A probability characteristic of crack intersecting with embedded microcapsules in capsule-based self-healing materials

1 Introduction

Various properties of composite materials will degrade under the coupling of loads and environmental factors over time. During their service, the plastic shrinkage, mechanical loading, and thermal effects will lead to deterioration and then micro/macro-cracks will appear in the material, resulting in a reduction in the load-carrying capacity and durability of the structural elements [1,2]. Cracks, whether self-generated or loading-induced, are generally difficult to detect and repair. Especially in the early stages, the internal cracks to be repaired or sealed in composite materials in service are therefore particularly necessary and urgent [3]. Inspired by the self-healing properties of living organisms, scientists began to study self-healing composite materials to improve the service life, stability, and safety of materials throughout their use [4]. Researchers have conducted a lot of experimental research work to figure out the mechanism and healing efficiency of self-healing composite materials and have divided the self-healing strategies of self-healing composites into two types [5]: natural healing and engineered healing. Natural healing is based on the material’s potential, and engineered healing is divided into two categories: automatic healing and active repairing.

White et al. [6] first proposed a structural polymer material with autonomous capsule-based crack healing capability and obtained up to 75% toughness recovery by fracture experiments, which inspired other researchers working in composite materials to design capsule-based self-healing systems (including ceramics, cementitious, and glass). The development of a self-healing system can effectively improve the reliability and longevity of materials, which can greatly reduce the loss of raw materials as well as the increase in costs, thus achieving a sustainable society [7]. Researchers have recently designed and developed a variety of new self-healing composites such as bio-based self-healing materials [8], elongated fibers filled with glue [9], spherical capsules filled with healing agents [10], self-healing microencapsulated coating materials [11], and hollow polypropylene fibers with methyl methacrylate as the restorative agent [12]. Microencapsulation or microencapsulation technology is a method in which a monomer or other substance with a specific function, separated from the outside world, is encapsulated as a core material in a tiny container. Self-healing composites with encapsulated healing agents have developed rapidly in the last decades. Compared to other self-healing technologies such as glass tubes, self-healing therapies based on microencapsulated self-healing agents are more likely to be commercialized soon [13]. However, more attention should be paid to the design (i.e., self-healing efficiency), preparation, and characterization of microencapsulation technology to promote the large-scale application of self-healing materials.

Many researchers have experimentally investigated the capsule-based self-healing properties of composite materials. Qian et al. [14] explored the construction technology of microbial self-healing concrete through engineering practice, indicating that the spray-dried fermentation bacterium method has great potential for the production of powder-based and capsule-based microbial healing agents for concrete. Zhang and Qian [15] designed and prepared a new core-shell carrier to prolong the survival time of microorganisms in concrete. Han et al. [16] found that specimens would be damaged by secondary extension of unhealed microcracks rather than repaired microcracks. Qian et al. [17] measured the variation in Ca²⁺ and CO 3 2 − in the crack zone with concentration and cumulative molar amount over time and performed theoretical calculations of the crack healing depth. In addition to the material properties, the dosage and size of the microcapsules embedded also affect the crack healing of the matrix. Wu et al. [18] found that the addition of bio-capsules at 5% of the cement mass could lead to the complete healing of cracks 150–550 µm wide. In addition, compared with reference samples, the permeability of samples containing bio-capsules was reduced by about two orders of magnitude. Kosarli et al. [19] directly correlated the capsule size with the reduction in healing efficiency and mechanical properties after the combination of self-healing systems and found that the healing efficiency was proportional to the capsule size, and the larger capsule could achieve 68% of the maximum load recovery so that an optimal capsule size exists.

The healing efficiency of capsule-based self-healing cementitious materials has been extensively studied by modelling methods and numerical simulations. The probability of cracks intersecting with capsules in a capsule-based self-healing system is of concern. When the intersection probability increases, the probability of capsules being ruptured and the efficiency of the healing agent released from the capsules will increase accordingly. Therefore, in addition to the traditional investigation of self-healing properties and mechanical characterization, the probability of cracks intersecting with capsules will help to design the self-healing material system with a higher self-healing efficiency [20]. Of course, this can be addressed experimentally, but it is expensive and takes more time to manufacture and test these materials, and it is difficult to progress by limiting the field of knowledge to experimental studies [21]. Therefore, if the various scenarios of cracks intersecting with microcapsules are simulated and analyzed, it is possible to determine the properties of microcapsule-based self-healing composites under many different conditions. To evaluate the efficiency of capsule-based self-healing cementitious materials, Fang et al. [22] derived analytical models considering the effect of capsule shell thickness for three common capsule shapes: spherical (SPH), cylindrical (CYL), and cylindrical with a spherical tip (CST). Pan and Gencturk [23] proposed engineered aggregates (EA) which have an internal brittle capsule carrying the healing agents and are covered with a cementitious coating and geometric probability methods were used to determine the probability of the intersection of cracks with different shapes with EA. For a simulation study of a given density and type of crack, the optimal shape and size of EA that can be effectively used for self-healing concrete are determined while the volume fraction of EA is minimized.

