Analysis and Control of the Molecular Weight and Chemical

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Ind. Eng. Chem. Res. 1997, 36, 1144-1150

Analysis and Control of the Molecular Weight and Chemical Composition Distributions of Polyolefins Made with Metallocene and Ziegler-Natta Catalysts Joa˜ o B. P. Soares,* Jung Dae Kim, and Garry L. Rempel

Ind. Eng. Chem. Res. 1997.36:1144-1150. Downloaded from pubs.acs.org by MIDWESTERN UNIV on 01/29/19. For personal use only.

Department of Chemical Engineering, Institute for Polymer Research, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

The final properties of polyolefins are closely related to their distributions of molecular weight (MWD) and chemical composition (CCD). Polyolefins produced with conventional heterogeneous Ziegler-Natta catalysts have broad MWDs and CCDs caused by the presence of several activesite types on these catalysts. These distributions can be adequately described as a weighted superposition of narrower distributions valid for each catalytic site type. With the advent of metallocene catalysts, it is now possible to synthesize polyolefins with narrow MWDs and CCDs, allowing a much better control over the MWDs and CCDs of these polymers. The selective combination of different metallocene types on a given support should permit the production of polyolefins with designed MWDs and CCDs from the knowledge of the MWDs and CCDs of polyolefins made with the individual metallocenes. This paper reviews some recent techniques used to analyze the MWDs and CCDs of polyolefins made with multiple-site-type catalysts and shows how to control the MWD of polyethylene made with a two-site-type silica-supported metallocene catalyst by varying the ethylene pressure during polymerization. Introduction The extensive use of polyolefins today as commodity polymers was made possible by the discovery by K. Ziegler and G. Natta in the early 1950s that transitionmetal salts of metals from groups IV and VIII could polymerize ethylene to high-molecular-weight polyethylene when combined with appropriate metal alkyls of a base from groups I-III. These Ziegler-Natta catalysts have evolved considerably from the low-active, lowstereospecific catalysts of the early 1950s and 1960s to the highly active, highly stereospecific catalysts used in modern polyolefin-manufacturing industrial plants. Industrial processes witnessed a similar evolution, from processes containing several unit operations for catalyst deash and atactic polypropylene extraction to the very simple process used today, where the polymer powder requires minimal postreactor processing. The most important polyolefins introduced with Ziegler-Natta catalysts were high-density polyethylene (HDPE), linear low-density polyethylene (LLDPE), and isotactic polypropylene (i-PP). Before the discovery of Ziegler-Natta catalysts, only low-density polyethylene (LDPE), containing both long- and short-chain branches, could be synthesized using free-radical processes. HDPE has few or no short-chain branches, and because of its great rigidity, it is used in structural applications. LLDPE is a copolymer of ethylene and R-olefins, and it shares the market (especially for film manufacture) with LDPE. Isotactic polypropylene is used for several injection molding and extrusion processes due to its excellent rigidity, toughness, and temperature resistance. One of the distinguishing characteristics of polyolefins made with heterogeneous Ziegler-Natta catalysts is that they have MWDs which are much broader than Flory’s most probable distribution expected for ionic chain growth homo- and copolymerization. For the case of copolymerization, these polyolefins also have broad * Phone: (519) 888-4567, ext. 3436. Fax: (519) 746-4979. E-mail: [email protected]. S0888-5885(96)00479-4 CCC: $14.00

