# Modeling of Peroxide Initiated Controlled Degradation of Polypropylene in a Twin Screw Extruder.

B. VERGNES [*]

In this work, experimental and theoretical studies of the free-radical initiated molecular weight degradation of polypropylene in a modular self-wiping corotating twin screw extruder have been investigated. Our objective was to build a model that would be able to predict the evolution of the average molecular weight along the screws, in relation to the processing conditions and the geometry of the twin screw extruder. Modeling the process involves resolving interactions occurring between the various flow conditions encountered in the extruder, the kinetics of the reaction and the changes in viscosity with changes in molecular weight. We have studied the influence of operating parameters such as the initial peroxide concentration, the feed rate and the screw speed on the degradation reaction. Good agreement was found between theoretical results and experimental values obtained by size exclusion chromatography measurements.

INTRODUCTION

Over the last twenty years, reactive extrusion has undergone considerable development and the number of publications and patents in the subject area has continually increased [1, 2]. The success of the reactive extrusion process is due to the fact that polymers can be directly modified in the molten state, during a single stage of transformation. Twin screw extruders provide great flexibility and are often used to perform the chemical modification of natural or synthetic polymers. The modular construction of screw-barrel blocks is particularly favorable for the design of screw profiles, which are specific to the studied reactions and include, if necessary, separated zones for melting, homogenization, reaction and devolatilization [3-5].

In comparison to reactions conducted in solution, reactions carried out in the extruder show an important influence of the temperature, viscosity and pressure [6]. In fact, reactive extrusion is a highly nonisothermal process and the temperature is generally high, relative to the dissipated power and the exothermy of the undergoing reactions. The viscosity of the system depends simultaneously on the temperature, the shear rate conditions and the molecular mass. Finally, the pressure depends mostly on the type of screw elements used (fully filled or starved zones).

In the modeling of the reactive extrusion process, we started by studying in a first step chemical reactions which do not modify in a significant way the system's viscosity (transesterification of an ethylene and vinyl acetate copolymer). Therefore, we could calculate separately the flow parameters and the reaction progress [7]. In the present work, we are interested in more complex reactions including sensitive modifications of the rheological behavior during the reaction, in connection with the evolution of molecular weight. Among the different possible reactions, we have chosen the peroxide induced degradation of polypropylene.

This type of reaction has been largely studied in the literature, because it is widely used in polymer industry to control the molecular weight distribution of polypropylene resins. Indeed, polypropylenes manufactured using classical industrial polymerization processes (using Ziegler-Natta catalyst systems) usually have excellent mechanical properties, but have also a high molecular weight and a broad polydispersity. This makes their processing difficult, in view of their high viscosity and elasticity [8]. In order to adjust their molecular weight distribution, polypropylene resins can be modified in reactive extrusion operations by means of organic peroxide initiated scission reactions. The principle of this reaction is as follows. The peroxide is decomposed at high temperature, leading to free-radicals. These free-radicals then attack the links C-H of polypropylene that have the weakest link energy and give rise to tertiary polymer macroradicals. There follows division of the macromolecular chain of the po lypropylene by [beta]-scission, which leads to the creation of polypropylene of reduced mass and of a new polymer radical, but this time of secondary type [9]. This mechanism has been proposed and explicated in numerous publications [10-14].

In the literature, there is little work devoted to the global approach of the coupling between flow and reaction. Tzoganakis et al. [11, 15] have proposed models for the peroxide induced degradation of polypropylene in a single screw extruder. In a first approach [11]. plug flow and isothermal conditions in the extruder were assumed. In a second step [15], residence time distribution and nonisothermal conditions were taken into account, and the viscosity was a function of the average molecular weight. More recently. Kim and White [16] have described a twin screw extrusion model allowing simulation of the peroxide induced degradation and maleation of polypropylene. Unfortunately, they do not give any information about the change in viscosity due to the reaction and its effect on the thermomechanical parameters.

In the present study, the kinetic scheme used to describe the changes of the molecular weight distribution was taken from the literature. Many studies, and particularly the work of Tzoganakis [17], have been made on the peroxide initiated degradation of polypropylene in the molten state, to build a kinetic model, which can predict the evolution of molecular mass. In the present work, the dependence of the viscosity with the molecular mass was characterized by studying different molecular weight polypropylenes. This allowed us to determine relationships between the viscosity, the shear rate, the temperature and the molecular weight. A theoretical model for predicting changes in molecular weight all along the extruder is presented. We also describe experiments carried out with a modular corotating twin screw extruder, which show clearly the influence of the operating parameters on the degradation reaction. Finally, we compare the results of the model to these experimental trials.

EXPERIMENTS

Materials and Processing Conditions

A commercial homopolymer (PP 3050 BN 1, Appryl) was used. The polypropylene has a melt flow index of 4.5 g/10 min (2.16 kg/230[degrees]C), an initial weight average molecular weight of 301,600 g/mol and a polydispersity index, [M.sub.w]/[M.sub.n] of 6.4. The peroxide used was 2,5-dimethyl-2,5-di(tert-butylperoxy)hexane (DHBP, Trigonox 101) provided by Akzo Chemie.

Reactive experiments were carried out on a corotating self-wiping twin screw extruder (model ZSK 30, Werner & Pfleiderer), which included twelve barrel sections and six corresponding heating zones. The first barrel (feeding hopper) was not heated. The centerline distance was 26 mm, the screw diameter. D, 30.85 mm and the L/D ratio was 37.5, where L is the total length of the screws. The PP polymer pellets were fed via a weighted feeding system. The peroxide was injected into the extruder by a volumetric piston feed pump. The DHBP was previously diluted in 1, 2, 4 trichlorobenzene (solution at 15%) to have a good precision on the pump flow rate, even for concentrations less than 1000 ppm (i.e. 0.1 wt%). The output rate of the piston pump was adjusted according to the flow rate of polypropylene and its regularity was controlled by the loss in weight of the beaker containing the peroxide. This method provided specific data on the real peroxide flow rate for use in simulations of the degradation reaction.

The extrusion setup and screw configuration used are presented in Fig. 1 and Table 1. The melting of the polymer was assured by kneading disc blocks and left-handed screw elements. The injection of DHBP was housed in a unfilled area, just after the melting zone. The reaction area first included two kneading disc blocks designed to improve the mixing and homogenization between free-radicals and polypropylene. Next were situated several right-handed conveying screw elements with decreasing pitches and finally a left-handed screw element situated just before the devolatilization zone (localized at barrel 10). At this level, the degradation reaction of polypropylene was stopped by the elimination by a vacuum pump of all volatile reactants. The melt was extruded through a converging plate and three parallel capillary dies (each with diameter of 3.5 mm and length of 10 mm). At the die exit, extrudates were quenched in water before being dried and cut into pellets. Samples were collected after about 20 min of stead y state extrusion conditions (stable die pressure and motor torque).

During the tests, the following parameters were varied: quantity of initiator concentration [I.sub.0] (0.0, 0.025, 0.05, 0.1, 0.2 and 0.4 wt%), feed rate Q (2.5, 10 and 20 kg/h), screw speed N (150, 225 and 300 rpm). The barrel regulation temperature [T.sub.R] was fixed at 170[degrees]C. We chose a relatively low barrel temperature such that the degradation reaction was still unfinished upon reaching the devolatilization zone. This choice allowed us to examine the influence of the feed rate and screw speed (i.e. the influence of the residence time and temperature along the screws) on the molecular weight measured at the die exit. Indeed, tests conducted at higher barrel regulation temperatures (greater than 220[degrees]C) did not permit us to characterize the influence of processing conditions, as the reaction was finished before the vent zone, due to the very short half-life time of peroxide at these temperatures. The data of the different experimental conditions are reported in Table 2.

Molecular Weight Distribution and Rheological Measurements

In order to have a precise knowledge of the reactive system's viscosity, the complex viscosity of different molecular weight polypropylenes was measured in a Rheometrics RMS 800 spectrometer, using parallel-plate geometry in oscillatory mode. These samples were previously prepared at 250[degrees]C in a laboratory twin screw extruder (Leistritz 30-34) by using various amounts of peroxide. The concentration range was 0.01 to 0.5 wt%. The samples were dried under vacuum, at 80[degrees]C during a 24-hour period, in order to eliminate all the residual peroxide. The molecular weight distributions of these differently degraded polypropylenes were analyzed by size exclusion chromatography (SEC). The samples were dissolved at 145[degrees], at a polymer concentration of 0.8 g/L in the 1,2,4 trichlorobenzene, which was used as a solvent with the addition of an antioxidant to prevent any degradation. The results obtained are presented in Fig. 2 and Table 3. We classically observed that increasing the peroxide concentrat ion decreased the presence of high molecular weight species and narrowed the molecular weight distribution. We observe also a slight decrease in the molecular weight between virgin and extruded polypropylene, due to some thermomechanical degradation during extrusion. Indeed, processing conditions were particularly severe (250[degrees]C, 300 rpm). It was no more the case for the experiments carried out at low temperature (170[degrees]C) on the ZSK 30.

