Principles of the Morphogenesis of Polyolefin Particles - Industrial

Ind. Eng. Chem. Res. 2005, 44, 2389-2404

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Principles of the Morphogenesis of Polyolefin Particles Zdeneˇ k Grof, Juraj Kosek,* and Milosˇ Marek Department of Chemical Engineering, Prague Institute of Chemical Technology, Technicka´ 5, 166 28 Prague 6, Czech Republic

In a previous paper (Grof, Z.; Kosek, J.; Marek, M. Morphogenesis model of growing polyolefin particles. AIChE J., accepted), we have introduced the model of the morphological evolution of polyolefin particles in catalytic polymerization reactors. The model considers the polyolefin particle to consist of a large number of microelements with viscoelastic interactions acting among them. Here we present the results of the systematic mapping from the parametric space of catalyst activity, reaction conditions, mass transport resistance, and viscoelastic properties of polyolefins into the space of possible morphological developments, such as the formation of hollow particles, particles with macrocavities, regular or irregular replication of particle shape during its growth, highly porous or compact particles, the formation of fine particles, etc. The predicted particle morphologies are compared with experimental findings. We focus on the effect of temperature on the morphogenesis of polyolefin particles and identify the reaction conditions leading to the disintegration of the growing particle into fine particles, which is the unwanted phenomenon observed in industrial reactors. The causes of different pore space morphologies of Ziegler-Natta and metallocene-born polyolefin particles are also investigated. 1. Introduction In a previous paper1 we have developed the morphogenesis model of the growing polyolefin particle and presented illustrative examples of the model’s capabilities. Illustrative examples comprised the formation of various morphological features, such as hollow particles, particles with macrocavities, fine particles, attrition from the particle surface, and imperfect replication of the particle shape. In this paper we perform the systematic mapping from the parametric space of the catalyst activity, the reaction conditions, the mass transport resistance, and the viscoelastic properties of polyolefins into the space of possible morphological developments. Such a systematic parametric study is essential for any model with uncertainty in the model parameters and is required for the semiquantitative comparison of modeling results with experiments. The morphology of the polymer particle is in the context of this paper considered to be its shape, the texture of its surface, and the spatially 3D distribution of pores and polymer. The results on the evolution of the particle morphology presented here are motivated by practical industrial problems. There is no generally accepted opinion as to what are the causes of the formation of particle agglomerates, the sheeting of polymer at reactor walls, and the formation of fine particles in industrial gas-dispersion reactors. For example, a number of hypotheses can be formulated about the origins of fine particles with equivalent diameter smaller than the arbitrary selected value of 125 µm2, namely: (i) fine particles are deactivated catalyst particles with a low content of the polymer, where the deactivation was caused either by the poisoning or by the overheating of catalyst/polymer particles in the early stages of their growth; (ii) fines are formed by particles colliding mutually or with * To whom correspondence should be addressed. Tel.: +420 220 443 296. Fax: +420 233 337 335. E-mail: Juraj.Kosek@ vscht.cz.

reactor walls, where the collisions could be either of the head-on type resulting in particle impact breakage or of the attrition type that could be a significant source of fines for particles with rough-textured surfaces; (iii) fines could arise by the replication of the fraction of catalyst particles with a small diameter injected into the reactor; and (iv) fines are formed by mechanical disintegration of vigorously growing catalyst/polymer particles due to the internal tension built up inside the particle. Although only the last hypothesis is systematically explored in this paper, it is important to keep other possible mechanisms of the formation of fine particles in perspective. First models of catalytic polymerization of olefins were concerned only with concentration and temperature fields in reactors or growing polymer particles.3 Gradually, capabilities allowing prediction of the molecular architecture of the produced polymer were introduced into the models, e.g., the distribution of chain lengths or the distribution of comonomer in polymer chains. At last, complications such as the effects of particle size distribution and residence time distribution in gas- or liquid-dispersion reactors were considered.4 Most modeling studies of growing polyolefin particles employ the effective-scale reaction/transport models with perfect spherical symmetry and with various descriptions of the mass and heat transport resistances.5-7 However, mathematical models attempting to describe the formation of polymer particles with complex spatially 3D structure on the mesoscale level (0.01-100 µm) are scarce.8 The spatially 3D structure of porous or multiphase materials can be digitally represented in several ways, e.g., by the concept of the reconstructed porous media or by the network diagram of connected cylindrical pores.9 These two concepts represent well the geometrical properties of mesostructured materials and are often employed in calculations of effective diffusivity or permeability of porous media. The disadvantage of these concepts is their limited applicability to the modeling of transformation processes of the mesoscopic structure.

10.1021/ie049106j CCC: $30.25 © 2005 American Chemical Society Published on Web 02/12/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005

Figure 1. Evolution of the particle consisting of microelements with mutual force interactions.

Therefore we have introduced in the previous papers10,11 the morphogenesis model based on the discretization of the mesostructured material into a number of microelements with mutual viscoelastic interactions and applied this model to the case of the growing polyolefin particle, cf. Figure 1. Thus we can model the actual physical evolution of the structure of polymer and pore phases rather than reconstructing the geometrical replica of the porous material. The morphogenesis model can be (after some generalization) employed to model other diagenetic processes, such as dynamics of fragmentation of catalyst carriers in catalytic polymerization of olefins, colloid aggregation of silica, or foaming of polymeric materials. The microelements in the morphogenesis model can be of various types; e.g., they can represent the catalyst carrier, the active catalyst sites, the polymer phase(s), and even the gas occupying the void space. However, here we restrict the model only to one type of microelement. The morphogenesis model considers relevant reaction, transport, and mechanical processes taking place in the porous polymer particle growing due to the catalytic polymerization of olefins. Our model is not bound to the spherical symmetry of the resulting particle shape. We exclude the detailed discussion of the fragmentation of catalyst carriers, although the importance of the early stages of particle growth is recognized and supported by the modeling results. The problem of the broad spatial scale is often involved in modeling on the mesoscale level. For example, the pore size of many porous materials spans the wide range from 0.01 to 100 µm. The simultaneous computer representation of the fine and coarse spatial scales is hardly feasible, and therefore the techniques of multiscale modeling have to be applied. The multiscale approximative approach is used also in this work. When considering the multigrain morphology of polymer particles,12 the number of grains in the particle is of the order of 106 or more. Therefore the microelements in our model represent the clusters of micrograins as the building blocks of the particle rather than the individual micrograins. Morphologies different from the welldeveloped multigrain structure are also well represented by our model. In fact, the perfect multigrain morphology with well-distinguishable micrograins is the limiting case of the morphology of polyolefin particles because the micrograins are often fused together and form thus large clusters with hardly recognizable grains. Another simplification of the model used in this paper is the simple effective-scale description of the transport of monomer in the growing particle, although the model considering the morphology-dependent transport of monomer(s) has been already developed in our group.13 The problem of industrial importance is also the control of particle shape, porosity, and the presence of

