Measurement Techniques and Methods for the Pressure Coefficient of Viscosity of Polymer Melts

Liao, Yang; Hu, Yeyuan; Tan, Yunxiang; Ikeda, Kosuke; Okabe, Ryoji; Wu, Ruifen; Ozaki, Ryota; Xu, Qingyan

doi:https://doi.org/10.1155/2023/2020247

Advances in Polymer Technology

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Review Article | Open Access

Volume 2023 | Article ID 2020247 | https://doi.org/10.1155/2023/2020247

Measurement Techniques and Methods for the Pressure Coefficient of Viscosity of Polymer Melts

Yang Liao,¹Yeyuan Hu,¹Yunxiang Tan,¹Kosuke Ikeda,²Ryoji Okabe,²Ruifen Wu,²Ryota Ozaki,²and Qingyan Xu¹

Academic Editor: Rafael Munoz-Espi

Received18 Jul 2023

Revised20 Nov 2023

Accepted01 Dec 2023

Published18 Dec 2023

Abstract

Viscosity is a prominent rheological property of polymer, which is affected by temperature, pressure, shear rate, molecular structure, and other factors. Despite the importance of the pressure effect, there remains a paucity of investigations on the dependence of pressure on viscosity compared with other factors, such as shear rate and temperature. Previous research has established that the correlation between pressure and viscosity is usually expressed by the pressure coefficient. In this paper, different measurement techniques and methods for the pressure coefficient of viscosity of polymer melts are reviewed and evaluated on the basis of published experimental data. The capillary rheometer with a pressurized exit chamber is widely employed because of its accuracy and simple use. Besides, the accuracy and relationship of pressure coefficients determined by different methods are discussed.

1. Introduction

Extreme processing conditions such as high pressure, high temperature, and high deformation rate often occur during polymer processing, such as injection molding, film blowing, and extrusion [1]. These processing conditions have an important influence on polymer viscosity. A great deal of previous researches studying viscosity has focused on temperature [2–4] and shear rate [5–7], while the investigations on the effect of pressure on viscosity are still quite scarce. This may be attributed to the inherent difficulties in high-pressure rheological measurements and the instable measurement results by different measurement methods. However, the pressure dependence of viscosity is of great importance in polymer processing, particularly in the selection of injection molding pressure [8–10]. As early as 1957, Maxwell and Jung [11] found that the viscosity of polymer could increase by one or two orders of magnitude when pressure increased from atmospheric pressure to more than 100 MPa. Westover [12] proved that the apparent viscosity of polyethylene increased 10-fold, and the viscosity of polystyrene increased over 100-fold when the hydrostatic pressure changed from 14 to 172 MPa. Therefore, the pressure dependence of melt viscosity cannot be ignored at high pressures.

Usually, the viscosity varies exponentially with pressure. The pressure coefficient β determined by the Barus equation can be used to express the pressure dependence of viscosity [13] as follows:where is the viscosity at pressure p, is the viscosity at atmospheric pressure, and p is the hydrostatic pressure of melt.

Values of pressure coefficients for various polymers have been reported, which are typically estimated to be of the order of 10⁻⁹–10⁻⁸ Pa⁻¹ [14, 15]. The specific pressure coefficient values of PE, PP, PS PMMA, PαMASN, PC, and ABS are shown in Table 1. The molecular structures of these polymers are displayed in Figure 1. Obviously, pressure coefficients of polymers with bulkier backbones, like PS or PαMSAN, are always larger than those of polymers with simple structures, such as PE or PP. It is well known that side groups or bulky backbone groups will stiffen the polymer chain and reduce its flexibility, resulting in a large free volume between molecular chains [10, 15]. When pressure increases, the amount of free volume available between polymer chains decreases. The less the free volume, the stronger the intermolecular interaction [16, 17]. As a result, a corresponding increase in intermolecular friction causes a subsequent rise in the viscosity.