The fundamental requirement for the self-healing process to work is that cracks need to meet the microcapsules and cause it to rupture. Besides the material properties, the size and volume fraction of microcapsules influence crack healing in the matrix. Understanding the crack and microcapsule interaction is critical in the development and design of structures made of self-healing materials. An important issue of self-healing efficiency in many studies is what the optimal parameters (size, dosage, and shape) of capsules containing healing agents must be to achieve the optimal self-healing effect. Zhang et al. [24] developed a probability density function model for different self-healing particles considering the geometric characteristics of cracks and validated it by large-scale sample testing. The model accurately evaluated and predicted the probability distribution characteristics of the healing agent particles on the crack surface. To guarantee the crack is well healed, a theoretical method for calculating the dosage of self-healing particles was also developed at a 95% confidence level. Zemskov et al. [25] considered two two-dimensional analytical models to calculate the probability of crack meeting the encapsulated particles and found that a completely random distribution required the placement of 20% fewer capsules compared to a stratified random distribution. However, for the first model with stratified capsule placement, it is difficult to calculate the hit probability of stratified capsule placement when the number of layers is large. To determine the exact dosage of capsules required to repair cracks, Lv et al. [26] proposed a theoretical solution for the exact dosage of capsules containing healing agents to be broken for different crack patterns via surveying the irregular crack patterns occurring on the composite surface and verified the accuracy of these probability values and the theoretical solutions via Monte Carlo simulations. Further, based on the given relationship between the length of elongated capsules and the average spacing of adjacent cracks, Lv et al. [27,28] obtained analytical solutions for the exact number of capsules required for different types of crack patterns, and verified the reliability of these analytical models by computer simulations. Katoueizadeh et al. [21] presented a novel model for simulating the characteristics of a micro-encapsulated composite and studied the probability of crack incidence for a different number of capsules to be intersected. However, the micro-encapsulated composite matrix was assumed to be segmented into a series of reproducible cells, each containing a microcapsule, and it was unrealistic and idealistic. Meanwhile, it is feasible that theoretical models and computer simulations on the cracks intersecting with capsules can be combined to predict the optimal parameters (size and dosage) of microcapsules of self-healing composites.

The engineering use of microcapsules technology in the design of self-healing materials is limited due to the difficulty of achieving the desired restorative effect with current microencapsulation technology and self-healing composites. One of the challenges is the agglomeration and an uneven distribution of microcapsule particles, which lead to reduce the possibility of the microcapsules to be ruptured and restrict the self-healing ability of microcapsules [29]. Further, the efficacy of self-healing materials in this region will be decreased by the agglomeration of microcapsules in the matrix. In this article, a two-dimensional micro-encapsulated model composite matrix is considered to be composed of reproducible and random cells and some microcapsules are randomly distributed in some cells. Such a model can overcome the agglomeration and inhomogeneous distribution of microcapsules. The qualitative characterization of a common crack that appeared on the surface of the material matrix, such as cementitious paste and polymer composites, intersecting with embedded microcapsules is investigated. A geometrical probability characteristic of the linear crack intersecting with microcapsules is developed and the analytical solution of the intersecting probability is obtained to express the self-healing efficiency of capsules embedded in terms of considering the dosage and size of microcapsules in the capsule-based self-healing composites. Meanwhile, the results of the theoretical model were verified via Monte Carlo simulation.

