# Transient Nanoscale Tribofilm Growth: Analytical Prediction and Measurement
## 1 Introduction  
Lubrication is crucial for the functional performance of a wide range of mechanical and biological interfaces. Most interfaces operate under a mixed regime of lubrication, meaning that a certain degree of direct interaction of stochastic solid contacting surfaces occurs. Therefore, the conditions endured by many interfaces require the formation of an intervening fluid film to extend beyond its mere bulk microscale rheology. Protective boundary-active additives are used to form surface-adhered/bonded tribofilms. These can contribute to a reduction in contact friction and prevent wear in the nanoscale interactions of surface asperities in mechanical as well as biological interfaces. Hitherto, despite significant improvements in lubricant technology, it is estimated that 23% of the world's total energy consumption is dissipated in tribological contacts [1]. Typically, additives form around 5-24 wt% of fully formulated lubricants for engineering applications [2, 3]. Depending on application, lubricants comprise a blend of hydrocarbon base stock, as well as detergents, dispersants, antioxidants, friction and viscosity modifiers, anti-foaming agents, pour point depressants, corrosion inhibitors and anti-wear additives [3]. In the formulation, the constituent physical chemistry enhances certain aspects of the lubricant system performance and can be adjusted to suit specific applications. Despite their crucial intended functions, the mechanisms through which most additives adsorb or bond to surfaces remain somewhat empirical, particularly due to their potential synergistic as well as antagonistic competition in occupying attachment sites onboundary surfaces [4]. The lack of a fundamental understanding is exacerbated by the significant difficulty associated with taking in situ measurements of additives' activation and tribofilm characterisation. Molybdenum-based additives were first patented in 1939 in the form of colloidal molybdenum disulphide (MoS\({}_{2}\)) as an anti-wear and friction modifying additive [5]. The use of oil soluble molybdenum dialkyldithiophosphate (MoDDP) was reported in 1955 as an alternative to zinc dialkyldithiophosphate (ZDDP) which was already in use [6]. Other forms of oil soluble molybdenum-based friction modifiers have also been used commercially, including molybdenum dithiocarbamate (MoDTC). The reaction process that transforms an oil soluble molybdenum-based molecule into a functional friction-reducing layer of MoS\({}_{2}\) is chemical and requires energy and suitable reactants for initiation. The Arrhenius equation has long been applied in chemistry to model the rate-limiting step in a reaction process [7]. Through modifying the Arrhenius equation, found in the seminal works of [PERSON] [8] and [PERSON] [9], the mechanical contribution to reaction rates can also be considered as an energy input. [PERSON] [10] and [PERSON] [11] have recently provided detailed summaries of the evolution of these concepts and their application in tribology through tribochemistry or mechanochemistry. The shear-induced thermal activation model has recently gained prominence following the work of [PERSON] et al. [12] on in situ activation of ZDDP via scanning Atomic Force Microscopy (AFM). Using a range of contact pressures and temperatures, [PERSON] et al. [12] recorded the change in surface height at intervals after scanning with an AFM tip over a surface submerged in a lubricant containing ZDDP. The tip was scanned parallel to the cantilever support in order to promote additive activation, and then operated in tapping mode over the scanned area and its surroundings to assess the formed tribofilm height. By evaluating the increased film height after intervals of time, they were able to determine an exponential film growth rate which they linked to an Arrhenius model using the generated contact pressure as the source of supplied mechanical energy.