Selection of Descriptors for Particle Shape ... - Wiley Online Library

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Selection of Descriptors for Particle Shape Characterization Mark L. Hentschel, Neil W. Page* (Received: 5 September 2002; accepted: 15 October 2002)

Abstract Conventional shape descriptors, formed from a ratio of two particle size measurements, are among the simplest of the many methods used for quantitative particle shape characterization. However, a significant limitation of using one of these shape descriptors is that its value is often not unique to a specific shape. Use of several different shape descriptors may circumvent this problem but, as particle size can be defined in a large number of ways, a similarly large number of shape descriptors can be defined. While some differ substantially, others are only subtly different, conveying similar information. Thus, it is not obvious which of the many

possible descriptors should be utilized. In this paper, two-dimensional particle shape descriptors obtained by image analysis of six different commercially sourced powders were considered. Techniques of cluster and correlation analysis were applied to assist in identifying redundant descriptors for shape characterization of these powder particles, allowing for efficient description of shape using a reduced set. It was found that at least two descriptors are required: aspect ratio and the square root of form factor. Significantly, each descriptor is most sensitive to a different attribute of shape: elongation and ruggedness, respectively.

Keywords: cluster analysis, conventional shape descriptors, particle shape

1 Introduction Particle shape has been demonstrated to influence physical and behavioral properties of powders including: compaction and sintering behaviour [1, 2]; packing and fluid interaction [3, 4]; powder flow rate, apparent and tap densities [1, 5]; angle of internal friction [6]; strength and deformation behaviour [7] amongst a host of others. The shape of particles produced by different processes of wear has also been used to assist in identifying the wear mechanism behind their generation [9, 10], and is thus useful for machine condition monitoring. Qualitative shape characterization is a relatively simple matter. Visual inspection of particles allows their description with reference to simple geometric shapes or standard particles [11]. However, analytic models of particulate behaviour require quantitative shape description, and importantly, quantitative descriptors with clear physical interpretations. Many techniques are available to quantitatively describe both the size and shape of powder particles [1 ± 3, 8, 12 ± 20]. These two *

M. L. Hentschel, Research Fellow; N. W. Page, Professor of Mechanical Engineering (corresponding author), School of Engineering, The University of Newcastle, NSW 2308 (Australia). E-mail: [email protected]

¹ 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

particle properties are often considered to be separate attributes, however, they cannot be completely decoupled as the concept of shape results from variation of the extent (linear size) of an object with direction. Likewise, for practical purposes, a single numerical parameter is rarely sufficient to describe the size of an object without an accompanying description of shape (isometric, acicular, etc.) [18]. Reflecting this mutuality, most procedures for quantitative shape description include a measure of object size at some stage of the process. For example, elongation of an object may be quantified by aspect ratio: the quotient of minimum linear size by maximum linear size. As shape is a dimensionless quality, the only absolute constraint is that size measures are combined such that a non-dimensional parameter results. Although particle shape is inherently a three-dimensional attribute, many characterization techniques utilize two-dimensional data. Often this is in the form of a projected or sectioned image of the particle under study, or a two-dimensional section of the field of radiation scattered by the particle [2, 3, 14, 15, 17, 19, 21]. Shape characterization by analysis of three-dimensional images is possible, although collection of accurate data and its reduction to meaningful parameters is considered problematic [22]. Hence, despite two-dimensional methods 0934-0866/03/0506-0025 $ 17.50+.50/0

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Table 1: Example of some two dimensional shape factors using particle size measures obtained from image analysis. Category refers to dimensions of numerator and denominator. Category 1 descriptors are a ratio of size measures with dimensions of length, etc. See Table 2 for size definitions. Shape factor

Most sensitive to:

Formula

Aspect ratio, AR

Elongation

AR ˆ

B L

1

Form factor, FF

Boundary irregularity

FF ˆ

4pA P2

2

Roundness, R

Elongation

Rˆ

Convexity, Con

Boundary irregularity

Con ˆ

strictly failing to describe all particle attributes, such methods have proven useful in practice provided the particles are at least approximately isometric, viewed in random orientation, and the properties of interest apply to a typical, rather than specific particle. Some common techniques for particle shape characterization that utilize two-dimensional image data, include fractal geometry [6, 9, 23 ± 26], and spectral (Fourier) decomposition of the particle profile [13, 16, 27]. However, both methods require relatively complex data processing to obtain the parameter of interest. A substantially simpler shape characterization technique is to combine two measures of particle size, xi and xj, as a ratio: so-called −conventional shape descriptors× [2]. Sij ˆ

xi : xj

…1†

For an irregularly shaped particle, a plethora of possible size measures can be defined, based on a geometric property (volume, surface area, distance between boundary features), or a behavioral feature (sedimentation time, sieve aperture size, etc.) of the particle [2, 3, 14, 15, 17, 19, 28]. Typically, particle size is quoted as an equivalent dimension: the dimension of a regular shape (typically a sphere) that would yield the equivalent feature measurement. In this work, only geometric measures of size are considered, as not all applications logically correlate with a common behavioral measurement. Three basic categories of geometric measures exist, including particle volume, surface area, and linear distances between particle features. For simplicity, this study only considers those geometric measures obtained by two-dimensional image analysis. In the context of this work, an advantage of the image analysis method is that it provides ready access to a number of different particle size measures for a single particle. Further, by its very nature it facilitates

