Investigation of slider flying characteristics and frequency response in helium-air gas mixtures

Abstract

A numerical model for the simulation of slider flying characteristics and frequency response in helium-air gas mixtures has been developed. The dynamic flying characteristics of the slider are obtained by simultaneously solving the Reynolds equation and the slider equilibrium equations. The contact force, friction force and adhesive force at the slider/disk interface are investigated using a slider/disk contact model. Frequency analysis shows that the second pitch mode of the slider is shifted to a higher frequency with an increase in the percentage of helium.

Appendices

Appendix 1: Slider/disk contact model

The contact model implemented in this paper includes contact forces, friction forces, and adhesive forces. Following Kogut and Etsion (2004), contact and friction forces were modeled for the case of contact between a flat and a rough surface including elastic, elastic–plastic, and plastic deformations of surface asperities. Adhesive forces between two rough surfaces were calculated considering the presence of a thin lubricant film (Suh and Polycarpou 2008). Both of the above models were combined and implemented in the present head/disk contact model. For the sake of completeness, we define the various forces in this section based on Kogut and Etsion (2004) and Suh and Polycarpou (2008).

The dimensionless contact force is given by Kogut and Etsion (2004) as

$$P^{*} (h^{*} ) = \frac{P(h)}{{A_{n} H}} = \frac{2}{3}\pi \beta K\omega_{c}^{*} \left( {\int_{{d^{*} }}^{{d^{*} + \omega_{c}^{*} }} {I^{1.5} } + 1.03\int_{{d^{*} + \omega_{c}^{*} }}^{{d^{*} + 6\omega_{c}^{*} }} {I^{1.425} } + 1.4\int_{{d^{*} + 6\omega_{c}^{*} }}^{{d^{*} + 110\omega_{c}^{*} }} {I^{1.263} } + \frac{3}{K}\int_{{d^{*} + 110\omega_{c}^{*} }}^{\infty } {I^{1} } } \right)$$

(A1)

where the superscript * denotes normalization with respect to σ (the standard deviation of surface heights); H is the hardness of the softer material; β is a roughness parameter ($\beta = \sigma \eta R$), where η is the area density of asperities and all asperities are assumed to have the same radius of curvature R); K($K = 0.454 + 0.41v$) is the hardness coefficient and v is the Poisson’s ratio of the softer material (Chang et al. 1988); ω _c is the critical interference at the inception of plastic deformation and d is the distance between the mean of the asperity heights and the smooth rigid flat.

In Eq. (A1), I ^b is a general form of the integrand

$$I^{b} = \left( {\frac{{\omega^{*} }}{{\omega_{c}^{*} }}} \right)^{b} \varphi (z^{*} )dz^{*}$$

(A2)

where $z(z^{*} = z/\sigma )$ is the asperity height measured from the mean of the asperity heights, and $\varphi^{*} (z^{*} )$ is the normalized Gaussian asperity height distribution

$$\varphi^{*} (z^{*} ) = \frac{1}{{\sqrt {2\pi } }}\left( {\frac{\sigma }{{\sigma_{s} }}} \right)\exp \left[ { - \frac{1}{2}\left( {\frac{\sigma }{{\sigma_{s} }}} \right)^{2} z^{*2} } \right]$$

(A3)

where σ _s is the standard deviation of asperity heights defined by Suh and Polycarpou (2008)

$$\sigma_{s} = \sqrt {\sigma^{2} - \frac{{3.717 \times 10^{ - 4} }}{{\eta^{2} R^{2} }}}$$

(A4)

The dimensionless friction force is given by Kogut and Etsion (2004) as

$$Q_{\hbox{max} }^{*} = \frac{{Q_{\hbox{max} } }}{{A_{n} H}} = \frac{2}{3}\pi \beta K\omega_{c}^{*} \left[ {0.52\int_{{d^{*} }}^{{d^{*} + \omega_{c}^{*} }} {I^{0.982} } + \int_{{d^{*} + \omega_{c}^{*} }}^{{d^{*} + 6\omega_{c}^{*} }} {\left( { - 0.01I^{4.425} + 0.09I^{3.425} - 0.4I^{2.425} + 0.85I^{1.425} } \right)} } \right]$$

(A5)

where I ^b is defined in Eq. (A2).

Finally, the dimensionless total adhesive force which accounts for the presence of a molecularly thin lubricant film of thickness t_l (mobile portion only) is given by Suh and Polycarpou (2008)

$$F_{s}^{*} (h^{*} ) = \frac{{F_{s} (h)}}{{A_{n} H}} = 2\pi \beta \frac{\varDelta \gamma }{\sigma H}\left( {\int_{ - \infty }^{{d^{*} - t\_l}} {J_{nc} } + \int_{{d^{*} - t\_l}}^{{d^{*} }} {J_{lc} } + 0.98\int_{{d^{*} }}^{{d^{*} + \omega_{c}^{*} }} {J_{ - 0.29}^{0.298} } + 0.79\int_{{d^{*} + \omega_{c}^{*} }}^{{d^{*} + 6\omega_{c}^{*} }} {J_{ - 0.321}^{0.356} } + 1.19\int_{{d^{*} + 6\omega_{c}^{*} }}^{{d^{*} + 110\omega_{c}^{*} }} {J_{ - 0.332}^{0.093} } } \right)$$

(A6)

where Δγ is the energy of adhesion, J _nc and $J_{b}^{c}$ denote the contributions of non-contacting and contacting asperities, respectively, and J _lc denotes the contribution of lubricated asperities in contact. The J-terms have the form:

$$J_{nc} = \frac{4}{3}\left[ {\left( {\frac{{\varepsilon^{*} }}{{d^{*} - z^{*} }}} \right)^{2} - 0.25\left( {\frac{{\varepsilon^{*} }}{{d^{*} - z^{*} }}} \right)^{8} } \right]\varphi^{*} (z^{*} )dz^{*}$$

(A7)

$$J_{lc} = \varphi^{*} (z^{*} )dz^{*}$$

(A8)

$$J_{b}^{c} = \left( {\frac{{\omega^{*} }}{{\omega_{c}^{*} }}} \right)^{b} \left( {\frac{{\varepsilon^{*} }}{{\omega_{c}^{*} }}} \right)^{c} \varphi^{*} (z^{*} )dz^{*}$$

(A9)

In Eqs. (A7, A8, A9), ɛ is the intermolecular distance that is typically between 0.3 and 0.5 nm Kogut and Etsion (2004).

