Investigation of the Relationship between the 2D and 3D Box-Counting Fractal Properties and Power Law Fractal Properties of Aggregates

Abstract

The fractal dimension $D_f$ has been widely used to describe the structural and morphological characteristics of aggregates. Box-counting (BC) and power law (PL) are the most common methods to calculate the fractal dimension of aggregates. However, the prefactor $k$, as another important fractal property, has received less attention. Furthermore, there is no relevant research about the BC prefactor ($k_{BC}$). This work applied a tunable aggregation model to generate a series of three-dimensional aggregates with different input parameters (power law fractal properties: $D_{f,PL}$ and $k_{PL}$, and the number of primary particles $N_P$). Then, a projection method is applied to obtain the 2D information of the generated aggregates. The fractal properties ($k_{BC}$ and $D_{f,BC}$) of the generated aggregates are estimated by both, for 2D and 3D BC methods. Next, the relationships between the box-counting fractal properties and power law fractal properties are investigated. Notably, 2D information is easier achieved than 3D data in real processes, especially for aggregates made of nanoparticles. Therefore, correlations between 3D BC and 3D PL fractal properties with 2D BC properties are of potentially high importance and established in the present work. Finally, a comparison of these correlations with a previous one (not considering $k$) is performed, and comparison results show that the new correlations are more accurate.

1. Introduction

Aggregates made of nano-sized spherical primary particles have complex and irregular structures. Aggregates of this kind occur in many practical applications, such as in colloidal, aerosol, or combustion systems [1,2,3]. The irregularity of the aggregates can influence their chemical and physical properties, by changing, for example, their surface area [4], light extinction efficiency [5], or aerodynamic behavior [6]. Since the aggregates can be considered as fractal-like structures [7], it is acceptable to quantify their irregularity by means of the fractal dimension $D_f$, as proposed by Mandelbrot [8]. In recent years, the fractal dimension has been investigated and used in order to characterize the morphology of particle systems and porous media [9,10,11], including building materials [12], as well as in the context of transport equations for fractal media [13]. Regarding the aggregates of primary particles, the box-counting (BC) [14] and the power law (PL) [15] methods are the most common methods used to calculate the fractal dimension.

Complementary to the fractal dimension is the prefactor $k$, as a second fractal property, which has though received much less attention than $D_f$. It is worth pointing out that for a given aggregate size, the spatial distribution of primary particles in aggregates is dependent on both the fractal dimension and the prefactor [16,17]. The constituent primary particles of aggregates become more concentrated in space with an increase in either of these two parameters. In the frame of a power law, the prefactor has an influence on how the aggregate mass is filling up the space, independently of its size, and on how the primary particles are packed [18]. The prefactor has been associated with the porosity and lacunarity of the aggregates [19]. The variation of the prefactor as a function of the fractal dimension and the packing density has been discussed in [16]. However, the power law method needs to collect data from many aggregates with distributed size to, then, provide mean fractal properties ($D_{f,PL}$ and $k_{PL}$) for the whole particle population [20]. In contrast, the box-counting method (BC) is a simple mathematical method that enables the determination of fractal dimensions on a single aggregate [21,22]. However, there are no studies with respect to the BC prefactor $k_{BC}$, which is not even defined and evaluated in many cases. Moreover, interrelations between BC fractal properties and PL fractal properties are missing.

Three-dimensional (3D) information about the aggregates can visually and comprehensively describe the morphological properties of aggregates. Tomography is widely applied to obtain 3D scanning of aggregates. The inner structure of aggregates can be directly accessed by optical coherence tomography or micro-computed tomography (μ-CT) [23]. Pashminehazar et al. [21] investigated the morphological properties of amorphous maltodextrin aggregates with highly porous and soft structures by X-ray μ-CT. The morphology and microstructure of diesel soot particulate matter were investigated by synchrotron soft X-ray tomography in [24]. Those authors calculated the fractal dimensions of agglomerates and diesel soot particulate matter by image analysis. However, when the aggregates are composed of nano-sized primary particles, their 3D data are difficult and expensive to obtain with the necessary resolution. It is, however, relatively easy and quick to get the 2D data of such aggregates by SEM or TEM. Light scattering and electron microscope methods for calculating the fractal dimension of fumed silica are compared in [25]. Quantitative analysis of the fractal dimension of soot aggregates by SEM and image processing techniques was performed by Chakrabarty et al. [26] to find the dependence of particle morphology on the electrical charging of particles. A new method was proposed in [27] to estimate the fractal dimension of individual soot aggregates, which can be applied to TEM images. Therefore, the development of a correlation between 2D and 3D fractal properties of aggregates would be highly beneficial, because then the 3D fractal properties of aggregates could be obtained from 2D fractal properties by means of the correlation. Wang et al. [28] proposed a 2D projection method to obtain the minimum overlapping between primary particles and built a correlation between 2D BC fractal dimension and the 3D power law fractal dimension. However, this correlation neglects the effect of the prefactor, considering the prefactor to be constant and equal to 1.

