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

Abstract

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.

Full text

Friction on Ice: How Temperature, Pressure, and Speed Control the Slipperiness of Ice
Rinse W. Liefferink ,1,* Feng-Chun Hsia ,1,2 Bart Weber ,1,2 and Daniel Bonn1
1Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands
2Advanced Research Center for Nanolithography, Science Park 106, 1098 XG Amsterdam, Netherlands
(Received 17 July 2020; revised 30 October 2020; accepted 9 December 2020; published 8 February 2021)
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.
DOI: 10.1103/PhysRevX.11.011025 Subject Areas: Materials Science, Soft Matter
I. INTRODUCTION
It is commonly believed that ice is slippery due to the
presence of a layer of liquidlike water on the surface of ice
which acts as a lubricant. However, the origin of this layer
and the resulting lubrication have been debated for more
than 150 years [1–13]. The lubricating layer that allows ice
skating has been attributed to pressure-induced [2] or
friction-induced [3] melting of the ice surface and to the
presence of a premelted layer of ice [4]. More recently,
authors have suggested that the diffusion of water mole-
cules over the ice surface is responsible for low ice
friction at high temperatures and low sliding speeds
[14]. Furthermore, reciprocated ball-on-ice friction mea-
surements performed using a tuning fork have recently
revealed that—during reciprocated sliding [15] on ice—a
lubricating, viscous mixture of liquid water and ice
particles dominates the frictional behavior [16]. In the
context of each of these proposed lubrication mechanisms,
the local contact pressure exerted at the slider-on-ice
interface is a crucial parameter that remains ill understood.
In this work, we therefore take a closer look at this local
contact pressure and show that (i) the hardness of ice displays
a strong temperature and strain rate dependence that, close to
melting, leads to rich ploughing behavior that is controlled by
the temperature, sliding speed, surface topography, and
surface geometry; (ii) friction on ice increases exponentially
with the local contact pressure, suggesting that this pressure
frustrates the mobility of the lubricating layer; (iii) in the
water-immersed sphere-on-artificial ice experiment, we
observe the onset of mixed lubrication at sliding speeds
above 1m=s, indicating that most of our ball-on-ice experi-
ments are likely boundary lubricated.
II. METHODS
To investigate the slipperiness of ice, we move a spherical
slider over an ice surface. The slider is clamped to a
commercial rheometer (Anton Paar, DSR 502), at a distance
of5mmfromtherotationaxis.Theimposedrotationspeed
and the measured torque can thus be converted into a sliding
velocity and a friction force, respectively. We vary the sliding
speed from 10−6up to 10−1m=s and measure the normal
force Nand friction force Fexerted at the slider-on-ice
interface. The ratio of these two forces gives the friction
coefficient μ. The setup is thermally isolated and cooled
with liquid nitrogen (temperature Tbetween −110 °C and
−15 °C) or a coolant liquid (Tbetween −15 °C and 0°C).
A flat ice surface is established by repeatedly adding a small
amount of demineralized water on top of the already-frozen
water. As the added water initially melts the top surface of
the ice, a smooth polycrystalline ice surface is formed. The
temperature is measured with an embedded thermocouple
close to the surface and controlled with the flow rate of the
cooling liquid.
As sliders, we use silicon-carbide spheres (from Latech),
soda-lime glass spheres (from SiLibeads), a sapphire sphere
*r.w.liefferink@uva.nl, he/him/his
Published by the American Physical Society under the terms of
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Further distribution of this work must maintain attribution to
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(from Edmund Optics), and a model ice skate (stainless
steel); see Table Ifor details. The microscopic surface
topography of the balls is measured by laser-scanning
confocal microscopy (Keyence VK-X1000) over an area of
208 by 208 μm with a lateral resolution of 138 nm=pixel
and 20 nm resolution in the height direction. We do not
observe significant changes in the surface topography of
the sliders after the friction experiments and therefore
conclude that the sliders do not wear during the friction
experiments. The surface roughness values listed in
Table Irefer to the root-mean-square (rms) height variation
Sqfrom the profilometry experiments, after subtracting the
curvature of the spheres. As the surface roughness is known
to influence the local contact pressure at interfaces, we vary
the surface roughness of the soda-lime sliders by inserting
them one at a time in a container with sandpaper walls and
shaking them for 2 hours to obtain a roughened surface. By
varying the sandpaper grits (P3000, P2500, and P150), the
resulting surface topography can be controlled (Sq¼222,
575, and 3077 nm, respectively). To approximate an ice-
skate-on-ice interface in the experiments, we cut a 5-mm
piece out of an actual ice skate. This model skate has a
width of 1.67 mm and a radius of curvature (along the
length) of 22 m. The front and back edges of the model
skate are rounded off.
To quantify the penetration hardness Phof the ice, we
perform indentation experiments in which a stainless-steel
sphere with radius R¼1.6mm is pushed onto the ice by a
tensile tester machine (ZwickRoell Z2.5, with a Z6FD1
load cell) at various temperatures and preset indentation
speeds vind, resulting in plastic deformation of the ice. The
indentation depth δand indentation force Nare measured
up to a maximum load of 80 N; see the Appendix A.To
quantify the penetration hardness, we divide the penetration
force by the projected area Ac¼πr2on which it acts [see
inset of Fig. 2(a)]. Since r2¼2Rδfor δ≪R, we can write
Ph¼ðN=2πRδÞ, which is averaged over the measured
indentation range from 25 to 75 N.
