1. Introduction
Cracks have an adverse effect on the strength of concrete, as the penetration of cracks can cause corrosion of steel bars. Generally, repair can be carried out through the carbonization of calcium hydroxide caused by carbon dioxide and water [
1,
2,
3]; accordingly, self-healing helps concrete to resist microcracks [
4,
5]. Self-healing concrete can be produced using silica particles [
6]. However, concrete self-healing is only effective for cracks less than 1 mm in size, and the repair influence is limited for large cracks. Many techniques have been reported to deal with larger cracks, including the use of sugar-coated concrete, microbial self-healing concrete, polypropylene macrofiber concrete, silicon-based polymers, polymeric and cementitious materials, epoxy grouting materials, and bacteria-based self-healing concrete [
3,
6,
7,
8,
9,
10,
11].
Recently, MICP has been comprehensively researched for repairing concrete cracks. Microbiologically induced calcium carbonate precipitation (MICP) is a biogeochemical process that induces calcium carbonate precipitation via selected microorganisms within the material matrix through different pathways, considered as a potential plugging agent in many environmental and engineering applications [
12]. The addition of selected microorganisms to cementitious materials is considered a cost-effective and eco-friendly method for microcrack repair. The most appropriate pathway for MICP is urea hydrolysis. Urease enzymes can be produced by microorganisms, and calcium CaCO
3 can be produced with the help of calcium ions [
11,
13]. The healing process caused by bacteria normally consists of some materials and bacteria used for the reaction [
14,
15]. Several kinds of bacteria have been used, depending on the application situation, to repair cracks via MICP [
16]. For example, the compressive strength of concrete repaired using Shewanella baltica increased after twenty-eight days of MICP repair, whereas the compressive strength of another sample did not change significantly [
17]. Urine-soluble bacteria have also been commonly used in previous research to repair concrete cracks [
16,
18,
19,
20]. Microbially induced calcium carbonate precipitation can be used to seal cracks because it has cementation ability and can improve the strength of concrete [
2,
21,
22]. The strength, toughness, and resistance of concrete can be improved using biocomposite metakaolin bacterial spores, the MICP method, and bio-inspired strategies under high fatigue loading [
2,
23,
24]. As well, Sun et al. [
25] considered glucose addition, microbial calcium carbonate, high urease activity, and
Bacillus subtilis to enhance the strength parameters and bioremediation efficiency of repaired concrete samples [
25,
26,
27].
More recently, many scholars have made a great effort to repair larger cracks through the MICP technique [
16,
28,
29]. Based on the MICP method, the precipitation of CaCO
3 can seal cracks without any addition. On the other hand, repairing cracks through the precipitation of CaCO
3 is not beneficial for larger cracks, and this is why a few scholars have explored other chemicals to seal large cracks [
30]. For example, Zhang et al. [
31] and Sun et al. [
32] suggested the use of polyvinyl alcohol (PVA) fibers and aluminum oxide to repair cracks 0.5 to 2.0 mm in size through microbially induced carbonate precipitation. Existing studies on crack repair did not consider the crack roughness. Furthermore, many studies on concrete crack repair only conducted qualitative examinations, which are not appropriate in practical engineering. In reality, theoretical quantification of rough crack repair with the MICP technique cannot be ignored, and it is of great importance to develop a new mathematical model for MICP rough crack repair. As well, control of the repair period and microbial metabolism using a mathematical model could allow for biomineralization, biofilm growth, on-demand adjustment, and successive bioremediation of building materials.
However, some scholars adopted a theoretical model approach to quantitively determine the repair effects of MICP. Most of prior investigations have selected theoretical methods about fine aggregate cementation based on the MICP technique. According to the MICP process, these methods can be categorized into four parts: the distribution and transport of biomass, biochemical reaction, MICP solidification, and CaCO
3 precipitation [
33,
34,
35,
36,
37,
38,
39]. These parts can affect the ultimate impact of biocementation. Bacteria would adhere to the crack surface during the repair by MICP. Biofilm is an extracellular polymeric substance, which is produced by attached bacteria [
40,
41,
42]. Previously conducted laboratory-based experiments do not considered biofilm, but this does not mean that no biofilm exists in experiments. Therefore, researchers of this study think that the biofilm growth should be considered for rough crack repair in a mathematical model for deeper understanding.
To assess the repair effects using a new mathematical model, biofilm growth, the distribution of suspended biomass, the amounts of precipitated CaCO
3, and the distribution of solutes were assessed using Python. Furthermore, crack repair experiments were performed to show the practicability and feasibility of the new model. Even if conventional precipitation of crystallization may have an influence on the repair effects, it is an inherent mechanism and the impact is negligible [
43]. Thus, the repair effect was taken as the main outcome of MICP in the tests. In the experiments, the values of sonic time, productive rates of CaCO
3, and absorbance of the leachate were obtained and matched with the calculated results. In the end, the results revealed the accuracy of the transport of suspended biomass, transport of solutes, CaCO
3 precipitation, and biofilm growth in the developed mathematical model.
