# Disjoining Pressure Derived from the Lennard–Jones Potential, Diffusion Equation, and Diffusion Coefficient for Submonolayer Liquid Film

## Abstract

**:**

## 1. Introduction

_{d}= A/6πh

^{3}, where A is the Hamaker constant and h is the lubricant film thickness. For a polar lubricant such as Z-dol, the measured polar surface energy γ

_{p}as a polynomial function of lubricant thickness h was additionally considered as a polar disjoining pressure Π

_{p}= ∂γ

_{p}/∂h. Marchon and Karis [12] proposed an advanced diffusion equation using the disjoining pressure Π

_{d}= A/6π(h + d

_{0})

^{3}and showed that the diffusion characteristics of the steplike boundary of a non-polar lubricant layer with a thickness of a few nanometers can be quantitatively evaluated by this equation. Here d

_{0}is the van der Waals (vdW) distance between diamond-like-carbon (DLC) substrate and PFPE lubricant film. This advanced disjoining pressure equation seems to be enlightened by the dispersive surface energy function of the form −A/12π(h + d

_{0})

^{2}proposed by Waltman et al. [13]. As PFPE lubricant film has been decreased to a monolayer thickness and the main fraction was bonded thereafter, a question arose whether the conventional diffusion equation based on the Poiseuille flow model and disjoining pressure for continuum medium can still hold for a submonolayer lubricant film.

^{−19}J. It was also found that the replenishment speed was reduced as the elapsed time increased to a few days. In that paper, the reflow speed of replenishment to a rectangular depleted scar was also investigated. Based on the general feature that replenishment time decreases proportionally to the square of the depleted scar width, a generic design chart for lubricant film was proposed. From this chart, we can evaluate the allowable width of scar in which 50% replenishment can be achieved after one revolution time of 8.3 ms (at 7200 rpm disk speed) as a function of the lubricant thickness and the value of A/μ (μ is the lubricant viscosity).

## 2. Disjoining Pressure for Submonolayer Liquid Film

#### 2.1. Analytical Model and Assumptions

_{m}, which is nearly equal to the diameter of gyration. When lubricant is a submonolayer film, its mean thickness h is smaller than h

_{m}. In the experiments, the density ρ

_{h}of the submonolayer is measured. Then, the submonolayer thickness h and its density ρ

_{h}are respectively defined as

#### 2.2. Disjoining Pressure Derived from Attractive Term in LJP

^{n}, the sum of the interactive energy between the liquid molecule and whole solid molecules is derived by integrating the effect of an annular region from z = d to ∞, and x = 0 to ∞. The integrated result is given by

_{s}is the number density of the DLC.

_{a}(d) due to only the attractive term of the LJP is written as

_{sa}(subscripts s and a denotes submonolayer theory and attractive term, respectively) due to the attractive term at a distance of d

_{0}is given by

_{m}. When the liquid film thickness h is larger than h

_{m}, we can obtain the disjoining pressure Π

_{a}for the continuous liquid film by replacing Ah/h

_{m}by A in Equation (5), as follows:

_{0}is considered to be the mean distance between the mean center position of liquid molecules at the contacting film surface and the mean center position of solid surface molecules and corresponds to the averaged sum of vdW radii of liquid molecules and solid surface molecules. The quantity Π

_{sa}in Equation (5) is always negative when h > 0, and becomes zero in the second order of h when h → 0. In contrast, Π

_{a}in Equation (6) becomes zero in the first order of h when h → 0. Further, it is unnatural that Π

_{a}approaches a constant value as h increases to infinity.

_{c}

_{1}for a continuous liquid film is obtained from Equation (6), by ignoring the second term, as follows:

_{0}[7,8,9,10,11]. Although a detailed derivation process of this equation is not known, Π

_{c}

_{1}always takes a positive value when A > 0. It takes a maximum value at h = 0 and becomes zero as h increases to infinity. Therefore, Equations (5) and (6) show completely different characteristics from Π

_{c1}. In submonolayer diffusion theory [17], the present author intuitively replaced the Hamaker constant A by Ah/h

_{m}because the Hamaker constant is proportional to the density of submonolayer film and used the disjoining pressure Π

_{s}

_{1}of the form

_{s}

_{1}in Equation (8) becomes zero in the first order of h as h decreases to zero. This seems to be more reasonable than Π

_{c}

_{1}in a submonolayer regime. Here Π

_{s}

_{1}is always positive when A > 0.

