The influence of material and process parameters on powder spreading in additive manufacturing

Abstract Additive manufacturing (AM) or 3D printing is beginning to mature from a rapid prototyping to an industrial production technology. However, there are still a lot of fundamental questions that must be addressed in order to make this leap forward. There are many different AM technologies; here, we focus on laser powder bed fusion (LPBF). A key step in LPBF is the initial spreading of the powder layer before it is melted in a solid object, via interaction with a laser. Ideally the powder should be spread as a dense, uniform layer. However, developing a spreading process that can produce a consistent layer, across the wide range of powders used, is a challenge for LPBF manufactures. Therefore, we investigate the influence of materials and process parameters on layer quality. To perform this study we perform computing simulations using the discrete particle method (DPM), a.k.a. discrete element method. This allows us to define metrics to evaluate the powder layer quality, allowing direct comparisons of different tools and parameters. We emulated the effect of the complex particle shape and surface roughness via rolling resistance and interparticle sliding friction. Additionally we investigated the effect of particle cohesion and type of spreading tool. We found that all these factors have a major, albeit sometimes surprising influence on the powder layer quality. In particular, more irregular shaped particles, rougher particle surfaces and/or higher interfacial cohesion usually, but not always, lead to worse spreadability. In general, there is a trade-off between material and process parameters. For example, increasing the spreading speed decreases layer quality for non- and weakly cohesive powders, but improves it for strongly cohesive ones. On the other hand, using a counter-clockwise rotating roller as a spreading tool improves the powder layer quality compared to spreading with a blade. For both tools, a unique correlation between the quality criteria uniformity and mass fraction is reported allowing an easily measured experimental value to be related to the layer quality. Finally, we showed that size-segregation occurs during spreading and this effects is able to explain some of our results.


