Data processing of membrane characterisation by permporometry

The porosity of a material indicates what is the volume of empty space with respect to the total volume. So if you know that a material is 25% porous, then it consists for 75% of ‘real material’, and the remaining quarter is ‘nothing’ or air. But in addition to this absolute value for the porosity, sometimes you would also like to know the size of the individual pores. For example when you want to remove solid particles from a liquid stream using a porous membrane. The (maximum) pore size of the membrane determines which particles the membrane can reject, and the (average) pore size determines the magnitude of the liquid stream passing the membrane. Here you would like to know the pore size distribution of the membrane – so the number of pores with a certain pore size. If you know this distribution, you will see immediately the size of the largest pore, en by using simple statistics you can calculate the average pore size.

By means of ultrafiltration it is possible to separate macromolecules and other components with a high molecular weight from solvents – such as water – and other small molecules. So-called mesoporous membranes, having pores with a size between roughly 2 and 100 nm, play a major role in ultrafiltration. When applying a driving force of 1 to 10 bars over a membrane to separate a mixture, the small molecules will pass and the larger ones will stay behind. In addition to concentrating of macromolecular solutions, you can stumble across ultrafiltration in food and dairy industry (to concentrate protein solutions), in pharmaceutical industry (recovery of antibiotics or enzymes), and to remove oil from (process) water – to name a few applications.

Mesoporous membranes can be characterised by using permporometry to determine their pore size distribution. Permporometry only measures the active pores, i.e. the pores that contribute actively to the separating function of a membrane. In this elegant characterisation technique, a part of the pores is being blocked by condensing a vapour inside these pores, and the gas permeating through the remaining open pores is measured. How is this being done in practice?

Permporometry setup
You bubble a flow of nitrogen through a vessel filled with liquid cyclohexane (or another organic solvent), so that the nitrogen flow becomes saturated with cyclohexane vapour, and guide this mixture along one side of the membrane. A second flow of nitrogen, similarly saturated with cyclohexane, will be guided along the other side of the membrane. The cyclohexane vapour condenses in all pores of the membrane, so that all of them are filled with cyclohexane. This ‘capillary condensation’ phenomenon follows the Kelvin equation, that relates the relative vapour pressure of cyclohexane to the maximum radius of the pore that has just been blocked. In a nitrogen flow saturated with cyclohexane vapour, the relative vapour pressure – i.e. the ratio between the current vapour pressure and the saturated vapour pressure – equals 1. The corresponding maximum radius of the blocked pores is infinite, so that indeed all pores have been filled with condensed cyclohexane. Subsequently, you lower the relative vapour pressure of cyclohexane stepwise by diluting the saturated flow of nitrogen with pure oxygen along one side of the membrane, and with pure nitrogen along the other side of the membrane, and by simultaneously reducing the nitrogen flows through the bubblers. The largest blocked pores will now open – entirely in line with Kelvin’s equation – and nitrogen and oxygen can flow through these open pores. As oxygen has been applied at one side of the membrane only, you can measure how much oxygen has permeated through the largest, now open pores by measuring the oxygen flow or concentration at the other side of the membrane, for example using a gas chromatograph or an oxygen sensor. When the relative vapour pressure decreases further, more and more pores will open, and the permeated oxygen flow can still be measured. This is in fact an accumulated oxygen flow through all pores with a radius larger than the radius that corresponds with the current relative vapour pressure. At maximum dilution flows and at gas flows through the bubblers that are reduced to zero, the relative vapour pressure is zero and no cyclohexane will condensate in the pores by capillary condensation. All pores are open now. Finally, a thin layer with a thickness of a few atoms will adhere to the pore walls, and desorbs eventually. The actual measurement has now been completed, and the obtained data have to be processed.
Permporometry schematic principle

But how is it possible to get a pore size distribution from this measurement? The secret is in the ‘accumulated oxygen flow’ that was measured above. You can convert the set relative vapour pressure with the Kelvin equation to a radius of the pores that are just open. When you plot the accumulated oxygen flow as function of the Kelvin radius, you can calculate the oxygen flow through pores with a certain radius by taking the first derivative of the plotted curve. This ‘oxygen flow per pore’ is a measure for the number of pores with a certain radius (or pore size, which is twice the radius) – and this leads to the desired pore size distribution.
Permporometry accumulated oxygen permeation vs. relative vapour pressure
Permporometry accumulated oxygen permeation and pore size distribution vs. Kelvin radius

A computer program (in TurboPascal) was written to process the permporometry data. At first, relevant data such as the nitrogen and oxygen flows (input), the measurement temperature and the accumulated oxygen flow (output) are read. The latter values are converted into accumulated oxygen permeation values that can be viewed graphically on-screen as a function of the relative vapour pressure or the Kelvin radius. Then the measurement data (accumulated oxygen permeation vs. Kelvin radius) are fitted according to a least-squares method. The noise is removed from the data by fitting these to a set of polynomial functions comparable to Savitsky-Golay’s way, where the number of points for the polynomial fit and the order of the polynomial can be set by the user. From the first derivative of the fitted values the pore size distribution is calculated, as well as the average pore size. The computer program shows this pore size distribution also graphically on-screen.

During his PhD research project at University of Twente between 1990 and 1994, Eddy Brinkman together with his colleagues characterised ceramic membranes using permporometry, in order to measure pore narrowing due to CVD-like infiltration techniques. The results of this investigation, dated just before the ‘real’ digital area, are too valuable to not disclose any further. Hence, you can find them on this website. The article ‘Permporometry study on the size distribution of active pores in porous ceramic membranes‘, published in 1993 in the Journal of Membrane Science, disclosed the backgrounds and (measurement) results of these characterisations. The software (TurboPascal) to process the permporometry data was written by Eddy Brinkman.