# Investigations of Adhesion under Different Slider-Lube/Disk Contact States at the Head–Disk Interface

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Theoretical Model

_{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 θ 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.

_{DLC-disk}, t

_{lub}and t

_{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 {h (x, y)}. z

_{TP}(x, y) represents the z coordinate of the undeformed DLC layer surface on the slider.

_{a}and the contact force F

_{c}satisfying

_{c}is the integration of pressure P (x, y) in the contact zone Ω

_{c}between the slider and the disk, while the adhesion force F

_{a}is the absolute value of the integration of pressure P (x, y) outside the contact area Ω

_{a}, which can be expressed, respectively, as

_{vdw}and P

_{rep}denote the vdW pressure and the repulsive pressure, respectively.

_{01}. If h ≥ 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

_{vdw}can be uniformly expressed as

^{−23}J/K and T is the absolute temperature T = 293.15 K. The prime on the summation indicates that the n = 0 term is assigned half weight [24].

^{8}m/s. ε

_{j}= ε

_{j}(iξ

_{n}) and μ

_{j}= μ

_{j}(iξ

_{n}) denote the dielectric permittivity and magnetic permeability of substance j evaluated at the imaginary frequency iξ

_{n}. ξ

_{n}= nξ

_{0}and ξ

_{0}≈ 2.5 × 10

^{14}rad/s at room temperature [24].

_{rep}at the HDI is written as [15]

_{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

_{a}, l

_{b}and l

_{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.

_{TP}(x, y) can be written as

_{4}(x, y) and u

_{6}(x, y) can be expressed as

_{4}and g

_{6}denote the Green functions for the normal elastic deformation on the surface of the slider and that of the disk, respectively.

#### 2.2. Numerical Method

_{01}:

_{vdW}+ P

_{rep}= 0 at h = z

_{01}. Therefore, z

_{01}satisfies

_{vdw}:

_{r}is larger compared with the TP height, the adhesive interaction at the HDI is small and the surface deformation can be neglected. Thus, h = lsinθ + z

_{r}− z

_{TP}can be taken as the initial guess of h. However, with the decrease in the nominal flying height z

_{r}, the adhesive interaction becomes remarkable and the initial estimate of h should take the surface deformation into account. In addition, jump-in and jump-out instabilities [29] may occur at the HDI. Near the jump-in and jump-out points, small changes in the nominal flying height may lead to the abrupt variation of the interface interaction between the head and the disk, which will influence the convergence of the numerical simulation. To solve this problem, the nominal flying height is substituted by the distance h(0, 0). Denoting h

_{c}= h(0, 0), the following equation is derived from Equation (11).

_{c}. To obtain the initial value of h at a preset nominal flying height z

_{rset}, the simulation procedure starts from a large central distance h

_{c}and solves Equation (19) using an initial value h = h

_{c}− [z

_{TP}(x, y) − z

_{TP}(0, 0)]. After achieving the convergent solution, z

_{r}can be inversely calculated by Equation (18). If z

_{r}< z

_{rset}, then the central distance is decreased by a preset interval Δh

_{c}, and the existing solution at h

_{c}is used as the initial estimate at h

_{c}− Δh

_{c}. The above process is repeated until the condition z

_{r}< z

_{rset}does not satisfy. Using the obtained solution as the initial guess at z

_{r}= z

_{rset}, and solving Equation (11), the desirable convergent solution can be acquired. The detailed flowchart for the computational procedure is presented in Figure 4, and the element of residuals R

_{r}, R

_{r}′ and that of the Jacobian Matrix J, J′ in discrete form can be written as

_{c}and j

_{c}stand for the grid nodes at x = 0 and y = 0, respectively. ${K}_{IJ}^{ij}={K}_{4IJ}^{ij}+{K}_{6IJ}^{ij}$, in which K

_{4}and K

_{6}denote the influence coefficients for the elastic deformation due to pressure on the surface of the slider and that of the disk, respectively.

_{4}and u

_{6}:

_{4}and u

_{6}can be expressed as

_{4}and K

_{6}can be derived from the frequency response functions ${\tilde{\tilde{g}}}_{4}$ and ${\tilde{\tilde{g}}}_{6}$ based on the convolution theorem and the fast Fourier Transform technique [30]. Then, the elastic deformations can be solved according to Equations (21) and (22) through the DC-FFT (Discrete convolution and fast Fourier transform) algorithm [30].

