1 Introduction

Binder jetting is a two-step additive manufacturing process that is suitable for the fabrication of complex metal or ceramic parts. During binder jetting, an ink is ejected from a print head to selectively bond the powder in the cross-section of a geometry. The individual powder layers are joined together by the ink to shape a green part layer by layer. After a curing or drying step, the loose powder is removed, and the green part is debinded. Subsequently, it is densified in a sintering process [1, 2]. Recently, binder jetting has been employed for applications in various sectors such as energy [3], chemical sciences [4], biomedical [5], and construction [5].

An advantage of the binder jetting process is the possible utilization of particle-loaded inks. Integrated nanoparticles in the ink can selectively change the material composition in the final part [6, 7], function as sintering additives [8], or improve the density and the strength of the green part [9, 10]. However, the penetration of the particle-loaded inks into the powder bed has hardly been investigated. To exploit the potential of particle-loaded inks and enable a stable binder jetting process, it is crucial to understand and control the distribution as well as the penetration depth of the particles in the powder bed.

The behavior of fluids in powder beds has been the subject of many studies. A promising approach to describe the penetration depth h is given by the Washburn equation [11], which is derived as

$$\begin{aligned} {h(t)=\displaystyle \sqrt{\frac{\gamma \cos (\theta )Rt}{2\eta }}}, \end{aligned}$$
(1)

with the pore radius R, the surface tension \(\gamma\), the penetration time t, the dynamic viscosity \(\eta\), and the contact angle \(\theta\) between the fluid and the solid. The equation displays the different parameters that influence the penetration behavior of the ink. While R is related to the powder, \(\eta\), \(\gamma\), and \(\theta\) depend on the fluid-mechanical properties of the ink and, consequently, are affected by the addition of particles. These variables are described in more detail in the following.

1.1 Fluid properties influenced by the particle addition

Viscosity and fluid-mechanical behavior

Adding particles to a fluid increases the viscosity and can even lead to a change in the rheological behavior [12]. The viscosity of particle-loaded inks is a function of their physical properties. Important factors included are the particle volume fraction \(\varPhi\), the particle shape, the original ink viscosity, and the interactions between the particles [12]. The rheological behavior changes with \(\varPhi\) [13]. For many applications, the influence of \(\varPhi\) can be divided into three regimes:

  • \(\varPhi \le 0.02\), where the viscosity increases linearly with a higher shear rate, and the rheological behavior is Newtonian

  • \(0.02 < \varPhi \le 0.25\), where the viscosity shows a higher dependence on \(\varPhi\), but the behavior is still of Newtonian type

  • \(\varPhi > 0.25\), where the regime is characterized by a rapid viscosity growth and a non-Newtonian behavior

Many studies dealing with binder jetting worked with inks showing a Newtonian behavior [4, 14,15,16]. Non-Newtonian fluids behave differently during the drop ejection process [17]. Due to the viscoelastic behavior and the change in viscosity, it can be assumed that non-Newtonian fluids would affect the penetration behavior. Since the viscoelastic behavior is not considered in the Washburn equation [18], the authors expect this to be valid only for Newtonian inks. This limits the feasible values of \(\varPhi\) as described above.

Surface tension

There are contradicting findings in the literature on how the surface tension is affected by the addition of particles to an ink. Tanvir and Qiao [19] stated that the molecular forces, such as the van der Waals forces between the particles, increase the surface tension. Dong and Johnson [20] observed two different effects. At low particle contents, a reduction of the internal energy occurred due to spontaneous adsorption of the particles at the surface. This resulted in a decrease in the surface energy of the ink. With an increase of \(\varPhi\), the capillary attractive forces between the particles led to a higher surface tension.

In contrast, Murshed et al. [21] explained the effect of a reduced surface tension with an increasing \(\varPhi\) by the Brownian motion. Kunchala and Kappagantula [9] investigated particle-loaded inks for binder jetting with different particle mass fractions. They also noticed a decrease of the surface tension at higher particle loads and attributed this to the disruption of the cohesive forces between the water molecules caused by the particles.