In this study, this limitation is removed. A particle-cluster tunable aggregation model is applied to generate a series of aggregates with various fractal properties (based on the power law and denoted by $D_{f,PL}$ and $k_{PL}$) and with various numbers of primary particles $N_p$. This is the modified polydisperse tunable sequential aggregation (MPTSA) model from [29]. Compared to cluster–cluster models, the particle–cluster tunable aggregation model can predict realistic fractal properties of aggregates accurately and quickly. Then, the projection method that has been introduced in [28] is applied to get 2D data of the generated aggregates. Further, the box-counting method is applied to obtain the respective fractal properties both in 2D (denoted by $D_{f,BC,2D}$, $k_{BC,2D}$) as well as in 3D ($D_{f,BC,3D}$ and $k_{BC,3D}$). Not only the fractal dimension, but also the fractal prefactor is evaluated in this frame. Finally, correlations between 3D BC fractal properties and 3D PL fractal properties with 2D box-counting fractal properties are established. Prospectively, such correlations can be used to reconstruct the spatial structure of aggregates based just on planar microscopy images.

2. Methods

2.1. Power Law (PL) Method

Using the power law method to estimate the fractal properties (fractal dimension $D_{f,PL}$ and prefactor $k_{PL}$) of the aggregates requires knowledge of several parameters, namely of the mean radius of primary particles ($R_p$), the radius of gyration ($R_g$), and the number of primary particles ($N_p$). The above parameters combine with the fractal properties to the power law relationship,

$$N_p=k_{PL}{\left(\frac{R_g}{R_p}\right)}^{D_{f,PL}}.$$

(1)

In Equation (1), gyration radius $R_g$ is one of the basic parameters to describe an aggregate since it is influenced by the spatial distribution of mass around the mass center of the aggregate. Thus, $R_g$ depends on both, the aggregate size and the mass distribution in it. The illustration of $R_g$ of an aggregate is shown in Figure 1.

In our present work, the constituent primary particles of aggregates are considered as monodisperse, so that $R_g$ can be calculated according to [30] by means of the relationship

$$R_g=\sqrt{\frac{1}{2N_p{}^2}{\displaystyle \sum }_{i=0}^{N_p}{\displaystyle \sum }_{j=0}^{N_p}{\left(R_i-R_j\right)}^2},$$

(2)

where $R_i$ and $R_j$ are the position vectors of the ith and the jth primary particles in the aggregate.

Equation (1) can be transformed into a logarithmic form:

$$\log \frac{R_g}{R_p}=D_{f,PL}\log N_p-k_{PL}.$$

(3)

According to Equation (3), plotting $N_p$ versus $R_g/R_p$ in logarithmic coordinates results in a linear regression that can be used to correlate $\log N_p$ and $\log (R_g/R_p)$. Then, the linear regression slope is fractal dimension $D_{f,PL}$ and the intercept is prefactor $k_{PL}$.

The power law method is an averaging method, which means that this method requires to collect data from a relatively large number of aggregates with distributed size and provides mean fractal properties ($D_{f,PL}$ and $k_{PL}$) of the whole particle population.

2.2. Box-Counting (BC) Method

The box-counting method is a relatively simple mathematical method that enables the determination of fractal properties on a single aggregate. This method can estimate the fractal dimension for aggregates with or without self-similarity. As discussed in Wang et al. [28], Strenzke et al. [31], and Pashminehazar et al. [21], the BC method can be used on both 2D and 3D aggregates.

The key point of the BC method is to create a grid with the same number of unit boxes ($n$) in each direction (for 3D, $x$, $y$ and $z$ directions, and one direction less for 2D). The number of boxes $n$ varies with the scaling factor, $\mathsf{\epsilon }$, as shown in Figure 2. Typically, $n$ has the value of a power of 2: it starts with 2 and ends with a limiting number ($Sn$). $Sn$ is related to the resolution, for example, the number of voxels in 3D X-ray scans or the number of pixels of 2D microscopy images. The effect of $Sn$ has been discussed in [28], $Sn$ = 512 having been selected for this work. Then, the BC fractal dimension (denoted by $D_{f,BC}$) is estimated by the dynamic relationship between the number of boxes that are occupied by the aggregate N(ε) and the scaling factor $\epsilon$:

$$D_{f,BC}=\underset{\epsilon \to 0}{\lim }\left(\frac{\log \left(N\left(\epsilon \right)\right)}{\log \left(\frac{1}{\epsilon }\right)}\right).$$

(4)

In Figure 2, L is the side length of the whole grid and δ is the size of one box. The definition and determination of $L$ have been investigated in [28], and in this work the same setting of $L$ as in [28] has been applied when using the BC method. The relationship of $L$ with respect to $\delta$ and $\epsilon$ is

$$L=n\delta =\frac{\delta }{\epsilon }.$$

(5)

As shown in Figure 2, $\delta$ cannot reach zero. Thus, it is required to count the changing number $N\left(\epsilon \right)$ of corresponding boxes with different side lengths, decreasing the value of $\epsilon$ for several times. Plotting $N\left(\epsilon \right)$ versus $1/\epsilon$ on a log-log plot gives a straight line with the least square method, and the absolute value of the slope of this line is the box-counting fractal dimension. Thus, $D_{f,BC}$ is calculated through the equation

$$y=D_{f,BC}x+k_{BC}.$$

(6)

where $y$ represents log($N\left(\epsilon \right)$), $x$ is log($1/\epsilon$), and $k_{BC}$ is the BC fractal prefactor. Comparing Equations (3) and (6), one can see that the slope of both equations represents the fractal dimension, whereas the intercept represents the prefactor. The relationship between BC fractal properties and PL fractal properties is discussed in Section 3.