## 3. Results and Discussions

_{j}(iξ

_{n}) shown in Ref. [24] are used and the magnetic permeability μ

_{j}is equal to 1. The constant B is taken to be 10

^{−76}J·m

^{6}[15]. Parameters related with TP and the thickness of layers are l = 8 μm, θ = 10

^{−4}rad, t

_{DLC-disk}= 0.5~3 nm, t

_{lub}= 0.5~3 nm, t

_{DLC-slider}= 1~2 nm, l

_{a}= 2 μm, l

_{b}= 10 μm, l

_{c}= 5~13 nm. For the TFC slider, the interaction at the HDI is mainly determined by the interaction between the TP and the disk. Therefore, for the sake of computational efficiency, the calculation domain is set to be −1.5l

_{a}≤ x ≤ 1.5l

_{a}and −1.5l

_{b}≤ y ≤ 1.5l

_{b}, and the mesh numbers are 512 × 512.

#### 3.1. Comparisons of Results Between Hamaker Theory and Lifshitz Theory

_{vdW}= -A

_{H}/(6h

^{3}), in which A

_{H}is the Hamaker constant and a typical value of A

_{H}= 10

^{−19}J [15] is assumed in the present study. Discrepancies are found between results predicted by these two theories, implying that it is unreasonable to set a value of the Hamaker constant directly.

#### 3.2. Analysis of The Adhesive Contact Characteristics at The HDI

_{a}, contact force F

_{c}, net force F and flying height FH as a function of the nominal flying height z

_{r}. In regard to the results of this figure, five sub-regions, I–V, can be divided. In region I, the slider flies sufficiently away from the disk, and the net force depends entirely on the adhesion force as no contact occurs. In this region, the magnitude of the net force (|F|) increases and the FH decreases with the nominal flying height. When the nominal flying height decreases to the value at point A in region II, the attractive interaction increases abruptly and the TP jumps into contact with the lubricant, as evidenced by the sudden decrease in FH to the equilibrium distance z

_{01}. If the nominal flying height is now increased, the net force follows the A’B branch as far as point B but then jumps to point B’, implying that the TP jumps out of contact with the lubricant. In region III, the slider is at the TP–lube contact state, and the net force is still determined totally by the adhesion force. As the nominal flying height decreases, it causes the magnitude of the net force to increase accompanied by a decrease in the FH. In region IV, the TP jumps from point C to point C’, at which it comes into contact with the DLC layer and the FH decreases to z

_{02}-t

_{lub}, and jumps out of contact from point D to point D’. z

_{02}stands for the equilibrium distance between two parallel surfaces having properties of the slider and the disk across the lubricant. In region V, the slider is at the TP–DLC contact state. During this full contact regime, both the adhesion force and the contact force increase with the decreasing nominal flying height, while the FH is almost unchangeable. Due to the combined effect of the adhesion force and the contact force, the magnitude of the net force increases firstly until reaching a maximum value and then decreases.

_{01}, at the TP–lube contact state when z

_{02}-t

_{lub}< FH ≤ z

_{01}, and at the TP–DLC contact state when FH ≤ z

_{02}-t

_{lub}. Second, jump-in and jump-out phenomena occur in regions II and IV, which can influence the stability of the slider and should be avoided during the operation of the HDD. Third, at the TP–DLC contact state, the adhesive interaction is remarkable compared with that at the flying or TP–lube contact states. The FH is difficult to be reduced further due to the strong repulsive interaction of the HDI at the small distance.

_{r}= 11 nm, 9 nm and 8 nm corresponding to the flying state, the TP–lube contact state and the TP–DLC contact state is shown in Figure 7 in conjunction with the distance h in the section of y = 0. It can be seen that the pressure is almost zero and the surface is basically undeformed at the flying state (for example z

_{r}= 11 nm). When the TP is in contact with the lubricant (for example z

_{r}= 9 nm), the adhesive interaction near the contact edge between the TP and the lubricant becomes remarkable. At the TP–DLC contact state (for example z

_{r}= 8 nm), the local pressure is positive at the contact center, together with the obvious surface deformation. Furthermore, strong attractive interactions near the TP–DLC contact edge and near the periphery of the TP–lube contact zone are observed. The direct contact between the TP and the disk can exacerbate the wear of the slider and/or the disk.

