1 Introduction

Head-disk clearance has been reduced to less than ~5 nm in current hard disk drives using thermal fly-height control (TFC) sliders to increase areal density. Atop the disk surface is a molecularly thin lubricant layer which is used to lower surface energy as well as to protect the interface against intermittent head-disk contacts and disk corrosion. With the continual decrease in the head-disk clearance, there is an increasing probability of interactions between the slider and the lubricant film, resulting in lubricant motion and displacement on the rigid disk surface. Thus, the behavior of the lubricant on the disk surface at such a small clearance plays an important role in establishing a stable head-disk interface (HDI), and it has been studied extensively [1–3]. In particular, lubricant can be transferred to the slider even in the absence of any head-disk contact, causing the problems of flying stiction [4] and lubricant depletion [5] on the disk surface. Various explanations [1, 3, 6–8] have been proposed to account for this phenomenon. Among these, lubricant evaporation from the disk surface and condensation onto the slider was believed to be the dominant mechanism of lubricant transfer at a relative large spacing without head-disk contact happening [6, 9].

Recently, a new experimental observation was reported that lubricant transfer from the disk to slider gets significantly enhanced when head-disk clearance is smaller than a certain value which was referred to as the “critical clearance” [10]. This phenomenon was interpreted as occurrence when the slider and disk are close enough to cause the lubricant film on the disk surface to become unstable due to increased attractive van der Waals interactions from the slider. This phenomenon could result in HDI instability as the spacing is further reduced. Properties of the lubricant film, such as film thickness, bonding ratio, molecular weight, molecular structure, etc., have been demonstrated to have important influence on the lubricant transfer behavior [11, 12]. Therefore, it is expected that the critical clearance and/or the lubricant transfer below the critical clearance can be reduced by improving the properties of lubricant film. However, the effects of lubricant properties on the critical clearance or lubricant transfer rate below the critical clearance are still not sufficiently studied experimentally.

In this article, we report on experimentally observed transfer of perfluoropolyether (PFPE) lubricant from the disk surface to the slider as a function of TFC slider clearance for different lubricant thicknesses, bonding ratios, and molecular structures. The latter is systematically varied in two ways: by increasing the number of hydroxyl (OH) end-groups per molecule from 2 (Zdol) to 4 (Z-Tetraol) (resulting in larger molecular polarity) [13] and by enhancing the stiffness of the molecular main chain (D-4OH) [14]. The effects of these properties of the lubricant film on the critical clearance and lubricant transfer below the critical clearance were studied in detail and the reasons behind the observed effects were discussed.

2 Experimental Details

2.1 Samples

All experiments were performed with “femto” TFC sliders (0.85 mm × 0.7 mm) on 95-mm-diameter media. The root-mean-square (RMS) roughness of the carbon surface on the disks was less than 0.4 nm as measured by a Veeco Dimension V atomic force microscope (AFM) with a standard probe for tapping mode. The typical scan size for these measurements was 5 μm × 5 μm with a scan rate of 0.5 Hz and 256 lines of resolution.

The PFPEs Fomblin Zdol-2500 and Z-Tetraol-2000 were commercially available from Solvay Solexis Inc., while another two kinds of PFPEs, A20H-2000 and D-4OH were synthesized by Moresco Corp. (Kobe, Japan). A20H is a PFPE lubricant with a cyclotriphosphazene at one end and a hydroxyl at the other end. The backbone of A20H is the same as that of Zdol. A 1:1 mixture of Zdol-2500 and A20H-2000 was used in the experiment, which provides longer durability and good performance at high humidity [15]. D-4OH is a novel end-functionalized PFPE lubricant based on the n-perfluoropropyleneoxide (CF2CF2CF2O) main chain which is stiffer than the Fomblin Z backbone (CF2O) p –(CF2CF2O) q used for Zdol and Z-Tetraol [14]. The chemical structures for the A20H, Zdol, Z-Tetraol, and D-4OH lubricants are presented in Fig. 1, and the parameters for the lubricant films used in the experiments are summarized in Table 1.

