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Computational prediction of PVC degradation during injection molding in a rectangular channel.

INTRODUCTION

One of the most common problems encountered when polyvinyl chloride (PVC) is not molded properly is the appearance of burn (degradation) marks. These burn marks are easily visible in materials that are translucent or of light coloration, usually varying in degree from yellow to brown, and in severe cases to completely black streaks. The kinetics of degradation of PVC has been extensively studied (1-10). However, the general consensus is that the rate is determined by the initial dehydrochlorination reaction. This is followed by a very fast unzipping reaction, which leads to conjugated double bonds in the backbone. These conjugated double bonds, or polyenes, vary from 5 to 25 units long and absorb visible light, causing the coloration changes seen during degradation. If the levels of degradation are high, a secondary process is observed during degradation. The polyene sequences react with one another, leading to a crosslinked network.

There have been several studies of the kinetics of PVC thermal degradation (2-5). Georgiev and Christov (2) presented a kinetic model of the thermal dehydrochlorination of PVC based on Lottka's scheme. This model includes three kinetic transmission regimes to a stationary state. Furthermore, the model was expanded to account for the influence of the HCl diffusion on the PVC dehydrochlorination kinetics. It has been confirmed that the released HCl has an autocatalytic action on the rate determining step, i.e., the initiation of the formation of these conjugated systems. Also, the model assumes that the propagation of the system length precedes the zip-reaction mechanism and does not depend on the hydrogen chloride concentration.

Thus, the proposed model considers two regimes of thermal dehydrochlorination, kinetic and diffusion, which can be divided depending on the conditions for diffusion of the released HCl. However, it is accepted that thermal dehydrochlorination without diffusion controls the process. In accordance with Lottka's model, thermal dehydrochlorination of PVC can be represented by the following scheme:

PVC [right arrow] A + HCl

A + HCl [right arrow] B + n HCl

S + HCl [right arrow] C

where A are defects in PVC macromolecules, which are formed with a constant rate v = [k.sub.0][PVC] in the absence of oxygen and light and at constant temperature. B and C are nonactive products that are formed as a result of the zip-reaction mechanism termination and the interaction of the released hydrogen chloride with the stabilizer, S, respectively.

The autocatalytic effect of HCl in the dehydrochlorination of PVC has also been studied in detail. Simon and Valko (3) studied the kinetics of PVC dehydrochlorination in an atmosphere of HCl. Furthermore, they also examined the effects of high conversion. Their results indicate that the activation energy and the pre-exponential factor of random elimination of HCl from PVC are identical with those for the inert atmosphere. However, it appears that HCl catalyzes the initiation step of the zip-reaction. Also, HCl causes the elongation of polyene sequences, mainly for temperatures below 220[degrees]C. Moreover, the results suggest that elongation occurs even for low concentrations of HCl in the polymer.

An interesting result from the study by Simon and Valko (3) was that the sum of the squares of deviations between experimental and calculated values of conversion using immediate growth was lower by about an order of magnitude than for the mechanism of gradual zip growth. These results further support that the dehydrochlorination of PVC in that atmosphere of HCl occurs via the mechanism of immediate zip growth. In addition, it appears that the autocatalytic effect in the dehydrochlorination is due to the interaction of HCl, polyene sequences and non-dehydrochlorinated parts of the polymer chain.

However, even though the kinetics of degradation has been extensively studied, there has been very little work in the area of computational models for the degradation of PVC during processing. Morrette and Gogos (11) presented one of the earliest studies on the viscous dissipation in capillary flow of rigid PVC and PVC degradation in 1968. Their computational work was based on the steady state, non-isothermal behavior of rigid PVC, flowing in capillaries of circular cross section. They considered the flow of polymer melt described by a Power-Law constitutive equation in their solution. Moreover, the effects of thermal degradation of PVC on its viscosity were also introduced in the equation of momentum and energy.

Typically, polymer processing involves flows through geometrically complex channels. Therefore, obtaining an analytical solution for real processing problems is very difficult, if not impossible. However, it is common to find that parts of the flow channels in various processing machines are similar and geometrically very simple. For instance, cylindrical channels are used as runner systems in injection molding or as extruder dies. Thus, one can examine the problem of viscous energy dissipation in the steady state pressure flow of PVC by focusing on the simpler sections, as considered by Morrette and Gogos (11), who examined the flow of melts through circular tubes. This simplification of the problem allows them to use the information on the temperature and velocity fields, which is coupled with the kinetics of thermal degradation of PVC, to yield an estimate of amount of degradation during tube flow.

Morrette and Gogos (11) indicated that there were several authors who have considered the problem of viscous energy dissipation in laminar flow of high viscosity fluids. Some of the highlights of their work include the work of Gee and Lyon, who treated the nonisothermal flow of non-Newtonian fluids. Gerrad et al. provided an improved computational technique, which included radial velocity components. However, their technique dealt only with Newtonian fluids. Finally, the work of Toor is significant because it considers the problem of polymer cooling due to the expansion of the fluids as it flows through the capillary.

A significant contribution from the work of Morrette and Gogos (11) was the introduction of polymer degradation kinetics. In their model. Morrette and Gogos assumed that degradation at any given high temperature started after an induction period. It is thought that during the induction time. HCl is probably consuming stabilizers added to the PVC. After the induction time the reaction proceeds at essentially a constant rate. The degradation was introduced in the calculations by changes in the consistency index (the viscosity measured at a shear rate equal to unity)

[eta][degrees](T,P) = A exp [[[[DELTA]E]/[RT]] + [beta]P] (1)

The system of differential equations was then solved using the Crank-Nicholson finite difference method and the pressure drop was calculated by using a fourth order Runge-Kutta technique. As indicated, the change in viscosity was also included whenever the residence-to-induction time was larger than unity. The change in viscosity was calculated from

d(ln[eta][degrees])/2.3dt = k = C[e.sup.-E/RT] (2)

Thus, most of the calculated degradation was located near the walls, where the residence time is large. However, the model provides only a general trend. The model did not have enough detail to determine the exact location of the degraded layers. The work presented here is directed at evaluating the location of the degraded layer by expanding on the finite difference model and introducing a computational tracer method to obtain the temperature and degradation history of different fluid elements during processing.

