In recent years,the injection molding the macro-component with a micro-channel is attracting more and more attention due to its high precision and cost efficiency. Generally speaking, micro-components can be classified into (i) components with micro-sized channels and (ii) components with volumes in the range of milligrams^[1]. In recent decades,macro-components with micro-sized channels have been widely used and concerned in many fields,such as micro-fluidic, medical,biotechnology,and consumer electronics^{[2, 3, 4, 5]}. The macro-sized component with low thickness-radius ratio micro-channel has been successfully produced. However,the flowing behaviors of the polymer melt in the macro-cavity are obviously different from those in the microcavity^[6]. Therefore,how to fabricate the macro-sized component with high thickness-radius ratio micro-channel by injection molding is still a considerable problem.

In recent years,conventional softwares (C-MOLD,Moldex3D,and Moldflow) are widely used and developed for the plastic injection molding simulation. These software packages are extremely useful to simulate the flowing behavior of the polymer melt in the macro-cavity of injection molded plastic component,but are no longer sufficient to describe all effects in the micro-injection molding due to the extremely small dimensions of the micro-parts. Therefore, the conventional softwares existing simulation packages for macro-scopic applications cannot simulate the micro-scopic aspects properly^[7]. When performing the analysis using the conventional injection molding software to simulate the flowing behavior of macro-sized component with micro-channels,the obtained macro-scopic data would not be suitable for modeling micro-scale flows. However,As micro-injection molding is attracting more and more attention nowadays,computer simulation is desired to predict micro-flowing behaviors. Therefore,the flowing behaviors in micro-cavity of macro-sized component with micro-feature were studied by employing an analytical method by Young^{[8, 9]}. This simulation approach is suitable for forecasting the flowing behavior in a micro-channel,but not in a macro-cavity. In addition,the pressure values at the inlets of the micro-feature can be underestimated or overestimated. For the complex macro-sized component with micro-feature,even the pressure data are unable to be obtained.

In this case,a simplified model is established to estimate the injection distance into the micro-channels of a macro-sized component. Firstly,the inlet pressure profile of micro-channel versus times is gained based on the theory of generalized Hele Shaw by combining the finite difference and finite element methods. Secondly,the analytic method is performed to observe the relationship between the flowing distance and the inlet pressure of the micro-channel under non-isothermal environment. Moreover,the influence of the injection rate,micro-channel size, heat transfer coefficient,and mold temperature on the flowing distance is studied based on the model.

Figure 1(a) shows a schematic description of a macro-sized mold with a micro-channel,where the dimension in the macro-mold is far large as compared with the micro-channel,and the effect of the flow resistance in the micro-channel on the flowing behavior of polymer melt in the macrocavity can be neglected. Therefore,in this study,the flowing behavior in the macro-cavity can be assumed as the generalized Hele Shaw flow^[10]. Correspondingly,the local flowing of the flow front near the micro-channel in the macro-cavity is described in Fig. 1(b). P₀ is the pressure at the inlet of the micro-channel,which will depend on the filled distance x from the point O to the flow front. Re is the Reynolds Number in the micro-channel,which is assumed to be very small because of the small dimension of the micro-channel. In addition,the micro-channel size is far small so that the surface tension and wall slip effect can be neglected,and the flowing behavior in the micro-channel can be considered as the pressure-driven one-dimensional laminar flow. The momentum equation in the micro-channel can be described by the following equation:

Fig. 1. Schematic description and simplified flow behavior of macro-mold with micro-channel.

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where s is the flowing distance along the flow in the micro-channel,r is the radial directions of the micro-channel,η and ν_s are the viscosity and the flowing velocity of the polymer melt, respectively,and P is the pressure in the macro-cavity.

In this study,the polymer melt is treated as a power law fluid,which can be expressed as the following equations:

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where n is the power law index,m varies with the polymer melt temperature,r is the share rate, a₀ and m₀ are the material constants,and T_s and T₀ are the polymer melt mean temperature and the reference temperature in the micro-channel,respectively.