Fig. 1
figure 1

The chemical structures of the PFPE lubricants used in this study

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Table 1 Lubricant parameters
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All PFPE films were applied to the carbon surfaces from their solutions in the CF3CF2CF2CF2OCH3 solvent (HFE-7100, 3 M Co. Ltd.) using a standard dip-coating methodology, with a typical PFPE concentration of 0.1 g/L. The thickness of the lubricant films was adjusted by varying the disk pull-out speed, and quantified using specular reflection Fourier transform infrared (FTIR) spectroscopy, calibrated to film thickness by X-ray photoelectron spectroscopy (XPS) as described in the Ref. [16].

To obtain a stronger chemical bonding, the Zdol-2500/A20H-2000 lubricated disks were exposed to ultraviolet (UV) irradiation (185 nm in wavelength) in a nitrogen-purged chamber. A set of lubricant films with increased bonding ratios (from 15.5 to 86.7%) were prepared by extending the UV exposure time from 0 s to tens of seconds. The bonded lubricant thickness was measured with FTIR after the removal of the mobile lubricant from the disk by the solvent (HFE-7100). The bonding ratio was obtained by normalizing the bonded lubricant thickness to the initial lubricant thickness.

2.2 Experimental Method

Lubricant transfer from the disk to slider was investigated by flying sliders on the disks as a function of PFPE type (Zdol/A20H, Z-Tetraol, and D-4OH), film thickness, and bonding ratio. The experiments were conducted on a Vena VS-90 spin-stand tester.

In order to set the head-disk clearance, the slider was first allowed to come into contact with the disk by adjusting the input power of the heater in the TFC sliders, using acoustic emission (AE) and/or laser doppler vibrometer (LDV) as contact detection schemes [17, 18]. After contact, the slider’s flying height was increased again to the desired value by reducing the input power of the heater, according to the calibrated power-spacing sensitivity factors of the TFC sliders used in the experiment, and then the slider was moved to an unperturbed part of the disk for an on-track flying test at a linear speed of ~23 m/s. After each test, the disk was immediately interrogated by a Candela optical surface analyzer (OSA), while the slider air-bearing surface (ABS) and deposit end (DE) was photographed under a microscope.

For characterizing the quantity of lubricant transfer, both the lubricant loss from the disk and PFPE pick-up by the slider were measured, so that they could verify each other. The volume of lubricant loss from the disk was quantified by measuring the disk lubricant depletion with OSA as described in the Refs. [1, 5]. This approach might give an overestimate of the lubricant loss volume, because slider-induced shear effects and air-bearing pressure might redistribute the lubricant on the disk. The volume of lubricant picked up by each slider was approximately calculated by multiplying the area of the PFPE deposit by the thickness of it. The thickness of the lubricant on the slider could be roughly estimated by the number of interference fringes [12]. It should be noted that only visible lubricant deposits under a microscope could be counted, so the lubricant pick-up volume might be underestimated using this method.

3 Results

Representative OSA images of disk surfaces after an increasing duration of head-flying on Z-Tetraol at different clearances are shown in Fig. 2. Light lines are seen in the middle of the OSA images for 1 and 0.5 nm clearances, but not seen in the OSA images for 2 or 3 nm clearances. These light lines in the OSA images stand for thinner lubricant and are just located under the position of slider thermal protrusion, so they are speculated to be mainly due to the lubricant loss from the disk. The moguls shown in the upper portion of the OSA images for 2 and 3 nm clearances should be caused by the vibration of slider side pad. Based on the data of lubricant thickness distribution from Candela OSA, the volume of lubricant loss under the slider thermal protrusion was calculated. Summary plots for the volume of Z-Tetraol lubricant loss from the test track versus flying time are presented in Fig. 3 for various values of clearance. It shows that the volume of lubricant loss grows with flying time and reaches a plateau gradually, but the initial growth rate of lubricant loss volume increases significantly as the clearance decreases from 2 to 1 nm. To clarify the relationship between lubricant loss and clearance, average lubricant loss rate for the first 30 min of flying (hereinafter referred to as lubricant loss rate) is calculated and plotted against the clearance in Fig. 4. It is seen that below ~2-nm clearance, a transition occurs where lubricant loss from the disk accelerates sharply. Such a clearance can be considered as the critical clearance for this lubricant film according to the definition in the Ref. [10]. However, the value of critical clearance may vary with lubricant properties, such as film thickness, bonding ratio, lubricant type, etc. Thus, the effects of some of these lubricant parameters on the critical clearance and the lubricant transfer rate below the critical clearance were further studied.

