Abstract
Aside from porosity, the degree of pore connectivity and tortuosity are important pore structure characteristics of cement-based materials to understand better the effect of microstructure on transport processes that influence the durability of these materials. Synchrotron X-ray computed microtomography is employed to provide information about the three-dimensional (3D) pore structure at submicron resolution (0.5 µm/voxel) using the SPring-8 facility in Hyogo, Japan. This paper presents a method to characterize the pore space of hardened cement pastes taken from different specimens of various ages (2, 7, and 28 days) of curing. By defining the pore threshold value on the basis of the transition point in the porosity-threshold dependency curve, the pores were distinguished from the solid matrix in the microtomographic images. Further, pore cluster multiple labeling was performed to gain information on the pore connectivity and the associated effective porosity. 3D random walk simulation in the largest percolating pore cluster was then conducted to evaluate the diffusion tortuosity. The results from this study provide indications of increasing diffusion tortuosity as pores that can be resolved by the X-ray imaging system decreased and became more disconnected. Sensitivity analysis to pore threshold value was performed to evaluate the robustness of the method.
Introduction
The pore structure characteristics of cement-based materials play a fundamental role in governing transport processes that influence the durability-based performance of these materials (Sugiyama et al., 2003; . For example, the pore connectivity and the tortuosity of pore space are important variables to understand and model the diffusion whereby aggressive species (e.g., chlorides, sulfates, etc.) from the environment penetrate concrete structures. In this study, we investigate the pore structure characteristics of hardened cement paste with particular focus on evaluating the pore connectivity and the diffusion tortuosity in three-dimensional (3D) pore space.
A variety of experimental techniques such as mercury intrusion porosimetry, scanning electron microscopy, optical microscopy, and gas adsorption have been used to characterize the pore structure of cement-based materials. Recently, thermoporometry or low-temperature calorimetry has also been used to examine the pore structure indirectly without any drying or special specimen preparation (e.g., see Bentz, 2006) . However, most of these techniques assume the pore geometry in the interpretation of results, and are either invasive or limited to two-dimensional (2D) information. One way to overcome these limitations is to use X-ray computed microtomography to examine directly the three-dimensional internal microstructure of the material in a non-invasive manner without requiring any specimen preparation typical of the other mentioned experimental techniques.
Microtomography with synchrotron radiation as the X-ray source has already been applied by various researchers (e.g., see Bentz et al., 2000; Bentz et al., 2002; Burlion et al., 2006; Gallucci et al., 2007; Helfen et al., 2005; Hitomi et al., 2004; Koster et al., 2006; Lu et al., 2006) in cement and concrete research. Synchrotron microtomography at the ESRF (European Synchrotron Radiation Facility) in Grenoble, France has been used by Bentz et al. (2002) and Helfen et al. (2005) to observe directly the three-dimensional percolation of capillary pore structure in hydrating cement pastes at a resolution of about 1 micrometer. Using the microtomographic images that are made available online (Bentz et al., 2002) from the NIST's "Visible Cement Data Set, " Koster et al. (2006) simulated the water permeability and water vapor diffusion through hardened cement pastes by converting first the extracted pore space into a transportation network of cylindrical tubes. More recently, at the SLS (Swiss Light Source) in Villigen, Switzerland, Gallucci et al., (2007) observed the 3D pore structure evolution of cement pastes at a resolution of about 0.7 micrometers.
In this paper, we present the results of the microstructure investigation carried out on different hardened cement pastes of various ages (2, 7, and 28 days) using synchrotron microtomography at SPring-8 in Hyogo, Japan, at a higher resolution of 0.5 µm per voxel. Three-dimensional random walk simulation is performed to evaluate the diffusion tortuosity through the complex pore space of hardened cement paste. To our knowledge, this study is the first attempt to quantify tortuosity from the extracted pore space in high resolution microtomographic images of cement pastes. The advantage of using random walk simulation is that we could directly measure tortuosity, a pore structure parameter that has a direct bearing on transport properties, without resorting to any assumption of the pore geometry. Furthermore, another significant feature of this study is the description of the linear attenuation coefficients of the different phases of interest in cement pastes. From this information, the relative brightness of the different phases is inferred and a pore threshold value is introduced to distinguish the pores from the solid matrix during segmentation. The sensitivity of the pore structure parameters to the pore threshold value, and the correlation between the effective porosity and tortuosity are also investigated.