The dimensionless net adhesive force W ^* for rough surfaces is given by Suh and Polycarpou (2008) as

$$W^{*} (h^{*} ) = \frac{W}{{A_{n} H}} = P^{*} (h^{*} ) - F_{s}^{*} (h^{*} )$$

(A10)

In Fig. 11, the adhesive force, contact force, friction force and net adhesive force are shown as a function of the separation between slider and disk (Tang et al. 2014). We observe that contact and friction forces become of importance at the slider/disk interface if the slider/disk separation decreases below 2.5 nm. These forces increase strongly as the separation between slider and disk decreases. At large spacing, the net adhesive force is an attractive force due to non-contacting asperities. This force becomes a repulsive force as the separation decreases further. Additional details for the implementation of the contact model can be found in Tang et al. (2014).

Fig. 11

Adhesive, contact, friction and net adhesive forces as a function of separation between slider and disk (Tang et al. 2014)

Appendix 2: Physical properties of gas mixtures

Following the calculation in our previous paper (Tang et al. 2014), the density ρ _m of the helium-air gas mixture was calculated using linear interpolation, i.e.

$$\rho_{m} = (1 - \alpha )\rho_{A} + \alpha \rho_{H}$$

(A11)

where, the subscript m denotes the helium-air gas mixture, α is the fraction of helium in the helium-air gas mixture; ρ is the density, and the subscripts H and A refer to helium and air, respectively.

The mean free path of the gas mixture was calculated using Bird’s method (Bird 1994), i.e.

$$\lambda_{m} = \frac{\alpha }{{\sqrt 2 \pi {\text{D}}_{\text{H}}^{ 2} {\text{n}}\alpha + \pi {\text{D}}_{\text{HA}}^{ 2} {\text{n}}(1 - \alpha )\sqrt {1 + M_{H} /M_{A} } }} + \frac{1 - \alpha }{{\sqrt 2 \pi {\text{D}}_{\text{A}}^{ 2} {\text{n}}(1 - \alpha ) + \pi {\text{D}}_{\text{HA}}^{ 2} {\text{n}}\alpha \sqrt {1 + M_{A} /M_{H} } }}$$

(A12)

where n is the number of molecules per unit volume; D _H and D _A are the molecular diameters for helium and air, respectively; M is the molecular weight; $D_{HA} = (D_{H} + D_{A} )/2$.

The viscosity of the gas mixture was determined using Poling’s method (Poling et al. 2001), i.e.

$$\mu_{m} = {\text{K}}_{\text{H}} ( 1 {\text{ + H}}_{\text{HA}}^{ 2} {\text{K}}_{\text{A}}^{ 2} )+ {\text{K}}_{\text{A}} ( 1 {\text{ + 2H}}_{\text{HA}} {\text{K}}_{\text{H}} {\text{ + H}}_{\text{HA}}^{ 2} {\text{K}}_{\text{H}}^{ 2} )$$

(A13)

with

$${\text{K}}_{\text{H}} = \frac{{\alpha \mu_{\text{H}} }}{{\alpha + (1 - \alpha )\mu_{\text{H}} {\text{H}}_{\text{HA}} [3 + 2({\text{M}}_{\text{A}} /{\text{M}}_{\text{H}} )]}}$$

$${\text{K}}_{\text{A}} = \frac{{(1 - \alpha )\mu_{\text{A}} }}{{(1 - \alpha ) + \alpha \mu_{\text{A}} {\text{H}}_{\text{HA}} [3 + 2({\text{M}}_{\text{H}} /{\text{M}}_{\text{A}} )]}}$$

$${\text{H}}_{\text{HA}} { = }\frac{{\sqrt {M_{H} M_{A} /32} }}{{ ( {\text{M}}_{\text{H}} {\text{ + M}}_{\text{A}} )^{ 3 / 2} }}U_{HA} \left( {\frac{{{\text{M}}_{\text{H}}^{ 1 / 4} }}{{\sqrt {\mu_{\text{A}} {\text{U}}_{\text{A}} } }}{ + }\frac{{{\text{M}}_{\text{A}}^{ 1 / 4} }}{{\sqrt {\mu_{\text{A}} {\text{U}}_{\text{A}} } }}} \right)^{ 2}$$

$$U_{i} = \frac{{[1 + 0.36T_{ri} (T_{ri} - 1)]^{1/6} }}{{\sqrt {T_{ri} } }}$$

In Eq. (A13), i represents HA (helium-air), H (helium), or A (air), respectively; ${\text{T}}_{\text{rH}} {\text{ = T/T}}_{\text{CH}}$, ${\text{T}}_{\text{rA}} {\text{ = T/T}}_{\text{CA}}$, ${\text{T}}_{\text{rHA}} { = }\left( {{\text{T/T}}_{\text{CH}} {\text{T}}_{\text{CA}} } \right)^{1/2}$; T is the gas temperature, while T_CH and T_CA are critical temperatures of helium and air, respectively. The parameters required in the equations are listed in Table 1, taken from Lide (2009) and Lemmon et al. (2000).

Table 1 Physical properties of helium and air