large voids (macrocavities) in polyolefin particles. Investigations of sections of polyolefin particles by electron microscopy imaging and computed microtomography revealed in many cases the existence of large pores with a characteristic size larger than 5 µm separating areas of lower porosity.14-16 A large content of voids in polymer particles (or in other words, low bulk density of the product) is undesirable since it increases the cost of shipping and storage of the polymer. The hollow and ruptured particles have smaller resistance to external mechanical forces and could thus disintegrate in the reactor or during downstream processing. On the other hand, the macrocavities or small well-connected channels are essential for maintaining the adequate transport of monomer and are often a catalyst design target.17 Moreover, the pore structure of the polymer particle significantly affects the dynamics of the degassing operation in the downstream processing of products, where the degassing unit could even limit the capacity of the production line. The quality of the porous structure also affects the penetration of air/steam in the catalyst deactivation operation and micromixing with optionally applied additives. Large pores in particles (here referred to as macrocavities) could be formed as a consequence of the following: (i) replication of the inhomogeneity of the porous structure of the catalyst carrier, e.g., of spraydried silica containing large pores, (ii) dynamics of the fragmentation of porous catalyst carriers resulting in partially disintegrated catalyst/polymer particles in the early stage of their growth, (iii) agglomeration of several small polymer particles in the reactor, e.g., of sticky particles at higher temperatures, (iv) clogging of pores by the polymer, (v) mass-transport limitations in the growing polymer particle resulting in the nonuniform concentration field and in the slow formation of polymer phase in some parts of polymer particles, e.g., in the center, (vi) uneven distribution of catalyst activity in the catalyst/polymer particle. This paper investigates only scenarios v and vi where the underlying mechanism in the terminology of our model is the nonuniform growth of microelements. The free-flowing polymer powder consisting of spherical particles is desired as the product at the reactor outlet.15 Gentle breakup of the porous catalyst carrier and controlled temperature are the prerequisites of the best morphology; hence the polymerization of highly active catalyst particles is often employed. Polymer particles replicate under gentle polymerization conditions not only the shape of catalyst particles but also the shape of their particle size distribution during the batch polymerization.2,18 The paper is organized as follows. First, the principal equations of the morphogenesis model are briefly summarized and the relation between viscoelastic properties of polyolefins and their molecular architecture is outlined to facilitate the discussion of model parameters and results. Results of the mapping from the parametric space of reaction conditions and architecture of catalyst particles into the space of morphologies of final polymer particles are then presented. Finally we discuss systematically the effect of temperature on the evolution of the morphology of polyolefin particles and the causes of different morphologies of Ziegler-Natta and metallocene-born polyolefin particles.

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2391

2. Mathematical Model The derivation of the mathematical model of morphogenesis was thoroughly discussed in our previous paper1 and is briefly outlined in the Appendix. Here we give the principal equations in the dimensionless form to facilitate the discussion of model parameters and results. The polymer particle is discretized into a number of spherical microelements of the same type. The translation movement of the ith microelement is governed by the equations

dXi ) Vi dτ

(

4 d Fi πRi3Vi 3

)

dτ

)

τ02 4

R0

(

+ ∑Fternary ) ∑j Fbinary ij ijk j,k

(1)

(2)

where τ is the dimensionless time; Xi, Vi, Ri, and Fi are the dimensionless coordinate of the center, the dimensionless velocity, the dimensionless radius, and the density of the ith microelement, respectively. For the sake of simplicity, the density Fi is considered to be constant. The characteristic time and length scales are τ0 and R0, where R0 is selected, for example, as the initial radius of microelements. The summations of forces Fij and Fijk are carried out over all binary and ternary interactions with connected neighbors of the ith microelement. The growth of the ith microelement is governed by the kinetic equation

Mcat,iCi dRi )Γ dτ R2

(3)

i

where Γ is the dimensionless growth factor related to the polymer yield rate AY in kgpol/(gcat h) by the relation Γ ) τ0AY/(3 × 3.6), Mcat,i is the dimensionless mass of catalyst in the ith microelement, and Ci is the dimensionless concentration of monomer surrounding the ith microelement, respectively. The monomer concentration ci is transformed into its dimensionless counterpart Ci as Ci ) ci/cbulk, where cbulk is the bulk concentration. The evolution of the concentration profile of monomer Ci in the growing polymer particle is governed via the simple effective-scale description

Ci )

Rpart sinh(Θsi/Rpart) si sinh Θ

(4)

where Rpart is the radius of the polymer particle, si is the distance of the ith microelement from the particle center, and Θ ) Θ0(Rpart,0/Rpart)1/2 is the Thiele modulus which decreases as the particle grows. Here Θ0 and Rpart,0 are the initial Thiele modulus and the initial radius of the particle at time τ ) 0, respectively. The initial value of the Thiele modulus is