Methods

Sample preparation and X-ray imaging of specimen
The cement pastes analyzed in this study were prepared using an ordinary Portland cement (OPC) that is designated as JIS R5210. By mass, the cement contains 67.80% of CaO, 21.30% of SiO 2 , 3.80% of Al 2 O 3 , 2.41% of Fe 2 O 3 , and 2.20% of SO 3 . The fineness of OPC in terms of Blaine specific surface area is 3200 cm 2 /g. The pastes were prepared at an initial water to cement ratio of 0.50 (mass basis). A cellulose-based viscosity admixture was also added in the mixture to prevent segregation. After mixing the cement paste, it was sealed for one day followed by curing in water until the predetermined ages, namely 2, 7 and 28 days, respectively. Accordingly, these hardened cement pastes of different ages of curing were referred to as OPC50-2d, OPC50-7d, and OPC50-28d, respectively. Note that in this way, the microstructure evolution was evaluated according to the age of the hardened cement pastes prepared from the single batch mix.
These samples were crushed and the crushed pieces were treated with acetone to suspend the hydration, followed by drying under vacuum. Then, these prepared samples were stored in a sealed container prior to the X-ray CT experimentation. It was assumed that the microstructure of the specimens is not altered significantly during the sample preparation as this is commonly practiced by many researchers (Japan Cement Association, 2008) . For exposure to the beam line, shards of about 1 mm at its greatest width and 1 mm in length were prepared to examine the internal pore structure with synchrotron X-ray computed microtomography.
Synchrotron X-ray computed microtomography is a three-dimensional imaging technique that is based on the 3D computed reconstruction of a sample from the 2D radiographic projections acquired at different angles around its axis of rotation. In principle, the synchrotron microtomography technique is similar to medical computed tomography (CT) scans or industrial micro-focus CT, but this technique achieves much higher spatial resolution by combining extremely bright, monochromatic synchrotron radiation with high-quality optics and X-ray detection. In addition, reconstruction artifacts such as the occurrence of beam hardening, which is commonly encountered in polychromatic CT systems, are also reduced (Ashbridge et al., 2003) .
In this study, we used the X-ray CT system through the BL20XU beamline at SPring-8 (Super Photon ring-8 GeV), which is the world's largest third generation synchrotron radiation facility, located at Hyogo, Japan. A schematic illustration of the X-ray imaging system at SPring-8 is shown in Fig. 1 . The system consists of an X-ray light source from the beam line, double crystal monochromator, high-precision rotation stage, and high-resolution X-ray image detector (Uesugi et al., 2001) . The transmitted images are detected by an X-ray imaged detector that consists of a thin scintillator, optic system and CCD camera. In this experiment, the effective pixel size in the transmitted image was 0.50 µm by 0.50 µm. Moreover, the beam energy was set to a value of 15 keV. For each specimen, 1500 X-ray projection images with an angle step of 0.12 degrees and an exposure time of 0.30 s each were obtained.
Tomographic reconstruction was done using a computer program in use at SPring-8, which employs the convolution back projection algorithm to generate the "slice" images. The slices refer to the individual microtomographic images, similar to a slice from a loaf of bread or a petrographic thin section, since the slices correspond to what would be observed if the slices were observed from the object in the scanned plane. Stacked up, these slices provide the reconstructed volumetric data of the scanned object. In this study, the reconstructed 3D-image data set was composed of 1300 contiguous grayscale images where each slice image contained 2000 x 2000 voxels (volume element or 3D pixel). The dimension of a voxel is a 0.5 x 0.5 micrometer-sized pixel with a thickness of 0.5 µm. In other words, the effective size of the cubic voxel is 0.50 µm in the CT image. The voxel intensities or grayscale values (GSV) in slices correspond to the CT values, which are proportional to the measured linear attenuation coefficient (LAC) of X-rays obtained from the CT reconstruction (see Fig. 2 ). It should be noted that the LAC defined in Beer-Lambert's law is strongly correlated with density, and is also a function of the mean atomic number and X-ray energy. In general, the darker voxels in the reconstructed grayscale image correspond to low density phases (e.g., air voids or pores), whereas the brighter voxels denote the high density phases (e.g., anhydrous cement). Table 1 
where w i is the fraction by weight of the ith atomic (or elemental) constituent;
is the mass attenuation coefficient at a given X-ray energy (E) obtained from the NIST database (Hubbell and Seltzer, 2004) .