5. Model Accuracy and Application
To judge the accuracy of the proposed model, the determination coefficient (
R2) has been generally used and is well known today. The
R2 value defines the goodness of method, that is an arithmetical approach for observing the accuracy of a technique in forecasting the real data sets. The determination coefficient (
R2) has been used to evaluate the performance of proposed reduction method. Larger value of
R2 indicates that the forecasting precision of the method is high. These matrices used the following Equation (28) as:
where,
A and
B are the targeted and output values, respectively, and
n is the number of specimens.
A comparison of targeted and output values is presented in
Figure 9, at the calculated and experimental stage. The constant of determination (
R2) between the calculated and experimental results shows a good crack repairing capacity of the proposed model. There is almost no remarkable dissimilarity between the calculated and experimental results (
Figure 9). Results show that the developed mathematical model is an appropriate tool to repair concrete cracks of rough surface.
MICP has a wide range of applications in the field of concrete and building material repairs. One area of research involves using MICP to produce self-healing concrete, reduce water adsorption, and fill cracks. Recently, Ahmad et al. [
60] repaired structural cracks by bacterial sustainable concrete treated with MICP. They found that the surface deposition of CaCO
3 reduced water absorption by 65–90%, depending on the porosity of the specimens. This, in turn, decreased the carbonation rate and chloride migration by about 25–30% and 10–40%, respectively. They also observed an increased resistance to freezing and thawing. Dai et al. [
49] used MICP method for surface treatment of concrete, evaluating the efficiency of the resulting improvement in both macroscale and microscale properties of the materials.
The developed model and MICP technique is also applied to repair rough cracks of old concrete surfaces.
Figure 10 shows the self-healing effects on concrete rough cracks after 28 days of water curing. It could be seen that the reference cases had mostly been repaired, while a few white powders appeared on the surface of the samples and the cracks were almost completely repaired. Some areas of rougher cracks did not heal completely as compared to less rough cracks as shown in
Figure 10a. As the unprotected bacteria had limited survival time to move easily in rougher cracks and cement-based materials, resulting in the poor repair effect, which indicates that the proposed model is more effective to repair cracks of less rough surfaces. The area repair rate of specimens was estimated by using the method presented by Zheng and Qian [
64]. The results showed that the cracks of rougher surfaces were less repaired, and the area repair rate of this group was 91%, while the area repair rate of less rough cracks was 98%, as clearly observed from
Figure 10b–d.
The proposed mathematical model based on MICP has a wide range of applications in the field of concrete repairs and building materials. One area of research involves using MICP technique to deposit films of CaCO3 on the surfaces to reduce water adsorption, produce self-healing concrete, or fill cracks. The proposed mathematical model can be used to calculate the distribution of calcium carbonate in the concrete rough cracks, in order to validate the practicability of the CaCO3 precipitation model, ureolysis, and solute transport model in the tests.
6. Conclusions
In the current research, a novel mathematical model was proposed by considering biofilm growth, the transport of solutes, geochemistry, CaCO3 precipitation, the transport of suspended biomass, and ureolysis. By using this model, the productive rates of CaCO3, concentrations of biomass, concentrations of solutes, and biofilm volume fractions were estimated via Python. The accuracy of the proposed model was judged by coefficient of determination (R2). Additionally, numerous rough concrete cracks were repaired through the MICP technique in the experiments to validate the usefulness and applicability of the proposed model. The following main conclusions are drawn:
- (1)
In different rough cracks, the concentrations of suspended biomass changed significantly, and the suspended biomass concentrations diminished slowly during the experiments. The suspended biomass concentration for the rougher crack (crack 10#) was higher. The experimental results were in line with the calculated results, which validated that the transport model of suspended biomass is feasible and precise.
- (2)
The biofilm volume fractions in the mathematical model decreased after increasing the distance from the crack inlet. The distributions of the concentrations of urea in the rough concrete cracks had similar laws, finally causing an increase in the productive rates of calcium carbonate. For the smoother cracks (4# and 6#), the reducing limits were higher after the daily cycle. In the smoother cracks, the productive rates of calcium carbonate were very similar to the experimental outcomes. The variation between the experimental and calculated outcomes was in an acceptable range.
- (3)
The value of sonic time in all cracks diminished after increasing the repair time. The smoother cracks had lower values of sonic time and showed excellent repair effects as compared to the rougher crack. As well, near the inlet, the sonic time values for cracks 4# and 6# were small due to the precipitation of CaCO3.
Overall, the practicability and feasibility of the new mathematical model is verified through crack repair experiments. The model provides a better technique to civil engineers that leverages repair period and microbial metabolism to impart a new repairing, adjustive, and sensing multifunctionality to concrete.