Substitute Eqs. (2) and (3) into Eq. (1),and then integrate Eq. (1). The flowing velocity of the polymer melt in the micro-channel varying with the pressure at the inlet of the microchannel can be written as

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where h is the radius of the micro-channel. At the instant time,t_t = t _t-1 + ∆t _t-1 for the discrete time domain. The flowing distance at the micro-channel can be described as

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In the above analysis,according to Eqs. (4) and (5),it can be seen that the flowing distance s of the micro-channel is related to the inlet pressure P₀(t _t-1) and the time interval ∆t _t-1. For the micro-channel,the cooling of the polymer melt can be tremendous due to the small dimension of the channel. Convection along the channel and heat conduction across the channel width are assumed. A quasi-like steady state is considered at each of the time step t_t in the microchannel. At the inlet of the micro-channel,the temperature is set to be the melt temperature T_m. The mold temperture is kept at a constant T_w. Under the above assumption,the heat transfer equation and the boundary conditions in the micro-channel can thus be written as

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where α,k,and h_tare the thermal diffusivity,the thermal conductivity of the polymer melt, and the heat transfer coefficient,respectively.

The solutions of Eq. (6) can be divided into two functions,T(r,s) = u(r,s) + T_w,where

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and

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The resulting temperature profile in the micro-channel can be determined as

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where

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The mean temperature T_sin the micro-channel can be obtained as

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The parameter λ_ncan be expressed as

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where J₀ and J₁ are the Bessel functions. Equation (10) can be further described as

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From Eq. (9),it can be seen that the mean temperature in the micro-channel is a function of s (the flowing distance) and v_s (the flowing velocity). The mean temperature will change along the channel. Concurrently,according to Eq. (4),the velocity (v_s) itself is a function of the mean temperature (T_s) through the variable β. In the calculation of the velocity at each time step, an assumed velocity is used to determine the temperature distribution along the micro-channel. The integral in Eq. (5) can thus be evaluated to calculate the value of β. Then,the updating velocity can be obtained by substituting β into Eq. (4). Iteration is necessary until the value of the velocity converges.

Based on the above analysis,the flow behaviors of the polymer melt in the micro-channel can be obtained if the pressure profile at the inlet of the micro-channel is decided. The inlet pressure in the micro-channel is related to the flowing process in the macro-channel. The flow behavior of polymer melt in the macro-channel can be expressed as the generalized Hele Shaw flow. Consequently,the continuity and momentum equations can be described as

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where is the fluidity,and 2H is the thickness of the macro-cavity.

The energy equation can thus be written as

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where v and u are the flowing velocity components in the x- and y-directions in the macrocavity,respectively. ρ,T,˙γ,η,and c_p are the melt density,melt temperature,shear rate,shear viscosity,and specific heat,respectively.

According to Eqs. (12) and (13),the control volume method is used to formulate the finite element equation for the pressure. The difference equation is determined for the temperature in the direction of macro-channel thickness. The volume flow in the macro-channel at each time interval is used to determine the polymer melt flow front^[11]. The pressures at each moment and each point in the macro-channel are obtained by solving both difference and finite element equations.

As shown in Fig. 1(b),the width W,length L,and thickness H of the macro-channel are 50 mm,200 mm,and 4 mm,respectively. The width of the micro-channel is set at the range of 20 mm to 100 mm.

The distance from the inlet of micro-channel to the gate of macro-channel is 60 mm. In this case,the Acrylonitrile Butadiene Styrene (ABS) is used to simulate the flow behavior of the polymer melt in the micro-channel. Table 1 shows the constants in the viscosity model.

Table 1. Constants in viscosity model

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Figure 2 shows the pressure profile with respect to the flowing time for different injection rates at the inlet of micro-channel. It can be found that the pressure at the inlet of the microchannel is proportional to the flowing time. However,there is variation in the slope of pressure curves for different flowing rates. Therefore,the flowing behaviors of polymer melt in the micro-channel can be decided if the inlet pressure is obtained.