and very often multimodal CCDs. It is now generally accepted that these broad distributions are caused by the presence of several types of active sites on the catalyst, although mass and heat transfer resistances can further broaden these distributions under certain polymerization conditions (Zucchini and Cecchin, 1983; Floyd et al., 1988; Soares and Hamielec, 1995a). From the knowledge of the MWDs and CCDs of polyolefins, one can estimate the number of site types present on the catalyst used for their production (Broyer and Abbott, 1982; Vickroy et al., 1993; Soares and Hamielec, 1995b,c; Soares et al., 1996). Soluble homogeneous Ziegler-Natta catalysts, such as the vanadium-based catalysts, are also active for olefin polymerization. They are commercially used mainly for the production of ethylene-propylene-diene monomer rubbers (EPDM). One of the advantages of these catalysts over the heterogeneous ones is that they produce polymers with narrow MWDs and CCDs. A narrow CCD is especially important for rubber production, since a broad CCD could lead to the formation of polymer fractions of high crystallinity, which is evidently undesirable for rubber applications. However, these catalysts are not applicable for the production of LLDPE, HDPE, and i-PP due to the low catalytic stability and inadequate stereochemical control (Tait, 1989). The technology for the production of polyolefins is again experiencing a revolution with the discovery of metallocene/aluminoxane catalysts. The most remarkable characteristics of these novel catalysts is that they permit the synthesis of polyolefins with narrow and well-controlled MWDs and CCDs at very high polymerization rates. Additionally, they can be adapted to existing polyolefin-manufacture industrial plants with only minor modifications (Langhauser et al., 1994). Polyolefins produced with metallocene catalysts are expected not only to compete with the polyolefins made with Ziegler-Natta catalysts but also to open entirely new markets for polyolefin resins. Several reviews have been published recently on metallocene catalysts (Gupta © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1145

et al., 1994; Huang and Rempel, 1995; Reddy and Sivaram, 1995; Soares and Hamielec, 1995d) and polymerization reactor engineering using metallocene catalysts (Hamielec and Soares, 1996). This article reviews how the MWDs and CCDs of polyolefins made with multiple-site-type catalysts can be described as a superposition of narrower distributions assigned to each site type. It will also be shown that the polyethylene produced with a combination of different metallocene catalysts will have a MWD that reflects the MWDs of the individual metallocenes. It has been found that the ethylene pressure during polymerization can be used to control the MWD of the formed polyethylene from broad and unimodal at low pressures to broad and bimodal at intermediate pressures to narrow and unimodal at high pressures. Mathematical Models for the MWDs of Singleand Multiple-Site-Type Catalysts For linear chains, Flory’s (1953) most probable distribution can be used to describe the instantaneous MWD of polyolefins made with single-site-type catalysts:

w(r) ) τ2r exp(-τr)

Figure 1. Representation of an experimental polypropylene chain length distribution (measured by gel permeation chromatography) into six of Flory’s most probable distributions. Experimental and predicted curves superimpose almost completely (Soares and Hamielec, 1995b).

(1)

where τ is the ratio of transfer-to-propagation rates and r is the polymer chain length. Some soluble metallocene and Ziegler-Natta catalysts produce a polymer that follows Flory’s distribution, and this is generally considered to be solid evidence of the single-site-type nature of the catalyst. Polymers that obey Flory’s distribution have a number-average chain length equal to 1/τ and a polydispersity index (PDI) of 2. The ratio of transferto-propagation rates, τ, can be calculated using the following generic equation:

τ)

Rβ RM RCTA RMAO + + + ) Rp Rp Rp Rp kM kCTA[CTA] kMAO[MAO] kβ + + (2) + kp[M] kp kp[M] kp[M]

where Rp, Rβ, RM, RCTA, and RMAO are the rates of propagation, β-hydride elimination, transfer to monomer, transfer to chain-transfer agent, and transfer to methylaluminoxane (MAO), respectively, kp, kβ, kM, kCTA, and kMAO are their equivalent rate constants, [M] is the monomer concentration, [CTA] is the concentration of the chain-transfer agent, and [MAO] is the MAO concentration. For the case of multiple-site-type catalysts, it has been proposed that each site type produces polymer chains that instantaneously follow Flory’s distribution (McLaughlin and Hoeve, 1988; Vickroy et al., 1993; Kissin, 1993; Soares and Hamielec, 1995b). In this case, the instantaneous MWD for the whole polymer will be a weighted sum of individual Flory’s distributions:

w j (r) )

∑i miwi(r) ) ∑i miτi2r exp(-τir)

Figure 2. Analytical TREF curves of several LLDPE resins (Stark, 1996).