Contact angle and capillary forces

Bai et al. [15] calculated the dynamic contact angle in the powder bed for inks with varying copper particle contents. They stated that a larger dynamic contact angle reflects the higher solid loading in the inks. In addition, they found that the capillary number ultimately determines the contact angle. The comparative impact of the viscous forces as opposed to the surface tension is given by this number. From their studies, they concluded that the increase of the particle load raises the capillary-driven penetration up to the amount of 15 m% of the particles. A further addition led to a reduction of the penetration depth resulting from an increased viscosity.

Accordingly, it can be assumed that the contact angle and the capillary forces depend on the surface tension and the viscosity of the ink and are, therefore, influenced by the particle load.

1.2 Penetration time

The penetration time represents the duration required for the drop to fully infiltrate the powder bed, leaving no liquid on the surface [22]. It is influenced by the porosity, the pore size, and the rheological properties of the ink [22] and, hence, by the particle load.

In addition, the ink penetrates in the z-direction (vertical) and spreads laterally on and in the powder bed [23, 24]. The spreading and the penetration occur at a similar time scale and can, therefore, not be treated as consecutive phenomena. With the optical tracking systems applied by various authors, the penetration time for a drop in the mm range can be measured directly [22, 24,25,26]. In binder jetting, the drop size is an order of magnitude smaller (\(\upmu\)m range), making it difficult to measure the dynamic drop infiltration.

Barui et al. [27] installed a time-resolved X-ray imaging method to investigate the real-time powder-ink interactions in binder jetting. They observed a linear relation between the infiltrated volume and the infiltration time. To predict the penetration behavior, they applied a model introduced by Denesuk et al. [28], which was adapted by Holman et al. [23] to account for an advancing drop in the lateral direction. This model is based on the Washburn equation 1 and expresses the time-dependent volume V(t) in a porous structure that is infiltrated by the ink. It can be written as follows:

$$\begin{aligned} V(t)= \int _{0}^{t} \frac{\kappa }{2}\left( \frac{\left( b(t)^{\prime }\right) ^2}{4\sqrt{(t)^{\prime }}} \right) \text {d}(t)^{\prime }, \end{aligned}$$
(2)

with

$$\begin{aligned} \kappa =\pi p \sqrt{\frac{\gamma \cos ({\theta }) R}{2\eta }}, \end{aligned}$$
(3)

where b(t) is the contact diameter of the drop at a given time, and p is the porosity in the powder bed [27]. To account for the total wetted volume in the powder bed, Barui et al. [27] added a porosity correction factor to include the wetted particles and the ink volume in the pores. However, they found a difference between the actual volume of the drops and the predicted volume of the ink that infiltrated the powder bed. The latter was five times as high as the volume deposited.

Other authors observed similar trends. They concluded that the use of the Washburn model requires modifications. In particular, the implementation of a different pore radius to consider an effective porosity that cannot be measured through standard measurement methods, such as pycnometry, was discussed. Hapgood et al. [22] introduced an effective porosity that excludes large macrovoids in loose powder beds with an inhomogeneous pore distribution. Applying this porosity to the Washburn equation resulted in a better fit with their experimental data. While the approach of Hapgood et al. [22] can be used for loose powder beds, Lehmann et al. [25] modified the Washburn equation by proposing a weighted radius that represents the effective porosity in the powder bed during binder jetting. This radius takes the volume fraction and the acting capillary pressure of the pores into account. The calculation with the adapted equation showed a good agreement for two different powders.

1.3 Intermediate conclusion and approach

Most of the studies outlined above are based on large drops to overcome resolution challenges during the optical analysis of the infiltration into the powder bed or on continuously printed drops into one position [27]. The actual penetration depth in a manufactured layer in the binder jetting process has not been investigated. Thus, the transferability of the approaches from the literature is not yet entirely given. In addition, no studies have evaluated the actual influence of the particle load in inks. Bai et al. [15] investigated different particle loadings in inks. However, the observed effects can only be attributed to the change in the fluid mechanical properties caused by the particles and not to the particle behavior in the powder bed.

