Equations for the growth of a distribution of small physical objects

Abstract

The correspondence between the discrete chemical rate equations, or statistical birth equations, and their continuous limit, the classical continuity equation, is investigated, for the growth of a distribution of small objects such as particles, droplets, or bubbles. The origins of the physical growth laws, n = β n s , with 8 = ,1/3,1/2, 2/3, and 1, are briefly summarized, where n is the number of atoms or molecules in the object. Moments of the discrete distributions are obtained directly by deriving and solving a set of coupled differential equations, and their accuracy is checked by numerical calculations. The moments are also derived from those of the temporal distributions, but this method is less accurate. For physical purposes, the validity of the continuity equation description at large n is established for all the growth laws except s =1, including the case when there is a time-dependent source at small n . The consequent possibility of coupling a microscopic description of growth at very small n to a continuous description at larger n is illustrated in a realistic model for growth from a monomer. For s = 1, the continuous description cannot be used because size fluctuations introduced by the discreteness are of the same order as the mean size. Distributions produced, which could be realized by small gas bubbles under certain conditions, are very sensitive to initial conditions. However, since their analytic form is known, problems involving time-dependent sources can also be solved in this case.