Computer simulation of thermoplastic injection molding

CCC 0272-9504/81/1001027/0

computer Simulation of Thermoplastic Injection Molding"

by P.G. Lafleur and M.R. Kamal** Chemical Engineering Department

McGill University Montreal, Canada

INTRODUCTION

njection molding is one of the most important in- I dustrial processes for the manufacturing of plastics objects. The injection molding cycle is composed of three stages: filling, packing, and cooling. During filling, the molten polymer, which is produced by the shearing action of a rotating screw combined with ex- ternal heating, is introduced into the mold. After filling, extra material is packed under high pressure to compensate for shrinkage as the material cools. After the packing is completed, cooling continues and the pressure decreases.

The ultimate properties of a molded part obtained by injection molding depend on material and process parameters. The resin properties, mold geometry, and molding conditions which give rise to the thermomech- anical history experimented by the resin, determine the microstructure distribution and, therefore, the ultimate properties of the molded article; as shown in Fig- ure 1. Accordingly, it would be desirable to establish the feasibility of predicting product properties by computer simulation schemes that incorporate relation- ships between resin properties, design parameters, molding conditions, microstructure, and product properties. A realistic simulation of the injection molding process should be based on a model capable of describ- ing moldability parameters such as pressure drop in the delivery system and cavity, melt-front advance ment during cavity filling, as well as characteristics of the final product such as residual stresses and orientation.

* Paper presented a t the 2nd World Congress of Chemical Engineering, Oct. 4-9. 1981. ** To whom all correspondence should be addressed.

In the present work, a brief review of existing models that have been used to simulate certain aspects of the injection molding process will be presented. The models will be treated in two respects: firstly, with regard to the potential to predict the thermomechani- cal history, particularly flow and thermal conditions during the process and, secondly, the potential to p re dict some of the properties of the molded article. Subse- quently, an integrated generalized approach will be proposed to deal with both of the above aspects of injection molding simulation.

MOLDABILITY MODELS

Conventionally, moldability is associated with the filling part of the molding cycle and it refers to the speed and ease with which a polymer can be fabricated ac- cording to a given set of specifications. Two of the most impor tan t engineering quantities, useful in estimating molding cycles and in the design of the mold, are the position of the melt front a t any time and the filling time. However, a complete mold filling simulation would require the calculation of the detailed velocity, temperature and pressure profiles throughout the mold, including the position and shape of the advanc- ing front.

Different models have been proposed with varying complexity, depending on the mathematical formulation of the flow equations, the nature of the constitutive equation and other assumptions relating to material properties, and the technique used for the treatment of the resulting system of equations.

Spencer and Gilmore (1) developed the first correla- tion to calculate the filling time associated with the molding process. They used an empirical equation for

8 ADVANCES I N PLASTICS TECHNOLOGY

MECHANICAL MICROSTRUCTURE MACHINE DESIGN

OPERATING CONDITIONS

ULTIMATE ' PROPERTIES

Fig. 1. Components of the Plastics Processing System

capillary flow to represent the flow characteristics coupled with a quasi-steady state approximation. They found that the filling time was directly proportional to the polymer melt viscosity, and inversely proportional to a power of the ram pressure. Comparison of their equation with observed values showed a tendency of underestimating the fill times. This discrepancy was probably due to the use of a Newtonian viscosity estimated at the highest temperature, the temperature of the polymer just before it enters the mold.

During the last decade, models treated the mold filling process as a problem of combined transport of momentum and energy. Harry and Parrott (2) coupled a one-dimensional flow analysis with a heat balance equation to study the filling of a rectangular cavity. A Power Law fluid and viscous heat generation were considered. The most serious limitation is related to the assumption of a constant pressure gradient in the direction of flow. For a non-isothermal flow, the result would be a conservative estimate of the pressure, as shown in Figure 2. In consequence, comparison with experimental data shows that the simulation accurately distin- guishes between the short shot and full shot conditions, but considerable error is noted in predicting the length of the short shot as well as the fill time for the full shot condition.

