## 2 Methodology
The tube is full of liquid nitrogen under different thermodynamic conditions. All the space outside of the tube was filled with stagnant gaseous nitrogen as a supercritical environment. Periodic boundary conditions (BCs) were individually applied in the x, y and z directions. About 32,697 nitrogen molecules and 17,640 gold atoms were involved in the simulation. The number of atoms is different under different working conditions. After setting up the model, the simulation was implemented under an NVE (i.e. constant atom number, constant volume and constant energy) ensemble to update all the nitrogen molecules with a time step of 0.01 ps using the classic parallel MD code [15]. Gaussian distribution at a given initial temperature was used to determine the initial velocities of all nitrogen atoms. In practice, we followed [PERSON]'s method [11] to divide the simulation into three stages, as illustrated in Figure 2. The main target of step one is to form a developed pipe flow. Firstly, energy minimization was carried out for the whole system. After 10,000 steps of relaxation, the liquid and environment reached the desired temperature and pressure. Then the volume force was applied to the liquid nitrogen for a period of time. With a periodic BC, the liquid in pipe will loop back to the left boundary when it flows out of the right boundary. This enables the formation of a developed pipe flow. For the second stage, the gaseous nitrogen of the environment in a specific temperature and pressure was simulated. Finally, the system was coupled together to continue the injection simulation for 100 ps. Note that we need to be especially careful when operating the coupling process since the bond of nitrogen may cross the boundary and overlapping atoms may appear. Figure 1: Schematic diagram of the supercritical nitrogen jet with periodic boundary conditions (BCs) in x, y, and z directions. There are two categories of force between nitrogen atoms: the bonded force between nitrogen atoms inside a nitrogen molecule and the non-bonded force between nitrogen atoms of different nitrogen molecules. The bond forces result from harmonic potential [16]:  
\[E(l)=K(l-l_{b})^{2} \tag{6}\]  
where \(l\) is the instantaneous bond length of a nitrogen molecular, \(l_{b}=1.1\,\mathbf{\dot{A}}\) is the equilibrium bond length of nitrogen and \(K=3.496\,\mathrm{eV}\) is the stretching force constant of the \(N\equiv N\) triple bond. The non-bond force between nitrogen atoms of different nitrogen molecules is the so-called van der Waals force, which is described by the Lennard-Jones (L-J) 12-6 potential:  
\[U(r)=\left\{ \begin{array}{l}4\varepsilon[(\frac{\sigma}{r})^{12}-(\frac{ \varepsilon}{r})^{6}]\,,r\leq r_{c}\\ 0\hskip 14.226378 pt,r>r_{c}\end{array} \right. \tag{7}\]  
where \(\varepsilon\) is the potential well depth, \(\sigma\) is the core distance at which the potential is equal to zero, \(r\) is the distance between two atoms and \(r_{c}=10\,\mathbf{\dot{A}}\) is the cut-off radius of force interaction. For N-Au pairs, the L-J parameters are determined using the Lorentz-Berthelot [17; 18] mixing rule. It is noteworthy that the hydrophilic and hydrophobic parameters are considered for the solid-liquid interaction to avoid the liquid accumulation near the outlet of the injector. The hydrophilic and hydrophobic factors are defined as Equation (8) [19]:  
\[ \begin{array}{l}U(r)=4\alpha\varepsilon[(\frac{\sigma}{r})^{12}-\beta(\frac {\sigma}{r})^{6}]\\ =4\varepsilon^{\prime}[(\frac{\sigma^{\prime}}{r})^{12}-(\frac{\sigma^{\prime }}{r})^{6}]\end{array}  \tag{8}\]  
\[\left\{ \begin{array}{l}\sigma^{\prime}=\sigma/\beta^{1/6}\\ \varepsilon^{\prime}=\alpha\beta^{2}\varepsilon\end{array} \right. \tag{9}\]  
where \(\alpha\) is the hydrophilic parameter, \(\beta\) is the hydrophobic parameter and \(\sigma^{\prime}\) and \(\varepsilon^{\prime}\) are actual parameters for the solid-liquid interaction. In summary, the potential parameters used for the force routine are listed in Table 1.  
Figure 2: Simulation processes of a supercritical nitrogen jet.