Transcript
Page 1: Computer simulations of crystallinity gradients developed in injection molding of slowly crystallizing polymers

Computer Simulations of Crystallinity Gradients Developed in Injection Molding of

Slowly Crystallizing Polymers

C. M. HSIUNG a n d M. CAKMAK*

Insti tute of Polymer Engineering College of Polymer Science a n d Polymer Engineering

University of A k r o n Akron , Ohio 44325-0301

A model is developed to simulate the crystallinity gradients developed in injection molding of slowly crystallizing polymers. In this model, effects of nonisothermal and stress-induced crystallization kinetics are taken into account through phenomenological relationships. Computer simulations included calcu- lations of the temperature, velocity, and pressure distributions as well as two dimensional crystallinity distributions in the final products. In addition, effects of various processing conditions: mold temperature, injection flow rate, and holding time are also included in the calculations. The crystallinity gradients obtained through computer simulations agree with the experimental re- sults obtained with poly (p-phenylene sulfide) under a variety of processing conditions.

INTRODUCTION culation of the distribution of both temperature and

omputer simulation efforts for variety of proc- C esses to incorporate crystallization generally suffer from lack of sound theoretical development to include the influence of stress field on the crystal- lization. There have been considerable experimental (1, 2) and simulation (3-6) efforts to investigate the thermomechanical aspects of mold filling of amor- phous polymers such as polystyrene. Some of these studies incorporated viscoelastic models into the simulation codes [7, 8). In a more recent study, Rigdahl ( 10) incorporated residual stress distribu- tion calculations into the calculation procedures.

Attempts to include crystallization into these models are rather limited: transient temperature and crystallinity profiles within a PET slab in contact with a cooling fluid were predicted (1 1). Sifleet (12), mathematically modeled the unsteady state heat transfer in a crystallizing polymer during quench- ing. He included temperature dependence of induc- tion times and kinetic processes, and was able to predict the final morphological distribution within the polymer. More recently, a model was proposed by Kamal (9) to treat the heat transfer with crystal- lization during injection molding process. This model incorporates experimentally determined crys- tallization kinetics parameters. It permitted the cal-

crystallinity in the molded parts. Later, (13) they adopted the White-Metzner ( 14) modification of Maxwell model as their viscoelastic model and non- isothermal crystallization model of Nakamura ( 15) to calculate the distribution of shear stress, normal stress, birefringence, crystallinity and tensile modu- lus. Most recently, Lafleur (16) and Kamal (17) pro- posed a structure-oriented model of the injection molding of viscoelastic crystalline polymer. They were able to describe moldability parameters such as pressure drop in the delivery system and cavity and melt front progression during cavity filling as well as predict the characteristics of the final product, such as residual stresses and crystallinity distribution (18).

All of these above efforts were concentrated on the polymers with fast crystallization characteristics and did not consider effects of stresses on the crystalliza- tion behavior. The new class of slowly crystallizing polymers including poly phenylene sulfide, poly aryl ether ketone family exhibit quite unique structural gradients along and across the flow directions. In our previous studies (19,20,29), we described the structures developed in three of these close poly- mers (PPS, PEEK, and PAEK) during the process of injection molding. The most interesting feature of these injection molded samples is that depending on processing conditions they can exhibit three main types of structural gradients: *To whom correspondence should be addressed.

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Computer Simulation of Crystallinity Gradients

a) uniformly amorphous (or possessing small crys- tallinities) parts at low mold temperature and high injection speed;

b) three layer structural gradient amorphous skin-stress crystallized ring layer-amorphous (or semi-crystalline) core at intermediate mold tempera- tures above the glass transition temperature or low injection speed;

c) uniformly crystalline parts obtained at mold temperatures well above the cold crystallization temperature where the maximum rate of thermal crystallization is expected and where structural de- velopment is primarily dominated by the thermally induced crystallization.

These structural features are a result of complex interplay between flow behavior and thermal effects such as fast cooling and crystallization. It is, there- fore, clear that in order to simulate the injection molding process of these slowly crystallizing poly- mers, we have to take into account not only the non-isothermal effects but also the stress effects on the crystallization kinetics.

MATHEMATICAL FORMULATION

Geometry

Here, a simple geometry of a slit mold of width W, length AL, and half-gap thickness H is considered. With its large width to thickness ratio, we can ne- glect the effect of transverse flow.

Governing Equations

The governing equations that consider the bal- ance of mass, momentum, and energy for polymer melt flow inside the slit mold can be represented as follows:

Continuity Equation:

Q = w J-:udy (1)

Momentum Balance:

and Energy Balance:

where Q represents the volumetric flow rate (typi- cally taken as a specified constant), x is the stream- wise direction, y the gapwise direction and t the time. Further, u denotes the velocity in the x- direction, 7 the shear viscosity, i. = I au /a y I the shear rate, P the pressure, T the temperature, with p , C, and k representing the melt density, specific heat and thermal conductivity.

