Statistical analysis

Table 3.4: Average CPU time, the average number of particles and associated standard deviations (over 10 realizations of each microstructure) in the generation of microstructures with several particle phases 3A, 3B and 3C (see Figure 3.33).

Example CPU Time (s) Number of particles Total volume fraction of particles (%) Mean Std. Dev. Mean Std. Dev.

3A 18.15 4.00 77 5 30

3B 20.10 7.96 65 1 18

3C 40.14 8.31 153 5 12.5

and is equal to the probability thatn points at positionsx1,x2, ...,xn are found in phasei, where〈(•)〉denotes an ensemble average, i.e., an average over all realizationsωof the ensemble.

This function can also be referred to as ann-point correlation function.

When then-point probability functionS_n⁽ⁱ⁾depends on the absolute positionsx1,x2, . . . ,xn, the medium is said to be statistically inhomogeneous. Assuming that the medium is statistically homogeneous and isotropic, the n-point correlation function depends only on the relative distances between points. Adopting the notation xj k = |xj k|and xj k =xk−xj, the 2-point correlation function can then be written as

S⁽ⁱ₂⁾(x1,x2)=S⁽ⁱ⁾₂ (x12). (3.29) When the system is statistically homogeneous, it is meaningful to define volume averages.

This suggests an ergodic hypothesis, i.e., the result of averaging over all realizations of the ensemble is equivalent to averaging over the volume for one realization in the infinite-volume limit. Thus, complete probabilistic information can be obtained from a single realization of the infinite medium, and ensemble averaging can be replaced with volume averaging when the volume tends to infinity, i.e.,

S⁽ⁱ_n⁾(x12,...,x1n)= lim

V→∞

1 V Z

V I⁽ⁱ⁾(y)I⁽ⁱ⁾(y+x12)···I⁽ⁱ⁾(y+x1n)dy. (3.30) It is possible to establish asymptotic properties ofS_n⁽ⁱ⁾and associated bounds that apply to any statistically inhomogeneous two-phase random medium. Forn=2, one has

xlim12→0S⁽ⁱ₂⁾(x12)=φi, lim

x12→∞S⁽ⁱ₂⁾(x12)=φ²_i, (3.31) whereφ_i denotes the volume fraction of phasei.

The geometrical-probabilistic significance of then-point probability function can be easily comprehended for a given microstructure. For statistically isotropic media, S⁽ⁱ⁾₂ may be interpreted as the probability that the two vertices of a line segment lie inVi when the line segment is randomly placed in the material domain.

Nearest neighbor function

The aforementioned statistical functions are defined for random media of arbitrary microstructure. However, statistical descriptors may be formulated to characterize specific types of microstructures, e.g., an isotropic media composed of identical spherical particles of radius,R, at number densityρ⁴, embedded in another phase.

One example is the so-called nearest neighbor function, denoted asHP. Despite multiple definitions, the one adopted here states that HP(r)dr is the probability that, at an arbitrary particle center in the system, the center of the nearest particle lies at a distance between r andr+dr. Another similar function, used in Pathan et al. (2017) and here denoted as H_P^∗, is defined such that H_P^∗(θ)dθis the probability that, at an arbitrary point particle center in the system, the center of the nearest particle lies at an angle betweenθandθ+dθ. BothHP(r) and H_P^∗(θ) are probability density functions, nonnegative for allr andθand normalize to unity. For statistically inhomogeneous media, both functions will also depend upon the location of the central particle.

Ripley’sK function

Another widely used function in the description of point distributions is called Ripley’s K function,K(r) (Buryachenko et al. (2003), Melro (2011), Pathan et al. (2017)). Also known as the

4The number densityρis defined as the number of particle centers per unit volume.

second-order intensity function, it is a tool used to analyze completely mapped spatial point process data. For the particular application envisaged here, the points are the coordinates of the particles’ center-of-mass.

According to Dixon (2014), Ripley’sK function is defined as the number of points expected to be located within a distancer of an arbitrary point divided by the number of points per unit area. It can be defined as

K(r)=ρ⁻¹E, (3.32)

whereρis the density of points,r is the distance, andEis the number of extra points within a distancer of the randomly chosen point. Given the locations of all events within a defined area of interest,Ω, whose area is denoted asA, the density of points can be estimated asρb=N/A, whereN is the observed number of points. Accordingly, an estimator forK(r) is given by

Kb(r)= A N²

X

i

X

j6=i

I(di j≤r)

w(li,lj) , (3.33)

whereAis the area of the region of interest,Nis the total number of particles,di jis the distance between pointsiandj, andI(x) is an indicator function having the value 1 if the conditionxis true and 0 otherwise.

