Abstract
Packing density of metal powders is an important aspect of additive as it directly impacts the physical and mechanical properties of printed products. In order to achieve the most efficient packing of a powder, different grades of that powder must be mixed together in such a way that we minimize the voids. Research has shown that packing the coarser grains first not only yields higher density powders but also decreases balling defects in the finished printed product. In this study, we developed a simple model that adequately predicts the volumetric fractions of different powder grades that can yield the highest powder density. The model accounts for the disparities between theoretical assumptions and experimental outcome, such as volume reduction. The model equations, based solely on void ratios and true specific gravity, will be validated experimentally and compared to other modeling efforts in literature to further prove the potency of the model.
Keywords
Introduction
Efforts to improve the methods of powder metallurgy have been in progress for centuries. In recent years, a heightened interest in metal powder-based additive manufacturing (AM) has emerged and a significant number of researches have been shifted in this direction. Some of these researches investigated powder size distribution as it directly affects the packing density , powder quality, as well as additive manufacturing capabilities. As known, the powder used in additive manufacturing can be produced via several processes such as gas atomization [1, 2], water atomization, plasma atomization, and plasma rotating among other methods. Each method produces a unique particle size range, and the selection is based on preference and cost option. However, to ensure the most efficient packing, the best particle size distribution that eliminates the most voids in the final product must be chosen. The importance of powder packing is generally appreciated because of its influence on the physical and mechanical properties of the 3D-printed parts, on shrinkage and density on sintering [3], and on the microstructure of the finished product, which may also contribute to the surface finish. Additionally, understanding powder bed properties and its interaction with laser is critical in predicting final part properties. Particle parameters such as powder density, particle size distribution , and shape play an important role in powder bed formation [4, 5].
In order to achieve the densest packing for a powder, different sizes of powder may be mixed together. In the mixing process, the smaller particles will fill in between the interstices of the larger particles, reducing the amount of voids and consequently improving the packing of the powder. Factors that may affect the packing density include loosening effect, wall effect, and wedging effect. In a coarse powder-dominant mixture, fine particles are not small enough to fit into interstitial spaces, which will cause loosening effect. When fine powder makes the bulk of the mixture, coarse particles may displace fine particles, but when they are not large enough to fill the spaces, they create voids causing wall effect. Also, when coarse particles are dominant, the wedging effect happens when a fine particle decreases the coordination number of the coarse particles by being in between two coarse particles, rather than in the interstitial space. When fine particles are dominant, the wedging effect occurs when coarse materials impede a layer of fine particle arrangement, creating voids too small for fine particles to fill. Other factors that may affect the packing density include shape and size of the particles, particle size distribution , and powder flowability.
Many models have been formulated and studied by researchers in attempts to predict how packing density can be improved. Particle packing models may be categorized into (1) discrete models which can be further categorized based on how many different classes of particles are involved in the mix. Discrete models are classified as binary mixture models such as Furnas model [6], ternary mixture models such as Goltermann or modified Toufar model [7], and multi-component mixture models; (2) continuous models; (3) computer simulation models; (4) design of experiment models; and (5) statistical modeling such as Monte Carlo computation. This research project falls under the category of discrete ternary mixture models.
Project Description
Assume, hypothetically, that there are several particle sizes in a system, where the packing of a larger size leaves a certain fraction of voids, and that each subsequent smaller size exactly fills the voids of the preceding size class, without increasing the overall volume of the system. Therefore, if infinitely small particles are successively introduced into the system, the packing density could theoretically reach 100%. Now, the question is how many combinations of different sizes are needed to yield the highest packing density ? For example, in a unimodal (for one size of sphere), the maximum theoretical density is π/(3√2) ≈ 0.74. In a bimodal (two different sizes of spheres), the theoretical highest density could reach 0.933. Thus, hypothetically, adding a third size of smaller diameter than the first two would increase the theoretical packing density (greater than 0.933), and so on. Therefore, introducing an infinite number of decreasing sphere sizes to the first matrix could result in a packing density of 1.0.
