4.1. Influence of the Parameters LT, DAR and DR on the Degree of Filling
The fact that the density increase as a function of DR in
Figure 3 barely differs from LT-DAR combination to LT-DAR combination shows that by varying the parameters LT, DAR, and DR, parts with different porosity and density, respectively, can be manufactured in a targeted manner. A part with a certain density can be manufactured for any combination of LT and DAR as long as the DR is accordingly adjusted. This observation has implications for the previous understanding of parameters in the APF. For example, according to Hentschel et al., a decrease in DAR leads to an increase in density and thus to an increase in mechanical properties [
10]. Or according to Charlon et al., a decrease in LT also leads to an increase in density and thus to an increase in mechanical properties [
8]. It was concluded that the DAR and LT must be reduced for high density and mechanical properties, but this is not necessarily the case when DR is taken into account. The three parameters LT, DAR, and DR are rather to be seen as a parameter set for controlling the degree of filling or density within a layer and must be adjusted to each other for a desired density. In this case, LT and DAR could remain constant at any level and only by adjusting DR, the same density can be reached compared to a reducing of LT or DAR. In summary, the observation from
Figure 3 contradicts the statement that high density and mechanical properties can only be achieved with low LT and low DAR.
Besides the free adjustability of the density, a clear linear region between the density and the DR below 1.02 g/cm
3 is noticeable in
Figure 3. This behaviour can be explained as follows: The outer contour of the specimens is defined via the perimeter and due to the surface sealing (see
Figure 2) no water can enter the microstructure during the density measurement. As a result, the total volume of the specimens remains constant. For the volume flow of the freeformer,
DR is assumed and under the assumption of a constant density of the discharged droplets,
applies. Therefore, with a linear increase in mass, the density of the specimens also increases linearly as long as the specimen is underfilled and additional mass can be added to the microstructure. This relationship remains as long as the total volume of the specimens is not significantly changed by the increase in DR.
It is noticeable that the bulk density of ABS (1.04 g/cm
3) is barely or not at all reachable. Not even when the DR is increased to any value within a LT-DAR combination, which is due to the fact that the density curves flatten out and the linear relationship between the density and the DR is lost. Specimens close to the bulk density of ABS also show significant overfilling as shown in
Figure 8 for the LT-DAR combination LT150DAR14 and no longer meets the requirements for additively manufactured parts.
This implies that, as in the FFF process, a part manufactured by the APF process always has a certain void volume in the microstructure, which is due to the way a part is additively manufactured. The individual rounded tracks must first fuse together and the necks grow between the tracks. However, the material solidifies before complete coalescence, resulting in partial voids between the tracks [
17,
18]. A further increase in the DR cannot fill these voids and will inevitably lead to overfilling. This state of overfilling is not desired and not acceptable from a procedural point of view.
4.2. The Determination of an Optimal Degree of Filling
In the present work, a state is defined as optDF if the void volume is small, the density high, and the external geometry is still accurately reproduced. This state should exist for each LT-DAR combination and should be located at the transition between underfilling and overfilling. On the basis of the 3D reconstructions of the surface structure in
Figure 4, underfilling and overfilling can be qualitatively comprehended. The large gaps, which could be seen between the tracks at a DR below 70%, reduce the density of the specimen. The specimen is underfilled because its microstructure would allow more material to enter. To fill up these gaps and voids in the microstructure and consequently increase the density, more material has to be introduced into the specimen by increasing DR. Theoretically, due to the gaps and the resulting depressions, the surface roughness should be higher at an underfilling than for an optDF. At a certain point, however, the void volume becomes so small that material does not flow exclusively into the voids, but also to the outside. Overfilling begins with the onset of waviness, as seen in
Figure 4e for LT200DAR14DR80 or in
Figure 8 for LT150DAR14DR35. This overfilling breaks the boundary condition of geometric accuracy, which is why optDF must be located at smaller DR. An overfilling does not lead to an even accumulation of material, as the nozzle, for example, shifts material excess in edges and leaves it at turning points. A theoretical increase in surface roughness can be derived from this behaviour. To summarise, surface roughness should be minimal at optDF compared to underfilling and overfilling.
