5.9 Phase changes under curved surfaces

 

When the interface is not a flat surface but a curved one, a pressure difference due to surface tension appears between the phases.

It is known from elementary physics that the so-called surface tension acts at the interface, the force tending to reduce the area of liquid at the interface to a minimum. It will be recalled that by definition the surface tension a is the name given to the force acting on unit length of the perimeter of the area; these forces, normal to the perimeter and tangent to the interface, are depicted by arrows in Fig. 5.7, which shows an elementary section of an interface. Since with a flat interface the surface tension forces, acting at all points of the perimeter of the section (Fig. 5.7a), lie in one plane, the resultant of these forces is evidently zero. Things are different with a curved interface. As can be seen from Fig. 5.7b, the surface tension forces, acting at various points of the perimeter, lie in different planes, and their resultant is no more equal to zero, the resultant evidently being the larger the greater the curvature of the phase interface is. It can be easily seen that the resultant force acts in the direction of the concave side of the curved surface. Thus, the phase under the concave side of the interface will be affected by an additional pressure due to surface tension.

 

Fig. 5.7.jpg

 

Fig. 5.7

 

Let us find how this force is related to the magnitude of the surface tension and to the curvature of the interface. Consider a small square on the curved surface (Fig. 5.8), and calculate the resultant of the surface tension forces acting on the sides AB and CD of the perimeter. The force acting on side AB is

 

                                                                                                                         (5.153)

 

where lAB is the length of side AB. By analogy, the force acting on side CD is

 

                                                                                                                         (5.154)

 

Fig. 5.8.jpg

 

Fig. 5.8

 

The resultant of the forces FAB and FCD is the sum of the projections of the vectors representing FAB and Fcd on the line OO', which is normal to the section of the surface at its centre. This resultant force is equal to

 

                                                      

 

where αI is the angle between the normal OO' and the radius of curvature of our section of the interface, in the plane of the square with the sides AB and CD ,drawn to point I. Inasmuch as

 

                                                                                                                       (5.155)

 

it is clear that

 

                                                                                                                         (5.156)

 

and, consequently, the resultant of the forces FAB and FCD will be

 

                                                                

 

Analogously, the resultant of forces FAD and FBC will be

 

                                                                

 

where an is the angle between the normal OO' and the radius of curvature of our section of the interface, in the plane of the square with the sides AD and BC, drawn to point II.

Hence, the resultant of the forces FAB, FBC, FCD, FAD, i.e. of all surface tension forces acting on the section of the interface, will be equal to:

 

                                                                                                     (5.157)

 

As can be seen from Fig. 5.8,

 

                                                   and                                            (5.158)

 

With account taken of relationships (5.158), Eq. (5.157) takes the following form:

 

                                                                                                           (5.159)

 

Dividing the force F by the area of the section considered (this area can easily be seen to be l2), we obtain the magnitude of the supplementary pressure due to surface tension forces:

 

                                                                                                               (5.160)

 

This equation, first derived by Laplace, relates the difference between phase pressures with surface tension and the curvature of the interface (it will be noted that the Laplace equation can also be derived in a purely thermodynamic way).

Present in Laplace's equations are two radii of curvature ρI and ρII. This can be easily justified. From analytic geometry it is known that the curvature of a surface is characterized by two radii that determine the curvature of this surface in two mutually perpendicular cross sections.

The curvature of an interface will be assumed to be positive when the phase of a greater density is arranged on the concave side of the interface, and negative when this phase lies on the convex side of the interface.

It follows from the Laplace equation that for a flat interface (i.e. for an interface whose radius of curvature is infinitely large) the supplementary pressure p* is equal to zero, as could be expected.

The curvature of a spherical interface is the same in any section:

 

                                                                                                                         (5.161)

 

where ρ is the radius of the sphere. From Eq. (5.161) it is clear that for a sphere the Laplace equation takes the form

 

                                                                                                                             (5.162)

 

Consider a spherical drop of liquid in equilibrium with its saturated vapor at some temperature T0. If the liquid-vapor interface (or boundary) were flat, the pressure of the liquid would be equal to the vapor phase pressure; this would be the pressure p0, the saturation pressure at temperature T0. However, for a spherical drop, the liquid phase is acted upon by a supplementary pressure due to surface tension forces. According to Poynting's equation (see Sec. 5.8), the presence of the supplementary pressure will cause an increase in the vapor phase pressure. If liquid phase pressure is denoted by pl, and the pressure of the vapor in the space above the drop of liquid by pv, then from Eqs. (5.148) and (5.149) it follows that for the case considered (a drop with radius ρ) we have:

 

                                                                                                          (5.163)

 

                                                                                                          (5.164)

 

If vapor pressure is low and, consequently, the specific volume is large compared with the specific volume of the liquid, so that the value of vl can be ignored in comparison with vv, these equations can be presented in the following form:

 

                                                                                                                     (5.165)

 

                                                                                                                     (5.166)

 

Finally, if vapor pressure is so low that the vapor can be considered as an ideal gas, then

 

                                                                 

 

and, consequently, from Eq. (5.166) we obtain:

 

                                                                                                             (5.167)

 

whence

 

                                                                                                                 (5.168)

 

 

Taking account of surface tension forces is essential in analyzing various phase change processes proceeding in the presence of a curved phase interface.