On the tortuosity-connectivity of porous materials

In this work, the transport equations of ionic species in concrete are studied. First, the equations at the porescale are considered, which are then averaged over a representative elementary volume. The so obtained transport equations at the macroscopic scale are thoroughly examined and each term is interpreted. Furthermore, it is shown that the tortuosity-connectivity does not slow the average speed of the ionic species down. The transport equations in the representative elementary volume are then compared with the equations obtained in an equivalent pore. Lastly, comparing Darcy’s law and the Hagen-Poiseuille equation in a cylindrical equivalent pore, the tortuosity-connectivity parameter is obtained for four di ﬀ erent concretes. The proposed model provides very good results when compared with the experimentally obtained chloride proﬁles for two additional concretes.


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 (2,3). 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 (1). 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. Bertolini (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.
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 This work focuses on the influence of the geometric complexity of the porous network on the different mass fluxes. First, the equations at the porescale are considered, which are then averaged over a representative elementary volume. This procedure shows the origin of the tortuosity-connectivity parameter used for modelling ionic transport in porous media. An explicit expression for the tortuosity-connectivity parameter is finally obtained by comparing Darcy's law and the Hagen-Poiseuille equation in a cylindrical equivalent pore. This expression can be written as a function of the pore radii, or more conveniently, as a function of the pore water content.

General considerations
In order to determine the differential equations which govern the transport problem at the macroscopic scale, the equations at the pore scale are integrated in a representative elementary volume (REV). The REV Ω is assumed to contain three phases, namely the liquid phase (the pore water), the gaseous phase (air) and the solid phase (concrete). Furthermore, it is assumed that the ionic species are transported solely through the liquid phase, and the solid phase is assumed to be inert. The REV can be seen as a whole of multiple elementary volumes. In this work, only homogeneous materials are considered, so that each elementary volume has the same properties. The characteristic length of the elementary volume is denoted as l, while the characteristic length of the REV is denoted as L. Furthermore, the size of the REV is chosen such that = l/L << 1. A representation of one elementary volume, projected onto a 2D plane, is depicted in figure 1. The volume of the liquid phase is denoted as Ω l , while the boundaries between the phases i and j are denoted as A i j . Finally, the unit vectors n i j are normal to the i − j interface pointing from the i−phase to the j−phase.  Throughout this work, the following notations are adopted: Note that equations 1-2 are related as follows: where φ l = Ω l /Ω is the pore water content.
Furthermore, let u be a quantity related to the l−phase and consider the following decomposition (19): ũ Ω l = 0 whereũ is the deviation of u with respect to the average u Ω l . From equation 5, the following relations may be deduced: Equation 6 states that the average of the average value of u in Ω l is simply the average itself, while equation 7 states that the deviation ofũ is equal toũ.
Moreover, the following theorems are used throughout this work (20,21,22) : where J is a vector quantity related to the l−phase, i refers to the g−phase and the s−phase and v li is the velocity vector of the l − i interface.
Equations 8-10 referred to the volume Ω l read: ∇u Applying equations 8-9 to u = 1, the following lemmas are obtained: 1 Using equations 14-15, equations 11-12 can be rewritten as: ∇u 3. The transport equations at the pore scale Denoting the concentration, mass flux and reaction terms of species i by u i (kg/m 3 ), J i (kg/m 2 /s) and r i (kg/m 3 /s), respectively, the transport equation at the pore scale is formulated as follows: where each term is expressed in units of mass per volume of pore water per second (kg/m 3 /s).
The total mass flux is the sum of the diffusive mass flux J i D , the advective mass flux J i A , the mass flux related to migration J i M and the mass fluxes due to temperature gradients and chemical activity J i T (5): where D (m 2 /s) is the diffusion coefficient at infinite dilution and solely depends on temperature, a (m/s) the advection velocity (i.e. the velocity of the pore water), z the valence, F (C/mol) the Faraday constant, R (J/mol/K) the gas constant, T (K) the temperature, Φ (V) the electric potential, γ (m 3 /kg) the activity coefficient and a the chemical activity. An explicit expression for the advection velocity a will be given later.

