**2. Fused deposition modelling**

Fused deposition modelling (FDM) is one of the most common polymer additive manufacturing techniques. FDM is an extension of hot melt extrusion (HME), which is an already established technique in the field, whereby thermoplastics are heated to their semi-molten state and extruded through a given orifice. However, HME can only be used to fabricate basic geometries, whereas FDM utilises a gantry system that allows a nozzle to move and extrude the semi-molten polymer in three-dimensions until a 3D print is fabricated.

however, most FDMs are not equipped with a transducer, and thus the shear rate will need to be determined semi-empirically. This can be achieved by: (i) performing an initial shear-rate viscosity measurement of the melt to obtain the power law index; (ii) knowing the speed of printing and nozzle diameter; and (iii) applying the rheological equations (Eqs. (1)–(4)).

**Figure 1.** Schematic of fused deposition modelling. The figure lists the components involved in the fabrication process,

app **<sup>=</sup>** \_\_\_ **4Q**

where *Q* is the volume flow rate, determined from the exit nozzle radius *r* and the speed of

*Q = r* **<sup>2</sup>** *v* (2)

For example, a printing speed of 50 mm/s and a nozzle diameter of 0.2 mm equates to a flow

analysis, as 103 s−1 is above the attainable shear rate performed by a rotational rheometer [3].

app(

For the true shear rate γ̇, the following equation should be used [4–6]:

app provides a relatable shear rate that is then examined using the rheometer, which is

\_\_\_\_\_ (**3***n +* **1**)

/s and consequently an apparent shear rate of ~1000 s−1. The apparent shear

to 103 s−1 [2]. Thus, a capillary rheometer is best suited for such

app of the nozzle can be semi-empirically determined using the fol-

Polymeric Additive Manufacturing: The Necessity and Utility of Rheology

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45

**r3** (1)

**<sup>4</sup>***<sup>n</sup>* ) (3)

The apparent shear rate γ̇

̇

as well as the rheological facets of interest measurable by rheometers.

̇ *=* ̇

extrusion *v* (i.e. printing speed) [1]:

lowing equation [1]:

rate of 6.3 mm3

typically in the order of 10<sup>2</sup>

rate γ̇

The similarities between FDM and HME are that both use high heat to achieve a semi-molten thermoplastic polymer, and that it is then forced through an orifice<sup>1</sup> . Hence, high temperature rheology and the shearing effect at the orifice, respectively, are of interest to both. Upon exiting the orifice, the thermoplastic is cooled until solidification, which again, is rheologically relevant. The rheological events of FDM are delineated in **Figure 1**.

#### **2.1. Nozzle flow and viscosity**

#### *2.1.1. Determining the shear rate from FDM parameters*

Knowing the ideal viscosity range can help in predicting whether the new melt formulation is extrudable. Said knowledge will prevent time-consuming and costly empirical trials, as well as mitigating nozzle blockage and consequently machine downtime. A straightforward approach is to compare the viscosity of the new formulation to that of a successfully extruded formulation (e.g. a commercial filament) using a rheometer. A dissimilar viscosity profile may not necessarily equate to an unextrudable melt, provided that they possess comparable viscosity at the operating shear rate; hence the shear rate of interest will need to be identified. Unlike HMEs,

<sup>1</sup> For HME, the orifice is typically called a die; for FDM it is the nozzle.

Furthermore, there are several polymeric AM technologies available for purchase by consumers that will allow manufacturing of goods to be achieved at home. However, despite the vast progression made, the technology is still in its infancy. Therefore, to make AM an essential

Rheology is a necessity for all polymer fabrication techniques. The characterisation can deliver extensive and reliable material information. The data is subsequently correlated to the process to maximise productivity. Rheology will further be a key component as new materials are formulated to advance the versatility of AM. In spite of this, rheology remains an underutilised tool. Thus, a chapter into how rheology can be utilised to help maximise AM efficiency is warranted. The chapter presents two of the commonly-used AM techniques in the field of polymers: fused deposition modelling (FDM) and stereolithography (SLA); and demonstrates the necessity of rheology thereto. As detailed herein, both technologies form 3D structures through disparate means, and albeit different, rheology is still an indispensable tool for both technologies. The chapter will conclude with a brief description of other AM techniques and