#### 2.3. Disjoining Pressure Derived from LJP

_{0}

^{3}in Equations (5) and (6) is why Π

_{sa}in Equation (5) and Π

_{a}in Equation (6) have negative sign compared with Π

_{s}

_{1}and Π

_{c}

_{1}and Π

_{a}does not tend to zero as h increases. It is considered that the existence of this term is a result of the derivation only from attractive term in the LJP. Therefore, let us next derive the disjoining pressure from the LJP including both attractive and repulsive terms.

_{0}, the LJP between a pair of molecules is given by

^{n}, the surface interaction energy in Equation (3) due to the attractive term in Equation (9) corresponds to the case where C = 2εr

_{0}

^{6}. Because the repulsive term in the LJP (9) is expressed as a potential w(r) = εr

_{0}

^{12}/r

^{12}, the interactive energy W

_{r}(d) due to the repulsive term can be obtained, by replacing C by −C r

_{0}

^{6}/2 in Equation (3) and using n = 12 and ${\pi}^{2}C{\rho}_{s}{\rho}_{l}=A$, as follows:

_{sr}due to the repulsive term in the LJP at vdW distance of d

_{0}is given by

_{sa}in Equation (5) and Π

_{sr}in Equation (11), the total disjoining pressure Π

_{st}derived from the LJP becomes

_{m}), replacing Ah/h

_{m}by A, the total disjoining pressure Π

_{t}is given by

_{t}has a constant value as h increases to infinity and this result is unnatural. Therefore, the second term in the square bracket must be zero and thereby we have a condition

_{0}in the LJP and mean vdW distance d

_{0}between the liquid film and solid surface. Note that without this relationship, the disjoining pressure Π

_{t}in Equation (13) takes a constant value even at infinite liquid thickness. This implies that the relationship in Equation (14) is a necessary condition for consistently modeling the mutual interactions expressed by the LJP between discrete molecules as a simple equation with respect to the averaged physical quantities between continuous solid and liquid film.

_{s}

_{2}including both attractive and repulsive terms becomes, from Equation (12),

_{c}

_{2}for a multilayer film is given by

_{s}

_{2}in Equation (15) for a submonolayer film always takes a positive value and becomes zero in the second order of h when h approaches zero. Because these features are consistent with Π

_{s}

_{1}, it can be said that Π

_{s}

_{2}corresponds to the rigorous solution of the previous disjoining pressure Π

_{s}

_{1}for submonolayer film. In Equation (16), Π

_{c}

_{2}also has a positive value in the entire range of film thickness similar to the conventional disjoining pressure Π

_{c}

_{1}. Here Π

_{c}

_{2}becomes zero in the first order of h as h decreases to zero and zero in the order of 1/h

^{3}as h increases to infinity. Therefore, it can be said that Π

_{c}

_{2}corresponds to the rigorous solution of the prevailing disjoining pressure Π

_{c}

_{1}. Although the derivation process of the conventional disjoining pressure Π

_{c}

_{1}is not clear to the best of the author’s knowledge, the present author thought that the validity of Π

_{c}

_{1}is not appropriate for submonolayer lubricant film [17]. This is because the disjoining pressure Π

_{c}

_{1}takes a maximum value at zero lubricant film thickness. From this analytical study, however, it is found that the rigorous disjoining pressure Π

_{c}

_{2}derived from the LJP becomes zero at h = 0. It is also found that Π

_{c}

_{1}can be approximated from the solution effectively used in a film thickness region h ≥ 0.5d

_{0}because the second term of Π

_{c}

_{2}in Equation (16) becomes negligibly small.

_{c}

_{1}, Π

_{c}

_{2}, Π

_{s}

_{1}, and Π

_{s}

_{2}normalized by A/6πd

_{0}

^{3}are plotted in Figure 2a,b as a function normalized film thickness h/d

_{0}. Here, it is assumed that d

_{0}= 0.3 nm and h

_{m}= 4d

_{0}= 1.2 nm. Figure 2b shows an enlarged illustration of Figure 2a to show the difference between Π

_{s}

_{1}and Π

_{s}

_{2}. From Figure 2a it is clear that Π

_{c}

_{2}takes a maximum value at h/d

_{0}= ~0.25 and then approaches zero in contrast to Π

_{c}

_{1}. In the region where h/d

_{0}> 0.5, Π

_{c}

_{1}is very close to the rigorous disjoining pressure Π

_{c}

_{2}. In addition, it is also found from Figure 2b that the disjoining pressure Π

