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## Sources of error when simulating random materials

Any model is only an approximation of reality. There are always shortcomings built into any model, which must be quantitatively addressed when assessing the accuracy of model results. Digital models, like the NIST cement hydration model, are no different. Here, the term "model" refers to a fundamental computer simulation model [12], where the microstructure is directly represented in the computer.

When one builds this kind of model of a random system, there are immediately two main sources of error built into the process. These two error sources are [13,14]: (1) statistical fluctuation, and (2) finite size effect. As mentioned in the introduction, statistical fluctuation errors come about simply because the system modelled is random, while finite size effect errors arise because the model is usually not as large as the real material. Digital models have a third important source of error, digital resolution. Here, the term "digital model" means a direct representation by a 2-D or 3-D digital image, where each pixel bears a label identifying it as a phase of the material. Each of these three sources of error will be discussed in turn for the percolation and transport property aspects of the cement paste microstructure model.

Any model of a random material can only represent a piece of the material. Finite size error comes about if the piece of the material is not big enough to be "typical" of a representative size sample. The terminology that is used in the composite material field is "representative elementary volume" [15,16]. Uchikawa has examined this concept for cement paste, mortar, and concrete, and has observed qualitatively, in 2-D images, that an area of 100 µm by 100 µm was representative for cement paste [17]. An example would be the usual rule for concrete samples, that they should be 3-5 times as big as the largest aggregate for them to be considered "typical." Empirical testing has determined that this is "big enough" for concrete. This concept has been dealt with very carefully for a limited class of problems, but the formalism is general [13,18]. For any model of a random material, the only way to tell if the model is "big enough" is by testing the property of interest for different size models. In our case, one uses bigger (smaller) unit cells with more (fewer) particles. The unit cells have periodic boundary conditions, which help greatly in reducing the effect of the surface by essentially removing the surface. This has been done for the cement hydration model. Models that are 100 µm by 100 µm by 100 µm in size, containing typically 2000 particles, have been found to be large enough to be representative. Larger models, containing more particles at the same digital resolution and with periodic boundary conditions, have not been found necessary for accuracy.

Since cement paste is a random material, the digital representation must also be random. This means that different realizations of the model will have some differences between them. Random numbers determine where the cement particles are originally placed in the unit cell, and how the hydration takes place [8]. As will be seen below, averaging a given percolation curve over different random numbers does not make much of a difference. The averaged curves become smoother than any one curve, but the qualitative behavior of the curve, i.e. where its percolation threshold is, does not change significantly. Other properties of the hydration process, like heat release curves, have also been shown to not vary much with statistical fluctuation [19].

The final source of error is digital resolution, which is unique to digital models [11,13,14]. For example, the model tries to represent a cement particle, which has an order of 1013 atoms for a typical 15 µm diameter C3S particle, by a fairly small number of pixels. In the model, the pixels dissolve and move around, resulting in digitally fragmented remainders, that have digitally fragmented surfaces. Another way of thinking about digital resolution error is to compare the pixel size to a intrinsic length scale in the problem. For example, if the smallest CH crystal of importance is 0.1 µm, and the pixel size is 0.4 µm, then obviously the smallest CH crystals can not be represented as single units and the model will fail to resolve these particles. This may not induce much error in predicted properties, however, if these small particles do not play a significant role in determining the properties of interest. The main purpose of this paper is to determine exactly how digital resolution error matters. Of the three sources of error discussed in this section, it turns out that the digital resolution error is the only important one for the cement hydration model. We investigate this effect by keeping the same physical size model, a cube 100 µm on a side, but with a different length per pixel, so that more pixels are used at finer resolution.

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