## 3 Results and Discussion
### Equilibrium MD Simulations
#### 3.1.2 Effects of the Defined Boundary on the Average Density and Pressure  
The thermodynamic properties of the confined system are calculated with various boundary definitions to demonstrate the impact of the boundary definitions on the property estimation. The average fluid density, \(\rho_{\rm avg}\), and pressure, \(P_{\rm avg}\), are used to interpret the nanochannel transport phenomena [30; 41; 42; 43]. These properties are coupled with the volume of fluid and the volume can be differently estimated depending on solid/fluid boundary. Assuming the microscopic solid/fluid boundary as a plane that is \(\delta_{\rm b}\) away from the innermost solid layers, the volume of confined fluid can be expressed as follows:  
\[V=A_{\rm c}(H_{\rm cc}-2\delta_{\rm b}) \tag{1}\]  
where \(A_{\rm c}\) is the cross-section area of confined fluid (xy plane), \(H_{\rm cc}\) is the center to center height and \(\delta_{\rm b}\) is the boundary location.  
Then, the average fluid density between the two boundaries is described as follows:  
\[\rho_{\rm avg}=\frac{n_{\rm fluid}}{A_{\rm c}(H_{\rm cc}-2\delta_{\rm b})} \tag{2}\]  
where \(n_{\rm fluid}\) is the number of fluid between the defined boundary.  
Figure 3: The density and pressure profiles of nano-confinement. A variable \(\delta_{b}\) is the distance between the innermost solid layer and the defined solid/fluid boundary. The channel height based on the defined boundary is \(H_{\rm fluid}\equiv H_{\rm cc}-2\delta_{\rm b}\).  
Similarly, the microscopic definition of the pressure of the confined fluid with boundary definition can be expressed as follows:  
\[P=\frac{1}{3A_{\text{c}}(H_{\text{cc}}-2\delta_{\text{b}})}\Bigg{(}2N\langle E_{ \text{k}}\rangle+\sum_{i}\sum_{<j}\textbf{F}_{\text{ij}}\tau_{\text{ij}}\Bigg{)} \tag{3}\]  
where \(\textbf{F}_{\text{ij}}\) is an interatomic force vector between i and j atoms and \(\textbf{r}_{\text{ij}}\) is a relative positional vector from i to j atom.  
As can be seen in Figure 4, the average fluid densities and pressure show non-linear variations with respect to the defined boundary. The non-linearity of the average properties are resulted by the dynamical structuring of fluid molecules near the solid surface, which induce the local variation of density and pressure profile. The different crystal orientations show a similar variation of \(\rho_{\text{avg}}\) and \(P_{\text{avg}}\). It is noted that the largest discrepancy of the average fluid density is 16% for the smallest channel studied (i.e., 3.47 nm). Considering the fact that the dense fluid generally has tiny compressibility, this discrepancy can substantially affect the interpretation and understanding of the physical behavior of the confined fluid. Also, the average properties for smaller channels are more sensitive to the definition of solid/fluid boundary indicating that defining solid/fluid boundary becomes more important for a small system. This is because the smaller channel has a larger surface-to-volume ratio and it enhances the effect of the structural inhomogeneity of fluid near the surface. Thus, this size effect can be roughly estimated by non-dimensional number: \(2\sigma_{\text{sf}}/H_{\text{cc}}\), which indicates the ratio of boundary uncertainty over the channel size.  
Figure 4: Effects of the defined solid/fluid boundary on the average thermodynamic properties of confined fluid. The average density with respect to the defined solid/fluid boundary of Ag-Ar contact plane of (**a**) (001) and (**c**) (111). The average pressure as a function of solid/fluid defined boundary for (**b**) (001) (**d**) (111) contact planes.