Pressure build-up during the packing stage of injection molding

Pressure Build-Up During the Packing Stage of Injection Molding

TAI-SHUNG CHUNG

Celanese Specialty Operations 86 Morris Avenue

Summit, New Jersey 07901

A thorough study of an isothermal fluid motion within the mold cavity during the packing stage is presented. The fluid is considered Newtonian, and its compressible behavior is assumed to obey the Spencer-Gilmore equation of state. Mathe- matical results indicate that the pressure built up during the packing stage is strongly dependent on the melt viscosity and the boundary of the cavity. The effects of pressure gradient and distribution, during this stage, on the shrinkage of the final products are also discussed.

INTRODUCTION

e packing stage of injection molding may be Th the most important stage affecting the final dimensions and quality of the molded parts. Since a plastic material expands when it is heated and con- tracts when it is cooled, warpage is therefore caused by uneven density and stress distribution within the molded articles. Sink marks and internal voids are also created as a result of insufficient material at a given spot. In order to minimize these deficiencies, additional material is forced into the mold cavity after the injection filling stage to compensate for the material shrinkage during the cooling stage. However, over-packing the mold cavity will cause flash at the parting line, stick during ejection, and excess stresses within the molded parts. An under- standing of the flow phenomena during the packing stage is, therefore, essential for the determination of the ultimate mechanical properties and quality of the finished products.

Compared to the studies on the injection mold filling stage, relatively little work has been done with regard to the packing stage. The first mathematical simulation of the packing stage was con- ducted by Kamal and Kenig (1, 2). They considered that the movement of the polymeric melt into the cavity was proportional to the difference between the injection pressure and the average pressure within the cavity. A more complicated and detailed analysis of the packing stage for rectangular geom- etries was later developed by Kamal, et al. ( 3 , 4 ) . They employed a linear pressure-density relation- ship to describe the compressibility of the molten polymer. Chung and Ryan (.5) extended their analyses employing different initial and boundary conditions, as well as a numerical scheme. Recently, Chung and Ide (6) studied the pressure built up during the packing stage in a disk cavity. A New-

tonian fluid was used, and its compressibility was assumed to obey the Spencer-Gilmore equation of state (7). Kamal and Lafleur (8) simulated the packing stage by incorporating a White-Metzner rheological model into their calculations. They found that non-Newtonian and viscoelastic effects play an important role during this stage.

Since most commercial injection molded parts are molded in thin cavities, and the fluid motion within cavities is always, at least, a two-dimensional flow; therefore, we attempt to extend the previous work to a rectangular cavity in this paper. The governing equation, as well as its solution for a two- dimensional flow during the packing stage, will be derived. The pressure variation and flow distribution which develop during the packing stage will be determined. In addition, a detailed insight on the effect of mold boundary on the melt flow will be presented.

ANALYSIS

The following assumptions are introduced for the mathematical modeling of the packing stage in thin rectangular cavities:

1. The fluid is Newtonian, and the flow is isothermal.

2. Velocity components, pressure, and density variations in the thickness direction are considered to be negligible.

3 . Inertial, body, and surface tension forces and viscoelastic effects are neglected.

4. The compressible behavior of the melt obeys the Spencer-Gilmore equation of state:

( p + w ) (; - ;) = RT

Because the deformation rate during the packing stage is small and the power-law model does not

772 POLYMER ENGINEERING AND SCIENCE, AUGUST, 1985, Vol. 25, No. 11

Pressure Build-Up During the Packing Stage of Znjection Molding

have the capability of predicting melt viscosities near the zero shear rate region, the Newtonian fluid assumption is assumed to be a good approximation for the melt viscosity behavior during this stage. The negligence of density variation across the thickness of the cavity is for the simplicity of calculations. Experimental results published by Kamal, et al. (9, 10) showed that polymeric density varies slightly across the thickness of injection molded parts. Therefore, this assumption is probably not valid if the filling and packing stages are not isothermal. Making use of the above assumptions, the relevant components of the equations of continuity and motion are:

Y = L

aP a a - + - p i i + - p v = 0 at ax dY

( 3 )

(4)