Berger and Gogos (3,4) and later on Wu, Huang and Gogos (5) treated the radial filling of a disk cavity. The transport equations for a onedimensional Power Law fluid were used, neglecting normal stresses, transient terms and axial heat conduction. They considered two cases, the full non-isothermal and the isothermal problem. From their results, they demonstrated that, for the prediction of filling times, the isothermal solution can be used under high pressures and corresponding short filling times; the same is true for relatively thick cavities. If the cavity has extensive thin sections, the full non-isothermal flow problem has to be considered.

The model proposed by Kamal and Kenig (6) presented an integrated mathematical treatment of the filling, packing and cooling stages of the injection molding cycle. For the analysis of the filling stage, equations of continuity, motion and energy were simplified, taking into account a onedimensional creeping flow (neglecting the transient term and the normal stresses). Viscous heating, latent heat effects, viscosity depen- dence on temperature and shear rate assuming the Power Law, and the variability of polymer density with temperature and pressure were taken into considera- tion. Numerical and experimental results were com-

.5 -

.4

.3 .

P' -

.2 - .1 .

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

r* Fig. 2. Pressure Profile at the End of the Filling

October 1981 9

TABLE I. COMPARISON OF FILLING TIME

Fillinn Time (Set)

P(PS1) Numerical Numerical

Experimental Non-Isothermal Isothermal 350 350 2.00 1.88 1.76 350 450 1.00 0.90 0.86 350 500 0.65 0.69 0.66 350 650 0.54 0.46 0.52 400 250 1.80 2.10 1.72 400 300 1.08 1.09 1.08 400 400 0.71 0.61 0.64 400 450 0.50 0.46 0.49

1 .o

.9

.8

.7

.6 R'

.5

.4

.3

.2

.1

t(sec)

1 , 1 1 , 1 , ,

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1

Fig. 3. Progression of the Melt Front

pared for the case of spreading radial flow in a semi- circular cavity. Table 1 shows a comparison between the filling times for isothermal and non-isothermal models versus experimental values obtained for a polyethylene resin at two different temperatures. For both temperatures, the isothermal analysis appears to be satisfactory for predicting the filling time and also the melt front displacement, as shown in Figure 3, except for the low pressure cases, where the isothermal model gives shorter filling time. This is probably due to the application of a low shear rate in the isothermal case, producing relatively small viscous energy dissipation. The difference between the non-isothermal model and the experimental results, in predicting the moldability

parameters for this model, is due to the fact that some of the terms which have been neglected in the formulation of the flow equation are very important in the entrance region of the mold.

Broyer, Gutfinger and Tadmor (7), as well as White (8), simulated the flow in a narrow gap mold with a HeleShaw type flow. They considered a two-dimensional flow, assuming a fully developed isothermal creeping flow in each of the directions. Timedependent terms in the momentum equations were neglected by assuming that the unsteady state effects were negligi- ble. By assuming that the theory of Hele-Shaw on steady Newtonian viscous flow in narrow parallel plates is applicable to small changes in flow area and moderate variation of temperature in the flow direction, Kuo and Kamal (9) extended the analysis to unsteady non-isothermal and non-Newtonian flows. Us- ing an analytical-numerical method with a generalized viscous model to calculate the melt front position and flow quantities, they reported differences between experimental and computed results with error less than 1570, as shown in Figure 4.