_{s}

_{1}that was proposed in the previous paper [17] is very close to Π

_{s}

_{2}where h/d

_{0}> 0.5. Because it is assumed that h

_{m}= 4d

_{0}, Π

_{s}

_{2}becomes equal to Π

_{c}

_{2}at h/d

_{0}= 4 as can be seen from Figure 2b. This means that the disjoining pressure for submonolayer theory is continuously connected with the conventional theory as the film thickness h exceeds the monolayer thickness h

_{m}.

## 3. Diffusion Equation Incorporating the Rigorous Disjoining Pressure

#### 3.1. Diffusion Flow Model for Submonolayer Film

_{m}. The height of the film surface changes with respect to time t and position x and, thus, can be expressed as h(x, t). If we consider a small rectangular element of ABCD (dx × dz) close to the film surface as shown in Figure 3, the flow velocity u(z + dz) on the upper plane A–B is approximated by the mean velocity of molecules at the surface of h(z + dz). Similarly, it is considered that the flow velocity u(z) can be approximated by the mean molecular velocity at the surface of h(z). If liquid molecules do not slip perfectly, u(z) and u(z + dz) can have different values, so that the mean velocity can have the first derivative with respect to z. Then, we can assume that the shear stress τ

_{z}acting on the z plane is proportional to du/dz, similar to a Newtonian flow. If we define this proportional constant as the viscosity μ

_{z}, we can write

_{z}is generally a function of z.

_{z}changes linearly from μ

_{0}to μ

_{m}for the sake of analytical simplicity, as follows:

_{z}and τ

_{z+dz}given by Equation (17) are applied to the lower and upper surfaces, respectively, of the element. The pressure p is caused by the ambient pressure, disjoining pressure, and capillary pressure. These pressures are a function of only x. Because p

_{x}dz and p

_{x}

_{+dx}dz are applied on the left- and right-hand sides of the element, as shown in Figure 2, we can obtain the basic equation of equilibrium of the form

#### 3.2. Derivation of the Diffusion Equation

_{s}

_{2}and approximated pressure p = −Π

_{s}

_{1}into Equation (25), we can obtain the diffusion equation for submonolayer liquid film of the form

_{s}

_{2}and Π

_{s}

_{1}are respectively given by

_{c}

_{1}into the conservation equation of the Poiseuille flow for multilayer liquid film, we can obtain the conventional diffusion equation of the form

_{c}

_{2}we can obtain

_{s}

_{1}and J

_{s}

_{2}become negative when h < ~d

_{0}/2, this implies that there is a limiting liquid thickness that cannot spread.

## 4. Comparison of the Diffusion Coefficient Between Theories and Experiment

#### 4.1. Theoretical Diffusion Coefficient

_{s}

_{2}(h) for submonolayer theory with disjoining pressure Π

_{s}

_{2}is written as

_{s}

_{1}(h) in Equation (29), the diffusion coefficient D

_{s}

_{1}(h) for Π

_{s}

_{1}becomes,

_{c}

_{1}(h) for the conventional disjoining pressure Π

_{c}

_{1}in Equation (7) is given by

_{c}

_{2}(h) for the rigorous conventional disjoining pressure Π

_{c}

_{2}in Equation (31) is

#### 4.2. Calculated Diffusion Coefficients and Comparison with Experimental Ones

_{m}= 2.1 nm is used in the calculations here unless otherwise noted. Marchon and Karis [12] used 0.317 nm for the vdW distance d

_{0}. Fukuda et al. [23] estimated that the force equilibrium distance of the non-bonded pair potential of a PFPE film-coated carbon surface was 0.46 nm based on coarse-grained molecular dynamics analysis. If we assume r

_{0}= 0.46 nm, we obtain d

_{0}= 0.29 nm. Mate [14] used d

_{0}= 0.3 nm when theoretically evaluating the diffusion profiles of a small droplet. Referring to these values, we use d

_{0}= 0.3 nm in the following calculation.

