# Effects of surface roughness on microinjection molding.

INTRODUCTION

Injection molding is one of the most common processes for cost-effective mass production of microplastic parts. Many researchers have carried out extensive numerical and experimental analysis on the optimization of the injection molding process parameters. However, not many of these investigations [1-4] were focused on the effects of microscale factors such as wall slip and mold surface roughness. When the dimensions of a part, and thus the cavity of the mold, are small, these microscale factors which are normally neglected in the analysis of conventional injection molding may play an important role in the flow behavior [5-8]. This article addresses the effects of mold surface roughness on the cavity filling of polymer melt in microinjection molding.

The following three methods were proposed by researchers for modeling the surface roughness effects on fluid flow in microchannels, conduits, and tubes:

* The use of some regular (normally rectangle) roughness elements [9] or irregular and randomly distributed roughness elements [10] in the computation domain.

* The roughness layer was considered as a porous medium layer (PML) [5, 11, 12] and analysis was performed based on the available PML theory.

* The roughness effect was included into the viscosity model [13, 14].

On the basis of these prior investigations, the following conclusions can be drawn:

* Higher roughness will enhance heat transfer while keeping the other conditions constant [10, 12, 15].

* The roughness effect on heat transfer is dependent on the Reynolds number of the flow [11, 12, 15].

* For fluid flow with low Reynolds number ranging from 0.001 to 10, the roughness has a negligible effect on pressure drop under isothermal conditions [9].

* When Reynolds number is above 100, the roughness affects the friction factor and hence the flow parameters [5].

However, the existing models of surface roughness have the following limitations:

* A number of researchers used the height of the roughness elements or the relative roughness to describe the magnitude of the roughness. However, in practice, the measurement of roughness is often expressed in Ra, Rq, or Rz, etc. There is no explicit relation between the relative roughness and these measurable values. This limits the applicability of their models from a practical standpoint.

* Although the PML was used to model the roughness effects by some researchers, the volume porosity and permeability of the roughness layer are difficult to determine in practical applications, and significant modification will be required for numerical implementation into the existing algorithms for injection molding.

* The fluids used in their investigations were assumed to be Newtonian and isothermal, without interaction between the flow field and the thermal field, and that the flow was assumed to be in a steady state. These assumptions are not suitable for microinjection molding.

* The Reynolds numbers of the flow were typically above 100. Only a few studies considered Reynolds number as low as 1 or even 0.01, which is normally encountered in microinjection molding.

* Although the roughness-viscosity model was used to account for the roughness effects in some studies, it was not sensitive to roughness at low Reynolds number. Moreover, the contribution of temperature effects was not taken into consideration.

As such, there are deficiencies in the existing approaches to account for the effects of surface roughness for microinjection molding, where the non-Newtonian nature of a polymer melt, and the heat transfer between the melt and the mold are important. In this investigation, a new model will be proposed which will take into consideration the conductivity and specific heat of the roughness layer, and the volume of the mold cavity as a function of surface roughness. A numerical procedure incorporating the surface roughness model is implemented by employing the finite volume and level set methods. Simulation on the melt flow injected into a microdisk cavity was performed and experimental investigations were conducted to verify the simulation results.

MATHEMATICAL MODELING OF SURFACE ROUGHNESS

A cavity having two parallel plates is used as an illustration (see Fig. 1a) to model surface roughness effect on the volume of the mold cavity. In practice, the distance between the roughness peaks of the lower and the upper walls (i.e., [H.sub.m]) is usually measured to represent the height of the cavity, as normal measurements are determined by the peaks of the roughness. Indeed, as shown in Fig. 1a, the values of [H.sub.m] are different at different measurement locations because of the randomness of surface roughness. The apparent height, [H.sub.app], of the cavity as shown in Fig. 1b, is often determined as the average of the values of [H.sub.m] at different measurement locations. In the conventional analysis of injection molding, [H.sub.app] is usually used to represent the height of the cavity and mold surface roughness is neglected. However, when mold surface roughness is comparable to [H.sub.app], the volume of the melt for filling the rough region (i.e. the region consisting of roughness peaks and valleys) will become significant and cannot be neglected. In such situation, ignoring surface roughness by employing [H.sub.app] could result in significant inaccuracy in the calculation of the cavity volume.

To consider the surface roughness effect on the volume of the mold cavity, the effective mold wall should be at the mean line of the surface roughness (see Fig. 1a). The mean line is generated by calculating a weighted average for each data point resulting in equal volume above and below the line. In this case, the volume of the cavity as defined by the mean line will correspond to the volume of the physical cavity. Thus, instead of [H.sub.app], the effective height, [H.sub.eff], which is defined as the distance between the mean line of the roughness elements at the lower and the upper walls (see Fig. 1c), should be employed in the analysis. Conceptually, [H.sub.eff] can be considered as the sum of [H.sub.app] and the lower and upper effective heights of the roughness layers, i.e., the average distance from the roughness peaks to the mean line at the lower wall and the upper wall, namely, [[delta].sub.lower] and [[delta].sub.upper] as shown in Fig. 1c.

[FIGURE 1 OMITTED]

To model the surface roughness effect on heat transfer, surface roughness can be considered as a homogenous roughness layer characterized by its thermal conductivity, specific heat, and effective height, [delta]. On the basis of a simple rule of mixture, the thermal conductivity within the roughness layer (either the lower or the upper) is modeled as:

K = [theta][k.sub.1] + (1 - [theta])[k.sub.2] (1)

where, [k.sub.1] and [k.sub.2] are the thermal conductivities of the melt and the mold, respectively, and [theta] is the volume fraction that will be occupied by the melt, which is a function of the specific roughness profile.

