Kinetics Of Propylene Polymerization Research Articles

Polymerization Kinetics of Propylene with the MgCl2-Supported Ziegler-Natta Catalysts—Active Centers with Different Tacticity and Fragmentation of the Catalyst

The catalytic activity and stereospecificity of olefin polymerization by using heterogeneous TiCl4/MgCl2 Ziegler-Natta (Z-N) catalysts are determined by the structure and nature of active centers, which are mysterious and fairly controversial. In this work, the propylene polymerization kinetics under different polymerization temperatures by using Z-N catalysts were investigated through monitoring the concentration of active centers [C*] with different tacticity. SEM was applied to characterize the catalyst morphologies and growing polypropylene (PP) particles. The lamellar thickness and crystallizability of PP obtained under different polymerization conditions were analyzed by DSC and SAXS. The PP fractions and active centers with different tacticity were obtained with solvent extraction fractionation method. The catalytic activity, active centers with different tacticity and propagation rate constant kp, fragmentation of the catalyst, crystalline structure of PP are correlated with temperature and time for propylene polymerizations. The polymerization temperature and time show complex influences on the propylene polymerization. The higher polymerization temperature (60 °C) resulted higher activity, kp and lower [C*], and the isotactic active centers Ci* as the majority ones producing the highest isotactic polypropylene (iPP) components showed much higher kp when compared with the active centers with lower stereoselectivity. Appropriate polymerization time provided full fragmentation of the catalyst and minimum diffusion limitation. This work aims to elucidate the formation and evolution of active centers with different tacticity under different polymerization temperature and time and its relations with the fragmentation of the PP/catalyst particles, and provide the solutions to the improvement of catalyst activity and isotacticity of PP.

Effect of Varying Hydrogen Concentration, External Donor Concentration, and Temperature on Propylene Polymerization Kinetics and Microstructure of Polypropylene Made with a 4th Generation Ziegler-Natta Catalyst

This paper presents the results of a detailed investigation on propylene polymerization kinetics and microstructural characterization of polypropylene samples made with a 4th generation Ziegler–Natta catalyst under varying hydrogen pressures, external donor concentrations, and polymerization temperatures. Hydrogen had a positive effect on the mmmm pentad fraction at low donor concentrations, in agreement with our previous Monte Carlo model. Estimated activation energies for activation, propagation, and deactivation were very close to the values previously reported by our group for a narrower set of experiments with the same catalyst.

EFFECTS OF HYDROGEN ON ACTIVE CENTERS IN PROPYLENE POLYMERIZATION WITH SUPPORTED ZIEGLER-NATTA CATALYST

The kinetics of propylene polymerization with a TiCl4/ Di / MgCl2( Di = internal donor) supported Ziegler-Natta catalyst was studied by measuring the number of active centers using 2-thiophenecarbonyl chloride as quenching agent. Influences of hydrogen and alkoxysilane type external donors on the number of active centers and chain propagation rate constant( k p) were investigated. The number and k p value of three groups of active centers that produce atactic,medium-isotactic and isotactic polypropylene( PP) fractions were determined. The catalyst activity was found to significantly increase with the addition of hydrogen,which was a result of simultaneous increases in the number of active centers and k p value. The three groups of active centers response to hydrogen differently. Hydrogen caused lower degree of increase in the number of active centers that produce atactic PP than the other two groups of active centers,and caused the highest degree of increase in the k p value of active centers that produce medium-isotactic PP. The hydrogen response of the active centers was also influenced by external donor. Increasing the bulkiness of alkyl groups in alkoxysilane donors leads to decreased hydrogen sensitivity,meanwhile the stereo- and regio-selectivities were evidently improved,implying that there are close relations between the hydrogen effect and the stereo-and regio-selectivities of the catalyst.A mechanistic model was proposed to explain the hydrogen effect.

Counting the number of active centers in MgCl2-supported Ziegler–Natta catalysts by quenching with 2-thiophenecarbonyl chloride and study on the initial kinetics of propylene polymerization

Abstract Using 2-thiophenecarbonyl chloride (TPCC) as a quenching agent, the number of active centers in propylene polymerization with MgCl2-supported Ziegler–Natta catalysts were determined by measuring the sulfur content of the quenched polymer. Under suitable conditions, the thiophenecarbonyl-labeled polymer chains were stable in the reaction system when TPCC/Al > 1. The number of active centers was found to increase in the first 5 minutes of propylene polymerization, accompanied by rapid decrease of the propagation rate constant kp. Diffusion barrier from the polymer covering the catalyst fragments is thought as the main reason of the rapid decay of kp in the initial stage.