The reliable application of particle-loaded inks in binder jetting requires an understanding of the behavior of these inks in the powder bed. A modeling approach to predict the penetration depth can significantly reduce the empirical effort to identify suitable process parameters for new material systems and enable the control of the particle distribution in the powder bed. Accordingly, this work aims to predict the penetration depth and to investigate the behavior of particle-loaded inks in the powder bed in binder jetting. The modeling approach for calculating the penetration depth was established, which considers the influences of the process parameters. Due to the difficulty of determining the penetration time, the model is based on a correlation between the ejected drop volume and the actual wetted volume in the powder bed using the sessile drop method. The latter does not require complex measurement systems to estimate the penetration time and can be applied to different powder-ink material systems. To validate the model, a method to measure the penetration depth of particle-loaded inks in the powder bed in binder jetting was introduced. The penetration depths of three different particle-loaded inks after one printed layer were determined and the distribution of the particles in the layer was evaluated.

2 Materials and methods

2.1 Preparation and characterization of the inks

Following the procedure proposed in Lehmann et al. [16], water-based inks with a Newtonian fluid behavior and a particle addition of 1 m%, 5 m%, and 10 m% were prepared. Polyvinylpyrrolidone (PVP; Luvitec 17, BASF, Germany) was simultaneously employed as a binder [29] and a dispersant [30] for the nanoparticles in the ink. PVP was dissolved in a water–isopropanol–ethylene–glycol mixture for 30 min using a magnetic stirrer to prepare the base ink.

To adapt the viscosity to the print head requirements, the ratio of water and ethylene glycol in the base ink was adjusted to the respective particle content. Sodium hydroxide (NaOH) was used for the modification of the pH value of the inks to achieve higher dispersion stability [31]. Graphite nanoparticles (Graphite Nanopowder, Nanografi, Turkey) with a mean diameter of less than 50 nm were added, respectively. These mixtures were homogenized by stirring for another 30 min and sonicated for 160 s with a 1 min cooling break after each 20 s interval.

The viscosity change over the particle load was demonstrated by adding particle contents from 0.1 to 5 m% and additionally 10 m% and 15 m% to the base ink without an adaption of the ratio of water and ethylene glycol.

The rheological properties of the inks were measured using a rotational rheometer (Kinexus lab +, Netzsch, Germany) equipped with a 40 mm plate-plate geometry. The sample gap was set to 0.1 mm. The viscosity was measured at a shear rate of \(80\,{\text {s}^{-1}}\) for 3 min, capturing 36 single-point measurements. Samples were kept at \(25\,^\circ \text {C}\) throughout the procedure.

The surface tensions of the inks were determined by the stalagmometric method according to Traube [32]. Deionized water was applied for the calibration. For each measurement, 20 drops were weighed, and four measurements were performed. The compositions and the properties of the inks used are given in Table 1.

Table 1 Composition, viscosity \(\eta\), and surface tension \(\gamma\) of the different inks; EG: ethylene glycol
Full size table

2.2 Characterization of the powder

For this study, spherical aluminum oxide (Al\(_2\)O\(_3\)) powder (BAK-40, Xtra GmbH, Germany) was employed. The morphology of the particles was captured by scanning electron microscopy (SEM; JSM-IT200 INTouchScope, JEOL, Japan) and is depicted in Fig. 1. The micrograph was taken with a secondary electron detector, a voltage of 15 kV, a working distance of 11 mm, and a magnification of 140.

Fig. 1
figure 1

SEM micrograph of the Al\(_2\)O\(_3\) powder

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The bulk density was determined using a mercury intrusion porosimeter (AutoPore V, Micromeritics Instrument Corporation, USA). The particle size distribution was evaluated with a laser diffraction analyzer (SALD-2201, Shimadzu, Japan). The properties of the powder are summarized in Table 2.

Table 2 Properties of the Al\(_2\)O\(_3\) powder used
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2.3 Contact angle, penetration time, and wetted volume based on the sessile drop method

The apparent contact angle and the penetration time were experimentally examined by the sessile drop method with a drop shape analyzer (DSA25E, Kruess Scientific, Germany) according to the procedure demonstrated in Lehmann et al. [25]. To imitate the powder bed for binder jetting, the powder was placed in a flat vessel and the top layer was recoated by a blade. To provide various drop volumes, needles with five different diameters (0.09 mm, 0.15 mm, 0.2 mm, 0.25 mm, 0.33 mm, and 0.41 mm) were applied. The drop volume \(V_\text {T}\) was calculated following the equation for a sphere, and the drop diameter \(d_\text {0}\) was obtained from the images recorded by the drop shape analyzer. The images were taken with a frame rate of 200 fps during the complete drop imbibition.

