## 2 Microbial Induced Carbonate Precipitation by _Sporosarcina pasteurii_
### Crystallization of CaCO\({}_{3}\)  
The dissolution or precipitation of CaCO\({}_{3}\) crystals in the mixed solution can be explained by Gibbs Free Energy Theory [22,23]. The following reaction, shown in Formula (1), occurs in the CaCO\({}_{3}\) solution:  
\[\text{CaCO}_{3}\rightleftharpoons\text{Ca}^{2+}+\text{CO}_{3}^{2-} \tag{1}\]  
According to Phase Transformation Theory, the direction of phase transformation is always from the phase with higher chemical potential to the phase with lower chemical potential, that is, towards the direction of a decrease in chemical potential. Therefore, chemical potential can be regarded as the driving force of the transformation of substances from one phase to another. For any chemical reaction, the Gibbs free energy can be expressed by the Van't Hoff Isothermal Equation, as shown in Equations (2) and (3) below:  
\[\Delta_{r}G_{m}=\Delta_{r}G_{m}^{\Theta}+RTln\mathfrak{q}_{\alpha} \tag{2}\]  
\[\mathfrak{q}_{\alpha}=\frac{\alpha_{\text{Ca}^{2+}\cdot\alpha}C_{3}^{2-}}{ \alpha_{\text{CaCO}_{3}}}\prod\alpha \tag{3}\]  
where \(\Delta_{r}G_{m}^{\Theta}\) is the change of the standard molar Gibbs free energy in the reaction, \(\mathfrak{q}_{\alpha}\) is the activity quotient, \(\alpha\) is the activity of the substance or ion in the reaction, \(R\) is the partial molar quantity and \(T\) is the temperature. When the reaction is in equilibrium, \(\Delta_{r}G_{m}=0\), \(\Delta_{r}G_{m}^{\Theta}=-RTln\mathfrak{q}_{\alpha}=-RTlnK_{sp}\), where \(K_{sp}\) is the solubility product. At this time, the ion concentration is the largest and the solution is saturated. Another expression of the Van't Hoff Isothermal Equation can be obtained by transforming Equation (2), as shown in Equation (4):  
\[\Delta_{r}G_{m}=RTln\mathfrak{q}_{\alpha}-RTlnK_{sp}=RTln\frac{\mathfrak{q}_{ \alpha}}{K_{sp}} \tag{4}\]  
When \(\mathfrak{q}_{\alpha}>K_{sp}\), the reaction in Formula (1) proceeds to the left, the mixed solution is supersaturated and CaCO\({}_{3}\) crystals are deposited. When \(\mathfrak{q}_{\alpha}<K_{sp}\), the reaction in Formula (1) proceeds to the right, the solution is unsaturated and no crystals are deposited. When \(\mathfrak{q}_{\alpha}=K_{sp}\), the solution is saturated. The dissolution or precipitation of CaCO\({}_{3}\) crystals in mixed solution also can be explained by Ion Activity Product Theory [24]. In the process of mineralization and deposition, tiny particles will be produced as the crystal nucleus; these tiny particles collide with each other continuously in the solution due to Brownian motion, resulting in the growth of CaCO\({}_{3}\) crystals on the surface of the crystal nucleus. In this process, supersaturation directly affects the formation of the CaCO\({}_{3}\) crystal nucleus and the growth of CaCO\({}_{3}\) crystals. The saturation coefficient (SI) of CaCO\({}_{3}\) reflects the trend of the deposition or dissolution of CaCO\({}_{3}\) crystals in the solution. The equation for \(SI\) is as follows:  
\[SI=\sqrt{\frac{IAP}{K_{sp}}} \tag{5}\]where \(IAP\) is the ion activity product, \(K_{sp}\) is the ion solubility product, \(IAP_{\text{CaCO}_{3}}=\alpha_{\text{Ca}^{2+}\cdots\text{Ca}_{\text{CO}_{3}^{2-}} },\alpha_{\text{Ca}^{2+}}\) is the activity of Ca\({}^{2+}\), \(\alpha_{\text{CO}_{3}^{2-}}\) is the activity of CO\({}_{3}\)\({}^{2-}\) and \(K_{sp\text{CaCO}_{3}}\) is the solubility product of CaCO\({}_{3}\). When \(SI>0\), the reaction tends to deposit CaCO\({}_{3}\) crystals; when \(SI=0\), the deposition and dissolution of CaCO\({}_{3}\) reach equilibrium; and when \(SI<0\), the reaction tends to dissolve CaCO\({}_{3}\) crystals. The crystallization of CaCO\({}_{3}\) occurs through the following processes: First, _Sporosarcina pasteurii_ secretes urease from its metabolic processes, which decomposes urea in the surrounding environment into NH\({}_{4}\)\({}^{+}\) and CO\({}_{3}\)\({}^{2-}\) so that the solubility product of Ca\({}^{2+}\) and CO\({}_{3}\)\({}^{2-}\) in the surrounding environment continues to accumulate and reach supersaturation.