Introduction
Laser powder bed fusion (LPBF) is an additive manufacturing (AM) technology. In contrast to subtractive or formative methods, objects are produced from three-dimensional digital models in a layer-bylayer fashion. It offers design flexibility and easy customisation that contributed to its rapid growth and wide utilisation in different industrial sectors [1][2][3]. Fig. 1 shows a schematic of the process. Parts are produced by spreading successive layers of powder material and solidifying selected parts by partially or fully melting them with a laser [2,4]. The powder spreading process is governed by the geometry, speed and material properties of the spreading tool. In addition, powder feedstock and powder characteristics play a major role for the powder layer quality, which in turn influences the final product properties and quality [5][6][7][8].
The discrete particle method (DPM) has been recently used to simulate the spreading process in AM. Despite its computational expense, it is a powerful tool for simulating granular materials and understanding particulate system phenomena that are either inaccessible or difficult to obtain from experiments. Early studies of Herbold et al. [9], Mindt et al. [10] and Parteli et al. [11] have used DPM to simulate the spreading process in LPBF. For example, Mindt et al. [10] investigated the influence of the blade gap height: the distance between the powder bed or base plate and the spreading blade, on the spread layer of spherical Ti-6Al-4V powder. They concluded that a blade gap height equal to or less than the maximum particle diameter would result in a reduced packing density. While Parteli et al. [11] simulated the spreading of PA12 powder using a multisphere method to model complex particle shapes with a counter-clockwise (cc) rotating roller as a spreading tool. They showed that the powder bed surface roughness is increased for a higher spreading speed and for powders with a wider size distribution. Parteli et al. [11] findings were confirmed by Haeri et al. [12], who used DPM to spread rod-like particles. They found a higher surface roughness and smaller volume fractions for increased spreading speed and particle aspect ratios. Another study by Haeri [13] investigated the optimization of the blade spreader geometry and found that using a super-elliptic edge profile would result in a better layer quality than a normal flat edge blade.
The flowability behaviour of spherical 316 L stainless steel particles during the spreading process was investigated by Chen et al. [14]. They used DPM to simulate the process with a blade as a spreading tool. They found that decreasing either sliding or rolling friction would decrease the dynamic repose angle and thus improve flowability. While Nan et al. [15] have reconstructed complex particle shape of 316 L stainless steel powder as a function of particle size. They investigated the period and frequency of transient jamming in powder spreading with a small gap height; they found relationships between particle properties, blade speed and gap height. Later, they studied 316 L stainless steel powder flow [16]. They showed that the mass flow rate through the gap, initially, increases linearly with the gap height until it reaches a limit beyond which the mass flow rate cannot be further increased.
Other studies have investigated particle cohesion influence on the spreading process [9,17]. For instance, Meier et al. [17] introduced a DPM model for cohesion and were able to predict the effective surface energy of Ti-6Al-4V. Then they performed a parametric study, highlighting the effect of cohesion on the spreading process. They found that powder layer quality decreases as particle size decreases i.e. cohesiveness increases [18]. Later, Han et al. [19] adapted the approach of Meier et al. [17] to calibrate the surface energy of Hastealloy X (HX) alloy. They found that a layer thickness of 40 μm produce a uniform powder bed spreading.
Recently, Chen et al. [20] investigated 316 L stainless steel powder layer packing density using a blade as a spreading tool. They found that there is a "stress-dip" region at the bottom front of the spreader, and identified three mechanisms that affect the packing density of the powder layer: (1) The "cohesion effect" causes particle agglomerates, (2) the "wall effect" creates vacancies in the powder layer and (3) the "percolation effect" leads to particle segregation. While Fouda et al. [21] performed a DPM simulation of an idealized system with monosized particles using a blade as a spreading tool. They showed that the powder layer packing fraction is always lower than the initial powder heap due to three mechanisms, shear-induced dilation, particle rearrangement and particle inertia.
The "spreadability" of a powder can be defined as the powder ability to spread under certain conditions to form a uniform and highly packed powder layer. Bad spreadability can lead to powder bed defects, segregation, non-uniform density and/or a loose particle packing, all of which have negative effects on the quality of the final product. Unfortunately, powders used in AM tend to behave differently under different conditions. In addition, the recycling of powder changes the material properties, both chemical and morphological ones. Powder properties also depend on powder storage, contamination and environmental effects [5,7,22,23]. Spherical particle shapes are favourable in terms of flowability and powder bed packing density [8]. Fig. 2a shows an SEM image of Ti-6Al-4V powder (produced by plasma rotating electrode) with spherical particles. However, non-sphericity is usually present due to satellites, fractured, adhered particles, etc., as shown in Fig. 3. Fig. 2b shows an example of a Ti-6Al-4V powder (produced by gas atomization) with satellites and wider particle size distribution. 1 The influence of the powder material and spreading process parameters on the spreadability have not been investigated enough in the literature. More specifically, the relationship between particle's shape, surface roughness, cohesiveness and process parameters has not been investigated yet.
In this work we perform a study of the influence of material and process parameters on the spreading process of a Ti-6Al-4V powder, using DPM simulations. The focus is on three material parameters: (i) The interparticle sliding friction as a measure of surface roughness: we assume that an increase in surface roughness causes a reduction in contact area, and thus an increased normal pressure, which causes plastic deformation of the asperities and thus sliding friction. Fig. 4a shows a schematics of the contact area between two rough solid surfaces. The apparent area A a is much smaller than the actual contact area A r , where only the highest asperities are in contact [24]. Fuchs et al. [25] have used nanoindentation to study the rolling, sliding and torsion of micro-sized silica particles. They showed that the measured sliding friction increases as the surface roughness increases. (ii) Rolling resistance as a simple measure to mimic particle's shape, as an approximation of the behaviour of aspherical particles (resembling small asperities) [24,26,27]. Fig. 4b illustrates how rolling resistance results from the imbalance of the normal reaction force f R n at the contact area when an external torque is applied. For example, the rolling behaviour of a polygonal particle can be modelled by a spherical particle with a coefficient of rolling friction μ Estrada r,eq ¼ c 2R [28], Fig. 4b. Wensrich et al. [27] have demonstrated that a complex particle shape can be captured by rolling friction coefficients of spherical particles, where "a value of around half of the normalised average eccentricity (equivalent rolling friction)" was considered as an appropriate amount to capture the effect of particle shape. (iii) Effective dry cohesion due to van der Waals interaction, as determined by the interfacial surface energy [9,25,29,30]. Where the standard trick of dry-coating particle surface with nanoparticles is one way to vary cohesion as we do by changing surface energy [31].
The process parameters investigated are the spreading speed and tool geometry. The effect of the gap height and the layer thickness are not considered in this study. A small gap height, and thus a thinner powder layer is usually preferable to achieve higher resolution, i.e., better adhesion between constitutive powder layers during the sintering/ melting process. Our parametric study can provide a guidance for the calibration of DPM, showing trends and importance of material and process parameters. A quantitative calibration of the model parameters according to a specific powder material is not performed here. However, we perform a rough calibration to show which parameters result in realistic material behaviour. This is done by measuring the static angle of repose and verify/compare our results with the literature, as a basis of our results on the spread powder layer quality. Fig. 5 shows a flow chart of DPM simulation, calibration and validation framework.
Thus, this paper aims to answer the following questions: (i) How to quantify the powder layer quality? (ii) What are the relations between material and process parameters? (iii) What are the effects of particle shape, roughness and cohesiveness on the spread powder layer quality for different process parameters?
This paper is divided into four more sections. In Section 2, we introduce the methods used, e.g. the DPM force law, the simulation setup, the calibration, etc. We then present and discuss the results in Section 3, before we conclude in section 4.

Methods
In Section 2.1, we introduce the force law used in the DPM. The exact DPM parameters used are detailed in Section 2.2. Then we describe the simulation setup in Section 2.3. In Section 2.4, we illustrate the design of the parametric study. We present a preliminary calibration in section 2.5. Finally, we define metrics to characterise the powder layer in Section 2.6.

Discrete particle method
The discrete particle method is used to simulate the spreading process. The interaction of N poly-disperse particles is modelled using the standard linear spring-dashpot model [32] for the normal force. The normal force (parallel to r ij ) is composed of a linear elastic, linear dissipative and a linear adhesive force: with a normal spring stiffness k n , damping coefficient η n , normal relative velocity δ : ij n and a linear adhesion force f adh ij . Each pair of particles i and j are in contact, if their overlap δ n ij is positive. In addition, particles can interact with the base or the powder bed and the spreading tool, which is constructed from polygonal shapes.
Many models exist in DPM to describe dry cohesion of small particles, the attractive force due to van der Waals interaction between particles close to each other or in contact. For simplicity, a linear elastic adhesive force law (acting opposite to the normal elastic repulsive force) is used, which was shown to yield the same bulk rheology as more complicated, more realistic non-linear models [33]: where k adh is the adhesion "stiffness" during unloading. The maximum adhesion force f adh max is defined identical to the pull-off force of the JKR representation of van der Waals interaction [29]: where γ is the surface energy and D eff ¼ D i D j D i þD j is the effective diameter of two particles i and j in contact or close proximity.
The tangential forces (sliding and rolling) are modelled using linear elastic and dissipative forces, where the rolling force is a virtual force, used to calculate the rolling torque. Both the tangential sliding force f s and rolling torque M r are assumed to have a yield criterion, truncating the magnitude of δ s and δ r (the sliding and rolling displacements, respectively) as necessary to satisfy: where μ s and μ r are the sliding and rolling  friction coefficients, respectively, usually assumed to be constant (Coulomb type). For the contact model used, the torque scales with the noncohesive particle diameter for a constant rolling friction coefficient, More details about the contact model can be found in [25,34,35].

