Methodology to Quantify the Water Content of Axisymmetric Cylindrical Cement-Based Material Samples Using Neutron Radiography

Abstract

Neutron radiography has been used to assess water content and transport in porous media like cement paste, mortar and concrete. This paper presents a methodology to evaluate water content profiles from neutron radiographs of axisymmetric cylindrical samples along the radial direction. Three examples are proposed as potential applications of this methodology: a drying mortar specimen, a sample of fluid cement paste during high-shear flow, and a steel-reinforced mortar specimen with a gap at the steel–mortar interface. In each case, a simulated neutron image is generated to represent experimental data. The proposed methodology is used to back-calculate the water content distribution of the original sample. The proposed approach can accurately quantify the distribution of water in all three theoretical cylindrical samples. For neutron radiographs created using a random distribution of neutron cross-section values for each constituent, emulating the experimental variability of the imaging process, the proposed method was able to quantify the distribution of water along the radial direction with an average error less than 1.5% for the drying mortar specimen, 3% for the cement paste sample during high-shear flow, and 4% for the reinforced sample with a gap at the steel–mortar interface.

1. Introduction

The water content and its spatial distribution are important in fresh and hardened cement-based materials. For mixtures in the fluid state, liquid phase separation and particle migration influence the rheology and pumpability of concrete [1,2,3]. During solidification, moisture movement occurs during settlement [4], and evaporation results in capillary stress development [5] and plastic shrinkage cracking [6]. In the hardened state, the water content can influence shrinkage [7,8], fluid and ionic transport (e.g., capillary sorptivity, hydraulic permeation, and ionic diffusion) [9,10,11,12], and freeze–thaw damage [13].

A range of experimental methods have been used to quantify the water content of cementitious materials including gravimetric measurements [13], humidity sensors [14], magnetic resonance spectroscopy [15], and various types of radiographic imaging [12,16]. This work focuses on the use of neutron radiography, a process by which an object is irradiated with a high-energy neutron beam, and a detector captures the remaining radiation that is not attenuated by the object, projecting a two-dimensional image [17]. The attenuation process is interpreted using the Beer–Lambert Law:

$$\mathit{ln}\left({\displaystyle \frac{I}{I_0}}\right)=-t{\Sigma }_{avg}=-t\left({SUM}_{i=1}^j\left(V_i{\Sigma }_i\right)\right),$$

(1)

where t is the pixel-wise thickness of the sample that the beam passes through, Σ_avg is the average macroscopic neutron cross-section of the specimen, Σ_i is the cross-section of each constituent i of the specimen (for a total of j constituents), V_i is the volume fraction of each corresponding constituent, I₀ is the intensity of the unattenuated radiation beam, and I is the detected intensity.

Neutron radiography specimens are usually cast and conditioned in tightly controlled conditions (e.g., temperature, humidity, specimen tilt and positioning relative to neutron detector [18]). This allows for the approximation, at the macroscopic scale, that the pore structure and material properties of the specimen are uniform over the region-of-interest (ROI) analyzed in the neutron images. Boundary conditions around the ROI can also often be idealized as analytical expressions. Cement paste and mortar samples are commonly chosen for radiography analysis as they can be reliably made homogeneous during casting and curing, while concrete specimens may have local variations due to the presence of coarser aggregates [19].

Neutron radiography studies of cement-based materials generally use Equation (1) to quantify the volume of evaporable water after drying or absorption tests [20], the degree of pore saturation [21,22], or the water associated with the degree of hydration of the cementitious paste [19]. The state-of-the art consensus [23,24,25] is that this technique is better suited for quantifying the water content of cementitious samples than other types of radiographic imaging such as X-ray radiography as described in the following paragraph.

X-ray radiography is widely used to identify features in porous media through changes in material density and atomic number [26,27,28,29]. However, X-ray images have a small contrast between saturated and unsaturated pores [30,31]. Contrasting agents may either alter the properties of the fluid or react with the cement matrix [32,33], depriving the material of its characteristics of interest. Neutron imaging, on the other hand, is particularly sensitive to water [18], as the large neutron cross-section of the hydrogen atom (for which the primary source in cement-based materials is water) leads to high neutron attenuation and scattering [34], providing a large contrast.

Neutron tomography, where multiple radiographs of the same object are taken at different angles and algorithmically reconstructed into a 3D projection, is another established analysis technique for cement-based materials [12,35]. While traditionally tomography did not have the temporal resolution to resolve rapid processes such as fluid flow or early-age moisture transport [30,36,37,38], recent developments have allowed for “ultra-fast” imaging, in the order of seconds, that enables such analysis even in asymmetric samples [39,40,41,42]. Still, not all neutron radiography setups are equipped to allow for ultra-fast tomography with sufficient spatial resolution and signal-to-noise ratio for accurate measurements. Therefore, obtaining water content profiles over time from individual radiographs of test samples is a desirable alternative to these novel neutron tomography techniques.

One current shortcoming of neutron radiography is that the composition of the sample is generally assumed to be uniform through its thickness (i.e., in the direction parallel to the incident neutron beams. As a result, it is not possible to profile the change in water content between the core and the edges of a sample in the direction of the incident beam. This issue is commonly circumvented with the use of prismatic specimens of constant thickness, and by only analyzing the change in average water content across the height of the material [19,43,44,45]. Figure 1a illustrates an example of this geometry. However, for many neutron radiography experiments involving cement-based materials, it would be convenient to identify different phases of a sample through its thickness, particularly when considering an axisymmetric cylindrical geometry. While cylindrical material samples have been used in neutron radiography observations of hydrogen-rich fluids in multi-layered systems [44,46], the present literature does not address the challenge of identifying such layers through the depth of the sample.