2 Binomial probability distribution

In general, the mass of capsules embedded in the matrix is relatively low and may be less than 5% of the total matrix mass [30]. In addition, capsule particles at very low volume fractions will mainly exhibit “randomness” [31]. Here it is generally assumed that the capsule particles are separate and randomly dispersed in the matrix material, and therefore a two-dimensional planar self-healing model composite is considered in which the embedded capsules may collide with the cracks. The extension of a crack after placing the capsules into the matrix will lead to two consequences. One is that the capsule contacts the crack, and the other is that the capsule does not contact the crack. Although the actual interrelationship between the capsule and the crack in the matrix is very complex, the capsules and the crack are assumed to be independent of each other in the development of the analytical model because each cell has adhered to at most one capsule. Therefore, independence is well-defined in the model. Based on this assumption, the event of the intersection between embedded microcapsules in the cell and a linear crack will follow a probability distribution, i.e., a binomial probability distribution [32]. The characteristics of the binomial distribution are as follows.

1. The experiment consists of X identical trials. Each trial results in one of two outcomes. For lack of a better name, one outcome is called a success, S, and the other a failure, F.

2. The trials are independent. The probability of success on a single trial is equal to p and remains the same from trial to trial. The probability of failure is equal to 1 − p.

3. We are interested in the number of successes X _i observed during the X identical trials, for X _i = 0, 1, 2..., X.

When the trials (Bernoulli trials in X independent replications) come from a large range, the success probability p does not change essentially across repeated experiments. When the number of trials is small, the probability of success p varies considerably between trials and does not conform to the binomial distribution. The binomial distribution is a discrete random distribution of the number of successes in binomial trials, each with a probability of success p; the probability of failure is 1 – p. In addition, when the random variable X _i indicating the number of successes in X trials conforms to the binomial distribution, then the mean value of X _i is expressed as follows:

The binomial probability distribution theory is employed here to develop a probabilistic model for the random occurrence of a crack encountering microcapsules in the two-dimensional capsule-based self-healing composite material.

3 Analytical characteristics of a crack intersecting with microcapsules

3.1 Crack pattern

Various crack forms may appear on the surface or inside the composite matrix under its service life due to the influence of external loads and environmental factors. In terms of appearance, some cracks are linear or quasi linear on the whole. Taking the cementitious material as an example, Figure 1(a) shows a quasi-parallel linear crack pattern on the surface of cementitious material and Figure 1(b) shows a common pattern of parallel cracks in bending concrete members. So, in this study, the crack pattern on the surface of material is considered to be linear crack pattern and crack width is negligible at the beginning of the crack.

Figure 1

Schematic of representative domain for parallel linear crack pattern: (a) orientational cracks pattern [33]; (b) crack pattern in reinforced concrete and fiber reinforced concrete elements subjected to tension [34].

3.2 Fracture surface of capsule-based self-healing composite

The fracture surface of epoxy resin containing randomly dispersed spherical polymer particles was experimentally examined to predict the self-healing efficiency of capsules by scanning electron microscopy (SEM) [35]. Figure 2(a) shows a self-healing composite material where the spherical capsules were randomly dispersed on the fracture surface. When the composite is subjected to loading and environmental factors, the microcracks appear, grow, and potentially meet the capsules, a necessary condition for self-healing working and healing agent releasing. Inspired by the dispersed capsules of the fracture surface, a two-dimensional material sampling region is partitioned into the mosaic cells and some microcapsules with healing agents randomly distributed in the cells. In other words, some cells have microcapsules and some do not. So, what is the intersection probability of randomly dispersed capsules with an emerging individual crack in the sampling region? In Section 3.3, a two-dimensional analytical model will be developed to formulate the intersection of cracks with randomly dispersed microcapsules and the process to calculate the intersection probability of randomly dispersed microcapsules in the matrix with linear cracks is shown.

Figure 2

Dispersed spherical capsules are distributed in the fracture surface: (a) experimental investigation (SEM) of a fracture surface of dispersed spherical capsules in self-healing composite material [35]; (b) the two-dimensional sampling region is partitioned into the cells and some microcapsules with the healing agent are randomly distributed in the cells.

Figure 3(a) shows an example of a single equivalent unit cell, including a microcapsule. In this schematic diagram, it can be seen that the microcapsule is located in the center of the small cell, and it is imagined that the three-dimensional self-healing composite is made up of many randomly distributed cubic cells of equivalent dimension L. Meanwhile, some microcapsules are randomly dispersed in the cells. That is, single equivalent cells of dimension L each with or without a microcapsule are shown in Figure 3(b). Similarly, the two-dimensional capsule-based self-healing composite can be developed as illustrated in Figure 2(b).

Figure 3

Fragmented self-healing composite embedded microcapsules with the healing agent: (a) example of a single equivalent unit; (b) single equivalent cells of dimension L with or without a single capsule.

