In the continued quest to reduce friction between mobile surfaces, that between the sole of a ski and snow stands out because of its long history of debate, the joy of skiing, as well as the notion that understanding of this complex issue may have broad, more general implications. In a seminal series of papers, Bowden and Hughes1, 2 examined the dependence of the time of descent of model and standard skis on, among other things, the nature of the materials used as “soles,” including poly(tetrafluoroethylene) (PTFE), better known as Teflon®, which was reported to be optimal for faster skiing. Numerous studies have been devoted since to elucidate the very parameters that influence the friction of matter on snow and ice. Unfortunately, these studies often yielded conflicting results. For instance, it has been advanced that hydrophobicity and thermal conductivity of the ski-sole material is critical.1–5 It also has been argued that the temperature of the track,1, 6, 7 the size and the shape of the snow grains8 are major factors influencing friction between a slider and snow.9 Furthermore, with respect to the sliding mechanism, concepts of pressure- or friction-induced formation of a thin film of water between both the ski sole and snow have been invoked.1, 2, 10–15 Accordingly, models have been developed to describe friction on snow based on the heat balance at the interface slider-water film-snow/ice.4, 12, 13, 16
Nonetheless, consensus seems to be that the total friction results from a combined effect of lubricated friction, dry friction, and capillary suction. The relative importance of these factors is thought to depend on the thickness of the lubricating water film and were schematically presented by Colbeck4 as a Stribeck curve,17 reproduced in Figure 1. At low water film thickness, friction is high due to excessive plowing. Increased lubrication at increasing film thickness lowers friction, until the water film is thick enough to cover the entire contact area and so-called “capillary suction” drag forces, resulting from the breakup of capillary bridges, increases friction again.
Inspired by the fact that in the past half century since the publications by Bowden and Hughes,1, 2 a wealth of new polymeric materials have been developed, and also because the issue of just what promotes reduced friction on snow remains largely unsolved, we embarked on a study to revisit the issue of “skiing faster,” by evaluating the performance of model skis that are equipped with polymeric soles of a broad range of chemical compositions and surface structures.
For the majority of the experiments, we adopted the approach of Bowden and Hughes1, 2 and constructed small-scale sliders [Fig. 2(a)] of a length of 25 cm and a width of 6 cm to match the width of a Nordic ski track. They were built with aluminum (alloy AlMg4,5MnZn) and a reflective flag was mounted on top for rate-detection purposes. Brass weights were fitted in a central position onto the slider; the average weight of the ensemble was 1.69 kg so that the surface pressure (ca. 1.3 kPa) is comparable to the surface pressure during actual skiing. It should be noted that at the temperatures where most experiments were conducted (around −3 °C), the influence of surface pressure on the friction coefficient has been reported to be negligible (up to ca. 10 kPa).6
To evaluate various ski-sole materials and structures, two methods for attaching them to the sliders were selected. To fit thin polymer films, the base of the aluminum sliders was covered with double-sided adhesive tape, to which the film was adhered and clamped around with aluminum strips to prevent snow from penetrating between the film and adhesive tape. In a selected series of structured ski bases, a 2-mm thick sheet of ethylene tetrafluoroethylene copolymer (ETFE, Symalit, Switzerland) with a glass mesh imprinted on its back side was glued to the slider base with a commercial epoxy glue (Agomet P 79, Angst & Pfister, Switzerland).
The model sliders were equipped with a library of polymer films covering chemical compositions ranging from ultrahydrophilic to ultrahydrophobic (Table 1). The polymer films were chosen to probe the extremes between adhesive, friction enhancing interactions, and repulsive friction-reducing phenomena between snow and the ski sole.
Table 1. Polymers Employed in Slider Soles, Ordered According to Decreasing (Approximate) Contact Angle; Surface Tension Values in Brackets are Estimations
Fine surface structures were created onto the soles along the sliding direction of the ski with a steel brush (TOKO, Switzerland), as practiced in traditional preparation of skis. For the introduction of coarser surface structures, an in-house-developed ski structuring device was used to create topological features on a slider surface in a well-defined manner. The device consists of a board onto which sand paper of the desired grain size (Brütsch/Rüegger AG, Switzerland) was mounted and a holder for the small sliders to move them under a given load and angle over the sand paper. An elastomeric film was inserted between the grinding board and the sand paper to ensure an even surface pressure distribution. The pressure exerted on the skis was 12.75 g/cm2 (1.25 kPa). The standard grinding procedure consisted of 10 strokes over the sand paper, whereby the latter was cleaned with an air gun after each stroke to remove wear particles.
