The effect of surface heterogeneity on pseudo-line tension and the flotation limit of fine particles

The contact angles θ for water at methylated quartz surfaces were measured using the sessile-drop technique to determine the advancing contact angle, and using a captive-bubble technique to examine the effect of bubble size on contact angle. No linearity between cos θ and $\text{1}\text{r}$ where r is the drop base radius, was observed for these systems as would be expected for an ideal system. In fact the pseudo-line tension decreased with decreasing bubble size. Also, the degree of quartz methylation effected a change in the pseudo-line tension. The pseudo-line tension increased from 0.4·10⁻⁶ N to 3.3·10⁻⁶ N with an increase in the fractional coverage of trimethylsilyl groups from 0.14 to 0.51 for large bubbles (bubble base diameter d > 0.34 mm), whereas the pseudo-line tension decreased from 2.2·10⁻⁷ N to 0.8·10⁻⁷ N with an increase in fractional coverage for small bubbles (bubble base diameter d = 0.06–0.2 mm).

The flotation limit of fine particles has been re-examined based on the effect of bubble size on contact angle, and a new surface chemistry-limited relationship describing the minimum particle size which can be floated is proposed:

where r_c is the critical bubble (drop) radius for which there is no effective attachment between solid surface and dispersed phase, 7_LV is the interfacial tension at the liquid-vapor interface, Δp is the density difference between the particle and the liquid and V is the bubble ascent velocity.