Flying characteristics in the free molecular region (influence of accommodation coefficients)

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1 Microsyst Technol (25) 11: DOI 1.17/s TECHNICAL PAPER Shigehisa Fukui Æ Hidekazu Shimada Æ Kiyomi Yamane Hiroshige Matsuoka Flying characteristics in the free molecular region (influence of accommodation coefficients) Received: 2 October 23 / Acceted: 9 Setember 24 / Published online: 16 July 25 Ó Sringer-Verlag 25 Abstract To examine molecular gas film lubrication (MGL) characteristics for sacings of several nanometers, the free molecular MGL equation, which assumes that the flow in the lubrication region is the free molecular flow and involves the Poiseuille flow rate and the shear stress coefficient for the free molecular flow region with surface accommodation coefficients as arameters, is established. The usefulness of the free molecular MGL equation for the sacings of less than several nanometers is shown by comaring with the numerically calculated characteristics of flying head sliders using the conventional exression of the MGL equation with accommodation coefficients of running disk and slider, a and a 1, as arameters. The static ressure generation is found to deend only on the accommodation coefficient of the running disk in the free molecular region. 1 Introduction As the sacing (flying heights) of the flying head sliders in magnetic disk storage devices is now less than 1 nm, accurate estimation of the flying characteristics of the sliders is required. Analyses of the ressures of the sliders with ultra small sacing are executed by solving the molecular gas film lubrication (MGL) equation, which is derived from the linearized Boltzmann equation (Fukui et al. 1995; Karniadakis et al. 2). Shear stresses are obtained by calculating the shear stress equation based on the molecular gas dynamics (Kang et al. 1999) (see S. Fukui (&) Æ H. Shimada Æ K. Yamane Æ H. Matsuoka Deartment of Alied Mathematics and Physics, Faculty of Engineering, Tottori University, 4-11 Minami, Koyama-cho Tottori, , Jaan fukui@dam.tottori-u.ac.j Tel.: Fax: Fig. 1). The validity of the MGL equation for several tens of nanometers to several nanometer sacings was examined by the use of the Direct Simulation Monte Carlo (DSMC) method (Huang et al. 1997; Fukui et al. 1998). Also, the influence of surface accommodation coefficient a, which determines the behaviors of the reflected molecules at the boundary walls, is an imortant factor to govern the flying characteristics (Kang et al. 1999; Huang et al. 2), and the values of the Poiseuille flow rate coefficient Q P and the Couette flow rate coefficient Q C for various a are interolated from the database values (Fukui et al. 199; Huang et al. 2). On the other hand, flows in the lubricating region with ultra small sacing are almost free molecular flows, in which the molecular collisions are negligible comared to collisions with boundary walls (Cercignani 1988; Sone 22). The limiting exression of the Poiseuille flow rate Q P for the free molecular region has been used only for the accommodation coefficient a = 1, that is, for diffuse reflection. Recently the numerical formulation for the ressure generation analyses even for rough surfaces in the free molecular flow region has been roosed based on the molecular gas dynamics (Fukui et al. 2). Thus, the MGL equation which can be used for analyzing normal stress (ressure) and shear stresses for such an ultra small flying height has been eagerly awaited to know the static and dynamic characteristics of the near contact sliders. In the resent study, the free molecular MGL equation is newly derived, by utilizing the obtained analytical exression of the flow rate coefficients, Q P and Q C, taking into account surface accommodation coefficients, a, for the free molecular limit, and it is used to calculate the static characteristics of a slider with ultra small sacing. 2 MGL equation in the free molecular region The MGL characteristics deend on the flow rate coefficient of the Poiseuille flow, Q P, and that of the Couette flow, Q C. The velocity distributions, therefore, the flow

86 where the subscrit fm denotes values in the free molecular flow, and the coefficients A QP and A QC are exressed as follows: A QP ð2 a Þð2 a 1 Þ 1 ð1 a Þð1 a 1 Þ ; ð4aþ Fig.

