Volumetric effects in the injection molding of polymers

Volumetric Effects in the Injection Molding of Polymers

A. I. ISAYEV

Polymer Engineering Center University of Akron Akron, Ohio 44325

and

T. HARIHARAN

Sibley School of Mechanical and Aerospace Engineering Cornell University

Ithaca, New York 14853

Density and shrinkage measurements have been performed in quenched and molded slabs from polystyrene (PS) and poly(methylmethacry1ate) (PMMA). Various processing conditions have been employed and their effect on density and shrinkage variation in the final parts, as well as volumetric aging us. elapsed time at room temperature, have been eluci- dated. A numerical simulation of the density variation in quenched parts and their aging has been performed by using first-order rate theory for volumetric changes in conjunction with solving the transient one-dimensional heat-conduction equation with a convective heat-transfer boundary condition at surface. A numerical simulation of the shrinkage in molded parts has been carried out by using the equation of state with a simultaneous solving of the governing equations for one- dimensional mold filling during the cavity filling stage followed by transient one-dimensional conduction during packing and cooling stages. Predicted results for density and shrinkage are compared with experimental data.

INTRODUCTION

olumetric effects in the injection molding of V polymers are due to complicated thermal and pressure histories during cavity filling, packing, and cooling stages (1). A rapid non-homogeneous cooling of polymeric melt through the glass transition temperature, T,, or melting point, T,, introduces nonequilibrium volumetric changes, orientation, and residual stresses in final products (2, 3 ) . Ac- cordingly, the density and shrinkage at any point of a molded part become function of space and time and, presently, there is no approach available for evaluating a priori the shrinkage of a molded part after demolding. Thus, moldmakers rely heavily upon experience and intuition in order to correctly design molds, compensating for an unavoidable shrinkage of the part. Evidently, in doing so, a few experimental trials involving an alteration of the mold will usually be needed. This is especially critical for the molding of high-precision parts for which dimensional specifications are to be strictly satisfied. Hellmeyer and Menges (4) were the first

to approach the modeling of shrinkage by making use of the equation of state. However, their multi- layer model for shrinkage during the holding-pressure phase has been found to be only in qualitative agreement with experimental data.

Concerning the density distribution in quenched and molded parts, the available data are highly contradictory. According to Siegmann, et a1 (S), the density at the center of a quenched Noryl (modified poly (phenylene oxide)) strip was about -5 percent higher than at the surface. On the other hand, for a quenched polystyrene (PS) strip, Greener and Kenyon (6) have found the density at the surface to be higher than at the center with the variation being about 0.1.5 percent. In addition, Moy and Kamal(7) have observed a slightly larger density at the center than at the surface of molded polyethylene strips. These latter investigators have also found that the melt temperature has little effect and that the density is practically uniform in the flow direction. Recently, Wust and Bogue (8) investigated the density in thin PS rods quenched at various cooling rates. They have assumed a uniform temperature

POLYMER ENGINEERING AND SCIENCE, MID-APRIL, 1985, Vol. 25, No. 5 27 1

h

throughout the cross section during quenching and have determined the density variation (aging) after quenching b y using a rate equation (9). In particular, in order to fit the density data right after quenching and during aging, two different time constants have been used.

In an attempt at addressing the rather complex issue of shrinkage and density in quenched and molded parts, the present investigation includes measurements and predictions of density and shrinkage at various processing conditions.

EXPERIMENTAL

Materials and Methods of Investigation The material that has been investigated is PS

Styron 678U (Dow). Injection molding has been performed on a 55-ton Boy machine with shot size of about 9.6 by 10-’mm”. The injection molding process was controlled by means of a MOOG con- troller (MOPAC-2 1). The cavity dimensions were O.12m by 0.04m by 0.00254m with gate size of 0.006m by 0.005m by 0.001m. Two flush-mounted pressure transducers were located in the cavity at distances of 0.015 and 0.105m from the gate. Si- multaneous readings of the cavity and injection pressure, ram position, and ram velocity were taken by means of a data acquisition system (MINC-23).

