Numerical simulation of the effect of fine fraction on the flowability of powders in additive manufacturing

Journal Pre-proof Numerical simulation of the effect of fine fraction on the flowability of powders in additive manufacturing Yifei Ma, T. Matthew Evans, Noah Philips, Nicholas Cunningham PII:

S0032-5910(19)30864-2

DOI:

https://doi.org/10.1016/j.powtec.2019.10.041

Reference:

PTEC 14808

To appear in:

Powder Technology

Received Date: 26 January 2019 Revised Date:

20 September 2019

Accepted Date: 9 October 2019

Please cite this article as: Y. Ma, T.M. Evans, N. Philips, N. Cunningham, Numerical simulation of the effect of fine fraction on the flowability of powders in additive manufacturing, Powder Technology (2019), doi: https://doi.org/10.1016/j.powtec.2019.10.041. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

Numerical Simulation of the Effect of Fine Fraction on the Flowability of Powders in Additive Manufacturing Yifei Maa , T. Matthew Evansa,∗, Noah Philipsb , Nicholas Cunninghamb a School

of Civil and Construction Engineering, Oregon State University, Corvallis, OR 97331, USA b ATI Specialty Alloys and Components, Albany, OR 97321, USA

Abstract Additive manufacturing (AM) is a rapid and flexible technique for the production of metal parts and prototypes from metal powders. The quality of parts manufactured using AM is a strong function of powder physical and mechanical properties. Previous work has shown that powders with a large fraction of fine particles produce better parts in terms of smooth finished surface and high mass density. However, an excessive fine fraction in the source powder causes serious flowability issues, leading to unexpected voids or discontinuities in the finished product. This effect of the fine fraction on the flowability of metal powder is widely encountered but poorly understood in the AM industry. This study presents a three-dimensional (3D) discrete element method (DEM) model to simulate the microscale mechanisms of powder flow considering the effects of van der Waals force. This microscale force has a negligible influence on the flowability of coarse grains, but the effect becomes the dominant factor governing the behavior of fine fractions (micrometer scale). The results show that the numerical model presented in this paper is capable of reproducing the experimental dependency of the powder flowability on the fine fraction. Moreover, it also successfully captures the characteristics of particle flow under the ∗ Corresponding author at School of Civil and Construction Engineering, Oregon State University, Corvallis, OR 97331, USA. Email address: [email protected] (T. Matthew Evans)

Preprint submitted to Powder Technology

October 15, 2019

influence of microscale van der Waals force. Keywords: discrete element method, additive manufacturing, flowability, van der Waals force

1. Introduction Metal additive manufacturing (AM), also known as three dimensional printing, is a fast developing technology for the production of a wide variety of solid components from powders [1, 2]. Most current metal additive manufacturing systems are of the powder bed fusion type, which involves various printing techniques: direct metal laser sintering (DMLS), electron beam melting (EBM), selective heat sintering (SHS), selective laser melting (SLM), and selective laser sintering (SLS). Regardless of the fabrication technique, a critical prerequisite before creating solid parts is to spread powders on top of a deposition bed (building platform). The efficacy of the manufacturing process is a strong function of the uniformity with which the metal powder is spread on the deposition surface, which in turn, depends on the gradation of the metal powder [3, 4, 5]. The correlation between particle size distribution (PSD), bed quality, and powder flowability is of great interest to metallurgists and engineers because choosing proper powder gradation and processing parameters can significantly enhance the quality of the AM product and reduce the cost. However, the ambiguity surrounding this issue primarily owes to a general inability to characterize the grain scale interactions. Recent studies have shown that the uniformity with which the powder can be spread on the depositional surface is a strong function of the grading of the input powder [6, 7, 8]. Specifically, experiments with stainless steel powder of three different PSDs suggests that a proper amount of fine particles can improve the properties of additive manufactured part in terms of the surface roughness, density and mechanical strength [9, 10]. That substantial fines content improves the 2

smoothness and increases the density of finished parts is not surprising. From previous work on concrete mix proportioning [11, 12] and earthwork construction [13], it is known that materials with a wider gradation are capable of achieving a denser packing. At the microscale, this is attributable to the fact that smaller particles will migrate to fill voids between larger particles [14]. However, an excessive amount of fine particles can significantly hinder the particle flow behaviors, leading to unexpected voids and defects [15]. Therefore, understanding the effect of fine fractions on the flowability of metal powder and determining the optimum amount of fine particles is of great interest to engineers. Indeed, prior experimental work on flow behavior of fine powders has been reported by researchers in different fields but the interpretations of the results are not consistent because of the complex physics and large number of governing factors [16, 17, 18, 19]. In general, the correlation between powder flowability and particle size depends on inter-particle adhesions [20] caused by inter-molecular forces including van der Waals (vdW) force, electrostatic charges, and capillary attractions. Of the many microscale forces, the van der Waals force is the most common because it occurs in all situations and plays a dominant role in determining the quality of the deposition bed in AM [21, 22, 23]. Unfortunately, experimental study of the spreading process under complex boundary conditions with various processing parameters is expensive and fails to provide an understanding of the fundamental underlying problem physics, because many of the governing behaviors are simply inaccessible in the laboratory [24]. To address this issue, numerical simulation tools have been developed to investigate the mechanical behaviors of the powder in the spreading process [25, 26, 27]. Considering the fact that the exact theoretical solution of the inter-particle cohesive force is complex and computationally expensive [28, 29], the van der Waals force is approximated using a simplified expression with a

3

cut-off distance to avoid a singularity when particles are near contact [30, 26]. It is rather problematic to simply use experimentally measured values for the numerical parameters, because no attempts have been made to realistically capture the behavior of the electron cloud or the atoms in the numerical models. Besides, the parameters experimentally obtained under a specific scenario may not be appropriate for other powders or loading configurations. In the present work, we propose a three-dimensional discrete element method model to simulate the effect of fine fractions on the flowability of dry metal powders. Through the use of discrete element method (DEM) simulations, we intend to capture the evolution of microscale forces during the particle spreading process in additive manufacturing and characterize the correlation between particle flowability and van der Waals force. An essential prerequisite of implementing this model is to properly calibrate the parameters governing the bulk powder behavior, i.e., the cut-off distance for van der Waals force. Hence, an experimental procedure to calibrate this microscale parameter is also introduced in this study. Then the quality of the deposition bed is evaluated in terms of volume fraction and surface roughness. The flowability is quantified using the force and potential energy ratios. Finally, a correlation between the powder flow behavior and amount of fine fractions is proposed.