tography (GPC). Figure 1 shows that the MWD of a polypropylene sample produced with a heterogeneous Ziegler-Natta catalyst can be described as a superposition of six of Flory’s most probable distributions. Peak broadening was estimated with narrow MWD polystyrene standards and considered to be negligible (Soares, 1994). For this catalyst-monomer system, it was found that the presence of hydrogen increased the rate of polymerization and at the same time significantly broadened the MWD of polypropylene, from a PDI of 2.5 without hydrogen to a PDI of 4.5 when hydrogen was present during polymerization. This seems to indicate that hydrogen activates site types that are dormant in its absence (Soares and Hamielec, 1996a).

(3)

where i indicates the site type and mi is the mass of the polymer produced on each site type. The conditions for using eq 3 to estimate the cumulative MWD of polyolefins made with multiple-site-type catalysts were discussed by Soares and Hamielec (1995b). When these conditions are met, one can use eq 3 to estimate the values of mi and τi from the experimental value of w j (r), as measured by gel permeation chroma-

Mathematical Models for the MWDs and CCDs of Single- and Multiple-Site-Type Catalysts Heterogeneous Ziegler-Natta catalysts produce copolymers of ethylene and R-olefins with very broad and generally multimodal CCDs, as illustrated in Figure 2. Temperature-rising elution fractionation (TREF) has been a technique fundamental to the elucidation of this phenomenon. It is now well-established that different site types produce copolymer chains with different

1146 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

Figure 4. LLDPE composition as a function of molecular weight; comparison of experimental data and model predictions (Soares et al., 1996).

Figure 3. Representation of a analytical TREF curve of a model LLDPE as a superposition of five Stokmayer’s bivariate distributions (Soares and Hamielec, 1995c).

average chemical compositions and comonomer sequence lengths (Soares and Hamielec, 1995e). Analogously to the approach adopted for describing the MWDs of linear homopolymers, the instantaneous MWDs and CCDs of linear binary copolymers made on single-site-type catalysts can be conveniently described by Stockmayer’s distribution (1945):

w(r,y) dr dy ) rτ2 exp(-rτ) dr

1

x2πβ/r

( y2βr) dy 2

exp -

(4)

where y is the deviation from the average molar fraction, F h 1, of monomer type 1 in the copolymer and

h 1)K β)F h 1(1 - F

(5)

K ) [1 + 4F h 1(1 - F h 1)(r1r2 - 1)]0.5

(6)

where r1 and r2 are the reactivity ratios for the copolymerization. For copolymers made with multiple-site-type catalysts, the overall instantaneous CCDs and MWDs can also be considered to be a weighted sum of the individual site-type distributions; therefore,

w j (r,y) )

∑i miwi(r,y)

(7)

This model has been used to qualitatively describe the analytical TREF curves of LLDPE and i-PP (Soares and Hamielec, 1995c), as shown in Figure 3. It should be possible to obtain polymerization kinetic parameters such as τ, r1, and r2 per site type by fitting eq 7 to experimental TREF and GPC curves, although the

nonidealities in TREF fractionation can make this estimation inaccurate. In a recent paper, Soares et al. (1996) applied some of these modeling concepts to propose a new methodology for estimating τ, F h 1, r1, and r2 per site type using a combination of GPC and chemical composition of GPC fractions measured by Fourier transform infrared spectroscopy. They were able to obtain τ and F h 1 for each catalytic site type for three samples of LLDPE made with heterogeneous Ziegler-Natta and Phillips catalysts. Figure 4 compares the experimental and predicted values for the curve of molecular weight vs chemical composition for one of the LLDPE samples. Mathematical Models for the MWDs, CCDs, and Long-Chain Branching of Single-Site-Type Catalysts Constrained geometry metallocene catalysts (CGCs) can be used to produce polyethylene and ethylene-Rolefin copolymers with long-chain branches (Lai et al., 1993; Montagna, 1995). These polyolefins have remarkable new properties, since they combine the good mechanical properties of polyolefins with narrow MWDs with the easy processability of branched polyolefins. It has been demonstrated (Soares and Hamielec, 1996b, 1997) that the molecular weight and chemical composition distributions for polymer populations of a different number of long-chain branches per chain, n, can be simply described by

w(r,y,n) )

