Development of runner reservoir and its effect on optical properties of small high-precision plastic injection molded parts

Adding reservoirs as extensions to a multi-cavity runner-system to regulate cavity fill-rate within small, high-precision optical parts during filling and fill-to-pack switch-over (F/P) was studied with the aid of simulations. This work aimed for a constant melt-front velocity inside the cavities with varying geometries. Three methods of reservoir designs were considered: first, using engineering intuition, second and third, using mass balance and mass and momentum balance equations, respectively. Eight reservoirs were designed and compared to two no-reservoir cases. For each case, 27 runs covering three levels of fill-rate, F/P, and packing pressure were simulated, resulting in 270 simulation runs. The quality variables of flow and thermally induced retardation and the average and standard deviation of volumetric shrinkage were considered and for each parameter, the minimum, best cases, occurred with a reservoir case. Thus, this study offers a proof-of-concept design for using reservoirs to improve molding of small, high-precision optical parts.


Introduction
Improving the quality and accuracy of injection molded parts has been an objective since the injection molding machine was developed in the nineteenth century [1]. Polymers, and their ability to be molded into an infinite number of shapes, allow for the manufacture of simple billiard balls, one of the original plastic products, and complex precision parts such as micro-camera lenses. In the optical field, the use of polymers versus traditional glass is highly desirable since polymers can be easily manufactured into complex shapes, which are lightweight and readily massproduced via injection molding and other methods [2]. About a hundred years after the first injection molding machines were introduced, the materials and process had advanced sufficiently to allow for the manufacture of precision plastic lenses with qualities that competed with their glass counterparts [3,4]. This advancement can be credited to several sources, including the development of optical polymers, the improvement of injection molding machines and machine control, the increased understanding of viscoelasticity and rheology of polymers, and the highquality standards of the injection mold. However, there is a limit to the quality improvements that can be obtained in the standard injection molding process due to the inherent characteristics of the molecular structure of polymers, which cause either geometric or birefringence defects as shown in Figure 1 [5]. To overcome these material and process shortcomings, many novel approaches have been pursued to further improve the quality of plastic optic parts. This work explores the approach of adding a sacrificial pressure and flow control reservoir to the mold to improve the part quality for optical applications.
In the injection molding process, the filling of the mold is mostly controlled by the melt fill rate or the speed of the injection screw as it pushes the polymer melt into the mold [1]. This correspondingly controls the melt-front velocity (MFV), or the speed of the polymer melt front as it travels into the mold [5].
For high-precision parts such as optical lenses, it is desired for the MFV to be held constant while the part cavity in the mold is filled [5]. This reduces the development of surface profile defects and improves optical properties by decreasing density variations throughout the part and preventing strong or nonuniform molecular orientation, which can lead to anisotropy, warpage, and birefringence [1]. For large, simple geometries, constant MFV can be achieved through screw injection speed profile control by the injection molding machine [5]. However, for small components with complex geometries such as optical lenses, this is very challenging and, in many cases, practically impossible. Typically, these difficulties arise due to very small part-to-runner/sprue volume ratio of compact optical lenses. The reservoir mold modification is a sacrificial runner extension added to a separate branch of a multi-component mold; a sketch of a possible implementation is shown in Figure 2. The principal idea behind this design is that, even under a constant injection speed and in the presence of F/P switch-over, the MFV in one branch (e.g. the part cavities) can be controlled by changing the geometry in another branch (i.e. the runner with the reservoir), since the melt will want to flow in the direction of least resistance. Thus, the first goal of introducing the reservoir is to provide a constant MFV control for the filling of the part cavities by diverting some portion of the melt into the reservoir during the filling stage without employing a highly complex injection speed profile. Adding a reservoir will require runner modification and additional mold machining. This study aims to identify the optimal reservoir design based on the process conditions and optical part geometry.