A CFD-PBM coupled model with polymerization kinetics for multizone circulating polymerization reactors

An Eulerian–Eulerian two-fluid coupling the population balance model (PBM) is developed to describe the gas–solid two-phase flow in a multizone circulating polymerization reactor (MZCR). The polymerization kinetics is also incorporated into the coupled model by using a user-defined function (UDF). The model is first used to predict the entire field in the MZCR with considering the polydispersity of solid phase. Furthermore, the temperature field is obtained numerically with considering polymerization reaction, and the effect of inlet gas temperature on the temperature field is also investigated. Finally, the model is adopted to distinguish between the flow behaviors in a circulating fluidized bed reactor (CFBR) and a MZCR. The simulated results show that the flow behavior in the MZCR with polydisperse solid phase is different from that with uniform particle size and the inlet gas temperature has great effects on the temperature distribution. Moreover, the simulation results also show that the differences in between CFBR and MZCR are mainly embodied in the temperature distribution. A CFD-PBM model is developed to describe the gas–solid two-phase flow in a multizone circulating polymerization reactor (MZCR). The polymerization kinetics is also incorporated into the CFD-PBM model. The entire fields in the MZCR are obtained numerically. The model is also used to distinguish the flow behaviors in between a circulating fluidized bed reactor (CFB) and a MZCR. ► We present a CFD-PBM model to describe the flow field in a polymerization MZCR. ► The CFD-PBM model incorporates propylene polymerization kinetics. ► The entire field in the MZCR with the solid PSD is predicted via the model. ► The model is used to distinguish between the flow behavior in MZCR and CFBR. ► The results show that the temperature field in a CFBR is different from that in a MZCR.

Effect of Hydrogen and External Donor on Propylene Polymerization Kinetics with a 4th‐Generation Ziegler‐Natta Catalyst

AbstractApparent kinetic rate constants and activation energies for catalyst activation, deactivation, and propylene propagation with a 4th‐generation Ziegler‐Natta catalyst are estimated. The rate of polymerization in the presence of donor and hydrogen is higher than in their absence. Catalyst activation and propagation rate constants increase with the addition of hydrogen, likely because hydrogen frees 2–1‐terminated dormant sites. The deactivation term was the highest without electron donor, presumably because the nonspecific sites deactivate faster than the specific sites. Apparent activation energies for catalyst activation, deactivation, and propagation are not significantly influenced by the presence of hydrogen or external donor, indicating that they do not affect these mechanisms substantially. magnified image

Polydentate phosphine oxides as external electron donors for titanium-magnesium catalysts for propylene polymerization

The possibility of using R n P(O)(CH2OR′)3—n (R = alkyl, R′ = methyl or acyl, n = 0–2) polydentate phosphine oxides as external electron donors for the titanium-magnesium catalysts for isotactic polypropylene synthesis is demonstrated for the first time. The kinetics of propylene polymerization in liquid monomer at 70°C and the isotacticity and molecular-weight characteristics of the resulting polypropylene are studied as functions of the nature of the substituents at the phosphorus atoms in the external donor and the molar ratio of the cocatalyst AlEt3 to the external electron donor. Among the compounds examined, isoamyldi(methoxymethyl)phosphine oxide (R = iso-Am, R′ = Me, n = 1) is the most efficient. The isotacticity index of the polypropylene (PP) synthesized on the titanium-magnesium catalyst with this external donor is as high as 94–95%, and the activity of the catalyst (Cat) in the absence of hydrogen is 5.0–6.5 (kg PP) (g Cat)−1 h−1. With the optimum combination, the activity of this catalyst is ≈5 (kg PP) (g Cat)−1 h−1 and the isotacticity index is 94%. These parameters are close to those obtained for propylene polymerization in the absence of hydrogen on the same titanium-magnesium catalyst with phenyltriethoxysilane (external donor used in the industrial synthesis of PP): the activity is 5.6 (kg PP) (g Cat)−1 h−1, and the isotacticity index is 95%. The introduction of hydrogen into the reaction zone makes it possible to efficiently control the molecular weight of PP, increases the catalyst activity by a factor of 1.5–2.5, and somewhat decreases the isotacticity index (from 94 to 91–92%).