Since we assume the fluid to be incompressible, Eq I represents a balance of mass, and because of the symmetry about the x-axis passing through the

midplane, we can rewrite it as:

h Q = 2 w / udy ( la)

0

Further, the inertia and gravity forces are ne- glected due to the large viscosities typical of poly- mer melts. And if we consider the flow to be lami- nar, then the Eq 2 gives the force in the streamwise direction between the viscous shear stress and the pressure gradient.

Finally, heat conduction along streamlines can be neglected in comparison with conduction across streamlines, thus Eq 3 indicate that the change in temperature as one follows a fluid particle is due to the net effect arising from the gapwise thermal con- duction and the viscous heating.

Boundary and Initial Conditions

At the mold inlet, we assume a constant melt temperature of To, and a fully developed velocity profile for a power law fluid in a slit die i.e.,:

s + 2 Q u(0, y. t ) = - - ( s + 1 ) [ 2 w h ) [' - ( '+I] (4b)

where s = 1 / u,, and un is the power law index.

would typically be required that at the cavity wall, Concerning conditions in the gapwise direction, it

U ( X, h , t ) = O

T ( X, h , t ) = T,

( 5 4

(5b)

and

where Eq 5a is the no-slip boundary condition. On the other hand, assuming symmetry at y = 0

about the x-axis (flow direction), it then follows that:

a u - ( x , O , t ) = O a Y

and aT -( x.0, t ) = 0 a Y

Finally, along the advancing melt front, the curva- ture and associated transverse flow due to fountain flow are neglected. The advancing melt front is con- sidered to be flat and progresses uniformly accord- ing to the mass balance equation ( E q la ) . Besides, the temperature along this front, x = xmf( t ) , is con- sidered to be uniform and equal to the calculated center line temperature at the streamwise location immediately upstream of the front (21). i.e.,:

and 2hw

A t = - - - - A x 9

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C. M . Hsiung and M . C a k m a k

Hence, we can model the fact that the polymer melt along the front is supplied by the hot core region (a concept from the fountain flow phenomenon),

The Effect of Crystallization

In Eq 3, if we consider the effects of crystalliza- tion, then the specific heat term should be ex- pressed as:

where C;, is the specific heat without considering crystallization, h, is the crystallization heat released by the sample, X, is the isothermal ultimate crystal- lization heat, and X is the fractional crystallinity i.e.,:

(9)

Hence, if we substitute Eq 8 into Eq 3, after some rearrangements, we get the new energy balance equation as following:

a2T pc; ( $ + u g ) - p , X, , [ at ax + u z ax) = k y ay + Ti2

Since rapid cooling occurs in the injection mold- ing process, we have to take this into account and use non-isothermal crystallization kinetics. Here we choose the Generalized Avrami equation (15):

where K = kT/", and k, is the isothermal rate con- stant, n, is the Avrami exponent, and tc, is the time to start crystallization.

As mentioned before, in slowly crystallizing poly- mers, stress induced crystallization plays a very important role in determining the structure develop- ment during injection molding process. In order to model the stress effects we rely on earlier experi- mental observations.

It has been observed qualitatively and to a limited extent quantitatively that the stress has an acceler- ating effect on the crystallization behavior. It re- duces the induction times and increases the rate of crystallization several orders of magnitude and de- pending on the type of stress field reduces the di- mensionality of the crystal growth to differing de- grees. In other words, it is well known that under no stress conditions three dimensional spherulitic be- havior occurs. When the stress is applied to the fluid, oriented structures such as shish-kebab, etc., which grow with lesser dimensional freedom are formed. Interplay between stress history and ther- mal history in non-isothermal process like injection molding essentially dictates the details of the final structure.

For polymers with relatively slow crystallization characteristics, we observed (19 ,ZO) that the effect of stress is more pronounced and clearly distin- guishable. Figure 1 schematically describes the thermal and stress field on the crystallization in a non-isothermal process.

Near the wall (Fig. 1 a) fast cooling takes place and stresses are generally high. This high stresses move the induction time envelope to shorter times and higher temperatures. The two curves do not inter- sect and as a result polymer vitrifies to form a glass. Figure l b describes the condition where slower cooling and lower stress result in intersection of these two curves and as a result crystallization starts. We call this type of crystallization under high stress and cooling rate: stress induced crystalliza- tion. This naturally involves the orientation of chains in the melt phase which accelerate the crystalliza- tion process. We have experimentally observed this behavior at the intermediate distances from the sur- face of the parts. If the polymer exhibits inherently very low crystallization behavior, core of the mate- rial may not crystallize at all and remain amor- phous. This is demonstrated in Fig. Ic where the CCT curve is basically that of under quiescent con- ditions and receded to the longer time regions which the cooling curve is not able to catch up. If, how- ever, sufficient thermal energy and time at eleva- ted temperatures is provided, the core can also crys- tallize. We observed this behavior at high mold temperatures.