1 r2n+1τ2n+2 × (2n + 1)! y2r 1 exp dy (8) exp(-rτ) dr 2β x2πβ/r

( )

Use of eq 8 permits a detailed description of the polymer populations produced with CGCs and should be useful for the determination of a property-structure relationship for these polymer resins. Production of Polyolefins with Designed MWDs with Metallocene Catalysts The techniques described in the last sections are useful to estimate the number of active sites present and the relative amount of polymer made on each site of a multiple-site-type catalyst under different polymerization conditions. The combination of single-site-type catalysts can be used to produce polyolefins with controlled MWDs and CCDs.

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1147

Although narrow MWD polyolefins are adequate for applications such as injection molding and precision injection, several other commercial uses such as extrusion molding, thermoforming, and production of films require broad MWDs. By the appropriate combination of different metallocenes, it should be possible to design multiple-site-type catalysts to make polymers with a specified MWD and CCD. Additionally, one or more types of metallocene catalysts can be supported on inorganic or organic carriers to form a single- to multiple-site-type heterogeneous catalyst that can be used in existing polyolefin-manufacture reactors (Welborn, 1987, 1991, 1992, 1993). The advantages of these catalysts over conventional heterogeneous ZieglerNatta catalysts is the better control over the number of active-site types present on the support. Another incentive for supporting metallocenes is the enhancement of stereochemical and regiochemical control of those catalysts and the need of a lower MAO/transition-metal ratio (Chien and He, 1991; Kaminsky and Renner, 1993; Soga and Kaminaka, 1993). Polymerizations of ethylene and ethylene-R-olefins using single and combined metallocene catalysts (soluble or supported) are under investigation in our laboratories. It is possible to evaluate the interactions between the different metallocene types in the combined catalyst from the knowledge of the MWDs and CCDs of polyolefins made with single-site-type metallocene catalysts. For instance, if no interaction occurs, each metallocene would behave in the combined catalyst as it would behave alone under the same polymerization conditions. Therefore, the MWDs and CCDs of polyolefins made with a combination of metallocene catalysts would be equal to the average (properly weighted according to the relative amount of polymer made with each metallocene type) of the MWDs and CCDs of polyolefins made with each metallocene catalyst alone. If each metallocene behaves as a single-site-type catalyst, then eqs 3 and 7 can be used to predict the instantaneous MWD and CCD of the combined catalyst. The polymerizations of ethylene using supported single- and two-type metallocene catalysts seem to support this hypothesis. By controlling the amount and type of metallocene catalyst and polymerization conditions, it has been possible to synthesize polyethylene with broad-to-narrow, unimodalto-bimodal MWDs. Experimental Procedure Ethylene was polymerized in a Parr miniautoclave reactor. Purified toluene (refluxed over metallic sodium under high-purity nitrogen atmosphere) was used as the diluent. Bis(cyclopentadienyl)hafnium dichloride (Cp2HfCl2) and rac-ethylenebis(indenyl)zirconium dichloride (Et[Ind]2ZrCl2) were used as the catalysts, and MAO was used as the cocatalyst. Et[Ind]2ZrCl2 was synthesized following the procedure described by Huang (1996), and Cp2HfCl2 was purchased from Aldrich. MAO was kindly donated by Albermarle Corp. as a 10 wt % solution in toluene. CP-grade ethylene was purchased from Linde and used without any further purification. Ultrahigh-purity (UHP) grade nitrogen was purchased from Linde and further purified by passing through deoxygen and molecular sieve beds. Nitrogen was used to provide an inert atmosphere for catalyst preparations and for purging the polymerization reactor. Figure 5 shows the polymerization reactor system used for the production of polyethylene. The reactor is operated in the semibatch mode, and it is equipped with

Figure 5. Polymerization reactor system.

