Combined effect of organic phosphate sodium and nanoclay on the mechanical properties and crystallization behavior of isotactic polypropylene

Combined effect of α‐nucleating agent (NA) sodium 2,2′‐methylene‐bis(4,6‐di‐tert‐butylphenyl) phosphate (NA11) and nanoclay (NC) on the mechanical properties and crystallization behavior of isotactic polypropylene (iPP) was investigated by mechanical testing, wide‐angle X‐ray scattering (WAXD), differential scanning calorimetry (DSC), polarized optical microscopy (POM), and scanning electron microscopy (SEM). The mechanical testing results indicated that the separate addition of NA11 and NC only increased the stiffness of iPP while the combined addition of NA11, NC, and maleic anhydride grafted polypropylene (PP‐g‐MA) simultaneously improved stiffness and toughness of iPP. Compared to pure iPP, the tensile strength, the flexural modulus, and impact strength of iPP composites increased 9.7, 38.6, and 42.9%, respectively. The result indicated good synergistic effects of NC, NA11, and PP‐g‐MA in improving iPP mechanical properties. WAXD patterns revealed NA11, and NC only induced the α‐crystals of iPP. SEM micrograph showed that the PP‐g‐MA could effectively improve the dispersing of NC in iPP. Finally, the nonisothermal crystallization kinetics of neat iPP and PP nanocomposites was described by Caze method. The result indicated that the addition of NA overcame the shortcoming of low crystallization rate of NC nanocomposites and maintained the excellent mechanical properties, which is another highlight of the combined addition of NAs and nanoclay. Meanwhile, the result showed that nuclei formation and spherulite growth of iPP were affected by the presence of NA and nanoclay. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2011     

The correlation between crystal structure and nucleation efficiency of a lithium (I) complex on isotactic polypropylene

In this work, 2,2′-methylene-bis-(4,6-di-t-butylphenylene)phosphate lithium (NA03) was synthesized and its crystal structural characterization was obtained by single crystal X-ray diffusion. The crystal data showed geometrically the cell parameter of NA03 matched with isotactic polypropylene (iPP), the a cell dimension was about two times to the value of cell edge of (010)_iPP. The disregistry was 2.89%, which was under the upper limit between the lattice matching spacing of host and guest crystals. Then NA03 was proved to be a highly effective nucleating agent for iPP through studying crystallization behaviors, crystallization morphologies, and mechanical properties of iPP nucleated with NA03. The outstanding nucleation efficiency could be attributed to the lattice matching between nucleating agent and iPP. © 2012 Wiley Periodicals, Inc. J Appl Polym Sci, 2012     

Charge storage behavior of isotropic and biaxially‐oriented polypropylene films containing α‐ and β‐nucleating agents

A study on the influence of the crystal modification (α and β) of isotactic polypropylene (i‐PP) films on the resulting electret properties is presented. Two commercial nucleating agents, sodium 2,2′‐methylene‐bis(4,6‐di‐tert‐butylphenyl)‐phosphate (NA11) and N,N′‐dicyclo‐hexyl‐2,6‐naphthalene‐dicarbox‐amide (NU100), were employed in this investigation. Isothermal charge decay was measured at 90°C. In hot pressed isotropic polypropylene films, no significant differences in the charge storage properties were observed for α‐ and β‐nucleated specimens. In addition, the article presents the influence of the nucleating agents at different concentrations on the PP‐film morphology of biaxially stretched films with respect to electret features. It was possible to prepare elongated cavities with the virtually insoluble NA11 additive during stretching, even at concentrations below 0.3 wt %. These films displayed slightly improved electret properties in comparison to stretched neat PP films due to generated cavities acting as barriers for the drift of charges. Various draw ratios were also studied for i‐PP films with 0.15 wt % NA11. © 2005 Wiley Periodicals, Inc. J Appl Polym Sci 99: 650–658, 2006     

Effect of DMDBS (3 : 2, 4‐bis(3,4‐dimethyldibenzylidene) sorbitol) and NA11 (sodium 2,2′‐methylene‐bis(4,6‐di‐tertbutylphenyl)‐phosphate) on electret properties of polypropylene filaments