These concepts were all taken into account in developing the model described below.

MATERIAL FUNCTIONS

Rheological Equation

Here a modified power law is used to include the dependence of power law index n, as function of shear rate, which is determined experimentally and incorporated into the calculations as a Look-Up- Table.

We choose to represent the effect of temperature on viscosity by an Arrhenius type expression:

tl"P t.ne tmne

sk,n shear zone core

(a) (b) ( C )

Fig. 1 . Relationship between cooling curve and continu- ous cooling transformation curve for material at three different positions in the mold.

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where T, = AE/R, A E is the activation energy of flow which can be determined experimentally, To is some reference temperature here we use our inlet melt temperature, and m, = TJ(T = To, + = 0).

Thermo-Physical Properties

Specific heat without considering crystallization C; was experimentally determined using a DuPont 9900 differential scanning calorimeter. This data was incorporated into calculation scheme as a Look- Up-Table. Thermal conductivity k is assumed to be constant 2.88 x 104erg/ cm s “C, and density p is assumed to be constant at 1.3 g/ cm3 (29).

Crystallization Kinetics

Induction Time t,

In order to determine when crystallization actu- ally starts, i.e., the values of t,,, we have to consider the induction time t,. Under non-isothermal condi- tions this is complicated. Sifleet (12) suggested that “during a non-isothermal process any amount of time spent at a temperature increased the relative completion of the induction time by the about of that time spent divided by the induction time at that temperature.” When this accumulated induction time factor (0) reaches 1.0, the induction time is assumed to be finished and crystallization begins. Mathematically, this is written as:

And crystallization start when 0 2 1.0. Note that t , is function of both temperature and shear stress i.e.,

Temperature dependence of this material function at quiescent state was reported by Sifleet (12). A parabolic functional form fits the data reasonably well. As for the effect of shear stress, Lagasse (26) and Chien (27) have reported some studies in polyethylene and PEEK, respectively (although they both only cover quite narrow a temperature range in the higher temperature part). Also, in the process of melt spinning, White and Spruiell (28) use a CCT (Continuous Cooling Transformation) curve to de- scribe the log t , vs. temperature relationship. They also considered the effect of stress on these curves. Since stress will increase both the crystallization kinetics and the equilibrium melting temperature, the CCT curve for the stressed melt will shift toward lower induction time and higher temperature com- pared to the quiescent state. Thus, for a give cooling rate, crystallization will occur at a higher tempera- ture for a stressed melt than for a quiescent one.

With all the studies mentioned above, mathemati- cally we can express the material function of induc- tion time as a parabolic function between log t , and temperature at quiescent state and shift toward lower induction time and higher temperature by the

ti( T, 7) .

influence of stress:

(134

(13b)

(13c)

2 log t,=logtrb+D(T-Tb)

T, = Tbq + 7 X E

log t rb = log t,,, - T X F

Again the constants involved in these expressions were estimated from literature.

Isothermal Ultimate Heat of Crystallization

This value was determined experimentally using differential scanning calorimetry as a function of temperature.

Non-isothermal rate constant K

Here we use a parabolic function to express the relation between log K and temperature at quies- cent state, the effect of shear stress is to shift this equation toward higher temperature and higher K values (15,22). Mathematically, we can use the fol- lowing equations to represent this effect:

( 1 4 4

(14b)

(14c)

2 log K = log K , - A( T - T,)

Tp = Tpq + 7 x B

logK, = logK,, + 7 x C

K , and Tp in the literature (23-25). The value of A was estimated from those data. Since there has been little quantitative description of the effects of shear stress on the crystallization kinetics of our material, the values of B and C were estimated (22).

Avrami Exponent n,

In general, higher Avrami exponents represent greater dimensionality in the crystal growth pro- cess. It has been experimentally observed that the morphology of injection molded crystalline polymer sample shows three dimensionally grown spherulitic structure in the low stressed core region, while the highly stressed skin layers are populated with crys- talline forms of grown with less than three dimen- sional freedom (e.g., shish-kebab, rod-like structures etc). Thus, in the stress induced crystallization ki- netics equations, this material function should de- pend on shear stress. Here, we use an arbitrary simple linear relationship between n, and r , with n, equal 3.0 at quiescent state ( r = 0) and as shear stress increase, n, decrease to approach 1.0 (Appendix A.3c).

COMPUTATION SCHEME

Filling Stage

With our simple slit geometry, finite difference method was found to be adequate. Here, however an implicit representation in time is used to allow for larger values in At.

It is well known that even in the mold filling stage polymer melt starts to cool rapidly on contact with the cold mold surface, which cause the frozen layer to form. This can be observed in our simulation

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model also. The regions less than a predetermined small velocity is assumed to be frozen and this de- termines the frozen layer thickness distribution at a given time interval.