Also, their computational technique was compared to experiments in order to verify the assumptions. It was observed that there was some deviation between experimental and computed values. A possible explanation for the difference is that the activation energy is dependent on the flow rate and that crystallization of PVC may induce orientation.

Their solutions for the velocity, temperature, and pressure profiles were obtained for both adiabatic flow and flow through a tube of constant wall temperature. Either case indicates that considerable heating of melt, due to viscous dissipation, can be achieved at moderate flow rates. Furthermore, thermal degradation also occurs in the capillary under certain conditions of temperature history and residence time of the fluid. However, the results presented by Morrette and Gogos also show a non-physical behavior at high flow rates. The velocity profiles at the higher flow rates show a localized maximum near the wall. Their explanation for this observation was related to a mass conservation, which leads to an axial acceleration near the wall. However, it is more likely that this may be related to the limited computing power available at the time.

Another interesting result from their work is related to the Bagley entrance correction. One can examine whether the correction is valid by plotting the pressure drop versus L/D. Thus, examination of their computational results shows that the Bagley correction is valid only for the non-isothermal case, at least for the pressure range studied.

Hence, Morrette and Gogos (11) developed a computational technique that was able to capture the viscous heating effect and the degradation of PVC flowing in a capillary. In 1973, Berger and Gogos (12) expanded the model by proposing a numerical simulation for the cavity filling process with PVC injection molding. Their work was directed at analyzing the flow in a disk. Their results showed that most of the total pressure drop is dissipated in the entrance of the tube and most of the viscous heating is generated there. Therefore, an important conclusion from their work is that the design of the gate and runner system is perhaps the most important factor.

Berger and Gogos (12) also examined the work of Kenig and Kamal prior to developing their model. Kenig and Kamal performed a numerical calculation of the unsteady heat conduction after the cavity has been filled. They observed that the numerical temperature stability depended on the choices of the constant values and the space and time increment employed in the numerical solution. Berger and Gogos also observed that the choice of [DELTA]t affected the thermal stability for their simulations.

Berger and Gogos studied the flow of a polymer melt at a temperature [T.sub.1] into a disk cavity with walls at [T.sub.0]. They assumed that the flow was pressure controlled. They also assumed that both u/r and [partial derivative]u/[partial derivative]r [much less than] [partial derivative]u/[partial derivative]x. This is a good assumption for cases where the melt front has advanced to large radial positions, but relatively poor for very short times in the cavity filling process. Moreover, they used the lubrication approximation technique to solve for the velocity profile. Therefore, the velocity is also a function of the radius. However, the assumption [partial derivative]u/[partial derivative]r [much less than] [partial derivative]u/[partial derivative]x is used only to eliminate a stress component in the momentum equation. The continuity and momentum equations are

[[rho]/r] [[partial derivative]/[partial derivative]r] (ru) = 0 (3)

[[partial derivative]P]/[[partial derivative]r] = -[[[partial derivative][[tau].sub.rx]]/[[partial derivative]x]] (4)

where the stress, [[tau].sub.rx], is given by the Power-Law constitutive equation

[[tau].sub.rx] = - 2m|[partial derivative]u/[partial derivative]x|[.sup.n - 1][[[partial derivative]u]/[[partial derivative]x]] (5)

The energy equation is given by

[rho][C.sub.v]([[[partial derivative]T]/[[partial derivative]t]] + u[[[partial derivative]T]/[[partial derivative]r]]) = K[[[[partial derivative].sup.2]T]/[[partial derivative][x.sup.2]]] - [[tau].sub.[theta][theta]][u/r] - [[tau].sub.rx][[[partial derivative]u]/[[partial derivative]x]] (6)

where the hoop stress, [[tau].sub.[theta][theta]], is also included in the equation. Therefore, the Power-Law constitutive model must also be modified to include the hoop stress. The viscosity is then given by

[eta] = 2m(T)|2(u/r)[.sup.2] + ([[partial derivative]u]/[[partial derivative]r])[.sup.2]|[.sup.(n - 1)/2] (7)

This problem was then solved by using a finite difference method similar to the one presented above. One of the major differences, however, is the construction of the finite difference grid. Since the average velocity decreases as the material moves away from the center of the plate, the distance between the nodes also decreases. The grid is, therefore, generated from a mass balance

[r.sub.ik.sup.2] = [r.sub.ik - 1.sup.2] + Q[DELTA]t/[pi]h (8)

The most interesting result from their simulation was the fact that the fill time did not change if the flow was taken to be isothermal or non-isothermal. This implies that there is no need to solve for the heat transfer if the only parameter of interest is the fill time. The simulations also captured the viscous heating effect and the formation of a "frozen" layer.

Therefore, the work of Gogos et al. (11, 12) presents a good starting point from which to develop the numerical technique to evaluate the thermal degradation during injection molding. Furthermore, it has been shown in studies of viscous heating that it is very difficult to make direct comparisons between numerical and experimental results. There are no techniques available to measure the actual melt temperature profile in the cavity without affecting the flow. However, since PVC changes color as it degrades, most researchers examine the properties of the final parts and make interjections about the conditions during processing. This approach is also used to compare the experimental and computational results presented here.