Based on the mechanical properties of the slider and
the ice and the surface topography of the slider, we use the
Tribology Simulator (from Tribonet [17]) to solve the
elastic-plastic contact equations through a numerical
model. Here, we make use that the ice surface has an
elastic modulus and Poisson’s ratio of 0.75 GPa and 0.33,
respectively [14]. The hardness is measured independently
as a function of temperature and velocity. As the surface
roughness of ice is relatively low [Sq¼61 nm, calculated
for the measured surface topography of Fig. 3(b), bottom]
and without long-range curvature, the surface topography
of the sliders dominates the contact calculations. Including
the ice topography raises the contact pressure only 4%, and,
therefore, the surface topography of ice can be neglected.
III. RESULTS
A. Temperature dependence
Figure 1shows the friction coefficient μas a function of
temperature for the two types of SiC spheres and the model
TABLE I. Mechanical and geometrical details of the sliders used in the friction experiments.
Material
Radius
(mm)
Roughness
(nm)
Hardness
(GPa)
Elastic
modulus (GPa)
Poisson’s
ratio (-)
Silicon carbide 0.75, 6.00 140 27.0 410 0.14
Soda-lime glass 1.84 98, 222, 575, 3077 5.7 65 0.22
Sapphire 1.59 28 21.6 2 0.29
Stainless steel ≈22 856 2.0 200 0.28
Small sphere
Ice
F
N
Big sphere
Ice
mR = 0.75 mm R = 6 m R 22 m
N
F
Skate
Ice
FN
5 mm
-100 -80 -60 -40 -20 0
Temperature °C
T ()
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(-)
Ice
τ
N
FIG. 1. Friction coefficient μas a function of the temperature T
for various sliders on ice. At a constant sliding speed vsof
0.38 mm=s, a small sphere (radius R¼0.75 mm, blue circles),
a big sphere (R¼6mm, red circles), and a model ice skate
(R≈22 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 ΔE¼11.5kJ=mol.
Close to the melting point, the friction coefficient increases rapidly
as the sliders start to plough throughthe ice. The error bars represent
the standard deviation. Inset: schematic illustration of the exper-
imental setup. A slider is clamped to a commercial rheometer
where, for an imposed rotation speed, the torque τand normal force
Nare monitored.
LIEFFERINK, HSIA, WEBER, and BONN PHYS. REV. X 11, 011025 (2021)
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skate. In agreement with earlier measurements [14], we find
that the temperature dependence of the friction coefficient
can be captured by an Arrhenius-type equation:
μ¼ceΔE=kBT;ð1Þ
with fitting parameter c¼1.5×10−4and activation energy
ΔE¼11.5kJ=mol. As reported in Ref. [14], this activa-
tion energy matches the activation energy for ice-surface
diffusion [18,19], suggesting that the diffusion of water
molecules over the ice surface plays an important role in ice
friction. For temperatures above −20 °C, the spherical
slider displays a friction coefficient that is higher than
the friction coefficient predicted by the Arrhenius equation
and increases with temperature up to the melting point of
ice. This increase in friction with temperature is the result
of ploughing friction; the slider plastically indents the ice in
the normal direction and consequently ploughs through the
surface in the lateral direction [20]. The pressure that the
slider exerts on the ice surface controls the magnitude of
the ploughing force. To further investigate the influence of
contact pressure and quantify the ploughing force, we vary
the contact pressure exerted by the slider by varying its
curvature.
B. Ploughing
The ploughing force can be calculated by considering
plastic indentation in the normal direction, which occurs
when the contact pressure exceeds the penetration hardness
Phof the ice. This penetration hardness decreases linearly
with increasing temperature [see Fig. 2(a)]upto−1.5°C
when pressure melting sharply decreases the hardness.
Now, during sliding, the sphere plastically indents the ice
with a depth δuntil the contact area Achas increased
enough to support the set normal force N[see inset of
Fig. 2(b) and Appendix B]. This indentation results in
scratching laterally into the ice with a ploughing area AP
and a ploughing force FP, which, consequently, results in a
ploughing friction coefficient [21,22]:
μP¼APPhðTÞ
N:ð2Þ
Based on the geometry shown in Fig. 2(b), this can be
written as [14,23]
μP¼4ffiffiffi
2
p
3π3=2Rffiffiffiffiffiffiffiffiffiffiffiffi
N
PhðTÞ
s:ð3Þ
R
N
δ
r
-0.5˚C
-10˚C
-10˚C
-20˚C
-12˚C
-90˚C
-100 -80 -60 -40 -20 0
Temperature °
CT ()
0
50
100
150
200
Penetration hardness Ph (MPa)
10-7 10-5 10 -3
vind (m/s)
0
100
200
Ph (MPa)
(a) (b)
R
A
P
δ
Ploughing
012345
Normal force N (N)
0
0.15
0.3
0.45
0.6
(-)
FIG. 2. (a) Penetration hardness Phof ice as a function of temperature, obtained from indentation experiments at a speed of
vind ¼3.8μm=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 δand force Nare monitored to calculate the Ph. The error bars, defined by the standard
deviation in the penetration hardness, are smaller than the symbols used. A linear decrease of Phwith temperature is found (black line)
up to −1.5°C when pressure-induced melting sharply decreases the hardness. Upper inset: Phversus vind for various temperatures.
(b) Friction coefficient μas a function of the normal force Nfor a small (radius R¼0.75 mm, blue open and filled circles) and large
(R¼6.00 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 Rindents the ice in the normal direction with a depth δand
cross section AP.
