Sliding friction on ice

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A.1 The penetration hardness of ice
The penetration hardness P h is quantified with an indentation experiment where a sphere with a radius of R = 1.6 mm is pushed into the ice surface, see Section 2.2 for more details. For an indentation velocity v ind of 3.8 µm/s the indentation depth δ as a function of the loading force N is plotted in Figure A.1 for several temperatures.
Close to the melting point, pressure melting occurs; the melting temperature of ice decreases with increasing pressure because the liquid-phase density is lower than the solid phase. The pressure that has to be exceeded to melt ice is described by the Clausius-Clapeyron equation [175]:  with L = 3.34 × 10 5 J/kg the latent heat of fusion, T 0 = 0°C the freezing point of water at a pressure of 1 bar, and ∆V = −9.05 × 10 −5 m 3 /kg the change in specific volume from solid to liquid. For temperatures higher than −1.5°C, the pressure necessary for pressure melting is lower than the penetration hardness. Therefore, the limiting pressure for −1.5°C up to 0°C in Figure 5.2 is described by the Clausius-Clapeyron equation.

A.2 Ploughing model
A spherical slider ploughing through ice, similar as ploughing through granular materials as described in Chapter 3, occurs when the contact pressure exceed the penetration hardness. In this plastic regime, the sphere indents into ice up to the contact area A c can support the normal force: A c = N P h (T) . This contact area, the projected area of contact in the normal direction, which is in contact with the ice surface, is A c = 1 2 πr 2 , with r the radius of the ploughing track. The final depth of indentation δ can be written, with the use of δ ≈ r 2 2R for δ << R, as δ = N πRP h (T) . Consequently, this indentation results in scratching ice with a ploughing area A P and a ploughing force F P = A P P h (T). The ploughing area is the cross-sectional area A P ≈ 4 3 rδ, and it can be rewritten as which results in a ploughing force of

Velocity-dependent ploughing model
To take into account the velocity dependency of the penetration hardness, as is shown in the inset of Figure 5.2(a) and fitted with Equation (A.1), the ploughing model has to be modified. Two velocities, and therefore two penetration hardnesses, are involved in ploughing: the indentation speed v ind in the normal direction, where the ice is indented by the slider; and the sliding speed v s in the tangential direction, the speed at which the final ploughing occurs. As the ploughing area A P is set by the indentation in the normal direction, the corresponding penetration hardness is at the indentation speed: The subsequent ploughing force is then based on the penetration hardness at the sliding speed and the calculated ploughing area A P : Based on the sphere-on-ice geometry, we can calculate the indentation speed corresponding to the sliding speed and subsequently calculate the related penetration hardness for the ploughing force. The ratio of the related speeds is: Tangential direction Normal direction

Soda-lime glass
To quantify the real contact area (RCA) and the average contact pressure P c of the spherical sliders on ice, we use the Tribology Simulator (from Tribonet  Figure A.3: Pressure P as function of the temperature T for glass spheres with surface roughnesses 98, 222, 575, and 3077 nm at a normal force of 500 mN. The smoothest sphere is mainly elastic where, for increasing surface roughnesses, the pressure increases until the plastic limit is reached. The dashed and solid lines are, respectively, the elastic Hertzian pressure and the penetration hardness P h of the ice. For the latter, the penetration hardness in the normal direction for the set sliding speed of 0.38 mm/s is used. Inset: RCA as a function of the normal force N. Indenpendent of the surface roughness, the real contact area increases linearly with the normal force. Therefore, the contact pressure is almost independent of the normal force.
In Figure A.3, the contact pressure as a function of temperature is given for glass spheres with surface roughness S q from 98 nm to 3077 nm. The dashed and solid lines represent, respectively, the elastic Hertzian contact pressure [176] and the plastic limit given by the penetration hardness as N/P h . For increasing surface roughnesses, the contact mechanics convert from a mainly elastic contact to a plastic contact. The RCA increases linearly, even for the relatively smooth sphere, with the normal force, as is given in the inset of Figure A.3. Therefore, the contact pressure is almost independent of the normal force.

Silicon carbide
In Figure A.4, the real contact area for the SiC spheres is given as a function of the normal force. Both spheres, with radii of 6.00 (red) and 0.75 mm (blue), have a mainly elastic contact with the ice surface at −50°C (open circles) and at −90°C (closed circles). The large sphere has a large RCA and, due to the finite size of the measured surface topography (208 by 208 µm), can only be calculated up to 400 mN.

Sapphire
The contact mechanics for a sapphire sphere on ice are given in Figure A.5. The contact is mainly elastic due to the low surface roughness (28 nm). Therefore, the RCA increases, as expected for an elastic Hertzian contact [176], sublinearly with the normal force. The RCA for a normal force of 500 mN is used to quantify the shear stress and contact pressure.

A.4 Ploughing on the microroughness scale
Ploughing not only occurs on the macroscale of the slider-on-ice contact; single asperities can plastically deform the ice and therefore plough through it tangentially. In Figure 5.3(b), the ploughing tracks that were left on the ice after a sphere slid over the ice surface are given. For a high surface roughness, S q = 3077 nm, the measured ploughing area (A P = 7.8 × 10 −11 m 2 ) results in a friction coefficient based on ploughing of µ P = 0.07 [for N = 0.21 N, penetration hardness in the normal direction P h = 194 MPa and using Equation (5.2)]. As the increase in the friction coefficient for the highest surface roughness relative to the lowest surface roughness is ∆µ = 0.24, the ploughing can only explain 30% of the increased friction. Based on the measured surface topography, a ploughing area of (A P = 12.2 × 10 −11 m 2 ) T = -50°C can be calculated [see gray area in the top panel of Fig. 5.3(b)]. As the orientation of the sphere on ice is not the same, a small difference is found compared with the measurements in both the ploughing track and the ploughing area. Now, the ploughing can explain 40% of the observed increased friction.
In Figure A.6, the increase of shear stress with increasing contact pressure where the ploughing is excluded is given. For the ploughing stress σ P = F P /RCA, a ploughing force based on the quantified plastic indentation of the surface topography is used. We can model the effective shear stress σ s − σ P as Equation (5.4) with σ 0 = 3.4 kPa and b = 2.6. Consequently, if ploughing is taken into account, the shear stress set by the mobility of the ice surface is, although smaller, still the main contribution to the friction force.