Cooling Stage In the cooling stage the calculation time intervals

are progressively increased to save memory and computation time. In this stage, the calculation of temperature and crystallinity follows the same pro- cedure as the filling stage. Except that, heat conduc- tion in the streamwise direction is included to cover the large cooling effect coming from the cold wall at the end of the mold.

RESULTS AND DISCUSSION

The theory and computation scheme presented in the previous sections have been applied to the filling and cooling stage of the injection molding of PPS as an example.

Effect of Mold Temperature

In the slowly crystallizing polymers, because their crystallization half times are of the same order of magnitude as the regular cooling time encountered in a typical injection molding process, they can ex- hibit unique structural gradients along and across the flow directions after demolding ( 1 9 , 2 0 ) . The main purpose of this simulation work is to repro- duce the three characteristic structure gradients that were found in our injection molded aromatic semi- crystalline polymer such as Poly p phenylene sul- fide. Figure 2 shows our calculated crystallinity dis- tribution results after filling stage and one minute of holding time. Figure 3 shows the corresponding optical photomicrographs of injection molded PPS cut perpendicular to flow direction at middle of the dumbbell bars. Five different mold temperatures ranging from below Tg. between Tg and T, (cold crystallization temperature), and above T, were used at three different injection flow rates. The three characteristic structure gradients we found in our experiments are shown here: at high injection flow rate (23.2 cm3/s) and low mold temperatures (T, = 20 and 70°C) the crystallinities are minimal and we get essentially amorphous samples. At highest mold temperature of 200°C which is near the material's maximum crystallization rate temperature (about 19O"C), highly crystalline parts with gentle crys- tallinity gradients from skin (lowest crystallinity) to core (highest crystallinity) are obtained. This latter type of structural gradient is typical of what has been observed in fast crystallizing polymers such as polyethylene, polypropylene and poly(oxymethy1- ene) (30-34) (Here we define the fast crystallizing polymers loosely as those polymers that cannot be quenched into amorphous form under normal proc- essing conditions). At intermediate mold tempera- tures and/or low injection flow rate, the three-layer- structures are observed in these slowly crystallizing

polymers. These are in two types: at lower mold temperatures we obtain amorphous skin-stress crys- tallized layer-amorphous core structure, and at higher mold temperature (about 150°C) the core region become thermally crystallized, and we obtain amorphous skin-stress crystallized layer-thermally crystallized core structure. In addition, we observed that the lowering of injection flow rate enhances the formation and growth of the intermediate stress crystallized layer.

Now we take a more quantitative look at our simu- lation results: Shown in Fig. 4a is the gapwise crys- tallinity distribution measured by DSC at different distances from the gate. The optical photomicro- graphs of these samples are shown in Fig 4c (29). Stress induced crystallized layer forms near the gate and extends along the flow direction for more than half of the sample's length. The area it covers gradu-

Tw= 2OoC

Tw= 7OoC

Tw=l 15OC

Tw=1500C

Tw=2000C

Q = 2 cm3/sec Q = 5.2crn3/sec

C. M . Hs iung a n d M . Cakrnak

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I

Fig. 2. Calculated crystallinity distribution of injection molded PPS after one minute holding time, showing the effect of mold temperature.

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30

Fig. 3. Optical photomicrographs of HMWPPS cut per- pendicular to flow direction at # 3 location showing the effects of mold temperature and injection speed.

(HMWPPS INJECTION FLOW RATE -5.2cm3/sac) - (Lorp Dumb bdl. Location 13)

30

2 5

2 0 s b .e 15

v

.- - - 0 - F 10 0

5

HMWPPS Tw=20'C

INJECTION FLOW RATE = 5.2cm'/sec

0-0 Loc.01 *--. Loc.02 LOC.03 LOC.#S

I 0 10 2 0 30 40 50 60 70 80 90 100

surface center Normolized Distance from Surface (W)

Fig. 4a. Crystallinity variation along the gapwise direc- tion at various distances from gate. (HMWPPS, T, = 20"C, injectionflow rate = 5.2 cm3/s).

ally decreases after # 3 location. Our simulation re- sult for this case is shown in Fig. 2, and a slicing of that graph perpendicular to flow direction which gives gapwise crystallinity distribution. This is

30

25 Tw=20"C

20

1 5

10

5

0

t Q=5.2cm3/sec

0-0 L o c . # l 0--. Loc.#2 A-.A LOc.#3 &-A L o c . # 5

-5

SURFACE CORE 0 10 20 3 0 40 50 60 70 80 90 1 0 0

NORMALIZED DISTANCE FROM SURFACE

Fig. 4b. Calculated gapwise crystallinity distribution of PPS at various distances from gate. (T, = 20"C, injection flow rate = 5.2 cm3/s).