EXPERIMENTAL

The materials used in this study were provided by the PolyOne Corp. Both M3800 and M4200 were evaluated in a series of injection molding experiments. A 700 ton HPM injection molding machine (700 MK II--WP - 80) with a maximum injection pressure of 25,000 psi was used to process the parts. A spiral mold was used to approximate the flow in a rectangular channel. A summary of the experiments and the results can be found elsewhere (13), as well as the physical properties, the rheological behavior, and degradation data. In this work, only the degradation data will be discussed along with an explanation of how it was used in the proposed model.

Degradation Data

The degradation experiments were conducted in the RMS 800 rheometer using the 25 mm parallel plates with the frequency held constant at 10 rad/sec and the strain set to 1%. The oven was preheated to the specified temperature and the sample was loaded. Time was measured from the time the oven was closed. The viscosity measurements started when the temperature of the sample reached equilibrium, approximately 5 minutes, determined by the normal force measurement. The mechanical properties were then measured over time. The limit on the highest possible temperature that can be examined is related to how fast a sample can be loaded and the first equilibrium data point obtained.

The results clearly indicate that the material undergoes mechanical changes as it degrades. Furthermore, these changes seem to occur after a delay, or the induction time. The transitions are easily detected by looking at the first and second derivatives of the material properties with respect to time, in particular in the phase angle (tan [delta]) curves. The phase angle is given by

tan [delta] = [G"/G'] = [[Viscous Modulus]/[Elastic Modulus]] (9)

in other words, the phase angle gives a ratio of the liquid-like state to the solid-like state of the material. A high tan [delta] indicates that the material is behaving more like a liquid, while a low tan [delta] indicates that the material is behaving more like a solid.

In order to quantify the induction or degradation time, it is important to determine the time at which the materials have undergone the same degree of degradation. The S-shape of the tan [delta] curves provides an easily determined parameter, the inflection point. Hence, for this study, the induction or degradation times were defined at the inflection point of the curves for the tan [delta]. It turned out that the inflection point was determined to be very near the point where tan [delta] is close to 1.0. At tan [delta] close to 1.0, the material has equal contributions from the fluid-like state and the solid-like state. Therefore, it was assumed that the inflection point represents a similar state for the materials at different temperatures, in other words, the materials had undergone the same amount of degradation by that time.

The degradation times at different temperatures can be used to calculate the activation energy for degradation. Applying some of the known information about the mechanism of the reaction, one can set the rate limiting step as the initial dehydrochlorination step. This is followed by a very fast zip reaction that produces the polyene. Also, the stabilizer combines with HCl very quickly. Thus, the autocatalytic effect can be neglected during the initial stages, where only small amounts of HCl have been generated. Furthermore, as a first approximation, the dehydrochlorination step can be considered a zero order reaction because, again, the concentration of Cl in the polymer does not change significantly. In other words, the production of degradation sites is of zero order. Also, since the zip reaction is much faster, one can also say the rate of polyene production is of zero order.

[[d[Cl[[up arrow].sub.initial]]]/[dt]] = [[d[P]]/[dt]] = k (10)

where [Cl[[up arrow].sub.initial]] represents the chlorine sites where the zip reaction starts from, [P] represents the polyene sites, and k is the reaction rate constant.

Assuming that the measured degradation times represent similar states of the material in terms of the polyene concentration, HCl evolved, and stabilizer used, one can then set the concentration of polyenes equal to [[P].sub.x], where x is the concentration of polyenes when crosslinking begins. After integration of the zero order kinetic equation, it follows that the degradation time is inversely proportional to the rate constant.

[t.sub.degradation] = [[P].sub.x]/k (11)

If one assumes that the rate constant follows a typical Arrhenius relationship, one can express it in terms of a pre-exponential factor, A, and an activation energy, Ea.

k = A * [e.sup.- [Ea/RT]] (12)

where R is the ideal gas constant and T is the temperature. Substituting into the equation and taking the log of both sides, one finds that

log ([t.sub.d]) = log([[P].sub.x]/A) + [Ea/RT] (13)

The data is used to calculate the activation energy for the degradation reaction. The activation energies for the two PVC systems are 27.0 kcal/mol for M3800 and 32.4 kcal/mol for M4200. The values obtained from the mechanical studies correlate very well with values reported by others, which vary from 17 to 33 kcal/mol (8, 14-16).

In addition to mechanical data, visualization of the degraded samples was conducted using grayscale images and imaging software. This data was collected in order to correlate mechanical changes to optical changes. A description of the setup used is found elsewhere (13).

Figure 1 shows the grayscale increase over time for M3800 and M4200. One of the problems encountered using this technique was the fact that M4200 is a transparent material with a blue coloration. Degradation of PVC is indicated by coloration changes from yellow to red to brown to black. Therefore, initial degradation of M4200 is indicated by a transition from blue to green, which cannot be distinguished after the image is converted to grayscale. Only high levels of degradation are distinguished, as shown in Fig. 1. Nonetheless, this technique provides useful information that can be used to analyze the parts produced in the injection molding trials.

[FIGURE 1 OMITTED]

The time for degradation determined from mechanical changes is indicated in Fig. 1. One can see that, indeed, the degradation times obtained from mechanical changes correlate with high degrees of degradation, indicated by the large grayscale change. Furthermore, since a pseudo-zero order reaction is being used to represent the kinetics of degradation, the grayscale at the time of degradation was taken to be equal to a concentration of 1.0. This simplifies the calculations since the rate of reaction becomes.

[kappa] = 1/[t.sub.deg] (14)

It is important to note that this simplification is valid only because of the pseudo-zero order reaction kinetics. This is shown from Eq 13, where [[P].sub.x] represents the degradation concentration where crosslinking begins.

