Quantifying pore size and density for membranes in the Knudsen and transitional-flow regimes*

Membranes with well-controlled nanoscale pores have interest for applications as diverse as chemical separations, water purification, and ``green'' power generation. For instance, membranes incorporating carbon nanotubes (CNTs) as through-pores have been shown to pass fluids orders-of-magnitude faster than predicted by theory.$^{\mathrm{1}}$ However, the efficient characterization of the pore size and density of membranes is an important area of focus, particularly for membranes fabricated from bulk nanotubes.$^{\mathrm{2}}$ Here, we report on a new technique to identify the pore size ($d)$ and number of open pores ($N)$ in membranes. A nanoporous membrane is characterized with a combination of pressure-driven gas flow, and electrical-conductance measurements in aqueous solution. For the conductance measurements, the electrical current passing through the membrane scales as $d^{2}N$. For pressurized gas flow, the scaling with molecular weight ($M)$ and gas viscosity ($\mu$) identifies the flow as either Poiseuille or Knudsen, scaling as either $d^{4}$\textit{N/$\mu$ } or $d^{3}N/M^{1/2}$, respectively. With this combination of measurements, the pore size and number of pores in the membrane can be calculated. We validate this technique using track-etched polycarbonate membranes and CNT membranes with known pores, and show that it can be used to count open pores and identify defects in CNT membranes.\newline 1) N. Biu, et al., Adv. Mat. (2016) 2) R. J. Castellano, et al., J. Applied Physics. (2015)