DPM parameters
The simulations of the spreading process are done using the opensource code MercuryDPM [36]. The same parameter values, see Table 1, were set for particle-particle, particle-substrate and particletool interactions.
The normal spring k n and damping η n constants are set such that the collision time t c = t g /200 with t g ¼ ffiffiffiffiffiffiffiffiffiffiffiffi ffi D 50 =g p and an intermediate restitution coefficient of ε = 0.4 (for no adhesion) is assumed. To study the effect of particle surface roughness and sphericity, we simulate the spreading process for varying values of interparticle sliding friction μ s and rolling friction μ r , as illustrated in section 2.4. To study the effect of particle cohesion, we simulate the spreading process for varying values of γ such that we can observe a wide range of particle bond numbers. E.g., for γ = 0.1 mJ/m 2 [17]: i g ≈36, 4, 0:8, for D min = 12, D 50 = 37, and D max = 79 μm, respectively; This is consistent with the observation that the effect of cohesion is only moderate in the data presented in Section 3 for γ = 0.1 mJ/m 2 (Bo 50 = 4). The model parameters are set to assure that the interaction is computationally stiff enough i.e. the particle overlap is well below 1% of particle diameter, preventing unrealistic bulk behaviour. Table 1 shows the main DPM simulation parameters. A log-normal particle size distribution (PSD) is used, fitted to the particle size distribution of Ti-6Al4V. The PSD is measured using laser diffraction with D 10 = 24 μm, D 50 = 37 μm and D 90 = 56 μm. Fig. 6 shows the PSD of Ti-6Al4V as implemented in simulation. It should be noted that we use the real particle size distribution of the powder material i.e. high polydispersity Dmax D min ≈6:6, which is not considered previously in the literature, to the best of our knowledge.

Simulation setup
We simulate a small part of the powder bed (width 1 mm), using periodic boundary condition in y-direction. The spreading tool is a blade (commonly used in AM machines) as shown in Fig. 7, moving from left to right at a constant speed v T . The substrate is assumed to be smooth. We insert particles in front of the spreader tool, at (x, y, z) ∈ [0.5,2.5] mm × [0,1] mm × [0,h] mm until the total particle volume   equals 0.7 mm 3 , which is sufficient material to create a powder layer of 7 mm length, 1 mm width and 0.1 mm height. After the particles are settled down and the system is relaxed, the simulation of the spreading process starts by moving the tool at a constant speed v T . Particles reaching the end of the powder bed (at x = 10 mm) get deleted. The simulation ends after spreading the particles in a layer where the tool gap is always set to H = 100 μm, which corresponds to about 2.7×D 50 in z-direction. We stop the simulation at time t max when the system is static i.e. the kinetic energy is very low. Fig. 7 shows the simulation setup, after inserted particles have settled down and during the spreading process.

Design of simulations
The aim of this study is to find the effect of material and process parameters on the spread powder layer quality. The parameter values considered for the spreading process simulations are shown in Table 2.
The upper limit of the interparticle sliding friction μ s is chosen to be 0.5, which is realistic for metal powders [37,38]. The rolling friction was varied as a simple measure to mimic particles non-sphericity [26,27] and was varied between 0.005 and 0.4.
The process parameters considered in this study are the spreading tool speed and the spreading tool geometry. The basic spreading tool velocity is chosen to be v T = 10 mm/s, which is low enough such that there are no inertia effects on the bulk behaviour of particles during spreading [18]. Two different spreading tools are considered, a blade Fig. 7 and a counter-clockwise (cc) rotating roller, Fig. 8 similar to the blade setup presented previously in Section 2.3. The roller radius is r roller = 0.5 mm and the angular velocity is w roller = − v T /r roller . The gap height is fixed at H = 100 μm, such that it is higher than the maximum particle diameter D max = 79 μm [10]. The variation of gap height is not considered here, where increasing the gap height will increase the spread layer packing height and fraction [15,16,18].
We use a full factorial design simulating the effect of four variables for two spreading tools geometry, Table 2.