The dynamic measurements were carried out over a frequency range of 0.1 to 100 rad/s and at five temperatures (185[degrees]C to 245[degrees]C). Time sweep tests (30 min) made at the different temperatures at a frequency of 1 rad/s showed that the complex viscosity and the storage and loss moduli were constant with time (18). Figure 3 shows the time-temperature superposition of the complex viscosity (temperature reference of 215[degrees]C). The results obtained showed the influence of the molecular weight upon the viscosity. We observed the usual decrease in viscosity with decreasing molecular weight and a pronounced shift towards more Newtonian behavior (19, 20). To characterize the influence of the modifications induced by the peroxide on the homopolymer's rheological behavior, a Carreau-Yasuda law was fitted to the master curves:

[eta]/[a.sub.T] = [[eta].sub.0][[1+[([lambda][gamma]*[a.sub.T]).sup.a]].sup.m-1/a] with [a.sub.T] = [e.sup.[E.sub.a]/R[1/T-1/[T.sub.0]] (1)

where [a.sub.T] is the shift factor, [E.sub.a] the activation energy and R the gas constant. [T.sub.0] is the reference temperature (215[degrees]C) at which the zero shear viscosity [[eta].sub.0], the characteristic time [lambda], the power-law index m, and the parameter a were determined.

For each value of the weight average molecular weight, the determination of the parameters [[eta].sub.0], [lambda], m, and a was done using a homemade identification program which minimized, using a least square method, the differences between the measured and calculated viscosities. The set of initial values was selected in function of the experimental values. If changing parameters of the initial set one at a time gave different solutions, the solution adopted was the set which gave the best minimization of the objective function. The result of the superposition of the experimental and calculated curves is shown in Fig. 3. Figure 4 shows the evolution of each factor as a function of the weight average molecular weight [M.sub.w]. We note that the zero shear viscosity varies with a 3.9 power of [M.sub.w]. For flexible polymer chains, at high molecular weights, the zero shear viscosity usually increases with a 3.4 power of the molecular weight [20, 21]. We have observed that, when increasing the molecular wei ght, characteristic time [lambda] increased while m and a decreased continuously. Finally, we have observed that the activation energy of the different degraded samples slightly decreased (44.9 to 39.3 kJ/mol) when increasing the molecular weight (62,970 to 301,600 g/mol).

THEORETICAL MODELING

Kinetic Scheme

To model the peroxide induced degradation of polypropylene by a free-radical initiator, it is assumed that:

- The peroxide is homogeneously distributed in the polymer,

- There is no thermo-oxidative degradation,

- All the free-radicals have the same dissociation energy.

- The secondary decomposition of the DHBP molecules are neglected.

The following mechanism has been proposed by numerous authors [10-14]:

- Free-radical initiation:

I[right arrow] [k.sub.d]2R (i)

- Hydrogen abstraction and chain [beta]-scission:

[P.sub.n] + R [right arrow] [k.sub.1] [P.sub.r] + [P.sub.n-r] + R (ii)

- Inter-molecular chain transfer:

[P.sub.n] + [[P.sup.*].sub.r] [right arrow] [k.sub.2] [P.sub.r] + [[P.sup.*].sub.s] + [P.sub.n-s] (iii)

- Thermal degradation:

[P.sub.n] [right arrow] [k.sub.3] [[P.sup.*].sub.r] + [[P.sup.*].sub.n-r] (iv)

- Termination by disproportionation:

[[P.sup.*].sub.n] + [[P.sup.*].sub.r] [right arrow] [k.sub.4] [P.sub.n] + [P.sub.r] (v)

I represents the free-radical initiator, [R.sup.*], the peroxide free-radicals, [P.sub.n], the polymer, [[P.sup.*].sub.n], the polymer macro-radical and [k.sub.i], the rate constants. Depending on the authors, the species balance can be expressed differently. In this paper, we have chosen the equations proposed by Krell et al. [14], who are alone to provide kinetic constants for all the reactions cited above. The changes of concentration of the different species with time are thus the following:

- Peroxide balance:

d[I]/dt = - [k.sub.d][I] (2)

- Initiation radical balance:

d[[R.sup.*]]/dt = 2f[k.sub.d][I] - [k.sub.1][[R.sup.*]] [[[sigma].sup.[infinity]].sub.n=2](n - 1)[[P.sub.n]] (3)

- Polymer balance:

d[P.sub.n]/dt = [k.sub.1][[R.sup.*]] [[[[sigma].sup.[infinity]].sub.t=n+1] [[P.sub.i]] - (n - 1) [[P.sub.n]]] - [k.sub.2][(n-1)[[P.sub.n]] [[[sigma].sup.[infinity]].sub.i=1] [[[P.sup.*].sub.i]] - [[[P.sup.*].sub.n]] [[[sigma].sup.[infinity]].sub.i=1](i - 1)[[P.sub.i]] - [[[sigma].sup.[infinity]].sub.i=1][[[P.sup.*].sub.i]] [[[sigma].sup.[infinity]].sub.j=n+1] [[P.sub.j]]] - [k.sub.3] (n -1)[[P.sub.n]] + [k.sub.4][[[P.sup.*].sub.n]] [[[sigma].sup.[infinity]].sub.i=1][[[P.sup.*].sub.i]] (4)

- Polymer macroradical balance:

d[[[P.sup.*].sub.n]]/dt = [k.sub.1][[R.sup.*]] [[[sigma].sup.[infinity]].sub.i=n+1] [[P.sub.i]] - [k.sub.2][[[[P.sup.*].sub.n]] [[[sigma].sup.[infinity]].sub.i=1](i = 1)[[P.sub.i]] - [[[sigma].sup.[infinity]].sub.i=1][[[P.sup.*].sub.i]] [[[sigma].sup.[infinity]].sub.j=n+1][[P.sub.j]]] + 2[k.sub.3] [[[sigma].sup.[infinity]].sub.i=n+1][[P.sub.i]] - [k.sub.4][[[P.sup.*].sub.n]] [[[sigma].sup.[infinity]].sub.i=1][[[P.sup.*].sub.i]] (5)

The parameter f, which appears in Eq 3 is called the efficiency of the peroxide. It is defined as:

f = number of R which causes [beta]-scission at time t/total number of primary R at time t (6)

In order to reduce the dimensions of the infinite set of the obtained coupled differential equations, the ith moment equations of the chain length distribution for the polymer ([Q.sub.i]) and polymer radicals ([Y.sub.i]) are introduced [22]:

[Q.sub.i] = [[[sigma].sup.[infinity]].sub.n=1] [n.sup.i]*[[P.sub.n]] (7)

[Y.sub.i] = [[[sigma].sup.[infinity]].sub.n=1][n.sup.i]*[[[P.sup.*].sub.n]] (8)

Introducing these quantities into Eqs 3 to 5 leads to the following system of only eight equations, than includes also Eq 2:

d[[R.sup.*]]/dt = 2f [k.sub.d][I] - [k.sub.1][[R.sup.*]]([Q.sub.1] - [Q.sub.0]) (9)

d[Q.sub.0]/dt = [k.sub.2] [Y.sub.0] ([Q.sub.1] - [Q.sub.0]) - [k.sub.3] ([Q.sub.1] - [Q.sub.0]) + [k.sub.4] [[Y.sup.2].sub.0] (10)

d[Q.sub.1]/dt = - 1/2 [k.sub.1][[R.sup.*]]([Q.sub.2] - [Q.sub.1]) - [k.sub.2][1/2[Y.sub.0] ([Q.sub.2] - [Q.sub.1]) - [Y.sub.1]([Q.sub.1] - [Q.sub.0])] - [k.sub.3]([Q.sub.2] - [Q.sub.1]) + [k.sub.4] [Y.sub.1] [Y.sub.0] (11)

d[Q.sub.2]/dt = [k.sub.1][[R.sup.*]][- 2/3 [Q.sub.3] + 1/2 [Q.sub.2] + 1/6 [Q.sub.1]] + [k.sub.2][[Y.sub.0](- 2/3 [Q.sub.3] + 1/2 [Q.sub.2] + 1/6 [Q.sub.1]) + [Y.sub.2]([Q.sub.1] - [Q.sub.0])] - [k.sub.3]([Q.sub.3] - [Q.sub.2]) + [k.sub.4] [Y.sub.0] [Y.sub.2] (12)