FRICTION ON ICE: HOW TEMPERATURE, PRESSURE, AND…PHYS. REV. X 11, 011025 (2021)
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Figure 2(b) indeed confirms that both a decrease of the
radius (red-filled compared to blue-filled circles) and a
decrease of hardness (blue-filled compared to blue open
circles) result in an increase in ploughing force; the
ploughing model captures the experimentally measured
variations in the friction coefficient without adjustable
parameters. This result is also reflected in the different
amounts of ploughing for different spherical sliders
observed in Fig. 1.
These insights into the phenomenon of ploughing trans-
late to the practice of ice skating. During ice skating, low
sliding friction is desired to achieve a high sliding speed,
but, simultaneously, high friction is required to enable
changing the sliding direction. Therefore, the blades of ice
skates have a large radius of curvature in the sliding
direction, R¼3–22 m, and sharp edges with a flat or
even negative radius of curvature along the width [24].
A low coefficient of friction can be expected if the skate is
perfectly aligned with the ice surface, but if the skate is
tilted, a quick increase of the friction coefficient is found
[6,25]. A tilt of the skate results in (deeper) indentation of
the ice and therefore an increase of the friction, particularly
in the direction perpendicular to the length of the skate
because sliding in this direction involves a larger ploughing
area. This larger ploughing force gives the skater the
opportunity to push forward and make turns. In Fig. 1,
the friction coefficient of a 5-mm section of a long skate
blade is measured as a function of temperature (black
squares). A large decrease of the friction coefficient with
increasing temperature can be found up to −8°C, where-
after the friction increases again due to ploughing. The
minimum friction for the model ice skate is found for
T¼−7.72.3°C with μ¼0.039 0.003.
Therefore, sliding on ice is largely temperature depen-
dent and can be captured with an Arrhenius-type equation
in the elastic regime. Close to the melting point, when the
slider plastically indents the ice surface, the friction
coefficient increases due to ploughing, where the magni-
tude of ploughing is set by (a) the hardness of the ice,
(b) the slider geometry (radius of curvature), and (c) the
exerted normal force.
C. Local contact pressure
Conventional liquid lubrication is essentially a competi-
tion between squeeze flow and sliding (or rolling) induced
entrainment of the lubricant. The squeeze flow is driven by
an externally applied normal force, which sets the local
pressure experienced by the lubricant. To investigate the
influence of this local contact pressure on the slider-on-ice
friction, we vary the microscopic surface topography of
the spherical slider; the sharper the roughness peaks on
the slider, the higher the local contact pressure [26].In
Fig. 3(a), we report the friction force as a function of
normal force, measured for glass spheres with surface
roughnesses Sqfrom 98 nm to 3077 nm. We find that the
smoothest sphere displays a friction coefficient that is equal
to that reported in Fig. 1at the corresponding temperature,
here set to −50 °C, and described by the Arrhenius
equation, Eq. (1). The spheres with higher surface rough-
ness, and therefore a higher contact pressure, display a
significantly higher friction coefficient. For T¼−30 °C,
−70 °C, and −90 °C, a qualitatively similar result is found.
To quantify the contact pressure Pc, we perform contact
calculations in which the mechanical properties of the
slider and the ice, and the measured surface topography of
the slider form the input. The interfacial gap, at each of the
in-plane coordinates defined by the topography, forms the
output of the calculation for a given normal force. Those
locations at which the interfacial gap is zero form the area
of real contact where, in addition, the local contact pressure
is quantified. In Fig. 3(c), we plot the measured surface
topography and the calculated area of real contact for
glass spheres with increasing roughness at a temperature
of −50 °C. We find that the relatively smooth spheres
[Sq¼98 nm; Fig. 3(c), left panel] primarily deform the ice
elastically at an average contact pressure of 35 MPa. This
result is independent of temperature because the elastic
modulus of the ice (and the slider) does not change
significantly with temperature. The situation is different
for balls with a relatively high surface roughness
[Sq¼3077 nm or higher; Fig. 3(c), right panel]. As the
surface roughness is increased above this level, the calcu-
lated average contact pressure increases up to 85 MPa,
which equals the hardness of the ice, indicating that
plasticity plays an important role in the contact formation
for these rougher spheres. The hardness of the ice decreases
linearly with temperature and limits the maximal contact
pressure; the contact pressure in this regime of plastic
deformation varies from 130 MPa at −90 °C to 70 MPa at
−30 °C (see Appendix C). Note that the contact pressure in
both the plastic and the elastic regime is almost indepen-
dent of the normal force because the area of real contact
increases linearly with normal force; see Appendix C.
Spheres that deform the ice plastically will plough
through the ice when tangentially loaded. In Fig. 3(b),
top, we plot the ploughing track that was left on the ice after
a sphere with high roughness, Sq¼3077 nm, slid over the
ice surface with a normal force of N¼0.21 N and a speed
of v≈5mm=s. In contrast, spheres with low roughness
Sq¼98 nm do not leave visible damage after sliding on
the ice [Fig. 3(b), bottom], as expected based on the fact
that the calculated average contact pressure for these
balls in contact with ice is smaller than the penetration
hardness of the ice. Although the plastification during
sliding increases the friction force, it only provides a small
contribution. The maximum friction due to ploughing,
represented by the arrow in Fig. 3(a), can only explain
30% of the observed variation in friction with roughness
(see Appendix D). Therefore, we measure and calculate
the interfacial shear stress σs, which is the friction force
LIEFFERINK, HSIA, WEBER, and BONN PHYS. REV. X 11, 011025 (2021)
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divided by the area of real contact at which the friction
force is generated.
