Guidelines to adjust particle size distributions by wet comminution of a bioactive glass determined by Taguchi and multivariate analysis

Abstract

A quaternary bioactive glass of high silica content was used as a model system for flexible design of powder characteristics obtained by ball milling. The dependence on milling time was used to demonstrate consistency with the expected correlation between comminution and cumulative kinetic energy of impacting balls. Additional experiments were based on Taguchi planning of simultaneous changes in milling time, balls-to-powder ratio and ethanol-to-powder ratio, with ethanol used as a process control agent (PCA) to prevent agglomeration and to seek greater flexibility in the design of powder distributions. Plausible physical mechanisms allowed us to obtain improved fitting by multivariate analysis, based on log-log scales. Normal distribution was found to be well-suited to describe the actual particle size distributions, which are very positively skewed. Weibull distributions provided good fitting, mainly by considering the three different contributions from particles in small, medium and large size ranges. These contributions are affected differently by ball milling parameters, as demonstrated by finer analysis. This yields suitable conditions for a flexible design of asymmetric powder size distributions (i.e. bimodal or skewed), in addition to a decrease in average particle size - both highly significant factors when designing glass-ceramic powders for robocasting and additive manufacturing.

Introduction

A wide variety of technologies require powder comminution to decrease mean particle size (PS), as well as controlled particle size distribution (PSD) and particle shape [1]. For example, powder processing may be needed to allow flexible geometric design of biomaterials, and to optimise their microstructural features which determine bioactivity. PS and PSD are also key parameters in optimising the rheological properties of ceramic suspensions [2], as well as specific particle size distributions for less common powder processing methods, such as additive manufacturing [3], robocasting or freeze casting [4]. Optimised green packing may even require suitable combinations of grain size ranges, such as bimodal distributions with a fraction of coarse particles for designed packing [5], or to prevent collapse of intended porosity [6].

Although narrow PSDs are usually more suitable for improving sintering ability than wider PSDs [7], other processing methods may yield different properties, depending on particle size distribution. For example, selective laser melting [8] may yield denser materials with wide PSDs, whereas a narrower PSD provides final products with higher tensile strength or greater hardness. PSD may also range from unimodal to multimodal. In a bimodal mixture of powders, small particles may fill in the porosity between the larger particles [9], leading to production of a green body with improved green density and flowability [10].

The asymmetry (skewness) of PSD may also be an important parameter in green processing, possibly playing a greater effect on green packing density than the polydispersity of particle distribution [11]. Thus, the normal distribution is insufficient [12], and one may need alternative functions such as the Weibull distribution [13], based on the analogy between powder comminution and fracture statistics of brittle materials.

Particle size is expected to decrease with increasing milling time, at least up to a comminution limit, when it is no longer possible to utilise the stored energy in the particles to initiate and propagate further cracking, as expected for the transformation from brittle to ductile behaviour, or when changes occur in particle shape rather than size reduction [14]. Ductile metallic powders may show even more complex dependence on milling time with initial coarsening, before the onset of effective comminution [15]. Thus, powder distributions and mean particle size with milling depend on a variety of parameters [16], possibly involving interactions of different parameters such as time, ball-to-powder ratio or stirring rate [17].

Nevertheless, one may assume a generic power law dependence, based on literature [16]:

$$D_m\propto t^{-n}$$

where D_m = particle size and t = time, as expected for a correlation between the energy required to create fracture surfaces, and the cumulative kinetic energy of impacting balls. Fracture surface and corresponding energy should vary with the reciprocal square of particle size ${\left(D_m\right)}^{-2}$, and considering the expected dependence of cumulative kinetic energy on milling time, the frequency of impacts (f) and number of balls per unit mass of powder (N_b), one obtains ${\left(D_m\right)}^{-2}={\left(D_{m,o}\right)}^{-2}=k{N}_b\mathit{ft}$, or:

$$D_m={\left[{\left(D_{m,0}\right)}^{-2}+{\mathit{kN}}_b\mathit{ft}\right]}^{-1/2}$$

Where k is a kinetic constant related to the mechanisms of fracture at the grain size level. After the initial stage, this is expected to evolve into a secondary regime:

$$D_m\approx {(k{N}_b\mathit{ft})}^{-1/2}$$

Indeed, this may also depend on other structural or mechanical changes in the milled powders, eventually leading to a comminution limit after a sufficiently long time.

Eq. (3) also predicts similar power laws for the dependence of mean size on time and ball-to-powder ratio (r_b). However, one cannot exclude efficiency losses by heat dissipation or other comminution limitations. Thus, the effective exponents of the power law dependences (n₁ and n₂) may deviate from the ideal − 1/2 value, i.e.:

$$D_m\propto {t^{n_t}(r_b)}^{n_b}$$

Process control agents (PCAs) may also exert significant effects on wet milling. For example, Baheti and co-workers reported that a unimodal particle size can be produced using wet milling rather than dry milling [18]. Thus, powder characteristics obtained by ball milling with a dispersing liquid may also depend on type and quantity of dispersing liquid [19]. PCAs may contribute to prevent powder agglomeration [17], to avoid the excessive welding of ductile powders rather than their fracture [20], or even to decrease the surface energy, thus assisting the formation of new fracture surfaces and a corresponding decrease in mean size. However, this latter effect is rarely found, except possibly on addition of a suitable surfactant [21]. The effects of increasing liquid-to-powder ratio are complex, potentially combining effects on viscosity with a risk of slurry flooding for excessive liquid quantities [22], damping the kinetic energy of the impacting balls, and possibly even structural changes or phase decomposition [23].

The nature of PCAs may be diverse (anionic, cationic, nonionic or zwitterionic), also implying differences in adsorption mechanisms in milled powders, with impact on the rheology of concentrated powder suspensions and their milling [24]. Heat dissipation may also interfere with PCA additives, mainly those with low boiling temperatures (e.g. methanol, ethanol). Thus, effects of the PCA and its quantity (r_a = PCA to powder ratio) must also be taken into account

$$D_m\propto {t^{n_t}(r_b)}^{n_b}{(r_a)}^{n_a}$$

Thus, the main purpose of this work is to model wet milling, using a bioactive glass as a representative material; this should allow one to draw guidelines to describe the effects of the main elements, namely time, balls-to-powder ratio and ethanol-to-powder ratio, with ethanol added as PCA. The resulting effects include changes in mean particle size, broadness of particle size distributions as described by standard deviation, and their asymmetry, described by a skewness parameter. This attempt to describe deviations from normal distributions was motivated by expected effects on rheology and flowability of milled powder suspensions [25], with corresponding impact on ceramic processing [26]. Detailed investigations were performed by the authors in a previous work “Robocasting: prediction of ink printability in sol‎–‎gel ‎bioactive glass”, which showed that the increase of balls-to-powder ratio led to an increase of the zeta potential, and of the dispersion of the suspension [26]. However, there is a clear general trend for the apparent ‎viscosity to increase with particle size reduction. Moreover, the elastic modulus (G´)‎ can be increased for finer particles by increasing balls-to-powder ratio [27].