Degradation During Processing

One of the observations from the injection molding is that a significant amount of degradation takes place in a very short period of time. For the test parts molded in this study, the typical injection time was on the order of one second or less and the cooling time was on the order of 30 seconds. Overall, the total cycle, from beginning of injection to the final part, was on the order of 40 seconds or less. During this time, a significant degree of degradation was observed for some of the processing conditions being tested. This is significant in the fact that, even applying some of the mechanisms presented in literature, there are no models that can predict such a fast degradation reaction. One of the possible reasons for this phenomenon may be an enhanced degradation kinetics due to shear during flow.

Designing an experiment to determine the effect of shear rate on the rate of reaction is non-trivial, considering that shear rate and temperature are highly coupled in polymeric systems. One well-known effect of shear during polymer processing is viscous heating. Depending on the processing conditions being tested, the temperature increase observed in some locations of the mold may be as high as 100[degrees]C. Consequently, some of the faster degradation rates may be attributed to the temperature increase during processing. However, performing some preliminary calculations indicates that just the temperature increase does not justify the high levels of degradation observed in the injection molded parts. Bacaloglu et al. (17) found that the shear stress degradation is a radical reaction, which is promoted by mechanically formed radicals.

Therefore, it is proposed in this work that an enhanced degradation mechanism may be responsible for the high levels of degradation observed during processing. In order to determine the effects of flow on the degradation kinetics, a simple modification of the degradation kinetics is proposed. As a first approximation, it is assumed that the reaction order can still be taken to be a pseudo-zero order reaction; however, it is assumed that the material becomes more sensitive to temperature during flow. This is represented on Fig. 2, where the degradation time is plotted against the temperature. The proposed mechanism is assumed to be more sensitive to temperature and, therefore, has a steeper slope or higher activation energy. Furthermore, changing the pre-exponential factor in the kinetic equation can change the relative location of the line. However, instead of discussing changes of the pre-exponential factor, it is more tangible to discuss the temperature where the enhanced degradation mechanism intercepts the line obtained from quiescent degradation studies. This provides some indication as to what temperature the degradation mechanism starts to deviate from quiescent studies and becomes more temperature sensitive.

In order to determine the activation energy and the intercept temperature, the enhanced degradation mechanism was used in conjunction with the computational model. The computational model was used to predict the degree of degradation obtained by varying both the activation energy and the intercept temperature. Figure 3 shows the calculated degree of degradation for just one of the sets of processing conditions used. The data was calculated at the location where the actual degree of degradation observed in the final part was 1.0. The visualization technique described in the previous section was used to determine the actual degree. The plane for the actual degree of degradation of 1.0 intersects the calculated degradation surface, giving a locus of points for the activation energies and intercept temperatures that can predict the correct degree of degradation. Figure 4 shows the curve for the possible combinations of activation energies and intercept temperatures.

[FIGURE 2 OMITTED]

Similarly, the calculations were repeated for another arbitrary set of processing conditions. The loci of points for activation energies and intercept temperatures for this set are also shown in Fig. 4. The activation energy and intercept temperature for the enhanced degradation mechanism was taken to be the point at which the two curves intersect because the activation energy and the intercept temperature satisfy these two processing conditions. Therefore, for all the calculations performed in this study, the activation energy was taken to be 65 kcal/mol and the intercept temperature was taken to be 200[degrees]C during polymer flow. This is important since it indicates that the effects of flow start to become significant at temperatures similar to the processing temperatures.

COMPUTATIONAL MODEL

One of the drawbacks of commercial computational packages is that detail in the thickness direction is sacrificed so that complex geometries can be examined (18-21). It would be highly computational intensive to try to increase the level of detail in the gap direction and examine complex geometries. Therefore, the work presented here offers a more detailed analysis in the gap direction by reducing the complexity of the geometry. The systems examined include typical sections found in complex geometries: flow in a tube, flow in a rectangular channel, and radial flow. The governing equations for these systems reduce to 2-D problems and are readily available in literature (22, 23).

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

By assuming laminar flow, both the flow and energy equation become 1-dimensional by neglecting any transverse flow. This assumption breaks down at the fountain flow region. However, this region can be treated separately and it does not affect the solution for the main flow problem (20).

In addition, it is important to note that the sprue also contributes to viscous heating and degradation of the material. Therefore, the sprue must be included in any numerical scheme designed to predict degradation during injection molding. This is accomplished by dividing the flow geometry into two separate sections: the sprue and the flow channel.

In this study, the sprue was assumed to be cylindrical. The continuity, momentum and heat transfer equations for the sprue, assuming incompressible flow, are:

[[[partial derivative][v.sub.z]]/[[partial derivative]Z]] = 0 (15)

[1/r][mu][[[partial derivative][v.sub.z]]/[[partial derivative]r]] + [[partial derivative]/[[partial derivative]r]]([mu][[[partial derivative][v.sub.z]]/[[partial derivative]r]]) - [[[partial derivative]P]/[[partial derivative]Z]] = 0 (16)

[rho][C.sub.v][[[partial derivative]T]/[[partial derivative]t]] = k([1/r][[[partial derivative]T]/[[partial derivative]r]] + [[[[partial derivative].sup.2]T]/[[partial derivative][r.sup.2]]]) + [mu]([[[partial derivative][v.sub.z]]/[[partial derivative]r]])[.sup.2] (17)

where [v.sub.z] is the axial velocity, [rho] is the density, [C.sub.v] is the specific heat, and k is the conductivity. It is important to point out that the convective heat transfer parameter [v.sub.z] dT/dz has been left out of Eq 17. However, convective heat transfer is not being neglected in the computational model presented here. Convective heat transfer is not solved explicitly, instead it is taken into account in the numerical scheme and will be discussed in the next section. The viscosity, [mu], is calculated from the WLF-Cross model

[mu]([dot.[gamma]], T) = [[[eta].sub.0](T)]/[1 + ([[[eta].sub.0](T)[dot.[gamma]]]/[[tau]*]])[.sup.1-n]] (18)

where,

[[eta].sub.0] (T) = [D.sub.1] exp([-[A.sub.1] (T - [T.sub.r])]/[[A.sub.2] + T - [T.sub.r]]) (19)

where [dot.[gamma]] is the shear rate, [tau]* is the characteristic stress, n is the shear thinning index, [[eta].sub.0] is the viscosity at a reference temperature, and [A.sub.1], [A.sub.2], [D.sub.1], and [T.sub.r] are the WLF parameters (Table 1).

