Why Percentile Diagnostics Matter
Particle-size measurements are often summarized by three values: D10, D50, and D90. These compact summaries are useful, but they do not describe every feature of a powder. Their interpretation depends on what is being counted or weighted, how the measurement was made, and whether the measurement covered the relevant size range. A precise-looking percentile can still be misleading if the sample was not representative or if particles remained clumped during measurement.
For dissolution modeling, a particularly important diagnostic question is whether the method resolves the small-particle tail. For particles of similar density and shape, smaller particles provide more external surface area per unit mass than larger particles. Thus, a modest fine fraction may matter even when it has little effect on D50. This tutorial focuses on evaluating particle-size data and its limitations, not converting a full distribution into dissolving surface area.
Laser diffraction commonly reports a volume-based distribution of equivalent spherical diameters inferred from an optical model. Sieving commonly reports a mass-based distribution of material passing specified apertures. These are not automatically interchangeable: the percentile refers to a different basis, and laser diffraction’s equivalent diameters depend on its measurement and inversion assumptions. When reading a result, record the method, weighting basis, units, and whether the reported cumulative curve is undersize (percent below a diameter) or oversize.
Reading the Percentiles and Their Spread
If the cumulative distribution is an undersize distribution, the usual ordering is \(D_{10}\leq D_{50}\leq D_{90}\). D50 describes the midpoint on the stated basis, not the arithmetic mean diameter. D10 and D90 show the locations of the lower and upper parts of the distribution, but they do not specify the full distribution’s shape. Two powders can share all three percentiles while differing in the tails or in the amount of material between them.
A simple spread measure is the span, \((D_{90}-D_{10})/D_{50}\). It is dimensionless and can help compare distribution breadths when measurements use the same method, basis, and processing. It does not reveal whether the distribution is unimodal, bimodal, or strongly tailed. The ratio \(D_{90}/D_{10}\) is another useful spread descriptor, provided both percentiles are reliable and positive.
Worked Example: Interpolating D50 and D90
A hypothetical laser-diffraction result reports an undersize volume distribution. The cumulative fraction is 38% at 20 µm, 72% at 45 µm, and 94% at 90 µm. Estimate D50 and D90 by linear interpolation in cumulative percentage against log diameter. This interpolation is appropriate only as a local approximation between the stated points; it assumes the reported cumulative curve is meaningful over those intervals.
For interpolation between diameters \(d_1\) and \(d_2\), use the fraction of the cumulative-percentage interval, \(f=(q-q_1)/(q_2-q_1)\), and interpolate linearly in log diameter:
For D50, the bracketing points are 38% at 20 µm and 72% at 45 µm. Thus \(f=(50-38)/(72-38)=12/34=0.3529\), and:
For D90, use 72% at 45 µm and 94% at 90 µm. The interpolation fraction is \(f=(90-72)/(94-72)=18/22=0.8182\), giving:
The resulting percentiles are approximately D50 = 26.63 µm and D90 = 79.3 µm. These estimates describe the stated volume-based curve; they are not number-based percentiles or exact values independent of interpolation method.
Why Resolving the Fine Tail Matters
The relevance of small particles can be understood with a limited geometric argument. For nonporous spherical particles of density \(\rho\) and diameter \(d\), one particle has area \(\pi d^2\) and mass \(\rho\pi d^3/6\). Their ratio is:
Under these simplifying assumptions, area per unit mass is inversely proportional to diameter. A particle ten times smaller therefore has ten times the external area per mass. This does not establish the total surface area of a real powder: shape, porosity, agglomeration, density variation, and the distribution itself matter. It does explain why a measurement that misses the fine tail may be inadequate for dissolution-related interpretation, even if D50 appears stable.
Worked Example: Checking Whether D10 Is Resolved
A hypothetical laser-diffraction method has a validated lower measurement limit of 3.0 µm for the powder and dispersion conditions used. The software reports D10 = 2.4 µm, D50 = 18 µm, and D90 = 60 µm. Assume the validation limit applies to this measurement and that values below it are not reliably quantified.
D10 = 2.4 µm is below the validated lower limit of 3.0 µm. The reported D10 therefore cannot be treated as a resolved percentile. It may reflect extrapolation or an unstable estimate at the edge of the instrument’s range. D50 and D90 are above the lower limit, but that alone does not guarantee they are reliable; the entire measurement and sample still require checks.
For scale, the ratio \(D_{90}/D_{10}\) calculated from the reported values is \(60/2.4=25\). This is only a provisional descriptive calculation: because D10 is not resolved, the ratio is not a trustworthy measure of spread. The appropriate report is that the fine tail is not adequately quantified by this method under these conditions, rather than presenting the ratio as definitive. For dissolution interpretation, the missing fine fraction is a material limitation because small particles can contribute disproportionately to external area per mass.
Sampling and Dispersion Diagnostics
A particle-size analyzer measures the portion presented to it, not an abstract perfectly mixed powder. Fine and coarse particles can segregate during handling, transport, or subdivision. A small aliquot may overrepresent one fraction, particularly if the batch is broad or contains aggregates. Replicate measurements should therefore begin with independently collected and prepared aliquots, not merely repeated readings of the same prepared suspension. Agreement across repeated readings of one aliquot does not rule out a sampling problem.
Dispersion is another potential source of error. If particles form agglomerates, an insufficiently dispersed measurement may treat clusters as larger entities. Increasing dispersion energy can reduce apparent size by separating agglomerates. But continued reduction in apparent size with increasing energy may also indicate that primary particles are being fractured. The goal is not to force the smallest possible result; it is to identify a preparation condition that disperses the sample reproducibly without changing the particles.
