International Communications in Heat and Mass Transfer 36 (2009) Contents lists available at ScienceDirect

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1 International Communications in Heat and Mass Transfer 36 (2009) Contents lists available at ScienceDirect International Communications in Heat and Mass Transfer journal homepage: Numerical simulation of residual stress and birefringence in the precision injection molding of plastic microlens arrays Can Weng a,, W.B. Lee a,s.to a, Bing-yan Jiang b a Advanced Optics Manufacturing Centre, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, P. R. China b School of Mechanical & Electrical Engineering, Central South University, Changsha, Hunan, P.R. China article info abstract Available online 16 December 2008 Keywords: Microlens array Residual stress and birefringence Precision injection molding Numerical simulation Microlens arrays are critical components in the assembly of display devices used in various telecommunication products. The optical quality of these plastic lenses is very sensitive to the presence of residual stress. In this paper the distribution of residual stress and birefringence of these microlens arrayshasbeeninvestigated.itisfoundthatmaximumresidualstressalwaysoccursintheregionsnearthe gate. The predicted trends and patterns of the residual stress and birefringence distribution agree well with the experimental results. The effects of six main processing parameters on maximum value of residual stress are studied. The noticeable discrepancy between the simulation and the experimental results are discussed. Crown Copyright 2008 Published by Elsevier Ltd. All rights reserved. 1. Introduction Communicated by W.J. Minkowycz. Corresponding author. addresses: Candice.Weng@polyu.edu.hk (C. Weng), jby@mail.csu.edu.cn (B. Jiang). Microlens arrays are used extensively in the display devices of various telecommunication devices from mobile devices to computer screens. The successful production of these lens arrays depends not only on a good optics design and accuracy in the making of the mold, but also lies in the control of the residual stress induced in the micro-molding process. The effects of all these factors from optical design, mold machining and injection molding are inter-related and contribute to the final quality of the molded components. A better understanding of the flow process and the distribution residual stress patterns is therefore important to predict their occurrences and to eliminate much of the trial and error in the process parameters optimization. Over the last 15 years, several research groups and industrial research laboratories have been focusing their attention on the development of fabrication technologies for microlens arrays [1]. Recently, some researchers tried to investigate the relationship between processing parameters and the final quality of injection molded microlens arrays using an empirical approach. Lee et al. [2] studied the injection molding processing condition effects on the replicability of the microlens array profile for three different polymeric materials: PS, PMMA and PC. Shen et al. [3] tried to describe the optical properties of the microstructure of a light guiding plate for micro injection molding and micro injectioncompression molding using PMMA plastic. Weng et al. [4] applied the Factorial Design method to identify key process parameters from a large number of potential parameters by considering the interaction of injection molded microlens arrays. McFarland and Colton did some pioneering work by applying a numerical simulation method for this purpose. They [5] used ANSYS 5.7 software to predict the influence of different processing parameters to the molding of micro lenses molded in PS, PC and PMMA. However, the simulations were based on simplified 2D models. In this paper, the residual stress and birefringence of precision injection molding for microlens arrays is simulated. The main purpose of this paper is to investigate the effects of the main processing parameters on the maximum value of residual stress. This has not been studied before in relation to precision injection molded microlens arrays. 2. Mathematical modeling During the micro injection molding process, the behaviour of the plastic melt is considered as a non-isothermal, non-newtonian laminar flow. Then the control volume finite element method is used to solve the mathematical models. The general tensorial form of transport equations for the non-isothermal, non-newtonian fluid is given as follows: Continuity equation: Aρ At + j ð ρν Þ=0 ð1þ /$ see front matter. Crown Copyright 2008 Published by Elsevier Ltd. All rights reserved. doi: /j.icheatmasstransfer 转载