Θ0 ) Rpart,0

x

AYFpart,0

3.6cbulkMMDef

(5)

where Fpart,0 is the initial density of the particle, MM is

the molecular weight of the monomer, and Def is the effective diffusivity. The constitutive equations describing the force interactions and the rules for the connectivity of microelements are described below. The generalization of model equations to more general cases, such as copolymerization or nonisothermal conditions, is obvious. 3. Viscoelastic Properties of Polyolefins The concept of mutual force interactions acting among individual microelements and their projections into force vectors was introduced by Grof et al.1 and is summarized in the Appendix. The viscoelastic character of the force is governed in our simulations by the simple Maxwell model consisting of a spring with elastic modulus E and a dashpot with viscosity η connected in series and described by the equation

deAB dσ σ )E dτ dτ τR

(6)

where σ ) (FAB/AAB) is the stress, eAB is the strain, τ is the time, and τR ) η/E is the relaxation time. The force FAB acts between two microelements A and B sharing the contact area AAB. The resistance against the deformation of the triplet of connected microelements A-V-B with the bonding angle R ) ∠AVB is proportional to the deviation from the original angle R0 at the start of the simulation (or at the time when the triplet A-V-B was formed):

FR/AAB ) -G(cos R - cos R0)

(7)

where G is the “bending” modulus. Here we discuss briefly the temperature dependence of viscoelastic properties, the stress-strain deformation characteristics, and the real viscoelastic behavior of polyolefins as the guidance for the selection of model parameters. 3.1. Stress-Strain Characteristics of Polyolefins. For the purpose of calculation of force interactions, the computational algorithm keeps the list of connected pairs and triplets of microelements. These lists are updated; i.e., new connections are created or existing connections are removed in each step of dynamic simulation. Let us describe the rules for creation/ deletion of bonds of microelements. Microelements A and B become connected when they touch each other. Two microelements become disconnected if the strain exceeds a maximum value emax, i.e., when

eAB )

|u| - u0 > emax u0

(8)

where u0 ) rA + rB is the equilibrium and u ) |u| is the actual distance between the microelements A and B. One of the connections A-V or B-V is disconnected in the triplet of the connected microelements A-V-B also if the deviation between the actual angle R ) ∠AVB and the initial angle R0 exceeds the limiting value Rmax:

|R - R0| > Rmax

(9)

The standard stress-strain characteristics of polymers measured at room temperature and at the slow rate of deformation are not suitable for the estimation of the maximum elongation strain emax and the maxi-

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Table 1. Typical Mechanical Properties of Commercial Grades of Polyolefins at Room Temperature polymer

density/kg/m3

tensile strength/MPa

modulus E/MPa

HDPE LLDPE PP

945-953 920 910

20-25 10 34-39

700-1100 300 1400-1800

mum bending angle difference Rmax at the reactor temperature required in the morphogenesis model. The parameters emax and Rmax are therefore chosen empirically, and this section provides some guidance for this selection. Let us note that the values of both emax and Rmax used in this work are relatively small because of the employed simple approximation of the behavior of polyolefins. The Young modulus and the elongation at break increase with the molecular weight of the polymer. The elastic modulus decreases with increasing temperature, and the tested material ruptures at larger strains. For example, the stress in the polypropylene sample at 2% strain obtained from the measurements of Ariyama19 at temperatures of 25, 50, and 75 °C was 28, 20, and 12 MPa, respectively. As the content of the comonomer in the ethylene copolymers increases, the tensile behavior at room temperature changes from necking and cold drawing typical of a semicrystalline thermoplastic to the uniform drawing and high recovery characteristics of an elastomer.20,21 Maximum strain emax allows a rough approximation of various types of tensile behavior (e.g., the necking or brittleness). Yield stress and the elastic modulus decrease with the enhanced comonomer content because the comonomer reduces the crystallinity and the size of crystallites, cf. Table 1. Ethylene homopolymer thus has the highest elastic modulus and yield stress. Incorporation of 2.8 mol % of octene comonomer results in the decrease of the modulus from 1.5 to 0.4 GPa reflecting the decrease in the crystallinity from 77 to 44.20 Tensile yield strain increases as the crystallinity of the polyethylene sample is reduced.22 Isotactic polypropylene (PP) is stiffer than the high-density polyethylene (HDPE) despite the fact that HDPE is more crystalline and its crystallites are stiffer than PP ones.23 HDPE is the toughest material in the family of polyethylenes. In practice, its superior toughness is exhibited only at relatively slow draw rates ( emax but because the deformation of the “bonding” angle |R - R0| in the triplet of the connected microelements exceeds the maximum deformation angle Rmax ) 5°. The effect of viscoelastic relaxation of stress characterized by the relaxation time τR is displayed in Figure 7. It is surprising that the fast growth at AY ) 100 kgpol/(gcat h) causes the formation of hollow particles and macrocavities, but no disintegration into fines is observed. Formation of fines is observed for AY ) 10 or 36 kgpol/(gcat h) at large relaxation times. The elastic limit formally corresponds to τR f ∞ and is shown for comparison purposes. In qualitative agreement with experimental observations of Llinas and Selo,43 the particle can disintegrate into fines also at relatively low reactor temperatures, when the polymer yield AY is low and the polymer is brittle. Hence it is suggested to run the polymerization in the optimum temperature window a few degrees below the sintering temperature.

Figure 7. The effect of polymer yield rate AY (in kgpol/(gcat h)) and the relaxation time τR of the Maxwell model on the morphology of the polyolefin particle at t ) 2 s after the start of its growth. Ternary interactions are characterized by the bending modulus G ) 0.2E, and the maximum elongation is set to emax ) 0.04. The rectangle is used for scaling and shows the initial size of the particle.