2.3. Aggregate Generation

Here, a tunable aggregation model is applied to generate a series of aggregates with various fractal properties and different numbers of primary particles. This is the modified polydisperse tunable sequential aggregation model (MPTSA) from [29]. In this aggregate generation model, the input parameters are the fractal dimension $D_{f,PL}$, the prefactor $k_{PL}$, the number of primary particles $N_P$, and the radius of primary particles $R_p$. The flowchart of the MPTSA model is shown in Figure 3.

The first primary particle is allotted to the center of the simulation space. Next, a point is selected on the surface of the first particle, and the second particle is placed adjacent to this point. As discussed in [32], there is a limitation of prefactor in the tunable sequential algorithm. So, the input fractal dimension $D_{f,PL}$ and prefactor $k_{PL}$ are tuned to $D_{f,t}$ and $k_t$(=1) by the following equation [32],

$$D_{f,t}=D_{f,PL}\left(\frac{\log \left(\frac{N_p}{1}\right)}{\log \left(\frac{N_p}{k_{PL}}\right)}\right).$$

(7)

Then, the third and subsequent particles are inserted one by one. The center of each additional primary particle is located on a sphere of radius $T$

$$T^2=\frac{P^2{R}_p{}^2}{P-1}{\left(\frac{P}{k_t}\right)}^{\frac{2}{D_{f,t}}}-\frac{0.6P{R}_p{}^2}{N-1}-P{R}_p{}^2{\left(\frac{P-1}{k_t}\right)}^{\frac{2}{D_{f,t}}}.$$

(8)

Here, $P$ ranges from 3 to $N_p$. The precise position on the sphere of radius $T$ is chosen to have point contact with the new primary particle without overlapping. The addition of primary particles is continued one by one until $P$ ≥ $N_p$.

For illustration, we first generated a series of aggregates using the MPTSA model with different fractal properties ($D_{f,PL}$ and $k_{PL}$) and the same number of primary particles ($N_p$ = 50), as shown in Figure 4. It can be seen from Figure 4 that both $D_{f,PL}$ and $k_{PL}$ can affect the structural and morphological properties of aggregates. When $D_{f,PL}$ or $k_{PL}$ increases gradually, the constituent primary particles tend to be more and more concentrated around the center point of the aggregate, which more and more resembles a sphere. However, the morphology of aggregates is more sensitive upon $D_{f,PL}$ than upon $k_{PL}$ (variation of $D_{f,PL}$ is from 1.8 to 3.0, whereas the variation of $k_{PL}$ is from 1.0 to 7.0). This is due to the fact that $k_{PL}$ is the prefactor, not the exponent of Equation (1).

3. Results and Discussion

3.1. Effective Range of Prefactor $k_{PL}$

The prefactor $k_{PL}$ is an important parameter to describe the structural and morphological properties of aggregates, as shown in Figure 4. Therefore, and with values of $D_{f,PL}$ being in the range of 1 to 3, the question arises about the effective range that values of the prefactor $k_{PL}$ may attain. For a first orientation, we summarized values of $k_{PL}$ for aggregates with different $D_{f,PL}$ from the literature in

Table 1. Values of $k_{PL}$ and $D_{f,PL}$ from previous work.

References	${\mathit{k}}_{\mathit{P}\mathit{L}}$	${\mathit{D}}_{\mathit{f},\mathit{P}\mathit{L}}$
Mountain and Mulholland [33]	5.80	1.90
Wu and Friedlander [18]	1.30	1.84
Puri et al. [34]	9.00	1.74
Sorensen and Roberts [16]	8.50	1.82
Ouf et al. [35]	2.44	1.78
Brasil et al. [36]	1.27	1.82

.

Table 1 shows a spread of $k_{PL}$ from 1.27 to 9.0. However, the variation of $D_{f,PL}$ is small in the previous research, and derivations are not always detailed and clear. Therefore, we have decided to test the effective range of $k_{PL}$ with the help of aggregates generated by the MPTSA model.

In the power law relationship (Equation (1)), the radius of gyration $R_g$ can describe the spatial mass distribution around the mass center of the aggregate. As a criterion for the effective range of $k_{PL}$, we compare the radius of gyration calculated by Equation (1), denoted by $R_{g,PL}$, to the radius of gyration from generated aggregates according to Equation (2), denoted by $R_{g,MP}$. Both $R_{g,MP}$ and $R_{g,PL}$ are obtained with the same input parameters. The comparison is quantified by the ratio

$$Ra=\frac{R_{g,PL}}{R_{g,MP}}.$$

(9)

If $Ra$ of the generated aggregate is greater than 99.99%, then $k_{PL}$ is assumed to have been in its effective range.

Next, we generated in the frame of this evaluation two groups of aggregates with different fractal dimensions and prefactors. In the first group, the number of primary particles was $N_p$ = 50, with $D_{f,PL}$ = 1.8:0.2:2.8 and $k_{PL}$ varying from 0.9 to 5.0 in steps of 0.1. As for the second group, it had $N_p$ = 300, $D_{f,PL}$ = 1.8:0.2:2.8, and $k_{PL}$ = 0.9:0.1:10. The radius of primary particles $R_p$ was constant and equal to 0.2 mm, but the absolute value of this variable has no influence on the results. The relationship of $k_{PL}$ and $D_{f,PL}$ with $Ra$, based on two different values of $N_p$, is shown in Figure 5.

