In-line rheological monitoring of fused deposition modeling

Fused deposition modeling (FDM^®), also known as fused filament fabrication (FFF), is a 3D printing technique that deposits strands of a molten polymer that solidify into the final, desired shape as described in previous works [1–4]. FDM is an appealing process with low capital investment that allows users to quickly and easily create complex designs unattainable by traditional manufacturing processes. Despite the benefits and widespread use of FDM, there are many opportunities for improvement, including part strength [1–9], resolution [10–15], and material availability [16–18].^1,2 Many of these problems could be improved with advances to FDM process control. Mani et al. from the National Institute of Standards and Technology (NIST) call out the need for more real-time measurements and in-line analysis of additive manufacturing to develop process understanding and predictive models that can be used for real-time quality control [19]. Turner and Gold propose that regulating flow rate through process monitoring and control algorithms is one of the greatest opportunities for improving FDM resolution [10].

A first step to implementing the control algorithms proposed within the NIST paper and article by Turner and Gold is the development of appropriate on-line sensors; Huang et al. similarly state that one of the largest gaps in the field of additive manufacturing is the lack of sensors capable of providing relevant, real-time data [20]. The only sensors and feedback controls in most FDM printers are proportional-integral-derivative temperature controllers for the thermistors of the nozzle and build platform. The motion, positioning, and extrusion are controlled with open-loop stepper motors in over 95% of entry-level FFF printers [21]. While open-loop stepper motors offer many benefits, they are prone to skipping steps where the lack of feedback control results in inaccurate positioning and speed [22]. Weiss et al., therefore, included an encoder on the X and Y gantries to monitor the trajectory and accuracy of the positional motors, they improved the accuracy by using a PI controller, and they recommend incorporating more closed-loop motion control into FDM printers [21].

To combat some of these control and resolution problems, high-end FDM companies experimentally optimize their print conditions for their filament materials by applying an empirical program during the acceleration phases of a print [23]. While effective for a limited set of materials, this experimental optimization makes it very difficult and time-consuming to introduce new materials. To combat resolution problems due to dimensional inaccuracies of the filament, the high-end printer companies have described many forms of area measurement sensors that could provide feedback to the drive system to correct for feed-rate errors [24]. Greeff and Schilling took flow rate control one step further by not only measuring filament diameter but also monitoring the extrusion speed and the motor gear speed through visualization and image analysis to detect motor errors and slippage in addition to filament inaccuracies; they incorporated a control algorithm to decrease the deposition speed when slippage occurred [15].

New in-line sensors, like the ones described by Greeff and Schilling, are critical for understanding the processing mechanisms of FDM and confirming theoretical, predictive models. Sbriglia et al. incorporated sensors into the FDM process by stopping the printer in the middle of a job to add accelerometers, which inform users about the frequency response of the system [25]. Bragg sensors and thermocouples were added using a similar approach to monitor strain and temperature during the FDM process [26], but such methods of adding sensors into a part cannot be used for real-time control. The most common type of sensor studied in the additive manufacturing literature has been the nondestructive infrared (IR) camera for monitoring material temperature evolution during processing. The heating and cooling of the powder bed has been studied for selective laser sintering [19,27], and the initial extrudate temperature and the subsequent cooling process have been studied for FDM [8,28–30]. IR measurements have become popular in research because the transient cooling process of the FDM extrudate has been shown to be a critical parameter for predicting the strength of FDM parts [1,5,8,9,31–33]. Similarly, Phan et al. used an IR camera to compare against predicted melt temperatures within the nozzle. By utilizing off-line rheological measurements as well as the power supplied to the stepper motor that drives the filament feeding gears, they predicted melt temperatures that showed similar trends to those measured by the IR camera [34].

The original research that is the focus of this journal article adds to the development of in-line sensors for FDM by incorporating a filament feed-rate encoder, a variety of thermocouples, and a custom pressure transducer to develop an in-line rheometer. Rheology is the most critical material property for most polymer melt and liquid processing methods, including extrusion, injection molding, coating, and FDM. Rheology influences the temperature of the melt through viscous dissipation [35–38], determines the material output and flow rate [35–37,39], determines the shear rate and velocity profiles along the material flow path [35,39,40], creates the pressure profiles that determine the force and torque requirements of the process [10,15,36,37,39,41], and can influence the final material shape and strength [9,32,42–44].

Due to the practical and theoretical importance of rheology, on-line and in-line rheometers have been developed and extensively studied for many polymer processing techniques, including extrusion [45–47], injection molding [48], and coating processing [49]. Dealy describes on-line rheometers as taking measurements on melt that has been redirected away from the processing melt stream while in-line rheometers, such as the in-line FDM rheometer proposed in this article, offer the benefit of measuring viscosity directly in the process line with minimal delay in response [50]. On-line (or in-line) rheometers are typically validated by applying the appropriate rheological corrections and comparing the on-line measurements against off-line rheological measurements (a similar approach is also used in this article to verify the FDM in-line rheometer). For example, Köpplmayr et al. have developed methods for simultaneous on-line shear viscosity and on-line extensional viscosity on an extruder. They compared the on-line measurements to rotational and capillary rheology after applying Rabinowitsch corrections to the on-line and capillary data, and they found the on-line extensional viscosity compared well with off-line Sentmanat extensional rheometer (SER) data [47]. Coates et al. incorporated in-line pressure transducers to obtain accurate viscosity measurements for injection molding, which they recommended using, to implement closed-loop controls [48].