Topographic analysis of the surfaces of the slider soles was conducted with a white light optical profilometer FRT MicroProf® using the software Mark III (Fries Research & Technology GmbH, Germany). The resolution was 1 μm in the x and y direction and 10 nm in the z direction. To eliminate deviations due to differences in (UV) light transparency of the polymers used, the analysis was performed on replicas made with a dimethyl siloxane polymer, PROVILnovo Light C.D.2 fast set (Heraeus Kulzer GmbH, Germany), instead of on the actual soles. The arithmetical mean roughness (Ra) was determined according to DIN EN ISO 4288.33 The cut-off length (a filter that Fourier transforms the surface profile and rejects values larger than the cut-off length) was set to 0.8 mm and five measurements were recorded per sample. The slope of the overall surface profile was set to 0° with a linear correction factor prior to the analysis. The direction of the surface analysis measurements was always perpendicular to the gliding direction of the sliders.
As it is well known that surface roughness has a significant influence on contact angle measurements,34–37 and processing techniques considerably affect the surface roughness of polymer films, contact angles were determined of the actual films used and not simply adopted from literature. A dynamic contact angle analysis was chosen over a static method to generate results of higher accuracy as in the latter the data can vary between the values of the advancing and the receding contact angles recorded in dynamic measurements.34 The dynamic experiments were conducted using an optical drop-shape analyzing system (G2/G40 2.05-D, Kruess GmbH, Germany). Distilled water was used and applied at a rate of 15 μl/min. A video camera recorded the growth and shrinkage of a drop with 40 images per analysis. Two measurements were performed: one for the advancing and one for the receding contact angle both perpendicular to the ski axis. To examine effects due to the sliding tests, samples were analyzed before and after sliding on snow.
Analysis of the data acquired was performed using the tangent Method 2 routine of the Krüss Drop Shape Analysis program (DSA version 22.214.171.124 for Windows 9x/NT/2000, 1997–2002 Krüss), which program utilizes a fourth-order polynomial function to fit both sides of the drop shape and calculates its angle with the base line. To avoid interference of pinning effects, the measured angle values were analyzed only after the border of the water droplet moved in a steady manner. If one side of the drop remained pinned, the values of that side were rejected.
To ensure constant conditions, all sliding experiments were performed at intermediate temperatures (−3 to −4 °C) in the indoor ski hall at Neuss, Germany [see Fig. 2(b)], with the exception of Nordic ski trials that were conducted on an outdoor Nordic ski track in Davos, Switzerland. To create reproducible Nordic tracks [Fig. 2(c)], the snow was prepared with conventional grooming machines (Pistenbully 100, Kässbohrer Geländefahrzeuge AG, Germany), complying with world-cup regulations. Grooming was commenced several days before the experiments were conducted. To generate a compact and homogenous snow cover, the snow is first processed with a snow tiller, a fast rotating (ca. 2000 rpm) cylinder with spikes. The various ice crystals in the snow are crushed to more or less spherical particles, resulting in a densely packed cover. A secondary effect in this process is heating up of the snow, which enhances sintering (solidifying) of the processed cover. On the day before the experiments, the grooming equipment pulled track plates behind the tiller and produces the Nordic ski tracks. Final grooming of the tracks was performed at least 6 h before the measurements were started. This rest period was necessary for the track to solidify and to gain additional mechanical strength. When a fresh track is used immediately after grooming, it deteriorates rapidly as the track gets carved out after only a few trials.
In a typical experiment, in one series, 20 small sliders ran down the same Nordic ski track. The time between each individual run in a series was about 30 s, and the interval between each series was about 5 min. The same track was not used for more than three series. The time of descent of the sliders was recorded with a precision of 1/1′000 of a second with an infrared light-barrier timing system (Alge RLS1n, Timy Alge, Austria, generously provided by TOKO).
It was found that repeated runs of the sliders down a track causes snow grains on its surface to be flattened and compressed, leading to an increase in the contact area of the ski base with the snow. In addition, after repeated use, slow erosion of the grooves and sides of the Nordic ski track was observed. Both phenomena result in increased friction that was found to lead to a – most fortunate – highly consistent, approximately linear increase of the time of descent in sequential runs of identical skis. Importantly, however, that time was reduced to its initial value when a new track was used. It was also established that a used track can be “refreshed.” This was concluded from the fact that the time of descent of a slider is reduced close to the initial value on a fresh track, if a slider equipped with sand paper was conveyed down a worn one prior to conducting the run. As a result of the simple linear increase of the time of descent due to the aforementioned track wear and its reproducible restoration with sand paper, in the following, data obtained in sequential runs were corrected by superposing a monotonous linear trend on the experimental data. These corrected data display a considerably reduced spread and a higher accuracy; therefore, this correction was applied throughout, unless stated differently.
