Once the image of the sample has been scanned, converted into a binary matrix, and processed using pixel connectivity, our script can determine a number of properties. These properties are based on the assumption that all connected components (connected white pixels) are “objects” representing the 2-D image generated by a random plane through a 3-D sample of mortar. Using information from the 2-D image, real 3-D properties can be estimated based on principles of stereology.
Air content
The simplest property of air-entrained mortar computed from a plane section is the volumetric air content. Air content is computed using the Principle of Delesse [5], a fundamental principle of stereology that states the following: if a structure containing objects is randomly sectioned, then the areal density A
A
of the profiles of those objects is equal to the volumetric density V
V
of the objects in the structure:
$$\begin{aligned} V_V = A_A \end{aligned}$$
(1)
The script computes the areal density of the air voids by counting the number of white pixels in the image and dividing by the total number of pixels. This value is equivalent to the air content in the sample.
Powers spacing factor and specific surface
In Powers’s 1949 paper [15] on the role of air in frost resistant concrete, Powers defined an air void spacing factor \(\bar{L}\) as the distance between the surface of a bubble and its sphere of influence, such that
$$\begin{aligned} \bar{L} = \left\{ \begin{array}{ll} \displaystyle \frac{3}{\alpha } \left[1.4 (\frac{p}{A} + 1)^{1/3} - 1\right] &{} \displaystyle \frac{p}{A} \ge 4.33\\ \\ \displaystyle \frac{p}{\alpha A} &{} \displaystyle \frac{p}{A} < 4.33 \end{array} \right. \end{aligned}$$
(2)
where α is the specific surface, defined as the ratio of the average void surface area to the average void volume, and p/A is the volumetric paste to air ratio, where p is the volume fraction of cement, water, and supplementary cementitious materials (if any). In this study, p is computed from the mix proportions.
Equation 2 shows that it is necessary to determine the specific surface of the air voids to compute the Powers spacing factor. In a discussion appended to [15], T.F. Willis showed that
$$\begin{aligned} \alpha = \frac{4}{l} = \frac{4n}{A} \end{aligned}$$
(3)
where l is the average chord length through the air voids along a line of traverse, n is the number of air voids intersected per unit length of traverse, and A is the air content.
The conventional technique for determining α is to use a stereoscopic microscope and employ either the linear traverse method or the point-count method described in ASTM C457. It is also possible to replicate the same action using automatic image analysis. For example, one could take lines of traverse on a matrix and determine the number of times an air void was encountered by counting the number of times the pixel value changes from 1 to 0 or from 0 to 1. This change in pixel value represents the movement between the paste/aggregate matrix and the air void.
However, a limitation to this method is that Eq. 3 assumes that all the air voids in the sample are convex in shape. Although this assumption should be valid for entrained air voids, which should be spherical, the assumption of convexity is not always true for entrapped air voids, which can be irregularly shaped. Here, we develop a more general equation of the specific surface of objects in a matrix. Beginning with two independent derivations by Saltykov [19] and Tomkeieff [25],
$$\begin{aligned} S_V = 2 I_L \end{aligned}$$
(4)
where S
V
is the surface density of objects within another object and I
L
is the number of intersects per length of random line of traverse.Footnote 1
The surface density is defined as the set of points at the interface between the objects and the matrix divided by the volumetric sum of the objects and the matrix. S
V
therefore is in units of [length]−1. In the case of air voids within a paste matrix,
$$\begin{aligned} S_V = \frac{S_{ap}}{V_p+V_a} \end{aligned}$$
(5)
where S
ap
is the surface of the air-paste interface, V
p
is the volume of the paste/aggregate matrix, and V
a
is the volume of the air voids. Note that S
V
is not the same definition as the specific surface α. Knowing that \(V_V = A_A\), we rewrite S
V
as follows:
$$\begin{aligned} \alpha = \frac{S_{ap}}{V_a} = S_V (\frac{V_p + V_a}{V_a}) = \frac{S_V}{V_V} = \frac{S_V}{A_A} \end{aligned}$$
(6)
Substituting Eq. 4 into Eq. 6,
$$\begin{aligned} \alpha = \frac{2 I_L}{A_A} \end{aligned}$$
(7)
If all the objects are convex, then \(I_L = 2 n\) and \(\alpha = 2n/A_A = 2n/V_V\), where n is the number of intercepted objects per unit length of traverse, since every object crossed by a line of traverse would intersect at the boundary of any convex object exactly twice. Therefore, for convex objects within a matrix, such as entrained air voids dispersed with a cement or concrete sample, the specific surface of the air voids presented in Powers’ 1949 paper (Eq. 3) is equivalent to Saltykov’s 1945 equation (Eq. 4) of surface density.Footnote 2
Many methods of stereology, such as Eqs. 3 and 4, were developed for the computing capabilities of the middle 20th century. Using image processing, it is possible to use stereological principles that were once considered to be impractical or useful for academic exercise only. For example, the specific surface of objects can be determined by directly computing the perimeter of those objects on a representative 2D plane. Measuring a perimeter is never straightforward, but if an image finely discretizes the object boundaries, the perimeter can be computed by counting pixels. It can be shown from Buffon’s 1777 needle problem that
$$\begin{aligned} B_A = \frac{\pi }{2} I_L \end{aligned}$$
(8)
where B
A
is the boundary length density of objects in a representative 2D plane, where “boundary length density” is defined as the perimeter of the objects divided by the area of the plane. Equating Eq. 7 and Eq. 4, we find that
$$\begin{aligned} \alpha = \frac{4}{\pi } \frac{B_A}{A_A} \end{aligned}$$
(9)
Eq. 9 shows that given the perimeter and area of objects on a representative 2D plane through a 3D sample, it is possible to determine the specific surface of those objects. Note that this formulation is more general than Eq. 3 which is used in ASTM C457, since Eq. 9 does not require the objects to be convex. However, because the objects in question area primarily spherical air voids, in our samples, Eq. 3 and Eq. 9 agree well with each other.
