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, C_{3}S) 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 _{3}S ^{
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 µm^{3
} 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
_{3}2 is red, C_{2}
S is blue, C_{3}A is bright
green, C_{4}AF is orange, gypsum is pale
green, C-S-H is yellow, CH is dark blue, and aluminate hydration products
(ettringite, monosulfoaluminate, and C_{3}
AH_{6}) are green).