## 2 Materials and Methods
### Test Sample Preparation  
In this study, a standard cylindrical specimen (\(\phi\) 50 mm \(\times\) 100 mm) with a diameter-to-height ratio of 1:2 was selected. As the main raw material of CBM, particle size distribution is one of the important parameters of aggregate. Considering the local effect caused by particle size, the aggregate particle sizes we used were all below 8 mm, and the particle size range of aggregate particles was 7 particle size ranges, namely 1-2 mm, 2-3 mm, 3-4 mm, 4-5 mm, 5-6 mm, 6-7 mm, and 7-8 mm. Based on the fractal theory [29] shown in Equations (1) and (2), four aggregate particle size distributions with different fractal dimensions were selected, and the final calculation results are shown in Figure 2 and Table 1.  
\[P_{i}=\frac{M_{i}}{M_{t}}=\frac{d_{i}^{3-D}-d_{\text{min}}^{3-D}}{d_{\text{max }}^{3-D}-d_{\text{min}}^{3-D}} \tag{1}\]  
Figure 1: The distribution of particle size: cement and nanobubbles: (**a**) Cement; (**b**) nanobubble. Figure 2: Aggregate particle size distribution with different fractal dimensions (Solid lines: Interval proportion; Dotted lines: cumulative proportion). In the above formula, \(D\) is the fractal dimension; d\({}_{i}\) is the pore size; d\({}_{\rm min}\) is the minimum particle size of the aggregate particles; d\({}_{\rm max}\) is the maximum particle size of the aggregate particles; M\({}_{i}\) is the mass of the particles between d\({}_{i}\) and d\({}_{\rm max}\); M\({}_{t}\) is the total mass of the particles; and P\({}_{i}\) is the mass fraction of the particle size distribution in the particle interval. It is worth noting that when the maximum particle size is much larger than the minimum particle size, \(d_{\rm min}^{3-D}\) tends to 0, so the above formula can be simplified to  
\[P_{i}=\frac{M_{i}}{M_{t}}=(\frac{d_{i}}{d_{\rm max}})^{3-D} \tag{2}\]  
In this test, the gas cylinder outlet pressure was set to 0.4 MPa, the ambient temperature was about 20 \({}^{\circ}\)C, and the outlet pipe diameter was about 0.7 cm. Under this environmental condition, the density of CO\({}_{2}\) is about 1.82 kg/m\({}^{3}\), so according to Formula (3), the theoretical maximum flow rate of gas entering the nanobubble machine is 1320 L/min. \[v=\sqrt{\frac{2(P_{1}-P_{2})}{\rho}} \tag{3}\]  
In the above formula, \(P_{1}\) is the cylinder pressure valve pressure, \(P_{2}\) is the atmospheric pressure, and p is the CO\({}_{2}\) density under standard conditions. According to [PERSON]'s law shown in Equation (4), at 20 \({}^{\circ}\)C and a standard atmospheric pressure, the maximum solubility of CO\({}_{2}\) is about 0.034 mol/L. \[C=k_{H}P \tag{4}\]  
In this formula, \(C\) is the solubility, \(k_{H}\) is the Henry constant, and \(P\) is the atmospheric pressure. Therefore, according to the molar flow calculation formula shown in Formula (5) and the molar concentration calculation formula shown in Formula (6), the theoretical maximum intake flow rate under saturation conditions is 15.32 L/min. \[n=\frac{Q_{CO_{2}}}{V_{CO_{2}}} \tag{5}\]  
where \(n\) is the molar flow rate, \(Q_{CO2}\) is the gas flow rate, and \(V_{CO2}\) is the volume of 1 mole of CO\({}_{2}\) gas. \[M=\frac{n}{Q_{w}} \tag{6}\]  
where \(M\) is the molar concentration and \(Q_{W}\) is the water flow rate. However, nanobubbles have a larger surface area and a higher gas-liquid interface ratio, which makes their contact with the liquid more complete, greatly increasing the solubility. In addition, considering the fugitive nature of the gas dissolution process, the CO\({}_{2}\) gas flow rate was set to 200 L/min in this test. By adjusting the air entrainment valve of the nanobubble machine, the CO\({}_{2}\) entrainment rate is set to 200 L/min. After the nanobubble machine shears the CO\({}_{2}\) gas into nanobubbles and dissolves them in water, nanobubble water is obtained.