As can be seen in Figure 5a,b, all the curves show the same trend: at the beginning, $Ra$ of the aggregates does not change with increasing $k_{PL}$, being on a plateau with $Ra=1$. Then, as $k_{PL}$ increases, all curves show an inflection point, after which $Ra$ decreases dramatically. When $N_p$ is the same, the main difference among the curves is in the length of their plateau regions; aggregates with a smaller $D_{f,PL}$ show a longer plateau with $Ra=1$ over $k_{PL}$. This means that the aggregates with smaller $D_{f,PL}$ have a broader effective range of $k_{PL}$. Besides, by comparing Figure 5a,b at the same $D_{f,PL}$, we can find that the effective range of $k_{PL}$ of aggregates with smaller $N_p$(=50) is narrower than in the case of larger $N_p$(=300). The horizontal axis coordinates of the inflection points on each curve are considered as the upper limit of the effective range of $k_{PL}$ under conditions specified by different $D_{f,PL}$ and $N_p$. Respective values are shown in

Table 2. Lower and upper limits of the effective range of $k_{PL}$ for different $D_{f,PL}$ and $N_p$.

${\mathit{D}}_{\mathit{f},\mathit{P}\mathit{L}}$	${\mathit{N}}_{\mathit{p}}$ = 100	${\mathit{N}}_{\mathit{p}}$ = 300
1.8	[0.9, 4.7]	[0.9, 9.7]
2.0	[0.8, 3.6]	[0.8, 6.9]
2.2	[0.7, 2.8]	[0.7, 4.5]
2.4	[0.6, 2.1]	[0.6, 3.1]
2.6	[0.5, 1.6]	[0.5, 2.1]
2.8	[0.4, 1.2]	[0.4, 1.4]

.

In Figure 5a, when $k_{PL}$ is larger than 4.7 at $N_p$ = 50 and $D_{f,PL}$ = 1.8, then the $Ra$ of the aggregates is less than 1. This is since with an additional increase of $k_{PL}$ (>4.7), the primary particles of these aggregates can no longer be concentrated further in space. So, $R_{g,MP}$ of these aggregates does not change with further increasing $k_{PL}$. For example, as shown in Figure 4, the structural and morphological characteristics of the aggregate with $D_{f,PL}$ = 1.8 and $k_{PL}$ = 4.7 are the same as those of the aggregate with $D_{f,PL}$ = 1.8 and $k_{PL}$ = 7.0, with $R_{g,MP}$ of these two aggregates being same and equal to 0.74 mm. However, $R_{g,PL}$ calculated formally from Equation (1) continues to decrease as $k_{PL}$ increases, namely from $R_{g,PL}$ = 0.74 mm for aggregates with $D_{f,PL}$ = 1.8 and $k_{PL}$ = 4.7, to $R_{g,PL}$ = 0.6 mm for aggregates with $D_{f,PL}$ = 1.8 and $k_{PL}$ = 7.0. Therefore, when $D_{f,PL}$ = 1.8 and $k_{PL}$ > 4.7, the ratio $Ra$ of the aggregates is less than unity, meaning that $k_{PL}$ has moved outside of its effective range.

As to the lower limit of $k_{PL}$, it has been determined by decreasing its value in steps of 0.1. This process stops when the MPTSA model ceases being able to generate the aggregate. Until then, the values of $Ra$ of the generated aggregates remain equal to 1. The minimum $k_{PL}$ at which aggregates can be generated is the lower limit of the effective range. Values for different $D_{f,PL}$ and $N_p$ are shown in Table 2.

In addition to the above method that presupposes the generation of agglomerates by means of the MPTSA algorithm, a much simpler, algebraic estimation of the limits of the effective range of $k_{PL}$ has also been implemented in the present work. According to Equation (1), $k_{PL}$ shows a negative relationship to $R_g/R_p$ under fixed $N_p$ and $D_{f,PL}$,

$$k_{PL}=N_p/{(R_g/R_p)}^{D_{f,PL}}.$$

(10)

Therefore, when $N_p$ and $D_{f,PL}$ are fixed and $R_g/R_p$ minimal, $k_{PL}$ takes its upper limit value. The lower limit of $k_{PL}$ occurs when the situation is conversed ($R_g/R_p$ at maximum value). The radius of gyration $R_g$ of an aggregate shows the mass distribution around the aggregate center of mass. In our present work, the radius of primary particles $R_p$ is constant at 0.2 mm. Therefore, when two aggregates with the same $N_p$ show different $R_g$, the lower value of $R_g$ indicates that the mass (primary particles) of the aggregate is more concentrated at the center of mass. So, the minimum $R_g/R_p$ is reached when the morphology of the aggregate is like that of a sphere, which is here assumed to happen for an aggregate with $D_{f,PL}$ = 3.0 and $k_{PL}$ = 1.0. On the contrary, the relative gyration radius $R_g/R_p$ of the aggregates is maximum when the primary particles of the aggregates are most dispersed, assumed here to be the case for aggregates with $D_{f,PL}$ = 1.7 and $k_{PL}$ = 1.0. Minimal $R_g/R_p$ (at $D_{f,PL}$ = 3.0 and $k_{PL}$ = 1.0) and maximal $R_g/R_p$ (at $D_{f,PL}$ = 1.7 and $k_{PL}$ = 1.0) of aggregates with different $N_p$ are calculated by Equation (1), the results are summarized in

Table 3. Minimal $R_g/R_p$ ($D_{f,PL}$ = 3.0 and $k_{PL}$ = 1.0) and maximal $R_g/R_p$ ($D_{f,PL}$ = 1.7 and $k_{PL}$ = 1.0) of aggregates with different $N_p$.

