Over the past 5 years, x-ray absorption techniques have been used successfully in non-destructive tests to determine relative density and moisture contents in materials such as cement pastes, mortars, and wood. For example, Bentz and Hansen [1] and Bentz et al. [2] used profiles of point measurements from an x-ray absorption system to extract fundamental data on water movement in paste samples at early ages. These measurements showed that water always flows from a coarser pore layer to a finer one, as predicted by the Kelvin-Laplace equation [3], whether the difference in porosity is due to the water to binder mass ratio (w/b) or varying particle size distributions. These results were then used to add the drying process into the National Institute of Standards and Technology (NIST) CEMHYD3D cement hydration and microstructure development model [4, 5].
It must be remembered, however, that the resulting measured signal combines information about the physical structure of the specimen and the random noise from the instrument used to do the sampling. To properly interpret the resulting signal, therefore, it is important to understand the uncertainty of the measurements. It is often assumed that the x-ray photon counting used in x-ray absorption is approximately a Poisson process in which the uncertainty is the square root of the number of counts [6, 7]. This uncertainty estimate accounts for the random noise expected for a point measurement. No published data confirming this assumption for x-ray absorption measurements have been found.
Moreover, if information about the mean physical features of a specimen is desired, a point measurement or one profile of point measurements may not be representative of the larger specimen, in which case the preferred methodology would include averaging several points or profiles together. In the case of an x-ray absorption system, the proper averaging procedure may vary depending on the materials being sampled, the specimen size, the sampling time period, and the intensity of the x-ray beam. Such information is not currently available for x-ray absorption systems. In this paper, the discussion focuses on experiments that were designed 1) to determine the uncertainty of x-ray absorption measurements and evaluate the utility of the Poisson estimate and 2) to determine how to properly average data to get a representative view of the specimen's mean relative density and composition variations. The materials considered in this study are water, epoxy, cement paste, and mortar (Table 1). The results discussed within provide an overview of the basic uncertainty of an x-ray absorption system for different machine settings and provide information about the reduction of random noise in the measurements by averaging data.
Technically speaking, what is commonly referred to as x-ray absorption is more correctly called x-ray attenuation, which includes photoelectric absorption, incoherent (Compton) scattering, pair production, coherent (Thomson, Rayleigh) scattering, and photodisintegration [8]. Of these, photoelectric absorption, Compton scattering, and pair production generally dominate the attenuation of the x-ray beam. Photoelectric absorption occurs when a photon interacts with an atom and the photon energy is completely transferred to an orbital electron, which is then ejected. The process by which the x-ray photon ejects an electron from an atom and an x-ray photon of lower energy is emitted from the atom is called Compton scattering and is important for materials with low atomic numbers. For pair production, an electron and positron are created with the annihilation of the x-ray photon. This process can be important for specimens with high atomic numbers. In this paper, the term "x-ray absorption" is used, as is commonly done, to describe the integrated effect of all the processes that attenuate the x-ray beam.
In x-ray absorption systems, x rays are produced by bombarding a metal target (e.g., tungsten) with electrons that are produced by heating a metal filament (e.g., tungsten). The x-ray beam is directed toward a specimen at a selected point. The amount of energy that is transmitted through the specimen is described by Beer’s Law (e.g.,[9]):
where I is the x-ray intensity leaving the specimen; I0 is the x-ray intensity of the beam entering the specimen; ρ is the specimen density; t is the specimen thickness; and µ / ρ is the mass absorption coefficient. For a specimen containing several materials, the effective µ / ρ is determined by summing the µ / ρ for each material multiplied by its mass. Because each attenuating process described above is dependent on atomic number, the amount of energy transmitted will depend not only on the specimen thickness but also on the specimen composition. In Eq. (1), this dependence on atomic number is accounted for in the mass absorption coefficient.
The x-ray energy that is transmitted through a specimen reaches a detector crystal that may be composed of NaI or a combination of cadmium, zinc, and tellurium (CZT). Software then processes the signal and determines how much energy has been transmitted and outputs this as x-ray "counts." The number of counts indicates the number of x-ray photons that have been collected by the detector. Because the amount of energy transmitted depends on specimen composition, the number of counts alone cannot be used reliably to relate the density of one specimen to that of another of equal thickness but with a different composition. However, for a given specimen, an increase in counts with time, or a positive "count difference" suggests that the specimen has become less dense. In the case of pastes and mortars after set has occurred, temporal count differences indicate changes due to the movement of water.
Work by Bentz et al. [10] with an x-ray absorption system ascertained the effects of shrinkage-reducing admixtures on self-desiccation of cement-based materials at early ages. Admixtures were found to accelerate the drying of bulk solutions while slowing the drying rate from cement paste specimens. Based on the change of counts with time (count differences), Bentz [11] concluded that cement pastes that contain fly ash lost more water and exhibited deeper drying fronts than those without fly ash. Hu and Stroeven [12] detected a complex internal moisture gradient within paste and mortar specimens with obvious top-down drying occurring only over a small zone near the specimen surfaces. Finally, Lura et al. [13] showed with x-ray absorption measurements that water transport from saturated lightweight aggregates to hydrating cement paste occurred over a distance of at least 4 mm.
The data in these previous studies show some scatter and without an estimate of the uncertainty of the measurements, it is difficult to determine if the variations seen are actual physical features of the specimen or random noise. To reduce scatter and to reveal the mean trends in the data, it may be necessary to average many data points together.
In addition, it is not clear if the results of these previous studies are applicable to larger mortar and concrete specimens for which the smaller paste and mortar specimens are intended to be a model. For example, the presence of an interfacial transition zone around aggregates and different particle packing in mortars and concrete may cause water movement in mortars and concretes that is different from that in cement pastes. Also, the mortar specimens tested by Hu and Stroeven [12] had a minimum dimension of 5 mm and a maximum aggregate size of 4 mm, violating a rule-of-thumb that to obtain a representative volume of a specimen, the ratio of the minimum specimen dimension to the maximum aggregate size should be five or larger [14]. For a mortar with a maximum aggregate size of 4 mm, the minimum horizontal specimen dimension should be at least 20 mm and for a concrete with a maximum aggregate size of 40 mm, the minimum specimen dimension should be at least 200 mm. Due to expected density and composition variations in these larger samples caused by the presence of aggregate, many points of the specimen would need to be measured and averaged to decipher the specimen’s mean structure. To properly interpret the counts and their temporal changes, it is necessary to understand the inherent uncertainty introduced into the measurements by the machine itself. Determining the best way to sample the mean features of larger specimens and determining the uncertainty of an x-ray absorption system’s measurements are the goals of this paper.