Similar but more complex rheological corrections have been applied to better understand the flow through capillary rheometers. Moshe et al. analyzed the transient behavior of capillary rheometers by accounting for pressure measurement errors, end effects, viscous dissipation, pressure effects, and compressibility [51]. Ansari et al. account for pressure effects, viscous dissipation, compressibility, and wall slip to model the pressure drop through the coni-cylindrical section of a capillary rheometer for low density polyethylene [52]. Different rheological corrections are critical for obtaining accurate viscosity measurements, depending on the system of interest. Despite all the on-line rheological studies for other polymer processes, no on-line or in-line rheometer has previously been developed for FDM printing.

Huang et al. state the importance of understanding the relationship between viscosity, in-line measurements, and additive manufacturing processes [20]. Similarly, a variety of other sources stress the importance of rheology to the FDM process [5,6,10,40,41,53–55]; nonetheless, there has not been any systematic study comparing material rheology to FDM outputs, such as mechanical properties, accuracy, shrinkage, roughness, or resolution. A variety of works have calculated pressure predictions through the FDM nozzle for a given viscosity profile [40,41,56–58]. Gilmer et al. model the relationship between the viscosity profile and FDM pressure and relate these to the practical problems of filament buckling and melt backflow [56]. Ramanath et al. have performed simulations to predict temperature and pressure profiles during the FDM process [40], but their analysis only used a power-law model, neglected viscous dissipation, and was never validated against real, in-line measurements. Despite the criticality of rheology, in-line rheometers have yet to be incorporated or studied on FDM to confirm the theoretical calculations or to study the influence of rheology on final properties.

To the authors' knowledge, this is the first work that describes the development of an in-line rheometer for the FDM process. This article describes the design of the custom pressure transducer and custom nozzle required for the in-line rheometer. The performance of the rheometer is validated against off-line rheological measurements as well as with an in-line comparison to a capillary rheometer. Finally, a variety of rheological corrections are considered and discussed, including end effects, non-Newtonian flow, viscous dissipation, pressure effects, and temperature corrections, which are described in Sec. III. While viscous dissipation was analyzed, it was found to be negligible, so viscous dissipation corrections were not applied to the data and are only discussed in the supplementary material [74]. Although a similar work attempted to measure the melt pressure based on the power sent to the feeding motor of an FDM printer [34], this approach is limited due to the nonlinear relationships between power, torque, and speed [22]; the approach was unable to be utilized as an accurate rheometer, so the original work presented in this article has developed a direct measurement of melt pressure, which is able to provide very accurate viscosity measurements.

An FDM printer (Lulzbot Taz 6 v 2.1, Loveland, CO) with a single extruder is used in this research. The printer is provided with a 0.5 mm diameter, brass nozzle as shown in Fig. 1. This printer model can achieve a maximum nozzle temperature of 298 °C, a maximum bed temperature of 120 °C, a maximum print speed of 200 mm/s (12 000 mm/min), a minimum layer height of 0.05 mm, and a maximum layer height of 0.5 mm. The printer is designed to process 3 mm diameter filaments, and its maximum print area is 280 × 280 × 250 mm.

A new nozzle system with multiple custom parts was produced to create the in-line rheometer on the printer hot end. First, a new nozzle was designed [Fig. 2(a)] with a pressure port hole for measuring the melt pressure. A load transfer column [Fig. 2(b)] was designed to be inserted into the pressure port within the nozzle [Fig. 2(c)] so that it is in direct contact with the melt. The load transfer column contains a thermocouple for measuring the melt temperature [Figs. 2(c) and 2(d)], and it transfers the load from the melt to a custom load transfer clamp, which is shown in Fig. 2(e). The clamp transfers the melt load to a 10-lb load cell (voltage from the load cell is then converted into melt pressure) as shown in the assembly of Fig. 2(f). The load cell is rigidly fixed to the nozzle system using a 1.5 in. long 6–32 screw as shown in Fig. 2(f).

The nozzles and load cell clamps were all produced from brass castings of lost wax patterns produced by a 3D printing process (Shapeways, New York, NY). The nozzle was designed to match the original printer nozzle as closely as possible in terms of both dimensions (Fig. 1) and materials (brass). To achieve the necessary accuracy of the critical dimensions [shown in Figs. 2(a), (c), and (d)], the custom nozzles were finished by drilling, reaming, and tapping with a 3-axis mill (EZ Trak DX, Bridgeport, Elmira, NY). The 10-lb load cell was an LCL-010 from Omega (Norwalk, CT), and all thermocouples used in this work were 2 m long, type K, glass braid insulated, 30 gauge thermocouples from Omega (Norwalk, CT). The melt thermocouple was attached to the load transfer column using a silver solder (SSQ-6, Muggy Weld, Olympia, WA) as shown in Figs. 2(c) and 2(d). Additional information about the design and assembly of all the in-line rheometer parts is provided in the supplementary material [74].

