## 1 Introduction  
Hard disk drive (HDD) storage, because of its cheaper cost per stored data, still plays an important role in the world of digital data and the demand for its storage density increases with the explosive growth of information in recent years [1]. Optimization of the head-media spacing (_HMS_), defined as the distance from the bottom of the recording element on the head to the top of the magnetic layer on the disk, plays a pivotal role in achieving this demand [2, 3]. Critical to the success of this effort is the minimization of the flying height (_FH_), which denotes the clearance between the closest points of the head slider and the disk [4]. However, decreasing the _FH_ through directly bringing the whole slider close to the disk may induce vibration and lead to a catastrophic crash of the slider [5]. To counteract such problems, a thermal flying height control (TFC) technology was proposed [6], which reduces the _FH_ by heating an element in the head to yield a small thermal protrusion (TP) around the recording element whilst the slider substrate flies steadily at a relatively large nominal flying height. The TFC technology has been widely used in HDD products [7], and has made a sub-2 nm _FH_ become reality [8]. In pursuit for much higher read/write capacity, a _FH_ of less than 0.5 nm may be required in the future [9]. To achieve this, the surfaces of the slider and the disk need to be atomically smooth. An extremely smooth slider and disk working under ultra-low _FH_ can lead to strong adhesive interactions at the head-disk interface (HDI), which may influence the dynamic properties of the slider and the read/write capacity of the disk [4; 8; 9; 10]. Furthermore, the possibility of contacts between the slider and the disk increases, and, as a result, the traditional flying recording mechanism may transform to the lubricant-contact [11; 12] or solid-contact [7] recording mechanisms. Hence, in order to mitigate the negative effects of adhesion at the HDI and promote further development of the TFC technology, it is necessary to model and predict the various possible contact states of the HDI with the TP geometry and the adhesive interaction taken into consideration. One of the main contributors to the adhesion of the HDI is the intermolecular interaction, which includes both the attractive portion and the repulsive portion [13]. The dominant source of the attractive portion is known as the van der Waals (vdW) interaction, and the repulsive portion is caused by the overlap of electron clouds of atoms at small interatomic distances, which is a short-range interaction and often characterized by the power potential [13]. The commonly used method to quantify the adhesive interaction resulting from intermolecular forces is the Hamaker theory, which gains the interaction force between macroscopic bodies through the pairwise summation of all molecular interactions by assuming that the number density and the interaction coefficient are constant over the volume of the bodies [14]. Following this theory, [PERSON] et al. [15] derived the interaction force between each triangular cell of the slider and the disk according to the Lennard-Jones pair potential [13], and found that the vdW force had a significant effect on the _FH_ when the slider fle below 5 nm, while the repulsive force dominated when the _FH_ was below 0.5 nm. [PERSON] [16] incorporated the [PERSON] theory into the dynamic analysis of the HDI and investigated the influence of vdW force on the flying characteristics. In [PERSON]'s [15] and [PERSON]'s [16] studies mentioned above, surface deformations caused by the interaction force are neglected. Investigations of the surface deformation governed by the Green function and the vdW force effects on _FH_ for the TFC slider-disk interface were conducted by [PERSON] et al. [9] with a hypothesis that the adhesion force did not change the Hertzian profile. However, the lubricant layer on the disk was not considered in the above studies. [PERSON] et al.