## 3 Results and Discussion
### Non-Equilibrium MD Simulations  
#### 3.2.1 Determination of Solid/Fluid Boundary with Microscopic Heat Flux Relations  
To examine the solid/fluid boundary with atomic accuracy, heat transferred MD simulation is conducted. The mathematical description of microscopic heat flux expression is expressed as follows [45]:  
\[\phi_{\text{z}}=\frac{1}{\Omega}\sum_{\text{i=1}}^{N}\left(\mathbf{v}_{\text{i}}E_ {\text{i}}+\frac{1}{2}\sum_{\text{i=1,j,j,i}}\mathbf{r}_{\text{ij}}(\mathbf{F}_{\text {ij}}\mathbf{v}_{\text{i}})\right) \tag{4}\]  
where \(\mathbf{v}_{\text{i}}\) is the velocity of atom i. \(E_{\text{i}}\), is the total energy. \(\mathbf{r}_{\text{ij}}\) is the position vector of atom i and j. \(F_{\text{ij}}\) is the force vector between atom i and j. Equation (4) includes a volume term, which is determined by the solid/fluid boundary. Inserting Equation (1) to Equation (4), we can derive Equation (5). Figure 5: **(a)** The local statistical significance near the solid/fluid boundary with normalized density profiles. **(b)** The normalized distribution of velocity magnitudes and Maxwell-Boltzmann distribution at the corresponding temperature. The location z = 0 is the most probable location of the innermost solid layer. \[\phi=\frac{1}{A_{\rm c}(H_{\rm cc}-2\delta_{\rm b})}\Bigg{(}\mathbf{v}_{\rm i}E_{\rm i}+ \frac{1}{2}\sum_{\rm i=1,j\
eq i}\mathbf{r}_{\rm ij}(F_{\rm ij}\mathbf{\cdot}\mathbf{v}_{\rm i })\Bigg{)} \tag{5}\]  
From the setup of MD simulation, the average heat flux in a steady state is given as 0.01 \(eV/ps\), which is the rate of thermal energy generating and removing by the local thermostats. Thus, the only unknown variable in Equation (5) is the solid/fluid boundary location, \(\delta_{\rm b}\), and thus, the solid/fluid boundary can be estimated. Figure 6 shows the calculated heat flux with respect to \(\delta_{\rm b}\) for Ag-Ar, Ag-He, and Cu-Ar system at 300 K. The calculated heat flux increases as the boundary definition shift to the liquid region. The relative difference between the calculated and the given heat flux is between \(-8\%\) to \(+12\%\) within \(\sigma_{\rm sf}\) shifting. This observation confirms the importance of the definition of the solid/fluid boundary for the quantitative analysis of heat transport. 0.01 eV/ps heat flux is archived at approximately \(\delta_{\rm b}/\sigma_{\rm sf}\) = 0.445 for Ag-Ar interface regardless of the channel heights. This result agrees with the location that recovers the local viscosity near the wall (i.e., \(\delta_{\rm b}/\sigma_{\rm sf}\) = 0.44) report by [PERSON] et al. [10]. In the case of Ag-Ar interface at \(T\) = 100 K, this location of solid/fluid boundary can considerably reduce the discrepancies of average density from 7.94% to 1.89% and average pressure from 8.81% to 1.75%. Figure 6: The estimated heat flux with respect to the location of the defined solid/fluid boundary (\(\delta_{\rm b}\)) for (**a**) Ag-Ar interface, (**b**) Ag-He interface, and (**c**) Cu-Ar interface with three different channel heights. The temperature maintained at T = 100 K for all systems. It is noted that the difference of LJ diameter of Ar and He is approximately 29%, but the estimated boundaries show only 9% difference (See Figure 6a,b) when it is normalized by \(\sigma_{\text{sf}}\). Similarly, the LJ diameter of Cu and Ag show 13% difference but the estimated boundary shows about 2% difference (See Figure 6a,c). This result implies that the LJ diameter of solid-fluid pair is one of the key factors for solid/fluid boundary. This variation of heat flux affects the estimation of thermal conductivity using [PERSON]'s law (so-called direct method [18; 39]). Then, not only quantitative analysis but also qualitative analysis can be disturbed by boundary definition.