Traditional ultrafiltration membranes, made from solvent casted polymers, have a tortuous path, sponge-like, pore structures that limit the resolution of separations. Such membranes are typically used to separate materials that are orders-of-magnitude different in size in dialysis, solute concentration, and buffer exchanges. The membranes are also microns thick and highly porous giving more than 100 um² of internal surface area for every um² of frontal surface ⁵. High internal surface area leads to flow resistance, sample loss, and clogging in flow-through filtration systems. Experimental thin (~50–100 nm) tortous path membranes made from cross-linked proteins ⁶ or polymers ⁶ have pore size cut-offs appropriate for nanofiltration (< 2 nm) rather than ultrafiltration (2 nm to 50 nm) ⁷.

To create idealized membranes for size-based separations, technologists have developed nanoporous membranes with well-defined and tunable pore sizes. Aluminum oxide and track-etched membranes are commercially available examples, but the pores in these membranes are too large (> 20 nm) for many biological separations. In addition, these membranes are very thick (6–10 um) so that transport is too slow for many purification processes ⁸. Flow through membranes created from aligned carbon nanotubes (CNTs) has been shown to exceed the predictions of viscous flow theory because of the smooth and hydrophobic walls of CNTs, however the pore sizes are limited to a small range between 2 – 6 nm that depends on the CNT formulation. Ultrathin silicon membranes (~10 nm) with defined pores (25 nm) were first created using an ion-beam drilling process that is far too slow for scale up ⁹. Recently, track-etched technology has been applied to SiN membranes to create 100 nm thick membranes with pore sizes that can be tuned between ~10–50 nm depending on the time allotted to etching ¹⁰, however this technique requires access to a cyclotron capable of heavy ion (Bi, Xe) bombardment and thus also faces significant large scale manufacturing challenges.

We compared water flow rates through pnc-Si to a formula derived by Dagan et al. ¹³ describing low Reynolds number flow through short pores accounting for entrance, exit and tube resistances (see Supplement). We used custom image processing routines (Fig. S1) to determine the distribution of pore sizes in a transmission electron micrograph (Fig. 2a) and then used the Dagan formula to calculate the predicted water flow through each pore in the micrograph. The theoretical hydraulic permeability for each membrane test was calculated by summing the flows through the individual pores and dividing by the total area imaged in the micrograph. An example of our analysis is shown in Fig. 2. Fig. 2a shows the pore number histogram for a particular membrane and Fig. 2b shows the predicted water flow through each pore in the image. Dagan’s formula has been previously validated for micron thick membranes with micron-sized pores ^{14, 15}. Here we find agreement between the formula and experiment for pnc-Si (Fig. 2c), indicating that continuum descriptions of fluids are appropriate for the analysis of flow through nanometer-thick membranes ¹⁵.

In Fig. 2d we compare the hydraulic permeability of a pnc-Si membrane with a 12 nm average pore diameter to the permeabilities of carbon nanotube (CNT) membranes reported in the recent literature to give high flow rates ^16–19. We also compare pnc-Si permeabilities to measurements for commercially available track-etched (TE) membranes assembled into the same centrifuge devices used for pnc-Si hydraulic permeability measurements. Direct comparisons between CNT, TE and pnc-Si membranes are appropriate given that all three membranes have well defined through-pores. Advances in manufacturing ²⁰ allowed Yu et al. to create CNT membranes with 3 nm pores and ~80% porosity and achieve hydraulic permeability values of 3.3 cm^3/(cm^2-min-bar) ¹⁹. CNT membranes achieve high hydraulic permeabilities despite being thicker (2 – 200 um) and having smaller pores (1–6 nm) than pnc-Si because the smooth and hydrophobic nanotube walls allow fluids to slip ^{18, 21, 22}. Unlike CNT membranes, pnc-Si is hydrophilic and the assumption of no-slip, highly viscous flow agrees with measurement (Fig. 2). Still the hydraulic permeability we measure for pnc-Si membranes with 1.4% porosity is 7 times higher than the highest porosity CNT membranes. Not surprisingly, we also found that the hydraulic permeability of pnc-Si membranes is >35× higher than 6 µm thick, hydrophilic TE membranes with slightly larger pore sizes (30 nm) and a 3 fold lower porosity (~ 0.5% porosity). Indeed the hydraulic permeability values we report for 15% porous pnc-Si (~40 ml/(cm²-min-bar)) appear to be the highest on record for a nanoporous membrane (pores < 100 nm), exceeding commercial ultrafiltration membranes (< 1 ml/(cm²-min-bar)) ⁷, high porosity nanoporous alumina (< 1 ml/(cm²-min-bar)) ²³, experimental block co-polymer membranes (~ 4 ml/(cm²-min-bar)) ²⁴, and 60 nm thick protein membranes (~ 15 ml/(cm²-min-bar)) ⁶.

