Measurement of pressure–volume–temperature diagrams based on simulated melt temperature and actual cavity pressure

Pressure–volume–temperature (pvT) diagrams illustrate the behavior of a polymer melt during injection molding and can be used to determine the optimal process parameter settings for producing quality parts. However, the measurements of pvT diagrams are costly and time-consuming and have limitations for process control. This study generated an economic pvT diagram based on sensor data and simulation analysis. To avoid the use of expensive infrared sensors and inaccurate pressure simulations, a pvT curve was constructed by combining the melting temperature, which was simulated using computer-aided engineering, and the actual cavity pressure. The specific volume of the polymer melt was then calculated using a modified Tait model. The pvT diagram could be used to investigate the changes in specific volume at various injection stages, which could be applied to quality prediction. In this study, a flat plate with uneven thickness was used as the research vehicle. The specific volume of each sensor position of the plastic component cooled to room temperature was calculated. Thereafter, a correlation analysis between the changes in specific volume and the shrinkage ratio of the plastic part was conducted. The experiment indicated that both variables were highly correlated. Therefore, monitoring changes in specific volume could predict part size. Additionally, the proposed measurements of the pvT diagram could be used to compute the cooling time, which greatly affects the quality and production efficiency of injection-molded parts. A 25-s cooling time setting demonstrated a stable molding quality compared with the traditional heat transfer formula (16 s) and the commercial simulation (17 s).


Introduction
Injection molding has attracted considerable attention because it consumes energy efficiently and can be used to manufacture products with complex shapes. During injection molding, the mold cavity is filled with plastic material preheated to a molten state and pressurized. The mold is then cooled to make the desired product. Due to the rheological behavior of processed polymer materials, the quality of plastic parts can be affected by the material properties, mold design, injection molding machine, and process parameters settings. Among the process parameters, temperature and pressure are two key factors that cause plastic parts to shrink or expand, which results in the over-or under-dimensioning of part geometries [1]. Moreover, uneven shrinkage can lead to part warpage or sink mark defects; thus, controlling part shrinkage is critical during injection molding [2]. The specific volume (i.e., the inverse of density) usually indicates how much a part shrinks or expands. A pressure-volume-temperature (pvT) diagram, which represents how the specific volume of plastic materials is affected by pressure and temperature, is an excellent method for optimizing the process parameters [3].
Polymer pvT properties can be measured through conventional measurement methods (e.g., the piston-die technique and confining fluid technique) and in-line techniques by using injection-molding machines or extruders. The in-line technique of measuring the pvT relationship by using an injection-molding machine is a potentially powerful tool for programming process parameters because normal conditions for injection molding can be met [4]. The pvT diagrams are usually generated using the modified Tait model [5]. Because the pvT relationship of polymers is the most essential factor for the shrinkage and warpage of polymer products, pvT data are crucial for numerical simulations. To obtain accurate predictions, three-dimensional numerical simulations were used in the present study to simulate the injection-molding process. The Cross-WLF model was used for assessing the viscosity, and the modified Tait model was used for assessing the pvT data. To demonstrate the application of pvT data in numerical simulations and to verify the reliability of pvT results through in-line measurements, Wang et al. [4] compared the theoretical dimensional deviation values of experimental and simulated results and found that the warpage predictions were relatively accurate, but considerable differences existed.
The application of the pvT properties of polymers in injection-molding process control has been studied for years. Process variations during injection molding can be attributed to several causes, including process pressure and temperature variations. Cavity pressure distribution and its repeatability considerably affect the quality of the molded part, especially the weight, dimensional stability, mechanical properties, and surface quality. The cavity pressure profile can be used to maintain high-quality products and control the machine during the injection-molding process. Reproducing the cavity pressure curve in each shot can also maintain optimal shot-to-shot consistency [6]. The cavity temperature can influence the production rate, the stability of the injection-molding process, and the quality of moldings such as surface quality, after-shrinkage, orientation, and residual stress. Therefore, a robust process control technique should be developed to achieve a high process capability for the cavity pressure and temperature [7,8]. Michaeli et al. [9] investigated the condition of the polymer melt in the pvT diagram, and the injection-molding quality could be controlled by selecting the appropriate holding pressure settings. Moreover, the mold and melt temperatures affected the stability of the injection-molding process. Wang et al. [10] demonstrated the potential of improving the part weight stability by adjusting the pressure settings according to the pvT relationship. Similarly, Hopmann et al. [11] adjusted the pressure settings to design the optimal holding pressure and time settings to minimize residual stress when fabricating optical lenses. Chang et al. [12] used infrared sensors and cavity pressure sensors mounted near the gate, far from the gate, and midway between the filling distances to generate pvT courses. Thereafter, Chang et al. optimized the holding conditions to minimize the specific volume differences that result in minimum shrinkage and warpage of the injection-molded parts. These studies demonstrate the feasibility of injection-molding quality monitoring and control based on pvT characteristics.