The systematic exploration of the effect of mass transport resistance inside the growing particle on its morphogenesis yields analogous results to those displayed in Figure 7. Generally, the mass transport resistance depends on reaction conditions (e.g., on pressure), on the type of the monomer, and on the distribution of pores and semicrystalline polymer phase inside the growing particle. As already mentioned above, the calculation of transport effects associated with the evolving structure of the particle is not the subject of this work. Let us also note that the shape of the particle is not perfectly replicated during its growth, especially in the case of significant mass transport limitation inside the particle or significantly nonuniform distribution of catalyst activity. Pater et al.44 conducted polymerization of propylene in the liquid-phase reactor and observed irregular texture and poor replication of the shape of catalyst particles at higher reaction rates corresponding to catalyst activities of 60-80 kgpol/(gcat h) achieved by the polymerization at higher temperatures (80 and 90 °C) and in the presence of hydrogen. In the absence of hydrogen, regularly shaped particles were obtained even at high temperatures. Debling and Ray17 investigated


Figure 8. Simple system of microelements with uneven growth, where R0 is the initial radius of all microelements.

the shape of the produced impact polypropylene particles in dependence on the content of the comonomer and found that perfectly spherical particles are produced only at low content of the comonomer. 5. Effect of Temperature on the Morphology of Polyolefin Particles The final morphology of particles depends on the interplay of processes causing the formation and relaxation of the stress in the particle during its growth. As explained by Grof et al.,1 the buildup of the stress is caused by uneven growth rates of microelements forming the particle. However, the condition of uneven growth is essential but not sufficient for the formation of complex morphological features because the spatial arrangement of microelements has to be taken into account. Both the buildup and the relaxation of the stress depend on temperature. The particle is considered to have a constant temperature during its growth; i.e., the effect of the dynamically evolving particle temperature is not systematically investigated here. The rate of the buildup of the stress depends on the rate of polymerization reaction and inherits its Arrhenius dependence on temperature as well as the effects of intraparticle transport of monomer and temperaturedependent sorption of monomer in the polymer. The rate of the stress generation is proportional to the strain rate de/dτ, cf. eq 6. Let us consider the simple system of nine microelements displayed in Figure 8, where the central microelement with the radius R2 ) r2/R0 is not growing and is assumed to remain in the center (although this situation is not mechanically stable as the attachment of the central microelement to one of the larger microelements would occur). The evolution of radii of microelements is governed by the solution of eq 3 for constant C1 and Mcat,1, 3

R1 ) r1/R0 ) x1 + 3ΓMcat,1C1τ

R2 ) r2/R0 ) 1 (11)

where the uneven growth rate of microelements having the radius R1 and R2 is apparent. The strain e can be defined as

e)

R1 - R2 R1 + R2

(12)

and the strain rate de/dτ is obtained by the differentiation of eq 12 with R1 and R2 substituted from eq 11:

de ) dτ (1 +

2ΓMcat,1C1 3

2 3

x1 + 3ΓMcat,1C1τ) (x1 + 3ΓMcat,1C1τ)

2

(13)

The maximum rate of the strain growth is at time τ ) 0, and the strain rate de/dτ gradually decays in later

Figure 9. The dependence of strain rate scaled as (de/dτ)T/ (de/dτ)T)323 on temperature T at times τ ) {0, 0.5, 1, 2}. The activation energy in eq 14 is set to Ea ) 40 kJ/mol, and the growth factor at T ) 323 K is Γ ) 1.

Figure 10. The dependence of strain rate (de/dτ) on time τ for several temperatures T ) {323, 348, 373} K. The activation energy in eq 14 is set to Ea ) 40 kJ/mol, and the growth factor at T ) 323 K is Γ ) 1.

stages of particle growth. Let us further consider the Arrhenius dependence of the growth factor on temperature

Γ ∼ e-Ea/(RT)

(14)

where Ea is activation energy and R is the gas constant. The strain rate de/dτ thus depends both on temperature T and time τ. As expected, the dependence on temperature at time τ ) 0 displayed in Figure 9 is Arrhenian. However for τ > 0 the dependence of de/dτ on temperature is non-Arrhenian and can even exhibit a maximum or can be monotonically decreasing. The elastic modulus E is decreasing with the growing temperature, cf. Table 2, and also affects the rate of the stress generation. The dependence of the strain rate de/dτ on time for the selected temperatures demonstrates that the fastest stress generation is expected in the early stages of particle growth, cf. Figure 10. Figures 9 and 10 also show that the strain rate de/dτ decays faster at higher temperatures. All significant morphological changes are going to occur in the early stage of particle growth, where the rate of relative displacement of microelements caused by the growth is large. However, the discussion in the previous paragraphs and Figures 9 and 10 would somewhat change if the absolute strain e˜ ) R1 - R2 is considered instead of the relative strain e defined by eq 12. In this case the strain rate would decay from (de˜ /dτ) ∼ Γ1/3 at τ f ∞. Our dynamic simulations are based on the relative strain e, and this could be one source of possible discrepancies

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in the time scale of dramatic morphological changes predicted by our model and those observed experimentally. The time scale of our simulations is actually too fast when compared to the real data. The use of relative strain e thus enhances the tendency of the model to fast generation of the stress only in the short initial stage of particle growth. Another probable source of the discrepancy in the time scale of our simulations and real experiments is the catalyst activation and the progressive fragmentation of the catalyst carrier causing the catalyst sites to become gradually accessible for the polymerization reaction. The polymer is more rigid and tough at lower temperatures, but it is more soft and ductile at higher temperatures. The relaxation time τR estimated from experimental data decreases only slightly between the temperatures 50 and 100 °C, cf. Table 2. The testing frequency is f ) 1 Hz. The single relaxation time τR cannot comprehensively describe the relaxation behavior of polymers. Moreover, the polymer produced at higher temperatures is going to have a lower average molecular weight because the chain transfer and other reactions terminating the growing polymer chain are typically more sensitive to temperature than the chain propagation. The lower molecular weight of the produced polymer is going to enhance the rate of stress relaxation. The limited amount of systematic measurements of viscoelastic data in dependence on temperature and molecular weight available in the open literature prevents the formulation of a more quantitative model. The evolution of the morphology of polymer particles depends on the stress generation and relaxation, cf. eq 6. The dependence of the stress generation on time seems to be the determining factor of the morphogenesis in the early stages of particle growth. At low temperatures the reaction is relatively slow and thus no significant stress is supposed to be accumulated in the particle provided the relaxation of the stress is sufficiently fast. Open and irregular morphologies are likely to occur at higher temperatures associated with a fast growth in the early stage of particle evolution. Therefore the prepolymerization of active catalyst particles at mild reaction conditions is sometimes employed to decrease the rate of strain generation and to obtain a more compact morphology of the particle. The relaxation of the stress becomes faster than its generation for longer times τ, and this results in the slow decrease of particle porosity during its growth, cf. experimental data reported by Han-Adebekun et al.45 The temperature also affects the stress-strain characteristics of polymers. Thus cracks could form in polymers at low temperatures not only because the stress is relaxed slowly but also because the polymer is more brittle. Hence polymer particles could form macrocavities14 and could disintegrate into fines even at low temperatures as reported by Llinas and Selo,43 cf. Figure 11. The maximum elongation strain emax and the maximum bending angle Rmax can be taken as the measure of the brittleness of the polymer material, but the quantification of their dependence on temperature is uncertain. Figures 11 and 12 display the effect of temperature on particle morphology. The activation energy Ea ) 40 kJ/mol of the polymerization reaction is considered, and the temperature scale corresponds to this activation energy. The temperature axis could be easily rescaled for different values of the activation energy. As the

