n 



for bubbles with radius r* = -2y/p (see Figure 12 - after Reference 25). In 

 order to grow, cavities with radii less than r* require free energy of activa- 

 tiont for the motion of an individual molecule of liquid past other molecules 

 into or away from the surface. However, those with radii larger than r* grow 



Figure 12 - Work Required for Reversible Formation 

 of a Vapor Cavity in a Liquid under Tension 



with decreasing free energy. Since the growth of such cavities is a statis- 

 tical process, cavities with radii less than r* will usually disappear without 

 reaching the critical size. On a statistical basis, however, there will be a 

 certain (small) number of cavities for which the thermal fluctuations are such 

 as to produce a cavity exceeding the critical radius. When this happens, the 

 supercritical cavity will grow until the pressure of the system rises to the 

 vapor pressure of the liquid. It has been shown 23 ' 26 ' 29 > that the rate of 

 formation of such cavities is proportional to e w m&x/ where k is Boltzmann's 

 constant and T is the absolute temperature of the process. The proportional- 

 ity factor is estimated from the theory of absolute reaction rates 29 to be 

 -pj- e~ &f o /kT where n is the number of molecules in the liquid, 4f * is the 

 free energy and h is Planck's constant. With these results, Fisher 25 writes 

 the equation for the rate of formation of bubbles of vapor in a mole of liquid 

 subjected to negative pressure p: 



dn 

 dt 



NkT -{if* + w mKx) lkT 

 h e 



TT-P 



{-K 



1 on x 



$/«■ 



f "Free energy of activation" is associated with the formation of a region of a new phase of a sub- 

 stance and is proportional to the surface energy, the bulk free-energy in the absence of stress or sur- 

 faces, and the strain energy. 26 