Similarly, the heat capacity (in term of density and the specific heat) of the roughness layer is modeled as:

[rho][C.sub.p] = [theta][[rho].sub.1][C.sub.p1] + (1 - [theta])[[rho].sub.2][C.sub.p2] (2)

where, [[rho].sub.1][C.sub.p1] and [[rho].sub.2][C.sub.p2] are the heat capacities of the melt and the mold, respectively.

Assuming a linear distribution on the relative volume between the mold and polymer melt materials, which is applicable for the various roughness profiles as depicted in Table 1 the volume fraction, [theta], can be expressed in a general form as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where, [[lambda].sub.1] is the ratio between the area that will be occupied by the melt and the total area in the top plane of the roughness layer (i.e., the plane of y = [[delta].sub.lower] or the plane of y = [H.sub.eff] - [[delta].sub.upper]). Similarly, [[lambda].sub.0] is the ratio between the area that will be occupied by the melt and the total area in the central plane (i.e., the plane of y = 0).

By using these three parameters, [delta], [[lambda].sub.0], and [[lambda].sub.1], the proposed model takes into account the heights, the spacing, the shape, and the irregularity of the roughness profile. Table 1 provides the values of [delta], [[lambda].sub.0], and [[lambda].sub.1] for some simple roughness profiles.

MICROFILLING SIMULATION

During the filling stage, the melt and air can be assumed to be incompressible, and the effects of gravity and surface tension can be neglected. Thus, the governing equations for fluid flow in the cavity can be written as: Continuity equation

[nabla] x [right arrow.[nu]] = 0 (4)

Momentum equation

[rho]([[[partial derivative][right arrow.[nu]]]/[partial derivative]t] + [right arrow.[nu]] x [nabla][right arrow.[nu]]) = -[nabla]p + [nabla] x ([eta][dot.[gamma].[bar]]), [dot.[gamma].[bar]] = [nabla][right arrow.[nu]] + ([nabla][right arrow.[nu]])[.sup.[tau]] (5)

Energy equation

[rho][C.sub.p]([[partial derivative]T/[partial derivative]t] + [right arrow.[nu]] x [nabla]T) = [nabla] x (k[nabla]T) + [eta][dot.[gamma].sup.2], [dot.[gamma]] = [square root of ([1/2][dot.[gamma].[bar]]:[dot.[gamma].[bar]])] (6)

where, [rho], [eta], [C.sub.p], and k are the density, viscosity, specific heat, and conductivity either of the polymer melt in the melt filled region or of the air in the unfilled region.

On the basis of the proposed model, the effects of surface roughness on the conductivity of the roughness layer can be easily included in the energy equation (Eq. 6) with minimum disturbance, i.e., in the roughness layer, [rho][C.sub.p] and k are calculated using Eqs. 2-4. Otherwise, [rho], [C.sub.p], and k are the density, specific heat, and conductivity either of the melt or of the air, respectively.

The viscosity of the non-Newtonian polymer melt, [eta], is modeled by the widely accepted seven-constant Cross-WLF model [1, 16-18]:

[FIGURE 2 OMITTED]

[eta](T, [dot.[gamma]], P) = [[[eta].sub.0](T, P)]/[1 + ([[eta].sub.0](T, P) x [dot.[gamma]]/[tau]*)[.sup.1-n]]

[[eta].sub.0](T, P) = [D.sub.1] x exp{[[A.sub.1] x (T - T*)]/[[A.sub.2] x (T - T*)]} (7)

where,

T* (P) = [D.sub.2] + [D.sub.3]P

[A.sub.2] = [~.A.sub.2] + [D.sub.3]P

n, [tau]*, [D.sub.1], [D.sub.2], [D.sub.3], [A.sub.1], [~.A.sub.2] are material dependent constants.

To track the movement of the melt front, a scalar variable [phi] which is the level set function, is used to identify the interface between the melt and the air. The zero-level set of [phi] indicates the melt front, which is transported by:

[[partial derivative][phi]/[partial derivative]t] + [right arrow.v] x [nabla][phi] = 0, |[nabla][phi]| = 1 (8)

Around the melt-air interface, the physical properties such as density and viscosity are calculated using:

[alpha] = (1 - h)[[alpha].sub.1] + h[[alpha].sub.2] (9)

where, [[alpha].sub.1] and [[alpha].sub.2] stand for the properties of melt and air, respectively, and where h is the heaviside function [19].

The initial and boundary conditions are given as:

u = [U.sub.in], T = [T.sub.melt] at the inlet; T = [T.sub.wall] on the mold wall (10)

where, [U.sub.in], [T.sub.melt], and [T.sub.wall] are velocity at the inlet, melt temperature at the inlet, and the mold temperature, which are assumed to be constant during the cavity filling.

