Deactivation Kinetic Model in Catalytic Polymerizations

62

Ind. Eng. Chem. Res. 1996, 35, 62-69

Deactivation Kinetic Model in Catalytic PolymerizationssTaking into Account the Initiation Step Martin Olazar,* Gorka Zabala, Jose´ M. Arandes, Ana G. Gayubo, and Javier Bilbao Departamento de Ingenierıá Quı´mica, Universidad del Paı´s Vasco, Apartado 644, 48080 Bilbao, Spain

A kinetic model based on the postulates of Langmuir-Hinshelwood has been proposed for the catalyst deactivation by coke in heterogeneous polymerizations. The originality of the model lies in the fact that deactivation is also taken into account during the initiation period by defining the activity as the useful fraction of active sites throughout the polymerization. The validity of the proposed deactivation kinetic model has been shown for the polymerization of gaseous benzyl alcohol in an ample range of operating conditions: catalysts (silica/aluminas and a HY zeolite) with different acidity and different porous structure, temperature, and monomer concentration. The results are compared with those of the previously defined model, where activity referred to the maximum polymerization rate. Introduction In previous papers (Olazar et al., 1987; Bilbao et al., 1987a), the validity of the kinetic models of LangmuirHinshelwood’s type has been proven in the polymerization reaction of gaseous benzyl alcohol on solid catalysts for obtaining thermostable poly(benzyl)s. In this way, the validity of the postulates of Clark and Bailey (1963a,b) and Maitı´ (1975) has been proven experimentally. These postulates assume that the catalyst active sites take part in the polymerization mechanisms in a manner similar to that established in conventional contact catalysis. Unfortunately, all of the catalysts used (silica/aluminas with different acidity and different porous structure and Y zeolites that are used industrially in catalytic cracking) have rapid deactivation by coke deposition, which is formed on the porous structure by degradation of the growing polymer chains. The kinetic study of deactivation in catalytic polymerization has an additional problem with respect to other reactions with rapid deactivation. In the initiation step, the deactivation occurs in parallel with the development of growing polymer chains. Consequently, it is difficult to quantify the decrease in activity in the initiation step, in which the polymerization reaction (also affected by an incipient deactivation) increases with time on stream. This situation, which is evident in the polymerization of benzyl alcohol over a wide range of operating conditions (Bilbao et al., 1987; Olazar et al., 1989, 1991), is also observed by McDaniel and Johnson (1986) in the polymerization of ethylene. In the initiation step of the polymerization, a situation is created that is similar to that raised by Agorreta et al. (1991) for reactions where processes of activation and deactivation of the catalyst take place simultaneously. This situation is relatively frequent in reactions on metallic catalysts due to changes in the oxidation state of the active sites or to the formation of new active species in the catalyst through the interaction of the catalytic surface with reactants and/or products (Hicks and Bell, 1984; Perti and Kabel, 1985; Laine et al., 1985). The initial deactivation and the activation of the catalyst can also take place simultaneously in acidic catalyst. Lukyanov (1990, 1991) observed the formation of strong acidic sites by the action of steam coming from the reaction medium in H-ZSM5 zeolites in the transformation of methanol into gasolines. 0888-5885/96/2635-0062$12.00/0

Nevertheless, in catalytic polymerizations, no catalyst activation process takes place in the initiation step, but the development of the growing chains does occur, so that a kinetic model in which activity increases with time on stream, reaches a maximum value (above unity), and then decreases cannot be accepted. Consequently, the kinetic models established for reactions with simultaneous activation-deactivation are not suitable for describing the deactivation in catalytic polymerizations. In a previous paper (Olazar et al., 1991), a data analysis method was proposed to calculate mechanistic kinetic equations of deactivation in polymerization reactions on solid catalysts. The method is an extension for these reactions of the deduction of kinetic equations of deactivation of the Langmuir-Hinshelwood type (Corella and Asuá, 1982; Corella et al., 1986). The method only quantifies the deactivation when the decrease in the polymerization rate is noticeable and requires the identification of the previous period of initiation, in which the polymerization rate increases with time on stream. The treatment proposed gives way to equations that have been proven to be valid for the reactor design (Bilbao et al., 1987b, 1989), but it has some disadvantages such as the following: (a) the deactivation during the initiation period is not taken into account; (b) the initiation period is erroneously defined as it is affected by the deactivation; and (c) the polymerization kinetics has a complex mathematical definition that requires, in addition to the rate expressions for the main and deactivation reactions, an equation (empirically deduced) for polymer evolution with time on stream during the initiation period (Olazar et al., 1989). In this paper, a kinetic model that takes into account the deactivation from zero time on stream, that is to say, including the initiation period, is proposed. Consequently, the effect of the time on stream is described with only one kinetic equation that fits the data of the three characteristic steps of catalytic polymerizations, which are initiation, developed polymerization, and rapid deactivation (Bilbao et al., 1987a; Olazar et al., 1989). Experimental Section Three catalysts have been studied (Table 1). The I-35 catalyst is a microporous silica/alumina (Aguayo et al., © 1996 American Chemical Society

Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 63 Table 1. Catalyst Properties surface pore real particle area volume density density catalyst (m2 g-1) (cm3 g-1) (g cm-3) (g cm-3) I-35 MZ-7P (I-35)p

286 120 216

Physical Properties 0.56 2.37 1.02 0.23 2.10 1.39 1.06 2.56 0.68

particle size (mm) +0.32 to -0.50 +0.15 to -0.20 +0.32 to -0.50

acidity (mg of n-butylamine/g of catalyst) catalyst I-35 MZ-7P (I-35)p

% Al2O3 12 33 12

pK + 6.8

pK + 4.8

Chemical Properties 48 35 110 46 26 18

pK+ 3.3

pK + 2.8

26 12 11

23 8 10

1987). The (I-35)p catalyst is prepared from I-35 catalyst according to the following steps: (1) by pelleting the I-35 once it was finely ground [in this way, a porous structure with mainly mesopores, which are the spaces among the catalyst particles agglutinated (Olazar et al., 1987), is obtained]; and (2) by fractioning the pellets up to the same particle size as that of the I-35 catalyst (size for which it has been proven that there are no diffusional limitations). The MZ-7P catalyst is a cracking zeolite of Akzo Chemie. Prior to the reaction, the catalysts are activated by calcination at 500 °C for 2 h. The experiments were carried out in a jet-spouted bed reactor in a discontinuous regime for the catalyst, under different reaction times and under the following reaction conditions: temperature, 250, 270, 290, and 310 °C; space time, 0.4 g of catalyst h/mol; partial pressure of benzyl alcohol at inLet (diluted with N2), between 0.06 and 1 atm. The reaction equipment used is outlined in Figure 1 (Olazar et al., 1994a). The reactor is of Pyrex glass and has the following geometric factors: column diameter, 0.12 m; base diameter, 0.02 m; conical section height, 0.20 m; angle, 30°; inlet diameter, 0.004 m. The height of the upper cylindrical section is 0.20 m. The reactor is installed in a convection oven. The circulating air is driven by blowers and heated by the resistance R2. From the tank T1, the benzyl alcohol is driven through a dispenser valve DV toward the reactor. Subsequently, it is mixed with a nitrogen stream whose flow rate is controlled by the mass flow meter-regulator FIR1, and it is vaporized and preheated by means of the preheater R1. The gaseous flow enters the lower section of the reactor. The non-reacted alcohol, together with the nitrogen (diluent) and the steam produced, leave the reactor through the upper section. The departing gaseous mixture is cooled in a condenser, and the resulting alcohol and steam are collected in tank T2. This tank is periodically emptied by means of the pneumatic valves NV7 and NV8. The nitrogen is recirculated. Valve V1, which is controlled by a vacuum meter, allows nitrogen to enter the circuit when the pressure decreases (the system is maintained under atmospheric pressure), and valve V2, which is controlled by a pressure meter, releases gases to the outside if the pressure in the circuit increases excessively. Total pressure in the circuit is around 1 atm. Partial pressure changes according to alcohol/nitrogen mixture. The preheater must vaporize and raise the temperature of the mixture to the set value. A routine for data acquisition and real time control of the equipment components has been prepared. We must insist on the contact regime of the reaction equipment in Figure 1. The jet-spouted bed is an original regime, also named the dilute spouted bed

