Contact Mechanics and Friction on Dry and Wet Human Skin

Abstract

The surface topography of the human wrist skin is studied using an optical method and the surface roughness power spectrum is obtained. The Persson contact mechanics theory is used to calculate the contact area for different magnifications, for both dry and wet condition of the skin. For dry skin, plastic yielding becomes important and will determine the area of contact observed at the highest magnification. The measured friction coefficient [M.J. Adams et al., Tribol Lett 26:239, 2007] on both dry and wet skin can be explained assuming that a frictional shear stress σ_f ≈ 15 MPa acts in the area of real contact during sliding. This frictional shear stress is typical for sliding on polymer surfaces, and for thin (nanometer) confined fluid films. The big increase in the friction, which has been observed for glass sliding on wet skin as the skin dries up, can be explained as resulting from the increase in the contact area arising from the attraction of capillary bridges. This effect is predicted to operate as long as the water layer is thinner than ∼14 μm, which is in good agreement with the time period (of order 100 s) over which the enhanced friction is observed (it takes about 100 s for ∼14 μm water to evaporate at 50% relative humidity and at room temperature). We calculate the dependency of the sliding friction coefficient on the sliding speed on lubricated surfaces (Stribeck curve). We show that sliding of a sphere and of a cylinder gives very similar results if the radius and load on the sphere and cylinder are appropriately related. When applied to skin the calculated Stribeck curve is in good agreement with experiment, except that the curve is shifted by one velocity-decade to higher velocities than observed experimentally. We explain this by the role of the skin and underlying tissues viscoelasticity on the contact mechanics.

Appendices

Appendix A: Capillary Adhesion; Role of Humidity

In Sect. 5, we studied the influence of relative thick water films (average film thickness of order ∼1−10  μm) on the skin-glass contact mechanics. Such film thickness prevail, for example, during drying of an originally flooded contact. Similar film thickness may prevail during sweating. Here, we consider instead the skin in equilibrium with humid air as a function of the air relative humidity. For most humidity this implies water film thickness of order nanometers.

Figure 16 shows the logarithm (with 10 as basis) of the relative contact area as a function of the relative humidity. If the elastic modulus of the stratum corneum would take the same value as in the wet (or 100% relative humidity) state, E ₀ ≈ 7  MPa, for all relative humidity, then capillary adhesion for small relative humidity would be strong enough to bring the surfaces into nearly complete contact. However, in reality the elastic modulus rapidly increases with decreasing relative humidity (see Fig. 17), and the influence of the capillary adhesion for relative humidity below 90% is negligible. This is illustrated in Fig. 16 with a calculation with E ₀ = 70  MPa, for which case the area of real contact is nearly independent of the relative humidity and determined by the applied load or pressure.

Fig. 16

The logarithm (with 10 as basis) of the relative contact area as a function of the relative humidity. If the elastic modulus of the stratum corneum would take its wet (or 100% relative humidity) value, E ₀ = 7  MPa, for all relative humidity’s, the capillary adhesion for small relative humidity would be strong enough to bring the surfaces into nearly complete contact. However, in reality the elastic modulus rapidly increases with decreasing relative humidity and the influence of the capillary adhesion for relative humidity below 90% is negligible. This is illustrated with a calculation with E ₀ = 70  MPa for which case the area of real contact is nearly independent of the relative humidity and determined by the applied load or pressure. For the applied pressure p ₀ = 6.83  kPa

Fig. 17

The logarithm (with 10 as basis) of the Young’s elastic modulus of the stratum corneum as a function of the relative humidity. The blue line is an approximate cubic spline fit to the experimental data. Based on experimental data obtained by Park and Baddiel, Papir, Koutroupi, Nikolopoulos, Gardner and Briggs and Yuan and Verma. Adopted from [41]

The calculation presented above (and in Ref. [15]) shows that if an elastic solid is soft enough, for a smooth hydrophilic interface there will be a strong increase in the contact area for some humidity range. This effect may be important in many situations, e.g., for rubber friction involving smooth hydrophilic interfaces, since the elastic modulus of rubber is nearly independent of the humidity and of similar magnitude as that of wet stratum corneum. This may contribute to the friction for, e.g., rubber wiper blades sliding on glass [17].

