## 1 Introduction
For example, they cannot efficiently pack and so exhibit intrinsically low viscosities, while also possessing the high molecular weights common to viscosity modifiers [24; 33; 34]. Recently, there have been a number of examples of branched polymers being used to modify viscosities in lubricant and heavy oil applications, including transportation [35; 36; 37; 38; 39]. However, polymer addition has been reported to lead to a reduction in oil/lubricant quality [40; 41]. In this respect, the importance of the oil solubility of these polymers has been shown to be important in minimising these issues and also improving traditional additives [42]. Thus, characterisation of the oils/formulations components has aided the development of polymeric viscosity modifiers by allowing the understanding of what intermolecular interactions may be promoted upon the introduction of an additive [43; 44; 45]. Furthermore, in specific relation to this study, as difunctional monomers are utilised to synthesise HB polymers, care must be taken to avoid cross-linking. This leads to the formation of insoluble gels, rather than the desired solvent-soluble HB materials [26]. This coupling can occur in conventional free radical polymerisation at very low difunctional monomer concentrations (\(<\)\(<\)10%) and at low monomer conversions (\(<\)20%) [46]. Catalytic chain transfer polymerisation (CCTP) is an industrially viable polymerisation control strategy which typically uses a cobalt organometallic complex as a chain transfer agent (CTA). The CTA is catalytic in nature, thus, very low-molecular-weight polymers (e.g., dimer/trimers) can be produced using only parts per million quantities of CTA with some monomers [47]. Recently, researchers including [PERSON], [PERSON], [PERSON] and [PERSON] all reported the use of catalytic chain transfer polymerisation (CCTP) to synthesise HB polymers [46; 48; 49; 50]. With difunctional monomers, CCTP prolongs HB gelation within a range of monomer types, provided that either (a) the branching monomer concentration is kept low (\(<\)5 wt%) [46] or (b) the level of overall conversion is limited to specific levels [48; 49; 50; 51; 52]. Additionally, CCTP-terminated methacrylate chains are terminated by a vinyl group, resulting in increased concentrations of vinyl chain ends when compared to other CTAs. This presents the possibility of further post-modification/polymerisation [53]. Furthermore, the low molecular weight CCTP oligomers of monofunctional methylacrylates have also been demonstrated to act as free radical CTAs via a beta-scission mechanism. This leads to a fragment of the CTA becoming an intrinsic part of the product HB structure [53]. Importantly, for lubricant applications, this will allow for the introduction of functionality that will promote base-oil solubility. This study investigated the synthesis and use of HB materials as viscosity/friction modifiers to evaluate if they deliver high levels of wear protection and improved low-temperature oil fluidity. Furthermore, the work reported in this manuscript conducts polymerisations mediated by species which are added to interact and influence the propagation of the chain end radical [54]. In fact, the HB polymers were synthesised via a previously unreported route, where lauryl methacrylate (LMA) dimers were produced using CCTP and then used as CTAs to control the polymerisation of DVB. This method generated high conversions of highly polydisperse, base oil soluble HB polymers via the inclusion of the lauryl moiety in the HB structure. Solvent/oil soluble hyperbranched divinylbenzene (hbDVB) was of interest as a friction modifier because it can (a) be delivered as part of a liquid formulation and (b) cross-link _in situ_ upon hot engine surfaces to produce protective films [55]. These HB polymers were then subjected to tribological testing and their performance was compared to industry-standard ethylene-propylene linear copolymers.