Different studies on porous media [47,48,49,50] have used cylindrical cores to observe the absorption of water through neutron radiography, but the quantification of the water content was limited to the longitudinal axis of the sample (see Figure 1b). Such an experimental design is not always practical for long-term experiments [24]. A simplified approach to quantifying the change in water content along the radial axis of axisymmetric cylindrical samples has been presented in [51,52,53]. The authors approximate the geometry of the drying sample as a pair of coaxial cylinders separated by a drying front (similar to Figure 1c, but with the core region wetter than the periphery, due to the drying conditions) and successfully identify this front through the thickness of the specimen at different measurement times. However, fully characterizing different cement-based material systems requires more specific information in the form of a radial water content profile, providing crucial data for the study of moisture transport, extrusion flow, and reinforcement corrosion scenarios, for instance. The approach in [51,52,53] cannot be extended to represent multiple layers of distinct water contents (i.e., multiple coaxial cylinders) as it would lead to an unreasonably large number of fitting parameters to be calculated.

Therefore, it is desirable to overcome the limitation in quantifying the spatial distribution of water across the thickness of a material specimen from its two-dimensional neutron radiograph. The process of back-calculating the water content distribution of a sample from a measurement of its properties—in this case, the macroscopic neutron cross section—is an inverse problem [54]. This paper proposes a practical methodology to solve this inverse problem for axisymmetric cylindrical samples with multiple phases across its thickness. This methodology is distinct from that described in [51,52,53] since it does not require binarizing the specimen cross-section into wet/dry regions, but instead enables quantifying the moisture content in the radial direction.

The methodology and some of its potential applications in cement-based material studies are demonstrated in three simulated example experiments to obtain (1) the pore water content (i.e., the degree of saturation) of a drying hardened mortar specimen, (2) the local volume fraction of water in a fresh cementitious mixture during pumping and/or extrusion, and (3) the width and water content of a gap along the steel–mortar interface of a reinforced specimen (i.e., a mortar specimen with embedded steel rebar at its center). These experimental procedures can provide crucial data to inform physics-based models describing cement-based materials, but are not feasible to execute with current neutron radiography analysis techniques. They can also be used as reference points for the design of other neutron radiography studies investigating internal moisture movement in cement-based materials, such as in the case of internal curing [36,55].

In these simulated experiments, virtual neutron radiographs are generated from known inputs for sample geometry and water content, as the solution to the process defined as the forward problem. This is equivalent to obtaining a neutron radiograph from a physical measurement. Then, the inverse problem is solved to calculate the geometry and water content of the heterogeneous sample from the output radiograph of the forward problem. The comparison between the two sets of results is used to further validate the use of the methodology. The proposed methodology is only applicable to axisymmetric cylindrical samples, which are widely used in characterization of cementitious systems. This sample geometry also allows for a simplified description of the cross-section in radial coordinates, with an analytical solution to the inverse problem. While more complex geometries can be solved through conventional inverse modeling algorithms [54], this is beyond the scope of this paper.

2. Materials and Methods

The approach used in this paper involves (1) obtaining the neutron radiograph of a sample (the forward problem), and (2) interpreting this radiograph to determine the radial composition of the sample (the inverse problem). The flowchart in Figure 2 illustrates the steps of the methodology and the relationship between the forward and inverse problems.

The neutron radiograph can be obtained through either a physical measurement or a virtually generated radiograph (Section 2.1). The modeled sample used as reference for this virtual radiograph is equivalent to a perfectly axisymmetric physical sample, unaffected by experimental variability. Using a modeled sample allows for increased flexibility in testing different experimental scenarios and material compositions, and for prescribing levels of artificial noise that can emulate the different sources of experimental error in physically measured radiographs. Therefore, the resulting virtual radiograph is analogous to an experimental radiograph obtained after image processing techniques are used to compensate for hardware limitations, beam hardening, and background and neutron scattering noise [30,56]. While analyzing the effect of each of these sources of variability on the application of the methodology to physically measured radiographs is relevant, such discussion is beyond the scope of this paper.

When using a modeled axisymmetric cylindrical sample, the solution to the forward problem consists of defining the geometry of the sample using concentric rings (a), identifying the composition and associated neutron cross-section of each ring (b), computing the thickness of each ring the beam passes through (c,d), using Equation (1) to compute the normalized beam intensity for each pixel (e), and using these pixel values to create the neutron image (f). The forward problem and the approach used to solve it are further detailed in Section 2.1.

The solution to the inverse problem for an axisymmetric cylindrical sample consists of using the normalized beam intensity from the radiograph (g) along with sample geometry to solve for the radial macroscopic neutron cross-section of the object. The geometry of the cross-section of the sample is defined by a series of concentric rings, each ring with a width that corresponds to the radiograph pixel size (h), and with chords corresponding to the thickness that the beam passes through (i). At the outer edge of the cylinder, the thickness is defined by the pixel size and sample radius. In this scenario, only the neutron cross-section of the outer ring is unknown. After back-calculating the neutron cross-section of the outer ring using the Beer–Lambert Law, the process is repeated for the neighboring rings until the center of the cylinder is reached (j). These neutron cross-sections are then used to segment the concentric rings into different layers (k) and determine their composition (l). The inverse problem and its solution are fully described in Section 2.2.

The methodology is initially validated through the analysis of a perfectly axisymmetric test sample. This is necessary to avoid confusion on whether any inaccuracies between the original radial water content (or, more broadly, radial composition) of the material, and the one predicted in the solution to the inverse problem, are due to geometric imperfections or to a flaw in the methodology itself. The authors propose the use of virtual cylindrical samples (Section 2.3) to understand the practical viability of the proposed methodology.

2.1. Solution to the Forward Problem

The process of creating a virtual radiograph is analogous to the experimental procedure used to obtain a physical one. An incident beam is projected from the neutron source with intensity I₀. The sample is placed between the neutron source and the detector, such that its curved face is orthogonal to the beam direction. The intensity of the incident beam after being attenuated by the sample (I) is a function of its thickness (t) and its average macroscopic neutron cross-section (Σ), as described by Equation (1). The sample thickness along its width is given by the length of the parallel chords crossing the circular cross-section of the cylinder, in the direction of the neutron beam (see Figure 3a).

