Capillary condensation

In

The Kelvin equation can be used to describe the phenomenon of capillary condensation due to the presence of a curved meniscus.^[2]

${\displaystyle \ln {\frac {P_{v}}{P_{sat}}}=-{\frac {2H\gamma V_{l}}{RT}}\ }$

Where...

${\displaystyle \ P_{v}}$ = equilibrium vapor pressure

${\displaystyle \ P_{sat}}$ = saturation vapor pressure

${\displaystyle \ H}$ = mean curvature of meniscus

${\displaystyle \ \gamma }$ = liquid/vapor surface tension

${\displaystyle \ V_{l}}$ = liquid molar volume

${\displaystyle \ R}$ =

ideal gas constant

${\displaystyle \ T}$ = temperature

This equation, shown above, governs all equilibrium systems involving

${\displaystyle \scriptstyle \Delta P}$

System A → P_v=0, no vapor is present in the system

System B → P_v=P₁<P_sat, capillary condensation occurs and liquid/vapor equilibrium is reached

System C → P_v=P₂<P_sat, P₁<P₂, as vapor pressure is increased condensation continues in order to satisfy the Kelvin equation

System D → P_v=P_max<P_sat, vapor pressure is increased to its maximum allowed value and the pore is filled completely

This figure is used to demonstrate the concept that by increasing the vapor pressure in a given system, more condensation will occur. In a porous medium, capillary condensation will always occur if P_v≠0.

Dependence on curvature

The

It is also worthy to mention that different pore geometries result in different types of curvature. In scientific studies of capillary condensation, the hemispherical meniscus situation (that resulting from a perfectly cylindrical pore) is most often investigated due to its simplicity.^[5] Cylindrical menisci are also useful systems because they typically result from scratches, cuts, and slit-type capillaries in surfaces. Many other types of curvature are possible and equations for the curvature of menisci are readily available at numerous sources.^[5][10] Those for the hemispherical and cylindrical menisci are shown below.

General Curvature Equation:

${\displaystyle \ H={\frac {1}{2}}({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}})}$

Cylinder: ${\displaystyle \ R_{1}=r}$ ${\displaystyle \ R_{2}=\infty }$

${\displaystyle \ H={\frac {1}{2r}}}$

Hemisphere: ${\displaystyle \ R_{1}=r}$ ${\displaystyle \ R_{2}=r}$

${\displaystyle \ H={\frac {1}{r}}}$

Dependence on contact angle

(${\displaystyle \ \theta }$ = 0^o) is hardly ever achieved. The

(${\displaystyle \ \theta }$ > 0^o). Also, in systems where ${\displaystyle \ \theta }$ = 0^o the radius of curvature is equal to the capillary radius.

${\displaystyle \ \theta }$

Non-uniform pore effects

Odd pore geometries

In both naturally occurring and synthetic porous structures, the geometry of pores and capillaries is almost never perfectly cylindrical. Often, porous media contain networks of capillaries, much like a sponge.^[11] Since pore geometry affects the shape and curvature of an equilibrium meniscus, the Kelvin equation could be represented differently every time the meniscus changes along a "snake-like" capillary. This makes the analysis via the Kelvin equation complicated very quickly. Adsorption isotherm studies utilizing capillary condensation are still the main method for determining pore size and shape.^[11] With advancements in synthetic techniques and instrumentation, very well ordered porous structures are now available which circumvent the problem of odd-pore geometries in engineered systems.^[3]

Hysteresis

Non-uniform pore geometries often lead to differences in adsorption and desorption pathways within a capillary. This deviation in the two is called a hysteresis and is characteristic of many path dependent processes. For example, if a capillary's radius increases sharply, then capillary condensation (adsorption) will cease until an equilibrium vapor pressure is reached which satisfies the larger pore radius. However, during evaporation (desorption), liquid will remain filled to the larger pore radius until an equilibrium vapor pressure that satisfies the smaller pore radius is reached. The resulting plot of adsorbed volume versus relative humidity yields a hysteresis "loop."^[2] This loop is seen in all hysteresis governed processes and gives direct meaning the term "path dependent." The concept of hysteresis was explained indirectly in the curvature section of this article; however, here we are speaking in terms of a single capillary instead of a distribution of random pore sizes.