Figure 6. Data acquisition and reactor control system.

a mass flowmeter and a temperature control unit comprising a cooling coil and an electric heater. The control is performed by a personal computer through analog-to-digital (A/D) and digital-to-analog (D/A) converters, as shown in Figure 6. Two independent proportional-integral (PI) control loops are used to control the flow of cold water in the cooling coil and the power input to the electric heater. Using this control loop, the bulk polymerization temperature is maintained to within (0.2 °C of the set point. For MWD control, it is fundamental to have precise control of the polymerization temperature, since temperature oscillations during polymerization will broaden the MWD significantly (Eskelinen and Sepa¨la¨, 1996). The reactor containing diluent is brought to the polymerization temperature under a continuous nitrogen purge. Premeasured amounts of supported catalyst slurry and MAO are introduced in the reactor via transfer needles under nitrogen pressure. MAO is always injected first to scavenge any impurities left in the reaction medium. The polymerization starts by pressurizing the reactor with ethylene, and the pressure is kept constant by feeding ethylene on demand to the reactor. The reaction rate is measured by monitoring the flow of ethylene to the reactor by the on-line mass flowmeter. The polymerization is terminated by closing the monomer feed line, depressurizing the reactor, and adding acidic methanol to the polymerization medium to deactivate the catalyst. The final product is carefully washed with ethanol, filtered, and dried in a vacuum oven overnight. Standard Schlenk techniques are used for supporting the catalyst under dry nitrogen atmosphere. Davidson 952 silica is calcinated at 500 °C for 5 h under UHP grade nitrogen flow. Calcinated silica (5.0 g) is placed in a Schlenk flask, and 50 mL of purified toluene is added. While continuously stirring using a magnetic

1148 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

Figure 7. MWDs of polyethylene produced with silica-supported Et[Ind]2ZrCl2 (lower MW peak), Cp2HfCl2 (higher MW peak), and Et[Ind]2ZrCl2 + Cp2HfCl2 (molar ratio Zr/Hf ) 0.36) at 40 °C.

bar at 50 °C, 10 g of MAO solution is added to the system dropwise. After addition of MAO, the system is kept at 50 °C for 3 h and then allowed to cool to room temperature. Using Schlenk-type fritted glass, the solids are recovered and washed with 20-mL aliquots of purified toluene 5 times. Under vacuum, the MAOtreated silica is dried, and finally, nitrogen is flowed through the free-flowing powder. The MAO-treated silica is stored in a glovebox and used for supporting metallocenes. For metallocene support, the same procedure used for supporting MAO is repeated, using the MAO-treated silica, a reaction time of 1 h, and a metallocene solution in toluene. Also, before the filtration step, ethylene is added to the reaction medium to prepolymerize the supported catalyst without adding any additional MAO. It is assumed that only the supported metallocene will be active for the prepolymerization (Hoel et al., 1994). The recovered supported catalysts form a free-flowing powder. The MWD of the formed polyethylene is determined using a Waters GPCV 150+. Standard analysis conditions for polyethylene are adopted: (1) column and sample compartment temperatures of 140 °C, (2) flow rate of mobile phase (trichlorobenzene) of 1.0 mL/min, (3) 2.0 g of Irganox 1010 added into 4.0 L of the mobile phase as antioxidant, (4) dissolution times over 5 h to eliminate polymer aggregates, and (5) use of a universal calibration curve determined with polystyrene narrow MWD standards. Discussion of Polymerization Results Figure 7 shows the MWDs of polyethylene produced at 40 °C with individually silica-supported Et[Ind]2ZrCl2 and Cp2HfCl2 and with their combination in a Zr/Hf molar ratio of 0.36. It seems evident from Figure 7 that the observed MWD bimodality of polyethylene made with the combined catalyst is due to the presence of the two different metallocenes. The MWD of polyethylene made with the combined catalyst is slightly narrower than the one expected by the direct combination of the MWDs produced with the individual metallocenes. This might indicate that the two metallocenes interact in the combined catalyst or, more likely, that the supporting conditions for the individual and combined metallocenes were slightly different, resulting in supported catalysts with somewhat distinct behavior.

