6 Lubrication Transformation and Nanoscale Thin Film Lubrication
6.3 Analysis of Thin Film Lubrication
6.3.1 Difficulties in Numerical Analysis of Thin Film Lubrication
It should be noted that the numerical calculation of sub-micron and nano-thin film lubrication will face many complex phenomena. These include the molecular structure of lubricant film, non-Newtonian properties, thin-film shearing, phase change, limit shear stress, yield failure,
k k surface roughness and solid particles contained in a lubricant film. Some have not been studied
yet and are not well understood.
As mentioned above, when a lubricant film is thin to the molecular scale, traditional assump- tions no longer apply. It is now widely recognized that the lubricant film thickness should be at least larger than the molecular length of the liquid so that continuum mechanics can effectively be applied to analyze the lubrication behaviors.
For a molecular smooth surface, the Reynolds equation based on continuum mechanics can be applied to films thicker than 30 nm. Some people believe that for the thin lubricant film, only a few correction factors simply need to be added and thus the Reynolds equation is still able to be used for analysis.
At high pressure, phase transition of a liquid will take place, that is, solidification. The solid- ified pressure of a lubricant is related to the lubrication film thickness and the shear rate. Hu Yuanzhong calculated using molecular dynamics simulation that the equivalent viscosity will increase with decrease of a nano liquid film thickness [13]. As the film thickness drops to a cer- tain extent, the liquid will lose the mobility, that is, the liquid-solid phase transition appears, and the phase transition pressure drops with a decrease in film thickness. Granick (1991) pointed out that the continuous shearing movement prevents the liquid from solidifying. The higher the shear rate, the higher the solidified pressure. This shows that the lubricant film solidification brings great difficulty for numerical analysis on thin film lubrication.
In addition to solidification, the rheology of a lubricant is another important feature. The existence of the limit of shear stress is a kind of rheological property. As shown in Figure 6.15, various rheological models proposed for the lubricant film are of the shear stress limit. The visco-plastic limit and the exponential limit are the common two. In the exponential limit mod- els, different values of indexnare also proposed, usuallyn=1.8 or 2.
As a rheological property, the limit shear stress should be related to pressure, temperature and so on. Hu Yuanzhong et al. showed that the limit shear stress will increase with decrease in film thickness [13]. The flow of thin liquid lubrication film is divided into two regions. In the region near the surface, liquid withstands much high shear stress. As the stress reaches the limit, a slip occurs in the liquid–solid interface.
Huang and Wen studied lubrication failure caused by the limit shear stress [14]. They pointed out that as the shear stress in the lubricant film reaches its limit, the fluid begins to yield and a slip exists inside or at the liquid–solid interface. With increase of the shear rate, the slip zone is gradually expanded, which leads to the loss of the load-carrying capacity of lubricant film that causes lubrication failure.
In addition, with decrease of the film thickness, the surface topography becomes very important. For thin film lubrication, whether to use the Reynolds equation or the simplified Navier–Stokes equations to analyze rough surface lubrication is still unclear. Elrod (1973) distinguished two kinds of roughness: Reynolds roughness and Stokes roughness. Ifl is the Figure 6.15 Several rheological models.
k k roughness peak wavelength andhis the film thickness, whenh/l≪1 it is Reynolds roughness,
suitable for thin film lubrication conditions. At this time, the Reynolds equation can be applied to calculate and analyze the lubrication problem. Whenh/l≫l Stokes roughness applies. If so, Navier–Stokes equations can be used. However, some people suggest using Navier–Stokes equations to analyze thin film lubrication of rough surfaces.
A lubricant film contains more or fewer solid particles, which may also cause difficulties in analysis of thin film lubrication. Experimental results show that even after a fine filtering, the solid particles contained in a lubricant are usually several times larger than the thickness. These solid particles affect the thin film lubrication performances and can even cause surface damage.
A lubrication model with solid particles has not yet been established.
In short, although the physical model and numerical analysis of thin film lubrication have gained wide attention, because of its complexity there is limited understanding of this phe- nomenon and its laws so far.