The average macroscopic neutron cross-section describes the ability of a material sample to attenuate the flux of neutrons through the combined effects of absorption and scattering [57]. It is determined through the assumption that the value of the neutron cross-section of a composite material is a weighted sum of the neutron cross-section and the volume fraction values of its constituents. The neutron cross-sections of these constituents are obtained by measuring samples of different thicknesses of each reference material in a particular radiography setup, with specific beam energies, collimation ratios, and detector positioning [58].

Neutron cross-sections are frequently taken as constant and linearly related to sample thickness, in radiography setups where the influence of beam hardening on the detected neutron intensity can be assumed to be negligible [19,22,58,59]. This is the case for the testing setup of the neutron radiography facility (NRF) of the Oregon State TRIGA^® Reactor (OSTR) in Corvallis, OR, USA, used as a reference in this study [60].

Table 1. Macroscopic neutron cross-section values of constituent materials used in Section 2.3.

Material	Macroscopic Neutron Cross-Section (mm−1)
Type I/II Ordinary Portland Cement (OPC)	0.0187 [61]
Water	0.1256 [58,59,61] 1
Ottawa Silica Sand (ASTM C109, C778)	0.0179 [58] 2
1018 Carbon Steel (ASTM A108)	0.0343 3

¹ Average of maximum (0.1304) and minimum (0.1208) values obtained for Σ_water. ² ASTM standards only used to specify sand composition, fineness, particle size and roundness. ³ Experimentally obtained following the procedure described in [58].

includes the measured baseline macroscopic neutron cross-section values for the materials used in the examples in the following section, measured at the NRF with an exposure time of 2 s.

As the partially attenuated beam (I) reaches the detector, its measured intensity values are converted into pixel intensity (grayscale) values to create the radiograph of the sample. This radiograph is the projection of the cylindrical geometry considered in this study, appearing as a rectangle. Each column of pixels of this rectangle corresponds to a different thickness of the cylinder that attenuates the incident neutron beam.

The first step (a) for solving the forward problem is defining the sample geometry and the composition of each of its radial layers. The sample is represented as a series of concentric rings in the circular cross-section. Each concentric ring has a corresponding macroscopic neutron cross-section Σ_n

$${\Sigma }_n={SUM}_{k=1}^l\left(V_k{\Sigma }_k\right),$$

(2)

where Σ_k is the macroscopic neutron cross-section of each constituent of the layer (for a total of “l” constituents), and V_k is the volume fraction of each constituent. The number of concentric rings is a function of the smallest unit of information used to generate the virtual radiograph. If the change in the composition of the concentric rings is determined through an experiment (e.g., the simulation in Section 2.3.1), then the number of rings is either extrapolated or interpolated from the collected data points. If the concentric rings are an idealized representation of the cross-section of the sample (e.g., the samples in Section 2.3.2 and Section 2.3.3), then any arbitrary quantity can be chosen to represent this smallest unit of information.

The second step (b) is to determine the Σ_k values, which is performed in two distinct ways and leads to the creation of a pair of analogous images. In the first one, the neutron cross-sections are directly extracted from Table 1, and the resulting image is expected to allow for a reconstruction of the original sample composition (i.e., the solution to the inverse problem) with minimal error. In the second one, a uniform probability distribution is assigned to the neutron cross-section values, ±4% from those described in Table 1. This is a stochastic representation of experimental variability in the measurements of the neutron cross-sections of each constituent, due to the combined effects of nonuniform energy distribution of the neutron beam, neutron scattering noise, and operator precision during image processing. The magnitude of this variability is estimated from the range of Σ_water values measured at the NRF setup and reported in the literature [58,59,61]. At every concentric ring (i.e., for every Σ_n value), a pseudorandom number generator in MATLAB [62] is used to select an Σ_k value within this range, for all material constituents of the ring. This second image is expected to provide some initial insight on the susceptibility of the reconstruction method to the variability inherent to physically measured radiographs. After obtaining these neutron cross-section values everywhere in the modeled sample, Equation (2) can be rewritten as a function of the axisymmetric layers:

$$ln\left({\displaystyle \frac{I_0}{I_r}}\right)={SUM}_{n=m}^q\left(t_{n,r}{\Sigma }_n\right),$$

(3)

where I_r is the intensity of the neutron beam after being attenuated by the sample, at a given radial position r, and t_n,r is the thickness of material of layer n that contributes to the attenuation at r. The innermost concentric ring at this position is labeled m, while the outermost ring of the cylindrical sample is q.

The third step (c) required to model the virtual image is to determine the thickness values inside the summation term of Equation (3). Considering the smallest unit of information defined in the first step of this process, the circular cross-section is divided by vertical chords separated by this distance. These chords represent the neutron beams passing through the material and being partially attenuated by the sample, with the thickness of material traversed, t_r, given by the trigonometric function:

$$t_r=2rsin\left({cos}^{-1}\left({\displaystyle \frac{x_r}{r}}\right)\right),$$

(4)

where r is the radius of the cylindrical cross-section, and x_r the horizontal position of the chord.

The fourth step (d) is to divide these chords into fractions that intersect each concentric ring, corresponding to the thickness of material of a particular layer that contributes to the overall neutron attenuation. Calculating these values requires two trigonometric expressions: one for the fraction of the chord that intersects the innermost circle, and one for the analogous fractions for all other concentric rings. The geometric relationships used for these equations are illustrated in Figure 3b. For any chord traced at location r along the width of the cross-section, intersecting an innermost ring m and all other rings external to it, the first expression is analogous to Equation (4), with the length t_m,r being:

$$t_{m,r}=2r_{m}sin\left({cos}^{-1}\left({\displaystyle \frac{x_r}{r_m}}\right)\right),$$

(5)

with r_m representing the radius of the innermost ring intercepted by this chord. For the rest of the intersecting rings, identified in ascending order from p = m + 1 to the outermost ring p = q, the fraction of the length of the chord that passes through them is always a function of the geometry of their neighboring inner circle:

$$t_{p,r}=2\left[r_{p}sin\left({cos}^{-1}\left({\displaystyle \frac{x_r}{r_p}}\right)\right)-r_{p-1}sin\left({cos}^{-1}\left({\displaystyle \frac{x_r}{r_{p-1}}}\right)\right)\right],$$

(6)

$$t_r=t_{m,r}+2\left\{{SUM}_{p=m+1}^q\left[r_{p}sin\left({cos}^{-1}\left({\displaystyle \frac{x_r}{r_p}}\right)\right)-r_{p-1}sin\left({cos}^{-1}\left({\displaystyle \frac{x_r}{r_{p-1}}}\right)\right)\right]\right\},$$

(7)

with r_p and r_p−1 being the radii of the concentric ring of interest and its inner neighbor, respectively. Figure 3b shows an example calculation for a three-layered axisymmetric cylindrical sample.