Perhaps somewhat surprisingly, in the elastic regime, σs
increases exponentially for increasing contact pressure Pc;
see inset of Fig. 4for −50 °C. Qualitatively similar results
are found for T¼−30 °C, −70 °C, and −90 °C; the lowest
roughness has a shear stress expected based on the
Arrhenius behavior, while increasing the contact pressure
up to the penetration hardness of the ice results in an
exponential increase of the shear stress. These results are
summarized in Fig. 4(triangles), where the contact pressure
and shear stress are normalized by, respectively, the
penetration hardness of the ice Phand the Arrhenius
temperature dependence of the friction coefficient
eΔE=kBT. The exponential increase of interfacial shear stress
with pressure is also known as piezo-viscosity; the viscos-
ity of a confined lubricant increases exponentially with the
mechanical pressure [27,28]. The viscosity ηis then
described empirically as
η¼ηref ePc
β:ð4Þ
Here, the pressure-viscosity parameter βsets the increase
of the viscosity with the exerted pressure starting from
the unconfined viscosity ηref. For sliding friction on ice, a
qualitatively similar process occurs; the shear stress
increases when the contact pressure on the mobile layer
is increased. From Fig. 4, we can model the shear stress as
σs¼σ0e
ΔE
kBTebPc
PhðTÞ;ð5Þ
with σ0¼2.1kPa and b¼3.4. The shear stress is set by
the mobility of the ice surface, which is decreased, or
Topography sliders
Area of real contact
0.8
1.6
2.4
3.2
4.0
0.0
µ
m
(a) (b)
Surface topography ice
(c)
Maximum
ploughing
50
012345
Normal force N (N)
0
0.5
1
1.5
2
Friction force F (N)
Sq = 98 nm
Sq = 222 nm
Sq = 575 nm
Sq = 3077 nm
-2
0
(
-2
0
(
m)
m)
FIG. 3. (a) Friction force as a function of normal force measured for glass spheres with surface roughness Sqof 98, 222, 575, and
3077 nm at a temperature of −50 °C and sliding speed of 0.38 mm=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 δin 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 δfor 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
T¼−50 °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.
FRICTION ON ICE: HOW TEMPERATURE, PRESSURE, AND…PHYS. REV. X 11, 011025 (2021)
011025-5

“frustrated,”for increasing contact pressures up to the plastic
limit. The piezo-viscous effect on the shear stress could be
interpreted as a result of confinement; the surface water
molecules become more strongly confined at the slider-on-
ice interface with increasing contact pressure. For nano-
confined water molecules, it has been observed that the
(apparent) viscosity increases when the gap size is decreased
to less than a nanometer [29–31]. Additionally, we include in
Fig. 4the measurements for the small (0.75-mm radius) and
large (6.00-mm radius) SiC spheres and a low-roughness
sapphire sphere (1.59-mm radius). For these three spheres,
the calculated shear stress and contact pressure based on
the measured friction force and surface topography
(T¼−50 °C and −90 °C; see Appendix C) match well with
the fit made for the glass spheres. A “slippery”state can
therefore only be reached when the exerted contact pressure
is sufficiently small, which is the case for a slider (or skate)
with a small surface roughness and a large curvature.
Overall, we observe an increase of the friction force
when the local contact pressure is increased. Next to,
perhaps, a minor contribution due to ploughing, the
increase of friction can be explained by a piezo-viscous
effect; for increasing contact pressure up to the plastic limit,
the shear stress increases exponentially.
D. Sliding speed
For a traditional lubricant—for example, a thick grease
in a journal bearing—the friction coefficient strongly
depends on the sliding velocity. As the sliding speed
increases, more lubricant is entrained into the contact
resulting in a pressure in the lubricant that can partially
support the external load: This process is known as mixed
lubrication. At yet higher sliding speeds, the friction may
increase with velocity because the lubricant forms a
continuous film that separates the solids and undergoes
Newtonian flow: Viscous dissipation within the lubricant is
responsible for the friction in the hydrodynamic lubrication
regime [32,33].
To investigate the slider-on-ice friction in the context of
lubrication, we perform friction experiments at velocities
ranging from 1μm=sto10 cm=s and find a nonmonotonic
relation between friction and sliding velocity at a temper-
ature of −20 °C [Fig. 5(a), red triangles]. This velocity
dependence of the friction can be fully explained using a
velocity-dependent ploughing model: During sliding, the
slider plastically indents the ice in the normal direction
at an indentation speed vind, which is a fraction of the
sliding speed vs(approximately 4%; see Appendix B).
Consequently, the indentation depth sets the ploughing area
AP, the projected cross-sectional area over which the slider
ploughs through the ice. Both during indentation and
(subsequent) ploughing, the velocity-dependent penetration
hardness of the ice controls the normal and tangential
pressure at the interface. Remarkably, the penetration
hardness is highly speed dependent; for increasing inden-
tation speed, the penetration hardness increases, as can be
seen in the inset of Fig. 2(a) for various temperatures. The
hardness of ice for temperatures up to −25 °C has been
studied before for various loading times when a sphere is
pushed into the ice [34–36] and for various impact
velocities with a short contact time when a steel sphere
is dropped onto the ice [37,38]. Although both measure-
ment methods and the definition of hardness vary, an
increase of the hardness with decreasing temperature and
increasing speed was also observed in these experiments.