3.3 Probability of a crack intersecting with microcapsules

Rule et al. [36] proposed an equation to estimate the number of capsules by simply dividing the total mass of the capsules by the approximate mass of the individual capsules. Normally, in the design and experimentation of self-healing composite materials, the dosage of microcapsules is often calibrated by mass. Nevertheless, for the convenience of calculation, the volume fraction and individual volume of microcapsules, as well as the number of microcapsule particles, were utilized as model parameters in the model development. These parameters can be combined with the density of the microcapsule to gain the mass of the incorporated microcapsules. The particle size of microcapsules ranges from nanometers to microns and even millimeters. For different types of microcapsules, there are different production techniques and processes, which determine the particle size of the microcapsules. Recent tests have shown that the radius of the capsule ranges from 550 to 3,600 μm [37]. At the same time, although the incorporated microcapsules also have multiple sizes, the average size of particles is selected as the model parameter. The influences of microcapsule size on the properties of composites had been reviewed from the viewpoint of nonlocal elasticity, strain gradient elasticity, or molecular dynamics [37,38,39,40].

The sampling region is divided into many equivalent cubic cells of dimension L with at most one capsule in each cell and those cells are irregularly and compactly distributed on the surface of the matrix and there are N cells along the edge of the specimen, as shown in Figure 2(b). The irregular and compact distribution of cells means that cells in each column are randomly dislocated in [0, L]. This irregularity can also lead to the random distribution of microcapsules on the surface of matrix and ensure the independence of the intersection of crack with microcapsules. Apparently, in the two-dimensional theoretical model, the size of the composite specimen is equal to N * L and there are N ² cells. This irregularity leads to the conclusion that the intersection probability can be calculated based on the following facts: for each microcapsule in the equivalent cell, the crack has individual intersection probability, as illustrated in Figure 4. A linear crack can be defined as a crack of negligible thickness compared to other dimensions. When a crack appears and grows on the surface of the matrix, it may intersect more than one microcapsule, which is directly correlated with the dosage and size of microcapsules. So, a geometrical probability model is developed in this section for the randomly distributed microcapsules, which gives an analytical model for the intersection of microcapsules with a liner crack in self-healing composites.

(2) S t = N 2 · L 2 ,

(3) L = S t N 2 2 ,

where S _t is the total area of the sampling region.

Figure 4

Schematic representation of whether the crack intersects the microcapsule in the equivalent cell.

The independence of the event of a crack hitting the microcapsules is well-defined in the model due to the randomness of the cells. In the model development, the thickness of the microcapsule wall is negligible, so the volume of the healing agent released from the microcapsules is equal to the volume of the microcapsules. Based on this assumption, the intersection or non-intersection between the microcapsule and the linear crack will follow a binomial probability distribution. As shown in Figure 4, the linear crack randomly passing through the equivalent cell region would have only two possible outcomes: success (Crack 2) or failure (Crack 1). Mathematically, if a linear crack meets the microcapsule, then the distance from the center of the microcapsule to the crack is less than or equal to the radius of the microcapsule. Otherwise, the crack will not hit the microcapsule. Figure 4 illustrates the “success” or “failure” outcomes whether a crack passing an equivalent cell intersects a capsule or not. Specifically, crack plane 1 represents “failure” outcome while crack plane 2 hits the capsule with “success.” For each microcapsule, an intersection occurs if and only if the crack position satisfies y ∈ [L/2 – D/2, L/2 + D/2], where D is the capsule diameter. As y ∈ [0, L], the intersecting probability for a given microcapsule can be determined as follows:

(4) p 1 = D L .

As shown in Figure 2(b), the sampling region of the matrix can be divided into N ² small cells, and these cells are randomly and tightly arranged and combined. Now assume that in such a sampling region of capsule-based self-healing composite, there are n (n < N ²) microcapsules randomly placed inside these cells and the center of the cell and the microcapsule is coincident. In other words, if n microcapsules are randomly dispersed in the N ² cells, the probability that each cell has a microcapsule is

(5) p 2 = n N 2 .

Thus, the probability that a linear crack passes through a microcapsule in the sampling region is

(6) p 3 = D L · n N 2 .