Fig. 2
figure 2

OSA Q-phase images of disk surfaces as a function of head flying time on Z-Tetraol at different clearances: 3, 2, 1, and 0.5 nm

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Fig. 3
figure 3

Lubricant loss volume from the test track for Z-Tetraol as a function of head flying time at various clearances: 3, 2, 1, and 0.5 nm

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Fig. 4
figure 4

Average lubricant loss rate in the first 30 min of head-flying as a function of clearance for Z-Tetraol (film thickness: 12 Å, bonding ratio: 55.5%)

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In order to probe the effect of PFPE bonding ratio on the lubricant transfer rate and critical clearance, Zdol/A20H-coated disks with the same film thickness but different bonding ratios (see Table 1) were used to conduct similar tests as mentioned above. Figure 5a shows the lubricant loss rate versus TFC clearance for the bonding ratios of 15.5, 60, and 86.2%. The similar “critical clearance” phenomenon for lubricant transfer, as seen in Fig. 4, is also observed regardless of the lubricant bonding ratio. However, the lubricant loss rate, especially below the critical clearance, is diminished by increasing the bonding ratio of PFPE film. Moreover, the critical clearance is also reduced from ~5.5 to ~3.5 nm as the bonding ratio rises from 15.5 to 86.2% (−0.33 nm/10% bonding). This can be seen more clearly in Fig. 6 which shows the critical clearance against the bonding ratio. These results were verified by measuring the lubricant pick-up on the sliders after 30-min flying. Compared with the lubricant loss rate, the lubricant pick-up rate reveals similar results in the relationship with both the clearance and lubricant bonding ratio (Fig. 5b). In particular, the critical clearances obtained from the two methods are almost the same, as shown in Fig. 6. The difference between the value of the lubricant pick-up rate and that of the lubricant loss rate is probably due to the intrinsic measurement error of the two characterization approaches for lubricant transfer as mentioned in Sect. 2. In addition, some of the PFPE molecules lost from the test track, especially through evaporation, may not deposit onto the slider, which also contributes to the difference between the lubricant pick-up rate and lubricant loss rate.

Fig. 5
figure 5

a Lubricant loss rate and b lubricant pick-up rate as a function of clearance for Zdol/A20H films with bonding ratios of 15.5, 60, and 86.2% (film thickness: 10.8 Å, flying time: 30 min)

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Fig. 6
figure 6

Critical clearance versus bonding ratio for Zdol/A20H film with a thickness of 10.8 Å. The results are compared using the measurement data from the disk lubricant loss and the lubricant pick-up by slider, respectively

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The dependence of the lubricant transfer on the lubricant thickness was also investigated using various thicknesses of Zdol/A20H films. The lubricant loss rates for three thicknesses of lubricant films are plotted against the clearance in Fig. 7a. It is seen that thinning the lubricant film can also decelerate the lubricant loss, especially below the critical clearance. As regards the lubricant accumulation or pick-up on the slider, similar results were obtained, as shown in Fig. 7b. Figure 8 presents the reproduced critical clearances from the measurements of both the lubricant loss and pick-up as functions of lubricant thickness. It shows that 0.6 nm reduction (from 1.4 to 0.8 nm) in film thickness squeezes the critical clearance from ~6 to below 3 nm, or ~0.5 nm/0.1 nm thickness. Therefore, the lubricant thickness on the disk surface not only affects the lubricant transfer amount as demonstrated previously [5, 12], but also plays an important role in determining the critical clearance as the slider is brought into close proximity to the disk.

Fig. 7
figure 7

a Lubricant loss rate and b lubricant pick-up rate as a function of clearance for Zdol/A20H films with thicknesses of 0.8, 1.08, and 1.4 nm (bonding ratio: 60%, flying time: 30 min)

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Fig. 8
figure 8

Critical clearance versus film thickness for Zdol/A20H film with a bonding ratio of 60%. The results are compared using the measurement data from the disk lubricant loss and the lubricant pick-up by slider, respectively