This theoretical consideration is in good agreement with the behaviour of the arithmetical mean height Sa over the DR for all LT-DAR combinations in
Figure 5. Each combination has a minimum of Sa and for LT200DAR11, LT200DAR14 and LT250DAR14 this minimum can be determined clearly. For LT150DAR14 and LT200DAR17, the behaviour in case of underfilling is unusual at first, but can be explained by the measurement method. Due to gaps between the tracks, in some cases the confocal microscope does not find a depth value or the value is not correct, leading to statistical variations in case of underfilling. Previously, a
good degree of filling was qualitatively identified on such surface structures as shown in
Figure 4, as Hentschel et al. conducted for example [
19]. The method used in this work can be considered more reliable.
Comparing the DR values at minimum Sa (32%, 45%, 75%, 113%, and 147%) from
Figure 5 for each LT-DAR combination with the corresponding density curves from
Figure 3, it is noticeable that the DR values are still in the linear region. This leads to the conclusion that the onset of overfilling is still in the linear region and that the density behaviour above the DR is therefore not suitable for finding optDF. The reason for this could be that although the total volume of the specimen is increased at the onset of overfilling, there is still some material being pressed into the voids.
In summary, for each LT-DAR combination, it is possible to determine a unique parameter set for manufacturing a part with optDF. For the investigated combinations used in this work, the DR for optDF can be taken from the green marked points in
Figure 5.
4.3. Model for Calculating a Desired Degree of Filling
An optDF is of great importance for the application of additive manufacturing processes such as the APF process. Until now, however, suitable parameters for the APF process could only be determined retrospectively, which made the finding of suitable parameters a laborious process. Especially, when deviating from proven standard parameters. For this reason, a model is proposed to calculate the degree of filling for combinations of LT, DAR, and DR in the linear region of density increase.
As described in
Section 2.1, the freeformer can only manufacture parts based on the file created by the slicing software, which contains information about the parameters and tracks used. For the APF process, there are two volume flows with regard to the tracks: one that results from the slicing software and one that is adjusted on the hardware side (freeformer) during the actual manufacturing process. These two volume flows are derived and equated in the following for inflexible volumes in order to be able to calculate the required parameters for a desired degree of filling.
The ideal volume of a droplet from the slicing software is illustrated in
Figure 9a. A slicer setting with a filling density of 100% is assumed. In this case, the distance between the tracks
T is equal to the track width
W and the track width
W of the ideal droplet can be determined using Equation (
1). The control of the freeformer is configured in such a way that the following droplet is deposited exactly at the distance of the track width
W. This results in a square base area for a ideal droplet as shown in
Figure 9a. With the help of the layer thickness
, the ideal volume can be calculated according to
The frequency at which the droplets are deposited one after the other within a track depends on the printing speed
c at which the build plate moves and the distance that must be covered before a new droplet is deposited. This distance is the track width
W, so the slicer software calculates the frequency according to
. If the maximum frequency is exceeded, the slicer software reduces the printing speed. Consequently, the ideal volume flow is calculated according to
As already noted in
Section 4.1, a certain void volume is always present in the APF process due to the round shape of the droplets in the droplet chain. This void volume is shown in
Figure 9b and must be taken into account for the real volume flow “requested” by the slicer software. The void volume is taken into account by the residual porosity factor
K, which is defined as follows:
Two extreme cases can be considered. At
there is no more air inside, so the ideal volume is completely filled and the density of the ideal volume corresponds to the bulk density of the material used. On the contrary, at
the ideal volume is exclusively filled with air and its density thus corresponds to that of air. The real volume of a droplet is the difference between the ideal volume and the void volume and thus the real volume flow requested by the slicer software is calculated according to
On the hardware side, there is also a volume flow. This volume flow of the freeformer occurring in the manufacturing process results from the frequency of the closable nozzle and the droplet volume:
The droplet volume results from the discharged plastic volume, which is considered as an inflexible volume. This volume can be calculated via the diameter of the plasticating cylinder
and a certain travel increment
L of the screw that conveys the volume through the nozzle. This travel increment
L is given by the DR and a reference travel increment
defined by
. According to Arburg, this reference travel increment
is 0.00115 μm %
−1. Overall, the volume flow of the freeformer is calculated according to:
On the hardware side, the frequency is regulated according to
. If the two volume flows from Equations (
7) and (
9) are set equal, the frequency can be shortened and the relation
is obtained between the parameters LT, DAR, and DR. The free variables of Equation (
10) are LT, DAR, DR and the desired porosity of the part, which is defined in K. One of the three parameters LT, DAR, or DR can therefore be calculated for a desired porosity. For example, for any LT-DAR combination, the DR for optDF can be calculated if the residual porosity at the state of optDF is known.