The weight function, w(l_i,l_j), provides an edge correction. To see the need for this correction term, consider some pointi and a given distancer. Summing the functionI(d_{i j}≤r) over all points excepti, the result is the number of points at a distance smaller or equal tor fromi. When the pointi is located near the boundary ofΩ, there will be fewer points than expected at a distanceror smaller fromi, as a significant part of the circle with center ati is outside the area of interestΩ. Thus, if part of the circle falls outside ofΩ, thenw(li,lj) is the fraction of the circumference of that circle that falls withinΩ. It has the value 1 when the circle is centered at li and passing through the point lj, i.e. it has a radius equal to di j and is completely inside the areaΩ. The effects of edge corrections are more important for larger because large circles are more likely to be outside the region of interest. For this reason, it is common practice to consider only values ofrless than one-half the shortest dimension ofΩ.

The information it provides about the medium is different from the one given by the nearest neighbor functions. While the latter only provides information about the short-range interaction between the particles, Ripley’s K function also provides some insight into the microstructure at distances beyond the vicinity of the particles.

For a homogeneous Poisson process, also known as complete spatial randomness, it can be shown that

K(r)=πr². (3.34)

If the spatial distribution under analysis provides a plot ofKb(r) below the Poisson curve, then it is likely that the microstructure exhibits some degree of regularity. On the contrary, ifKb(r) is above the Poisson curve, the distribution is likely to present clustering. A stair-like pattern for Kb(r) points to a regular pattern. The difference between a Poisson distribution and the actual spatial distribution in the region of interestΩcan be conveniently studied using an alternative function,L(rb ), defined as

L(rb )= sKb(r)

π −r. (3.35)

In this case, peaks of positive values in a plot ofL(rb ) would indicate clustering while negative troughs indicate regularity at distancer.

Comparison: 2D microstructure

In this section is performed a comparison between a real microstructure and the corresponding computationally generated microstructure through AMINO.

The real microstructure is a carbon-fiber-reinforced metallic composite taken from Kim et al.

(2001) (see Figure 3.34(a)). The original micrograph is post-processed using the open-source software ImageJ (Schneider et al., 2012) (see Figure 3.34(b) and 3.34(c)), being the extracted particle volume fraction approximately 67%.

Using the ellipses fitted by ImageJ to determine the geometrical descriptors, the particle phase is modeled as containing ellipses whose major axis follows a normal distribution N(0.0526,0.0027²). The ratio between the major and minor axes follows a normal distribution N(1.2029,0.0887) and the angle that the ellipses make with positivexsemi-axis follows another normal distributionN(1.6386,0.9360) rad. The computational microstructure is normalized to have a unitary side length. Five realizations of this microstructure are generated through AMINO and shown in Figures 3.34(d)-3.34(h). The average CPU time, the number of particles, and the corresponding standard deviations are shown in Table 3.5 for 10 realizations.

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 3.34: Comparison between real and computationally generated two-dimensional microstructures through AMINO: (a) Micrograph of a carbon-fiber-reinforced metallic composite taken from Kim et al. (2001); (b)(c) post-process of micrograph using the open-source software ImageJ (Schneider et al., 2012); (d)-(h) computationally generated microstructure.

Table 3.5: Average CPU time, number of particles and corresponding standard deviations (over 10 realizations of each microstructure), as well as, total volume fraction of the particle phase, in the generation of a legal configuration (Ψlim=0).

CPU Time (s) Number of particles Total volume fraction of particles (%) Mean Std. Dev. Mean Std. Dev.

49.29 5.04 369 1 67

0 2 4 6 8 10 r/R

−0.020

−0.015

−0.010

−0.005 0.000 0.005 0.010 0.015 0.020

L(r)

Sample #1 Sample #2 Sample #3

Sample #4 Sample #5 Real

(a)

1.0 1.5 2.0 2.5

r/R 0.00

0.05 0.10 0.15 0.20 0.25 0.30

Hp(r)

Sample #1 Sample #2 Sample #3

Sample #4 Sample #5 Real

(b)

0 2 4 6 8

r/R 0.40

0.45 0.50 0.55 0.60 0.65 0.70

S(1) 2

(r)

Sample #1 Sample #2 Sample #3

Sample #4 Sample #5 Real

(c)

Figure 3.35: Statistical comparison between real and computationally generated microstructures (see Figure 3.34): (a) Difference between Ripley’sK function and a Poisson point process; (b) Nearest neighbor function; (c) 2-point probability function. Ris defined as the mean major semi-axis of the ellipses in the real micrograph.

In Figure 3.35, the previously introduced statistical descriptors are employed to compare the real and computationally generated microstructures. An excellent agreement between real and computational statistics supports the ability of AMINO to generate high-fidelity microstructures.