Muster and Prestidge [33] and Danilov et al. [34] stated that the apparent contact angle can be measured when the maximum spreading radius of the drop is reached. The apparent contact angle was determined from the first image showing the largest spreading radius. The measurement was performed for at least ten drops.

To determine the penetration time of a drop, the number of frames from the first impact on the powder bed to the complete penetration of the drops was counted and divided by the frame rate. The maximum spreading diameters \(d_\text {max}\) of the single drops deposited onto the powder on micrographs were captured by a light microscope (MM40, Nikon, Japan) after the complete infiltration. They were distinguishable from the powder bed due to the black color of the ink and the white color of the powder. The formed powder granules were dried in a drying furnace (VT 6060 M, Thermo Fisher Scientific, USA) at \(80\,^\circ \text {C}\) for 8 h and retrieved from the loose powder. The penetration depths, which equal the heights of the granules in the z-direction, were measured for ten granules using the light microscope. An example of a formed powder granule is shown in Fig. 2.

Fig. 2
figure 2

Light micrograph of a powder granule with the measured penetration depth; z: penetration direction

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The wetted volume \(V_\text {P}\) was determined by using the equation for a spherical cap, which best approximates the shape of the granule in the powder bed, and can be described as

$$\begin{aligned} V_{\text {P}}=\frac{1}{6}\,h\,\pi \,\left( 3\,\left( \frac{d_{\text {max}}}{2}\right) ^{2}+h^2\right) . \end{aligned}$$
(4)

All measured values are listed in Table 5 in the Appendix.

2.4 Method for the measurement of the penetration depth of a particle-loaded ink

2.4.1 Requirements

To investigate the influence of the particles in more detail, it is beneficial to keep the surface tension, the viscosity, and thus the contact angles approximately constant for inks with different particle contents. For this purpose, the base inks were adjusted to ensure that the resulting particle-loaded inks were in the same viscosity range. The viscosity values of the base inks are presented in Table 3.

Table 3 Viscosity values of the base inks for the three different particle-loaded inks
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The higher the particle load was, the lower was the viscosity of the base ink. To be able to measure the penetration depth of the particle-loaded inks, it is necessary to ensure that only one layer is measured. In addition, the measurements must reflect the entire binder jetting process. It is important that the powder layer to be printed on is wetted by the ink to represent the actual process. A similar saturation in the process ensures the comparability of the different inks.

2.4.2 Fabricated green parts and measurement method to determine the penetration depth

Fig. 3
figure 3

Method to measure the penetration depth of particle-loaded inks in (1) a green part; (2) SEM images over the height of the green part (z-direction) indicate (3) the beginning and (4) the end of the penetrated layer by the presence and the absence of particles, respectively

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The green parts for the validation were fabricated with a binder jetting test rig (VTS128, Voxeljet Technology, Augsburg, Germany). A piezoelectric print head (Spectra SL-128 AA, Fujifilm Dimatix, Japan) with 128 individual nozzles, a nozzle diameter of 50 \(\upmu\)m, and a nozzle spacing of 508 \(\upmu\)m was integrated into the machine. The drop size \(d_\text {0,BJ}\) was optically analyzed using the monitoring setup according to Lehmann et al. [16] with a frame rate of 100,000 fps. The preparation and the slicing of the digital data were performed using the machine software (Rapix3D, Forwiss, Germany). Cuboids with the dimension of 10 \(\times\) 10\(\,\times\) 4.5 mm\(^3\) were fabricated with a layer height of 150 \(^\upmu\)m. The first 15 layers were printed with the base ink (see Table 1). For the subsequent layer, the respective particle-loaded ink was employed. This was followed by 14 layers with the base ink. The printed cuboids were dried in the furnace at 80 \(^{\circ }\)C for 8 h. The printing parameters are listed in Table 4.

Table 4 Parameters used during the binder jetting process
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To ensure the same saturation for the particle-loaded layers and to compensate for possibly clogged nozzles, the printing angle (angle between the print head and the x-axis) was adjusted for the particle-loaded layers. The measurement method for the penetration depth is exemplary depicted in Fig. 3. The completed cuboids were cut lengthwise and perpendicular to the printing direction (1) and micrographs were taken with an SEM along the z-direction and were obtained with a secondary electron detector, an acceleration voltage of 3 kV, a working distance of 9 mm, and a magnification of 800. The micrographs were stitched together (2) and the distance from the beginning of the particle-loaded layer to the end in the z-direction was determined visually by the presence of particles in the micrograph. The end was defined by the absence of particles in the micrograph. At least 15 measurement values per layer were obtained for each green part. In the particle-loaded layer, the particles in the ink were clearly visible in between the powder particles in the area of the ink (3). In the layers printed by the base ink, no particles could be detected. Without particles, the surface of the ink appeared smooth (4).