Near the end of filling the mold, the injection molding machine control switches from flow control to pressure control (F/P switch-over) [1]. This provides protection from pressure spikes for the machine and prevents defects like flashing from occurring on the part [1]. In a traditional mold the final portion to be filled is at the end of the part. Thus, there is a change in MFV near the end of filling the part. This leads to the development of variable molecular orientation within the part and therefore, shrinkage variation (warpage) and birefringence defects [5]. Since molds are typically oriented and designed in such a way that the part cavity is the last geometry filled, these defects occur within the part geometry. However, if there is still volume to be filled in a sacrificial reservoir, the part cavity can be filled completely prior to the F/P switch-over. The second goal of the reservoir is to act as a 'damping cushion' to reduce pressure and melt velocity changes at the F/P switch-over from negatively affecting the part quality.
The concept of using the runner system as a means of flow control has been previously studied with regard to artificially balancing unbalanced multicavity molds [21][22][23][24][25]. During the early days of injection molding simulations, Wang et al. developed a programme for runner design [21]. The algorithm driving this programme started with isothermal and Power-Law assumptions and using the balance laws to determine relations between unbalanced runners by holding the pressure drop between channels as equal and relating the volumetric fill rates between channels [21]. In this way the researchers were able to specify flow restrictors for runners, which would result in the target part cavities to fill at the same time [21]. The authors also took into consideration the thermal effects that were overlooked by their assumptions, by doing an iterative correction process [21]. Work on artificially balancing unbalanced molds continued into the 90s with improved algorithms, which took into account thermal effects, runner  geometries that are not round, other flow phenomena, and more advanced iterative optimisation techniques [22][23][24][25]. All the methods worked off from the Power-Law assumption and started with the fundamental balance laws.
The goal of this project is to develop a method for designing an effective reservoir, which can provide, for the part cavity, constant cavity flow rate control, F/P switch-over point damping, and complete material packing. This method will be based on engineering intuition and the use of the fundamental balance laws and the Power-Law constitutive equation and will be verified using commercial injection molding simulation software Moldex3D TM [26].

Method
As is described in Figure 3, the more specific aims in adding a reservoir cavity are to: ▪ Slow down the fill rate as the melt reaches and enters the gate area. ▪ Provide flow control to ensure that the lens cavity is completely filled and packed with a near constant melt-front velocity. ▪ Minimise the transition effect of filling to packing switch-over point (SOP).
To achieve this, the cross-sectional area of the reservoir is varied to compensate for the changing cross-sectional area of the lens gate and lens cavity geometry, and to control the melt-front velocity so that it is constant through the filling of the lens cavity.
The initial approach to develop a reservoir was a 'guess-and-check' engineering intuition approach. With this approach, changes in the reservoir design were made based on examination of previous simulation results and how the material was supposed to flow. Since the lens parts used in the initial testing were geometrically complicated and due to the complex flow properties of polymers, the reservoir to be optimised for a particular lens was very difficult. An approach using a simple mass balance was considered next. Finally, to provide a more detailed methodical approach a mathematical procedure using mass and momentum balance equations was proposed.

Derivations
The following two derivations were completed with guidance from references [27] and [28] and are more fully expanded in reference [18]. Table 1 gives a brief glossary of the notation used. Figure 4(a) shows the geometry of interest. The channel/sub-runners leading to the parts are designated as channels 1 and 2, and the reservoir segment is designated as channel 3. Therefore, notation shown in Table 1 with a 1, 2, or 3 subscript corresponds to that channel. Since channels 1 and 2 are mirrored copies, they are assumed to be identical in these derivations.
