# Transient Nanoscale Tribofilm Growth: Analytical Prediction and Measurement
## 1 Introduction
[PERSON] and [PERSON] [13] investigated the formation of ZDDP tribofilm using a Mini Traction Machine (MTM) to provide the necessary input mechanical energy and interferometry to measure the tribofilm growth. They adjusted the slide-roll ratio and applied contact load, thus the resulting shear and contact pressure, respectively. They concluded that shear, not normal pressure should be used as the main source of supplied mechanical energy, also when using the Arrhenius equation as a predictive tool. [PERSON] and [PERSON] [13] used the [PERSON]'s reaction path concept [14] to demonstrate that induced shear not only lowers the activation energy barrier for the chemical reaction of boundary active additives, but also increases the potential barrier for the reversal of the reaction. [PERSON] et al. [15] showed that for the case of dimethyl disulfide on a copper substrate in Ultra-High Vacuum (UHV), a Surface Force Apparatus (SFA) can be used to activate the molecules through shear and bond them to the surface. They made full use of the UHV environment by employing angle-resolved X-ray Photoelectron Spectroscopy (XPS) to track the chemical composition of the surface following the SFA activated chemical reaction. The [PERSON] [8] and [PERSON] [9; 14] models apply a force-distance term (work done: \(F\Delta x\)) as the mechanical input energy, whilst [PERSON] et al. [12] used the pressure volume term (\(p\Delta V\)) and [PERSON] and [PERSON] [13] used a shear stress-volume term (\(\tau\Delta V\)). The pressure and shear stress terms effectively represent the characteristic forces applied in the contact, whilst the volume terms (referred to as the activation volume) are necessary to provide a dimensionless exponential term in the Arrhenius equation. [PERSON] and [PERSON] [13] and [PERSON] [10] provided experimental evidence that shear stress is the most pertinent characteristic force expression for the mechanical contribution to activation. The activation volume can be considered for vertically orientated molecules to be the area of the surface occupied by the molecule multiplied by the activation length [11]. Previous studies have suggested that a molecular area between 20 and 400 A\({}^{2}\) and an activation volume between 7.8 and 180 A\({}^{3}\) are applicable for various molecules, including ZDDP, fatty acids and base polymers [13; 16; 17; 18; 19; 20]. A number of studies have measured the rate of growth of tribofilms with both fully formulated and partially formulated lubricants [21; 22; 23; 24; 25; 26; 27]. These studies have shown that there is a slow initial growth period, followed by an increased growth rate and finally a steady state region with a minimal increase or decrease in tribofilm thickness with time. This is particularly noteworthy in the study of [PERSON] et al. [12] who fitted a linear relationship to the initial growth stage and modelled the subsequent faster growth rate with the Arrhenius model. They noted a \"limiting value\", beyond which the growth appears to cease and some small patches of the tribofilm are removed from the edges of the scan area under extreme AFM contact loads (600-1700 nN). An important point to note is that neither the slow initial growth rate nor the limiting value can be explained by the Arrhenius model alone. [PERSON] and [PERSON] [28] used the data from the experimental work of [PERSON] et al. [12] and applied a simple wear term to explain the observed plateauing of the growth rate. This wear rate term was included with the Arrhenius growth model and showed good agreement with the experimental results, although the underlying physical nature of the wear mechanism was not discussed in any detail. Many papers [29; 30; 31] have shown that wear at the nanoscale does not follow the behaviour demonstrated at larger scales described by classical wear models [32].