The final dimensions of one of the custom nozzles are shown in Fig. 3. It is critical to accurately measure the nozzle dimensions highlighted in Fig. 3 because the length and diameter of the nozzle tip are used to calculate the shear rate and shear stress [Eqs. (1) and (2)], and the area of the pressure port hole is used to convert the load cell force to a pressure (described in Sec. II E). The diameters of the nozzle tip and pressure port were initially measured via optical microscopy while the nozzle length was obtained by measuring the length of calibration pins inserted into the nozzle. These three dimensions were verified using x-ray computed tomography (Phoenix v|tome|x s, General Electric, Boston, MA) as shown in Fig. 3, which confirmed the manual measurements as well as the diameter uniformity of the bores. X-ray scans were run at 210 kV and 180 mA, creating images with an 8 μm voxel size.

Once designed and connected, the load cell voltage was collected in real time by transferring the voltage through a signal conditioner to a data acquisition device (DAQ), which transfers the data to a personal computer. Additionally, to measure the filament extrusion rate, the DAQ was counting the pulses from both the stepper motor that drives the filament feeding gears and an encoder in direct contact with the filament. Finally, four thermocouples (including the melt thermocouple shown in Fig. 2) were measuring temperature data, which were collected through a temperature input module. The system of controllers, sensors, conditioners, and data acquisition devices is sketched in Fig. 4. The DAQ is an NI USB-6210 from National Instruments (Austin, TX) with eight differential, 16-bit analog input channels, four digital input channels, four digital output channels, and two counters. The DAQ collected voltage data from the load cell at its fastest sampling rate of 250 000 Hz.

The temperature input module is an NI USB-9162 from National Instruments and converts the thermocouple voltage into temperature. An intrusive thermocouple monitors the apparent polymer melt temperature (Fig. 2), which is of critical importance as later discussed. Another thermocouple is in contact with the load cell and secured between the load transfer clamp and the load cell clamp; it is used to apply an in-line temperature correction to the load cell data as described in Sec. II E. The other two thermocouples do not directly influence the in-line rheometer, but they are important for monitoring the 3D printing process. The third thermocouple is suspended in the environment, parallel to the nozzle but located about 2 in. away from the nozzle to monitor the environmental temperature during a print. The last thermocouple is in contact with the top of the build plate as shown in Fig. 4 to monitor and confirm the accuracy of the build plate temperature.

The printer's filament drive is a NEMA 17 stepper motor (type MS17HD6P4150-1, Moons' Industries, Shanghai, China); it steps in increments of 830 pulses/mm of filament. A pulse is sent to the DAQ counter from the FDM microcontroller every time the motor is commanded to step, providing a measure for the commanded extrusion rate, but not necessarily the actual extrusion rate since the filament may slip in the drive during high pressure applications. Therefore, a filament displacement encoder was mounted to the printer to verify the volumetric flow rate (and subsequently, the shear rate) and account for skipped motor steps or filament slippage. The filament displacement encoder was provided by Mosaic Manufacturing (Toronto, Canada). An encoder housing and mounting system was 3D printed and attached to the printer frame so that the encoder is in constant contact with the 3 mm filament as depicted in Fig. 4. The filament encoder provides 7.53 pulses/mm of filament, which is not high enough resolution for high-speed measurements but successfully provides confirmation of the flow rate and shear rate through a moving average filter. It was experimentally determined that the encoder measures shear rates within ±5% of those measured by the stepper pulses when neither the motor is skipping steps nor the filament is slipping. The accuracies of the volumetric flow rates calculated from both the stepper motor and the encoder were verified by measuring the mass of printed material as described in the supplementary material [74]. The testing showed that both stepper motor and encoder were accurate with roughly the same tolerances as the cross-sectional area of the acquired filaments (about 4%). The consistency of the volumetric flow rate including variations in filament diameter is one of the more significant sources of error in both viscosity characterization and FDM process control as described in the supplementary material [74].

Matlab R2015b (Mathworks, Cambridge, MA) receives all the data from both the DAQ and temperature input module, then processes and saves the data. The data are collected using the Data Acquisition Toolbox and are analyzed using the Signal Processing Toolbox as well as the Statistics and Machine Learning Toolboxes. Temperature values are collected for all four thermocouples from the temperature input module every 0.5 s. Data are collected from the DAQ buffer every 4 ms, during which the DAQ has collected 1000 signals from each channel. The 1000 voltage values from the load cell were averaged together to provide a single voltage measurement every 4 ms, which was found to be an effective filtering method known as oversampling [22]. A shear rate is calculated from both the stepper motor and encoder counters as described in Sec. III. A low pass Butterworth filter was applied to the stepper motor shear rate data to reduce high frequency noise. The filter was a first-order Butterworth filter with a normalized cutoff frequency (cutoff frequency divided by sampling rate) of 0.01, which was tuned to prevent overshoot while maintaining acceptable time delays. Additional filtering is applied to set the shear rate to zero when it falls below a critical value of 10 s⁻¹ (the printer was observed to have difficulty in maintaining shear rates below 20 s⁻¹).

