Friction on Ice: How Temperature, Pressure, and Speed Control the Slipperiness of Ice

We present sphere-on-ice friction experiments as a function of temperature, contact pressure, and speed. At temperatures well below the melting point, friction is strongly temperature dependent and follows an Arrhenius behavior, which we interpret as resulting from the thermally activated diffusive motion of surface ice molecules. We find that this motion is hindered when the contact pressure is increased; in this case, the friction increases exponentially, and the slipperiness of the ice disappears. Close to the melting point, the ice surface is plastically deformed due to the pressure exerted by the slider, a process depending on the slider geometry and penetration hardness of the ice. The ice penetration hardness is shown to increase approximately linearly with decreasing temperature and sublinearly with indentation speed. We show that the latter results in a nonmonotonic dependence of the ploughing force on sliding speed. Our results thus clarify the complex dependence of ice friction on temperature, contact pressure, and speed.

Popular Summary

Ice friction—critical to winter sports, glacier movement, and transportation risks—is generated by an interface that includes many discrete contact points due to the surface irregularities on the ice and slider. Understanding how this extended interface impacts the slipperiness of ice is a major scientific challenge that remains difficult to address because the interface is buried between two bulk materials. We have overcome this challenge and discovered that ice friction is low because of mobile ice molecules at the surface, whose mobility—and thus slipperiness—can be suppressed by a high contact pressure or a low temperature.

We use a novel combination of experimental and numerical techniques to enable a rich understanding of the interplay between local contact pressure, sliding velocity, ice temperature, and ice friction. In our experiments, we drag various sliders over an ice surface at varying speeds, all the while tracking the forces on the object and the ice temperature. Close to the melting point, we find that the slipperiness is disrupted by a plastic deformation of the ice surface, controlled by the hardness of the ice and the surface geometry, which results in a sharp increase of friction.

We conclude that it is the high mobility of molecules in the outermost layer of the ice combined with the exceptional hardness of ice close to its melting point that cause its slipperiness. Ice friction can be minimized by curtailing the contact pressure, a factor already controlled by ice skaters: The optimal ice skate has a smooth bottom (low pressure) for low friction and sharp edges (high pressure) for grip.

Figure 1

Friction coefficient $\mu$ as a function of the temperature $T$ for various sliders on ice. At a constant sliding speed $v_s$ of $0.38\mathrm{mm}/\mathrm{s}$, a small sphere (radius $R=0.75\mathrm{mm}$, blue circles), a big sphere ($R=6\mathrm{mm}$, red circles), and a model ice skate ($R\approx 22\mathrm{m}$, width 1.67 m, and length 5 mm; black squares) are slid over an ice surface at a normal force of 2.5 N. Far from the melting point, the friction coefficient follows an Arrhenius temperature dependence with an activation energy of $\mathrm{\Delta }E=11.5\mathrm{kJ}/\mathrm{mol}$. Close to the melting point, the friction coefficient increases rapidly as the sliders start to plough through the ice. The error bars represent the standard deviation. Inset: schematic illustration of the experimental setup. A slider is clamped to a commercial rheometer where, for an imposed rotation speed, the torque $\tau$ and normal force $N$ are monitored.

Figure 2

(a) Penetration hardness $P_h$ of ice as a function of temperature, obtained from indentation experiments at a speed of $v_{\mathrm{ind}}=3.8\mu \mathrm{m}/\mathrm{s}$. Indentation is performed with a sphere pushed into the ice at various temperatures; see inset (left bottom) for a schematic illustration. The indentation depth $\delta$ and force $N$ are monitored to calculate the $P_h$. The error bars, defined by the standard deviation in the penetration hardness, are smaller than the symbols used. A linear decrease of $P_h$ with temperature is found (black line) up to $-1.5°\mathrm{C}$ when pressure-induced melting sharply decreases the hardness. Upper inset: $P_h$ versus $v_{\mathrm{ind}}$ for various temperatures. (b) Friction coefficient $\mu$ as a function of the normal force $N$ for a small (radius $R=0.75\mathrm{mm}$, blue open and filled circles) and large ($R=6.00\mathrm{mm}$, red filled circles) SiC spherical slider. The ploughing model [lines, Eq. (3)] matches the observed friction coefficient. Inset: schematic illustration of ploughing in ice. The spherical slider of radius $R$ indents the ice in the normal direction with a depth $\delta$ and cross section $A_P$.

Figure 3

(a) Friction force as a function of normal force measured for glass spheres with surface roughness $S_q$ of 98, 222, 575, and 3077 nm at a temperature of $-50°\mathrm{C}$ and sliding speed of $0.38\mathrm{mm}/\mathrm{s}$. The smoothest sphere displays a friction equal to that reported in Fig. 1, which can be described by the Arrhenius equation [Eq. (1)]. For increasing surface roughness, a higher friction force is measured. (b) Surface topography and corresponding ploughing depth $\delta$ in the ice after a sphere with the highest (top) and lowest (bottom) roughness slides over it at a normal force of 0.21 N. The calculated plastic indentation depth $\delta$ for a normal force of 0.21 N is added in light gray in the insets. (c) Surface topography (top) and calculated area of real contact (bottom) for the same glass spheres at $T=-50°\mathrm{C}$ at a normal force of 0.5 N. A transition from primarily elastic contact for a smooth slider towards elastic-plastic contact for a rough slider can be observed.

