Next: Summary
Up: Microstructure models in
Previous: Artificial image models
It is desirable to generate porous microstructures based on actual physical and chemical processes. This is often very difficult because of the complexity of these processes. Natural materials like sandstone are not made under a controlled laboratory environment, so our quantitative knowledge of the processes involved is weak. However, manmade materials such as concrete (5 billion tonnes per year worldwide) and ceramics (including bricks), and various gels are manufactured according to well-defined processes, so in principle, it should be possible to simulate these processes by a 3-D microstructure model. The success of the model is in fact a test of our understanding of the processes.
Consider the case of the porous material concrete. It is made up of cement, water, sand and pebbles (aggregates). It is formed by the hydration of cement, the most common variant of which, known as portland cement, consists of mainly calcium silicates with minor amounts of aluminate, sulphate, and ferrite phases [116]. When water is mixed with the cement, the various phases of the cement undergo hydration reactions, each at a different rate and interacting with each other. The initial viscous mixture of liquid and particulates grows into a rigid solid that keeps increasing its strength as the hydration progresses, which can continue for months. The cement paste (cement plus water) turns into a solid matrix, in which the sand and pebbles are embedded.
The main geometrical feature that must be understood about concrete microstructure in order to be able to optimize concrete properties is the evolution of the cement paste microstructure during hydration, as the cement paste matrix governs the properties of concrete. This is a microstructure made up of unhydrated cement grains, reaction products, and water-filled pore space. The starting cement grains have an average size of about 20-50 µm, so that the length scale that initially characterizes the primary cement paste pores is of the order of micrometers. These pores do, however, become as small as a tenth of a micrometer, as hydration progresses. There are smaller secondary pores present, called gel pores, inherent in the main reaction product, amorphous calcium silicate hydrate. Their diameters are of the order of tens of nanometers [78,116], but we will ignore them in considering the primary cement paste microstructure. Cement paste is a thus a porous material whose solid phase is not uniform. This has a sensitive effect on many concrete properties like elastic moduli and thermal conductivity.
Models have been made to simulate the evolution of the cement paste microstructure from a mixture of water and cement grains to the final hydrated product [54,117,118]. These models only incorporate some of the relevant cement chemistry and physics. The amounts and volume of reactants and products are correctly handled. The randomness of the original multi-phase composite cement particles is realistically taken into account by using 2-D scanning electron microscope digital images of real particles as a basis for constructing 3-D particles [118]. The randomness in the growth process and the topology of the various reaction products are also realistically simulated.
Figure 16 shows an SEM micrograph of a real cement paste, compared to the model equivalent [54,117]. The gray scales indicate the different phases. The darkest gray pixels contain other minority phases, including gel pores, which are not shown. The pores are black. For calculating composite properties like elastic moduli, the different phases in the solid framework must be identified, as they all have different elastic moduli. Color pictures, that reveal more details of the various stages of cement hydration can be found in Refs. [117,119].
Figure 16: Showing a real hydrated cement paste (right), and its model equivalent (left). The different gray levels indicate the principal different solid phases of unhydrated cement and its reaction products. Porosity is black. The darkest gray level contains minor phases that are not shown.
Properties that have been computed using the various methods described in Section 3 of this chapter include the connectivity of both the solid and pore phases of hydrated cement [54], diffusivity of the pore space [62], and how the cement paste matrix in concrete is modified by the nearby presence of aggregates [119,120]. Comparison with experimental data has been quite favorable.
Another example of a microstructural development model has to do with the high temperature sintering of powders into ceramics and metals. The powder particles change shape, and the powder compact densifies in order to minimize surface energy [121]. This process has been simulated by a cellular automaton model that minimizes the surface area of a digital image of particles [122,123]. A simple algorithm that transfers pixels from areas of high curvature to areas of low curvature captures the essence of the process. It is clear that by moving mountains to fill valleys, surface area is reduced, which is the main driving force for sintering. To implement the model on a digital image requires a simple algorithm to measure curvature, which is illustrated in Fig. 17. The solid pixels in this figure are shown with heavy black lines and the pore space pixels with thin black lines. A circular template (shown in gray) is centered at the point of interest on the surface. The local curvature is estimated by counting the number of pore pixels in the circular template. It is intuitively obvious that a flat surface would have 50% pore pixels in the circle, with less than 50% for negative curvature, and more than 50% for positive curvature. It can be proved mathematically [123] that this procedure is asymptotically exact, in the limit when the template radius is much smaller than the local radius of curvature. Other ways of measuring curvature in a digital image are described in Ref. [124]. The algorithm is applicable in 3-D as well, but it only gives the average of the two principal radii of curvature [123].
Figure 17: Showing the circular template algorithm in 2-D. Thick black lines denote solid pixels, thin black lines denote pore pixels. The circular template is shown in gray.
Figure 18 shows the evolution (from left to right) of a collection of circular grains as the curvature and therefore surface area is reduced by the algorithm. The collection of grains is gradually becoming a circle, which has the minimum perimeter of a given surface area of any finite area shape in 2-D [125]. This growth model can be applied directly to a digital image that has been acquired experimentally.
Figure 18: Showing the evolution of a collection of circular particles (left to right) under the sintering algorithm described in the text.
The sintering model described does not contain all of the relevant dynamics. When material is removed from a high curvature surface, it must be transported to the low curvature surface, either by vapor transport through the pores, surface diffusion, volume diffusion, or diffusion through any grain boundaries. There are also elastic forces that arise from the tendency of the particles to coalesce to minimize surface area [126,127].