Figure 11. Effect of temperature on the particle morphology (at t ) 2 s) for low polymer yield rate AY (in kgpol/(gcat h)) with Arrhenius dependence on temperature. The relaxation time τR is decreasing with temperature. Ternary interactions were computed with G ) 0.2E, Rmax ) 10°.

Figure 12. Effect of temperature on the particle morphology (at t ) 2 s) for high polymer yield rate AY (in kgpol/(gcat h)) with Arrhenius dependence on temperature. The relaxation time τR is decreasing with temperature. Ternary interactions were computed with G ) 0.2E, parameter Rmax ) 2.7°, 6.2°, and 10.5° for emax ) 0.04, 0.09, and 0.15, respectively.

temperature increases, we also decrease the relaxation time from τR ) 1.3 s at 50 °C to τR ) 0.75 s at 80 °C. The Thiele modulus Θ characterizing the mass transport resistance in the growing particle is larger at higher temperatures and thus enhances the uneven growth of microelements. The particle formed with a high value of Θ is likely to be more porous and is going to exhibit poor replication of particle shape and could even form fines in the initial stages of particle growth. However, once the initial stage of particle growth at the high temperature is over, the stress relaxation becomes the dominant factor for the evolution of the morphology and the porosity of the particle gradually decreases. At high temperatures, the polymeric material is more soft and more sensitive to changes in the shape caused by shear and normal stresses arising in the polymer. The mass transport limitations could be more significant in the case of partial melting of the polymer, cf. ref 45, where the viscoelastic relaxation of the stress becomes the dominant mechanism in morphogenesis. Han-Adebekun et al.45 attributed the sintering of the particle pore structure of low crystallinity ethylene-propylene random copolymers obtained at higher temperatures to the softening or partial melting of the polymer phase during


the polymerization. The increase of the monomer transfer resistance due to softening and partial melting could be also responsible for the observed rate decrease in the high-temperature region, when the particle porosity becomes low.46 The predictions of our morphogenesis model are in agreement with videotaped ethylene polymerization on chromium-silica catalyst recorded by Niegisch et al.,47 who observed the one-time explosive growth of particles at elevated pressures, and this explosive growth can give rise to the generation of fines and poor replication of particle shape.48,49 However, the slow growth following the initial explosive growth replicates well the shape of the nascent particle. A comprehensive experimental investigation of the effect of temperature on the morphology of particles formed by polymerization in liquid propylene was performed by Pater et al.,44 who quantified the morphological properties only as the bulk density of the polymer powder. However, the bulk density is the combination of internal porosity, polymer density, particle shape, and particle size distribution. Therefore the estimation of particle porosity from their experimental data is difficult. Powders with irregular shape of particles could have larger bulk densities.50 It is also difficult to judge the internal morphology of particles from the texture of the particle surface. At low temperatures Pater et al.44 obtained particles with high bulk density and a small amount of pores on the particle surface and the particles consisted of smooth-textured compact agglomerates of spherical granules. A gradual transformation into a more open structure with the irregular “spongelike” particle surface and with lower bulk density was observed with the increasing temperature. When the low-temperature prepolymerization step was employed, the final morphology was determined by the prepolymerization step and was independent from the conditions in the main polymerization. Hence the initial polymerization rate plays an important role in the morphogenesis. 6. Onionlike Structure of Polyolefin Particles Let us examine the sensitivity of the predicted morphology to the number of microelements acting as the constituents of the polymer particle. Therefore we start the dynamic simulation of the morphogenetic process from the same initial conditions (i.e., circle), but with approximately 2700 microelements of initial radius R0 ) 1 µm instead of approximately 600 microelements with initial radius R0 ) 1 µm employed in previous simulations. Hence the initial radius of the polymer particle Rpart,0 is larger than 24 µm. The distribution of catalyst activity in microelements is uniform and the monomer transport limitation is parametrized by the evolving Thiele modulus Θ according to eq 4. The results obtained with the larger number of microelements are summarized in Figure 13, where most predicted morphologies are analogous to those obtained in Figure 12 with a smaller number of microelements. However, the details of particle morphology are better resolved in Figure 13. For small values of the maximum elongation emax, we observe the formation of the annulus of the void space separating the outer (faster growing) shell of the particle from its interior. Thus the formation of large pores (cavities) positioned in the azimuthal direction is possible.

Figure 13. Morphology predicted with a large number of microelements. Effect of temperature on the particle morphology (at t ) 1 s) for high polymer yield rate AY (in kgpol/(gcat h)) with Arrhenius dependence on temperature. The relaxation time τR is decreasing with temperature. Ternary interactions were computed with G ) 0.2E, parameter Rmax ) 1.3°, 2.7°, 6.2°, and 10.5° for emax ) 0.02, 0.04, 0.09, and 0.15, respectively.