2. Model setup The DEM code PFC 3D [31, 32] is employed in this analysis to model the flowability of granular materials. PFC 3D uses an explicit central difference time integration scheme to simultaneously solve Newton’s equations of translational and rotational motion for all particles. DEM is thus well suited to model granular interactions during powder flow. The accuracy of the simulations is influenced by the choice of the contact force model [33, 34]. Implementation of

4

the contact model in this study is realized by incorporating a van der Waals force model in addition to a linear elastic contact law with rolling resistance. The inter-granular forces, including normal and shear contact forces, rolling resistant torques, and cohesive van der Waals forces are considered. All of these forces can be categorized into two groups: one is associated with mechanical contact of the particles while the other corresponds to micro-interactions which occur over a short separation distance. 2.1. Mechanical contact forces The mechanical contact can be envisioned as a series of elastic springs distributed over the contact point between a pair of particles in contact, as shown in Figure 1. In addition to the normal and shear contact forces, rotary torques can be transmitted through the contact as well. These properties can be described by the following microscale parameters: the normal and shear contact stiffnesses, kn and ks [F/L]; the friction coefficient µ; the normal and shear damping coefficient ηn and ηs [F/(L · T −1 )]; and the rolling resistance stiffness kr [F · L/deg]. Since some of the microscale parameters cannot be directly measured or calculated, it is more convenient to specify the bulk material properties, such as apparent modulus Ec , as the input parameters instead. Therefore, the normal contact stiffness is estimated from [32],

kn = 2Ec (Ra + Rb )

(1)

where Ra and Rb are the radii of the two particles in contact. Assuming compression positive, the contact model relates the contact forces and particle overlap through,

Fn = kn δn + ηn δ˙n

and

5

∆Fs = ks ∆δs + ηs δ˙s

(2)

and follows Coulomb’s law of friction,

|Fs | ≤ µFn

(3)

where Fn and Fs denote the normal and shear contact forces, respectively; δn is the shortest distance between two particles (negative value indicates a gap at the contact and positive represents an overlap) and δs is the slip between the pair of particles. The normal and tangential damping coefficients, ηn and ηs , are material constants related to the mass, stiffness and coefficient of restitution (COR). Generally, the COR can be determined by experiments and thus,

ηn = 2βn

p m c kn

and

ηs = 2βs

p

mc ks

(4)

where mc is the effective inertial mass of the contact; βn and βs are the ratios of the damping constant to the critical damping constant in the normal and shear directions, respectively. Given the coefficient of restitution, critical damping ratios can be determined according to [35], ln(COR) β = −q ln2 (COR) + π 2

(5)

Experimental results show that COR is normally in a range of 0.6∼0.8 for metal powder at the micrometer scale [36]. In this study, we assume COR=0.6 and thus βn = 0.16. For simplicity, we also assume βn /βs = 1.0. It has been noted that the effect of rolling resistance at contacts is of great importance for many granular applications [37]. Particle rotation in granular materials, no matter dry or wet, would be exaggerated without considering the rolling resistance contributed from particle angularity and interlocking mechanisms [38]. Therefore, a rotary spring as shown in Figure 1(c) is implemented to provide a torque to counteract rolling at the contact. The increment of rolling

6

resistance moment ∆Mr can be determined from,

∆Mr = −kr ∆θr

(6)

where kr is the stiffness of the rotary spring that can be determined from kr = 2 ks Req and Req = Ra Rb /(Ra + Rb ); and θr is the relative rolling angle. Rolling

slip occurs when Mr > µr Req Fn . Then the total torque at a contact is defined as [39],

Mr =

   

Mr ≤ µr Req Fn

Mr

   µr Req Fn

(7)

Mr > µr Req Fn

where µr is the rolling resistance coefficient. [Figure 1 about here.] The linear elastic spring and dashpot model is capable of accounting for mechanical contact forces, including normal and shear forces, viscous damping forces and rolling resistance torques; however, this model becomes problematic if significant nonlinear behaviors or large overlaps are present. In this study, the particles are typically subjected to gravitational or small external load conditions, so the observed inter-granular overlaps are less than 0.5% of the grain radius. The impact of the overlaps on the flowability of the particles is negligible under these conditions [40]. Considering the computational efficiency, a linear spring-dashpot model is applied in this study. It has been shown that this model performs as well as the nonlinear Hertz-Mindlin model in capturing the characteristics of particle flow if the parameters are properly chosen, especially when the details of collision forces are irrelevant [41, 42].

7

2.2. Van der Waals force Van der Waals forces are electric dipole forces formed by the electron cloud and the nucleus; they exist in all matter and are responsible for the attractive force between molecules. Generally these are considered weak forces and are negligible when studying macroscale behaviors of materials. However, the grain size of powders investigated in this study can be as small as several micrometers. The magnitude of van der Waals force becomes comparable to the inter-particle contact forces caused by gravity as the particle size reduces. During particle spreading or repose angle tests, the particle flow is mostly triggered by gravity. Thus, understanding the effect of the weak forces on the flowability of the powder particles as a function of particle size becomes necessary for further investigation of the particle spreadability in AM. [Figure 2 about here.] Interactions between two particles can be described by a simple mathematical model, the Lennard-Jones potential U (z), see Figure 2. The interaction force then can be calculated by the derivative of the potential energy. The analytical solution of the interaction potential [43] is complex and computationally expensive. In our analysis, the particles are sufficiently large compared to the size of molecules that the expression of the van der Waals force simplifies to [44]:

FvdW (z) = −

AReq 6z 2

(8)

where A is the Hamaker coefficient, considered as a material constant (A ∈
10−19 ∼ 10−20 J ) [44]; Req is the equivalent particle radius as previously defined; and z is the shortest distance between particle surfaces. The force calculated from Eq. (8) becomes problematic when the distance

8

z is approaching zero. This issue has been addressed in the literature by implementing a minimum cut-off distance [45, 30, 24]. In existing models, the attractive force remains constant when it is within the minimum cut-off distance. However, this assumption is inaccurate when investigating flowability issues because it overestimates the particle overlaps which is closely related to particle flow behaviors. This discrepancy is critical, especially in “soft contact” models [46] where the contact forces stem from particle overlaps. Therefore, we assume that the equilibrium state where attraction is balanced by repulsion is obtained at zeq = 0, as shown in Figure 3. The attractive force linearly reduces to zero when particle separation distance is within z0 . The shape of the total inter-particle force is thus comparable to that between real molecules (the dashed line shown in Figure 2). [Figure 3 about here.] With the hypothesis of zeq = 0, the interaction force becomes zero if two particles are in perfect contact state (no gap or overlap). the maximum attractive force is obtained at z = z0 , defined as the cut-off distance. Experimental results of the van der Waals force between two half-spaces suggest that the equilibrium distance zeq is roughly between 0.1 ∼ 0.5 nm and the cut-off distance is z0 = 1.2zeq [47]. However, these experimental results cannot be directly applied in this model because the formulation of van der Waals force is different in the case of particle-particle contact. Additionally, no attempt has been made in this study to simulate the actual molecule nor the electron cloud. Therefore, the cut-off distance reported in literature is not comparable to z0 in the discrete element model. We consider z0 a parameter that needs to be calibrated according the behavior of the metal powder. Details of this calibration will be introduced in Section 2.3. The van der Waals force drops off quickly as the particle separation increases. To speed up the simulations, we set a maximum 9