In probability theory, the binomial distribution is the probability of the number of occurrences of the desired outcome in X independent Bernoulli trials with only two outcomes, where the probability of success per trial is p. In the microcapsule model, the outcome of each intersection of a random linear crack with a microcapsule is independent. The probability that a random linear crack passes through the sampling region and intersects the microcapsule is the outcome p ₃ (equivalent to the probability of success per trial p in the binomial distribution). The other case is 1 – p ₃. So the probability of a randomly occurring crack hitting "m" capsules in Figure 2(b) is consistent with the definition of Bernoulli's test. Then, the probability P(m) of a crack intersecting exactly m microcapsules is calculated as shown in equation (7).

(7) P ( m ) = N ! ( N − m ) ! m ! ( p 3 ) m ( 1 − p 3 ) N − m = N ! ( N − m ) ! m ! D L · n N 2 m 1 − D L · n N 2 N − m ,   m < N .

Similar to the previous equation, the probability of a crack intersecting at least m capsules can be calculated by adding up the probabilities of P(m) and greater than m. Thus, the cumulative probability of a crack intersecting at least m microcapsules is calculated as follows:

(8) P = ∑ i = m N N ! ( N − i ) ! i ! ( p 3 ) i ( 1 − p 3 ) N − i = ∑ i = m N N ! ( N − i ) ! i ! D L · n N 2 i 1 − D L · n N 2 N − i ,   m < N .

Equation (8) expresses the cumulative intersecting probability of a crack intersecting at least m microcapsules, which can be compared with the average probability obtained from the later simulations. Specifically, the average number of microcapsules to be intersected in the simulation is compared with the calculated number of microcapsules at approximately 50% of the cumulative probability of occurrence.

4 Verification via Monte Carlo simulation

Experimental research is necessary to verify the proposed theoretical model and results and bond the theoretical analysis to be consistent with the actual phenomenon. In this study, both the appearance of capsules and the patterns of cracks are simplified to present a novel two-dimensional capsule-based self-healing model composite material whose surface is paved by reproducible and random cells and some microcapsules are randomly dispersed in those cells to investigate the rupture behavior of capsules forced by growing cracks. For the proposed theoretical model, it is difficult to experimentally compare with the analytical solution because of the artificial design of the microcapsules embedded in the cells of the self-healing system. Generally speaking, the size and compatibility of the capsules, the healing agent in the capsules, and the cracks induced in the sample all lead to a large gap between the experimental results and the theoretical solution. On the one hand, the complexity and uncertainty of quantitative experimental studies of the required dosage and size of microcapsules is a great challenge. On the other hand, significant discrepancies between the experimental results and the proposed analytical solutions are expected. Even though some exploratory experiments have been carried out to verify our models, those have just proven our anticipation. In addition, limiting the field of knowledge to experimental studies limits the applicability of self-healing materials. Hence, it is highly expensive to verify the intersection probability between cracks and microcapsules in a capsule-based self-healing system using experimental methods, such as X-ray computer tomography [41].

It is necessary to verify the accuracy of the above theoretical model before putting it into an application. Computer simulation may be a very popular solution in the field of materials science research to validate the proposed model. In fact, the development of mathematical models and using numerical simulations are two strategies to study the quantitative relationship of a crack intersection with capsules in self-healing materials, respectively. Specifically, for our work, analytical model on the intersection relationship was developed from the perspective of geometric probability, while computer simulation using the random numbers (or pseudo-random numbers) to reconstruct the growth of crack and the distribution of capsules in a virtual sampling region was implemented via Monte Carlo simulation. In other words, computer simulation is a numerical test methodology. Therefore, the accuracy of theoretical analysis and results can be verified by numerical experiments.

Monte Carlo simulation idea is used to develop the corresponding computer simulation algorithm program to simulate the relationship between the spatial location of cracks and microcapsules in the matrix and the number of intersections where the crack meets the capsules. The accuracy and reliability of the probability of crack intersection with capsules of different sizes will be verified. Monte Carlo method is based on a probabilistic model, and the results obtained through simulation experiments, according to the process depicted in this model, are used as approximate solutions to the problem. In simple terms, it means that the probability calculation method in computer simulation is independent of that in the analytical model probability formula. Therefore, this section begins with a computer modelling technique to simulate the probability of the intersection of the randomly dispersed capsules with an emerging linear crack. The following assumptions are made to verify the accuracy of the previously proposed model:

1. Each cell is square and has a dimension of L.

2. The microcapsules are disc-shaped and independent.

3. Once a microcapsule with diameter D is embedded in the cell, its center was assumed to be coincided with the center of the cell.