Appendix B: Mixed Lubrication; Approximate Mapping of Elliptic Contact on Cylinder Contact

In Sect. 6.1 we have shown how one can (approximately) map the sliding dynamics of a sphere on a lubricated substrate on a problem of a cylinder sliding on the same substrate, which is useful from a computational point of view, but also interesting from a conceptional point of view. This type of mapping should be even more accurately for a Hertz elliptic contact with the sliding direction orthogonal to the major elliptic axis (see Fig. 18).

Fig. 18

Hertz contact region for sphere on flat (left) and ellipse on flat (right)

We determine the radius of the cylinder and the load on the cylinder so that (a) the average Hertz contact pressure are the same, and (b) so that the condition $r_{\rm e} =w \sqrt(2/3)$ is satisfied, where w is the width of the Hertz infinite rectangular contact strip for the cylinder, and r _e an effective radius of the elliptic contact region defined by:

$$ \frac{1}{r_{\rm e}^2} = \frac{1}{2} \left(\frac{1}{a^2}+\frac{1}{b^2} \right) $$

The latter condition guaranty the same fluid squeeze-out time for an infinite rectangular sheet (width w) as for an elliptic disk (with the major and minor axis a and b) (both rigid and with identical applied squeezing pressures). For the special case of a circular contact region, a = b = r ₀ so that r _e = r ₀ and the condition above reduces to $r_0 =w \sqrt(2/3)$ which was used in Sect. 6.1.

Assume that the interfacial separation h(x, y) between the (undeformed) surfaces is of the form

$$ h=\frac{x^2}{2 R_1}+\frac{y^2}{2R_2} $$

We define the effective radius R _e = (R ₁ R ₂)^1/2. Thus for a sphere (radius R _s) in contact with the flat R ₁ = R ₂ = R _s and R _e = R _s. The Hertz elliptic contact mechanics can be expressed using the complete elliptic integral of the first K(x) and the second E(x) kind, which are easy to evaluate using the integral representations

$$ K(x)= \int_0^{\pi /2} d\phi \left(1-x^2{\rm sin}^2\phi \right )^{-1/2}, $$

and

$$ E(x)= \int_0^{\pi /2} d\phi \left(1-x^2{\rm sin}^2\phi \right)^{1/2}. $$

Defining

$$ F_1^3 = \frac{4}{\pi e^2} \left(\frac{b}{a}\right)^{3/2} \left( [(a/b)^2E(e)-K(e)][K(e)-E(e)]\right)^{\frac{1}{2}} $$

where

$$ e = \left(1-{\frac{b^2}{a^2}}\right)^{1/2} $$

we can write the average contact pressure $\bar{p}_{\rm e}$ and the radius r _e as [31]

$$ \bar{p}_{\rm e} = \frac{2}{3 \pi} \left(\frac{6F_{\rm N}{E^*}^2}{R_{\rm e}^2}\right)^{1/3} F^{-2}_1, \,\, \,\, \,r_{\rm e}= \left(\frac{3F_{\rm N}R_{\rm e}}{4 E^*}\right)^{1/3} G_1 $$

where

$$ G_1 =F_1 \left [\frac{1}{2}\left(\frac{b}{a}+\frac{a}{b}\right )\right ]^{-1/2} $$

Thus, we get

$$ R_{\rm c} = C_{\rm e} R_{\rm e}, \,\, \,\, \,f_{\rm N} = C_{\rm e}' \left(\frac{E^* F_{\rm N}^2}{R_{\rm e}}\right)^{1/3} $$

(B1)

where

$$ \begin{aligned} C_{\rm e} &= C F_1^3 \left [\frac{1}{2}\left(\frac{b}{a}+\frac{a}{b}\right)\right ]^{-1/2}\\ C_{\rm e}' &= C' F_1^{-1} \left [\frac{1}{2}\left(\frac{b}{a}+\frac{a}{b}\right)\right ]^{-1/2} \end{aligned} $$

In Fig. 19, we show C _e/C and C _e′/C′ as a function of a/b.