2 214 C. Weng et al. / International Communications in Heat and Mass Transfer 36 (2009) Nomenclature A 1, A ~ 2 coefficients related to temperature b half-gap thickness C B stress-optical constant C v specific heat at constant volume D Dt material or substantial derivative D 1 viscosity coefficient under zero shear rate and glass transition temperature D 2 glass transition temperature D 3 coefficient related to pressure g body force vector h thickness of the microlens array n non-newtonian index P pressure P in injection pressure at the gate Q in injection volumetric flow rate at the gate q heat flux vector S rate of heat generation T temperature T w mold wall temperature Vˆ specific volume ν velocity vector ν x, ν y, ν z velocity x, y, z Cartesian coordinate Greek symbols gradient operator η 0 viscosity at zero shear rate ρ density σ residual stress δ retardation τ extra stress tensor τ shear stress at the transition between Newtonian and power law behavior Momentum equation: ρ Dν = jp + j τ + ρg Dt ð2þ Energy equation: DT AP ρc v = j q T Dt AT ˆV ðj νþ+ τ : jν + S : ð3þ The well-known 7-constant Cross-Williams-Landel-Ferry (WLF) [6] is adopted here to describe the behavior of polymer melts. It has the form: ηðt;p; γ Þ= η 0 ðt;p η 0 1+ η 0 γ τ T 1 n ð4þ Þ= D 1 exp A 1 T T T A 2 + ðt T T ; TNT 4 ð5þ Þ A 2 = ~ A 2 + D 3 P T T = D 2 + D 3 P η 0 ðt;pþ= ; T T T ð8þ Moving boundary and initial conditions: 8 at the flow front : P =0 >< at the mold wall boundaries : AP Az =0; v x = v y =0 at the gate : P = P in or Q = Q in ; v y =0; Av ð9þ >: x Az =0 ð6þ ð7þ Energy boundary and initial conditions: ( at the geometry center ðz =0Þ: AT Az =0 at the mold wall boundaries ðz = bþ: T = T w 3. 3D numerical simulation ð10þ The structure designed for this study is a 4 4 microlens array with a microlens diameter of 2 mm. Polycarbonate (PC) was selected as the material for the microlens array molding. Fig. 1 shows the microlens array structure and mesh modeling for the 3D numerical simulation. Simulations were conducted on the precision injection molding conditions according to the Fractional Factorial method with the six molding parameters at two levels, as shown in Table 1. The Moldflow Plastics Insight (MPI )v.6.2commercialsoftware was used to carry out the simulation. The maximum value of residual stress was calculated using the maximum retardation obtained. δ σ = : ð11þ h C B 4. Experimental verification Molding experiments on the microlens array were performed on a Battenfeld HM 250/60 injection molding machine. The custom-built photoelastic residual stress analysis system was used to obtain the experimental residual stress. Fig. 2 shows the simulated and experimental maximum values of residual stress in the final items. They present a similar response to every change in processing conditions, while there is a noticeable discrepancy between the numerical simulation and the experimental results from a quantitative point of view. 5. Effects of processing factors The experimental validation proves that MPI predicted residual stress and birefringence well, in tendency. The processing factors affecting the residual stress and birefringence in precision injection molded microlens arrays are then further investigated by employing this software. Table 2 summarizes the processing conditions used in the study. 6. Results and discussion Fig. 3 shows the effect of the melt temperature on the maximum value of residual stress in precision injection molding for microlens arrays. It was found that the maximum value of residual stress decreased as the melt temperature increased. A higher melt temperature can lead to a lower viscosity, which results in a lower cavity pressure and lower shear stress. Then, it will bring less residual stress to the final molded products. The effect of the mold temperature on the maximum value of residual stress is shown in Fig. 4. The maximum value of residual stress is lowest when the mold temperature is 90 C. Except at that special point of 90 C, the maximum value of residual stress decreased as the mold temperature increased. The relationship between the flow rate and the maximum value of residual stress is shown in Fig. 5. It is found that the maximum value of residual stress increases as the flow rate increases. A higher flow rate will lead to higher shear stress, and then bring higher flow-induced residual stress to the final item. Fig. 6 shows the effect of packing pressure on the maximum value of residual stress. The maximum value of residual stress is found to