Fused deposition modelling (FDM) is one of the most common polymer additive manufacturing techniques. FDM is an extension of hot melt extrusion (HME), which is an already established technique in the field, whereby thermoplastics are heated to their semi-molten state and extruded through a given orifice. However, HME can only be used to fabricate basic geometries, whereas FDM utilises a gantry system that allows a nozzle to move and extrude

The similarities between FDM and HME are that both use high heat to achieve a semi-molten

rheology and the shearing effect at the orifice, respectively, are of interest to both. Upon exiting the orifice, the thermoplastic is cooled until solidification, which again, is rheologically

Knowing the ideal viscosity range can help in predicting whether the new melt formulation is extrudable. Said knowledge will prevent time-consuming and costly empirical trials, as well as mitigating nozzle blockage and consequently machine downtime. A straightforward approach is to compare the viscosity of the new formulation to that of a successfully extruded formulation (e.g. a commercial filament) using a rheometer. A dissimilar viscosity profile may not necessarily equate to an unextrudable melt, provided that they possess comparable viscosity at the operating shear rate; hence the shear rate of interest will need to be identified. Unlike HMEs,

. Hence, high temperature

the semi-molten polymer in three-dimensions until a 3D print is fabricated.

thermoplastic polymer, and that it is then forced through an orifice<sup>1</sup>

relevant. The rheological events of FDM are delineated in **Figure 1**.

*2.1.1. Determining the shear rate from FDM parameters*

For HME, the orifice is typically called a die; for FDM it is the nozzle.

instrument, further research is needed, including new material formulations.

how rheology is still relevant.

44 Polymer Rheology

**2.1. Nozzle flow and viscosity**

1

**2. Fused deposition modelling**

**Figure 1.** Schematic of fused deposition modelling. The figure lists the components involved in the fabrication process, as well as the rheological facets of interest measurable by rheometers.

however, most FDMs are not equipped with a transducer, and thus the shear rate will need to be determined semi-empirically. This can be achieved by: (i) performing an initial shear-rate viscosity measurement of the melt to obtain the power law index; (ii) knowing the speed of printing and nozzle diameter; and (iii) applying the rheological equations (Eqs. (1)–(4)).

The apparent shear rate γ̇ app of the nozzle can be semi-empirically determined using the following equation [1]:

$$
\dot{\mathbf{y}}\_{\text{app}} = \frac{4\mathbf{Q}}{\pi \,\mathrm{r}^3} \tag{1}
$$

where *Q* is the volume flow rate, determined from the exit nozzle radius *r* and the speed of extrusion *v* (i.e. printing speed) [1]:

$$\mathcal{Q} = \pi r^2 \upsilon \tag{2}$$

For example, a printing speed of 50 mm/s and a nozzle diameter of 0.2 mm equates to a flow rate of 6.3 mm3 /s and consequently an apparent shear rate of ~1000 s−1. The apparent shear rate γ̇ app provides a relatable shear rate that is then examined using the rheometer, which is typically in the order of 10<sup>2</sup> to 103 s−1 [2]. Thus, a capillary rheometer is best suited for such analysis, as 103 s−1 is above the attainable shear rate performed by a rotational rheometer [3]. For the true shear rate γ̇, the following equation should be used [4–6]:

$$
\dot{\lambda} = \dot{\lambda}\_{\text{app}} \left( \frac{(3n+1)}{4n} \right) \tag{3}
$$

whereby *n* is the power law index obtained using the power law model from a viscosity-shear rate test:

$$
\eta = \, k \, \dot{\mathbf{y}}^{\ast -1} \tag{4}
$$

**<sup>P</sup>** *<sup>=</sup>* **<sup>2</sup>***l* ̇ \_\_\_\_\_*<sup>w</sup>*

**Figure 2.** Illustration depicting buckling and no buckling conditions.

mulation is shear-thinning in order to avoid premature extrusion.

**2.2. Extrudate swelling and viscoelasticity**

sity of rheology once-more.

