General methods for determining the droplet size distribution in emulsion systems

We present a general method that allows us to figure out the size distribut ion of an isolated collection of droplets of dilute emulsion system using n uclear magnetic resonance pulsed gradient spin echo measurements. We show t hat the expression to obtain the volume fraction distribution function is e quivalent to a Fredholm integral equation of the first kind. We prove, usin g the Dirac notation, that a solution of this equation can be easily found if its kernel has a complete biorthogonal system of eigenvectors. Two numer ical procedures are discussed. The first, termed indirect, is based on the expansion of the unknown distribution function in the eigenfunctions of the kernel. The second one, called direct, uses the properties of shifted Lege ndre polynomials to integrate numerically the integral equation and evaluat es the unknown distribution by means of a constrained least square procedur e. The computational limits are analyzed. To extract the distribution's for m directly by experimental data we have constructed a generating function u sing the shifted Jacobi polynomials. The procedures have been tested on sim ulated and experimental data and appear to be a powerful and flexible metho d to obtain the size distribution function directly by the experimental dat a. (C) 1999 American Institute of Physics. [S0021-9606(99)51702-9].