In addition to molding, a free-quenching experiment has been performed on PS strips having dimensions 0.0508m by 0.0508m by 0.00254m and 0.0762m by 0.0762m by 0.0038m together with PMMA strips having dimensions 0.0508m by 0.0508m by 0.0028m and 0.0508m by 0.0508m by 0.0064m. These strips have been annealed be- fore quenching. Procedures for annealing and quenching have been described earlier (3).

The density distribution in the quenched and molded strips has been determined by means of a density-gradient column at temperature 25°C. The column was filled with an aqueous solution of sodium bromide. Measurements of gapwise-average density were made on samples having planar dimensions of 0.005m by 0.005m which were cut from the strips by using a low speed diamond saw. In addition, slices about 0.00038m thick were cut at different distances from the surface in order to determine the gapwise distribution of density. Ini- tially, the samples were wet thoroughly by dipping them into the sodium bromide solution. After making sure that the samples were free of air bubbles sticking to the surface, they were added to the gradient column. Time was allowed for the samples to come to equilibrium. Measurements on the annealed samples showed that the thick and thin sam- ple comes to equilibrium at approximately 35 and 70 min., respectively. Volume shrinkage of the molded strips has been determined as (V,, - V,)/V,, where V, is the nominal volume of the cavity and V, is the volume of molded strip. The latter has been measured by using Archimedes’ principle.

Birefringence has been measured on molded PS strips by using an Olympus polarizing microscope.

Hariharan

The procedure for such measurements has been described earlier (3).

THEORETICAL CONSIDERATIONS AND COMPARISON WITH EXPERIMENT

Modeling of Density in Quenching A numerical simulation of the gapwise density

variation in the quenched parts during quenching and aging has been performed by using first-order rate theory for specific volume (9) as follows:

dV v - v, d t ~ T , V ~ R

where V and V, are, respectively, the actual and equilibrium volume at current temperature T, T R is a relaxation time at reference temperature T R and u T , ~ is a shift factor determined from a modified form of the WLF equation:

(1) - = -~

where C1 and Cz are constants and T, is the effective temperature defined as

(3) T, = T + ( V - V,)/Act, T, = Tz + ( V - V,)/Act,

T 2 T2 T < T.2

where T2 is considered to be the “true” Tg in the equilibrium thermodynamic sense for the second order transition at which free volume ceases to exist and Aa is the difference between the liquid, ctL and glass, (YG volumetric expansion coefficients. The equilibrium volume is defined as:

V, = Vi - ct~(Ti - T ) , T 2 Tz

V, = Vi - ctL( Ti - T2) - ctc( T2 - T ) , (4) T < T2

where Ti is the initial temperature and Vi is the volume at Ti. It should be noted that during quenching, the quantities V, V,, T R , uT,V and T, are depend- ent upon gapwise location and time. In order to evaluate them, the temperature variation, T( y , t ) , in the gapwise direction and with time is needed. The latter has been obtained by solving the one- dimensional heat conduction equation with a convective heat-transfer boundary condition at the surface:

aT d2T ( 5 ) _ -

at - a a y 2 with

(6) \ I

h[T(b, t ) - T,] -k dT(b t ) =

dY where T, is the bath temperature, b the half-thickness of the strip, k the thermal conductivity, and a the thermal diffusivity. A heat-transfer coefficient h = 1470 J/m2.so . K has been calculated for steady

272 POLYMER ENGINEERING AND SCIENCE, MID-APRIL, 1985, Yo/. 25, NO. 5


natural convection of a vertical plate in water (10). Equation 5 with initial and boundary conditions (6) has been solved using a fourth-order Runge-Kutta scheme together with Eqs 2 to 4. The mate'rial constants used in the calculations are given in Table 1. The values of a L , a G , and Tz for PS have been calculated from the P-V-T data of Hellmeyer and Menges (4). For PMMA, the values have been taken from (11). The time constant TR for PS was calculated at 130°C from stress-relaxation data presented by Aklonis and Tobolsky (12, 13). Relaxation time at any temperature has been taken to be the time at that temperature for 3G(t) (the shear-stress relaxation modulus) to reach a value of 10' Pa. The resulting calculated value for TR of 0.0021 sec. agrees well with the relaxation time of 0.002 sec. measured by Patterson, et al (14) by means of Photon Correlation Spectroscopy. The relaxation time for PMMA has been taken to be TR = 0.008 sec., based upon the results presented by Patterson, et a1 (15).