2 86 where the subscrit fm denotes values in the free molecular flow, and the coefficients A QP and A QC are exressed as follows: A QP ð2 a Þð2 a 1 Þ 1 ð1 a Þð1 a 1 Þ ; ð4aþ Fig. 1 Pressure and shear stress in free molecular region rates deend on the reflection characteristics at the surfaces. Figure 2 shows the two tyical reflections; diffuse reflection (a = 1) and secular reflection (a =). The mixed reflection called the Maxwell tye reflection ( < a < 1) is used for the more general case. The MGL equation taking into account the surface accommodation coefficients of the disk and slider, a and a 1, is as follows: d Q P ðd;a ;a 1 Þ PH 3 dp K d f Q CðD;a ;a 1 ÞPHg ð1þ D=6 where P(= / a ) and H(= h/h ) are nondimensional ressure and sacing; D is the inverse Knudsen number defined as D ð2= ffiffiffi Þ=Kn: The left-hand side of Eq. 1 is the Poiseuille flow term, and the right-hand side is the Couette flow term. The coefficients, Q P and Q C, are Poiseuille and Couette flow rate coefficients, and a and a 1 are the accommodation coefficients of the disk and the slider, resectively. For the continuum flow Q P /(D/6) and Q C are both unity, and for the sli flow aroximation, an analytical exression for the sli flow is emloyed. The coefficients Q P and Q C are obtained by enormous numerical calculation of an integro-differential equation based on the molecular gas dynamics. In the free molecular flow region, the coefficients, Q P and Q C, are relaced by those in the free molecular limit, Q Pfm and Q Cfm Q P! Q Pfm D 1 ffiffiffi A QP log D Q C! Q Cfm D A QC ð2þ ð3þ A QC a ð2 a 1 Þ 1 ð1 a Þð1 a 1 Þ ð4bþ The detailed rocedure to obtain the coefficients in the free molecular limit is exlained in the Aendix. In the free molecular limit, the macroscoic velocity rofiles of the Poiseuille flow and the Couette flow are not deendent on the coordinate z, or velocities u are constant everywhere in the sacing. The velocity distribution functions f everywhere in the sacing are simly the sum of that of the lower boundary and that of the uer boundary, each of which is the Maxwell distribution function determined by each boundary temerature and velocity. Finally, Eq. 1 is reduced to the following equation, which we call the free molecular MGL equation : d dp log DH2 D ffiffiffi 6 a 2 a K dðphþ ð5þ Equation 5 shows that ressure generation in the free molecular region deends only on the accommodation coefficients of the disk surface, a. This is because the deendency of the accommodation coefficient, a 1,onthe Poiseuille flow and that on the Couette flow is the same. In the lubrication theory, the mass conservation is considered as the sum of the Poiseuille flow and the Couette flow. Therefore, in the static lubrication equation, Eq. 5, the accommodation coefficient of the slider a 1 is canceled out and that of the disk, a, remains. On the contrary, in the dynamic lubrication theory, as the time deendent squeeze term has no accommodation coefficients, both the accommodation coefficients a and a 1 remains in the dynamic lubrication theory. 3 Shear stresses in the free molecular region Using the static ressure calculated by Eq. 1 or Eq. 5 in the free molecular limit, shear stress, a zx, is obtained by the following equation: Fig. 2 Tyes of reflection at boundary: (a) diffusive reflection (a = 1) and (b) secular reflection (a = )

