Compositionally Driven Viscometric Behaviors of Poly (Alkyl Methacrylates) in Lubricating Oils

Viscosity index (VI) and shear stability index (SSI) are standard methods used in the lubricant industry to determine temperature-viscosity dependen-cy and resistance to product degradation, respectively. A variety of oil-soluble polymers, including poly(alkyl methacrylates) (PAMAs) are routinely used to control these properties in fully-formulated liquid lubricants. In this report, we use reversible addition-fragmentation chain transfer (RAFT) polymerization to precisely target identical degrees of polymerization in a family of PAMAs with varying lauryl, hexyl, butyl, ethyl, and methyl groups. Then, ex-panding on previous methodology reported in the literature, we establish structure property relationships for these PAMAs, specifically looking at how intrinsic viscosity [η] and Martin interaction parameters K M relate to VI and SSI characteristics. While the intrinsic viscosity [η] is associated with the volume of macromolecules at infinite dilution, the parameter K M reflects the hydrodynamic interactions of polymer chains at actual polymer concentrations in lubricating oils. In this paper, we show that the dependence of VI on the non-dimensional concentration c/c* (or c[η]) can be presented in a form of master curve with shift factors proportional to K M that decreases with increasing size of alkyl groups. This finding implies that even in the dilute regime, the coil-expansion theory used to explain the effect of macromolecules on VI should be complemented with the idea of hydrodynamic interactions between polymer molecules that can be controlled by the choice of alkyl chains in the family of PAMAs.


Introduction
A powerful and widely-applied strategy for reducing frictional losses between two moving mating surfaces is to separate the surfaces by means of lubricating oil. In this way, frictional penalties due to solid-to-solid contacts are greatly reduced by hydrodynamic lubrication [1] [2]. To maintain the load-bearing characteristics which lead to this fluid film, the lubricant must maintain a minimum viscosity across its entire range of operating conditions, including those occurring at high temperatures and high shear environments. While these minimum viscosity requirements at higher operating temperatures are critical for maintaining durability and efficiency, at lower operating temperature regimes, the natural increase in the lubricant's viscosity is counterproductive to overall operational efficiency. One strategy then, to improve overall efficiency, has been to use lubricants that have been designed to have weak viscosity-temperature dependence and a reduced average viscosity across their entire operational temperature range. This reduced average viscosity has been shown to be an effective formulation strategy for the improvement of overall operational efficiencies in automotive and industrial lubricant applications [3] [4] [5] [6].
Within the automotive and industrial lubrication field, one of the most widely used metrics for assessing a lubricant's viscosity-temperature dependence is the viscosity index (VI), determined by comparing a fluid's kinematic viscosity at 40˚C to that of a known reference fluid with identical kinematic viscosity at 100˚C. Fluids with higher viscosity indices have reduced viscosity-temperature dependence. A highly impactful strategy to impart increased viscosity index to a lubricating oil is by the addition of soluble polymer components known as viscosity index improvers (VII). The general nature of these polymers is known to have a significant impact on the VI contribution, and previous research has used coil-expansion theory to provide mechanistic insights into how VII's impart increased VI [7] [8] [9] [10]. These previous investigations have largely sought to describe the VI impact of polymers treated at levels well below the theoretical overlap concentration. In higher treat applications, an interesting reversal of the VI impact is observed with certain types of VIIs wherein additional polymer content results in decreases to the viscosity index. Developing a more complete understanding of the mechanistic behaviors that lead to this reversal as well as what implications this behavior might have to other fundamental properties such as shear stability is the primary aim of this work.