3D image processing and analysis
Although numerous routines such as smoothing and noise suppression can be used to process the reconstructed slice, minimal image processing was conducted in order to maintain the direct correlation of grayscale intensity (GSV) in the image to the measured LAC of the original reconstructed slice. In this study, image normalization or contrast stretching was applied to the whole stack of the reconstructed slices to even out the brightness and contrast variation between the slices, as well as, to enhance the contrast in the said images. In the normalization process, the set of images was also down-sampled from 16-bit to 8-bit grayscale images resulting in a smaller image file size. SLICE, a basic 3D-image analysis program developed in SPring-8, was also used for the segmentation and cluster labeling of the normalized reconstructed grayscale images (Nakano et al., 2006) . To avoid possible edge effects and reduce the computing time, a region of interest (ROI) was selected, and a cubic volume of interest (VOI) was obtained therefrom. In this study, the VOI was limited to 200 x 200 x 200 voxels, i.e., a cubic VOI with physical dimension of 100 µm, for the analysis of quantitative data from the microtomographic images of cement paste. It has been previously stated by Uchikawa (1988 Uchikawa ( , 1989 ) that 100 micrometers is a representative size for studying cement pastes.
The pore space was then extracted from this VOI by thresholding the slices on the basis of their gray level histogram. Although several approaches to pore segmentation are mentioned in the literature (e.g., see Bentz et al., 2002) , which is appropriate for the given task is still an open discussion. For this study, a global thresholding method, the simplest and usual method of segmentation, was used to identify the pore voxels from the solid matrix. Global thresholding is done by visiting each pixel (or voxel) in an image, and setting the pixel to a grayscale value (e.g., GSV = 255 for an 8-bit grayscale image) if its value is above the given threshold value, and to another grayscale value (e.g., GSV = 0 for an 8-bit grayscale image) if the pixel value is below or equal to the threshold value. Accordingly, the lower bound of GSV associated with pore voxel was set to 0 while the upper bound was set to the pore threshold value on the basis of the transition point in the segmented porosity-threshold dependency curve, as described in Fig. 3 . At this transition point, the segmented porosity started to increase rapidly wherein the boundary between the pore and the solid matrix is most likely to be segmented as pore space . The output from this pore segmentation is a set of binary images where the pores and solid matrix are imaged as black (GSV = 0) and white voxels (GSV = 255), respectively.
After pore segmentation, the cluster multiple labeling technique described in Hoshen and Kopelman (1976) was applied to identify and label the connected pore clusters using the 6-point connectivity rule. In this connectivity analysis (Ikeda et al., 2000) , two voxels are considered to be connected when a pore voxel shares a common face with another voxel. This set of connected voxels forms a so-called cluster. On the other hand, if the pore voxels are in contact only at the vertex or edge, then the voxels are considered to be disconnected and belong to different clusters. The output from the pore cluster labeling enumerates the number of pore clusters and their sizes (number of voxels) including their end-to-end coordinates in the cubic image system.
3D random walk simulation
In this study, the concept of diffusion tortuosity is introduced as a pore structure-transport parameter associated with the 3D-microgeometry of the pore space in cement-based materials. Random walk simulations were applied to measure the tortuosity of the pore space in the volume of interest (VOI). This technique has also been applied to measure the tortuosity from microtomographic images of geologic materials (Nakashima et al., 2004; Nakashima and Kamiya, 2007) . It is based on the concept that the time-dependent diffusion coefficient D(t) associated with the random Brownian motion of molecules can be used to probe the geometry of porous media (Sen, 2004) . The self-diffusion coefficient of molecules in a bulk medium is time-independent but becomes reduced and time-dependent in a confining geometry. With a long 6  11  16  21  26  31  36  41  46  51  56  61  66  71  76  81  86  91  96  101  106  111  116  121  126  131  136  141  146  151  156  161  166  171  176  181  186  191  196  201  206  211  216  221  226  231  236  241  246  251 6  11  16  21  26  31  36  41  46  51  56  61  66  71  76  81  86  91  96  101  106  111  116  121  126  131  136  141  146  151  156  161  166  171  176  181  186  191  196  201  206  211  216  221  226  231  236  241  246  251 Transition point: pore threshold value (ThV) diffusion time, the self-diffusion coefficient approaches zero in a closed pore space. However, the long-time behavior of the diffusion coefficient in the porous media could approach a limiting value as diffusing molecules probe the connectivity and tortuosity of an open pore network. Diffusion tortuosity is therefore defined as the ratio of the self-diffusion coefficient (D o ) of non-sorbing walkers in free space to the long-time self-diffusion coefficient (D ∞ ) of these walkers in pore space (Promentilla and Sugiyama, 2007a; 2007b) . The operational definition of
lattice space is based on the following equation (Nakashima and Kamiya, 2007) :
where 2 r and t D refer to the mean square displacement of the walkers and dimensionless lattice walk time, respectively. Note that the symbol denotes the average over all the initial and final positions of the walkers.