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By comparing the results in Figs. 5 and 7, it is found that the bonding ratio variation induces greater change in the lubricant transfer rate than the film thickness variation does, especially below the critical clearances. As an example, the lubricant loss rates at 3 nm clearance as functions of bonding ratio and film thickness are re-plotted in Fig. 9. It shows that the lubricant loss rate increases significantly from ~0.5 to ~9.5 k μm3/h as the bonding ratio deceases from 0.862 to 0.155, or −1.2 k μm3/h per 0.1 (or 10%) bonding, while the lubricant loss rate grows modestly as the film thickness increases from 0.8 to 1.4 nm at a smaller rate of 0.7 k μm3/h per 0.1 nm thickness. For the convenience of comparison, 0.1 (or 10%) change in bonding is assumed to be equivalent to 0.1 nm variation in film thickness, since they are almost the same fraction of their corresponding test ranges or available ranges. As for the PFPE bonding ratio, its maximum available range is from 0 to 1 (or 100%). Regarding the film thickness, it is mainly restricted by application requirements. Generally, the thickness of functionalized PFPE film is less than its monolayer thickness, otherwise, the lubricant film will dewet. The monolayer thickness for Zdol/A20H used in this study is about 1.8 nm [19]. In addition, the lubricant film cannot be too thin in order to keep enough film coverage and low surface energy. So the thickness for Zdol/A20H can vary from 0.8 to 1.8 nm, approximately, i.e., 1 nm range for change at most. Therefore, in the range of bonding ratio and film thickness discussed above, the bonding ratio plays a more important role in determining the lubricant transfer rate, especially below the critical clearance. On the contrary, the critical clearance is more sensitive to lubricant thickness according to the results shown in Figs. 6 and 8.

Fig. 9
figure 9

Lubricant loss rate as functions of bonding ratio (upper curve, film thickness: 10.8 Å) and film thickness (lower curve, bonding ratio: 60%) for Zdol/A20H at 3 nm clearance in 30-min head-flying

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Finally, the influence of lubricant type or molecular structure on the lubricant transfer behavior at a decreasing clearance was preliminarily studied using Zdol/A20H, Z-Tetraol, and D-4OH. A composite graph for the lubricant loss rate versus clearance is presented in Fig. 10 for the three kinds of lubricants. Here, the thickness and bonding ratio of the Zdol/A20H film are 1.08 nm and 60%, respectively, while the parameters for Z-Tetraol and D-4OH are listed in Table 1. Figure 10 shows that the lubricant loss rates of Zdol/A20H and Z-Tetraol get dramatically increased when the clearances are less than ~4 and ~ 2 nm, respectively, while the lubricant loss rate for D-4OH remains at a very low level throughout the test clearances. This indicates that Z-Tetraol has a smaller critical clearance than Zdol/A20H, although the Z-Tetraol film used in the experiment is slightly thicker. For the D-4OH film, the critical clearance is hard to determine, because no significant increase of the lubricant loss rate is seen in the test range of the clearance. There are two possible reasons for this phenomenon. One may be that the loss rate of D-4OH is still very small even below the critical clearance, the other may be that the critical clearance for the D-4OH film is smaller than 1 nm (the lower limit of the test range). No matter which reason is true, the similar effect is obtained that the lubricant transfer rate at ultra-small clearance can be dramatically reduced by using D-4OH. It is seen in Fig. 10 that the lubricant loss rate falls remarkably in the order: Zdol/A20H > Z-Tetraol > D-4OH. The results discussed above indicate that both the lubricant transfer rate and critical clearance can be reduced by increasing the number of OH end-groups per PFPE molecule (from Zdol/A20H to Z-Tetraol), i.e., increasing the polarity of lubricant molecule. In addition, these results suggest that using the lubricant with a stiffer main chain (from Z-Tetraol to D-4OH) can further suppress the lubricant transfer especially at ultra-small clearances. However, the dependence of critical clearance on the main chain stiffness of PFPE molecules needs further verification.

Fig. 10
figure 10

Lubricant loss rate as a function of clearance for Zdol/A20H, Z-Tetraol, and D-4OH

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4 Discussion

The experimental results discussed above demonstrate that the critical clearance for lubricant transfer as well as the transfer rate below the critical clearance is dependent on the properties of the lubricant film on the disks, such as film thickness, bonding ratio, and molecular structure. In particular, the critical clearance is a strong function of the film thickness and molecular polarity. The lubricant transfer rate below the critical clearance relies more on the lubricant bonding ratio and molecular flexibility.