In summary, in all the models mentioned, the melt front region (fountain effect), the transient and inertial terms are neglected, as well as the viscoelastic nature of the polymer melt. However, the models still give good representation of the flow characteristics related to the filling of the mold cavity. A recent report by

I I I I I 1 I

20 - - - E 0 u c m In 0 - 10 m

c ._

- .- 2

CASE 1 P.E. 251 1

Tb = 177°C

Tm = 26.7"C

Experimental Photography

Computation

_----

- .-

0 I 1 I I I I .o .5 1 .o 1.5 2.0 2.5 3.0

t ime(sec) Fig. 4. Progression of Flow Front

10 ADVANCES IN PLASTICS TECHNOLOGY

Huang, Gogos, and Schmidt (10) presents a model using the Marker-and-Cell numerical technique to solve the transient problem involved with cavity filling. Isothermal, incompressible flow was assumed in conjunction with the Power Law model. The complete two- dimensional momentum equations were used and the results were in good agreement with experimental data, but with an average error comparable to the prev- ious models in predicting flow characteristics.

The influence of the elastic behavior of the polymer melt in the injection mold filling process has been studied by Wang, et al. (1 1). If one considers only their comparison of pressure gradient for elastic and inelastic models, as shown in Table 2, it might be concluded that pressure is little affected by the nature of the model. Thus, the fluid elasticity appears to have a small

Fig. 5. Distribution of Properties

X Resin EX 2 (2059°C) 0 -

0 /---

-O- 0 I 0 - centre

J 0.02 I I I I I I I 10 20 30 40

DISTANCE FROM SURFACE (40 um)

TABLE 11. ROLE OF ELASTICITY

Pressure Gradient (106 dynelcm2) t(sec) Elastic Inelastic

0 7.18 7.18 0.25 13.82 13.61 0.50 18.89 18.67 0.75 24.95 24.71

influence upon the shear stress and the associatedpres- sure gradients required to maintain the flow. This r e sult is consistent with the observation that the simpler models mentioned above have proven fairly successful in predicting cavity pressure during filling.

In conclusion, it appears justifiable to state that for purposes of simulating injection mold filling to obtain data regarding filling times and the position of the melt front, satisfactory results are obtainable with isothermal models, under certain conditions, or by using a simple onedimensional creeping flow equation and a purely viscous constitutive equation.

PRODUCT QUALITY MODELS

The above models are mainly concerned with the elu- cidation of the filling behavior and the thermomech- anical history of the resin during various stages of the injection molding process. Thus, they might be considered as moldability models which could be useful in the analysis, optimization, and control of variables like filling time, pressure and temperature distributions during molding and the minimization or avoidance of flash and short shots. They also could give significant information regarding viscous heating and the possible degradation of the melt during processing.

Another important objective of simulation studies must be the prediction of the ultimate properties of the molded articles. For this purpose, it would be necessary to develop product quality models that relate the ther- momechanical history of the moldability models to the development of microstructure, mainly crystallinity, morphology, and orientation in the molded part, and ul- timately relate the final properties of the molded articles to the distribution of microstructure. In injection

molding, the optical and mechanical properties are very important for various commercial applications. A variety of studies have attempted to relate the distributions of these properties in molded articles to microstructure.

Kamal and Moy (12) employed several experimental techniques to characterize the microstructure (crystallinity and orientations distributions) of injection molded polyethylene parts. Their experimental results were employed in conjunction with an existing mechanical model to calculate the distribution of tensile moduli in the moldings. Figure 5 shows that experimental and calculated results exhibit good agreement.

Oda, White and Clark (13) reported, in an experimental study on polystyrene melts, that orientation development during non-isothermal polymer processing operations involving vitrification can be predicted by a knowledge of the stress field prior to solidification and by the application of stress-optical laws where the birefringence can be related to the stress field through the stress optical coefficient. Dietz, White and Clark (14) applied this idea to orientation distributions in injection molded parts. They obtained calculated birefringence values based on an extremely simplified flow model. The predicted values were up to 15% lower than the measured ones. Similar work has been reported by Kamal and Tan (15) for the injection molding of polystyrene using a relation developed by Wales (16) to relate the birefringence to the shear stress at the wall. Since the shear stress is a linear function of the pressure gradient, they showed that the residual stress in injection molded parts, as indicated by the distribution of birefringence in the moldings, may be estimated from a knowledge of the pressuretime variation at various points in the cavity during the injection molding cycle.