A rectangular flow channel represented the spiral mold. The equations for a rectangular channel are readily available if the width is taken to be much larger than the thickness. If this is the case, then the problem reduces to that of flow between parallel plates. The equations of continuity, momentum, and heat transfer for an incompressible fluid are:

[[partial derivative][v.sub.z]]/[[partial derivative]Z] = 0 (20)

[[partial derivative]/[[partial derivative]y]]([mu][[[partial derivative][v.sub.z]]/[[partial derivative]y]]) - [[[partial derivative]P]/[[partial derivative]Z]] = 0 (21)

[rho][C.sub.v][[[partial derivative]T]/[[partial derivative]t]] = k[[[[partial derivative].sup.2]T]/[[partial derivative][y.sup.2]]] + [mu]([[[partial derivative][v.sub.z]]/[[partial derivative]y]])[.sup.2] (22)

The convective heat transfer component has again been left out of the energy equation. As indicated previously, the convective heat transfer component is taken into account in the numerical scheme developed.

Prior to developing the finite difference scheme to solve the unsteady flow problem aforementioned, the problem was changed to dimensionless form. The dimensional analysis of the problem leads to some important dimensionless groups that characterize the injection molding process. The non-dimensional equations for the flow in a tube are

[1/[epsilon]]f([dot.[gamma]],T)[[[partial derivative][omega]]/[[partial derivative][epsilon]]] + [[partial derivative]/[[partial derivative][epsilon]]](f([dot.[gamma]],T)[[[partial derivative][omega]]/[[partial derivative][epsilon]]]) - ([R/L])[Re.sub.R][[[partial derivative][pi]]/[[partial derivative][xi]]] = 0 (23)

[[[partial derivative][THETA]]/[[partial derivative][tau]]] = ([1/[epsilon]] [[[partial derivative][THETA]]/[[partial derivative][epsilon]]] + [[[[partial derivative].sup.2][THETA]]/[[partial derivative][[epsilon].sup.2]]]) + Br * f([dot.[gamma]],T)([[partial derivative][omega]]/[[partial derivative][epsilon]])[.sup.2] (24)

where the dimensionless variables are:

[epsilon] = [r/R] [xi] = [z/L] [THETA] = T/[T.sub.cool] [omega] = v/[v.sub.0]

[tau] = [[t * k]]/[[rho] * [C.sub.v] * [R.sup.2]] [pi] = P/[[v.sub.0.sup.2] * [rho]] [Re.sub.R] = [[rho][v.sub.0]R]/[eta] (25)

The non-dimensional equations for the rectangular channel are

[[partial derivative]/[[partial derivative][epsilon]]](f([dot.[gamma]],T)[[[partial derivative][omega]]/[[partial derivative][epsilon]]]) - ([H/L])[Re.sub.H][[[partial derivative][pi]]/[[partial derivative][xi]]] = 0 (26)

[[[partial derivative][THETA]]/[[partial derivative][tau]]] = [[[[partial derivative].sup.2][THETA]]/[[partial derivative][[epsilon].sup.2]]] + Br * f([dot.[gamma]],T)([[[partial derivative][omega]]/[[partial derivative][epsilon]]])[.sup.2] (27)

where the dimensionless variables are:

[epsilon] = y/H [xi] = z/L [THETA] = T/[T.sub.cool] [omega] = v/[v.sub.0] (28)

[tau] = [t * k]/[[rho] * [C.sub.v] * [H.sup.2]] [pi] = P/[[v.sub.0.sup.2] * [rho]]

where L is the length of the mold and H is the mold thickness.

In both cases the function f([dot.[gamma]], T) represents the Cross-WLF constitutive equation used to evaluate the viscosity at a given shear rate and temperature. Furthermore, dimensional analysis leads to the following dimensionless groups: [Re.sub.R], [Re.sub.H], the Reynolds number: and Br, the Brinkman number.

[Re.sub.R] = [[[rho][v.sub.0]R]/[eta]] [Re.sub.H] = [[[rho][v.sub.0]H]/[eta]] (29)

Br = [[eta][v.sub.0.sup.2]]/[[T.sub.cool]k] (30)

The Brinkman number provides a ratio of the energy generated by viscous heating to the energy lost by conduction. Therefore, when the Brinkman number is high, viscous heating must be taken into account in the energy equation. Typically, the Brinkman number is high because of the high viscosity of polymeric materials and the high speeds used during injection molding. Thus, the viscous heating component has been taken into account in the energy equation.