This observation is in qualitative agreement with our
findings for a broad temperature and indentation speed
domain. However, the linear dependence of the hardness on
temperature in a broad domain from −110 °C almost up to
melting that we report here, to the best of our knowledge,
has not been observed before.
As the velocity domain during ice skating is broad, from
standing still up to moving at about 30 m=s, the velocity
dependence of the hardness of the ice is of key importance.
Thus far, most calculations of friction on ice used either a
T = -50°C
0 50 100
P
c (MPa)
0
20
40
s (MPa)
0 0.2 0.4 0.6 0.8 1
Normalized contact pressure P
c/P
h (-)
0
2
4
6
8
10
Normalized shear stress s/e E/k bT (Pa)
104
FIG. 4. Normalized shear stress σs=eΔE=kBTas a function of the
normalized contact pressure Pc=Phfor various sliders, surface
roughnesses, and temperatures at a sliding speed of 0.38 mm=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
Sqof 98, 222, 575, and 3077 nm, where upward, right, down, and
left-pointing triangles are measurements at T¼−90 °C, −70 °C,
−50 °C, and −30 °C, respectively. The blue, red, and cyan circles
correspond to, respectively, a small SiC (R¼0.75 mm), a large
SiC (R¼6mm), and a sapphire sphere (R¼1.59 mm) at T¼
−90 °C for closed and −50 °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 T¼−50 °C
and a sliding speed of 0.38 mm=s.
LIEFFERINK, HSIA, WEBER, and BONN PHYS. REV. X 11, 011025 (2021)
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constant or linear dependence of the hardness on velocity
[10,39–42]. The ploughing force is set by both the
penetration hardness in the normal direction (at vind) and
by the penetration hardness in the tangential direction (at
vs). Consequently, we can write, for the friction coefficient,
μP¼4ffiffiffiffiffiffiffi
2N
p
3π3=2R
PhðT;vsÞ
PhðT;vindÞ3=2:ð6Þ
For the data shown in Fig. 5(a), with a set sliding
speed vs, the corresponding indentation speed vind for the
spherical slider with radius Rand average normal force N
can be calculated directly (see Appendix B). Therefore,
without adjustable parameters, the ploughing contribution
can now be calculated for the −20 °C data and, as shown
with the red line in Fig. 5(a), this calculation is in
reasonable agreement with the measured friction coeffi-
cient; the ploughing model captures the nonmonotonic
dependence of friction on sliding speed.
At T¼−50 °C and −90 °C [Fig. 5(b)], we find velocity
dependencies that cannot be described based on ploughing.
This result is expected as, at low temperatures, the
penetration hardness of the ice increases and the ice can
accommodate the normal force through elastic deforma-
tions. At −90 °C (blue markers in Fig. 5), we observe—in
agreement with earlier measurements [14]—velocity
strengthening friction; the friction coefficient increases
logarithmically from a friction coefficient of μ≈0.55 at
μm=s speeds up to μ≈0.9for speeds on the order of cm/s.
A logarithmic increase with speed has been described
before for Eyring processes; a stress (or force) can
effectively decrease the Arrhenius energy barrier and
therefore influence the rate of the process; the Arrhenius
process for the ice surface is the diffusive motion of the
weakly bonded surface water molecules. In such so-called
stress-augmented systems, the relation between the applied
stress, or force, and the velocity is logarithmic [43,44], like
we observe here. The −50 °C case seems to be in between
the behavior of the −20 °C and the −90 °C cases, sharing
some of the features of both. A detailed (quantitative)
understanding of these observations is not available yet.
E. Substrate
As the large velocity dependence of ice friction is often
attributed to water lubrication, we finally investigate the
role of water lubrication in our friction experiments. We
replace the ice surface with a material that has similar
mechanical properties: high-density polyethylene (HDPE,
elastic modulus 1.1 GPa, surface roughness of 207 nm;
from Simona). In Fig. 5(c), the dry (open circles) and water-
lubricated (closed circles) friction coefficients, measured at
a normal force of 1 N, are plotted as a function of sliding
speed. The significant decrease of the water-lubricated
friction coefficient observed at sliding speeds higher than
1m=s indicates the onset of mixed lubrication. At larger
sliding speeds, which we cannot reach using our current
experimental setup, elastohydrodynamic lubrication is
expected to occur. These measurements suggest that, at
least up to sliding speeds of 1m=s, the slipperiness of ice is
not the result of mixed or hydrodynamic lubrication from a
liquid water film. However, we note that the onset of mixed
0
0.04
0.08
0.12
(-)
0
0.5
1
(-)
10-6 10-4 10-2 100
vs (m/s)
0
0.1
0.2
(-)
(a) Plastic regime
(b) Elastic regime
(c) Artificial ice
Ploughing model
Dry
Wet
T =
T = C
T =
-20°C
-90°
-50°C
FIG. 5. Friction coefficient μas a function of the sliding speed vs
for a smooth glass sphere (surface roughness Sq¼98 nm). All
measurements were performed with increasing and decreasing
sliding velocity to confirm that hysteresis was absent. (a) At
−20 °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 °C (green triangles) and
−90 °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.
FRICTION ON ICE: HOW TEMPERATURE, PRESSURE, AND…PHYS. REV. X 11, 011025 (2021)
011025-7

lubrication can also depend on the surface chemistry and
would occur at lower speeds if the contact pressure was
reduced.