#### 3.3. Effects of TP Height on the Interaction Force and FH at the HDI

_{r}for the case of HDI without using the TFC technology. It is observed that the adhesion force at the HDI without TP is larger than that at the HDI with TP, and this discrepancy is more remarkable when the FH becomes smaller, demonstrating the necessity of using the TFC technology at ultra-low flying heights. The effects of the TP height on the adhesion force, contact force and FH at a fixed nominal flying height are presented in Figure 9. Transitions of the slider operation state from the flying state (the left grey region) to the TP–lube contact state (the middle yellow region) and then to the TP–DLC contact state (the right pink region) with the increase in the TP height are observed. During the flying state, the adhesion force increases and the FH decreases if the TP height is increased. When the TP comes into contact with the lubricant, the increased TP height results in a decreased FH and an almost unchangeable adhesion force. If the slider is at the TP–DLC contact state, the adhesion force and contact force increase but the FH cannot be obviously reduced by increasing the TP height on one hand, and on the other hand, the contact zone and contact pressure increase, as depicted in Figure 10. The increase in the contact pressure and contact zone can affect the friction and wear of the HDI and, as a result, may influence the read/write accuracy. Therefore, excessive TP height should be avoided.

#### 3.4. Effects of Lubricant Thickness on the Interaction Force and HMS at the HDI

_{r}= 11 nm, 10 nm and 7 nm, respectively. At z

_{r}= 11 nm, the slider is at the flying state over the entire range of the investigated lubricant thickness. A decrease in the HMS and slight increase in the adhesion force with the decreasing lubricant thickness are found. When z

_{r}= 10 nm, the TP is in contact with the lubricant at 1 nm ≤ t

_{lub}≤ 3 nm, at which the variation trends of the HMS and adhesion force with the lubricant thickness are the same as for the case of the flying state. However, if t

_{lub}< 1 nm, the adhesion force increases remarkably because of the contact between the TP and the DLC layer. From the perspective of avoiding large adhesion force, the lubricant thickness should not be small. When z

_{r}= 7 nm, the slider operates at the TP–DLC contact state for the lubricant thickness ranging from 0.5 nm to 3 nm. It is found that the HMS cannot be reduced further through decreasing the lubricant thickness.

