## Abstract

Injection molded semi-crystalline thermoplastic parts show variable morphologies across their thickness. Process parameters such as injection speed, mold temperature, and melt temperature, play an important role in forming these morphologies. The heat and shear history has a great effect on the crystallization process of the semi-crystalline plastics. In this study, different available crystallization models were used to predict the crystallinity distribution, and capabilities of the models compared. Based on the results from these models, a more realistic model, which considers stress relaxation during the crystallization process, is proposed.

Original language | English (US) |
---|---|

Pages (from-to) | 497-506 |

Number of pages | 10 |

Journal | Journal of Reinforced Plastics and Composites |

Volume | 21 |

Issue number | 6 |

DOIs | |

State | Published - 2002 |

## All Science Journal Classification (ASJC) codes

- Ceramics and Composites
- Mechanics of Materials
- Mechanical Engineering
- Polymers and Plastics
- Materials Chemistry

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**On the prediction of crystallinity distribution in injection molded semi-crystalline thermoplastics**. In: Journal of Reinforced Plastics and Composites. 2002 ; Vol. 21, No. 6. pp. 497-506.

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**On the prediction of crystallinity distribution in injection molded semi-crystalline thermoplastics.** / Guo, Jianxin; Narh, Kwabena A.

Research output: Contribution to journal › Article › peer-review

TY - JOUR

T1 - On the prediction of crystallinity distribution in injection molded semi-crystalline thermoplastics

AU - Guo, Jianxin

AU - Narh, Kwabena A.