The values for the constants C1 and Cz are given by Rusch (9). The values of thermal conductivity and thermal diffusivity, a, for PS and PMMA have been taken from Ref: 16. An attempt at extending this rate theory to an evaluation of the specific volume during molding has proven unsuccessful. In particular, at the instant of pressure release, the value of V - V, becomes negative, such that Eq 1 becomes inappropriate. Thus, for the shrinkage modeling during molding, an instantaneous response of the specific volume to temperature and pressure variation has been assumed.

Modeling of Shrinkage in Injection Molding In an attempt at modeling shrinkage in injection

moldings, we have proceeded to incorporate the equation of state with one-dimensional cavity-filling simulation. Following (1 7), the pressure, velocity, and temperature field during cavity filling have been determined by employing the equations of motion, energy, and continuity:

(7)

Q = W lE udy (9)

where x and y are respectively the streamwise and gapwise coordinates, u is the streamwise velocity, T the temperature, A = -dP/ax the pressure gradient, Q the specified constant volumetric flow rate, b the half-gap thickness, W the width, and r] is the shear velocity, assumed to be of the form

(10) where + = du/dy is the shear rate and A, n, and T, are material constants. The boundary conditions are as follows:

r](+, T)= A+"-' exp( T,/T)

Table 1. Properties of Polystyrene and PMMA Used in Modeling

PS PMMA

L Y ~ , m3/kg. O K 5.2 x 10-7 4.7 x 10-7 00, rn3/kg. O K 1.9 x 10-7 1.2 x 10-7

9.95 x at 8.685 x at Vi. m3/kg 130°C 130°C T2, "C 70 50 k i J/(s.rn. OK) 0.13 0.1 9 LY, m2/sec 6.13 x lo-* 1.095 X 1 0-7 c1 12.4 16.7

c2, "C 100°C 105°C

TR, sec 1 3OoC 130°C

41 at TR =

0.0021 at TR =

55 at TR =

0.008 at TR =

u(x, kb, t ) = 0,

T(0, y, 0) = To, T(x, kb , t) = T,, (11)

= o d u x , 0, t ) dY

where x = 0 corresponds to the inlet and T,, and T,, denote the mold and melt temperatures, respectively. By combining Eqs 7 and 9 as given in (17), an integral equation for A vs. Q can be derived which has been integrated by numerical quadrature, once the energy equation (8) has been solved by tridiagonalization. The following set of material constants for PS has been used (18):

n = 0.36, T, = 6000"K,

A = 0.027 kg/(m.s.) - (SJ

p = 940 kg/m3, cp = 2.05 x lo3 J/(kg."K),

k = 0.122 J/(m.s."K)

A theoretical approach to predict volume shrinkage of molded parts at various processing conditions has been based upon the specific volume history which a polymeric melt passes through during all stages of the injection molding operation including cavity filling, packing, and cooling stages. It has been assumed that the specific volume responds instantaneously to temperature and pressure varia- tions. The theoretical shrinkage, S, has been determined as follows

(13) s=-

with Vf denoting thespecific volume of PS melt at room temperature, Vj being the initial specific volume of melt. During injection molding, the polymer melt undergoes severe pressure and temperature changes in a short time. Since the initial specific volume does not correspond to a uniform temperature and pressure, it has been takenas some average specific volume. Accordingly, Vj has been calculated via

- vi - Vf vi

POLYMER ENGINEERING AND SCIENCE, MID-APRIL, 1985, Vol. 25, NO. 5 273

A. I . lsayev and T . Hariharan

where V ( t ) is the specific volume averaged in the gapwise direction at any time ( t ) for the particular cross section being considered, such that