*2.1.3. Further considerations*

where *l* is the length of the tube flown through, *r* is the filament radius, and γw is the wall shear rate. Thus, as the pressure is proportional to viscosity and shear rate, reducing the two rheological factors can help mitigate filament buckling; and thereby demonstrating the neces-

Prior to extruding, the nozzle is heated to the desired printing temperature, wherein a portion of the filament is housed. The filament should exhibit an appreciable yield strength, whereby flow is resisted at high temperatures until the designated pressure is applied; and thereby preventing 'premature extrusion'. For this reason, the melt should exhibit shear-thinning characteristics at elevated temperatures, whereby the viscosity is high at low shear rates and resists, for example, gravity; but decreases with increasing shear rate. Conversely, a melt with Newtonian flow characteristics possesses no yield strength, and consequently will prematurely extrude, which can result in print failure if not addressed promptly. Therefore, it is necessary to perform viscosity-shear rate measurements and confirm whether the new for-

Extrudate swelling is a frequently encountered phenomenon, and of great interest in polymer processing. The phenomenon occurs in contemporary processes such as hot melt extrusion [12], injection moulding [13] and electrospinning [14], and also reported for fused deposition modelling [15]. Extrudate swell, or die swell, occurs when polymers pass through an orifice with a smaller diameter. The polymer is constrained with energy that is elastically stored as it enters the nozzle, whereafter the energy is released upon exiting the nozzle, leading to a

*<sup>r</sup>* (5)

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The best fit<sup>2</sup> to the data gives the power law index *n*, which is a dimensionless value between 0 and 13 . From the above equations, the true shear rate at the nozzle wall can be obtained, and therefore, viscosity-shear rate tests can be performed at the relatable shear rate range.

Although a capillary rheometer covers the ideal shear rate found in FDM, and exhibits the same flow behaviour to that found within the nozzle (i.e. Poiseuille flow), a rotational rheometer can be used if a capillary rheometer is not accessible. A shear rate test can be performed up to the instrument's shear rate limit, and the experimental data can then be fitted with a rheological model to predict the viscosity at higher shear rates. Examples of curve fitting models include the power law model, Williamson model, Cross model and Carreau-Yasuda model. Note that the oscillatory mode extends the shear rate limit of the rotational mode, however, the former provides the complex viscosity. If the Cox-Merz rule [7] is upheld for the melt formulation, then the complex viscosity can be converted into the steady-state viscosity, and subsequently curve fitted.

In addition to the above rotational rheometer analysis, large amplitude oscillatory shear (LAOS) measurements can be conducted to investigate the performance of the formulation. LOAS is regarded as a more complex analysis, however, it can be more revealing than its counterparts: small amplitude oscillatory shear (SAOS) and medium amplitude oscillatory shear (MAOS). In the former, the sample measured is subjected to large deformations, which is more reflective of the deformation polymers sustain during most polymer processing techniques; and the analysis is more sensitive to polymer architecture and consequently deformation. LAOS has been used to predict wall slip [8, 9] and polymer morphology, with regards to orientation, during extrusion [10]. Such an approach is a subject of interest for the author, and is currently under investigation.

#### *2.1.2. Filament buckling*

An additional consideration with too high a viscosity is filament buckling. The filament acts as the piston that drives the extrusion process. If the filament is not extruded at the desired rate it can apply backpressure to the ensuing filament, and in turn causes it to buckle. A critical stress limit σ<sup>c</sup> exists that the filament can be subjected to, of which above this value the filament will buckle, and consequently rendered inadequate. Hence, the critical stress must be greater than the pressure P imparted thereupon to drive the extrusion process.

The pressure required to drive the filament through the nozzle needs to be greater than the filament critical stress by a factor of 1.1 [11], as depicted in **Figure 2**.

The factor of 1.1 accounts for the difference between the nozzle and the filament diameter. The dependence of pressure on viscosity is given in the following equation for an ideal flow:

<sup>2</sup> Typically performed by taking the slope from a double-log plot (i.e. log Viscosity vs log Shear Rate).

<sup>3</sup> Note that since *n* is between 0 and 1, the true shear rate is greater than the apparent shear rate.

**Figure 2.** Illustration depicting buckling and no buckling conditions.

$$\mathbf{P} = \frac{2l\eta \,\dot{\mathbf{r}}\_w}{r} \tag{5}$$

where *l* is the length of the tube flown through, *r* is the filament radius, and γw is the wall shear rate. Thus, as the pressure is proportional to viscosity and shear rate, reducing the two rheological factors can help mitigate filament buckling; and thereby demonstrating the necessity of rheology once-more.

#### *2.1.3. Further considerations*

whereby *n* is the power law index obtained using the power law model from a viscosity-shear

therefore, viscosity-shear rate tests can be performed at the relatable shear rate range.

