Lucas-Washburn Relation

Mechanistic Action

The theoretical model assumes that the complex void structure of paper can be represented by simplified cylindrical capillaries. When liquid contacts the paper, the capillary pressure pulls the fluid into these pores, resisting the viscous force of the liquid itself. In accordance with the model, the penetration rate decreases over time as the liquid travel distance increases, creating a square-root dependence on time.

Highly porous coatings with small capillary radii generate high capillary pressures, which can draw liquid in very quickly over short distances. Coating formulators use this relationship to choose the size of pigment particles, like calcium carbonate and clay, to create a pore network that matches the ink viscosity. This careful selection ensures that the printed image remains sharp and vibrant.

Performance Threshold

Applying this equation requires acknowledging the non-ideal nature of paper structures, which deviate from straight cylinders due to pore tortuosity and varying pore diameters. Despite these limitations, the formula provides a valuable guide for formulating coatings with specific pigment and binder ratios. Controlling the pore structure ensures that ink solvent penetrates at a rate that allows the ink to set without causing dry-back defects.

This maintains the desired balance between print gloss and ink drying times.