Where p is the density across the mold thickness, which is independent of z , and U and 6 are the average velocities in x and y directions, respectively. Figure I defined the coordinate system and the mold geometry. Integrating Eqs 3 and 4 triple with the appropriate boundary conditions yields the average velocities ti and 6. Substituting them into Eq 2 gives,

( 5 )

This equation may be rewritten with the aid of Eq 1 as

at

Equation 6 is the general governing equation for a two-dimensional isothermal packing. It is different from the equations derived by Kamal, et al. (3 , 4) and Chung and Ryan (5). The initial and boundary conditions for p are,

1 . t = 0 p = p(x, y) (7) 2. t > 0 p = p l ( t ) at 1x1 5 a, y = 0 (8)

- 2 h

- 2

Fig. I . Rectangular mold cauity.

As suggested from the previous work (4-6), the initial density or pressure distribution for packing may be calculated from the pressure distribution which exists at the end of the filling stage. The initial pressure distribution may be easily determined by means of a conformal mapping transfor- mation which was proposed by Ryan and Chung, as follows (1 1);

(12) P L - P

P = P o - P L

where Po is the known pressure measured at gate at the end of the filling stage. P denotes the transformed value of the y coordinate, which is defined as follows for x > 0:

2

2 P = - cosh-' [2 sin a

(({sin( 2) cosh( 2) + sin( 5)) + {cos($ sinh(2))l)li2

2

- ({sin(g) cosh(2) - sin(%))

+ { cos( g) sinh( 2))l)7] POLYMER ENGINEERING AND SCIENCE, AUGUST, 1985, Vol. 25, No. 11 773

Tai-shung Chung

and for x = 0: where

Bi = B(1 , j )

c1 = C(1, j )

cz = C(2, j )

Bz = B(2 , j )

and

where B(i, j ) and C(i, j ) are defined in Eqs 16 and 17, respectively. A similar equation can be derived for the points along y = 0, or ( i , 1). Since only density at gate is given, the calculation always starts in the positive x-direction. The mold dimensions, material constants, and rheological parameters for the calculation are given in Table 1 (14).

RESULTS AND DISCUSSION Figures 2 and 3 show the calculated pressure

distributions along the center line of the cavity for a low-density polyethylene and a polystyrene resin, respectively. The initial pressure distribution for packing was determined by E q 12. The pressure development of the amorphous polystyrene is seen to be quite similar to that of the low-density polyethylene. The pressure gradient is highest near the gate, but becomes essentially uniform at a distance from the gate. The pressure gradient decreases with time, and the pressure distribution in the mold cavity tends to become more uniform. These phenomena are in agreement with the previous theoretical work (5, 6), as well as the experimental results published by Kamal, et al. using density measurement on the injection molded polyethylene bars (9, 10). Comparing these calculated results to the previous experimental data and theoretical studies, reported in the literature (3,5,9), confirms that the non-linear pressure variation observed in the published experimental results was due to the uneven temperature profile within the cavity after the mold filling stage. In addition, both3gure.s show that pressure near the gate decreases at the begin- ning of packing, and then increases as a function of time. This is due to the fact that there is a radial flow near the gate area. Similar phenomena have been reported in the injection molding of disks (6).

Figure 4 illustrates the pressure variation along the cavity wall, y = 0, for these two materials. Except in the early stage of packing, the pressure ratios, PIP,, along y = 0 line seem to increase almost

sin( 2) sin( z)

Substituting the pressure calculated from E q 12 into the Spencer-Gilmore equation determines the initial density distribution, p ( x , y), within the cavity. The first boundary condition, E q 8, implies that the pressure variation with time, P I ( t ) , is known during this stage. The other boundary conditions arise from the symmetry about the centerline and from the fact that there is no velocity across cavity walls.

