The PSD of each sample was evaluated using a Sympatec Helos/Rodos laser diffraction particle-size analyzer (Sympatec Inc.,
Lawrenceville, NJ) with dry dispersion capability. Small bulk samples (0.5–5.0 g) were introduced to the dry dispersion feeder
system using a vibratory feed tray. The dispersion pressure was optimized (0.5–4.0 bar) to reasonably break up any loose agglomerates
in the sample. The laser diffraction system, equipped with the appropriate lens for the sample's particle-size range was used
to measure the PSD of the sample in 31 channels. The cumulative PSD and common PSD statistics (D10, D50, D90, D[4,3]) were
calculated using the Sympatec Windox 4.0 software.
Powder flow testing.
The powder-flow behavior of each sample was determined using a Schulze ring shear tester model RST-XS [Dietmar Schulze, Wolfenbuttel,
Germany]. In general, the RST-XS medium-sized annular cell (31.37 cm3) was used to conduct each yield locus test because it is suitable for most pharmaceutical testing where bulk quantities are
often limited. Where appropriate, the large-sized cell was used for powders with particles larger than 500 μm. Each sample
was equilibrated in a closed environment at 20 ± 2 °C and 50 ± 2% relative humidity for 16–24 h before testing to minimize
the effects of moisture uptake. Although the influence of humidity on powder-flow properties or PSD were not explicitly explored
in this work, the adhesion force between contacting particles is known to often increase with absorbed water, thus inhibiting
flow performance (5).
The equilibrated powder was spooned into the test cell and carefully leveled with the top of the annulus to form an unconsolidated
powder bed with minimal voids. The cell was placed on the driving axle of the tester and fitted with the lid to which the
shear force tie rods and normal loading rod were connected. The steady-state shear stress of each powder was measured as the
powder was subjected to a series of applied normal loads ranging from 1 to 2.6 kPa. Before each load, the powder was sheared
to a steady state at 4 kPa, which helped create a consistent initial powder bed condition (i.e., bulk density). From the yield
locus plot, the ratio of the principle consolidation stress (σ1) to the unconfined yield strength (FC) was used to calculate the FFC value.
The laser-diffraction analysis provided 31 channels of PSD for each powder, which allowed detailed PSD to be constructed.
The 10th, 50th, and 90th percentiles of the distribution were calculated for each material. In addition, the VMD (D[4,3])
calculated by the Sympatec Windox software was used as an input for the models. These four parameters were selected because
they are commonly used PSD statistics. The response variable, FFC, was subdivided into four categories related to the powder's
relative flow performance based on the laboratory's experience with powders (see Table II).
Table II: Rating categories for flow-function coefficient (FFC) values.
Two statistical methods were applied for correlating the PSD and FFC values: partial least squares (PLS) and tree-based regression.
The PLS method generates a linear model describing the FFC in terms of the PSD values. The output is a linear model containing
weighted linear combinations of PSD percentiles, which is used to calculate the FFC value. Preliminary statistical analysis
of the data suggested a logarithmic relationship between the FFC and PSD values. This is not surprising because many properties
of bulk powder and compacts are related via logarithmic relationships (e.g., compact tensile strength and solid fraction) (1). This is consistent with other research
demonstrating empirical relationships between powder-flow parameters such as flow function or angle of internal friction and
a single particle-size parameter. Kohler and Schubert report a model for describing a relationship between the inverse flow
function (1/FF) and the median particle-size diameter (D50) for fine alumina powders (5). Podczeck and Miah found that the
angle of internal friction of unlubricated powders is dependent on particle size and shape (6).
A tree-based model (7) was also used to generate splitting criteria for the powder-flow performance category based on one
or more of the PSD statistics (e.g., if distribution statistic X is < Y μm, then powder flow is classified as poor). Tree-based modeling is often used in data mining investigations such as this
where the most important inputs for describing a response must be identified.
In each method, four common measurements provided in particle-size analysis (D10, D50, D90, D[4,3]) were used to build the
models. Initial results showed that there was no significant advantage to using additional PSD statistics to parameterize
the model. The four-parameter model, therefore, was pursued; these distribution statistics are typically readily available
as outputs from particle-size analysis.