# Surface Tension and Capillary Pressure

When a liquid is in contact with another medium, be it liquid, vapor, or solid, a force imbalance occurs at the boundaries between the differ­ent phases. For example, a liquid molecule surrounded by other liquid molecules will not experience any resultant force since it will be attracted in all directions equally. However, if the same liquid molecule is at or near a liquid-vapor interface, then the resultant molecular attraction of a liquid molecule on the surface would be in the direction of the liquid, since forces between the interacting gas and liquid molecules are less than the forces between the liquid molecules. It is basically due to the asymmetry of the force field acting on a molecule on the surface tending to pull it back to a higher density region or phase. If a liquid is bounded by its own vapor, then the force in the surface layer is directed into liquid because, in gen­eral, the liquid is denser than the vapor. As a result of this effect, the liquid will tend toward the shape of minimum area and behave like a rubber membrane under tension. In this context, if the surface area of the liquid is to be increased, then negative work must be done on the liquid against the liquid-to-liquid molecular forces. Any increase in the surface area will require movement of molecules from the interior of the liquid out to the surface. The work or energy required to increase the surface area can be obtained from the following relation, which is also the definition of surface tension

$\sigma = \left (\frac{\partial G }{\partial A} \right )_{T,p}$

where G is the surface free energy, and A is the surface area. Equation is valid for solid-liquid, solid-vapor, liquid-vapor, and liquid-liquid interfaces. Liquid-liquid interfaces are present between two immiscible liq­uids, such as oil and water. From this expression, the surface tension can be described as a funda­mental quantity which characterizes the surface properties of a given liquid. Additionally, the surface tension is referred to as free energy per unit area or as force per unit length. Surface tension exists at all phase interfaces; i.e., solid, liquid and vapor. Therefore, the shape that the liquid assumes is determined by the combination of the interfacial forces of the three phases. The surface in which interfacial tension exists is not two-dimensional, but three-dimensional with very small thickness. In this very thin region, prop­erties differ from the bordering bulk phases.

It is worthwhile to discuss surface tension from the molecular perspective in order to understand the mechanism of surface tension for different substances. Surface tension can be considered as the summation of two parts: one part is due to dispersion force, and the other part is due to specific forces, like metallic or hydrogen bonding (Fowkes, 1965). Surface tension force in a nonpolar liquid is due only to the dispersion force; therefore, the surface tension for a nonpolar fluid is very low. For a hydrogen-bonded liquid, surface tension is slightly higher because the surface tension is due to both dispersion forces and hydrogen bonding. The surface tension for a liquid metal is highest because the surface tension is due to a combination of dispersion forces and metallic bonding, and metallic bonding is much stronger than hydrogen bonding. The surface tensions for different liquids are quantitatively demonstrated in Table 7.1.

Table 7.1 Surface tensions for different liquids at liquid-vapor interface.

Types of liquid Liquid Temperature (°C) Surface tension (mN/m)
Nonpolar liquid Helium -271 0.26
Nitrogen -153 0.20
Hydrogen-bonded liquid (polar) Ammonia -40 35.4
Water 20 72.9
Metallic liquid Mercury 20 484
Silver 1100 878

It is generally necessary to specify two radii of curvature to describe an arbitrarily-curved surface, RI and RII, as shown in Fig. 7.11. The surface section is taken to be small enough that RI and RII are approximately constant. If the surface is now displaced outward by a small distance, the change in area is

Figure 7.1 Arbitrarily curved surface with two radii curvature RI and RII
$\Delta A = \left ( x+dx \right )\left ( y+dy \right )-xy$

If $dxdy\approx 0$, then

ΔA = ydx + xdy

The work required to displace the surface is:

δW = σ(xdy + ydx)

Displacement acting on the area xy over the distance dz also creates a pressure difference p across the surface – capillary pressure (pcap). The work attributed to generating this pressure difference is