NUMERICAL SOLUTION Equation 6 has to be solved numerically. Using a

modified unconditional stable explicit numerical scheme proposed by Saul'yev (12, 13), a finite difference grid was constructed, and a finite difference equation at each grid point ( i , j ) was developed. Due to symmetry, only half of the mold cavity is simulated. The spatial and time increments are denoted as Ax, A y , and A t , respectively. Thus, E q 6 may be expressed as follows:

(1 - 2B) B (1 + 2B) /)?:' = P7.j + (1 + 2B)

where

(17)

Ax = A y (18)

h 2 R T P 2 ( p , + &)At C =

37Ax2(po - ~ y j ) ~

and

The most difficult parts during the simulation are the calculation of p at points (1, 2), ( l , j ) , and ( i , 1) in the first time increment. To overcome these, Ax is chosen to be equal to half of the gate size so that p (2, 1) = p (1, 1). For i = 1 (x = 0), E q 15 becomes

1

Table 1. Parameters in the Simulation of Equation 6 ~

Parameter Polyethylene Polystyrene

0.25 2.5 15.0 0.25

403 3.4 x lo4 1.1 4286 2.9647 x lo6 3.281 9 x 1 Og 3.4 x 107

f, (1 .o + 200t)

0.25 2.5 15.0 0.25

6.895 X lo4 1.21 655 7.9978 x lo5 1.8616 X lo9 6.895 x lo7

f, (1 .o + 200t)

433

774 POLYMER ENGINEERING AND SCIENCE, AUGUST, 1985, Vol. 25, NO. 11

Pressure Build-Up During the

t

Fig. 2. Axial pressure variation for polyethylene.

I 1

0 5 10 I5 v ["I

Fig 3. Axial pressure variation for polystyrene.

equally in each time increment for each material. Figures 5 and 6 show that the isobaric lines within half of the mold cavity vary with time for the polyethylene and the polystyrene resins, respectively. At the onset of packing, the isopressure lines near the gate follow a semicircular shape, and they gradually change to bedome parallel to the x-axis at points distant from the gate. Once the packing is initiated, material is forced into the cavity and, as a result, the transverse pressure distribution is al- tered. The ratio of the pressure near the center line of the cavity to the pressure at the gate increases faster than that near the wall of the cavity. A similar result was also observed by Kamal, et al. using density measurement on molded parts (9, 10). This behavior may partly be due to the fact that these two regions are at different distances from the gate, so that the pressure drops from the gate to these two regions differ. It may also be due to the influence of the walls of the mold cavity. Since the lowest pressure regions within the cavity are lo- cated at the corners furthest away from the gate, the shrinkage, sink marks, or warpage may likely occur in these spots. On the other hand, the high stresses or pressure gradient remaining at the gate area may cause parts to craze and fail in service. Since the polyethylene used in this analysis has a lower melt viscosity than the polystyrene, the pressure distribution for polyethylene within the cavity

Packing Stage of Injection Molding

0 . 0 2 5 Sec

5 -

------ 0.015

0;- ' I I . 8 I 1.0 1 . 2 .4 .6

x/b Fig. 4 Pressure variation along Y = 0.

C: I : 0 .025 see . PI ( 0 . 0 ) : 6.0

I < m ,

Fig. 5 lsopressure lines within half of the canityfiw polyetlaylene

tends to become more uniform than that for polystyrene.

In order to further understand the fluid motion accompanying the packing stage, Figs. 7 and 8 give the velocity in the machine direction as a function of x at various axial positions for the polyethylene and polystyrene resins, respectively. The average axial velocity V is given by the following expression:

- h2 aP 311 dy

u = - - -

Near the gate, the magnitude of V is highest at the center line of the cavity and rapidly decreases in the x- direction to the cavity wall. Since E9 20 is not applicable to calculate the velocity at the wall, no velocity at the walls is given. In fact, there is no fluid motion at the walls if a non-slip assumption is employed. When the axial distance is slightly larger than the width of the cavity, melt near the wall moves faster than that near the center line of the cavity. This interesting phenomenon is attributed to the influence of the cavity corner. In other

POLYMER ENGINEERING AND SCIENCE, AUGUST, 1985, Yo/. 25, No. 1 7 775

Tui-shung Chung

5 -

c : t : 0.025 y e . PI 10.0) 6 . 0

Polyethylene t = 0 . 0 1 5 Sec

o , , , I 1

Fig. 6. lsopressure lines within halfof the caoity for polystyrene.

Fig 7. Aceruge axial velocity in the machine direction as a function of distance from the gate

words, since we imposed the boundary conditions that no melt is allowed to flow across the walls, the compressed polymeric melt in the corner has to move in the machine direction in order to dissipate pressure. This phenomenon disappears at a distance away from the corner, and the axial velocity in the machine direction becomes uniform across the width of the cavity. The difference in the magni- tudes of the axial velocities between these two polymers comes from the fact that their packing pressures are different.

