# Transient Nanoscale Tribofilm Growth: Analytical Prediction and Measurement
## 4 Discussion  
The stress-dependent film growth rate is predicted by the [PERSON] shear-promoted thermal activation model as:  
\[\Gamma_{growth}=\Gamma_{0}e^{-\frac{\Delta E_{act}-\Delta E_{unch}}{k_{B}T}} \tag{1}\]  
where \(\Gamma_{growth}\) is the growth rate with the unit: m\({}^{3}\)s\({}^{-1}\), \(\Gamma_{0}\) is referred to as the pre-exponential term, \(\Delta E_{act}\) is the internal activation energy, given as the energy barrier in the absence of any applied stress, \(k_{B}\) is the Boltzmann's constant (\(1.38064852\times 10^{-23}\) JK\({}^{-1}\)), \(T\) is the absolute temperature and \(\Delta E_{mech}\) is the expended mechanical energy as:  
\[\Delta E_{mech}=\tau\Delta V \tag{2}\]  
where \(\tau\) is the mean value of stress component affecting the activation barrier, in this case the contact shear stress and \(\Delta V\) is the activation volume. \[\Gamma_{0}=vV_{m} \tag{3}\]  
where \(v\) and \(V_{m}\) are described by [PERSON] et al. [12] as the attempt frequency (i.e., the rate of interactions between an additive molecule and the target surface) and the molar volume (the volume occupied by one mole of the growth species) respectively. In the current application, \(v\) and \(V_{m}\) should be defined rather differently as the mechanical input energy is produced through shear rather than application of pressure. Therefore, the attempt frequency, \(v\), becomes the number of molecules sheared by the AFM tip per second. This can be estimated by the area occupied by a single molecule on the surface, the footprint tip contact area and its sliding speed. The volume, \(V_{m}\), is therefore no longer the molar volume, but the molecular volume (i.e., the volume of the product produced by the reaction of one molecule). \(V_{m}\) is, therefore, regarded as the product of the area occupied by a molecule and the molecular-chain length. Applying these definitions and the conditions of the experiment to the Arrhenius shear-promoted thermal activation model and adjusting for the best fit to the experimental data, yields the results shown in Figure 5. The best fit was found by applying the following conditions: \(\Delta E_{act}=0.695\) eV, \(\Delta V=160\times 10^{-30}\) m\({}^{3}\), \(v=1\times 10^{15}\)\(\frac{1}{s}\) and \(V_{m}=1.40\times 10^{-27}\) m\({}^{3}\). However, as Figure 5 shows, the Arrhenius shear-promoted thermal activation model clearly does not predict a limiting value or any initial state of transience. Therefore, it is rather inadequate for the evaluation of full tribofilm forming characteristics. MoDTC and ZDDP are polar molecules and can adsorb to metallic surfaces through van der Waals interactions [3]. This provides for a high viscosity weakly adsorbed near surface layer [34], which can be considered as the mechanism for the initial stages of tribofilm formation, whereas there would not yet be a significant bond or performance benefit to the additive film formation in the case of MoDTC and ZDDP. This is thought to be the form an additive film takes through application of thermal energy alone. [PERSON] et al. [12] showed that for ZDDP such a layer is formed to a depth of approximately 10 nm but was easily removed using the scanning conditions that were later shown to induce film growth. From the state of an adsorbed layer, sufficient energy input is required to form a more permanent structure and a stronger bond to form with the target surface (a process commonly referred to as 'activation'). [PERSON] et al. showed that, in UHV tribometry with a SFA, molecules bonded in a similar structure which can be sheared into activation [15]. This proposition is supported by the work of [PERSON] and [PERSON] [13], among others, at the microscale of tribometry. It is hypothesised that with adequate shear the polar attachment of the additives forms an anchor-point while the free ends are forced over and laid in the direction of shear [46].