Although a capillary rheometer covers the ideal shear rate found in FDM, and exhibits the same flow behaviour to that found within the nozzle (i.e. Poiseuille flow), a rotational rheometer can be used if a capillary rheometer is not accessible. A shear rate test can be performed up to the instrument's shear rate limit, and the experimental data can then be fitted with a rheological model to predict the viscosity at higher shear rates. Examples of curve fitting models include the power law model, Williamson model, Cross model and Carreau-Yasuda model. Note that the oscillatory mode extends the shear rate limit of the rotational mode, however, the former provides the complex viscosity. If the Cox-Merz rule [7] is upheld for the melt formulation, then the complex viscosity can be converted into the steady-state viscosity, and subsequently curve fitted. In addition to the above rotational rheometer analysis, large amplitude oscillatory shear (LAOS) measurements can be conducted to investigate the performance of the formulation. LOAS is regarded as a more complex analysis, however, it can be more revealing than its counterparts: small amplitude oscillatory shear (SAOS) and medium amplitude oscillatory shear (MAOS). In the former, the sample measured is subjected to large deformations, which is more reflective of the deformation polymers sustain during most polymer processing techniques; and the analysis is more sensitive to polymer architecture and consequently deformation. LAOS has been used to predict wall slip [8, 9] and polymer morphology, with regards to orientation, during extrusion [10]. Such an approach is a subject of interest for the author, and

An additional consideration with too high a viscosity is filament buckling. The filament acts as the piston that drives the extrusion process. If the filament is not extruded at the desired rate it can apply backpressure to the ensuing filament, and in turn causes it to buckle. A criti-

filament will buckle, and consequently rendered inadequate. Hence, the critical stress must be

The pressure required to drive the filament through the nozzle needs to be greater than the

The factor of 1.1 accounts for the difference between the nozzle and the filament diameter. The dependence of pressure on viscosity is given in the following equation for an ideal flow:

greater than the pressure P imparted thereupon to drive the extrusion process.

Typically performed by taking the slope from a double-log plot (i.e. log Viscosity vs log Shear Rate).

Note that since *n* is between 0 and 1, the true shear rate is greater than the apparent shear rate.

filament critical stress by a factor of 1.1 [11], as depicted in **Figure 2**.

exists that the filament can be subjected to, of which above this value the

to the data gives the power law index *n*, which is a dimensionless value between

. From the above equations, the true shear rate at the nozzle wall can be obtained, and

*<sup>n</sup>−***<sup>1</sup>** (4)

rate test:

46 Polymer Rheology

The best fit<sup>2</sup>

0 and 13

*= k* ̇

is currently under investigation.

*2.1.2. Filament buckling*

cal stress limit σ<sup>c</sup>

2

3

Prior to extruding, the nozzle is heated to the desired printing temperature, wherein a portion of the filament is housed. The filament should exhibit an appreciable yield strength, whereby flow is resisted at high temperatures until the designated pressure is applied; and thereby preventing 'premature extrusion'. For this reason, the melt should exhibit shear-thinning characteristics at elevated temperatures, whereby the viscosity is high at low shear rates and resists, for example, gravity; but decreases with increasing shear rate. Conversely, a melt with Newtonian flow characteristics possesses no yield strength, and consequently will prematurely extrude, which can result in print failure if not addressed promptly. Therefore, it is necessary to perform viscosity-shear rate measurements and confirm whether the new formulation is shear-thinning in order to avoid premature extrusion.

#### **2.2. Extrudate swelling and viscoelasticity**

Extrudate swelling is a frequently encountered phenomenon, and of great interest in polymer processing. The phenomenon occurs in contemporary processes such as hot melt extrusion [12], injection moulding [13] and electrospinning [14], and also reported for fused deposition modelling [15]. Extrudate swell, or die swell, occurs when polymers pass through an orifice with a smaller diameter. The polymer is constrained with energy that is elastically stored as it enters the nozzle, whereafter the energy is released upon exiting the nozzle, leading to a radial expansion of the melt that consequently results in an extrudate diameter greater than that of the nozzle (**Figure 3**). This event is significant to FDM as it affects print resolution [16]. In addition, it affects print surface topography, which in the case of tissue engineering may influence biological properties. Thus, predicting the degree of extrudate swelling can help to avoid undesirable prints. The size of extrudate swelling is positively affected by shear rate and pressure, and exhibits a negative correlation to temperature and nozzle length. As these are FDM parameters that can be controlled, they can be exploited to minimise extrudate swelling once their effects thereto have been elucidated.

