Figure 3: Schematic diagram of the digital image to random conductor network mapping used to compute the electrical conductivity of the cement paste microstructural model.
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The relative diffusivity of the model cement paste is calculated after hydration and after each leaching cycle, by mapping the underlying lattice structure of the model onto a 3-D electrical conductor network [6]. This is accomplished as shown in Fig. 3 by connecting adjacent pixels with a bond whose conductance is determined by the phases assigned to the two pixels. For cement paste, both the capillary pore space and the C-S-H gel phase will provide pathways for the transport of diffusing ions, so both must be included in this mapping process (Fig. 3). The C-S-H phase is assigned a conductance 1/400th of that of the capillary pore space, a factor determined by calibration against experimental results [6]. The CH phase and unreacted cement are assigned conductances of zero as it is assumed that no transport occurs through these phases. The network of conductors is completed by attaching one-pixel-thick electrodes to the top and bottom faces of the three-dimensional cubic volume of cement paste.
Figure 3: Schematic diagram of the digital image to random conductor network mapping used to compute the electrical conductivity of the cement paste microstructural model.
Using a conjugate gradient relaxation algorithm [7], this network of conductors is solved to determine the equivalent conductivity of the 3-D system, which relates to the diffusivity by the Nernst-Einstein relation [6]
where Do is the diffusivity of the ions being considered in free water, σo is the pore solution conductivity, and D and σ are the measured diffusivity and electrical conductivity of the 3-D system. Having obtained the value of σ / σo from the conjugate gradient solution, D/Do, the relative diffusivity, is obtained using the Nernst-Einstein relation.