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Determination of the moisture
3.1. Transport of ions in the liquid phase
The transport model is based on the observation that the transport of ions only occurs
in the liquid phase. Hence, no equation has to be developed for the solid or gaseous
phase. The conservation equation for an ionic species i in the liquid phase at
pore scale is given by the following microscopic equation
[13]:
 | (52) |
The quantity ci is the concentration, ji is the ionic flux, and ri is a source/sink term accounting for the homogeneous chemical reactions [39] between ions in solution. The bulk equation is obtained from averaging this equation over the REV, following the procedure that lead to Eqs. (13) and (34):
 | (53) |
The average of the time derivative leads to:
 | (54) |
The first integral on the right-hand side of Eq. (54) contains a term that accounts for the velocity of the solid/liquid interface. While this interface may possibly move as a result of some dissolution/precipitation chemical reactions, it will do so very slowly. It can thus be neglected. The other integral involves the movement of the liquid/gas interface. It is similar to the first integral in Eq. (14). While it was used in the mathematical development of the moisture transport, it is assumed that it has only a small effect on the ionic transport and can thus be neglected, simplifying Eq. (54) to:
 | (55) |
The average of the divergence is given by:
 | (56) |
The last integral on the right-hand side of Eq. (56) accounts for the ionic flux crossing the liquid/gaseous interface. The value of this flux is zero because ions do not go into the gaseous phase. The other integral, related to the flux of ions across the solid/liquid interface, will be used to model the various chemical reactions involving those phases. Accordingly,
Eq. (56) can be reduced to:
 | (57) |
Substituting Eqs. (55) and (57) in Eq.
(53) and averaging the term 
by 
(see Eq.
(9)), one finds:
 | (58) |
The next step consists of writing the proper flux expression at the microscopic level (ions in bulk electrolyte) and averaging it over the REV. Due to the charged nature of ions, this expression has to consider the electrical coupling between ionic particles. Furthermore, because the pore solution of cement-based materials is highly concentrated, it deviates from the ideal behavior of a dilute solution, requiring consideration of the chemical activity. Finally, the movement of the fluid itself will have an impact on the movement of ions. All these physical phenomena can be taken into account through the extended Nernst-Planck model to which is added an advection term [18]:
 | (59) |
The parameter Diμ is the self-diffusion coefficient [30] of species i in diluted, free water conditions, γi is the chemical activity coefficient, ψ is the electrical potential, zi is the valence number of the ion, F is the Faraday constant, R is the ideal gas constant, and T is the absolute temperature. The terms on the right-hand side of Eq. (59) are associated with diffusion, electrical coupling between the ions, chemical activity effects, and water transport, respectively.The integration of the flux over the REV, similar to the procedure followed in Eqs. (9), (11), (12),
and (13) leads to:
 | (60) |
The next steps consist in averaging the various gradients and variables in Eq. (60).The average of the concentration gradient is given by
[12] and [13]:
 | (61) |
The quantity τL is the tortuosity of the aqueous phase. It is a purely geometrical factor, accounting for the complexity of the paths the ions must travel through in liquid space. It is a function of the water content θL because it is related to the volume of liquid in the pore space, and its value is less than one
[12]. The parameter x˚ was first
encountered in Eq. (40).To evaluate the surface integrals in Eq.
(61), one has to refer to the double-layer theory [31]. Fig. 2 shows the cross-section of a pore and the schematic shape of the concentration profile along its radius. The solid bears an electrical surface charge σsolid. It is neutralized by charges of the opposite sign in two different zones, the Stern and the diffuse layers, bearing, respectively, σstern and σdiff charge per unit area. The electrical balance respects the following expression:
The external limit of the Stern layer, called the outer Helmholtz plane or the shear plane, separates the solid from the aqueous phase, in which ionic diffusion occur. The aqueous phase is divided in the diffuse layer and the free water zone, where ions do not feel the effect of the solid/liquid interface. A recent study by Revil
[32] showed that ionic transport may occur in the Stern
layer. But it was also mentioned in the paper that this phenomenon is negligible
with respect to transport in the bulk pore when the pore solution is highly
concentrated, as it is the case in cement-based materials. Consequently, only the ionic
transport in the aqueous phase is considered in this paper. Finally, the description of
the cross-section of the pore is complete by considering a gaseous phase at the center
of the pore, when the latter is not saturated [33].
Fig. 2. Concentration and potential profile across a pore near the solid/liquid interface according to the double-layer theory.