4A pL2 pD P

Category

2 1

comparison between calculated shape descriptors and observable particle features. If two measures of particle size are available, an elementary descriptor of particle shape can be obtained by combining them as a ratio Eq. (1). The value of this ratio relative to that of a reference shape (usually a sphere) indicates the degree to which the particle under study differs from the reference shape. However, given that particle size can be defined in many different ways, a correspondingly large number of conventional shape descriptors are also possible. Each descriptor will be most sensitive to a specific attribute of shape, depending upon the size measures selected. If more than two measures of size are available, the question arises then, as to which combination provides the best description of shape. Some examples of conventional shape descriptors are listed in Table 1. These descriptors have the advantage of being simple functions of image properties, with easily conceptualized physical interpretations. For example, decreasing values of aspect ratio denotes more elongate particles. However, conventional shape descriptors suffer three main disadvantages. Firstly, is the issue of deciding which size measures are most appropriate to use when defining certain descriptors. For instance, is average Feret diameter or maximum Feret diameter more appropriate in the definition of roundness? A case could conceivably be made for either, but of course the resulting values obtained will depend on the choice. The second problem is a practical issue relevant to characterization techniques incorporating particle perimeter. For many particles, the measured boundary length (perimeter) is dependent on the length scale at which it is investigated (image resolution), i.e. perimeter is not an absolute measurement [6, 23, 25, 26, 30]. Despite this, shape descriptors incorporating perimeter may still be useful so long as the particles being compared are all

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Part. Part. Syst. Charact. 20 (2003) 25 ± 38 Table 2: Image measurements for calculation of two-dimensional shape descriptors. Measurement

Method of measurement

Feret diameter vertical, dF, vert horizontal, dF, horiz maximum, L (dF,max ) orthogonal to maximum, L90 minimum, B (dF,min ) orthogonal to minimum, B90 mean, D (dF,mean ) median, Fmed (dF,median )

distance between bottom and top-most image pixels distance between right and left-most image pixels maximum Feret measured for all orientations Feret diameter 90 8 to maximum Feret minimum Feret measured for all particle orientations Feret diameter 90 8 to minimum Feret mean of Feret diameters for all orientations median of Feret diameters for all orientations

Minimum area bounding rectangle length, a breadth, b Chord length vertical, dC,vert

longest Feret diameter of the pair (dF, dF,90 8) whose product is a minimum over all orientations shortest Feret diameter of the pair (dF, dF,90 8) whose product is a minimum over all orientations

maximum, Cmax (dC,max ) orthogonal to maximum, Cmax 90 (dC,max,90) minimum, Cmin (dC,min ) orthogonal to minimum, Cmin, 90 (dC,min,90) mean, Cav (dC,mean ) median, Cmed (dC,median ) Martin×s diameter, MD

distance between boundary pixels which intersect a vertical line passing through the particle centroid distance between boundary pixels which intersect a horizontal line passing through the particle centroid maximum chord length for all particle orientations length of chord 90 8 to the maximum minimum chord length for all particle orientations length of chord 90 8 to the minimum mean of chord length for all orientations median of chord length for all orientations length of the chord which most closely bisects the particle area

Perimeter, P

sum of the distance between successive boundary pixels

Area, A

total number of pixels in the particle image

horizontal, dC,horiz

of similar size, and are imaged on the same scale (magnification), as comparative perimeter lengths will still be indicative of gross particle shape. However, perhaps the primary disadvantage of conventional shape descriptors is that the inverse problem is illposed: description of particle shape (regeneration) based on the magnitude of a given conventional shape descriptor is generally not possible. Luerkens [16], and Russ [29] both provide examples of two-dimensional shapes that are visually quite different, but yield identical values for a given conventional shape descriptor. Clearly, this lack of uniqueness is not a desirable circumstance, as a procedure utilizing such a descriptor to numerically specify shape effects may predict an identical response for substantially different particle shapes. Podczeck [31] has addressed the problem of non-uniqueness by combining four conventional shape descriptors that each quantify the difference between the shape under investigation and a simple geometric shape. A fifth parameter, C0, describing the number of characteristic corners of the shape was then added to this combination to obtain a new shape descriptor, NS.