Multiple active site Monte Carlo model for heterogeneous Ziegler‐Natta propylene polymerization

AbstractIn this study, the kinetics of propylene polymerization catalyzed with the fourth heterogeneous Ziegler‐Natta catalyst is studied. More than one type of active site is present in the propylene polymerization based on an analysis of the GPC curves. A multiple active site kinetic model (MSmodel) is proposed by using Monte Carlo technique. Good agreements in the polymerization kinetics are achieved for fitting the kinetic profiles with the MSmodel. In addition, the MSmodel is used to describe the dynamic evolutions of the active sites and their effects on the propylene polymerization. The simulated results indicate that different types of active sites have different polymerization kinetics and the site type can affect the propylene polymerization kinetics. © 2009 Wiley Periodicals, Inc. J Appl Polym Sci, 2010

Modeling of the propylene polymerization catalyzed by single‐/multi‐active site catalyst: A Monte Carlo study

AbstractIn the present study, a model is established to describe the propylene polymerization kinetics catalyzed by the typical catalysts with single‐/multi‐active site type in a liquid phase stirred‐tank reactor using the Monte Carlo simulation method, regardless of the mass and heat diffusion effects within the polymer particles. Many kinetic data, including polypropylene yield, concentration transformation of catalyst active sites, number–average molecular weight, etc., are obtained by the model. The simulated kinetic results are found to be in agreement with the reference ones obtained in a population balance model. Furthermore, the comparisons of the kinetic data between the polymerization catalyzed by the catalyst with single‐active site type (typically silica‐supported metallocene) and the catalyst with multi‐active site type (typically MgCl2‐supported Ziegler‐Natta catalyst) have been studied using the model. Especially, the effects of hydrogen on the polymerization are studied using the model. The studied results show that the theory of catalyst active site can be used to explain the different propylene polymerization kinetics catalyzed by the typical catalyst with single‐/multi‐active site type. In addition, the role of hydrogen in the propylene polymerization needs to be emphasized. © 2008 Wiley Periodicals, Inc. J Appl Polym Sci, 2008

Study of the morphology and kinetics of novel Ziegler–Natta catalysts for propylene polymerization

AbstractThe morphological and kinetic characteristics of novel Ziegler–Natta catalysts were studied. Catalysts were prepared by Borealis Polymers Oy using a new synthesis technique (emulsion technology). Video microscopy was used to study the growth of single catalyst particles during polymerization in the gas and liquid phases. The distribution of single particle activity was very narrow in the catalyst without external support and was rather broad in the the silica‐supported catalyst. Video microscopy of molten polymer particles allowed observation of the process and degree of fragmentation of the catalyst particles. A correlation between the activation period during the initial stage of polymerization and catalyst fragmentation was found. Fragmentation was faster and more uniform with the catalyst without external support than with the silica‐supported catalyst. Scanning electron microscopy provided information on morphology evolution and shape replication of the catalyst particles. With the catalyst without external support, good shape replication was observed, and compact and spherical particles were formed. With the silica‐supported catalyst, shape replication was poor, and nonspherical porous polymer particle were formed. Modeling of the kinetics of propylene polymerization was done using a simple three‐step reaction scheme neglecting mass and heat transport effects. © 2005 Wiley Periodicals, Inc. J Appl Polym Sci 98: 2191–2200, 2005

Kinetics of Propylene Polymerization Using Bis(2-phenylindenyl)zirconium Dichloride/Methylaluminoxane

The kinetics of propylene polymerization in toluene solution by bis(2-phenylindenyl)zirconium dichloride/methylaluminoxane at 20 °C were investigated. As the structure and properties of elastomeric polypropylenes produced by these catalysts depend sensitively on the reaction conditions, a detailed study of the kinetics was carried out to evaluate the influence of these parameters on the polymerization behavior. Studies of the solubilities and mass-transfer rates reveal that dissolved atactic polypropylene has little effect on propylene solubility but influences the mass-transfer rate of propylene into solution. The rates of propylene polymerization reach a maximum after 10−20 min and then decrease. The decrease in rate over time is faster at higher monomer concentrations. Catalyst activity was negligible at [Al]/[Zr] = 1000 but constant from [Al]/[Zr] = 2500 to [Al]/[Zr] = 10 000. Analysis of molecular weights as a function of monomer concentration reveal β-hydride elimination to be the primary chain-tran...