In addition to the equations describing the system, the boundary conditions must also be specified before the problem can be solved. The flows in the sprue and the mold were taken to be symmetric, therefore, for the sprue the boundary conditions are

v(R) = 0 [[[partial derivative]v]/[[partial derivative]r]]|[.sub.r=0] = 0 [[[partial derivative]T]/[[partial derivative]r]]|[.sub.r=0] = 0 (31)

- k[[[partial derivative]T]/[[partial derivative]r]]|[.sub.r=R] = h*(T(R) - [T.sub.cool]) (32)

and for the mold, the boundary conditions are

v(H) = 0 [[[partial derivative]v]/[[partial derivative]y]]|[.sub.y=0] = 0 [[[partial derivative]T]/[[partial derivative]y]]|[.sub.y=0] = 0 (33)

- k[[[partial derivative]T]/[[partial derivative]y]]|[.sub.y=H] = h * (T (H) - [T.sub.cool]) (34)

where h represents an overall heat transfer coefficient for the mold. The heat transfer coefficient was determined by running the simulation with different values and comparing the final surface temperature with the experimental surface temperature after the part was ejected. It was determined that the best fit value was 400 J/[m.sup.2] sec K.

FINITE DIFFERENCE PROCEDURE

The system of equations was solved using a typical forward finite difference scheme, where the flow field is divided into a finite number of nodes. The distance between nodes is given by how much the flow front moves in a given time interval [DELTA]t,

[DELTA]t = [t.sub.injection]/N (35)

where N is the number of nodes along the flow length. Since the velocity in the channel is constant, the spacing between nodes is also constant,

[DELTA]x = [v.sub.0] [DELTA]t (36)

where [v.sub.0] is the average velocity in the mold.

However, since the volume of the sprue is typically much smaller than the volume of the part itself, two different time steps must be used for the sprue and the mold in order to use the constant volumetric flow rate. This allows one to include more axial nodes in the sprue section. The time steps for the mold, of course, are multiples of the time steps in the sprue. For instance, the time step in the mold may be twice as large as in the sprue. Thus, during calculations, the computations in the mold are performed after two iterations in the sprue.

The injection molding process itself was computed via a finite difference approach similar to that proposed by Gogos et al. (11, 12). However, because the injection process is inherently unsteady, a "stop-and-go" approach was used to modify the simulation (24). The "stop-and-go" approach has been used to solve for the complex flow of mixtures through chromatographs. The basic principle of this scheme is that the material is allowed to flow for a small time [DELTA]t at which point the velocity profiles and temperature profiles are calculated. The velocity profiles are calculated assuming a pseudo steady-state condition, while the temperature profiles are calculated by allowing heat transfer to take place for [DELTA]t amount of time. The basic finite difference equations, derived from the dimensionless equations, for the velocities and the temperatures in the mold are

[[omega].sub.i] = [1/[[f.sub.i+1] + [f.sub.i]]]([f.sub.i+1][[omega].sub.i+1]+[f.sub.i][[omega].sub.i-1]) - [[[DELTA][[epsilon].sup.2]]/[[f.sub.i+1] + [f.sub.i]]][H/L] [Re.sub.H] [[[partial derivative][pi]]/[[partial derivative][xi]]] (37)

[[THETA].sub.i] = [[[THETA].sub.i.old] + [DELTA][tau]{[[[[THETA].sub.i+1] + [[THETA].sub.i-1]]/[[DELTA][[epsilon].sup.2]]] + Br*[f.sub.i]([[[omega].sub.i+1] - [[omega].sub.i-1]]/[2[DELTA][epsilon]])[.sup.2]}]/[1 + [[2[DELTA][tau]]/[[DELTA][[epsilon].sup.2]]] (38)

These equations are slightly modified at the boundaries. Similar equations for the sprue can be derived. In these equations, the index "i" represents the nodes in the gap direction.

The melt front advancement is calculated in the following manner: 1) one knows that the melt front (fountain flow region) moves by one node in the amount [DELTA]t (that is how [DELTA]t is defined); 2) a "fresh" set of material is positioned at the inlet; 3) the material moves from the inlet to next set of nodes; 4) the velocity and temperature profiles are calculated; 5) the material moves to the next set of nodes; 6) the process continues until the material has reached the melt front; 7) the melt front is moved by one node again; and 8) a "fresh" set of material is positioned at the inlet and the calculations repeated.

Therefore, by calculating the changes from the inlet to the melt front every time the melt front advances, one is in essence calculating the convective heat transfer contribution in the energy equation. If this were not included, material could "freeze" during flow. This is because, without including the heat input from new material coming in, there would be a significant loss of energy as the material flows into the cavity. This would lead to a significant temperature drop near the melt front region, eventually leading to "freezing."

In addition to implicitly including the convective heat transfer component in the numerical scheme, a mass balance is also implicitly calculated in the numerical solution. A pressure drop term that must be calculated in order to solve for the velocity profile. The pressure term is calculated by setting a mass balance,

[H.[integral] (0)]v(y)dy = [v.sub.0] * H (39)

This integral is calculated using a trapezoid rule, giving a solution for the pressure term,

[DELTA][[epsilon].sup.2] * [H/L] * [Re.sub.H] * [[[partial derivative][pi]]/[[partial derivative][xi]]] = [[[[omega].sub.1]/2] - [1/[DELTA][epsilon]] + [summation][1/[f.sub.i+1] + [f.sub.i]]([f.sub.i+1][[omega].sub.i+1]+[f.sub.i][[omega].sub.i-1])]/[[1/2] + [summation][1/[f.sub.i+1] + [f.sub.i]]] (40)

This numerical scheme was then used to develop a Visual Basic computer program that could be executed from within Microsoft Excel[TM]. Integration with Excel allowed the generated data to be directly imported into spreadsheets for easy accessibility.