Fig. 19

The factors C _e/C and $C_{\rm e}^{\prime}/C'$ (where C ≈ 0.567 and C′ ≈ 0.429) as a function of the ratio a/b between the ellipse major and minor axis

Appendix C: Stribeck Curve for Skin; Role of Shear Thinning and Flow Factors

In Sect. 6.2, we have calculated the Stribeck curve for a hard sphere sliding on lubricated skin and compared to experiments where the skin where lubricated with silicon oil of different viscosity’s. The calculated Stribeck curve is similar to the measured one but shifted by a factor ∼20 to higher sliding velocity than observed. We have presented a tentative explanation for this related to the viscoelastic nature of the human skin and underlying tissues. In this section, we show that the deviation between experiment and theory cannot be attributed to inaccuracy in the fluid flow factors or shear thinning of the silicon oil since both effects gives very small changes in the Stribeck curve. We attribute this insensitivity of the Stribeck curve to the flow factors and shear thinning to the small contact pressure and large (average) interfacial separation even in the boundary lubrication region.

The fluid flow at the interface between solids with surface roughness is a very complex problem. However, if the surface roughness gradient is not too large one may average the basic equations of fluid flow over the surface roughness and obtain effective equations for smooth surfaces. In the effective equations enters the pressure and shear flow factors ϕ_p and ϕ_s which depends on the (average) local interfacial separation $\bar{u}.$ For smooth surfaces ϕ_p = 1 and ϕ_s = 0 which we refer to as trivial pressure and shear flow factors. Similarly in the expression for the frictional shear stress enters three frictional shear stress factors ϕ_f, ϕ_fs and ϕ_fp which are also functions of $\bar{u}.$ For smooth surfaces ϕ_f = ϕ_fp = 1, and ϕ_fs = 0, which we refer to as trivial friction factors. In Ref. [29] one of us have shown how all these functions can be calculated in an approximate but accurate way, and we used this theory in the numerical calculations of the Stribeck curve presented in this article.

Figure 20 shows the calculated Stribeck curve including all fluid flow and frictional shear stress factors (red curve), and for trivial fluid flow and frictional shear stress factors (blue curve). There is negligible difference between the two curves and we conclude that an accurate knowledge of the fluid flow and frictional shear stress factors is not necessary for the present problem due to the large average separation between the surfaces.

Fig. 20

The red line is the calculated Stribeck curve including all fluid flow and frictional shear stress factors, while the blue line is with trivial fluid flow and frictional shear stress factors. For the fluid viscosity η = 1  Pa s. See text for details (Color figure online)

Figure 21 shows the calculated Stribeck curve for fluids with the (low-shear rate) viscosity’s $\eta_0 = 0.01, 0.1, \ldots, 100 \,{\rm Pa s}$ including shear thinning. The shear thinning is described by using the effective viscosity [29, 33, 34]

$$ \eta = \frac{\eta_0}{1 + (\eta_0 /B) \dot \gamma^n} $$

where (using SI units) B = 10⁴ and n = 0.8. The shear rate $\dot \gamma = v/u$ (where u is the film thickness). This expression for the effective viscosity describes accurately the shear thinning of silicon oils of different viscosity’s [33, 34]. Not that shear thinning start at lower shear rate as the low-shear rate viscosity η₀ increases. Thus, shear thinning becomes important when $\dot \gamma > \dot \gamma_{\rm c} = (B/\eta_0)^{1/n}.$ For example, for η₀ = 1  Pa s shear thinning start when $\dot \gamma > \dot \gamma_{\rm c} \approx 10^5 \,{\rm s}^{-1}.$ Note that including shear thinning has a negligible influence on the Stribeck curve. This is again due to the low contact pressure and large average interfacial separation in the present case.

Fig. 21

The calculated Stribeck curve for fluids with viscosities $0.01, 0.1,\ldots , 100 \,{\rm Pa\,s}$ including shear thinning. See text for details