[phi] = x - [X.sub.ini] at t = 0 (11)

where, [X.sub.ini] is the initial position of the polymer melt in the cavity.

u = 0 on the melt-wetted wall;

[partial derivative]u/[partial derivative]y = 0 on the air-welled wall (12)

[partial derivative]u/[partial derivative]x = 0; [partial derivative]T/[partial derivative]x = 0 at the axisymmetric axis (13)

[FIGURE 3 OMITTED]

The governing equations (Eqs. 4-6) and the level set equation (Eq. 8) may be expressed in the general form:

[[partial derivative]/[partial derivative]t]([bar.[rho]][phi]) + [[partial derivative]/[partial derivative]x]([bar.[rho]][u.sub.j][phi]) = [[partial derivative]/[[partial derivative][x.sub.j]]]([GAMMA][[partial derivative][phi]/[[partial derivative][x.sub.j]]]) + S (14)

For the continuity equation (Eq. 4), [bar.[rho]], [phi], [GAMMA] and S correspond to 1, 1, 0, and 0, respectively; for the momentum equation (Eq. 5), they correspond to 1, [u.sub.i], [eta]/[rho], and -[1/[rho]][nabla]p + [[[partial derivative][u.sub.j]]/[[partial derivative][x.sub.i]]] [[partial derivative]/[[partial derivative][x.sub.j]]] ([eta]/[rho]); for the energy equation (Eq. 6), they correspond to [rho][C.sub.P] T, k, and [eta][dot.[gamma].sup.2]; for the level set equation (Eq. 8), they correspond to 1, [phi], 0, and 0.

The finite volume method of Partankar [20] is used to solve the general equation (Eq. 14). A fixed and staggered grid is used, with the scalar variables stored in the centers of the control volumes, while the velocities are located at the control volume faces. The coupling between velocity and pressure is handled by the SIMPLER algorithm, and the diffusion-convection effect in the momentum equations is modeled by the power-law scheme. The fully implicit scheme is used to discretize the transient term. The upwind scheme is used to model the convection of the level set equations. The resulting algebraic equations are solved using the TriDiagonal Maxtrix Algorithm.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Numerical Results and Discussion

The developed numerical procedure was applied to polymer filling (in this case Polyoxymethylene, POM) of a microdisk cavity, taking into consideration the surface roughness effects. Figure 2 shows the schematic of filling of a disk cavity. The diameters of the injection inlet and the disk are [d.sub.1] = 0.4 mm and [d.sub.2] = 8 mm, respectively. The upper wall is assumed to be smooth and the lower wall is rough. The roughness profile of the lower wall is assumed to be the same as the third profile as shown in Table 1 except explicit mentioned otherwise. Polymer melt enters with a constant and uniform velocity, i.e., constant volume flow rate, pushing air out through the outlet. It is assumed that the melt flow is axisymmetric during the filling of the disk cavity, therefore only a two-dimensional axisymmetric analysis is performed. 40 x 200 grids in the x and radial direction respectively as shown in Fig. 2 are used to discretize the computational domain.

[FIGURE 6 OMITTED]

Table 2 gives the density, thermal conductivity, and specific heat used in the simulation for POM, air, and the mold material, respectively. To enhance the convergence of the numerical solution but without sacrificing accuracy, the viscosity of air was assumed to be 1.0 Pa s. For POM, the constants in the cross-WLF model are given in Table 3.

[FIGURE 7 OMITTED]

The first validation of the numerical model is based on the mass conservation principle. The development of the melt front in the radial direction of the disk cavity can be obtained analytically as:

r(t) = [square root of ([[d.sub.in.sup.2][H.sub.eff] + [d.sub.1.sup.2][U.sub.in]t]/[4[H.sub.eff]])] (15)

where, [d.sub.ini] is the initial position of the melt front, which is 0.6 mm, and [d.sub.1] is the diameter of the inlet, which is 0.4 mm. Figure 3 shows the analytical and simulated development of the radial flow length of the molded disk, where the inlet velocity, [U.sub.in], is 2 m/s, the melt temperature ([T.sub.melt]) 453 K, the wall temperature ([T.sub.wall]) 323 K, and the average roughness (Ra) 0. It can be observed that the analytical and the simulation results are in good agreement as required.

[FIGURE 8 OMITTED]

To investigate roughness effect on heat transfer, the melt and the mold temperature are set at 453 and 323 K, respectively, [H.sub.eff] = 0.4 mm, [U.sub.in] = 2 m/s, and the average roughness (Ra) is varied. Figures 4-7 show the profiles of the velocity, temperature, viscosity, and shear rate of the melt at the radial position of 2.0 mm for a filling volume of 13 [cm.sup.3]. Figure 4 shows that the higher the roughness, the larger is the maximum velocity at the vicinity of the center of the cross-section. This is because for the same effective thickness, the higher the surface roughness will result in a higher effective conductivity of the roughness layer. This will lead to a lower temperature in the roughness layer, and thus lower melt temperature (see Fig. 5) and higher melt viscosity (see Fig. 6) next to the wall. It is therefore also expected that the higher surface roughness will result in a higher maximum shear rate next to the surface roughness layer (see Fig. 7). Figure 8 shows the pressure development at the center of the bottom surface of the cavity as a function of surface roughness. As expected, with the same effective thickness and the same filling volume, higher pressure is required to fill the cavity with higher surface roughness.