Fig. 2. Pressure of polymer melt at inlet of micro-channel at various injection

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The velocity in the micro-channel with respect to the flowing time for different injection rates and mold temperatures is shown in Fig. 3. The radius of the micro-channel and the heat transfer coeffient are h=100 µm and h_t=10 000 W/(m²·℃),respectively. It can be found that the calculation of the velocity has oscillation in the beginning because of the instability of the pressure gradient when the polymer melt just reaches the micro-channel. As the cooling time of polymer melt and the flowing distance increase,the melt flow rate reduces gradually. As the polymer melt temperature approaches the transition temperature,the flowing velocity decreases sharply to zero and the melt stops flowing,which makes the micro-channel not be filled. It is worth noting that,for different injection rates,the flowing velocity decreases to zero almost at the same time. Therefore,the flowing distance is proportional to the injection rates. In addition,the flowing time decreases with the decreasing of the mold temperature. For instance,when the mold temperature is 80 ℃,the flowing time of the polymer melt in the micro-channel is 4.05× 10^-4 s; when the mold temperature decreases to 60 ℃,the flowing time in the micro-channel decreases to 3.48× 10^-4 s.

Fig. 3. Flowing velocity profiles of melt in micro-channel at various mold temperatures and injection rates.

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The flowing distance in the micro-channel with respect to the radius of micro-channel at the different mold temperatures is shown in Fig. 4. The injection rate and heat transfer coefficient are Q=60 cm³/s and h_t=10 000 W/(m²·℃),respectively. It can be seen that the flowing distance increases dramatically with respect to the increase of the radius of the micro-channel due to the decreasing of the melt cooling rate. In fact,from Eq. (4),it also can be seen that the flowing velocity will increase with the increase of the radius of micro-channel. In addition, the mold temperature has a large effect on the melt flowing distance because of the influence on the heat transfer distance. For example,under the same radius of 100 µm,when the mold temperature is 80 ℃,the flowing distance is 173.3 µm; when the mold temperature decreases to 40 ℃,the flowing distance decreases to 130.5 µm. This indicates that it can effectively increase the flowing distance of the micro-channel by increasing the mold temperature.

Fig. 4. Flowing distance in micro-channel with various radii and mold temperatures.

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Figure 5 presents the flowing distance in the micro-channel with respect to the mold temperature and the heat transfer coefficient. The injection rate is 60 cm³/s. From Fig. 5,it can be found that,when the heat transfer coefficient is constant,the flowing distance increases with the increase of the mold temperature as described previously. In addition,the flowing distance decreases with the increase of the heat transfer coefficient because of the melt faster cooling rate. For comparison,the simulation and experiment results of the mold temperature versus the flowing distance in the micro-channel obtained by Huang and Young^[12] are provided in Fig. 6,respectively. From Figs. 5 and 6,It can be seen that the simulation results of the mold temperature versus the flowing distance in this case are in consistent with the tendency of the experiment results provided by Ref. ^[12]. In addition,compared with the simulation results in Fig. 6,the simulation curves in Fig. 5 are close to the tendency of the experiment results shown in Fig. 6. It is indicated that the analytical method and model used in this case can better simulate the flowing behavior in the micro-channel.

Fig. 5. Distribution of angle θ as function of rotation angle φ for (6,6) and (10,10) SWCNTs under different field intensities.

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Fig. 6. Flowing distance in micro-channels for different mold temperatures[12].

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In this case,the flow behaviors in the injection molded component with a micro-channel are analyzed by combining the numerical simulation method and the analytical solution. The effects of the injection rate,radius of the micro-channel,heat transfer coefficient,and mold temperature on the flowing behavior in the micro-channel are studied. The results indicate that the flowing distance in the micro-channel decreases with the increase of the heat transfer coefficient,and increases with the increases of the injection rate,radius of the micro-channel, and mold temperature.