The fifth step (e) is to aggregate the Σ_n values calculated for every concentric ring that forms the specimen cross-section, along with all terms of t_r for every chord traced along the width of the cylinder, to solve Equation (3) for a dimensionless I/I₀ that is repeated along the height of the sample (due to axial symmetry).

Finally, the sixth and final step (f) is to convert this relative measure of beam intensity into pixel intensities to generate the radiograph of the simulated sample. This virtual radiograph is generated without extrapolating data from its corresponding virtual sample. This results in the smallest unit of information used to model the sample (i.e., the sample resolution) to be always smaller than or equal to the smallest unit of information that can be directly retrieved from the neutron radiograph (i.e., the radiograph resolution). Therefore, when creating the virtual neutron radiograph, the I/I₀ array corresponding to the sample must be transformed into a shorter array of relative intensity values corresponding to the radiograph.

While this can be achieved through interpolation (e.g., using a linear function), the difference in resolution between the object and the image might be problematic due to the constraint that the outer edge of the outermost pixel of the radiograph must match the boundaries of the sample. This geometric requirement may lead to the central pixel of the radiograph to be smaller than the specified image resolution. Although this approximation can be a source of error for the central value of the image, it should not impact the interpretation of the water content profiles since in all three examples the core of the samples is assumed to have a relatively uniform water distribution. Therefore, any outlier values at the center of the cross-section can be ignored without meaningful loss in accuracy. These interpolated energy intensity values are normalized to greyscale, pixel intensity values, then repeated along the height of the sample to represent the curved face of the cylinder irradiated by the neutron beams.

2.2. Solution to the Inverse Problem

The process of back-calculating the composition of the axisymmetric cylindrical sample and determining its water content distribution resembles the one used to generate a virtual radiograph—with most of the steps being similar, only taken in the reverse order.

The first step (g) is to convert the greyscale pixel intensity values from the neutron radiograph into the normalized energy intensity values (I/I₀). In a radiograph obtained through a physical measurement, this is performed using flat field and dark field images as references [22]. The virtual radiograph obtained through the solution to the forward problem does not require this step since the radial I/I₀ profile is already provided.

The second step (h) assumes a circular cross-section of the cylinder composed of concentric rings with a width of one pixel. The rationale behind this choice is that the pixel is the smallest unit of information that can be directly obtained from a radiograph, regardless of the actual resolution of the sample. Therefore, this results in the maximum number of unique layers that can represent the image. As a consequence of the sixth step of the forward problem, the innermost concentric ring is slightly narrower than one pixel, allowing the radius of this cross-section to match the radius of the cross-section of the sample.

The third step (i) consists of tracing a set of vertical chords through the centerline of each pixel, analogously to the forward problem (step (c) in Figure 2). These chords are divided into fractions (using Equations (5) and (6)), where the fraction of each chord length represents the thickness of material in each layer that attenuates the neutron beam.

The fourth step (j) starts by entering the values corresponding to the chord length fractions and the normalized measured intensities, at each radial position corresponding to the centerline of the pixels of the radiograph, in Equation (3). The only remaining unknowns in the equation are the neutron cross-sections of each concentric ring. If the chord that passes through a certain pixel intersects multiple rings, then Equation (3) cannot be solved since it has multiple unknown Σ_n values. This applies to all chords except for the one passing through the outermost pixel of the cross-section, which only intersects the outermost ring (m = q). As such, Σ_q is the only unknown quantity, which can be determined. With this value, Equation (3) for the neighboring inner ring (m = q − 1) now only has one unknown, Σ_q−1. Using this approach, it is possible to iteratively determine Σ_n for all the concentric rings.

The fifth step (k) is using the back-calculated neutron cross-sections to segment the different layers of the sample and identify its different features. Even when the composition of the specimen is unknown or the layers have very distinct constituents (as in the reinforced mortar example detailed in the next section), this allows for a qualitative analysis of the image.

The sixth and final step (l) is relating the Σ_n values of each layer to their respective composition by solving Equation (2) for a Σ_k or V_k value of interest. In most cementitious systems, the cylindrical sample is only made from a cement-based paste or mortar, and it is assumed to be radially homogeneous. In these cases, the original mixture design is known, and the cross-section of the constituents has been measured at the neutron radiography setup (as discussed in Section 2.1). Therefore, the only remaining variable in Equation (2) is V_water as it changes over time and along the radius of the sample. Depending on the experiment that the neutron radiographs correspond to, this volume fraction of water is associated with the degree of pore saturation or the degree of hydration of the cementitious paste [19,20,21,22].

The authors propose the use of virtual cylindrical samples (Section 2.3) to understand the practical viability of the proposed methodology. This is necessary to avoid confusion on whether any inaccuracies between the original radial water content (or, more broadly, radial composition) of the material, and the one predicted in the solution to the inverse problem, are due to geometric imperfections or to a flaw in the methodology itself.

2.3. Simulated Experiments

The proposed simulations are computational models that represent a physical object—in this case, an axisymmetric sample of cement-based material used to obtain a neutron radiograph, with the experiments consisting of changing the model parameters to simulate the effect of the testing conditions on the real system, similar to a “digital twin” [63]. Using simulated experiments is particularly useful to assess the accuracy of the method of interpreting neutron radiographs described in Section 2.2. Since the virtual radiographs are perfectly axisymmetric and their pixel intensity values are theoretical (obtained from the Beer–Lambert Law), the outputs of the inverse problem must match the inputs of the forward problem with minimal error if the proposed methodology is mathematically sound, as further discussed in Section 3.