(Epstein, 1991), which is obtained by expansion of the spouted bed in exclusively conical contactors (Olazar et al., 1992; San Jose´ et al., 1992, 1993). The jet-spouted bed is especially interesting as a reactor in catalytic polymerizations due to the fact that vigorous movement of the solid is obtained, which allows for the treatment of both sticky solids without the fusion of particles and mixtures of different particle size with low segregation (Olazar et al., 1993, 1994a). Due to these qualities, the jet-spouted bed has already been used in the exploitation of wood and agroforest residues (Olazar et al., 1994b) and in the gasification of bituminous coals (Uemaki and Tsuji, 1986, 1991). It has been proven that the jet-spouted bed reactor has better performance in the polymerization of benzyl alcohol than the fluidized bed reactor used in previous studies (Bilbao et al., 1987a, 1989; Olazar et al., 1989). Steady state operation can be carried out with a continuous catalyst feed without plugging problems at the reactor exit, as well as with lower dilution of the catalyst with an inert solid (silica gel). The cyclic movement of the particles into the rector, peculiar to the spouted beds (Figure 1), favors thermal equilibrium and perfect mix conditions of the catalyst covered with polymer, which means that temperature uniformity is achieved. Gas inlet velocity must be suitable (on the order of 5-10 m/s). Bed voidage in the steady state reaches a constant value in the range between 0.80 and 0.95. Kinetic Model The steps proposed for the mechanism are based on the theoretical postulates of Clark and Bailey (1963a,b) and Maitı´ (1975) and assume that in polymerization reactions there is an active site intervention similar to that of contact catalysis. 1. Adsorption and monomer activation step: During the initiation period of the polymerization, this reaction step will not be in equilibrium: kM

k1

r

M+L\ yk z M1 98 M1* M1

2. Reaction step (propagation) kp

M1 + M1* 98 M2* kp

M1 + M2* 98 M3* l kp

M1 + Mn-1* 98 Mn* kp

(n - 1)M1 + M1*98 Mn* 3. Termination 3.1. Spontaneous desorption kd

Mn* 98 Pn + L 3.2. Desorption by monomer km

Mn* + M1 98 Pn + L + M1

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Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996

Figure 1. Diagram of the reaction equipment.

3.3. Deactivation kc

bMn* 98 Cn + (b - 1)L kc

bMn* + (1 - b)L 98 Cn

for b g 1 for b e 1

In this last step, the parameter b indicates the number of polymer chains irreversibly adsorbed that finally occupy a catalyst active site. A value of b > 1 corresponds to a situation in which the polymer chains that constitute the coke and block the active sites are partially inert. This inert coke fraction will consequently be displaced by the growing polymer chains. This situation and the displacement of the inert coke will be easier in catalysts with meso- and macroporous structure. A value of b < 1 corresponds to a deactivation that occurs simultaneously by active site blockage and by pore blockage. This situation is more probable in microporous catalysts. An active site balance, in which the number of sites occupied by growing polymer chains, [Mn*], is neglected, gives

[N] - [Cn] ) [M1] + [L]

(1)

Activity is defined as the fraction of active sites that are not blocked by coke:

a)

[N] - [Cn] [N]

(2)

This expression of activity exclusively refers to the catalyst state and is the commonly used definition in kinetic studies of deactivation (Froment and Bischoff, 1979; Corella and Asuá, 1982; Butt and Petersen, 1988;

Gayubo et al., 1993a,b). With this definition, the error introduced in the modeling by considering the activity as a ratio of reaction rates, which implies erroneous attribution of the influence of the development of the growing polymer chains (initiation step) in the catalyst deactivation, is eliminated. By using eqs 1 and 2, the disappearance rate of the activated monomer is expressed as a function of the partial pressure of the monomer, PA:

d[M1] ) kMrPA[N]a - kMrPA[M1] - kM1[M1] dt

(3)

The general expression for the kinetic equation of deactivation is deduced:

da ) kc′[M1]2b dt

-[N]

for b g 1

da ) kc′[M1]2bL1-b ) dt kc′[M1]2b{a[N] - [M1]}1-b

(4)

-[N]

for b e 1 (5)

The polymerization rate is related to the adsorbed monomer concentration, [M1], as follows:

dP ) kd′[M1]2 + km′[M1]3 + kc′[M1]2b dt

for b g 1 (6)

dP ) kd′[M1]2 + km′[M1]3 + kc′[M1]2b{a[N] dt for b e 1 (7) [M1]}1-b where

Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 65

kc′ ) kc(2kpk1)b

(8)

kd′ ) 2kdkpk1

(9)

km′ ) 2kmkpk1

(10)

