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### Introduction

Concrete is a composite material. It is made up of, at first sight, a cement paste matrix and aggregate grains of various sizes, ranging from the very smallest sand grains of diameter 100 µm, to the large aggregates of diameter 10 mm to 20 mm. However, upon closer examination, one finds a thin layer of matrix material surrounding each aggregate grain, call the interfacial zone (ITZ), where the cement paste matrix is different, usually more porous, than the bulk of the surrounding cement paste matrix. The ITZ has an average width approximately equal to the median cement particle size [1], and arises mainly from the "wall effect," where cement particles are constrained by the aggregate surface to pack less efficiently in the ITZ [2], although other minor mechanisms may play a role [3]. Typical widths of the ITZ are in the range 10 µm to 30 µm. Figure 1 gives a 2-D schematic view of a concrete with two sizes of aggregate to show the type of microstructure that must be considered.

So to attempt to represent concrete properly as a composite material, at least three phases must be considered, consisting of matrix, aggregates, and the ITZ regions. Assigning each of these phases a different transport parameter, diffusivity or conductivity, then results in a complicated composite transport problem. Here conductivity refers to either thermal or electrical conductivity. In the diffusivity problem, which is the main problem of interest of these three to concrete [4], the aggregates have diffusivities of zero, while the ITZ and the matrix have in general different and non-zero diffusivities. The language of conductivity will be used throughout the remainder of this paper, but the diffusivity problem is exactly mathematically analogous, along with several other physical problems [5,6]. The equivalent elastic problem is of interest as well, but is outside the scope of this paper [7,8].

Of course, the real problem is more complicated still. The ITZ region in fact has a gradient of properties, since the porosity is a gradient from the aggregate surface outwards [2,9,10]. The dilute limit, with a single spherical aggregate surrounded by a spherically symmetric gradient of properties, can be handled exactly [11,12,13,14]. But the real microstructure of concrete, with a wide size distribution of aggregates each surrounded by overlapping gradients of properties, is too difficult to treat analytically, by numerical methods, or by effective medium theory (EMT). However, it has been shown that a multi-scale model can be used in order to map this very complicated microstructure into a simpler, but still complicated, microstructure, like that shown in Fig. 1, where the ITZ regions can be treated theoretically as having uniform properties [4,11,15]. This multi-scale, multi-step approach [4,11,15] assigns the best value of ITZ thickness, which is the same surrounding all aggregates, and conductivity, which is the same for all ITZ regions, to match the real material. Once this multi-scale procedure has been carried out, one ends up with a system as shown in Fig. 1, where the ITZ regions have uniform properties.

To compute the overall conductivity of the system shown in Fig. 1, random walk simulations have been performed [4,16,17]. Uncorrelated mathematical walkers (points) are thrown down at random, and then undergo random walks. Walkers that initially land in the aggregates do not move, and are not counted. A "clock" is maintained for each walker. The walkers move at different speeds depending on which phase they are in. The slope of the average root-mean-squared distance vs. time curve is then used to extract the overall conductivity or diffusivity. These are accurate and simple, but time-consuming, computations. The hope is to use some version of EMT to replace the random walk simulations [18]. This is done to reduce the computer time that is necessary to evaluate this step of the multi-scale model [15], so that the model becomes more widely used. However, the existence of accurate simulations is still required in order to validate the EMT results.

This idea was tested in earlier work, using a form of differential effective medium theory (D-EMT). This previous application of D-EMT [15] agreed fairly well with the random walker computations, but was handicapped by having to use an arbitrary parameter that was fit to the result of random walk simulations. The point of the present paper is to derive a new kind of D-EMT that has no adjustable parameters. After introducing standard D-EMT, and deriving this new kind of D-EMT, the results of this new D-EMT are compared with the results of random walk computations on various concrete models [4,15,16], and are found to agree better with the simulations than did the old D-EMT results.

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