## 3 Results and Discussions
### Crystal Growth Mechanisms of CaCO\({}_{3}\) Nanoparticles in Chitosan Hydrogels
Amorphous regions are noticed inside and on the edge of the calcite nanoparticles. (**g**,**h**) TEM images of the nanocomposite coating in mantis shrimp dactyl clubs, showing the combination of chitin macromolecules and hydroxyapatite nanocrystals. Images are taken from [16] with permission. \[F=-Af=-AD\frac{\partial c}{\partial x} \tag{1}\]  
\[\frac{\partial c}{\partial t}=D\left[\frac{\partial^{2}c}{\partial x^{2}}+\frac{ \partial^{2}c}{\partial y^{2}}+\frac{\partial^{2}c}{\partial z^{2}}\right] \tag{2}\]  
where \(F\) is the flux, \(f\) is the flux through a unit area, \(A\) is the area over which diffusion occurs, \(D\) is the diffusion coefficient, \(c\) is the concentration of the diffusing substance, and \(x\), \(y\), \(z\) represent the spatial diffusion distances. [PERSON]'s second law of diffusion describes how particles tend to move from an area of higher concentration to an area of lower concentration. Based on the diffusion experiments in this study, it can be simplified into one dimensional case. The diffusion equation can be as follows:  
\[\frac{\partial c}{\partial t}=D\frac{\partial^{2}c}{\partial x^{2}} \tag{3}\]  
Since the volume of the NaHCO\({}_{3}\) solution is much larger than the thin chitosan film, this can be considered as a semi-infinite solid scenario. By solving the above equation of (3), the final concentration of CO\({}_{3}^{2-}\) ions as a function of time and distance can be describe as follows:  
\[c(x,t)=\frac{c^{\prime}}{2}[1+erf(\frac{x}{\sqrt{4 Dt}})] \tag{4}\]  
where \(c^{\prime}\) is the concentration of CO\({}_{3}^{2-}\) in the original solution, and \(erf\) is the error function. The form of the error function is expressed as follows:  
\[erf(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-x^{2}}dx \tag{5}\]  
\[z=\frac{x}{\sqrt{4 Dt}} \tag{6}\]  
Figure 4:0 Schematic of in situ growth of CaCO\({}_{3}\) nanocrystals inside chitosan hydrogels via the double diffusion method. Calcium ions are diffused into the 3D-printed chitosan hydrogel first, until an equilibrium is reached. Samples are then immersed in NaHCO\({}_{3}\) solution. As the CO\({}_{3}^{2-}\) is introduced, calcite crystals start growing at the surface first. Microcrystals are formed on the surface, while crystal sizes are limited to \(\sim\)100 nm inside the hydrogel due to the confinement of the chitosan molecular network. According to the above solution for diffusion equations, the concentration of CO32- ions on the surface (\(x=0\)) is independent of time, which correspond to the stable size of calcite crystals grown on the surface initially. The concentration of ions increases as a function of time, which explains why nanoparticles grow from 5 min to 6 h. Interestingly, in our experiments, as the diffusion time kept growing, no obvious changes in size and shape of the calcite crystals were observed. This indicates that diffusion is no longer the main controlling mechanism of crystal growth. Additional thermodynamic driving force is needed to overcome the energy barrier. Given the good solubility of chitosan hydrogel, the diffusion principle is applied to initially introduce Ca2+ into the hydrogel, ensuring uniform distribution in the network. Subsequently, CO32- is introduced, creating a concentration gradient between the internal hydrogel environment and the NaHCO3 solution, establishing a double diffusion system. The hydrogel surface, being the first to contact the solution and lacking a three-dimensional network to restrict crystal growth, allows for the rapid formation and growth of CaCO3 particles to a micrometer size. Inside the hydrogel, the network pores limit crystal growth. Over time, as Ca2+ diffuses into the solution and CO32- diffuses into the hydrogel, CaCO3 nucleates heterogeneously in the fiber network, progressing from the surface to the interior. Once nucleation occurs, the nuclei continue to adsorb Ca2+ and CO32- to promote crystal growth. However, the limited space and dense distribution provided by the three-dimensional network physically restrict and locally inhibit the crystal growth process.