δW = Δpxydz = pcapxydz

From the geometry of Fig. 7.11, it follows that

$\frac{x+dx}{R_{I}+dz} = \frac{x}{R_{I}}$

or

$dx = \frac{xdz}{R_{I}}$

Similarly,

$dy = \frac{ydz}{R_{II}}$

For the surface to be in equilibrium across this differential change, the two expressions for the work must be equal:

i.e.,

$\sigma \left ( \frac{xydz}{R_{I}}+\frac{xydz}{R_{II}} \right ) = \Delta pxydz$

The pressure difference between two phases becomes

$p_{cap} = \Delta p = \sigma \left ( \frac{1}{R_{I}}+\frac{1}{R_{II}} \right ) = \sigma (K_{1}+K_{2})$

where K1 and K2 are curvatures of the surface. This expression is called the Young-Laplace equation, and it is the fundamental equation for capillary pressure. It can be seen that when the two curvature radii are equal, in which case the curved surface is spherical, eq. can be reduced to

$p_{cap} = \Delta p = \frac{2\sigma }{R}$

In addition to the surface tension at a liquid-vapor () interface discussed above, surface tensions can also exist at a solid-liquid interface () and solid-vapor interface (); this can be demonstrated using a liquid-vapor-solid system as in Fig. 7.12. The contact line is the locus of points where the three phases intersect. The contact angle, , is the angle through the liquid between the tangent to the liquid-vapor interface and the tangent to the solid surface. The contact angle is defined for the equilibrium condition. In 1805, Young published the basic equation for the contact angle on a smooth, insoluble, and homogeneous solid:

$cos\theta =\frac{\sigma _{sv}-\sigma _{sl}}{\sigma _{lv}}$

which follows from a balance of the horizontal force components (Faghri, 1995), as shown in Fig. 7.12. Several theories have been proposed to explain the mechanism of dropwise condensation. The first model, proposed by Eucken (1937), has been supported by many experimental studies, such as that by McCormick and Baer (1963). It states that liquid droplets form only heterogeneously at nucleate sites; if they are formed with a radius exceeding that of equilibrium, they will continue to grow and then join with surrounding droplets. Once the mass of the condensate reaches a critical point, it will be removed from the surface by gravitational forces or by drag forces produced by the surrounding gas. As droplets are removed, the surface is wiped clean of condensate and the process restarts at the nucleate sites. This periodic cleaning constitutes the advantage of dropwise condensation over filmwise condensation, as there is no resistance to heat transfer through the condensate when the condensate layer is removed; and thus the heat transfer rate increases greatly. The second approach postulates that between drops there exists a thin and unstable liquid film on a solid surface. As the condensation process continues and the thin film grows thicker, the film reaches a critical thickness – estimated to be in the order of 1m – at which point it breaks up into droplets. Condensation then continues in the dry areas between the recently-ruptured droplets, and on top of the already-formed droplets. The majority of new condensate does occur on the wall surface because there is less resistance to heat conduction than if the new condensate formed on the already-existing droplets. These new condensate droplets are then drawn to the neighboring droplets by surface tension effects, producing a new thin film. This film will then grow and rupture at the critical thickness, and the process will repeat continuously. Dropwise condensation takes place if the condensate cannot wet the surface. When the contact angle, θ, is greater than 90°, the condensate cannot wet the surface and dropwise condensation occurs. The criterion for dropwise condensation is the critical surface tension which is characteristics of the surface alone. If the surface tension between liquid-vapor interface is greater than , dropwise condensation occurs (Shafrin and Zisman, 1960). Critical surface tensions for selected solid surfaces are given in Table 7.2. It can be seen that the critical surface tensions for all solids listed in Table 7.2 are below the surface tension of water at 1 atm (σlv = 58.91x10 − 3N / m ). Therefore, a metal surface, on which film condensation usually occurs, can be coated with another substance with lower critical surface tension to promote dropwise condensation.

Table 7.2 Critical surface tension for selected solid surfaces (shafrin and Zisman, 1960)

Solid Surface σlv(10 − 3N / m)
Kel-F ® 31
Nylon 46
Platinum with perfluorobutyric acid monolayer 10
Platinum with perfluorolauric acid monolayer 6
Polyethylene 31
Polystyrene 33
Polyvinyl Chloride 39
Teflon ® 18
• Reprinted with permission from American Chemical Society.