Kinetics of propylene polymerization by hafnocene diamide catalyst

The kinetics of propylene polymerization initiated by ansa-metallocene diamide compound rac-(EBI)Hf(NMe 2 ) 2 (EBI = C 2 H 4 -(indenyl) 2 , rac-1) were investigated. The rac-1 compound could be directly utilized for catalyst formations without converting to a dihalide or dialkyl complex in the presence or absence of methylaluminoxane (MAO). The MAO-free system rac-1/AlR 3 /[Ph 3 C][B(C 6 F 5 ) 4 ] (2) is much more effective than the rac-1/MAO catalyst. The activity of the rac-1/Al(iBu) 3 /2 system is much higher than that of the rac-(EBI)HfCl 2 /MAO or rac-(EBI)ZrCl 2 /MAO catalyst, and almost same as that of the rac-(EBI)Zr(NMe 2 ) 2 /Al(iBu) 3 /2 catalyst under similar conditions. The alkylation of rac-1 to rac-(EBI)HfR 2 by using AlR 3 needs more time than the corresponding zirconocene analogue. The activity increases by a factor of 7 by increasing the aging time from 1 min to more than 4 h. The activity of the rac-1/AlR 3 /2 catalyst is very sensitive to the type and concentration of AlR 3 , and decreases in the order: Al(iBu) 2 H > Al(iBu) 3 > AlEt 3 . > AlMe 3 . The catalyst keeps high activity in a narrow range of the [Al]/[Hf] ratio. In addition, the activity is influenced by the concentration of 2, and as a result, the maximum activity is observed when 2/rac-1 = 0.7. The activity of the rac-1/AlR 3 /2 catalyst is also sensitive to the polymerization temperature. The activation energies for the initiation and overall reactions are calculated as 7.61 and 7.14 kcal/mol, respectively. The properties of polymer such as isotacticity (as [mmmm]), molecular weight (MW), molecular weight distribution (MWD), melting temperature (T m ), and crystallinity are similar level with those obtained with the rac-(EBI)HfCl 2 /MAO system. The MW and isotacticity of the polymer produced by MAO-free system decreases monotonically as T p increases, and MWD becomes narrow from 2.90 to 2.10 when T p increases from 30 to 90°C because of the compositional homogeneity of the polymer produced at high T p , which is demonstrated by fractionation of the polymer. Both MW and [mmmm] values of polymers decrease as aging time and anion concentration increase.

Kinetics of propylene polymerization by hafnocene diamide catalyst

The kinetics of propylene polymerization initiated by ansa-metallocene diamide compound rac-(EBI)Hf(NMe2)2 (EBI = C2H4-(indenyl)2, rac-1) were investigated. The rac-1 compound could be directly utilized for catalyst formations without converting to a dihalide or dialkyl complex in the presence or absence of methylaluminoxane (MAO). The MAO-free system rac-1/AlR3/[Ph3C][B(C6F5)4] (2) is much more effective than the rac-1/MAO catalyst. The activity of the rac-1/Al(iBu)3/2 system is much higher than that of the rac-(EBI)HfCl2/MAO or rac-(EBI)ZrCl2/MAO catalyst, and almost same as that of the rac-(EBI)Zr(NMe2)2/Al(iBu)3/2 catalyst under similar conditions. The alkylation of rac-1 to rac-(EBI)HfR2 by using AlR3 needs more time than the corresponding zirconocene analogue. The activity increases by a factor of 7 by increasing the aging time from 1 min to more than 4 h. The activity of the rac-1/AlR3/2 catalyst is very sensitive to the type and concentration of AlR3, and decreases in the order: Al(iBu)2H > Al(iBu)3 > AlEt3 > AlMe3. The catalyst keeps high activity in a narrow range of the [Al]/[Hf] ratio. In addition, the activity is influenced by the concentration of 2, and as a result, the maximum activity is observed when 2/rac-1 = 0.7. The activity of the rac-1/AlR3/2 catalyst is also sensitive to the polymerization temperature. The activation energies for the initiation and overall reactions are calculated as 7.61 and 7.14 kcal/mol, respectively. The properties of polymer such as isotacticity (as [mmmm]), molecular weight (MW), molecular weight distribution (MWD), melting temperature (Tm), and crystallinity are similar level with those obtained with the rac-(EBI)HfCl2/MAO system. The MW and isotacticity of the polymer produced by MAO-free system decreases monotonically as Tp increases, and MWD becomes narrow from 2.90 to 2.10 when Tp increases from 30 to 90°C because of the compositional homogeneity of the polymer produced at high Tp, which is demonstrated by fractionation of the polymer. Both MW and [mmmm] values of polymers decrease as aging time and anion concentration increase. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 75: 843–855, 2000