Worked Example: Separating Sampling Variability From Dispersion Effects
A hypothetical powder is tested using two independently prepared aliquots at a fixed, initially selected dispersion condition. Their reported percentiles are: aliquot A, D10 = 5 µm, D50 = 28 µm, D90 = 95 µm; aliquot B, D10 = 4 µm, D50 = 24 µm, D90 = 72 µm. Assume both aliquots used the same method, weighting basis, and validated measurement range.
The D50 difference is \(28-24=4\) µm relative to the average of the two medians, \((28+24)/2=26\) µm. The relative difference is \(4/26\times100=15.4\)% . For D90, the difference is \(95-72=23\) µm relative to the average, \((95+72)/2=83.5\) µm, or \(23/83.5\times100=27.5\)% . These discrepancies suggest meaningful aliquot-to-aliquot variability, especially in the upper part of the distribution. With only two aliquots, they do not identify the cause or establish a general precision estimate.
The next diagnostic is to prepare additional independent aliquots using a controlled sampling procedure and compare their results. Separately, measure aliquots over a justified range of dispersion conditions. If increasing dispersion energy consistently lowers D90 while the result reaches a reproducible plateau, agglomerate breakup is plausible. If size keeps falling without a plateau, or microscopy shows fractured primary particles, the condition may be changing the material. Sampling and dispersion should be investigated as distinct factors rather than attributing every shift to the instrument.
Method-Specific Checks and Failure Modes
For laser diffraction, check that the reported optical and dispersion settings are appropriate for the material and that obscuration or concentration is within the method’s validated operating range. An unsuitable optical model, multiple scattering at excessive concentration, or inadequate dispersion can distort the inferred distribution. Inspect the raw cumulative curve, measurement-range limits, and replicate results rather than relying only on exported percentiles. A smooth curve or several decimal places is not proof of accuracy.
For sieving, check that the sieve apertures bracket the sizes of interest, that the mass balance is acceptable, and that the procedure does not cause loss of fines or breakage. Material adhering to equipment, incomplete passage, or inconsistent shaking can affect the retained masses. Sieve results are binned by aperture, so the data may not support a finely resolved percentile without an explicit interpolation assumption. A reported D10 far below the smallest relevant aperture is not justified by the measurements.
In either method, a percentile near or beyond a measurement boundary deserves particular caution. A percentile calculated by extrapolating past the measured range is model-dependent, not directly observed. Also inspect whether the cumulative curve is monotone and whether the reported percentiles obey \(D_{10}\leq D_{50}\leq D_{90}\). Violations can indicate a reporting, sorting, or definition error. If distributions are multimodal, three percentiles may conceal distinct particle populations; inspect the fuller distribution and, where useful, independent images or measurements.
Common Mistakes and Practical Alternatives
- Calling D50 the mean. D50 is a median on the stated cumulative basis. It equals a mean only in special distributions, not by definition.
- Comparing unlike bases as if identical. A volume-based laser-diffraction D50 and a mass-based sieve D50 are not automatically equivalent. State the method and basis, and treat cross-method agreement as a question to test.
- Assuming three percentiles describe the whole distribution. D10, D50, and D90 do not uniquely specify tails or modes. Inspect the curve or bin data when those features matter.
- Treating dispersion energy as a harmless adjustment. More energy can break agglomerates, but may also fracture particles. Use evidence such as a reproducible plateau and particle imaging to distinguish these possibilities.
- Accepting a precise value outside the validated range. Numerical precision does not make an unresolved percentile valid. Report the range limitation and avoid interpreting extrapolated values as measurements.
Laser diffraction offers a continuous inferred distribution and is often convenient for broad ranges, but its results depend on optical and dispersion assumptions. Sieving gives direct mass fractions retained or passed through apertures and can be useful for coarser particles, but its resolution is limited by the selected sieve series and procedure. Neither method is universally superior. The appropriate method is the one validated for the material, relevant size range, and intended interpretation; using complementary methods can help investigate discrepancies, while differences in weighting basis must remain explicit.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- A volume-based undersize curve places 10% of the measured volume at or below 6 µm. What does this say about D10, and what does it not say about particle number?
- A method’s validated lower limit is 2 µm, but the software reports D10 = 1.5 µm. How should this percentile be interpreted?
- Why might D90 decrease when dispersion energy increases? What additional evidence would help distinguish deagglomeration from particle fracture?
- Between two hypothetical percentiles, the cumulative fraction rises from 40% at 10 µm to 80% at 40 µm. Using log-diameter interpolation, estimate D60.
- Why can D10, D50, and D90 be insufficient for assessing a multimodal powder?
Solutions. (1) D10 = 6 µm on the stated volume basis: 10% of measured volume is at or below that diameter. It does not state that 10% of particles by number are below 6 µm. (2) It is below the validated range and should be treated as unresolved, not as a trustworthy measured percentile. (3) Increased energy may separate agglomerates, lowering apparent size; continued reduction, lack of a plateau, or imaging evidence of broken primary particles would raise concern about fracture. (4) The interpolation fraction is \(f=(60-40)/(80-40)=0.5\). Thus \(D_{60}=10(40/10)^{0.5}=20\) µm. (5) Three percentiles do not determine the shape or number of modes, and a multimodal distribution can have the same three percentile values as a distribution with a different shape.