A capillary rheometer is the simplest method of predicting the degree of swelling. The material is extrudate through a capillary die with a similar configuration to that of the FDM nozzle, and the swell ratio *B* is defined as the ration between extrudate diameter *Dext* and die diameter *Ddie* [17]:

$$\mathbf{B} = \frac{D\_{\rm out}}{D\_{\rm die}} \tag{6}$$

where *τw* is the wall shear stress; and *G0*

concern [13].

stress σ0

removed at *t1*

stress; σ - stress; γ- strain; γ<sup>r</sup>

FDM [21], given the latter's low diameter tolerance.

*2.2.1. Analysing extrudate swelling through creep recovery*

= η0

zero-shear viscosity. The zero-shear viscosity can be determined by a rheological mathematical model (for example the Williamson or Cross Model) following a viscosity-shear rate test. In a modified FDM, the wall shear stress was acknowledged to induce swelling, and accordingly a lower extrusion speed was opted for to limit extrudate swell [19]. This corresponded with another study that found increasing the extrusion speed increased the filament diameter, again due to extrudate swelling, but also due to time-dependent deformation [20]. Similarly, a slower hot melt extrusion rate is once-more favoured for fabricating filaments suitable for

The relaxation time λ is another rheologically-derived parameter that has been proven to correlate well to extrudate swelling. The relaxation time can be obtained through various rheological tests, including from a steady shear rate measurement and curve fitting the data to the Carreau model; or by performing an oscillatory frequency sweep [22, 23]. The relaxation time is directly proportional to the ratio of extrudate swell, therefore, a shorter relaxation time is indicative of improved melt stability and of a polymer that is less susceptible to extrudate swelling [24–27]. Furthermore, the lower the relaxation time in contrast to the deformation time (e.g. time spent deformed in the die or nozzle) then extrudate swelling will be of less

Creep and creep recovery experiments are two-halves of an experiment. First, a constant

material, the strain generated, and the strain recovery is instantaneous to the application and removal of the stress, respectively. However, polymeric materials, which display viscoelastic deformation, convey a different response. Under the constant stress, part of the polymer strains instantly, whereas another part of the polymer deforms at a slower rate under the action of the stress; hence the term 'creep'. Similarly, in the recovery phase, a part of the material recovers instantly, another slowly recovers, and a final part does not recover completely,

and hence, the polymer remains permanently deformed [28] (**Figure 4**).

**Figure 4.** Schematic delineating the possible material responses to a creep test. (t- time; t0

recoverable strain).

is applied to the sample and the shear deformation is measured. The stress is then

and the recovery of the deformation is observed in creep recovery. In an elastic

/λ, where λ is the relaxation time and η0

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Polymeric Additive Manufacturing: The Necessity and Utility of Rheology

is the

49


The swelling phenomenon can also be predicted using a rotational rheometer. Tanner et al. demonstrated that the above equation is correlated to both the wall shear stress and zeroshear viscosity [18]:

$$\frac{D\_{\rm out}}{D\_{\rm div}} = \left[\mathbf{1} + \frac{\mathbf{r}\_w^2}{2\,G\_0^2}\right]^\wedge \tag{7}$$

**Figure 3.** Cross-sectional view of the nozzle portraying extrudate swelling. The schematic illustrates that the extrudate diameter (Dext) is greater than the diameter of the die (or nozzle) (Ddie) once the melt exits the orifice.

where *τw* is the wall shear stress; and *G0* = η0 /λ, where λ is the relaxation time and η0 is the zero-shear viscosity. The zero-shear viscosity can be determined by a rheological mathematical model (for example the Williamson or Cross Model) following a viscosity-shear rate test. In a modified FDM, the wall shear stress was acknowledged to induce swelling, and accordingly a lower extrusion speed was opted for to limit extrudate swell [19]. This corresponded with another study that found increasing the extrusion speed increased the filament diameter, again due to extrudate swelling, but also due to time-dependent deformation [20]. Similarly, a slower hot melt extrusion rate is once-more favoured for fabricating filaments suitable for FDM [21], given the latter's low diameter tolerance.