break down distance zmax = Req /4 , beyond which the interaction force reduces to zero [26]. Considering that z0 is normally in nanometer scale while particle size in this study is in micrometer scale, the force at break down distance to the force at cut-off distance ratio is about O(10−7 ). It is thus safe to neglect the interactions when z > zmax without influencing the macroscale behavior of particles. 2.3. Cut-off distance calibration For a given particle size, the magnitude of van der Waals force is a function of the separation distance. The effect of this force on the flowability of the powder is thus strongly influenced by the cut-off distance z0 . Numerical simulations of repose angle tests are performed with a particle assembly (the particle size distribution, PSD, is shown in Figure 4) to qualitatively show the effect of the cut-off distance on flowability. The cut-off distance is set to be z0 = 0.5 nm and 1.0 nm, respectively, while the rest of the parameters remain the same, see Table 1. As shown in Figure 5, at z0 = 0.5 nm, the cohesion attributed to van der Waals force is so significant that the particles clump and fail to flow, while when z0 is increased to 1.0 nm, the particles exhibit a good flowability. Therefore, it is critical to carefully calibrate z0 before performing subsequent spreading simulations. [Figure 4 about here.] [Table 1 about here.] [Figure 5 about here.] In this study, we calibrate z0 by identifying the critical size of particles that are cohesive to flat walls. Three powders (labeled as Pda, Pdb, and Pdc) with different particle size distributions are used for calibration. The sieve analysis

10

results and the cumulative particle size distribution curves are shown in Table 2 and Figure 6, respectively. The physical properties of the powders are shown in Table 3. The powders are deposited under gravity on a smooth polycarbonate board and then gently tapped off while the board is held horizontal to the ground surface. The board we used in this study is reasonably smooth that the length scale of the asperity is much less than the mean particle size of the steel powder. A scanning electron microscope (SEM) stub with double sided tape was placed on the polycarbonate to sample only the particles that adhere to the board. Then SEM micrographs are taken as shown in Figure 7 to measure the particle size distributions. We use a MATLAB script that exploits the sphericity of the metal particles to identify individual particles from these two dimensional images. [Table 2 about here.] [Table 3 about here.] [Figure 6 about here.] [Figure 7 about here.] To increase the accuracy, the SEM micrographs are first converted to binary images. Then spheres are identified and marked with red circles, as shown in Figure 7. Some of the nonspherical particles (marked with green square in Figure 7) and particles near the edge are ignored. Figure 8 summarizes the particle size distributions of the particles that adhere to the flat surface. Regardless of the original particle size distributions of the assemblies, the cohesive particles are bounded by particle size d ' 27 µm, which means the cohesive force has the most significant influence on particles with d . 27 µm. No particles with d > 27 µm are identified. Moreover, the distribution of particle sizes is discontinuous near d ∼ 20 µm in all three cases, 11

which could be interpreted as a sign of transitioning of micro-scale mechanism that governs the cohesive force. Theoretically, the van der Waals force dominates the cohesion when particle size d . 20 µm, while other factors, such as surface roughness, become more significant when particle size d ? 20 µm.

Therefore, we assume that d ∼ 20 µm is the maximum particle size at which the potential energy associated to van der Waals force is balanced by the work done by gravity when kinetic energy is negligible. Note that this condition holds under particle-wall contacts where Req is equal to the particle radius, assuming the radius of the wall is infinity. [Figure 8 about here.] Considering the interaction between a single particle of radius r and a flat wall, see Figure 9, the particle is cohesive to the wall only when the potential energy associated to the van der Waals force satisfies the following condition,

EvdW ≥ Wg

(9)

where EvdW is the potential energy that equivalent to the shaded area in Figure 9 and Wg is the work done by gravity when the particles separate. At the initial state, the particle has perfect contact with the wall, namely, the distance from the wall to the particle is zero. The particle-wall interaction force is thus zero according to Figure 3. As the separation distance increases, the total interaction force increases first and then decreases until z > zmax . The work done by the gravity during the process can be calculated as,

Wg =

4 3 πr ρg · zmax 3

(10)

Note that the maximum separation distance is assumed to be zmax = r/4.

12

Therefore, Eq. (9) can be expressed as,

EvdW

Ar = + 12z0

zˆmax

Ar 4 dz ≥ πr3 ρg · zmax 2 6z 3

(11)

z0

The parameter z0 can thus be expressed as a function of particle radius r and Hamaker coefficient A, z0 ≤

3Ar 4πρgr4 + 8A

(12)

As previously mentioned, we assume the maximum radius of cohesive particles due to van der Waals force is r ' 10 µm. The Hamaker coefficient depends on many physical and chemical properties of the material, which are difficult to determine theoretically or experimentally [48]. In general, the Hamaker coefficient for a metal powder is in the range from 1 × 10−19 to 3 × 10−19 J [49]. We simply choose A = 1.6 × 10−19 J in this study. Therefore z0 can be estimated as z0 . 0.71 nm. We set z0 = 0.7 nm to perform subsequent simulations. [Figure 9 about here.] It has been shown that the electrostatic forces may become comparable to the van der Waals force in magnitude [50], thus contributing to the attraction between the particle and the flat surface. However, theoretical and experimental studies [51] suggest that this condition only occurs when the particles carry a sufficient amount of charge or an external electric field is applied. Considering that neither of these conditions are satisfied in this study, Eq. (12) is sufficient to estimate the cut-off distance without accounting the effect of electrostatic forces.

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3. Numerical Simulations 3.1. Powders A metal powder with a particle size 45 µm < d < 150 µm has been used in additive manufacturing. The spreading test with this powder has exhibited satisfying flowability such that the surface of the deposition bed is smooth and no apparent voids can be observed, see Figure 10. Since the objective of this study is to identify the effect of fine fraction on powder flowability, this coarse powder is considered as a baseline material. In our preliminary experimental study, powders with d < 45 µm, 10 µm < d < 30 µm , and 15 µm < d < 45 µm show hindered flow behaviors. In general, powders with d < 20 µm are removed before spreading. Therefore, a fine fraction of particle size d in the range from 20 µm to 40 µm with different volume fractions Ψ is mixed with the baseline powder to investigate the effect on flowability. The particle size distribution of the baseline powder is obtained through sieve analysis. According to the sieve analysis results, a synthetic granular material is numerically modeled using discrete element method. Figure 4 shows the particle size distribution curve of the baseline powder; a good match with the experiment is obtained. For the fine powder, we assume the particle size satisfies uniform distribution. [Figure 10 about here.] The microscale parameters of the model are calibrated to match the material properties of the metal powder used in additive manufacturing, see Table 1. However, the elastic modulus of the particles is set to be 10 MPa, instead of 90 GPa as listed in Table 3, for two considerations. First, the efficiency of DEM simulation of particle spreading process is primarily governed by the critical time step defined as,