In addition, the pressure coefficients show large variations even for the same polymer. For example, some investigations show that the pressure coefficient decreases with increasing shear rate, while shear rate independent values have also been observed [19–21, 24]. The same holds for the effect of temperature. Both temperature-dependent [19, 21, 25] and independent [14, 20, 26] pressure coefficients have been reported. There is no doubt that the different measurement techniques and determination methods of pressure coefficient are the critical reasons for its great variations.

However, the existing pressure coefficient measuring procedures often require a large sample volume and a long test period. With the advancement of rheology, an increasing number of researchers are interested in small-scale testing, with the goal of enhancing data production efficiency through high-throughput experimentation (HTE) [27]. The HTE method has lately been used for viscosity measurement [28], interfacial rheology [29], material design [27], catalytic olefin polymerization [30], and other fields. Tammaro and Maffettone [31] have made a lot of cutting-edge work in characterizing polymer viscosity with custom-made microcapillary rheometers (μCR) [31–33]. Tommaro’s innovative gas-pressurized multipass μCR is able to execute tests in parallel with only a few milligrams of material to reduce test time [33]. Furthermore, the ability to adjust the pressure in the exit chamber allows for high-pressurized rheological experiments, opening up a new study avenue for measuring the pressure coefficient of viscosity.

In this paper, a concise review is given about the characteristics of various measurement techniques and methods for the pressure dependence of viscosity. The accuracy and relationship of pressure coefficients determined by different methods are also discussed on the basis of published experimental data.

2. Measurement Techniques

Since Maxwell and Jung’s [11] ground-breaking work on the pressure dependence of viscosity in the 1950s, numerous measurement techniques have been developed, which can be primarily divided into two categories: one involves rheometers based on drag flow, and the other involves rheometers based on pressure-driven flow [34].

2.1. Drag Flow Rheometers

Drag flows are generated by sandwiching the sample between two parallel plates and moving one plate relative to the other [20, 34]. Rotational rheometers, rotating cylinder viscometers, and sliding plate rheometers are common types of drag flow rheometers. Pressurized versions of these rheometers are designed to investigate the effect of pressure on viscosity. In 1967, Hellewege et al. [35] investigated the rheological behavior of polystyrene using a high-pressure rotational viscometer with pressure up to 150 MPa. The results demonstrated that the viscosity of polystyrene was much more reliant on pressure compared to polyolefins. Also, they found the rheological behavior of melt at low shear rates is consistent with the linear viscoelastic theory. Cogswell et al. [36] used a pressurized Couette–Hatschek rotating cylinder viscometer capable of operating at pressure ranging from atmospheric pressure to 175 MPa. They discovered that polymers with benzene rings as the main chain had larger parachor (V) values and that the viscosity of polymer became much more sensitive to pressure as V increased. However, it is laborious to load samples and time-consuming to clean the instrument due to its complex structure.

In 1999, Koran and Dealy [20] designed a high-pressure sliding plate rheometer capable of running at pressure up to 70 MPa and temperature up to 225°C to investigate the effect of pressure on the viscosity of polymer melts. An inert liquid that has no effect on polymers is used as the pressurized medium instead of gas. Because gas may dissolve in the melt under high pressure and thus affect the rheological properties of polymer melts [37]. In this design, the deformation of the melt is independent of pressure; thus, the pressure and the deformation rate can be controlled separately. The shear stress is measured using a shear stress transducer mounted in the middle of a fixed plate, as shown in Figure 2. The pressure is obtained using a hand pump, which can pump the inert fluid into the cavity. It is believed that the pressure of the melt is equivalent to that of the inert liquid. This high-pressure sliding plate rheometer was later used to investigate the effect of pressure on the viscosity of HDPE and PS by Park and Dealy [38] and Park et al. [39]. Since the test is often carried out under a parallel disk with a diameter of 25 mm and a gap of 1 mm, the sample volume is about 0.5 cm³.