The last two treatments of stresses and birefringence are essentially based on a pure shear flow assumption. Birefringence is obtained in the shear plane parallel to the main flow-direction, the only important stress is the shear stess. However, in injection molding, the flow is not a pure shear flow, and to calculate the bire fringence in all the relevant planes, the complete stress field development should be considered. However, the results of these simple analysis give a fair representa-

October 1981 11

tion of the experimental data, especially away from the gate where the flow is reasonably close to a shear flow.

For more accurate and general simulation, viscoelas- ticity must be considered. Wang, et al. (17) have developed a detailed model incorporating a viscoelastic constitutive equation. The numerical scheme employed a finiteelement technique to solve the relevant system of equations. The proposed model yields predictions of residual stresses, orientation and birefringence in all planes, taking into account the effects of unsteady, non-isothermal flow during filling and non-isothermal shear-rate-dependent stress relaxation during cooling. For this purpose they have employed a constitutive equation developed by Leonov (18), which is based upon the thermodynamics of irreversible processes. Figure 6 shows some of the results that they obtained in predicting the maximum birefringence for amorphous molded parts. Their model is systematically underpredicting the maximum birefringence, except for the gate region. A part of this underprediction may be attributed to neglecting the packing stage. Their study was restricted to amorphous polymer.

5 -

1 2 X

2.5

a -;in

I I I I

I 0 ' 5 10

x, cm

Fig. 6. Maximum Birefringence I

INTEGRATED GENERALIZED MODEL FOR MOLDABILITY AND PRODUCT QUALITY

The injection molding process can be fully described in terms of the equations of conservation of mass, momentum and energy. Each stage of the process is governed by a set of initial and boundary conditions coupled with simplifying assumptions. For example, the filling can be considered as an incompressible flow, but certainly not the packing stage. In general ten- sorial formulation the conservation equations are as follows:

Continuity: + p(v 07 = o +

p - ( v * 3 + Z Dv' Dt Momentum: p - = - v

The momentum equations relate the velocity vector v to the stress tensor+nd are valid for any fluid. There is no restriction as to whether the fluid is Newtonian or non-Nev.tonian, viscoelastic or inelastic. Therefore, a f lox field is completey specified once the components of v andTare known as functions of the independent variables (time and position). Thus, there are nine un- knowns, three components of velocity and six independent components of stress. Only four equations are available to relate these functions: one continuity and three dynamic equations. The additional required information is embodied in the constitutive equations which relate the stress field to the flow field, usually through the rate-of-deformation tensor, and thereby define the type of fluid for which a solution is being sought.

There is no constitutive equation that can represent the full range of flow and deformation encountered in polymer processing operations. The generalized non- Newtonian fluid equation allows for a nonlinear rela- tionship between shear stress and shear rate, but still fails to predict the normal stresses commonly observed in shear flows. The Maxwell model, which is a viscoelastic constitutive equation, can predict transient behavior but gives a Newtonian viscosity in shear flow. I t has been shown that the Phan-Thien model (19) gives a good fit of the first normal stress difference and a part of the shear viscosity curve (20).

A generalized model of the injection molding process must have the capacity to cope with both amorphous and crystalline polymers. In semicrystalline polymers, the degree of crystallinity will influence the degree of orientation and will give rise to a variety of morphological zones in the molded parts. Thus, a realistic model should include a crystallization kinetic equation to follow the development of the crystallinity. Figure 7 gives a comparison between experimental data and two

+

Fig. 7. Kinetics of Isothermal Crystallization

A

__-- - -- -_

XC

t

1. AVRAMI X ~ = ~ O ~ [ ~ - C - Z '"1

2. MODIFIED AVRAMI

EQUATION FOR

Xc = x A , o ~ [ l - C - ~ ~ ~ ~ ] + Xs,oo0 xt

12 ADVANCES IN PLASTICS TECHNOLOGY

forms of the Avrami equation, used to predict the d e gree of crystallinity for isothermal conditions. To be helpful in the analysis of the injection molding process, these models need to be extended to the non-isothermal conditions wevailing, due to the large cooling rates.

