# Disjoinig Pressure

When a liquid film on a solid surface becomes very thin, the intermolecular attractive force between the molecules in the liquid and those in the solid surface turns to pull the liquid back to the liquid film. For a flat liquid film in which capillary pressure is absent, the pressure in the liquid film, ${{p}_{\ell }}$, is changed by an amount, pd (pd < 0), referred to as disjoining pressure which causes the liquid pressure to be less than the vapor pressure ( ${{p}_{\ell }}<{{p}_{v}}$). At equilibrium, the liquid pressure becomes ${{p}_{\ell }}={{p}_{v}}+{{p}_{d}}$; in this case, the disjoining pressure is negative when the solid draws the liquid into the film. The disjoining pressure is a product of long-range intermolecular forces composed mainly of molecular (dispersion) and electrostatic interactions. They can be effective as long-range forces when the intermolecular spacing in the liquid film is about 0.2 nm to 10 nm. These forces are present even in nonpolar liquids because they are quantum-mechanical in origin. A pressure gradient is generated within the thin layer of liquid that covers the solid section in contact with the vapor. The disjoining pressure becomes more important for ultra-thin liquid films, because the liquid-vapor interface is closer to the solid surface. As a result, the attractive forces from the solid to liquid will have a greater effect on the liquid-vapor interface. As an example, Fig. 2 illustrates the variation of disjoining pressure with the liquid film thickness for CCl4 on glass at 77 ºC.

The dependence of disjoining pressure, pd, on liquid film thickness, δ, can be obtained by analyzing the molecular forces and the reversible work required to establish a liquid microlayer (see Section 2.6.4): ${{p}_{d}}(\delta )=-\frac{2{{A}_{H}}}{\pi (n-2)(n-3)}{{\delta }^{3-n}}$ (1)

where AH is a constant, and n is the exponent in the long-range interaction potential φ(r) = − Cφ / rn that characterizes the force interaction between the liquid and solid molecules. This long-range interaction potential can be represented by the second term of the Lennard-Jones 6-12 potential. This results in the exponent in the long-range potential, n, equaling 6. Thus eq. (1) becomes ${{p}_{d}}(\delta )=-\frac{{{A}_{H}}}{6\pi }{{\delta }^{-3}}$ (2)

A more generalized relationship between the disjoining pressure and the film thickness can be expressed as (Potash and Wayner, 1972)

 pd = − A'δ − B (3)

where A' and B are constants that characterize the molecular and electrostatic interactions. Several components of the disjoining pressure are distinguished: dispersion (molecular), adsorption, electrostatic, and structural (Derjaguin, 1955; 1989). Values of dispersion constants strongly depend on liquid types. For ammonia, for example, ${A}'=1\times {{10}^{-21}}$ J. The nature of this phenomenon produces increasing (absolute value) pressure with decreasing film thickness. The pressure is considered to be positive for repulsion and negative for attraction of the surface film. Due to the extremely high values of pd in ultra-thin films, the liquid transport can be significant; thus the role of disjoining pressure in evaporation can be essential, particularly for low-temperature fluids. Disjoining pressure is one of the fundamental phenomena that affect the formation of thin evaporating films and the magnitude of the contact angles.

When a wetting liquid is in contact with a solid, the liquid forms a curved liquid/vapor interface, as shown in Fig. 3. When the liquid has a high wetting capability, the liquid spreads on the solid wall to form an extended meniscus. The extended meniscus can be divided into three regions: the equilibrium thin film, microfilm region, and intrinsic region. In the equilibrium thin film, the liquid molecules adhere to the solid, and the interfacial temperature is equal to the wall temperature. No evaporation occurs in this region because the liquid/vapor interfacial equilibrium temperature is elevated to the wall temperature due to the disjoining pressure effects. Virtually all of the evaporation occurs in the microfilm region. In this region the disjoining and capillary pressure significantly affect its shape. When the wall temperature is greater than the saturation temperature for a given vapor pressure, the interfacial temperature lies between these two values, i.e., Tsat(pv) < Tδ < Tw. In the intrinsic region, the effect of disjoining pressure is negligible and surface tension effect is dominating. The liquid flow that continually feeds the evaporating thin film is driven by a pressure gradient. The pressure gradient near the equilibrium region is primarily caused by changes in disjoining pressure as the film thickens. While the film thickens, the curvature of the surface increases to a maximum. Once the surface curvature is at its maximum, it starts to decrease, and both the disjoining pressure gradient and the change in curvature drive the flow. As the film thickens further, the disjoining pressure effects become negligible and curvature change alone drives the flow. As the film thickness increases further, the evaporation rate drops to zero, and the curvature stays at a constant value. This region of the interface can be referred to as the meniscus, and is the fourth region labeled in Fig. 3. The ultra-thin film phenomena and the effects of disjoining pressure are very important in two-phase micro/miniature devices such as micro heat pipes (Khrustalev and Faghri, 1994), rotating miniature heat pipes (Lin and Faghri, 1999), and pulsating heat pipes (Zhang and Faghri, 2002).

## References

Derjaguin, B.V., 1955, “Definition of the Concept of and Magnitude of the Disjoining Pressure and its Role in the Statics and Kinetics of Thin Layers of Liquid,” Kolloidny Zhurnal, Vol. 17, pp. 191-197.

Derjaguin, B.V., 1989, Theory of Stability of Colloids and Thin Films, Plenum, New York, NY.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Khrustalev, D., and Faghri, A., 1994, “Thermal Analysis of a Micro Heat Pipe,” ASME Journal of Heat Transfer, Vol. 116, No. 1, pp. 189-198.

Potash, M., and Wayner, P.C., 1972, “Evaporation from a Two-Dimensional Extended Meniscus,” International Journal of Heat and Mass Transfer, Vol. 15, pp. 1851-1863.

Lin, L., and Faghri, A., 1999, “Heat Transfer in the Micro Region of a Rotating Miniature Heat Pipe,” International Journal of Heat and Mass Transfer, Vol. 42, pp. 1363-1369.

Zhang, Y., and Faghri, A., 2002, “Heat Transfer in a Pulsating Heat Pipe with Open End,” International Journal of Heat Mass Transfer, Vol. 45, No. 4, pp. 755-764.