According to scholars that have worked in this area, a three-component mixture is ideal to achieve the maximum density of powder. Thus, the goal of this project is to develop a simple model that truly represents a three-sized powder packing density , and to generate a simple formula for the volumetric fraction of each particle class required to yield highest density of a mixture. In the proposed model, the packing density of a powder sample is defined as a ratio of the solid volume to the total volume of the sample:
where
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\( V_{s} \) = Volume of solids,
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\( V_{v} \) = Volume of voids,
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\( V_{t} \) = Total Volume = volume of solids plus volume of voids, and
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\( e \) = Void ratio (porosity ).
Methodology
As aforementioned, the highest packing density of a mono-sized sphere in an unbound space cannot be more than \( \frac{\pi }{3\surd 2} \approx 0.7404 \), which means the solid material fills only 74% of the volume space. Consider a container of unit volume with mono-sized spheres (s1) crushed into powder such that it the powder occupies approximately 74% of the volume, and the air above is the remainder of the volume. Let the solid volume be \( a = \frac{\pi }{3\surd 2} \), so that the volume of voids is \( \left( {1 - a} \right) \) as shown in Fig. 1.
Assume we fill up the volume space V2 with another sphere (s2) of smaller diameter such that it reaches the maximum capacity. Based on the theorem from above, filling this column to full capacity would mean a packing density of \( a = \frac{\pi }{3\surd 2} \). To find the packing density (Φ) of the mixture, we must find the new solid volume:
For s1, \( \Phi _{1} = \frac{{V_{s1} }}{V} = \frac{{V_{s1} }}{1} = V_{s1} = a \); and for s2, \( \Phi _{2} = \frac{{V_{s2} }}{{V_{2} }} = \frac{{V_{s2} }}{1 - a} = a \), which implies \( {\text{V}}_{s2} = a\left( {1 - a} \right) \).
So, the total solid volume, \( V_{s} = V_{s1} + V_{s2} = a + a\left( {1 - a} \right) = 2a - a^{2} = 0.933 \).
This means that the use of two different sizes of spheres increases the theoretical highest density from 0.74 to 0.933. Thus hypothetically, adding a third size of smaller diameter than the first two would increase the theoretical packing density (greater than 0.933), and so on. Therefore, introducing an infinite number of decreasing sphere sizes to the first matrix could result in a packing density of 1.0. It should be noted that the values 0.74 and 0.933 obtained for single sphere packing and the packing of two spheres, respectively, are considered as upper bounds, implying that the actual densities that can be achieved in practice are much lower. Several factors contribute to this difference. These factors are shape, size, and random packing of spheres. Thus theoretically, we can keep adding smaller particles to fill in all the voids. But in practice, this is not possible due to the above-stated reasons.
Packing Density for Ternary Powder Mixtures
The packing mechanism in the proposed model is that the smaller particles are introduced into the interstices of the larger packed particles, reducing the voids in the system. Now, assume a stack of powder samples with decreasing sizes (average particle diameter) in a vertical container. Let ri be the volume fraction of the coarser material of any two consecutive powder samples.
The void ratio \( e \) of component \( i\varvec{ } \) is defined as the ratio of the volume of voids (\( V_{vi} \)) to the bulk volume (\( V_{i} \) which is the volume of the space that sample \( i \) occupies) (Fig. 2):
In a unit volume system, the solid volume for component 1:
Similarly, \( e_{2} = \frac{{V_{v2} }}{{V_{2} }} = \frac{{V_{v2} }}{{V_{v1} }} = \frac{{V_{v2} }}{{e_{1} }} \), which implies \( V_{v2} = e_{1} e_{2} \)
The solid volume for component 2,
The solid volume for component 3,
Consider a binary mix of maximum density. Let \( r_{i} \) be the volume fraction of the larger of two consecutive particle sizes, \( d_{i} \). Thus, the fraction of the smaller of two consecutive particle sizes, \( d_{i + 1} \), that will exactly fill the voids of the larger component is \( 1 - r_{i} \). Suppose another set of smaller particles of size \( d_{i + 2} \) can be introduced into the interstices of the larger particles of size \( d_{i + 1} \), and then the total volume of each component size will be given by a series of terms of decreasing magnitude. The first term is \( r_{i} \) and the second term is \( 1 - r_{i} \). The common ratio between two consecutive components would therefore be \( \left( {\frac{{1 - r_{i} }}{{r_{i} }}} \right) \). We call the larger of any two consecutive components as the coarse component, and hence the volume fractions would be termed the coarse volume fraction. We define a geometric series for the coarse volume fractions as