The load cell outputs an analog voltage from 0 to 10 V, so a calibration curve was created to convert voltage into pressure. A calibration fixture was cast in brass; the design is the same as the load transfer clamp shown in Fig. 2(e), but the 0.5 mm recess is replaced with a 3.1 mm diameter column with a platform with two holes at the top. A mass is hung from the calibration fixture via a nylon string threaded through the two holes as shown in Fig. 5 such that the nylon does not contact either side of the load cell. The mass was converted to a pressure using the area of the pressure port hole (dimensions shown in Fig. 3). The calibration was carried out by hanging the following masses from the calibration fixture: 0, 0.541, 1.127, 1.668, and 2.287 kg. The procedure with all five masses was repeated four times at four temperatures ranging from 21 to 100 °C. Temperature was recorded with the thermocouple which was clamped against the load cell, and the 20 data points were used to create a least-squares fit for the calibration curve for pressure as a function of voltage and temperature. The coefficient of determination (R²) was greater than 0.99, and an example of a pressure versus voltage calibration curve is presented in Fig. 8.

Both polycarbonate (PC) and high-impact polystyrene (HIPS) filaments were analyzed on the in-line rheometer. The filaments were PC Transparent from Ultimaker (Cambridge, MA) and HIPS Natural from eSun (Shenzhen, China), which were both chosen based on their reported diameter tolerances: 2.85 ± 0.05 mm for PC and 2.95 ± 0.05 mm for HIPS. Any diameter deviations will result in a shear rate error proportional to the squared diameter error as indicated by Eqs. (1) and (2), so these filaments were chosen because they had some of the best tolerances on the market. The glass transition temperatures of the PC and HIPS filaments are 112 and 98 °C, respectively, as measured by differential scanning calorimetry (Q2000, TA Instruments, New Castle, DE) with a 20 °C/min temperature ramp rate.

The validation of the in-line FDM rheometer (FDMRheo) was performed by printing into open space, meaning that the nozzle was stationary and raised about 30 cm above the build platform to avoid any exit pressure effects that could build up between the nozzle and the platform. Nine shear rates were measured at three temperatures for each material as shown in Table I to account for shear rate and temperature dependencies. Each shear rate sweep was repeated twice, and results from both tests are reported in this article. In order to achieve the melt temperatures listed in Table I, the nozzle temperatures were set to higher temperatures as shown in Tables SIV–SVII of the supplementary material [74] (for example, the nozzle temperature of the printer was set to 259 °C to achieve a polymer melt temperature of 250 °C within the nozzle); the nozzle settings were determined using the intrusive melt thermocouple. The measurements were repeated with two nozzle lengths (both with the same nozzle diameter and conical entrance) to account for end effects as described in Sec. III C. The corrections were all applied to the results from the 1.72 mm length nozzle because this nozzle more closely resembles the standard printer nozzle.

To allow enough time for equilibration, each shear rate was applied for approximately 25 s before stepping up to the next shear rate. A 5 s delay was applied between each shear rate to observe the full pressure equilibration process as shown in the example in Fig. 6 for all shear rates from 30 to 650 s⁻¹ for HIPS at 225 °C.

Both rotational rheology (RotRheo) and capillary rheology (CapRheo) were performed for both PC and HIPS at the same temperatures described in Table I to validate the performance of the FDMRheo. A DHR1 rotational rheometer (TA Instruments, New Castle, DE) and a Rosand RH10 capillary rheometer (Malvern Panalytical, Westborough, MA) were used. Frequency sweeps from 460 to 1 rad/s with three points per decade on a logarithmic scale were performed on the RotRheo. The Cox–Merz rule was assumed to apply [59]. A strain of 4% was used for PC, and a strain of 2% was used for HIPS to test within the linear viscoelastic region as determined by strain sweeps. Due to the high frequencies explored in this study, an inertia correction was applied to minimize the effects of inertia, since the raw phase angle was above 100° for most of the data above 100 rad/s. The accuracy of the correction was validated by testing a 200 Pa · s Newtonian polybutene viscosity standard (Cannon Instrument Company, State College, PA), which was confirmed to maintain its Newtonian viscosity up to 460 rad/s.

Shear rate sweeps were done with three points per decade on a logarithmic scale for both materials on the CapRheo. PC was characterized with a 20 mm length die with a 1 mm diameter from 100 to 1000 s⁻¹ at all three PC temperatures listed in Table I as well as with a 20 mm length die with a 2 mm diameter from 10 to 460 s⁻¹ at 270 °C to observe the Newtonian plateau. HIPS was characterized with a 20 mm length die with a 2 mm diameter from 6.5 to 1000 s⁻¹ at all three HIPS temperatures listed in Table I. The Rabinowitsch and pressure effect corrections (see Secs. III D and III E) were applied to the CapRheo results, but the Bagley correction was found to be negligible for the CapRheo experiments. All measurements on both RotRheo and CapRheo were repeated twice, and both specimens are reported in this article.

The sensor data must be converted into an apparent shear rate and a shear stress at the wall to calculate the apparent viscosity. First, a volumetric flow rate, Q, is calculated based on the stepper motor pulses (due to the much higher resolution, the stepper counts were used by default rather than the filament encoder counts unless there was a shear rate discrepancy >5%, indicating skipped steps or filament slip) as described in the following equation:

where $V˙$ is the velocity of the filament, A is the area of the filament, v is the measure of pulses per second, $ζ$ is the pulses per mm (830 pulses/mm for the stepper motor), and R_fil is the radius of the filament.