Disjoining pressure, arising from interaction energies of molecules in a film being different from that in the bulk, is an important physical parameter determining the properties and behavior of thin liquid film on a solid surface [12]. It is defined as the negative derivative of the Gibbs free energy per unit area with respect to film thickness [20–22]. In terms of the disjoining pressure, the dramatic enhancement of lubricant transfer below the critical clearance has been attributed to the instability of the lubricant film induced by van der Waals forces between the head and lubricant [10, 23]. The “effective” disjoining pressure of the lubricant film in the HDI is given by [12, 24]:

$$ \Uppi_{\text{eff}} \left( {h,d} \right) = \Uppi_{\text{disk}} \left( {h,\infty } \right) - \Uppi_{\text{head}} \left( {h,d} \right) $$
(1)

where h is lubricant thickness on the disk, and d represents the spacing between the head and disk carbon surface. Пdisk is the disjoining pressure exerted on the lubricant film by the disk surface in a free space without the head, while Пhead stands for the conjoining pressure exerted on the lubricant film by the presence of the head surface. In the case of OH-terminated PFPE films, Пdisk and Пhead consist of both dispersive (d) component and polar (p) component. Regarding the polar component, \( \Uppi_{\text{disk}}^{\text{p}} \) includes hydrogen bonding (structural component) and dipole–dipole interaction, while \( \Uppi_{\text{head}}^{\text{p}} \) contains only the dipole–dipole interaction. For the PFPE lubricants, the dipole–dipole interaction with the disk or head surface is very weak compared to the dispersive or hydrogen bonding interaction [25], so its contribution is ignored in the following calculation of the effective disjoining pressure. Thus, Eq. 1 can be rewritten as:

$$ \Uppi_{\text{eff}} \left( {h,d} \right) = \Uppi_{\text{disk}}^{\text{d}} \left( {h,\infty } \right){ + }\Uppi_{\text{disk}}^{\text{p}} \left( {h,\infty } \right) - \Uppi_{\text{head}}^{\text{d}} \left( {h,d} \right) $$
(2)

where \( \Uppi_{\text{disk}}^{\text{d}} \) and \( \Uppi_{\text{head}}^{\text{d}} \) are the dispersive components of Пdisk and Пhead, respectively. The expressions of them are as follows [12]:

$$ \Uppi_{\text{disk}}^{\text{d}} \left( {h,\infty } \right) = \frac{{A_{\text{DLA}} }}{{6{{\uppi}}\left( {h + d_{0} } \right)^{3} }} $$
(3)
$$ \Uppi_{\text{head}}^{\text{d}} \left( {h,d} \right) = \frac{{A_{\text{LAH}} }}{{6{{\uppi}}\left( {d - h - d_{0} } \right)^{3} }} $$
(4)

with A DLA and A LAH being the effective Hamaker constants for the lubricant/disk interactions and the lubricant/head interactions through the air gap, respectively. d 0 represents the non-bonding distance of closest approach between the lubricant molecule and the underlying carbon surface. Unlike their dispersive counterpart, the polar component \( \Uppi_{\text{disk}}^{\text{p}} , \) mainly originating from the hydrogen bonding between the polar end-groups of the PFPE lubricant and the active polar sites on the carbon surface and from the consequent molecular layering effect, can not be derived from a simple potential. It usually oscillates and decays with increasing film thickness. In a pure phenomenological viewpoint, the film thickness dependence of the polar disjoining pressure, \( \Uppi_{\text{disk}}^{\text{p}} \), can be approximated by a sinusoidal function in the limit of low thickness [3, 6], as below:

$$ \Uppi_{\text{disk}}^{\text{p}} \left( {h,\infty } \right) = \frac{{{{\uppi}}\gamma_{0} }}{{h_{0} }} \cdot \sin \left( {{{\uppi}}\frac{h}{{h_{0} }}{ + }\alpha } \right) $$
(5)