The advantage of a drag flow rheometer is that the generated flow is rheologically simple. In other words, the sample is subjected to uniform pressure and shear rate, making direct data processing without any corrections [10, 22]. Besides, it is beneficial to investigate the rheological properties of polymer melts at low shear rates or transient flows [15]. However, its applicability is restricted by the relatively small shear rate ranges and intricate mechanical configurations [10]. Therefore, in order to study the actual rheological behavior of polymers at high shear rates close to polymer processing, rheometers based on pressure-driven flow are more widely used.

2.2. Pressure-Driven Flow Rheometers

Slit rheometer and capillary rheometer are the two primary types of pressure-driven flow rheometers, which use the pressure difference between the entrance and exit of the flow geometry as the driving force. The difference between the two rheometers lies in the geometry of the die, where the slit is rectangular, and the capillary is circular. The high viscosities of polymers make it necessary to apply significant pressures to drive the flow through the die [34]. The effect of pressure on viscosity may cause nonlinear pressure distribution along the die (see Figure 3) [15, 40, 41]. Therefore, the nonlinearity of the pressure profile can be used to determine the pressure dependence of viscosity.

2.2.1. Traditional Standard Rheometers

For slit rheometer, pressure transducers can be flush mounted on the surface of a rectangular die, making a direct measurement of pressure profile. In 1983, Laun [42] designed a high-pressure slit rheometer with a pressure of up to 200 MPa by attaching the slit die to the barrel of the capillary viscometer (see Figure 4). The height and diameter of the barrel are 200 and 12 mm, respectively, which means that the volume of the loaded sample is about 23 cm³. The impact of pressure on viscosity and entrance pressure loss can be determined by data from three pressure transducers mounted along the die and one in the barrel. Hay et al. [41] presented an approximate treatment of a fluid flowing through a slit die. They found that the pressure coefficient estimated by the curvature of the pressure profile is unreliable if viscous heating is not taken into account. Kadijk and Van Den Brule [21] measured the pressure-dependent viscosity of PP, ABS, and PS by a slit viscometer. Results showed that the pressure coefficient for ABS and PS was pressure-independent but declined with pressure for PP.

For capillary, however, the curvature of the circular die is too high to mount pressure transducers. Although the pressure profile cannot be directly measured, it can be deduced from the Bagley plot, whose curvature is used to estimate the pressure-dependent viscosity by Dudvani and Klein [43]. Nevertheless, the nonlinearity of the pressure profile can also be caused by many other factors, such as the effects of viscous heating [41, 44], wall slip [45, 46], and molecular reorientation [47, 48]. As a result, the pressure coefficient calculated from the nonlinear pressure profile is relatively reliable only for polymers with large pressure-dependent viscosities but unreliable if these effects are disregarded [10].

2.2.2. Rheometers with Counter-Piston Devices

Another direct measurement of pressure-dependent viscosity can be achieved by two simple modifications of pressure-driven rheometers. One is adding a counter piston to the downstream of the capillary die; another is adding a pressurized exit chamber. The schematic of the first modification can be seen in Figure 5(a). It first appeared in the 1950s when Maxwell and Jung [11] mounted the die between two cylinders. The plunger in each cylinder was hydraulically operated to pressurize the melt. Similarly, Westover [12] and Ito et al. [49] employed comparable devices with pressures as high as 170 and 100 MPa, respectively. In 1978, Karl [50] reduced the pressure on one piston to generate a flow through the capillary according to the back pressure principle. The maximum pressure of this device can reach 500 MPa. Mackley and Spitteler [51] used a multi-pass rheometer that could operate at steady and oscillatory modes to study the pressure-dependent rheology of LLDPE. By using servo-hydraulic control, the two pistons of the rheometer can run separately or jointly, allowing the measurement not only at a constant shear rate but also at constant shear stress. Similar to the counter-piston device, Park et al. [22] designed a counter-pressure nitrogen rheometer. As Figure 5(b) depicts, the upper gas chamber and counter-pressure reservoir are sealed and filled with nitrogen. Nitrogen pressures in the chamber and reservoir can be separately regulated by precision pressure calibrators. A sample of about 12 cm³ is required because the height of the barrel is 70 mm and the diameter is 15 mm. Apart from the capillary rheometer, a counter piston is also used in slit rheometer [21]. Son [52] reported a dual-piston rheometer with two chambers connected by a capillary or a slit. One piston moved at a constant speed while the other was kept under control at a constant pressure. However, it takes time and requires numerous attempts to loosen and tighten the adjustment screw valve in order to obtain a flow curve at a constant downstream pressure. In summary, the typical characteristic of the rheometer with a counter-piston device is independent control of melt pressure at both ends of the die, which allows constant shear stress control. Additionally, the pressure dependence of viscosity can be accurately measured in common ranges of shear rates and pressures in polymer processing. Besides, the sample consumption is small because polymer melt can flow back and forth in two chambers. However, the use of this precise pressure control device is limited due to its high cost and difficult operation and maintenance.