Kinetics of propylene polymerization initiated by rac ‐Me 2 Si(1‐C 5 H 2 ‐2‐Me‐4‐ t Bu) 2 Zr(NME 2 ) 2 /MAO catalyst

The kinetics of propylene polymerization initiated by ansa-metallocene diamide compound rac-Me2Si(CMB)2Zr(NMe2)2 (rac-1, CMB = 1-C5H2-2-Me-4-tBu)/methylaluminoxane (MAO) catalyst were investigated. The formation of cationic active species has been studied by the sequential NMR-scale reactions of rac-1 with MAO. The rac-1 is first transformed to rac-Me2Si(CMB)2ZrMe2 (rac-2) through the alkylation mainly by free AlMe3 contained in MAO. The methylzirconium cations are then formed by the reaction of rac-2 and MAO. Small amount of MAO ([Al]/[Zr] = 40) is enough to completely activate rac-1 to afford methylzirconium cations that can polymerize propylene. In the lab-scale polymerizations carried out at 30°C in toluene, the rate of polymerization (Rp) shows maximum at [Al]/[Zr] = 6,250. The Rp increases as the polymerization temperature (Tp) increases in the range of Tp between 10 and 70°C and as the catalyst concentration increases in the range between 21.9 and 109.6 μM. The activation energies evaluated by simple kinetic scheme are 4.7 kcal/mol during the acceleration period of polymerization and 12.2 kcal/mol for an overall reaction. The introduction of additional free AlMe3 before activating rac-1 with MAO during polymerization deeply influences the polymerization behavior. The iPPs obtained at various conditions are characterized by high melting point (approximately 155°C), high stereoregularity (almost 100% [mmmm] pentad), low molecular weight (MW), and narrow molecular weight distribution (below 2.0). The fractionation results by various solvents show that iPPs produced at Tp below 30°C are compositionally homogeneous, but those obtained at Tp above 40°C are separated into many fractions. © 1999 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 37: 737–750, 1999

Isospecific Polymerization of Propylene By ansa-Zirconocene Diamide Compound Cocatalyzed by Mao

The kinetics of propylene polymerization initiated by racemic ethylene-1,2-bis(1-indenyl) zirconium bis(dimethylamide) [rac-(EBI) Zr(NMe2)2(rac-1)] cocatalyzed by methylaluminoxane (MAO) were studied. The polymerization behaviors of rac-1/MAO catalyst investigated by changing various experimental parameters are quite different from those of rac-(EBI) ZrCl2 (rac-2)/MAO catalyst, due to the differences in the generation procedure of cationic actives species of each metallocene by the reaction with MAO. The activity of rac-1/MAO catalyst showed maximum when [Al]/[Zr] is around 2000, when [Zr] is 137.1 μM, and when polymerization temperature is 30°C. The negligible activity of rac-1/MAO catalyst at a very low MAO concentration seems to be caused by the instability of the cationic active species. The meso pentad values of polymers produced by rac-1/MAO catalyst at 30°C are in the range of 82.8% to 89.7%. The rac-1/MAO catalyst lost stereorigid character at the polymerization temperature above 60°C. The molecular weight of polymer decreased as [Al]/[Zr] ratio, polymerization temperature, and [Zr] increased. The molecular weight distributions of all polymers are in the range of 1.8–2.3, demonstrating uniform active species present in the polymerization system.