Polypropylene (PP) composite filaments containing two different nucleating agents—DMDBS (3 : 2, 4‐bis(3,4‐dimethyldibenzylidene) sorbitol) and NA11 (sodium 2,2′‐methylene‐bis(4,6‐di‐tertbutylphenyl)‐phosphate) were melt spun to modify polymer electrostatic charging characteristics. Sample filaments were charged with a corona instrument and their surface potentials were measured. Initial surface potential as well as potential stability was monitored through an accelerated decay procedure. NA11 was found to be more efficient as an electret additive leading to a 50% increase in charge stability. Filaments with DMDBS exhibited a faster decay. Charging at elevated temperatures resulted in enhanced charge density and stability for both additives. The fiber microstructure was examined by Wide Angle X‐ray Diffraction and Differential Scanning Calorimetry. Rather than reducing the crystal sizes, X‐Ray diffractograms suggest that the crystal size increases with the addition of nucleating agents, while the degree of crystallinity appears to remain unaltered. © 2013 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 130: 2068–2075, 2013     

Structural characterization of α‐ and β‐nucleated isotactic polypropylene

Modification of isotactic polypropylene (iPP) with two nucleation agents, namely 1,3:24‐bis(3,4‐dimethylobenzylideno) sorbitol (DMDBS) (α‐nucleator) and N, N′‐dicyclohexylo‐2,6‐naphthaleno dicarboxy amide (NJ) (β‐nucleator), leads to significant changes of the structure, morphology and properties. Both nucleating agents cause an increase in the crystallization temperature. The efficiency determined in a self‐nucleation test is 73.4 % for DMDBS and 55.9 % for NJ. The modification with NJ induces the creation of the hexagonal β‐form of iPP. The addition of DMDBS lowers the haze of iPP while the presence of NJ increases the haze. Copyright © 2004 Society of Chemical Industry     

Variation of non‐isothermal crystallization behavior of isotactic polypropylene with varying β‐nucleating agent content

The non‐isothermal crystallization behavior, the crystallization kinetics, the crystallization activation energy and the morphology of isotactic polypropylene (iPP) with varying content of β‐nucleating agent were investigated using differential scanning calorimetry (DSC) and scanning electron microscopy (SEM). The DSC results showed that the Avrami equation modified by Jeziorny and a method developed by Mo and co‐workers could be successfully used to describe the non‐isothermal crystallization process of the nucleated iPPs. The values of n showed that the non‐isothermal crystallization of α‐ and β‐nucleated iPPs corresponded to a tridimensional growth with homogeneous and heterogeneous nucleation, respectively. The values of crystallization rate constant showed that the rate of crystallization decreased for iPPs with the addition of β‐nucleating agent. The crystallization activation energy increased with a small amount (less than 0.1 wt%) of β‐nucleating agent and decreased with higher concentration (more than 0.1 wt%). The changes of crystallization rate, crystallization time and crystallization activation energy of iPPs with varying contents of β‐nucleating agent were mainly determined by the ratio of the content of α‐ and β‐phase in iPP (α‐PP and β‐PP) from the DSC investigation, and the large size and many intercrossing lamellae between boundaries of β‐spherulites for iPPs with small amounts of β‐nucleating agent and the small size and few intercrossing bands among the boundaries of β‐spherulites for iPPs with large amounts of β‐nucleating agent from the SEM examination. Copyright © 2010 Society of Chemical Industry     

A novel β‐nucleating agent for isotactic polypropylene

The nucleating ability of p‐cyclohexylamide carboxybenzene (β‐NA) towards isotactic polypropylene (iPP) was investigated by differential scanning calorimetry, X‐ray diffraction, polarized optical microscopy and scanning electron microscopy. β‐NA is identified to have dual nucleating ability for α‐iPP and β‐iPP under appropriate kinetic conditions. The formation of β‐iPP is dependent on the content of β‐NA. The content of β‐phase can reach as high as 96.96% with the addition of only 0.05 wt% β‐NA. Under non‐isothermal crystallization the content of β‐iPP increases with increasing cooling rate. The maximum β‐crystal content is obtained at a cooling rate of 40 °C min^–1. The supermolecular structure of the β‐iPP is identified as a leaf‐like transcrystalline structure with an ordered lamellae arrangement perpendicular to the special surface of β‐NA. Under isothermal crystallization β‐crystals can be formed in the temperature range 80–140 °C. The content of β‐crystals reaches its maximum value at a crystallization temperature of 130 °C. © 2012 Society of Chemical Industry     