A 4A 2A P ;H ˆ ;H ˆ ;H ˆ BB90 2 pB290 3 BB90 4 L H4 H1 NS ˆ C0 ‡ H2 H3 H1 ˆ

NS ˆ C0 ‡

2AP LBB90

4A2 : pBB390

…2†

H1±3 describes the difference between the studied shape and a square, circle, and triangle respectively, while H4 quantifies elongation. Particle size measurements are as defined in Table 2. Podczeck [31] subsequently demonstrated NS was better able to differentiate between model shapes (circles, ellipses, parallelograms, trapezoids, etc.) than individual conventional shape descriptors. Although an encouraging result, two main disadvantages of NS can be identified. Firstly, is that NS comprises six measures of particle size, which are all independent for a completely arbitrary shape. Although some correlation between size measures is expected in practice, this relatively large number of degrees of freedom increases the likelihood

28 of quite different shapes having equivalent values of NS. A potentially greater limitation on the use of NS though, is that a large part of the success at discriminating between model shapes can be attributed to C0. Podczeck [31] suggests C0 be determined either by visual inspection of the particle outline or calculated from the number of turning points in the particle waveform. However, an objective definition as to what constitutes a particle corner for realistic rugged particle profiles is elusive: a choice must be made as to both the magnitude of deviation from the particle profile that constitutes a corner, and the −sharpness× of a corner (see also [20, 29]). As NS is an additive function of C0, the value obtained will strongly depend on the criteria used to define a corner, and as such, it is difficult to assure the objectivity of NS. However, the approach of Podczeck [31] does recognize that, even for the hypothetical cases illustrating nonuniqueness of a given conventional shape descriptor [16, 29], there is generally another shape descriptor formed from a combination of different size measurements that is able to discriminate between the shapes. Thus, it appears that if a sufficient variety of particle size measurements are available, it should still be possible to uniquely identify each different shape using more than one conventional shape descriptor. In the most general case of a completely arbitrary shape, this may require a large number of descriptors, as the size measurements can be completely independent. It is important to bear in mind, however, that the shape assumed by powder particles is governed by definable physical laws, depending upon conditions and method of production. For a given process, variation in particle shape will depend upon the uniformity of conditions and reproducibility that can be achieved. Thus, it is expected that relatively strong correlations will exist between at least some geometric and behavioral particle features, particularly under controlled conditions of commercial manufacturing processes. As such, many shape descriptors will provide similar information when applied to manufactured powders, potentially allowing for a substantial reduction in the number required to specify shape if redundancies can be identified. Elimination of redundancies then allows for parsimonious description of shape using a limited number of conventional shape descriptors with sensitivity to all major shape features present. This work discusses an approach which may be used to identify this reduced set of descriptors. As the most general case is extremely broad, the range of particle shapes considered here is restricted to those typical of powders available from commercial sources. Similarly, in the interests of simplicity, attention is focused on conventional shape descriptors formed from particle size measurements derived from analysis of projected

Part. Part. Syst. Charact. 20 (2003) 25 ± 38

particle images, defined according to Eq. (1). Techniques of cluster and correlation analysis are applied to a number of shape descriptors to seek groups of similar descriptors, and identify descriptors representative of each group.

2 Particle Shape Characterization by Analysis of Two-Dimensional Images Analysis of images firstly requires a representative digital image of an individual particle to be acquired. Commonly, either a projected view, or an image of an exposed cross-section is used. Stachowiak [9] has compared values of the particle shape descriptor boundary fractal dimension when calculated from both sectioned and projected particle images. In the majority of cases, it was reported that projected images gave a significantly lower value of fractal dimension, with no simple quantitative relation between the two. The lower ruggedness indicated by projection was attributed to masking effects of surface protuberances and re-entrant features. Stachowiak [9] suggested however, that fractal dimension calculated from sectioned particle images relates to the surface fractal dimension (surface features), while projected images produce data related to particle angularity (gross particle features). In this work, projected particle images were analyzed. This is predominately because not all particle crosssections accurately represent the particle as a whole. Particles with extended, or distributed features (such as the dendritic copper particles illustrated in Figure 1a) furnish a particularly good example of this. While images of some particle cross-sections in Figure 2 clearly shows the dendritic structure, other particles have only been sectioned through a dendrite −arm× and are not representative of the complete particle. Further, in many cases surface detail is either excluded from the image, or isolated from the main particle body. By contrast, all surface features remain joined to the main particle body in projected images. While overlying material may obscure some surface features, it is felt that projected images provide a more complete description of the entire particle. Further, subjective choices as what constitutes a representative particle image are avoided. Practical issues relevant to the measurement of particle features by image analysis are discussed elsewhere [29, 45].

3 Experimental 3.1 Powders Six different powders were used in this study, spanning a range of particle shapes. All powders were sourced from

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commercial suppliers so that variation in particle shape between powders is typical of different types of feedstock used in powder metallurgical processes. A narrow sieve size fraction of each powder (75 ± 106 mm) was selected for study to reduce possible influence of particle shape changing with size [2, 14]. Details of the powders are summarized in Table 3. Micrographs of representative particles for each powder are provided in Figure 1.