IV. DISCUSSION AND CONCLUSION
Altogether, the speed dependence of sliding on ice
depends strongly on the contact regime, elastic or plastic
deformation. When the contact of the slider on ice is mainly
elastic, as observed for low temperatures and smooth
spherical sliders, the observed friction can be linked to
the mobility of confined water. However, for a plastic
contact, the friction is set by the amount of ploughing,
which largely depends on the hardness, the slider geometry,
and exerted normal force.
One interesting observation that merits discussion is that
during ploughing, tracks and debris particles can be formed
when the temperatures and contact pressures are high.
Under these conditions, the dynamics of ice debris particles
are expected to become important, particularly if the sliding
motion is reciprocated on a relatively small section of the
ice. Indeed, we have observed that when our sphere is made
to oscillate over the same surface area (an option that is
readily available on the rheometer) at −5°C, the frictional
response does not a steady state after 2 minutes. This was
measured with a smooth glass sphere oscillating at a
frequency of 20 Hz, with an amplitude of 100 μm and
normal force of 2 N.
Another point is that the chemical nature of the slider can
be of importance for the frictional behavior. In winter
sports, hydrophobic coatings are used to reduce the friction
[8,45]. Although sliding on snow, which is a soft porous
media of ice and water, is very different than sliding on
an ice surface, an influence of the wetting properties could
be expected. In our study, the sliders (Table I) are all
hydrophilic, and this may explain why there is little
variation in the friction that was measured with the various
materials. In this context, it would be interesting to conduct
similar ice friction experiments with hydrophobic materials
in the future.
The thermally activated diffusive motion of surface
molecules could also be interpreted as a result of the
presence of a premelted (quasi)liquid water layer. This
liquidlike layer, starting from one bilayer up to 45 nm,
grows above a critical temperature, which has been
experimentally reported in the range of −70 °C up to
−2°C [12,13,46–49]. However, in the given temperature
domain, we measure a continuous decrease of the friction,
independent of the presence or thickness of a liquidlike
water layer. Therefore, we interpret the measured Arrhenius
behavior of the friction coefficient as a result of ice-surface
diffusion.
In the mid-20th century, frictional melting of the ice was
already suggested as an explanation for the slipperiness of
ice [3]. The heat locally generates a lubricating water film
that, with increasing sliding velocity, eventually results in a
full water film that separates the surfaces (aquaplaning). We
observe that ice remains highly slippery at speeds as low as
1μm=s for −20 °C; therefore, ice remains slippery down to
very low sliding speeds, where the rate at which energy is
injected into the interface becomes negligible compared to
that at higher sliding speeds. This result indicates that the
friction coefficient is not very sensitive to frictional heating.
We interpret that, for the given microsurface and macro-
surface geometry, the slipperiness up to a speed of at least
1m=s is not the result of mixed or hydrodynamic lubri-
cation. Additionally, the slipperiness does not vary signifi-
cantly when a silicon carbide or a glass slider is used,
although the thermal conductivity of these materials differs
by 2 orders of magnitude.
In summary, temperature, pressure, and speed each have
an important impact on ice friction, largely through the
hardness of the ice. This hardness increases with decreasing
temperature and increasing strain rate (indentation speed).
On the other hand, the contact pressure exerted at the slider-
on-ice interface is set by the slider topography and
geometry. When this contact pressure approaches the ice
hardness, ploughing friction becomes dominant. This
ploughing friction depends on the sliding speed because
the rate at which the slider indents the ice in the normal
direction and ploughs through the ice in the tangential
direction varies with the sliding speed and the speed-
dependent hardness. Alternatively, at contact pressures
significantly below the ice hardness, no ploughing occurs,
and the friction is adhesive in nature. In this elastic regime,
ice friction is low and set by the mobility of the confined
water at the slider-on-ice interface. Ice friction in this
regime is inversely proportional to the mobility of water
molecules at the free ice surface, which can be viewed as an
activated process with an Arrhenius temperature depend-
ence. Increasing the local contact pressure exerted at the
slider-on-ice interface leads to increased confinement and
an exponential increase in interfacial shear stress.
Ice friction is thus low due to the high mobility of the
water molecules at the slider-on-ice interface at temper-
atures close to the ice melting point. This slipperiness can
be suppressed by increasing the local contact pressure
towards the ice hardness. It is the exceptionally high
hardness of ice, close to its melting point, that enables
the slipperiness of ice and distinguishes ice from other
solids. In practice, this means that the optimal ice skate is
very smooth and has sharp edges. When the smooth surface
makes contact with the ice, the contact pressure, and
therefore the sliding friction, is low. When the skate is
tilted, the sharp edge plastically penetrates the ice, leading
to high ploughing friction that enables grip, which is
necessary to accelerate and turn.
ACKNOWLEDGMENTS
R. W. L. thanks Shell for financial support (PT 67354).
B. W. acknowledges funding from the Netherlands
LIEFFERINK, HSIA, WEBER, and BONN PHYS. REV. X 11, 011025 (2021)
011025-8

Organization for Scientific Research (NWO) VENI Grant
No. VI.Veni.192.177.