In the injection-molding process, cooling accounts for most of the cycle time; thus, an appropriate cooling time can greatly affect the quality of the parts and the manufacturing efficiency. The cooling time is often determined using part thickness, thermal diffusivity, and melt, mold, and ejection temperatures; however, the settings are approximated based on the heat transfer equation (Eq. [1]). For instance, the cooling time for a plate-like part of uniform thickness can be estimated as follows [13]: where α is the thermal diffusivity, T 1 is the melt temperature, T 0 is the mold temperature, and T s is the average part temperature at ejection.
The formula provides an initial estimate of the cooling time but does not consider dynamic changes in cavity temperature during cooling, which can result in different shrinkage rates. This situation can cause a significant difference between the estimated cooling time and the necessary cooling time for various quality requirements, which must be adjusted using operators.
The cooling time can also be determined by purchasing specialized hardware sensors or simulation software. For example, Gordon et al. [14] used a multivariate sensor installed on a flex bar mold system to estimate the ejection temperature. These sensors can provide valuable data but add complexity and costs to the mold. Software simulations for analyzing the steady-state injection-molding cooling time are limited due to the reliance on inadequate material properties [15]. Infrared thermography can be used to measure part temperatures in a plastics manufacturing environment and determine the dimensional stability [16].
The distribution of part thickness can affect the cooling time; thicker sections may require more cooling, typically through a single cooling rate in the coolant channel. Therefore, a local mold-temperature control design for part thickness is necessary to minimize the variation in shrinkage and the cooling time [17]. Furthermore, conformal cooling systems are often used to efficiently reduce warpage and cooling time [18].
Different cooling times affect part shrinkage. The longer the cooling time is set, the less shrinkage the final part undergoes [19,20]. However, the duration of minimum cooling time to produce a stable shrinkage is unknown and requires optimal quality and efficiency.
An increasing number of sensor applications have captured data to analyze the relationship between changes in the polymer melt during injection molding and product quality.
Most of the sensors used in injection molding are pressure and temperature sensors, and the information obtained from the sensors is a curve of pressure or temperature versus time. For generating pvT diagrams, more postprocessing is required to determine the relationship between pressure and temperature. This study uses temperature and pressure to determine pvT relationships for polymer melts during injection molding. For quality prediction, the specific volume change rate is obtained by calculating the specific volume of polymer melt from the moment of mold opening to the cooling to room temperature. By using the specific volume rate, the optimum cooling time that meets the quality of the part can be determined. Temperature sensors are usually categorized into thermocouple temperature sensors and infrared temperature sensors. Thermocouple sensors are widely used; however, their response rate is low, and the sensor can only detect the surface temperature of the polymer melt. Infrared temperature sensors can detect the temperature of the polymer melt but are expensive and prone to damage [21,22]. To prevent the use of costly infrared sensors, this study employed computer-aided engineering (CAE) to simulate temperature to generate the pvT diagram. Finally, a correlation analysis was conducted between the part shrinkage rate and specific volume change rate from the opening of the mold to the cooling to room temperature. If there is a high correlation, an optimal cooling time for the part quality can be determined indirectly.

Generation of pvT diagrams
The proposed method for generating pvT diagrams consists of four steps: (1) input and preprocessing of actual cavity pressure, (2) input and preprocessing of simulated temperature, (3) calculation of specific volume by using the modified Tait model, and (4) plotting of pvT diagrams.