3.2 Image Capture Samples for image analysis were prepared following the procedure described by de Silva and Creasy [32]. A piece of double-sided adhesive tape was mounted on a glass microscope slide, then placed on the underside of a sieve one size larger than the largest particle size (125 mm sieve for 75 ± 106 mm size fraction). A sample of powder was

Fig. 1: Scanning electron micrographs of the powders, illustrating the range of particle shapes. (a) Dendritic copper. See also Figure 2. (b) Irregular copper (MM). (c) Irregular copper (US). (d) Spherical copper. (e) Stainless steel. (f) Aluminum. See also Table 3.

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Table 3: Powder details. See Figure 1 for micrographs of powder particles. Designation

Particle shape [11]

Manufacturing method

Source

Dendritic copper

Dendritic

Electrolytic deposition

Irregular copper ( MM )

Irregular

Water atomization

Irregular copper ( US )

Irregular

Water atomization

Spherical copper

Spheroidal

Gas atomization

Stainless steel

Spheroidal

Gas atomization

Aluminum

Spheroidal

Gas atomization

MicroMet Hamburg, Germany. MicroMet Hamburg, Germany. U. S. Bronze Powders Flemington, USA. Makin Metal Powders Lancashire, England. Anval Nyby Powder Torsh‰lla, Sweden. Alloys International Melbourne, Australia.

Fig. 2: Laser scanning confocal micrograph of sectioned dendritic copper powder particles. Image dimensions are 300  300 mm. See also Figure 1a.

sprinkled over the sieve, to form an array of nonoverlapping particles, ensuring individual particles were analyzed rather than pseudo-agglomerates of overlapping particles. This also helps to avoid small particle bias, as only one particle occupies a defined area (the sieve aperture). If particles were simply dusted onto the slide, more small particles would fit into a fixed field of view [29]. Ideally, particles should be viewed in a completely random orientation. However, the microscope slide imposes a boundary condition, resulting in a tendency for particles to be viewed normal to their plane of maximum stability [3]. It is expected this will cause most bias for strongly elongate particles, though sharp facets or isolated protrusions on the particle surface may also

have an effect [33]. Visual inspection of the morphology of particles used in this work (Figure 1) shows they are approximately isometric, although some particles have features that may affect their orientation. For example, the stainless steel particles are unlikely to balance on an −attached× sphere. As particles are held in place by an adhesive layer rather than them resting upon an impenetrable surface, these effects should be reduced, although the effectiveness of this is not known. Projected particle images were obtained using a light microscope equipped with a digital camera. The powder covered slide was illuminated from beneath to provide a projected view with little surface detail but good contrast between slide background and the particle boundary. This situation is advantageous for image processing (thresholding) to obtain a binary image of the particle, as changes in pixel intensity occur mainly at the particle boundary and not within the interior. From each slide, sufficient images were acquired to provide 250 individual particle images. This procedure was repeated on another three samples prepared in an identical fashion, yielding a total of 1 000 particles for analysis. For each sub-sample, close agreement was found between distributions of measured values. Binary images of individual particles were generated using Scion Image [34], with further processing for measurement of various particle dimensions (Feret diameters, area, perimeter, etc.) carried out using custom written software implemented in Matlab [35]. This involved firstly, identifying pixels on the boundary of an individual particle, and their representation as a coordinate set. This boundary profile was then rotated through 180 8, in one degree increments about the particle centroid (calculated assuming constant density). At each increment, the Feret diameter (F) and the length of the chord that passed through the centroid (C) were measured in two orthogonal directions (vertical and horizontal). Upon completion of rotation, various sum-

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mary statistics of the resulting distributions were calculated: maximum, minimum, mean, median, etc. In total, 17 measures of particle size were recorded for each particle image, as listed in Table 2. For comparison with conventional shape descriptors defined by Eq. (1), the boundary fractal dimension of the projected particle image was also calculated, using the Euclidean distance mapping (EDM) method [24, 36]. Key features of the algorithm are described below. 1. Identify boundary pixels using four-point connectivity. 2. For each pixel in the image, calculate the distance to the nearest boundary pixel. 3. Select all pixels at a distance less than a particular value, l, from the boundary. 4. Calculate the area covered by the selected pixels A(l). The corresponding perimeter estimate is: P(l) ˆ A(l)/(2l). (2l is the width of the −strip× covering the boundary). 5. Repeat 3 and 4 for different values of l. From the resulting set of l, P(l) data, boundary fractal dimension was calculated by regression of Eq. (3). P(l) ˆ Ml(1

FD)

.