Three simulated experiments are conducted, as illustrated in Figure 4. The first experiment consists of a mortar cylinder with its evaporable water content following a radial distribution, analogous to a drying experiment. The second experiment consists of a fresh cementitious sample in a confined cylinder with an interfacial “lubrication layer” to simulate extrusion or pumping. The third experiment consists of a mortar sample with a concentric reinforcing steel bar surrounded by a radial gap (simulating longitudinal cracking along the steel–concrete interface) that is assumed to be filled with air, water (water-filled crack) or mortar (control case with no cracking). This enables the interfacial debonding to be simulated which has implications on the mechanical and corrosion behavior of reinforcing steel.

The following subsections detail the features of the forward problems, as well as their respective solutions, for each of these cases. Since the problems are idealized as axisymmetric, only the radial properties are plotted to compare the virtual and reconstructed objects. Such a representation is equivalent to using a single row of pixels, consisting of the average pixel intensity values of each pixel column of the ROI that represents the sample, as the input for each inverse problem.

2.3.1. Drying of Mortar

This example simulates a hydrated mortar specimen with height of 150 mm and diameter of 20 mm. The sample is initially at 90% RH, then it is exposed to a 50% RH and 23 °C environment, and its radial RH profile is measured after 3, 45, and 600 days. The mixture design consists of a water:cement:sand ratio of 0.42:1:2.35 (in mass), with the volume fraction of entrained air inside the hardened mortar being 5%. The macroscopic neutron cross-sections of water, cement, and sand at the NRF setup are described in Table 1. After the raw materials are mixed, water and cement react to form solid hydrates and pore space, with the latter being filled with some amount of water depending on their degree of saturation. The Powers–Brownyard model [64] is then used to estimate the final composition of the specimen after complete (100%) hydration, consisting of a solid matrix (sand and hydrates) and porosity (air voids and pore space from hydration). Assuming complete specimen maturity, such that the role of ongoing hydration during the experiment is negligible, is common in studies on drying and transport in cement-based materials [24,65,66,67,68].

Calculating the neutron cross-sections along the radius of the specimen requires outlining the contributions of each of these phases, as shown in Equations (8) and (9)

$${\Sigma }_{solids}={\displaystyle \frac{\left({\Sigma }_{sand}{V}_{sand}+{\Sigma }_{cement}{V}_{cement}+{\Sigma }_{hydrates}{V}_{hydrates}\right)}{\left(V_{sand}+V_{cement}+V_{hydrates}\right)}},$$

(8)

$${\Sigma }_{porosity}=\left({\Sigma }_{water}S\left(r,t\right)\right),$$

(9)

where the neutron cross-section of the solid hydrates (Σ_hydrates) is assumed to be equivalent to that of a mixture of water and unhydrated cement with the same overall elemental composition [58]. Considering that the amount of chemically bound water in the gel solids is 0.23 g per gram of hydrated cement [69], Σ_hydrates is calculated through a weighted volume fraction sum just like the one in Equation 8, with an average value of 0.0756 mm⁻¹ (ranging between 0.0726 and 0.0786 mm⁻¹ for the case with artificial noise). Therefore, the only unknown is S(r,t), the degree of pore saturation as a function of both radial position and time of drying exposure.

Drying experiments of cement-based materials have been successfully simulated through finite element modeling in the literature [65,66,67]. The model used in this example includes a validated coupled moisture transport and heat transfer analysis, with its governing equations, boundary conditions, and modeling and meshing parameters described in [70]. Appendix A provides a brief description of the modeling approach. The COMSOL Multiphysics 6.0 finite element software [71] is used to simulate the nonlinear diffusion of humidity within the material after exposure to the drying environment, as well as its desorption isotherm curve (Figure 5a) that relates the radial pore RH profile to the pore degree of saturation profile, S(r,t) (Figure 5b) [72].

Since the mortar is assumed to be homogeneous, the axisymmetric layers of the sample are separated by the change in degree of saturation presented in Figure 5b. The S(r,t) values are then interpolated or extrapolated to match the desired object resolution. This results in a series of concentric rings with width equal to the chosen size for the smallest unit of information of the sample, each with a characteristic neutron cross-section.

$${\Sigma }_n={\Sigma }_{solids}\left(V_{sand}+V_{cement}+V_{hydrates}\right)+{\Sigma }_{porosity}\left(V_{air}+V_{hydration pores}\right),$$

(10)

With this geometry established, the vertical chords passing through the sample are traced as described in Section 2.1. Similarly, the chord fractions corresponding to the thickness of each layer that attenuates the neutron beams intersecting the cross-section are determined through Equations (5) and (6). The model then has all the necessary information to allow Equation (3) to be solved for I/I₀ at every position along the radius of the specimen, at each of the three times of exposure, and then convert this radial profile to greyscale intensity values for every pixel of the virtual radiograph.

2.3.2. High-Shear Flow of 3D-Printed Cement Paste

This example demonstrates how this methodology can be used to help quantify the flow-induced particle migration in cement-based materials during the pumping of concrete or the extrusion-based additive manufacturing of cement paste and mortar, which affects both their rheology and the heterogeneity of the hardened material [73,74,75,76]. The geometry shown in Figure 4b represents this behavior, with a slip/lubricating layer (LL) with high water content at the interface of the pipeline or nozzle, and partially sheared zones that separate the LL from the bulk material layer that undergoes plug flow [77,78].