It is assumed that the concentration profiles at the liquid/gas interface is flat (see Fig. 2). Consequently, the second integral in Eq. (61) is negligible because there is no concentration gradient along the radius at the liquid/gas interface. The situation is different for the first integral because of the concentration gradient along the radius at the solid/liquid interface caused by the electric charge at the surface of the solid. Simple double-layer calculations made with the Gouy–Chapman model
[31] are shown on Fig. 3. They emphasize that increasing the ionic strength of a solution in the vicinity of a charged surface decreases dramatically the thickness of the double layer. Since cementitious materials bear a highly charged solution, the double layer extends over a very small region. Outside this region, the concentration profile is unaffected by the surface charge. Following this, the term ∫x˚(gradci·nLS)dS in Eq. (61) is neglected, leaving:
 | (63) |
>
Fig. 3. Concentration profiles of a 1–1 electrolyte near a charged surface calculated with the Gouy–Chapman double-layer model. The calculations were made with a surface potential of 25 mV. The Debye length κ−1 is indicated.
Then one needs to average the term in Eq. (60) concerned with the electrical coupling between the ions. According to the procedure for averaging a product [12] and [13], one can write:
 | (64) |
where the quantities topped by ˚, called deviations, are defined as:
 | (65) |
It is assumed that the deviations lead to small terms, which allows to neglect the deviation product in Eq. (64):
 | (66) |
Following the same procedure as the one used for the concentration gradient, the average of the potential gradient gives:
 | (67) |
Fig. 2 shows a potential profile across the section of a pore. According to the
double-layer models [31], it has a shape similar to the
concentration profile; that is, it is disturbed near the solid/liquid interface but tends to a flat profile toward the center of the pore. And like the concentration profiles shown on Fig. 3, increasing the ionic strength of the solution reduces the thickness of the area where the gradient of potential is different from zero. Accordingly, the integrals are neglected, assuming again that the electrical phenomena near the interface do not affect ionic movement. Eq. (67) thus simplifies to:
 | (68) |
 | (69) |
The same approach is used to average the chemical activity term in Eq. (60). The same assumptions concerning the deviations, as well as those concerning the effect of the electrical phenomena at the solid/liquid interface, lead to:
 | (70) |
It is assumed that the term 
corresponds to the chemical activity coefficients calculated with the average concentrations
. For simplicity,
is approximated by
:
 | (71) |
Finally, the advection term in Eq. (60) is averaged as:
 | (72) |
The term in Eq. (72) containing the deviations is called the dispersive flux [13] and [34]. It is shown in the previous references that it can be written under a Fickian form:
 | (73) |
where Ddisp is called the coefficient of advective dispersion and is due to fingering, not diffusion. Consequently, this term can be added to the ionic diffusion term that would then exhibit a new diffusion coefficient being the sum of the classical one plus the coefficient of advective dispersion. When the fluid is in movement under the effect of a water content gradient, as described in the preceding section, the velocity is relatively
weak. In that case, the dispersion term can be neglected [35], leading to:
 |
(74) |
 | (75) |
The diffusion coefficient at the macroscopic level Di is defined as:
 | (76) |
To simplify the expression, let:
 | (77) |
 | (78) |
 | (79) |
Eq. (79) is now inserted in the averaged mass conservation Eq. (58) to yield the macroscopic ionic transport equation:
 | (80) |
To simplify this equation, the integral must be expressed in a manner that is more friendly to a further numerical analysis. The term (ji·nLS) gives the amount of ions crossing the solid/aqueous phase interface, as a result of dissolution/precipitation or ion exchange reactions. It is possible to express it differently by performing the averaging operation on the ions in the solid phase [13]. The conservation equation at the microscopic scale for the ions in solid phase is:
 | (81) |
where the subscript s designates the solid phase. Contrary to Eq. (52), it is assumed that no chemical reactions occur within the solid phase because all precipitation/dissolution phenomena are taking place at the solid/aqueous phase interface. Averaging Eq. (81) over the REV leads to:
 | (82) |
where θs is the volumetric fraction of the solid phase and nSL is an outward (to the S phase) unit vector on the solid/aqueous phase interface (designated as SSL). The integral in Eq. (82) has the same value as the one in Eq. (80) but with an opposite sign because the ions coming out of the aqueous phase are being bound by the solid phase. Furthermore, the flux jis within the solid is zero because there is no ionic movement in this phase. This allows one to write:
 | (83) |
Substituting Eq. (83) in Eq. (80) gives:
 | (84) |