The relaxation time λ is another rheologically-derived parameter that has been proven to correlate well to extrudate swelling. The relaxation time can be obtained through various rheological tests, including from a steady shear rate measurement and curve fitting the data to the Carreau model; or by performing an oscillatory frequency sweep [22, 23]. The relaxation time is directly proportional to the ratio of extrudate swell, therefore, a shorter relaxation time is indicative of improved melt stability and of a polymer that is less susceptible to extrudate swelling [24–27]. Furthermore, the lower the relaxation time in contrast to the deformation time (e.g. time spent deformed in the die or nozzle) then extrudate swelling will be of less concern [13].

#### *2.2.1. Analysing extrudate swelling through creep recovery*

**Figure 3.** Cross-sectional view of the nozzle portraying extrudate swelling. The schematic illustrates that the extrudate

radial expansion of the melt that consequently results in an extrudate diameter greater than that of the nozzle (**Figure 3**). This event is significant to FDM as it affects print resolution [16]. In addition, it affects print surface topography, which in the case of tissue engineering may influence biological properties. Thus, predicting the degree of extrudate swelling can help to avoid undesirable prints. The size of extrudate swelling is positively affected by shear rate and pressure, and exhibits a negative correlation to temperature and nozzle length. As these are FDM parameters that can be controlled, they can be exploited to minimise extrudate

A capillary rheometer is the simplest method of predicting the degree of swelling. The material is extrudate through a capillary die with a similar configuration to that of the FDM nozzle, and the swell ratio *B* is defined as the ration between extrudate diameter *Dext* and die diameter

> *Dext Ddie*

The swelling phenomenon can also be predicted using a rotational rheometer. Tanner et al. demonstrated that the above equation is correlated to both the wall shear stress and zero-

> *<sup>=</sup>* [**<sup>1</sup>** *<sup>+</sup> <sup>w</sup>* **2** \_\_\_ **2** *G***<sup>0</sup> 2**] **1** ⁄**6**

*Dext Ddie* (6)

(7)

swelling once their effects thereto have been elucidated.

*B =* \_\_\_

\_\_\_

*Ddie* [17]:

48 Polymer Rheology

shear viscosity [18]:

diameter (Dext) is greater than the diameter of the die (or nozzle) (Ddie) once the melt exits the orifice.

Creep and creep recovery experiments are two-halves of an experiment. First, a constant stress σ0 is applied to the sample and the shear deformation is measured. The stress is then removed at *t1* and the recovery of the deformation is observed in creep recovery. In an elastic material, the strain generated, and the strain recovery is instantaneous to the application and removal of the stress, respectively. However, polymeric materials, which display viscoelastic deformation, convey a different response. Under the constant stress, part of the polymer strains instantly, whereas another part of the polymer deforms at a slower rate under the action of the stress; hence the term 'creep'. Similarly, in the recovery phase, a part of the material recovers instantly, another slowly recovers, and a final part does not recover completely, and hence, the polymer remains permanently deformed [28] (**Figure 4**).

**Figure 4.** Schematic delineating the possible material responses to a creep test. (t- time; t0 - onset of stress; t1 – Endpoint of stress; σ - stress; γ- strain; γ<sup>r</sup> recoverable strain).

In the context of extrudate swelling, a creep and creep recovery experiment is analogous to the events that result therein, hence, the test is more closely related to extrudate swell than any other test measurable in a standard rheometer [29]. From a qualitative perspective, a polymer that displays a larger recovery following removal of the stress will indicate a tendency to exhibit a larger extrudate swelling. Conversely, little or no strain recovery is attributed to damping of the applied load [30].

For an experimental quantification in predicting extrudate swelling using a creep recovery test: typically the recoverable compliance *Jr* is determined, which is positively correlated to extrudate swelling [29, 31, 32]. After the stress is removed, the ratio between recoverable strain γ<sup>r</sup> as a function of recovery time *tr* , and stress applied σ0 gives the recoverable compliance [33]:

$$J\_r(\sigma\_{\vartheta}, t\_{\vartheta}, t\_{\star}) = \chi\_r(\sigma\_{\vartheta}, t\_{\vartheta}, t\_{\star}) / \sigma\_{\vartheta} \tag{8}$$

*G* (*t*) *=* (*t*)/ (9)