1. Input and preprocessing of actual cavity pressure: The measured cavity pressure profile was sampled at 1 kHz and averaged over every 10 continuous points to eliminate measurement noise. The average pressure was then used in the modified Tait model as the pressure value (P). 2. Input and preprocessing of simulated temperature: This study used the commercial simulation software Mold-ex3D to predict the polymer melt temperature profile that corresponded with the position of the pressure sensor, which detected the actual cavity pressure in Step (1).
The temperature was then used as the temperature value (T) in the modified Tait model. 3. Calculation of specific volume by using the modified Tait model: The modified Tait model for calculating the specific volume ( V ) is listed in Eqs. (2)(3)(4)(5)(6)(7)(8), where V 0 is the reference specific volume at a given condition (e.g., 5 , and b 6 are the fitting parameters; V t is the abrupt volumetric change of a semicrystalline material around the melting point; T t is the abrupt change in viscosity of the material around the transition temperature; and b 7 , b 8 , and b 9 are the material parameters, which equal 0 for amorphous materials [23]: 4. Plotting of pvT diagrams: The simulated temperature and actual cavity pressure of the polymer melt flowing along the cavity were calculated in Step 3 to generate the corresponding specific volumes, which were then plotted as a pvT diagram. Changes in the specific volume of polymer melt could be predicted using pvT diagrams, which could then reveal the final dimensions of the injection-molded parts. Figure 1 shows a typical pvT diagram, where V 1 and V 0 are the specific volumes of polymer melt when the mold opened in the cooling stage and when the mold cooled to room temperature, respectively. The difference between V 1 and V 0 was defined as the specific volume difference (ΔV), and the rate of change (r v ) could be calculated using Eq. (9). Therefore, the greater the change rate of the specific volume of a plastic part that is cooling to room temperature, the greater the volumetric shrinkage of the plastic part, which may lead to a dimensional reduction in the quality of the part. The shrinkage ratio (r s ) of the part size can be calculated using Eq. (10), where d m and

Correlation analysis of specific volume change rate and shrinkage rate
Pearson correlation and Spearman's rank correlation analysis were used in the study to evaluate the specific volume change rate obtained from the pvT diagram and the shrinkage rate of plastic parts. Higher correlations that corresponded with various sensing locations were used for polynomial regression analysis, and a second-order polynomial function was used to generate the relationship between the two objects. The Pearson correlation analysis formula is shown in Eq. (11), using which 25 pairs of data on the specific volume change rate (X i ) and the actual shrinkage rate (Y i ) are calculated to generate the correlation coefficient (r). r ranged between − 1 and 1, and an absolute value of r greater than 0.75 indicated a high correlation. Conversely, an absolute value of r less than 0.3 indicated a low correlation. For Spearman's rank correlation analysis, the rank correlation coefficient ρ shown in Eq. (12) was significant at values greater than 0.4. The Z score in this study was set to α = 0.05. For the evaluation of the secondorder polynomial regression analysis shown in Eq. (13), the fitting criteria R 2 shown in Eq. (14) was used: 3 Experimental setup  Fig. 1 Typical pressure-volume-temperature diagram reveals change in specific volume when mold opens and cools at room temperature plate had a rectangular hole. The diameter of the sprue of the injection mold was between 3.5 and 6 mm, and the thickness of the fan gate was between 1.5 and 2 mm. The cooling system consisted of four 8-mm-diameter linear channels, and two channels each were used for the male and female mold plates. The processed polymer material acrylonitrile butadiene styrene was an amorphous material (PA756, Chi-Mei, Tainan, Taiwan). Figure 2 displays the measurement position of the part width, where W1, W2, and W3 represent the injection-molding quality of the 2-mm-thick area, and W4 and W5 represent the quality of the 2.5-and 1.5-mmthick areas, respectively. The width was measured using a three-coordinate measuring machine (CRYSTA-Apex S 7106, Mitutoyo Corporation, Kawasaki, Japan). This study used a high-precision all-electric injection-molding machine (S2000i100B, Fanuc Corporation, Shibokusa, Japan) with a maximum clamping force of 1000 kN, screw diameter of 28 mm, a maximum injection pressure of 240 MPa, and a maximum injection speed of 500 mm/s. An oil-heating mold-temperature control device (YBMI-200-20, Taiwan Yann Bang Electrical Machinery, Taichung, Taiwan) was used for the experiment. Figure 2 presents the positions of seven pressure sensors installed in the injection mold, where P1, P2, and P3 measure the pressure history at the bottom of the sprue, front of the fan gate, and back of the fan gate, respectively; P4 and P5 measure the pressure history when the 1.5-mm-thick area is filled; P6 measures the pressure history when the 2.5-mmthick area is filled; and P7 measures the pressure history when the 1.5-mm-thick area is filled. The pressure sensors used in this study are of the pin type (SSB04KN10 × 08H, Futaba, Mobara, Japan). For the comparison with the simulated temperature of the polymer melt, this study also used two infrared sensors to accurately measure the polymer melt temperature (EPSSZL-4.0 × 50.25 P050, Futaba, Mobara, Japan.) The pin-type temperature sensors can be jointly installed with pressure sensors. Therefore, the locations of the two temperature sensors mounted on the female mold plate, P3 and P7, were used to measure the near-gate and far-from-gate polymer melt temperature history, respectively. The pressure and temperature signals were collected using a data acquisition card (USB-6343 DAQ, National Instruments, Austin, TX, USA).