The third sum in eqs 6 and 7 corresponds to coke deposited in the catalyst. Calculation of the Kinetic Parameters In Figures 2 and 3, the experimental results of the polymer amount deposited (internal coke included) have been plotted for the three catalysts at 270 and 310 °C, respectively. Each curve corresponds to one partial pressure of benzyl alcohol in the feed. The results of Figures 2 and 3 and those corresponding to other operating temperatures have been fitted to the equations of the kinetic model previously described. The partial pressure of the monomer in the reaction medium has been calculated as a function of time on stream:

PA )

PAo + PAs ) PAo(1 - XA/2) 2

(11)

where the monomer conversion, XA, is

XA )

MA dP/dt MP FAo

(12)

The kinetic parameters calculated by fitting the experimental results to the proposed kinetic model by means of the Marquardt method (Marquardt, 1963) are as follows:

I-35 catalyst b ) 1.5 kc′ ) 3.6 × 1032 exp(-37600 ( 400/T)

(13)

kd′ ) 8.2 × 1019 exp(-21600 ( 300/T)

(14)

km′ ) 2.4 × 1014 exp(-15300 ( 200/T)

(15)

(I-35)p catalyst b ) 1.5 kc′ ) 7.6 × 1012 exp(-15800 ( 200/T)

(16)

kd′ ) 1.8 × 1010 exp(-11100 ( 100/T)

(17)

km′ ) 10.4 exp(-1600 ( 100/T)

(18)

MZ-7P zeolite catalyst b ) 1.3 kc′ ) 3.6 × 1014 exp(-17900 ( 200/T)

(19)

kd′ ) 8.6 × 107 exp(-8400 ( 100/T)

(20)

km′ ) 1.9 × 1020 exp(-23100 ( 300/T)

(21)

Figure 2. Values of polymer deposited vs time on stream, at 270 °C, for different values of partial pressure of the monomer: dashed lines, experimental values; solid lines, calculated with the proposed kinetic model.

A mechanistic kinetic model such as the one proposed in this paper, despite it accurately representing the experimental results, runs the risk that its parameters may lack the physical meaning aimed at in the model and that they are mere results of an empirical fitting. This subject has been the object of analysis in relevant literature (Butt, 1980). A criterion that must be fulfilled by the kinetic parameters of the different catalysts used in the same reaction is the one called the compensation effect (it is also called the isokinetic effect, theta rule, or linear free energy relation) (Corma et al., 1993). This

66


Figure 4. Study of the compensation effect of the calculated kinetic parameters.

Figure 5. Study of the correlation between the entropy change and the enthalpy change of the monomer chemisorption.

Figure 3. Values of polymer deposited vs time on stream, at 310 °C, for different values of partial pressure of the monomer: dashed lines, experimental values; solid lines, calculated with the proposed kinetic model.

criterion requires a linear relationship between the logarithm of the frequency factor and the activation energy. In Figure 4 the fulfillment of the compensation effect for the kinetic parameters obtained in this paper is shown. A theoretical basis for the fulfillment of the compensation effect is established on the basis of the linear relationship between the entropy change and the enthalpy change, as has been established by Everett (1950) for systems with physical adsorption. This linearity is not fulfilled for catalytic reactions that imply a chemical

adsorption step, but the aforementioned thermodynamic properties must be in a given range of values. As is shown in Figure 5, the values of the entropy change and enthalpy change of the chemisorption equilibrium obtained in this paper are in the region corresponding to the thermodynamic data of most chemisorption data (Butt, 1980). The validity of the kinetic model and the kinetic parameters calculated is shown in Figures 2 and 3, in which the adequacy between the calculated values of polymer quantity deposited vs time on stream and the experimental results is observed at 270 and 310 °C for the three catalysts studied. In the same way, the data corresponding to the other temperatures studied, 250 and 290 °C, also fit the model adequately. In Figure 6, the calculated values of the polymerization rate by catalyst unit mass, rP, has been plotted vs time on stream for 270 °C, where

rP )

1 dP W dt

(22)

