Prediction of the maximum flow length of a thin injection molded part

3.1 Spiral flow tests

The results related to the length of each molded sample are summarized in Table 1 for the tests performed with 1 s injection time.

Figure 2 shows the cavity filling length versus the mold temperature, with three injection pressures, 13, 20, and 40 MPa. Both the mold temperature and the injection pressure influence the cavity filling length.

Figure 2:

Length measured on the samples produced with different mold temperatures and injection pressures. The dashed lines represent the final length predicted of Equation 23 with h/H=0.38 and β=1.

The mold temperature increase induces an increase of the cavity filling length, obviously due to a later solidification. The increase of the injection pressure also induces an increase of the cavity filling length. For the test conducted with a mold temperature of 150°C and injection pressure of 40 MPa, the cavity resulted to be always completely filled. This gives an indication of the fact that the no-flow condition is reached at a temperature between 140 and 150°C.

Figure 3A shows the optical micrographs of the samples obtained with 140°C mold temperature and different injection pressures at 10 mm from the gate.

Figure 3:

(A) Optical micrographs of the samples obtained with T_m=140°C, t_inj=1 s and different injection pressures, taken at 10 mm from the gate position. (B) Optical micrographs of the samples obtained with T_m=140°C, t_inj=1 s and different injection pressures, along the whole sample length, from the gate to the tip.

The samples are characterized by a skin core morphology, with the oriented structures (the colored bands in Figure 3A) close to the sample surface and un-oriented structures in the core. The formation of the oriented structures is due to the interplay between the flow field and the solidification: under strong flow and fast cooling, typical of the areas close to the sample surface, the formation of oriented structures can be observed, whereas under weak flow and slow cooling, the polymer chains have enough time to form more isotropic structures, namely the spherulites [12].

Figure 3B shows the optical micrographs of the whole sample obtained with 140°C mold temperature and different injection pressures. Figure 4A shows the distribution of the thickness of the oriented layers along the flow path. Concerning the distribution along the flow path, it is noticed that the oriented structures are mainly located in the central part of the sample. The sample tips are characterized by un-oriented structures, because in that area, the flow is not enough to assure the formation of oriented structures [12]. Close to the gate, the thickness of the layer characterized by oriented structures is smaller with respect to the central part of the samples, as in this area the temperature is closer to the temperature of the piston (200°C as mentioned in the experimental section).

Figure 4:

(A) Thickness of the oriented layers versus the flow path length, evaluated for the samples obtained with T_m=140°C, t_inj=1 s and different injection pressures. (B, C) Herman’s factor versus the flow path length, evaluated for the samples obtained with T_m=140°C, t_inj=1 s and different injection pressures. (B) Analysis of the peak λ=1256 cm⁻¹. (C) Analysis of the peak λ=1220 cm⁻¹.

The samples obtained with P_inj=13 MPa and P_inj=20 MPa show the highest values of the thickness of the oriented layers at about 10 mm distance from the gate. The sample obtained with P_inj=40 MPa shows the highest values of the thickness of the oriented layers after a distance of about 17 mm from the gate.

The orientation distribution inside the samples was measured from the FTIR spectra, which allow the determination of the Herman’s factor. This factor varies between −0.5 and 1.0, with −0.5 representing the perpendicular orientation with respect to the direction of the flow. For the case of parallel orientation with respect to the flow, Herman’s factor ranges from 0 to 1, with 0 corresponding to random orientation and 1 to the perfect alignment with respect to the flow direction (the reference axis). The oriented structures, in the injection molding process, correspond to fibrils, crystals that mainly grow along the flow direction, with limited growth in the direction perpendicular to the flow [13]. Figure 4B and C show the distribution of the Herman’s factor along the flow path. Two peaks of the FTIR spectra were analyzed, at a wave number (λ) of 1256 and 1220 cm⁻¹, the first one related to the crystalline phase, the second one related to the amorphous part.