3 r ZX H 1 2Z H W P ðd; a ; a 1 Þ 1 V D H W CðD; a ; a 1 Þ ð6þ where P(= / a ) is ressure, H(= h/h ) is sacing, Z(= z/h ) is osition across the sacing, V U= ffiffiffiffiffiffiffiffiffiffi 2RT is disk seed and (= h / l) is the asect ratio of the lubrication region. The coefficients, W P and W C, are the shear stress coefficients of Poiseuille flow and Couette flow, resectively, and are exressed as follows: W P! W Pfm 1 1 A W P 1 ð2z=dþ ð7þ W C! W Cfm D ffiffiffi A WC ð8þ where z is the variable equivalent to the z coordinate, and the coefficients A WP and A WC are as follows: a ð2 a 1 Þ A WP 1 ð1 a Þð1 a 1 Þ ; ð9aþ A WC a a 1 1 ð1 a Þð1 a 1 Þ ð9bþ Therefore, the shear stress in the free molecular region is exressed as follows: r ZXfm H 1 2Z ð2 A WP Þ V P ffiffiffi A WC ð1þ H The shear stress deends on the accommodation coefficients of both the disk and the slider. 4 Flow-rate coefficients and shear stress coefficients in the free molecular limit 87 Flow-rate coefficients in the free molecular region, Q Pfm (Eq. 2) and Q Cfm (Eq. 3), are comared to rigorous calculations of Q P and Q C for ultra small values of D in Fig. 3a and b. Figure 3a shows the relationshi between the Poiseuille flow rate Q P and the inverse Knudsen number D. As far as the accommodation coefficient is concerned, as the Poiseuille flow rate in the free Fig. 3 Flow rates, Q P and Q C vs the inverse Knudsen number, D: (a) Poiseuille flow rates, Q P and (b) Couette flow rates, Q C

4 88 molecular flow Q Pfm is symmetric with the accommodation coefficients, a and a 1, shown in Eq. 4a, the flow rate is also symmetric with a and a 1. For various combinations of a and a 1, the Poiseuille flow rates in the free molecular limit, Q Pfm are found to be good aroximations. Figure 3b shows the relationshi between the Couette flow rate, Q C, and the inverse Knudsen number D. The Couette flow rate in the free molecular limit, Q C, is found to be anti symmetric concerning a and a 1. Also in this case the Couette flow rates in the free molecular limit, Q Cfm are the good aroximations. Shear stress coefficients in the free molecular region, W Pfm (Eq. 7) and W Cfm (Eq. 8), are comared to the rigorous calculation results of W P and W C for ultra small values of D in Fig. 4a and b. Figure 4a shows the relationshi between the shear stress caused by the Poiseuille flow, W P, and the inverse Knudsen number D. The Poiseuille shear stress coefficient in the free molecular limit, W Pfm ; is found to be anti symmetric concerning a and a 1. Also in this case, the W Pfm values are the good aroximations. Figure 4b shows the relationshi between the shear stress caused by the Couette flow, W C, and the inverse Knudsen number D. As the Couette shear stress coefficient in the free molecular flow, W Cfm is symmetric with the accommodation, a and a 1, shown in Eq. 9b, therefore, the flow rate is symmetric with a and a 1. For various combinations of a and a 1, the Couette shear stress coefficient in the free molecular limit, W Cfm aroximation. is a good Fig. 4 Shear stress coefficients, W P and W C vs inverse Knudsen number, D: (a) shear stress coefficients, W P and (b) shear stress coefficients, W C Fig. 5 Plane inclined sliders 5 MGL calculations of ressure and shear stress To examine the validity of the flow rates in the free molecular limit, Q Pfm and Q Cfm ; and the shear stress coefficients in the free molecular limit, W Pfm and W Cfm ; ressure and shear stress distributions are calculated for a lane inclined slider shown in Fig. 5 and comared with calculations using Q Pfm ;Q Cfm ;W Pfm,andQ Cfm ; and the DSMC method. Figure 6a and b show the ressure distributions for lane inclined sliders having minimum sacing, h,of 1 and 1 nm, and slider inclinations, h 1 / h,of2.in the analysis, the accommodation coefficient at the disk surface, a, is fixed to 1, which means that the molecules reflect diffusively at the disk surface, whereas that at the slider surface, a 1, varies from.2 to 1 as a arameter. Calculations are executed using (1) the free molecular MGL equation [MGL fm ], (2) the conventional MGL equation [MGL], or (3) the DSMC method (Huang et al. 1997). Figure 6a shows that ressure distributions do not deend on a 1, which can be anticiated from Eq. 5. For h = 1 nm, the MGL results and DSMC results agree well, whereas the free molecular MGL results differ from the aforementioned results, because the 1 nm sacing is not sufficient to

distributions for h =1nm be treated as a free molecular aroximation.