Similarly, Fig. 9 illustrates the changes in the transverse velocity U with axial distance. Here, U is defined as:

- h2 dP u = - - -

377 ax The magnitude of U decreases sharply with axial distance, reaches a minimum, gradually increases,

I I I

0 0 2 0 4 0 6 0 .8 1 0 X/b

Fig 8 Aoerage axial velocity in the machine direction as a function of distance from the gate

1 0 5 10 IS 0

y "-1

Fig. 9 Aoerage transoerse velocity in the transoerse direction.

and finally drops to zero at the end of the mold cavity. This non-linear velocity profile is consistent with the previous analyses. In other words, this phenomenon is actually due to the effect of the corner on the fluid motion. The flow pattern of the melt changes from a one-dimensional radial flow near the gate area to a two-dimensional flow away from the gate. Since pressure gradient in the transverse direction reaches a minimum close to this transition region (as shown in Figs. 5 and 6), the transverse velocity in this region is, therefore, smaller than that in the rest of the mold cavity.

ACKNOWLEDGMENTS The author wishes to thank Drs. John A. Flint

and Keith T. O'Brien at Celanese for their useful comments and corrections. In addition, since part of this work is an extension of his thesis, the author also desires to express his utmost appreciation to

776 POLYMER ENGINEERINGAND SCIENCE, AUGUST, 7985, Vol. 25, No. 12

Pressure Build-Up During the Packing Stage of Injection Molding

Prof. M. E. Ryan at the State University of New York at Buffalo for his guidance and encourage- ments during that period of time.

NOMENCLATURE U

b h L = Mold length. P = Pressure. P,, P I ( t )

packing stage. R

state. T = Temperature. t = Time. U

u V 6 W

x, y, z = Cartesian coordinates. P = Transformed Y coordinate. PL

cavity. 17 = Melt viscosity. P = Density.

= Half width of gate. = Half width of rectangular cavity. = Half thickness of rectangular cavity.

= Pressure at gate at the end of filling stage. = Known pressure variation at gate during

= Constant in Spencer-Gilmore equation of

= Velocity in x- axis. = Average velocity in x- axis. = Velocity in y- axis. = Average velocity in y- axis. = Constant in Spencer-Gilmore equation of

state.

= Transformed value of y at the end of mold

Po

pI

= Constant in Spencer-Gilmore equation of

= Known density variation at gate during the state.

packing stage.

REFERENCES

1. M. R. Kamal and S. Kenig, Polym. Eng. Sci., 12, 294 (1972). 2. M. R. Kamal arid S. Kenig, Polym. Eng. Sci., 12, 302 (1972). 3. M. R. Kanral, Y. Kuo, and P. H . Doan, Polym. Eng. Sci., 15,

4. Y. Kuo and M. R. Kamal, Proc. Int. Conf., Polymer Process-

5 . T. S. Chung and M. E. Ryan, Polym. Eng. Sci., 21, 271

6. T. S. Chung and Y. Ide,]. A p p l . Polym. Sci., 28,2999 (1983). 7 . R. S. Spencer arid G. D. Gilmore, J. A p p l . Phys. 21 , 523

8. M. R. Kamal and P. G. Lafleur, SPE ANTEC Tech. Papers,

9, V. Tan and M. R. Kamal, J . Appl . Polym. Sci., 22, 2341

10. M. R. Kamal and F. Moy, J , A p p l . Polyin. Sci., 28, 1787

11. M. E. Ryan and T. S. Chung, Polym. Eng. Sci., 20, 642

12. V. K. Saul’yev “Integration of Equations of Parabolic Type by the Method of Nets”, Macmillan, New York (1964).

13. B. Carnahan, H. A. Luther, and J. 0. Wilkes, “Applied Numerical Methods”, John Wiley and Sons, Inc., New York (1969).

14. E. C. Bernhardt, “Processing of Thermoplastic Materials”, Reinhold, New York (1960).

863 (197.5).

ing, p. 329, MIT, Cambridge, Mass. (Aug. 1977).

(1981).

( 1 950).

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Pressure build-up during the packing stage of injection molding