d[Y.sub.0]/dt = [k.sub.1] [[R.sup.*]] ([Q.sub.1] - [Q.sub.0]) + 2 [k.sub.3] ([Q.sub.1] - [Q.sub.0]) - [k.sub.4] [[Y.sup.2].sub.0] (13)

d[Y.sub.1]/dt = 1/2 [k.sub.1][[R.sup.*]]([Q.sub.2] - [Q.sub.1]) + [k.sub.2][- [Y.sub.1]([Q.sub.1] - [Q.sub.0]) + 1/2 [Y.sub.0]([Q.sub.2] - [Q.sub.1])] + [k.sub.3]([Q.sub.2] - [Q.sub.1]) - [k.sub.4] [Y.sub.0] [Y.sub.1] (14)

d[Y.sub.2]/dt = [k.sub.1][[R.sup.*]][1/3 [Q.sub.3] - 1/2 [Q.sub.2] + 1/6 [Q.sub.1]] + [k.sub.2][[Y.sub.0](1/3 [Q.sub.3] - 1/2 [Q.sub.2] + 1/6 [Q.sub.1]) - [Y.sub.2]([Q.sub.1] - [Q.sub.0])] + [k.sub.3][2/3 [Q.sub.3] - [Q.sub.2] + 1/3 [Q.sub.1]] - [k.sub.4] [Y.sub.0] [Y.sub.2] (15)

This system introduces the moment [Q.sub.3], which can be correlated with the first three moments by using the closure method proposed by Hulburt and Katz [23]:

[Q.sub.3] = 2 [Q.sub.2]/[Q.sub.1] [Q.sub.0] [2 [Q.sub.2] [Q.sub.0] - [[Q.sup.2].sub.1]] (16)

The different average molecular weights are thus given by the following equation:

[M.sub.average] = [[[sigma].sup.[infinity]].sub.t=1] [n.sub.i][[M.sup.K].sub.i]/[[[sigma].sup.[infinity]].sub.t=1] [n.sub.i][[M.sup.k-1].sub.i] = [m.sub.0] [Q.sub.k] + [Y.sub.k]/[Q.sub.k-1] + [Y.sub.k-1] (17)

where k = 1 for the number average molecular weight, [M.sub.n], and k = 2 for the weight average molecular weight, [M.sub.w]. [m.sub.0] represents the molecular weight of the propylene monomer.

To solve the problem, the majority of authors, with the exception of Krell et al. [14], take only into account the free-radical initiation (i) and chain scission reactions (ii) and neglect the contributions from transfer (iii), thermal degradation (iv) and termination reactions (v). In this work, we decided to use the model of Krell et al. [14] and thus its kinetic constants, which are presented in Table 4. Consequently, we could solve the equation [2] and the moment equations [9-15] without neglecting any reaction. A Runge-Kutta method with adaptive stepsize control was used to solve the system of equations [18]. The quasi steadystate assumption for the peroxide radicals, which tells us that the free-radicals are in small number and that this number is constant, led us to calculate the expression of [k.sub.1][[R.sup.*]] thanks to Eq 9 (recently, Huang et al. [24] proposed an estimation of the chain scission rate constant [k.sub.1] using both deterministic and stochastic models). This kinetic model, which ca n accurately predict the reduction of the molecular weight, has a single variable parameter, f, the initiator efficiency of peroxide (Eq 6).

Reactive Twin Screw Extrusion Model

Degradation reaction was simulated using a one-dimensional global model developed for calculating the polymer flows in corotating self-wiping twin screw extruders. The model used is described into details in a previous paper [25]. It allows to calculate the profile along the screws of the main flow parameters such as the pressure, mean temperature, residence time, shear rate, viscosity and filling ratio by using a local one-dimensional approach.

The flow in screw elements (partially or totally filled right-handed and left-handed screw elements) is computed using cylindrical coordinates, in which the channel section is perpendicular to the screw flights. We consider the main flow along the screw channel. The flow path along the screws follows an eight-shaped pattern, and is composed of a succession of flows along the C-shaped chambers and flows through the intermeshing area between the adjacent screws. Thus, pressure/flow rate relationships are developed for these two categories of flows. In this simplified 1D approach, the section of the channel is considered as rectangular, with a constant width. The flows in kneading disks are modeled by considering only the peripheral flow around a disk. Owing to the geometry and the relative barrel velocity, this flow is characterized by a pressure peak located just before the tip of the disk. As the tips of the adjacent disks are staggered, the pressure profiles are also staggered, which creates an axial pressu re gradient, parallel to the screw axis and pushing the material downstream. This axial pressure gradient is determined by staggering the adjacent pressure profiles and adjusting the pressure level to match the imposed flow rate in the axial direction.

These elementary models are linked together to obtain a global description of the flow field along the extruder. It is assumed that the melting is instantaneous and takes place before the first restrictive element of the screw profile. As the screws are starve fed, the filling ratio of the system is not known. So, the computation has to start from the die and to proceed backwards. But, as the final product temperature is unknown, an iterative procedure is used. Starting from an arbitrary value of exit temperature, the software computes the successive pressures and temperatures in each element, until the first restrictive element is encountered, in which the melting is assumed to take place. Convergence is achieved when, at this location, the temperature equals the melting temperature of the product. Otherwise, the exit temperature is modified and the computation restarted.

The modeling of the reaction requires knowledge of the residence time spent inside each screw element and of the local temperature. We assume that the restrictive elements (left-handed kneading discs and screw elements) and the right-handed screw elements under pressure are completely full. For a sub-element, the mean residence time is assessed on the basis of the ratio of the free volume occupied by the molten polymer over the total flow rate. The calculation of average temperature is based on a local thermal balance, including the dissipated power and the heat transfer towards barrel and screws.

As seen before, the modeling of the flow parameters proceeds from the die towards the hopper, because the points at which the pressurization of the material begins are not known, the screws being starved. However, the calculation of the reaction progress of a chemical reaction must proceed in the opposite direction, for neither the final reaction rate, nor the residence time within the extruder are known. The calculation of the reaction begins at the place specified by the user, in the present case at the peroxide injection point. It is assumed that the peroxide is immediately homogenized. The coupling between the chemical reaction and thermo-mechanical parameters has been implemented as follows, assuming that the evolution in viscosity due to the chemical reaction progress does not modify in a significant way the filling ratio along the screws:

- We start a first calculation without any coupling, with the rheological and physical properties of the virgin material (i.e. the polypropylene alone). During this backward calculation, we identify points where the pressurization of the polymer begins.

- Then, we begin a second calculation, in downstream direction, which takes into account the interactions between the reaction conversion rate, the rheological changes and the flow parameters.

When the program meets a point where pressurization starts, it sets up a first pressure profile by coupling the reaction progress and the thermo-mechanical calculation. Coming out of a restrictive element (left-handed screw elements or blocks of kneading disks) or of the die, two cases are possible (Fig. 5):

- Either a positive pressure is found, which means that the filled length must be decreased and the point where the pressurization begins must then be replaced forward (case no. 1).

- Or a negative pressure is found, and then we must bring back the point where the pressurization begins (case no. 2).

After having modified the place where pressure starts, the program recalculates a new pressure profile, then adjusts progressively the starting point to obtain the desired exit pressure, i.e. 0 [plus or minus] [epsilon]. When the starting point is in a partially filled right-handed screw element, the program cuts out the C-shaped chambers of this screw element, until it finds the fraction of the C-shaped chamber corresponding to the required precision. Similarly, if the starting point is in a block of right-handed kneading disks, this block must be cut out to find the precise spot where pressurization must start. The convergence of the calculations is evidently more or less rapid, depending on the degree of precision required for the value of the exit pressure. In the following simulations, we required a precision equal to [10.sup.-3] MPa at the exit of each restrictive element and of the die, which corresponded to between 20 and 25 iterations to find the exact points where the pressurization began [18].

The principle of the calculation of the reaction progress in screw elements or in blocks of kneading disks is as follows (Fig. 6):

- We calculate the residence time [t.sub.i] in the element i.

- Knowing the temperature [T.sub.i] at the entrance of this element, we establish the variation of the corresponding molecular weights at this temperature and during this time, [delta][M.sub.i], using Eqs 9 to 17.

- We solve the Stokes equations taking into account the effect of the reaction progress on the viscosity, and thus we can calculate the local pressure gradient [delta]P along element i, corresponding to the throughput Q.

