A-to-Z Guide to Thermodynamics,
Heat & Mass Transfer, and Fluids Engineering

### Gas-Liquid and Liquid-Liquid Interfaces

Intermolecular forces existing between molecules are responsible for the condensation of vapors to form the liquid state. Liquid surfaces tend to contract so that the surface to volume ratio is a minimum. The surface is in a state of tension which may be readily explained in terms of these forces (Fowkes, 1962; Wu, 1969). The molecules in the bulk experience forces from neighbouring molecules which are, on average, equal in all directions. For a molecule in or near the surface, these forces will not balance and the molecule will experience a pull towards the bulk. Many molecules will leave the surface, which is consequently more sparsely occupied than the internal layers so that the average spacing is slightly greater than the minimum, giving rise to the contraction.

An alternative way of looking at it is that in order to bring a molecule from the bulk to the surface work must be done against the attraction. Hence, there is an excess Free Energy associated with the surface. It is usual to define the excess surface free energy as the work which must be done on the surface, isothermally and reversibly, to expand the area by unit amount. The same considerations hold for interfaces formed between immiscible liquids. Fowkes (1962) has developed a theory for interfacial tension in terms of the contributions of the polar and dispersion interactions. Many methods, both static and dynamic, are available for measuring surface and interfacial tensions. (See Surface and Interfacial Tension.)

As a consequence of the tendency of liquid surfaces to contract, an excess pressure must exist in the interior of a bubble or a drop. If r1 and r2 are the principal radii of curvature of an interface x, y and it is expanded parallel to itself by a displacement dz, then the work done against the surface tension is σd(xy). Where σ is the excess free-surface energy. Also, if the pressure difference is p = p − po, then the work done by the pressure is p(xy)dz. For equilibrium,

(1) Since

(2) (3) for a spherical drop and 4σ/r for a spherical bubble. This expression is known as the Young-Laplace equation.

Another consequence of the curvature of an interface is that outside vapor pressure is higher than over the corresponding flat surface. Energy must be expended to transfer liquid from a planar surface to a droplet since the free energy of the droplet will increase as its surface area increases. Let the radius of the droplet be r and it is increased to (r + dr) by the transfer of dn moles of liquid from a plane liquid surface, whose vapor pressure is po, to the droplet where the vapor pressure is pr. The surface free energy of the droplet will increase by 8πσrdr. Assuming the vapor behaves as a perfect gas, this free-energy change is also equal to dn RT ln (pr/po). Hence,

(4) where

(5) then

(6) where ρ is the density of the liquid, Vm is the molar volume of the liquid and M is the molar mass.

The above expression is known as the Kelvin equation. It explains many important effects which occur when small volumes of materials are involved. Examples of these phenomena are:

1. Droplets in a vapor atmosphere. Below a critical size, the difference in vapor pressure outside the droplet compared with a flat surface means that small droplets will evaporate. As the droplet decreases in size, the equilibrium vapor pressure increases further until the drop vanishes. Thus in a population of droplets of different radii surrounded by vapor, the large droplets grow at the expense of the smaller ones.

2. Bubbles in a liquid. In this case, the vapor pressure inside small bubbles is greater than that inside large ones. This explains the phenomenon of superheating. A liquid can only boil if vapor pressure is sufficiently large for bubbles to grow. If preexisting bubbles of suitably large radius are present (nuclei), the liquid can vaporize into the bubbles which then grow and initiate boiling. If no nuclei exist, the temperature of the liquid must be increased until equilibrium vapor pressure in the smallest bubbles is great enough to allow them to grow. In general, superheating does not occur since bubbles arising from dissolved air or the presence of dust particles act as nuclei.

3. Solid particles in a vapor atmosphere. Small solid particles may be treated as spheres having an equivalent radius. Hence the vapor pressure over small crystals is greater than over large ones, and in the atmosphere of this vapor small crystals sublime and larger crystals grow. This is an example of Ostwald ripening. Small crystals also melt at temperatures lower than the bulk melting point.

4. Crystals in saturated solution. An equivalent expression can be written for the solubility of solid crystals in solution.

(7) where Cr and C are the solubilities of the crystals in the bulk, respectively. Dissolution and redeposition of material will take place and large crystals will grow at the expense of smaller ones.

5. Capillary condensation. This phenomenon is also described by the Kelvin equation. If a narrow capillary is completely wet by a liquid, i.e., the contact angle formed by the liquid on the solid capillary wall is zero, vapor will condense in the capillary at vapor pressure lower than that at which it would condense on the free liquid surface. The liquid in the capillary presents a concave interface where the vapor pressure is lower than over a flat surface so that the capillary continues to fill. The pores in a porous solid may be treated as capillaries having an effective value of the radius.

### Work of Adhesion between Liquids

Intermolecular forces acting within a liquid give rise to its surface tension and, in a similar way, the interactions across a liquid-liquid boundary result in the observed interfacial tension. Harkins (1952) has introduced the concept of work of cohesion and adhesion. If a column of liquid A of unit cross-sectional area is divided by a plane as illustrated in Figure 2, two new surfaces are created and work is done against the forces of cohesion

(8)  Figure 1. Figure 2.

Similarly if liquid A is in contact with liquid B, an interfacial tension sAB exists between them. If the two liquids are separated, an amount of work must be done and two surfaces having free energies σAV and σSV result.

Hence,

(9) Some values of the work of adhesion for some liquids against water are given in Table 1. It can be seen that values of interfacial tension usually lie between those of the individual surface tensions. However, the last three values depart from the rule. This is because these molecules contain a hydrophilic moiety which is attracted to the water and causes the molecules in the interface to reorient. The interfacial tension is thus reduced and there is a correspondingly high value of the work of adhesion.

Table 1. Some values of works of adhesion and cohesion Table 2. Spreading coefficients at 20°C of liquids Table 3. Initial versus spreading coefficients ### Spreading of One Liquid on Another

When two immiscible liquids such as oil and water are placed in contact, drops are formed which float on the surface as shown in Figure 3. For equilibrium, As σA increases, ΘA decreases and becomes equal to zero, when Under these conditions, the drop spreads spontaneously. Figure 3.

In terms of the work of adhesion, this relation becomes The spreading coefficient, S, is defined by: The spreading coefficient may vary with time as the media become mutually saturated. Some values of spreading coefficients are given in Table 2 and Table 3.

#### REFERENCES

Fowkes, F. M. (1962) DETERMINATION OF INTERFACIAL TENSIONS, CONTACT ANGLES, AND DISPERSION FORCES IN SURFACES BY ASSUMING ADDITIVITY OF INTERMOLECULAR INTERACTIONS IN SURFACES J. Phys. Chem., 66:382 .

Harkins, W. D. (1952) The Physical Chemistry of Surface Films. Reinhold, New York.

Wu, S. (1969) J. Coll. Int. Sci. 31:153.

#### References

1. Fowkes, F. M. (1962) DETERMINATION OF INTERFACIAL TENSIONS, CONTACT ANGLES, AND DISPERSION FORCES IN SURFACES BY ASSUMING ADDITIVITY OF INTERMOLECULAR INTERACTIONS IN SURFACES J. Phys. Chem., 66:382 .
2. Harkins, W. D. (1952) The Physical Chemistry of Surface Films. Reinhold, New York.
3. Wu, S. (1969) J. Coll. Int. Sci. 31:153.