Time of Descent
The times of descent, te, of the sliders fitted with the various polymer soles on the above described well-defined slope are presented either in the form of a box–whisker plot38 or as their minimum value where so stated.
To account for differences in, among other things, the weight of the sliders and the length of tracks, the following concept of a “dimensionless time of descent,” t*e, is introduced. Figure 3 schematically depicts a slider of mass m moving down a distance s along a track of length s0 and slope α. The gravitational force F = mg with the gravitational constant g is decomposed in a normal force perpendicular to the slope and a component parallel to the slope . Assuming Coulomb friction and neglecting air drag (deemed justified in view of the compact size of the slider and the relative low speeds involved), the frictional force Ff is proportional to the normal force FN, assuming a constant friction coefficient . The equation of motion for the slider then follows from the force balance parallel to the slope:
Selecting the track length s0 as a characteristic distance and as a characteristic time, and defining a dimensionless time and distance as and , respectively, leads to the following dimensionless equation of motion:
Expressed in dimensionless quantities, the equation of motion of the experiment only depends on the slope α and the quantity of interest, that is, the friction coefficient μ. Therefore, experimental times of descent te measured on tracks of different track length but of similar slope, using sliders of different weight, can be compared using the dimensionless time of descent, t*e, defined as:
The advantage of this approach is demonstrated in Table 2, in which results are presented that were obtained with a small slider and a Nordic ski both equipped with the same base material linear-low density PE (LLDPE) on vastly different track lengths. The experiments with the Nordic skis were conducted by a professional athlete, according to industry standards. Because of the large difference in track length (small sliders 35 m and Nordic skis 142 m), the absolute times of descent in the two tests are, of course, vastly different. Gratifyingly, however, when expressed in the dimensionless time of descent, a meaningful comparison of the two test methods is obtained.
Table 2. Average Actual Time of Descent, te, and the Dimensionless Time of Descent, t*e, of Small Model Sliders (12 runs) and Nordic Skis (3 runs) Fitted with Identical LLDPE Sole Material. The Test Field was in Davos, Switzerland. The Snow Temperature Varied from −11 to −9 °C and the Air Temperature from −12 to −7 °C
In a first set of experiments, the friction between snow and miniature skis equipped with as-received, “flat” films of perfluoroalkoxy copolymer (PFA), LLDPE, ETFE, poly(vinylidene fluoride) (PVDF), poly(ethylene terephthalate), and polyimide (PI) with a thickness of 100–250 μm was investigated. The films were relatively smooth, without visible texture on their surface. Advancing and receding contact angles were measured perpendicular to the gliding direction before and after the sliding test. The ski track used had a length of 40 m, the air temperature was −3 °C, and the snow temperatures varied from −3.4 to −4 °C; for each base material, eight measurements of the time of descent were recorded. The dimensionless times of descent are shown in Figure 4 versus the advancing and receding contact angles determined on soles before the sliding experiments. On cursory examination of the data presented, there appears—not unexpectedly—to be some correlation between the (receding) contact angle, or polarity, of the material of the ski sole and the time of descent; that is, the more polar material exhibits a higher friction with snow, confirming the well-known notion that hydrophobic polymers glide faster over snow than their hydrophilic counterparts. No distinct correlation was observed between the time of descent and the advancing or receding contact angle measured after sliding on snow, possibly due to wear-induced surface roughness and dirt accumulation during the experiment.
In a second set of experiments, polymer soles of different surface chemistry, again ranging from hydrophilic to hydrophobic, were structured using a steel brush as it is practiced in traditional ski preparation and detailed in the above Experimental section, resulting in a unidirectional topography along the axis of the slider (see inset Fig. 5). The dimensionless times of decent of these structured small sliders are compared with those fitted with corresponding reference “flattened” films. The latter were produced by pressing the as-received films between two highly polished steel plates with flat PI separators at elevated temperatures (about 20 °C below the melting temperature of the polymer films). The arithmetical roughness values Ra of both the flattened and the structured films determined before the sliding tests are indicated in Figure 5. The relatively high Ra values of the “flat” hydrophobic PTFE, PFA, ultrahigh molecular weight PE (UHMW PE), and LLDPE films probably stems from some surface damage on removal of the PI film and poor flowability during hot pressing of the ultrahigh molar mass polymers PTFE and UHMW PE. The surface roughness of the slider soles after brushing, of course, depends on the wear resistance of the polymers used and, hence, varied widely for the films used in this study.