The uncorrected apparent shear rate, $Γ$⁠, is then calculated by the following equation [35,39]:

where R_noz is the radius of the nozzle tip. Note that $Γ$ is corrected for non-Newtonian effects into the corrected shear rate, $γ˙$⁠, as later described in Sec. III D. Shear stress at the wall, $τw$⁠, is calculated based on the following equation [35]:

where $ΔP$ is the pressure drop measured by the load cell and L_noz is the length of the nozzle tip. Apparent viscosity, $ηapp$⁠, is then calculated using an average $Γ$ and an average $ΔP$ for each shear rate step as shown in Eq. (4). Since the pressure and shear rate take time to equilibrate, the data for each shear rate step were averaged after the first, transient 25% of each step, so that only the equilibrated data were considered.

A rheological model was fitted to all the rheology data collected by each rheometer for both PC and HIPS to interpolate and extrapolate the viscosity, $η$⁠, to other temperatures and shear rates. Therefore, the Cross-Williams–Landel–Ferry (WLF) model was chosen, which combines the shear rate dependent Cross model as shown in Eq. (5) and the temperature dependent WLF model for predicting the zero-shear viscosity, $η0$⁠, as shown in Eq. (6) [39,51].

where $τ∗$ is the critical shear stress at which the viscosity profile moves from Newtonian to shear thinning, n is the power-law index (which is fit to experimental data), T* is a reference temperature (the glass transition temperature is used in this article), and A₁, A₂, and D₁ are fit to experimental data. In addition to comparing the Cross-WLF curves produced by the various rheometers (Fig. 10), the Cross-WLF models are also used to correct for pressure effects and temperature inaccuracies as described in Sec. III F.

To account for end effects (end effects include entrance effects from the coni-cylindrical taper prior to the nozzle tip as well as exit effects), the Bagley correction [35,60] was applied by measuring the viscosity with the two nozzles listed in Table I: the standard nozzle has an L/R of 6.49 while the longer nozzle has an L/R of 14.11. The entrance pressures were calculated by extrapolating the pressure to a zero L/R. The entrance pressures were then subtracted from the raw pressure data of the standard nozzle to calculate the corrected pressure. Entrance pressure is a function of temperature and shear rate, so the Bagley correction was applied across all the measurements as shown in Table I. Entrance pressure was found to be very significant, ranging from 5% to 35% of the pressure drop for PC and 8–45% of the pressure for HIPS, making entrance pressure the most critical of all the rheological corrections explored in this work.

The shear thinning nature of the polymers causes the cross-sectional velocity profile to vary from the parabolic flow pattern of Newtonian liquids that is assumed in Eq. (2). To account for this, the Rabinowitsch correction was applied according to the following equation [58]:

Equation (7) adjusts the cross-sectional flow field by considering a power-law model [8,53,58]. The viscosity is then corrected by using the following equation:

The high pressures seen in many plastics processing techniques cause a reduction in the free volume of the melt [61–63], causing an exponential increase in viscosity at elevated pressures [35,51,61,64]. Viscosity pressure dependency can be accounted for by applying a pressure shift factor, $αp$⁠, using the Barus equation as shown in the following equation [52,65]:

where β is the pressure coefficient and p is the pressure in the FDMRheo nozzle. The pressure shift factor can then be applied as shown in Eqs.(10) and (11) to obtain the zero-pressure viscosity, $ηp0$⁠, and the corrected shear rate, $γ˙p0$⁠, which provide the corrected values to compare off-line rheological results obtained at zero pressures.

The β values were taken to be typical values from the literature: $β=0.03MPa$ for PC [64,65] and $β=0.034MPa$ for acrylonitrile butadiene styrene (ABS) [65], which is assumed to be the same as HIPS. The calculated pressure corrections indicate that FDMRheo pressure effects are real but minimal.

Viscous dissipation was found to be negligible based on the simulations described in the supplementary material [74], but the melt temperature as monitored by the intrusive melt thermocouple was found to deviate from the desired temperature in certain scenarios. First of all, it was observed that the nozzle temperature setting on the printer was not an accurate representation of the polymeric melt as discovered from the intrusive melt thermocouple readings, so the nozzle settings were adjusted as shown in Tables SIV and SV of the supplementary material [74] to achieve the desired melt temperatures listed in Table I. Once the nozzle settings were tuned to achieve the desired melt temperatures, the real-time temperature in the FDMRheo was observed to deviate from the desired melt temperature by no more than 2 °C in most cases as shown in Tables SIV–SVII [74]; however, in one extreme case for PC at 290 °C, the melt temperature could only reach 281 °C due to the maximum temperature settings of the printer. The viscosity and shear rate values were therefore shifted to account for these temperature inaccuracies.

The temperature shift factor, $α$⁠, is calculated based on the temperature dependent WLF model [Eq. (6)] by calculating the predicted viscosity at the measured melt temperature, $η0(Tmeas)$⁠, and the predicted viscosity at the desired melt temperature, $η0(Tdes)$⁠, as shown in the following equation:

The viscosity and shear rate are then corrected using the following equations:

As discussed later in the article, the temperature inaccuracies were relatively negligible in all cases other than the FDMRheo data for PC at 290 °C.

To validate the accuracy of the pressure measurements acquired by the FDMRheo, the nozzle and FDMRheo setup was removed from the FDM printer and attached to the CapRheo as shown in Fig. 7. The nozzle connected to the capillary rheometer had a nozzle length of 1.36 mm long (all other results in this report are based on a nozzle with a 1.72 mm length, and the 3.74 mm length was only used to apply the Bagley correction). Despite the slightly shorter length, the pressure is measured in the same exact way, making this a valid approach for verifying the FDMRheo pressure against the CapRheo pressure transducer.