The system volumetric fill rate is assumed to be set and held constant, therefore, it can be related to the three channels by a volume balance or: It would be ideal to fill the lens parts with a constant melt-front velocity, thus we prescribe a value for the average velocity term, v 12 for the lens channels. Further, the pressure drop though the length of each of the channels is held to be equal, thus the pressure  relation yields: The geometry shown in Figure 4(a) is made up of cross-sections that can be approximated by two shapes: First, the runner portion of channels 1 and 2 are circular (Figure 4(b)), second, the cross-sectional shape of the gate and part regions are roughly rectangular (cf. Figure 4(c)) along the material flow direction. The reservoir channel, i.e. channel 3, is taken to be circular (cf. Figure 4(b)) with varying diameter. It is important to note that the reservoir geometry is taken to be circular based on ease of calculations, since a circular cross section can be defined by one geometry parameter. Based on prior research by Isaev et al. on converting between cross-sectional shapes in polymer flows via hydraulic radius/diameter [29], it is assumed that this work is relevant to most cross-sections and to convert to another cross-sectional shape, such as a rectangle, will be a straightforward final step in the calculation process.
Circular regions/segmentshagen-poiseuille equation Figure 4(b) shows a rough schematic of a representative control volume. These derived equations are relevant for the runner regions leading to the parts in channels 1 and 2 and the entirety of channel 3 since the reservoir is designed as a circular channel. The corresponding assumptions and definitions are: ▪ Only non-zero term of the rate-of-deformation is zr ▪ Considering the generalised Newtonian flow, only t zr is nonzero and t zr = t zr (r) There are two boundary conditions: ▪ Symmetry at r = 0 which requires that the flow front slope equals zero ▪ No-slip wall condition at r = R, thus, v z (r = R) = 0 Based on the assumptions given, every term in the continuity balance (cf. Equation 1, Supplemental Materials) and the r and u direction momentum balances (cf. Equation 2 Supplemental Materials) go to zero. Reordering and applying the Power-Law assumptions (cf. Equation 4, Supplemental Materials) yields the second order differential equation: This second order differential equation can be solved by integrating twice and applying the boundary conditions yielding velocity in the z direction as a function of r: Average velocity through the tube can be found: Volumetric fill rate through the tube can be determined by integrating over the cross-sectional area: Rectangular regions/segments Figure 4(c) shows a rough schematic of a representative control volume. These derived equations are relevant for the gate and part cavity regions in channels 1 and 2. The corresponding assumptions and definitions are: There are two boundary conditions: ▪ Velocity profile is at a maximum value at the centre of the velocity profile thus, After applying the assumptions and geometry, every term in the continuity balance (cf. Equation Using the boundary condition this equation can then be integrated twice to yield the velocity in the z direction as a function of x: Average velocity though the channel can be calculated: Volumetric fill rate through the tube then is two times the integration over half the cross-sectional area: Application of the derived equations With the derivations given for the circular and rectangular tube flows, a piecewise solution can be found for the geometry given. Figure 5(a) shows roughly how the lens sub-runner, gate, and part are segmented to allow for the piecewise solution and Figure 5(b) shows an example of the cross-sectional area measurements on a lens part. Generally, the part sub-runner will have a circular geometry and then the gate and part regions will be rectangular. The following two sections will describe the piecewise solutions for the two cases, namely, (i) circular runner, channels 1 and 2 and circular channel 3 (reservoir) and (ii) rectangular gate and part cavity, channels 1 and 2 and circular channel 3 (reservoir). With the solutions given for the circular and rectangular tube flow it would be easy to determine the solution for other cases as needed.
. The known parameters are: Q, v 12 , d 12 , L 12 , d 3 or l 3 , n, m. . The parameters to be solved for are: The parameters are described in Table 1 and shown in Figure 4. Subscripts after the parameter indicates the respective channel, 1, 2, or 3. First, the volumetric fill rate of channels 1 and 2 can be found from the prescribed melt-front velocity v 12 by combining the relations in Equations (5) and (6): Then, from the volumetric fill rate the pressure drop can be determined by reordering Equation (6): 12 (1 + 3n)Q 12 pn Equation (1) can then be used to determine the volumetric fill rate of channel 3: (13) Combining the relations in Equations (5) and (6) yield a relation for the melt-front velocity of channel 3: Finally, using the relation presented in Equation (6) either the diameter or the length of the channel 3 (reservoir) segment can be determined. The equation used will be determined by which of the two parameters has been prescribed in the initial set up.