Figure 4

Normalized shear stress ${\sigma }_s/e^{\mathrm{\Delta }E/k_{B}T}$ as a function of the normalized contact pressure $P_c/P_h$ for various sliders, surface roughnesses, and temperatures at a sliding speed of $0.38\mathrm{mm}/\mathrm{s}$. The solid line is a fit using Eq. (5). Triangles from light green to dark green correspond to glass spheres with a surface roughness $S_q$ of 98, 222, 575, and 3077 nm, where upward, right, down, and left-pointing triangles are measurements at $T=-90°\mathrm{C}$, $-70°\mathrm{C}$, $-50°\mathrm{C}$, and $-30°\mathrm{C}$, respectively. The blue, red, and cyan circles correspond to, respectively, a small SiC ($R=0.75\mathrm{mm}$), a large SiC ($R=6\mathrm{mm}$), and a sapphire sphere ($R=1.59\mathrm{mm}$) at $T=-90°\mathrm{C}$ for closed and $-50°\mathrm{C}$ for open markers. The error bars represent the standard deviation in the measured friction force. Inset: shear stress as a function of the contact pressure for various glass sliders with increasing surface roughness at $T=-50°\mathrm{C}$ and a sliding speed of $0.38\mathrm{mm}/\mathrm{s}$.

Figure 5

Friction coefficient $\mu$ as a function of the sliding speed $v_s$ for a smooth glass sphere (surface roughness $S_q=98\mathrm{nm}$). All measurements were performed with increasing and decreasing sliding velocity to confirm that hysteresis was absent. (a) At $-20°\mathrm{C}$ (red triangles), a nonmonotonic dependence of the friction on the sliding speed is found, which can be understood based on ploughing [Eq. (6), red line]. (b) At $-50°\mathrm{C}$ (green triangles) and $-90°\mathrm{C}$ (blue triangles), velocity strengthening of the friction is observed, which can be qualitatively described as a result of a stress-augmented thermal process; the stress exerted by the slider at the interface decreases the effective activation barrier, resulting in a logarithmic increase of the stress with the rate or velocity. (c) Dry (open markers) and water-lubricated (solid markers) friction on artificial ice (HDPE) (using the same glass slider) at room temperature and at a normal force of 1 N. The error bars represent the standard deviation in the measured friction force. In panel (c), the error is of the order of the symbol size.

Figure 6

Indentation depth $\delta$ as a function of the force $N$ for an indentation speed of $3.8\mu \mathrm{m}/\mathrm{s}$ captured with a hardness test for various temperatures. A plastic, irreversible, loading curve is observed; after indenting up to a maximum force of 80 N, during retraction the force quickly drops. During loading, the indentation increases linearly with the indentation force, where the slope $\mathrm{\Delta }\delta /\mathrm{\Delta }N$ is inversely related to the penetration hardness. For increasing temperature, we observe a larger slope and therefore a lower penetration hardness.

Figure 7

Penetration hardness in the normal direction (red) and tangential direction (black) as a function of sliding speed for ice at a temperature of $-20°\mathrm{C}$ based on Eq. (a2). The calculated indentation speed corresponding to the set sliding speed is given in the inset and as top axes for sliding a glass sphere ($R=1.84\mathrm{mm}$) at a normal force of 2.5 N over ice at $-20°\mathrm{C}$, Eq. (b5). The indentation speed is around 4% of the sliding speed, and consequently, the penetration hardness in the normal direction is smaller than the penetration hardness in the tangential direction.

Figure 8

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 $0.38\mathrm{mm}/\mathrm{s}$ is used. Inset: RCA as a function of the normal force $N$. Independent 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.

Figure 9

Contact mechanics of the SiC spheres sliding on ice and RCA as a function of normal force $N$ for a sphere with a radius of 6.00 mm (red) and 0.75 mm (blue) at temperatures of $-50°\mathrm{C}$ (open circles) and $-90°\mathrm{C}$ (closed circles). The dashed lines represent the elastic Hertzian pressure. The solid and dotted lines are, respectively, the plastic limit set by the penetration hardness at temperatures of $-50°\mathrm{C}$ and $-90°\mathrm{C}$. Inset: surface topography (left) and calculated area of real contact (right) for the SiC spheres.

Figure 10

Contact mechanics of a sapphire sphere (radius of 1.59 mm) sliding on ice and RCA as a function of the normal force N at temperatures of $-50°\mathrm{C}$ (open circles) and $-90°\mathrm{C}$ (closed circles). The dashed line represents the elastic Hertzian pressure, and the solid and dotted lines are, respectively, the plastic limit set by the penetration hardness at temperatures of $-50°\mathrm{C}$ and $-90°\mathrm{C}$. Inset: surface topography (left) and calculated area of real contact (right) for the sapphire sphere.

Figure 11

Normalized effective shear stress $({\sigma }_s-{\sigma }_P)/e^{\mathrm{\Delta }E/k_{B}T}$ as a function of the normalized contact pressure $P_c/P_h$ for various sliders, surface roughnesses, and temperatures at a sliding speed of $0.38\mathrm{mm}/\mathrm{s}$. Here, the effective shear stress is based on the measured friction force, excluding the ploughing contribution. The plastic indentation of the sphere in the ice surface, mainly for high surface roughnesses, results in a ploughing area $A_P$ and ploughing force $F_P=A_P{P}_h$ [Eq. (2)]. The ploughing can explain up to 40% of the observed increased friction. The dashed line is a fit using Eq. (5). The same symbols and colors are used as in Fig. 4. Inset: effective shear stress ${\sigma }_s-{\sigma }_P$ as a function of the contact pressure $P_c$ for various glass sliders with increasing surface roughness at $T=-50°\mathrm{C}$ and a sliding speed of $0.38\mathrm{mm}/\mathrm{s}$.