Preliminary calibration
A calibration of the DPM model parameters is required to accurately predict the bulk behaviour of a powder material. Here, we have not attempted to match the rolling, sliding friction coefficients and surface energy to any specific Ti-6Al-4V powder material. The current work's aim is to show the qualitative effect of those parameters on the powder layer quality. For an accurate calibration, the particle and contact properties should be chosen to match the static and dynamic angle of repose, the cohesive strength, and the apparent density of the powder materialall tests under low confining stress [39], as relevant for powder spreading. Note that the calibration can be done efficiently by taking advantage of a Bayesian calibration procedure [40].
So far, we have performed only one preliminary calibration measurement: we have simulated a static angle of repose (AOR) test, calculated the AOR, similar to Meier et al. [17] and compared the results to experimental AOR values of Ti-6Al-4V powder material. The simulation setup for measuring the AOR is illustrated in Fig. 9. The opening diameter was set to 0.4 mm and the side length of the square base was set to 1.25 mm. Meier et al. [17] and Han et al. [19] show that the size of the base has only little influence on the measured AOR. For some cases (mostly high frictional) of weakly cohesive particles (Bo 50 = 4), particles did not flow through the 0.4 mm opening, so a larger one of 0.8 mm is used. As most cases of strongly cohesive particles (Bo 50 = 15) did not flow through either the 0.4 mm and 0.8 mm openings, 1 mm one is used instead. However, the highly frictional and strongly cohesive ones did not flow even at 1 mm opening. Fig. 10a, b and c show the numerically measured AOR for Bo 50 = 0, 4 and 15, respectively. We see that the AOR increases as interparticle friction and cohesion increase. The white space in Fig. 10c indicates no-flow.
Meier et al. [17] measured the static angle of repose (AOR) of Ti-6Al-4V powder material for calibration. They have focused on the cohesion effect in terms of the effective surface energy, ignoring the effect of  rolling resistance, only considered rolling dissipation and particle sizes between D 10 and D 90 i.e. low polydispersity. Following the same approach for measuring the AOR, including interparticle rolling friction and high polydispersity, we obtain a qualitative comparison with their results. They have found that the AOR is approximately 41 ∘ with μ s = 0.4 and γ = 0.1 mJ/m 2 matches the experimentally measured AOR. This corresponds to our results for the same parameters with approximately μ r = 0.05, the red circle in Fig. 10b. A top view snapshot of the powder layers with μ s = 0.4, μ r = 0.05 and Bo 50 = 0, 4 and 15 are illustrated in Fig. 12a, b and c, respectively. These snapshots show that the results are qualitatively comparable with Meire et al. Fig. 8e, g and h, respectively, [18]. In addition, we have measured the AOR of another Ti-6Al-4V powder material with same PSD used in our simulations, relatively spherical particles shape and some satellite particles [41]. We find that the average measured value of AOR is approximately 44 ∘ , Fig. 11. If we assume that the surface energy γ = 0.1 mJ/m 2 and small rolling friction μ r = 0.05 (since the particles are relatively spherical) then the corresponding sliding friction μ s is between 0.5 and 0.2, Fig. 10b. This way we have provided a basic (rough) calibration with two reference Ti-6Al-4V powder materials. We then investigate how the changes in the powder material characteristics will influence the powder layer quality, by varying the values of interparticle friction and cohesion as presented in Section 2.4. Which is one of the main objectives of this paper. It should be noted that AOR is usually used for calibration and powder flowability assessment. However, It is not a sufficient or standalone measure. Powder flowability is one of the terms used to determine powder quality. Although this does not relate directly to powder "spreadability". We discuss further powder "spreadability" assessment and present a simple verification in Section 3.5.

Powder layer characterisation
In Sections 2.6.1 and 2.6.2, we define two different measures to quantify the powder layer characteristics, namely, mass fraction and uniformity, at the end of the simulation, i.e. after spreading.

Powder layer mass fraction MF
To evaluate the powder layer quality, we define the spread powder layer mass fraction for an assumed volume fraction. Assuming that the layer volume under consideration is V layer = 7 × 1 × 0.1mm 3 , and  assuming a theoretical volume fraction VF = 100%, we can calculate the maximum particle mass needed to achieve VF, where ρ p is the particle density and m layer = 7 × 1 × 0.1 × 10 −6 × 4430 = 0.003101 kg. Then we can define the spread layer mass fraction MF as where m SL is the total mass of remaining particles after the spreading process within the considered layer volume V layer . It should be noted that the maximum volume fraction that can be achieved is about 64% for random close packing.

. Powder layer uniformity and porosity
We use a micro-macro transition method, coarse-graining (CG) [42], to characterise the spread powder layer uniformity. We extract the solid volume fraction (ϕ) from the discrete data using CG. This method has the advantage that the fields produced satisfy mass and momentum balance exactly even near the boundaries. Here we only use the macroscopic solid volume fraction ϕ after spreading, where V i is particle volume. Here, we use a Gaussian coarse-graining function ψ of width (standard deviation) w = 40μm and a cut-off w c = 3w. The width was chosen to be approximately the maximum particle radius [43]. More details of the CG method are beyond the scope of this paper and the interested reader is referred to [42][43][44]. Height integration in z-direction yields a spatial distribution field of depthaveraged powder layer solid volume fraction in xy-directions, where H is the gap height (here also the expected, optimal layer thickness). The spatial distribution of the depth-averaged solid volume fraction ϕ x, y ð Þcan be used as a quantitative measure of the powder layer quality and uniformity, as illustrated in the CG figures in Section 3.3. However, we need scalar values for a comprehensive comparison. We define the coefficient of variation (cv) for each ϕ x, y ð Þ distribution, we obtain a scalar value that can be defined as a measure of spread powder layer uniformity. The coefficient of variation is defined as the ratio of the standard deviation σ to the mean μ ϕ of ϕ x, y ð Þ: where the mean value μ ϕ and standard deviation σ of the solid volume fraction ϕ x, y ð Þare defined as where k = 100 × 100 is the number of sampled (square) grid points in x and y directions, respectively. Non-uniform layers have a high cv, while relatively uniform ones have a low cv. This will be illustrated in   Fig. 13 shows an example of the spatial and probability distributions of the solid volume fraction ϕ.

Results and discussion
First, we discuss powder layer defects that reduce powder layer quality in section 3.1. Then, we present and discuss the measured powder layer mass fractions MF and uniformity cv in Section 3.2 and Section 3.3, respectively. We illustrate particle size segregation in Section 3.4. Finally, we present a verification in Section 3.5.