4. The number of microcapsules is given and the positions of microcapsules are randomly fixed and the linear cracks appear randomly.

5. The sampling region of the composite matrix is divided into multiple cells adjacent to each other.

6. The arrangement of cells is random and unrelated to the adjacent cells.

4.2 Results and discussion

The program is set to K = 1,000 executions to calculate the simulated intersection probability, and the theoretical probability values P(m) from equation (7) are verified by comparing them with the results obtained independently through computer programming simulations. The simulated parameters N = 20, D/L = 0.3, and n = 220 are set in the model. As shown in Figure 5, the orange bars illustrate K(m) after the program has been executed K times, and the number of intersections is m each time and the plotting curve presents the theoretical values for the given variable values. It can be seen that the binomial distribution graph is tending to be right-skewed due to p ₃ = 0.165, which is less than 0.5. The curve gradually approximates a normal distribution as the sampling area (NL)² increases. Both the simulated and theoretical results reach their maximum when m = 3. At this point, the theoretical value P(3) = 0.2388 and the simulated probability P ₁(3) = 0.246 from equation (9). Figure 5 shows a good agreement between the theoretical values and the simulations data.

Figure 5

Comparison of simulation results with theoretical values for the intersection probability.

Figure 6 shows the convergence of the simulated values of intersection number of microcapsules for different D/L. The curves indicate the average number of intersections obtained after K executions of the program. After the executions of the simulated program, the average values m of a crack intersecting with microcapsules eventually approximates the theoretical values obtained from equation (1). By the way, the representative volume element (RVE) used in the program exists the boundary rigidity since there is no way to realize the periodic boundary in the MATLAB simulation program. Therefore, in order to eliminate the influence of this boundary, the crack range to a certain distance from the boundary is used to reflect the periodicity and homogeneity of this material structure. After reducing the boundary of RVE by 0.5, the executions were performed 1,000 times and found that when D/L = 0.3, m = 4.812, the expected intersection number calculated by equation (1) is 4.8; when D/L = 0.7, m = 11.036, the expected intersection number calculated by equation (1) is 11.1434. The two results are consistent.

Figure 6

Convergence of the simulated intersection number m for different D/L.

Table 1 presents the theoretical probability values for N ² = 400, L = 1, and m = 6. As shown in Figure 7, the vertical line represents the simulated value, indicating the average number of intersections after 1,000 iterations. The curves in Figure 7 show the theoretical probability of the intersection of the crack with m microcapsules for a given N, n, and D/L. All the curves have a peak point, and it is known from probability theory that the highest peak in the binomial distribution plot is in the vicinity of m = N * p ₃ (i.e., equation (1)). The two results are compared and analyzed, and it is found that the vertical line and the curve peak point results are close. Hence, the accuracy of the theoretical model can be well verified. As the D/L increases, the number m of intersections between microcapsules and cracks increases when the hitting probabilities reach the maximum as shown in Figure 7(a). Such tendency is also true for the dosage of microcapsules embedded as shown in Figure 7(b). Equation (7) can analytically express the influence of the size and dosage of microcapsules embedded on the probability of a crack hitting the microcapsules.

Table 1

Theoretical probability table for N ² = 400, L = 1, and m =6

n	D = 0.1	D = 0.3	D = 0.5	D = 0.7	D = 0.9
	P(m)
400	0.0089	0.1916	0.037	0.0002	0
350	0.0048	0.1786	0.0863	0.0035	0
300	0.0023	0.1418	0.1496	0.0242	0.0005
320	0.0032	0.1589	0.1244	0.0122	0.0001
280	0.0017	0.1226	0.1712	0.0432	0.0022
250	0.0009	0.092	0.1902	0.0863	0.0116
220	0.0005	0.0626	0.1858	0.1398	0.04
200	0.0003	0.0454	0.1686	0.1712	0.0746

Figure 7

Comparison of the average intersection number obtained from the simulation (vertical line) with the theoretical equation (curve): (a) for various D/L; (b) for various n; and (c) for various N ².

For a given dosage of microcapsules, if a crack requires a fixed number of microcapsules to heal, then the intersection probability can be used to determine the capsule size. As shown in Table 2, the theoretical probability values for the intersection of randomly occurring cracks with the capsule (the number of intersections is 6) are calculated. The theoretical probability value reaches its maximum when the capsule diameter is 0.3. In agreement with the trend in Figure 7(a), the maximum probability value is found at the probability curve at D/L = 0.3 when the number of intersections is 3–6. In this case, microcapsules with a diameter of 0.3 are preferred. Furthermore, the number of microcapsules intersected by cracks also increases with the increase in sampling area (NL)², but the maximum intersecting probability will reduce as shown in Figure 7(c).