Figure 8. MWDs of polyethylene produced with silica-supported Et[Ind]2ZrCl2 (lower MW peak), Cp2HfCl2 (higher MW peak), and Et[Ind]2ZrCl2 + Cp2HfCl2 (molar ratio Zr/Hf ) 0.36) at 50 °C.

Figure 9. MWDs of polyethylene produced with silica-supported Et[Ind]2ZrCl2 + Cp2HfCl2 (molar ratio Zr/Hf ) 0.36) at 40 °C as a function of ethylene pressure.

Figure 8 shows similar results for a polymerization done at 50 °C. Since the area under the GPC curve is proportional to the weight of the polymer, the relative amount of polyethylene made on Cp2HfCl2 sites is smaller than that obtained at 40 °C. This can be explained by assuming that either the activation energy for Et[Ind]2ZrCl2 sites is higher or that Cp2HfCl2 sites deactivate faster at a higher polymerization temperature. It is clear, however, that MWD of the whole polymer can be controlled by varying the polymerization temperature. Figure 9 shows the effect of varying the polymerization pressure at 40 °C on the MWD of the combined catalyst. Surprisingly, the MWD varies from broad and unimodal at the lower pressure (5 psig) to broad and bimodal at intermediate pressures (50-150 psig) and finally to narrow and unimodal at the higher pressure (200 psig). By examining the lower molecular weight (MW) peak, it seems that the MW for Et[Ind]2ZrCl2 is not greatly affected by the ethylene pressure. This indicates that the ratio of transfer-to-propagation rates, τ, does not depend on the monomer concentration, and therefore, the leading chain-transfer mechanism for Et[Ind]2ZrCl2 under these polymerization conditions seems to be chain transfer to monomer. According to eq 2,

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1149

τInd )

kM kp

(9)

For Cp2HfCl2, the MWD shifts to higher molecular weight averages when the monomer pressure is increased, which indicates that β-hydride elimination and eventually transfer to MAO (notice that no chaintransfer agent was used in these experiments) is the controlling transfer step. According to eq 2,

τHf )

kMAO[MAO]

kβ kp[M]

+

kp[M]

(10)

The fact that the MW peak related to Et[Ind]2ZrCl2 is not significantly affected by the ethylene pressure and that the MW peak related to Cp2HfCl2 shifts to high MW values at increasing ethylene pressures permits the control of the MWD by only varying the ethylene pressure in the reactor, which is a very versatile way of controlling the MWD shape and MW averages. It can also be seen from Figure 9 that the relative amount of polyethylene made on Cp2HfCl2 sites is inversely proportional to the ethylene pressure. The MW peak associated with Cp2HfCl2 sites becomes increasingly smaller as the ethylene pressure increases, and at 200 psig, the MWD becomes unimodal. At this pressure, apparently only a small amount of polyethylene is produced on the Cp2HfCl2 sites, as attested by the high MW shoulder on the MWD. Since the polymerization rate using Ziegler-Natta and metallocene catalysts is generally assumed to be directly proportional to the monomer and catalyst concentrations, the relative amounts of polyethylene made on each site type should be independent of the ethylene pressure. Although there is insufficient experimental data at present to propose a definite explanation for this observation, some reasonable hypotheses can be made. It might be that Cp2HfCl2 sites deactivate faster than Et[Ind]2ZrCl2 sites at higher monomer pressure, therefore altering the relative amounts of polymer produced on each site type. This deactivation might be caused by an increasing level of impurities in the reactor (introduced as contaminants in the ethylene feed) at higher ethylene pressures. It is also reasonable to assume that higher polymerization rates at higher ethylene pressures lead to the formation of hot spots inside the catalyst particles and that Cp2HfCl2 sites are more sensitive to this temperature increase. Since most of the hot spots should occur at the initial stages of polymerization (Dube´ et al., 1997), a significant fraction of Cp2HfCl2 sites can become inactive early in the polymerization. The results shown in Figures 7 and 8 seem to support this hypothesis, since the relative amount of polymer produced by Cp2HfCl2 at 50 °C is less than the one produced at 40 °C, which might indicate that Cp2HfCl2 is more sensitive to thermal deactivation than Et[Ind]2ZrCl2. Diffusional resistances could also be used to explain the observed behavior. Since Cp2HfCl2 sites produce high-MW chains and since MW increases with increasing monomer pressure, it might also be possible that the Cp2HfCl2 sites are encapsulated by high-MW polyethylene chains at higher ethylene pressures and thus become inaccessible for polymerization. Figure 10 shows the results of a similar set of experiments performed at 50 °C. Similar trends are observed, but now the MWD becomes unimodal at the lower pressure of 150 psig. This is consistent with both hypotheses of site deactivation and encapsulation by polymer chains.