[FIGURE 9 OMITTED]

On the basis of the proposed model, surface roughness effect is a function of the roughness profile and the equivalent height of the roughness layer. Figure 9 shows the predicted pressures for filling the cavities with the triangle profile (i.e., the first profile shown in Table 1) and the rectangle profile (i.e., the third profile shown in Table 1). The two profiles have the same Ra (i.e., 40 [micro]m). It can be observed that higher pressure is required for filling the cavity with the triangle profile. This is because for the same Ra, the equivalent height of the roughness layer ([delta]) of the cavity with triangle profile is twice that of the rectangle profile. Similarly, if cavities have the same [delta] (i.e., 80 [micro]m) but different roughness profiles, the roughness effect may also be different. Figure 10 shows the predicted pressure for filling the cavities with the same [delta] but with different profiles (i.e., the triangle profile and the rectangle profile). It can be observed that the difference in pressure for filling the cavities with these two different roughness profiles is very small. Therefore, the surface roughness effect is significantly dependent on the equivalent height of the roughness layer and less dependent on the specific roughness profile.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

EXPERIMENT

To verify the effectiveness of the proposed roughness model and the simulation results, experiments were performed on a 5-ton microinjection molding machine (JMV-015S-5t) using a three-plate mold. A microdisk insert of 8-mm diameter that provides the lower wall of the disk cavity was employed. The disk insert was machined using EDM such that each half of the disk has different surface roughness but with the same mean line (see Fig. 11). On one half, Ra is 5.1 [micro]m and on the other half, Ra is 1.8 [micro]m. Figure 12 shows the surface roughness profile of the insert measured along the A-B direction as shown in Fig. 11 using a Talyscan 150 dual gauge system. It can be observed that, although the two halves have different surface roughness (e.g. Ra), they have almost the same mean line. The measured roughness profile of the upper wall of the cavity is shown in Fig. 13. It can be observed that the upper wall is very smooth (i.e., Ra = 0.1 [micro]m) and its roughness can be neglected as compared to the lower wall. Therefore, the disk cavity employed in the experiment is equivalent to that in the simulation (see Fig. 2). The effective cavity thickness of 250 [micro]m was employed. The material employed was POM (Ultraform W2320 003). Four cartridge heaters were used to heat up the micromold and the mold temperature was monitored using two thermocouples. The radial flow length on each half of the molded part was measured using a ROI OMIS II series optical microscope.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

Figure 14 shows the molded parts of different sizes obtained by varying injection rate with the mold temperature of 323 K and the melt temperature of 453 K. As expected, the melt flows faster in the smoother half of the cavity than it does in the rougher half.

In the simulation, for simplicity, it was judged that the rectangle profile (i.e., the third profile as shown in Table 1) can be used to describe the roughness profile measured as shown in Fig. 12. The effective height of the roughness layer ([delta]) is 4 [micro]m on one half, and 11 [micro]m on the other half. The injection rate, mold temperature, and melt temperature were set as the same as those in the experiment. 160 x 200 grids in the x and radial direction respectively are used to discretize the computational domain.

Figure 15 shows the volumes of material on each half of the cavity versus the total volumes filled for both experimental and simulation results. It can be observed that the simulation results are in good agreement with the experimental results. The volume differences between the two halves predicted by the simulation are not significantly different from those obtained from experiment. Therefore, the proposed roughness model can effectively model the effects of surface roughness on micro injection molding.

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

CONCLUSION

The mold surface roughness effects on microinjection molding were investigated. Its effects on the effective dimensions of the mold cavity and heat transfer were modeled. Subsequently, by employing the finite volume method and the level set method, a numerical procedure incorporating the surface roughness model proposed was implemented. Finally, simulation and experiment were conducted for molding of a microdisk. In summary, the following conclusions were obtained:

* Mold surface roughness has a significant effect on the volume of the mold cavity. Ignoring mold surface roughness may lead to significant inaccuracy in predicting the melt front position and filling pressure. Therefore, the effective dimension, which is a function of mold surface roughness, should be used in the analysis of microinjection molding.

* The surface roughness has a significant effect on heat transfer between the melt and the mold, and thus influences the profiles of velocity, temperature, viscosity, and shear rate of the melt, as well as the filling pressure.

* Surface roughness effect is significantly dependent on the equivalent height of the roughness layer and less dependent on the specific roughness profile.

* The simulation results and the experimental results are in good agreement. The proposed model could model the mold surface roughness effects on microinjection molding.

ACKNOWLEDGMENTS

The authors would like to thank MOLDFLOW Pty. Ltd. for their cooperation.

REFERENCES

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2. J. Zhao, R.H. Mayes, G. Chen, H. Xie, and P.S. Chan, Polym. Eng. Sci., 43(9), 1542 (2003).

3. Y.K. Shen and W.Y. Wu, Int. Commun. Heat Mass Transfer, 29(3), 423 (2002).

4. N.S. Ong and Y.H. Koh, Mater. Manuf. Process, 20, 1 (2005).

5. C. Kleinstreuer and J. Koo, J. Fluids Eng., 126, 1 (2004).

6. K.M. Awati, Y. Park, E. Weisser, and M.E. Mackay, J. Non-Newtonian Fluid, 89, 117 (2000).

7. R.D. Chien, W.R. Jong, and S.C. Chen, J. Micromech. Microeng., 15, 1389 (2005).

8. C.A. Griffiths, S.S. Dimov, E.B. Brousseau, and R.T. Hoyle, J. Mater. Process. Technol., 189, 418 (2007).

9. Y.D. Hu, C. Werner, and D.Q. Li, J. Fluids Eng., 125, 871 (2003).

10. G. Croce and P.D. Agaro, Superlattices Microstruct., 35, 601 (2004).

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12. J. Koo and C. Kleinstreuer, Int. J. Heat Mass Transfer, 48, 2625 (2005).

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H.L. Zhang, N.S. Ong, Y.C. Lam

School of Mechanical and Aerospace Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Republic of Singapore

Correspondence to: N.S. Ong; e-mail: mnsong@ntu.edu.sg

Contract grant sponsor: Nanyang Technological University, Singapore.