This study used Moldex3D 2020 Studio (R4OR version, CoreTech System, Zhubei, Taiwan) for mold flow simulation, which provides a three-dimensional simulation of polymer melt compression in the barrel, allowing the accurate prediction of injection pressure. Moreover, a model of a full mold base with a hybrid mesh for simulation was used. The total number of mesh elements was approximately 10 million.  Fig. 3a and b illustrate the cavity pressure distributions relative to cooling times set to 25 and 30 s, respectively. The results indicated that the residual pressure distribution of the pressure sensors P3-P7 was uniform in both settings. Thus, a minimum cooling time of 25 s is suggested.

Effect of cooling time on shrinkage rate
This study validated the injection-molded parts at various cooling times. Five injection-molded part samples corresponding to various cooling time settings were placed 3 days prior to the dimensional measurements of W1, W2, W3, W4, and W5. The results indicated that longer cooling time settings benefitted part shrinkage and resulted in more consistent dimensions between the part and cavity. Figure 4 shows the shrinkage rates of widths relative to various cooling times. Longer cooling times produced lower shrinkage rates. Additionally, W1, W2, and W3, which were located near the gate or in the middle of the flow path, had a sufficient packing effect and a relatively lower shrinkage rate than W4 and W5. All widths remained stable at 25 s of cooling except for a thickness of 2.5 mm, which required more cooling. Therefore, the optimal cooling time was 25 s, which is in good agreement with the recommended value obtained after referencing the residual pressure distribution among sensors.

Calculation of specific volume by using pvT diagrams
In this study, the specific volume of the polymer melt was calculated using pvT diagrams, which required the actual cavity pressure and the simulated temperature of the polymer melt. However, due to the lack of accurate information regarding the machines, molds, processed resins, auxiliary devices, environmental noise, and human factors that affect the quality of molded parts, simulated molding differs considerably from actual molding. Other sources of error, such as check-ring leakage, mold rigidity, and the simplicity of the mathematical model, can also cause discrepancies. In the simulation, the process parameter settings were adjusted to meet a pressure profile that was consistent with actual conditions [24]. For example, Fig. 5 shows the actual and simulated pressure profiles at a cooling time of 10 s, which were similar. The simulated temperature was then obtained under these settings. Three temperatures along the thickness in the simulation were recorded: the average temperature, peak temperature, and central temperature. In the case of P3, 11 sensing nodes were placed at a thickness of 2 mm in the simulation, and the predicted temperatures were calculated to obtain the average, peak, and central temperatures.
