## 1 Introduction  
Concrete and cement form the basis of many important materials used in the construction of modern buildings and engineering structures. Many believe that it would be difficult to develop another building material that can completely replace concrete in the future [1]. Meanwhile, since concrete is a material whose tensile strength is lower than its compression strength, it is unavoidable for cracks to develop in a concrete structure. In Japan, guidelines have been established for evaluating cracks that are narrower than certain thresholds to ensure that they do not cause structural or durability problems [2]. Even though fine cracks do not immediately degrade the safety performance of a concrete structure, they not only allow corrosive elements such as CO\({}_{2}\) and Cl\({}^{-}\) to infiltrate concrete, but also cause an increase in water permeability, which challenges the durability [3]. Repeated infiltration of corrosive elements can accelerate concrete deterioration due to the widening of cracks [4] and irreparably damage the safety performance of a structure [5, 6]. Therefore, it is important to prevent fine cracking at an early stage. Meanwhile, the self-healing phenomenon has been confirmed to occur in concrete in an environment with a continuous supply of water, where part of a particular small crack in a structure is filled as a result of the rehydration of cement particles and precipitation of calcium carbonate (CaCO\({}_{3}\)) [7]. The products of self-healing are closely related to hydrates such as the C-S-H hydrate, ettrinigite, and calcium hydroxide (Ca(OH)\({}_{2}\)) [8,9], as well as the calcium carbonate that is newly generated on the crack surface. In the self-healing mechanism of concrete, a carbonate compound, CaCO\({}_{3}\), which is not readily dissolved in water, is generated by the reaction between Ca\({}^{2+}\) in concrete and CO\({}_{3}\)\({}^{2-}\) dissolved in water [10]. The reaction is summarized by the following Equations (1)-(3) [10]:  
\[\mathrm{H_{2}O+CO_{2}\Leftrightarrow H_{2}CO_{3}\Leftrightarrow H^{+}+HCO_{3}}^{ -}\Leftrightarrow 2H^{+}+CO_{3} \tag{1}\]  
\[\mathrm{Ca^{2+}+CO_{3}}^{2-}\Leftrightarrow CaCO_{3}\left(\mathrm{pH_{water}>8}\right) \tag{2}\]  
\[\mathrm{Ca^{2+}+HCO_{3}}^{-}\Leftrightarrow CaCO_{3}+H^{+}\left(7.5<\mathrm{pH_ {water}<8}\right) \tag{3}\]  
The cracked part is also restored as a result, as reported for cracks that were narrower than 0.1 mm in earlier studies [10, 11, 12]. Researchers have also explored various approaches to promote self-healing, such as the use of bacteria, crystalline admixtures, and superabsorbent polymers [13, 14, 15, 16, 17, 18, 19, 20, 21]. [PERSON] et al. reported that a large amount of CaCO\({}_{3}\) precipitate was produced in the surface layer and inside microcracks because the self-healing of cement-based composite materials generated carbon dioxide gas in micrometer-sized ultrafine bubbles, which flowed into microcracks under a continuous water supply [1,22,23]. Crystals of CaCO\({}_{3}\) can be divided into three types: calcite, vaterite, and aragonite. The majority of the CaCO\({}_{3}\) produced in concrete may be considered to be calcite: Ca(OH)\({}_{2}\) is first generated among the hardened bodies of cement, which then combines with CO\({}_{3}\)\({}^{2-}\) ions in water in the pores to form CaCO\({}_{3}\) in the form of calcite [24,25]. Meanwhile, vaterite is a hexagonal crystal, whose pore-filling effect is better than that of other CaCO\({}_{3}\) crystals because of its large volume [25, 26, 27, 28, 29]. Based on reports of pore-filling contributing to the water tightness and strength enhancement of concrete, the generation of self-healing materials with a crystal structure that is denser than that of calcite is expected to be possible [24]. The formation of the three types of crystal forms of CaCO\({}_{3}\) is controlled by adjusting the temperature or pH [24,25].