4. Computational Methods

4.1 Problem Definition

The problem being considered is that of a completely water- saturated porous hardened cement paste. A concentration gradient of dissolved ions exists across the sample, so that there is a net diffusive flow of ions through the water-saturated pore space. A steady state in regards to any adsorption-desorption phenomena is assumed to have been established, so that the net flow is truly diffusive, and independent of time [3]. Under these conditions, the Nernst-Einstein relation connects the electrical conductivity of the material with its diffusivity [2,3]. Fig. 2 illustrates the physical content of this relation. If D_o is the diffusivity of the ions being considered in free water, and _o is the conductivity of the solution in the pore space, then the result of the Nernst-Einstein relation is that D / D_o = / _o, where D and are the measured diffusivity and electrical conductivity of the fluid-saturated material. Diffusion coefficients can be measured experimentally by the application of a concentration gradient, c, and electrical conductivities by the application of a potential gradient, V, as indicated by the arrow. The quantity D/D_o is also sometimes called the diffusibility [3], or the relative diffusivity [8]. The latter term is used in this paper.

4.2 Random Conductor Network

The computational approach taken in this paper is to exploit the relationship given in eq. (2), by converting the digital- image model into a random conductor network, and then computing the conductivity using one of two conductivity algorithms that were developed for simple lattice problems.

The method of converting the digital-image model into a conductor network is as follows, and is schematically illustrated in Fig. 3, which shows the resulting conductor network superimposed on an original random 2-d image. After a digital image cement paste model is generated, a one-pixel-thick electrode is "glued" on opposing faces of the unit cube. A network of nodes is created, with one node at the center of each pixel. Conductors with conductance _ij are then set up that connect the nodes in nearest-neighbor pixels i and j, which themselves have conductivities _i and _j. The conductance _ij is defined as the series combination of _i and _j, , where _i is the conductance of one half of pixel i. This means that , where d is the edge length of one pixel. If pixels i and j are both capillary pore space pixels, for example, then _i = _j = 1, so that _ij = d. If either pixel i or pixel j are cement or pore product pixels, then _ij = 0, since either _i or _j is zero.

The surface product (C-S-H) pixels are taken to have a small non-zero conductivity, because of the surface product's continuous gel micropores. The diffusivity results were all obtained using _C-S-H = 0.0025, which was based on experimental data from chloride ion diffusivity measurements [3,22]. Assigning a bulk conductivity to the C-S-H phase is an approximation, since these pores are small enough that the flow of diffusing ions or the movement of charged particles in them probably differs greatly from bulk processes [3], and thus depends on the ion considered. For example, it is known that cesium (Cs⁺) ion diffusivities are systematically smaller than chloride ion diffusivities measured in the same material [3], so a different value for _C-S-H would be needed to describe cesium ion diffusion. A conductor connecting a C-S-H pixel node to a capillary pore space node is then given a conductance of d/200.5, in accordance with eq. (3). The electrodes are considered to have infinite conductivity, which results in the value of _ij being 2 _j d when pixel i is on the electrode, and pixel j has finite conductivity _j.

Fig. 3 shows the five different values of conductances used. No connection indicates a zero conductance. The thin dashed lines are 0.0025 d, the thin solid lines are d/200.5, and the thick solid lines have conductances d. The zig-zag line denotes a conductor-electrode connection, having a conductance of either 2d or 2(0.0025)d, for a capillary or a C-S-H pixel connected to the electrode, respectively. Attaching an electrode is especially useful when simulating AC problems, where the electrode interface can be an important part of the total measurement [23,24,25,26,27]. However, the attached electrode is not necessary when just considering the response of the material, in which case an electric field can be maintained just using the periodic boundary conditions [28].

4.3 Conductance Algorithms

Once the conductor network is built, its effective conductance is computed using one of two efficient methods. If the average connectivity of the nodes is small enough, around 1.5 bonds per node on the average, then the Fogelholm algorithm can be used. This algorithm was first written in LISP by Fogelholm [29] for 2-d problems, and has been extended to three and higher dimensional problems using a program written in C [30]. This algorithm systematically reduces the network down to two nodes, with the conductance of the last remaining conductor being the equivalent conductance of the entire network. It is very efficient, partly because the equivalent conductivity is obtained without having to solve for the electric potential at every point. However, the speed of the algorithm decreases extraordinarily with the average number of connections per node. In Ref. [30], the problem considered was the computation of the conductivity close to the percolation threshold, p_c = 0.249, where p is the fraction of bonds remaining, for bond percolation [20] on the simple cubic lattice. We have found that the algorithm, however, becomes unacceptably slow for p < 0.29 for the same problem solved on a 100³ cubic lattice.

The Fogelholm algorithm is still useful for the cement paste problem, since there are many ranges of porosity that yield an effective p in the right range. However, most of the results reported in this paper were obtained using a second algorithm, a conjugate gradient relaxation algorithm [31].

The conjugate gradient relaxation algorithm solves the complete electrical problem of the voltage distribution in a random material across which a potential difference is applied. The output of the algorithm is the voltage at every node, from which the total current and thus equivalent conductance is calculated. The input is an initial voltage distribution, usually taken to be 1 and 0 at the two electrodes, and linearly interpolated at nodes in between. The voltages are then cyclically updated until Kirchoff's laws are satisfied at each node within some preset finite precision [31]. At low porosities, the conjugate gradient algorithm is slower than the Fogelholm algorithm, since it gives all the voltage information as well as the equivalent conductance, but it can handle any degree of connectivity, and eventually becomes the faster algorithm as the Fogelholm algorithm's speed decreases much more rapidly with increasing porosity.