Elsevier

Powder Technology

Volume 383, May 2021, Pages 348-355
Powder Technology

Powder grinding and nano-particle sizing: Sound, light and enlightenment

https://doi.org/10.1016/j.powtec.2021.01.059Get rights and content

Highlights

  • The particle size range of interest must guide the choice of base in the PSD curve.

  • The volume-based median particle size tolerates changes in measurement methods.

  • The DLS method is vulnerable to agglomeration in polydisperse size distributions.

  • Despite high solids concentration EAS promotes dispersion even for very fine sizes.

  • X-ray diffraction data strongly agrees with BET surface area-based particle size.

Abstract

In this work, a sub-micron quartz powder produced by high energy grinding was used to explore the information contained in the particle size distribution (PSD) curves obtained by electro-acoustic spectroscopy (EAS) and dynamic light scattering (DLS). Results show that although the customarily volume-based PSD curve and its median value (d50 = 135 nm) are more amenable to differences in measurement phenomena and sample preparation methods, the area and number-based representations are more sensitive to the presence of finer particles (35 and 65 nm, respectively for the number and area-based curves obtained by EAS). These values were supported by the crystallite size calculated from X-ray diffraction (d = 44 nm) and the equivalent spherical diameter calculated from specific surface area measurements (d = 43 nm). Thus, a reliable description of the particle size of a powder requires the use of complementary techniques, suggested by the envisioned application or the particular stringency of a given processing step.

Introduction

Once upon a time, there was one single way of measuring the size of a particle, big or small, grain or gravel. Everybody had the sieve of their choice, that they could shake at will, and could see that particles that were larger than the space between the threads of the sieve's mesh were retained, while those that were smaller passed through. So, that space was taken as the size of the particles, no matter how irregular, and everybody was happy.

But then, as technology crept up into man's everyday life, people became aware that particle size and shape were important beyond the number that represented it and had deep implications on the way powders flow (rheology) and interact (reactivity). Changes in the starting particle size distribution could be responsible for important changes in the processing behaviour and, in turn, affect the final properties of manufactured products [1,2].

Powders of small particle size present a large surface area, which, for instance, is desirable in the photocatalytic processes used to remove contaminants from drinking water [3], in the desulfurization of commercial fuels [4], and to improve the efficiency of sunscreens and their appearance (texture, colour, lustre) [5]. Powder metallurgy also makes use of very fine nano-sized particles to improve particle packing and reduce porosity in the final product [6]. And, in the ceramics industry, oxide powder particles of varied shape and size, from the millimetre down to the nanometre, are omnipresent in the aqueous slurries that are used in the preparation of pastes, slips, glazes, inks and pigments [7].

So, the road to happiness became a winding path and everybody realised that, to make further progress (and profit), particle size needed to be correctly measured and size distributions adequately described. And the quest began to find the means, in a simple and easy to interpret way, to ascribe a representative size and size distribution to any group of particles of different sizes, possibly with irregular shapes (far from spherical) and rough (not smooth) surfaces.

The obvious starting point was that mesh sieve used originally. Fig. 1 illustrates how a particle irregular in shape would pass through the space left in between the parallel threads in the mesh. That distance can be taken as the equivalent Feret diameter [1] and all particles that pass through have a size lower than that.

But there are other, perhaps more convenient, ways to assign a specific size to an irregularly shaped particle. Geometrically speaking, the sphere is the ideal shape because it can be described by a single parameter, its diameter. Easier said than done, because different equivalent spherical diameters are obtained depending on which particle property is chosen, as also shown in Fig. 1. If the size (diameter) of a particle is doubled, its linear Feret equivalent increases proportionally, whereas if its surface is doubled the equivalent spherical diameter increases by ~41% (the surface area increases with the square of the diameter) and by ~26% if its volume is doubled (the volume increases with the cube of the diameter) [8]. And there are cases (e.g. rods, platelets) in which the use of a sphere as a model will produce a gross estimate of a dependable particle size [1,2,[8], [9], [10]].

Nevertheless, the use of a model sphere has another important advantage, which is that the equivalent spherical diameter is always the same regardless of the particle's shape, orientation or movement. And so, the equivalent spherical diameter has become the universal attribute to describe the size (and size range) of any powder particle system. In characterization methods in which the particles adopt one single orientation in space, such as in filtering, sedimentation or microscopy, there will be one single equivalent spherical diameter. But there are applications sensitive to particle orientation (e.g. those involving slips, emulsions, etc.) and so, fostered by the increasing allure of ever decreasing particle sizes, new advanced measurement techniques were introduced in which particles are suspended and kept moving, hence with translation and rotation (i.e. dispersed), and the measuring equipment uses the way they interact (scattering) with light (e.g. laser diffraction) or sound (e.g. electro-acoustic spectroscopy) and views them as a cloud.