The resulting dynamic problem, consisting of the continuity equation, the energy equation, the dynamic equations and the set of constitutive equations, forms a system of multiple, simultaneous, coupled, non-linear partial differential equations which can only be solved by a numerical technique. In the particular case of the filling stage, a free surface due to the transient nature of the operation is involved. For regular wall bound- aries (ex. disk or rectangular cavities), it is possible to use a simple finite-difference method like the Marker- and-Cell (M.A.C.) technique, which can handle free surface and transient flow (21). This technique can also be extended for the treatment of the packing and cooling stages, where one can assume that there is no flow but that the properties of the Markers are still timedependent.

In view of the above, it is considered that a generalized integrated model of the injection molding process that could be useful both for moldability and product quality purposes should have the following ingred- ients: (1) Viscoelastic constitutive equation that can de- scribe both the viscous and elastic behavior of the material in both the liquid and solid states with approp- riate allowance for relaxation phenomena. (2) Crystallization kinetics that take into considera- tion isothermal and non-isothermal effects with due at- tention to morphological phenomena. (3) Integration of the filling, packing, and cooling stages.

REFERENCES

(1) R.S. Spencer & G.D. Gilmore, J. Coll. Sci., 6, 118 (1951). (2) D.H. Harry & R.G. Parrott, Polym. Eng. Sci., 10, 209 (1970). (3) J.L. Berger & C.G. Gogos, Technical Papers, 29th Annual

Technical Conference, Society of Plastics Engineers, 17, 8 (1971).

(4) J.L. Berger & C.G. Gogos, Polym. Eng. Sci., 13, 102 (1973). (5) P.C. Wu, C.F. Huang, and C.G. Gogos, Polymer Engineering

(6) M.R. Kamal & S. Kenig/Polymer Engineering & Science, 12,

(7) F. Broyer, C. Gutfinger, and Z. Tadmor, Trans. of the Society

(8) J.L. White, Polymer Engineering & Science, 15, 44 (1975). (9) Y . Kuo and M.R. Kamal, AiChE Journal, 22, 661 (1976).

& Science, 14, 223 (1974).

294 (1972).

of Rheology, 19, 423 (1975).

1

(10) C.F. Huang, C.G. Gogos, and L. Schmidt, Paper 94e, 71st Annual AiChE Meeting, November 15, 1978.

(11) K.K. Wang, e t al., Injection Molding Project, College of Engineering, Cornell University, Progress Report #6. September 1979.

(12) M.R. Kamaland F. Moy, VIII Internationl Congresson Rheo- logy, Naples, September 1, Vol. 3. P. 143 (1980). GogosiPolymer Engineering & Science, 14, 223 (1974).

(13) K. Oda, J.L. White, and E.S. Clark, Polymer Engineering & Science, 18, 53 (1978).

(14) W. Dietz, J.L. White, and E.S. Clark, Polymer Engineering & Science, 18, 273 (1978).

(15) M.R. Kamal and V. Tan, Polymer Engineering & Science, 19, 558 (1979).

(16) J.L.S. Wales, I.J. van Leeuwen, and R. van der Vijyh, Polymer Engineering & Science, 12, 358 (1972).

(17) K.K. Wang, e t al., Injection Molding Project, College of Engineering, Cornell University, Progress Report #7, October 1980.

(18) A.I. Leonov, Rheology Acta, 15, 85 (1976). (19) N. Phan-Thien, Journal of Rheology, 22, 259 (1978). (20) W.K.W. Tsang, Ph.D. Dissertation, McGill University, March

1980.

October 1981 13

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Computer simulation of thermoplastic injection molding