Comparison: 3D microstructure

In this section the previous comparison is extended to the three-dimensional case in some sense, being attempted to generate a high-fidelity 3D computational microstructure from a cross-sectional micrograph.

The real microstructure is a PC/ABS polymer blend, whose micrograph is taken from Dong et al. (1993) (see Figure 3.36(a)). In this case, the particle phase corresponds to ABS ellipsoidal domains with a total volume fraction equal to 30%. It is also important to emphasize that the micrograph cut plane is perpendicular to the injection molding direction, being the main elongation of the ABS particles therefore hidden (aligned with the direction perpendicular to the paper). As in the previous analyses, the original micrograph is post-processed using the open-source software ImageJ (Schneider et al., 2012) (see Figures 3.36(b) and 3.36(c)). However, the accurate extraction of 3D microstructure descriptors from 2D micrographs is quite challenging in itself. On the one hand, the morphology observed is strongly dependent on the orientation angle and location of the plane cut (e.g., Bärwinkel et al. (2016)). On the other hand, only the apparent values of the particles’ geometrical parameters and associated distributions can be directly extracted from the micrograph. For instance, the ABS volume fraction extracted from the post-processing of the micrograph shown in Figure 3.36(a) is equal to 26%, a significant difference relative to the actual volume fraction of 30%.

Given that overcoming these challenges is out of the scope of the present work, a simple modeling approach is assumed here. In the first place, given that the ABS domains are known to have approximately an ellipsoidal shape, the associated cross-sections observed in the 2D micrograph are approximated to ellipses in the post-processing (see Figure 3.36(c)). Pertaining the known injection molding direction, the ellipsoidal smallest semi-axes, a2 and a3, are assumed to be in the plane of the micrograph, and the largest semi-axis,a1, oriented along the out-of-the-plane direction. The dimensions a2 and a3 are sampled directly from the joint experimental probability distribution that is found by fitting the ellipses to the particles found in the micrograph. Concerning the hidden dimension,a₁, it is assumed that the ratioa₁/a₂is equal to the apparent ratioa₂/a₃, being here considered thata₁≥a₂≥a₃.

The computational microstructure is normalized to have a unitary side length and ten microstructures are generated using the proposed approach. Four equally-spaced cross-sections of one of the generated microstructures are shown in Figures 3.36(e)-3.36(g).

Figure 3.37 presents the full 3D microstructure and the aforementioned cross-sections. The average CPU time, the number of particles, and the corresponding standard deviations are presented in Table 3.6.

Table 3.6: Average CPU time, number of particles and corresponding standard deviations (over 10 realizations of each microstructure), as well as the total volume fraction of the particle phase, in the generation of a legal configuration (Ψ_lim=0).

CPU Time (s) Number of particles Total volume fraction of particles (%) Mean Std. Dev. Mean Std. Dev.

1089.85 144.37 990 65 30

(a) (b) (c)

(d) (e) (f) (g)

Figure 3.36: Comparison between real and computationally generated two-dimensional microstructures through AMINO: (a) Micrograph of a carbon-fiber reinforced metallic composite taken from Kim et al. (2001); (b)(c) post-process of micrograph using the open-source software ImageJ (Schneider et al., 2012); (d)(g) computationally generated microstructures.

(a) (b)

(c) (d)

Figure 3.37: 3D computationally generated microstructure with four equally-spaced cross-sections matching Figure 3.36: (a) section atz=1; (b) section atz=0.8; (c) section atz=0.6;

(d) section atz=0.4. It is assumed thatz is the injection molding direction.

0.0 0.5 1.0 1.5 2.0 r/R

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

S(1) 2

(r)

Section #1 Section #2 Section #3

Section #4 Section #5 Real

Figure 3.38: Comparison between the 2-point correlation function of the real micrograph and several cross-sections of a three-dimensional computational generated microstructure (see Figure 3.36). R denotes the average length of the largest axis of the elliptical cross-sections in the real micrograph.

Figure 3.38 presents a comparison between the 2-point correlation functions describing the real micrograph and the several cross-sections of the three-dimensional computational microstructure. Given the uncertainty associated with the apparent microstructure descriptors and the simplicity of the adopted approach, a reasonable agreement is obtained between experimental and computational results. According to Wu et al. (2018), the area fraction extracted from a representative micrograph can be an acceptable estimate of the actual volume fraction. Moreover, the 2-point correlation function is expected to yield the area fraction for r =0. Attending to the fact that the real ABS volume fraction is known to be 30%, it can be concluded that the 2-point correlation function of the real micrograph is actually a lower bound, yielding approximately 26% for r = 0 together with one of the computational cross-sections (section #3). The 2-point correlation function of the remaining cross-sections yield values closer to 30% forr=0, as expected.