Representing Particle-Size Distributions

Why the Weighting Basis Matters

A particle-size distribution is not a single list of diameters with one obvious average. It can describe the fraction of particles by count, the fraction of solid volume, or the fraction of solid mass found at each size. Those descriptions can look very different for the same powder: a small number of large particles may contain most of its mass, while numerous fine particles may contribute substantially to its exposed surface area. Since interfacial mass transfer depends on surface area, selecting a distribution or representative diameter without checking its weighting can alter a predicted dissolution rate.

The discussion below uses the interfacial mass-transfer result for spherical particles. The focus is how a distribution represents particle sizes and how its shape affects a rate prediction; the model is an initial-rate comparison, not a complete dissolution simulation.

From Number Fractions to Volume and Mass Fractions

For a discrete representation, let \(d_i\) be the diameter of size class \(i\), and let \(n_i\) be the number of particles in that class. The number fraction is \(f_{N,i}=n_i/\sum_j n_j\). Under a spherical-particle approximation, the volume of one particle is \(\pi d_i^3/6\). Thus its class volume is proportional to \(n_i d_i^3\), and the volume fraction is

If particle density is common across size classes, each class mass is its volume times that density, so \(f_{M,i}=f_{V,i}\). If density varies by class, the mass fraction instead uses \(n_i\rho_i d_i^3\), where \(\rho_i\) is that class's density. In that case, volume and mass weighting need not agree.

For any chosen weighting, a weighted arithmetic mean is the sum of diameters multiplied by their fractions. These means answer different questions. The number-weighted mean describes the average diameter of a randomly selected particle. The volume-weighted mean describes the average diameter when a random element of solid volume is assigned the diameter of its particle. Neither is automatically the best diameter for predicting dissolution.

Moments, Surface Area, and the Sauter Mean

Define the number moment of order \(k\) as \(M_k=\sum_i n_i d_i^k\). For spherical particles, total solid volume is \((\pi/6)M_3\), and total surface area is \(\pi M_2\). Their ratio is therefore

The diameter \(d_{32}\), called the Sauter mean diameter, is the diameter of a sphere with the same surface-area-to-volume ratio as the distribution. It is a useful summary for an initial dissolution-rate comparison because it preserves the ratio of total surface area to total solid volume. It is not generally equal to the number mean or the volume-weighted arithmetic mean.

For a fixed mass of solid with common density \(\rho\), total particle surface area is \(6M_{\mathrm{solid}}/(\rho d_{32})\). Under the assumptions of the interfacial mass-transfer model, the initial molar dissolution rate is \(k_m(C_s-C)\) times this surface area, where \(k_m\) is the mass-transfer coefficient, \(C_s\) the interfacial saturation concentration, and \(C\) the bulk concentration. If \(k_m\) and \(C_s-C\) are shared across the compared powders, their initial rates at equal solid mass are inversely proportional to \(d_{32}\).

Width and Multimodality

A distribution's width matters because surface area and solid volume scale differently with diameter: area scales as \(d^2\), while volume scales as \(d^3\). Consequently, two powders with the same ordinary mean diameter can have different \(d_{32}\) values. A single mean suppresses information about spread and tails; a multimodal distribution can be especially poorly summarized when a small-particle mode and a large-particle mode have very different contributions to number, volume, and area.

The relation to volume weighting makes the effect precise. The volume fraction in class \(i\) is \(f_{V,i}=n_i d_i^3/M_3\), so

Thus \(d_{32}\) is the reciprocal of the volume-weighted mean of reciprocal diameter. It is not the volume-weighted arithmetic mean, which is \(d_{43}=\sum_i f_{V,i}d_i\). Since \(1/d\) is convex for positive \(d\), Jensen's inequality gives \(\sum_i f_{V,i}/d_i \geq 1/\sum_i f_{V,i}d_i\), or \(d_{32}\leq d_{43}\). Equality holds for a single diameter; for a spread distribution, replacing it with \(d_{43}\) generally underestimates surface area per unit volume.

For a continuous number-weighted distribution with density \(f_N(d)\), the corresponding moments are \(M_k\) proportional to \(\int_0^\infty d^k f_N(d)\,{\rm d}d\), with \(f_N\) normalized to integrate to one. The same moment ratio \(d_{32}=M_3/M_2\) applies. A continuous representation can preserve smooth variation, while a discrete class representation can preserve distinct modes; either is useful only if its weighting and size range match the underlying measurement and the intended calculation.

Choosing and Interpreting a Representation

For initial surface-area-driven dissolution, \(d_{32}\) is more directly relevant than a generic average, but even \(d_{32}\) cannot describe every dynamic consequence of a distribution. The rate assumptions can fail if particle shape differs materially from spheres, if agglomeration changes accessible area, if size classes have different surface properties, or if the mass-transfer conditions vary across the powder. A changing concentration driving force or substantial particle shrinkage also means the initial-rate comparison cannot be carried forward unchanged over time.

A practical check is to ask whether the reported weighting basis matches the scientific question, whether density is effectively common, and whether the chosen summary preserves the quantity the model uses. If a number-based mean is substituted for \(d_{32}\), the result is not justified merely because both values have units of length. Likewise, a volume- or mass-weighted curve should not be treated as a number distribution. For a visibly broad or multimodal powder, retaining the distribution is usually more informative than asserting that one mean diameter represents all particles.