Ice Adhesion. Группа авторов
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where τc is the shear strength necessary to shear the asperity contact and Ac is the actual area of contact between the asperties of mating surfaces.
Bowden proposed that the actual area of contact, Ac, between two mating surfaces is a function of the applied normal load, FN, and the hardness, H, of the softer of the two materials [134].
Figure 1.13 (a) Surfaces are never completely flat, but rather possess asperities. In the dry friction regime, in the absence of a lubricating liquid, two mating surfaces are in contact at their asperities, (b) Ice possesses a liquid-like layer at its surface which acts as a lubricating liquid during friction. The thickness of the liquid-like layer, h, affects the coefficient of friction, µ, and leads to three friction regimes. Adapted from [133].
Therefore, the coefficient of friction can be expressed as
This regime, in the absence of a lubricating liquid, is known as dry friction. This regime is characterized as the work necessary to break solid-solid bonds, which is dependant on the applied normal force and the hardness of the two surfaces. The frictional force in this regime is independant of the relative sliding speed of the two mated surfaces [128, 135, 136].
Next consider the case where a lubricating layer is present at the solid-solid interface with the thickness of only a few molecular layers. This regime is known as boundary lubrication [135]. Boundary lubrication will occur on ice when the temperature, T, is below the melting temperature, Tm, within the ice-solid contact zone, and the thickness of the lubricating layer, h, is much smaller than the roughness of the surface, R [137].
Everywhere in the contact zone:
Within this regime, ice’s liquid-like layer reduces the solid-solid contact between the surfaces. This lessened contact results in the coefficient of friction typically being lower in the boundary friction regime than within the dry friction regime [135]. As shown in Figure 1.13(b), within the boundary friction regime the coefficient of friction will decrease with increasing lubricating layer thickness, as the solid-solid contact is further decreased.
The boundary friction regime on ice gives way to mixed friction as the temperature rises to the melting temperature at some points within the contact zone. The thickness of the liquid-like layer is increased by the amount of melt water generated by the frictional contact. However, within the mixed friction regime, the overall thickness of the liquid-like layer will remain thinner than the characteristic roughness of the surfaces in contact.
At some points in the contact zone:
As shown in Figure 1.13(b), the trend of decreased coefficient of friction with increased thickness of the lubricating layer continues from the boundary lubrication regime. The load on the sliding solid surface will be partially supported by the surface asperities and partly by the lubricating layer. An increase in the thickness of the lubricating layer thus reduces the solid-solid adhesion. However, a wetting lubricant, such as the liquid-like layer of ice, will lead to the formation of capillary bridges between asperities. This phenomenon is shown in the magnified section of Figure 1.13(b). The capillary bridges that form between asperities exert a drag force on the slider, resulting in additional frictional resistance [138]. Hence, a global minimum is manifested in the coefficient of friction versus film thickness graph of Figure 1.13(b). As film thickness increases within the mixed friction regime, more and more capillary bridges are introduced between the ice and the slider.
As the temperature everywhere within the contact zone increases to the melting temperature, the mixed friction regime gives way to the hydrodynamic regime. In this mode of sliding, the thickness of the lubricating layer is greater than the characteristic roughness of the asperities.
Everywhere in the contact zone:
As the lubricating layer is now supporting the entire applied load and solid-solid contact is no longer present, the shearing of solid-solid bonds no longer contributes to the friction force of the system. Further, the actual area of contact is now equal to the surface area of the slider [135]. The force of friction in this regime arises from shearing of the liquid lubricating layer. Thus, the frictional force can be described as
where τl is the shear strength of the lubricating layer (or an “effective” shear stress developed from shearing of the lubricating layer). The shear strength of the lubricating layer can be expressed as
where η is the viscosity of the lubricating layer and v is the velocity of the slider relative to the ice. Note that when the lubricating liquid is wetting and the slider surface is porous or rough, as in the case of water on ice, capillary drag forces should also be considered as in the mixed friction regime.
As shown in Figure 1.13(b), as the film thickness increases and becomes fully hydrodynamic, the coefficient of friction also increases. An inspection of Figure 1.13(b) as a whole shows that there exists an optimal lubricating film thickness for each slider system in order to minimize frictional forces. Further, the transitions between the regimes are smooth, as indicated by the dashed lines. Finally, the lubricating film between a slider and ice has been shown to become thicker toward the trailing end. Thus, a slider could simultaneously exhibit all the types of friction discussed in this section (save for dry friction) [139].
1.4.2 The Origin of Ice’s Liquid-Like Layer
Since Faraday’s 1859 suggestion of a liquid-like layer existing on ice, there has been much speculation whence this film arrives [129]. This subsection will present the major theories for the presence of the liquid-like layer, which is still under debate. For a more-thorough review of the subject, the interested reader is referred to the work of Petrenko and Whitworth [140]. The formation of an inherent liquid-like layer on ice has been suggested over the years to be derived from: electrostatic interactions [141, 142], surface free energy minimization [143, 144], subsurface pressure melting [145, 146], and the vibration and rotation of surface molecules [147–150].
Pressure melting has been used to explain the low friction coefficient of ice since it was first suggested by Reynolds in 1900 [131]. The theory being that external pressure applied to the surface of ice will lead to a depression of the melting point temperature, and thus melting of the ice and an increase in the lubricating liquid-like layer thickness. However, while pressure melting may contribute to the formation of a lubricating layer between ice and a solid slider at temperatures close to the melting point, this phenomenon cannot explain ice’s low friction at lower temperatures. Calculations show that the