## 2 Mercury Intrusion Porosimetry (MIP)  
The principle for MIP is simple, and it allows rapid detection speed across a wide range of pore throat sizes. MIP consists of intruding the cementitious sample with mercury by applying external pressure. Thanks to the non-wetting nature of mercury, the required pressure can be converted into pore sizes according to the Washburn equation (Equation (2)).  
\[r=-\frac{2\gamma\cdot cos\theta}{p}, \tag{2}\]  
where \(r\) is the pore entry radius, \(p\) is the pressure, \(\gamma=0.48\) N/m is the surface tension of mercury, and \(\theta\) is the wetting contact angle of mercury. The technique has been connected with some shortcomings (e.g., possible damage to the samples, pore shape assumption, toxicity) [13; 14], but it still allows a valuable indication of the pore structure of cementitious materials.  
MIP studies generally report the determination of the intrudable volume, pore entry size distribution, critical pore entry size and threshold pressure. The threshold pressure denotes the largest accessible pore size that mercury can ingress into. The accessible pore entry size distribution, \(PESD(r)\), is derived from the cumulative intruded volume [15], and it can be calculated as in Equation (3):  
\[PESD(r)=\frac{p}{r}\frac{dV_{t}}{dp}, \tag{3}\]  
where \(V_{T}\) is the intruded volume of mercury.  
A single mercury intrusion-extrusion cycle of MIP is generally applied and interpreted following the cylindrical-shaped model. The cumulative volume of mercury is not fully accountable to a singular pore size. Whereas the required pressure is determined by the pore entry size, the intruded volume corresponds to the entire cavity [15]. After the first cycle, a volume of mercury remains permanently trapped, identifying the corresponding cavity volume of the so-called ink-bottle pores. These pores require higher pressure than the one corresponding to their cavity size, which depends on the narrower pore entry size. The application of multiple cycles of MIP could help in the differentiation between pore entry volume and pore cavity volume.  
Moreover, hysteresis is noted between the entering volume during intrusion and the exiting volume during retraction [16]. Normally, only one intrusion is used for describing the pore entry size distribution. Several hypotheses have been suggested to explain this hysteresis. The most accepted ones relate to the presence of ink-bottle pores, different contact angles for intrusion and retraction, and pore connectivity [14; 15]. Convenient simultaneous analysis of intrusion and retraction hemicyes can provide additional information.  
The complex nature of the microstructure of cementitious materials requires complex geometrical models for interpreting MIP data. The fractal theory [5] provides an approach to describe the geometry of the pore structure by the fractal dimension of its surface, with values varying between 2.0 and 3.0. The higher value for the fractal dimension indicates the higher complexity of the surface. In practical terms, for a similar pore size range, a higher surface fractal dimension can be interpreted as a higher roughness. When applied to the analysis of MIP results, a linear relationship between pressure and mercury saturation can be constructed [17], and the fractal dimension is calculated as the slope of this relationship.