${\mathit{N}}_{\mathit{p}}$	Minimal ${\mathit{R}}_{\mathit{g}}/{\mathit{R}}_{\mathit{p}}$	Maximal ${\mathit{R}}_{\mathit{g}}/{\mathit{R}}_{\mathit{p}}$
5	1.71	2.58
50	3.68	9.99
100	4.64	15.01
150	5.31	19.06
200	5.85	22.57
250	6.30	25.74
300	6.69	28.65

.

Then, minimal $R_g/R_p$ and maximal $R_g/R_p$ of the aggregates with different $N_p$ are substituted into Equation (10), and upper and lower limits of the effective range are obtained for different $D_{f,PL}$, respectively. The lower and upper limits of the effective range of $k_{PL}$ that have been estimated in this way are shown in

Table 4. Upper limit and lower limit of effective range of $k_{PL}$ from the simplified estimation without aggregate generation.

${\mathit{N}}_{\mathit{p}}$	${\mathit{D}}_{\mathit{f},\mathit{P}\mathit{L}}$ = 1.8	${\mathit{D}}_{\mathit{f},\mathit{P}\mathit{L}}$ = 2.0	${\mathit{D}}_{\mathit{f},\mathit{P}\mathit{L}}$ = 2.2	${\mathit{D}}_{\mathit{f},\mathit{P}\mathit{L}}$ = 2.4	${\mathit{D}}_{\mathit{f},\mathit{P}\mathit{L}}$ = 2.6	${\mathit{D}}_{\mathit{f},\mathit{P}\mathit{L}}$ = 2.8
5	[0.91, 1.90]	[0.75, 1.71]	[0.62, 1.54]	[0.52, 1.38]	[0.43, 1.24]	[0.35, 1.11]
50	[0.79, 4.78]	[0.50, 3.68]	[0.32, 2.84]	[0.20, 2.19]	[0.13, 1.68]	[0.08, 1.30]
100	[0.76, 6.31]	[0.44, 4.64]	[0.26, 3.41]	[0.15, 2.51]	[0.09, 1.85]	[0.05, 1.36]
150	[0.74, 7.42]	[0.41, 5.31]	[0.23, 3.80]	[0.13, 2.72]	[0.07, 1.95]	[0.04, 1.40]
250	[0.73, 8.33]	[0.39, 5.85]	[0.21, 4.11]	[0.11, 2.89]	[0.06, 2.03]	[0.03, 1.42]
300	[0.72, 9.10]	[0.38, 6.30]	[0.20, 4.36]	[0.10, 3.02]	[0.05, 2.09]	[0.03, 1.44]

.

Comparing Table 2 and Table 4, we can find that the upper limit of $k_{PL}$ obtained by use of the MPTSA model is close (slightly smaller) to the results of the simplified estimation. The lower limit of $k_{PL}$ in Table 2 is nearly equal to the lower limit value of $k_{PL}$ for aggregates with the smallest $N_p$ (= 5) in Table 4.

3.2. Relationship between BC Fractal Properties and PL Fractal Properties

In this section, we generated a series of aggregates with different fractal properties ($D_{f,PL}$ and $k_{PL}$) and $N_p$ by the MPTSA model, with $N_P$ varying from 100 to 300 in steps of 50 and $D_{f,PL}$ = 1.8:0.2:2.8. The investigated range of $k_{PL}$ for each $D_{f,PL}$ is shown in

Table 5. Realized ranges of $k_{PL}$ for aggregates with different $D_{f,PL}$.

${\mathit{D}}_{\mathit{f},\mathit{P}\mathit{L}}$	Lower Limit ${\mathit{k}}_{\mathit{P}\mathit{L}}$	Upper Limit ${\mathit{k}}_{\mathit{P}\mathit{L}}$
1.8	0.9	6.3
2.0	0.8	4.6
2.2	0.7	3.4
2.4	0.6	2.5
2.6	0.5	1.8
2.8	0.4	1.3

. Those ranges correspond to the ranges for $N_P$ = 100 from Table 4, being more restrictive in comparison to the ranges for aggregates with a larger number of primary particles. Consequently, all the generated aggregates are safely within the effective range of $k_{PL}$ values. The primary particles of generated aggregates are monodispersed in the present work, with the radius of primary particles formally set at 0.2 mm. To capture stochastic variations, each aggregate is generated five times with the same input parameters.