PC was processed through the CapRheo/FDMRheo setup at 270 °C, and the test was repeated four times at shear rates ranging from 100 to 4000 s⁻¹ depending on the test. The results of the CapRheo pressure versus FDMRheo voltage for all four tests are combined in Fig. 8(a) and compared against the FDMRheo calibration curve. The continuous results from one of the CapRheo/FDMRheo tests are shown in Fig. 8(b). The results from both Figs. 8(a) and 8(b) show very good agreement between the CapRheo and FDMRheo pressures, confirming the accuracy of the FDMRheo against the standard pressure transducer of the CapRheo. The slight differences between the CapRheo and FDMRheo data shown in Fig. 8 are all within the accuracy of the FDMRheo load cell (The multiple calibration curves obtained for the FDMRheo indicated a 95% confidence uncertainty window of ±0.12 MPa at 0 MPa, which increases to ±0.165 MPa at 5 MPa.). All the data from this experiment are tabulated in Table SII in the supplementary material [74].

All the viscosity versus shear rate data are compared between the three rheometers in Fig. 9, and representative viscosity values are compared between the three rheometers in Table II. All the corrections described in Sec. III have already been applied to the FDMRheo and CapRheo data shown in Fig. 9 and Table II (note that for Table II, the data are based on equations that have been fitted to the corrected data to compare viscosity values at the same shear rates, which provides a good comparison with coefficients of determination of >0.99 for all HIPS data and 0.87–0.98 for all PC data). The FDMRheo measurements show excellent agreement with both off-line rheometers. The FDMRheo data nearly identically match the CapRheo data for HIPS, and the FDMRheo data fall between the RotRheo and CapRheo data for PC. Additionally, the deviation between the FDMRheo and the off-line rheometers is on the same order and even smaller than the deviation between the rotational and capillary rheometers, further validating that the in-line rheometer is equally as effective as the off-line rheometers for providing representative viscosity curves.

For a direct comparison between the final data between the three rheometers, the corrected viscosities at all corrected shear rates are shown in Table SIII in the supplementary material [74]. The raw data (temperatures, uncorrected pressures, uncorrected viscosities, etc.) for HIPS and PC at both nozzle lengths are tabulated in Tables SIV–SVII in the supplementary material [74]. Based on the high accuracy of the FDMRheo, it could potentially be combined with the interesting work from Phan et al. [34] to improve the accuracy of their pressure measurements, which could enable a more easily transferable in-line rheometer based on power measurements from the stepper motor that drives the filament gears.

A Cross-WLF model was fitted to the rheological data for the three rheometers using a least-squares fit. To compare the models, the viscosity was plotted over a very wide shear rate range at an interpolated temperature of 260 °C as shown in Fig. 10. The HIPS viscosity curves match well for all three rheometers; for PC, the FDMRheo and RotRheo data are in good agreement, but the CapRheo Cross-WLF model has an inaccurate profile as seen in Fig. 10 and indicated by the fitted Cross-WLF data in Table III. The major difficulty in fitting a Cross-WLF model to CapRheo experiments is that the measurements do not typically extend to low enough shear rates to observe the Newtonian plateau, which likely caused some of the error in this fit.

The Cross-WLF coefficients for both materials from the different rheometers are shown in Table III. A₂ was kept constant at 51.6 K, and T^* was fixed as the glass transition temperature, but D₁, A₁, τ^*, and n were fitted to all the data from the corresponding rheometer and material. All the data, except for the PC CapRheo data, match relatively well. Note from Eq. (6) that D₁ and A₁ counter one another in terms of their influence on the zero-shear viscosity; for example, when observing the HIPS data, D₁ is largest for the FDMRheo data, but A₁ is also largest for the FDMRheo, which balances the large D₁ value and provides similar viscosity results as the other rheometers as shown in Fig. 10. The fits for τ^* and n match very closely for all the rheometers, except in the case of the PC CapRheo data, which is at least partially caused by the small shear rate range of the CapRheo, focusing on high shear rates. The CapRheo data itself were actually very accurate and comparable to the other results, as displayed in Fig. 9, indicating that the Cross-WLF fit would be acceptable if a second capillary die with a larger diameter was used to extend the measurements to low shear rates; however, the CapRheo data were collected at relatively high shear rates to match the most common conditions measured on a CapRheo. One benefit of the FDMRheo over CapRheo is that the FDMRheo enables much faster testing (particularly at low shear rates) because the melt volume is much smaller in the FDMRheo, enabling more rapid pressure equilibration at each shear rate, which can be observed by comparing the pressure curves in Fig. 6 (FDMRheo) and Fig. 8(b) (CapRheo). Additionally, the FDMRheo can collect more measurements in a single test because there is no limitation in material volume like there is in the barrel of the CapRheo. The CapRheo offers the benefit of easily testing raw material in the pellet form, but the FDMRheo is an appropriate rheometer for analyzing FDM filaments.