Injection molding is one of the most common processes for cost-effective mass production of microplastic parts. Many researchers have carried out extensive numerical and experimental analysis on the optimization of the injection molding process parameters. However, not many of these investigations [1-4] were focused on the effects of microscale factors such as wall slip and mold surface roughness. When the dimensions of a part, and thus the cavity of the mold, are small, these microscale factors which are normally neglected in the analysis of conventional injection molding may play an important role in the flow behavior [5-8]. This article addresses the effects of mold surface roughness on the cavity filling of polymer melt in microinjection molding.

The following three methods were proposed by researchers for modeling the surface roughness effects on fluid flow in microchannels, conduits, and tubes:

* The use of some regular (normally rectangle) roughness elements [9] or irregular and randomly distributed roughness elements [10] in the computation domain.

* The roughness layer was considered as a porous medium layer (PML) [5, 11, 12] and analysis was performed based on the available PML theory.

* The roughness effect was included into the viscosity model [13, 14].

On the basis of these prior investigations, the following conclusions can be drawn:

* Higher roughness will enhance heat transfer while keeping the other conditions constant [10, 12, 15].

* The roughness effect on heat transfer is dependent on the Reynolds number of the flow [11, 12, 15].

* For fluid flow with low Reynolds number ranging from 0.001 to 10, the roughness has a negligible effect on pressure drop under isothermal conditions [9].

* When Reynolds number is above 100, the roughness affects the friction factor and hence the flow parameters [5].

However, the existing models of surface roughness have the following limitations:

* A number of researchers used the height of the roughness elements or the relative roughness to describe the magnitude of the roughness. However, in practice, the measurement of roughness is often expressed in Ra, Rq, or Rz, etc. There is no explicit relation between the relative roughness and these measurable values. This limits the applicability of their models from a practical standpoint.

* Although the PML was used to model the roughness effects by some researchers, the volume porosity and permeability of the roughness layer are difficult to determine in practical applications, and significant modification will be required for numerical implementation into the existing algorithms for injection molding.

* The fluids used in their investigations were assumed to be Newtonian and isothermal, without interaction between the flow field and the thermal field, and that the flow was assumed to be in a steady state. These assumptions are not suitable for microinjection molding.

* The Reynolds numbers of the flow were typically above 100. Only a few studies considered Reynolds number as low as 1 or even 0.01, which is normally encountered in microinjection molding.

* Although the roughness-viscosity model was used to account for the roughness effects in some studies, it was not sensitive to roughness at low Reynolds number. Moreover, the contribution of temperature effects was not taken into consideration.

As such, there are deficiencies in the existing approaches to account for the effects of surface roughness for microinjection molding, where the non-Newtonian nature of a polymer melt, and the heat transfer between the melt and the mold are important. In this investigation, a new model will be proposed which will take into consideration the conductivity and specific heat of the roughness layer, and the volume of the mold cavity as a function of surface roughness. A numerical procedure incorporating the surface roughness model is implemented by employing the finite volume and level set methods. Simulation on the melt flow injected into a microdisk cavity was performed and experimental investigations were conducted to verify the simulation results.

MATHEMATICAL MODELING OF SURFACE ROUGHNESS

A cavity having two parallel plates is used as an illustration (see Fig. 1a) to model surface roughness effect on the volume of the mold cavity. In practice, the distance between the roughness peaks of the lower and the upper walls (i.e., [H.sub.m]) is usually measured to represent the height of the cavity, as normal measurements are determined by the peaks of the roughness. Indeed, as shown in Fig. 1a, the values of [H.sub.m] are different at different measurement locations because of the randomness of surface roughness. The apparent height, [H.sub.app], of the cavity as shown in Fig. 1b, is often determined as the average of the values of [H.sub.m] at different measurement locations. In the conventional analysis of injection molding, [H.sub.app] is usually used to represent the height of the cavity and mold surface roughness is neglected. However, when mold surface roughness is comparable to [H.sub.app], the volume of the melt for filling the rough region (i.e. the region consisting of roughness peaks and valleys) will become significant and cannot be neglected. In such situation, ignoring surface roughness by employing [H.sub.app] could result in significant inaccuracy in the calculation of the cavity volume.

To consider the surface roughness effect on the volume of the mold cavity, the effective mold wall should be at the mean line of the surface roughness (see Fig. 1a). The mean line is generated by calculating a weighted average for each data point resulting in equal volume above and below the line. In this case, the volume of the cavity as defined by the mean line will correspond to the volume of the physical cavity. Thus, instead of [H.sub.app], the effective height, [H.sub.eff], which is defined as the distance between the mean line of the roughness elements at the lower and the upper walls (see Fig. 1c), should be employed in the analysis. Conceptually, [H.sub.eff] can be considered as the sum of [H.sub.app] and the lower and upper effective heights of the roughness layers, i.e., the average distance from the roughness peaks to the mean line at the lower wall and the upper wall, namely, [[delta].sub.lower] and [[delta].sub.upper] as shown in Fig. 1c.

[FIGURE 1 OMITTED]

To model the surface roughness effect on heat transfer, surface roughness can be considered as a homogenous roughness layer characterized by its thermal conductivity, specific heat, and effective height, [delta]. On the basis of a simple rule of mixture, the thermal conductivity within the roughness layer (either the lower or the upper) is modeled as:

K = [theta][k.sub.1] + (1 - [theta])[k.sub.2] (1)

where, [k.sub.1] and [k.sub.2] are the thermal conductivities of the melt and the mold, respectively, and [theta] is the volume fraction that will be occupied by the melt, which is a function of the specific roughness profile.