In a simple cubic lattice, the mean square displacement is computed as follows:
where n is the number of walkers, and x, y, and z are the 3D coordinates of the walker's position at time t D for the ith walker. Accordingly, the diffusion tortuosity ( D τ ) can be defined as follows:
Note that the diffusion tortuosity is known to be related to the electrical formation factor (F), a measured quantity from electrical resistivity measurement, such that (Latour et al., 1995; Sen, 2004) :
where Φ e is the effective porosity (accessible) of the porous medium and F is the electrical formation factor, which is the ratio of the electrical conductivity of the electrolyte in the bulk ( 0 σ ) to that of the porous medium ( σ ).
A 3D random walk algorithm implemented in Mathematica was employed to compute the mean-square displacement of random walkers in a simple cubic lattice with mirror boundaries (Nakashima and Kamiya, 2007) . In random walk simulation, the walker migrates on discrete voxels that correspond to the pore space. A pore voxel is chosen randomly as the starting position of the lattice walk trial at an integer time equal to 0. The walker then executes a random jump to one of the nearest voxels and the time of lattice walk is incremented by a unit integer time after the jump. If the randomly selected voxel is a solid voxel, the jump is not performed but the time is still incremented by one unit. This describes a restricted motion analogous to that of the "ant in the labyrinth" to simulate the diffusion of a particle in a disordered media as described in percolation theory (Stauffer and Aharony, 1994) . Figure 4 describes an example of the reconstructed image of the specimen from OPC50-2d and its corresponding image after the image normalization process. As shown in Fig. 4c , the normalization cut-off parameters were set to improve the contrast in the image by scaling the measured LAC to produce an 8-bit grayscale image with intensity values that range from 0 to 255. Consequently, the GSV in the image is directly related to the measured LAC according to this equation:
Results and discussion

Microtomographic images
In the microtomographic image, the background or surrounding air is imaged as very dark voxels in contrast to the object. From the visual examination of the image (see Fig. 5 ), it can be inferred that the two distinct peaks correspond to the surrounding air and the imaged specimen. As expected, the measured average LAC of air space is around 0 cm -1 , i.e., the X-ray attenuation coefficient associated with the first peak. In addition, the second peak, which is around 10 cm -1 , may be associated with the porous solid matrix of which the hydrated cement products are the main constituent. Note that the higher the LAC of the associated phase, the higher the relative CT image brightness of this phase will be. From Table 1 , it can be inferred that the brighter voxels in the image, i.e., the measured LAC of 30 to 40 cm -1 and more, are most likely associated with either the anhydrous cement clinker or high-density mineral phases.
Porosity, pore connectivity and tortuosity
To facilitate the discussion on characterizing pore structure from the microtomographic images, the VOI was extracted from the original data set as shown in Fig.  6 . In this figure, each volumetric image of the hardened cement paste consists of 200 stacked slices of the specified region of interest (200 x 200 voxels). It is assumed that the said VOI with dimension of 100 µm is a representative size for percolation studies (Garboczi and Bentz, 2001) , but this requires further investigation in the future. For the purpose of brevity, representative results from the 3D image analysis of these VOIs were only shown in this paper. Figure 7 describes the VOI after pore segmentation and cluster multiple labeling. In Figs.  7a, 7c , and 7e, the VOIs of OPC50-2d, OPC50-7d, and OPC50-28d are shown respectively as stacks of binary images in which the pores are imaged as black voxels while the "solid matrix" are imaged as white voxels. The segmented porosity was quantified by just counting the total number of black voxels and then dividing this number by the total number of voxels in the VOI. Using a cluster labeling technique, the largest percolating pore cluster (imaged as black voxels) was identified and differentiated from the smaller isolated pore clusters and dead-end pores (imaged as gray voxels) as shown in Figs. 7b, 7d, and 7e. Although these black voxels are apparently disconnected in 2D, the pores are in fact connected to form a large percolating cluster or network in 3D. Note that the criterion we used to define percolation here is when this pore cluster contains voxels that are connected to the six faces of the cubic VOI. In other words, percolation means that any external species can enter one face of the VOI and exit any other face through the said pore cluster. Figure 8 illustrates the 3D micro-geometry of the pore structure associated with the originally segmented porosity and the corresponding largest percolating pore cluster.