In Table 2, the calculated values of maximum rP, (rP)M,


initiation period was not considered and activity was defined starting at the initiation period:

a′ )

rP (rP)M

(23)

The reaction without deactivation (assuming a ) 1 for any time on stream) has been simulated using the kinetic model. The fictitious initiation time, ti, corresponding to the maximum polymerization rate at those hypothetical states (this rate would be constant from ti), has been calculated. In Table 2 the influence of deactivation on this parameter is observed. The real values of the time on stream corresponding to the maximum polymerization rate, tM, obtained by considering the deactivation from the beginning of the reaction, are noticeably smaller than the values of ti calculated without considering the deactivation during the initiation period. In Figure 7, the decrease in activity (defined by eq 2) with time on stream has been plotted for the three catalysts studied and for the partial pressure of benzyl alcohol corresponding to the maximum deactivation for each catalyst at 270 °C. It is appreciated that deactivation during the initiation period is all the more important, the more active the catalyst is. The activity has decreased in this period up to 0.77 for the I-35 catalyst, 0.94 for the MZ-7P zeolite catalyst, and 0.97 for the (I35)p catalyst. In Figure 8, to describe the decrease in the activity, the greater realism of the model proposed in this paper (with activity a defined by eq 2) with respect to the model previously proposed [with activity a′ defined by eq 23 (Olazar et al., 1991)] is shown. The data correspond to the I-35 catalyst at 310 °C and PAo ) 0.65 atm. Conclusions

Figure 6. Values of polymerization rate vs time on stream, calculated with the proposed kinetic model, at 270 °C, for different values of partial pressure of monomer.

and the corresponding values of catalyst activity, aM, and time on stream, tM, are set out for the I-35 catalyst. It is appreciated that as the maximum production rate is reached, the catalyst already presents an activity decrease with respect to the fresh catalyst. In this situation, the activity presents a value of 0.77 for 310 °C and PAo ) 0.65, which are the conditions corresponding to the maximum deactivation for I-35 catalyst in the initiation step. This decrease in activity depends on the reaction conditions. This situation is not taken into account in the kinetic model previously proposed (Olazar et al., 1989, 1991), in which the deactivation during the

It has been proven that the deactivation during the initiation period can be important, so that the more active the catalyst is, the more important it will be. Consequently, the deactivation during the initiation period cannot be neglected in the kinetic model. This circumstance is of great importance in the design of the reactor that operates with a continuous catalyst feed, where the initiation time is an important fraction of the residence time of the catalyst particle in the reactor. The model proposed will allow for the rigorous quantification of the deactivation when the operation is carried out in a continuous regime for the catalyst as the use of empirical equations to quantify the polymerization rate during the initiation period will be avoided. The validity of the proposed deactivation kinetic model has been shown in an ample range of operating conditions: catalysts with different acidity and different porous structure, temperature, and monomer concentration. As the model is based on the postulates of LangmuirHinshelwood, its fulfillment is evidence of the additional interest of these postulates for catalytic polymerizations. On the other hand, from the b parameter of the model, information is deduced on the stoichiometry of the relationship between the coke that deactivates the catalyst and its active centers. The value of b > 1 [b ) 1.5 for the I-35 and (I-35)p catalysts and b ) 1.3 for the MZ-7P zeolite catalyst] means that a fraction of coke irreversibly deposited in the catalyst pore is inert and does not block active sites. A smaller value of b for the zeolite-based catalyst could be due to its porous struc-

68


Table 2. Calculated Values of the Maximum Polymerization Rate, (rp)M [g of polymer (g of catalyst)-1 min-1], the Corresponding Activity, aM, the Corresponding Time on Stream, tM (min), and the Fictitious Initiation Time, ti (min) PAo (atm) temp (°C) 250