Similarly, the heat capacity (in term of density and the specific heat) of the roughness layer is modeled as:

[rho][C.sub.p] = [theta][[rho].sub.1][C.sub.p1] + (1 - [theta])[[rho].sub.2][C.sub.p2] (2)

where, [[rho].sub.1][C.sub.p1] and [[rho].sub.2][C.sub.p2] are the heat capacities of the melt and the mold, respectively.

Assuming a linear distribution on the relative volume between the mold and polymer melt materials, which is applicable for the various roughness profiles as depicted in Table 1 the volume fraction, [theta], can be expressed in a general form as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where, [[lambda].sub.1] is the ratio between the area that will be occupied by the melt and the total area in the top plane of the roughness layer (i.e., the plane of y = [[delta].sub.lower] or the plane of y = [H.sub.eff] - [[delta].sub.upper]). Similarly, [[lambda].sub.0] is the ratio between the area that will be occupied by the melt and the total area in the central plane (i.e., the plane of y = 0).

By using these three parameters, [delta], [[lambda].sub.0], and [[lambda].sub.1], the proposed model takes into account the heights, the spacing, the shape, and the irregularity of the roughness profile. Table 1 provides the values of [delta], [[lambda].sub.0], and [[lambda].sub.1] for some simple roughness profiles.

MICROFILLING SIMULATION

During the filling stage, the melt and air can be assumed to be incompressible, and the effects of gravity and surface tension can be neglected. Thus, the governing equations for fluid flow in the cavity can be written as: Continuity equation

[nabla] x [right arrow.[nu]] = 0 (4)

Momentum equation

[rho]([[[partial derivative][right arrow.[nu]]]/[partial derivative]t] + [right arrow.[nu]] x [nabla][right arrow.[nu]]) = -[nabla]p + [nabla] x ([eta][dot.[gamma].[bar]]), [dot.[gamma].[bar]] = [nabla][right arrow.[nu]] + ([nabla][right arrow.[nu]])[.sup.[tau]] (5)

Energy equation

[rho][C.sub.p]([[partial derivative]T/[partial derivative]t] + [right arrow.[nu]] x [nabla]T) = [nabla] x (k[nabla]T) + [eta][dot.[gamma].sup.2], [dot.[gamma]] = [square root of ([1/2][dot.[gamma].[bar]]:[dot.[gamma].[bar]])] (6)

where, [rho], [eta], [C.sub.p], and k are the density, viscosity, specific heat, and conductivity either of the polymer melt in the melt filled region or of the air in the unfilled region.

On the basis of the proposed model, the effects of surface roughness on the conductivity of the roughness layer can be easily included in the energy equation (Eq. 6) with minimum disturbance, i.e., in the roughness layer, [rho][C.sub.p] and k are calculated using Eqs. 2-4. Otherwise, [rho], [C.sub.p], and k are the density, specific heat, and conductivity either of the melt or of the air, respectively.

The viscosity of the non-Newtonian polymer melt, [eta], is modeled by the widely accepted seven-constant Cross-WLF model [1, 16-18]:

[FIGURE 2 OMITTED]

[eta](T, [dot.[gamma]], P) = [[[eta].sub.0](T, P)]/[1 + ([[eta].sub.0](T, P) x [dot.[gamma]]/[tau]*)[.sup.1-n]]

[[eta].sub.0](T, P) = [D.sub.1] x exp{[[A.sub.1] x (T - T*)]/[[A.sub.2] x (T - T*)]} (7)

where,

T* (P) = [D.sub.2] + [D.sub.3]P

[A.sub.2] = [~.A.sub.2] + [D.sub.3]P

n, [tau]*, [D.sub.1], [D.sub.2], [D.sub.3], [A.sub.1], [~.A.sub.2] are material dependent constants.

To track the movement of the melt front, a scalar variable [phi] which is the level set function, is used to identify the interface between the melt and the air. The zero-level set of [phi] indicates the melt front, which is transported by:

[[partial derivative][phi]/[partial derivative]t] + [right arrow.v] x [nabla][phi] = 0, |[nabla][phi]| = 1 (8)

Around the melt-air interface, the physical properties such as density and viscosity are calculated using:

[alpha] = (1 - h)[[alpha].sub.1] + h[[alpha].sub.2] (9)

where, [[alpha].sub.1] and [[alpha].sub.2] stand for the properties of melt and air, respectively, and where h is the heaviside function [19].

The initial and boundary conditions are given as:

u = [U.sub.in], T = [T.sub.melt] at the inlet; T = [T.sub.wall] on the mold wall (10)

where, [U.sub.in], [T.sub.melt], and [T.sub.wall] are velocity at the inlet, melt temperature at the inlet, and the mold temperature, which are assumed to be constant during the cavity filling.