Hysteresis in capillary condensation has been shown to be minimized at higher temperatures.^[12]

Accounting for small capillary radii

Capillary condensation in pores with r<10 nm is often difficult to describe using the Kelvin equation. This is because the Kelvin equation underestimates the size of the pore radius when working on the nanometer scale. To account for this underestimation, the idea of a statistical film thickness, t, has often been invoked.^[3][4][5][6] The idea centers around the fact that a very small layer of adsorbed liquid coats the capillary surface before any meniscus is formed and is thus part of the estimated pore radius. The figure to the left gives an explanation of the statistical film thickness in relation to the radius of curvature for the meniscus. This adsorbed film layer is always present; however, at large pore radii the term becomes so small compared to the radius of curvature that it can be neglected. At very small pore radii though, the film thickness becomes an important factor in accurately determining the pore radius.

Capillary adhesion

Bridging effects

Starting from the assumption that two wetted surfaces will stick together, e.g. the bottom of a glass cup on a wet counter top, will help to explain the idea of how capillary condensation causes two surfaces to bridge together. When looking at the Kelvin equation, where relative humidity comes into play, condensation that occurs below P_sat will cause adhesion.^[2] However it is most often ignored that the adhesive force is dependent only on the particle radius (for wettable, spherical particles, at least) and therefore independent of the relative vapor pressure or humidity, within very wide limits.^[2] This is a consequence of the fact that particle surfaces are not smooth on the molecular scale, therefore condensation only occurs about the scattered points of actual contacts between the two spheres.^[2] Experimentally, however it is seen that capillary condensation plays a large role in bridging or adhering multiple surfaces or particles together. This can be important in the adhesion of dust and powders. It is important to note the difference between bridging and adhesion. While both are a consequence of capillary condensation, adhesion implies that the two particles or surfaces will not be able to separate without a large amount of force applied, or complete integration, as in sintering; bridging implies the formation of a meniscus that brings two surfaces or particles in contact with each other without direct integration or loss of individuality.

Real-world applications and problems

Atomic-force microscopy

Capillary condensation

Pores that are not of the same size will fill at different values of pressure, with the smaller ones filling first.[2] This difference in filling rate can be a beneficial application of capillary condensation. Many materials have different pore sizes with ceramics being one of the most commonly encountered. In materials with different pore sizes, curves can be constructed similar to Figure 7. A detailed analysis of the shape of these isotherms is done using the Kelvin equation. This enables the pore size distribution to be determined.^[2] While this is a relatively simple method of analyzing the isotherms, a more in depth analysis of the isotherms is done using the BET method. Another method of determining the pore size distribution is by using a procedure known as Mercury Injection Porosimetry. This uses the volume of mercury taken up by the solid as the pressure increases to create the same isotherms mentioned above. An application where pore size is beneficial is in regards to oil recovery.^[13] When recovering oil from tiny pores, it is useful to inject gas and water into the pore. The gas will then occupy the space where the oil once was, mobilizing the oil, and then the water will displace some of the oil forcing it to leave the pore.^[13]

See also

• Adsorption
• Atomic-force microscopy
• BET theory
• Capillarity
• Curvature
• Capillary bridges
• Disjoining pressure
• Kelvin equation
• Self-assembled monolayers
• Sintering
• Microelectromechanical systems
• Meniscus (liquid)
• Sol-gel
• Colloid
• Ceramic

External links

• A Practical Guide to Isotherms of Adsorption on Heterogeneous Surfaces
• Langmuir/BET Derivation

References