N1 - Funding Information: Guo Jianxin Mechanical Engineering Department, New Jersey Institute of Technology, University Heights, Newark, NJ 07102 Narh Kwabena A. Mechanical Engineering Department, New Jersey Institute of Technology, University Heights, Newark, NJ 07102 04 2002 21 6 497 506 Injection molded semi-crystalline thermoplastic parts show variable morphologies across their thickness. Process parameters such as injection speed, mold temperature, and melt temperature, play an important role in forming these morphologies. The heat and shear history has a great effect on the crystallization process of the semi-crystalline plastics. In this study, different available crystallization models were used to predict the crystallinity distribution, and capabilities of the models compared. Based on the results from these models, a more realistic model, which considers stress relaxation during the crystallization process, is proposed. sagemeta-type Journal Article search-text On the Prediction of Crystallinity Distribution in Injection Molded Semi-Crystalline Thermoplastics JIANXIN GUO* AND KWABENA A. NARH** Mechanical Engineering Department, New Jersey Institute of Technology, University Heights, Newark, NJ 07102 ABSTRACT: Injection molded semi-crystalline thermoplastic parts show variable morphologies across their thickness. Process parameters such as injection speed, mold temperature, and melt temperature, play an important role in forming these morphologies. The heat and shear history has a great effect on the crystallization process of the semi-crystalline plastics. In this study, different available crystallization models were used to predict the crystallinity distribution, and capabilities of the models compared. Based on the results from these models, a more realistic model, which considers stress relaxation during the crystallization process, is proposed. INTRODUCTION SEMI-CRYSTALLINE POLYMERS EXHIBIT unique structure gradients in the injec- tion molded parts. Processing parameters such as melt and mold temperatures, injection speed, holding time, as well as the geometry of the mold have a signifi- cant effect on the morphology gradient [12]. Although there have been several studies to simulate the injection molding process, efforts to simulate the injection molding of semi-crystalline polymers are rather limited because of lack of realistic models of the crystallization kinetics that incorporate the influence of stress or strain. There have been several theories that dealt with stress induced crystalliza- tion [3]. However, it is very difficult to get the parameters involved in the theories, hence, they are difficult to apply to the real process. Lafleur and Kamal [4] used a MAC (Maker-and-Cell) type finite difference method for a non-isothermal This revised paper was presented in its original form at the 58th Annual Technical Conference (ANTEC 2000) held in Orlando, FL, May 711, 2000 and the copyright is held by the Society of Plastics Engineers. *Current address: Polymer Processing Institute, New Jersey Institute of Technology, University Heights, Newark, nnNJ 07102. **Author to whom correspondence should be addressed. 497Journal of REINFORCED PLASTICS AND COMPOSITES, Vol. 21, No. 6/2002 0731-6844/02/06 049710 $10.00/0 DOI: 10.1106/073168402026829 2002 Sage Publications viscoelastic fluid, and considered a kinetic equation of crystallization. They were able to predict the melt front progression as well as the residual stress and crystallinity distribution. Since a fast crystallization polymer was considered in their model, the stress effect was not included. Hsiung and Cakmak [5] developed a practical model for the simulation of slow-crystallizing polymers. Based on the experimental data on semi-crystallizing polymers, they proposed a computational model of crystallization kinetics, which includes the effect of stress. They used that model successfully to simulate the crystallinity distribution across the thick- ness, as well as the frozen layer development during the filling stage of injection molding. Han and Wang [6] predicted the volume shrinkage of slow-crystallizing thermoplastics in injection molding. The crystallization kinetics and its effect on the viscosity were considered, but flow-induced crystallization was neglected. Considering the enhancement of crystallization kinetics as a consequence of flow as well as during solidification, Titomanlio et al. [7] simulated the injection molding process on the basis of Lord and Williams model. Although it is claimed that crystallization may occur in the filling stage, none of the above models took into account the redistribution of the induction time index, which determines the onset of the crystallization process, as well as the redistribu- tion of the crystalline entities during the filling stage. And the models also ignored the effect of stress relaxation on the kinetics of crystallization. Isayev and Hieber [8] used a Leonov constitutive model and calculated idealized one dimensional flow, and non-isothermal relaxation, after cessation of the flow. Their result showed that the shear stress relaxes dramatically following the cessation of flow. It remains frozen in the skin layer but relaxes in the core region. The relaxation of stress during the cooling stage implies a decreased effect of stress on the kinetics of crystallization after filling. In this paper, some aspects of stress-induced crystallization in the injection molding process, such as the induction time and crystallinity distribution in the filling stage, and stress relaxation during the cooling stage, are considered. These aspects will affect the final structural morphology of the injection molded product. GOVERNING EQUATIONS Since the morphology changes rapidly across the thickness, the main concern here is the crystalline distribution across the thickness. A simple slit mold geom- etry with half thickness H is considered. Assuming that the width of the mold is much larger than the thickness, the transverse flow across the width can be neglected. The fountain flow phenomenon is also not considered because it only affects a small portion of the flow in the front area. In accordance with the above assumptions, the flow in the mold is simplified into one-dimensional flow. The governing equations for such a flow are as follows: 498 JIANXIN GUO AND KWABENA A. NARH Momentum equation: (1) Continuity equation: (2) Energy equation: (3) where x is the flow direction, z is the gapwise direction, t is time, and T is tempera- ture. Q represents the volumetric flow rate. u is the velocity in the x direction, the shear