Spread powder layer defects
We observe several different powder layer defects that affect the powder layer quality, i.e., increase layer porosity. Those defects include empty patches and vacancies, which are caused by particle drag. This occurs when particles are forced forward by the spreading tool, keeping other particles from flowing through the gap. This can either be due to interlocking/clogging for highly frictional (rough) particles, or agglomeration/sticking for strongly cohesive particles. Fig. 14 and Fig. 15 show that particle interlock, particle drag and particle agglomerates occur for both the blade and the counter-clockwise (cc) rotating roller. Whereas strongly cohesive particles stick on both tools, Fig. 15c shows that for the cc rotating roller.
Particle drag during the spreading process was also reported experimentally by Foster et al. [45] and Abdelrahman et al. [46].

Powder layer mass fraction (MF)
Next, we try to understand the collective effect of three different material parameters on the layer quality, as quantified by the spread layer mass fraction MF, for different spreading speeds, using the blade in Fig. 16 or the cc rotating roller in Fig. 17. The major findings and observations are summarized next, while more details are discussed in the following subsections.
The spread layer mass fraction is displayed in Fig. 16 and Fig. 17 for the blade and cc rotating roller, respectively. Each subplot shows the dependence of MF on the rolling and sliding friction, accumulating the results of 42 simulations. The plots are arranged in a 3 × 3 matrix, where each row shows a different speed, v T = 10, 50, 100 mm/s. and each column shows a different mean cohesiveness, Bo 50 = 0, 4, 15. In general, orange indicated very good (close to optimal) layer quality, yellow   Fig. 16 and Fig. 17) are displayed in Fig. 18 (blade) and Fig. 19 (cc rotating roller). From this representationjust by looking at the dominant color in a subplotwe can identify parameter combinations that lead to good or bad results in layer quality. Generally, for either tool, we see that as interparticle friction μ s and μ r increase, MF decreases. For large values of μ s , an increase in rolling friction, μ r , decreases MF, whereas for small values of μ s , μ r has only little influence. In other words, MF decreases as particle roughness and non-sphericity increase. In addition, increasing spreading speed typically decreases layer quality and MF. Likewise, increasing particle cohesiveness, Bo 50 , for constant spreading speed, reduces MFwith some exceptions, as discussed in detail in Section 3.2.1 for the blade and in Section 3.2.2 for the cc rotating roller. Brika et al. [47] have investigated powder flowability of three different Ti-6Al-4V powders. They concluded that powders with more spherical particles result in higher-quality powder layers, while the presence of fine particles   : (a,b,c,f) Empty layers due to global particle drag and particle interlock MF < 10%, (d, e, i) layers with empty patches due to local particle drag 10% < MF < 30% and (g, h) relatively good dense layers MF > 30%. reduces layer quality (which contributes to powder cohesion). Similar results were reported by Sutton et al. [48]. They have reviewed common methods used to characterise AM powders and concluded that the presence of agglomerates and irregular particles shape reduce layer quality and uniformity. While Pleass et al. [49] have observed completely empty layers for strongly cohesive particles i.e. no powder was deposited during the spreading process. Where the powder forms a pile and get dragged in front of the spreading blade tool. These conclusions agree well with some of our findings, which will be further discussed in this section. Comparing the two spreader geometries, unlike the blade tool, the cc rotating roller compacts the powder during the spreading process. This results always in a higher spread layer mass fraction, with MF max of about 60%, while for the blade MF max is only about 50%.
For some cases, e.g., Fig. 16a and Fig. 17a, we see that at low μ s , as μ r increases, MF is almost unaffected. In contrast, at low μ r , as μ s increases, MF reduces considerably. This can be related to particle segregation, as will be discussed in detail in section 3.4.

