## 1. Introduction

The importance of membranes in microfluidic systems is reflected by numerous applications, including the detection of chemical reagents and gases, drug screening, cell culture, protein separation, chemical synthesis at the small-scale, and electrokinetic and hydrodynamic fluid transport [

1,

2]. In particular, when the role of the membrane is to act as a gas-liquid interface, effective gas absorption and minimal gas leakage is required [

3]. The key issue for the combination of membranes and microfluidics is the sealing of the interfaces to avoid leakage, especially in the case of gases. The most convenient strategies to do this include (i) fabrication of the membrane as an integral part of the silicon chip; and (ii) exploiting the permeation properties of certain polymers by fabricating the chips directly from these materials [

2]. In the case of biological applications this last option has been adopted in most of the cases by using polydimethylsiloxane (PDMS) chips; it is well known that this polymer possesses high oxygen permeability [

4]. However, when chemical or temperature resistance is needed, the use of silicon chips is preferred.

In recent years, at the University of Twente, an elegant fabrication method has developed, which is based on a combination of anisotropic etching of silicon and so-called corner lithography, to create small nanoapertures of approximately 80–100 nm in the form of a three-dimensional fractal geometry [

5,

6]. Fractal geometry describes disciplines that consider symmetry-broken structures where, after a magnification, the shape appears identical. In other words, the magnified piece is almost a copy of the whole. The important features of the mentioned fabrication process are the possibility to easily scale up to the wafer-level and the ability to tailor the number of apertures and, thus, control the diffusion of gases through them. These apertures are distributed on the corners of pyramids which are part of a 3D fractal structure that can be replicated. This fact adds an additional advantage that could not be achieved in other 2D materials that are easily integrated with silicon microfluidics, such as porous silicon or anodized alumina. The 3D fractal structure can be embedded in a microchannel, which results in a larger interfacial area and a higher surface to volume ratio.

The hypothesis in the presented research is that the concept of membranes made of 3D fractals containing nanoapertures (also referred to as nanonozzles in this work) will result in effective gas permeation. To test this, the 3D fractal structure containing nanoapertures is integrated in a microfluidic channel and gas permeation through the pores is measured in order to quantify the diffusion of the gases oxygen and carbon dioxide through the fractal membrane. The measured values are validated with a model that considers the combination of the viscous and molecular flow regimes, the latter being the dominant mechanism for diffusion through the nanonozzles.

## 3. Results and Discussion

Figure 8 shows two different sets of experimental points obtained for Chip 1 for permeation of oxygen versus mean pressure, calculated as the average between the total pressure in the feed side and permeate side, and

Figure 9 corresponds to the permeation of oxygen and carbon dioxide in Chip 2. To evaluate the permeation through the pores of the fractal geometry two possible permeation mechanisms occurring simultaneously can be considered: molecular flow (or Knudsen) and viscous flow (or Poiseuille) [

11,

12]. All the permeation experiments show that there is no significant increase in the permeance with the mean pressure. Thus, as a first approximation, the viscous flow through these small nanoapertures could be considered negligible versus Knudsen flow [

13].

The different flow regimes can be described by the dimensionless Knudsen number: Kn = λ/d, λ being the mean free path of the molecule and d the diameter of the pore. According to the Kn number, the gas flow behaviour can be divided into viscous (Kn < 0.01), transition (0.01 < Kn <1) and molecular (Kn > 1) flow regime. In our case, the Knudsen number is between 0.76 and 0.84, which indicates that we are dealing with transition flow.

The transition flow through an ultrathin nanosieve membrane on top of a microsieve membrane, (see

Figure 5) was described by Unnikrishnan et al. as a linear addition of viscous and molecular fluxes [

12]. The nanosieve membrane had a thickness of 45 nm with circular nanopores of 120 nm supported on top of a microsieve membrane made of straight cylindrical pores of 6 μm, with a length of 80 μm. The flow through the membrane was described as a series resistance model where the total pressure drop (Δ

P_{total} [Pa]), i.e., the resistance, is the sum of the pressure drop through the micropores (Δ

P_{micro} [Pa]), and the pressure drop through the nanopores (Δ

P_{nano} [Pa]), see Equation (2)). The total flow (

Φ_{total} [mol/s]), is the same as the flow through the micropores (

Φ_{micro} [mol/s]), and the flow through the nanopores (

Φ_{nano} [mol/s]), due to the conservation of mass (see Equation (3)):