_{s}

_{1}(h), D

_{s}

_{2}(h), D

_{c}

_{1}(h), D

_{c}

_{2}(h), and Waltman’s experimental values. The common parameter A/6πμ

_{m}of theoretical diffusion coefficients is determined so that D

_{s}

_{1}(h), D

_{s}

_{2}(h), D

_{c}

_{1}(h), and D

_{c}

_{2}(h) can coincide with the experimental value of D = 7.69 × 10

^{−13}m

^{2}/s at h = 1.29 nm. Because the actual viscosity ratio is not known, we assume that r

_{μ}= μ

_{m}/μ

_{0}= 1. The effect of r

_{μ}on the diffusion coefficient functions is small as shown in Figure 5.

_{s}

_{2}of submonolayer diffusion theory plotted by a black solid line shows excellent agreement with experimental values plotted by triangular symbols. The dashed line indicating D

_{s}

_{1}is very close to D

_{s}

_{2}in the region h > d

_{0}and can be used as a good approximated solution for D

_{s}

_{2}. The black dotted line shows D

_{s}

_{1}when d

_{0}= 0.2 nm and r

_{μ}= 0.3 are used as in the same conditions as Waltman compared D

_{s}

_{2}with the experimental data [18]. We note that D

_{s}

_{2}tends to deviate from the experimental data to the upper side in the lower thickness region. In his paper, A/6πμ

_{m}was determined so that D

_{s}

_{1}(h) can coincide with the experimental diffusion coefficients at h = 0.25 and 0.51 nm. Therefore, D

_{s}

_{1}(h) at h = 7.4, 9.7, 12.9 nm deviates to the lower side of the experimental values to some extent. From these results it is clear that D

_{s}

_{2}and D

_{s}

_{1}can agree well with experimental values when a reasonable value of the vdW distance of 0.3 nm is used for calculation.

_{c}

_{2}and D

_{c}

_{1}, depicted respectively by blue solid and dashed lines, do not decrease until h decreases to ~0.4 nm and show characteristics completely different from the experimental values. This is because D

_{c}

_{1}takes a maximum value at h = 3d

_{0}= 0.9 nm and D

_{c}

_{2}is very close to D

_{c}

_{1}. The difference between D

_{c}

_{2}and D

_{c}

_{1}is less than 5% in the region h > d

_{0}, similarly to the relationship between D

_{s}

_{2}and D

_{s}

_{1}. The blue dotted line indicates D

_{c}

_{1}when d

_{0}= 0.2 nm. In this case D

_{c}

_{1}takes a maximum value at h = 0.6 nm and thus deviates further from the experimental results. From the comparison between theory and experiment in Figure 5, it can be said that the submonolayer diffusion equation including the decrease in density can evaluate the experimental diffusion coefficient function. It is also confirmed that the submonolayer diffusion equation using disjoining pressure derived from the LJP is more valid for the kinetics of submonolayer liquid film.

_{μ}(=μ

_{m}/μ

_{0}) and h

_{m}on the diffusion coefficient function for submonolayer theory are shown in Figure 5 when d

_{0}= 0.3 nm. Figure 5a shows D

_{s}

_{1}and D

_{s}

_{2}for various values of r

_{μ}when h

_{m}= 2.1 nm. Here r

_{μ}is changed as 0.5, 1, and 2. The solid and dashed lines depicting D

_{s}

_{2}and D

_{s}

_{1}for r

_{μ}= 1, respectively, are the same as shown in Figure 4. Here D

_{s}

_{2}for r

_{μ}= 0.5 depicted by closed circles is almost the same as that for r

_{μ}= 1, whereas D

_{s}

_{2}for r

_{μ}= 2 depicted by closed squares is slightly larger than D

_{s}

_{2}for r

_{μ}= 1 in the region where h < 0.4 nm. For r

_{μ}= 0.5, D

_{s}

_{1}(open circles) becomes slightly smaller than that for r

_{μ}= 1 in a range of small h value and D

_{s}

_{1}for r

_{μ}= 2 (open squares) becomes slightly larger than that for r

_{μ}= 1 in a range of large h values.