Tracer Method

In addition to the finite difference solution, a computational tracer method was developed to determine the temperature and degradation history of different fluid elements. The method works by inserting a set of elements at the inlet at different times during the filling process and calculating their position and velocity at different times. The temperature of these elements is determined directly from their position in the mold at any given time by interpolation between nodes. In essence one can image the computational method having two different domains: a fixed domain and a flowing domain. The fixed domain, the finite difference method, is used to calculate the velocity and temperature fields at any given time during processing. The flowing domain, or tracers, is allowed to flow relative to the velocity field and the temperatures are calculated relative to the temperature field. Thus, the tracers allow the temperature history of the material to be determined, which can then be used to calculate the degradation concentration. In the numerical scheme presented here, a new set of tracers was inserted at 2% intervals during filling. In other words, a new set of tracer elements was inserted every time the melt front advanced by 2% of the nodes in the axial direction. The tracer elements were located at every node in the thickness direction. The flow sheet for the program is shown in Fig. 5.

Since the flow velocity near the center is faster than the average velocity, some of the tracer elements will reach and pass the melt front. In order to avoid this, an algorithm simulating the fountain flow was developed. If a tracer element moved past the flow front, the element was moved closer to the wall and its axial position was set to be just behind the flow front. Thus, material near the center moved towards the walls as flow progressed, similar to what happens in the fountain flow. The algorithm developed to calculate how the material moves towards the wall was based on circulation zone at the melt front. It was assumed that the tracers rotate around the point where the velocity is equal to the average velocity, in order to maintain a mass balance. If the node where [omega] = 1.0 is M, the node at the wall is N, and the node at the center line is 0, then the new location X' of a tracer located in node X is given by.

[FIGURE 5 OMITTED]

X' = M + [[(M - X)]/[M - O]] (N - M) (41)

Degradation Model

Coupled with the tracer method, the computational model also investigated polymer degradation. Several studies regarding the kinetics of degradation of PVC have identified several factors that can affect the degradation kinetics of PVC. However, degradation is strongly dependent on the temperature. In addition, degradation depends on HCl concentration, a catalyst to the reaction. However, the effects of HCl are minimized by the use of stabilizers. Therefore, the degradation kinetics were taken to be of zero order in this study

[Dc]/[dt] = k (42)

where C represents the amount of chlorine evolved (or the formation of polyene sequences) and k is the reaction constant.

Assuming that the rate constant follows a typical Arrhenius relationship, the amount of degradation that occurs in an amount [DELTA]t of time for a zero order kinetic equation is given by

C([DELTA]t) = [k.sub.ref] * exp[??][E.sub.A](1/[T.sub.ref] - 1/T)[??] * [DELTA]t (43)

where both the activation energy, [E.sub.A], and the reference rate constant, [[kappa].sub.ref], can be adjusted for the enhanced degradation mechanism. This technique was used to find the enhanced degradation parameters, where several combinations for [E.sub.A] and [k.sub.ref] ([T.sub.intercept]) were used until the calculated degradation values were close to the experimental degradation levels for two typical processing conditions. The parameters that best fit the two experimental results were [E.sub.A] = 65 kcal/mol and [T.sub.intercept] = 200[degrees]C. These values were then used for all of the processing conditions examined in this study.

Finally, because degradation is a function of temperature, the model also includes the cooling time. Heat transfer was assumed to take place only in the thickness direction. The equations used are similar to those used during filling, however no viscous heating occurs because the system is static. In addition, since there is no flow, the standard kinetics for degradation, as measured from rheological studies, was used to calculate the amount of degradation during cooling. The flow sheet is shown in Fig. 5.

[FIGURE 6 OMITTED]

RESULTS

The computer program developed directly calculates the velocity, temperature, and shear rate profiles of the polymer melt during injection. This information was stored in spreadsheets at different times during injection. The results for the profiles at the end of injection are presented here. The term "end of injection" refers to the point in time right before flow ceases. Also, the results shown here represent the results for the numerical solution where 60 nodes across the gap were used.

An important result from the temperature profiles is that one can generate a correlation between the injection speed and the maximum temperature in the mold at the end of fill. All of the injection molding conditions were simulated in order to determine the maximum temperature during filling. Figure 6 shows the viscous heating results for M3800 as a function of injection speed for various shot sizes. One can see that the effect of shot size on the maximum temperature during fill is marginal. Similarly, Fig. 7 shows the maximum temperatures during fill as a function of injection speed for different inlet melt temperatures. It is interesting to note that even though the inlet temperature range is about 15[degrees]C, the range for the maximum temperature in the mold is only 5[degrees]C. This indicates that the differences in melt temperature at the inlet are diminished as a result of viscous heating during processing. Figure 8 shows the results for all of the processing conditions for M3800. One can see that all of the data falls within a very narrow band of injection speeds and maximum temperatures. This result appears to corroborate the idea that injection speed is the most significant parameter since it turns out that the maximum temperature is mostly a function of the injection speed.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Similar results were obtained for M4200. Combining the data for M3800 and M4200 (Fig. 8) shows that M4200, being a higher-viscosity material, tends to be more susceptible to viscous heating than M3800. One can clearly see that M4200 has a higher dependence on the injection speed, indicated by the slope of the line. Therefore, if the degradation observed in the molding trials is only thermal in nature, it is expected that M4200 will show burn marks at lower injection speeds than M3800.

Degradation Comparison

The most significant parameter calculated in the program was the degree of degradation. The tracer method allows the degradation and temperature history of different fluids element to be tracked over time.

[FIGURE 9 OMITTED]

The distribution of tracer elements in the mold at the end of the process is shown in Fig. 9. An interesting result from the tracer distribution is that the frozen layer can be inferred. The frozen layer is clearly delineated by the higher concentration of tracers some distance away from the wall. The velocity distribution right before the end of fill is shown in Fig. 10. Again, the frozen layer is clearly indicated near the wall. As indicated, each tracer has a temperature associated with it and Fig. 10 shows the temperature distribution in the mold at the end of filling and before cooling. One can clearly see that the maximum temperatures are observed some distance away from the wall, near the boundary of the frozen layer. Furthermore, each of these tracers has a degree of degradation associated with it. A degradation distribution in the mold at the end of the process is shown in Fig. 10. Again, one can see that there is a clear correlation between the maximum temperature observed and the degree of degradation. The maximum degradation is observed to be at a height of y/H = 0.8.

