Fractal analysis of porous structure and permeability of separation membranes

Pore structures of membranes are routinely characterized by imaging techniques. However, it remains a challenge to quantify the images, and ultimately, to correlate the calculated geometric quantities with membrane performance. In this study, we present systematic fractal analysis of images of membrane pore structures. Both pore-area ($D_f$) and tortuosity fractal dimensions ($D_T$) were obtained from in-plane and in-thickness cross-sectional SEM images, respectively. Analysis of symmetric PVDF membranes reveal that the best method of extracting fractal dimension value is the box counting method (BCM) with box size range between minimum (${\lambda }_{\min }$) and maximum (${\lambda }_{\max }$) pore size. Importantly, the permeability of the membranes can be accurately calculated using geometric parameters (${\lambda }_{\max }$, porosity ($\varphi$), $D_f$, and $D_T$) that are all obtained from the images. We further applied the methodology to two asymmetric PES membranes by approximating the membranes as having separate layers each with different pore structures. The geometric parameters including fractal dimensions were determined for these layers, which allowed estimations of the permeability of these layers. Using resistance-in-series approach, the calculated overall membrane permeance matches well with the experimental results, with the first 2–3 $\mu$ m layer contributing over 66 % resistance for both membranes.