_{m}on diffusion coefficient function D

_{s}

_{1}. Because the D

_{s}

_{1}is overlapped with D

_{s}

_{2}and is difficult to distinguish from D

_{s}

_{2}in Figure 5a, the same results of D

_{s}

_{1}for h

_{m}= 2.1 nm are plotted in Figure 5b. When h

_{m}= 1.29 nm and r

_{μ}= 0.5, D

_{s}

_{1}is comparatively plotted by a solid line. From a comparison between the open circle and solid line, we note that D

_{s}

_{1}decreases only slightly in the range h > 0.3 nm and increases slightly in the range h < 0.3 nm when h

_{m}is decreased by 61%. The effect of h

_{m}on the diffusion coefficient is very small. Accordingly, it can be said from Figure 4 and Figure 5a,b that the diffusion coefficients D

_{s}

_{1}and D

_{s}

_{2}are hardly influenced by r

_{μ}and h

_{m}if the physical parameter Ar

_{μ}/μ

_{m}h

_{m}is determined so that the theoretical diffusion coefficient coincides with the experimental value at a representative liquid film thickness.

_{m}in such a way that they coincide with the experimental diffusion coefficient at an arbitrarily chosen film thickness of h = 1.29 nm after specifying d

_{0}and h

_{m}. Table 1 lists the value of A/μ

_{m}determined in this manner for four different diffusion coefficients and effective bulk viscosities corresponding to typical values of the Hamaker constant when d

_{0}= 0.3 nm. The Hamaker constant A = 0.47 × 10

^{−19}J is the value for the dispersive surface pressure in the DLC–PFPE film–air layer calculated from Lifshitz theory [24]. Because the Hamaker constant calculated from Lifshitz theory is based on the assumption that each medium is continuous material, the effective Hamaker constant for the dispersive surface energy γ

_{d}(h) of molecularly thin lubricant film is quantified experimentally by fitting a mathematical expression of the form to the measured surface energy function.

_{0}is the surface energy of bulk lubricant and A* is an effective Hamaker constant [25,26,27]. Waltman et al. [26] evaluated the effective Hamaker constant of Z-Tetraol (molecular weight is 2000) to be 1.9 × 10

^{−19}J and used this value in [18].

^{−19}J. Therefore, the bulk viscosities were calculated for the Hamaker constant values of 0.47 × 10

^{−19}and 1.9 × 10

^{−19}J in Table 1. The effect of polar surface energy on diffusion coefficient function was ignored here because it seemed to have a minor effect compared with the dispersive component, particularly on disjoining pressure [26,27].

_{m}and the estimated effective viscosity for D

_{c}

_{1}and D

_{c}

_{2}and those for D

_{s}

_{1}and D

_{s}

_{2}are almost the same values within three digits. Compared with conventional theory, submonolayer theory estimates a smaller viscosity by one fifth even when r

_{μ}= 1. This is because the effective Hamaker constant for submonolayer theory is evaluated to a reduced value owing to the decreased liquid density.

_{s}

_{1}and D

_{s}

_{2}, the ratios of the estimated bulk viscosity μ

_{m}at r

_{μ}= μ

_{m}/μ

_{0}= 1, 0.5, and 2 become 1, 0.54, and 1.75, respectively. This relationship between r

_{μ}and μ

_{m}is reasonable. From the estimated bulk viscosities for h

_{m}= 1.29 and 2.1 nm, we note that the effective viscosity can be estimated as inversely proportional to the effective molecular thickness h

_{m}.

^{−19}J and r

_{µ}= 0.5, the estimated bulk viscosity is 0.107 Pa∙s for h

_{m}= 2.1 nm and 0. 186 Pa∙s for h

_{m}= 1.29 nm. When we use A = 1.9 × 10

^{−19}J [26], however, the bulk viscosity is evaluated to be 0.751 Pa∙s. If we use A = 4.8 × 10

^{−19}J [27], we can obtain µ

_{m}= 1.9 Pa∙s. Because the bulk viscosity of Z-Tetraol is estimated to be around 1 Pa∙s, it can be said that the submonolayer diffusion theory results in the more realistic relationship between Hamaker constant and bulk viscosity compared with the conventional diffusion theory. Because the submonolayer diffusion function can agree well with the experimental one in a wide range of submonolayer film thickness h by using the effective Hamaker constant, it is inferred that the second derivative of the polar surface energy with respect to film thickness is negligibly small compared with that of the dispersive surface energy in the submonolayer thickness region [26,27].

^{−19}J is assumed, whereas that of a polar PFPE film of Z-dol is estimated to be twice as large as the bulk viscosity. Therefore, it can be said that submonolayer diffusion theory is more appropriate for evaluating the spreading characteristics of non-polar submonolayer liquid film than the conventional theory.

_{m}is identified by the experimental diffusion coefficient at a certain film thickness.