(3)

Where M is a positive constant, and FD is the boundary fractal dimension. Most authors do not provide full details of the exact method employed to calculate FD, however it appears that generally, this proceeds by firstly taking the natural logarithm of each side of Eq. (3) to linearize it, and then applying a straight line regression to the data set ln(l), ln(P(l)) to estimate FD. However, as noted by Klinkenberg [37], this procedure is incorrect, as the objective function minimized is not appropriate to least-squares regression (see [38] for example). The proper leastsquares estimate requires iterative, non-linear regression to fit the power law of Eq. (3). The effect of linearization on calculated values of FD was investigated in this work as part of program validation. Three different Koch Island fractals of known fractal dimension (1.2618, 1.5, 1.6131), generated using Winfract [39], were used for validation. When applied to these mathematical fractals, it was found that values of FD calculated using transformed data and straight-line regression, differed by less than 1.2% with those obtained from non-linear least-squares regression of Eq. (3). The close agreement between linear (incorrect) and non-linear (correct) regression in this instance is most likely due to the relatively small range of step sizes (l) used in this work, which span only two orders of magnitude. Given the small discrepancy between the two

methods, it was felt that use of the linearized method of fractal dimension calculation would not result in a significant loss of accuracy in this work. Linear regression was preferred as it is substantially easier to implement, requiring only simple matrix manipulation rather than iterative procedures necessary for non-linear regression. However, it is worth stating explicitly that this −approximation× is unlikely to hold if l is varied over several orders of magnitude, and is only adopted here for computational convenience. Using the algorithm described above, and linear regression of the l, P(l) data set, resulted in calculated values of FD that agreed with theoretical values of the Koch Islands to within 2%.

4 Analysis 4.1 Selection of Suitable Shape Descriptors As illustrated in Table 1, the ratio of size measures used to form a conventional shape descriptor can be categorized according to dimensions of numerator and denominator in the ratio, i.e. 1 corresponds to length, and 2 corresponds to length squared. The basic measures of particle size listed in Table 2 comprise 16 measurements with dimensions of length, and one of length squared (area). Thus, 90 unique Category 1 shape descriptors can be defined, considering two measures of particle size and making no distinction between inverse combinations (such as L/B and B/L). To include projected particle area, Category 2 descriptors must be considered. This appreciably increases the number of descriptors that must be considered. Accepting only multiplicative functions for numerator and denominator, size measures can be selected in four different combinations. Area and one linear size measure squared, e.g. A/L2. This provides 16 possible descriptors. (ii) Area and two linear size measures, e.g. A/(L  B). In this case there are 90 possible shape descriptors. (iii) One linear measurement squared and two others e.g. (D  B)/L2. 560 shape descriptors can be defined. (iv) Four different linear size measurements, e.g. (D  B)/ L  Cmax). 1 820 shape possible descriptors result. (i)

In total, 2 486 unique Category 2 shape descriptors can be defined. As this number is unwieldy, only Category 1 shape descriptors are considered here, with area includp ed by taking the square root ( A). This provides 17 measures of size from which, 136 conventional shape descriptors can be defined according to Eq. (1). This

32 number, while certainly an improvement, is still somewhat larger than desired. To seek further reductions, a preliminary investigation of the 17 size measures was conducted to identify possible redundancies. This was performed using cluster analysis.

4.2 Cluster Analysis Cluster analysis is a technique for recognition of patterns in multivariate data [40 ± 44], primarily used to identify similar individuals within a population. Consider a population of m individuals labeled Xi (i ˆ 1 to m) upon each of which, n variables, xj (j ˆ 1 to n) are measured. The n measurements may be used as coordinates to represent the position of the i-th individual (Xi), in n-dimensional space. Individuals that are similar in terms of the measured variables will lie close together in this n-dimensional space. For the case of n ˆ 1, 2, 3, clusters (groups of similar individuals) can be identified visually by simply plotting the data. However, other methods are required for higher dimensional space (more variables). This more general case is addressed by cluster analysis. In this technique, individuals are grouped into clusters on the basis of their similarity (discussed below), to form a new virtual individual (the cluster) whose properties are representative of the constituents. A common method of cluster analysis is agglomerative hierarchical clustering, in which all individuals are initially considered to be the single member of a cluster. Similarity between all clusters is then calculated, and the most similar two are merged into a single cluster. This process is repeated, successively merging clusters until all individuals are contained in a single cluster. After each iteration, the number of clusters reduces by one. Alternative clustering methods are also available; see, for example [40, 43]. Importantly, the order in which clusters are formed and the individuals they contain, often indicates patterns within the data set as a whole, and may identify relations between individuals (a subpopulation). Usually this is done in conjunction with external information, to search for a common link between members of the cluster. For example, a cluster may comprise individuals processed by a method that was different to all other individuals in the population. Clustering of these individuals would suggest that processing method affects properties of the individual. Similarity can be quantified in a variety of ways. Euclidean distance between individuals is a common similarity measure, though again, others are possible [40, 43]. However, as a cluster must eventually contain more than one individual, inter-cluster distance is not unique: each set of two individuals (one drawn from either

