# Transient Nanoscale Tribofilm Growth: Analytical Prediction and Measurement
## 4 Discussion
Essentially, it is thought that the shearing action in combination with the available thermal energy would force the free ends of molecules into proximity with each other or with the solid surface, forming covalent bonds and thus promote the formation of a tribofilm. The requirement for a molecule to be adsorbed to the surface prior to activation by the sliding tip provides a physical restriction to the chemical reaction rate predicted by the Arrhenius model. The finite coverage of molecules adsorbed to the surface should also be considered unless a large period of time is elapsed between successive scanning passes for a film covering the surface to stabilise. Therefore, the rate of adsorption must be taken into account in the modelling of the activation process. [PERSON] is credited with the first mathematical model for surface adsorption [47; 48; 49]. The Langmuir model considers the formation of an adsorbed monolayer onto a surface from a gas as a function of the adsorption, desorption rates and surface coverage through a simple expression as:  
\[\frac{d\theta}{dt}=K_{A}(1-\theta)-K_{D}\theta \tag{4}\]  
where \(K_{A}\) is the rate of adsorption, \(K_{D}\) is the rate of desorption, \(\theta\) is the fraction of the area covered by a deposited layer and \(t\) is time. Figure 5: Comparison of the predictions of the Arrhenius shear promoted thermal activation model with the experimental measurements. A Langmuir adsorption model has been applied previously to the case of lubricant additive film formation by [PERSON] and [PERSON] [50; 51]. They investigated the effect of concentration, temperature and stirring on the adsorption of ZDDP on steel and determined values of \(K_{A}\) and \(K_{D}\) for a range of cases. Although the Langmuir model allows for a limiting growth rate at the onset of the scanning process, it does not provide an explanation for the limiting growth observed at the culmination of the experiments. A possible reason for the reduction in the growth rate and for the initial transience is provided through inclusion of a multi-molecular layer adsorption model. The first multi-molecular layer adsorption model was provided by [PERSON], [PERSON] and [PERSON] [52] and have since been named after them as the so-called BET model. Under the proposed conditions, additive molecules would have the ability to adsorb in layers upon each other in the form of a stack. Desorption can also occur at all points within the stack such that a steady state condition is always reached with a finite number of layers. Under these conditions the shear experienced by each molecule at each adsorption site may be considered as inversely proportional to the number of molecular layers resident there [53; 54]. A BET multi-molecular layer adsorption model can be applied and solved analytically for a finite maximum number of layers. For the current case, an initial condition at which there would be no adsorbent coverage on the surface was applied. In the time between successive passes of the AFM tip, the growth on any layer \(x\) can be described as:  
\[\theta_{n,x}=\theta_{n-1,x}+\int_{n-1}^{n}\theta_{x-1}K_{Ax}(1-\theta_{x})- \theta_{x}\sum_{i=x}^{1}K_{Di}dt \tag{5}\]  
where \(n\) denotes the descrete time step, \(\sum_{i=x}^{1}K_{Di}\) accounts for the probability that any molecule in the stack can desorb to reduce the coverage at layer \(x\) and the growth rate would be limited by the fraction coverage of the layer \(x-1\), multiplying through by \(\theta_{x-1}\)[55]. Applying the multi-molecular layer adsorption model to the Arrhenius equation yields:  
\[\Gamma_{growth}=\sum_{x=1}^{x_{max}}(\theta_{t,x}-\theta_{t,x+1})v\
u_{m}e^{- \frac{\Delta x_{max}-\frac{\gamma}{\gamma}AV}{k_{B}T}} \tag{6}\]  
where \(x_{max}\) is the maximum number of molecules in a stack accounted for in the simulation study. Equation (6) shows that the effective shear stress term in the mechanical energy is reduced as the number of adsorbed layers increases.