Although size is a continuous variable, it is measured in size classes, such as the mesh sizes between two consecutive sieves in a sieving set. As the range (width) of each class decreases, the representation gets closer to a continuous distribution and a cumulative distribution curve can be, and often is, calculated. The directly-measured weight basis found in a set of sieves, or the directly-measured number basis used on the microscope, however, is uncommon for most of those modern analysis techniques. Instead, an endless number of different equivalent spherical diameters are recorded (statistical diameters), from which a particle size distribution (PSD) can be calculated and plotted as a probability, or frequency, distribution curve, most frequently on a volume basis (Fig. 2). The cumulative curve literally represents the percentage of the sample with particle sizes below (finer than) or above (coarser than) each specific value and the frequency curve can be regarded as the derivative (differential curve) of the cumulative curve [2,8,9].

Jointly, those curves provide, in a simple and concise way, the major attributes to describe the particle system by, namely, the median (midpoint or middle value, obtained from the cumulative curve), the modes (most frequent value or values, or peak diameters in the differential curve) and the dispersion range (the variability) of the size distribution [1,8,9].

Although a normal (Gaussian) distribution has identical values for mode, median and mean, in real PSD curves they do not necessarily have the same value and two curves with completely different shapes can have the same values for mode, median and mean. The median is less affected than the mean by the distribution fringes (outliers) and, as a description of the distribution, is sturdier than the mean [2,8,9].

In the example illustrated in Fig. 2 two modes might be identified in the distribution (two clear peaks in the differential curve, at 66 and 194 nm; these are the most frequent particle sizes) and the distribution would be said to be bimodal [1]. The median divides the sample in two halves, 50% of the sample is finer, and the other 50% is coarser than the median value, which is customary represented by d50 (read from the cumulative curve, at 135 nm) and frequently misleadingly called average particle diameter. In the same line of reasoning, the values of d10 and d90 are used jointly with d50 to express the dispersion range of the PSD.

The mean is the weighted centre of the differential curve and, as a numerical mean, requires the foreknowledge of the number of particles. As such, as most modern particle sizing techniques do not count or assign size to individual particles, it cannot be extracted from the PSD curve and has to be calculated. There are ways of bypassing the need to know the number of particles and obtain a meaningful mean, namely from the ratio volume-to-surface of the particles (e.g. as described in a number of technical standards [10,11] and other relevant literature [1,9]). Various such algorithms are frequently used in the dedicated software in the measurement equipment.

The importance of the basis (e.g. number, area, volume, weight) used to construct the distribution curves can be highlighted by considering a sample that contains ten spheres from the same material, whose diameters vary from 1 to 10 μm. Based on the number of particles with the same diameter, one for each, they all have the same relative importance (1/10 or 10%). The distribution would be considered broad and both the mean and the median particle sizes are 5.5 μm. However, if the surface area or the volume are the basis of the calculation, the largest sphere is the most important, as the area depends on the square of the diameter and the volume (and the weight, related to volume through the density) on the cube of the diameter. In other words, one single particle of 10 μm occupies the same space (volume) as one thousand particles of 1 μm and has the same surface area as one hundred particles of 1 μm. In both cases, the differential curve peaks at 10 μm (mode) but mean and median have different values: respectively 6.2 μm and 7.8 μm for the area-based distribution, and 6.7 μm and 8.3 μm for the volume-based distribution.

If the measurement technique is based on light scattering, very popular among those with adequate resolution at the sub-micron size (nanoparticles, oligomers, proteins, viruses, etc.), the equipment records the intensity of the light scattered by the particles (which depends on d6). In the example above, the light scattered by the smallest particle (1 μm) will be 0.0001% of that scattered by the largest one (10 μm) and both the mean and the median get further displaced towards higher sizes.

The dynamic light scattering (DLS) method uses very diluted suspensions (from ~0.0001 to 1.0 vol%, controlled by the measured signal/noise ratio for the particular material being characterized) of particles in Brownian motion and estimates their sizes from the particles diffusivity (velocity) determined by the Doppler effect (hence, the method is also known as photon correlation spectroscopy, PCS). Then some software is used to translate the measured data (intensity) into volume, surface area or number. Regardless of the degree of mathematical manipulation, the particle size distribution is overestimated in favour of larger particles [[12], [13], [14], [15]].

