2. Digital-Image-Based Microstructural Model

2.1 Cement Hydration

The microstructure of cement paste is known to be complex [10]. That is not surprising, since cement paste is formed from a disordered aqueous suspension of irregularly-shaped cement particles, which undergo random growth due to hydration reactions. Since the original cement particles have a wide size distribution and an average size of 15-20 micrometers [11], the complex microstructure of cement paste extends over many length scales, from small fractions of a micrometer to tens of micrometers. Neglecting chemical details, which, admittedly, is an over- simplification, the reactive growth process that cement particles undergo to produce cement paste can be thought of in the following simple way [12]. The solid cement particles supply calcium ions to the surrounding water via dissolution of surface layers. These ions then react with silica-rich surfaces of cement particles to form solid reaction products (surface products) covering the cement particles, or spontaneously nucleate in the pore space to form crystals (pore products), which can then grow further by accretion. In cement paste, the main surface product, calcium silicate hydrate, is denoted C-S-H, and the main pore product, calcium hydroxide, is denoted CH, where the usual cement chemistry shorthand notation is C=CaO, Ss=SiO2, H=H₂O, A=Al₂O₃, and F=Fe₂O₃. The reason that cement hydration can produce a rigid solid from a viscous suspension of cement particles in water is that the hydration reaction products have a larger volume than the solid reactants. As the hydration process is nearly a constant total volume process, the reaction products can fill in the initially water-filled pore space, eventually forming a rigid solid backbone capable of bearing mechanical loads. It is convenient to define the following volume ratios. _S is the ratio of the volume of surface products produced to the volume of cement reacted, and _P is the analogous ratio for the pore products. The total volume expansion factor is defined as _T = _S + _P. Typical ranges for these parameters, for various types of portland cements, are 1.6 < _S < 1.9, and 0.4 < _P < 0.7 [13]. These parameters include the reaction of tricalcium silicate (C₃S), dicalcium silicate (C₂S), and the less abundant aluminate phases. The very small amount of ferrite phases present in cement were ignored in Ref. [13]. Somewhat surprisingly, for a variety of cements, the value of _T is fairly constant, around 2.3 ± 0.1 [13]. In the simulation model to be described next, _S is taken to be 1.7, and _P is taken to be 0.61, so that _T = 2.31. These particular values are those realized in the hydration of pure C₃S cement [14].

2.2 Microstructural Model

In the last decade, a number of random growth or aggregation models have been developed. These models, which employ very simple random growth rules, have been shown to produce complex aggregated structures, often with fractal morphology. Two examples are the diffusion-limited aggregation (DLA) [15] and Eden models [16]. In light of these models, it is not unreasonable to suggest that the complex microstructure of cement paste might be simulated using a few, relatively simple growth rules, which are repeated many times. The model used in this paper represents a realization of this approach.

The model operates on square or cubic arrays of pixels, typically of edge length equal to 500 pixels in 2-d, and 100 pixels in 3-d, where each pixel is assigned to a single phase, such as pore space or cement. Initially, a specified number of cement particles (about 2000 in 3-d) are randomly placed in the unit cell such that no two particles overlap, simulating the mixing process. Periodic boundary conditions are used to eliminate any artificial edge effects at the cell walls. The particles may be from any size distribution, within the resolution limits of the unit cell (1 to 100 pixels in 3-d). Model particle shapes, like circles in 2-d or spheres in 3-d, may be used, but since the model is based on a digital image representation of the cement particles, digitized micrographs of actual particle shapes can also be used as a starting point in 2-d or 3-d [17,18]. All the simulations described in this paper are 3-d simulations.

The model operates by the iteration of cycles. Each cycle consists of three steps: dissolution, diffusion, and reaction. Figure 1 schematically describes the growth process.

In the dissolution step, any cement pixels in contact with a water-filled pore space pixel attempt to take a step in a random direction. The pixels whose step lands them in the pore space dissolve, and each such pixel turns into a random diffuser. The pixels whose random step would land them in a solid phase are not allowed to move, and so remain at their original location, undissolved. The number of pixels that dissolve are counted, and the correct number of extra diffusing pixels are added at random locations within the pore system, replacing pore space pixels, to account for the correct amount of surface and pore product formation. More precisely, if n pixels dissolve from off the cement surfaces, (_P n) pore product and [(_S - 1) n] extra surface product diffusing pixels are added to the system at random locations in the pore space in order to achieve the correct volume of hydration products arising from the reaction of the n dissolved cement pixels.

  

Figure 1: Schematic diagram of cement paste microstructural development algorithm.

During the diffusion/reaction steps, the dissolved pixels move by executing random walks throughout the pore space. Surface product pixels continue to move in this random fashion until they encounter a cement surface, at which point they react and attach to this surface. Once surface products are present, diffusing surface product pixels can react and attach to these surfaces as well. For any given step taken by a diffusing pore product pixel, however, there is a non-zero probability that it can nucleate at its present location. This probability decreases exponentially as the number of diffusing pore product pixels decreases [19]. After a pore product cluster has been nucleated, other diffusing pore product pixels can aggregate onto the cluster upon contact. When all diffusing pixels have reacted or nucleated, the cycle is complete, and the next cycle begins with a new dissolution step. Microstructural development is complete when all cement pixels have reacted, or when all remaining cement pixels are covered by surface product, and therefore, are no longer available for dissolution. In 3-d simulations, usually up to 90% of the original cement can be hydrated using these simple rules. The degree of hydration achieved after any completed cycle is determined from analysis of the microstructure. Degree of hydration () is defined as the fraction of the original cement that has been reacted, so that = 0 when the cement particles are first mixed with water, and attains a value of 1 when hydration is complete. Another parameter that is easily calculated in the model is the water:cement (w/c) ratio, which is the weight ratio of water to cement in the initial cement-water mixture, a parameter often quoted in the cement literature . If f is the solid volume fraction, and 1-f is the water volume fraction, then the water:cement ratio w/c is given by w/c = (1 - f) / (3.2 f), where 3.2 is the specific gravity of portland cement. The original porosity of a cement/water mixture, defined as the volume fraction of pore space, is then also equal to f.