## 5. Conclusions

- The disjoining pressure for a submonolayer liquid film model derived from the LJP and diffusion equation for submonolayer liquid film incorporating the new disjoining pressure is a rigorous mathematical solution for the disjoining pressure and diffusion equation proposed previously. It was found that the previous submonolayer theory is a good approximated expression of the rigorous submonolayer theory in the region of film thickness above d
_{0}. - From the axiomatic condition that the disjoining pressure must be zero at infinite film thickness, the mean vdW distance d
_{0}of the liquid–solid interface must be 0.637 times the molecular equilibrium distance in LJP. - The conventional disjoining pressure for a multilayer continuous liquid film including the vdW distance d
_{0}cannot be derived from only the adhesive term of the LJP. The disjoining pressure rigorously derived from the LJP contains an additional term. Although the disjoining pressure derived from the LJP gives almost the same value as the conventional version in the region of film thickness larger than 0.5d_{0}, it exhibits different behavior for a film thickness smaller than 0.5d_{0}and becomes zero at zero film thickness in contrast to the conventional disjoining pressure that has a maximum value at zero film thickness. It is considered that the disjoining pressure derived from the LJP is more reasonable. - Four diffusion coefficients D
_{s}_{2}(rigorous submonolayer theory), D_{s}_{1}(approximated submonolayer theory), D_{c}_{2}(rigorous continuous layer theory), and D_{c}_{1}(conventional continuous layer theory) have been compared with Waltman’s experimental diffusion coefficients by identifying the multiplication factor A/μ_{m}from the experimental diffusion coefficient value at 1.29 nm film thickness. When the presumable value of 0.3 nm is used for d_{0}, D_{s}_{2}and D_{s}_{1}showed excellent agreement with the experimental data in the entire range of measured film thickness although D_{s}_{1}exhibited a slightly higher value in the region of film thickness less than d_{0}. In contrast, D_{c}_{2}and D_{c}_{1}showed quantitatively and qualitatively different behaviors from the experimental data. This is because D_{c}_{2}and D_{c}_{1}take maximum values at a film thickness of ~3d_{0}. As a result, it can be concluded that the spreading, diffusion, and replenishment behaviors of submonolayer liquid film can be evaluated by submonolayer theory if the amplification factor A/μ_{m}is identified from an arbitrarily chosen experimental value.

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | Hamaker constant for a multi-layer lubricant film coated on a solid surface [J] |

D | Diffusion coefficient [m^{2}/s] |

O-xyz | Rectangular coordinate system for lubricant flow analysis |

d | Mean distance between the liquid film and solid surface [m] |

d_{0} | Mean van der Waals distance between the liquid film and solid surface [m] |

h | Local thickness of the liquid film [m] |

h_{m} | Thickness of the monolayer of liquid film [m] |

p | Pressure in the liquid film [Pa] |

r_{μ} | = μ_{m}/μ_{0} |

t | Time [s] |

u | Local velocity within the liquid film [m] |

Π | Disjoining pressure [Pa] |

α | = r_{μ} − 1 |

μ_{z} | Local viscosity of the submonolayer liquid film at a height of z [Pa·s] |

μ_{0} | Effective viscosity at the solid surface [Pa·s] |

μ_{m} | Viscosity of the liquid film for monolayer and multilayer liquid film [Pa·s] |

