# On the Tortuosity-Connectivity of Cement-Based Porous Materials
## 1 Introduction  
The durability of concrete, which can be defined as its resistance to weathering action, chemical attacks and other degradation processes, is one of the most significant areas of research interest. Specifically, one of the most serious causes of durability problems affecting reinforcing steel is the introduction of chloride ions. The penetration of this aggressive ionic species into reinforced concrete induces steel corrosion and leads to premature deterioration of structures exposed to marine or de-icing salt environments [1, 2]. As a result, most modern codes of practice have adopted strict maximum levels for chlorides permitted in concrete structures, among others the Spanish Standard of Structural Concrete EHE-08 [3]. Therefore, studying the processes involved in transport of ionic species in concrete is of utmost benefit. Accurate prediction of the behaviour of concrete in different environments has been a challenge until now. The means by which ions penetrate into concrete at a given exposure time are complex and depend on several variables of the material, particularly on the moisture of the material. [PERSON] [4] pointed out that, among the different penetration mechanisms, four should be considered, namely diffusion, migration, absorption/capillary suction and permeation. An enormous variety of mathematical models has been proposed over the years with the main goal to explain and predict the penetration of ionic species into concrete [5, 6, 7, 8, 9, 10]. The most simple models are generally based on the diffusion equation with an apparent diffusion coefficient which depends on the material as well as on the concentration of the ionic species [11, 12, 13, 14]. The description of the various processes involved, requires the consideration of various transport mechanisms coupled with the effect of the interaction between the ion and the pore surface, which is very challenging. Despite this, some authors have recently proposed advanced approaches aiming at modelling the features of chloride transport in difficult and variable exposure conditions. Thus, [PERSON] et al. [15] studied how different types of cation influence the diffusive process ofchloride ions in concrete. [PERSON] et al. [16] proposed a 3D chemo-hygro-thermo mechanical model for reinforced concrete to simulate physical processes related to corrosion of steel bars. In a more recent work, [PERSON] et al. [17] have proposed a model that couples the ionic diffusion process with the concrete microstructure evolution. The problem is further complicated when dealing with heterogeneous and porous materials, such as concrete. Indeed, the complex geometry of the porous network of cement-based materials is tortuous in nature. The geometry of the pathways of this materials for the penetration of aggressive agents is very complicated, even further due to its heterogeneity and particularly the higher porosity of the interface zone can facilitate the ingress of external aggressive agents and the development of deleterious chemical reactions. [PERSON] et al. [18] found a notably higher effective diffusion coefficient of chloride ions in the interface zone (6 to12 times greater) than in the bulk cement paste. So, the detailed modelling of cement-based materials microstructure requires to consider different levels of approximation, which would greatly complicate the modelling. In addition, micro-models consider a large number of parameters which are usually difficult to calibrate through specific tests [17]. Seeking to more practical approaches, in previous works, a constant tortuosity factor has been adopted which assumes that all the effects arising from pore orientation, connectivity, size variation, etc., can be encompassed by a mean value valid for all pore sizes [19; 20; 21]. This is obviously a gross assumption since it depends on the implicit assumption that the effects of pore geometry and structure are the same for all pore sizes, even so, if the pore structure is characterized in detail, a sufficiently accurate aproach can be made.