Table 2

Geometrical parameters of microcapsules in the simulated self-healing system

Number of cells (N 2)	Cell side length (L)	Capsule diameter (D)	Embedded number of microcapsules (n)	Default intersection number of microcapsules (m)	Theoretical probability (P(m))
400	1	0.1	320	6	0.0032
		0.3	320		0.1589
		0.5	320		0.1244
		0.7	320		0.0122
		0.9	320		0.0001

The cumulative probability distribution (curve) and the average number of intersections obtained from 1,000 simulations (vertical line) are shown in Figure 8. The theoretical calculations using equation (8) for different values of D/L and n, respectively, as well as the average number of intersections obtained from the simulations are plotted in Figure 8. It shows that the average number of simulated intersecting capsules occurs at approximately 50% of the cumulative probability.

Figure 8

Comparison of the average intersection number obtained from simulation (vertical lines) and the analytical formulas (inverse S-curves): (a) for various D/L and (b) for various n.

As illustrated in Figure 9, the horizontal axis denotes the area of the embedded microcapsules for D = 0.3 and picks its values of 19 and 25, respectively, and the perpendicular coordinates of A, B, and C are the intersection number m. The intersection probability of both points B and C are given as P(m) = 0.2, and the probability of points A is given as P(m) = 0.05. If different amounts of microcapsules are embedded in the self-healing material, they both ensure an average probability of hitting 20% for capsules with a diameter of 0.3, n = 267 for point B and n = 347 for point C. Based on the above findings, for n = 267, m = 3 and n = 347, m = 5, the fewer the number of microcapsules embedded in the self-healing material, the fewer the number of microcapsules m intersecting the cracks. It can also be seen from point A that the lower the intersecting probability, the lower the number of capsules that intersect with the crack. Reversely, given a target level P of cracks to be repaired and the number m of microcapsules are required to repair a single crack, it is possible to determine the number of microcapsules to be incorporated.

Figure 9

Comparison of the intersection number m with the given healing target and number of the embedded microcapsules.

Finally, the number of microcapsules intersecting a crack at the maximum probability obtained from the theoretical formula (i.e., equation (7)) is compared with the simulated results with D/L = 0.1, 0.3, 0.5, 0.7, and 0.9, respectively. As can be seen from Figure 10, the simulated values are completely coincidental with the theoretical ones. Therefore, the accuracies of the geometrical probability model on a linear crack intersecting with microcapsules which are randomly dispersed in the capsule-based self-healing composite can be verified.

Figure 10

Comparison of the theoretical and simulated number of microcapsules intersected by a linear crack as the intersection probability reaches a maximum for D/L.

5 Conclusion

In this study, the rupture behavior of capsules forced by growing cracks in two-dimensional capsule-based self-healing materials is investigated. An analytical model is proposed from the viewpoint of geometrical probability characteristics to express the probability of the embedded microcapsules stimulated by linear cracks in a self-healing model composite. In addition, simulation algorithms were developed to verify the accuracy of the theoretical probability results. When comparing the results calculated by the theoretical model with those obtained independently through computer-programmed simulations, good agreement was found between the theoretical data and the simulations. The intersection probability values calculated from the proposed theoretical formulation give acceptable predictions and the theoretical formulation predicts the most probable number of intersecting microcapsules.

The effect of the amount and the size of the microcapsules on the intersecting probability are analytically expressed. Furthermore, if the number of microcapsules embedded are given and a crack requires a fixed number of microcapsules to heal, the optimal size of microcapsules could be determined based on the maximum intersection probability. Given a target level of cracks to be healed, it is possible to determine the number of microcapsules to be incorporated. However, the higher number of capsules may lead to a decrease in the overall performance of the material matrix. Therefore, in order to ensure the performance of the composites, a relatively appropriate number and an optimal size of microcapsules should be used to improve the healing efficiency. Therefore, the application of self-healing composites should consider both the number of capsules and capsule size in a certain area to ensure that the probability of the intersection of cracks encountering the capsule reaches an optimal value. In the future, we can study the probabilistic model of self-healing material microcapsules from a three-dimensional perspective.