The thickness of the LL has been experimentally reported to vary by several orders of magnitude (10⁻² to 10⁻⁵ m) depending on the high-shear flow conditions, including but not limited to the flow geometry, flow pressure, solid particle size distribution and volume fraction of the suspension, and the rheology of the suspension [3,78,79]. Furthermore, limited observations of the LL in rheological devices emulating pipe flow [3,73] have hinted that the observed thickness of this layer may locally vary in the axial and radial directions of a cylindrical sample. While a few numerical studies have proposed particle distribution profiles in cement-based materials undergoing this process [80,81,82], these results have yet to be validated with experimental data. In contrast with the previous example, there is not yet a reliable modeling technique to accurately replicate this behavior.

The authors propose that this methodology is adequate to quantify the material heterogeneity during steady-state high-shear flow, since the current literature shows that the different physical processes that contribute to the formation of the LL are axisymmetric. Therefore, the radial water content profile of a cylindrical sample under such flow conditions should reflect the different material layers formed due to flow-induced particle migration. Using the average pixel intensities of each pixel column of the sample radiograph as the input values for the inverse problem (as described in Section 3) is a way to mitigate the local variation in particle distribution and provide a comparison value for the thickness of the LL predicted by the numerical models.

A cement paste mixture flowing through a nozzle with an internal diameter of 15 mm is modeled with an idealized perfectly axisymmetric geometry to represent a similar scenario to the one expected for the physical system [83]. This flow diameter justifies an estimation for the thickness of the LL for pastes on the higher end (10⁻⁴ m); the chosen value is 180 μm for modeling convenience, since it is twice the size of the nominal pixel size of the NRF camera. The circular cross-section of the sample is then divided into four concentric rings: an 180 μm thick LL representing the fully sheared material, two 180 μm thick layers corresponding to the partially sheared material, and an internal bulk layer subjected to unsheared (plug) flow.

The chosen cement paste mixture has a water-to-cementitious materials ratio (w/c) of 0.30 by mass, with a corresponding volume fraction of water (V_water) of 0.486, which is typical for 3D printing applications [84]. The layered profile due to particle migration is modeled with a LL of V_water,S = 0.857, and two partially sheared layers of V_water,P1 = 0.8 and V_water,P2 = 0.667, respectively. The resulting volume fraction of water of the internal bulk layer (V_water,B) is then calculated following the principle of conservation of mass, expressed by Equation (11),

$$V_{water,B}={\displaystyle \frac{\left\{\left(A\right)\left(V_{water}\right)-\left[\left(A_S\right)\left(V_{water, S}\right)+\left(A_{P1}\right)\left(V_{water, P1}\right)+\left(A_{P2}\right)\left(V_{water, P2}\right)\right]\right\}}{A_B}},$$

(11)

where A_S, A_P1, A_P2, and A_S are the cross-sectional areas for each layer, as indicated by their respective subscripts, and A is the total area inside the nozzle. The value of V_water,B in this example is 0.402.

These volume fractions are used in Equation (2) to calculate the macroscopic neutron cross-sections of these concentric rings. The chords passing through the pixels along the width of the sample, as well as the chord fractions intersecting each layer, are determined using Equations (5) and (6). Finally, the forward problem is solved by computing the greyscale pixel values for each pixel of the virtual neutron radiograph using Equation (3).

2.3.3. Reinforced Mortar with Radial Gap at the Steel–Mortar Interface

This example considers a cylindrical mortar sample with a cylindrical reinforcing steel bar along its central axis. There is a gap at the steel–mortar interface that is also modeled as perfectly cylindrical, as shown in Figure 4c. It is well established that the steel–concrete interface plays a significant role in the corrosion of embedded reinforcement [85]. Cracking along the steel–concrete interface is a concern as it can cause local variability in the water content and chemical composition of the pore solution within these cracks, affecting how corrosion would initiate and progress. These cracks can also serve as sites for crevice corrosion, which is highly dependent on the crack thickness and length [86]. Therefore, accurate determination of crack geometry and its water content along the steel concrete interface is crucial.

The modeled mortar specimen has a diameter of 30 mm and the same mixture proportions as the sample from Section 3.1. The pores of this specimen are assumed to be fully saturated. The central reinforcing steel has a diameter of 9.36 mm, approximately corresponding to the size of #3 rebar, with its baseline macroscopic neutron cross-section is described in Table 1. Finally, the gap at the steel–mortar interface has a width of 450 μm, representing a crack along the rebar. This crack is significantly larger than the interfacial zones between mortar and steel, establishing a sharp steel–crack–mortar interface [87]. The simulated experiment considers three distinct conditions for this gap: filled with mortar (representing no cracking), filled with water (saturated crack), and filled with air (dry crack). The thickness values associated with each layer at each pixel of the image are calculated using Equations (5) and (6), for all three scenarios. Finally, all virtual neutron radiographs are generated by solving Equation (3) for I/I₀ at every pixel position along the radius of the specimens.

3. Results

Two sets of virtual neutron radiographs are generated for each of the scenarios described in Section 2.3.1, Section 2.3.2 and Section 2.3.3: in the first one, the radiographs are created with constant Σ_k values (from Table 1); in the second one, the radiographs are created with artificial noise (in the form of a distribution of Σ_k values). Their greyscale pixel values are represented as line plots relating the normalized pixel intensities of each radiograph to their radial position (i.e., horizontal distance from the symmetry axis), as exemplified in Figure 6.

In the first set of results, the effect of material variability on the neutron radiographs is not considered. As expected, the back-calculated water content profiles in each of the three simulated examples exactly match the original profiles of the virtual samples, which confirms the soundness of the inverse solution.

In the second set of results, the difference in resolution between the virtual samples and the virtual radiographs influences the magnitude of the error between the original and back-calculated water content profiles. When both resolution values are the same, the solution to the inverse problem is accurate, with the average percentage error for the back-calculated Σ_n profile (i.e., the average of the percentage error values between the original and back-calculated Σ_n values, at each radial position corresponding to one pixel of the virtual radiograph) being negligible (10⁻¹⁰%). This is expected since the type of artificial noise added to the virtual samples does not affect their axial symmetry; therefore, the accuracy of the reconstruction should not change. The resulting back-calculated water content profile is the same as the one in the virtual sample after the addition of noise.