Fig. 4c. Optical photomicrographs of HMWPPS cut per- pendicular to flow direction at # 1 - # 5 locations.

shown in Fig. 4b. A comparison of Fig. 4a and Fig. 4b shows striking similarity.

Here, we compare the results obtained under dif- ferent injection flow rates. The experimentally measured gapwise crystallinity distributions are shown in Fig. 5a for samples molded at 5.2 cm3/s (corresponding photomicrographs are shown in the middle column of Fig. 3) . Simulation results are shown in Fig. 2 (middle column). Slicing of these graphs perpendicular to flow direction provides the gapwise crystallinity distribution shown in Fig. 5b. As shown in these figures: our model reproduces the

t

0-0 Tm=20C 0-0 Tm=70C A-A Trn-l l5C A-A Tm=150C 0-0 Tm=200C

0 10 20 30 40 50 60 7 0 80 90 100 surface center

NORMALIZED DISTANCE FROM SURFACE (%)

Fig. 5a. Crystallinity variation along the gapwise direc- tion of PPS at # 3 location. (injection flow rate= 5.2 cm3 / sj.

3 5 ,

0-0 Tw=20O0C A-A Tw=150°C

A-A Tw=115'C *-* TW=7O'C 0-0 Tw=20°C

-5 1 ' 0 10 20 30 40 50 60 70 80 90 1 0 0

sur face core

NORMALIZED DISTANCE FROM SURFACE

Fig. 5b. Calculated gapwise crystallinity distribution of PPS at # 3 location. (injectionflow rate = 5.2 cm3/s).

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C. M. Hsiung and M . Cakmak

most important features at each mold temperature. The three layer structure start from the 20°C with amorphous core (7% in absolute crystallinity is about the lowest we could obtain in PPS). When the mold temperature increases, the intermediate stress crystallized layer become higher in crystallinity and broader in width, and its location shifts toward the surface. When the mold temperature rise above 1 15 "C, the ordinary thermally induced crystalliza- tion (crystallization in the absence of stress) start to become effective in the core region. Finally when the mold temperature reaches 200°C, a uniformly crystalline sample is formed.

The gapwise crystallinity distributions of the sam- ple molded at higher injection flow rate (23.2 cm3/s) are shown in the Fig. 6a. Their corresponding pho- tomicrographs are shown in Fig. 7. Simulation re- sults are shown in the right hand side column of Fig. 2. Widthwise slices of these graphs give us the corresponding gapwise crystallinity distribution in Fig. 6b. A comparison of Fig. 6a and Fig. 6b shows that: at 20 and 70°C both results give uniformly amorphous samples. At 200°C. both results give uniformly crystalline samples. Finally, at the inter- mediate mold temperatures of 115 and 150°C. both

30 t

5 t

Wmb MI, Locdbn #3)

0-0 Trn=20C 0-0 Tm=70C A-A Tm-115C A-A Tm-15OC 0-0 Trn=200C

Fig. 7. Optical photomicrographs of HMWPPS cut per- pendicular to flow direction at five different distances from gate showing the effects of mold temperature.

simulation and experimental results give the char- acteristic three-layer structures. However, at 1 15°C mold temperature, experimentally observed crys- 0

surface center tallinity levels are higher than those predicted by the model calculations. This is expected considering the severity of some of the assumptions we made

0 10 20 30 40 50 60 70 80 90 100

NORMALIZED DISTANCE FROM SURFACE (%)

Fig. 6a. Crystallinity variation along the gapwise direc- tion of PPS at # 3 location. (injection flow rate= 23.2 building the model. cm3/s].

Effect of Injection Flow Rate

The injection flow rate is an important parameter 35 I

\

\

Q=23.2cm3/sec

0--0 TW=ZOOT A-A Tw=150"C

A-A Tw=11S0C 0-0 T w ~ 7 0 - C 0-0 Tw=2O0C

in that it can be changed rapidly and it can affect the final product properties dramatically as we have shown in earlier publications (19,20). There are sev- eral factors that are responsible for the effects of injection flow rate. The first one is the formation of frozen layer during the filling stage, which reduce the effective channel thickness and increase the shear rate in the remaining fluid core region. Be- sides, the stresses are increased with slower filling due to increase of viscosity be longer cooling during flow. Additionally, mold filling time increases at lower injection flow rates. Since this is also the time that polymer is flowing, reducing injection flow rate can place the polymer melt longer shearing time.

Figure 8 gives the simulation results of crys-

-5 0 10 20 30 40 50 60 70 80 90 100

surface core

NORMALIZED DISTANCE FROM SURFACE

Fig. 6b. Calculated gapwise crystallinity distribution of PPS at # 3 location. (injection flow rate = 23.2 crn3/s).

