Measurement of emulsion droplet sizes using PFG NMR and regularization methods

The droplet size distributions of emulsions have been measured using pulsed field gradient (PFG) nuclear magnetic resonance (NMR) for many years. This technique finds particular application with emulsions that are concentrated and/or opaque, since such emulsion systems are difficult to characterize by other methods. Most studies employing PFG techniques assume a lognormal form when extracting the droplet size distribution from the experimental data. It is clearly desirable to retrieve a droplet size distribution from the experimental data without assuming such a functional form. This is achieved for the first time using regularization techniques. Regularization based on the distribution area and on its second derivative are compared and assessed along with the following techniques for selecting the optimal regularization parameter: the L-curve method, generalized cross validation (GCV), and the discrepancy principle. Regularization is applied to both simulated data sets and experimental data. It is found that when the experimental error can be estimated accurately, the discrepancy principle with area regularization is the best approach. When the error is not known the GCV method, with second derivative regularization and allowing only nonnegative values, is most effective.