The parameters are described in Table 1 and shown in Figure 4. Subscripts after the parameter indicates the respective channel, 1, 2, or 3. First, the volumetric fill rate of channels 1 and 2 can be found from the prescribed melt-front velocity v 12 by combining the relations in Equations (9) and (10): Then, from the volumetric fill rate the pressure drop can be determined by reordering Equation (10): Equation (1) can then be used to determine the volumetric fill rate of channel 3: Now, with the volumetric fill rate of channel 3 and the pressure drop relation, Equations (14) and (15) or (16) can be used to determine the melt-front velocity of channel 3 and the diameter or length of channel 3 (reservoir) for that segment.
The circular/circular or rectangular/circular solutions are used for each segment. Since the volumetric fill rate of the system and the melt-front velocity of channels 1 and 2 are held constant, the segments are related but can be calculated independently.

Geometry
The three approaches for reservoir design were developed into ten alternative geometry cases. One branch of a radially symmetric mold was used with ten variations in the reservoir geometry centrally located between the two-part cavities (cf. Figure 4(a)). Figure 6 shows the different geometry cases. Table 2 gives the basic geometry of the lens part studied. Figure 7 shows the design approach for the engineering intuition approach. The different engineering intuition reservoir approaches (Cases 1, 2, and 3) varied three length dimensions. In Figure 6, the first dimension in the label of the engineering intuition reservoirs corresponds to the second narrow region length described in Figure 7. The first and second wide region lengths were held equal and were given as the second dimension in the engineering intuition name after the 'T'. For example, reservoir Case 1 has a second narrow region length of 1 mm and a first and second wide region length of 1 mm.
To allow for optical simulations meshing of the part was completed in Rhino TM 5 [30] so the part can be meshed with fully solid hexahedron or prism elements. The runner system and reservoir were meshed in Moldex3D [26] with the default boundary layer meshing. Table 3 shows the simulation processing parameters used. For each geometry case three processing parameters were varied at three levels thus 27 runs were completed for each of the ten geometry cases.
The geometry parameters of the reservoirs designed via calculating the momentum and mass balance equations are given in the Supplemental Materials. The reservoirs were all assumed to be circular in cross-section and the specified diameters for each segment are given in each table. To complete the equations a constant value for the reservoir segment length, volumetric fill rate, and melt-front velocity were specified. The volumetric fill rate was taken from the simulation processing parameters and the melt-front velocity was determined from simulation results from preliminary simulation runs. Case 7 (cf. Figure 6) shows the reservoir design for an alternate version of the momentum balance equation for Case 4 with 20 mm/s fill rate. The reservoir segment length and melt-front velocity parameters were adjusted slightly, which helps to show the corresponding change the reservoir design. For Case 8 (cf. Figure 6) the parameters for the reservoir are determined considering only a mass or volume balance. For this case only the reservoir segment length needed to be specified.