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Fig. 8. Calculated crystallinity distribution of injection molded PPS after one minute holding time, showing the effect of injection flow rate.

tallinity distribution at the end of one min holding time for samples molded under six different injec- tion flow rates (from 1 to 23.2 cm3/s1 and three different mold temperatures (20, 115, and 150°C). The most obvious effect is that the stress crystal- lized layer can be greatly enhanced be reducing the injection flow rate (increasing in crystallinity and broadening in thickness]. This effect is more pro- nounced at lower mold temperatures. As the mold temperature increases, thermally induced crystal- lization (crystallization under no stress but under non-isothermal conditions) increasingly take part in the crystallinity development.

Figure 9 shows the cross-sectional view of a series of samples molded at different injection flow rates. It

is very similar to our simulation results given in top row of Fig. 8. Both figures show uniformly amor- phous sample at the highest injection flow rate and well defined three layer structure with thick stress crystallized layer at the lowest injection flow rate. Figure 1 O a shows another experimental result about

Fig. 9. Optical photomicrographs of HMWPPS cut per- pendicular to flow direction at # 3 location showing the effect of injection speed.

Fig. 10a. Optical photomicrographs of HMWPPS cut per- pendicular to flow direction at # 1 - # 5 locations, showing the effect of injection speed mold temperature = 20°C.

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C. M . Hsiung and M. Cakrnak

Tw= 115'C Q= 5 2 cm31sec

Holding Osec lOsec 30sec 1 mrn 2min 4mtn Time -- - ---- -

Fig. lob. Optical photomicrographs of HMWPPS cut per- pendicular tof low direction at # 1 - # 5 locations, showing the effect of injection speed mold temperature = 15°C.

the effect of injection flow rate at low mold tempera- ture of 20°C. Here we note that with lower injection flow rate, we are not only getting higher crystallinity and thicker crystallized layer, the length of this crystalline layer increases along the flow direction. Finally, Fig. I O b shows similar experimental results obtained at a higher mold temperature (150°C). As indicated earlier, the effect of injection flow rate does not appear to be significant here.

Effect of Holding Time

Holding time is defined as the period between the time polymer melt ceases to flow and the time that part is ejected from the mold. The effect of holding time become appreciable only at high mold temper- atures where thermally induced crystallization dom- inates in the formation of structure. At 20 and 70°C mold temperatures the cooling rate is so fast that no appreciable crystallinity develops even in the core regions after the filling stage. On the other hand, at 200°C the thermal crystallization rate is so fast that material reaches its ultimate crystallinity within the first few seconds of cessation of flow. Only at the mold temperature of 115 and 150°C (esp. 150°C) holding time plays a role in changing the ultimate product properties as we have shown in earlier publication (19).

The simulation results for crystallinity distribu- tion at holding times range from 0 to 4 min in Figs. 1 1 and 12. At 115"C, Fig. 1 I shows that after a holding of about 1 min, virtually no further crys- tallization takes place. Because the temperature of the material in the mold has cooled to a temperature that crystallization rate is undetectable. On the other hand, at 150°C (Fig. I2), crystallization can proceed until crystallinity finally reaches its maximum value. Here this takes about 4 min. As for the influ-

T,= 1 1 5 ' ~ Q= 23.2 cm3/sec

Holding 0 sec 10 sec 30 sec 1 min 2min 4 rnin I--- 7--

Time -

I , ; I

Fig. 1 1 . Calculated crystallinity distribution of injection molded PPS after various holding time. (T, = 11 5").

Fig. 12. Calculated crystallinity distribution of injection molded PPS after various holding time. [T, = 150°C).

ence of injection flow rate, it was found that samples molded under lower injection flow rate already ex- hibit some crystallinity at the end of filling stage. On the other hand, the crystallinity of those samples molded under high injection flow rate show no crys- tallinity at all at the end of filling stage. Crystallinity develops first near the wall of the samples at 150°C, then proceeds inwards towards the core. Figure 13 shows the cross-sectional view of samples molded at different holding times. Figure 14a give the corre- sponding experimentally measured crystallinity dis- tribution. Slicing of our graphs in Fig. 12 at their mid point give the crystallinity distribution shown in Fig. 14b . Both results confirm the same growth pattern described above.

Frozen Layer As we mentioned above, the concept of frozen

layer is crucial in explaining the influence of injec- tion flow rate on the structure development during injection molding process. PPS exhibits a Tg at about 85°C. This was not used as the defining factor to determine the location of the frozen layer boundary

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80

Fig, 13. Optical photomicrographs of PPS cut perpendic- ular topow direction at # 3 location showing the effect of holding time.

Tw=ZQ'C R -0.1

.

35

30

Y 25 2 2 20

% .- -

0 15

10

5

/A-A MMWPPS Tm= 1 50°C HJECTION FLOW RATE - 23.~crnJ/nc

Location Y3

A

0-0 lmin @-• 4min A-A 20min

0 '0-0-0

10 20 30 40 50 60 70 8 0 90 100 face Center

Normalized Distance from Surface (%)

Fig. 14a. Crystallinity variation along gapwise direction of MMWPPS at # 3 tocation showing the effect of holding time.

