A Conceptual Model for Ionic Transport in Cement-based Materials in Conditions of Externally Applied Electric Field
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TICAL BACKGROUND For simulating the profiles of ionic concentrations and ionic distribution in a cement-based system exposed to external applied current density (e.g. stray current flow) during the hydration process, the Nernst-Planck equation is generally used [11-13]: (1) where Dk (t) and Dk are the time dependent and the initial diffusion coefficient, zk is the charge number of the kth ionic species (k=1…N), F=9.648 x 10-4 C/mol is the Faraday constant, R = 8.314 J mol-1 K-1 is the ideal gas constant, T=298 K is the absolute temperature, and is the electrostatic potential. The time dependent diffusion coefficient (D(t)) is calculated using the following equation [15]: (2) where Do is the reference diffusivity after 28 days of hardening, to is the reference time (i.e. 28 days period), n is an aging factor.
In this case, the concrete is assumed to be representing a saturated pore medium where there are no chemical reactions between ionic species both in the liquid and solid phases. The electrostatic potential at any point in the concrete can be then solved using the Poisson equation [16]: (3) where o = 8.854 x 10-12 C V-1 m-1 and o=78.3 are permittivity of a vacuum and relative permittivity of water respectively at 25o C. The initial conditions, the boundary conditions, and all other relevant input data are summarised in Table 1 below: Table 1 Initial and boundary condition [13] Variables Boundary conditions x=0 x=L y=0 y=H Initial conditions Charge number Apparent diffusion coefficient (m2/s)
Potassium (mol/m3) 200 200 J=0 J=0 200 1 1.957x10-10
Sodium (mol/m3) 100 100 J=0 J=0 100 1 1.334x10-10
Hydroxide (mol/m3) 300 300 J=0 J=0 300 -1 5.260x10-10
Electrostatic potential (V) 0 2 and 20 y=0 y=0 0 N/A N/A
Figure 1 Two-dimensional model, schematic of ion migration test, electrostatic potential distributions for 20 Volts.
In this numerical model, a periodic boundary condition was applied, in which the ionic species concentration at the inlet boundary will transport out in direction to the opposite (outlet boundary) and these iterations will proceed until the chosen time frame elapses.
DISCUSSION Figure 1 reveals a two-dimensional model for concrete of 40 mm x 40 mm size, containing aggregates, cement pate and relevant pore solution (K+, Na+, OH− as main ions in the pore solution), a schematic representation of the applied field and ion migration test are also depicted in Fig.1. In this model, the aggregate phase is considered to be impermeable. External electric potential of 2V (i.e. electrostatic potential gradient, =50 Volts/m) and 20Volts (i.e. =500 Volts/m) were applied on the two sides of the concrete specimen to simulate ionic species migration for 4 hours. The electrostatic potential distribution in the concrete specimen with external electric field 20 Volts can be seen in the Figure 1c. During the electro-kinetic process,
positive charge (potassium and sodium ions) migrates towards the negative electrode (cathode), whereas negative charge (hydroxide ions) migrates towards the positive
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