ρ_{l} | Density of the bulk liquid [kg/m^{3}] |

ρ_{h} | Density of the submonolayer liquid film [kg/m^{3}] |

## References

- Xu, J.; Shimizu, Y.; Furukawa, M.; Li, J.; Sano, Y.; Shiramatsu, T.; Aoki, Y.; Matsumoto, H.; Kuroki, K.; Kohira, H. Contact/clearance sensor for HDI subnanometer regime. IEEE Trans. Magn.
**2014**, 50, 114–118. [Google Scholar] [CrossRef] - Kurita, T.; Shiramatsu, K.; Miyake, A.; Kato, M.; Soga, H.; Tanaka, S.; Saegusa, S.; Suk, M. Active flying-height control slider using MEMS thermal actuator. Microsyst. Technol.
**2006**, 12, 369–375. [Google Scholar] [CrossRef] - Canchi, S.V.; Bogy, D.B. Slider-lubricant interactions and lubricant distribution for contact and near contact recording conditions. IEEE Trans. Magn.
**2011**, 47, 1842–1848. [Google Scholar] [CrossRef] - Marchon, B. Lubricant Design Attributes Subnanometer Head-Disk Clearance. IEEE Trans. Magn.
**2009**, 45, 872–876. [Google Scholar] [CrossRef] - Chen, Y.-K.; Murthy, A.N.; Pit, R.; Bogy, D.B. Angstrom scale wear of the air-bearing sliders in hard disk drives. Tribol. Lett.
**2014**, 54, 273–278. [Google Scholar] [CrossRef] - Matthes, L.; 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, 39. [Google Scholar] [CrossRef] - de Gennes, P.G. Wetting: Statics and dynamics. Rev. Mod. Phys.
**1985**, 57 Pt 1, 827–863. [Google Scholar] [CrossRef] - Oron, A.; Davis, S.H.; Bankoff, S.G. Long-scale evolution of thin liquid films. Rev. Mod. Phys.
**1997**, 69, 931–980. [Google Scholar] [CrossRef] - Mate, C.M. Application of disjoining and capillary pressure to liquid lubricant films in magnetic recording. J. Appl. Phys.
**1992**, 72, 3084–3090. [Google Scholar] [CrossRef] - Karis, T.E.; Tyndall, G.W. Calculation of spreading profiles for molecularly-thin films from surface energy gradients. J. Non-Newton. Fluid Flow
**1999**, 82, 287–302. [Google Scholar] [CrossRef] - Tyndall, G.W.; Karis, T.E.; Jhon, M.S. Spreading profiles of molecularly thin perfluoro-polyether films. Tribol. Trans.
**1999**, 42, 463–470. [Google Scholar] [CrossRef] - Marchon, H.; Karis, T.E. Poiseuille flow at a nanometer scale. Europhys. Lett.
**2006**, 74, 294–298. [Google Scholar] [CrossRef] - Waltman, R.J.; Tyndall, G.W.; Pacansky, J. Computer-modeling study of the interactions of Zdol with amorphous carbon surfaces. Langmuir
**1999**, 15, 6470–6483. [Google Scholar] [CrossRef] - Mate, C.M. Spreading kinematics of lubricant droplets on magnetic recording disks. Tribol. Lett.
**2013**, 51, 385–395. [Google Scholar] [CrossRef] - Scarpulla, M.A.; Mate, C.M.; Carter, M.D. Air shear driven flow of thin perfluoropolyether polymer films. J. Chem. Phys.
**2003**, 118, 3368–3375. [Google Scholar] [CrossRef] - Ono, K. Replenishment speed of depleted scar in submonolayer lubricant. Tribol. Lett.
**2013**, 52, 123–195. [Google Scholar] [CrossRef] - Ono, K. Diffusion equation for spreading and replenishment in submonolayer lubricant film. Tribol. Lett.
**2015**, 57, 13. [Google Scholar] [CrossRef] - Waltman, R.J. Z-Tetraol reflow in heat-assisted magnetic recording. Tribol. Online
**2016**, 11, 50–60. [Google Scholar] [CrossRef] - Israelachvili, J.N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, UK, 1992. [Google Scholar]
- Mate, C.M. Anomalous diffusion kinetics of the precursor film that spreads from polymer droplets. Langmuir
**2012**, 28, 16821–16827. [Google Scholar] [CrossRef] [PubMed] - Ma, X.; Gui, J.; Grannen, K.J.; Smoliar, L.A.; Marchon, B.; Jhon, M.S.; Bauer, C.L. Spreading of PFPE lubricants on carbon surfaces: effect of hydrogen and nitrogen content. Tribol. Lett.
**1999**, 6, 9–14. [Google Scholar] [CrossRef] - Ma, X.; Gui, J.; Smoliar, L.; Grannen, K.; Marchon, B.; Bauer, C.L.; Jhon, M.S. Complex terraced spreading of perfluoropolyalkylether films on carbon surfaces. Phys. Rev. E
**1999**, 59, 723–727. [Google Scholar] [CrossRef] - Fukuda, M.; Zhang, H.; Ishiguro, T.; Fukuzawa, K. Structure-based coarse-graining for inhomogeneous liquid polymer systems. J. Chem. Phys.
**2013**, 139, 054901. [Google Scholar] [CrossRef] [PubMed] - Matsuoka, H.; Ohkubo, S.; Fukui, S. Correction expression of the van der Waals pressure for multilayered system with application to analyses of static characteristics of flying head sliders with an ultrasmall spacing. Microsyst. Technol.
**2005**, 11, 824–829. [Google Scholar] [CrossRef] - Waltman, R.J.; Deng, H.; Wang, G.J.; Zhu, H.; Tyndall, G.W. The effect of PFPE film thickness and molecular polarity on the pick-up of disk lubricant by a low-flying slider. Tribol. Lett.
**2010**, 39, 211–219. [Google Scholar] [CrossRef] - Waltman, R.J.; Newman, J.; Guo, X.-C.; Burns, J.; Witta, C.; Amo, M. The effect of UV irradiation on the Z-Tetraol boundary lubricant. Tribol. Online
**2012**, 7, 70–80. [Google Scholar] [CrossRef] - Tani, H.; Mitsutome, T.; Tagawa, N. Adhesion and friction behavior of magnetic disks with ultrathin perfluoropolyether lubricant films having different end-groups measured suing pin-on-disk test. IEEE Trans. Magn.
**2013**, 49, 2638–2644. [Google Scholar] [CrossRef]