Part. Part. Syst. Charact. 20 (2003) 25 ± 38

cluster) will have different similarities. Thus, an overall measure of inter-cluster similarity is required that is representative of similarity between pairs of individuals. In the present study, this was achieved by using the cluster centroid to represent overall cluster properties, then calculating the distance between cluster centroids to assess similarity. This representation was preferred as it incorporates information from all individuals in the cluster, but only requires a single distance to be calculated between clusters. Further details on cluster analysis, including alternative clustering techniques and distance metrics, are available elsewhere [40 ± 44]. Typically, cluster analysis is used to identify sub-populations within a large set of individuals using a number of variables measured on each individual. Thus, in most applications, the individuals are physical entities, and variables are mathematical parameters. However, in this application, the converse situation applies: the set of variables (xj) comprise a sample of powder particles, while the individuals (Xi) are various shape descriptors, as it is similarity between shape descriptors that is sought. A large number of particles (variables) were used to ensure the range of shape descriptors (individuals) is representative of each powder, defining both typical particle shape and its variation. As the results obtained for a given powder are strictly applicable to that shape only (e.g. dendritic), some generality was sought by studying several different powders that span an appreciable range of particle shapes (Figure 1). Thus, if similar clustering patterns are observed for a range powders, this suggests the overall results can be applied to that general class of particles, i.e. commercially manufactured powders. It is worth pointing out that shape descriptors identified as being optimal, may not necessarily be so for classification of particles drawn from another general class (e.g. wear particles). However, it is suggested that the techniques described in this study may prove beneficial if similarly applied to a representative sample from such a general class.

4.3 Identification of Redundancies in Size Measures Cluster analysis was initially applied to identify similarities within the set of 17 particle size measures. As the measures of size have intrinsic differences, each was standardized prior to clustering. This was performed by subtracting the mean value of the variable set and dividing by the standard deviation, to provide a new set of variables with a mean value of zero and a standard deviation of unity [42]. Standardization was adopted so variables which are inherently larger (e.g. P) than others (e.g. B), do not dominate the results, i.e. the distance between unstandardized values of, say, P and B will be

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Fig. 3: Dendrogram illustrating clustering behaviour of standardized measures of particle size. The ordinate axis scale represents (standardized) Euclidean distance between merging individuals or clusters. (a) Dendritic copper. (b) Irregular copper (MM). (c) Irregular copper (US). (d) Spherical copper. (e) Stainless steel. (f) Aluminum.

much larger than between Cav and B, simply because P is much larger than B. Standardization also ensures homoscedastisity, so variables with intrinsically large variances have the same importance as those with small variances. For instance, it is expected that L will have a larger variance than D. Thus, small changes in D are more significant than an equivalent change in L. Results of the particle size cluster analysis are presented graphically as a dendrogram [42, 44], in Figure 3. Figure 3 indicates the following sets of size measurements cluster early: 1. 2. 3. 4.

p Cav and A D and Fmed L, B90, a, and Cmax B, b, and L90

Some useful insights on relations between variables are provided by the nature of variables in each cluster. The second cluster suggests (standardized) D and Fmed are practically indistinguishable, indicating the distribution of Feret diameters for a single particle is symmetrical about the mean value. A similar relation also appears to hold for Cav and Cmed. Although Cmed usually clusters later, it always joins the cluster containing Cav. Of possibly greater interest though, is the relation suggested by clusters 3 and 4: that L is usually perpendicular to B. Further, the presence of Cmax in the cluster with L indicates they are approximately coincident. Considering L and B are usually perpendicular, the presence of L and Cmax in the same cluster suggests a similar relation may hold for chord lengths. The results generally support this, as Cmin and Cmax,90 join with

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cluster 4 and, Cmin,90 joins cluster 3, although the relation between the two is not as strong as for Feret diameter. Thus, the size measures may be divided into six main groups: I. B, L90, b, Cmin, and Cmax,90 II. L, B90, a, Cmin,90, and Cmax p III. Cav, Cmed, and A IV. D and Fmed V. P VI. MD The cluster analysis results show P usually clusters fairly early and could be included in the group containing D and Fmed. Despite this, it was decided to retain P separately as, along with area, it is a fundamentally different measure of particle size. In comparison, all of the various measures of chord length and Feret diameters are conceptually similar in that they represent a straight-line distance from one point on the particle outline to another. With groups of similar size measures identified, it is required to choose one variable most representative of each. This was accomplished by calculating the correlation coefficient between standardized variables. The average value of the correlation coefficient with all other members in the particle size groups defined above is given in Table 4. For the size measures of Group I, B has the highest average correlation with all other members, and thus was selected as most representative. For Group II, the average correlation of B90 is slightly higher than L. However, given that there is only a slight difference between average correlations, and the correlation is very high, L was preferred, as it is defined independent of

other measures of particle size. Further, L is a more common measure to use in formation of shape descripp tors. Similar logic was followed in the preference of A, over Cav, as representative of Group III, with the additional consideration that it describes a different fundamental particle feature. In the choice between D and Fmed to represent Group IV, D was favored out of deference to popular definition of shape descriptors, in which the average is more often used to describe central tendency. For all powders studied, cluster analysis identifies MD as the single most distinct size measure, always remaining separate until the penultimate or final cluster. Despite this MD was not given further consideration, as the value obtained for a single particle is quite often not unique (see also [16]). In contrast to Feret diameter and chord