Similarly, for any two consecutive particle sizes (components), we can define the weight fraction of the coarser material as
The weight fraction of component \( i \) can be defined in terms of the true specific gravity and the solid volume of component \( i \) as
Thus, the percent composition of the larger particles that are filled with smaller sizes is given by
The apparent specific gravity \( G_{a} = \left( {1 - e} \right)G \), where G is the true specific gravity. Thus, \( G_{a1} = \left( {1 - e_{1} } \right)G_{1} \) and \( G_{a2} = \left( {1 - e_{2} } \right)G_{2} \). So
The third term of the weight fraction series, \( \left( {1 - \omega_{1} } \right)\left( {\frac{{1 - \omega_{2} }}{{\omega_{2} }}} \right) \), simplifies to
Calculating the bulk volume for a three-component mixture,
Equation (13) represents the total volume of a system that is made up of three different sizes and that are each stacked upon another as separate layers. If the particles of the first component, \( d_{1} \), are relatively large, and all the other sizes (i.e. \( d_{2} ,d_{3} , \ldots \)) are very small (ideally, infinitely small), then the voids created as a result of the \( d_{1} \)-sized particles will be exactly filled, and the final volume of the system will be the volume of the first component (largest sized particles). Therefore, if each smaller sized particle exactly fills the voids created by the previously packed larger particle component, the total volume will just be \( v_{1} \). The reduction in volume as a result of filling the voids of component 1 with components 2 and 3 is equivalent to \( \left( {V - v_{1} } \right) \). This will be the ideal case. However, for an actual system, the decrease in volume will be less than the ideal case, as components 2 and 3 may not exactly fill the voids created by component 1. This decrease in volume can be expressed as
\( k_{d} \) is a reduction factor ranging from 0 and 1. It can only be determined experimentally. Factors that affect the value of \( k_{d} \) include size ratio, particle shape, loosening effect, wedging effect, etc. If the small particles are infinitely small (i.e. \( d_{i + 1} \ll d_{i} \)) that they exactly fill the voids, then \( k_{d} \) will be unity. Also, if all the particles are of the same size as the first component, then \( k_{d} \) will be zero. The determination of \( k_{d} \) will be discussed in detail later. The total volume of the mixture then becomes
The total weight:
Substituting Eqs. 8–12 into Eq. 16, the total weight can be expressed as
After simplifying, we get
Percent Composition
Let \( d_{i} \) be the average particle diameter of component \( i \) in the mixture composition. For a three-component mixture, we define \( d_{1} \) = Coarse material; \( d_{2} \) = Medium/intermediate material; and \( d_{3} \) = Fine material. From the above-derived formula for \( V_{f} \), we can calculate the volume of each component, and consequently the proportion by volume of each component needed in the mixture to achieve the maximum density of the system:
The packing efficiency (\( \phi ) \) which is the maximum possible packing degree can be expressed as
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Three Conditions for Density Calculation:
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(1)
Equal void ratios: \( if e_{1} = e_{2} = e_{3} = e \), then
Volume fractions and total volume:
Packing efficiency:
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(2)
Same true specific gravity: \( if G_{1} = G_{2} = G_{3} = G \), then
Volume fractions and total volume:
The packing efficiency:
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(3)
Equal void ratios and equal specific gravities:\( if e_{1} = e_{2} = e_{3} = e \); \( and G_{1} = G_{2} = G_{3} = G \), then
Volume fractions and total volume:
Packing efficiency:
Determination of kd
As aforementioned , kd can only be determined experimentally. To the best of our knowledge, there are no or little data on metal powder to determine the value of kd. However, Furnas [6] conducted an experiment on angular materials using different size ratios of different mixtures, and in each case determined the corresponding contraction in volume that resulted. With this data (shown in Fig. 3), we can generate a quadratic fitting and generated an equation for the parameter kd:
where \( k \) is the ratio between consecutive sizes.