Air void perimeters
To use Eq. 9, it is necessary to compute the boundary length, or perimeter, of those air voids on a representative 2D plane.
An estimation of the perimeter of the air voids can be obtained by taking each connected component from the image and determining the equivalent circumference, assuming that each connected component is a circular intersection of the spherical air voids with the plane. However, this method is problematic because of the imperfections related to sample preparation. For example, there are cases where two air voids are in very close proximity to each other. Either sample grinding/polishing and/or imperfect placement of white powder on these air voids causes the voids to appear connected together on the surface. When two or more air voids appear to come in contact, it becomes difficult to separate these air voids as distinct objects through automatic image processing, especially if the objects are not always regular in shape. Therefore, assuming each connected component is a discrete, circular intersection would lead to an underestimation of the true perimeter of the air voids.
We developed a more accurate method of computing air void perimeters by identifying the boundary pixels and computing the distance between each neighboring boundary pixel. These boundary pixels were determined by identifying white pixels that neighbored black pixels. The length of the boundary was determined by identifying the manner in which successive white boundary pixels connect to each other. We designated a 4-connected neighborhood as described in Sect. 3 to define boundary pixel connectivity. Once the boundary pixels were identified, the straight-line Pythagorean distances between the centers of successive boundary pixels were computed to estimate the perimeter of each object.Footnote 3 This perimeter value was used to compute the specific surface in Eq.9.
Reconstruction of the air void size distribution
To study the effect of air-entraining agents on the air void distribution, it is desirable to estimate the size distribution of the air-entrained bubbles. In general, the process of estimating the 3D size distribution of objects using the apparent sizes, or profiles on a 2D plane is called “unfolding” or “reconstruction.” Sphere reconstruction can be accomplished by taking either a parametric or non-parametric approach. In a parametric approach, a distribution function representing the apparent circle sizes is first assumed and then transformed into a sphere distribution function. An analytical solution to this transformation was first published in 1925 by S. D. Wicksell [29]. Another noteworthy solution was published in 1955 by Reid [17], who included a transformation between spheres and chord lengths as well.
For an excellent review of stereological properties of air voids, including parametric methods, we refer the reader to Snyder et al. [24].
A non-parametric approach to sphere reconstruction does not assume an analytical form of the circle size distribution. Rather, the method operates on the collected data directly. The first mention of a non-parametric reconstruction method was included in the 1925 Wicksell paper, but the solution was greatly modified and expanded by Scheil [21] in 1931. In 1934, Schwartz [22] modified Scheil’s method, which was later improved on by Saltykov in 1958 [20]. Saltykov’s method, which is fully described by Underwood [26, 27], was chosen for this analysis and is briefly described here.
Consider a sphere of radius R. A rule of geometric probability states that if a large number of planes intersect the sphere, the profile circles will have a probability distribution ϕ(r) such that
$$\begin{aligned} \phi (r) = \frac{r}{R \sqrt{R^2 - r^2}} \end{aligned}$$
(10)
where r is the radius of each profile circle. Equivalently, ϕ(r) is the probability that a large quantity of identically sized spheres in a sample is randomly sectioned once. Note that the form of the curve of Eq. 10 increases asymptotically to \(r = R\). However, as expected, the area under the curve is equal to unity since ϕ(r) is a probability density function. An alternative representation of the curve is a histogram, where each bin corresponds to a class of profile radii. Figure 4a, b compare the smooth and discrete versions of Eq. 10.
Consider now a more realistic situation in which a sample is composed of a polydispersed system of spheres. For each discrete group, or class of spheres, the corresponding profile sizes will vary according to Eq. 10. Because there are multiple sphere sizes, the total distribution of profile sizes is a summation of the histograms corresponding to each sphere class. In order to compute the sphere sizes from an image of circular profiles, the histogram of profiles corresponding to the largest sphere class is subtracted from the total histogram, revealing the profiles of all spheres smaller than the largest sphere class. This “histogram stripping” method is repeated for the next-largest sphere class until only the smallest class remains.
If the diameter of the largest spheres is D
m
, then the class width \(\Delta\) is defined as \(\Delta = D_m/k\), where k is the number of classes. If i is the profile class number and j is the sphere class number, Saltykov’s expression relating the number density of profiles to the number density of spheres is
$$\begin{aligned} N_A(i,j) = N_V(j) \Delta [\sqrt{j^2 - (i-1)^2} - \sqrt{j^2 - i^2}] \end{aligned}$$
(11)
where N
A
is the number of profiles per unit area and N
V
is the number of spheres per unit volume. For a discrete sphere distribution, Eq. 11 is applied separately for each class. The result is a reconstruction of the 3D sphere distribution. Saltykov originally solved for N
V
in terms of N
A
and \(\Delta\) in the form of a table of coefficients using a maximum of 15 classes. However, it is no longer necessary to use this table because the computation can be performed rapidly on a computer using any desired number of classes.
One known problem that can arise with this classical reconstruction method involves negative binning. Because of sampling statistics and other practical measurement problems, a negative number of spheres from a smaller class may be computed. This phenomenon occurs when the measured number of profiles in a class is lower than its expected value and occurs most frequently with the smallest spheres in a sample [26]. In general, the problem can be solved by using an expectation-maximization algorithm [6][10], error bars [12], and/or histogram smoothing. In our image analysis, we avoided negative binning by choosing a large enough bin width that avoided gaps in the histogram. For our images, using a bin width = 4 μm was large enough to avoid negative binning during reconstruction.