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Computational studies experimentally determine the relaxation modulus G(t) to obtain the damping function [40–42] for their numerical swell predictions. The damping function has been found to correspond well with swell results empirically determined by a capillary rheometer [34, 43]. Transferring these computational studies, from extrusion dies to FDM nozzles, will indeed enhance FDM productivity. Additionally, determining damping behaviour of polymers is of general interest as it provides insight into the molecular structure thereof [44, 45]; which is not only helpful in understanding polymer behaviour under deformation, and thereby relevant to many fabrication techniques, but also can help in understanding, for example, polymer disintegration in a solvent medium. Therefore, rheological analysis delivers information that will be of interest beyond the FDM process-

The final stage of the FDM process is the layer-by-layer deposition of the filament. In this stage, the first layer is deposited and adheres onto the build platform. Subsequent layers are deposited thereupon, whereby adjacent layers adhere together until the 3D print is completed. The bonding quality determines the final properties of the 3D print; for example, poorly adhered layers exhibit weak mechanical properties. The layer bonding is referred to as sintering, which in polymers is driven by viscous sintering. Hence, viscosity plays another key role at this stage. There are numerical models used for predicting filament coalescence between two layers using viscosity measurements [46, 47], however, depending on the material used or printing parameters, the theoretical model may underestimate the neck growth

Most FDM printers have the option of controlling the temperature of the build platform. Ideally, the temperature should be high enough to ensure that viscous sintering can be achieved, and thereby adhesion. Below a critical sintering temperature sintering is negligible [46]. Equally, the build plate temperature should also ensure that the material possesses sufficient strength to maintain its structural integrity, particularly as layers are deposited above. Thus, a dynamic cooling ramp, via a rotational rheometer, can be utilised to examine the cooling evolution of the newly formulated material, and compared to that of an already successfully printed melt. Such a test can be incorporated to directly follow either a steady- or dynamic-shear test to determine whether shearing influences the solidification process, due

Finally, as adherence plays a vital role in FDM, and the printing parameters can affect polymer adhesive properties [49], this presents a potential to perform a tack test. Although not strictly rheology, a tack test allows one to determine the tack, or 'stickiness', properties of a material, which can be performed at elevated temperatures on some rotational rheometers. Information such as pull-off force, and mode of failure (i.e. cohesive failure, adhesive failure, or both) can be obtained. Furthermore, a tack test can be preceded by a shearing test, where

ing stage.

**2.3. Filament deposition: layer bonding and cooling**

achieved between adjacent models [48] (**Figure 6**).

the effects of shearing on tack properties can be measured [50].

to polymer chain dis-entanglement.

#### *2.2.2. Analysing extrudate swelling through stress relaxation*

Stress relaxation is another rheological test that can be employed to understand melt viscoelasticity, and verily the effects of stress relaxation characteristics on extrudate swell have been investigated [34–37]. A stress relaxation experiment entails applying a strain to a previously stress-free material and measuring the stress decay at this fixed strain (**Figure 5**). This test is used to determine whether the stress will dissipate within the processing technique time scale. In addition to extrudate swelling, stress relaxation may also help to explain flow warpage<sup>4</sup> [38], and other flow distortions.

Stress relaxation measurements can be made using step-strain rheology. Here, the molten polymer is subjected to an abrupt strain γ at time t0 , typically in the order of 20 ms, and the stress σ needed to keep this deformation is recorded as a function of time [39] (**Figure 5**). The strain applied should be in the linear-viscoelastic region. The relaxation modulus *G* can be simply determined from this measurement:

**Figure 5.** Schematic delineating the stress response to a stress-relaxation test.

<sup>4</sup> This is referred to as 'flow warpage' to differentiate it from 'drying warpage', where the former results in warped (i.e. bent) extrudates; whereas the latter results in a warped print due to inhomogeneous cooling.

$$\mathbf{G}\left(\mathbf{t}\right) = \sigma(\mathbf{t})/\mathbf{y} \tag{9}$$

Computational studies experimentally determine the relaxation modulus G(t) to obtain the damping function [40–42] for their numerical swell predictions. The damping function has been found to correspond well with swell results empirically determined by a capillary rheometer [34, 43]. Transferring these computational studies, from extrusion dies to FDM nozzles, will indeed enhance FDM productivity. Additionally, determining damping behaviour of polymers is of general interest as it provides insight into the molecular structure thereof [44, 45]; which is not only helpful in understanding polymer behaviour under deformation, and thereby relevant to many fabrication techniques, but also can help in understanding, for example, polymer disintegration in a solvent medium. Therefore, rheological analysis delivers information that will be of interest beyond the FDM processing stage.