3 Modeling approach

Since the measurement of the penetration depth is complex and time-consuming, a theoretical prediction is required to easily adapt the process parameters and to conduct material studies in a time-efficient way. Lehmann et al. [25] have shown that the Washburn equation combined with the sessile drop method is a suitable technique to investigate the penetration depth for binder jetting material systems. However, the penetration time needed for the Washburn equation can only be determined in situ with very high effort. Therefore, instead of the penetration time, the actual wetted volume \(V_\text {P}\) in the powder bed is correlated with the respective drop volume \(V_\text {T}\) within this study. In this way, a regression can be applied to infer \(V_\text {P}\) from different drop sizes. In the binder jetting process, \(V_\text {P}\) is assumed to be equal to the volume of a cylinder \(V_\text {Z}\) with the following equation:

$$\begin{aligned} V_{\text {Z}}=\pi \left( \frac{d_{\text {max}}}{2}\right) ^{2}h, \end{aligned}$$
(5)

with the maximum spreading diameter of the drop on the powder bed \(d_{\text {max}}\). This geometry was chosen under the assumption that if the diameter is known, the penetration depth can be calculated.

3.1 Empirical model

In Fig. 4, \(V_\text {P}\) of the powder granules is plotted over \(V_\text {T}\) from the different nozzle sizes for the three particle-loaded inks.

Fig. 4
figure 4

Plot of the wetted volume \(V_\text {P}\) over the respective drop volume \(V_\text {T}\) from the experimental data of the sessile drop method and a linear regression of the data for the three different particle-loaded inks

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According to the linear relation of the formation of the wetted volume in the powder bed [27], a linear regression was performed for each ink. Subsequently, the slopes were determined. From a physical point of view, if there is no drop volume, no wetted volume can occur. Thus, considering the standard equation for a linear regression

$$\begin{aligned} y=mx+b, \end{aligned}$$
(6)

with the slope m and the intercept with the y-ordinate b, b was set equal to 0. \(R^2\) gives the goodness-of-fit for each regression and shows that there is a high agreement of the regression with the measured data. Applying these regression models, the \(V_\text {P}\) for any \(V_\text {T}\) can be determined for the respective particle-loaded ink.

However, to estimate \(V_\text {P}\) during the binder jetting process, the actual drop volume \(V_\text {0}\) must initially be determined. On the one hand, it depends on the drop volume \(V_\text {drop}\) of a single drop from the print head. On the other hand, it correlates with the process parameters. Due to the spreading on the powder bed, the drops overlap during the process to ensure sufficient saturation in the green part. Therefore, it is assumed that the liquid of several drops is present at the deposition area of one drop. A process model to calculate \(V_\text {0}\) is presented in the following.

3.2 Process model to determine \(V_\text {0}\)

The model aims to calculate \(V_\text {0}\) in the area of the maximal spreading diameter of a drop on the powder bed. Since the print head moves relatively slowly (20 mm/s) compared to the velocity of the drops (about 2–2.5 m/s [16]), the shifted drop deposition position from one nozzle is neglected and the drop deposition position on the powder bed is assumed to be directly under the nozzle. Figure 5 depicts the schematic concept of the process model.

Fig. 5
figure 5

Schematic illustration of the process model to describe the printing of drops during the binder jetting process and the effect of overlapping drops in the printed area; the overlapping areas depend on the process parameters

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The print head moves at the printing velocity v over the powder bed and the print head ejects drops with a frequency f from the single nozzles. The nozzles have a distance \(d_\text {n}\) of 508 nm. The print head angle \(\alpha\) can be varied to change the saturation. \(V_\text {drop}\) can be calculated by the following equation for the volume of a sphere:

$$\begin{aligned} V_{\text {drop}}=\frac{4}{3}\pi \left( \frac{d_{\text {0,BJ}}}{2}\right) ^{3}. \end{aligned}$$
(7)

Depending on these parameters, the overlapping areas A alter. The number of overlapping drops can be derived from the sum of the overlapping areas and the area of the main drop.