The detail of the evolution of the “onion” structure reported, for example, by Galli et al.15 for the polypropylene and Zhou et al.51 for the polyethylene particles is presented in Figure 14. A number of the resulting morphologies predicted in spatially 2D simulations are highly porous and contain large macrocavities and fractures in the particle. However, the morphologies predicted in spatially 3D simulations with similar model parameters are more compact due to the larger coordination number of microelements in three dimensions. 7. Pore Space Morphologies of Ziegler-Natta and Metallocene-Born Polyolefin Particles It has been reported that the porosities of polyolefin particles obtained from supported metallocenes are lower than those obtained under the same conditions with traditional Ziegler-Natta catalysts.52,53 The enhanced incorporation of comonomers into polymer chains by metallocenes can be an important factor in this respect. Moreover, lower melting points at similar density and crystallinity and smaller sizes of crystallites have been observed with metallocene-born polymers. Let us discuss three possible hypotheses explaining this observed difference in porosity. 7.1. Hypothesis A. Active catalyst species could detach from the surface of the catalyst carrier and could diffuse through the polymer encapsulating the catalyst carrier in the micrograin, as suggested by Naik and Ray.53 Alternatively, active catalyst species could be transported not by diffusion but by the convection of the polymer to the interface of micrograins with the pore phase, and the presence of active catalyst sites at the surface of micrograins is then suggested to be responsible for the gradual reduction of the particle porosity. The direct experimental verification of this hypothesis is not available because the experimental detection of catalyst constituents by electron microscopy techniques is not yet sensitive enough to detect the active catalyst sites scattered in the polymer phase. But it is believed that a number of metallocene catalysts are not firmly attached to the surface of the catalyst carrier.

2400


Figure 14. Evolution of the onionlike structure, polymer yield rate AY ) 50 kgpol/(g emax ) 0.02.

cat

h), bending modulus G ) 0.2E, maximum strain

Figure 15. Effect of the relaxation time τR (in the Maxwell model) on the resulting morphology. Polymer yield rate AY ) 100 kgpol/(gcat h), ternary interactions computed with G ) E, Rmax ) 5°, emax ) 0.04, Θ0 ) 9.3.

Cecchin et al.54 also suggested that the copolymer phase in the impact polypropylene is formed at the surface and not in the interior of the polypropylene micrograins. Impact polypropylene is a heterophasic polymer where the polypropylene (the major component) is the continuous phase, and the elastomeric phase (ethylene-propylene rubber) is uniformly dispersed within the matrix. The usually accepted opinion about the second stage of the impact polypropylene process is that the copolymer formed inside the homopolymer micrograins is immediately squeezed out from the densely packed, crystalline regions of PP, where it grows and migrates to the interstitial porous zones located around the homopolymer micrograins.55 However, Cecchin et al.54 suggest that the fragments of catalyst are convected by the polymer phase from the interior to the surface of the polypropylene micrograins, where they sustain the reaction. 7.2. Hypothesis B. Polymer products made on two different catalysts could have almost the same density and crystallinity, but other properties could be different (e.g., the distribution of branching, the presence of the high molecular weight tail in the molecular weight distribution, the distribution of crystalline domains in the semicrystalline polymer). Different viscoelastic properties of polymers produced by the Ziegler-Natta and the metallocene catalysts36,34 could then also explain the observed reduction of particle porosity in the case of metallocenes, cf. Figure 15. Viscoelastic properties of polymers can thus be another reason for the observed small porosities of some metallocene-born polyolefins. 7.3. Hypothesis C. Distribution of catalyst activity also affects the porosity of the polymer particle. Different activities of two used catalysts or different initial distribution of the catalyst on the support could result in different patterns of fragmentation. Thus we might

Figure 16. Effect of the nonuniform distribution of catalyst activity in microelements on particle porosity. Uniform distribution of the monomer concentration is considered, AY ) 100 kgpol/(gcat h), emax ) 0.20, G ) E, Maxwell model with τR ) 1 s. Parameter β determines the broadness of the distribution of catalyst activity. The relative activity of each microelement is 10X/β, where X is the random number with normal distribution.

end up with a nonuniform distribution of catalyst activity in the growing particle. We believe that the experimental mapping of the distribution of catalyst activity immobilized on fragments of the catalyst support is attainable by the current microscopy techniques. The effect of the nonuniform activity of catalyst on the resulting particle porosity is illustrated in Figure 16. Here the microelements remain connected by the colddrawn fibrils even when e > emax. However, the elastic modulus E is significantly decreased when e > emax, resulting also in the decrease of the attractive force. Thus the capability of our model to simulate the morphogenesis of cobweb structures is demonstrated. The formation of the cobweb structures could be simulated by the introduction of the cold-drawing between the receding microelements. The simulation algorithm is thus modified in the following way: When the strain e of the connected pair of microelements is larger than the parameter emax, then the elastic modulus E for this


particular connection is reduced significantly instead of removing this connection. The experimental evidence of the cobweb morphologies is comprehensive. HDPE synthesized with a highactivity Cr-supported catalyst or with TiCl4/ MgCl2/SiO2 catalyst could exhibit tangled, wormlike structures of about 1 µm in diameter and up to 10 µm in length.2 Similarly Minkova et al.56 observed that as the “crust” of polymer/catalyst particle disintegrates during its growth, the micrograins could remain connected by a network of drawn fibrils that extend over a distance of about 20 µm and have a diameter of approximately 0.1 µm. This type of the structure is called the “cobweb” and was observed also in the ethylene copolymerization with 1-hexene50 and in the ethylene polymerization with Ti-Mg catalyst on the polystyrene support.57 It has been suggested that the formation of the wormlike and the cobweb morphology is a common feature of catalysts with high efficiency, while the globular morphology is obtained at low catalyst efficiency. Particles with the worms or the cobweb fibrils are typically highly porous, and the polymerization reaction is not retarded by the monomer diffusion effects. Nooijen58 reported that cobweb structures in PE particles form when the catalyst precursor is not activated by the cocatalyst prior to the introduction of the monomer into the slurry reactor. 8. Conclusions The evolution of polyolefin particles in catalytic reactors leads to a broad range of possible particle morphologies due to the interplay of reaction and transport processes with viscoelastic properties of the resulting polymer. A viable methodology for the prediction of the morphology of polyolefin particles was developed by Grof et al.1 and adopted here for the construction of the systematic mapping from the parametric space of reactor conditions and architecture of the catalyst particle into the space of morphologies of resulting polyolefin particles. Such a systematic mapping is necessary given the uncertainty of some model parameters and given the wealth of experimental observations. The most dramatic changes of particle morphology were observed in the initial stage of particle growth in agreement with experimental observations. Several hypotheses explaining the differences in pore space morphologies of Ziegler-Natta and metallocene-born particles were studied. The effect of temperature on the evolution of the morphology of polyolefin particles was thoroughly discussed and used to address the industrially important problem of the formation of fine particles in the gas-dispersion polymerization reactors. The onionlike and the cobweb morphologies of polyolefin particles were presented as the minor byproducts of extensive parametric studies. The model employed in this article was intentionally kept simple. Some important aspects of particle morphogenesis, such as the kinetics of catalyst activation, the evolution of particle temperature including the possibility of overheating, the pore clogging by the polymer, the detailed discussion of the fragmentation of catalyst carriers, and the formation of impact polypropylene, can be also analyzed by the advanced morphogenesis model considering the transport of monomer in the evolving structure of the porous particle. Further extensive modeling and experimental work is required