14

r m ∆tc = 2 Keff

(13)

where m is the particle mass and Keff represents the effective stiffness of a linear spring in a spring-mass system. Knowing that Keff is a function of the interparticle contact stiffness, the critical time step is thus inversely proportional to the elastic modulus of the particle. Given realistic elastic modulus E = 90
GPa, the critical time step is approximately ∆tc ∼ O 10−11 s which is too small that the simulation will become extremely slow. Therefore, the elastic modulus is scaled to E = 10 MPa so that the critical time step is acceptable as
∆tc ∼ O 10−8 s. Second, in the soft contact DEM approach, a slight overlap is allowed at physical contacts between rigid particles. It has been shown that if the overlap is within 0.5% of the particle’s radius then the flow behavior is independent of the particle stiffness [40]. At E = 10 MPa, the maximum overlap ratio is 0.02% under static equilibrium condition. Therefore, the flow behavior of the metal powder can be properly captured while the critical time step is acceptable by assigning E = 10 MPa in this study. Another numerical strategy to decrease simulation time is upscaling the particle mass. This is accomplished by artificially increasing particle density [52] or particle size [53]. Mass scaling in this manner is appropriate for quasi-static simulations in the absence of (or with reduced) gravity. However, true gravity is required for the simulations discussed herein. Furthermore, scaling the mass changes the particle’s inertia, which can become important for dynamic simulations. Therefore, we elect instead to scale particle stiffness to achieve the same end. 3.2. Spreading test setup The spreading configuration is prepared by randomly generating particles within a cube of length L = 1400 µm. Then the particles are deposited under 15

gravity until equilibrium is achieved, see Figure 11. Spreading is simulated by imparting a constant velocity V = 25 mm/s to a rigid blade to spread the powder on a deposition bed with width W = 1400 µm. The periodic boundary condition is applied in the y direction to approximate a large spreading platform. Thus the total number of particles necessary for the spreading simulation is limited. The deposition bed is simplified by using a rigid wall element that exerts normal and tangential forces on the particles. Eq. (8) is used to determine the van der Waals force between particles and the bed. We consider the bed an infinite large particle and R → ∞ to approximate the case when spreading particles on top of another layer of deposited particles. The gap between the bottom of the blade and deposition bed surface is a constant dg = 3.0d50 , where d50 = 89 µm. The length of the bed is long enough to accommodate all particles as they spread such that no interactions occur between particles and +x boundary. It has been widely acknowledged that the spreading quality is associated with many factors such as the properties of the bed, the gap dg , the spreading velocity V , the geometry of the blade, and ambient humidity [4]. In this study, we primarily focus on the effect of fine fractions on the powder spreadability. Hence, all other factors remain unchanged in the simulations. We add a fine powder of different volume fraction with 20 µm < d < 40 µm to the baseline powder to investigate how the fine particles can influence the overall particle flow behaviors and quality of the deposition bed. [Figure 11 about here.] 3.3. Bed quality and powder flowability characterization In order to consistently measure the quality of the deposition bed, a sampling region with dimensions of lb × wb is chosen that excludes the starting and ending part of the bed to reduce the boundary effect, see Figure 11. The sampling region starts from the edge of the particle pile and ends at 1400 µm away 16

from the back edge of the blade when spreading ends. The length lb is determined empirically such that the distribution of the particles is representative of the whole deposition bed. Then, the local volume fraction ψ, coordination number α, surface roughness ξ, and projection index P I are measured from the sampling region to quantify the quality and characteristics of the deposition bed. Meanwhile, dimensionless force ratio γf , defined as the ratio of total van der Waals force to the total weight of the powder, and potential energy ratio γe , defined as the ratio of total potential energy contributed from van der Waals force to that contributed by gravity, are measured from the pile of powder ahead of the blade during spreading to quantify the flowability. The local volume fraction ψ is defined as the volume of particles bounded within a sub-domain divided by the total volume of the sub-domain. The subdomain is defined by length lb , width wb , and height hb , see Figure 11. When a particle intersects with the boundary of the sub-domain, the volume of spherical cap outside of the sub-domain is subtracted from the total volume of particles. Therefore, the local volume fraction could be calculated from, Pm ψ=

Vi +

Pn

(Vj − Vj,cap ) − Voverlap × 100% lb wb hb

(14)

where Vi is the volume of particle i located within the sub-domain; Vj is the volume of particle j intersects with the sub-domain; Vj,cap is the volume of the spherical cap located outside of the sub-domain; Voverlap is the total volume of particle overlaps. By definition, The volume fraction also represents the density of the deposition bed, and reflects the average void ratio. However, calculation of ψ is sensitive to the ratio of hb to average particle size d. We set hb = 2.0d50 in this work to do all the relevant calculations. The coordination number α of a particle is determined by counting the number of particles that are in contact with it. In DEM simulations, the coordination

17

number is measured using spherical measurement volumes. Before the simulation starts, a layer of measurement spheres with radius rm = hb /2 that covers the whole area of the sampling region is generated as shown in Figure 11. We take the average coordination number obtained from all measurement spheres to represent the arrangement of particles on the deposition bed, namely, high value indicates formation of clumps while low isolated individual particles. The surface roughness ξ of the deposition bed is determined using a shining light tracing technique where a grid of light sources is created at h → ∞ while the deposition bed is at h = 0. The lighted area corresponds to the sampling region, thus equivalent to the area covered with measurement spheres. The resolution of the grid is defined by nx × ny , where nx is the number of columns in x direction while ny is the number of rows in y direction. Each center of the grid is considered a light source that shoots light beams towards the top surface of the deposition bed. The elevation of the intersection point between the light beam and the particle is recorded. The surface roughness is then calculated using the normalized standard deviation of the elevations through, s

PN

ξ=

¯ 2 − h) /d50 N −1

i=1 (hi

(15)

¯ is the average of the where hi is the elevation of the intersection point and h elevations; N is the total number of light beams that calculated from N = nx × ny . The accuracy of the measurement is dependent of the resolution of the light source grid. Sensitivity analysis shows that ξ is a function of N . When N is too high, ξ becomes too sensitive to the particle size; when N is too low, ξ cannot capture the characteristics of the voids near the surface. In this study, nx and ny are determined through,

nx = 2lb /d50

and

18

ny = 2wb /d50

(16)

Powder flowability is a critical process parameter that characterizes the flow behavior of a powder. In this study, the force and potential energy ratios are introduced to quantify the flowability of the powder during the spreading test. The force ratio γf and energy ratio γe are defined as, P γf =

Fi,vdW Wp

P and

γe =

Ei,vdW Eg

(17)

where Fi,vdW and Ei,vdW is the maximum van der Waals force determined from Eq. (8) at z0 and potential energy associated to the ith contact calculated from Eq. (11), respectively; Wp and Eg is the total weight and potential energy assoP ciated to gravity, respectively, where Eg = mi ghi , mi is the mass and hi is the elevation of the ith particle. Since particle flow mostly occurs during spreading, particles ahead of the blade are taken into consideration when calculating the force and energy ratios, see Figure 11(b). According to the definitions in Eq. (17), both γf and γe are not necessarily constant through the whole spreading process. In this study, we quantify the flowability of the powder with these ratios during the spreading tests when the powder is deposited under gravity. According to the definitions of the properties, a good quality of deposition bed or good flowability of the powder is usually represented by high ψ, α and low ξ, γf , γe . To quantify the spatial uniformity and degree of clumping of the fine particles, we calculate the projection index (P I) [54]. For a homogeneous spatial distribution, P I = 1.0. A higher degree of inhomogeneity will yield a larger value of P I. To calculate P I, we first project the spatial locations of fine particles onto the deposition bed plane (convert 3D data to 2D). Then we project all the 2D coordinates (xi , yi ) to a straight line of angle θ passing the origin of the axis (convert 2D coordinates to 1D). We define this straight line as a new