where h 0 is the dewetting thickness of the lubricant film, γ0 is the oscillatory amplitude of the polar component of the surface energy on the lubricant film, and α is the phase shift. They are determined by the curve fit to the experimental data of the lubricant film surface energy reported in the Refs. [19, 26, 27]. The values of the parameters in the above equations are listed in Table 2, for Zdol/A20H, Z-Tetraol, and D-4OH. Here, only the effective Hamaker constant A LAH is unknown, but it can be estimated using the equation \( A_{\text{LAH}} /12{{\uppi}}d_{0}^{2} { = }W_{\text{LH}}^{\text{d}} { = }2\sqrt {\gamma_{\text{L}}^{\text{d}} \gamma_{\text{H}}^{\text{d}} } \) [28]. Where \( W_{\text{LH}}^{\text{d}} \) is the dispersive component of the adhesion work between the bulk lubricant and the head surfaces, \( \gamma_{\text{L}}^{\text{d}} \) and \( \gamma_{\text{H}}^{\text{d}} \) are the dispersive contributions to the surface energies of the bulk lubricant and the head, respectively. For the three types of lubricants used in the study, their values of d 0 and \( \gamma_{\text{L}}^{\text{d}} \) are similar: d 0 ≈ 0.29 nm, \( \gamma_{\text{L}}^{\text{d}} \) ≈ 14 mJ/m2. Thus, A LAH is mainly dependent on the head surface energy \( \gamma_{\text{H}}^{\text{d}} \) and approximated to be 1.6 × 10−19 J if \( \gamma_{\text{H}}^{\text{d}} \) = 45 mJ/m2 for CNx-coated surface. But, \( \gamma_{\text{H}}^{\text{d}} \) may vary from head to head due to the difference in material composition and processing. To analyze the effect of the head surface energy, both a smaller (1.0 × 10−19 J) and a larger (4.0 × 10−19 J) values for A LAH were used in the simulation to accommodate the known Hamaker constants of materials (0.4–4 × 10−19 J) [28].

Table 2 Parameters used to calculate the effective disjoining pressure of the lubricant films
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Based on Eqs. 2–5 and the parameters in Table 2, the effective disjoining pressure Пeff of the lubricant film can be calculated. Figure 11 shows the simulated Пeff as a function of head-disk clearance (C = d – h − d 0) for the Zdol/A20H films with different thicknesses. In Fig. 11a and b, a larger and a smaller values for A LAH are used, respectively, but only dispersive contributions to Пeff are taken into account, i.e., \( \Uppi_{\text{disk}}^{\text{p}} \, = \,0 \). To clarify the effect of the polar interaction, Fig. 11c shows the simulation results of Пeff with \( \Uppi_{\text{disk}}^{\text{p}} \) considered, using a larger value of A LAH as a comparsion with Fig. 11a. The results show that, as the clearance is reduced to below several nanometers, the effective disjoining pressure holding the lubricant to the disk surface starts to decrease dramatically and eventually becomes negative. Once the condition, Пeff < 0, is satisfied, there will be a net attractive pressure of the lubricant film toward the head instead of the disk surface, which makes the lubricant film unstable and may accelerate the lubricant transfer through bridging or evaporating. So the clearance at which Пeff = 0 can be considered as the critical clearance for lubricant transfer.

Fig. 11
figure 11

Пeff as a function of clearance and of film thickness for Zdol/A20H. The results are compared using a \( \Uppi_{\text{disk}}^{\text{p}} = 0 \), A LAH = 4 × 10−19 J, b \( \Uppi_{\text{disk}}^{\text{p}} \, = \,0\, \), A LAH = 1 × 10−19 J, and c \( \Uppi_{\text{disk}}^{\text{p}} \ne \, 0 \), A LAH = 4 × 10−19 J

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It is also seen that a thinner lubricant film has a larger Пeff and thus obtains a smaller critical clearance regardless of the value for A LAH or whether \( \Uppi_{\text{disk}}^{\text{p}} \) is considered. This result qualitatively agrees with the experimental observations shown in Fig. 8. If ignoring the polar interaction, the critical clearance for Пeff = 0 can be simply derived from Eqs. 2–4 and expressed as below:

$$ C_{\text{crit}} = \left( {\frac{{A_{\text{LAH}} }}{{A_{\text{DLA}} }}} \right)^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}} \left( {h + d_{0} } \right) $$
(6)

In terms of this equation, the critical clearance scales linearly with the lubricant thickness and decreases with reducing the ratio of A LAH to A DLA, as illustrated in Fig. 11a and b. When the polar interaction, \( \Uppi_{\text{disk}}^{\text{p}} , \) is taken into consideration as shown in Fig. 11c, the effective disjoining pressure for all the films with the thickness from 0.8 to 1.4 nm is significantly raised, the critical clearances of them are reduced consequently, and the variation of critical clearance with film thickness becomes smaller. These results suggest that the polar interaction between the PFPE flim and the disk surface is important for the film stability especially at ultra-small spacing, but it hardly changes the dependence of critical clearance on the film thickness in the thickness range we studied.