5 89 Fig. 6 Pressure distributions and shear stress distributions for a = 1 with a 1 as a arameter (MGL, free molecular MGL and DSMC results): (a) ressure distributions for h = 1 and 1 nm and (b) shear stress distributions for h =1nm be treated as a free molecular aroximation. Figure 6b shows the distributions of the shear stress at the slider surface, r zx slider, and that at the disk surface, r zx disk, for the same configuration as Fig. 6a and the minimum sacing h of 1 nm. The calculation results of (1) MGL fm, (2) MGL, and (3) DSMC agree Fig. 7 Pressure distributions and shear stress distributions for a 1 =1 with a as a arameter (MGL, free molecular MGL and DSMC results): (a) ressure distributions for h = 1 nm and (b) shear stress distributions for h =1nm well, and the shear stresses deend on both the accommodation coefficient at the slider, a 1, and that at the disk, a. Figure 7a and b show the ressure distributions for lane inclined sliders having minimum sacing, h,of 1 nm, and the accommodation coefficient at the disk surface a 1 is fixed to 1, which means that molecules reflect diffusively at the slider surface, whereas that at the disk surface, a, varies from.2 to 1 as a arameter. Figure 7a shows that the ressure distributions largely deend on a. The MGL results and DSMC results agree well. Figure 7b shows the distributions of the shear stress at the slider surface, r zx slider, and that at the disk surface, r zx disk, for the same configuration as Fig. 6a and the minimum sacing h of 1 nm. The

6 81 calculation results of (1) MGL fm, (2) MGL, and (3) DSMC agree well, and also in this case the shear stresses deend on both the accommodation coefficient at the slider, a 1, and that at the disk, a. 6 Conclusions To examine MGL characteristics for sacings of several nanometers, the free molecular MGL equation, which assumes that the flow in the lubrication region is the free molecular flow and involves the Poiseuille flow rate and the shear stress coefficient for the free molecular flow region with surface accommodation coefficients as arameters, is established. Using the newly derived free molecular MGL equation, ressure and shear stress distributions are calculated. For sacings of several nanometers, the free molecular MGL equation for ressure and shear stress are useful, and ressure generation deends only on the accommodation coefficient of the disk, a, whereas the shear stress deends on the accommodation coefficients of both the disk and the slider. Aendix Deduction of Eqs 2, 3, 7, and 8 Velocity rofile V X and shear stress r X for the lubrication flow based on the Boltzmann equation can be obtained by solving the erturbed quantity / of the linearized Boltzmann equation with the aroriate boundary condition. The linearized Boltzmann equation is exressed as f þ 1 ð / þ x þ 2f k X V X Þ ð11þ As the similarity solution, the quantity / is exressed as / X e / þ f X / 1 ðz; f Z Þ: ð12þ The quantities / and / 1 can be obtained by solving the following differential equations: / b f 1 1 ð / k 1 þ 2V X Þ / where b is the nondimensional ressure gradient: b e@p=@x ð14þ ð15þ The boundary condition for / 1 is as follows when the Maxwell-tye boundary condition is emloyed: / 1 j fz > ð1 a Þ/ 1 j fz \ þ2a V for disk surface Z = ð16þ / 1 j fz \ ð1 a 1Þ/ 1 j fz > for slider surface Z = H ð17þ After some tedious calculations, the solution / 1 is obtained as solutions: / 1fZ> for f Z > and / 1fZ\ for f Z \: The velocity rofile V X is calculated by integrating in the velocity sace V X Z þ Z Z ðf 2 X / 1EÞdf X df Y df Z ðf 2 X / 1 fz> EÞdf X df Y df Z ðf 2 X / 1 fz> EÞdf X df Y df Z where E is exressed as follows: E 3=2 exð f 2 X f2 Y f2 Z Þ ð18þ ð19þ The velocity rofile V X is given as the linear combination of the velocity rofile of the Poiseuille flow V XP and that of the Couette flow V XC ; therefore, the mass flow rate is given as q X q XP þ q XC : ð2þ Nondimensional mass flow rates for the Poiseuille flow, Q P, and for the Couette flow, Q C, are defined as q XP Q P ðd; a ; a 1 Þ h 2 ðd=dxþ= ffiffiffiffiffiffiffiffiffi 2RT q XC qhu=2 and Q C ðd; a ; a 1 Þ ð21þ As the inverse Knudsen number D aroaches zero for the free molecular flow, the asymtotic exression of Q P and Q C is as follows: Q P! Q Pfm D 1 ffiffiffi A QP log D Q C! Q Cfm D A QC ð2þ ð3þ The shear stress r ZX in the flow direction can be obtained by the following definition and solutions: / 11z> for f z > and / 11z\ for f z <