19

axis. The new 1D coordinates of the points are noted xi,θ ,

xi,θ = xi cos θ + yi sin θ

(18)

Then denote vi,θ as the distance between each pair of adjacent points in xi,θ . Here we define a new variable, squared coefficient of variation SCV for vi,θ ,  SCVθ =

σv,θ µv,θ

2 (19)

where σv,θ and µv,θ are the standard deviation and mean of vi,θ , respectively. Finally, the projection index is calculated as, P179◦ PI =

SCVθ 180

θ=0

(20)

4. Simulation results and discussion 4.1. Flowability transition with fine fraction Ψ The baseline material introduced in Section 3.1 is mixed with a fine powder of 20 µm < d < 40 µm. The volume fraction Ψ of the fine powder varies from 0 to 4%. Then the spreading test is performed with this new mixture. The simulation results show that the baseline material where Ψ = 0 exhibits a good flowability that no clear sign of particle clumping and voids can be observed after the spreading as shown in Figure 12(a). Then the volume fraction of the fine powder is increased to 2% and the deposition bed shows similar apparent properties, see Figure 12(b). An obvious flowability transition is first observed when the fine fraction reaches Ψ = 4%. The particles are clumped by the cohesive van der Waals forces and they move together with the blade during the spreading process. Meanwhile, individual particle flow is hindered, leading to a low flowability. Thus a non-homogeneous powder bed is left behind after the

20

blade spreads across the deposition region. Near the beginning of the deposition bed, some of the particles tend to roll away from the pile when Ψ = 0, also indicating a good flowability. Note that the movement of the particles behind the blade is dominated by gravitational force. At Ψ = 2% or 4%, the particles are more compacted than the free rolling behaviors are restrained by the cohesive forces. However, the flow behavior of individual particles near the end of the deposition bed is similar and independent of Ψ because the lateral force exerted by the blade governs the movement. [Figure 12 about here.] 4.2. Bed quality and powder flowability In this study, the quality of the deposited bed and powder flowability will be quantified using the properties defined in Section 3.3. Figure 13 summarizes the variation of the volume fraction, coordination number, and surface roughness with respect to the volume percentage of fine fractions. Since these properties are not applicable if the deposition bed is highly non-homogeneous when Ψ ≥ 4.0%, the spreading tests are performed with Ψ varying form 0 to 3.8% by approximately 0.5%. The simulation results are summarized in Figure 13. When fine fractions are introduced into the baseline particle assembly, the solid volume fraction ψ of the deposition bed slightly increases with Ψ up to Ψ ∼ 1.5%. Afterwards, ψ consistently decreases with Ψ when Ψ>1.5%, suggesting that the volume of voids within the deposition bed is increasing with large amount of fine particles, see Figure 13(a). Therefore, a small amount of fine particles mixed with the baseline material can slightly reduce the total volume of voids, thus yielding a relatively denser powder bed. However, this tendency reverses once the amount of fine powder exceeds a threshold, Ψ ∼ 1.5% in this study. Compared to the baseline material, the solid volume fraction is reduced

21

by about 5.56% at Ψ = 3.8% and increased by about 0.9% at Ψ ∼ 1.5%. The densest deposition bed is obtained at Ψ ∼ 1.5%. Another important property of the deposition bed is the surface roughness which is directly related to the quality of the finished product in additive manufacturing. Figure 13(b) shows the simulation results at different fine fractions. The dependency of surface roughness ξ on fine fraction Ψ is consistent as predicted from ψ − Ψ relationship, namely, a low surface roughness corresponds to a high solid volume fraction. Hence, the minimum surface roughness is obtained at Ψ ∼ 1.5% where the deposition bed yields the maximum volume fraction. In powder spreading, the surface roughness is closely related to the amount of voids on the bed surface. When Ψ < 1.5%, adding fine particles to the matrix can fill those voids between the baseline particles, leading to a significant decrease of the roughness by approximately 18% relative to the baseline powder. However, the continued addition of fine particles results in the effect of cohesive forces becoming significant and the particles start to clump, creating even more voids among clumps than those filled by the fine particles. As a result, the surface roughness increases as adding more fine particles until the powder eventually loses all the flowability as shown in Figure 12(c). [Figure 13 about here.] Interestingly, the coordination number does not increase monotonically with increasing fine fraction. The coordination number rather decreases at 1.5% . Ψ . 2.5%. This phenomenon seems to contradict previous observations that the more fine particles the more clumps formed during spreading. In fact, the coordination number increases as the fine particles are attractive to the baseline particles, leading to a more compacted assembly. Meanwhile, the large clumps would hinder the flow behaviors and create voids between clumps especially when the fine fraction is significantly high. This phenomenon stems from two 22

mechanisms: (1) increment of rotational inertia and interlocking due to angularity; and (2) creation of clusters from fine particles. At 1.5% . Ψ . 2.5%, this first mechanism governs the particle behaviors and the number of interactions between particles decreases. When Ψ > 2.5%, the second mechanism becomes dominant where the coordination number increases linearly with the fine fraction and the assembly exhibits a very loose structure. Figure 14 schematically shows the transitioning of micro-structures at different levels of fine fractions. It is interesting to note that as fine fractions increase in the mixture, the small particles will not fill the voids between large particles as normally observed in mixing two sizes of granular materials [55, 56, 57]. The fine particles would rather clump into large agglomerates under the effect of attractive van der Waals force due to the relatively low self-weight. As a result, the void increases with the amount of fine fractions as shown in Figure 14(d). The observation presented in Figure 14(a) - (d) suggests that the evolution of the micro-structure is related to the ratio of coarse to fine particle number, λ, see Figure 14(e). As λ . 2, the flowability of the assembly is significantly hindered, suggesting that the baseline particles are “floating” in the fine fraction. The projection index is calculated and the results are summarized in Figure 15. As the fine fraction increases, the degree of inhomogeneity also increases almost linearly, suggesting that the clustering of fine particles becomes more significant. However, this method of using projections to describe the homogeneity of the fine particle distributions is not necessarily accurate because the third dimension (z dimension) is ignored. As a result, P I may overestimates the clustering of the particles. Nevertheless, considering the thickness of the deposited particles is small, i.e., ∼3 times of average baseline particle size or ∼5 times of average fine particle size, the projection index method is applicable to reflect the degree of clustering of fine particles in this study.

