Temporal behavior of the number density of particles during Ostwald ripening

The kinetic behavior of the number of particles per unit volume or area, N, is examined theoretically for different dimensionalities of the dispersed phase (d) and the diffusion field (D). For particles that are either perfectly spherical (d=3) or circular (d=2) a kinetic equation for N is presented which is valid for D=3, 2 and 1. The temporal exponents are also presented in terms of d and D. It is shown that the conventionally accepted kinetic law for N, i.e. N proportional to t(-m), where m=d(3+d-D), must be replaced by an equation containing two terms. This equation has the form N=At-m + Bt(-p), where A and B are constants related to the physical parameters of the system, and p=(d+1)/(3+d-D). The second term is not negligible with respect to the first, and arises because the volume fraction is not constant during Ostwald ripening. This equation is evaluated for self-consistency by the analyses of data on four systems: 1. Coarsening of gamma' (Ni3Al) precipitates in an Ni-Al alloy (d=D=3); 2. The results of a computer modeling experiment (d=D=2); 3. Coarsening of 'islands' in a diblock copolymer film (d=D=2); 4. Coarsening of droplets of succinonitrile on a quartz substrate (d=3; D=2). Self-consistency is tested by comparing values of A predicted using data on the kinetics of growth of the average particle with those obtained from the analysis of data on the temporal behavior of N. Self-consistency also demands that B be non-zero, and that different estimates of B obtained from different methods of data analysis be nearly equal. All the data sets examined fulfill these self-consistency requirements. Moreover, the values of B for the gamma'/Ni-Al alloy are in excellent agreement with that calculated using reasonably well-known physical constants. The importance of the second term in the equation for N can also account for the discrepancy that is often observed between values of m obtained from plots of log N versus log t and the values expected on the basis of conventional theory. (C) 1997 Elsevier Science S.A.