- We solve the thermal balance taking into consideration the effect of reaction heat [H.sub.R] on the average temperature. This term is a function of the system's activation enthalpy [delta]H (estimated to 130 KJ/kg) and of the degree of chemical reaction progression [[delta].sub.x] (here [[delta].sub.x] is equal to the relative variation of the molecular weight in one sub-element):

[delta]T = [H.sub.R]/[rho] [C.sub.p] [Q.sub.c] where [H.sub.R] = [rho] [Q.sub.c] [delta]H [[delta].sub.x] (18)

- Prior to assessing the variation in the molecular weight inside the next screw element i + 1, one must evaluate the position of the actual reaction progress along the kinetic curve at the temperature [T.sub.i+1] of this downstream element. One must thus determine the reaction time required moving from [T.sub.i] temperature curve to [T.sub.i+1]. This time [[t.sup.*].sub.i] is established by inverting the equations that allow us to calculate the variation of the molecular weights, knowing that the conversion rate to be considered is the cumulated conversion rate and that the temperature dependent parameters are calculated at the temperature of the downstream element [T.sub.i+1]. As no analytical solution is available, the approximation of [[t.sup.*].sub.i] must be carried out by dichotomy. Upon entering the element i + 1, the reaction progress is equal to [M.sub.i+1] = [delta][M.sub.i+1] [delta][M.sub.i+1], where [delta][M.sub.i+1] is equal to the variation of the conversion rate from time [[t.sup.*].sub.i] to [[t.sup.*].sub.i] + [t.sub.i+1]. The same method is then repeated moving downstream, element by element, and into the die.

Our modeling approach is different from that of Kim and White [16], who simulated the peroxide induced degradation and maleation of polypropylene, by using the software Akro-co-Twin Screw(R). They also used a nonisothermal approach based on continuum mechanics, and an iterative procedure for coupling temperature, reaction kinetics and viscosity changes. However they first calculate a pressure profile, proceeding from the die towards the hopper and considering the molecular weight as constant. They then calculate the evolution of temperature and knowing the reaction kinetics, a first profile of reaction rate, which is then used to recalculate the evolutions of the viscosity, pressure and filled zones. A new temperature can thus be estimated and the same method is repeated until the flow parameters do not vary significantly along the screws (criteria of convergence). But unfortunately, as we said before, they do not give any information about the change in viscosity due to the reaction and its effect on the th ermomechanical parameters.

RESULTS AND DISCUSSION

General Considerations

Figure 7 represents the evolution along the screw axis of the temperature, the pressure, the cumulated residence time as well as the weight average molecular weight. As we have already stated, the calculation of reaction starts at the place where the reactants have been injected and stops at the level of the vent zone. Then, from the vent zone to the die, only the thermal degradation reaction is taken into consideration. It is worthwhile to mention that the calculations have been made with adiabatic conditions for the screws and a heat transfer coefficient towards the barrel [h.sub.b] identical for virgin and degraded polypropylene, and regardless of the experimentally observed conditions ([h.sub.b] = 500 W/[m.sup.2]/K). We can see that the pressure decreases during the reaction. This decrease is more obvious if the quantity of injected free-radicals is greater. This decrease is linked to the reaction, which causes a substantial fall in the viscosity, which is reflected in the pressure gradient.

Similarly, we notice that the temperature of the material after reaction falls as far as the temperature for virgin polypropylene. This decrease is due to the fact that the dissipated power decreases if the viscosity decreases. At high concentrations of peroxide, the dissipated power becomes quickly very low and the average temperature depends essentially on the heat transfer. The temperature is then equal to the barrel regulation temperature. On the contrary, as far as the residence time is concerned, no significant variation has been observed. The minimal residence times measured with colored pellets were nearly identical for virgin PP and degraded PP. Nevertheless, we have observed a slight decrease of minimal residence times when increasing peroxide concentration. The average residence time modeled varies very little with DHBP concentration. The postulate on which we based the coupling between reaction and flow parameters has been checked: the variations of viscosity due to the degradation reaction do no t modify significantly the filling ratio along the screws.

Influence of Concentration of Peroxide

As explained previously, the peroxide efficiency f is an unknown parameter of the model. In the literature, values between 0.6 and 1 are often used for peroxide concentration lower than 0.1 wt% (11, 14, 26). In the present work, we decided to define f as a function of peroxide concentration. Efficiency values were obtained by fitting the model to the experimental weight average molecular weight measured by SEC. We chose [M.sub.w] instead of [M.sub.n] because the SEC's results are more reliable and accurate for the weight average than for the number average molecular weight. The results are presented in Fig. 8. We observe that f decreases when the peroxide concentration increases. The range of f values is coherent with those found in the literature. In the next sections, when we will study the influence of processing conditions, f will be kept constant and just defined according to the amount of peroxide.

Influence of Throughput

The change in mass average molecular weight along the screws for three different throughputs (at constant screw speed, N = 225 rpm) is presented in Fig. 9. We observe that the degradation is more rapid and more severe when the feed rate is low. This is principally due to a higher residence time, which increases from 40 s at 20 kg/h to 120 s at 2.5 kg/h. The experimental measurements made at the die exit show a very good agreement with the predictions of the model for the two studied peroxide concentrations (0.1 and 0.2 wt%).

Influence of Screw Speed

The influence of screw speed at constant feed rate is more difficult to foresee, because screw speed has an opposite effect on the two parameters which control the reaction: an increase in rotation speed decreases the residence time but increases the product temperature, due to a higher viscous dissipation. As observed in Fig. 10, despite a reduction of residence time from 65 to 40 s (when the screw speed is increased from 150 to 300 rpm), the polypropylene is more degraded at high rotation speed. This is due to the fact that, all along the screws, the temperatures are higher at higher screw speed (Fig. 11). Once again, it can be seen in Fig. 10 that the predictions of the model are confirmed by the SEC's measurements made at the die exit.

Other experimentations have been carried out to check the validity of the theoretical model, using another machine (Werner & Pfleiderer ZSK 58) and another polymer (polypropylene copolymer). In all cases, agreements between computed and measured molecular weights are correct [18]. A general comparison is proposed in Fig. 12. The agreement concerning the weight average molecular weight is pretty good. For the number average molecular weight, the model leads to an overestimation, mainly at high peroxide concentration. Nevertheless, the order of magnitude is correct.

CONCLUSIONS

We have proposed a modeling of the peroxide induced degradation of polypropylene in a corotating twin screw extruder. Using a kinetic scheme taken from the literature and a rheological model accounting for the changes in viscosity with the weight average molecular weights, we have developed a specific method to couple the computation of the thermo-mechanical parameters of the flow along the screws with the reaction's progress. Once the efficiency of the peroxide is assumed, we are able to predict the changes in average molecular weights along the screw profile, accounting for the processing conditions (feed rate, rotation speed, and barrel temperature). Experiments carried out with different machines provided results in good agreement with the predictions of the model. Future work in progress will include the computation of the whole molecular weight distribution, instead of only the average values.

ACKNOWLEDGMENTS

This study was partially granted by Elf Atochem Company. We are grateful to Cerdato (Serquigny, France) for the experimental facilities and to GRL (Lacq, France) for the SEC analyses of the samples. FB was supported by a grant from French "Ministereb de l'Enseignement Superieur et de la Recherche."

(*.) To whom correspondence should be addressed.

REFERENCES

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In this work, experimental and theoretical studies of the free-radical initiated molecular weight degradation of polypropylene in a modular self-wiping corotating twin screw extruder have been investigated. Our objective was to build a model that would be able to predict the evolution of the average molecular weight along the screws, in relation to the processing conditions and the geometry of the twin screw extruder. Modeling the process involves resolving interactions occurring between the various flow conditions encountered in the extruder, the kinetics of the reaction and the changes in viscosity with changes in molecular weight. We have studied the influence of operating parameters such as the initial peroxide concentration, the feed rate and the screw speed on the degradation reaction. Good agreement was found between theoretical results and experimental values obtained by size exclusion chromatography measurements.

INTRODUCTION

Over the last twenty years, reactive extrusion has undergone considerable development and the number of publications and patents in the subject area has continually increased [1, 2]. The success of the reactive extrusion process is due to the fact that polymers can be directly modified in the molten state, during a single stage of transformation. Twin screw extruders provide great flexibility and are often used to perform the chemical modification of natural or synthetic polymers. The modular construction of screw-barrel blocks is particularly favorable for the design of screw profiles, which are specific to the studied reactions and include, if necessary, separated zones for melting, homogenization, reaction and devolatilization [3-5].

In comparison to reactions conducted in solution, reactions carried out in the extruder show an important influence of the temperature, viscosity and pressure [6]. In fact, reactive extrusion is a highly nonisothermal process and the temperature is generally high, relative to the dissipated power and the exothermy of the undergoing reactions. The viscosity of the system depends simultaneously on the temperature, the shear rate conditions and the molecular mass. Finally, the pressure depends mostly on the type of screw elements used (fully filled or starved zones).