Results and discussion
For each of the ten mold geometry cases, 27 simulation runs were completed. Complete simulation Figure 6. The mold with various reservoirs used in this study. The parts are green, the runners are blue, and the melt inlet is red. This mold is a simplified version of a radially symmetrical mold with multiple branches.  results for all cases and processing parameters can be found in the Supplementary Materials. Table 4 gives the minimum values achieved for retardation and volumetric shrinkage out of the 27 runs for each case. Further, Figure 8 shows the plotted minimum values. For each run the retardation or volumetric shrinkage given is the average value through the entire lens parts. The flow induced retardation and thermally induced retardation are the retardations caused by flow induced and thermally induced residual stresses, respectively. Figure 8(a) shows the lowest average flow induced retardation occurs with Cases 2 and 7, where Case 2 is the Intuitive Design 4mmT1mm and Case 7 is the Alternate Momentum Balance 20 mm/s. The average flow induced retardation (cf. Figure 8(a)) is greater than the average thermally induced retardation (cf. Figure 8(a)), but they are not significantly different meaning both thermal and flow induced optical defects are relevant for the total retardation (cf. Figure  8(c)). The cases with a reservoir all have significantly lower average thermally induced retardation than the cases without a reservoir (i.e. Cases-1 and 0). The lowest average thermally induced retardation occurs with Case 2, Intuitive Design 4mmT1mm. Following these trends, the average total retardation is the smallest with Cases 2 and 7. Table 4 shows that the smallest average flow induced retardation occurred with higher fill rate of 99 or 60 mm/s, earlier F/P switch of 90% or 95%, and lowest packing pressure of 50% of the pressure at the end of filling. This matches the flow induced retardation and the thermally induced retardation except for the fill rate. The minimum thermally induced retardation occurred with a high fill rate for the no-reservoir Cases 1 and 2, but with the lowest fill rate of 20 mm/s for all the reservoir cases.
These results make sense when considering polymer viscosity and molecular alignment. Rapid filling promotes fluidity via a reduction in conductive heat loss and high shear heating. Then, early F/P switchover and low packing means that the forces on the polymer as it nears the end of the cycle and is cooling are low. Since the viscosity of the polymer melt is  Table 4. Minimum and optimal properties for each of the ten geometry cases and the corresponding input parameters. increasing as the melt cools, the low pressure prevents molecular alignment of the solidifying material. Considering the average volumetric shrinkage results in Figure 8(d), Case 2 (Intuitive Design 4mmT1mm) is predicted to have the lowest average volumetric shrinkage value and Case 1 (Intuitive Design 1mmT1mm) is predicted to have the minimum standard deviation volumetric shrinkage (cf. Figure 8(e)). The average volumetric shrinkage and the standard deviation of volumetric shrinkage throughout the lens parts are minimum at opposite extremes of the processing settings. The lowest standard deviation values all occurred at the lowest/ earliest values of fill rate, F/P switch-over, and packing pressures tested. However, the minimum volumetric shrinkage values occur at the highest/ earliest values of the processing parameters with the only exception being the F/P switch-over which alternated between 95% and 100% for the different cases.
These results are expected. Increased filling rate, earlier F/P switch-over, and increased packing pressure all yield increased force on the polymer melt, therefore decreasing the volumetric shrinkage overall. However, since the part will not cool homogeneously throughout the cavity these increased forces will cause non-homogeneous packing of the material in the part cavity, thus, the inverse relation of the process parameters with respect to the average and standard deviation of volumetric shrinkage.
Considering these minimum values the simulations show that some of the reservoir designs can yield improved part qualities and could benefit the manufacturing of high-quality optical components. Focusing on the simulation cases with the most significant improvement for the reservoir case and the best of the two no-reservoir case, the quality variables are plotted versus the processing parameters for the 27 runs completed for each of the cases. Figure 9(a,b) show the average flow induced retardation for Cases-1 and 7, as two representative cases. Complete simulation results for all cases and processing parameters can be found in the Supplementary Materials. The trends are similar, following the results of all the cases, and show that the flow induced retardation tend to be lower with the combination of higher fill rate, lower F/P switch-over, and lower packing pressure. This is likely due to a higher melt temperature because of stronger shear heating, the lower pressure experienced by the part with a lower F/P switch-over, and lower packing pressure value since lower pressure reduces the amount of molded-in residual stress. The difference due to fill rate could be attributed to shear heating effects allowing the melt to relax more effectively, and