V( t ) = 1 6’ V( y, t)dy b The lower limit of integration in E q 14 corresponds to the time at which the melt front reaches the location being considered during cavity filling. The upper limit, tp denotes the time at which the pressure at this location goes to zero. Tosalculate the specific volume variation with time, V(t), the temperature and pressure variation must be known as the polymer melt cools. In particular, the temperature profile during the cavity filling stage has been obtained by simultaneous solving of Eqs 7 to 9. The temperature profile during the packing and cooling stages has been obtained by solving the one-dimensional heat diffusion ( E q 5) with the following conditions on T( y, t ) :

T(b, t ) = T, where the initial distribution T(y, 0) = T(y, tfiu) corresponding to the time tf,ll has been obtained at the end of the cavity filling simulation. With T( y, t ) so determined and p ( t ) being measured at two locations within the cavity, the corresponding values of V( y, t ) in E q 15 have been obtained by using P-V-T data from the literature (4) where plots of the specific volume of PS has been presented as a function of temperature at various pressures. Then B(t) and V, have been evaluated by trapezoi- dal quadrature at the streamwide locations corresponding to the two flush-mounted pressure transducers (located 0.015 and 0.105m from the gate). The resulting comparison with the measured total volumetric shrinkage has then been done by aver- aging the above calculated results at the two trans- ducer locations.

RESULTS AND DISCUSSION Typical theoretical results for the gapwise distri-

bution of density at different time during the quenching of PS and PMMA strips are shown in Fig. 1 . At small times, the density at the surface is higher than at the center whereas at large times, the density becomes uniform throughout the cross section. This latter prediction is in contrast to experimental measurements of the gapwise density distribution shown in Fig. 2 which indicates that the final density at the surface is higher than in the interior. This latter result, however, is apparently due to the presence of compressive stresses at the surface. On the other hand, the free-volume concept has been found to give no variation of density in the gapwise direction for the present quenching conditions. Therefore, the gapwise density distribution arises from the combined effects of the free-volume variation during rapid cooling and the thermal stresses, with the final distribution being the result of the competition of both factors. In Fig. 2a is also shown

t

1 .m L 0 0.5 1 .o

Y/ b

1 1.1501 I

0 0.5 1.0

Y/b Fig. 1 . Theoretical gapwise density distribution during quenching of PS (a) and PMMA (b) strips with thickness of 0.0038 and 0.006m respectively, from 130 to 23°C at dvferent times:

1-0 sec; 2-10 see; 3-100 see; 4-210 see for PS. 1-0 sec; 2-15 sec; 3-150 see; 4-300 see for PMMA.

the density distribution for PS strips quenched from an initial temperature of 170°C to a bath temperature of 23 or 0°C. It is seen that a lower bath temperature introduces a slightly lower density. At a low bath temperature the difference between Tg and the bath temperature is large and the cooling rate when a polymer passes through Tg is high. Thus, in agreement with the rate theory (9) one would expect a higher specific volume or a lower density. In other words, the molecules pack better at a higher bath temperature. On the other hand, the initial specimen temperature has not been found to be an important factor affecting density. This is in accordance with the present modeling of density. That is, at sufficiently high temperatures, the response of the specific volume to the temperature decrease is almost instantaneous due to a small

274 POLYMER ENGINEERING AND SCIENCE, MID-APRIL, 1985, Vol. 25, No. 5


l . l g O r

1 .a44 L 1

0 05 1.0

Y'b

l o O0' 1 0

1.185 0 0.5 1.0

Y h Fig 2. (a) Experimental gapwise density distribution of PS strips with thickness of 0.0038m quenched from 170 to 23°C (curve I) or to 0°C (curve 2) 24 hours after quenching. (b) Experimental gupwise density distribution of PMMA strips with thickness of 0.006m quenched from 170 to 23"C, 8.5 hours (curve 1 ) and 24 hours (curve 2) after quenching.

relaxation time for volume relaxation. On the other hand, increasing the specimen thickness is seen to slightly increase the density in the interior but not at the surface. Moreover, from Fig. 2b it follows that the density distribution curve is shifting up- ward with elapsed time after quenching. This effect is due to aging. In particular, Fig. 3 shows the experimental and theoretical results for aging in quenched PMMA strips. The theoretical rate of density increase with elapsed time after quenching is seen to be lower than experimentally observed. Unlike in the case of PMMA strips, it is observed that the density variation for PS strips was practically absent after 8 h of elapsed time.