APPENDIX A: QUANTIFICATION AND FIT
OF THE PENETRATION OF ICE
The penetration hardness Phis quantified with an
indentation test, where a sphere is pushed into the ice
surface. The indentation depth δas a function of the force N
is monitored as shown in Fig. 6for several temperatures. As
discussed in Sec. II of the main text, the penetration and
error can be calculated based on the slope of these graphs:
Δδ
ΔN¼1
2πRPh
;ðA1Þ
with R¼1.6mm the radius of the indenter. The penetra-
tion hardness is fitted with a polynomial regression on
the variables temperature T(in °C) and the logarithm of
the indentation speed lnðvindÞwith, respectively, 1 and
3 degrees:
PhðT;vindÞ¼P00 þP10TþP01 lnðvind Þ
þP11Tlnðvind ÞþP02 lnðvindÞ2
þP12Tlnðvind Þ2þP03 lnðvindÞ3:ðA2Þ
The fit parameters found, with a resulting coefficient of
determination of R2¼0.8885, are P00 ¼8.041 ×108,
P10 ¼−3.337×106,P01 ¼1.465×108,P11 ¼−2.645×105,
P02 ¼8.936 ×106,P12 ¼−6.282 ×103, and P03 ¼
1.792 ×105. For a constant indentation speed of
vind ¼3.8μm=s, as used in Figs. 2(a) and 6, this fit results
in a penetration hardness that linearly decreases with
temperature as Ph¼ð−1.01Tþ19.2Þ×106.
Close to the melting point, pressure melting occurs; the
melting temperature of ice decreases with increasing pres-
sure 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 [50]:
Pm¼L
T0ΔVT¼−13.5×106T; ðA3Þ
with L¼3.34 ×105J=kg the latent heat of fusion, T0¼0°
C the freezing point of water at a pressure of 1 bar, and
ΔV¼−9.05 ×10−5m3/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 Fig. 2(a) is described by the Clausius-
Clapeyron equation.
APPENDIX B: PLOUGHING MODEL
When sliding a sphere on ice, ploughing will occur
when the contact pressure exceeds the penetration hard-
ness. In this plastic regime, the sphere indents into the ice
up to the contact area Accan support the normal force:
AC¼½N=PhðTÞ. This contact area, the projected area of
contact in the normal direction, which is in contact with the
ice surface, is Ac¼1
2πr2, with rthe radius of the ploughing
track. The final depth of indentation δcan be written, with
the use of δ≈ðr2=2RÞfor δ≪R,asδ¼½N=πRPhðTÞ.
Consequently, this indentation results in scratching ice with
a ploughing area APand a ploughing force FP¼APPhðTÞ.
The ploughing area is the cross-sectional area AP≈4
3rδ,
and it can be rewritten as
AP¼4ffiffiffi
2
p
3π3=2R
N3=2
PhðTÞ3=2;ðB1Þ
which results in a ploughing force of
FP¼4ffiffiffi
2
p
3π3=2R
N3=2
ffiffiffiffiffiffiffiffiffiffiffiffi
PhðTÞ
p:ðB2Þ
With μP¼FP=N, we get Eq. (3).
1. Velocity-dependent ploughing model
To take into account the velocity dependency of the
penetration hardness, as is shown in the inset of Fig. 2(a)
and fitted with Eq. (A2), the ploughing model has to be
020406080
Force N (N)
0
0.1
0.2
0.3
0.4
0.5
Indentation depth (mm)
-100
-80
-60
-40
-20
0
Temperature (°C)
FIG. 6. Indentation depth δas a function of the force Nfor an
indentation speed of 3.8μm=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
Δδ=ΔNis inversely related to the penetration hardness. For
increasing temperature, we observe a larger slope and therefore a
lower penetration hardness.
FRICTION ON ICE: HOW TEMPERATURE, PRESSURE, AND…PHYS. REV. X 11, 011025 (2021)
011025-9

modified. Two velocities, and therefore two penetration
hardnesses, are involved in ploughing: the indentation
speed vind in the normal direction, where the ice is indented
by the slider; and the sliding speed vsin the tangential
direction, the speed at which the final ploughing occurs. As
the ploughing area APis set by the indentation in the
normal direction, the corresponding penetration hardness is
at the indentation speed:
AP¼4ffiffiffi
2
p
3π3=2R
N3=2
PhðT;vindÞ3=2:ðB3Þ
The subsequent ploughing force is then based on the
penetration hardness at the sliding speed and the calculated
ploughing area AP:
FP¼4ffiffiffi
2
p
3π3=2R
PhðT;vsÞ
PhðT;vindÞ3=2N3=2:ðB4Þ
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
vind
vs¼δ
r¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N
2πR2PhðT;vindÞ
s:ðB5Þ
This nonlinear equation can be numerically solved to
yield an indentation speed vind for a given sliding speed vs.
For a glass sphere sliding over ice at −20 °C, the resulting
indentation speed as a function of the sliding speed is given
in the inset of Fig. 7; the indentation speed is, in general, a
fraction of the sliding speed, around 4%. Consequently, for
a given sliding speed, we can calculate the penetration
hardness in the normal and tangential directions; see Fig. 7.
Finally, the friction force and friction coefficient can be
calculated based on Eq. (B4), where the indentation speed
is based on numerically solving Eq. (B5).