Isothermal crystallization and melting behavior of polypropylene with lanthanum complex of cyclodextrin derivative as a β‐nucleating agent

A novel highly active β‐nucleating agent, β‐cyclodextrin complex with lanthanum (β‐CD‐MAH‐La), was introduced to isotactic polypropylene (iPP). Its influence on isothermal crystallization and melting behavior of iPP was investigated by differential scanning calorimeter (DSC), wide‐angle X‐ray diffraction (WAXD), and polarized light microscopy (PLM). WAXD results demonstrated that β‐CD‐MAH‐La was an effective β‐nucleating agent, with β‐crystal content of iPP being strongly influenced by the content of β‐CD‐MAH‐La and the isothermal crystallization temperature. The isothermal crystallization kinetics of pure iPP and iPP/β‐CD‐MAH‐La was described appropriately by Avrami equation, and results revealed that β‐CD‐MAH‐La promoted heterogeneous nucleation and accelerated the crystallization of iPP. In addition, the equilibrium melting temperature (T) of samples was determined using linear and nonlinear Hoffman‐Weeks procedure. Finally, the Lauritzen‐Hoffman secondary nucleation theory was applied to calculate the nucleation parameter (K_g) and the fold surface energy (σ_e), the value of which verify that the addition of β‐CD‐MAH‐La reduced the creation of new surface for β‐crystal and then led to faster crystallization rate. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2011     

Effects of substituted aromatic heterocyclic phosphate salts on properties,crystallization, and melting behaviors of isotactic polypropylene

Nucleation effects of 2,2′‐methylene‐bis (4,6‐di‐tert‐butylphenyl) phosphate metal salts as a nucleating agent for isotactic polypropylene (iPP) were investigated with differential scanning calorimeter and polarized optical microscope, and their effects on mechanical, optical, and heat resistance properties of iPP were also studied. The results showed that monovalent metal salts of substituted aromatic heterocyclic phosphate such as sodium salt, lithium salt, and potassium salt had a good performance. With 0.2 wt % of sodium salt, lithium salt, or potassium salt incorporated into iPP, the crystallization peak temperature of iPP could be increased by 13.5, 13.6, and 15.0°C, respectively; the mass fraction of crystallinity of iPP could be increased by about 5%; and crystallization rate was enhanced increasingly. Meanwhile the tensile strength and flexural modulus of iPP could be increased by about 10 and 30%, respectively, and the clarity and heat distortion temperature of iPP could also be improved significantly. But bivalent and trivalent metal salts of substituted aromatic heterocyclic phosphate had little effect on properties of iPP. Meanwhile the morphology study showed that the addition of monovalent sodium salt could decrease the spherulite size of iPP significantly. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 100: 4868–4874, 2006     

Influence of different β‐nucleating agent on crystallization behavior,morphology, and melting characteristic of multiwalled carbon nanotube‐filled isotactic polypropylene nanocomposites

To obtain isotactic polypropylene (iPP) nanocomposites with high β‐crystal content, TMB5, calcium pimelate and calcium pimelate supported on the surface of nano‐CaCO₃ were used as β‐nucleating agent and MWCNT filled β‐nucleated iPP nanocomposites were prepared. The effect of different β‐nucleating agent and MWCNT on the crystallization behavior and morphology, melting characteristic and β‐crystal content of β‐nucleated iPP nanocomposites were investigated by DSC, XRD and POM. The results indicated that addition of MWCNT increased the crystallization temperature of iPP and MWCNT filled iPP nanocomposites mainly formed α‐crystal. The β‐nucleating agent can induce the formation of β‐crystal in MWCNT filled iPP nanocomposites. The β‐nucleating ability and β‐crystal content in MWCNT filled β‐nucleated iPP nanocomposites decreased with increasing MWCNT content and increased with increasing β‐nucleating agent content due to the nucleation competition between MWCNT and β‐nucleating agents. It is found that the calcium pimelate supported on the surface of inorganic particles as β‐nucleating agent has stronger heterogeneous β‐nucleation than calcium pimelate and TMB5. The MWCNT filled iPP nanocomposites with high β‐crystal content can be obtained by supported β‐nucleating agent. POLYM. COMPOS., 36:635–643, 2015. © 2014 Society of Plastics Engineers     