In the modeling of the reactive extrusion process, we started by studying in a first step chemical reactions which do not modify in a significant way the system's viscosity (transesterification of an ethylene and vinyl acetate copolymer). Therefore, we could calculate separately the flow parameters and the reaction progress [7]. In the present work, we are interested in more complex reactions including sensitive modifications of the rheological behavior during the reaction, in connection with the evolution of molecular weight. Among the different possible reactions, we have chosen the peroxide induced degradation of polypropylene.

This type of reaction has been largely studied in the literature, because it is widely used in polymer industry to control the molecular weight distribution of polypropylene resins. Indeed, polypropylenes manufactured using classical industrial polymerization processes (using Ziegler-Natta catalyst systems) usually have excellent mechanical properties, but have also a high molecular weight and a broad polydispersity. This makes their processing difficult, in view of their high viscosity and elasticity [8]. In order to adjust their molecular weight distribution, polypropylene resins can be modified in reactive extrusion operations by means of organic peroxide initiated scission reactions. The principle of this reaction is as follows. The peroxide is decomposed at high temperature, leading to free-radicals. These free-radicals then attack the links C-H of polypropylene that have the weakest link energy and give rise to tertiary polymer macroradicals. There follows division of the macromolecular chain of the po lypropylene by [beta]-scission, which leads to the creation of polypropylene of reduced mass and of a new polymer radical, but this time of secondary type [9]. This mechanism has been proposed and explicated in numerous publications [10-14].

In the literature, there is little work devoted to the global approach of the coupling between flow and reaction. Tzoganakis et al. [11, 15] have proposed models for the peroxide induced degradation of polypropylene in a single screw extruder. In a first approach [11]. plug flow and isothermal conditions in the extruder were assumed. In a second step [15], residence time distribution and nonisothermal conditions were taken into account, and the viscosity was a function of the average molecular weight. More recently. Kim and White [16] have described a twin screw extrusion model allowing simulation of the peroxide induced degradation and maleation of polypropylene. Unfortunately, they do not give any information about the change in viscosity due to the reaction and its effect on the thermomechanical parameters.

In the present study, the kinetic scheme used to describe the changes of the molecular weight distribution was taken from the literature. Many studies, and particularly the work of Tzoganakis [17], have been made on the peroxide initiated degradation of polypropylene in the molten state, to build a kinetic model, which can predict the evolution of molecular mass. In the present work, the dependence of the viscosity with the molecular mass was characterized by studying different molecular weight polypropylenes. This allowed us to determine relationships between the viscosity, the shear rate, the temperature and the molecular weight. A theoretical model for predicting changes in molecular weight all along the extruder is presented. We also describe experiments carried out with a modular corotating twin screw extruder, which show clearly the influence of the operating parameters on the degradation reaction. Finally, we compare the results of the model to these experimental trials.

EXPERIMENTS

Materials and Processing Conditions

A commercial homopolymer (PP 3050 BN 1, Appryl) was used. The polypropylene has a melt flow index of 4.5 g/10 min (2.16 kg/230[degrees]C), an initial weight average molecular weight of 301,600 g/mol and a polydispersity index, [M.sub.w]/[M.sub.n] of 6.4. The peroxide used was 2,5-dimethyl-2,5-di(tert-butylperoxy)hexane (DHBP, Trigonox 101) provided by Akzo Chemie.

Reactive experiments were carried out on a corotating self-wiping twin screw extruder (model ZSK 30, Werner & Pfleiderer), which included twelve barrel sections and six corresponding heating zones. The first barrel (feeding hopper) was not heated. The centerline distance was 26 mm, the screw diameter. D, 30.85 mm and the L/D ratio was 37.5, where L is the total length of the screws. The PP polymer pellets were fed via a weighted feeding system. The peroxide was injected into the extruder by a volumetric piston feed pump. The DHBP was previously diluted in 1, 2, 4 trichlorobenzene (solution at 15%) to have a good precision on the pump flow rate, even for concentrations less than 1000 ppm (i.e. 0.1 wt%). The output rate of the piston pump was adjusted according to the flow rate of polypropylene and its regularity was controlled by the loss in weight of the beaker containing the peroxide. This method provided specific data on the real peroxide flow rate for use in simulations of the degradation reaction.

The extrusion setup and screw configuration used are presented in Fig. 1 and Table 1. The melting of the polymer was assured by kneading disc blocks and left-handed screw elements. The injection of DHBP was housed in a unfilled area, just after the melting zone. The reaction area first included two kneading disc blocks designed to improve the mixing and homogenization between free-radicals and polypropylene. Next were situated several right-handed conveying screw elements with decreasing pitches and finally a left-handed screw element situated just before the devolatilization zone (localized at barrel 10). At this level, the degradation reaction of polypropylene was stopped by the elimination by a vacuum pump of all volatile reactants. The melt was extruded through a converging plate and three parallel capillary dies (each with diameter of 3.5 mm and length of 10 mm). At the die exit, extrudates were quenched in water before being dried and cut into pellets. Samples were collected after about 20 min of stead y state extrusion conditions (stable die pressure and motor torque).

During the tests, the following parameters were varied: quantity of initiator concentration [I.sub.0] (0.0, 0.025, 0.05, 0.1, 0.2 and 0.4 wt%), feed rate Q (2.5, 10 and 20 kg/h), screw speed N (150, 225 and 300 rpm). The barrel regulation temperature [T.sub.R] was fixed at 170[degrees]C. We chose a relatively low barrel temperature such that the degradation reaction was still unfinished upon reaching the devolatilization zone. This choice allowed us to examine the influence of the feed rate and screw speed (i.e. the influence of the residence time and temperature along the screws) on the molecular weight measured at the die exit. Indeed, tests conducted at higher barrel regulation temperatures (greater than 220[degrees]C) did not permit us to characterize the influence of processing conditions, as the reaction was finished before the vent zone, due to the very short half-life time of peroxide at these temperatures. The data of the different experimental conditions are reported in Table 2.

Molecular Weight Distribution and Rheological Measurements

In order to have a precise knowledge of the reactive system's viscosity, the complex viscosity of different molecular weight polypropylenes was measured in a Rheometrics RMS 800 spectrometer, using parallel-plate geometry in oscillatory mode. These samples were previously prepared at 250[degrees]C in a laboratory twin screw extruder (Leistritz 30-34) by using various amounts of peroxide. The concentration range was 0.01 to 0.5 wt%. The samples were dried under vacuum, at 80[degrees]C during a 24-hour period, in order to eliminate all the residual peroxide. The molecular weight distributions of these differently degraded polypropylenes were analyzed by size exclusion chromatography (SEC). The samples were dissolved at 145[degrees], at a polymer concentration of 0.8 g/L in the 1,2,4 trichlorobenzene, which was used as a solvent with the addition of an antioxidant to prevent any degradation. The results obtained are presented in Fig. 2 and Table 3. We classically observed that increasing the peroxide concentrat ion decreased the presence of high molecular weight species and narrowed the molecular weight distribution. We observe also a slight decrease in the molecular weight between virgin and extruded polypropylene, due to some thermomechanical degradation during extrusion. Indeed, processing conditions were particularly severe (250[degrees]C, 300 rpm). It was no more the case for the experiments carried out at low temperature (170[degrees]C) on the ZSK 30.

The dynamic measurements were carried out over a frequency range of 0.1 to 100 rad/s and at five temperatures (185[degrees]C to 245[degrees]C). Time sweep tests (30 min) made at the different temperatures at a frequency of 1 rad/s showed that the complex viscosity and the storage and loss moduli were constant with time (18). Figure 3 shows the time-temperature superposition of the complex viscosity (temperature reference of 215[degrees]C). The results obtained showed the influence of the molecular weight upon the viscosity. We observed the usual decrease in viscosity with decreasing molecular weight and a pronounced shift towards more Newtonian behavior (19, 20). To characterize the influence of the modifications induced by the peroxide on the homopolymer's rheological behavior, a Carreau-Yasuda law was fitted to the master curves:

[eta]/[a.sub.T] = [[eta].sub.0][[1+[([lambda][gamma]*[a.sub.T]).sup.a]].sup.m-1/a] with [a.sub.T] = [e.sup.[E.sub.a]/R[1/T-1/[T.sub.0]] (1)

where [a.sub.T] is the shift factor, [E.sub.a] the activation energy and R the gas constant. [T.sub.0] is the reference temperature (215[degrees]C) at which the zero shear viscosity [[eta].sub.0], the characteristic time [lambda], the power-law index m, and the parameter a were determined.