Blade spreader
Zooming into the MF results using the blade tool, in Fig. 16, most parameter combinations lead to bad layer quality (small MF). Only in Fig. 16a and b yellow/green dominate on bottom and left, whereas in all other subplots MF values indicate bad layer quality. This means that the layer quality is better, the smaller the sliding friction and it improves a little for smaller rolling friction. From Fig. 16a, b and c we see that increasing particle cohesiveness from Bo 50 = 0 to 4 barely affects MF, while it is deteriorating for strong cohesion, Bo 50 = 15. On the other hand, we obtain relatively high MF for non-and weakly cohesive particles (Bo 50 = 0, 4, respectively) in two cases: (i) when particle roughness is relatively low (low μ s ), even for low particle sphericity (high μ r ), (ii) when the particle sphericity is high (low μ r ) even for high particle roughness (high μ s ).
For strongly cohesive particles Fig. 16c, the effect of interparticle friction on MF is different. We see that μ r has a major negative influence on MF. However, as μ s increases, MF increases. The reason can be due to the fact that as μ s increases, contacts between particles decreases reducing the effective cohesion. Similar effects have been previously observed numerically and experimentally. Savkoor et al. [50] have investigated the influence of the tangential forces on adhesive contacts. They concluded that the contact area between particles decreases as the tangential force increases. In addition, Fuchs et al. [25] have used nanoindentation to study the rolling, sliding and torsion of micro-sized silica particles. They showed that, as the surface roughness increases, the sliding friction increases and the sliding adhesion decreases. Fig. 20 show a top view of the spread powder layer for strongly cohesive particles Bo 50 = 15. It illustrates that powder layer quality decreases as μ s decreases due to particle agglomerates increase. Fig. 16d, e and f show MF for non-, weakly and strongly cohesive particles, Bo 50 = 0, 4, 15, respectively, with spreading speed v T = 50 mm/s. Similarly, Fig. 16g, h and i with spreading speed v T = 100 mm/s. We clearly see that, for non-or weakly cohesive particles, increasing the spreading speed v T for all cases has reduced the MF compared to the lower spreading speed v T = 10 mm/s. MF max is between 30 and 20% at v T = 50 mm/s and MF max = 20% at v T = 100 mm/s, compared to MF max ≈ 50% at v T = 10 mm/s. Qualitatively, this means increasing spreading speed will reduce layer packing fraction. (c) Empty layer due to global particle drag and particle interlock MF < 10%, (b, f, i) layers with empty patches due to local particle drag 10% < MF < 30% and (a, d, g, e, h) relatively good dense layers MF > 30%.  For strongly cohesive particles, increasing the spreading speed has the opposite effect. MF slightly increased at high μ r and low μ s compared to the lower spreading speed v T = 10 mm/s, as seen from the contour lines. Fig. 21 show the same effect in a top view of spread powder layer with different spreading speeds v T = 10, 50, 100 mm/s for strongly cohesive particles. A possible explanation is that the high shear rate resulting from higher spreading speed broke the interlocking in front of the spreading tool caused by particle agglomerates and high μ r , allowing particles to flow.
To provide a basis for the possibility of particle interlocking, we will estimate the range of inertial numbers I, to predict the flow state and the quantify dynamic effects in the system, for two different speeds, as illustrated in Fig. 21a (v T = 10 mm/s) and Fig. 21c (v T = 100 mm/s). The inertial number is defined as where d p is the particle diameter, ρ p is the particle density, γ : is the shear rate and P is the pressure. Usually three regimes are distinguished: (i) quasi static flow (I < 10 −3 ), (ii) dense flow (10 −3 < I < 10 −1 ) and (iii) collisional flow (I > 10 −1 )however, the boundaries between regimes are not sharp.
First we estimate the inertial number of a single average particle at the free surface: where D 50 is the average particle diameter, is the average single particle pressure, γ : ¼ vT H is the shear rate through the gap (assuming homogeneous shear), m 50 is the average particle mass, g is the gravitational acceleration, v T is the spreading tool speed and H = 100μm is the rather thin gap height. Larger gaps will only result in smaller I < I D , i.e. an upper limit.  Next we estimate the (smaller) inertial number I gap for the base powder layer in the gap (assuming homogeneous shear): which is smaller due to the weight of particles, neglecting possible vertical compression effects in the gap due to the tool. Finally, we estimate the inertial number I Z of the particles under the pile (with height Z pile ) in front of the spreading tool, in a thin shear zone of thickness ∝H, at the bottom of the pile: due to the weight of the particles in the pile. Since the shear rate in the bulk is typically even smaller, I Z resembles an upper limit estimate for the bulk. If the slip layer at the base, as observed in some cases, is localized (thickness ∝D 50 ), the estimate would be I > I Z (not shown), so that I Z represents the lower limit for particles within, and close to, the gap.
Comparing the lower limits, I Z 10 = 0.0660 < I Z 100 = 0.5482, we conclude that for the lowest spreading speed (v T = 10 mm/s), one could expect dense flow particle interlock, but not for the more dynamic flow state at the highest spreading speed (v T = 100 mm/s), where inertia effects can destroy particle interlock. Not considered in this estimate is the effect of larger gap heights that will strongly reduce the possibility for interlocking, whereas material parameters like friction and cohesion will work in favor of particle interlock in the gap. In the bulk of the pile, the inertial numbers can be much smaller, I ≪ I Z , due to very small, possibly vanishing shear rates in the dense flow state. In fact, inside the bulk, we either observe circular (slow) flow patterns, or completely blocked solid-like piles (field data not shown in this paper).

Roller spreader
Simulations with the cc rotating roller (Fig. 17) generally show a higher MF than simulations with the blade spreader. However, the main qualitative dependencies on the material properties remain the Fig. 24. The correlation between cv and MF. The dashed green and blue horizontal lines represent the layer uniformity, cv u , and vacancies, cv m , limits, respectively. The solid black curve represent the fitting function given by Eq. (14), for parameters see main text. The solid blue and green lines represent the power laws in Eq. (14) i.e. the first and second terms, with power laws p 1 = 0.5 and p 2 = 3, respectively. same; similar to the blade spreader. Fig. 17a and b show that the interparticle friction effect on MF is almost the same for non-and weakly cohesive particles. For strongly cohesive particles Fig. 17c, we also see similar behaviour as a blade spreader, low sphericity (high μ r ) has major negative influence on MF. As for the blade, MF increases as μ s increases; however, the effect is much more pronounced: at the highest values of μ s and μ r , a higher spread layer mass fraction MF is obtained, compared to the case for a blade spreader. Fig. 17d and e show MF for non-and weakly cohesive particles, respectively, at v T = 50 mm/s. Similarly, Fig. 17g and h at v T = 100 mm/ s. We see lower values of MF for almost all cases compared to lower spreading speed at v T = 10 mm/s. Unlike the case of the blade spreader, we see a higher dependency on μ s than on μ r at v T = 50 mm/s. Beside that at low μ s , MF is higher for weakly cohesive particles compared to non-cohesive ones; which can be seen at v T = 100 mm/s as well. It seems that the combined effect of particle cohesiveness and roller compaction allowed particles to adhere better to each other and to the substrate, compared to non-cohesive particles. Fig. 17f and i show MF for strongly cohesive particles at spreading speed v T = 50 and 100 mm/s, respectively. At high μ r and low μ s , increasing the spreading speed v T has significantly increased MF compared to lower v T = 10 mm/s. Surprisingly, MF increases at the higher limit of interparticle friction μ s and μ r compared to intermediate values. This indicates a lower and an upper limit of particle roughness (μ s value) at which particles sphericity (μ r value) has an effect on the spread powder layer quality when using a roller. Fig. 22 show top view of spread layer using a roller, illustrating the effect of increasing spreading speed v T . In addition, there is a waving effect on the spread layer for weakly and strongly cohesive particles at low μ s when v T = 100 mm/s e.g. Fig. 22f.