It is assumed that the largest percolating pore cluster would most likely contribute to the macroscopic transport property of the cement paste. Thus, random walk simulation in the largest percolating cluster was performed to determine the pertinent diffusion tortuosity. A random walk simulation was also performed in free Fig. 4 Example of reconstructed slice before and after normalization process. space (i.e., porosity = 100%) of the same VOI size to check the reliability of the results. Figure 9a shows the sample trajectory of a walker in the pore space. The dimensionless mean square displacement (MSD) averaged over 100,000 walkers is plotted against the time (t) of lattice walk (see Fig. 9b ). In free space where unconstrained diffusion can be observed, the mean square displacement of walkers was observed to be directly proportional to the time of the lattice walk. In contrast, the curve of MSD against time for the pore space indicated restricted diffusion resulting in a reduced self-diffusion coefficient of the walkers as they probe the tortuosity of the pore space. The slope of MSD against time would correspond to the inverse (diffusion) tortuosity of the pore space. As expected, the constant slope observed in free space yielded a diffusion tortuosity equal to one. On the other hand, the computed long-time diffusion tortuosity was estimated to be 13, 111, 167 for OPC50-2d, OPC50-7d and OPC50-28d, respectively.
OPC50-2d
The results of the analysis from the VOI of OPC50-2d, OPC50-7d and OPC50-28d are summarized in Fig. 10 . Note that the degree of pore connectivity is defined as the volume fraction of the largest pore cluster in the pore space (i.e., the segmented porosity). If all the pores in the segmented porosity are connected to each other, then the degree of connectivity is one. By multiplying the degree of connectivity to the segmented porosity, we defined the effective porosity that is percolating in three orthogonal directions. For example, the largest percolating pore in OPC50-2d is 87% of the segmented porosity (0.21) that corresponds to an effective porosity of 0.18. The other 13% is either isolated pores or dead-end pore clusters.
Although this study was limited in its extent, indications were found that the porosity and the degree of pore connectivity decrease as the age of hardened cement paste increases. Likewise, the diffusion tortuosity tends to increase in a 3D pore space that is becoming more disconnected as the age of cement paste increases. These Visualization of output from pore segmentation and cluster labeling. The pore voxels are imaged as black and the solid matrix is imaged as white. After cluster labeling, the pore voxels that are not connected to the largest percolating pore cluster are imaged as gray voxels.
preliminary findings seem reasonable and consistent with the common knowledge on pore microstructure evolution in hardened cement pastes as hydration progresses through time. For example, Powers et al. (1959) have demonstrated the depercolation of capillary pores with a corresponding decrease of permeability in cement pastes as the capillary pores decrease in volume and size and start to become disconnected with increases in the degree of cement hydration. However, some doubt as to the significance of the findings from this study may exist since the current spatial resolution (0.50 micrometers per voxel) of the 3D image is quite low relative to the sizes of the pores that could be critical to the transport properties in the hardened cement pastes. Note that the lower end of the pore size range that can be detected from synchrotron microtomography is determined by the spatial resolution of the X-ray imaging system whereas the upper end can be as large as possible depending on the specimen size. One might argue that porosity, pore connectivity, and tortuosity may be influenced by the pore size that can be resolved in the X-ray imaging system. As these limitations are related to the digital resolution, there have been studies on the effect of such resolution on pore percolation in digital images obtained from either microstructure models (Ye, 2005; Garboczi and Bentz, 2001; Navi and Pignat, 1996) or microtomographic studies (e.g., see Galucci et al., 2007) . As expected, an increase in pixel (or voxel) size, i.e., a decrease in the digital image resolution, results in a decrease in the calculated connectivity of the pore network. Despite this sensitivity to resolution, it is interesting to note that the digital resolution near 1 micrometer per voxel appears to be reasonable based on the rough agreement of the computer models with the experimental evidences as reported in Garboczi and Bentz (2001) . Furthermore, Koster et al. (2006) suggest that a voxel size of 1 micrometer can be scaled down to 20 nm without changing its morphological shape because of the self-similarity of pore space in the microtomographic images. The information obtained from microtomographic studies can then be interfaced with computer models that allow extrapolations to finer pore size. Therefore, it was conjectured that the current resolution of the microtomographic images is adequate to study the 3D micro-geometry of the pore structure in accordance to the physical length scale of interest that mainly controls the transport properties of cement paste.