For each value of the weight average molecular weight, the determination of the parameters [[eta].sub.0], [lambda], m, and a was done using a homemade identification program which minimized, using a least square method, the differences between the measured and calculated viscosities. The set of initial values was selected in function of the experimental values. If changing parameters of the initial set one at a time gave different solutions, the solution adopted was the set which gave the best minimization of the objective function. The result of the superposition of the experimental and calculated curves is shown in Fig. 3. Figure 4 shows the evolution of each factor as a function of the weight average molecular weight [M.sub.w]. We note that the zero shear viscosity varies with a 3.9 power of [M.sub.w]. For flexible polymer chains, at high molecular weights, the zero shear viscosity usually increases with a 3.4 power of the molecular weight [20, 21]. We have observed that, when increasing the molecular wei ght, characteristic time [lambda] increased while m and a decreased continuously. Finally, we have observed that the activation energy of the different degraded samples slightly decreased (44.9 to 39.3 kJ/mol) when increasing the molecular weight (62,970 to 301,600 g/mol).

THEORETICAL MODELING

Kinetic Scheme

To model the peroxide induced degradation of polypropylene by a free-radical initiator, it is assumed that:

- The peroxide is homogeneously distributed in the polymer,

- There is no thermo-oxidative degradation,

- All the free-radicals have the same dissociation energy.

- The secondary decomposition of the DHBP molecules are neglected.

The following mechanism has been proposed by numerous authors [10-14]:

- Free-radical initiation:

I[right arrow] [k.sub.d]2R (i)

- Hydrogen abstraction and chain [beta]-scission:

[P.sub.n] + R [right arrow] [k.sub.1] [P.sub.r] + [P.sub.n-r] + R (ii)

- Inter-molecular chain transfer:

[P.sub.n] + [[P.sup.*].sub.r] [right arrow] [k.sub.2] [P.sub.r] + [[P.sup.*].sub.s] + [P.sub.n-s] (iii)

- Thermal degradation:

[P.sub.n] [right arrow] [k.sub.3] [[P.sup.*].sub.r] + [[P.sup.*].sub.n-r] (iv)

- Termination by disproportionation:

[[P.sup.*].sub.n] + [[P.sup.*].sub.r] [right arrow] [k.sub.4] [P.sub.n] + [P.sub.r] (v)

I represents the free-radical initiator, [R.sup.*], the peroxide free-radicals, [P.sub.n], the polymer, [[P.sup.*].sub.n], the polymer macro-radical and [k.sub.i], the rate constants. Depending on the authors, the species balance can be expressed differently. In this paper, we have chosen the equations proposed by Krell et al. [14], who are alone to provide kinetic constants for all the reactions cited above. The changes of concentration of the different species with time are thus the following:

- Peroxide balance:

d[I]/dt = - [k.sub.d][I] (2)

- Initiation radical balance:

d[[R.sup.*]]/dt = 2f[k.sub.d][I] - [k.sub.1][[R.sup.*]] [[[sigma].sup.[infinity]].sub.n=2](n - 1)[[P.sub.n]] (3)

- Polymer balance:

d[P.sub.n]/dt = [k.sub.1][[R.sup.*]] [[[[sigma].sup.[infinity]].sub.t=n+1] [[P.sub.i]] - (n - 1) [[P.sub.n]]] - [k.sub.2][(n-1)[[P.sub.n]] [[[sigma].sup.[infinity]].sub.i=1] [[[P.sup.*].sub.i]] - [[[P.sup.*].sub.n]] [[[sigma].sup.[infinity]].sub.i=1](i - 1)[[P.sub.i]] - [[[sigma].sup.[infinity]].sub.i=1][[[P.sup.*].sub.i]] [[[sigma].sup.[infinity]].sub.j=n+1] [[P.sub.j]]] - [k.sub.3] (n -1)[[P.sub.n]] + [k.sub.4][[[P.sup.*].sub.n]] [[[sigma].sup.[infinity]].sub.i=1][[[P.sup.*].sub.i]] (4)

- Polymer macroradical balance:

d[[[P.sup.*].sub.n]]/dt = [k.sub.1][[R.sup.*]] [[[sigma].sup.[infinity]].sub.i=n+1] [[P.sub.i]] - [k.sub.2][[[[P.sup.*].sub.n]] [[[sigma].sup.[infinity]].sub.i=1](i = 1)[[P.sub.i]] - [[[sigma].sup.[infinity]].sub.i=1][[[P.sup.*].sub.i]] [[[sigma].sup.[infinity]].sub.j=n+1][[P.sub.j]]] + 2[k.sub.3] [[[sigma].sup.[infinity]].sub.i=n+1][[P.sub.i]] - [k.sub.4][[[P.sup.*].sub.n]] [[[sigma].sup.[infinity]].sub.i=1][[[P.sup.*].sub.i]] (5)

The parameter f, which appears in Eq 3 is called the efficiency of the peroxide. It is defined as:

f = number of R which causes [beta]-scission at time t/total number of primary R at time t (6)

In order to reduce the dimensions of the infinite set of the obtained coupled differential equations, the ith moment equations of the chain length distribution for the polymer ([Q.sub.i]) and polymer radicals ([Y.sub.i]) are introduced [22]:

[Q.sub.i] = [[[sigma].sup.[infinity]].sub.n=1] [n.sup.i]*[[P.sub.n]] (7)

[Y.sub.i] = [[[sigma].sup.[infinity]].sub.n=1][n.sup.i]*[[[P.sup.*].sub.n]] (8)

Introducing these quantities into Eqs 3 to 5 leads to the following system of only eight equations, than includes also Eq 2:

d[[R.sup.*]]/dt = 2f [k.sub.d][I] - [k.sub.1][[R.sup.*]]([Q.sub.1] - [Q.sub.0]) (9)

d[Q.sub.0]/dt = [k.sub.2] [Y.sub.0] ([Q.sub.1] - [Q.sub.0]) - [k.sub.3] ([Q.sub.1] - [Q.sub.0]) + [k.sub.4] [[Y.sup.2].sub.0] (10)

d[Q.sub.1]/dt = - 1/2 [k.sub.1][[R.sup.*]]([Q.sub.2] - [Q.sub.1]) - [k.sub.2][1/2[Y.sub.0] ([Q.sub.2] - [Q.sub.1]) - [Y.sub.1]([Q.sub.1] - [Q.sub.0])] - [k.sub.3]([Q.sub.2] - [Q.sub.1]) + [k.sub.4] [Y.sub.1] [Y.sub.0] (11)

d[Q.sub.2]/dt = [k.sub.1][[R.sup.*]][- 2/3 [Q.sub.3] + 1/2 [Q.sub.2] + 1/6 [Q.sub.1]] + [k.sub.2][[Y.sub.0](- 2/3 [Q.sub.3] + 1/2 [Q.sub.2] + 1/6 [Q.sub.1]) + [Y.sub.2]([Q.sub.1] - [Q.sub.0])] - [k.sub.3]([Q.sub.3] - [Q.sub.2]) + [k.sub.4] [Y.sub.0] [Y.sub.2] (12)

d[Y.sub.0]/dt = [k.sub.1] [[R.sup.*]] ([Q.sub.1] - [Q.sub.0]) + 2 [k.sub.3] ([Q.sub.1] - [Q.sub.0]) - [k.sub.4] [[Y.sup.2].sub.0] (13)

d[Y.sub.1]/dt = 1/2 [k.sub.1][[R.sup.*]]([Q.sub.2] - [Q.sub.1]) + [k.sub.2][- [Y.sub.1]([Q.sub.1] - [Q.sub.0]) + 1/2 [Y.sub.0]([Q.sub.2] - [Q.sub.1])] + [k.sub.3]([Q.sub.2] - [Q.sub.1]) - [k.sub.4] [Y.sub.0] [Y.sub.1] (14)

d[Y.sub.2]/dt = [k.sub.1][[R.sup.*]][1/3 [Q.sub.3] - 1/2 [Q.sub.2] + 1/6 [Q.sub.1]] + [k.sub.2][[Y.sub.0](1/3 [Q.sub.3] - 1/2 [Q.sub.2] + 1/6 [Q.sub.1]) - [Y.sub.2]([Q.sub.1] - [Q.sub.0])] + [k.sub.3][2/3 [Q.sub.3] - [Q.sub.2] + 1/3 [Q.sub.1]] - [k.sub.4] [Y.sub.0] [Y.sub.2] (15)

This system introduces the moment [Q.sub.3], which can be correlated with the first three moments by using the closure method proposed by Hulburt and Katz [23]:

[Q.sub.3] = 2 [Q.sub.2]/[Q.sub.1] [Q.sub.0] [2 [Q.sub.2] [Q.sub.0] - [[Q.sup.2].sub.1]] (16)

The different average molecular weights are thus given by the following equation:

[M.sub.average] = [[[sigma].sup.[infinity]].sub.t=1] [n.sub.i][[M.sup.K].sub.i]/[[[sigma].sup.[infinity]].sub.t=1] [n.sub.i][[M.sup.k-1].sub.i] = [m.sub.0] [Q.sub.k] + [Y.sub.k]/[Q.sub.k-1] + [Y.sub.k-1] (17)

where k = 1 for the number average molecular weight, [M.sub.n], and k = 2 for the weight average molecular weight, [M.sub.w]. [m.sub.0] represents the molecular weight of the propylene monomer.