The transport equations at the macroscopic scale
In order to obtain the equations at the macroscopic scale, the microscopic transport equation needs to be integrated in the REV. Thus, equation 24 is first integrated in the elementary volume Ω: In the following, each term will be treated separately.

The time derivative
The average of the time derivative is obtained by applying Reynolds' transport theorem (equation 8) to the first term on the left-hand side of equation 24: It should be noted that the velocity of the l − s interface is not necessarily equal to zero. Indeed, if the ionic species react to the cement matrix, the boundary between the l−phase and the s−phase may be deformed. However, for the sake of simplicity, it is assumed that this deformation does not alter significantly the transport of the ionic species, which simplifies equation 26 to:

The divergence of the mass fluxes
The average of the divergence of the total mass flux is obtained by applying the averaging theorem (equation 10) to the second term of the left-hand side of equation 18: Since the ionic species are assumed to be transported solely through the l−phase, and assuming that the temperature is constant within the elementary volume, the following boundary conditions can be applied: The assumption of a constant temperature within the elementary volume, which will be adopted in the rest of this paper, can be justified by the high thermal conductivity of concrete when compared with the slow penetration of the ionic species and the pore water (5). Moreover, it was found in (5) that temperature plays an important role in the transport of ionic species in concrete, while temperature gradients do not. Now, the average of the divergence of the mass fluxes are obtained by means of equation 28 and the boundary conditions 29-32:

The transport equations at the macroscopic scale
Substituting equation 27 and equations 33-36 into equation 25, the transport equations at the macroscopic scale are obtained: The velocity v lg represents the total velocity of the l − g interface, while a at the boundary is the velocity of the interface due to the movement of the l−phase. In other words, unlike a, v lg accounts for the velocity of the interface due to the evaporation and condensation of the pore water. The velocity v lg can thus be interpreted as the sum of the velocity at the l − g interface a| lg and the velocity of the boundary due to evaporation and condensation v lg,vap : Combining equations 14 and 38, the variations of the pore water content with time can be expressed as follows: where the second surface integral of equation 39 represents the change of φ l with time due to evaporation and condensation of the pore water. Mainguy et al. (23) showed that when concrete is subjected to drying processes the pore water is evaporated at the surface of the material, and the evaporation within the material can be ignored. Therefore, the effects of evaporation and condensation within the elementary volume are neglected, so that v lg,vap = 0.
Equation 37 can now be rewritten as: or with respect to volume Ω l :

The mass fluxes
Explicit expressions for the mass fluxes are given in equations 20-23, except for the advective mass flux, which depends on the advection velocity a. The advection velocity is not known a priori and depends strongly on the pore water content φ l . In this work, a is modelled by means of Darcy's law, based on research work (5): where k l (m 2 ) is the permeability of the porous medium, ν l (Pa · s) is the dynamic viscosity of the pore water, and p c (Pa) is the capillary pressure. The permeability of the porous medium relative to the pore water depends on the pore water content, the dynamic viscosity of the pore water depends on the concentration of the ionic species and on temperature, and the capillary pressure depends on the pore water content, temperature and porosity (5). As the porosity can change due to chemical reactions of the ionic species with the cement matrix, and those reactions depend directly on the concentration of the ions, the capillary pressure may be expressed as a function of the concentration, rather than the porosity.
The advective mass flux (21) at the pore scale now reads: The averages of the mass fluxes are obtained by applying the average theorem (equation 9) and the decomposition, as shown in equation 4 to the fluxes defined in equations 20, 22-23 and 43: where the subscript j refers to the g− and s−phases.
Note that, in equation 44, there is no need to calculate the average of the diffusion coefficient. Indeed, D solely depends on temperature which is assumed to be constant in the elementary volume. Furthermore, it should be noted that equation 45 was obtained by assuming that the deviations of u i within the elementary volume are sufficiently small so that the dynamic viscosity ν l can be assumed constant in the elementary volume. This approximation can be expressed as follows: Substitution of the expressions 44-48 into equation 40 yields the final transport equation in the elementary volume:

The dispersion terms
In this subsection, it is shown that the dispersion terms of equation 50 can be ignored in the REV. To that end, consider a quantity m related to the l−phase. According to equation 4, m can be decomposed as: Since the microscopic quantity m may vary significantly over a distance equal to the characteristic length l, the magnitude of the deviation of m with respect to the average m Ω l can be estimated as follows: Furthermore, the magnitude of a quantity at the macroscale only varies significantly over distances larger than L, which can be expressed as: Applying those approximations to equation 50, and noting that the diffusion coefficient can change within the REV due to temperature variations, the final transport equation in the REV is obtained: whereΩ,Ω l andÂ are related to the REV.

The surface integrals
From equation 54, it may be observed that all the mass fluxes in the REV have the form: where J i is a mass flux of species i and f is a scalar.
In order to interprete the terms on the right-hand side of equation 55, consider an elementary volume with only one straight cylindrical pore. The REV can then be thought of as a volume with a single cylindrical pore. In such case, the second term on the right-hand side of equation 55 is zero. Therefore, the first term can be interpreted as the mass flux through the cylindrical pore. The second term corrects the mass flux by accounting for the complex geometry of the porous medium as shown below.

The surface integrals at the pore scale
The second term on the right-hand side of equation 55 is often associated with the tortuosity-connectivity of the porous medium. Generally, the more tortuous the porous material is, the slower is the penetration of the ionic species. However, it should be noted that the term which accounts for the geometry of the porous network does not slow down the speed of the ionic species. To illustrate this, the simple diffusion equation at the pore scale for some ionic species i is considered: where ∆ is the Laplace operator.
Two kinds of pores are analysed, namely a straight cylindrical pore and a tortuous pore. The dimensions of the pores were chosen such that the length and volume of each pore were equal.
A concentration of u i = 1kg/m 3 was imposed on the left boundary of the pores. The remaining boundaries are taken to be the l − s interface, where a zero mass flux is imposed. The value of the diffusion coefficient was chosen such that: where t sim refers to the total simulation time, and d the length of the pore.  In conclusion, the tortuous pore may be represented by means of a straight pore of equal volume, orientated in the direction of the net mass flux, and of length equal to the length of the tortuous pore projected onto the direction of the net mass flux. The latter will be referred to as the equivalent pore. The mass flux in the tortuous pore can then be expressed as the sum of the diffusive flux in the equivalent pore and a term in the form of a surface integral which accounts for the geometry of the tortuous pore, as stated in equation 55.

The surface integrals in the REV
In the REV, the porous network can be significantly more complex than the previously defined tortuous pore. An equivalent pore of length L and volume φ l can be constructed, oriented in the direction of the net mass flux. The mass flux averaged over the equivalent pore is then equal to the mass flux averaged over that part of the REV which contains pore water: whereΩ eq refers to the volume of the equivalent pore.
Note that the average of the mass flux in the equivalent pore, as defined in equation 58, accounts for the tortuosityconnectivity of that part of the elementary volume which contains pore water. The last term reduces the mass flux in the direction parallel to the equivalent pore. Furthermore, the direction of the equivalent pore is parallel to the direction of ∇ u i Ω l . Taking this into account, equation 58 can be approximated as: where σ is a dimensionless positive function which accounts for the reduction of the mass flux in the direction of the equivalent pore.
Finally, the average of the mass flux in the REV is obtained by multiplying equation 59 by φ l :

The final transport equation in the equivalent pore
Based on the results obtained in section 4.6.2, the transport equation in the equivalent pore related to the REV is given by: Finally, defining the tortuosity-connectivity parameter as τ = 1 − σ, the final transport equation is obtained:

Darcy versus Hagen-Poiseuille flows
The tortuosity-connectivity parameter can be determined by comparing the Darcy flux with the Hagen-Poiseuille fluxes in a cylindrical pore. The following analysis, it should be noted, is valid only for homogeneous materials. Consider a homogeneous material, partially saturated with pore water which is assumed to be continuously and uniformly distributed. Then the equivalent pore related to the porous network which contains pore water can be chosen to be a cylindrical pore of constant section. In that case, the Darcy and Hagen-Poiseuille fluxes must coincide, which leads to the expression of the tortuosity-connectivity as a function of the permeability of the porous medium.
Based on the obtained results, the Darcy flux in the equivalent pore can be expressed as: wherek l = τk l is now a macroscopic function which describes the permeability of the complex and tortuous porous network.
The Hagen-Poiseuille flux in the equivalent pore reads: where r is the radius of the pore.
Setting equations 63 and 64 equal, the tortuosity-connectivity parameter can be written as a function of the permeability function: The tortuosity-connectivity for a fully saturated homogeneous material can be calculated from equation 65 with r = R, where R is the radius of the saturated equivalent pore associated to a fully saturated REV: The radii r and R are related according to the following expression: where φ is the porosity of the REV.
The tortuosity parameter can now be expressed as a function of the pore water content: The permeability is often expressed as the product of an intrinsic permeability coefficientK (m 2 ) and a permeability function relative to the l−phasek rl , which takes values between 0 and 1. Equation 68 can then finally be rewritten as:

Calculation of the tortuosity-connectivity parameter
Experimental measurements of the tortuosity-connectivity would require determining the ttortuosity-connectivity for various values of the pore water content, which should be distributed uniformly within the material. Methods to obtain a uniformly distributed pore water content in concrete can be found in research work (24,25,26).
According to equation 69, the tortuosity-connectivity parameter of the porous medium can be calculated if its permeability is known. The permeability is often modelled by means of the van Genuchten model: where e is a material constant.
The expression for the tortuosity-connectivity parameter becomes then: In order to calibrate this model with experimental results, Equation 71 1 was used in this study, where the material constant e and the tortuosity-connectivity in saturated conditions τ(φ) were measured for four different concretes, by fitting Darcy's law to experimentally obtained data.
The four concretes were designed in acordance with usual mixes used in agressive environments to improve the impermeability and resistance against penetration of ionic species. Concrete dosages are referred to as mat1, mat2, mat3 and mat. The cement type used for materials mat1 (without additions), mat2 (with addition of 20% of fly ash) and mat4 (with addition of of 10% of silica fume) is I 42.5 R/SR according to the RC-16 Standard (? ). The cement employed for material mat3 contains 66% blast furnace slag and its type is known as III/B 42.5 L/SR, in agreement with the RC-16 Standard (? ). The properties of the materials are described in detail in (28). The dosages of the materials are given in  The tortuosity-connectivity function of the materials, obtained from the experimental results, is plotted in figure 4. It is shown that the tortuosity-connectivity parameter decreases significantly for lower pore water contents, meaning that the mass fluxes are significantly reduced in the direction of the net mass flux. In saturated conditions, the tortuosity-connectivity functions reach their maxima. It can be observed that the tortuosity-connectivity undergoes hysteresis, which is due to the fact that the pore water can get trapped in ink-bottle shaped pores when the concrete is subjected to drying conditions (5). It is worth noting that this phenomenon has been found significantly greater for mat1 and mat2, probably due to a significantly higher amount of bottleneck pores, being the two mixtures with the higher total porosity as can be seen in (28). Similarly, mat1 and mat2 presented the greater value of the tortuosity-connectivity parameter. In the case of mat3 and mat4, the tortuosity-connectivity parameter obtained was considerably lower, with also a lower total porosity and average pore size (28) than the rest. This is coherent with the results obtained by (28) for differential thermal analysis, mercury intrusion porosimetry, water and gas per-meability and mechanical properties of The materials, in which the mixtures could be divided into two groups according to its similar behaviour, presenting better performance for mixtures mat3 and mat4.  Using these tortuosity-connectivity functions, the diffusion coefficient for ionic species τ(φ l )φ l D i can be obtained. Such coefficients were measured for chloride ions and for several pore water contents. As can be deduced from figure  4, the diffusion coefficient τ(φ l )φ l D i increases with the pore water content.

