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# Modelling Approach

In this section, we first derive an equation for determining the replacement level needed to ensure adequate water for "complete" curing of the concrete. In this case, complete curing means that the cement reaches the maximum degree of hydration that is possible, given the space limitations for forming hydration products in low w/c ratio systems. Then, we proceed to briefly describe the microstructural model of concrete which will be used to determine the relative proximity of the paste to the water sources (the LWFA surfaces).

The volume of water per cubic meter of concrete needed to be supplied by the LWFA depends on the mixture proportions of the concrete in the following manner. Let CS denote the chemical shrinkage occurring during the hydration of the cement; typically, this value is on the order of 0.06 kg H2O per kg cement hydrated [7,8]. The amount of needed water will depend on this quantity, as well as the cement content, Cf in kg cement/m3 concrete, and the w/c ratio for the mixture proportions. For w/c ratios below 0.40 (typical of a HPC), complete hydration can not be achieved and the maximum degree of hydration, max, can be estimated as (w/c)/0.40. Then, the volume of water, Vwat, that is "consumed" during hydration due to chemical shrinkage is given by:

 (1)

where is the density of water (1000 kg/m3). Denoting the porosity of the LWFA by LWFA and its saturation (0-1) by S, the total volume fraction of LWFA needed, VLWFA, is given by:
 (2)

The ratio of this quantity to the volume fraction of fine aggregate in the original mixture proportions is the required fractional replacement by LWFA. This equation assumes that all of the water in the LWFA will be readily accessible to the surrounding cement paste, a topic addressed in more detail below.

For example, consider a typical HPC mixture with the following characteristics: cement content of 500 kg/m3, w/c=0.3, and fine aggregate volume fraction of 0.30. For w/c=0.3, the maximum potential degree of hydration is 0.75. Substituting all the appropriate values into equation 1, 0.0225 m3 water/m3 concrete are needed to ensure that the capillary porosity in the cement paste is water-filled at the maximum degree of hydration. Assuming a porosity of the LWFA of 0.15 and complete saturation (S=1.0), one calculates VLWFA to be 0.15, so that 50 % of the fine aggregate (on a volume basis) needs to be replaced by saturated LWFA.

In addition to providing the necessary volume of water, a further issue to be addressed is the proximity of the cement paste requiring the water to the surfaces of the LWFA. Conceptually, this is similar to the "protected paste volume concept" for air entrained concrete [9,10,11], where one is interested in the volume of cement paste within a given distance of an air void surface. For our purposes, this question can be addressed using a previously developed 3-D continuum microstructural model of concrete [12]. In this model, the aggregates are represented by impenetrable spherical or ellipsoidal particles and each aggregate particle is surrounded by a soft penetrable shell representing the interfacial transition zone. For the current study, we are not specifically interested in the interfacial transition zones, but instead adapt the code to surround only the saturated LWFA particles with a shell of variable thickness. Then, by systematic point sampling [12], we can determine the volume fraction of paste contained within these shells and hence the relative proximity of the cement paste to the additional water sources.

It should also be noted that analytical equations exist [6,13,14] for estimating these paste volume fractions directly from the aggregate particle size distribution. The complete equations for doing this have been provided in [13], based on the original development of Lu and Torquato in [6]. Application of these equations to the case of partial replacement of the fine aggregate by LWFA considers only two components of the system: the cement paste with volume fraction Vpaste, and the saturated LWFA. Therefore, we correct the LWFA volume fraction used in the analytical equations by dividing it by the sum ( VLWFA+Vpaste), thereby keeping the ratio VLWFA/Vpaste the same for both the measured system and the analytical calculation. These equations will thus only provide approximate values of the cement paste fraction within a given distance of a LWFA surface, as they effectively ignore the probability that a point within a given distance from the aggregate surface could lie within a normal weight aggregate, as well as within the cement paste. This approximation will worsen as more of the normal weight aggregates are of the same size as the LWFA. Here, the accuracy of this approximation will be evaluated quantitatively for the two systems (described below) considered in this study.

In this preliminary study, two aggregate gradations were investigated based on the limits of the ASTM C 33 [15] aggregate specification. In both cases, the coarse aggregate particle size distribution (PSD) followed the coarse limit curve for a nominal size range of 12.5 mm to 4.75 mm (maximum aggregate size of 19.0 mm) and a coarse to fine aggregate volume ratio of 1.5:1 was assumed. For the first study, the fine aggregate PSD followed the coarse limit of the ASTM C 33 specification and a volume fraction of aggregate (Vagg) of 0.75 was used. In the second case, the fine limit of the ASTM C 33 fine aggregate specification was used. In the latter case, because of the higher surface area of the aggregate (nearly twice that of the first case), the volume fraction of aggregate was reduced to 70 %. Because of the fineness of the aggregate, over one million aggregate particles were needed to simulate a 3-D model concrete 30 mm on a side (27,000 mm3 in volume). In both cases, replacement of 25 %, 50 %, and 100 % of the normal weight fine aggregate by its saturated LWFA counterpart on a volume basis was simulated by randomly assigning the desired proportion of the fine aggregate to be lightweight during the aggregate placement process. The fraction of the cement paste within a given distance of the LWFA surfaces was then determined for the two distributions for each of the following distances: (10, 20, 30, 40, 80, 100, 150, and 200) µm. As the capillary pore space in the cement paste depercolates during curing, the water transport will be effectively limited to distances on the order of 100 µm to 200 µm [8,16].

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