In the same way, the effective disjoining pressure as a function of head-disk clearance for different types of lubricants with the same thickness is also simulated and shown in Fig. 12. The simulation results show that Пeff increases in the order Zdol/A20H < Z-Tetraol < D-4OH, and the critical clearance decreases in the same order accordingly, which agrees with the experimental results in Fig. 10 to some extent. Figure 12a and b reveal that the critical clearance can be reduced by using the lubricant with a larger value of A DLA or by decreasing the value for A LAH (e.g., by lowering the head surface energy), which can be interpreted using Eq. 6. Taking \( \Uppi_{\text{disk}}^{\text{p}} \) into account, as illustrated in Fig. 12c, increases the effective disjoining pressure of the three kinds of lubricants and makes the critical clearances of them become smaller, especially for D-4OH which has a much higher value of \( \Uppi_{\text{disk}}^{\text{p}} \). Nevertheless, the order of the critical clearances in these lubricants is not essentially changed.

Fig. 12
figure 12

Пeff versus clearance for Zdol/A20H, Z-Tetraol, and D-4OH with the same film thickness of 1.1 nm: a \( \Uppi_{\text{disk}}^{\text{p}} = \,0 \), A LAH = 4 × 10−19 J, b \( \Uppi_{\text{disk}}^{\text{p}} = \,0 \), A LAH = 1 × 10−19 J, and c \( \Uppi_{\text{disk}}^{\text{p}} \ne \, 0 \), A LAH = 4×10−19 J

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Since the actual lubricant-disk and lubricant-head interactions are dependent upon many factors that are not considered here, such as surface roughness, slider design, air-bearing pressure, shear stress, etc., the modeling data to be discussed in Figs. 11 and 12 are interpreted on a relative basis to isolate the lubricant effect. For example, the air shear-induced lubricant moguls which have larger local thickness and hence smaller disjoining pressure probably lead to earlier lubricant instability resulting in the transfer of lubricant from the disk to slider [12]. In addition, lubricant evaporation and condensation can also contribute to the lubricant transfer [6]. Therefore, the critical clearances obtained from this model are shown to be smaller than the experimental results.

The effect of the lubricant properties on the lubricant transfer rate at or just below the critical clearance, as shown in Figs. 7 and 10, can not be simply attributed to the effective disjoining pressure, because the Пeff for all lubricant films is nearly zero at that situation. However, besides the value of disjoining pressure, its negative slope with respect to the film thickness, known as the lubricant film stiffness, is also able to influence the lubricant stability on a disk and its transfer from a disk to a slider [29, 30]. The lubricant film stiffness expresses its ability to not be easily disturbed, so the effective stiffness of lubricant film (−dПeff/dh) can be considered as a measure of the resistance to the lubricant transfer from the disk to the slider here. Figure 13 shows the effective disjoining pressure-film thickness curves for Zdol/A20H films at different head-disk spacings d. For the lubricant films with thicknesses of 0.8, 1.0, 1.2, and 1.4 nm, Пeff is zero at their corresponding critical spacings (or clearances), but the slope of these curves is different. It is seen that a thinner lubricant film has a bigger slope of the Пeff curve when Пeff = 0, i.e., thinner lubricant film behaves stiffer at the critical clearance and consequently shows lower transfer rate. This coincides with the experimental results qualitatively. Compared with the results in Fig. 13a, using a smaller value for A LAH in Fig. 13b does not change the slop of these curves very much, indicating that the head surface energy has less influence on the effective stiffness of the lubricant film at the critical clearance. However, the polar interaction between the lubricant and disk surface plays an important role in the lubricant film stiffness. Figure 13c reveals that, when taking into account the polar disjoining pressure, the slope of the Пeff curves becomes much larger, nevertheless its dependence on the film thickness does not change.