APPENDIX C: CONTACT MECHANICS
To quantify the real contact area (RCA) and the average
contact pressure Pcof the spherical sliders on ice, we use
the Tribology Simulator (from Tribonet [17]). Based on the
surface topography of the slider and the mechanical
properties of the slider and the ice surface, the simulator
Tangential direction
Normal direction
10-6 10-4 10-2 100
Sliding speed vs (m/s)
0
50
100
150
200
250
300
350
400
Penetration hardness Ph (MPa)
10-6 10-4 10 -2
Indentation speed vind (m/s)
10-6 10-4 10 -2 100
vs (m/s)
10-6
10-4
10-2
100
vind (m/s)
FIG. 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 °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
(R¼1.84 mm) at a normal force of 2.5 N over ice at −20 °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.
T = -50 °C
Rough
Smooth
-100 -80 -60 -40 -20 0
Temperature T (°
C)
0
50
100
150
200
Pressure P (MPa)
0 500 1000
N (mN)
0
0.02
0.04
RCA (mm
2
)
FIG. 8. Pressure Pas function of the temperature Tfor 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 Phof the ice. For the latter, the penetration hardness in
the normal direction for the set sliding speed of 0.38 mm=sis
used. Inset: RCA as a function of the normal force N. Indepen-
dent 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.
LIEFFERINK, HSIA, WEBER, and BONN PHYS. REV. X 11, 011025 (2021)
011025-10

solves the elastic-plastic contact equations through a
numerical model.
The plastic limit is set by the penetration hardness of the
ice in the normal direction, calculated for the set sliding
speed vs¼0.38 mm=s and temperature T, with the use of
Eqs. (A2) and (B5). In Fig. 8, the contact pressure as a
function of temperature is given for glass spheres with
surface roughness Sqfrom 98 nm to 3077 nm. The dashed
and solid lines represent, respectively, the elastic Hertzian
contact pressure [51] and the plastic limit given by the
penetration hardness as N=Ph. 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 Fig. 8. Therefore,
the contact pressure is almost independent of the nor-
mal force.
In Fig. 9, 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.
The contact mechanics for a sapphire sphere on ice are
given in Fig. 10. The contact is mainly elastic due to the low
surface roughness (Sq¼28 nm). Therefore, the RCA
increases, as expected for an elastic Hertzian contact
[51], sublinearly with the normal force. The RCA for a
normal force of 500 mN is used to quantify the shear stress
and contact pressure.
APPENDIX D: CONTRIBUTION OF 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 Fig. 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, Sq¼3077 nm, the measured ploughing
area (AP¼7.8×10−11 m2) results in a friction coefficient
based on ploughing of μP¼0.07 [for N¼0.21 N, pen-
etration hardness in the normal direction Ph¼194 MPa
and using Eq. (2)]. As the increase in the friction coefficient
for the highest surface roughness relative to the lowest
surface roughness is Δμ¼0.24, the ploughing can only
explain 30% of the increased friction. Based on the
measured surface topography, a ploughing area of
(AP¼12.2×10−11 m2) can be calculated [see gray area
in the top panel of Fig. 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
0 200 400 600 800 1000
Normal force N (mN)
0
0.01
0.02
0.03
0.04
0.05
0.06
Real contact area RCA (mm 2)
Topography contact area
N = 250 mN
N = 500 mN
R = 6.00 mm
R = 0.75 mm
0.8
0.0
1.6
2.4
3.2
4.0
2.0
0.0
4.0
6.0
8.0
10.0
m
µµ
m
FIG. 9. Contact mechanics of the SiC spheres sliding on ice and
RCA as a function of normal force Nfor a sphere with a radius of
6.00 mm (red) and 0.75 mm (blue) at temperatures of −50 °C (open
circles) and −90 °C (closed circles). The dashed lines represent the
elastic Hertzian pressure. The solid and dotted lines are, respec-
tively, the plastic limit set by the penetration hardness at temper-
atures of −50 °C and −90 °C. Inset: surface topography (left) and
calculated area of real contact (right) for the SiC spheres.
0 100 200 300 400 500
Normal force N (mN)
0
0.01
0.02
0.03
0.04
0.05
0.06
Real contact area RCA (mm 2)
Topography Contact area
N = 500 mN
R = 1.59 mm
0.8
0.0
1.6
2.4
3.2
4.0
µ
m
FIG. 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 °C (open circles) and −90 °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 °C and −90 °C. Inset: surface topography (left) and calcu-
lated area of real contact (right) for the sapphire sphere.
FRICTION ON ICE: HOW TEMPERATURE, PRESSURE, AND…PHYS. REV. X 11, 011025 (2021)
011025-11

track and the ploughing area. Now, the ploughing can
explain 40% of the observed increased friction.
In Fig. 11, the increase of shear stress with increasing
contact pressure where the ploughing is excluded is given.
For the ploughing stress σP¼FP=RCA, a ploughing force
based on the quantified plastic indentation of the surface
topography is used. We can model the effective shear
stress σs−σPas Eq. (5) with σ0¼3.4kPa and b¼2.6.
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.
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T = -50 °C
0 50 100
P
c (MPa)
0
20
40
s - P (MPa)
0 0.2 0.4 0.6 0.8 1
Normalized contact pressure P
c/P
h (-)
0
2
4
6
8
10
Normalized shear stress (s - P)/e E/kbT (Pa)
104
FIG. 11. Normalized effective shear stress ðσs−σPÞ=eΔE=kBTas
a function of the normalized contact pressure Pc=Phfor various
sliders, surface roughnesses, and temperatures at a sliding speed of
0.38 mm=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 APand
ploughing force FP¼APPh[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 σs−σPas a function of the contact
pressure Pcfor various glass sliders with increasing surface
roughness at T¼−50 °C and a sliding speed of 0.38 mm=s.
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