General methods for determining the droplet size distribution in emulsion systems

Abstract

We present a general method that allows us to figure out the size distribution of an isolated collection of droplets of dilute emulsion system using nuclear magnetic resonance pulsed gradient spin echo measurements. We show that the expression to obtain the volume fraction distribution function is equivalent to a Fredholm integral equation of the first kind. We prove, using the Dirac notation, that a solution of this equation can be easily found if its kernel has a complete biorthogonal system of eigenvectors. Two numerical 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 Legendre polynomials to integrate numerically the integral equation and evaluates the unknown distribution by means of a constrained least square procedure. The computational limits are analyzed. To extract the distribution’s form directly by experimental data we have constructed a generating function using the shifted Jacobi polynomials. The procedures have been tested on simulated and experimental data and appear to be a powerful and flexible method to obtain the size distribution function directly by the experimental data.