To compare the simulated temperature with the actual temperature, high-resolution infrared temperature sensors were installed at positions P3 and P7. Figure 6 shows the actual and simulated temperature profiles at a cooling time of 10 s. The infrared sensor at P3 demonstrated a relatively high peak value compared with the simulated central temperature before and after mold opening. Conversely, the infrared sensor at P7 had a relatively low peak compared with the simulated central temperature before and after mold opening. In particular, the peak temperature of the infrared sensor was close to the simulated central temperature. This    Figure 7a shows the pvT diagram generated using the actual pressure and CAE average temperature when the cooling time was set to 10 s. P4 and P5 were almost identical in the pvT diagram because they were both located at a thickness of 2 mm, which did not affect the temperature and pressure. However, at a thickness of 2 mm, the near-gate P3 was effective at packing pressure and resulted in a differential course on the pvT plot. P6 at the thickest part (2.5 mm) had a temperature above 100 °C when the relative atmospheric pressure was zero, which indicated that the cooling rate was slow and resulted in a higher mold-opening temperature and larger specific volume. Therefore, a shorter cooling time resulted in a larger variation in specific volume with respect to thickness location. Longer cooling times favored a consistent specific volume distribution (Table 2). Figure 7b presents the pvT diagram generated using actual pressure and CAE peak temperature when the cooling time was set to 10 s. In this case, the peak temperature (250 °C) was considerably higher than the maximum average temperature (225 °C). However, the peak temperature dropped rapidly due to its location near the solid layer, which resulted in rapid heat loss. The curves in the pvT diagram for various sensing locations were similar when using the average temperature and the peak temperatures of CAE. The curve for P6 differed because the peak temperature of the thicker section was still high when the relative pressure was zero. Regarding the specific volume, a longer cooling time favored a consistent specific volume distribution, and thicker sections (e.g., P6) yielded a higher specific volume (Table 3). Figure 7c shows the pvT diagram generated using actual pressure and CAE central temperature when the cooling time was set to 10 s. The maximum central temperature was approximately 240 °C, and the rate of decline was relatively slow. Moreover, the pvT plots in Fig. 7b and c are similar. The pressure distribution at each sensing location was more consistent with the CAE central temperature, particularly during the filling stage. Conversely, during the packing stage, the pressure distribution at each sensing location was more consistent with the CAE peak temperature. Table 4 shows similar specific volume results during mold opening as Tables 2 and 3. In conclusion, pvT plots generated using the CAE average, peak, and central temperatures produced similar results. The curve that was thicker and farther from the gate location P6 was relatively different from other curves, and the effect of cooling time on the pvT plot was relatively significant before mold opening. Changes in the cooling time resulted in specific volume and temperature differences; the longer the cooling time was, the lower the temperature and specific volume were when the mold was opened. Figure 8a and b present the resulting pvT plots for P3 and P7 at the actual pressure and infrared temperature, respectively. At P3, the temperature after the packing phase was approximately 200 °C, which was higher than the simulated average temperature (175 °C) and peak temperature (140 °C), resulting in a larger specific volume when the mold was opened. Before mold opening, the temperatures at P3 and P7 were approximately 100 °C and 60 °C, respectively, and the thinner part (P7) cooled faster than the thicker part (P3). Table 5 also shows that the part at P3 had a larger specific volume when the mold opened.

Correlation analysis of specific volume change rate and shrinkage rate
Tables 6, 7, and 8 indicate that the Pearson and Spearman's rank correlation coefficients indicated a high correlation between specific volume change rate and actual shrinkage rate, except for P3. Consequently, a second-order polynomial regression analysis could describe the mathematical relationship between the specific volume change rate and the actual shrinkage rate, where R 2 ranges from 0.86 to 0.94. When considering the correlation coefficient between the specific volume change rate and the actual shrinkage, the CAE central temperature was the most suitable for calculating the specific volume change rate. Additionally, the specific volume change rate was consistent with the infrared temperature. Therefore, the following evaluations are based on the use of CAE central temperature. Table 9 presents the results of the correlation coefficients between the change in specific volume and the actual shrinkage rate, in which the actual temperature was measured using the infrared sensor. Both P3 and P7 demonstrated a high correlation with a large increase compared with the specific volume change rate calculated using the CAE central temperature. Moreover, a second-order polynomial regression between the predicted specific volume change rate and the actual shrinkage rate yielded a better fit. Figure 9a-e present the second-order polynomial curve fitting of the shrinkage rates of W1, W2, W3, W4, and W5 versus the specific volume change rates of P2, P4, P5, P6, and P7, respectively. The results indicated that the width shrinkage rate can be predicted using the specific volume change rate produced by the nearby pressure sensor.    