## 3 Results and Discussion  
### Effect of the Crosslinking Reaction on the Thermo-Mechanical Properties  
Uniaxial tensile loading simulations were conducted for various temperatures, ranging from 300 K to 390 K. The BC microstructure models were equilibrated at each temperature for 1 ns, using the NPT dynamics, and then were frozen immediately at 0.1 K for 0.5 ns, using the NVT dynamics. The tensile simulation near zero temperature was performed for each BC model until the overall strain reached 0.15% at a constant strain rate of 10\({}^{6}\) s\({}^{-1}\). The stress tensor (\(\sigma\)) of the BC unit cell was estimated from the virial theorem, as follows [26]:  
\[\sigma=\ \frac{1}{V}\left(-\sum_{i}^{N}m_{i}(v_{i}v_{i}^{T})+\ \frac{1}{2}\sum_{i}^{N} \sum_{j\
eq i}^{N}\tau_{ij}F_{ij}\right) \tag{1}\]  
where \(V\) and \(N\) are the system volume and the number of constituent atoms, \(m_{i}\) and \(v_{i}\) are the atomic mass and the velocity of atom \(i\), and \(r_{ij}\) and \(F_{ij}\) are the relative distance and the atomic force between atoms \(i\) and \(j\), respectively. Because all of the components in \(\sigma\) were extracted at 0.1 K, where the kinetic term was eliminated without changing the direction of the traction vector during the deformation, the stress was equivalent to the Cauchy stress [27]. In addition, the stiffness tensor (\(C\)) of the BC unit cell model was calculated as follows:  
\[\sigma=C\epsilon \tag{2}\]  
where \(\epsilon\) represents the applied strain tensor. The tensile simulation was performed for each direction, and the stiffness results were averaged assuming an isotropic medium. The thermoelastic behavior characteristics of the models, for four different temperatures, are shown in Figure 3a. At room temperature, an increase in the BC stiffness was clearly observed as the crosslinking density increased (from 5.77 GPa to 6.27 GPa). This result was straightforward because the load transfer between the BC constituent molecules became more efficient and dominant as the individual MMA liquid became polymerized in the system. Notably, the magnitude of elastic stiffness at 330 K is quite similar to the one observed at 300 K. The reinforcement on the stiffness with the progress of crosslinking reaction is also valid. The similarity between these two temperature conditions agrees very well with the results observed in the experiment [28; 29]. At temperatures above 360 K, the stiffness of the BC decreased considerably, and the deterioration of stiffness became more pronounced with increasing temperature. The sharp decrease of the elastic stiffness of BC material at temperatures above 360 K is in agreement with the experimental observations [28; 29]. Furthermore, it is interesting to find that the influence of the crosslinking density varies greatly with increasing temperature. This implies that the mechanical properties of the BC material after being stabilized in the body are significantly different compared to when the BC material is placed in a high exothermic state by a crosslinking reaction in the surgical environment. Meanwhile, the structural stability of the BC material can also be evaluated from the change in the free volume between polymer chains. Figure 3b,c shows the change in free volume of the BC unit cell model at each stage of temperature. The free volume was calculated based on the Connolly surface, where the probe radius was set as 2 A. As shown in the results, there was a rapid increase in free volume at temperature of 360 K and above, which causes a softening of the BC material [28]. To evaluate the thermal stability of the PMMA-based BCs at room temperature, mean squared displacement (MSD) curves were obtained at 300 K using the following equation:  
\[\mathrm{MSD}=\ \frac{1}{3\mathrm{N}}\sum_{i=0}^{\mathrm{N}-1}\left\langle \left|\overset{\rightarrow}{\mathrm{R_{i}}}(t)-\overset{\rightarrow}{\mathrm{R_ {i}}}(0)\right|^{2}\right\rangle \tag{3}\]  
where \(\overset{\rightarrow}{\mathrm{R_{i}}}(t)\) indicates the position vector of atom \(i\) at time \(t\).