viscosity, the shear rate, P the pressure, and , CP and k are the density, specific heat, and thermal conductivity, respectively. is the crystallization rate, Cc the heat of crystallization. By integrating Equation (1) and substituting into Equation (2), we can simplify the two equations into one: (4) where (5) The pressure gradients can be directly obtained from Equation (4). The fluid velocity can thus be obtained from the known pressure gradient: (6) Because polymer melts are non-Newtonian, the iteration method should be used to solve Equations (4)(6). The rheological model used in the calculation is a modified power law model. In accordance with the pre-simulation result indi- cating that a very small amount of crystallinity is formed in the filling stage, the effect of crystallinity on the viscosity is not considered. Crystallinity Distribution in Injection Molded Semi-Crystalline Thermoplastics 499 & & u P z z x = - 0 2 h Q w udz= 2 2 2P c T T T C u k C t x z + = + + && P Q x S - = 0 2 h S w zdz= & h z P z u dz x = - BOUNDARY CONDITIONS At the inlet, it is assumed that the melt has a constant temperature T0. The pres- sure gradient is calculated according to the constant flow rate. On the cavity wall, a non-slip condition is employed: (7) Also, symmetric boundary conditions are applied at the center line. Following the same treatment of temperature in the melt front as that of Hsiung and Cakmak [5], the temperature along the front is considered to be uniform and equal to the calculated center line temperature at the streamwise location immedi- ately upstream of the front: (8) The time step is selected in such a way that in each step, the melt front advances one element along the flow direction. CRYSTALLIZATION KINETICS For the purpose of comparison, different models were used to predict the gapwise crystallinity distribution. The Crystallization Model in Quiescent State In this model, the effect of shearing stress on the crystallization kinetics is neglected. The crystallization of polymer in the quiescent state has received exten- sive study, and the parameters in the model can be easily obtained from the litera- ture. The differential form of Nakamura's equation [9] is used: (9) where is relative crystallinity, and K(T) is the nonisothermal crystallization rate constant; n is the Avrami index. Stress-Induced Crystallization Model of Hsiung and Cakmak [5] A parabolic function between log K and the temperature at quiescent state is 500 JIANXIN GUO AND KWABENA A. NARH 0 and wu T T= = ( , , ) ( ,0, )mf mfT x z t T x x t= - ( 1)/ ( )(1 )[ ln (1 )] n nd nK T dt - = - - - used. The effect of shear stress is to shift this equation toward higher temperature, and higher K values: (10a) (10b) (10c) where is the shear stress. A, B, C are parameters that were estimated from litera- ture. Stress Relaxation Model At the end of the filling stage, some of the polymer material in the mold is still in the molten state. Although it has experienced high shear stress during filling, the stress will decrease because of stress relaxation of the polymer during the holding period. In the molten state, the stress will relax rapidly, usually the relaxation time is on the order of 101 103 seconds, depending on temperature. In the solid state, such as that of the frozen layer, the stress will take some time to relax and its effect on the kinetics of crystallization will remain until it has completely relaxed. The following simple one parameter model is used to describe the stress relaxation of the polymer: (11) where is the relaxation time of the material, which is a function of temperature. 0 is the shear stress at the end of the filling. The relaxation time as a function of temperature can be expressed by the well-known WLF relation [10]: (12) where C1 and C2 are material constants. Ts is the reference temperature, and 0 is the relaxation time at Ts. Unfortunately, the WLF relation is only valid in the temperature range of Tg < T < Tg + 100C. The injection molding temperature is out of this range. And the relaxation information of most materials is generally not available near the processing temperature. To demonstrate the effect of stress relaxation on the crys- tallization kinetics, a set of trial values of the constants in the WLF equation was used. Crystallinity Distribution in Injection Molded Semi-Crystalline Thermoplastics 501 2 log log ( )p pK K A T T= - - p pqT T B= + log logp pqK K C= + / 0 te-= 1 0 2 ( ) log s s C T T C T T - = - + - THE INDUCTION TIME Unlike the low molecular weight materials, semi-crystalline polymers do not crystallize immediately when the temperature reaches the crystallization tempera- ture. The time needed for the start of crystallization is the induction time. For non-isothermal crystallization, Sifleet et al. [12] proposed the following equation to determine the induction time: (13) where ti(T) is the isothermal induction time as a function of temperature. Crystalli- zation begins when the accumulated induction time index reaches unity. The shear stress will not only increase the rate of crystallization, but also decrease the induction time. Hsiung and Cakmak [5] proposed a parabolic func- tion between log ti(T) and temperature at quiescent and stress states. The same model was used in this study. During the filling stage, the polymer melt undergoes large shearing deforma- tion. The shear stress will influence both the induction time and crystallization kinetics. Therefore, the redistribution of induction time index during the filling stage must be considered: (14) The right side of Equation (14) can be thought of as the accumulation rate of the induction time index. The second term on the left side of Equation (14) is the convection of because of flow. CRYSTALLINITY REDISTRIBUTION DURING FILLING STAGE In the filling stage, high stress may induce some extent of crystallization. An extra equation of crystalline transportation caused by the flow should be consid- ered: (15) where is the crystallization rate as expressed by Equation (9). 