Powder layer uniformity and porosity
Dense, uniform powder layers are required to achieve high quality products with low porosity. Previously, we illustrated layer defects which reduce layer uniformity and increase porosity. In this section, we study the spatial distribution of the powder layer solid volume fraction, ϕ x, y ð Þ, where it is utilized to evaluate powder layer uniformity.
In Fig. 23, we show the solid volume fraction ϕ for three cases, that are representative of the typical kind of powder layers we obtain after spreading: (i) a uniform layer, (ii) empty patches and (ii) a nearly empty layer. We further plot the probability distribution for each case and determine its coefficient of variation cv. The cases shown use a blade spreader at v T = 10 mm/s with weakly cohesive particles, Bo 50 = 4. We see good powder layers with a homogeneous narrow normal distribution at low interparticle friction; as the interparticle friction increases the spread powder layer uniformity decreases, with empty patches indicated by the dark blue regions in the contour plot of ϕ. For very high interparticle friction, the probability distribution shows a high peak at zero. Predictably, we observe the highest coefficients of variation for nearly empty layers, and the lowest for uniform layers; thus, we aim to use the coefficient of variation as a measure of layer uniformity and we will see if and how it correlates with MF. Fig. 24a and b show the correlation between cv and MF, for a blade and a cc rotating roller, respectively. Each row shows one spreader velocity, increasing from top to bottom. The value of cv u = 0.2 is set as an upper limit for a uniform layers. For increasing cv > cv u the layer uniformity decreases, where layer vacancies and empty patches occur, up to the upper limit of cv m = 0.9; for cv > cv m , we see severe particle interlock and drag causing empty layers. In Fig. 24, we see that the data for different Bo 50 accumulate on a master curve, sometimes well above the upper, in between, or well below the lower limits. The former case are bad packings, whereas the latter are the good quality layers. Thus, we propose the following function to fit the data for a blade and cc rotating roller at v T = 10 mm/s First assuming d 1 = 20 and p 1 = 0.5, which reasonably fits the large cv > cv m data, we get p 2 = 3 ± 0.4, d 2 = 20.5 ± 1 or p 2 = 3 ± 0.2, d 2 = 23.5 ± 1, for a blade or cc rotating roller, respectively. This master curves fit all these data pretty well and differ only slightly in the d 2 coefficient, meaning that the roller produces slightly better layers with higher MF, or equivalently lower cv. The assumed criteria for d 1 and p 1 result from fitting the data with high cv ≥ cv m (the solid blue line in Fig. 24) with the term which gives d 1 ≈ 20 and p 1 ≈ 0.5. Then the master curve is assumed as proposed in Eq. (14). The solid green line in Fig. 24 is the term with p 2 = 3, d 2 = 20.5 or p 2 = 3, d 2 = 23.5 for a blade or cc rotating roller, respectively.  In more detail, for a blade spreader, Fig. 24a, we see that powder layers with high MF are uniform, mostly for non-and weakly cohesive particles, Bo 50 = 0 and 4, respectively. In contrast, for strongly cohesive particles Bo 50 = 15, powder layers either end up close to the low uniformity limit, cv u , or are of very bad quality. Increasing the spreading speed from v T = 10 mm/s to v T = 50 and 100 mm/s, decreases layer uniformity significantly in almost all cases.
For a cc rotating roller, Fig. 24b, we see similar qualitative results as for a blade spreader when changing the parameters. However, increasing the spreading speed has considerably improved powder layer quality for strongly cohesive particles Bo 50 = 15.

Particle size segregation
Polydisperse particles segregate when they are in motion. Fig. 25 shows the fraction of particles of a given diameter remaining in the power layer after spreading. I.e., a value of 60% indicates that 40% of the particles have been dragged off the plate by the spreading tool. The data shown is for a blade spreader at v T = 10 mm/s. In Fig. 25a μ r = 0.005 is fixed and μ s is varied. While in Fig. 25b μ s = 0.1 is fixed and μ r is varied. The columns show different bond numbers, from left to right Bo 50 = 0, 4 and 15, respectively. We can clearly see that, when μ r is fixed and as μ s increases Fig. 25a, the retained  fraction of small particles increases and the retained fraction of large ones decreases, indicating particle segregation (large particles are more likely to be dragged away by the spreader). In the reverse case, when μ s is fixed and as μ r increases Fig. 25b, we do not see much difference in the retained particle fraction. Thus, μ s has larger effect on size-segregation during the spreading process than μ r . For strongly cohesive particles, the behaviour is similar, but less pronounced, due most likely to the formation of particle agglomerates and a higher dependency of MF on μ r , as mentioned in the previous sections. In addition, It should be noted that the effect of μ r and μ s on particle size segregation is comparable for powder layers within the same range of MF. Fig. 27 shows the powder pile during the spreading process with a blade. Only very large (D = 75-79 μm) and very small particles (D = 12-24 μm) are made visible, to be able to see particles migration. Fig. 27a shows the powder pile at low μ r and high μ s for noncohesive particles, we see that mainly large particles are accumulated at the tip of the powder pile. While at low μ s Fig. 27b, both large and small particles are accumulated at the tip of the powder pile. Similar effect can be seen for strongly cohesive particles Fig. 27c,d, however with less frequency due to particle agglomerates. Similar behaviour is obtained when using a cc rotating roller as a spreading tool, see Fig. 26 and Fig. 28.