In the further course of evaluation, a projection method proposed by [28] is applied to get 2D data for the generated aggregates. Then, both 3D and 2D box-counting methods are applied to estimate 3D BC fractal properties ($D_{f,BC,3D}$ and $k_{BC,3D}$) and 2D BC fractal properties ($D_{f,BC,2D}$ and $k_{BC,2D}$) of the generated aggregates. Next, aggregates generated with different $N_p$ (=100, 200 and 300) and $D_{f,PL}$ (=1.8 and 2.8) are chosen to investigate the relationships between $D_{f,BC,3D}$ and $D_{f,BC,2D}$ with $k_{PL}$. The averages of $D_{f,BC,2D}$ and $D_{f,BC,3D}$ over the five realizations are shown in Figure 6 for the selected aggregates. Furthermore, averages ($D_{f,BC,2D}$ and $D_{f,BC,3D}$) over each entire aggregate series (with $N_p$ = 100:50:300) are also plotted in Figure 6 against $k_{PL}$.

All the curves in Figure 6 show the same trend, namely of BC fractal dimensions increasing with increasing $k_{PL}$. This is due to the fact that with the increase of $k_{PL}$, the distribution of primary particles becomes more and more concentrated (as shown in Figure 4). In the BC method, the number of boxes ($N$) occupied by aggregates is larger when the primary particles of aggregates are more concentrated [28]. According to Equation (4), $N$ and BC fractal dimensions show a positive relationship. Therefore, the BC fractal dimension increases as $k_{PL}$ increases.

Moreover, the trend in the variation of $D_{f,BC,2D}$ with $k_{PL}$ in Figure 6d ($D_{f,PL}$ = 2.8 and BC in 2D) is slightly different from the other three figures (Figure 6a–c). In Figure 6d, $D_{f,BC,2D}$ initially increases with increasing $k_{PL}$ (0.4 to 0.8), but then the value of $D_{f,BC,2D}$ starts fluctuating around 1.91 as $k_{PL}$ further increases. This is because the calculation of the 2D BC fractal dimension of the aggregates is based on their projection. The purpose of projection method in this work is to get the least overlapping between primary particles (the maximum projected area of the aggregates) [28]. The morphology of the aggregates is though close to spherical when aggregates with a high fractal dimension and prefactor are considered [27] (i.e., $D_{f,PL}$ = 2.8 and $k_{PL}$ > 0.8). Therefore, the 2D maximum projection area of these aggregates is almost constant with increasing $k_{PL}$, and the same holds for $D_{f,BC,2D}$ values since these are directly affected by the projection area (positive relationship). Therefore, the $D_{f,BC,2D}$ value of the mentioned kind of aggregates floats around 1.91. In less obvious but analogous way, it can be seen from Figure 6c that the $D_{f,BC,2D}$ values of fluffy aggregates also float around 1.91 when the value of $k_{PL}$ is large enough (> 5.5, in this case). In addition, the values of averages ($D_{f,BC,2D}$ and $D_{f,BC,3D}$) over the entire aggregate series (with $N_p$ = 100:50:300) are generally close to the values for primary particle number in the middle of the series ($N_p$ = 200).

3.3. Correlation between 3D BC Fractal Properties and 2D BC Fractal Properties

It is hard or even impossible to obtain the 3D fractal properties of aggregates composed of very small primary particles or nanoparticles by X-ray µ-CT, because of limitation in the spatial resolution of this imaging method. However, the 2D fractal properties of such aggregates can easily be retrieved by SEM or TEM. Therefore, a correlation between 2D and 3D fractal properties is necessary to be established. In this section, the correlation between 2D and 3D BC fractal properties is discussed first. Furthermore, the correlation between 2D BC and 3D PL fractal properties is discussed in the next section.

In Figure 7, the relationship between $k_{BC,2D}$ and $k_{BC,3D}$ for the aggregates with various $D_{f,PL}$ (=1.8:0.2:2.8) is shown. Values of $k_{BC,2D}$ and $k_{BC,3D}$ have been averaged over all $N_p$ (from 100 to 300 in steps of 50) of the entire aggregate series and then over five realizations.

As shown in Figure 7, $k_{f,BC,3D}$ increases with $k_{BC,2D}$ for any value of power law fractal dimension. All data points can, thus, be described by one and the same power regression,

$$k_{BC,3D}=1.0262{\mathrm{e}}^{0.3714k_{BC,2D}}\left({\mathrm{R}}^2=0.9705\right).$$

(11)

The average values of $D_{f,BC,3D}$ (over five iterations) and $k_{BC,3D}$ for the aggregates with different $D_{f,PL}$ are plotted in Figure 8. From Figure 8 we can find that the value of $D_{f,BC,3D}$ is linearly increasing with $k_{BC,3D}$. The respective linear regression for all the aggregates is

$$D_{f,BC,3D}=0.3585k_{BC,3D}+0.0423\left({\mathrm{R}}^2=0.9994\right).$$

(12)

A combination of Equations (11) and (12) can be used to obtain 3D BC fractal properties ($k_{BC,3D}$ and $D_{f,BC,3D}$) from a given 2D BC prefactor $k_{BC,2D}$ or, by additionally involving the later Equation (17), from a given 2D BC fractal dimension $D_{f,BC,2D}$.

3.4. Correlation between 2D BC Fractal Properties and PL Fractal Properties

The relationship between $k_{PL}$ and $k_{BC,2D}$ for aggregates with various $D_{f,PL}$(=1.8:0.2:2.8) is shown in Figure 9. Values of $k_{BC,2D}$ have been averaged over all $N_p$ (from 100 to 300 in steps of 50) of the entire aggregate series and then over five realizations.