23

[Figure 14 about here.] [Figure 15 about here.] Figure 16 shows that the force ratio γf and energy ratio γe are indeed linearly correlated with the fine fraction Ψ. It should be noted that the flowability transition, namely, the whole pile of particles moves together with the blade as shown in Figure 12(c), occurs at Ψ ∼ 4% where γf,c = 262 and γe,c = 6.4×10−4 . The ratios are evaluated at the beginning of the spreading tests. In fact, the flowability of a powder is related to both the physical properties of the material itself, as well as the specific processing conditions of the handling system. Hence, at different stages of the spreading process, the flowability of the powder changes with the size of the powder pile ahead of the blade. However, according to Eq. (17), γf is independent of the geometry of the powder pile. That said, γf roughly remains a constant during the whole spreading process as shown in Figure 17(a). On the other hand, the potential energy ratio increases as the powder pile ahead of the blade grows smaller, see Figure 17(b). It has been noted in the literature [58] that the powder flowability quantified by repose angles decreases when reducing the amount of material used in the test. From this point of view, γe is more appropriate than γf to characterize the flowability of the powder because it takes the state of the powder into considerations. It is interesting to note that the powder can still be spread on the deposition bed even though γe exceeds the critical value. That is because the flow behavior of the powder is also sensitive to the force applied by the blade. When the particle pile becomes smaller, gravity alone cannot trigger the particle flow because the effect of van der Waals force is growing stronger; however, the force exerted by the blade is more significant and becomes the dominant factor of powder flow. [Figure 16 about here.]

24

[Figure 17 about here.]

5. Discussion The effect of particle shape on powder rolling is implicitly considered by applying the rolling resistance µr . Considering the fact that powder particles are mostly spherical, see Figure 7, it is sufficient to use this simplified method by ignoring angularity and complex interactions between particles, e.g., interlocking. However, if complex particle geometries are involved in the simulations, it would be necessary to model realistic particle shapes with advanced techniques, e.g., clumping model [59, 25, 60]. The flowability of the powder decreases as increasing the fine particle fractions because of two mechanisms: (1) the cohesive van der Waals force tends to attract particles resulting in a more compacted assembly. The friction forces between particles are thus slightly increased, preventing the particles from rolling or sliding. This mechanism is more obvious when fine fraction Ψ . 1.5% , where the volume fraction increases with Ψ; (2) when Ψ & 1.5%, the fine particles are cohesive to larger particles to form a cluster. The relative sliding and rolling of these clusters are hindered due to the increment of the rotational inertia and angularity. Additionally, the interlocking interactions also reduce the flowability of the particles/clusters, creating a very loose structure with low volume fractions. This phenomenon has also been experimentally confirmed in other studies [61]. The authors showed that the void ratio of a clay-sand mixture decreases with clay content up to about 20%, after which further addition of fines lead to a linear increment of the void ratio. Therefore, an optimum fine fraction exists that can improve the density of the deposition bed and thus enhance the quality of the AM products. The numerical model in this study shows that a relatively small volume fraction (4%) of fines can have a dramatic effect

25

on spreading properties. It is unclear if this value is realistic when compared to real metal powders. While the model captures the fundamental physics of powder spreading, more comparisons with actual metal powder spreading are necessary to fine-tune the model. Force ratio γf and energy ratio γe are introduced in this study to evaluate the flowability of the powder. At the beginning of the sampling region, they both increase with fine fraction Ψ, suggesting that the flowability is decreasing with the amount of fine particles. However, γf remains roughly a constant through the whole spreading process while γe increases as the blade moves forward. Consequently, the flowability evaluated by γe takes consideration of the geometry of the particle pile, yielding a better estimation of the powder flowability than γf during the spreading process. Furthermore, the critical potential energy ratio γe,c can be identified from Figure 16(b), which is defined as the value of γe at the behavioral threshold. It is useful to get a general idea about the relative flowability of the powder by comparing γe with γe,c . Nevertheless, γe,c is not an intrinsic material property because the external force exerted by the blade can also change the flow behaviors. Note that force ratio and energy ratio have their own strengths in evaluating the flowability. Force ratio does not consider the pile geometry explicitly; thus, it could be used to estimate the flowability before spreading test. In this study, the flowability of the powder is inversely proportional to the potential energy ratio γe . For a given particle assembly, the total potential energy associated with the van der Waals force can be calculated from Eq. (11) as, EvdW,assembly

N X



 ARi,eq + = 12z0 i=1

zˆ i,max

 ARi,eq  dz 6z 2

(21)

z0

where Ri,eq is the equivalent radius of the particles associated to the ith contact; zi,max = Ri,eq /4; and z is the separation distance between the two contact 26

bodies; and N is the total number of contacts. Note that z0 and A are both constant. Hence, both the potential energy and van der Waals force of the assembly are linearly related to the equivalent radius Req , which is governed by the particle size distribution of the powder. In addition, the potential energy associated to gravity is proportional to the weight of the powder, namely, Eg ∼ 3 h · Req . Substituting these two conditions into Eq. (17) gives us,

2 flowability ∼ 1/γe ∼ h · Req

(22)

where h is the height of the powder. This is a simplified expression of the flowability neglecting the effect of capillary forces, electrostatic force, external loads, and dynamic effects. Consequently, Eq. (22) suggests that the particle size plays an important role in governing the powder flow behavior under such circumstances. It also implies that pile-ups and avalanches are key elements of the spreader design and its performance for marginal powders. In industrial processes, it is not straightforward to quantify the flowability using γe . According to Eq. (21), calculating γe requires knowledge of two microscale parameters: the total number of contacts N and the equivalent radius Req for each contact. Direct measure of these microscale parameters is nearly impossible. However, the relationships between macroscale material properties and microscale parameters could be determined numerically or empirically, namely, N = f (W ) and Req = f (PSD). Further study is required to find: (1) the correlation between the total contact number N and weight of powder W ; and (2) the equivalent radius Req as a function of the particle size distribution. Once the correlations are benchmarked with real materials, it is possible to predict the flowability for a given powder of different PSDs. In this study, the equivalent radius Req is calculated for each particle assembly with different fine fractions. Then, distribution density function could be calculated, see the

27

histograms in Figure 18(a). For simplicity, we only show two cases with Ψ = 0 and 3.8%, respectively. With more fine content, the distribution is skewed to the smaller Req end. The probability density function is fitted using the log-normal distribution function [62],

f (x|µ, σ) =

1 √



xσ 2π

exp

−(ln x − µ)2 2σ 2

 (23)

where µ and σ are the mean and standard deviation of logarithmic values. The average equivalent radius can be calculated as,

¯ eq = exp(µ + σ 2 /2) R

(24)

For different fine fractions, we can calculate the average equivalent radius for each case and the results are summarized in Figure 18(b). The equivalent radius decreases with increasing fine fractions that can be formulated by a linear func¯ eq at different tion. Therefore, we can extrapolate the function and determine R fine fractions, which could be further implemented to determine the relative flowability of the powder. [Figure 18 about here.] Existing numerical frameworks for modeling the behaviors fine powder lack the knowledge of correlation between material properties and modeling parameters, e.g., the cut-off distance z0 . To address this shortcoming, we developed a series of tests to estimate z0 for a specific powder. However, this method neglects the influence from other factors, such as capillary effect due to humidity and electrostatic force due to additional electrical charges. As such, the accuracy becomes questionable especially when the test is performed in humid environment or the deposition bed has a static electrical charge. Nevertheless, the accuracy of the test results could be improved by properly controlling the 28

test environments. Therefore, we could still use this technique to approximate the the cut-off distance by assuming the van der Waals force is the governing factor of adhesions between particle and flat surface.

