Because of the non-linear dependence of volumetric flow rate on pore size …
where t is the membrane thickness and r is the pore radius, large pores contribute disproportionately to the total flow through a membrane than small pores. The magnitude of this phenomenon can be appreciated by using the Dagan equation to calculate the flow per pore in an image (method and figure adapted from Gaborski et al., ACS Nano 2010; 4:6973-6981). The total flow is the integral under this curve multiplied by the ratio of the total membrane area to the imaged area. The pores in red in the pore size histogram account for all the flow to the right of the vertical line in the flow curve. Fewer than 20% of the pores account for more than half of the area under the curve.
One of the consequences of this phenomena is that when filtering a mixed population of nanoparticles some of which are larger than all of the pores, the largest pores in the membrane pores will clog first. This effectively leaves behind a smaller pore membrane. Karl Smith (PhD ’17) was the first to think about this as potentially useful as a means to make smaller pore membranes out of larger ones (Smith, et al, (2017), Sep Purif Technol; 189:40-47) – just mix in some particles to ‘knock out’ the pores you don’t want to contribute to your processes.
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