(a)

(b)

2.2.3. Rheometers with Pressurized Exit Chambers

For the second modification, a conical or needle valve is used to regulate the pressure at the exit of the die; thus, a hydrostatic pressure can be established in the capillary. The structure is schematically depicted in Figure 6. It comprises two pressure transducers and a valve on the downstream chamber. The overall pressure drop is changed by adjusting the valve as the piston descends at a constant speed. This method provides apparent viscosity data at a constant shear rate, and the constant shear rate pressure coefficient can be obtained by direct analysis of the data. Baker and Thomas [53] used a rheometer with a pin for adjusting the back pressure to study the effect of hydrostatic pressure on the flow properties of various polymer melts (PC, ABS, PP, HDPE) from ambient pressure to 200 MPa. Driscoll and Bogue measured the viscosity of PS at pressure as high as 124 MPa and shear rate between 1 and 100 s⁻¹ by use of a back pressure-regulated capillary rheometer with a needle valve [54]. They also developed a model based on a temperature–pressure–shear-dependent time constant and a pressure-dependent elastic modulus. Later, more investigations using a piston-driven capillary rheometer with a conical valve or a needle valve were reported by Chakravorty et al. [55], Laun [56], Carreras et al. [57], Sedlacek et al. [19], Hausnerova et al. [58], Hausnerova et al. [59], Park et al. [22], Cardinaels et al. [16], Couch and Binding [14], Binding et al. [24], Raha et al. [60], Li et al. [61], and Aho and Syrjälä [10]. The main benefits of this approach are the improved accuracy and slight modifications to the standard capillary rheometers. So, the volume of the loaded sample is the same as the standard capillary rheometer. However, like the inherent drawbacks of all capillary rheometers, which are manifested in the uneven pressure and shear rate in the die as well as the unavoidable entrance pressure drop, varying pressure complicates data processing. Despite the shortcomings involved, this method is commonly reported in the literature because it strikes a reasonable balance between complexity and accuracy [10].

(a)

(b)

3. Determination of Pressure Coefficient

3.1. Pressure Coefficient Calculated by Nonlinear Pressure Profiles

For traditional capillary rheometry, the pressure coefficient can be inferred from Bagley plots, which are plots of pressure drop versus the ratio of length over diameter (Figure 7). Hatzikiriakos and Dealy [45] noted that the data followed straight lines at low shear rates but did not lie on straight lines at high shear rates.

The force of a microsegment fluid in the capillary is investigated. If it is assumed that there is no wall slip and the flow is fully developed steady-state laminar flow, the momentum balance equation along the capillary axis is as follows:where is the wall shear stress at pressure p and R is the radius of capillary. According to Equation (1), shear stress is exponentially dependent on pressure, which can be expressed as follows:where is the wall shear stress at atmospheric pressure, and is the viscosity at atmospheric pressure. According to Equations (2) and (3), one gets the following:

Integral of the pressure of melt and capillary length can be expressed as follows:where is the pressure at the entrance of die, is the pressure at the exit of die, and L is the length of die. Assuming the exit pressure is atmospheric pressure, the following equation can be obtained, that is,where D is the diameter of die. Then, the entrance pressure can be written as follows:

By expanding the series of Equation (7) to the quadratic term, one can obtain the following:

Due to the entrance pressure loss, the relation is as follows:holds, where is the total pressure drop and is the entrance pressure loss. Then Equation (8) can be simplified as follows:

According to Equations (8) and (10), the pressure coefficient can be then written as follows:

Therefore, the pressure coefficient can be obtained according to Equation (11) by parabolic fitting of the Bagley plot. This method has been used by Laun [42], Dudvani and Klein [43], Penwell and Porter [62], Van Puyvelde et al. [63], and Kamal and Nyun [64].

3.2. Pressure Coefficient for Zero Shear Rate Viscosity

The pressure coefficient for zero shear rate viscosity is obtained from a series of zero shear rate viscosity data of melt at different pressures, which can be expressed as follows:where is the viscosity at zero shear rate. The value of is relatively stable and does not vary with shear stress and shear rate. Therefore, it can be taken as the thermodynamic property of polymer melts. However, it is challenging to obtain such data in the experiment [10].

3.3. Pressure Coefficient at Constant Shear Rate

The constant shear rate pressure coefficient is defined as follows:where p represents the hydrostatic pressure of melt. The experiment is carried out at a certain piston speed that corresponds to a constant apparent shear rate, and the exit pressure can be regulated by adjusting the valve of the counter-pressure chamber [52]. The wall shear stress is calculated from the pressure difference between the entrance and exit, taking the entrance pressure loss into account. The entrance pressure loss can be obtained directly from the orifice experiment or can be inferred from Bagley plots. The arithmetic mean of entrance and exit pressures is taken as the hydrostatic pressure of the melt in the capillary [10, 15]. As shown in Figure 8, the pressure coefficient is the slope of the straight line for the semi-log plot of viscosity and mean pressure. However, the pressure coefficient obtained by this method is not a real thermodynamic property because it varies with shear rate and temperature [14–17]. As Figure 9 shows, pressure coefficients decrease with the increasing shear rate and tend to a constant value in the power law region. Guo et al. [17] found that the relationship between pressure coefficient and temperature is similar for PET. Nevertheless, it is commonly used in literature due to its simple measurement and easy use.

3.4. Pressure Coefficient at Constant Shear Stress

The constant shear stress pressure coefficient is defined as follows:where p is the hydrostatic pressure of melt. Rheometers with counter-piston devices are often used to obtain constant shear stress pressure coefficients because pressures at the entrance and exit of the die can be controlled separately. As the pressure drop ΔP remains constant, the experiment is conducted at constant shear stress. The pressure coefficient can be calculated from the semi-log plot of viscosity and mean pressure (see Figure 10). Cardinaels et al. [16] noticed that the straight lines under varied shear stress are practically parallel, implying the constant shear stress pressure coefficient does not dependent on shear stress. As Figure 11 illustrates, the pressure coefficients of three polymers only show a slight fluctuation with shear stress, even in the small shear rate range deviating from the power law region. Therefore, in contrast to the constant shear rate pressure coefficient, the pressure coefficient at constant shear stress can be viewed as a thermodynamic property of the polymer melt.

Nevertheless, there is a correlation between pressure coefficient at constant shear rate and constant shear stress [37], which can be expressed as follows:

In the Newtonian regimes, viscosity is a constant, which means is equal to . However, the viscosity is dependent on the shear rate in the power law region, where viscosity can be written as follows:

Therefore, Equation (15) becomes the following:

The relationship between and is supported by Cardinaels et al. [16] and Park et al. [22].