Fig. 13
figure 13

Пeff versus film thickness at different spacings d which are specified to make the Zdol/A20H films with the thicknesses of 0.8, 1.0, 1.2, and 1.4 nm just at their corresponding critical clearances: a \( \Uppi_{\text{disk}}^{\text{p}} = \,0 \), A LAH = 4×10−19 J, b \( \Uppi_{\text{disk}}^{\text{p}} = \,0 \), A LAH = 1×10−19 J, c \( \Uppi_{\text{disk}}^{\text{p}} \ne \, 0 \), A LAH = 4×10−19 J

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The effective disjoining pressure-film thickness curves for different types of lubricants with the same film thickness are shown in Fig. 14. These results indicate that the effective stiffness of the lubricants at their corresponding critical clearances increases in the order Zdol/A20H < Z-Tetraol < D-4OH, no matter which value is used for A LAH or whether the polar interaction is considered. Thus, the lubricant transfer rate at or below the critical clearance should decrease in the order Zdol/A20H > Z-Tetraol > D-4OH, which gives an explanation to the experimental observations in Fig. 10. Similar to the results in Fig. 13, the polar interaction between the lubricant and the disk also significantly increases the effective stiffness of the lubricant films at the critical clearances (Fig. 14c), but decreasing the value for A LAH does not (Fig. 14b).

Fig. 14
figure 14

Пeff versus film thickness for Zdol/A20H, Z-Tetraol, and D-4OH at their corresponding critical clearances: a \( \Uppi_{\text{disk}}^{\text{p}} = \,0 \), A LAH = 4 × 10−19 J, b \( \Uppi_{\text{disk}}^{\text{p}} = \,0 \), A LAH = 1 × 10−19 J, c \( \Uppi_{\text{disk}}^{\text{p}} \ne \, 0 \), A LAH = 4 × 10−19 J

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The enhanced disjoining pressure and film stiffness for Z-Tetraol and D-4OH should benefit from their molecular structure. Both of the two lubricants have more OH end-groups than Zdol or A20H, which gains larger adhesion with the disk surface. Moreover, D-4OH has a stiffer main chain, so the work required for the segmental motion of D-4OH molecules perpendicular to the disk surface is larger, hence further reducing the lubricant disturbance or transfer [26].

Unfortunately, the model based on the effective disjoining pressure does not show the relationship of lubricant bonding ratio with the critical clearance or lubricant transfer rate. However, an increased fraction of the bonded lubricant has been demonstrated to lower the surface energy, especially the polar surface energy, of the lubricant film on the disk surface [31, 32], which suggests that the increase in lubricant bonding ratio can enhance the disjoining pressure or adhesion between the lubricant and the underlying carbon surface, making the lubricant film more stable. This might be a reason for the reduction of the critical clearance for the lubricant film with a larger bonding ratio. In addition, it is thought that only mobile lubricant molecules are able to move freely on the disk surface and have the chance to be picked up by the flying slider. Therefore, increasing the lubricant bonding ratio reduces the mobile component of the lubricant film, hence decreasing the amount or probability for the lubricant molecules to be transferred to the slider. Below the critical clearance, the lubricant transfer volume may rely more on the supply of the mobile molecules, because the lubricant transfer rate is much higher and the effective disjoining pressure is similar (near zero). On this basis, the effect of the bonding ratio on the lubricant transfer rate below the critical clearance, as shown in Figs. 5 and 9, can be explained.

5 Conclusions

Based on the experimental results and discussions above, the following conclusions can be drawn.

  1. (1)

    The lubricant transfer from the disk to slider is significantly enhanced when the slider flies below a critical clearance. Both the value of the critical clearance and the lubricant transfer rate below the critical clearance are shown to decrease by reducing the film thickness, increasing the lubricant bonding ratio, or increasing the molecular polarity or main chain stiffness.

  2. (2)

    In the range of lubricant thickness and bonding ratio we studied, the critical clearance is more sensitive to the lubricant thickness, while the lubricant transfer rate below the critical clearance is more dependent on the lubricant bonding ratio.

  3. (3)

    The lubricant stability or critical clearance is primarily determined by the effective disjoining pressure of the film in the HDI. The influences of the lubricant properties on the lubricant transfer rate below the critical clearance can be partially interpreted by the effective stiffness of the lubricant film.

  4. (4)

    In the range of the lubricant thickness we studied, polar interactions between the lubricant and the underlying carbon surface are shown by the simulation results to enhance the effective disjoining pressure and the stiffness of the lubricant film, which results in smaller critical clearance and lower lubricant transfer rate.

  5. (5)

    Decreasing the intermolecular interaction of the head on the lubricant film by lowering the head surface energy should be also an effective way to reduce the critcial clearance.

These results will help guide the lubricant design for the HDI with ultra-small spacing. It is believed that in a real HDI system, many considerations need to be taken into account, so a combination of the approaches discussed above is probably required.