Figure 10. MWDs of polyethylene produced with silica-supported Et[Ind]2ZrCl2 + Cp2HfCl2 (molar ratio Zr/Hf ) 0.36) at 50 °C as a function of ethylene pressure.

Conclusions The MWDs and CCDs of polyolefins made with multiple-site-type catalysts, such as heterogeneous Ziegler-Natta catalysts, can be conveniently described as a superposition of narrower distributions assigned to each individual site type on the catalyst. Several mathematical methods are available to obtain these individual distributions from the knowledge of the overall (experimental) MWDs and CCDs of these polyolefins. These mathematical modeling techniques permit the estimation of kinetic polymerization parameters per site type and can be used to enhance the understanding of these complex catalytic systems. This mathematical analysis can be reversed to design the MWDs and CCDs of polyolefins made with a combination of single-site-type catalysts, such as metallocene catalysts, from the knowledge of the individual distributions for each metallocene type. It has been shown that ethylene pressure can be used to control the shape of the MWD and the molecular weight averages of polyethylene produced with a two-site-type silicasupported metallocene catalyst, from narrow-to-broad and from unimodal-to-bimodal distributions. This is a very convenient way of controlling the MWD, since the same catalyst can be used with just minor adjustments on the polymerization conditions. Several aspects of this and similar catalytic systems are currently under investigation in our laboratories and will be the subject of a future publication. Nomenclature [CTA] ) concentration of the chain-transfer agent F h 1 ) average molar fraction of monomer type 1 in the copolymer kβ ) rate constant of β-hydride elimination kCTA ) rate constant of transfer to the chain-transfer agent kM ) rate constant of transfer to the monomer kMAO ) rate constant of the chain transfer to MAO kp ) rate constant of propagation K ) defined in eq 6 m ) mass fraction of the polymer [M] ) monomer concentration [MAO] ) MAO concentration n ) number of long-chain branches per polymer chain r ) chain length r1, r2 ) reactivity ratios

1150 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 Rβ ) rate of β-hydride elimination RCTA ) rate of transfer to the chain-transfer agent RM ) rate of transfer to the monomer RMAO ) rate of chain transfer to MAO Rp ) rate of propagation w(r) ) weight chain length distribution for single-site-type catalyst, eq 1 w j (r) ) weight chain length distribution for multiple-sitetype catalyst, eq 3 w(r,y) ) weight chain length and chemical composition distribution for single-site-type catalyst, eq 4 w j (r,y) ) weight chain length and chemical composition distribution for multiple-site-type catalyst, eq 7 w(r,y,n) ) weight chain length, chemical composition and long-chain branching distribution for single-site-type catalyst, eq 8 y ) deviation from average fraction of monomer 1 in the copolymer Greek Letters β ) defined in eq 5 τ ) ratio of transfer to propagation rates, eq 2 Subscripts Hf ) Cp2HfCl2 i ) catalytic site type Ind ) Et[Ind]2ZrCl2

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Received for review August 1, 1996 Revised manuscript received October 8, 1996 Accepted October 9, 1996X IE960479X

X Abstract published in Advance ACS Abstracts, February 15, 1997.