6.3.2 Tichy’s Thin Film Lubrication Models
In the 1990s, Dr Tichy of Rensselaer Polytechnic Institute studied thin film lubrication and proposed three physical mathematical models and gave some numerical results for thin film lubrication.
6.3.2.1 Direction Factor Model
In 1993, in a symposium on tribology at Tsinghua University, Tichy presented the direction factor model and numerical solutions [15]. He believed that thin film lubrication is composed of micro-contact areas, as shown in Figure 6.16. The film thickness of the micro-contact areas is about 1–10 nm, where the lubricant molecular size is about 1 nm. The micro-contact struc- ture is made of the polar molecules of the lubricant adsorbed on the surface. Near the solid surface the lubricant layer is of quasi-solid characteristics. Far away from the surface, the lubri- cant still has liquid characteristics. Tichy proposed that the direction factor is a unit vector whose direction follows the direction of molecular orientation. It can represent the structure of the lubricant film. He then proposed three parameters of material to describe the rheological performance:
1. Conventional viscosity:this characterizes the viscous resistance of the liquid part of the lubricant film. Its role in lubrication is the same as common viscosity in conventional lubrication.
2. Direction viscosity:this causes a flow resistance due to the direction variation of a direction factor.
3. Elastic modulus:in this case, a lubricant layer near the surface appears to have the charac- teristics of a quasi-solid due to a direction factor attaching to the surface.
With the rheological model of Figure 6.16, Tichy calculated the lubrication problems of a wedge-shape slider. The calculation results show that when the lubricant film was perpendicular
Figure 6.16 Direction factor model.
k k to the moving direction of the direction factors, viscous resistance is quite large. When the
lubricant film is fixed along the moving direction of the direction factors, there is a significant drop in the viscous resistance. Then, the resistance can be expressed by the common viscosity.
6.3.2.2 Surface Layer Model
Tichy (1995) proposes a surface layer model of thin film lubrication according to the fact that the viscosity near the surface is several orders larger than that of the bulk lubricant. The model consists of conventional viscosity, the surface layer thickness and the viscosity of the surface layer of the three material parameters. In Tichy’s view, the micro-structure of the lubricant molecules near the surface can be regarded as a highly rigid viscosity surface layer. There are two kinds of viscosity along the direction of the lubricant film thickness.
Ifzis the coordinate across the film thickness,his the film thickness,𝛿is the surface layer thickness,𝜂0is the conventional viscosity and𝜂sis the surface layer of viscosity, then
𝜂=𝜂s z≤𝛿 or z≥h−𝛿
𝜂=𝜂s 𝛿 <z<h−𝛿, (6.4)
where𝜂0and𝜂scan be measured with a viscometer, while the𝛿value can be calculated based on the lubricant’s molecular structure and the surface energy.
According to the surface layer model, Tichy derived a modified Reynolds equation and applied it to solve the wedge-shaped slider problem of thin film lubrication. His calculations showed that the surface layer thickness and the surface layer viscosity are the main parameters to affect the performances of lubrication. Their rise will increase the load-carrying capacity, but decrease the friction coefficient.
6.3.2.3 Porous Surface Layer Model
Tichy (1995) treated the molecular structure of the lubricant film close to the surface as a layer of porous coating, which is attached to the solid surface. Three material parameters used in the model are the conventional viscosity, the porous layer thickness and the pore-like parameter.
Among them, the porous layer thickness can be calculated by the lubricant molecular structure and the surface energy, while the hole-like parameter is measured by the experiments.
According to Darcy’s law, Tichy determined the properties of the porous layer. He then derived a modified Reynolds equation and applied it to obtain the load-carrying capacity, the friction coefficient and the relationship between the porous layer thickness and the pore-like parameter of a wedge-shaped slider lubrication.
It should be noted that the above several models are not perfected due to a lack of extensive experimental data.
Zhang Chaohui et al. proposed a corrected formula of the viscosity for the nano-thin film based on the experiments [16]. Furthermore, their computational results are generally consis- tent with their experimental results.