Sensitivity to pore threshold value
As in any tomographic imaging technique, the robustness of the results is influenced by the spatial resolution of the X-ray imaging system. Aside from the inherent statistical noise, the measured LAC in the CT image depends on the restricted size of the voxel. Finer pores with a voxel size of 0.5 µm or smaller cannot be detected in an appropriate manner. As each voxel in the reconstructed image represents the attenuation properties (LAC) of a specific material volume, we could always find partial volume effects (PVE) at the boundary between the pore and hydrated cement products (e.g., ettringite, portlandite, CSH, among others). In other words, if the voxel is actually composed of a number of different substances, then the resulting GSV (or measured LAC) represents some average of their properties. Furthermore, blurring at the interface of the material and pore due to fixed spatial resolution indicates that the voxel can be affected by the GSV voxel of the surrounding area. For example, if in the analysis, air has a GSV of 23 and the solid matrix has a GSV of 67, small pore voxels may show up as small features that are darker than the matrix; but may not ever achieve a grayscale of 23, and may not even reach the midpoint grayscale of 45. Thus, the interpretation of the pore threshold value to extract the pore from the matrix is not necessarily straightforward as compared to that of surrounding air (background) and the solid object (foreground) in the image.
To examine the uncertainty involved in the calculation of pore connectivity and tortuosity, we conducted a limited sensitivity analysis wherein the pore threshold value was varied within the range of GSV that corresponds to the values between the transition point and the inflection point, i.e., where a distinct peak in the respective VOI histogram is observed (see Fig. 3 ). This peak in said VOI was assumed to be most likely associated with the GSV (or LAC) of the hydrated cement pastes that may actually contain finer pores. In addition, it is also possible to have pore voxels with linear attenuation coefficients that are very close to those of the hydrated cement products. This has been shown before by Gallucci et al. (2007) based on closer examination of microtomographic images wherein increases of contrast resolution loss are observed with decreases in pore size. Figure 11 summarizes the results of this analysis for the VOI of OPC50-2d, OPC50-7d and OPC50-28d, respectively.
Porosity-tortuosity correlation
Results from this study also provide indications of the strong correlation between the effective porosity Φ e and the measured diffusion tortuosity (τ D ) in hardened cement paste.
Since an effective porosity of 1.0 yields a diffusion tortuosity of 1, a power law function was used to define the correlation between effective porosity and diffusion tortuosity, such that:
where m is the fitting parameter. Note that Eq. 7 is analogous to Archie's law, which correlates the electrical tortuosity or the so-called formation factor to the measured porosity of rocks. Archie's law, proposed in 1942, is an empirical relation that relates the porosity of brine-saturated rock to its electrical conductivity (Garboczi, 1990) . It can be written in the form of the dimensionless electrical formation factor (F), such that:
where 0 σ is the electrical conductivity of the brine, σ is the overall conductivity of the brine-saturated rock, treated as an effective uniform continuum, a and m are constants that can vary between rock formations, and φ is the measured porosity (which is usually referred to as total porosity in the classical definition of Archie's law).
Compared with Eq. 5, Archie's law can be expressed with a that is equal to 1.0 and replacing the porosity ( φ ) with the effective porosity ( e φ ) such that: 
This means that the formation factor contains the effective porosity and tortuosity of the porous medium.
Using the data in either Fig. 10 or Fig. 11 , parameter m in Eq. 7 was estimated from nonlinear regression analysis using the Levenberg-Marquardt algorithm. It is assumed that the data in Fig. 11 are a set of possible values that relate the effective porosity and diffusion tortuosity regardless of the age of cement paste. It is worth noting that in every case, there is a sudden change in the degree of connectivity or tortuosity as the porosity crosses the level of about 20%. This is in agreement with the previous experimental and computer modeling studies done in capillary porosity depercolation (e.g., see Powers, 1959; Bentz and Garboczi, 1991; Garboczi and Bentz, 2001) .