To solve the problem, the majority of authors, with the exception of Krell et al. [14], take only into account the free-radical initiation (i) and chain scission reactions (ii) and neglect the contributions from transfer (iii), thermal degradation (iv) and termination reactions (v). In this work, we decided to use the model of Krell et al. [14] and thus its kinetic constants, which are presented in Table 4. Consequently, we could solve the equation [2] and the moment equations [9-15] without neglecting any reaction. A Runge-Kutta method with adaptive stepsize control was used to solve the system of equations [18]. The quasi steadystate assumption for the peroxide radicals, which tells us that the free-radicals are in small number and that this number is constant, led us to calculate the expression of [k.sub.1][[R.sup.*]] thanks to Eq 9 (recently, Huang et al. [24] proposed an estimation of the chain scission rate constant [k.sub.1] using both deterministic and stochastic models). This kinetic model, which ca n accurately predict the reduction of the molecular weight, has a single variable parameter, f, the initiator efficiency of peroxide (Eq 6).

Reactive Twin Screw Extrusion Model

Degradation reaction was simulated using a one-dimensional global model developed for calculating the polymer flows in corotating self-wiping twin screw extruders. The model used is described into details in a previous paper [25]. It allows to calculate the profile along the screws of the main flow parameters such as the pressure, mean temperature, residence time, shear rate, viscosity and filling ratio by using a local one-dimensional approach.

The flow in screw elements (partially or totally filled right-handed and left-handed screw elements) is computed using cylindrical coordinates, in which the channel section is perpendicular to the screw flights. We consider the main flow along the screw channel. The flow path along the screws follows an eight-shaped pattern, and is composed of a succession of flows along the C-shaped chambers and flows through the intermeshing area between the adjacent screws. Thus, pressure/flow rate relationships are developed for these two categories of flows. In this simplified 1D approach, the section of the channel is considered as rectangular, with a constant width. The flows in kneading disks are modeled by considering only the peripheral flow around a disk. Owing to the geometry and the relative barrel velocity, this flow is characterized by a pressure peak located just before the tip of the disk. As the tips of the adjacent disks are staggered, the pressure profiles are also staggered, which creates an axial pressu re gradient, parallel to the screw axis and pushing the material downstream. This axial pressure gradient is determined by staggering the adjacent pressure profiles and adjusting the pressure level to match the imposed flow rate in the axial direction.

These elementary models are linked together to obtain a global description of the flow field along the extruder. It is assumed that the melting is instantaneous and takes place before the first restrictive element of the screw profile. As the screws are starve fed, the filling ratio of the system is not known. So, the computation has to start from the die and to proceed backwards. But, as the final product temperature is unknown, an iterative procedure is used. Starting from an arbitrary value of exit temperature, the software computes the successive pressures and temperatures in each element, until the first restrictive element is encountered, in which the melting is assumed to take place. Convergence is achieved when, at this location, the temperature equals the melting temperature of the product. Otherwise, the exit temperature is modified and the computation restarted.

The modeling of the reaction requires knowledge of the residence time spent inside each screw element and of the local temperature. We assume that the restrictive elements (left-handed kneading discs and screw elements) and the right-handed screw elements under pressure are completely full. For a sub-element, the mean residence time is assessed on the basis of the ratio of the free volume occupied by the molten polymer over the total flow rate. The calculation of average temperature is based on a local thermal balance, including the dissipated power and the heat transfer towards barrel and screws.

As seen before, the modeling of the flow parameters proceeds from the die towards the hopper, because the points at which the pressurization of the material begins are not known, the screws being starved. However, the calculation of the reaction progress of a chemical reaction must proceed in the opposite direction, for neither the final reaction rate, nor the residence time within the extruder are known. The calculation of the reaction begins at the place specified by the user, in the present case at the peroxide injection point. It is assumed that the peroxide is immediately homogenized. The coupling between the chemical reaction and thermo-mechanical parameters has been implemented as follows, assuming that the evolution in viscosity due to the chemical reaction progress does not modify in a significant way the filling ratio along the screws:

- We start a first calculation without any coupling, with the rheological and physical properties of the virgin material (i.e. the polypropylene alone). During this backward calculation, we identify points where the pressurization of the polymer begins.

- Then, we begin a second calculation, in downstream direction, which takes into account the interactions between the reaction conversion rate, the rheological changes and the flow parameters.

When the program meets a point where pressurization starts, it sets up a first pressure profile by coupling the reaction progress and the thermo-mechanical calculation. Coming out of a restrictive element (left-handed screw elements or blocks of kneading disks) or of the die, two cases are possible (Fig. 5):

- Either a positive pressure is found, which means that the filled length must be decreased and the point where the pressurization begins must then be replaced forward (case no. 1).

- Or a negative pressure is found, and then we must bring back the point where the pressurization begins (case no. 2).

After having modified the place where pressure starts, the program recalculates a new pressure profile, then adjusts progressively the starting point to obtain the desired exit pressure, i.e. 0 [plus or minus] [epsilon]. When the starting point is in a partially filled right-handed screw element, the program cuts out the C-shaped chambers of this screw element, until it finds the fraction of the C-shaped chamber corresponding to the required precision. Similarly, if the starting point is in a block of right-handed kneading disks, this block must be cut out to find the precise spot where pressurization must start. The convergence of the calculations is evidently more or less rapid, depending on the degree of precision required for the value of the exit pressure. In the following simulations, we required a precision equal to [10.sup.-3] MPa at the exit of each restrictive element and of the die, which corresponded to between 20 and 25 iterations to find the exact points where the pressurization began [18].

The principle of the calculation of the reaction progress in screw elements or in blocks of kneading disks is as follows (Fig. 6):

- We calculate the residence time [t.sub.i] in the element i.

- Knowing the temperature [T.sub.i] at the entrance of this element, we establish the variation of the corresponding molecular weights at this temperature and during this time, [delta][M.sub.i], using Eqs 9 to 17.

- We solve the Stokes equations taking into account the effect of the reaction progress on the viscosity, and thus we can calculate the local pressure gradient [delta]P along element i, corresponding to the throughput Q.

- We solve the thermal balance taking into consideration the effect of reaction heat [H.sub.R] on the average temperature. This term is a function of the system's activation enthalpy [delta]H (estimated to 130 KJ/kg) and of the degree of chemical reaction progression [[delta].sub.x] (here [[delta].sub.x] is equal to the relative variation of the molecular weight in one sub-element):

[delta]T = [H.sub.R]/[rho] [C.sub.p] [Q.sub.c] where [H.sub.R] = [rho] [Q.sub.c] [delta]H [[delta].sub.x] (18)

- Prior to assessing the variation in the molecular weight inside the next screw element i + 1, one must evaluate the position of the actual reaction progress along the kinetic curve at the temperature [T.sub.i+1] of this downstream element. One must thus determine the reaction time required moving from [T.sub.i] temperature curve to [T.sub.i+1]. This time [[t.sup.*].sub.i] is established by inverting the equations that allow us to calculate the variation of the molecular weights, knowing that the conversion rate to be considered is the cumulated conversion rate and that the temperature dependent parameters are calculated at the temperature of the downstream element [T.sub.i+1]. As no analytical solution is available, the approximation of [[t.sup.*].sub.i] must be carried out by dichotomy. Upon entering the element i + 1, the reaction progress is equal to [M.sub.i+1] = [delta][M.sub.i+1] [delta][M.sub.i+1], where [delta][M.sub.i+1] is equal to the variation of the conversion rate from time [[t.sup.*].sub.i] to [[t.sup.*].sub.i] + [t.sub.i+1]. The same method is then repeated moving downstream, element by element, and into the die.

Our modeling approach is different from that of Kim and White [16], who simulated the peroxide induced degradation and maleation of polypropylene, by using the software Akro-co-Twin Screw(R). They also used a nonisothermal approach based on continuum mechanics, and an iterative procedure for coupling temperature, reaction kinetics and viscosity changes. However they first calculate a pressure profile, proceeding from the die towards the hopper and considering the molecular weight as constant. They then calculate the evolution of temperature and knowing the reaction kinetics, a first profile of reaction rate, which is then used to recalculate the evolutions of the viscosity, pressure and filled zones. A new temperature can thus be estimated and the same method is repeated until the flow parameters do not vary significantly along the screws (criteria of convergence). But unfortunately, as we said before, they do not give any information about the change in viscosity due to the reaction and its effect on the th ermomechanical parameters.