502 JIANXIN GUO AND KWABENA A. NARH 0 ( ) It i dt t t T = t 1 ( , )i t t u t x t T + = t u t dx + = & & COMPUTATION PROCEDURE The computation procedure is composed of two separate stages: filling and holding. For simplicity the packing stage will not be considered separately and will be considered as a part of the holding stage. The pressure gradient and velocities can be directly obtained from Equations (4) and (6). The gap thickness is divided into many sublayers in order to get accurate velocity and temperature distribution. As mentioned above, the iteration method is used to compute the velocity distribution. The time derivative of the temperature in the energy equation (3) is approxi- mated using backward finite difference method, while heat convection and viscous dissipation terms are evaluated using the result from the previous time step. Finite element method is used to discretize the conduction term in an implicit manner. Since only the heat conduction along the thickness is considered, the finite element equations do not couple with the elements in the flow direction. Explicit finite difference method is used to solve Equations (14) and (15): (16) (17) SIMULATION RESULTS AND DISCUSSION To illustrate the effectiveness of our model, PPS was selected. Hsiung and Cakmak formulated all the parameters involved in the stress induced crystalliza- tion kinetics for this material. Figure 1 shows the distribution of induction time index at the end of filling for various processing conditions. It is obvious that in some areas in the mold cavity, the induction time index has reached unity and crys- tallization began in the filling stage. Figure 2 shows the crystallinity distribution at the end of filling. It is seen that even at high injection speed, the crystallinity gener- ated in the filling stage is very small, the maximum is about 0.2%. The crystallized area is near the wall in the stagnation zone of the flow. Because of the small amount of crystallinity generated during the filling stage, it is possible to neglect its effect on the viscosity of the polymer melt. Although there is not much crystallinity generated during the filling stage, the accumulated induction time index will certainly increase the crystallization kinetics and, hence, affect the final crystallinity distribution. Crystallinity Distribution in Injection Molded Semi-Crystalline Thermoplastics 503 t t t t I t t t t u t x + = + - t t t t t u x + = + - & 504 JIANXIN GUO AND KWABENA A. NARH Figure 1. Induction time index distribution at the end of filling. Figure 2. Crystallinity distribution at the end of filling. Figure 3 shows the simulation result of three different models. The quiescent crystallization model did not show the crystallinity difference in different layers. It under-predicts the crystallinity in the intermediate layer which is most affected by the stress. The stress induced model by Hsiung and Cakmak [5] which did not consider the stress relaxation effect, has over-predicted the crystallinity distribu- tion, especially near the core area. In this area, the stress relaxes rapidly. Two stress relaxation models were used in the simulation. The first model assumes that the stress has completely relaxed if the polymer is still in the molten state at the end of filling and the stress in the frozen layer remains unchanged; in the second model, the stress follows the WLF relaxation model. The constants in the WLF model were chosen by trial-and-error, using typical values for some amorphous thermoplastics (for demonstration C1 = 88.6, C2 = 110.6, To = 280 and o = 10 s were used [12]). The results of the above two models are also included in Figure 3. The completely stress relaxed model shows a sharp decrease of crystallinity at the interface of frozen layer and molten region. However, the distri- bution in this model is somewhat closer to the experiment results. The WLF relax- ation model in this case shows the same trend as the experimental results. It is evident from these results that by choosing a suitable relaxation model, stress induced crystallization kinetics could be simulated more accurately. CONCLUSIONS Crystallization kinetics in the injection molding process is a very complex phenomenon. The redistribution of the induction time index and the crystallinity Crystallinity Distribution in Injection Molded Semi-Crystalline Thermoplastics 505 Figure 3. Comparison of results of different models. generated during the filling stage should be considered in modeling this phenom- enon. The induction time index and the crystallinity generated in the filling stage will affect the final morphology of the injection molded part. The shear stress resulting from high injection speed would relax during the holding and cooling stages, and a suitable stress relaxation model should be incor- porated in the model. ACKNOWLEDGMENT Financial support for this work was provided by the Multi-Lifecycle Engi- neering Research Center, of NJIT, which is funded by the New Jersey Commission of Science and technology (NJCST). REFERENCES 1. Hsiung, C. M. and Cakmak, M., J. Appl. Polym. Sci., 17, 125147 (1993). 2. Hsiung, C. M. and Cakmak, M., J. Appl. Polym. Sci., 17, 149165 (1993). 3. Eder, G., Janeschitz-Kriegl and Liedauer, S., Prog. Polym. Sci., 15, 629714 (1990). 4. Lafleur, P. G. and Kamal, M. R., Polym. Eng. Sci., 26:1, 92102 (1986). 5. Hsiung, C. M. and Cakmak, M., J. Appl. Polym. Sci., 31:19, 13721385 (1991). 6. Han, S. and Wang, K. K., Inter. Polym. Process., 12:3, 228237 (1997). 7. Titomanlio, G., Speranza, V. and Brucato, V., Inter. Polym. Process, 12:1, 4553 (1997). 8. Isayev, A. I., and Hieber, C. A., Rheologica Acta, 19:2, 168182 (1980). 9. Nakumura, K., Katayama, K. and Amano, T., J. Appl. Polym. Sci., 17, 10311042 (1973). 10. Williams, M. L., Landel, R. F. and Ferry, J. D., J. Amer. Chem. Soc., 77, 3707 (1955). 11. Sifleet, W. L., Dinos, N. and Collier, J. R., Polym. Eng. Sci., 13:1, 1016 (1973). 12. Osswald, T. A. and Menges, G., Materials Science of Polymer for Engineers, pp. 5358, Hanser Publishers, NT (1996). 506 JIANXIN GUO AND KWABENA A. NARH 1. Hsiung, C. M. and Cakmak, M. , J. Appl. Polym. Sci. , 17 , 125 - 147 ( 1993 ). 2. Hsiung, C. M. and Cakmak, M. , J. Appl. Polym. Sci. , 17 , 149 - 165 ( 1993 ). 3. Eder, G. , Janeschitz-Kriegl and Liedauer, S. , Prog. Polym. Sci. , 15 , 629 - 714 ( 1990 ). 4. Lafleur, P. G. and Kamal, M. R. , Polym. Eng. Sci. , 26 : 1 , 92 - 102 ( 1986 ). 5. Hsiung, C. M. and Cakmak, M. , J. Appl. Polym. Sci. , 31 : 19 , 1372 - 1385 ( 1991 ). 6. Han, S. and Wang, K. K. , Inter. Polym. Process. , 12 : 3 , 228 - 237 ( 1997 ). 7. Titomanlio, G. , Speranza, V. and Brucato, V. , Inter. Polym. Process , 12 : 1 , 45 - 53 ( 1997 ). 8. Isayev, A. I. , and Hieber, C. A. , Rheologica Acta , 19 : 2 , 168 - 182 ( 1980 ). 9. Nakumura, K. , Katayama, K. and Amano, T. , J. Appl. Polym. Sci. , 17 , 1031 - 1042 ( 1973 ). 10. Williams, M. L. , Landel, R. F. and Ferry, J. D. , J. Amer. Chem. Soc. , 77 , 3707 ( 1955 ). 11. Sifleet, W. L. , Dinos, N. and Collier, J. R. , Polym. Eng. Sci. , 13 : 1 , 10 - 16 ( 1973 ). 12. Osswald, T. A. and Menges, G. , Materials Science of Polymer for Engineers , pp. 53 - 58 , Hanser Publishers , NT ( 1996 ).