Then, the transition flux is a linear addition of the viscous and molecular fluxes:

where

F_{i} represents the flow conductance, [mol/s Pa] viscous or molecular, either through the micropores or nanopores, that is represented by the following equations; in the case of molecular flow through the nanopores:

where

F_{i-molecular} is the flow conductance in the molecular flow regime through nanopores, i.e., [mol·s

^{−1}·Pa

^{−1}],

A is the total membrane surface area, see

Table 1 [m

^{2}],

ε is the porosity,

M is the gas molecular weight [kg·mol

^{−1}],

R is the gas constant [J·mol

^{−1}·K

^{−1}], and

T is the temperature [K]. The term (1 +

t/

d), where

t is the thickness of the pore and

d the diameter of the pore, is the Clausing function that considers the collisions of gas molecules with the walls of the pore and it is related to its geometry [

14].

In the case of viscous regime, the flow conductance can be described by:

where

P is the arithmetic mean pressure, and

η is the viscosity [Pa s] of the gas used,

t is the thickness of the pore and

r is the radius of the pore. The term (1 + 8

t/3

πr) describes the frictional losses experienced by the gas due to interaction with the pore surface and (1 −

f(

ε)) quantifies the influence of flow through the neighbouring pores on the flow through a single pore and, in the case of the straight cylindrical pores, was estimated as 0.9743 for the nanosieves [

12].

The system that we have to simulate is more complex compared to the one presented by Unnikrshnan et al. Our system has a 3D structure made based on 3rd-generation fractals emerging from the 2D membrane (SOI-wafer). Additionally, it is difficult to estimate the real size of all nanoapertures due to the fact that some could be closed, or only partially opened. Thus, to apply the model described above we have made several assumptions and simplifications, to obtain an estimation of the theoretical flux useful for a preliminary validation of our experimental data. We represent our membrane system in a similar way, as in the case of the ultrathin nanosieve [

12], made of micropores and nanopores. According to IUPAC, the apertures in the porous membranes can be divided into three main groups regarding their size: micropore (pore size not exceeding 2 nm), mesopore (size in the range of 50 nm to 0.05 μm), and macropore (larger than 0.05 μm). However, following the nomenclature used in the article of Unnikrshnan et al. [

12], in our case the micropores correspond to the different fractal generations of micrometer size and the nanopores correspond to the 3rd-generation nanonozzles with a thickness

t = 67 nm (the remaining SiO

_{2} after removing Si

_{3}N

_{4}, see

Section 2.1) and an average diameter

d = 100 nm. Considering the size of the micropores, in the range of 1 to 20 microns, we assume that the pressure drop accounts only through the nanoapertures. The porosity of the chip was calculated as the ratio of the total open nanopore area over the total channel area (see

Table 1).

The calculated values for the transition flow in the case of Chip 1 are presented as dashed lines in

Figure 8. The estimation of the permeance is around 4 times higher in the case of Chip 1 and 9 times for Chip 2 (not presented in

Figure 9). These differences could be attributed to the difficulties in the estimation of the real size of the pore apertures, which could vary from 50 to 150 nm. The theoretical values show that the contribution of the viscous flow to the total flow is between 14% and 20% depending on the mean pressure. This agrees well with our initial observation about the importance of the molecular flow over viscous. In case of the Knudsen diffusion mechanism, the permeation of the gases depends on the molecular weight and is inversely proportional to the square root of

M (see Equation (5)). Thus, diffusion of smaller molecules is faster compared to larger molecules and, accordingly, the ideal selectivity of O

_{2} over CO

_{2} could be calculated as the square root of 44 over 32, which results in a value of 1.2, whereas we found a slightly higher value of 1.6 in our experimental data (

Figure 9).

The permeation values obtained, in the order of 10

^{−7} mol/m

^{2}·s·Pa, are high in comparison with polymeric materials, such as polydimethylsiloxane (PDMS). Considering a permeability value for PDMS of 620 Barrer for oxygen [

4], a membrane film of just 1–4 micrometers would be required to obtain the same permeation flux as the nanonozzles presented here. However, such thin PDMS membranes cannot be handled.

Table 2 shows the experimentally-obtained fluxes of O

_{2} and CO

_{2} in two chips as a function of P

_{mean} and their corresponding theoretical thicknesses of the PDMS film.

The gas permeation of a fractal membrane is worse than, for example, anodized alumina membranes with 200 nm pores {Cooper, 2003 #30}. However, it would be very difficult and time consuming to integrate alumina membranes in the microfluidic chip. Moreover, there is a risk of membrane breaks or cracks in the presence of high pressure and leaks through the system. Depending on the reagents, the fabrication of anodized alumina can be very expensive.