However, for the noisy radiographs with a lower resolution than the simulated samples, the change in radiograph resolution affects the accuracy of the back-calculated water content. The cement-based material samples in all simulated experiments are generated with a resolution of 1 μm. The neutron radiographs corresponding to these samples (the solution to the forward problem) are then created with different pixel sizes, all larger than 1 μm. A baseline radiograph resolution of 90 μm is selected to match the resolution of the physically obtained radiographs at the NRF. A finer resolution of 15 μm, corresponding to a planned radiography setup at the NRF, is also selected to assess its effect on the methodology. In Section 3.2 and Section 3.3, with specimens with sharp composition changes, a coarser resolution of 180 μm is used to test the limitations of the methodology in distinguishing thin material layers.

A hundred radiographs are created for each neutron radiograph resolution setting, each time with a different random noise distribution. The effects of changing the resolution of the virtual radiographs are distinct for each of the simulated experiments and are further detailed in the following subsections. A summary of the contributions of different error sources to the results (i.e., the back-calculated neutron cross-section values, and their associated average and maximum percentage error values when compared to the original neutron cross-sections) of each simulated experiment is presented in

Table 2. Summary of different error contributions to the accuracy of the methodology for different simulated experiments.

Experiment	Error Source	Average Error	Maximum Error
Section 2.1/Section 2.2/Section 2.3	No noise	<10−10%	<10−10%
Section 2.1	Noise + 15 μm resolution	1.12%	6.26%
Section 2.1	Noise + 90 μm resolution	1.10%	5.03%
Section 2.2	Noise + 15 μm resolution	1.66%	26.0%
Section 2.2	Noise + 90 μm resolution	1.94%	42.8%
Section 2.2	Noise + 180 μm resolution	2.17%	38.2%
Section 2.3	Noise + 15 μm resolution	1.52%	255%
Section 2.3	Noise + 90 μm resolution	1.72%	134%
Section 2.3	Noise + 180 μm resolution	2.59%	267%

.

3.1. Drying of Mortar

A comparison between the Σ_n values used to generate the virtual radiographs (with artificial noise) for the drying samples, and those back-calculated in the solution to the inverse problem is shown in Figure 7 for two image resolutions of 15 μm and 90 μm, respectively. In all three reconstructed Σ_n profiles (at each time of drying), regardless of the chosen image resolution, the average percentage error for the neutron cross-section values is about 1%, with the maximum absolute percentage error being close to 5%.

As described in Section 2.3.1, Equations (9) and (10) are used to convert the neutron cross-section profiles presented in Figure 7 into the water content (degree of saturation, S) profiles shown in Figure 8. The saturation profiles in each case have more noticeable differences with the change in radiograph resolution.

3.2. High-Shear Flow of 3D-Printed Cement Paste

An iteration of the reconstructed Σ_n and V_water profiles for this experiment, considering radiograph resolutions of 15, 90, and 180 μm, is plotted in Figure 9. The neutron cross-section and water content curves only significantly differ at the radial positions corresponding to the transitions between material layers, with the average percentage error of the back-calculated profiles ranging 1–3%.

This behavior is expected in the presence of artificial noise, since the iterative nature of the methodology means that the calculation of Σ_n for a layer is affected by the previously calculated neutron cross-section values. Therefore, if there is a sudden change in water content between adjacent layers, a larger variation in the Σ_n value of the sample at that location due to noise will be over-compensated in the solution to the inverse problem and lead to visible “spikes” in the reconstructed curves (see Figure 9b,c,e,f). The error of the back-calculated profiles immediately decreases when the following adjacent pixels have a similar water content, returning to the 1% range that is also seen in the results in Section 3.1.

3.3. Reinforced Mortar with Radial Gap at the Steel–Mortar Interface

The neutron cross-section profiles for the three cases of this simulated experiment, considering radiographs with resolution of 15, 90, and 180 μm, are shown in Figure 10 (radial gap filled with mortar), Figure 11 (with water), and Figure 12 (with air–empty gap).

4. Discussion

In the drying example (Section 3.1), the error values in Figure 7 and Figure 8 show that the solution to the inverse problem closely matches the inputs used to create the virtual radiograph. The six-fold difference in the image resolution barely affects the error of the reconstructed Σ_n profiles. The back-calculated S values from the 15 μm resolution images have a slightly larger variation from the COMSOL data obtained from finite element analysis than those obtained from the 90 μm case, even though the difference is less than 1%. This may indicate that, for the type of noise simulated in these experiments, radiographs with coarser image resolution can average out more intense variations in the neutron cross-section of the object.

When larger portions of the sample are assumed to be homogeneous, there does not appear to be a significant reduction in accuracy, matching observations from fluid transport experiments using neutron radiography [88,89]. The increase in the maximum error values for the 15 μm radiographs at all measurement times aligns with this hypothesis. There is also an increase in the average percentage error as the samples are exposed to longer drying periods, from 0.46% to 0.62%. This might be explained by smoother water content profiles being proportionally more affected by the presence of artificial noise than steeper ones, even if to a limited extent.

In the high-shear flow example (Section 3.2), the average and maximum percentage error values in Figure 9 provide evidence that increasing the radiograph pixel size leads to less accurate results. The order of magnitude of the maximum error values suggests that this change in average accuracy is mostly due to the significantly differing values at the outer edge of the reconstructed sample cross-sections. These discrepancies are less pronounced in the neutron radiographs with higher resolution because there are more data points corresponding to the LL and the partially sheared layers, and therefore the methodology has more iteration steps to average out the outlier w/c values (“spikes”) in the reconstructed water content profile. Accurately quantifying the distribution of moisture in these layers would aid in the validation and further development of a governing equation to describe the high-shear flow behavior of the material.