4.4. Validation of the Model
The model derived in the last chapter can be validated by comparing the calculated residual porosities with those determined experimentally. For the five parameter sets where optDF was reached, the density was measured by using the principle of Archimedes and the porosity using a µCT. These values can be found in
Table 4. Since the residual porosity factor
K refers to an representative volume element (RVE) from
Figure 9 while the measured values are determined global, the assumption is made that the entire volume of the part is uniformly composed of individual RVEs. The measured density can be converted into a porosity by using the following equation
where
represents the density of the specimen,
represents the bulk density of ABS, and
represents the density of air in the voids. Since the density of air is much smaller than of ABS, it can be neglected. Consequently, a residual porosity factor
K is given for the density measurement according to the principle of Archimedes and µCT.
By using Equation (
10), the residual porosity factor
K can be calculated by reforming the equation and use LT, DAR and DR according to the five parameter sets at optDF from
Table 4.
Figure 10 compares the three residual porosity factors determined theoretically and experimentally. The difference between the two experimental methods can be explained as follows: For density measurement, small air bubbles that adhere to the rough surface of the specimen can reduce the density and thus increase the porosity. However, this influence is considered to be small. In the case of porosity determination with µCT, the threshold between material and air influences the size of the porosity. This influence was in the order of ±1%. Despite these method-related errors, the measurements are in good agreement with each other.
If the two experimental residual porosity factors are compared with the theoretically determined one, a certain deviation is noticeable. This deviation could be explained by the fact that process-related variations of LT, DAR, and DR are not considered in the model. In the manufacturing process, a certain deviation between real and ideal LT and DAR can be assumed due to the positioning accuracy of the axes, as well as a variation of DR. Hirsch et al. have shown that the parameter DR varies during the manufacturing process [
3].
Figure 10 shows how an absolute DR deviation of ±1% during the manufacturing process in relation to the DR setting affects the residual porosity factor
K. Taking this process related deviation and the method-related errors into account, the theoretically determined residual porosity factors are in good agreement with the experimentally determined ones. The model can therefore be used to calculate suitable parameter sets for a desired degree of filling. Since the influence of nozzle temperature was not investigated, it is not considered in the model. Further investigations would be useful.
Another aspect can be derived out of
Figure 10. The magnitude of the deviation of the residual porosity factor
K decreases with increasing DR. This effect can be attributed to the change in volume flow, which varies in intensity depending on the DR. With Equation (
9), the ratio of two volume flows is given by
. For the LT-DAR combination LT150DAR14, a DR that increases from 32% to 33% within a layer would fill this layer by 3.13% more. In contrast, for the LT-DAR combination LT250DAR14, a DR variation from 147 to 148% would only cause an increase in volume flow of 0.68%. Therefore, for small DR, process-related deviations in DR have a higher impact on the degree of filling than for a large DR.