Preliminary verification
Current methods used to assess powder flowability do not relate directly to powder "spreadability" in LPBF. Recently, Cordova et al. [41] have used two wiper blades "applicator tools" designed by the Netherlands Aerospace Centre (NLR) to manually access powder "spreadability" for the applications in LPBF. Both tools have the same blade profile. The first one "open tool" spread the powder pile directly over a metal strip, while in the second tool "funnel tool" the powder is spread through a funnel like shape. They have quantified a thin layer (100 μm) quality of four different powder materials and concluded that the "open tool" is better to access the "spreadability" of moisturised powder. In another study, Ahmed et al. [51] have used a simple method to investigate powder layer uniformity. They have spread a powder layer manually with a blade over a glass slide covered by Emery paper. Here, we have also used a simple method to obtain a qualitative comparison of powder layer quality between a blade spreader and cc rotating roller: Powder samples of equal weight were prepared and placed in front of each tool, using a small container box to form a pile of powder. Then the powder was spread manually over a paper tape. The powder used in both experiments is Ti-6Al-4V with measured AOR ≈44 ∘ . Fig. 29 illustrates the design of those tools and the corresponding layers spread. We see that a cc rotating roller produces a better quality layer than a blade. This shows that same powder material can produce different powder layer quality by changing the process parameter, in this case the spreading tool geometry. Further experiments using same type powder materials (e.g. Ti-6Al-4V) with different characteristics (e.g. particle shape, particle size, etc) will provide more qualitative verification/validation of our simulation results covering a wide range of material parameters that should include many real powder materials.

Conclusions
We have simulated the spreading process in AM with the discrete particle method (DPM) and characterized the powder layer quality. We have shown that different powder layer defects can reduce the spread powder layer quality. Those include particle drag, interlock and agglomerates, all of which lead to empty patches that reduce packing fraction, uniformity and thus increase layer porosity. Powder layer defects are more likely to occur due to either high rolling friction, μ r , high sliding friction, μ s , strong particle cohesion, Bo 50 , or the combined effects of those three material characteristics.
When using a blade as a spreading tool, for non-and weakly cohesive particles (Bo 50 = 0 and 4, respectively), we obtain relatively uniform layers with high layer mass fractions at either low μ r (even for high μ s ) or low μ s (even for high μ r ). For strongly cohesive particles (Bo 50 = 15), μ r has a major negative influence on layer mass fraction and uniformity, while μ s has a surprising positive effect: we obtain relatively high layer mass fractions and good uniformity in the limit of low μ r or high μ s .
When using a counter-clock wise rotating roller as a spreading tool, better packed and more uniform layers were obtained for almost all cases due to a constructive compression/shear effect. Similar to the blade tool, for strongly cohesive particles at low μ r or high μ s , we observe good layer quality.
Increasing the spreading speed has reduced the layer quality for non-and weakly-cohesive particles, for both tools. In contrast, for strongly cohesive particles, the layer quality slightly improved with speed for a blade spreader, but significantly improved for a cc rotating roller.
For both tools, it was shown that μ s has more influence than μ r on particle size segregation, which is at the origin of some of the nonintuitive trends we observe.
We have performed a simple experiment to obtain a qualitative comparison of powder layer quality between a blade spreader and counter-clockwise (cc) rotating roller. It was shown that a cc rotating roller provide a better layer quality than a blade.
The present work has provided insight into the combined effects of powder material and process parameters on the spreading process in AM. We did not consider the influence of two process parameters in our current study, namely the gap height and the ambient gas drag, which will be included in future work. Although a comprehensive experimental validation is missing in our current work; nevertheless, we have performed a preliminary calibration and a simple experimental verification.
Further research will focus on experimental calibration and validation of the spreading process for (i) the powder during the process cycle (virgin, used, recycled, etc) and (ii) using the same type powder materials with different characteristics. For calibration, the particle and contact properties should be chosen to match the static and dynamic angles of repose, the cohesive strength, and the apparent density of the powder materialall tests under the relevant low confining stress during spreading. For validation, the mass fraction of the spread powder layer should be measured from experiments and compared with simulations. All this can be done for virgin and re-used particles. Finally, a microscope can be used to investigate the powder layer uniformity, in order to obtain a solid volume fraction distribution, and thus the coefficient of variation. For that purpose, we are developing an experimental setup to accurately control the process parameters and quantify the layer quality.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Mohamad Yousef Shaheen is a PhD candidate at the Multi-Scale Mechanics group at the University of Twente. He is also a member of the MercuryDPM developers team since 2017. He holds a BSc in mechanical engineering and completed his MSc studies as part of the joint Erasmus Mundus MSc program "MathMods", where he obtained a double major MSc degree in mathematical engineering and computational materials science. His PhD research focuses on the numerical and experimental process optimization of laser powder bed fusion (LPBF) technology. He utilizes existing numerical models/tools and develops new ones to predict materials processability for LPBF. He also aims to validated the simulation results with experiments.
Anthony Thornton is an applied mathematician whose research focuses on combined theory, experiments and simulations to model granular systems. He is best known for his work on modelling particle-size segregation in dense granular flows and micro-macro transition methods. He cofounded the open-source simulation code MercuryDPM and his research has applications in many areas including 3D-printing, pharmaceutical, food science, mining, particle technology and geophysics. In 2015 he co-founded the UT spin-off company MercuryLab, whose aim is to make MercuryDPM accessible for industry utilisation and 2017 he gained the title Professor of 'Granular Materials'.