[phi] = x - [X.sub.ini] at t = 0 (11)

where, [X.sub.ini] is the initial position of the polymer melt in the cavity.

u = 0 on the melt-wetted wall;

[partial derivative]u/[partial derivative]y = 0 on the air-welled wall (12)

[partial derivative]u/[partial derivative]x = 0; [partial derivative]T/[partial derivative]x = 0 at the axisymmetric axis (13)

[FIGURE 3 OMITTED]

The governing equations (Eqs. 4-6) and the level set equation (Eq. 8) may be expressed in the general form:

[[partial derivative]/[partial derivative]t]([bar.[rho]][phi]) + [[partial derivative]/[partial derivative]x]([bar.[rho]][u.sub.j][phi]) = [[partial derivative]/[[partial derivative][x.sub.j]]]([GAMMA][[partial derivative][phi]/[[partial derivative][x.sub.j]]]) + S (14)

For the continuity equation (Eq. 4), [bar.[rho]], [phi], [GAMMA] and S correspond to 1, 1, 0, and 0, respectively; for the momentum equation (Eq. 5), they correspond to 1, [u.sub.i], [eta]/[rho], and -[1/[rho]][nabla]p + [[[partial derivative][u.sub.j]]/[[partial derivative][x.sub.i]]] [[partial derivative]/[[partial derivative][x.sub.j]]] ([eta]/[rho]); for the energy equation (Eq. 6), they correspond to [rho][C.sub.P] T, k, and [eta][dot.[gamma].sup.2]; for the level set equation (Eq. 8), they correspond to 1, [phi], 0, and 0.

The finite volume method of Partankar [20] is used to solve the general equation (Eq. 14). A fixed and staggered grid is used, with the scalar variables stored in the centers of the control volumes, while the velocities are located at the control volume faces. The coupling between velocity and pressure is handled by the SIMPLER algorithm, and the diffusion-convection effect in the momentum equations is modeled by the power-law scheme. The fully implicit scheme is used to discretize the transient term. The upwind scheme is used to model the convection of the level set equations. The resulting algebraic equations are solved using the TriDiagonal Maxtrix Algorithm.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Numerical Results and Discussion

The developed numerical procedure was applied to polymer filling (in this case Polyoxymethylene, POM) of a microdisk cavity, taking into consideration the surface roughness effects. Figure 2 shows the schematic of filling of a disk cavity. The diameters of the injection inlet and the disk are [d.sub.1] = 0.4 mm and [d.sub.2] = 8 mm, respectively. The upper wall is assumed to be smooth and the lower wall is rough. The roughness profile of the lower wall is assumed to be the same as the third profile as shown in Table 1 except explicit mentioned otherwise. Polymer melt enters with a constant and uniform velocity, i.e., constant volume flow rate, pushing air out through the outlet. It is assumed that the melt flow is axisymmetric during the filling of the disk cavity, therefore only a two-dimensional axisymmetric analysis is performed. 40 x 200 grids in the x and radial direction respectively as shown in Fig. 2 are used to discretize the computational domain.

[FIGURE 6 OMITTED]

Table 2 gives the density, thermal conductivity, and specific heat used in the simulation for POM, air, and the mold material, respectively. To enhance the convergence of the numerical solution but without sacrificing accuracy, the viscosity of air was assumed to be 1.0 Pa s. For POM, the constants in the cross-WLF model are given in Table 3.

[FIGURE 7 OMITTED]

The first validation of the numerical model is based on the mass conservation principle. The development of the melt front in the radial direction of the disk cavity can be obtained analytically as:

r(t) = [square root of ([[d.sub.in.sup.2][H.sub.eff] + [d.sub.1.sup.2][U.sub.in]t]/[4[H.sub.eff]])] (15)

where, [d.sub.ini] is the initial position of the melt front, which is 0.6 mm, and [d.sub.1] is the diameter of the inlet, which is 0.4 mm. Figure 3 shows the analytical and simulated development of the radial flow length of the molded disk, where the inlet velocity, [U.sub.in], is 2 m/s, the melt temperature ([T.sub.melt]) 453 K, the wall temperature ([T.sub.wall]) 323 K, and the average roughness (Ra) 0. It can be observed that the analytical and the simulation results are in good agreement as required.

[FIGURE 8 OMITTED]

To investigate roughness effect on heat transfer, the melt and the mold temperature are set at 453 and 323 K, respectively, [H.sub.eff] = 0.4 mm, [U.sub.in] = 2 m/s, and the average roughness (Ra) is varied. Figures 4-7 show the profiles of the velocity, temperature, viscosity, and shear rate of the melt at the radial position of 2.0 mm for a filling volume of 13 [cm.sup.3]. Figure 4 shows that the higher the roughness, the larger is the maximum velocity at the vicinity of the center of the cross-section. This is because for the same effective thickness, the higher the surface roughness will result in a higher effective conductivity of the roughness layer. This will lead to a lower temperature in the roughness layer, and thus lower melt temperature (see Fig. 5) and higher melt viscosity (see Fig. 6) next to the wall. It is therefore also expected that the higher surface roughness will result in a higher maximum shear rate next to the surface roughness layer (see Fig. 7). Figure 8 shows the pressure development at the center of the bottom surface of the cavity as a function of surface roughness. As expected, with the same effective thickness and the same filling volume, higher pressure is required to fill the cavity with higher surface roughness.

