What is the Contact Angle?

Figure 2-1 shows drops of two different liquids (one wetting, one non-wetting) resting on a flat solid surface. It also shows a drop of a wetting fluid entering a pore in the solid (from the side so that gravity does not affect the force drawing the liquid into the pore.

Figure 2-1: a) a drop of wetting fluid, b) a drop of non-wetting fluid,
c) a wetting fluid entering a closed-end pore

The ability of a specified liquid to wet a specified solid is characterized by the contact angle for a drop of that liquid resting on a horizontal flat surface of that solid (as measured through the liquid).
    For a wetting fluid the contact angle is < 90^o, so the drop flattens out and spreads on the solid (Fig. 2-1a)
    For a non-wetting fluid the contact angle is > 90^o, so the drop "beads up" to have minimum contact with the solid (Fig. 2-1b)
    A wetting fluid will compress the gas in a closed-end capillary as the interfacial tension pulls the liquid into the pore (Fig. 2-1c)

Figure 2-2 shows how the contact angle is related to the vectors of force along the various surfaces that meet at the shoreline. The angles are taken in a plane that is normal (perpendicular) to both the flat solid surface and to the shoreline.

cos _SLG = (_SL - _SG) / _LG

How Fast Does Liquid Penetrate a Pore?

t_L = 8 (f_tort L)² / (D_{pore LG} cos _SLG)

Here we have included a tortuosity factor, f_tort, one for straight culinders and larger than one for more complex geometeries.

The equation indicates that a liquid will penetrate the pores of a solid rapidly if the solid has large pores and if the liquid has a low viscosity, a large surface tension, and a low contact angle with the solid. Since a clump of particles is simply a solid with many tortuous and interconnected pores, the same factors will lead to rapid penetration of liquid into a clump.

If the pore is closed so that the gas cannot escape (and the gas has negligible solubility in the liquid) then the gas will be compressed until it has a pressure equal to the maximum that the surface tension of the liquid can apply,

P_trap = 4 _LG cos _SLG / D_pore

High pressures can be generated in small diameter pores or in clumps of submicron powders. At such high pressures the gas may dissolve in the liquid and then diffuse out of the pore into the bulk liquid, gradually reducing the volume of the bubble to zero.

EXAMPLE: Water wets quartz ( = 0) and has a high surface tension ( = 72.8 mJ/m²) at 20^o C. To what pressure will it compress the gas trapped in a pore of radius 1 m?
    P_trap = 4 * [72.8 x 10^-3] * 1 / [1 x 10^-6]
    => 291 x 10³ J/m³ or 2.91 x 10⁵ Pa or 2.87 atm.

The Value of Wetting Agents

Molecules of a wetting agent can be represented as A-B, where A is a region that is incompatible with the liquid and and adsorbs on or attaches to the solid surface and B is a region that is compatible with (dissolves readily in) the liquid. In the figure, A is represented by the head of the arrow and B by the tail. The wetting agent is normally dissolved in the liquid, where it is present as micelles (clusters of A-B molecules with all the B groups facing outward).

Figure 2-3: a) early stage, most wetting agent in micelles
b) mid-stage, some wetting agent in micelles, some adsorbed
c) end stage, maximum coverage by adsorbed wetting agent

When a solid clump is dropped onto this solution liquid, the A-B molecules adsorb on the solid so as to make the surface more compatible with the liquid. This allows an advance of the shoreline across the surface of the solid, depleting the liquid near the shoreline of wetting agent. The rapidity of advance of the liquid into a pore (or clump) may be determined by the rate at which micelles can diffuse up to the advancing liquid front to replace the adsorbate that has been removed from the solution there through adsorption onto the solid.