Kinetic Study of Propylene Polymerization Using Et(H4Ind)2ZrCl2/Methylalumoxane Catalysts

The kinetics of propylene polymerization with Et(H 4 Ind) 2 ZrCl 2 /methylalumoxane (MAO) catalysts have been investigated at 0-50 °C and 1 atm of propylene pressure. The effects of polymerization temperature, catalyst concentration, and catalyst/cocatalyst mole ratio on the catalyst activity and on the properties of the polypropylene products were studied. By assuming the coexistence of two dynamic equilibria between catalytically active species and two inactive species, which are the products of zirconocene dimerization and excessive zirconocene-MAO complexation, a kinetic model has been developed. This model could satisfactorily explain most of experimental results.

Kinetics of propylene polymerization with a non-supported heterogeneous Ziegler-Natta catalyst—effect of hydrogen on rate of polymerization, stereoregularity, and molecular weight distribution

The effect of hydrogen on the kinetics of polymerization of propylene catalysed with a non-supported, heterogeneous Ziegler-Natta catalyst, was studied using an integrated measurement methodology of polymerization rate determination and polymer characterization using size exclusion chromatography, temperature rising elution fractionation, and 13C nuclear magnetic resonance. It was unequivocally shown that the presence of hydrogen during the polymerization of propylene activates catalytic sites that are dormant when hydrogen is absent, increasing the polymerization rate and broadening the distributions of molecular weight and stereoregularity.

Polymerization of propylene using a prepolymerized high-active Ziegler-Natta catalyst. I. Kinetic studies

The kinetics of propylene polymerization using a prepolymerized high-active TiCl3 catalyst with an Et2AICI cocatalyst are investigated. The effect of various parameters such as AI/Ti ratio, pressure, temperature, hydrogen, and polymerization time on the rate of polymerization and yield are examined. The dependency of these parameters on the polymerization rate are studied. It is found that the variation at the rate of polymerization with the Et2AICI cocatalyst concentration complies with a Langmuir–Hinshelwood type of relationship. The overall activation energy of polymerization calculated from the Arrhenius plot was found to be 11.6 kcal/mol. © 1996 John Wiley & Sons, Inc.

Study of hydrogen effect in propylene polymerization on (with) the MgCl 2-supported ziegler-natta catalyst—part 2. Effect of CS 2 on polymerization centres

The effect of CS 2 on the catalytic system TiCl 4/MgCl 2/DBPH-AlEt 3/TEPS in propylene polymerization in the presence and absence of hydrogen was studied. It was shown that CS 2 poisons active centres in both cases by a similar way. The yield of polypropylene decreases with the CS 2 effect, but the isotactic centres poison to the greatest degree. Results obtained by the study of polymerization kinetics, fractional comosition and molecular mass distribution of polymers permit a conclusion that there is not any differentiation between initial and “hydrogen” centres, i.e. these centres have the same nature. An observed hydrogen activation effect is explained by the revival of “dormant” centres, resulting from 2-1 propylene insertion in the polymer chain in the chain transfer reaction with H 2. Seemingly the competition between the polymerization rate retarding by irregular monomer insertion and the activation by hydrogen takes place. It leads to non-stationary kinetics of propylene polymerization in the presence of hydrogen.

Kinetics of propylene polymerization in the initial acceleration stage

AbstractThe kinetics of propylene polymerization catalyzed over a superactive and stereospecific catalyst for the initial build‐up period was investigated in slurry‐phase. The catalyst was prepared from Mg(OEt)2/benzoyl chloride/TiCl4 co‐activated with AlEt3 in the absence or presence of external donor. Despite a very fast activation of the prepared catalyst the acceleration stage of polymerization could be identified by the precise estimation of polymerization kinetics for a very short period of time after the commencement of polymerization (ca. 2 min). The initial polymerization rate, (dRp/dt)0 extrapolated to the beginning of the polymerization was second order with respect to monomer concentration. The dependence of initial polymerization rate on the concentration of AlEt3 could be represented by Langmuir adsorption mechanism. The initial rate was maximum at about Al/Ti ratio of 20. The activation energy for the initiation reaction was estimated to be 14.3 kcal/mol for a short‐time polymerization. The addition of a small amount of p‐ethoxy ethyl benzoate (PEEB) as an external donor increased the percentage of isotactic polymer, which was obtained after 120 s of polymerization, to 98% and the initial polymerization rate decreased sharply as [PEEB]/[AlEt3] increased. © 1994 John Wiley & Sons, Inc.