3 5 r I Tw=150°C

Q=23.2cm3/aec holding time H-H 4 min.

0-0 2 rnin.

A-A 1 min. A-A 30 sec.

0-0 10 sec. 0-0 0 sec.

- 5 1 ' ' ' ' ' ' ' ' ' 1 0 10 20 30 40 5 0 60 70 80 90 100

surface core

NORMALIZED DISTANCE FROM SURFACE

Fig. 14b. Calculated gapwise crystallinity distribution of PPS at # 3 location after various holding time.

with molten material. This was determined by the R factor (limiting velocity) which was set to 0.1 cm/s. This is necessitated due to the fact that the crystal- lization that occur at much higher temperatures causes the cessation of flow. So in our simulation this definition was not left to whether the polymer

vitrify or crystallize but it was left to a critical limit- ing velocity [ R ) which determines the location of the boundary between the melt and frozen region. Here we present some simulation results on the progres- sion of the frozen layers during the filling stage under different mold temperatures and injection flow rates. Figure 15 shows the results at 20 and 115°C mold temperatures and three different 9. In each graph, we show the corresponding regions of frozen layers as the melt fronts reach four different loca- tions during the mold filling process. As expected, the frozen layer thickens during the filling stage. These data also indicate that as the mold tempera- ture increases the frozen layer becomes thinner. This is a result of slower cooling rates experienced at higher mold temperatures. Figure 16 compares the final frozen layer locations at the end of filling stage for three different flow rates. A s the injection

Fig. 15. Progression of frozen layer at four dvferent melt front positions.

z 0

-20 0 20 40 60 80 100 120 140 160 180 200 220 gate end

STREAM WISE DIRECTION

Fig. 16. Loci of frozen layer a tpn i sh of moldplling.

POLYMER ENGJNEERING AND SCIENCE, MID-OCTOBER 1991, Vol. 31, NO. 19 1301

Page 11: Computer simulations of crystallinity gradients developed in injection molding of slowly crystallizing polymers

C. M . Hsiung and M. Cakrnak

flow rate is increased, frozen layers become thinner. This is due to longer cooling time during filling at the lower flow rates. As the frozen layer thickens during filling stage, the velocity profile change and their corresponding locus of maximum shear rate also moves. This is shown in Fig. 17 where the loci of the maximum shear rate are shown with curves drawn with symbols. However, since crystallization is induced by the shear stress (not the shear rate), the locus of maximum shear stress is of interest here. Our formulation shows that shear stress at each x position (along the flow direction) increases monotonically with y (the gapwise distance from the center toward the wall). Hence, the loci of maxi- mum shear stress, in effect, is at the boundaries of frozen layers. In Fig. 18, we overlap the results of Fig. 15 with their corresponding final crystallinity distributions. Since the mold temperature is low,

FROZEN LAYER & LOCUS OF MAXIMUM SHEAR RATE - 100

90 80

6 70 I- y 60

50 40

30 0 20

10

Q

WAU.

Tw=ZO'C Q4.2 R =0.1

___L-

260 -

300 gate end

STREAM WISE DIRECTION

Fig. 17. Loci of frozen layer and maximum shear rate when melt front reach four different positions.

Fig. 18. Progression of frozen layer and the resulting crystallinity distribution in thepnal product.

stress induced crystallization is the dominating fac- tor. In these figures, the main crystalline zone al- ways sit inside a region through which the progress- ing frozen layer had passed during the filling stage.

Thermomechanical History

Here we list some characteristic velocity, pressure and temperature fields that prevail in our simula- tion study. Figure 19 shows 6 velocity distributions at the end of filling stage. In Fig. 19 the lowest contour level were set equal to 0.7 cm/s, which is our R value in defining the boundary of the frozen layer. Hence, we can clearly see frozen layer regions in those graphs.When the mold temperature is in- creased, there are two factors that will affect the velocity profile. One is the lowering of the viscosity which increases the velocity and the other is the reduction of the frozen layer thickness at a given position (from the gate), time and injection speed. As Fig. 19 indicates that the influence of frozen layer seems to overcome that of the reduced viscos- ity, i.e., as mold temperature increases, velocity de- creases slightly.

Figure 20 shows the cavity pressure distribution change with the progression of melt front in the cavity. As expected, increase in both flow rate and mold temperature causes the pressure to decrease. But the effect of flow rate decreases with increasing mold temperature, while the effect of mold tempera- ture decrease with increasing flow rate.

Finally, we list some temperature distribution at the end of filling stage. Figure 21 show the tempera-

Tw= 2OoC

r 1

Fig, 19. Calculated velocity distribution of PPS melt in- side the mold at finish of f i l l ing stage.