On the other hand, the electro-acoustic spectroscopy (EAS), based on the interaction of particles with sound waves, is capable of characterizing concentrated suspensions and emulsions (~40 vol%) using the measured attenuation (energy dissipation) of incident sound at specific wave frequencies caused by the presence of particles [[16], [17], [18]]. For such measurement techniques the likely difference between the densities of particles and suspending fluid is an extra source of chagrin. The degree of attenuation due to that density difference depends on the volume of each particle and, therefore, the PSD curves can be best presented on a volume (or weight) basis [[16], [17], [18]]. Due to the interaction with sound waves, particles are continuously kept dispersed, but this technique is also very sensitive to changes (inaccuracy) in the solids content, as well as to entrapped air bubbles.

Thus, although there is no absolute single value for the measurement of a particle's size, it is obvious that PSD curves should only be compared when determined based on the same assumptions, i.e. the same particle property. Nevertheless, differently based PSD curves bring to evidence details of the distribution that are only apparent in a specific basis. For instance, if the emphasis of the characterization is on the smallest particles (e.g. contamination), a number-based PSD curve might be more realistic, or an area-based PSD, which also favours the finer particles. The latter will certainly be more adequate if the envisioned application depends on surface area (e.g. catalysis). On the contrary, the customary volume-based PSD curve is ideal when undesired large particles are to be detected (e.g. additive manufacturing or ink-jet printing [19]). These might be breadcrumbs, but they still show the way.

Rather than a stand-alone technique, any size measurement method is best used as a complement to others. Especially in the characterization of polydisperse powders more than one technique should be used, despite the different values that each technique might produce. The characteristics of the powder and the way they interfere with the particular measurement technique may provide explanations for those differences [18]. In the event that those values are found to be similar, a greater confidence can be associated to the characterization results [2,8] and passed on to the whatever purpose the characterization was carried out for.

This having been clarified, the problem remains because the average mathematical brain readily interprets median, mean and mode as average, and average as arithmetic (population) mean, which is by default based on number. Nowadays, as the winding path to enlightenment takes us deeper into the Nanosizeland, powerful technology is looming over everywhere and casting an unsuspecting longer shadow, hastening the pace but shortening the sight, so it is very easy to take the tree for the forest.

Going back to the increasing allure of ever decreasing particle sizes, one common purpose of powder size characterization is the evaluation of the grinding limit or grinding efficiency. Grinding efficiency can be defined in many a way but it is usually understood as the relationship between the energy supplied to the grinding operation and the fineness of the ground product. When the amount of final product is of interest, the specific grinding energy (J/g) can be used to gauge the grinding efficiency, along with the grinding time. For a given final particle size and constant grinding parameters, the grinding efficiency increases when the grinding time or the specific grinding energy decrease.

Studies can be found in the literature in which the grinding efficiency is expressed through the PSD, generally in terms of the volume-based median particle size (d50) as a function of the specific grinding energy [20,21]. The graphic trend observed is that of a descending curve, as the expended energy increases as the particle size decreases. Frequently, however, the calculation basis is not reported, which, in studies comparing for instance grinding media materials (or sizes) or wear, or mill velocity, leaves the reader to assume that, at best, all results are referred to the same basis. Nevertheless, consensus has it that information valuable for the grinding process (e.g. efficiency, technical limits, even operational costs) can be withdrawn from the analysis of PSD curves regardless of the basis in which they are constructed. Alternatively, the fineness of the ground product can be expressed in terms of its specific surface area (BET). But measuring specific surface area is, yet again, threading murky waters, as the method is very sensitive to sample preparation, particularly sample degassing conditions. Ultimately, the reported grinding efficiency is affected by the characterization of the ground powder, depending on which technique (BET, PSD) is selected to do that [22,23].

In Nanosizeland, this is when experimental results might stop making sense and all hell breaks loose. Particle size is far too important to let hard to interpret results go unnoticed and other measurement techniques are called for [24,25]. One such technique is the evaluation of the additional peak broadening (i.e. beyond inherent instrumental peak width) in X-ray diffraction (XRD) due to size effect, which can be used to estimate crystallite sizes of less than 1 μm and as low as 1 nm [[26], [27], [28]]. In such cases, particularly for non-ductile, brittle materials like ceramics, the broadening effect due to size is more significant than that of microstrain, which can safely be ignored.