In Figure 9, $k_{BC,2D}$ is seen to increase with increasing $k_{PL}$; however, the growth rate of $k_{BC,2D}$ decreases as $k_{PL}$ increases. As pointed out in Section 3.2, 2D BC fractal properties of aggregates are influenced by the 2D projection area [28], being positively interrelated. And when the morphology of the aggregates with higher $k_{PL}$ or $D_{f,PL}$ has approached that of a sphere (as shown in Figure 4), the projection area of these aggregates changes only slightly with further increase in $k_{PL}$. Therefore, the rise of $k_{BC,2D}$ with $k_{PL}$ flattens up at larger $k_{PL}$ or $D_{f,PL}$. Here, an exponential function can be used for regression,

$$k_{BC,2D}=a{e}^{-b{k}_{PL}}+c.$$

(13)

In Equation (13), the curves with different $D_{f,PL}$ have different values of $a$, $b$, and $c$, as summarized in

Table 6. Fitted values of $a$, $b$, and $c$ corresponding to different $D_{f,PL}$.

${\mathit{D}}_{\mathit{f},\mathit{P}\mathit{L}}$	$\mathit{a}$	$\mathit{b}$	$\mathit{c}$
1.8	−1.182	0.551	5.223
2.0	−1.084	0.891	5.203
2.2	−0.928	1.244	5.210
2.4	−0.845	1.825	5.209
2.6	−0.805	3.311	5.197
2.8	−0.752	3.931	5.205

.

Then, correlations between, first, $D_{f,PL}$ and $a$, and second, between $D_{f,PL}$ and $b$ are developed as follows:

$$a=0.4389D_{f,PL}-1.9419\left({\mathrm{R}}^2=0.9508\right),$$

(14)

$$b=0.0363D_{f,PL}{}^{4.5778}\left({\mathrm{R}}^2=0.9877\right).$$

(15)

The average value of $c$ = 5.208 is used to represent this parameter.

Combining Equations (13)–(15), the correlation between $k_{BC,2D}$ and power law fractal properties is obtained:

$$k_{BC,2D}=\left(0.4389D_{f,PL}-1.9419\right)e^{-0.0363D_{f,PL}{}^{4.5778}{k}_{PL}}+5.208.$$

(16)

The averages of $D_{f,BC,2D}$ and $k_{BC,2D}$ over five realizations are plotted in Figure 10 for aggregates with different $D_{f,PL}$. As shown in Figure 10, $D_{f,BC,2D}$ increases linearly with $k_{BC,2D}$, according to the regression

$$D_{f,BC,2D}=0.3693k_{BC,2D}-0.0076\left({\mathrm{R}}^2=0.999\right).$$

(17)

Equations (16) and (17) are very important. Combining these two equations enables to predict power law fractal properties ($D_{f,PL}$ and $k_{PL}$) of aggregates from their 2D box-counting fractal properties ($k_{BC,2D}$ and $D_{f,BC,2D}$), the determination of which from microscope images is fast and easy in practice.

Therefore, the reliability of these two correlations is tested by a new series of aggregates generated by the MPTSA model. Here, three different values of $D_{f,PL}$ are used, namely $D_{f,PL}$ = 1.9, 2.3, and 2.7. The input number of primary particles $N_p$ varied from 100 to 300 in steps of 50. The prefactor $k_{PL}$ of the aggregates takes values from 0.9 to the upper limit of its effective range for each $D_{f,PL}$ (according to Table 5). The primary particles are still monodispersed, and the radius of primary particles is kept same as for the previously generated aggregates. Each aggregate with the same input parameters is generated five times. Then, the aggregates that have been generated in 3D are projected onto a 2D plane by the projection method from [28], and the 2D BC method is applied to estimate the 2D BC fractal properties for those projections. Then, the averages of $D_{f,BC,2D}$ and $k_{BC,2D}$ for each aggregate are calculated over five realizations. Substituting $k_{BC,2D}$ and $D_{f,BC,2D}$ into Equations (16) and (17), values of power law fractal properties ($k_{PL}$ and $D_{f,PL}$) are finally calculated. Examples of calculated results for aggregates with $D_{f,PL}$ = 1.9 are summarized in

Table 7. Predicted $D_{f,PL}$ and $k_{PL}$ of aggregates with input $D_{f,PL}$ = 1.9.

Input ${\mathit{k}}_{\mathit{P}\mathit{L}}$	Calculated ${\mathit{D}}_{\mathit{f},\mathit{P}\mathit{L}}$	Calculated ${\mathit{k}}_{\mathit{P}\mathit{L}}$
1.0	1.97	0.79
1.5	2.08	0.97
2.0	1.61	4.06
2.5	2.27	1.02
3.0	2.20	1.46

.

In Table 7, there is a notable difference between the input fractal parameters and the calculated values. This is due to the difficult inversion of Equations (16) and (17) for given $k_{BC,2D}$ and $D_{f,BC,2D}$. This is done by numerical optimization, which is though confronted with several flat and similar optima.