Effects of melt structure on crystallization behavior of isotactic polypropylene nucleated with α/β compounded nucleating agents

In this study, the melt structure of isotactic polypropylene (iPP) nucleated with α/β compounded nucleating agents (α/β‐CNA, composed of the α‐NA of 0.15 wt % Millad 3988 and the β‐NA of 0.05 wt % WBG‐II) was tuned by changing the fusion temperature T_f. In this way, the role of melt structure on the crystallization behavior and polymorphic composition of iPP were investigated by differential scanning calorimetry (DSC), wide‐angle X‐ray scattering (WAXD) and scanning electron microscopy (SEM). The results showed that when T_f = 200°C (iPP was fully molten), the α/β‐CNA cannot encourage β‐phase crystallization since the nucleation efficiency (NE) of the α‐NA 3988 was obviously higher than that of the β‐NA WBG‐II. Surprisingly, when T_f was in 179–167°C, an amount of ordered structures survived in the melt, resulting in significant increase of the proportion of β‐phase (achieving 74.9% at maximum), indicating that the ordered structures of iPP played determining role in β‐phase crystallization of iPP nucleated with the α/β‐CNA. Further investigation on iPP respectively nucleated with individual 3988 and WBG‐II showed that as T_f decreased from 200°C to 167°C, the crystallization peak temperature T_c of iPP/3988 stayed almost constant, while T_c of iPP/WBG‐II increased gradually when T_f < 189°C and became higher than that of iPP/3988 when T_f decreased to 179°C and lower, which can be used to explain the influence of ordered structure and α/β‐CNA on iPP crystallization. Using this method, the selection of α‐NA for α/β‐CNA can be greatly expanded even if the inherent NE of β‐NA is lower than that of the α‐NA. © 2014 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2015 , 132, 41355.     

Effect of orotic acid as a nucleating agent on the crystallization of bacterial poly(3‐hydroxybutyrate‐co‐3‐hydroxyhexanoate) copolymers

The kinetics of crystallization induced by orotic acid (OA) and boron nitride (BN) as nucleating agents were investigated for bacterial poly(3‐hydroxybutyrate‐co‐3‐hydroxyhexanoate)s (P(HB‐co‐HH)s) containing from 0 to 18% HH monomer units. The nucleation efficiency of these two chemicals was investigated by differential scanning calorimetry (DSC) and polarized optical microscopy (POM). It was found that both orotic acid and boron nitride are able to nucleate the crystallization of PHB. In the case of P(HB‐co‐HH) copolymers, orotic acid showed an outstanding nucleating effect. The comparison of half‐crystallization times shows that for P(HB‐co‐10% HH), the crystallization initiated by orotic acid is more than three time faster than the one induced by boron nitride (t_1/2BN/t_1/2OA(60°C) = 3.7 and t_1/2BN/t_1/2OA(90°C) = 4.5). According to the fact that orotic acid is a biodegradable, biocompatible and a nontoxic chemical, this nucleating agent is a promising solution for PHAs used in medical applications such as implants. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

Controlled synthesis of low‐molecular‐weight polyethylene waxes by titanium–biphenolate–ethylaluminum sesquichloride based catalyst systems

Soluble complexes of titanium(IV) bearing sterically hindered biphenols, such as biphenol, 1,1′‐methylene di‐2‐naphthol, 2,2′‐methylene bis(4‐chlorophenol), 2,2′‐methylene bis(6‐tert‐butyl‐4‐ethyl phenol), and 2,2′ ethylidene bis(4,6‐di‐tert‐butyl phenol), were prepared and characterized. These catalyst precursors, formulated as [Ti(O∧O)X₂], were active in the polymerization of ethylene at high temperatures in combination with ethylaluminum sesquichloride as a cocatalyst. The ultra‐low‐molecular‐weight polyethylenes (PEs) were linear and crystalline and displayed narrow polydispersities. The catalytic polymerization leading to PE waxes in this reaction exhibited unique properties that have potential applications in surface coatings and adhesive formulations. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci 104: 1531–1539, 2007     

