Abstract
Settling calculations can help choose suspension variables for testing. This note uses an idealized particle model to examine size, density and viscosity, then identifies the checks needed for real microspheres. The calculation does not predict stability in a concentrated paste.
Estimate the dilute settling timescale
For an isolated sphere settling slowly in a Newtonian liquid, the Stokes estimate is v = Δρ g d²/(18 μ), where Δρ is the particle-liquid density difference, d is diameter and μ is dynamic viscosity [1]. It is a screening model, not a model of a concentrated, yield-stress cement.
Worked example: assume Δρ = 2000 kg/m³, μ = 0.10 Pa·s and g = 9.81 m/s². A 20 µm sphere gives v = 4.36 µm/s, equivalent to approximately 15.7 mm/h. Doubling diameter to 40 µm multiplies the estimate by four. Tenfold viscosity gives one-tenth the velocity under these assumptions.
The example directs the first experiments: compare size, liquid viscosity and rest time separately. Sintered spray-dried HA microspheres can also change surface area, pores and aggregation [2]; a common diameter does not guarantee common behavior.
| Illustrative model | Calculated settling | Interpretation |
|---|---|---|
| 20 µm; 0.10 Pa·s | 15.7 mm/h | Reference model point |
| 40 µm; 0.10 Pa·s | 62.8 mm/h | Fourfold diameter-squared effect |
| 20 µm; 1.0 Pa·s | 1.57 mm/h | Tenfold viscosity effect |
| Concentrated paste | Do not use these values | Measure settling and yield behavior |
Test the real suspension
1. Inspect the particle interior
Compare intact spheres with cross-sections or fractured particles. Hollow and porous spheres can have different effective density and liquid uptake from dense spheres. Determine whether mixing or extrusion breaks the spheres into a different size distribution.
2. Follow concentration through the container
Use a fixed fill height and record the settled layer and supernatant over the relevant handling period. Sample top and bottom to compare solids content. Visual clarity alone can miss a concentration gradient that changes the delivered composition.
3. Check reversibility and delivery
Apply a defined redispersion motion after the hold period, then compare first and last delivered portions. Increasing viscosity is helpful only if the suspension remains mixable and deliverable. Assess force and composition together; easy extrusion of particle-poor liquid is a failure.
Development decision
Use the simple model to prioritize variables, then select the formulation from measured settling, redispersion and delivery behavior. If particles fracture, qualify the post-mixing particle population as well as the original powder.
References
[1] Jillavenkatesa A, Dapkunas SJ, Lum L-SH. Particle size characterization. Gaithersburg (MD): National Institute of Standards and Technology; 2001. NIST Special Publication 960-1. doi:10.6028/NBS.SP.960-1.
https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication960-1.pdf
[2] Wang AJ, Lu YP, Zhu RF, Li ST, Xiao GY, Zhao GF, et al. Effect of sintering on porosity, phase, and surface morphology of spray dried hydroxyapatite microspheres. J Biomed Mater Res A. 2008;87(2):557-562. doi:10.1002/jbm.a.31895.
https://pubmed.ncbi.nlm.nih.gov/18306315/