Figure 4 shows that excluding the region very close to the gate, the orientation decreases from the gate, where the Herman’s factor is close to 1, for the samples obtained with an injection pressure of 20 and 40 MPa, to the tip, where the Herman’s factor decreases down to 0.2. The highest orientation was measured at the same distance from the gate characterized by the highest thickness of the oriented layer (see Figure 3). The sample obtained with an injection pressure of 1 MPa shows smaller values of the Herman’s factor, mainly due to the decrease of the flow intensity at lower injection pressures.

Indeed, orientation can be easily related to the flow intensity: the shorter the distance from the gate, the higher the flow intensity, the higher the orientation [14]. Very close to the gate, because of the high temperature, the relaxation time is short and thus the orientation can be low. Far from the gate, the flow becomes less intense, because of the pressure decrease along the flow path; thus, the polymer chains can crystallize in almost quiescent condition. As a result, close to the tip, poorly oriented structures and spherulites can be observed.

3.2 Estimation of polymer rheology from the time evolution of flow length

All the samples produced in this work were obtained without holding stage; this means that the shrinkage due to the material solidification and crystallization was not compensated. This procedure allows analyzing the effect of the injection pressure on the flow length with the aim of modelling the filling process. The filling of the material into the cavity was analyzed by recording the piston path and correlating this movement to the flow length inside the cavity (Equation 2), as mentioned in the experimental section. Because during the experiments the machine performs a control of the injection pressure, the flow rate changes to keep constant the injection pressure during the experiment, and thus, the flow can be considered as a pressure-driven flow. Figure 5A shows the flow length (L_sper) inside the cavity recorded during the experiments performed with T_m=140°C, t_inj=1 s, and different injection pressures.

Figure 5:

(A) Flow length recorded during the experiments performed with T_m=140°C and t_inj=1s, with different injection pressures. The full lines represent the predicted evolution of the flow length, L_c, with s=1/2. (B) Comparison between the viscosity values predicted by the power-law based models proposed in this paper, the viscosity experimental data at 160°C, and the values of the viscosity evaluated by Cross-WLF at zero pressure.

The flow length, as expected, increases with the injection pressure.

In order to determine the rheology of a polymer from the spiral flow tests, one has to model the history of the length at a certain injection pressure. For a polymer melt filling, a rectangular cavity, having a width W and a thickness H, the relationship among the stress, τ_xy, gradient along the thickness (y) and the pressure drop along the cavity length (x) is given by Equation 3.

The integration along the thickness y gives

with τ_xy=0 for y=0 (namely at the midplane).

Under the hypothesis of a viscous fluid and viscosity described by a power law, the stress is given by Equation 5:

where η₀ is the consistency index, n is the flow behavior index, and v_x is the velocity along the flow direction (x). Thus, for y >0,

On integrating and using the no-slip boundary condition (y=H; v_x=0), it is possible to obtain the velocity profile along the cavity thickness (y) as a function of the pressure drop (in the following, the reciprocal of the flow index will be adopted: m=1/n):

It is possible to evaluate the flow rate Q by the Equation (8):

During any increment of time dt, the fluid volume injected into the channel is [15], [16]

Assuming a pressure profile linearly decreasing with x and indicating with P the difference between the injection pressure P_inj and the pressure at the tip, it is possible to relate the flow length, L(t), to the pressure drop:

In the model considered in this work, sketched in Figure 6, the polymer flows with a constant temperature inside the central part of the cavity. The section through which the melt fills the cavity reduces during the process, as the formation of a frozen layer close to the cavity surface takes place. The temperature at the interface between melt and solid, located at a distance δ from the mold walls, is equal to the so-called no-flow temperature, T_f. The section available to the flow will be indicated as h(t).

Figure 6:

Sketch of the cavity thickness decrease due to the formation of the frozen layer.

Thus, Equation (11) has to be rewritten with h(t) instead of H.

Barrie [17] proposed the following relationship to describe the formation of the frozen layer during the cavity filling, also applied by other authors [18]:

C and s are constants and Barrie found that s=1/3 for many polymers. At the solidification time t_f, δ=H/2, thus, it is possible to calculate the value of the constant C.