PY - 2002

Y1 - 2002

N2 - Injection molded semi-crystalline thermoplastic parts show variable morphologies across their thickness. Process parameters such as injection speed, mold temperature, and melt temperature, play an important role in forming these morphologies. The heat and shear history has a great effect on the crystallization process of the semi-crystalline plastics. In this study, different available crystallization models were used to predict the crystallinity distribution, and capabilities of the models compared. Based on the results from these models, a more realistic model, which considers stress relaxation during the crystallization process, is proposed.

AB - Injection molded semi-crystalline thermoplastic parts show variable morphologies across their thickness. Process parameters such as injection speed, mold temperature, and melt temperature, play an important role in forming these morphologies. The heat and shear history has a great effect on the crystallization process of the semi-crystalline plastics. In this study, different available crystallization models were used to predict the crystallinity distribution, and capabilities of the models compared. Based on the results from these models, a more realistic model, which considers stress relaxation during the crystallization process, is proposed.

UR - http://www.scopus.com/inward/record.url?scp=0036266707&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036266707&partnerID=8YFLogxK

U2 - 10.1177/0731684402021006829

DO - 10.1177/0731684402021006829

M3 - Article

AN - SCOPUS:0036266707

VL - 21

SP - 497

EP - 506

JO - Journal of Reinforced Plastics and Composites

JF - Journal of Reinforced Plastics and Composites

SN - 0731-6844

IS - 6

ER -