6. Conclusion The effect of fine fractions on the flowability of metal powder in additive manufacturing is numerically investigated using DEM simulations in this work. The van der Waals force implemented in this framework is an explicit numerical solution of particle separation distance, cut-off distance, and particle radii. The experimental procedure for calibrating the model parameter, cut-off distance, is introduced and validated. Then powder spreading is numerically modeled with various fine fractions. The quality of the deposition bed and powder flowability are quantified. Specifically, the flowability is shown to be inversely proportional to the amount of fine particles and square of particle size. According to the numerical simulation results, the following conclusions can be drawn from this study: • A DEM approach is investigated to simulate the powder spreading process in additive manufacturing while considering the effect of van der Waals inter-particle forces. The numerical results show that the framework is capable of capturing the characteristics of powder spreading and reproducing the fundamental physics behind powder flow. • A small amount of fine particles added to the baseline material can slightly improve the quality of the deposition bed after spreading in terms of volume fraction and surface roughness. At Ψ . 1.5%, the volume fraction is increased by about 0.9% while the surface roughness is improved by 18% compared to that of the baseline material. At Ψ > 1.5%, the bed quality decreases with the amount of fine particles. The optimum quality 29

of the deposition bed is obtained at Ψ ' 1.5%, where the volume fraction reaches the maximum and the surface roughness minimum. • The flowability of the powder consistently decreases with fine fraction Ψ. The powder shows a flowability transition, where the particles clump and move together with the blade, at Ψ ' 4.0%. The flowability can be quantified by both the force ratio γf and energy ratio γe when the powder is in the same state (pile size) and subject to low level of external forces. During spreading, the powder state changes, namely, the total amount of powder ahead of the blade decreases as the blade moves. The energy ratio γe is more appropriate to describe the powder flowability at a specific stage during the spreading because it takes the geometry of the powder pile into consideration while γf does not. The flowability can be qualitatively evaluated using Eq. (22). The DEM framework introduced in this study could be further extended to include complex factors influencing powder spreading behaviors in additive manufacturing. For instance, non-spherical particles could be considered by implementing the clump technique as introduced in [26]; long-range interactions such as capillary effect and electrostatic force could be coupled with the van der Waals force model to investigate the mechanisms of powder flow under working conditions. Once the DEM framework is properly calibrated and benchmarked, not only the flow behaviors of a specific powder under given processing conditions could be estimated, but also the processing parameters could be optimized by performing sensitivity analysis. Moreover, the numerical framework presented in this study provides an opportunity to address the issue of choosing proper powder gradation in additive manufacturing, e.g., what is the optimum fine fraction and how is the flowability correlated with amount of fine fraction? In future works, multiple powders with different PSDs could be investigate using

30

the developed framework to explore the universal correlation between PSD and powder flow behaviors. This research provides metallurgists and engineers for the first time with a predictive and proactive tool to estimate the effect of fine fractions on the flowability of bulk powder in additive manufacturing. By providing engineers with a quantitative, science-based understanding of the performance of fine powder during spreading, the decision-making process will be guided. This study will enable comprehensive analysis of the powder spreading in additive manufacturing, including the quality of the deposition bed and the flowability of the powder. Thus, engineers will have the necessary information to make crucial decisions when choosing the proper powder gradation or deciding the optimum processing parameters. The results will allow assessment of the flowability of a given powder in a way that is not currently possible.

Acknowledgments The work described in this manuscript has been funded by ATI Specialty Alloys and Components (Grant No. PO375783) and the Oregon Metals Initiative (Grant No. C2015107). TME was also partially supported by the U.S. National Science Foundation (Grant No. CMMI-1538460). This support is gratefully acknowledged.

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List of Figures 1

2 3 4 5 6 7 8 9 10

11 12 13 14

15 16 17 18

Schematic representation of particle interactions in the mechanical contact model. (a) normal; (b) shear; (c) rotation. The normal and shear models are characterized by a linear spring and viscous dashpot, with no force being transmitted for particles not in contact. A torsional spring defines the rotation model. . . . . Potential energy and force between two molecules. . . . . . . . . Interaction force between two particles. . . . . . . . . . . . . . . Particle size distribution of baseline material. Bar plot shows the distribution of particle numbers. . . . . . . . . . . . . . . . . . . Effect of cut-off distance z0 on the flowability of fine powders. . Cumulative volume percentage vs. particle size. Note that Pdb and Pdc are very similar in terms of particle size. . . . . . . . . Identify cohesive particles by image processing. . . . . . . . . . . Particle size distribution of adhesive particles. . . . . . . . . . . van der Waals force and potential energy associated with particlewall contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deposition bed of metal powder with 45 µm < d < 150 µm spread on a smooth substrate. Along the direction of spreading, the surface is smooth and homogeneous. . . . . . . . . . . . . . . . . Spreading test setup. . . . . . . . . . . . . . . . . . . . . . . . . . Flowability transition with different fine fractions Ψ. . . . . . . . Characterizations of the deposition bed as a function of fine fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three dimensional representation of micro-structures with different fine fraction levels. In (a) - (d), blue spheres represent the baseline particles while green spheres are fine particles. The red box shows the boundary of the representative elementary volume. The dashed line in (e) indicates the flowability transition. . . . . Variation of P I index with fine fraction Ψ. . . . . . . . . . . . . Force and potential energy ratios with Ψ. The flowability transition occurs at Ψ = 4%. . . . . . . . . . . . . . . . . . . . . . . . Variation of force and potential energy ratio during spreading. The legends represent different fine fractions. . . . . . . . . . . . Equivalent radius at different fine fractions. The probability den¯ eq is shown in (a), sity function and the method to determine R and the averaged equivalent radius for each fine fraction is shown in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

40 41 42 43 44 45 46 47 48

49 50 51 52

53 54 55 56

57

s

n

kn

n

s

kr

ks

(a)

(b)

(c)

Figure 1: Schematic representation of particle interactions in the mechanical contact model. (a) normal; (b) shear; (c) rotation. The normal and shear models are characterized by a linear spring and viscous dashpot, with no force being transmitted for particles not in contact. A torsional spring defines the rotation model.

40

Repulsion

Force

z

zeq z0

Potential energy Attraction Figure 2: Potential energy and force between two molecules.

41

Repulsion

zeq

z0

zmax z

Attraction

vdW force

Figure 3: Interaction force between two particles.