Figure 6 illustrates the total overlapping drops in the x-y-plane.

Fig. 6
figure 6

Overlapping drops in the x-y-plane; the bold circle in the center indicates the main drop. The dashed circles overlap with the main drop. The distances between the centers of two circles \(s_\text {x}\) in the x-direction, \(s_\text {y}\) in the y-direction, and \(s_{\psi }\) depend on the process parameters

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The red line marks the main drop and the red dashed line marks the drops that are overlapping with the main drop. Depending on the distances between the centers of two circles \(s_\text {x}\) in the x-direction, \(s_\text {y}\) in the y-direction, and \(s_{\psi }\) at the angle \(\psi\) to the y-direction, the number of overlapping drops and the respective overlapping areas change.

These distances vary with the process parameters and were defined as follows:

$$\begin{aligned} s_\text {x} & = \frac{v}{f}, \end{aligned}$$
(8)
$$\begin{aligned} s_\text {y} &= d_\text {n} \sin {(\alpha )}, \end{aligned}$$
(9)

and

$$\begin{aligned} s_{\psi } = \frac{d_\text {n}\sin {(\alpha )}}{\cos {(\psi )}}, \end{aligned}$$
(10)

with

$$\begin{aligned} \psi =\arctan {\left( \frac{d_\text {n}\sin {(\alpha )}f}{v}\right) }. \end{aligned}$$
(11)

Due to the simultaneous infiltration and horizontal spreading of the drop on the powder bed, the \(d_{\text {max}}\) of the drop is not equal to the drop diameter \(d_\text {0,BJ}\). \(d_{\text {max}}\) was determined following the model according to Jung and Hutchings et al. [35] as

$$\begin{aligned} d_{\text {max}} = \beta \; d_\text {0,BJ}, \end{aligned}$$
(12)

with

$$\begin{aligned} \beta = \left( \frac{8}{\tan (\frac{\theta }{2})(3+\tan ^{2}(\frac{\theta }{2}))}\right) ^\frac{1}{3}, \end{aligned}$$
(13)

with the equilibrium spreading factor \(\beta\) and the apparent contact angle \(\theta\) as defined in Muster and Prestidge [33] and Danilov et al. [34].

The overlapping areas A are calculated by the equation for the intersection area of two circles [36]. For this, the equation for one circle intercept

$$\begin{aligned} A_{\text {intersection}}= r^2 \,\frac{\delta - \sin {(\delta )}}{2} \end{aligned}$$
(14)

was used with

$$\begin{aligned} \delta = 2 \arccos {\biggl (\frac{a}{2r}\biggr )}, \end{aligned}$$
(15)

where a is the x-coordinate of the intersection point, \(\delta\) the opening angle of the corresponding circle sector, and r the radius of the circle. Two of these circle intercepts add up to the total overlapping area.

For the calculation, a is replaced by the distance between two drops (in this case, the distances \(s_\text {x}\), \(s_\text {y}\), and \(s_{\psi }\) from the Eqs. 8 to 10). The number of the intersecting drops \(n_\text {drops}\) in each direction varies with the process parameters. Therefore, the sum of the intersecting areas in the x-direction is calculated as follows:

$$\begin{aligned} \begin{aligned} A_{\text {x}}=\sum \limits _{n=1}^{N} \biggl (\frac{d_\text {max}}{2}\biggr )^2 \Biggl (2 \arccos {\biggl (\frac{n\,s_{\text {x}}}{d_\text {max}}\biggr )} \\\ -\sin {\biggl (2 \arccos {\biggl (\frac{n\,s_{\text {x}}}{d_\text {max}}\biggr )\biggr )}\Biggr )}, \end{aligned} \end{aligned}$$
(16)

with

$$\begin{aligned} N< \frac{d_\text {max}}{s_{\text {x}}}. \end{aligned}$$
(17)

N reflects only whole drops. If \(N \ge d_\text {max}/s_{\text {x}}\), there is no more overlap of the drops.

Analogously to Eqs. 16 and 17, the determination of \(A_\text {y}\) and \(A_\psi\) is performed by the calculation in the y-direction and at the angle \(\psi\) to the y-direction.