Most interestingly, an initial comparison between the times of descent for sliders equipped with the structured and “flat” films reveals that the chemical nature of the ski sole appears to be only of minor importance when compared with the influence of their surface structure (Fig. 5). For instance, indeed, a slider equipped with a smooth film of the hydrophilic polyamide (PA) 6,6 did not slide (i.e., infinite time of descent), but when brushed along the gliding direction, it performed virtually as well as sliders fitted with structured hydrophobic polymers, such as polyethylene (!).
A plot of the shortest dimensionless time of descent of all small sliders versus surface roughness Ra of the different polymer soles is presented in Figure 6. The data in this figure suggest a strong correlation between t*e and Ra, in particular for relatively flat surfaces and especially for ski soles made of hydrophilic polymers. However, at increasing roughness, Ra > 0.2 μm, the frictional properties of the sliders become increasingly independent of the chemical composition of the sole. It actually emerges that optimum friction, leading to the fastest descent, of all ski-sole materials studied is predominantly determined by their surface roughness and not by their chemical composition.
As ski soles used in Nordic skiing competitions typically have a high roughness (Ra of 2.5–12.5 μm),39 which exceeds that introduced with the steel brush in the above study (cf. Fig. 6), it was deemed desirable to extend the roughness range of the small slider soles to higher values. For these experiments, ETFE soles were selected, since this polymer displayed a rather strong dependence of friction on surface roughness. These soles were structured with the home-built structuring device, using sand paper of different grain sizes (25–200 μm). Two types of textures were selected. One was a linear structure oriented parallel to the ski axis, similar to those applied in traditional ski preparation. The other structure was of no preferred orientation, which was created by circular grinding movements with the sand papers. In addition, the data obtained with the two structures that were used in the previous experiments with ETFE soles, that is, the untreated and steel-brushed surfaces were added to complete the experimental set. The results, shown in Figure 7, display two distinct trends. For ski soles with oriented structures, a gentle, slight increase of the dimensionless time of descent with increasing surface roughness Ra > 1 μm is observed. By contrast, ski soles with unoriented structures exhibit a much steeper increase of t*e with surface roughness, to the point that the slider fitted with the sole with the roughest unoriented structure (created with 200 μm grain size sand paper) did not glide at all.
The experiments presented so far revealed that if the surface roughness of the sole of sliders is in the optimum range, the difference between oriented and “random” surface textures is minimal. To further substantiate this finding, a set of measurements were performed, involving ski soles with surface structures that exhibited distinctly different orientations. For this purpose, soles were prepared in a two-step process. First, they were grinded with a milling cutter and, subsequently, structured with sand paper using structuring device detailed above. The initial grinding of the skis yielded a base structure with a wave-like pattern with a “peak-to-valley” distance of about 10 μm. Ra values of these structures were in the range of 1 μm or lower (due to the filter applied; cf. Experimental). Thereafter, structures with different orientations were introduced on top of this base pattern. An illustrative image of the topography of a surface with a structure parallel to the sliding direction on top of the base structure is shown in Figure 8.
Five different “top” patterns were created in this fashion: in addition to the aforementioned parallel and unoriented structures, features perpendicular to and at 45° with the sliding direction were applied, as well a cross-hatched structure alternating by 45° to the gliding direction, with sand papers of different grain sizes. Results obtained with the sliders fitted with the various structured ETFE soles are compared with those of only the base pattern in Figure 9. The grain sizes of the sand paper used are indicated on the x-axis; the open symbol on the right-hand side of the graph indicates the slider with the base structure only. The data in this figure reveal that the effect of the orientation of the fine “top” structure is most pronounced for coarse structures of a high surface roughness. When approaching the above discussed optimum surface roughness of about 1 μm, the influence of the orientation of the applied structures becomes less important, consistent with the above reported results. On the other hand, the results presented in Figure 9 also reaffirm the common knowledge that gliding properties of ski soles with structures oriented parallel to the sliding direction invariably are superior.40
A summary of the above experimental findings is presented in Figure 10, in which the dimensionless times of descent of small-scale sliders equipped with soles of a broad spectrum of polymers and featuring a wide range of surface structures are collected.
