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Over the past few years, the mechanisms of ionic transport in cement systems have received a great deal of attention. Chloride-related damage to an increasing number of concrete structures has led to intense research efforts dedicated to chloride ingress in porous materials. Most published reports on the subject have strongly emphasized the intricate nature of the problem. Given the number of parameters involved, analytical models have proven inadequate to describe the ionic transport process. Numerical modeling is therefore required. The first generation of models only considered the chloride ion. The latest models consider other species besides the chloride that are present in pore solution, allowing better representation of the physical and chemical mechanisms. STADIUM® belongs to this latest generation.
STADIUM® is a multi-ionic transport model based on a split operator approach that separates ionic movement and chemical reactions. Ionic transport is described by the extended Nernst-Planck equation applied to unsaturated media. This equation accounts for the electrical coupling between ionic species, chemical activity, transport due to water content gradient, and temperature effects.
where ci is the concentration [mmol/L], w is the water content [m3/m3], Di is the diffusion coefficient [m2/s], zi is the valence number of the ionic species i, F is the Faraday constant [96488.46 C/mol], y is the electrodiffusion potential [V], R is the ideal gas constant [8.3143 J/mol/°K], T is the temperature [°K], gi is the activity coefficient, and Dw is the water diffusivity [m2/s]. Eight ionic species are considered: OH-, Na+, K+, SO42-, Ca2+, Al(OH)4-, Mg2+, and Cl-.
The ionic transport equation is coupled to Poisson’s equation, which gives the electrical potential in the material as a function of the ionic profile distribution:
where e [C/V/m] is the medium permittivity and N is the number of ions in the pore solution.
To account for water flow in the presence of water content gradients in unsaturated materials, the above-mentioned equations are coupled to Richard’s equation:
This diffusion-type equation gives the distribution of water content within the material. The effect of water on ionic movement is modeled by adding an advection term to the extended Nernst-Planck equation.
Finally, the temperature distribution in the material is calculated from the classical heat condition equation:
where r is the density of the material [kg/m3], Cp is the specific heat of the material [J/kg/°C], and k is the heat conductivity [W/m2/°C].
This system of nonlinear equations is solved using a numerical algorithm with all equations solved simultaneously. The spatial discretization of this coupled system uses a finite element approach using the standard Galerkin procedure. An Euler implicit scheme is used to discretize the time-dependent part of the model. The nonlinear set of equations is solved with the Newton-Raphson algorithm. This second-order algorithm gives a good convergence rate and is robust enough to handle the electrical coupling between the ionic species as well as the non-linearity coupling between the ionic flux and water movement.
The second module in STADIUM® is a chemical equilibrium code. Following the transport step, the chemical equilibrium module checks for equilibrium between the concentrations at each node of the finite element mesh and the different phases of the hydrated cement paste: calcium hydroxide, calcium silicate hydrates, ettringite, and monosulfates. The equilibrium of each phase is modeled according to:
where M is the number of solid phases, N is the number of ions, Km is the equilibrium constant (or solubility constant) of the solid m, ci is the concentration of the ionic species i, gi is its chemical activity coefficient, and nmi is the stoichiometric coefficient of the ith ionic species in the mth mineral. If the solution is not in equilibrium with the paste, solid phases are either dissolved or precipitated to restore equilibrium. Solid phases can also be formed when aggressive species penetrate into the porous network of the material: ettringite, gypsum, Friedel’s salt, hydrated sodium sulfate, and halite.
These variations of solid phases lead to local variations of porosity, which tend to have local effects on the material’s transport properties. STADIUM® takes this phenomenon into account in its transport module.
A specialized version of STADIUM® called STADIUM® IDC was specifically developed to analyze the results of migration tests. It uses the electrical currents measured during a migration test to evaluate the diffusion coefficients of the ionic species present in the material's pore solution.
The model has been thoroughly validated through extensive laboratory testing. Figure 2 shows the reproduction of total chloride profiles measured on two w/c 0.45 CSA Type 10 concrete samples exposed to a 3% NaCl solution for one and eight months respectively.
The model was also validated for other types of chemical degradation such as exposure to sulfate solutions and pure water. In Figure 3, the capacity of STADIUM® to predict simultaneous degradation phenomena is shown. Cement paste samples (w/c 0.6, CSA Type 10 cement) were immersed in a sodium sulfate solution and analyzed after six months for total calcium and sulfur content. Microprobe measurements showed a gypsum peak due to the presence of sulfate as well as calcium loss near the exposed surface due to portlandite (Ca(OH)2) dissolution and decalcification of C-S-H. In both cases, the model reproduces the measurements.
Figure 2. Chloride ingress simulations of laboratory samples (w/c 0.45 CSA Type 10 concrete)
Figure 3. Validation experiment: cement paste exposed to sulfate solution (w/c 0.6, CSA Type 10 cement)
STADIUM® also demonstrates proven ability to reproduce field measurements of existing structures. The following profile was measured on a concrete parking structure located in Canada (see Figure 4). The structure was 20 years old and had been exposed to high chloride levels. The model successfully reproduced the actual state of the structure and predicted its remaining service life.
Figure 4. Long-term chloride ingress simulation on a parking structure
Web site: http://www.simcotechnologies.com/