RESULTS AND DISCUSSION

General Considerations

Figure 7 represents the evolution along the screw axis of the temperature, the pressure, the cumulated residence time as well as the weight average molecular weight. As we have already stated, the calculation of reaction starts at the place where the reactants have been injected and stops at the level of the vent zone. Then, from the vent zone to the die, only the thermal degradation reaction is taken into consideration. It is worthwhile to mention that the calculations have been made with adiabatic conditions for the screws and a heat transfer coefficient towards the barrel [h.sub.b] identical for virgin and degraded polypropylene, and regardless of the experimentally observed conditions ([h.sub.b] = 500 W/[m.sup.2]/K). We can see that the pressure decreases during the reaction. This decrease is more obvious if the quantity of injected free-radicals is greater. This decrease is linked to the reaction, which causes a substantial fall in the viscosity, which is reflected in the pressure gradient.

Similarly, we notice that the temperature of the material after reaction falls as far as the temperature for virgin polypropylene. This decrease is due to the fact that the dissipated power decreases if the viscosity decreases. At high concentrations of peroxide, the dissipated power becomes quickly very low and the average temperature depends essentially on the heat transfer. The temperature is then equal to the barrel regulation temperature. On the contrary, as far as the residence time is concerned, no significant variation has been observed. The minimal residence times measured with colored pellets were nearly identical for virgin PP and degraded PP. Nevertheless, we have observed a slight decrease of minimal residence times when increasing peroxide concentration. The average residence time modeled varies very little with DHBP concentration. The postulate on which we based the coupling between reaction and flow parameters has been checked: the variations of viscosity due to the degradation reaction do no t modify significantly the filling ratio along the screws.

Influence of Concentration of Peroxide

As explained previously, the peroxide efficiency f is an unknown parameter of the model. In the literature, values between 0.6 and 1 are often used for peroxide concentration lower than 0.1 wt% (11, 14, 26). In the present work, we decided to define f as a function of peroxide concentration. Efficiency values were obtained by fitting the model to the experimental weight average molecular weight measured by SEC. We chose [M.sub.w] instead of [M.sub.n] because the SEC's results are more reliable and accurate for the weight average than for the number average molecular weight. The results are presented in Fig. 8. We observe that f decreases when the peroxide concentration increases. The range of f values is coherent with those found in the literature. In the next sections, when we will study the influence of processing conditions, f will be kept constant and just defined according to the amount of peroxide.

Influence of Throughput

The change in mass average molecular weight along the screws for three different throughputs (at constant screw speed, N = 225 rpm) is presented in Fig. 9. We observe that the degradation is more rapid and more severe when the feed rate is low. This is principally due to a higher residence time, which increases from 40 s at 20 kg/h to 120 s at 2.5 kg/h. The experimental measurements made at the die exit show a very good agreement with the predictions of the model for the two studied peroxide concentrations (0.1 and 0.2 wt%).

Influence of Screw Speed

The influence of screw speed at constant feed rate is more difficult to foresee, because screw speed has an opposite effect on the two parameters which control the reaction: an increase in rotation speed decreases the residence time but increases the product temperature, due to a higher viscous dissipation. As observed in Fig. 10, despite a reduction of residence time from 65 to 40 s (when the screw speed is increased from 150 to 300 rpm), the polypropylene is more degraded at high rotation speed. This is due to the fact that, all along the screws, the temperatures are higher at higher screw speed (Fig. 11). Once again, it can be seen in Fig. 10 that the predictions of the model are confirmed by the SEC's measurements made at the die exit.

Other experimentations have been carried out to check the validity of the theoretical model, using another machine (Werner & Pfleiderer ZSK 58) and another polymer (polypropylene copolymer). In all cases, agreements between computed and measured molecular weights are correct [18]. A general comparison is proposed in Fig. 12. The agreement concerning the weight average molecular weight is pretty good. For the number average molecular weight, the model leads to an overestimation, mainly at high peroxide concentration. Nevertheless, the order of magnitude is correct.

CONCLUSIONS

We have proposed a modeling of the peroxide induced degradation of polypropylene in a corotating twin screw extruder. Using a kinetic scheme taken from the literature and a rheological model accounting for the changes in viscosity with the weight average molecular weights, we have developed a specific method to couple the computation of the thermo-mechanical parameters of the flow along the screws with the reaction's progress. Once the efficiency of the peroxide is assumed, we are able to predict the changes in average molecular weights along the screw profile, accounting for the processing conditions (feed rate, rotation speed, and barrel temperature). Experiments carried out with different machines provided results in good agreement with the predictions of the model. Future work in progress will include the computation of the whole molecular weight distribution, instead of only the average values.

ACKNOWLEDGMENTS

This study was partially granted by Elf Atochem Company. We are grateful to Cerdato (Serquigny, France) for the experimental facilities and to GRL (Lacq, France) for the SEC analyses of the samples. FB was supported by a grant from French "Ministereb de l'Enseignement Superieur et de la Recherche."

(*.) To whom correspondence should be addressed.

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Screw Profile (From Hopper to Die). Length (mm) 14 84 56 40 56 28 28 20 42 28 60 14 14 84 Pitch (mm) 14 42 28 20 14 KB [*] KB [*] -20 [**] 42 28 20 KB [*] KB [*] 28 45/5 90/5 45/5 90/5 Length (mm) 130 126 10 126 56 70 70 Pitch (mm) 20 14 -20 [**] 42 28 20 14 (*.)KB 45/5 means kneading discs block of 5 elements, with a staggering angle of 45[degrees] (right-handed). (**.)A negative pitch corresponds to a left-handed screw element. Experimental Conditions. [I.sub.0] (wt%) [I.sub.0] (wt%) Q (kg/h) N (rpm) Target Measured Influence of peroxide concentration 0.025 [*] 0.028 10 225 0.050 [*] 0.055 10 225 0.100 [*] 0.101 10 225 0.200 [*] 0.194 10 225 0.400 [*] 0.378 10 225 Influence of feed rate 0.100 0.116 2.5 225 0.200 0.215 2.5 225 0.100 0.960 20 225 0.200 0.198 20 225 Influence of screw speed 0.100 0.102 10 150 0.200 0.192 10 150 0.100 0.101 10 300 0.200 0.191 10 300 (*.)The experimental weight average molecular weight of the samples collected during these experiments are further used to determine the variation of the peroxide efficiency value with the concentration of peroxide. SEC Results: Average Molecular Weights of Polypropylenes. Sample [I.sub.0](wt%) [M.sub.n](g/mol) [M.sub.w](g/mol) [M.sub.z](g/mol) 1 Virgin PP 47,050 301,600 1,125,000 2 Extruded PP 46,070 271,200 779,000 3 0.01 45,870 209,300 482,300 4 0.02 40,590 190,500 425,800 5 0.06 36,210 135,600 278,100 6 0.10 34,730 114,800 224,800 7 0.15 33,570 104,900 202,600 8 0.25 29,140 87,480 175,500 9 0.35 27,170 78,760 148,300 10 0.50 25,220 62,970 112,600 Sample [M.sub.w]/[M.sub.n] 1 6.41 2 5.89 3 4.56 4 4.69 5 3.74 6 3.31 7 3.12 8 3.00 9 2.90 10 2.50 Kinetic Constants, From Krell et al. [14]. Kinetic Constant Pre-exponential Factor A k = A exp (-[delta]E/RT) Chain transfer [k.sub.2] 5.49 x [10.sup.3] Thermal degradation [k.sub.3] 1.57 x [10.sup.-1] Termination [k.sub.4] 3.5 x [10.sup.8] Peroxide decomposition [k.sub.d] [*] 1.98 x [10.sup.12] Kinetic Constant Activation Energy k = A exp (-[delta]E/RT) [delta]E/R [K] Chain transfer [k.sub.2] 5998 Thermal degradation [k.sub.3] 9383 Termination [k.sub.4] 10 Peroxide decomposition [k.sub.d] [*] 14,947 (*.)Data from Tzoganakis et al. [11].

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Author: | BERZIN, F.; VERGNES, B.; DUFOSSE, P.; DELAMARE, L. |
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Publication: | Polymer Engineering and Science |

Article Type: | Brief Article |

Geographic Code: | 1USA |

Date: | Feb 1, 2000 |

Words: | 6913 |

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