Measured values of Martin×s diameter, MD, [mm]: 61.5, 61.9, 61.9, 62.0, 62.4, 62.5, 63.0, 63.1, 74.1, 75.2, 75.3, 75.5, 173.5 average: 73.9 mm, max.: 173.5 mm, min.: 61.5 mm range: 112 mm (150% of the average) Fig. 4: Illustration of the non-uniqueness of Martin×s diameter (irregular copper (US) powder particle).

Table 4: Average value of correlation coefficient for each size measure with other group members. The table is divided into three sections according to the groups identified. The upper section is Group I size measures; the middle Group II; and the lower section is Group III size measures. Dendritic copper

Irregular copper ( MM )

Irregular copper ( US )

Spherical copper

Aluminum

Stainless steel

B L90 b Cmin Cmax,90

0.83 0.80 0.81 0.63 0.76

0.82 0.78 0.81 0.64 0.74

0.83 0.79 0.80 0.69 0.74

0.96 0.93 0.93 0.91 0.94

0.84 0.79 0.81 0.71 0.79

0.94 0.90 0.93 0.89 0.91

L B90 a Cmax Cmin,90

0.91 0.91 0.90 0.90 0.73

0.91 0.91 0.90 0.90 0.75

0.93 0.93 0.92 0.93 0.80

0.95 0.96 0.95 0.95 0.91

0.96 0.96 0.95 0.95 0.89

0.94 0.95 0.94 0.95 0.86

0.93 0.95 0.89

0.94 0.96 0.91

0.93 0.95 0.89

0.98 0.99 0.97

0.93 0.95 0.89

0.98 0.99 0.97

p

A Cav Cmed

35

Part. Part. Syst. Charact. 20 (2003) 25 ± 38

length, for which a single measurement is also generally not unique, the distribution of values obtained for MD is generally neither smooth nor continuous, with substantial variation in magnitude. Thus, summary statistics of the distribution (mean, median, etc.) are of little value. In the present work, it was found that large variation in values of MD could potentially be obtained for a single particle. If chords that divide the particle in to areas equal to within 1% are accepted as candidates for MD, 14 chords satisfy this requirement for the irregular copper powder particle (US) in Figure 4. The length of these chords is listed next to Figure 4. This practical example illustrates the non-uniqueness and wide distribution of MD for a single elongate

particle. Given that the range of values spans 150% of the average for the particle of Figure 4, MD was not considered further in this work. Thus, the following five variables have been identified to use in formation of p conventional shape descriptors: D, L, B, P, and A.

4.4 Identification of Redundancies in TwoDimensional Shape Descriptors With five size measures, ten unique conventional shape descriptors can be defined according to Eq. (1). Boundary fractal dimension was also included to ascertain whether it provides shape information not conveyed by

Fig. 5: Dendrogram illustrating clustering behaviour of standardized shape descriptors (see Table 5 for definitions). (a) Dendritic copper. (b) Irregular copper (MM). (c) Irregular copper (US). (d) Spherical copper. (e) Stainless steel. (f) Aluminum.

36

Part. Part. Syst. Charact. 20 (2003) 25 ± 38

Table 5: Shape descriptors. S1 ˆ

D L

S7 ˆ

P pL

B D p B p S8 ˆ p 2 A

S2 ˆ

S3 ˆ

p 2 A p D p

S9 ˆ

pB P

pD P p 2 pA S10 ˆ P S4 ˆ

conventional shape descriptors. The complete set of shape descriptors considered in the second phase of this study is listed in Table 5. All descriptors defined using Eq. (1) were scaled to yield a value of unity for a circle, with the combination arranged such that deviations from circularity generally result in values less than unity. Commensurate with this, the inverse of fractal dimension was used so increasingly rugged boundaries produce a shape descriptor approaching 0.5 (1  FD  2). Cluster analysis was again performed on standardized variables for each powder separately, seeking similarities in the eleven shape descriptors. Results are shown in the dendrograms of Figure 5. In this case, the clustering patterns identify two main groups, comprising: 1. S3, S4, S10, S11 2. S1, S2, S5, S6, S7, S8, S9 . This immediately suggests that at least two shape descriptors are required to characterize the shape of powders produced by commercial manufacturing methods. However, a strong positive of this technique is, if a more precise description of shape were required, it is apparent the variables S7 and S8 would be appropriate, as these are seen to join Group 2 quite late. In order to determine the most representative of each group identified above, correlation between descriptors was again considered. The average value of the correlation coef-