Applying all the corrections described in Sec. III enabled excellent agreement between the three rheometers as shown in Fig. 9, but it is important to understand which corrections are the most critical and which could potentially be neglected. The influence of each correction can be visualized in Fig. 11 where the raw data for PC were modified by applying the corrections in the following order: Bagley end effects → Rabinowitsch → Temperature inaccuracies → Pressure effects. PC at 270 °C [Fig. 11(a)] is taken as the representative case, and PC at 290 °C [Fig. 11(b)] is also examined because this sample had a much larger temperature inaccuracy relative to the other samples. The influence of the corrections on HIPS was similar to those of PC that are shown in Fig. 11(a).

Accounting for the entrance effects with the Bagley correction is clearly the most critical correction that must always be applied to accurately measure viscosity. The Rabinowitsch correction consistently causes a slight shift in the data, and it is a very simple correction to apply if viscosity is measured at multiple shear rates. The temperature correction is typically negligible, but it becomes very significant in the case of PC at 290 °C, so it is necessary to monitor temperature accuracy with the intrusive melt thermocouple. The pressure correction is negligible except for at elevated shear rates (i.e., when the pressure is highest).

Wall slip, which typically occurs above a critical wall shear stress [52], was neglected due to the relatively low wall shear stresses observed in this work, ranging from 0.005 to 0.2 MPa. Moreover, the excellent agreement between the in-line and off-line rheometers indicates that slip is not a significant source of error for HIPS and PC at the shear rates studied. Other materials may begin to slip at elevated shear stresses, which can be monitored by looking for the onset of stick-slip, which will show up as fluctuations in the real-time pressure data; the stable pressure data (Fig. 6) further indicated that slip did not occur during any of the tests described in this article.

The following paragraphs are incorporated to illustrate how the FDMRheo addresses some of the shortcomings that are common among on-line and in-line rheometers. Dealy describes the most important requirements of process rheometers to be: negligible leakage from the rheometer, no process interference, fast sample renewal rate, and a robust design [50]. Dealy continues to mention other important considerations, including preventing friction interference and maintaining measurement accuracy by accounting for nonisothermal effects, temperature uniformity, shear rate dependency, and by avoiding material degradation and build-up [50]. The following paragraphs describe how well these requirements have been met with the proposed design.

The load transfer column was designed to fit into the pressure port with tight tolerances, but a slide fit was used to prevent any frictional forces from interfering with the measurement, which results in the possibility of melt leakage around the load transfer column. A series of 3–4 g parts were printed over a range of shear rates and temperatures to quantify the amount of leakage; during printing, negligible leakage was observed while printing most of the parts, but some leakage was observed during high pressure prints. When observed, the melt leakage was collected and weighed: the melt leakage was found to weigh between 0 and 14 mg, which corresponded to 0–0.4% of the weight of the actual part. These amounts of leakage have no influence on the process or part quality, but it means the nozzle needs to be cleaned after printing a handful of parts or after printing a large part. Leakage could be further minimized through higher precision machining or by making the load transfer column and nozzle out of the same materials for better thermal expansion matching. Another potential concern about melt leakage is the viscous resistance from the melt surrounding the load transfer column that could limit the velocity of the transfer column pressing into the load cell. Viscous calculations based on the Couette flow indicate that instantaneous pressure changes would be resisted, causing a time delay on the order of 1 s. Practically, this does not influence steady-state measurements and should not add additional time delays, considering the already existing delays that can be caused by natural melt compressibility and relaxation [51]. These response times of about 1 s are fast compared to commercial in-line and on-line rheometers, some of which can have response delays on the order of minutes [66].

The pressure port and load transfer column are observed to create negligible process interference. Although the pressure port creates a slight recess (the recess varies from about 0.2 to 1 mm, depending on the pressure in the melt stream) from the circular cross section of the melt stream [as shown in Fig. 2(c)], estimations using the Hagen–Poiseuille equation indicate that changes in the total pressure drop due to the pressure port are negligible. In contrast, Lodge has indicated the importance of flush mounted pressure transducers to avoid such errors [67]; however, Lodge's intended application was for pressure transducers mounted directly in the small diameter, high shear rate region of interest where any error in shear rate has a profound effect on the measurement. In this FDM application, the pressure port is mounted in the large diameter section above the critical 0.5 mm diameter tip as shown in Fig. 2(c); therefore, estimates indicate that changes in total pressure due to the recessed pressure port will be small, on the order of 0.15%. Similar reasoning can be applied to understand the influence of how the thermocouple may disrupt the flow through the process; the pressure drop caused by the thermocouple is estimated to be smaller than that caused by the recess in the pressure port. The negligible influence of the pressure port and thermocouple is further validated by the excellent agreement between the in-line and off-line rheological measurements after applying the Bagley correction.

Another potential concern caused by the recess from the pressure port is the ability for melt stagnation and degradation. Since collecting the data displayed in Fig. 9, 25 parts were printed with a total printing time of more than 21 h. The load transfer column was then removed, and the melt was purged: no discoloration of the melt was observed, indicating no visual degradation. Additionally, the change between polymers of different colors (white to black and back to white) was very responsive without a long tailing edge of residence time distribution and without any residual black or white material coming out during a later print. Next, the printer was run for an additional 10 h, and the shear rate sweeps were repeated for PC at all three temperatures both before and after the 10 h of printing. The results for PC at 250 °C are shown in Fig. 12. The results overlay nearly perfectly with no trend of either increasing or decreasing pressure, indicating that the measurement is very consistent without any build-up of the degraded polymer.