Figure 10 shows the specific volume change rate with CAE central temperature with respect to various cooling times. The results indicated that a longer cooling time reduces the variation in the specific volume change rate. Except for W1 and W4, the specific volume remained stable when the cooling time was not less than 25 s. Furthermore, Fig. 11a shows similar results for the actual warpage rate. W1 near the gate area exhibited a relatively high temperature, which resulted in a slower cooling rate and a longer cooling time to reach stable dimensions. More time was also required for the thickest area, W4 (2.5 mm), to reach a stable size. When considering the part quality and production efficiency, this study recommends a cooling time of 25 s. Although W5 exhibits a lower specific volume change rate than W4 in Fig. 10, the opposite is true of the shrinkage rate in Fig. 11a. The calculation of the shrinkage rate differed from the calculation of the specific volume change rate. The reasons for the difference in the specific volume change rate and shrinkage rate are manifold. For example, the specific volume change rate corresponded to linear shrinkage, and the shrinkage rate corresponded to volumetric shrinkage. The shrinkage of the plastic part was nonlinear, and the true size could not be predicted in the pvT diagram. Additionally, part warpage can make width measurements difficult. Figure 11b presents the CAE simulation of the volumetric shrinkage rate relative to the various cooling times. The results revealed that the volumetric shrinkage rate of W4 was relatively high and that of W5 was low, and this trend was consistent with that shown in Fig. 10. The volumetric shrinkage rate at W2 and W3 was greater than that at W1, which was also consistent with the results shown in Fig. 11a. Therefore, when the cooling time was set to 25 s or higher, the volumetric shrinkage rate was stable. However, the cooling time recommended by CAE is only 17 s, which can result in unstable shrinkage of injection-molded parts (Fig. 11b).
When the cooling time was calculated using the traditional method (Eq. 1), a thickness of 2.5 mm resulted in a melt temperature of 220 °C, mold temperature of 60 °C, ejection temperature of 70 °C, thermal diffusivity coefficient (α) of 0.12 mm 2 /s, and a recommended cooling time of 16 s. These results were consistent with the CAE results and led to a greater-than-actual shrinkage of the width.

Conclusion
This study generated a pvT diagram by using the actual pressure signal and the simulated temperature of a polymer melt in a mold cavity. Moreover, expensive infrared temperature sensors were not used, and the cavity pressure exhibited relatively high accuracy in the CAE simulation. For example, the specific volume change rate of a flat plate with uneven thickness can be calculated to estimate the shrinkage rate of the part size. Additionally, a cooling time that ensures a high quality of the part is recommended based on the pvT information. The results can be summarized as follows: 1. This study verified that polymer melt temperature predicted using CAE software can be replaced with actual cavity pressure to generate a precise pvT diagram. The central temperature at a high thickness (e.g., 2.5 mm) is relatively consistent with the actual temperature, which was verified using an infrared sensor. 2. The calculation of the specific rate change rate in the pvT diagram of the CAE simulated temperature indicated that all the sensing positions at P4, P5, P6, and P7 are highly correlated with the actual shrinkage rate. Regarding P3 near the gate area, the correlation is moderate to high. Furthermore, the CAE central temperature can best estimate the specific volume by using the modified Tait model. 3. The correlation between the specific volume change rate and the shrinkage rate is high when the actual temperature was measured using the infrared sensor or the CAE simulated temperature. Specifically, the correlation between the two variables at P3 near the gate area greatly improved when the actual temperature was used. 4. Experimental results indicated that a longer cooling time stabilizes the specific volume change rate of plastic parts. Additionally, the lower specific volume change rate indicates a lower width shrinkage rate as the injection-molded part cools to room temperature. 5. A second-order polynomial regression model describing the relationship between the specific volume change rate and the actual shrinkage rate is established to predict the part size by using the proposed economic pvT diagram. 6. A cooling time of 25 s produces a stable condition.
Similarly, the width shrinkage rate at a cooling time of 25 s or longer is stable. Therefore, a cooling time of 25 s is recommended when considering the part quality and productivity. Both the traditional formula and CAE Data availability The authors confirm that the data supporting the findings of this study are available within the article.

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Competing interests
The authors declare no competing interests.