#### 3.5. Effects of DLC Thickness on the Interaction Force and HMS at the HDI

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Dawit, Z.S.; Polina, V.K.; Pyung, H. Experimental and direct numerical analysis of hard-disk drive. J. Mech. Sci. Technol.
**2018**, 32, 3507–3513. [Google Scholar] - Samad, M.A.; Rismani, E.; Yang, H.; Sinha, S.K.; Bhatia, C.S. Overcoat Free Magnetic Media for Lower Magnetic Spacing and Improved Tribological Properties for Higher Areal Densities. Tribol. Lett.
**2011**, 43, 247–256. [Google Scholar] [CrossRef] - Marchon, B.; Pitchford, T.; Hsia, Y.-T.; Gangopadhyay, S. The Head-Disk Interface Roadmap to an Areal Density of Tbit/in2. Adv. Tribol.
**2013**, 2013, 1–8. [Google Scholar] [CrossRef] [Green Version] - Katta, R.R.; Polycarpou, A.A. Microtribodynamics of Magnetic Storage Hard Disk Drives; Springer Science and Business Media LLC: New York, NY, USA, 2013; pp. 2244–2256. [Google Scholar]
- Vakis, A.I.; Polycarpou, A.A. Head-disk interface nanotribology for Tbit/inch2recording densities: Near-contact and contact recording. J. Phys. D Appl. Phys.
**2010**, 43, 225301. [Google Scholar] [CrossRef] - Schultz, B.E. Thermal fly-height control (TFC) technology in Hitachi hard disk drives. Hitachi Glob. Storage Technol. San Jose
**2007**, 1–4. [Google Scholar] - Hua, W.; Liu, B.; Yu, S.; Zhou, W. Contact recording review. Microsyst. Technol.
**2010**, 16, 493–503. [Google Scholar] [CrossRef] - Vakis, A.I.; Polycarpou, A.A. Optimization of thermal fly-height control slider geometry for Tbit/in2 recording. Microsyst. Technol.
**2010**, 16, 1021–1034. [Google Scholar] [CrossRef] - Ono, K. Effect of van der Waals Forces in a Near Contact Head-Disk Interface. IEEE Trans. Magn.
**2008**, 44, 3675–3678. [Google Scholar] [CrossRef] - Suh, A.Y.; Polycarpou, A.A. Adhesive contact modeling for sub-5-nm ultralow flying magnetic storage head-disk interfaces including roughness effects. J. Appl. Phys.
**2005**, 97, 104328. [Google Scholar] [CrossRef] - Liu, B.; Zhang, M.; Yu, S.; Hua, W.; Wong, C.H.; Zhou, W.; Man, Y.; Gonzaga, L.; Ma, Y. Towards fly- and lubricant-contact recording. J. Magn. Magn. Mater.
**2008**, 320, 3183–3188. [Google Scholar] [CrossRef] - Man, Y.; Liu, B.; Ng, K.K.; Yu, S.; Sinha, S.K.; Lim, S.C. Investigations of Light Contact and Lube-Surfing State with Electrical Current. IEEE Trans. Magn.
**2014**, 50, 1–4. [Google Scholar] [CrossRef] - Israelachvili, J.N. Intermolecular and Surface Forces, 3rd ed.; Elsevier: Santa Barbara, CA, USA, 2011. [Google Scholar]
- Hamaker, H. The London—van der Waals attraction between spherical particles. Physica
**1937**, 4, 1058–1072. [Google Scholar] [CrossRef] - Wu, L.; Bogy, D.B. Effect of the Intermolecular Forces on the Flying Attitude of Sub-5 NM Flying Height Air Bearing Sliders in Hard Disk Drives. J. Tribol.
**2002**, 124, 562–567. [Google Scholar] [CrossRef] - Trinh, T.D. Tribological Performance of the Head-Disk Interface in Perpendicular Magnetic Recording and Heat-Assisted Magnetic Recording. Ph.D. Thesis, University of California, San Diego, CA, USA, 2019. [Google Scholar]
- Yu, N.; Polycarpou, A.A. Adhesive contact based on the Lennard–Jones potential: A correction to the value of the equilibrium distance as used in the potential. J. Colloid Interface Sci.
**2004**, 278, 428–435. [Google Scholar] [CrossRef] [PubMed] - Prokopovich, P.; Starov, V. Adhesion models: From single to multiple asperity contacts. Adv. Colloid Interface Sci.
**2011**, 168, 210–222. [Google Scholar] [CrossRef] - Matthes, L.M.; Brunner, R.; Knigge, B.; Talke, F.E. Head Wear of Thermal Flying Height Control Sliders as a Function of Bonded Lubricant Ratio, Temperature, and Relative Humidity. Tribol. Lett.
**2015**, 60, 1–10. [Google Scholar] [CrossRef] - Brunner, R.; Tyndall, G.W.; Waltman, R.J.; Talke, F.E. Adhesion Between Surfaces Separated by Molecularly Thin Perfluoropolyether Films. Tribol. Lett.
**2010**, 40, 41–48. [Google Scholar] [CrossRef] [Green Version] - Lai, T.; Huang, P.; Cai, Y. Adhesion Reduction of Diamond-Like Carbon Films Based on Different Contact Geometries by Using an AFM. J. Adhes.
**2014**, 92, 18–38. [Google Scholar] [CrossRef] - Lifshitz, E.; Hamermesh, M. The theory of molecular attractive forces between solids. Perspect. Theor. Phys.
**1992**, 2, 329–349. [Google Scholar] [CrossRef] - Ninham, B.W.; Parsegian, V.A. van der Waals Interactions in Multilayer Systems. J. Chem. Phys.
**1970**, 53, 3398–3402. [Google Scholar] [CrossRef] - White, L.R.; Dagastine, R.R.; Jones, P.M.; Hsia, Y.-T. van der Waals force calculation between laminated media, pertinent to the magnetic storage head-disk interface. J. Appl. Phys.
**2005**, 97, 104503. [Google Scholar] [CrossRef] - Peng, W.; Crone, R.M.; Jones, P.M.; Hsia, Y.T. Effect of van der Waals force on air-bearing flying characteristics at ultra-low fly heigh. IEEE Trans. Magn.
**2006**, 42, 2483–2485. [Google Scholar] [CrossRef] - Dagastine, R.R.; Bevan, M.; White, L.R.; Prieve, D.C. Calculation of van der Waals Forces with Diffuse Coatings: Applications to Roughness and Adsorbed Polymers. J. Adhes.
**2004**, 80, 365–394. [Google Scholar] [CrossRef] - Derjaguin, V.B. Theorie des Anhaftens kleiner Teilchen. Prog. Surf. Sci.
**1992**, 40, 6–15. [Google Scholar] [CrossRef] - Stanley, H.M.; Etsion, I.; Bogy, D.B. Adhesion of Contacting Rough Surfaces in the Presence of Sub-Boundary Lubrication. J. Tribol.
**1990**, 112, 98–104. [Google Scholar] [CrossRef] - Wu, J.-J. Numerical analyses on elliptical adhesive contact. J. Phys. D Appl. Phys.
**2006**, 39, 1899–1907. [Google Scholar] [CrossRef] - Liu, S.; Wang, Q. Studying Contact Stress Fields Caused by Surface Tractions with a Discrete Convolution and Fast Fourier Transform Algorithm. J. Tribol.
**2001**, 124, 36–45. [Google Scholar] [CrossRef] [Green Version] - Yu, N.; Polycarpou, A.A.; Hanchi, J.V. Elastic contact mechanics-based contact and flash temperature analysis of impact-induced head disk interface damage. Microsyst. Technol.
**2007**, 14, 215–227. [Google Scholar] [CrossRef]