7 811 r ZX Z Z Z þ f Z f X ðf X / 1 ÞEdf X df Y df Z f Z f X ðf X / 1fZ \ÞEdf X df Y df Z f Z f X ðf X / 1fZ >ÞEdf X df Y df Z ð22þ Also the shear stress r ZX is the linear combination of the shear stress caused by the Poiseuille flow, r ZXP ; and that caused by the Couette flow, r ZXC : By introducing the shear stress coefficients for the Poiseuille flow, W P, and for the Couette flow, W C, which are the coefficient comared with the shear stresses for continuum r ZXPcon and r ZXCcon : r ZX r ZXPcon W P þ r ZXCcon W C ð23þ where r ZXPcon H 2 e dp 1 2Z H and r ZXCcon 1 D V H ð24þ As the inverse Knudsen number D aroaches zero for the free molecular flow, the asymtotic exression of W P and W C is as follows: W P! W Pfm 1 1 A W P 1 ð2z=dþ ð7þ W C! W Cfm References D ffiffiffi A WC ð8þ Cercignani C (1988) The Boltzmann equation and its alications. Sringer, Berlin Heidelberg New York Fukui S, Kaneko R (199) A database for interolation of Poiseuille flow rates for high Knudsen number lubrication roblems. ASME J Tribol 112:78 83 Fukui S, Kaneko R (1995) Molecular gas film lubrication (MGL). In: Bhushan B (ed) Handbook of micro/nanotribology. CRC Press, Boca Raton, Fukui S, Yamane K (1998) Monte Carlo direct simulations of stetye gas bearings with nanometer sacings. Adv Inf Storage Syst 9: Fukui S, Yamane K (21) Molecular gas film lubrication in ultrahigh Knudsen number Regime new formulation of MGL roblems for free molecular flow. Proceedings of the International Tribology Conference Nagasaki 2, vol III, Huang W, Bogy DB (1997) Three-dimensional direct simulation Monte Carlo method for slider air bearing. Phys Fluids 9: Huang W, Bogy DB (2) The effect of accommodation coefficient on slider air bearing simulation. ASME J Tribol 122: Kang SC, Crone RM, Jhon MS (1999) A new molecular gas lubrication theory suitable for head-disk interface modeling. J Al Phys 85: Karniadakis GE, Beskok A (2) Micro flows fundamentals and simulations. Sringer, Berlin Heidelberg New York Sone Y (22) Kinetic theory and fluid dynamics. Birkhäuser, Basel

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