The effect of molding conditions upon the de- pendence of the gapwise-averaged density on distance from the gate can be seen from Fig. 4 . In

CL- t

11601 1

- 5 0 5

logt, hour Fig. 3. Theoretical (curves) and experimental (symbols) gapwise atieraged density tis. ehpsed time after quenching 0.0028m (curve I , 0) und 0.006m (curve 2, A) thick PMMA quenched from 130 to 23 ' C.

particular, increasing the melt temperature or packing pressure noticeably increases the density (Figs. 4a and 4b) whereas the flow rate is seen to have little effect (F ig . 4c) . Evidently, at high melt temperature or packing pressure more melt can be forced through the gate into the cavity, leading to higher density. Further, it is noted that at low packing pressures and melt temperatures the density is found to be uniform in the longitudinal direction. However, the density does depend on the distance from the gate at the highest packing pressure or melt temperature with the density being higher at the gate apparently due to a high degree of packing there. In particular, the pressure during the packing stage has been found to decay toward the cavity end. In all cases, it is noted that the maximal variation of density is no more than 0.12 percent.

The level of the packing pressure has been found to highly affect the frozen-in gapwise birefringence distribution in molded strips (F ig . 5). In particular, major changes have been observed near the gate (F ig . 5 a ) where there appear two maxima of birefringence across the gap-one near the surface and the other in the core region. Evidently, the packing pressure is responsible for the appearance of this second maximum. In particular, whereas the position of the maximum near the surface practically does not vary with the packing pressure, its height significantly increases. On the other hand, position of the second maximum slightly moves toward the center with increasing packing pressure and its height also increases. The appearance of this second maximum can be explained as follows. Whereas the birefringence maximum near the wall is formed

POLYMER ENGINEERING AND SCIENCE, MID-APRIL, 1985, Vol. 25, NO. 5 275

1.065

% * 0, 1 .Ob6

4'

1.063

t- 6 (a)

4 ?2 C

I a

- a Y 2

-a -

0

0 --

1

0 0.5 1.0

1 ,065

m

Q-

1.W6 li

1 ,063

X/l (b)

-

3 C -\ I

-

--

Y/b I Fig. 5. Gapwise distribution of birefringence An at cross section

1.065-

% \o

Q-

m 1.066

1 .Oh3

X / l presstires: 0 -200; A -500; 0 -1000 p s i . T,, = 227"C, Q = 36 cin'/scc, T, = 30' C.

(C)

during cavity filling, after the cavity is filled the cooling continues and material in the cavity undergoes contraction, creating space for additional material. If the packing pressure is imposed after filling, additional material is supplied. Accordingly, the deformation process in the material continues. This deformation process, however, proceeds under high stresses which in turn introduce higher orientation. The layers adjacent to the frozen-in surface layer have high relaxation times due to a further decrease in temperature whereas the material in the core region is still hot. Development of normal stresses in the layers adjacent to the

0 0.5 1.0 frozen-in surface layer will be retarded in compar-

-

J

located far from the gate (Fig. 5b). However, this effect is difficult to interpret due to the closeness of this cross section to the cavity end.

Shown in Fig. 6 are typical pressure-time traces obtained by two pressure transducers located in the cavity. In particular, three stages of pressure change can be distinguished in thisfigure. The first stage corresponds to the cavity filling characterized by a fast rise in pressure. The second and third stages correspond to packing and cooling, respectively. These experimental pressure-time traces together with the simulated temperature field have been used to calculate volume shrinkage.