In our initial attempts to measure the hydraulic permeability of pnc-Si, we discovered that membranes are impermeable to water when one side of the membrane is left dry (Fig. 3). We adopted several strategies to overcome water impermeability of the wet/dry configuration, including the use of high pressures (> 1 ATM), ozone treatment of membranes to decrease contact angles from ~70 degrees to less than 15 degrees (Figure S3), and lowering surface tension 2–3 fold by adding surfactants (0.2 wt% SDS or 0.2 wt% Triton X-100) or using ethanol instead of water. We also switched from centrifuge-generated pressure to a simple pressure cell because we suspected centrifugal forces were quickly removing fluid droplets from the membrane backside and stopping flow. High water permeability was observed when we immersed the membrane in water throughout the experiment or added a hygroscopic polymer, polyvinylpyrrolidone (PVP), to the membrane backside. The most reasonable explanation for impermeability of the wet/dry arrangements is that water cannot wick through the pores to wet the backside of a membrane. This is somewhat surprising given that capillary forces should easily drive water to fill 15 nm long pores. While we cannot confirm that the hydrophilic contact angles we measure on the top surface of pnc-Si chips also apply to pore walls, the fact that rapid transmembrane diffusion occurs when the membranes are immersed in water ^{11, 12} does suggest that water has no difficulty entering pores without applied pressure. Thus we suspect that the meniscus in a filled pore cannot navigate the obtuse angles at the pore exit and pressures compatible with pnc-Si membranes ¹¹.

To examine the performance of pnc-Si membranes in size-exclusion separations, we filtered gold nanoparticles in a size ladder ranging between 5 nm and 30 nm in diameter. To avoid filtrate dilution from the immersing fluid in the centrifuge set-up, we used a simple pressure cell for which flow could be initiated by the addition and removal of 5 ul of water on the membrane backside. The pressure cell was operated at 10 PSI until it passed 100 µL of a 200 µL starting sample through membrane chips from three different wafers (F, G, H). Absorbance values in the retentate and filtrate were measured to determine concentrations and these were normalized to the starting concentration to calculate sieving coefficients. Results show that each pnc-Si membrane exhibited a sharp cut-off with no detectable transmission of the larger particle and significant transmission (40–85%) of the smaller particle (Fig. 3A). Most notably, membranes from wafer F allow more than 80% transmission of 5 nm particles while fully blocking the larger 10 nm nanoparticle (Fig. 3A and 3B). Membranes from wafer G also exhibited a resolution of 5 nm and a cut-off between 10 nm and 15 nm.

