## 3 Results and Discussion
### Mechanism Analysis  
There are two main reasons why the electrodeposition patterned copper foam with micro/nanostructures exhibits excellent performance during the solidification of the water-based cool storage phase-change material. First, as shown in Figure 6, the Cu foam surface is relatively smooth and flat, while the electrodeposition-patterned copper foam has more micro/nanostructure. The electrodeposition-patterned copper foam has many micro/nanostructures, which provide abundant crystal nucleation sites during the solidification of material. With an increase in the deposition time, more micro/nano structures are produced, which is beneficial for nucleation. Figure 7 shows the solidification process of water in a test tube for the blank group and CF#300s. In the blank group, a crystal nucleus was first generated at the wall on the right side of the test tube, as shown in the photo at 20 ms. After 40 ms, the crystal nucleus grew and diffused to the left side of the test tube. After 60 ms, the crystal nucleus covered the entire test tube. When CF#300s was added to water, it was determined that ice crystals first formed on the surface of copper foam, as shown in the photo at 20 ms. After 40 ms, the entire copper foam was covered with ice, and ice quickly spread to the surrounding area. Finally, solidification was completed after 60 ms. This result demonstrates that copper foam can promote the formation of crystal nucleus. Second, after the modification of electrodeposition, the micro-nanostructures grown on the copper foam skeletons increased the surface roughness of the copper foam. This made it easier for solid phase crystals to be adsorbed on the surface of the structure, which, to some extent, also eliminate the supercooling degree of phase-change materials with high latent heat. [PERSON] et al. [21] have  
Figure 5: Supercooling and delay time during the solidification process (**a**) supercooling; (**b**) delay time. Figure 6: SEM images of different samples (**a**) CF#0s; (**b**) CF#195s; (**c**) CF#255s. demonstrated that the larger roughness of Al surfaces produced smaller supercooling. Our experimental results confirm this conclusion. In classical nucleation theory, the heterogeneous ice nucleation rate, \(Ja\), can be predicted by [22]:  
\[Ja=c\cdot\exp[-\frac{\Delta G_{A,het}+\Delta G^{*}_{het}}{kT_{het}}] \tag{4}\]  
where \(k\) is the Boltzmann constant, \(T_{het}\) is the temperature of the system under heterogeneous ice nucleation, and \(c\) is the kinetic constant [23], respectively. \(\Delta G_{A,het}\) and \(\Delta G^{*}_{het}\) are the diffusion activation energy and total free energy of heterogeneous nucleation, which can be defined as [24; 25; 26; 27; 28]:  
\[\Delta G_{A,het}=\frac{kET_{het}^{2}}{(T_{het}-T_{R})^{2}} \tag{5}\]  
\[\Delta G^{*}_{het}=\Delta G_{hom}[(1-\cos\theta^{*})^{2}(2+\cos\theta^{*})/4] \tag{6}\]  
where \(\Delta G_{hom}\) is the homogeneous nucleation energy of ice, and \(\theta^{*}\) is the apparent contact angle. The constant values of \(E\) = 892 K and \(T_{R}\) = 118 K were experimentally determined and could be adapted to liquid water between the temperature of 150 and 273 K [29]. The relationship between \(\Delta G_{hom}\) and supercooling \(\Delta T\) can be expressed as:  
\[\Delta G_{hom}=\frac{16\pi}{3}\frac{\gamma^{3}T_{eq}^{2}}{\left(\rho_{ice}h_{ f}\Delta T\right)^{2}} \tag{7}\]  
where \(\gamma\) is the surface tension, \(T_{eq}\) is the phase change equilibrium temperature, \(h_{f}\) represents the latent heat of solidification of water, and \(\rho_{ice}\) is the density of ice. According to [PERSON]'s model [30], which considered a factor of roughness \(f\):  
\[\cos\theta^{*}=f\cos\theta \tag{8}\]  
where \(\theta\) is the intrinsic contact angle. The roughness factor is directly proportional to surface roughness. The apparent contact angle is essential for the total free energy of heterogeneous nucleation of ice crystals; thus, we try to obtain this parameter using a 4-\(\mu\)L water droplet experiment.