Influence of a particulate nucleating agent on the quiescent and flow-induced crystallization of isotactic polypropylene

Flow induced crystallization of commercial isotactic polypropylene (iPP) and its blends with sodium 2,2′-methylene bis-(4,6-di-tert-butylphenyl) phosphate (also known as NA11) is studied by means of in-situ time resolved small-angle X-ray scattering (SAXS). The isothermal crystallization at 145 °C (i.e. well below melting temperature of polymer) is performed after the application of steady shear to probe the anisotropic structure formation. In order to separate the influence of shear rate and shear time on polymer crystallization, four different shear conditions (60 s⁻¹ for 1 s, 30 s⁻¹ for 2 s, 15 s⁻¹ for 4 s and 6 s⁻¹ for 10 s) are applied while maintaining the same imposed strain in the polymer melt. Further the effect of different concentration of nucleating agent on the crystallization kinetics of iPP is examined both under quiescent and shear flow conditions. For instance, under quiescent condition, the crystallization half-time (τ_1/2) decreases with the increasing concentration of nucleating agent in the polymer. Under shear flow conditions, our observations are as follows: In the case of neat iPP, τ_1/2 decreases significantly at higher shear rates (≥30 s⁻¹). Compared to the neat iPP, for the same concentration of NA11 in the NA11/iPP blends differences in τ_1/2 with the increase in applied shear rates are significantly smaller. In other words, the crystallization kinetics is dominated by the amount of nucleating agent in the NA11/iPP blends as opposed to shear rates in the neat iPP. The present study shows that the critical value of shear rate required for chain orientation in the molten polymer is lower in the presence of the nucleating agent compared to neat iPP. The self-nucleation process investigated with the aid of differential scanning calorimetry (DSC), indicates that the nucleating efficiency of NA11 on iPP is around 60%.     

Non‐isothermal crystallization of polypropylene with sorbitol‐type nucleating agents at cooling rates used in processing

Sorbitol‐type nucleating agents used as clarifiers for polypropylene (PP) show a complex interplay of phase and crystallization behavior. To study the crystallization behavior, cooling measurements were performed by fast scanning calorimetry using rates between 10 and 4000 K s^?1. These rates correspond to cooling rates used in processing. These measurements were combined with conventional differential scanning calorimetry (DSC) measurements at 10 K min^?1. The optical properties were characterized by haze measurements. For this investigation the commercially available clarifiers 1,3:2,4‐bis(3,4‐dimethylbenzylidene)sorbitol and 1,2,3‐tridesoxy‐4,6:5,7‐bis‐O‐[(4‐propylphenyl)methylene]nonitol were added to PP in various amounts up to 0.6 wt%. At relatively slow cooling rates only a single crystallization process was observed. In contrast, fast cooling leads to a complex crystallization behavior with up to four different crystallization processes. It was found that the temperature of the main crystallization process during fast cooling correlates with the optical properties from haze measurements. Finally, the cooling rate dependence of the different crystallization processes is discussed in terms of nucleating efficiency of the clarifiers. We conclude that the results of conventional DSC measurements cannot be extrapolated in a simple way to describe the nucleation activity of nucleating agents at cooling conditions relevant during processing. © 2018 Society of Chemical Industry     

Joint effects of molecular structure and crystal morphology of organophosphate monovalent salts on nucleated isotactic poly(propylene)

The nucleation ability of organophosphate monovalent salts in isotactic poly(propylene) (iPP) was elucidated from the crystalline morphology and molecular structure with different size of cationic radius (r) of substitute parts. Differential scanning calorimetry result showed that the degree of crystallinity of nucleated iPP increased linearly with the increasing of r following the order of lithium (NA-10), sodium (NA-11), potassium (NA-12), ammonium (NA-13) salts of 2,2′-methylene- bis-(4,6-di-t-butylphenylene) phosphate. Moreover, small-angle X-ray scattering result displayed a large increase in the periodical length of nucleated iPP, which arises from an increment of both lamella thickness and amorphous thickness. Further, at 0.15 wt% concentration, the haze values of nucleated iPP presented a decrease tendency in the order of iPP/NA-10 (16.7 %), iPP/NA-11 (15.1 %), iPP/NA-12 (14.6 %), iPP/NA-13 (14.8 %), and their flexural strength was increased by 26.2 %, 30.8 %, 31.4 % and 31.7 %, respectively, as comparison to virgin iPP. These results demonstrated that the nucleating ability of these nucleating agents increased with the increase of r of substitute parts.     

