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<ul><li><p>Computer Simulations of Crystallinity Gradients Developed in Injection Molding of </p><p>Slowly Crystallizing Polymers </p><p>C. M. HSIUNG a n d M. CAKMAK* </p><p>Insti tute of Polymer Engineering College of Polymer Science a n d Polymer Engineering </p><p>University of A k r o n Akron , Ohio 44325-0301 </p><p>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. </p><p>INTRODUCTION culation of the distribution of both temperature and </p><p>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. </p><p>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- </p><p>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). </p><p>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. </p><p>1372 POLYMER ENGINEERING AND SCIENCE, MID-OCTOBER 1991, Vol. 31, No. 19 </p></li><li><p>Computer Simulation of Crystallinity Gradients </p><p>a) uniformly amorphous (or possessing small crys- tallinities) parts at low mold temperature and high injection speed; </p><p>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; </p><p>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. </p><p>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. </p><p>MATHEMATICAL FORMULATION </p><p>Geometry </p><p>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. </p><p>Governing Equations </p><p>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: </p><p>Continuity Equation: </p><p>Q = w J-:udy (1) </p><p>Momentum Balance: </p><p>and Energy Balance: </p><p>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. </p><p>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 </p><p>midplane, we can rewrite it as: </p><p>h Q = 2 w / udy ( la) </p><p>0 </p><p>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. </p><p>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. </p><p>Boundary and Initial Conditions </p><p>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.,: </p><p>s + 2 Q u(0, y. t ) = - - ( s + 1 ) [ 2 w h ) [' - ( '+I] (4b) </p><p>where s = 1 / u,, and un is the power law index. </p><p>would typically be required that at the cavity wall, Concerning conditions in the gapwise direction, it </p><p>U ( X, h , t ) = O </p><p>T ( X, h , t ) = T, </p><p>( 5 4 </p><p>(5b) </p><p>and </p><p>where Eq 5a is the no-slip boundary condition. On the other hand, assuming symmetry at y = 0 </p><p>about the x-axis (flow direction), it then follows that: </p><p>a u - ( x , O , t ) = O a Y </p><p>and aT -( x.0, t ) = 0 a Y </p><p>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.,: </p><p>and 2hw </p><p>A t = - - - - A x 9 </p><p>1373 POLYMER ENGINEERING AND SCIENCE, MID-OCTOBER 1991, VOI. 31, NO. 19 </p></li><li><p>C. M . Hsiung and M . C a k m a k </p><p>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), </p><p>The Effect of Crystallization </p><p>In Eq 3, if we consider the effects of crystalliza- tion, then the specific heat term should be ex- pressed as: </p><p>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.,: </p><p>(9) </p><p>Hence, if we substitute Eq 8 into Eq 3, after some rearrangements, we get the new energy balance equation as following: </p><p>a2T pc; ( $ + u g ) - p , X, , [ at ax + u z ax) = k y ay + Ti2 </p><p>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): </p><p>where K = kT/", and k, is the isothermal rate con- stant, n, is the Avrami exponent, and tc, is the time to start crystallization. </p><p>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. </p><p>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. </p><p>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. </p><p>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. </p><p>These concepts were all taken into account in developing the model described below. </p><p>MATERIAL FUNCTIONS </p><p>Rheological Equation </p><p>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. </p><p>We choose to represent the effect of temperature on viscosity by an Arrhenius type expression: </p><p>tl"P t.ne tmne </p><p>sk,n shear zone core </p><p>(a) (b) ( C ) </p><p>Fig. 1 . Relationship between cooling curve and continu- ous cooling transformation curve for material at three different positions in the mold. </p><p>1374 POLYMER ENGINEERING AND SCIENCE, MID-OCTOBER 1991, VOl. 31, NO. 19 </p></li><li><p>Computer Simulation of Crystallinity Gradients </p><p>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 </p><p>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). </p><p>Crystallization Kinetics </p><p>Induction Time t, </p><p>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: </p><p>And crystallization start when 0 2 1.0. Note that t , is function of both temperature and shear stress i.e., </p><p>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. </p><p>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 hig...</p></li></ul>