## 2 Methods
### Theoretical Model
The previous Lifshitz-based model [24; 26], which constructed the relationship between the vdW pressure and the dielectric function for two  
Figure 1: Schematic showing of the head–disk interface (HDI): (**a**) Flying state, (**b**) Thermal protrusion–lubricant (TP–lube) contact state, and (**c**) Thermal protrusion–diamond–like carbon (TP–DLC) contact state. Figure 2: Illustration of the adhesive contact for the simplified HDI. parallel layered systems, can be extended to predict the vdW interaction between the TFC slider and the disk according to the Derjaguin approximation [27], as the local curvature radius of the TP is usually on the micron scale which is much larger than the nanometer-sized _FH_[8; 19]. It is assumed that the equilibrium distance, at which the net interaction force vanishes, between two parallel surfaces having properties of the slider and the disk across the air gap is denoted by \(z_{01}\). If \(h\geq z_{01}\), the slider flies away from the disk and the intermediate medium is air, while if \(h<z_{01}\), the TP is in contact with the lubricant or the DLC layer and the intermediate medium between the slider and the disk can be considered as lubricant. Whether the slider operates at the flying state, the TP-lube contact state or the TP-DLC contact state, the vdW pressure \(P_{\rm vdw}\) can be uniformly expressed as  
\[P_{\rm vdW}(x,y)=\left\{\begin{array}{ll}-\frac{kT}{8\pi h^{3}}\sum\limits_{n =0}^{\infty}\,^{\prime}\int_{n}^{\infty}dr\,r^{2}\cdot\left[\frac{\frac{ \Delta_{31}^  
where,  
\[ \begin{array}{c}\Delta_{ij}^{E^{\prime}}=\frac{\varepsilon_{i}s_{j}^{\prime}- \varepsilon_{j}s_{i}^{\prime}}{\varepsilon_{i}s_{j}^{\prime}+\varepsilon_{j}s_ {i}^{\prime}},\ \Delta_{ij}^{F^{\prime}}=\frac{\mu_{i}s_{j}^{\prime}-\mu_{j}s_{i}^{\prime}}{ \mu_{i}s_{j}^{\prime}+\mu_{j}s_{i}^{\prime}}\\ s_{j}^{\prime}=\sqrt{r^{2}+r_{n}^{\prime}\left(\frac{\varepsilon_{j}}{ \varepsilon_{5}}-1\right)},\ r_{n}^{\prime}=\frac{2\eta_{i}\varepsilon_{n} \sqrt{\varepsilon_{5}}}{c}\end{array}  \tag{9}\]  
The repulsive interaction between molecules is a short-range interaction, which is less influenced by the adjacent atoms, and the pairwise additivity applies. Under the hypothesis that the lubricant layer does not affect the contact force and the solid contact area [28], the repulsive interaction force per unit area \(P_{\text{rep}}\) at the HDI is written as [15]  
\[P_{\text{rep}}(x,y)=\left\{ \begin{array}{ll}\frac{B}{45\pi\hbar^{3}},&if\ h(x,y)\geq z_{01} \ \ \ (a)\\ \frac{4}{45\pi\hbar^{3}_{1}},&if\ h(x,y)<z_{01}\ \ \ \ (b)\end{array} \right. \tag{10}\]  
where \(B\) is a constant. The influence of the TP-lube contact or the TP-DLC contact on the lubricant thickness is neglected. The same assumptions with [PERSON]'s study [28], that the force at the lubricant surface can be transmitted to the solid and the topography of the lubricant surface is the same as the solid surface, are made. According to the geometrical relationship shown in Figure 2, the following equations can be constructed. \[h(x,y)=l\sin\theta+z_{r}-z_{TP}(x,y)+u(x,y) \tag{11}\]  
\[h_{l}(x,y)=h(x,y)+h_{\text{lub}} \tag{12}\]  
where \(u(x,y)=u_{4}(x,y)+u_{6}(x,y)\). \(u_{4}(x,y)\) and \(u_{6}(x,y)\) denote the surface elastic deformation in the \(z\) axis of the DLC layer on the disk and that of the DLC layer on the slider, respectively. It is stipulated that the deformation is positive if the disk deforms downward or the slider deforms upward, otherwise it is negative. The geometry of the TP is modeled approximately as an ellipsoid, as shown in Figure 3. \(l_{\text{a}}\), \(l_{\text{b}}\) and \(l_{\text{c}}\) denote the three hemi-axial lengths of the ellipsoid. The coordinate \((x,y,z)\) for the surface of the ellipsoidal TP satisfies the following equation. \[\left(\frac{x\cos\theta-z\sin\theta}{l_{a}}\right)^{2}+\left(\frac{y}{l_{b}} \right)^{2}+\left(\frac{x\sin\theta+z\cos\theta}{l_{c}}\right)^{2}=1 \tag{13}\]  
Then, \(z_{TP}(x,y)\) can be written as  
\[z_{TP}=\left\{