3.5. Pressure Coefficient Calculated by Superposition Method

Considering the effect of temperature and pressure on viscosity, a shift factor is usually given by [40, 57] the following:where R′ is the gas constant, E is the activation energy, is the reference temperature, and is the reference pressure. Here, we only discuss the pressure coefficient at a single temperature, which is defined by the following:where is a horizontal shift factor exhibiting the response of pressure to time, which is larger than the vertical shift factor [16, 22]. The process of determining the pressure coefficient by time–pressure superposition method is similar to that of determining the activation energy by time–temperature superposition method [14, 52]. As Figure 12 depicts, by multiplying the shear rate and dividing the viscosity by the shift factor , flow curves of viscosity as a function of pressure can be shifted to a master curve, which can be interpreted by the following:

(a)

(b)

To determine the pressure coefficient, the shift factor should be obtained first. As Figure 13(a) illustrates, double logarithmic plots of viscosity and shear rate at different pressures are constructed. The viscosity curve at elevated pressure can shift to the curve at a reference pressure, and the horizontal shifting distance is ln . Once the shift factors under a series of pressures are calculated, the pressure coefficient can be obtained from the slope of the semilogarithmic plot of shift factor and pressure (see Figure 13(b)).

(a)

(b)

Cardinaels et al. [16] handled data from the rheometer with a pressurized exit chamber using the superposition method. However, since the mean pressure of melt cannot be directly controlled, substantial data processing is necessary. Therefore, viscosity curves at various fixed exit pressures should be recalculated to those at mean pressures. Son [52] used a double-piston rheometer where the mean pressure can be directly controlled to avoid the drawback of extensive data handling. Park et al. [22] used the superposition method to handle data from the high-pressure sliding plate rheometer. Viscosities at different pressures fall on a master curve at low shear rates ranging from 1 to 100 s⁻¹, implying that this method applies to the transition region from Newtonian to the power law region.

3.6. Pressure Coefficient Calculated from PVT Data

Pressure, volume, and temperature are crucial parameters in polymer processing, and the dependence of polymer viscosity on pressure and temperature can be obtained by PVT data [65]. Utracki has made great efforts to study the effect of pressure on viscosity based on the free volume method [66, 67]. The pressure-dependent viscosity treated as a function of free volume fraction has been proposed [68, 69], which can be written as follows:where is either the zero-shear viscosity or viscosity at constant shear stress, and are adjustment parameters, is an empirical value which is small and can be neglected here, h is the free volume fraction as a function of temperature and pressure: h = h(T, P). Utracki [67] described the linear relationship of 1/h and pressure at a given temperature:where is a characteristic parameter that can be calculated by fitting the PVT data to a modified Simha–Somcynsky equation of state (S–S) [70, 71]. Therefore, according to Equations (21) and (22), the pressure coefficient can be calculated by the following:

However, the pressure coefficient obtained by PVT data is not reliable for all polymers. Sedlacek et al. [68] reported that the master curves of zero-shear viscosity versus free volume fraction for PP and PS exhibit slight deviations due to the failure of zero-shear viscosity fitting. The PVT measurements should also be carried out in a wide range of temperature and pressure to obtain viscosity as precise as possible. Besides, can be predicted only in a single phase, and the treatment should change if a phase transition like crystallization happens [67].

4. Evaluation of Different Measurement Techniques and Methods for Pressure Coefficients

4.1. Evaluation of Different Measurement Techniques

For drag flow rheometers, a sample is subjected to uniform pressure and shear rate, resulting in direct data processing without any corrections. However, constrained by low shear rates, it cannot reach the high shear rates required for polymer processing. Therefore, pressure-driven rheometers that can operate at high shear rates are more widely used. Nevertheless, a sample is subjected to uneven pressure and shear rate, and the entrance pressure loss is not negligible, which requires appropriate corrections to correct pressure (Bagley correction) and shear rate (Rabinowitsch correction) [45, 72, 73].