[FIGURE 10 OMITTED]

The calculated effect of injection speed on the degree of degradation is shown in Fig. 11. This figure clearly indicates the increase in degradation at the higher flow rates and correlates very well with the experimental profiles (Fig. 12). The computational model not only captures the increase in degradation, but also predicts the location of the degrade layer relative to the surface of the part. The results show that the degradation is not a surface phenomenon, and it is directly related to the high-temperature layer that develops during flow.

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

The calculated degradation data can be directly compared with the experimental data (25). The degradation profile along the length of the part is shown in Fig. 13. The calculated values show good agreement with the experimental results for the processing conditions shown. Also, the calculated values capture the effect of different injection speeds on the degree of degradation.

[FIGURE 14 OMITTED]

The effect of shot size on the degree of degradation is shown on Fig. 14. This figure shows a large discrepancy between the calculated values and the experimental values. As indicated previously, the experimental results show very little variation with respect to the shot size. The calculated values, however, show a significant dependence of the degree of degradation on the shot size. A possible explanation for this result may be related to the unsteady nature of injection molding. One of the assumptions used throughout the calculations is that the injection speed is constant during injection. In reality, however, the machinery has to overcome inertia during the initial stages before reaching the desired injection speed. Therefore, during actual injection molding experiments, there exists a period of time where the injection speed is accelerating to the desired value, typically called the response time for the equipment. The response time for very large machines, such as the one used for the experiments performed in this work, may vary from 0.2 sec to 0.5 sec, which is of similar magnitude as the injection time. Thus, it is very likely that the unsteady nature of the process itself masks the differences associated with different shot sizes. One would expect higher degradation for longer shot sizes owing to the longer residence time at higher temperatures. This effect is seen on the computational results.

The effect of inlet melt temperature on the degree of degradation is shown in Fig. 15. As can be seen, both the experimental and computational results show very little dependence on the melt temperature. This is because viscous heating tends to minimize the melt temperature effect. During injection the maximum temperature in the melt varies slightly for the different inlet melt temperatures.

[FIGURE 15 OMITTED]

Finally, the effect of material viscosity is shown in Fig. 16. One can see a significant discrepancy between the experimental and the calculated values for M3800. On the other hand, M4200 shows higher degradation and considerably better agreement between the experimental and calculated values. This is probably because M4200 shows a higher temperature increase for the same processing conditions. Therefore, since the degradation is taken to be only thermal, it is expected that the calculated values for M4200 should have a higher degree of degradation.

Comparison as a Function of Maximum Temperature During Processing

Figure 17 shows the experimental data for M3800 in terms of the maximum temperature. One can see that the data shows very little variation, with most of the data falling on a single curve. Figure 17 also shows the experimental data for M4200. The data for M4200 also shows very little variation and falls within a single curve. Combining the results for M3800 and M4200, Fig. 17 shows a very interesting result. Although the degradation of M3800 and M4200 show a difference relative to the injection speed, it appears that significant degradation for both materials starts around the same temperature. Thus, this result shows that the degradation of PVC may be primarily thermal since both materials show significant degradation when the maximum temperature during injection is higher than approximately 250[degrees]C.

[FIGURE 16 OMITTED]

[FIGURE 17 OMITTED]

CONCLUSIONS

The results for the spiral flow mold show very good agreement between the computational predictions and the actual injection molded parts. There are some discrepancies with respect to the shot size or flow length. However, it is likely that these differences can be attributed to the unsteady nature of the experiment, where inertia has to be overcome during the early stages of injection. It is probable that the accelerating regime masks the effects of shot size. Nonetheless, the computational results show very good agreement with respect to the degradation profile along the length of the part, the maximum degradation levels observed, and the location of the degraded layer relative to the surface. More important, however, the results show that the degradation for M3800 and M4200 occurs at approximately the same maximum temperature of 250[degrees]C. Therefore, this result confirms that the degradation of PVC is thermal in nature and independent of the molecular weight of the material. One would expect that different grades of PVC would also show considerable degradation if the maximum temperature during injection was higher than 250[degrees]C. Of course, the different grades of PVC would require different processing conditions because of the difference in viscous heating sensitivity. Higher-viscosity materials attain the higher temperatures at lower injection speeds than lower-viscosity materials.
Table 1. WLF Parameters for PVC Materials.

Parameter n (-) [tau]* (Pa) [D.sub.1] (Pa.s) [A.sub.1](-)

M-3800 0.3994 4.607E + 04 3.182E + 16 42.9
M-4200 0.3783 6.058E + 04 3.230E + 14 35.1

Parameter [A.sub.2] (K) [T.sub.r] (K)

M-3800 51.6 353
M-4200 51.6 360


[c] 2004 Society of Plastics Engineers

Published online in Wiley InterScience (www.interscience.wiley.com).

DOI: 10.1002/pen.20125

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JOSE L. GARCIA (1), KURT W. KOELLING (1), and JAMES W. SUMMERS (2)

(1) Department of Chemical Engineering

The Ohio State University

Columbus, OH 43210

(2) PolyOne Corporation

Avon Lake, OH 44012
COPYRIGHT 2004 Society of Plastics Engineers, Inc.
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Author:Garcia, Jose L.; Koelling, Kurt W.; Summers, James W.
Publication:Polymer Engineering and Science
Date:Jul 1, 2004
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