Model Verification
We tested the proposed model with Furnas model [6] for a three-component mixture. The paper cited above includes an example problem where limestone, fine sand and cement are mixed together. The Furnas model is used to calculate the maximum density, along with the corresponding volume and volume fraction of each component. Void ratios (ei), specific gravities (Gi) and average diameters of the components (di) were given as follows (Table 1):
For this example, Furnas [6] does not consider the volume contraction/reduction parameter, thus in demonstrating the utility of his model, and as such, for comparison purposes, our \( k_{d} \) value will be kept as 0.
These volumes computed with the proposed model are quite closely related to the values obtained with Furnas’ model [6], which, respectively, are 1.39, 0.493, 0.234 and 0.0986. The proposed model gives a total volume of 0.825 (Vf), compared with 0.8260 for the Furnas model. The proportion by volume of each component required to produce the maximum density of the system is compared as follows (Table 2):
The values shown above reveal that the proposed model can predict the proportions by volume required to produce concrete of maximum density. Not only does the proposed model accurately calculates the proportions of concrete compositions, but it can also be used in several other applications where different powder proportions are required to create a maximum density mixture.
Conclusions
The proposed model proves to be very powerful in its versatility of application. The derivations shown suggest that for any powder sample, we can determine the maximum packing density of powder to be used for any application, notably 3-D printing. This flexibility of the model is very valuable for industrial purposes. Even though research suggests that a ternary mixture is the best to achieve maximum density of a given powder type, the model can be derived for any powder mixture with multiple number of components. Not only can the model be applied to powder mixtures of the same material but this model can be applied to mixtures with different components, as with the concrete example problem that was used for model verification. The following conclusions can be drawn:
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1.
The size composition of a powder mixture is a very important factor in determining the spreadability of the powder mixture and the density of the 3D-printed product.
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2.
To attain the most efficient packing, different sizes of the powder must be mixed together in such a way that it minimizes the voids, by allowing the smaller particles fill in between the interstices of the larger particles.
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3.
A model that predicts the volume fraction of each powder size becomes necessary to predict the maximum possible powder density.
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4.
Researchers have studied the problem of packing spheres into containers over the years, but to the best of our knowledge there is little success that has been achieved in the mathematical development of the laws of powder packing density .
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5.
In this research project, we have developed a simple model that adequately predicts the volumetric fractions of powder mixture that can yield the highest powder density. The model accounts for the disparities between theoretical assumptions and practical outcome, such as volume reduction. The model equations are based solely on void ratios and true specific gravity.
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6.
The proposed model is a discrete model which assumes that each class of particle will pack to its maximum density in the volume available.
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7.
Although the proposed model is used to model ternary mixtures, it can be extended to include multicomponent mixtures.
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8.
For generalization purposes, the model assumes that the particles being packed are perfect spheres, and that the size ratio between successive sizes is constant for the entire system.
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9.
According to Furnas [6], for a maximum density of three-component sizes, the size ratio must be very small.
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10.
The proposed model accurately predicted the proportions of concrete compositions for maximum packing density . It can also be used in several other applications where different powder proportions are required to create a maximum density mixture.
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11.
Although the model is derived for metal powder mixtures, it can be applied to mixtures of other systems such as asphalt concrete , paint, rubber, coal storage, etc.
Recommendations and Future Work
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1.
The proposed model was derived with the aid of some assumptions; future developments of this model should focus on generating experimental results to further validate it and verify its practical applications .
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2.
The volume reduction factor kd can be evaluated only experimentally. Its empirical equation was obtained by Furnas limited data on granular material [6]. Experimentation on metal powder may show a more refined and accurate relation.
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3.
Experimental work should include factors that directly impact the packing density such as shape and size of the particles, particle size distribution (size ratio), wall and loosening effects, “Fine grain dominant,” and “coarse grain dominant” mixtures.
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© 2019 The Minerals, Metals & Materials Society
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Abu-Lebdeh, T., Damptey, R., Lamberti, V., Hamoush, S. (2019). Powder Packing Density and Its Impact on SLM-Based Additive Manufacturing. In: TMS 2019 148th Annual Meeting & Exhibition Supplemental Proceedings. The Minerals, Metals & Materials Series. Springer, Cham. https://doi.org/10.1007/978-3-030-05861-6_33
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DOI: https://doi.org/10.1007/978-3-030-05861-6_33
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