42

15

100 Simulation Experiment

particle number percentage (%)

80 70 10 60 50 40 5 30 20

cummulative volume percentage (%)

90

10 0 20

40

60

80

100

120

140

160

0 180

Figure 4: Particle size distribution of baseline material. Bar plot shows the distribution of particle numbers.

43

5.00

ing Group, Inc. Model

PFC3D 5.00

7 Itasca Consulting Group, Inc. Academic Model

l alls (9716) ball

ll acets (189) facets

ometry

(a) z0 = 0.5 nm

Z Y

X Z Y

(b) z0 = 1.0 nm XFigure 5: Effect of cut-off distance z0 on the flowability of fine powders.

44

cumulative volume percentage (%)

100 90 80 70 60 50 40 30

Pda Pdb Pdc

20 10 20

30

40

50

60

70

80

Figure 6: Cumulative volume percentage vs. particle size. Note that Pdb and Pdc are very similar in terms of particle size.

45

(a) Pda

(b) Pdb

(c) Pdc

Figure 7: Identify cohesive particles by image processing.

46

10 60

40 5 20

0

5

10

15

20

25

0 30

10 60

40 5 20

0 0

(a) Pda

5

10

15

(b) Pdb 100

80 10 60

40 5 20

0 0

5

10

15

20

25

cummulative volume percentage (%)

15

volume percentage (%)

100

80

volume percentage (%)

volume percentage (%)

80

0

15

0 30

(c) Pdc Figure 8: Particle size distribution of adhesive particles.

47

20

25

0 30

cummulative volume percentage (%)

100

cummulative volume percentage (%)

15

0 z

FvdW

FvdW

z0

r G

z max z

Figure 9: van der Waals force and potential energy associated with particle-wall contact.

48

Figure 10: Deposition bed of metal powder with 45 µm < d < 150 µm spread on a smooth substrate. Along the direction of spreading, the surface is smooth and homogeneous.

49

z

periodic boundary

y

(a) Before spreading

x blade

measurement spheres

V

deposition bed W

sampling region

dg

(b) During spreading

lb

(c) After spreading nx  n y

light source grid

wb

zoom-in side view hb

Figure 11: Spreading test setup.

50

PFC3D 5.00 ©2017 Itasca Consulting Group, Inc. Academic Model

PFC3D 5.00 ©2017 Itasca Consulting Group, Inc. Academic Model

PFC3D 5.00 ©2017 Itasca Consulting Group, Inc. Academic Model

(a) Ψ = 0

Z

(b) Ψ = 2% Y

X

Z

(c) Ψ = 4% Y

X

Figure 12: Flowability transition with different fine fractions Ψ.

Z Y

X

51

53

0.21

52.5

0.2

52

0.19

51.5 0.18 51 0.17 50.5 0.16

50

0.15

49.5 49

0.14 0

0.5

1

1.5

2

2.5

3

3.5

4

0

(a) Volume fraction

0.5

1

1.5

2

2.5

3

(b) Surface roughness

6.2

6

5.8

5.6

5.4 0

0.5

1

1.5

2

2.5

3

3.5

4

(c) Coordination number Figure 13: Characterizations of the deposition bed as a function of fine fractions.

52

3.5

4

(a) Baseline

(b) Ψ < 1.5%

(c) 1.5% < Ψ < 2.5%

(d) Ψ > 2.5%

14 12 10

flowability transition

8 6 4 2 0 0

5

10

15

20

(e) variation of coarse to fine particle number ratio λ with fine fraction ψ. Figure 14: Three dimensional representation of micro-structures with different fine fraction levels. In (a) - (d), blue spheres represent the baseline particles while green spheres are fine particles. The red box shows the boundary of the representative elementary volume. The dashed line in (e) indicates the flowability transition.

53

1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 0.5

1

1.5

2

2.5

3

3.5

Figure 15: Variation of P I index with fine fraction Ψ.

54

4

500

450

400

350

300

250

200 0

5

10

15

20

(a) van der Waals force to body force ratio 10-4

12 11 10 9 8 7 6 5 0

5

10

15

20

(b) Potential energy ratio Figure 16: Force and potential energy ratios with Ψ. The flowability transition occurs at Ψ = 4%.

55

270 260 250

baseline 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 3.8%

240 230 220 210 200 190 1

2

3

4

5

6

(a) Force ratio γf at different blade locations 10-4

9 8.5 8

baseline 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 3.8%

7.5 7 6.5 6 5.5 5 1

2

3

4

5

6

(b) Potential energy ratio γe at different blade locations Figure 17: Variation of force and potential energy ratio during spreading. The legends represent different fine fractions.

56

10 baseline 3.8% fine

8

6

4

2

0 0

20

40

60

80

100

(a) Probability density function of equivalent radius for two gradations. 18.5 18 17.5 17 16.5 16 15.5 15 14.5 0

0.5

1

1.5

2

2.5

3

3.5

4

(b) Average equivalent radius with different fine fractions. Figure 18: Equivalent radius at different fine fractions. The probability density function and ¯ eq is shown in (a), and the averaged equivalent radius for each fine the method to determine R fraction is shown in (b).

57

List of Tables 1 2 3

Microscale parameters. . . . . . . . . . . . . . . . . . . . . . . . . Sieve analysis results of metal powders. (∗ Weight of pan is corrected to account error.) . . . . . . . . . . . . . . . . . . . . . . . Physical properties of metal powders. . . . . . . . . . . . . . . .

58

59 60 61

Table 1: Microscale parameters.

Parameters 3 Density ρ (kg/m ) Coefficient of restitution Elastic modulus E (MPa) Local damping ratio Normal to shear stiffness ratio Friction coefficient µ Rolling resistance µr Cut-off distance z0 (nm)

59

5500 0.6 10.0 0 2.0 0.35 0.2 0.5 and 1.0

Table 2: Sieve analysis results of metal powders. (∗ Weight of pan is corrected to account error.)

Sieve No.

Opening (µm)

100 140 200 230 270 325 400 500 635 Pan∗ Total

149 105 74 63 53 44 37 25 20 -

60

Weight (%) Pda Pdb Pdc 0.9 0.0 0.8 0.0 0.7 4.4 6.1 12.3 26.9 25.8 10.0 11.9 12.4 26.7 26.8 27.3 12.6 8.3 8.7 36.0 21.7 19.7 100 100 100

Table 3: Physical properties of metal powders.

Parameters Elastic modulus E (GPa) Density ρ (kg/m3 )

61

90 5500

10 z

baseline 3.8% fine

periodic boundary

y

(a) Before spreading

x blade

measurement spheres

V

8

6 deposition bed

4 W

sampling region

dg

2 (b) During spreading

0

0

20

40

60

80

100

18.5 18 17.5

lb

(c) After spreading nx  n y

wb

17 16.5

light source grid

16 15.5 15

zoom-in side view

hb

14.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Highlights: • • • •

Powder flowability can be hindered by the van der Waals force effect. Force ratio and potential energy ratio can be used to quantify powder flowability. Spreadability increases and then decreases with adding fines. The flowability decreases with increasing fine fraction.