to fully understand and describe polymer morphology produced with supported catalysts. The morphogenesis model is formulated and coded in three spatial dimensions, but we have mostly presented results obtained by two-dimensional simulations because they are more illustrative. The extrapolation of the simulated morphogenetic behaviors from two dimensions into three dimensions has to be done cautiously because of the larger number of connections with the neighboring elements in the 3D case. Dynamic simulations with the number of microelements on the order of 104 are reasonably fast. However, the default initial conditions sufficient for illustrative purposes were represented by a circle with approximately 600 regularly ordered microelements of the same diameter in the 2D case. The morphogenesis model can be subjected to many refinements. Particularly, the viscoelastic properties of the polymer were approximated by the simple Maxwell model with limited capabilities to quantitatively represent the experimentally measured viscoelastic characteristics. The consideration of the relaxation spectrum and of its temperature dependence is one of the possible improvements in this direction. Another problem is the somewhat arbitrary definition of the parameters emax and Rmax representing in a simple way the yield condition in the stress-strain characteristics of the polymer at reactor temperature. Measurements of the stressstrain characteristics at higher than room temperature are scarce, and therefore we consider correlating both emax and Rmax with the sintering temperature or other measurable quantities in the future. The dynamics of the fragmentation of catalyst carriers in the early stage of particle growth was not directly addressed yet, but the effect of the resulting distribution of catalyst activity in the particle on its morphogenesis was clearly demonstrated. The nonuniform distribution of catalyst activity in the particle is at least as important as the mass transport limitations in the context of particle morphogenesis. The method of the start-up of the particle growth controls the development of catalyst activity as well as polymer powder morphology as systematically studied by Nooijen58 for several different ways of introduction of the cocatalyst into the ethylene polymerization in a slurry. Some features of the evolution of particle morphology can be described in alternative ways, as was demonstrated in the example of different porosities of ZieglerNatta and metallocene-born polyethylene particles. The interplay of the stress generation, stress relaxation, and temperature allows for a rich range of possible morphological developments. Acknowledgment The support from the Czech Grant Agency (Projects 104/02/0325 and 104/03/H141) is acknowledged. Appendix. Formulation of Mathematical Model Let us consider a growing polyolefin particle discretized into a number of small spherical microelements with elastic or viscoelastic interactions acting among individual microelements. The ith microelement is characterized by the position of its center xi, velocity vi, radius ri, and monomer concentration ci in the fluid phase surrounding the ith microelement. The translational movement of each microelement is governed by

2402


a kinematic equation and Newton’s equation of momentum

dxi ) vi dt d(mivi) dt

)

+ ∑Fternary ∑j Fbinary ij ijk j,k

(A.1)

(A.2)

where mi ) (4/3)πFiri3 is the mass of the ith microelement and the density of each microelement Fi is considered to be constant in time. Summations of forces F on the right side of eq A.2 are carried out over all binary and ternary interactions between the ith microelement and its connected neighbors. The rate of the growth of the ith microelement depends on the monomer concentration ci at the surface of this microelement:

dri dmi ) 4πFiri2 ) kmcat,ici dt dt

(A.3)

where mcat,i is the mass of catalyst in the microelement. Let AY be the polymer yield rate at reactor bulk temperature in kgpol/(gcat h). Then the rate constant k ) AY/(3.6cbulk), where cbulk is the bulk concentration of monomer in the reactor. The mathematical model of polymer particle evolution consists of the set of differential eqs A.1-A.3. The constitutive equations describing the force interactions, transport of monomer, phase equilibria at the interface between polymer and pore phase, and the rules for connectivity of microelements have to be specified. The magnitudes of binary and ternary force interactions required in eq A.2 are calculated by simple elastic or viscoelastic constitutive equations and then projected into force vectors. The elastic model is the direct implementation of Hooke’s law with the stress between microelements A and B dependent linearly on the strain eAB:

FAB/AAB ) EeAB

(A.4)

where E is the elastic modulus, AAB is the contact area, and FAB is the magnitude of binary interaction force. A positive value of FAB results in attraction and a negative one in repulsion of microelements. The strain eAB is defined as

eAB ) (|u| - u0)/u0

(A.5)

where u0 ) rA + rB is the equilibrium distance and u ) xB - xA is the vector connecting microelements A and B. Ternary interactions for the triplet of connected microelements A-V-B with the bonding angle R ) ∠AVB are described by eq 7. The contact area AAB is, for the sake of simplicity, selected for both the binary and ternary interactions as AAB ) (1/2)π(rA2 + rB2), although for ternary interaction the microelements A and B do not have to be in direct contact. The viscoelastic interactions of connected microelements of semicrystalline polyolefins are approximated by the Maxwell model (eq 6). The Maxwell model describes particularly well the relaxation of stress in viscoelastic material. When the connection of microelements A and B is subjected to constant strain (deAB/dt