Routine X-ray diffraction work requires crystallite sizes larger than 1 μm (individual particles may contain several crystallites), so that the inherent instrumental peak width can be dealt with by reliable simulation (fitting) of experimental XRD peaks (Gauss, Lorentz, Voigt, etc.). However, for crystallite sizes within “size effect range” (< 1 μm), size estimation from peak broadening requires a number of geometrical considerations and rather laborious fitting techniques [28] and, to make things worse, can be comparable to the inherent instrumental broadening. Such size effect is well-known and frequently used, but the fact that it is related to “additional” broadening of the X-ray peaks, despite being duly referred to as such, is also often neglected. Thus, when this is not accounted for, crystallite sizes determined in this way for common nano-powders solely from their corresponding X-ray diffraction patterns frequently return a value in the range of 15–30 nm, whatever the powder [25,29,30].

The best way, and maybe the easiest way, to guarantee that the instrumental broadening is indeed accounted for is to use, as reference in the calculation, the breadth of the peak generated by the same instrument from a specimen (of the same material) that exhibits no broadening beyond inherent instrumental peak width (i.e. for which the crystallite size is safely larger than 1 μm). Then, it is just a matter of using one of the various calculation methods based on Scherrer's Eq. (1) to obtain the average crystallite size, τ.τ=Kλβτcosθ

In Eq. (1), K is a dimensionless number close to unity (a shape factor ranging from 0.81 to 1.39), λ is the radiation wavelength, θ is the Bragg angle and βτ is the peak additional broadening, commonly measured as the peak breadth at half height (called the full width at half its maximum intensity, FWHM, in radians) [27,31]. When using the XRD of the reference sample and the FWHM method (for which a value of K of 0.90 is usually taken [26]) to calculate the true additional broadening, one just needs to replace βτ in Scherrer's equation by the difference, for each pair of equivalent peaks in the corresponding XRD patterns, between the values obtained for the actual sample and for the reference.

Although numerous studies can be found in the literature, on grinding and on powder synthesis for a variety of applications, that are guided by the final product fineness, the need remains for a sturdier methodological approach to help interpret the effect of process variables in the characteristics of the final product. This work describes such an effort, carried out on fine ground quartz particles using the corresponding PSD obtained by dynamic light scattering (DLS) and electro-acoustic spectroscopy (EAS), bracketed by the information extracted from electron microscopy, X-ray diffraction and gas absorption, used as additional characterization techniques.

Section snippets

Experimental

The raw material, supplied by Brasilminas, was a natural quartz powder from Bahia, Brazil, 99.7% pure with 2.65 g/cm3 density [32]. This powder was wet ground in a zirconia lined stirred media mill (Netzsch, Labstar LS01) using 400 μm diameter Yttria-stabilized zirconia grinding spheres (75% of the chamber volume) and deionized water (milli-Q, Direct-Q, Merck, 0.05 ± 0.01 μS/cm electric conductivity at 25 °C and pH 9.0). For better grinding efficiency [21,33], suspensions were prepared with

Results and discussion

The as-received (AR) quartz powder presented, when characterized by EAS, volume-based particle sizes d50 ≈ 3.5 μm, d90 ≈ 11 μm and d99 ≈ 25 μm (Fig. 3). Fig. 3 also shows how the particle size was reduced upon grinding and, more importantly, how very different are the PSDs for the same powder, depending on which basis is considered in the calculation.

The customary volume-based differential curve, which was already shown in Fig. 2, is clearly shifted towards higher size values. The corresponding

Conclusions

Different particle size characterization techniques, with their own limitations and complexities, use different interaction phenomena and rely on different powder properties. Different representation bases (number, area and volume of particles) highlight different particle size ranges. In this work, the characterization of a sub-micron quartz powder produced by high energy grinding was used to explore the information contained in the PSD curves obtained by EAS and DLS and represented in those

CRediT authorship contribution statement

W.F. Camargo: Investigation, Data curation, Methodology, Validation, Writing - original draft. P.Q. Mantas: Visualization, Formal analysis, Writing - review & editing. A.M. Segadães: Methodology, Visualization, Writing - review & editing. R.C.D. Cruz: Conceptualization, Formal analysis, Methodology, Resources, Supervision, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The general financial support received from CAPES/PROSUC (tuition fees) and the Secretaria de Desenvolvimento Econômico, Ciência e Tecnologia from the Rio Grande do Sul State, Brazil (SDECT/RS) is gratefully acknowledged. Authors very much appreciate the assistance from the technical staff of the Ceramic Materials Institute at the University of Caxias do Sul (IMC-UCS) and from the senior microscopist Marta Ferro at the Aveiro Institute of Materials (CICECO).

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