## 2 Methods  
### Theoretical Model  
An illustration of the multilayered construction of a typical HDI under the dry air and steady-state condition is shown in Figure 1, in which the head slider applies the TFC technology and comprises of the Al\({}_{2}\)O\({}_{3}\)-TiC substrate and the DLC layer, while the disk is an assembly consisting of the glass or aluminum substrate, NiP underlayer, CoCr magnetic layer, diamond-like carbon (DLC) layer and perfluoropolyether (PFPE)-type lubricant layer. The size of the thermal protrusion (TP) is determined by the power of the heating element in the TFC slider. Three states, that is the flying state, the thermal protrusion-lubricant (TP-lube) contact state and the thermal protrusion-diamond-like carbon (TP-DLC) contact state, may present, as shown in Figure 1. The symbol \(\theta\) denotes the pitch angle, \(z_{r}\) is the spacing between the trailing edge of the slider and the lubricant layer, which also denotes the flying height for the case without TP and is referred to as the nominal flying height. _FH_ is the nearest distance between the head and the disk, namely, the flying height. _HMS_ represents the distance from the bottom of the slider to the top of the magnetic layer on the disk, which is the sum of the _FH_, lubricant thickness and DLC thickness of the disk. For the sake of computational efficiency, only the CoCr magnetic layer (layer 1) overlaid with layer 4 of DLC and layer 5 of lubricant is modelled, as presented in Figure 2. This model is reasonable, as reported by [PERSON] [24] that the relative error of the vdW force per unit area predicted by the model considering the first two layers of the disk and that the full multilayered construction of the disk, is less than 10%. In regard to symbols in Figure 2, the layers of the slider and the medium air are numbered by 2, 6 and 3, respectively. The thicknesses of the DLC layer and the lubricant layer on the disk and the DLC layer on the slider are denoted as \(t_{\text{DLC-disk}}\), \(t_{\text{lub}}\) and \(t_{\text{DLC-slider}}\), respectively. The coordinate system is established at the center of the TP, which is at a distance of \(l\) from the trailing edge of the slider \(h\) stands for the distance between the slider surface and the disk lubricant layer, while \(h_{1}\) denotes the distance between the slider surface and the disk DLC layer. The minimum value of \(h\) (\(x,y\)) is the flying height _FH_, that is \(FH=\min\ \left\{h\ (x,y)\right\}\). \(z_{\text{TP}}\ (x,y)\) represents the \(z\) coordinate of the undeformed DLC layer surface on the slider. The net interaction force \(F\) at the HDI caused by the molecular interaction includes both the adhesion force \(F_{\rm a}\) and the contact force \(F_{\rm c}\) satisfying  
\[F=F_{\rm c}-F_{\rm a} \tag{1}\]  
The contact force \(F_{\rm c}\) is the integration of pressure \(P\left(x,y\right)\) in the contact zone \(\Omega_{\rm c}\) between the slider and the disk, while the adhesion force \(F_{\rm a}\) is the absolute value of the integration of pressure \(P\left(x,y\right)\) outside the contact area \(\Omega_{\rm a}\), which can be expressed, respectively, as  
\[F_{\rm c}=\iint\limits_{\Omega_{\rm c}}P(x,y)\mathrm{d}x\mathrm{d}y \tag{2}\]  
\[F_{\rm a}=-\iint\limits_{\Omega_{\rm a}}P(x,y)\mathrm{d}x\mathrm{d}y \tag{3}\]  
Pressure \(P\left(x,y\right)\) is the interaction force per unit area of the HDI, which is positive inside and negative outside of the contact zone and can be given by  
\[P(x,y)=P_{\rm vdW}(x,y)+P_{\rm rep}(x,y) \tag{4}\]  
where \(P_{\rm vdw}\) and \(P_{\rm rep}\) denote the vdW pressure and the repulsive pressure, respectively. The vdW attractive interaction between macroscopic bodies can be obtained by the Lifshitz theory [22], as described in the introduction.