**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 6.**Variations of force and flying height with the nominal flying height under l

_{c}= 10 nm, t

_{lub}= t

_{DLC-disk}= t

_{DLC-slider}= 2 nm: (

**a**) Adhesion force F

_{a}, Contact force F

_{c}, Net force F and (

**b**) Flying height FH.

**Figure 7.**Pressure and distance distributions in the section of y = 0 at different operation states under l

_{c}= 10 nm, t

_{lub}= t

_{DLC-disk}= t

_{DLC-slider}= 2 nm: (

**a**) Pressure P and (

**b**) Height h.

**Figure 9.**Force and flying height versus TP height under t

_{lub}= t

_{DLC-disk}= t

_{DLC-slider}= 2 nm, z

_{r}= 10 nm: (

**a**) Adhesion force F

_{a}, Contact force F

_{c}and (

**b**) Flying height FH.

**Figure 10.**Pressure distribution in the section of y = 0 for different TP heights under t

_{lub}= t

_{DLC-disk}= t

_{DLC-slider}= 2 nm, z

_{r}= 10 nm.

**Figure 11.**Force and head–media spacing versus the lubricant thickness under t

_{DLC-disk}= t

_{DLC-slider}= 2 nm: (

**a**) Adhesion force F

_{a}, Contact force F

_{c}and (

**b**) Head-media spacing HMS.

**Figure 12.**DLC thickness versus adhesion force and HMS for (

**a**) flying height of 11 nm; (

**b**) flying height of 10 nm; (

**c**) flying height of 7 nm; (

**d**) DLC thickness versus contact force for flying height of 7 nm, under l

_{c}= 10 nm, t

_{lub}= 2 nm.

Material | Elastic Modulus (GPa) | Possion Ratio |
---|---|---|

CoCr | 130 | 0.3 |

DLC | 280 | 0.24 |

Al_{2}O_{3}-TiC | 450 | 0.3 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Jiang, L.; Yang, W.; Ma, C.; Yu, Q.
Investigations of Adhesion under Different Slider-Lube/Disk Contact States at the Head–Disk Interface. *Appl. Sci.* **2020**, *10*, 5899.
https://doi.org/10.3390/app10175899

**AMA Style**

Zhang Y, Jiang L, Yang W, Ma C, Yu Q.
Investigations of Adhesion under Different Slider-Lube/Disk Contact States at the Head–Disk Interface. *Applied Sciences*. 2020; 10(17):5899.
https://doi.org/10.3390/app10175899

**Chicago/Turabian Style**

Zhang, Yuyan, Ling Jiang, Weixu Yang, Chenbo Ma, and Qiuping Yu.
2020. "Investigations of Adhesion under Different Slider-Lube/Disk Contact States at the Head–Disk Interface" *Applied Sciences* 10, no. 17: 5899.
https://doi.org/10.3390/app10175899