1382 POLYMER ENGINEERING AND SCIENCE, MID-OCTOBER 1991, YO/. 31, NO. 19

Page 12: Computer simulations of crystallinity gradients developed in injection molding of slowly crystallizing polymers

Computer Simulation of Crystallinity Gradients

Q = 2 cm3/sec Q = 5.2 cm3/sec Q = 23 2 cm3isec

I I -- - - - - -

Fig. 20. Calculated pressure profile along flow direction at various melt front location.

Q = 5.2 cm3/sec Holding Time 0

Tw= 2OoC Tw= 11 5'C Tw= 2OO0C

Fig. 21. Calculated temperature profile of PPS material in the mold atfinish of filling stage. fQ = 5.2 cm3/s)

ture distribution at different mold temperatures for the injection flow rate of 5.2 cm3/s. Figure 21 shows that increasing mold temperature reduces the cool- ing rate during the mold filling stage. Finally, Fig. 22 shows that at mold temperature of 20°C and at a holding time of 10 sec. the material cools to below its glass transition temperature.

model that can predict not only the thermo- mechanical history that prevail in the injection molding process but also the resulting structure de- velopment specifically crystallinity gradients inside the final products. More specifically, we try to repro- duce the three types of structural gradients found in our injection molded products of the slowly crystal- lizing aromatic polycondensates esp. PPS. In these materials, stress induced crystallization plays a cru- cial role in determining the final structure of the injection molded products.

The agreement between simulated and the experi- mentally determined crystallinity gradients are gen- erally good. Our model successfully predicted the effect of major operating variables such as injection flow rate, mold temperature, and holding time on the crystallinity gradients.

We also found that the stress crystallized region is situated in the area swept by the frozen layer boundary during the filling stage.

AF'PENDII: Xaterial fnnctions

0 CONCLUSION '-

The main purpose of this simulation work was to construct a structure-oriented injection molding

Tw= 2OoC Q 5.2 cm3/sec

Holdlng T h e = 0 Haldlng Tlme = 5 seC Holdlng Tlme = 10 pec

1- -- -----7

Fig. 22. Calculated temperature profile of PPS material in the mold at various holding time. (Q = 5.2 cm3/s , T, = 20°C).

0 . 7 }

o.6 t U \ 0

0- 0

U.

00-o,, 0.5 I 1E-41E-31E-21E-1 1 10 100 1000 1E4 1E5

SHEAR RATE (SEC-') Fig. A l . Dependence of power law index as function of shear rate.

POLYMER ENGINEERING AND SCIENCE, MID-OCTOBER 1991, Yo/. 31, NO. 19 1383

Page 13: Computer simulations of crystallinity gradients developed in injection molding of slowly crystallizing polymers

C. M . Hsiung and M . Cakmak

A-2 Thermal Physical Proper t ies :

(a) thermal conduct iv i ty k = 2.88 x 10' (erg/cm s 6 . C )

(b) dens i ty p = 1.3 (g/cms)

(c) s p e c i f i c heat v i thout considering c r y s t a l l i e a t l o n Cp' :

3E7 I

(c) Avrmi exponent nc :

2E7 1

A N 0 I 1 E 7 ' 1

0 50 100 150 200 250 300 350

TEMPERATURE ("C) Fig. A2. Dependence of Cp' o n temperature.

A- 3 Crystallization kmetxcs

(a) isothermal ultimate heat of crystalllzatlon A m

1 0 0 , I n

DSC

2

TEMPERATURE ( " C ) Fig. A3. Dependence of isothermal ultimate hea t of crys- allization o n temperature and a typical DSC trace of PPS.

(b) rate constant R:

l a g K = log H, - A II (T -T,)2 ~ (13%) T, i T,, + i I B ( 1 3 ~

( 1 3 4 log Kp = log K,, + i x C

A = 4 . 5 I 10.' ( ' C - 2 )

Tpq = 19O'C

K p q i 4.1 ( ~ 4 . 1 )

B = 3.0 * 10-5 ('C cm>/dyne)

c = 9 5 I 10-7 (cto/dyne)

1.000 I

0.800

0.200

0.000

TEMPERATURE( "C) Fig. A4. Calculated rate constant K as f unc t ion of tern. perature a n d shear stress a n d a typical DSC trace of PPS.

0.0 ' 0.0 S.OE7 1 .OEB

SHEAR STRESS (dyne/cm?

(d) Induct ion time tI:

log tI= log t I b + D x (T - Tb)? ~ (14a)

Tb = Tbq + 7 x E

log tIb= log tIbq- 7 x F (14b)

(14c)

D = 1.0 10-4 ( s c - 2 )

Tbp = 19o'c

tIbq = 10 sec

E = 3.0 x 10-5 ('C cms/dyne)

F = 2.6 x 10-8 (cmz/dyne) Fig. A 5 . Dependence of Aurarni exponent o n shear stress.

ACKNOWLEDGMENT

This work was supported by U.S. National Science Foundation Grant # DMC 8858303.

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