**Figure 2.**(

**a**) Comparison of disjoining pressure Π

_{s}

_{1}of Equation (8), Π

_{s}

_{2}of Equation (15), Π

_{c}

_{1}of Equation (7), and Π

_{c}

_{2}of Equation (16); (

**b**) Enlarged view of (

**a**).

**Figure 3.**Coordinate system and forces acting on a small liquid element in a submonolayer film near a liquid surface.

**Figure 4.**Comparison of various theoretical diffusion coefficient functions D

_{s}

_{1}(h), D

_{s}

_{2}(h), D

_{c}

_{1}(h), and D

_{c}

_{2}(h) (h

_{m}= 2.1 nm and r

_{μ}= 1) and experimental data [18].

**Figure 5.**Effects of the viscosity ratio and monolayer thickness on diffusion coefficient functions. (

**a**) Effects of viscosity ratio μ

_{m}/μ

_{0}on D

_{s}

_{1}(h) and D

_{s}

_{2}(h); (

**b**) Effects of monolayer thickness h

_{m}on D

_{s}

_{1}(h).

**Table 1.**Estimated physical parameter values calculated from the experimental diffusion coefficient at h = 1.29 nm.

Diffusion Coefficient | D_{c}_{1} | D_{c}_{2} | D_{s}_{1} | D_{s}_{2} | |||||
---|---|---|---|---|---|---|---|---|---|

h_{m} [nm] | 2.1 | 2.1 | 2.1 | 1.29 | 2.1 | ||||

r_{μ} (=μ_{m}/μ_{0}) | 1 | 1 | 1 | 0.5 | 2 | 0.5 | 1 | 0.5 | 2 |

A/μ_{m} [×10^{−19} m^{3}/s] | 0.432 | 0.432 | 2.385 | 4.384 | 1.362 | 2.528 | 2.385 | 4.385 | 1.362 |

μ_{m} [Pa∙s] (A = 0.47 × 10^{−19} J) | 1.089 | 1.089 | 0.197 | 0.107 | 0.345 | 0.186 | 0.197 | 0.107 | 0.345 |

μ_{m} [Pa∙s] (A = 1.9 × 10^{−19} J) | 4.403 | 4.402 | 0.797 | 0.433 | 1.395 | 0.751 | 0.796 | 0.433 | 1.395 |

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**MDPI and ACS Style**

Ono, K.
Disjoining Pressure Derived from the Lennard–Jones Potential, Diffusion Equation, and Diffusion Coefficient for Submonolayer Liquid Film. *Surfaces* **2018**, *1*, 122-137.
https://doi.org/10.3390/surfaces1010010

**AMA Style**

Ono K.
Disjoining Pressure Derived from the Lennard–Jones Potential, Diffusion Equation, and Diffusion Coefficient for Submonolayer Liquid Film. *Surfaces*. 2018; 1(1):122-137.
https://doi.org/10.3390/surfaces1010010

**Chicago/Turabian Style**

Ono, Kyosuke.
2018. "Disjoining Pressure Derived from the Lennard–Jones Potential, Diffusion Equation, and Diffusion Coefficient for Submonolayer Liquid Film" *Surfaces* 1, no. 1: 122-137.
https://doi.org/10.3390/surfaces1010010