) 0), then the stress in the connection decays exponentially, σ ) σ0 exp(-Et/η) ) σ0 exp(-t/τR), where σ0 is the initial stress at t ) 0 and τR ) η/E is the relaxation time. The Maxwell model also allows for permanent deformation. The disadvantage of the pure elastic interaction described by eq A.4 is that the repulsive force may not be sufficiently large to prevent microelements from interpenetrating to a large extent because eq A.4 is valid only for small strains. The elastic model for binary interactions thus has to be modified to (i) nonlinearly increase the repulsion of spherical microelements A and B as the strain is becoming more negative due to the increasing contact area AAB, (ii) attract microelements A and B stretched apart and connected by the neck of the polymer for eAB > 0, and (iii) behave according to Hooke’s law for small absolute values of strain eAB. Therefore the following simple equation satisfying the above listed requirements is proposed to calculate the magnitude of the binary interaction force instead of eq A.4:

eAB FAB/AAB ) E 1 + eAB

(A.6)

The modified description of elastic interactions given by eq A.6 substituted into eq A.4 can be used in the rederivation of the specific form of the Maxwell model with the evolution of the magnitude of force FAB given by

(

)

d(FAB/AAB) (E - (FAB/AAB))2 deAB (FAB/AAB) ) dt E dt η (A.7) Here the strain rate (deAB/dt) is evaluated as

deAB u0 du/dt - u du0/dt ) where dt u2 0

du (xB - xA)‚(vB - vA) ) (A.8) dt u and (du0/dt) ) drA/dt + drB/dt, and u ) |u| is the distance of microelements A and B. Let us proceed with the projection of the magnitude of force interactions into vectors required in eq A.2. Let u ) xB - xA be the vector connecting microelements A and B, and let FAB be the magnitude of the binary interaction force acting along the connection u. Then the force acting on the microelement A is the vector FAB

FAB ) FAB u/|u|

(A.9)

and the force acting on B has the opposite direction, FBA ) -FAB. The force with magnitude FR representing the resistance against the change of the angle ∠AVB in the triplet of connected microelements is projected in the direction of unit vectors wa and wb. Let us define the vectors a ) xA - xV and b ) xB - xV. The unit vector wa (or wb) is perpendicular to a (or b) and coplanar with vectors a and b, that is,

wa‚a ) 0, wa‚(a × b) ) 0, |wa| ) 1 wb‚b ) 0, wb‚(a × b) ) 0, |wb| ) 1 (A.10) Unit vectors wa and wb are oriented into the convex


angle R spanned by vectors a and b. Forces acting on microelements A and B, i.e., FAVB and FBVA, have to be compensated by the force FVAB acting on the microelement V in order to prevent the translation movement of the triplet A-V-B

FAVB ) FRwa, FBVA ) FRwb, FVAB ) -(FAVB + FBVA) (A.11) Model equations can be transformed into dimensionless form. Let us introduce the dimensionless parameters of the ith microelement: the position Xi, the velocity Vi, the radius Ri, the concentration of surrounding monomer Ci, and the mass of catalyst Mcat,i, as well as the dimensionless time τ defined as

Xi ) xi/R0, Vi ) viτ0/R0, Ri ) ri/R0 Ci ) ci/cbulk, Mcat,i ) mcat,i/mcat,0, τ ) t/τ0 (A.12) where R0, τ0, cbulk, and mcat,0 are the characteristic length, the time, the monomer concentration, and the mass of catalyst. The characteristic length R0 is, for example, the typical or initial radius of microelement. Particularly, substitution of dimensionless variables into eq A.3 gives the growth rate of the dimensionless microelement radius dRi/dτ:

Mcat,iCi dRi kmcat,0cbulkτ0 Mcat,iCi ) ) Γ (A.13) dτ F 4πR 3 R2 R2 i

0

i

i

where the dimensionless parameter Γ characterizes the growth, cf. the discussion below eq 3. List of Symbols AY ) polymer yield rate, kgpol/(gcat h) AAB ) contact area between microelements, m2 ci ) concentration of monomer in the ith microelement, mol/ m3 cbulk ) concentration of the monomer, mol/m3 Ci ) dimensionless concentration, 1 Def ) effective diffusivity, m2/s e ) strain, 1 emax ) maximum strain, 1 E ) elastic modulus, Pa E′ ) storage modulus, Pa E′′ ) loss modulus, Pa Ea ) activation energy, J/mol f ) frequency, s-1 Fij ) binary interactions force, N Fijk ) ternary interactions force, N G ) “bending” modulus, Pa k ) reaction rate constant, m3 mol-1 s-1 mi ) mass of the ith microelement, kg mcat,i ) mass of catalyst in the ith microelement, kg mcat,0 ) characteristic mass of catalyst, kg MM ) molecular weight of the monomer, kg/mol Mcat,i ) dimensionless mass of catalyst, 1 ri ) radius of the ith microelement, m Ri ) dimensionless radius of the ith microelement, 1 R ) gas constant, J/(molK) Rpart ) radius of the particle, m R0 ) characteristic length, m s ) radial coordinate, m t ) time, s T ) temperature, K u ) vector connecting microelements, m u0 ) equilibrium distance between microelements, m

vi ) velocity of the ith microelement, m/s Vi ) dimensionless velocity, 1 xi ) position vector of the ith microelement, m Xi ) dimensionless position, 1 R ) value of the “bonding” angle, rad R0 ) equilibrium value of the “bonding” angle, rad Rmax ) limiting value of the “bonding angle”, rad δ ) phase lag of stress and strain, rad η ) viscosity, Pa s Γ ) dimensionless growth factor, 1 Fi ) density of the ith microelement, kg/m3 σ ) stress, Pa τ ) dimensionless time, 1 τ0 ) characteristic time, s τR ) relaxation time, s Θ ) Thiele modulus, 1 Θ0 ) initial Thiele modulus, 1 ω ) angular frequency, rad/s

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Received for review September 14, 2004 Revised manuscript received December 6, 2004 Accepted December 7, 2004 IE049106J

Principles of the Morphogenesis of Polyolefin Particles - Industrial

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