Three main pressure-driven rheometers are often used, which can be classified as traditional standard rheometer, counter-piston rheometer, and rheometer with exit pressurized chamber. For a traditional rheometer with a slit die, pressure transducers can be mounted along the die to directly measure the pressure of melt. However, it requires a pressure transducer with high sensitivity, and the cleaning of the die is not convenient. For traditional rheometer with capillary die, however, pressure transducers cannot be mounted on the die owing to its high curvature. Considering this situation, a series of dies with different ratios of length and diameter are used to infer the pressure distribution along the die. For the counter-piston rheometer, the constant pressure drop between two ends of the die can be achieved by adjusting the hydraulic system, resulting in a direct measurement of the pressure coefficient at constant shear stress. However, the operation of a hydraulic system to keep constant mean pressure is not easy, which limits the widespread use of this device. For a rheometer with an exit pressurized chamber, only minor modification is required, and it is easy to operate. Besides, it provides a compromise between accuracy and complexity. Therefore, it is most reported in the literature. Table 2 summarizes the features of different measurement techniques.

4.2. Evaluation of Different Methods for the Pressure Coefficient

Although the method of calculating pressure coefficient by nonlinear pressure profile is simple and direct, it is not necessarily reliable. There are some inherent drawbacks of this method:

Pressure coefficients of polymers are typically of the order of 10⁻⁹–10⁻⁸ Pa⁻¹. So, the pressure profile measured is almost linear for those polymers with small pressure coefficients, especially when the sensitivity of the pressure sensor is not high. If the nonlinearity becomes dominated, the parabolic approximation is no longer accurate because it will make the pressure coefficient overestimated. Besides, many factors contribute to the nonlinearity, such as viscous heating, wall slip, and molecular reorientation. If these factors are ignored, the calculated pressure coefficients will have experimental errors. Propagation of experimental and fitting errors should also be noted [15].

The pressure coefficient for zero shear rate viscosity can be taken as the thermodynamic property of polymer. However, zero shear rate viscosities at different pressures for various polymers are not easily obtained in the experiment.

The pressure coefficient at a constant shear rate can be directly calculated by the semi-log plot of viscosity and mean pressure. However, it is not a real thermodynamic property of polymer because it varies with shear rate, pressure, and temperature. It is exciting that the pressure coefficient at constant shear rate can be related to that at constant shear stress by power index, which can be regarded as the thermodynamic property. Therefore, it is commonly used in literature due to its simple measurement and easy use.

Time–pressure superposition method is often used, too. The pressure coefficient obtained by this method is almost the same as that at constant shear stress because the superposition method is equivalent to shifting the viscosity curves at constant shear stress.

The free volume concept can also be used to explain the effects of pressure on polymer viscosity. However, the pressure coefficient obtained by PVT data is not reliable for all polymers, and it is not clear under what conditions this method is reliable. Besides, the prediction of viscosities at different temperatures and pressures is valid only for a single phase. Table 3 summarizes the pressure coefficients calculated from different methods.

5. Conclusions and Prospects

In this paper, different measurement techniques and methods of determining pressure coefficient are reviewed and compared. It can be concluded that pressure-driven rheometers are more often used to study the effect of pressure on viscosity, particularly rheometer with exit pressurized chamber because it provides a reasonable compromise between accuracy and complexity.

Besides, the accuracy of pressure coefficients obtained by the superposition method and at constant shear rate and constant shear stress are relatively high. The pressure coefficient at constant shear stress, which is equivalent to the pressure coefficient obtained by the superposition method, is regarded as the thermodynamic property of polymer melts. What is more, pressure coefficients at constant shear rate and constant shear stress can be related to each other by power index. Therefore, pressure coefficients obtained by these three methods are interconnected. Nevertheless, pressure coefficients obtained by nonlinear pressure profiles and PVT data are less reliable due to the inherent limitation of the methods.

Many researchers have put their efforts into investigating the effect of pressure on viscosity, but somehow, the experimental data is still few compared with other external factors such as stress, shear rate, and temperature. That remains to be investigated. Besides, with the development of HTE, it is possible to measure the pressure dependence of viscosity quickly by a microcapillary rheometer, using only a little quantity of polymer. Therefore, future work can focus on small-scale measurement to meet the demands of HTE, which can be further used for artificial intelligence analysis.

Data Availability

The data supporting this review are from previously reported studies and datasets, which have been cited. The processed data are available.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

This work is supported by a grant provided by the Mitsubishi Heavy Industries.

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