## 2 Theory  
### Capillary Flow Porometry (CFP) Principles  
The capillary flow porometry technique, also known as fluid-fluid displacement porometry (FDP), mainly comprises two similar techniques, gas liquid displacement porometry (GLDP), and liquid-liquid displacement porometry (LLDP) (Figure 1).  
The CFP technique is based on the well-known Young-Laplace (Equation (1)) equation that governs the pressure difference at the interface between two immiscible fluids [11].  
Figure 1: Young–Laplace-based techniques.  
Gas liquid displacement porometry (GLDP) is a form a displacement technique in which both the differential gas pressure and the gas flow rates are simultaneously measured (first on a wet sample, followed by a similar measurement in a dry one). The technique has a typical lower size range of 50 nm for a maximum applied pressure around 14 bar, which restricts the application of the technique to the characterization of MF membranes or the upper limit of UF ones. To analyze membranes with pores smaller than 50 nm (as usually found in typical UF and all NF membranes), the air-liquid interface requires too much pressure, allowing for membrane structure damage. In this case, liquid-liquid displacement porosimetry (LLDP) circumvents this drawback by making use of a liquid-liquid interface [11] inside the pores. Then, the main difference between the two techniques is the use of a gas-liquid-solid versus a liquid-liquid-solid interface, the latter of which results in a substantial reduction of the surface tension of the corresponding interface and, accordingly, the necessity of using smaller pressures to analyze such small pores. As a result, the combination of GLDP and LLDP flow based techniques allows for the accurate characterization of membranes in across the entire range of MF and UF and, even, partially that of NF.  
Both techniques (Figure 2) consist of a pressurized fluid (gas or liquid) that is introduced forcefully inside the pores of a pre-wetted (by means of a proper wetting liquid) filter medium. Operationally, the procedure consists of gradually increasing the applied pressure so that pores of decreasing sizes are opened in each step. When this fluid pressure overcomes the capillary intrusion pressure of the largest pore, the displacement fluid can penetrate into those pores and push out the wetting liquid. The pore size \(d_{p}\) (\(m\)) of the pores open to flow at each applied pressure, \(\Delta P\) (\(Pa\)), is calculated by the Young-Laplace equation:  
\[\Delta p=\frac{4\gamma cos\theta}{d_{p}} \tag{1}\]  
where \(\gamma\) (\(n\)/\(m\)) is the surface tension of the wetting liquid, and \(\theta\) is the contact angle between the liquid and capillary wall. To obtain reliable results, the value of \(\cos\theta\) must be assumed to be equal to one (perfect wetting) for both techniques. The pressure at which the displacement fluid starts to flow through the pore is called a bubble point [12; 13; 14] (GLDP) or droplet point (LLDP). The pore size at the bubble/droplet point is termed as the maximum pore size of the sample filter. When the pressure is further increased, the displacement fluid flows through smaller pores.  
Figure 2: (a) Young–Laplace based techniques for gas liquid displacement; (b) liquid displacement.