Structure modification of isotactic polypropylene by bi‐component nucleating systems

The efficiency of heterogeneous nucleation of isotactic polypropylenes (iPP) with various isotacticity and MWD is estimated by an increase of the crystallization temperature (T_cr). A bi‐component nucleating system, composed of a pimelic acid and a calcium stearate, was added to the polymeric matrix in the molten state. The relative increase of T_cr is found to be independent on the structure stereo‐regularity of iPP. On the contrary, dependent on the polymer structure, the investigated nucleating system influences in a different way the creation of the β‐phase. The form of the DSC melting curves, for the β‐phase rich iPP samples, is discussed in terms of various preliminary cooling conditions. The transformation from a multi‐modal form of the DSC melting peak to a simple one is related to the lowering of the cooling rate and to the increase of the heating rate.     

Comparative study on the crystallization behavior of β‐isotactic polypropylene nucleated with different β‐nucleation agents —effects of thermal conditions

In this study, the crystallization behavior of the β‐isotactic polypropylene (β‐iPP) samples nucleated by a rare earth based β‐nucleating agent (β‐NA) WBG‐II and a metal salts compound β‐NA NAB83 (denoted as WPP and NPP, respectively) under different cooling conditions were comparatively investigated. The thermal conditions such as the cooling rate, isothermal crystallization temperature, isothermal crystallization time, and the subsequent cooling to room temperature. The results of WAXD, SEM, and nonisothermal crystallization reveal that under the same processing conditions, the crystallite size of NPP is smaller, which arrange more compactly as compared with WPP. Meanwhile, NPP has shorter crystallization rate and higher β‐nucleation selectivity, but WPP can crystallization at wider temperature range. The results of isothermal crystallization showed that NPP has higher selectivity and higher β‐nucleation efficiency, which favors the formation of high proportion of β‐phase at the isothermal crystallization temperature of 110–130°C with and without subsequent cooling; WPP has lower selectivity, which can only induce high content of β‐phase under isothermal crystallization without subsequent cooling to 25°C. In tuning the crystallization behavior and the properties of β‐PP, the joint influence of the efficiency and selectivity of the β‐NA, and the thermal conditions should be taken into consideration. © 2013 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2014 , 131, 40115.     

Crystallization morphologies and non-isothermal crystallization kinetics of isotactic polypropylene modified by α/β compounded nucleating agents

采用偏光显微镜(POM)及示差扫描量热(DSC)法考察了3种α/β复合成核剂NA40/NABW、NA40/HHPA-Ba、NA40/PA-03对成核等规聚丙烯(iPP)的结晶形态及非等温结晶动力学的影响。对成核iPP结晶形态的研究结果表明：α/β复合成核剂的加入能够减小iPP的球晶尺寸。影响α/β复合成核剂成核iPP结晶形态的主要因素是ΔT_Cp（ΔT_Cp为成核iPP结晶峰值温度与iPP结晶峰值温度的差值）,即复合体系中ΔT_Cp较大的成核剂在iPP结晶过程中起主导作用,最终的结晶形态与单独添加这一成核剂时iPP的结晶形态相类似;当两种成核剂的ΔT_Cp接近相同时,两者竞争成核,成核iPP的结晶形态表现为两种成核剂共同作用的结果。因此,通过改变α/β复合成核剂的复合比例即改变两种成核剂的添加浓度,进而改变其ΔT_Cp,可以得到结晶形态完全不同的iPP。采用Caze法对非等温动力学进行了研究,结果表明：添加α/β复合成核剂能够提高iPP的结晶温度,缩短半结晶时间。复合成核剂成核iPP的结晶行为也同样受成核剂ΔT_Cp的影响,复合成核iPP的Avrami指数接近于复合体系中ΔT_Cp较大的成核剂单独添加时iPP的Avrami指数。