The effect of molding conditions upon shrinkage in the molded PS strips are shown in Fig. 7, from which it is seen that shrinkage is mainly affected by melt temperature (Fig. 7u) and packing pressure (Fig. 7b). One might expect larger shrinkage at higher melt temperatures due to a higher volume contraction associated with a larger difference between melt and mold temperature. However, at high melt temperatures the viscosity is low and additional material can be supplied into the cavity during the packing stage. This additional material can offset the contraction in volume due to cooling. Thus, a dominant role by the packing pressure can lead to a situation in which shrinkage decreases with increasing melt temperature, as observed experimentally (Fig. 7u) . Further, the packing pressure increase is seen to be the most effective way of decreasing the volume shrinkage (Fig. 7b). On the other hand, flow rate has almost no effect on shrinkage, as can be noted from Fig. 7. Besides experimental data on shrinkage, Fig. 7 also shows results on shrinkage modeling. These latter results are found to be in quantitative agreement with the experiment. In particular, the maximal value of shrinkage in the PS molded strips is seen to be about 6 percent which is much higher than the density variation (<O. 12 percent). This indicates that shrinkage in injection molding is mainly associated with changes in geometrical dimensions of the molded part.

vi 3 -

1 -

CONCLUSIONS Measurements of the density distribution in

freely quenched PS and PMMA strips have been performed at different quenching conditions and


POLYMER ENGINEERING AND SCIENCE, MID-APRIL, 1985, Vol. 25, NO. 5

T I R E , I E C

Fig. 6. Pressure-time traces at two locations in the caoity corresponding to the two-flush mounted pressure transducers, located at distances of0.015in (ctime I ) and 0.105m (ciirue 2) from the gate. Processing conditions are: P = 500 psi, Q = 36 cm3/sec, To = 220"C, T, = 30°C.

1 I HI i 210 260

I #

0 rx, m P, psi

- 4 o---- -------__________ ----__ 1 =-----------

* I

I 1

0 30 60

Q , cd/sec Fig. 7. Experimental (daslzerl curve) and tlreoretical (solid curve) shrinkage os. inelt teinperatwe (a), packing pressure (h) , oolu- inetricflozc: rate (c)

(a) P = 500, Q = 36 cin3/sec, T, = 30°C. (1)) To = 220"C, Q = 72 cm3/sec, T, = 30°C. (c ) T,, = 200"C, P = 500 psi, T, = 30" C.

277

A. 1. Isayev and T. Hariharan

strip thicknesses by using a density gradient column. The density has been found to be higher at the surface than in the interior. This effect is apparently due to a dominant role by compressive residual stresses at the surface. Increasing the bath temperature or strip thickness has been found to decrease the final density whereas the initial temperature has essentially no effect. For PMMA strips, the density has been found to increase with elapsed time after quenching. For PS strips, practically no change in density has been observed for elapsed times larger than 8 h. Modeling the density has been performed by using a first-order rate theory for specific volume in conjunction with solving the transient one-dimensional heat-conduction equation with a convective heat-transfer boundary condition at the surface. In contrast to experiment, this simple modeling shows no gapwise distribution of density in the quenched strips at large times due to the fact that the modeling has not accounted for the development of residual stresses during quenching. On the other hand, the modeling has been found capable of describing an experimentally observed increasing density during the aging of PMMA strips.

Measurements of density, birefringence, and shrinkage have been carried out for molded PS strips. Increasing the packing pressure and melt temperature have been found to increase the density and decrease shrinkage with the flow rate having little effect. Maximal variation in the measured density and shrinkage have been 0.12 and 6 percent, respectively. A dramatic increase in the gapwise maximum of birefringence in the core region have been observed near the gate with increasing packing pressure. Modeling of shrinkage has been carried out by using the equation of state for specific volume and simultaneously solving the governing equations for one-dimensional mold filling followed by the one-dimensional transient conduction equation during the packing and cooling stages.

Predicted and experimental results for shrinkage have been found to be in quantitative agreement.

ACKNOWLEDGMENTS This work has been performed as part of the

Cornell Injection Molding Program which is supported by the NSF under Grant MEA 82-00743 and by an industrial consortium. It was completed after A. I. Isayev joined The University of Akron, where it was supported by the Polymer Engineering Program.

1.

2 3. 4.

5.

6.

- I .

8

9 10

11

12

13

14

1.5

16

17

18

REFERENCES I. I. Rubin, “Injection Molding,” p. 275. Wiley, New York,

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Volumetric effects in the injection molding of polymers