3 C. Weng et al. / International Communications in Heat and Mass Transfer 36 (2009) Fig. 1. The structure of designed microlens array and mesh modeling. decrease as packing pressure increases. Higher packing pressure in an appropriate range can make the product denser and prolong the solidification time of the melt, and make the pressure transportation more even and complete. As a result, it can bring lower residual stress on the final product. The effect of the packing time on the maximum value of residual stress is shown in Fig. 7. There is no systematic relationship found between the packing time and the maximum value of residual stress. Fig. 8 shows the effect of the cooling time on the maximum value of residual stress. The maximum value of residual stress is found to decrease as the cooling time increases. Retardation distribution obtained from MPI numerical simulation results can then express residual stress distribution. Pictures of the residual stress patterns from numerical simulation and experiments are shown in Fig. 9, according to the processing conditions in Table 1. Obviously, the maximum residual stress always occurs in the regions near the gate due to the relatively large shear stress caused by the sharp geometrical change when the melt flows into the cavity. Residual stress distributions from numerical simulation and experiments express a matching agreement. When comparing the simulated and experimental results, it has to be mentioned [7] that it is noticeable that the simulation tools only work adequately from a qualitative point of view. The inadequacy could be attributed to the following limitations involved in the commercial package. Firstly, the rheological data used in current packages are obtained from macroscopic experiments. Secondly, widely used simulation tools use the no-slip boundary. While Gennes [8] states that a polymer melt will exhibit a non-zero tangential velocity at a melt-smooth metal interface, in contrast to the commonly imposed no-slip condition. Thirdly, another complicated boundary factor in the simulation is the surface tension. For conventional injection molding, surface tension would exert little influence over the polymer melt front advancement for the imposed pressure is much higher. Focusing on increasingly smaller flow channels, the driving Table 1 Processing conditions used for precision injection molding Injection conditions A B C D E F pressure produces, however, an ever-shrinking force. At some threshold, the forces relating to imposed pressure and velocity become small enough that the surface tension is no longer negligible. 7. Conclusions Fig. 2. Comparison of simulated and experimental residual stress. In this study, the residual stress and birefringence of precision injection molding for microlens arrays has been studied by Table 2 Processing conditions Processing conditions Factors under investigation Set no. Melt temperature ( C) Mold Flow temperature rate ( C) (cm 3 /s) Packing pressure (MPa) Packing time (s) Cooling time (s) Melt 270, 275, 280, temperature ( C) 285, 290, 295, 300 Mold temperature ( C) , 90, 100, 110, Flow rate (cm 3 /s) 6 6 3, 4.5, 6, , 9 Packing , 65, 75, pressure (MPa) 85, 95 Packing time (s) , 4, 6, 6 8, 10 Cooling time (s) , 15, 20, 25

4 216 C. Weng et al. / International Communications in Heat and Mass Transfer 36 (2009) Fig. 3. Effect of melt temperature on maximum value of residual stress. Fig. 6. Effect of packing pressure on maximum value of residual stress. Fig. 4. Effect of mold temperature on maximum value of residual stress. Fig. 7. Effect of packing time on maximum value of residual stress. Fig. 5. Effect of flow rate on maximum value of residual stress. Fig. 8. Effect of cooling time on maximum value of residual stress. simulation and injection molding experiments. There is a good agreement in the change pattern of the residual stress and birefringence distribution obtained by numerical simulation software. Maximum residual stress always occurs in the regions nearthegate.theeffectsofthemainprocessingparameterson the maximum value of residual stress in precision injection molded microlens arrays were investigated by employing 3D numerical simulation. Fig. 9. Residual stress distributions from MPI and experiments. a) Residual stress distribution from MPI and experiments in No. 1 processing condition. b) Residual stress distribution from MPI and experiments in No. 2 processing condition. c) Residual stress distribution from MPI and experiments in No. 3 processing condition. d) Residual stress distribution from MPI and experiments in No. 4 processing condition. e) Residual stress distribution from MPI and experiments in No. 5 processing condition. f) Residual stress distribution from MPI and experiments in No. 6 processing condition. g) Residual stress distribution from MPI and experiments in No. 7 processing condition. h) Residual stress distribution from MPI and experiments in No. 8 processing condition.

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6 218 C. Weng et al. / International Communications in Heat and Mass Transfer 36 (2009) Fig. 9 (continued).

7 C. Weng et al. / International Communications in Heat and Mass Transfer 36 (2009) Acknowledgements The authors would like to express their sincere thanks for the financial support provided by the Research Committee of the Hong Kong Polytechnic University. References [1] H. Ottevaere, R. Cox, H.P. Herzig, et al., Comparing glass and plastic refractive microlenses fabricated with different technologies, Journal of Optics. A, Pure and Applied Optics 8 (7) (2006) S407 S429. [2] B.K. Lee, D.S. Kim, T.H. Kwon, Replication of microlens arrays by injection molding, Microsystem Technologies 10 (6 7) (2004) [3] Y.K. Shen, H.J. Chang, C.T. Lin, Study on optical properties of microstructure of lightguiding Plate for micro injection molding and micro injection-compression molding, Materials Science Forum (2006) [4] C. Weng, W.B. Lee, S. To, The influences of process parameters on residual stress on injection moulded microlens array, Proceedings of the 10th Anniversary International Conference of the European Society for Precision Engineering and Nanotechnology, vol. 2, Zurich, Switzerland, 2008, pp [5] A.W. McFarland, J.S. Colton, Production and analysis of injection molded microoptic components, Polymer Engineering and Science 44 (3) (2004) [6] M.L. Williams, R.F. Landel, J.D. Ferry, The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids, Journal of the American Chemical Society 77 (14) (1955) [7] V. Piotter, K. Mueller, K. Plewa, et al., Performance and simulation of thermoplastic micro injection molding, Microsystem Technologies 8 (6) (2002) [8] P.G. de Gennes, Wetting: statics and dynamics, Reviews of Modern Physics 57 (3) (1985)