Liquid Bridges

For hysterisis to be seen, there must be a physical difference in the adsorption and desorption isotherms, that is, show a reversal, but not a direct reversal. In the case of water liquid vapour on synthetic opal, it is a physical effect of the size and shape of the pores in the solid (there will be some chemisorption due to SiO₂ having some sites which attract certain charged ions). The effect observed is the adsorption and desorption on the mesoporous surface (20 - 500Å) as well as in the gaps between the opal spheres.

If a sample of alumina is run with a N₂ isotherm (at 77K), the isotherm is a type 2. If the alumina is crushed and the N₂ isotherm repeated, the result is a type 4 with clear hysterisis (the hysterisis region corresponds to the filling of the mesopores). In the simplest form of hysterisis, the effect is a result of capillary condensation. It would be expected that with an increase of surface area (macro to mesopores), there is a corresponding increase in the uptake of gas for a given p/p₀.

When p_g > vapour pressure, condensation occurs with a resultant surface tension between the gas - liquid interface being curved in a capillary. A second aspect is that the vapour pressure on the concave side on the interface is lower than that above the plane surface. This is quantified in the Kelvin equation

r₁ and r₂ are the radii of curvature of the interface - this is not the radius of the capillary. V_l is the molar volume of the liquid.

The radii of curvature (the spherical / hemispherical meniscus) will take one of three shapes :

1. spherical (r₁ = r₂)
2. oval (r₁¹ r₂)
3. cylindrical (r₂ = ¥)

For mesopores, the radii of curvature (which is related to the pore dimensions) are small enough for p_vap to be significantly lower than p^*. In macropores p_vap » p^* so the adsorption is negligable. For micropores, the Kelvin equation does not apply.

Consider a simple pore, cylindrical and open at both ends.

The meniscus is cylindrical, i.e. r₂ = ¥, r₁ = r. Therefore for adsorption at a cylindrical surface

For desorption, the meniscus (dotted lines, diagram 6b) is hemispherical, i.e. r₁ = r₂ = r / cosθ. Therefore

Given the two differing mechanism, each corresponding to a different p/p*, it is simple see how the hysterisis will occur.

In a real structure, the model is far more complex than above (for example, ink bottles and long channels linking larger voids) but the same form of analysis is applicable. Since r₁ and r₂ are related to the size of pore, it is possible to obtain some information on pore size can be extracted from the detailed shape of the isotherm.

Practical Measurement

The method employed for the determination of the adsorption isotherm was that of using a microbalance and subjecting a known mass of the solid to gas or liquid vapours. This was performed using a Hiden microbalance (diagram 7).

A sample is pre-weighed and placed into the reaction vessel which is then degassed to 2 bar at 80°C. The vessel is thermostated to the correct temperature using a water bath and water jacket. After thermal equilibrium is attained, the liquid vapour is introduced at a constant and measured rate with the weight of the sample taken at regular intervals via a data logging device. At p/p₀ ≈ 1, the vapour introduction is stopped and the system left.

This will give the adsorption isotherm.

The process is then reversed with pressure being reduced in a controlled way and weight loss recorded. This will give the desorption isotherm. Depending on the pore size and structure, the adsorption and desorption branches of the isotherm may show hysteresis.

Diagram 7. Hiden microbalance. Used with permission.

By a simple calculation, it is possible to see the degree of infill a sample is capable of taking. Infill here is defined as the total pore volume of all void space.

Possible problems with the microbalance.

Whilst in practice the microbalance can provide an absolute measurement of the amount of vapour the sample will adsorb (and fill into the void spaces), the hysteresis loops may not follow the theoretical path which would be expected (left). This is due to the "ink bottle effect" (diagram 9) or bridge coalescence.

Diagram 9. The ink bottle effect.

The "ink bottle" effect is due to capillary condensation. This is the process where the vapour condenses in a capillary at a vapour pressure below that at which it would condense on the free liquid surface. This can be described by the Kelvin equation in the form of (16)

16

where γ = surface tension, V = volume of liquid and r = radius of meniscus.

A simple method of explaining the theory of capillary condensation is the consideration of the conical tube placed in a vapour of pressure p > p" (diagram 10). The vapour will immediately condense at the bottom of the tube which will fill with liquid. As long as the radius of curvature of the liquid meniscus is small enough (so the vapour pressure p" above it is less than p_o and p is > p"), the tube will continue to fill. When p/p#, equilibrium is attained and condensations ceases.

Diagram 10. Capillary condensation in cones.

With porous materials, the theory is slightly different as the shape of meniscus is not normally known.

Consider a small section of a close packed array (diagram 12). Here, the spheres are all of a uniform size and shape. Here the vapour adsorption occurs on the active sites (see below) and approaches a saturation point (this may or may not be a monolayer). At the top end, bulk liquid condenses on a flat solid surface.

It is not normal to talk of "active sites" for physisorption as essentially all sites are active, albeit with a varied level of activity due to the Van der Waals forces in operation between the gas and all sites.

Diagram 12. A small section of a close packed array

If (as in diagram 12) all the pores have the same radius, the liquid will condense in those pores as soon as p reaches the equilibrium value, p₁. This corresponds to the blue line in diagram 11. The curve rises steeply as each pore can fill almost to the top at the same value of p. When the pores are almost full, p must be increased to enable further condensation. In practice, the pores and spheres will not be an even size and will therefore have differing values of p, with the smaller ones filling first. These difference in pressures in the adsorption and desorption isotherms gives rise to the hysterisis loop (diagram 8). If the adsorption and desorption isotherms are identical, then hysterisis cannot said to have occured.

As p is increased a point is reached (p₁) where the equilibrium curvature is shown in diagram 9 with the pores filling as shown (a). If p is increased further, the curvature can become flatter and the pore fills to a point given in (b). If p is reduced, the liquid in that pore can remain in equilibrium with the vapour until the point p₁ is reached again.

If the opal was purely constructed of hard, totally solid spheres, it is unlikely that there would be hysterisis (all of the adsorption sites would be identical, the difference is due to the opal being mesoporous ("holes" of between 20 and 500Å) and to a small extent, micropores (< 20Å).