Multiscale Contact Mechanics with Application to Seals and Rubber Friction on Dry and Lubricated Surfaces

Abstract

Fluid leakage out of mechanical equipment such as gearboxes, hydraulic systems, or fuel tanks could cause serious problems and thus should be avoided. Seals are extremely useful devices for preventing such fluid leakages. We have developed a theoretical approach for calculation of the leak rate of stationary rubber seals and the friction force for dynamic seals. The theory is based on a recently developed theory of contact mechanics, which we briefly review. To test the theory, we have performed both simple model experiments and experiments on engineering seal systems. We have found good agreement between the calculated and measured results, and hence our theory has the potential to improve the future design of efficient seals.

We briefly review the processes that determine rubber friction on lubricated smooth and rough substrate surfaces. We present experimental friction results for lubricated surfaces, obtained using a simple Leonardo da Vinci setup. The data is analyzed using the Persson rubber friction and contact mechanics theory.

Appendices

Appendix 1

When a hard ball is in contact with a smooth rubber surface, a circular contact region forms with a radius r ₀ that depends on the applied normal load and the adhesive ball–rubber interaction. When the ball is removed from the substrate, the radius r ₀(t) of the contact region decreases with time. The bond-breaking process can be considered as an opening crack propagating with velocity $v=\left|{\overset{.}{r}}_0(t)\right|$ (see Fig. 44a). For a viscoelastic material such as rubber, the energy to propagate an opening crack at a finite speed v can be much larger than for an adiabatic (infinitely slowly) moving crack. This effect is a result of energy dissipation inside the rubber because of the time-dependent stress field from the moving crack. For an infinitely long crack moving with velocity v one usually writes the energy per unit area to propagate the crack as [9, 64, 79–81]:

Fig. 44

Pull-off (a) and sliding (b) of a hard, smooth ball in contact with a smooth rubber surface

$$ G\left(v,T\right)={G}_0(T)\left[1+f\left(v,T\right)\right], $$

(20)

where f(v, T) is the viscoelastic enhancement factor. Note that $f\to 0$ as $v\to 0$, so G ₀ is the energy per unit area to form the crack surfaces in the adiabatic limit. For an infinitely long crack in an infinitely extending media, f may increase by a factor of about 1000 as v increases from zero to a high value. For a particular rubber and counter-surface combination, G(v, T) is often measured by pulling a rubber strip in adhesive contact with the counter-surface [81].

We assume that the viscoelastic loss function $\mathrm{I}\mathrm{m}{E}^{-1}\left(\omega \right)$ is maximal for $\omega ={\omega}_{\mathrm{c}}$. When an opening crack propagates at some speed v, the time-dependent deformations of the rubber at distance r from the crack tip are characterized by the deformation frequency $\omega =v/r$. Thus, most of the viscoelastic energy dissipation occurs in a region centered a distance $r\approx v/{\omega}_{\mathrm{c}}$ away from the crack tip.

Let us now consider opening cracks of finite size, for example, a circular opening crack with radius r ₀(t) formed during removal of a ball from the substrate. If ${r}_0>v/{\omega}_{\mathrm{c}}$ one can still (approximately) use expression (20) to calculate the energy and force necessary to remove the ball (which depends on the pull-off speed). However, if the ball has a small enough radius (such that the radius of the contact region r ₀ is small enough), or if the pull-off speed is large enough (so the crack tip velocity v is large enough), then this inequality is not valid and the viscoelastic enhancement factor decreases (B. Persson, unpublished). This important fact is usually overlooked in the interpretation of adhesion experiments.

When a hard ball slides on the surface of a rubber block, an opening crack is formed at the back (see Fig. 44b). In this situation too, when calculating the contribution of viscoelastic energy dissipation to the friction force, one needs to take into account the possibility that f is reduced because of finite-size effects. This is particularly important for sliding of a rubber block on a hard rough substrate, where the substrate asperities act on the rubber block in the same way as on the ball in Fig. 44b. Here, the linear size of the contact regions may be very small (e.g., of micrometer size) and the value of f(v, T) derived or measured for long cracks may give a much larger viscoelastic enhancement factor than the actual factor, particularly for high sliding speeds (B. Persson, unpublished).

Note that for rubber friction on very rough surfaces such as road surfaces, the cut-off length λ ₁ (related to the wavenumber cut-off ${q}_1=2\pi /{\lambda}_1$) is typically abound 1 μm. Because the slip velocity in tire applications is about 1 m/s and the maximum of $\mathrm{I}\mathrm{m}{E}^{-1}\left(\omega \right)$ is typically for ${\omega}_{\mathrm{c}}={10}^6\kern0.5em {\mathrm{s}}^{-1}$ _, we obtain ${r}_0=v/{\omega}_{\mathrm{c}}\approx 1\kern0.5em \upmu \mathrm{m}$. Thus, in tire applications or for road surfaces the reduction in f (and in the adhesive contribution to friction) because of finite-size effects may not be very important.

Appendix 2

In this appendix we briefly describe hydration lubrication [55, 56], which is the hypothesis that hydration shells surrounding charges act as lubricating elements in boundary layers, resulting in extremely low sliding friction in aqueous media, (e.g., between mica surfaces) [75]. This highly fluid behavior under extreme confinement and shear is in direct contrast to the performance of nonassociating simple liquids (e.g., octamethylcyclotetrasiloxane (OMCTS), cyclohexane, or toluene) under similar conditions. These nonassociating liquids undergo liquid-to-solid phase transition, resulting in a finite yield stress once confined to six to nine molecular layers. It has been suggested that the lubricity of hydrated films of ${\mathrm{Na}}^{+}$ relies upon two factors: (i) the capacity of water to retain its bulk fluidity under confinement and in hydration layers around charged species, and (ii) the strong binding of water molecules within the hydration shell around ${\mathrm{Na}}^{+}$. The latter supports an applied load, therefore preventing primary minimum contact (which is adhesive and leads to high shear forces) between the mica surfaces. The shear lubricity of these load-bearing hydrated films has been rationalized in terms of the rapid kinetics of exchange of water molecules within hydration spheres with adjacent water molecules, as well as the rapid rotational dynamics and diffusivity of the water molecules within the thin film. The bulk water exchange rate, k _ex, for a water ligand in the primary hydration sphere of ${\mathrm{Na}}^{+}$ is approximately ${10}^9\kern0.5em {\mathrm{s}}^{-1}$. In addition, the rotational relaxation time of water molecules (about ${10}^{-11}\kern0.5em \mathrm{s}$ in bulk water) is thought to be a factor in the persistent fluidity of the confined hydration layers.

In general, two conditions must be satisfied in order for hydration lubrication to take place [56]: (i) ions must remain bound to the shearing surfaces under confinement, and (ii) the surface-bound ions must retain their hydration shell under confinement and applied load. When these conditions are fulfilled, hydration lubrication is mediated by the (thermodynamically) bound and fluid-like hydration layers attached to interfacial charged species. With respect to contact mechanics and friction, we can qualitatively consider the hydrated ${\mathrm{Na}}^{+}$ ions as slippery elastic balls. Note that hydration lubrication is also likely to occur in more complex, strongly hydrated interfaces, such as occur in synovial joints and mucosal surfaces.

Finally, we note that at very large applied pressure p the grafted lipid molecules may be removed from the contact region. The result is increased friction, which may be similar to that found for dry surfaces.