4.5. Optimal Degree of Filling and Different Residual Porosity
In the two chapters above, a first volume flow modelling of the APF process was proposed and for the validation the different measured residual porosity was taken into account. This difference is only noticeable by the requirement for a maximum part density while maintaining geometric accuracy, since, as discussed in
Section 4.1, equal part densities can be achieved for each LT-DAR combination. Based on the 3D reconstructions from the µCT, it was possible to determine not only how large the residual porosity is at optDF but also what the shape and distribution of the porosity is. In
Figure 6, the appearance of the voids for the five parameter sets is visible. It is noticeable that almost no voids occur within a layer for each of the parameter sets. This observation is in good agreement with the expectation of optDF. Upon reaching this state, there are no gaps between the tracks not only on the top of the surface, but also in the inner layers. Only in the transition from perimeter to filling small voids can be found. However, the reason for the different residual porosity depending on the parameter sets can be found between the interfaces of two layers. Depending on the parameters LT and DAR, the shape of the droplet inside a part and the associated void volume seem to be different.
If the model is used for a LT-DAR combination to determine the DR at optDF, the residual porosity factor
K is needed. Based on the measured densities from
Table 4 at optDF, the dependence of the residual porosity factor
K on LT and DAR is given in
Figure 11.
Related to the earlier predictions of Charlon et al. or Hentschel et al. that an increase in density occurs as LT or DAR decreases, fails at this point [
8,
10]. In
Table 5, three explanations for the difference in residual porosity are considered.
1. An intuitive explanation might be that the total amount of possible void locations correlates with the residual porosity. This was estimated under “Contact surface” by calculating the total contact surface between the tracks within the layers and between the layers. This explanation does not fit the behaviour of the residual porosities from
Table 4. In
Figure 6, it can be seen that the size of the voids varies depending on the parameter set, which is probably the reason why the explanation “Contact surface” does not work.
2. Another possible explanation was taken by McDonagh et al. [
12]. They notice that the merging of the tracks is enhanced for a 10% larger DAR compared to the measured DAR of a droplet chain. After previous investigations, the DAR of the droplet chain can be determined by using Equation (
1) and empirical equations: For the machine parameters from
Table 3, the length of the droplet is given by
and the width of the droplet is given by
. As seen in
Table 5 under “Compensate merging”, the ratio from DAR to DAR of the droplet chain does not correspond to 1.1, so this explanation cannot explain the difference in residual porosity either.
3. In this work, a new explanation is proposed. For example, for the investigated parameter set LT150DAR14DR32, the droplet length according to the above empirical equation is 199 μm and the desired LT value is only 150 μm. Assuming the droplet has a similar shape to the available RVE from
Figure 9, it will approximately form as a sphere. Otherwise, a force acts on the droplet and deforms it. As a result, the same volume spreads in a more adapted way and fills the available space under less void formation. The more a droplet has to deform, the more voids can be compacted. If a ratio is calculated from the droplet length and the LT, and from the droplet width and the track width, the average of these two ratios can be used as a criterion for the deformation of a droplet. The correlation between the “Deformation factor” and the residual porosity from
Table 4 is in good agreement. Further investigations are useful in the future for validating this explanation.
Finally, the existing approach for parameter determination is compared with the new approach presented in this work. With regard to the parameters DR, DAR and LT, the existing approach is as follows [
6,
12]: With a suitable nozzle temperature, droplet chains at a desired DR are generated and the droplet length as well as the DAR of a droplet chain must be measured. The LT should be approximately correspond to the droplet length. Starting from the measured DAR of a droplet chain for the DAR in the slicing software, several specimens with varying DAR are manufactured. This approach is illustrated in
Figure 12b for an constant LT and DR. If the DAR is decreased, the density of a specimen increases until a desired degree of filling is reached. Although a part with an optDF can be achieved, this does not necessarily have to be an optDF with low residual porosity. Using this approach, for example, Hirsch et al. conclude that the lower the DAR value, the higher the density and the mechanical properties [
3].
In comparison,
Figure 12a shows the approach used in this work. The DR is varied for constant DAR and LT. In this case, an optDF can be determined for one LT-DAR combination and searched for further optDF on other combinations. However, a more appropriate approach for parameter identification is the determination of the residual porosity factor
K as a function of LT and DAR and the use of the proposed model.