MODELING IONIC TRANSPORT IN CEMENT SYSTEMS

As previously mentioned, the numerical results yielded by the NIST CEMHYD3D model were implemented in another model called STADIUM^{12, 13}. This latter model has been developed to predict the transport of ions in unsaturated porous media. The model also accounts for the effect of dissolution/precipitation reactions on the transport mechanisms.

The description of the various transport mechanisms relies on the homogenization technique. This approach first requires writing all the basic equations at the microscopic level. These equations are then averaged over a Representative Elementary Volume (REV) in order to describe the transport mechanisms at the macroscopic scale^{19, 20}.

In this model, ions are considered to be either free to move in the liquid phase or bound to the solid phase. The transport of ions in the liquid phase at the microscopic level is described by the extended Nernst-Planck equation²¹ to which is added an advection term. After integrating this equation over the REV, the transport equation becomes:

 

$$\begin{multline}\frac{\partial \bigl( (1-\phi)C_{is} \bigr) }{\partial t} + \fra... ...{\partial \ln \gamma_{i}}{\partial x} - C_{i} V_{x} \right) = 0 \end{multline}$$

where the uppercase symbols represent the variables averaged over the REV. In equation (2), C_i is the concentration of the species i in the aqueous phase, C_is is the concentration in solid phase, θ is the volumetric water content (expressed in m³/ m³ of material), D_i is the diffusion coefficient, z_i is the valence number of the species, F is the Faraday constant, R is the ideal gas constant, T is the temperature of the liquid, Ψ is the electrical potential, γ_i is the chemical activity coefficient and V_x is the velocity of the fluid. Equation (2) has to be written for each ionic species present in the system.

To calculate the chemical activity coefficients, several approaches are available. However, models such as those proposed by Debye-Hückel or Davies are unable to reliably describe the thermodynamic behavior of highly concentrated electrolytes such as the hydrated cement paste pore solution. A modification of the Davies equation described in reference²² was found to yield good results.

The Poisson equation is added to the model to evaluate the electrical potential Ψ. It relates the electrical potential to the concentration of each ionic species²³. The equation is given here in its averaged form:

$$\frac{\partial }{\partial x} \left( \theta \tau \frac{\partia... ...ht) + \theta \frac{F}{\epsilon} \sum_{i=1}^{N} z_{i} C_{i} = 0$$ (2)

where N is the total number of ionic species, ε is the dielectric permittivity of the medium, in this case water, and τ is the tortuosity of the porous network.

The velocity of the fluid, appearing in equation (2) as V_x , can be described by a diffusion equation when its origin is in capillary forces present during drying/wetting cycles²⁴:

$$V_{x} = - D_{w} \frac{\partial \theta }{\partial x}$$ (3)

where D_w is the non-linear water diffusion coefficient. This parameter varies according to the water content of the material²⁴.

To complete the model, the mass conservation of the liquid phase must be taken into account²⁴:

$$\frac{\partial \theta}{\partial t} - \frac{\partial }{\partial x} \left( D_{w} \frac{\partial \theta}{\partial x} \right) = 0$$ (4)

As can be seen, moisture transport is described in terms of a variation of the (liquid) water content of the material. It should be emphasized that the choice of using the material water content as the state variable for the description of this problem has an important implication on the treatment of the boundary conditions. Since the latter are usually expressed in terms of relative humidity, a conversion has to be made. This can be done using an adsorption/desorption isotherm²⁴.

The first term on the left-hand side of equation (2) (in which C_is appears), accounts for the ionic exchange between the solution and the solid¹⁹. It can be used to model the influence of precipitation/dissolution reactions on the transport process. More information on this procedure can be found in reference¹².

The transport of ions and water in unsaturated cement systems can be fully described on the basis of equations (2) to (5). Previous experience¹² has shown that most practical problems can be reliably described by seven different ionic species (OH⁻, Na⁺ , K⁺, SO₄²⁻, Ca²⁺, Al(OH)₄⁻ and Cl⁻ ) and five solid phases (CH, C-S-H, ettringite, gypsum and hydrogarnet).

The input data required to run the model can be easily obtained. The initial composition of the material (i.e. its initial content in CH, ettringite, etc.) can be easily calculated by considering the chemical (and mineralogical) make-up of the binder, the characteristics of the mixture and the degree of hydration of the system²⁵.

The model also requires determining the initial composition of the pore solution and the porosity of the material. Samples of the pore solution of most hydrated cement systems can be obtained by extraction using the technique described by Longuet et al.²⁶. The total porosity of the material can easily be determined in the laboratory following standardized procedures (such as ASTM C642)²⁷.

Some information on the transport properties of the material is also required to run the model. The ionic diffusion properties of the solid can be determined by a migration test¹². The water diffusion coefficient of the material can be assessed by nuclear magnetic resonance imaging²⁴.