu at the School of Material Science and Engineering, ECUST, for DSC analysis. Financial support from the National Natural Science Foundation of China (NSFC), Grants 20676031 and 20876051, is acknowledged. ’ NOMENCLATURE a, b, c = model parameters b0, bT, bP, bt, bTP = significant factors E = apparent activation energy ΔG = Gibbs free energy difference between the amorphous and crystalline phases per unit volume ΔH = enthalpy of crystallization per mass, J/g ΔH0 = enthalpy change of crystallization between entirely crystalline and noncrystalline phases per mass, J/g k = rate constant K0 = pre-exponential factor K = constant in the Scherrer equation n = Avrami exponent P = pressure, MPa R = gas constant, J/(mol.K) t = SC-CO2-treatment time, h T = temperature, K xi = nondimensional variables in significant level analysis a, β = model parameters δϕc = residue between model value and experimental values of crystallinity ϕc = crystallinity ϕc,A = crystallinity in the Avrami equation ϕc,lim = limiting crystallinity ϕc,mod = modeled value of crystallinity σ = energy per surface unit necessary to form the interface between the amorphous phase and the nucleus σϕc = standard deviation of modeled values of crystallinity ’ REFERENCES (1) Reiter, G.; Sommer, J.-U. Polymer Crystallization; SpringerVerlag: Berlin, 2003. (2) Allcock, H. R.; Lampe, F. W.; Mark, J. E. Contemporary Polymer Chemistry, 3rd ed.; Pearson Education, Inc: Upper Saddle River, NJ, 2003. (3) Hiemenz, P. C.; Lodge, T. P. Polymer Chemistry, 2nd ed.; CRC Press: Boca Raton, FL, 2007. (4) Chiou, J.; Barlow, J.; Paul, D. Polymer crystallization induced by sorption of CO2 gas. J. Appl. Polym. Sci. 1985, 30, 3911–3924. (5) Mizoguchi, K.; Hirose, T.; Naito, Y.; Kamiya, Y. CO2-induced crystallization of poly (ethylene terephthalate). Polymer 1987, 28, 1298– 1302. (6) Beckman, E.; Porter, R. S. Crystallization of bisphenol a polycarbonate induced by supercritical carbon dioxide. J. Polym. Sci., Part B: Polym. Phys. 1987, 25, 1511–1517. 10514
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