Whereas further improvement is desirable at this point, the ratio $R_g/R_p$, which is an important parameter for the morphological analysis of aggregates, can be applied to test the predicted values from Equations (16) and (17). It is recalled that Wang et al. [28] have recently established an original correlation between 2D BC fractal dimension and PL fractal dimension. This correlation, however, neglected the influence of $k_{PL}$ and kept this parameter constant (=1). The correlation is

$$D_{f,PL}=0.2015D_{f,BC,2D}^{4.079}.$$

(18)

Predicted results ($D_{f,PL}$ and $k_{PL}$) from Equations (16) and (17) are substituted to Equation (1) to calculate $R_g/R_p$ of the new series of aggregates ($D_{f,PL}$ = 1.9, 2.3, and 2.7, $N_p$ = 100, 200, and 300). For the sake of comparison, $D_{f,BC,2D}$ of the new generated aggregates are substituted to Equation (18) to estimate their $D_{f,PL}$. Then, keeping $k_{PL}$ as constant and equal to 1, another $R_g/R_p$ is estimated by means of $D_{f,PL}$ predicted from Equation (18). Finally, the two kinds of $R_g/R_p$ are compared in Figure 11 based on three $N_p$ (= 100, 200, and 300). $R_g/R_p$ calculated from prediction results of the equations in this research (Equations (16) and (17)) are denoted by “present”, $R_g/R_p$ calculated from the correlation of the previous work (Equation (18)) are denoted by “previous”. In addition, the standard $R_g/R_p$ which is calculated from the input parameters ($D_{f,PL}$, $k_{PL}$, and $N_p$) of the MPTSA model is also shown in Figure 11 (dotted lines). The R-square analysis represents the deviation of the predicted $R_g/R_p$ (present or previous) to standard $R_g/R_p$.

As shown in Figure 11, when both $D_{f,PL}$ and $N_p$ are small ($D_{f,PL}$ = 1.9 and $N_p$ = 100), the difference between present predicted results (Equations (16) and (17)) and previous predicted results (Equation (18)) is insignificant, the R² of the two sets of results to the standard (input, reference) data being 0.923 and 0.929, respectively. However, when $D_{f,PL}$ or $N_p$ increases, the R² of previous results decreases significantly. Especially when the aggregates with $D_{f,PL}$ = 2.7 and $N_p$ = 300 are considered, the R² of previous results reaches a very low value of 0.439. However, the changes in $D_{f,PL}$ or $N_p$ hardly affect the accuracy of the present results, which are based on predictions from Equations (16) and (17). In Figure 11, the minimum R² of present results is equal to 0.868 when $D_{f,PL}$ = 2.7 and $N_p$ = 100.

4. Conclusions

This study aimed to investigate the fractal properties of aggregates made of primary particles. To enable this investigation, synthetic aggregates have been generated by an appropriate numerical method (MPTSA model) with various input parameters. Special emphasis was set on the variation of power law prefactor $k_{PL}$. Not every value of this parameter is reasonable, so that its so-called effective range had to be first determined. Therefore, we introduced a ratio $Ra$ between two differently derived radii of gyration $R_g$ as a criterion that can be used to judge whether aggregates are within the effective range of $k_{PL}$ or not. Where $R_{g,MP}$ is estimated from generated aggregates (aggregates produced with the help of the MPTSA model) and $R_{g,PL}$ is obtained from the power law. Conceivable deviation of $Ra$ from unity means out-of-range values of $k_{PL}$. A simplified method that works without aggregate generation is also proposed as an alternative to this rigorous approach.

In the main part of the work, a series of aggregates (with different fractal properties $D_{f,PL}$ and $k_{PL}$, and also with different number of primary particles $N_p$) is generated by the MPTSA model. Values of $k_{PL}$ are taken from their effective ranges based on different $N_p$ and power law fractal dimension $D_{f,PL}$. Next, a projection method proposed by [28] is applied to retrieve 2D information for the generated aggregates. Both the 3D BC fractal properties ($D_{f,BC,3D}$ and $k_{BC,3D}$) and the 2D BC fractal properties ($D_{f,BC,2D}$ and $k_{BC,2D}$) of the generated aggregates are calculated by the box-counting (BC) method.

Considering that the 3D fractal properties of aggregates composed of very small particles are hard to gain by means of X-ray µ-CT, or with the help of other tomographic methods, due to limited physical resolution, tractable pathways for their determination from more easily accessible information would be of great importance. Such information could be the 2D fractal properties of those aggregates, because 2D fractal properties can easily be extracted from SEM or TEM images. To this purpose, novel correlations between 3D BC fractal properties and 3D PL fractal properties with 2D BC fractal properties have been established with the help of synthetic aggregates. In addition, one more series of synthetic aggregates has been generated to validate the correlation between 2D BC and 3D PL fractal properties. The validation results are compared with the results of a previous correlation (not considering the variability of $k_{PL}$) as well as with results from the used input data as a benchmark. Present and previous results meet similarly well the benchmark only when both $D_{f,PL}$ and $N_p$ are small ($D_{f,PL}$ = 1.9 and $N_p$ = 100). However, our new correlations are more accurate for aggregates with higher $D_{f,PL}$ or $N_p$.

The generated aggregates in this work had monodispersed primary particles. Therefore, aggregates with polydisperse primary particles need to be also investigated, which is planned for near future.