The total amount according to Fig. 6 results via the following equation:

$$\begin{aligned} A = 2A_\text {x} + 2A_\text {y} + 4A_\psi . \end{aligned}$$
(18)

Since the area for one drop is known, the number of drops infiltrating the area A can be calculated from Eq. 18 with

$$\begin{aligned} n_\text {drops} = 1 + \frac{A}{\pi \bigl (\frac{d_\text {max}}{2}\bigr )^2}. \end{aligned}$$
(19)

This leads to the actual drop volume

$$\begin{aligned} V_\text {0} = n_\text {drops}\,V_\text {drop}. \end{aligned}$$
(20)

The calculated values are listed in Table 6 in the Appendix.

3.3 Validation and discussion

By multiplying the calculated \(V_\text {0}\) with the slope from the experimental regression in Sect. 3.1, \(V_\text {P}\) can be determined. If this volume is equated with that of the cylinder \(V_\text {Z}\), the penetration depth can also be derived.

To validate the model, the penetration depths of three inks with particle loads of 1 m%, 5 m%, and 10 m% were measured.

In Fig. 7, the values from these experiments are compared with the data obtained from the model. The wetted volumes and the penetration depths are plotted over the particle contents.

Fig. 7
figure 7

Comparison of the experimentally determined values for \(V_\text {Z, exp}\) and \(h_\text {exp}\) with the data obtained from the model for \(V_\text {P, model}\) and \(h_\text {model}\)

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The ink with a particle load of 1 m% shows a measured penetration depth of 807 \(^\upmu\)m with a standard deviation of 55 \(^\upmu\)m. The particle-loaded ink with a content of 5 m% penetrated to a depth of 1039 \(^\upmu\)m with a standard deviation of 154 \(\upmu\)m, and the ink with a particle load of 10 m% infiltrated up to 1,220 \(\upmu\)m with a standard deviation of 59 \(\upmu\)m. The penetration depth increases with a higher particle load. The ink with a particle load of 10 m% had a slightly higher viscosity, which can explain the less pronounced increase between 5 m% and 10 m% compared to that between 1 m% and 5 m%. However, this does not elucidate the trend of the rise. The values of the model for 1 m% concur with the experiments. The values for 5 m% show a deviation from the experimentally determined values. The agreement improves for 10 m% but does not attain the same level as observed for 1 m%. The model reflects the changes in the printing angle. Thus, the calculated values of 1 m% and 5 m% are the same, and for 10 m%, the wetted volume and the penetration depth increase. However, this does not explain the trend of the rise for the measured data, since the same angles were used for 1 m% and 5 m%.

The difference between the values of the model and the measured values can be attributed to a change in viscosity during the infiltration process. This becomes clear when considering the SEM images in Fig.  8a, b.

Fig. 8
figure 8

SEM images showing a the upper area of the measured layer with accumulations of particles and reflection during the measurement due to the conductivity of the carbon particles and b the lower part of the measured layer with a reduced amount of particles; the marked areas highlight the amount of particles

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It can be noticed that in the top area of the particle-loaded layer, the particles accumulated. In Fig. 8a, three areas are marked where the high concentration of particles is clearly visible. The particles and the binder in the ink are mostly present on the surface of the powder particles. This indicates an excellent wetting of the Al\(_2\)O\(_3\) powder particles by the ink. In addition, the attraction of the carbon particles to the Al\(_2\)O\(_3\) surface can be attributed to attractive forces [37]. As a result of the accumulation, the particle content in the ink decreased the deeper the ink penetrated into the powder bed. In Fig. 8b, only a few particles are detectable (see marked areas). Figure 9 shows the top area of a particle-loaded layer.

Fig. 9
figure 9

Micrograph of a green part with the last layer printed with a particle-loaded ink; the particle accumulation at the top is clearly visible

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No further layer was printed, so that the accumulation of particles on the surface is clearly visible. While on this top area, the entire surface of the powder particles is wetted with the ink, the ink penetrates into the layer on the surfaces along the pores. Therefore, it can be assumed that the top layer adsorbs a large part of the particles in the build-up direction, while the particles are randomly distributed within the layer.