#### **2.3. Filament deposition: layer bonding and cooling**

**Figure 5.** Schematic delineating the stress response to a stress-relaxation test.

bent) extrudates; whereas the latter results in a warped print due to inhomogeneous cooling.

In the context of extrudate swelling, a creep and creep recovery experiment is analogous to the events that result therein, hence, the test is more closely related to extrudate swell than any other test measurable in a standard rheometer [29]. From a qualitative perspective, a polymer that displays a larger recovery following removal of the stress will indicate a tendency to exhibit a larger extrudate swelling. Conversely, little or no strain recovery is attrib-

For an experimental quantification in predicting extrudate swelling using a creep recovery

to extrudate swelling [29, 31, 32]. After the stress is removed, the ratio between recover-

, *tr*) *= r*(**<sup>0</sup>**

Stress relaxation is another rheological test that can be employed to understand melt viscoelasticity, and verily the effects of stress relaxation characteristics on extrudate swell have been investigated [34–37]. A stress relaxation experiment entails applying a strain to a previously stress-free material and measuring the stress decay at this fixed strain (**Figure 5**). This test is used to determine whether the stress will dissipate within the processing technique time scale. In addition to extrudate swelling, stress relaxation may also help to explain flow

Stress relaxation measurements can be made using step-strain rheology. Here, the molten

stress σ needed to keep this deformation is recorded as a function of time [39] (**Figure 5**). The strain applied should be in the linear-viscoelastic region. The relaxation modulus *G* can be

This is referred to as 'flow warpage' to differentiate it from 'drying warpage', where the former results in warped (i.e.

, *t***0**

is determined, which is positively correlated

, *tr*)/**<sup>0</sup>** (8)

, typically in the order of 20 ms, and the

gives the recoverable

, and stress applied σ0

uted to damping of the applied load [30].

*Jr*(**<sup>0</sup>**

able strain γ<sup>r</sup>

50 Polymer Rheology

warpage<sup>4</sup>

4

compliance [33]:

test: typically the recoverable compliance *Jr*

as a function of recovery time *tr*

*2.2.2. Analysing extrudate swelling through stress relaxation*

[38], and other flow distortions.

polymer is subjected to an abrupt strain γ at time t0

simply determined from this measurement:

, *t***0**

The final stage of the FDM process is the layer-by-layer deposition of the filament. In this stage, the first layer is deposited and adheres onto the build platform. Subsequent layers are deposited thereupon, whereby adjacent layers adhere together until the 3D print is completed. The bonding quality determines the final properties of the 3D print; for example, poorly adhered layers exhibit weak mechanical properties. The layer bonding is referred to as sintering, which in polymers is driven by viscous sintering. Hence, viscosity plays another key role at this stage. There are numerical models used for predicting filament coalescence between two layers using viscosity measurements [46, 47], however, depending on the material used or printing parameters, the theoretical model may underestimate the neck growth achieved between adjacent models [48] (**Figure 6**).

Most FDM printers have the option of controlling the temperature of the build platform. Ideally, the temperature should be high enough to ensure that viscous sintering can be achieved, and thereby adhesion. Below a critical sintering temperature sintering is negligible [46]. Equally, the build plate temperature should also ensure that the material possesses sufficient strength to maintain its structural integrity, particularly as layers are deposited above. Thus, a dynamic cooling ramp, via a rotational rheometer, can be utilised to examine the cooling evolution of the newly formulated material, and compared to that of an already successfully printed melt. Such a test can be incorporated to directly follow either a steady- or dynamic-shear test to determine whether shearing influences the solidification process, due to polymer chain dis-entanglement.

Finally, as adherence plays a vital role in FDM, and the printing parameters can affect polymer adhesive properties [49], this presents a potential to perform a tack test. Although not strictly rheology, a tack test allows one to determine the tack, or 'stickiness', properties of a material, which can be performed at elevated temperatures on some rotational rheometers. Information such as pull-off force, and mode of failure (i.e. cohesive failure, adhesive failure, or both) can be obtained. Furthermore, a tack test can be preceded by a shearing test, where the effects of shearing on tack properties can be measured [50].

**Figure 6.** Schematic depicting the evolution of neck size during polymer sintering. The larger the sintering neck size formed the better the adhesion between adjacent layers.
