Wittmann et al. 3, 4 were perhaps the first to consider representing a cement or concrete microstructure numerically within a computer with the development of their "numerical concrete." A concrete microstructure consisting of aggregates in a cement paste matrix was generated and mapped onto a finite-element grid, allowing for the computation of thermal, hygral, and mechanical stress distributions. Each aggregate particle could be mapped onto one or more finite elements, so that this model represented the concrete at the subparticle level.
At the level of cement paste, pioneering work in directly representing the cement paste microstructure at the cement particle level was performed by Jennings and Johnson, 5 who developed a continuum representation based on spherical cement (tricalcium silicate, C3S) particles enveloped by hydration shells of calcium silicate hydrate (C-S-H) gel, whose thickness, increased over time. In addition, calcium hydroxide (CH) crystals were allowed to nucleate and grow in the continuum pore space. This model is classified as being of the continuum type, in that each particle can be described by its centroid location and a set of radii, corresponding to the unhydrated core and one or more shells of hydration products (representing inner and outer C-S-H product, 6 for example). A somewhat similar approach was formulated by van Breugel, 7, 8 who, by accounting for the volumes of embedded cement particles and other morphological aspects of the hydrating cement paste system, was able to predict the hydration behavior, explicitly considering the cement particle-size distribution, its chemical composition, water-to-cement (w/c) ratio, and temperature. An example of the graphical output produced by this model, and quite similar to that first produced by the Jennings/Johnson model, can be found in Fig. A1. This approach is currently undergoing further development by other research groups. 9
A second type of continuum model being used to generate cement microstructures is based on the mosaic method. 10 Here, a two-dimensional space is divided by a set of intersecting lines (planes would be used in three dimensions), and the resultant polygonal shapes taken to represent unhydrated cement particles and hydration products. Although multiple discrete phases can be modeled easily using this technique, simultaneously modeling multiple continuous or percolated phases, such as CH, capillary porosity, and C-S-H gel in cement paste, may present a computational challenge. In summary, continuum models can provide valuable quantitative information, such as the effects of particle size on hydration kinetics, but one finds it difficult to analyze such a microstructure to directly compute transport and elastic properties, such as can be easily computed for Wittmann's numerical concrete.
Fig. A1. Example of graphical output from the model of van Breugel. Image shown is a two-dimensional slice from a three-dimensional spherical computational volume (unhydrated cement cores are dark blue, inner C-S-H product is red, outer C-S-H project is yellow, and water-filled space is light blue). (Courtesy of K. van Breugel.)
An alternate approach to continuum-based models has been the development of so called digital-image-based models 2, 11 These models operate at the subparticle level as each cement particle is represented as a collection of elements (pixels). Cement hydration can then be simulated by operating on the entire collection of pixels using a set of cellular-automata-like rules, 12 as illustrated in Fig. A2. This allows for the direct representation of multisize, multiphase, nonspherical cement particles. In two dimensions, a processed SEM image (as described in Panel B) can be used as direct input into the hydration model. The model has evolved from one based simply on the hydration of C 3S 13 to one that considers all of the major phases present in cement. 14 Recently, a similar two-dimensional model that emphasizes ion concentrations and diffusion processes has been developed.15 Because of the underlying pixel representation of these microstructures, mapping the microstructure onto a finite-difference or finite-element grid becomes trivial, because there can be a simple one-to-one mapping between pixels and finite elements. Thus, properties such as percolation, 11 diffusivity, 16 complex impedance, 17 and setting behavior 2 are easily computable. The major limitation of digital-image-based models is perhaps one of resolution. Because each pixel is typically 1 µm3 in volume, features smaller than this cannot be resolved. Fortunately, most cement particles are between 1 and 70 µm in diameter, so that a given cement can be very accurately represented in a computational volume of 200 x 200 x 200 pixels. In addition, models at the micrometer level have been successfully integrated with others at the manometer (C-S-H gel) and millimeter (mortar or concrete) level to provide a quantitative description of concrete microstructure that spans 7 orders of magnitude in scale. 18, 19
Fig. A2. Illustration of various steps in the digital-image-based cement hydration model showing, from bottom to top, initial cement particles in water (black), highlighting (white) of all cement particle surfaces in contact with water, generation of one-pixel diffusing species, and hydrated images at ~32% and 76% hydration, respectively (C 32 is red, C2 S is blue, C3A is bright green, C4AF is orange, gypsum is pale green, C-S-H is yellow, CH is dark blue, and aluminate hydration products (ettringite, monosulfoaluminate, and C3 AH6) are green).