therefore lower shear induced molecular orientation with the higher fill rates. Figure 10 shows this trend of increased shear rate with increased fill rate, which is similar for all the geometry cases. This increased shear rate indicates increased shear heating of the polymer melt and therefore increased fluidity and molecular relaxation. Figure 9(c,d) show the average thermally induced retardation for Cases 0 and 2, as two representative cases. Between cases the trends match, and the trends also match those of the average flow induced retardation. As is expected, Figure 9(e,f) shows the same trends for the average total retardation for Cases-1 and 7, as was seen for the average flow induced retardation and average thermally induced retardation. Again, complete simulation results for all cases and processing parameters can be found in the Supplementary Materials. Figure 11 show the average volumetric shrinkage and standard deviation volumetric shrinkage for the minimum cases with and without a reservoir. For the average volumetric shrinkage Cases-1 and 2, as two representative cases, are shown in Figure  11(a,b). While the magnitudes are different the trends match between all the cases and show that the minimum average volumetric shrinkage occurs when the highest packing pressure and highest fill rate are used. The later F/P switch-over (100%) also yields lower average volumetric shrinkage although the difference is less significant. These results are expected since the higher injection rate caused higher injection pressure and the higher packing pressure cause more force to be experienced by the melt and therefore increased density and reduced volumetric shrinkage. The standard deviation of the volumetric shrinkage for Cases 0 and 1, as two representative cases, are plotted in Figure 11(c,d) showing the trends of the standard deviation of volumetric shrinkage for all 27 runs. There is more variation than in the trends of the standard deviation, but the lowest results occur with the lowest fill rate, earlier F/P switch-over, and lowest packing pressure values.
These results show that the minimum retardation value and minimum volumetric shrinkage results occur at opposite ends of the processing parameters. Thus, there will have to be a tradeoff between these quality variables. The standard deviation of volumetric shrinkage can be related to warpage due to the nonuniform volumetric shrinkage. The minimum standard deviation of volumetric shrinkage trend is more similar to the minimum retardation trends, which could lead to process settings that yield high-quality parts that have low retardation and low warpage. In this way, these simulations indicate that reservoirs could be used to improve injection molding of plastic optical parts. Figure 9. Average flow induced, thermally induced, and total retardation versus processing parameters of F/P switch-over and packing pressure for the three fill rate levels.  . Shear rate versus processing parameters of F/P Switch-over and packing pressure for the three fill rate levels. Figure 11. Average and standard deviation of volumetric shrinkage versus processing parameters of F/P switch-over and packing pressure for the three fill rate levels.

Conclusion
The use of a sacrificial reservoir as an extension of an injection mold runner with optical polycarbonate (PC) materials was explored with simulations. Three different methods of reservoir designs were considered. The first method used engineering intuition to determine the geometry, while the second and third methods were based on mass balance and a combination of mass and momentum balance, respectively. Using these three methods eight reservoirs were designed and simulated and compared to two no-reservoir cases. For each of the 10 geometry cases, 27 runs covering three levels of injection fill rate, F/P switch-over, and packing pressure were simulated.
The quality variables of flow and thermally induced retardation and the average and standard deviation of volumetric shrinkage of lens parts were considered.
Depending on the quality variables considered, the best case occurred with different reservoir designs. Among all the reservoir cases, the Case 2 design approach (Intuitive Design 4mmT1mm) yielded the best overall or the second-best results for each of the quality variables considered. For the momentum and mass balance approaches, Case 7 (Momentum Balance 20 mm/s (Alternate)) yielded the best results for most of the quality variables.
For each of the geometry cases the quality variables versus the three processing parameters were compared. The trends for the quality variables agreed among the cases. The lowest volumetric shrinkage occurred with high fill rate, late F/P switch-over, and high packing pressure, which was the inverse processing parameters for the lowest retardation and standard deviation volumetric shrinkage results. In this way a tradeoff must be made between these quality variables. Since overall volumetric shrinkage can be Figure 11 Continued compensated with proper cavity dimensioning, the optical property of retardation and standard deviation of volumetric shrinkage, which can be directly related to warpage, are more significant to optical part quality. Thus, the reservoirs offer a method to improve both the retardation and warpage defects in injection molded optical parts.