In contrast to pnc-Si, commercially available cellulose (MW cut-offs 3k and 100k) and PES membranes (MW cut-offs 3k, 30k, 50k, and 100k) did not pass any nanoparticles between 5 nm and 30 nm. We quantified losses by multiplying concentrations in the rententate and filtrate by the recoverable volumes in each compartment and compared this to the amount of starting material. For pnc-Si, losses were within the experimental uncertainty deriving from pipetting and measurement errors (<5%), while losses were significant (~12%) for PES membranes. Since no detectable quantities of nanoparticles passed into the filtrate and the PES membranes appeared pink after the experiment, the lost particles are presumably embedded in PES membranes. We also performed separation experiments on a protein size ladder containing largely globular proteins with the exception of a linear myosin control protein (Fig. 3d). Table 1 lists the unreduced and reduced sizes of the proteins used in the ladder. The unreduced sizes are relevant to the filtration process, while the reduced sizes explain the molecular weights of monomers seen on the reducing gels. The results indicate complete retention of myosin and β-galactosidase (17 nm) and a nearly undiluted transmission of the carbonic anhydrase (~ 4 nm). The transition between full transmission and full complete retention takes place between olvalbumin (~ 6.5 nm) and phosphorylase B (8.3 nm) again suggesting a resolution of better than 5 nm for proteins as with nanoparticles.

Using dynamic light scattering, we confirmed that membranes from wafer G could be used to ‘purify’ 10 nm particles from a mixture of 10 and 15 nm nanoparticles (Fig. 3C). Filtering a mixture of 10 nm and 15 nm stock largely recovers the light scattering profile of 10 nm nanoparticles with only a slight shift that presumably arises from some smaller particles in the 15 nm sample entering the filtrate. It is important to note that each nanoparticle stock is not homogenous and that light scattering tends to broaden the distribution relative to the actual size of the nanoparticles. We established this by comparing the sizes of nanoparticles in the 10 nm and 15 nm stocks as measure by light scattering to the sizes of nanoparticles measured by electron microscopy (Table S1). Thus the ‘purification’ of the 10 nm particles from the mixture is implied by fact that the shifted spectrum of the filtered mixture closely resembles the original 10 nm stock spectrum.

High transmission of materials just below a cut-off is not typical of ultrafiltration membranes ^{7, 25, 26} and is likely enabled by the thinness of pnc-Si. Advanced sieving theories ²⁷ demonstrate that diffusion can significantly boost the concentration of filtrate materials for ultrathin membranes compared to purely convective transport. The ratio of transport by convection vs. diffusion is known as the Peclet number: where V is the average solvent velocity in a pore, L is the membrane thickness and D is the diffusion coefficient of the species being transported. For 15 nm thick pnc-Si, Pe ~ 0.2, assuming a typical monomeric protein (D = 7.5×10⁻⁷ cm²/s) and a transmembrane velocity of 0.1 cm/s. By comparison Pe ~ 7 for filtration at similar velocities with a commercial ultrafiltration membrane having a 500 nm thick skin. Thus even at high flow rates diffusion is the dominant transport mechanism for molecularly thin pnc-Si membranes and this fact should help to increase filtrate concentrations compared to commercial ultrafiltration membranes.

It is important to emphasize that while manufacturing is currently limited to making small membrane devices, the use of highly scalable semiconductor manufacturing allows large numbers of those devices to be made. The membranes used for the current work were produced on 4" wafers that contained more than 80 membrane chips. Continued optimization of manufacturing and chip design since the completion of this work is now resulting in more than 400 similar devices being produced on a 6" wafer and less than twice the cost (Figure S3). Similarly while the membranes used in the current work were limited to ~1% active membrane area which is sufficient to characterize the intrinsic properties of the material, advances in manufacturing are now producing membranes with ~10% active area which increases the total volumetric flow at ~ 500 ul/min at 1 atmosphere of pressure. Continued advances in manufacturing, including the use of [110] crystalline silicon to allow vertical etches through the support wafer (rather than the [100] silicon currently used) should allow the fraction of chip area occupied by free-standing membrane to increase at least another 4 fold. Since only standard semiconductor manufacturing processes are used in the production of pnc-Si, the volume of wafers produced each day is also highly scalable. Thus while the prospects for scaled-up manufacturing of many of the nanoengineered membranes appearing in the literature are dim, large scale manufacturing is very realisitic, and partially achieved already, for pnc-Si.

Supplement