Closure equations
As shown in (5), the terms in equation 62 related to migration and chemical activity can generally be ignored. For the sake of simplicity, the same assumptions are made in this paper, yielding: The capillary pressure p c Ω l was modelled by means of the capillary model proposed in (5): where A (MPa) is the capillary modulus, and a and b are material constants.
The reaction term r i Ω l was modelled by means of the Langmuir adsorption equation, which in this case can be expressed as (5): where k eq (m 3 /kg) is a constant which describes the equilibrium between the free ions and the bound ions, while û i Ω lΩ  Once the tortuosity-connectivity parameter had been calibrated with concretes widely studied by the authors and its good fit had been verified to adequately simulate the experimental results obtained, the model was tested by comparing the numerical simulations with experimental chloride profiles obtained from two different concretes. The latter were obtained from two different mixtures (referred to as mat5 and mat6) selected to represent the two types of behaviour observed in the previous experimental campaign. This way, in the new experimental campaign the reference concrete mat5 is a mixture without additions representing the group mat1 and mat2, and mat6 is a mixture containing 10% of silica fume, representing the group with the best general durable perfomance mat3 and mat4. Both dosages were based on Portland cement. The water-cementitious materials ratio were 0.40 and 0.45 for the concrete with silica fume. More details are given in table 2. These materials were tested under several boundary conditions in order to experimentally simulate half a year of service life of a concrete subjected to high mountain environment with melting salts, which is a highly aggresive environment with chlorides. The initially fully saturated concrete samples were subjected to five different aggressive environments for 180 days. The first 75 days, the samples were exposed to a chloride solution of 100g/l at a temperature of 1 • C. This phase simulates the winter season in which chloride solutions are sprayed over the roads and concrete structures in order to avoid the forming of ice patches on the road surface. As a result, the chloride ions start to penetrate into the concrete. On day 76, a thin layer (2cm) of fresh water was poured onto the exposed surface at a temperature of 10 • C, and was removed 30 days after. This phase corresponds to a rainy spring. The rain eliminates chlorides from the concrete structures. In order to simulate a drier summer, for the next 60 days, the concrete was dried at a relative humidity of 75% and temperatures of T = 16 • C and T = 5 • C (30 days each). During that season, the chloride ions are transported towards the surface, where they start to precipitate. Finally, the samples were exposed once more to a chloride solution of 100g/l at temperature 3 • C for 15 days. A schematic representation is given in

Numerical modelling
In order to account for the relative humidity and temperature, the transport model was coupled to differential equations based on Darcy's law (to solve for the pore water content) and on Fourrier's law (to solve for temperature). The coupled differential equations were solved by means of the finite element method. A detailed description of the formulation can be found in (5). The model was then calibrated for the materials as described in (5).
The numerical results versus the experimental chloride profiles are plotted in figures 5-7 for material mat5 and figures 8-10 for material mat6. The vertical axis represents the chloride profiles expressed in kg/m 3 of concrete, while the horizontal axis represents the penetration depth expressed in mm. It can be observed that a good fit is obtained as numerical simulations reproduce the experimental results with a significantly good fit. The capillary model enabled modelling of the pore water flow, accounting for the properties of the microstructure of the material. The microstructure was characterized by the tortuosity-connectivity parameter which was quantified in function of the pore radii for different materials.

Conclusions
The study carried out in this work led to the following conclusions: 1. In order to obtain the transport equations at the macroscopic scale, the microscopic equations were integrated over the REV by means of the well known averaging technique. 2. It was shown that the dispersion terms which arose from the averaging procedure can be ignored. 3. The surface integrals which arose from the averaging technique were interpreted. It was shown not only how they are related to the tortuosity-connectivity of the porous network, but also how they influence the mass fluxes of the ionic species. 4. The transport equations in the REV were compared with the equations in an equivalent pore, oriented in the direction of the net mass flux. 5. An explicit expression of the ttortuosity-connectivity parameter was obtained by comparing Darcy's law and the Hagen-Poiseuille equations for the equivalent pore. 6. The proposed model accurately fits the results obtained from experimental tests performed on concretes in non-saturated conditions.