S5 ˆ

B L

S11 ˆ

p 2 A S6 ˆ p L p

1 FD

ficient for each descriptor and others in the cluster are provided in Table 6. From Table 6, it is apparent S10 best describes the first group and S5 the second. Comparison of Tables 5 and 1 identifies S5 as aspect ratio, and S10 as the square root of form factor. In the following, the more common −aspect ratio× (AR) will be used to refer to S5, and p similarly S10 shall be referred to as −root form factor× ( FF). Group 1 contains descriptors sensitive to boundary p irregularity (ruggedness). The presence of FF in the same group as boundary fractal dimensionp(S11) offers a substantial advantage, as calculation of FF is much simpler than fractal dimension. It is noticeable that all members of Group 2 contain either, or both L and B. Along with the presence of aspect ratio, this indicates these descriptors are most strongly influenced by particle elongation. Thus, each of the two descriptors identified by this process conveys information on a different attribute of shape: AR characterizes particle elongation, p whereas FF describes ruggedness. Both shape attributes are likely to be important in describing the physical behaviour of individual particles and their interaction in an assemblage. However, rather than seeking to combine both, it is suggested these descriptors be retained separately, with each used to describe one particular component of a granular system response to a given process or conditions. For the case of commercially manufactured

Table 6: Average value of correlation coefficient for each shape descriptor (Table 5) with other group members. The table is split with Group 1 shape descriptors in the upper portion, and Group 2 descriptors in the lower. Dendritic copper

Irregular copper ( MM )

Irregular copper ( US )

Spherical copper

Aluminum

Stainless steel

S10 S11 S4 S3

0.87 0.81 0.76 0.60

0.88 0.79 0.78 0.69

0.88 0.79 0.80 0.69

0.93 0.88 0.87 0.87

0.88 0.81 0.77 0.75

0.94 0.90 0.90 0.88

S5 S2 S1 S6 S9 S8 S7

0.89 0.84 0.82 0.74 0.70 0.72 0.57

0.88 0.85 0.83 0.76 0.75 0.75 0.63

0.89 0.87 0.84 0.80 0.79 0.79 0.63

0.96 0.94 0.93 0.92 0.92 0.87 0.90

0.93 0.91 0.89 0.86 0.87 0.82 0.81

0.92 0.89 0.88 0.86 0.85 0.80 0.73

37

Part. Part. Syst. Charact. 20 (2003) 25 ± 38

powders, the overall response, R, of a material model incorporating a dependency on particle shape would p include both AR and FF. Algebraically, this can be written: R ˆ R(S, X), where S is a matrix whose elements Sij are shape descriptors, and X describes all other variables of interest. R can then be decomposed into two components, Relongation ˆ R(AR, X) and Rruggedness ˆ p R( FF, X), with the overall response R ˆ R(Relongation, Rruggedness). In general, recombination of individual components to obtain the overall response may not be trivial, as some form of interaction is to be expected. However, a positive feature of the present method of selecting shape descriptors, is that they should be relatively independent (as each describes a different shape attribute). Hence, for a response dependent on shape alone, superposition of component responses may be valid.

5 Summary and Conclusions A methodology has been presented for identifying a reduced set of conventional shape descriptors that allow parsimonious description of particle shape. This entailed the use of cluster analysis to identify similarities in firstly particle size measures, and secondly amongst particle shape descriptors. It was found that particle shape for a range of commercially manufactured powders could be efficiently described using two conventional shape descriptors: aspect ratio (AR) for elongation and root p form factor ( FF) for ruggedness. It is recommended that shape descriptors identified using this process are retained separately rather than seeking to combine them into a single parameter. Given that many physical properties will be influenced by a number of shape attributes, it is felt that it may be beneficial to model the response to each attribute separately using the appropriate shape descriptor.

6 Acknowledgements The authors gratefully acknowledge scholarship support for MLH through the Australian Research Council Small Grants Scheme. We also wish to extend our warmest thanks to Dr. Simon Iveson for helpful comments while proofreading this manuscript.

7 Nomenclature a p

A

length of the smallest possible rectangle that bounds the particle of projected particle image square root of projected area

AR b B B60 C0 Cav Cmax Cmax,90 Cmed Cmin Cmin,90 D FD p FF Fmed L L90 MD P P(l) NS S x l

aspect ratio breadth of the smallest possible rectangle that bounds the particle breadth (minimum Feret diameter) length of the Feret diameter 908 to B number of characteristic corners mean chord length maximum (longest) chord length length of the chord 908 to Cmax median chord length minimum (shortest) chord length length of the chord 908 to Cmin mean Feret diameter boundary fractal dimension square root of form factor median Feret diameter length (maximum Feret diamter) length of the Feret diameter 908 to L Martin×s diameter perimeter of the projected particle image perimeter estimate using step size l combination of conventional shape descriptors general term representing any conventional shape descriptor general term for any of the particle size measures step size used for perimeter estimate P(l)

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