270

290

310

(rp)M aM tM ti (rp)M aM tM ti (rp)M aM tM ti (rp)M aM tM ti

0.02

0.06

0.12

0.18

0.22

0.30

0.40

0.50

0.65

0.09 1 9.2 10.5 0.06 1 10.0 10.9 0.03 1 10.2 13.6 0.02 1 15.2 18.8

0.38 0.98 5.2 6.0 0.35 0.97 6.0 9.4 0.24 0.97 7.6 10.0 0.15 0.98 8.4 11.2

0.70 0.98 2.8 5.2 0.84 0.95 3.6 9.2 0.68 0.94 4.4 9.2 0.47 0.94 5.2 10.3

0.91 0.98 2.0 3.6 1.25 0.93 2.8 6.8 1.12 0.89 3.6 6.8 0.82 0.88 4.4 10.0

1.02 0.97 2.0 3.0 1.48 0.90 2.8 5.5 1.37 0.90 2.8 6.0 1.06 0.87 3.6 8.4

1.19 0.97 1.6 2.8 1.88 0.91 2.0 5.2 1.82 0.81 2.8 5.7 1.48 0.84 2.8 6.9

1.13 0.97 1.2 2.0 2.26 0.88 1.8 3.6 2.32 0.84 2.0 4.4 1.91 0.85 2.0 6.8

1.14 0.97 1.2 1.6 2.57 0.90 1.4 3.3 2.72 0.80 1.8 3.9 2.35 0.78 2.0 4.5

1.53 0.97 1.0 1.4 2.94 0.89 1.2 2.8 3.21 0.81 1.4 3.7 2.85 0.77 1.6 4.4

Nomenclature

Figure 7. Decrease in activity (defined by eq 2) with time on stream for the three catalysts studied and for the partial pressure of benzyl alcohol corresponding to the maximum deactivation for each catalyst at 270 °C.

Figure 8. Comparison of the decrease in the activities defined by the model proposed in this paper (a, eq 2) and by the previously proposed model (a′, eq 23) (Olazar et al., 1991).

ture with a greater proportion of micropores, which are more sensitive to coke deposition. Acknowledgment This work was carried out with financial support from the University of the Basque Country/Euskal Herriko Unibertsitatea (Project No. 069.310-EB144192).

a ) catalyst activity defined as the fraction of active sites that are not blocked by coke (eq 2) a′ ) catalyst activity defined as ratio of reaction rates (eq 23) b ) number of polymer chains irreversibly adsorbed that end up by occupying a catalyst active site Cn ) irreversibly adsorbed polymer molecule of n monomer units (coke) E ) activation energy (kcal mol-1) FAo ) molar flow of monomer at reactor inlet (mol min-1) ∆Ho ) enthalpy of the chemisorption (kcal mol-1) kc, kd, km ) rate constants of deactivation, spontaneous desorption, and desorption by monomer, respectively kc′ ) apparent deactivation rate constant [(g of polymer) min-1 (adsorbed monomer molecule)-2b] kd′ ) apparent rate constant of spontaneous desorption [(g of polymer) min-1 (adsorbed monomer molecule)-3] km′ ) apparent rate constant of desorption by monomer [(g of polymer) min-1 (adsorbed monomer molecule)-2] koc, kod, kom ) frequency factors of kc′, kd′, and km′ constants, respectively kMl, kMr ) rate constants of monomer desorption and adsorption kp ) propagation rate constant k1 ) rate constant of monomer activation on the active site L ) free active site M, M1 ) monomer molecule in the gas phase and adsorbed M1*, M2*, ..., Mn* ) active species, monomer, dimer, ..., polymer of n monomer units, respectively MA, Mp ) molecular weights of the monomer and the structural unit of the polymer N ) total number of active sites by unit weight n ) number of monomer molecules in the polymer P ) polymer weight (g) PA ) average partial pressure of benzyl alcohol between the inlet and the outlet of the reactor, respectively (atm) PAo ) partial pressure of benzyl alcohol at the inlet of the reactor (atm) PAs ) partial pressure of benzyl alcohol at the exit of the reactor (atm) Pn ) polymer of n monomer units R ) gas constant (kcal mol-1 K-1) rP, (rP)M ) polymer formation rate and polymer formation rate at t ) tM [g of polymer (g of catalyst)-1 min-1] ∆So ) entropy of the chemisorption (cal mol-1 K-1) T ) temperature (K) t ) time on stream (min) ti ) initiation time when there is no deactivation (min) tM ) time on stream for the maximum polymerization rate (min) W ) catalyst weight (g) XA ) conversion referred to monomer


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Received for review February 25, 1995 Accepted August 10, 1995X IE9501283

X Abstract published in Advance ACS Abstracts, November 15, 1995.

Deactivation Kinetic Model in Catalytic Polymerizations

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