[FIGURE 9 OMITTED]

On the basis of the proposed model, surface roughness effect is a function of the roughness profile and the equivalent height of the roughness layer. Figure 9 shows the predicted pressures for filling the cavities with the triangle profile (i.e., the first profile shown in Table 1) and the rectangle profile (i.e., the third profile shown in Table 1). The two profiles have the same Ra (i.e., 40 [micro]m). It can be observed that higher pressure is required for filling the cavity with the triangle profile. This is because for the same Ra, the equivalent height of the roughness layer ([delta]) of the cavity with triangle profile is twice that of the rectangle profile. Similarly, if cavities have the same [delta] (i.e., 80 [micro]m) but different roughness profiles, the roughness effect may also be different. Figure 10 shows the predicted pressure for filling the cavities with the same [delta] but with different profiles (i.e., the triangle profile and the rectangle profile). It can be observed that the difference in pressure for filling the cavities with these two different roughness profiles is very small. Therefore, the surface roughness effect is significantly dependent on the equivalent height of the roughness layer and less dependent on the specific roughness profile.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

EXPERIMENT

To verify the effectiveness of the proposed roughness model and the simulation results, experiments were performed on a 5-ton microinjection molding machine (JMV-015S-5t) using a three-plate mold. A microdisk insert of 8-mm diameter that provides the lower wall of the disk cavity was employed. The disk insert was machined using EDM such that each half of the disk has different surface roughness but with the same mean line (see Fig. 11). On one half, Ra is 5.1 [micro]m and on the other half, Ra is 1.8 [micro]m. Figure 12 shows the surface roughness profile of the insert measured along the A-B direction as shown in Fig. 11 using a Talyscan 150 dual gauge system. It can be observed that, although the two halves have different surface roughness (e.g. Ra), they have almost the same mean line. The measured roughness profile of the upper wall of the cavity is shown in Fig. 13. It can be observed that the upper wall is very smooth (i.e., Ra = 0.1 [micro]m) and its roughness can be neglected as compared to the lower wall. Therefore, the disk cavity employed in the experiment is equivalent to that in the simulation (see Fig. 2). The effective cavity thickness of 250 [micro]m was employed. The material employed was POM (Ultraform W2320 003). Four cartridge heaters were used to heat up the micromold and the mold temperature was monitored using two thermocouples. The radial flow length on each half of the molded part was measured using a ROI OMIS II series optical microscope.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

Figure 14 shows the molded parts of different sizes obtained by varying injection rate with the mold temperature of 323 K and the melt temperature of 453 K. As expected, the melt flows faster in the smoother half of the cavity than it does in the rougher half.

In the simulation, for simplicity, it was judged that the rectangle profile (i.e., the third profile as shown in Table 1) can be used to describe the roughness profile measured as shown in Fig. 12. The effective height of the roughness layer ([delta]) is 4 [micro]m on one half, and 11 [micro]m on the other half. The injection rate, mold temperature, and melt temperature were set as the same as those in the experiment. 160 x 200 grids in the x and radial direction respectively are used to discretize the computational domain.

Figure 15 shows the volumes of material on each half of the cavity versus the total volumes filled for both experimental and simulation results. It can be observed that the simulation results are in good agreement with the experimental results. The volume differences between the two halves predicted by the simulation are not significantly different from those obtained from experiment. Therefore, the proposed roughness model can effectively model the effects of surface roughness on micro injection molding.

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

CONCLUSION

The mold surface roughness effects on microinjection molding were investigated. Its effects on the effective dimensions of the mold cavity and heat transfer were modeled. Subsequently, by employing the finite volume method and the level set method, a numerical procedure incorporating the surface roughness model proposed was implemented. Finally, simulation and experiment were conducted for molding of a microdisk. In summary, the following conclusions were obtained:

* Mold surface roughness has a significant effect on the volume of the mold cavity. Ignoring mold surface roughness may lead to significant inaccuracy in predicting the melt front position and filling pressure. Therefore, the effective dimension, which is a function of mold surface roughness, should be used in the analysis of microinjection molding.

* The surface roughness has a significant effect on heat transfer between the melt and the mold, and thus influences the profiles of velocity, temperature, viscosity, and shear rate of the melt, as well as the filling pressure.

* Surface roughness effect is significantly dependent on the equivalent height of the roughness layer and less dependent on the specific roughness profile.

* The simulation results and the experimental results are in good agreement. The proposed model could model the mold surface roughness effects on microinjection molding.

ACKNOWLEDGMENTS

The authors would like to thank MOLDFLOW Pty. Ltd. for their cooperation.

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H.L. Zhang, N.S. Ong, Y.C. Lam

School of Mechanical and Aerospace Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Republic of Singapore

Correspondence to: N.S. Ong; e-mail: mnsong@ntu.edu.sg

Contract grant sponsor: Nanyang Technological University, Singapore.

TABLE 1. The values of [delta], [[lambda].sub.0], and [[lambda].sub.0] for some simple roughness profiles. Roughness profile [[lambda].sub.0] [[lambda].sub.1] [THETA] 0.5 1 0.5y/[delta] + 0.5 0.75 1 0.25y/[delta] + 0.75 0.5 0.5 0.5 0.75 0.75 0.75 5/6 1/6 2y/3[delta] + 1/6 The average roughness (Ra) is defined as the average deviation from the mean line or arithmetical average. TABLE 2. Properties of the POM melt, air, and mold material. Material [rho] (kg/[m.sup.3]) K (W/mK) [C.sub.p] (J/kgK) POM melt 1153 0.14 2101 Air 1.0 0.037 1.0 Mold (steel) 7800 29 460 TABLE 3. The constants in the cross-WLF model for POM. Material n T* (Pa) [D.sub.1] (Pas) [D.sub.2] (K) POM 0.382 2.29 x [10.sup.5] 7.54 x [10.sup.12] 223 Material [D.sub.3] [A.sub.1] [~.A.sub.2] (K) POM 0 28.5 51.6

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Author: | Zhang, H.L.; Ong, N.S.; Lam, Y.C. |
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Publication: | Polymer Engineering and Science |

Article Type: | Technical report |

Geographic Code: | 1USA |

Date: | Dec 1, 2007 |

Words: | 4334 |

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