Therefore, it is possible to obtain h(t):

Substituting Equation 15 in Equation 12 and integrating, the melt advancement with time can be obtained as follows:

Equation 16 provides a relationship between time during which the melt fills the cavity and the flow path length in the spiral-shaped rectangular section cavity. When t=t_f, the equation gives the final value of the final length of the sample.

Some authors adopt an exponent s=1/2 in Equation 13 [19], in this case the melt advancement can be obtained as follows:

Again, when t=t_f, the equation gives the final value of the length of the sample.

The relationships given by Equations 16 and 18 allow evaluating the rheological parameters of a resin comparing the predicted flow path lengths with the experimental ones shown in Figure 5. In particular, it is possible to obtain values of η₀ and t_f, for selected values of 1/m. It has to be pointed out that the values of the solidification time t_f evaluated from Equations 16 and 18 are close to 1 s for all the considered values of 1/m.Table 2 summarizes the results of η₀ and 1/m.

Table 2 shows also the values of the deviation, evaluated as sum of the absolute values of the differences among the experimental flow length and the calculated ones at each time. The values of η₀ depend on the injection pressure: the higher the injection pressure is, the higher the values of η₀. This result is consistent with the fact that the pressure increase induces an increase of the viscosity [20], [21], [22]. Is it possible to compare the values of the η₀ found applying the aforementioned models to the values estimated adopting the model reported in the literature to describe the rheological behavior of the iPP adopted in this work. The model proposed in the literature is a Cross-Williams-Landel-Ferry (Cross-WLF) equation, which takes also into account the effect of pressure on viscosity [9]. For the temperature, pressure, and shear rate ranges considered in this work, the parameters of Cross-WLF are reported in the literature [23]. In particular, the Cross-WLF model for high shear rate can be approximated by a power law having the same exponent (1/m) of Cross-WLF and the consistency index equal to the Cross-WLF viscosity evaluated with a shear rate equal to 1 s⁻¹; these values are reported in Table 2. Both the rheological data, those obtained from Cross-WLF and the experimental ones, are referred to temperatures of 160 and 170°C, namely, intermediate values between the injection temperature and the mold temperature. In order to consider the effect of pressure, the rheological data are calculated at the arithmetic average between the maximum pressure (equal to P_inj) and the minimum pressure (equal to zero), namely, at P_inj/2.

The minimum values of the deviation were obtained with 1/m=0.50, for s =1/3, and 1/m=0.38, for s=1/2. Figure 5B shows the comparison among the viscosity values predicted by the power law models (adopting the values of 1/m and η₀, which correspond to the minimum value of the deviation), the viscosity experimental data at 160°C, and the values of the viscosity evaluated by Cross-WLF. The comparison shown in Figure 5B confirms that the values of the viscosity are reasonably well predicted from Equation 18, with s=1/2. Additionally, the Cross-WLF model and the power law model with s=1/2 proposed in this paper show very close values and both correctly predict the experimental data at 160°C. The shear rate characteristic of the tests ranges from 300 to 5000 s⁻¹; this confirm the reliability of the power law model proposed in this paper in the description of the viscosity behavior.

Figure 5B shows the comparison among experimental and calculated flow path lengths obtained selecting the values of η₀ and 1/m that give the minimum deviation for s=1/2 (Equation 18).

The behavior of L(t) is reasonably well described by the model reported in this paper; this is a further confirmation that the flow behavior is correctly predicted by the model with s=1/2.

3.3 Prediction of maximum flow length

When the rheology of the material (namely, 1/m and η₀) are known, Equations 17 and 19 allow the determination of the flow length, which can be reached during an injection molding test. The only unknown becomes the solidification time, t_f, whose determination requires the solution of the thermal problem. Several analysis have been proposed in the literature for the determination of the solidification time into a long rectangular cavity (e.g. by Wissbrun [24] or Richardson [25]). The phenomena to be taken into account are axial convection, transverse conduction, and viscous dissipation. The former two are kept into account by the Graetz number, the latter by the Brinkman number.

For the cases under analysis in this work, the Graetz and Brinkman numbers can be expressed as follows:

α is the thermal diffusivity, T_f is the temperature at the boundary of the frozen layer, namely the no-flow temperature. As mentioned above, T_f was estimated in the range from 140 to 150°C. In the following, we will assume T_f=147°C.

