First-order correction to classical nucleation theory: A density functional approach

It is shown that the classical expression for the change in grand potential of a system on formation of a cluster of radius R is modified by a factor [1-(2w+6 delta(T))/R], to first order in 1/R, where w is a correction due t o the nonzero compressibilities of liquid and vapor (near the triple point, w is approximately equal to the product of liquid compressibility and surf ace tension), and delta(T) is the coefficient in the expression relating th e surface tension of the droplet, gamma(R), to the planar surface tension, gamma(infinity), i.e., gamma(R)=gamma(infinity)(1-2 delta(T)/R). An express ion for delta(T) is derived involving the pair and triplet correlation func tions and the density profile of the planar surface. This complements the e xpression for delta(T) involving the pair distribution function derived by Blokhuis and Bedeaux; the equivalence of the two expressions in the low den sity limit is demonstrated. Calculations of delta(T) and w are performed us ing mean-field density functional theory for the Yukawa potential and an r( -6) potential, as well as using the square-gradient approximation. delta(T) is found to be negative for all conditions investigated; its magnitude dep ends on the potential used, and tends to increase with increasing temperatu re. However, the ratio delta(T)/w is found to be relatively insensitive to potential and to temperature, being between about -1.2 and -1.5 for the con ditions investigated. The effect of using a weighted density approximation in place of the local density approximation for the hard-sphere part of the potential is estimated in a square-gradient approximation and found to be small. (C) 1999 American Institute of Physics. [S0021- 9606(99)51437-2].