The difference in the penetration depth can be attributed to the varying viscosities of the base inks of the respective particle-loaded inks. The effect of reducing the particle content in an ink is presented in Fig. 10. The viscosity is plotted over the particle content.

Fig. 10
figure 10

Dependence of the viscosity on the particle content in an ink; different particle contents were added to the base ink (cf. Table 1)

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The base ink used had a constant water-to-ethylene-glycol ratio to enable the measurement of the influence of the decrease of the particle load. An addition of particles led to a linear increase in the viscosity, which is consistent with observations in the literature [38].

The three particle-loaded inks used for the fabrication of the green parts were prepared with different water-to-ethylene-glycol ratios in the base inks. When particles accumulate and remain at the top of the layer, they are extracted from the ink, which penetrates deeper into the powder bed. This results in the absence of the particle fraction responsible for the viscosity increase of the base ink in the deeper areas of the layer. Consequently, the viscosity decreases with further penetration.

At the same time, according to the Washburn equation 1, a lower viscosity of the ink leads to a deeper penetration into the powder bed. Thus, the effect is cumulative.

It can be stated that the model shows very good agreement for low particle loadings, while for higher particle contents a deviation due to the viscosity changes during the penetration is caused. However, compared to the studies of Barui et al. [27], Hapgood et al. [22], and Popovich et al. [39], a much more accurate statement can be made about the actual wetted volume. Another advantage of the model is that it also represents the powder and ink characteristics. The penetration behavior changes with the powder and the ink properties. The sessile drop method takes this into account in the calculation. Šikalo et al. [40] found that a higher viscosity produced a smaller maximum spreading of a drop, while Charles-Williams et al. [41] explained that the drop spreading and the infiltration process are competing processes. By including the maximum diameter of the ink on the powder bed in the model, both phenomena are considered equally.

To use particles to influence the properties in the green part or to change the composition, it is important that the distribution can be controlled. As has been shown, the penetration depth increases with an increased particle content. For a homogeneous distribution, the layer thickness could be reduced or the viscosity of the base ink could be increased. In the latter case, however, there is a conflict between the penetration depth and the printability, since more viscous inks are more difficult to process [16, 42]. Another possibility is to reduce the drying time by changing the drying conditions with a heating lamp so that a too-deep infiltration is avoided and the saturation in one layer can be adjusted [43].

4 Conclusions

Particle-loaded inks exhibit different penetration behaviors in binder jetting. In this study, a model was introduced that allows the determination of the actual wetted volume in the powder. The presented model represents an easily implementable method to predict the penetration depth of binders into powder beds. Through the empirical model using the sessile drop method, the influencing variables of the powder bed and the rheological properties of the ink are incorporated. The ejected drop volume \(V_\text {T}\) was correlated with \(V_\text {P}\) obtained from the sessile drop experiments. From the linear regression, any \(V_\text {T}\) can be correlated with the respective \(V_\text {P}\). The process parameters can be included via the process model, which facilitates the transferability to different binder jetting systems. The model was validated by a measuring method based on SEM images. The particles were detectable in the images and a stitching in z-direction of the images enabled the measurement of the penetration depth of the particles. The prediction of the penetration depth using the empirical model and the process model shows a good agreement with the measured values. It is possible to determine the actual wetted volume \(V_\text {P}\) in the binder jetting process. The model can be used to predict the penetration depth for one layer in binder jetting without using complex measuring systems, such as X-ray, during the process.

The findings of this study can be summarized as follows:

  • The actual wetted volume \(V_\text {P}\) in the powder bed can be estimated by the empirical model on the basis of the sessile drop method.

  • The established process model provides the actual drop volume \(V_\text {0}\) ejected in the area of the maximum spreading diameter of one drop.

  • The prediction of the penetration depth using the empirical model and the process model coincide with the measured values.

  • The particles accumulate at the top of the layer, which leads to a viscosity decrease of the remaining ink and thus to a higher penetration depth.

The non-uniform distribution of particles in a layer is critical for a controlled change of material properties. To ensure that the particles are evenly distributed, the layer thickness is required to be sufficiently small and the rheological properties of the ink must be adjusted accordingly. Reducing the layer thickness improves the balance with the particle accumulation. When the layer is sufficiently thin, the subsequent layer begins right after the particle accumulation ceases. This can be evaluated by the sessile drop method. The process stability could be increased by considering an additional drying step.