The radiograph pixel size of 180 μm was included to illustrate a scenario where the resolution of the image is close to the length scale of the sample where significant changes in composition exist. As seen more clearly in Figure 9e,f, this mismatch may lead to large discrepancies in the predicted water content of the LL of the sample. While the presence of these outlier values is likely to restrict the use of the methodology to directly quantify the particle migration in such scenarios, the reconstruction method can still clearly identify the unsheared layer, even in the lowest-resolution case. Identifying the width of the bulk layer and its water content, as well as having an estimate for the water content of the LL, would provide unprecedented data for the validation of analytical expressions describing the rheology and particle distribution of the cement-based materials, which are currently analyzed only using pressure-flow rate relationships [3,82].

However, as the extent of the flow-induced particle migration in solid suspensions (such as cement-based materials) is still an open question, the heterogeneities of the sample cross-section may not be as significant as the ones in the idealized geometry described in Section 2.3.2. If the interface between the bulk layer and the edge of the pipe/nozzle is only a few pixels wide, the methodology might not be able to accurately resolve the water content in this region of the sample cross-section. Therefore, the image resolution and noise removal capabilities of experimental neutron radiography setups are likely to play a determining role in the feasibility of this application.

Finally, in the reinforced mortar example (Section 3.3), the methodology clearly allows for the segmentation of the mortar, the rebar, and the material filling the gap, in all three cases. While a gap filled with a small amount of water could be harder to identify due to having an equivalent neutron cross-section between the values for steel and mortar, the relatively large value of Σ_water means that the saturation threshold for reliably identifying the gap would be small, even after the addition of artificial noise.

Similar to the high-shear flow example, the reconstructed Σ_n values at the layer transition regions are very contrasting from the ones used to generate the radiograph, due to the change in material composition. In this simulated experiment, the neutron cross-sections even reach unphysical, negative values for some of the pixels at the edges of the radial gap. The scenario for the sample with an empty gap shows this behavior more frequently since Σ_air is taken as 0 mm⁻¹, meaning that any negative variations in the pixels near the gap lead to back-calculated “negative” neutron cross-sections. Still, these large discrepancies are also outliers in these profiles, as the solution to the inverse problem leads to an average error of less than 1% for the remainder of the cross-section.

The change in radiograph pixel size does not have a significant effect on the average accuracy of the reconstruction of the scenario without a gap, but the error slightly increases with a decrease in resolution when the gap is filled with water or air. This is likely because the presence of the gap adds a second material transition boundary to the sample cross-section, where larger discrepancies are expected to occur. As illustrated in Figure 11 and Figure 12, these outlier back-calculated values become more frequent with a decrease in image resolution. Since the error associated with these values is orders of magnitude larger than the error for the rest of the cross-section, a few additional outliers can noticeably affect the averages listed in the figures. When excluding them from the calculations, the average percentage error was less than 1% for all scenarios for the gap, at all resolution settings.

The radiograph pixel size does not seem to affect the presence of largely inaccurate predicted values at the edges of the gap in either of the three scenarios. However, the radiograph resolution can play a significant role in the final step of the solution to the inverse problem by providing sufficient data points at the center of the gap, where the neutron cross-section of the material in the gap can be more accurately quantified. This would enable a quantitative analysis of the sample, where the outlier values at the edges of the gap could be filtered out and the composition of the material inside it could still be determined. The outlier values themselves are still useful to clearly delimit the width of the radial gap, another crucial parameter for corrosion modeling.

5. Conclusions

A methodology was developed to calculate the water content of axisymmetric cylindrical cement-based samples using neutron radiography measurements. Three experimental geometries (corresponding to drying, extrusion flow, and cracking at the matrix-reinforcement interface scenarios) were used to demonstrate the potential applications of this approach. Virtually generated radiographs were used to better quantify the accuracy of the methodology.

When the neutron radiographs were created using a single neutron cross-section value for each constituent, the reconstructed water content profiles for all three examples exactly matched the presumed water content of the virtual samples. This confirms the reasonableness of the inverse problem solution (i.e., the method used to back-calculate the radial neutron cross-section of the sample). For the neutron radiographs created using a random distribution of neutron cross-section values for each constituent (i.e., each pixel location represents stochastic material variability), varying within ±4% from the neutron cross-sections in Table 1, the accuracy of the methodology for each simulated experiment is outlined in the following paragraphs.

In the drying example, the methodology was able to quantify the macroscopic distribution of water along the radial direction after 3, 45, and 600 days of exposure with less than 1.5% average error. In the high-shear flow example, the water content associated with the LL sharply increased at the edge. The methodology was still able to accurately determine the water content profile of the samples, provided that the pixel size is smaller than the thickness of the LL. The average error of the back-calculated profiles was smaller than 3% even at the lowest resolution (180 μm). In the reinforced mortar example, the methodology was able to quantify the distribution of water and the difference in material composition along the radial direction. The average error of the back-calculated composition profile was lower than 4% for all resolution settings and materials filling the gap. The transition between materials was evident. This may be useful to estimate the width of the gap between the matrix and the reinforcement, and whether it is filled with air or water.

Future work is planned to evaluate the robustness of this approach with physically obtained radiographs. In addition to investigating the properties of cement-based specimens, the study will assess the effects of specific variability sources on the back-calculated results, particularly beam hardening and neutron scattering effects. The impact of experimental limitations on the assumption of perfect axial symmetry, including but not limited to cylinder casting heterogeneities, cylinder-beam alignment, and control of boundary conditions, must also be investigated to ensure the applicability of the methodology to physical experiments. The authors believe cylinder tilting and out-of-axis issues can be controlled with precise, motorized sample mounts; out-of-roundness issues due to imperfect cylinder fabrication might need further assessment. Furthermore, the assumption that the image processing techniques effective for prismatic samples can be reliably applied to the axisymmetric cylinders used in the modeling of the virtual radiographs also demands direct validation with experimental data.

This developed methodology has the potential to enable the water content profile of fresh and hardened cement-based materials to be determined with high temporal and spatial resolution, expanding the current experimental design options for neutron radiography testing. The resulting insights will not only improve the reliability of the methodology, but also broaden its utility in advancing the study of dynamic moisture transport and related phenomena in cement-based materials, including, but not limited to, those described in the examples presented in this paper.