Before discussing the above results, it is important to recognize that the roughness range of the slider soles explored (Ra = 0.05 to 10 μm) is much smaller than the (macroscopic) contact spot size between a ski sole and snow, which has been reported to be of the order of 100–200 μm.15, 41, 42 Therefore, in all experiments shown in, for instance, Figure 10, the microscopic contact area between ski sole and snow is dominated by the roughness of its sole as the ski is sliding over the macroscopic contact spots. It, therefore, seems reasonable to argue that for sliders with a sole of a low surface roughness complete lubrication occurs in these localized contact spots—as long as the thickness of the water layer generated by frictional heat is large compared with the surface roughness of the ski sole. In this roughness regime, capillary suction, advanced by Colbeck,4 applies, according to which capillary attraction exerts a frictional force due to “liquid bridging” between slider and snow grains. Naturally, capillary suction is more pronounced for polar surfaces, which explains the longer times of descent observed for relatively smooth hydrophilic surfaces when compared with their smooth hydrophobic counterparts. On increasing surface roughness of the slider soles, the wetted contact area where capillary suction is active is reduced and the friction coefficient decreases accordingly.40 However, when that roughness is of the order of the thickness of the friction-induced water film, increased plowing into the snow will occur, which, in turn, increases the friction coefficient. It is to be expected that plowing will be more severe when the relief structures are not aligned in the ski gliding direction, which is consistent with the large difference in performance between sliders with unoriented and oriented structured soles.
According to the above reasoning, a slider should experience the lowest friction when the surface roughness is of the order of the lubricating water layer thickness. At this optimum roughness, the “microscopic contact area” is favorably reduced, without too many protrusions of the sole penetrating through the lubricating water film, and thus minimizing the plowing forces. For the prevailing experimental conditions in this work, the water layer thickness has been estimated to be of the order of 0.1–1.2 μm,4, 7 which is in reassuring agreement with the roughness of the slider soles for which minimum times of descent are observed (Fig. 10).
Naturally, it is to be expected that the optimum surface roughness at which a ski sole experiences the lowest friction coefficient depends on environmental conditions. Clearly, at higher snow temperatures, the water film generated by frictional heat will be thicker and, correspondingly, the optimum surface structure should be of a higher roughness. By contrast, at lower temperatures, the water layer will be thinner, which would suggest the use of a smoother surface to prevent plowing.
In this study, polymer slider soles of a wide range of chemical compositions and surface structures were explored for their tribological properties on snow at near constant conditions (snow temperature −2.6 to −4 °C). An optimum surface roughness of the soles is detected – in the range of 0.5–1 μm – for which friction is minimal, essentially independent of the surface topology, and, most remarkably, for which the influence of the chemical composition of the sliding surface becomes virtually negligible. Outside this optimum roughness range, two distinctly different trends are apparent. Reducing the surface roughness to lower values of Ra, that is, smoother soles, resulted in an increase of the time of descent, most pronounced for hydrophilic materials. Indeed, sliders fitted with relatively smooth soles of polar polymers, such as PA 6,6, ETFE, and PVDF, do not glide at all, while they performed in a satisfactory fashion when featuring a roughness of the aforementioned dimensions. On the other hand, on increasing the roughness of the soles in excess of the optimum range, the time of descent again increases, now, however, strongly depending on the orientation of the surface structure. For instance, for ski soles made of ETFE, fitted with rough linear structures parallel to the gliding direction, the increase in time of descent is modest, but when a structure of the same roughness but with no preferred orientation is applied, the sliders simply do not descend.
The above findings suggest that capillary suction is the dominant friction mechanism for “flat” ski soles and plastic deformation of snow due to plowing by protrusions on the slider base dominates the friction behavior of “rough” ski soles. Importantly, here, the terms “flat” and “rough” relate to the roughness of the ski sole relative to the thickness of the thin water layer that is generated by frictional heat and that may assist in lubricating gliding of the ski. As the thickness of this water layer depends on the actual environmental conditions, it is expected that the optimal roughness of slider soles for skiing fast varies with the temperature of the snow, such that optimal skiing at elevated temperatures requires soles of a high surface roughness, while under colder conditions smoother bottoms should be more beneficial. Gratifyingly, the above view is in excellent accord with general empirical experience in the skiing industry.39, 43
Finally, it should be noted that it was established that repeated use of a snow track causes wear of it. While a systematic study of that phenomenon permitted a simple correction of the times of descent of sliders, it has not escaped the attention of the authors that the observations regarding track wear and concomitant increase in friction between skis and tracks puts into question the fairness of the line-up in skiing competitions.
The authors gratefully acknowledge M. Jufer, J. Glück, and U. Raunjak from TOKO, Mammut Sports Group AG for experimental and financial support. The authors also thank T. Giesbrecht and M. Simonet for their help during field experiments, and the mechanical workshop of the Department of Materials at ETH Zurich for their assistance with manufacturing essential equipment.