Both Gz and Br depend on time through the value of h. It is easy to find that

is independent on time.

Being, in our experiments, P of the order of 10⁷ Pa, α of 10⁻⁷ m²/s, k of 10⁻¹ W/mK, (T_inj–T_f) of 10² K, and m of 10⁰, the order of magnitude of Gzx−1Brx is 10⁻¹, indicating that the viscous dissipation can be neglected at a first approximation.

A solidification criterion can be determined by considering a macroscopic energy balance in the cavity and comparing the convective heat entering the cavity, E_conv, with the conductive heat lost toward the cavity surface, E_cond.

The order of magnitude of the former can be evaluated as

and the order of magnitude of the latter as

where δ=H−h2.

When E_conv is larger than E_cond, it can be assumed that solidification cannot take place (and indeed h remains close to H). As soon as E_cond overcomes E_conv, h decreases quickly, and solidification happens. Considering Equations 23 and 24, the ratio between the two heat quantities is

in which

Solidification takes place when the E_cond and E_conv become comparable, namely when

β being a constant (of order of magnitude 1).

On this basis, the value of the final flow length is immediately drawn from Equation 20.

To validate the results and hypotheses of the method proposed in this work, numerical simulations of the injection molding tests were performed by a commercial software. Numerical results and comparisons between predictions and experimental results are reported in Appendix 1, Figures 9–12.

By adopting Equation 28, a description of the final length reported in Figure 2 can be obtained by selecting a suitable value of β and of h/H at solidification. If β is taken as 1, then only the value of h/H at solidification has to be chosen. In Figure 2, the predictions (dashed lines) of the flow lengths are obtained by adopting the values of 1/m and η₀ obtained by WLF at 160°C (last line of Table 2) and h/H=0.62. The description is satisfactory, with a value of h/H absolutely reasonable. Obviously, a different value would be found with a different value of β. For instance, for β=0.1, the optimal value of h/H would be 0.34.

It can be observed that the dependence of L_f upon P as provided by Equation 28 is the same resulting from Equations 17 and 19, so that it can be written as follows:

Equation 28 outlines the dependence of the dimensionless time at solidification tfαH2 on the dimensionless temperature Θ. In particular, if one assumes, as above, that soon before solidification h/H takes the same value for all the tests carried out in this work, then it becomes clear that

where ε is a constant.

The values of t_f calculated by Equation 30 for the tests carried out in this work are reported in Figure 7; the values of β and hH are the same as above (1 and 0.62, respectively).

Figure 7:

Estimated solidification times for the tests carried out in this work. T_inj=200°C, T_f=147°C.

Figure 7 shows that the solidification time t_f at 140°C is close to 1 s. This validates the results reported in Table 2.

3.4 The oriented layer

As indicated in the literature, specifically for this material, the fibrils forms when the amount of work overcomes a critical value, w_c, which is of the order of 10 MPa [23].

For the model adopted in this work, which assumes the flow of a power-law fluid in a rectangular cavity, the rate of specific work, given by the product between stress and shear rate, can be expressed as

where y is the distance from the mold wall.

If L(t) is taken from Equation 18, then the amount of work at each specific location y at the end of the process can be written as

where

and b is equal to t/t_f.

Assuming that the fibrillar layer forms when w=w_c, considering that the work decreases on increasing the distance from the mold wall, then the thickness of the oriented layer can be determined as

Assuming that w_c=10 MPa [23], the fraction of oriented layer reported in Figure 8 is obtained. It can be noticed that in the range of pressure adopted in this work, the fraction is weakly dependent on pressure, as also described by the experimental data reported in Figure 3. The maximum fraction of the oriented layer is underestimated by the model by about 15–20% (0.63 instead of 0.75 at 13 MPa, 0.67 instead of 0.8 at 20 MPa, and 0.73 instead of 0.9 at 40 MPa). However, also considering the simplicity of the model, the predicted values can be surely considered good.

Figure 8:

Fraction of oriented layer obtained from Equation 34 for different injection pressure.