Results from the regression analysis are summarized in Fig. 12 . The computed value of m in the effective porosity-tortuosity correlation is very close to the power law exponent of 2. In terms of the formation factor, this corresponds to the cementation factor m of 3, which is on the high side but still in the range of values encountered in some geologic materials. This is consistent with the concept that the formation factor (electrical tortuosity) is directly related to the diffusivity (diffusion tortuosity) of the material pore structure according to the Einstein relation. There have been reports (Garboczi, 1990) about reasonable success estimating the diffusivity of hardened cement paste using this relation along with experimental measurements, provided that the pore structure is the only geometric constraint to transport, and any interactions among ionic species, and between ionic species and pore walls, are considered insignificant. This implies that tortuosity is not just a "fudge" factor (Grathwohl, 1998) in solving transport equations but a physical parameter that can be measured independently from the 3D micro-geometry of pore structure in hardened cement paste. Diffusion tortuosity can thus be used as a parameter to relate the pore structure to the transport properties in cement-based materials.
Conclusions
Synchrotron microtomography allows us to examine the 3D micro-geometry of hydrated cement systems with submicron spatial resolution. The three-dimensional imaging of the pore space allows us to measure the pore connectivity and diffusion tortuosity, which clearly demonstrates the advantage of this technique over the 2D microscopic imaging technique. To our knowledge, this study, which uses a spatial resolution that corresponds to a voxel size of 0.5 µm, offers by far the highest resolution thus far reported for studying the cement paste microstructure.
By employing 3D image analysis and random walk simulation, relevant pore structure parameters from the 3D micro-geometry, such as porosity, pore connectivity and diffusion tortuosity, were quantified. Given the pores that can be resolved in the X-ray imaging system, the porosity and degree of pore connectivity tend to decrease while diffusion tortuosity tends to increase as the age of hardened cement paste increases. However, caution must be taken with regards to the interpretation of results as this method is also sensitive to the specified pore threshold value. Preliminary analysis of the robustness of the results by varying the pore threshold value at a reasonable range was investigated. In addition, the effective porosity-tortuosity correlation was shown as a power law function analogous to Archie's law.
As in any other techniques for characterizing pore structure, synchrotron microtomography has its limitation, in particular as regards its spatial resolution, which is quite low relative to the reported capillary pore size range that controls the transport processes in hardened cement pastes. Nevertheless, the results from this study were consistent with the previous experimental and computer modeling studies reported in the literature. This implies that the current "rough" resolution used in this study could have extracted the percolating 3D pore space that plays an important role in the transport processes in cement pastes. As this technique is the only noninvasive 3D imaging technique available so far, this kind of research could contribute to the clarification of the relationship between the 3D microstructure and transport properties of cement pastes. Diffusion tortuosity could be used as a robust microstructure parameter that has a direct bearing on transports properties (e.g., permeability, diffusivity, etc.) as it can be evaluated directly from a complex pore network without requiring any assumption about the pore geometry in three dimensions. Therefore, future studies along these lines are being considered to further understanding of the 3D micro-geometry of pore structure as it influences the transport properties of cement-based materials.
Acknowledgments
Part of this present research work was funded by the Japan Society for the Promotion of Science (Research No.: 196561060 7, Representative: Sugiyama Takafumi) . The synchrotron radiation experiments were performed at the BL20XU beamline at SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2007A1951-NL-np) . The authors wish to acknowledge SPring-8 for letting them use the synchrotron radiation facilities, with particular thanks to the staff of BL20XU. The authors are also thankful to the three anonymous reviewers for their valuable comments and insights in improving the content of the paper. The first author acknowledges Dr. Yoshito Nakashima and Dr. Masayuki Uesugi for their kind assistance in the use of the Mathematica program and SLICE, respectively. He also acknowledges the support of Prof. Tamon Ueda, the group leader of the COE project of which this research was part of. He is grateful to the financial support given by the COE program of the Graduate School of Engineering, Hokkaido University, and the JSPS Postdoctoral Fellowship for Foreign Researchers in the conduct of this study. Fig. 12 Effective porosity-tortuosity correlation using power law model. The correlation coefficient (r 2 ) from using either data of Fig. 10 or Fig. 11 is 0.96.

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