Perhaps, the largest practical limitation of the design is its industrial robustness. The load cell protrudes away from the nozzle, occupying space around the nozzle and potentially conducting heat away from the nozzle. The load cell could be damaged if it hits against something or if excessive pressure is applied. While the design is excellent for research purposes and has provided consistent results, a more space efficient and durable design would be best for widespread use. These problems could be solved, and the design could become commercially viable by designing a smaller pressure transducer to be integrated directly into the nozzle.

Viscosity temperature dependence is very important for polymer melt in-line rheometers, and it was discovered that the in-line thermocouple is critical to the success of FDM in-line rheometers because the melt temperature setpoint is typically not the same as the actual melt temperature. As done in this paper, new materials should be analyzed at multiple temperatures to quantify the viscosity temperature dependence [for example, the WLF equation shown in Eq. (6) can be used] so that temperature errors can be corrected by shifting viscosity to a reference or desired temperature. Interestingly, it was observed that at high shear rates (∼500 s⁻¹), melt temperatures begin to decrease with increasing shear rate as displayed in Tables SIV–SVII, indicating insufficient melting and nonuniform temperatures across the nozzle. The in-line rheometer is, therefore, an excellent tool for studying heat transfer and high-speed limitations during the FDM process. Although the high flow rate limitations were not significant for the shear rates explored in this paper, heat transfer limitations are of both practical and theoretical interest, so future work will use the in-line rheometer to further explore this concept while extruding at higher shear rates.

The other important temperature consideration is the fact that the nozzle is not directly heated; it is heated by conduction with the heating element in the melting section of the hot end. Therefore, the nozzle material is cooler at its exit tip than at its entrance (as validated through manual temperature measurements), which may explain why many printers use a short nozzle with a small length to radius (L/R) ratio in the nozzle tip. Short L/R ratios of about 7 and 15 were used in this paper to resemble a typical printer nozzle and to prevent the melt from cooling through the length of the nozzle. While shorter length nozzles are better for both temperature uniformity and lower pressure drops (lower pressure drops prevent inaccurate flow rates due to skipped steps from the stepper motor or filament slippage in the gears), they result in smaller lengths of fully developed flow and a greater influence of end effects as made apparent by the Bagley corrections in Fig. 11. Due to the excellent agreement between all the in-line and off-line data after the Bagley correction, the authors found that L/R ratios of 7 and 15 enable a successful in-line rheometer.

The Bagley correction must be obtained by using two nozzles of differing L/R ratios, which is practically time-consuming to do on a regular basis. This could be overcome by predicting the entrance pressure. Although entrance effects have been modeled via many approaches in the literature [47,57,68–73], the authors are not aware of any analytical expressions that can be applied in real time that are universally accepted for describing the entrance pressure for all polymer melts. Therefore, the Bagley procedure is the most universally accurate method for analyzing the entrance pressure. Real-time corrections can be applied by fitting a curve to the entrance pressures obtained from the Bagley correction. The Bagley procedure could therefore be performed once for each material of interest, and the correction could then be applied for all future testing. The supplementary material [74] describes how the Bagley entrance pressures can be fitted to an equation for real-time analysis and control.

By accounting for entrance pressure, the in-line rheometer showed excellent accuracy when printing into open space; however, exit pressures must be accounted for when printing actual parts. When printing with small layer heights, elevated pressures are observed due to the pressure drop experienced between the nozzle and the previous layer. Additional work is on-going to measure and predict this exit pressure. Once the exit pressures are characterized, the in-line rheometer can be used for accurate viscosity measurements throughout an entire print. The real-time viscosity analysis will enable a better understanding of the FDM processing window and the role of polymer rheology in process optimization. In-line rheology and pressure measurements can also be used for real-time process control and predictions of quality metrics, such as strength and part density.

The in-line FDM rheometer (FDMRheo) designed in this work has been found to provide very accurate viscosity measurements as confirmed against off-line rheometers. The FDMRheo can collect data across a wide range of temperatures and shear rates to generate a successful Cross-WLF model for analyzing continuous viscosity curves as a function of temperature, shear rate, and pressure. The sensor for measuring the filament feed rate as well as the thermocouple for measuring melt temperature were both critical for the deployment and accuracy of the in-line rheometer. Entrance effects were the most significant correction for obtaining an accurate viscosity, so the Bagley correction should be applied to allow the FDMRheo to be used for real-time process control of the FDM process. For example, a control scheme could be developed to optimize the printing speed while maintaining pressures and viscosities within the ideal processing window. The FDMRheo is suitable for analyzing the viscosity of new, 3D printable materials to more rapidly introduce new materials to the market; the vision is that the rheometer can enable automatic process optimization and quality assurance using physics-based models for weld fusion (i.e., interlayer strength), residual stress, print density, and shrinkage.

This work was funded jointly by Saint-Gobain Research North America and the University of Massachusetts, Lowell. The authors would also like to thank David Shackleford of Saint-Gobain for his insightful discussions on rheological flow, Steve Johnston of UMass Lowell for his recommendation of the CapRheo/FDMRheo experiments, and Brian Beauvais and Phil Desroches of Saint-Gobain for their support in performing the x-ray computed tomography experiments.