Coil-Expansion Theory and Recent Work
Coil-expansion theory has a long history in polymer research with two encompassing methods accounting for polymer expansion: continuous variation and discrete step. The continuous variation theory describes a scenario where polymer size scales by some proportionality with temperature [11] [12]. Alternatively, a discrete step-change in polymer size has also been observed, where the polymer radius of gyration or other measure of its size remains constant until a R. A. Patterson et al. Advances in Chemical Engineering and Science significant step-change to the coil size is induced by a factor such as temperature, solubility, or concentration [13] [14].
For many decades, the Selby coil-expansion concept was the dominating theory on polymer behavior in lubricating oils, relating expansion to parameters such as VI [7]. Selby describes the complexity involved with comparing polymer dynamics and viscosity index as originally defined by Dean and Davis [8] as the practice for determining viscosity-temperature relationships [9]. Therein, the expansion of hydrodynamic volume with improved solvency and increasing temperature was described. This has since taken hold as the mechanism of choice used to describe polymer systems in commercial lubricating oils.
Although the coil-expansion model has typically been used to rationalize VI improvements, it must be noted that not all polymers follow this model. Specifically, significant differences are observed between the solution behavior of poly(alkyl methacrylates) (PAMAs) and polyisobutylene (PIB) [15] [16] [17] [18] [19]. In a recent contribution to the conversation, Covitch and Trickett verified that PAMAs follow the coil-expansion theory, but other polymers, such as ethylene-propylene copolymers (olefin copolymers, or OCPs), did not behave in the same manner [10]. In addition to determining intrinsic viscosity in mineral oil, they used small-angle neutron scattering (SANS) as a tool to measure the radius of gyration of macromolecules in various deuterated solvents at different temperatures and concentration regimes. Their work showed that the presence of slightly collapsing polymer chains can still lead to VI improvement. In contrast to the temperature-dependent increase in radius of gyration measured for PA-MAs, it was shown that OCPs decreased in size at elevated temperature, but nonetheless had a positive VI effect. Ultimately, their work concludes that the Selby coil-expansion mechanism does not explain behavior in mineral oils for all polymer types.
Ramasamy, et al. [20] used molecular modeling (Large Atomic/Molecular Massively Parallel Simulation, or LAMMPS) to further confirm Covitch's conclusions that PAMAs followed the coil-expansion theory and OCPs did not. Here, they suggested that the oxygen present in the PAMA was the contributing factor in the coil expansion. The OCP, which does not contain oxygen, does not expand with heating. In simulations of a PAMA and a theoretical "oxygen-free" PAMA, they observed a temperature-dependent increase in coil size in the PAMA, in contrast to coil contraction in the oxygen-free model.
Willett, et al. [21] examined the VI phenomenon in additional chemistries to those referenced above. This work included polybutene in addition to PAMA and OCP and identified four cases for VI trends with concentration: Case 1-VI increases with concentration, linear or non-linear; Case 2-VI increases then plateaus or decreases; Case 3-VI does not increase or decrease; Case 4-VI decreases. Their analysis attributed many of these behaviors to expansion, contraction, interaction, and combinations of such.
It is clear from recent works that a coil-expansion theory may not fully describe VI effects for all polymer systems. This theory is based on isolated coil solution behavior and does not consider the inter-chain hydrodynamic interactions occurring in engine oils. These interactions are quite significant at the concentration regimes encountered in liquid lubricants. Therefore, more detailed account for both, coil expansion and inter-molecular interactions is needed to aid in formulating engine oils with desirable viscosity index. In the work presented herein, we will discuss the contribution of coil volume and hydrodynamic interactions to viscosity of dilute and semi-dilute solutions of a well-defined family of PAMAs.

Polymer Synthesis
To study solely the effects of polymer solubility on VI and SSI, PAMAs with the same degree of polymerization (DP) and varying compositions were targeted.
Conventional free radical polymerization (or use of a thiol chain transfer agent) is typically used to synthesize PAMAs, which does not allow for precise targeting of a specific polymer chain length. The inclusion of a suitable chain transfer agent, such as a thiocarbonylthio compound typically used in reversible addition-fragmentation chain transfer (RAFT) polymerization, provides significant control over the polymer composition [22]. Therefore, butyl-2-methyl-2-[(do decylsulfanylthiocarbonyl)sulfanyl] propionate (CTA) was used to mediate the polymerizations. The five copolymers prepared were a poly(lauryl methacrylate) homopolymer (LMA), poly(hexyl methacrylate-co-lauryl methacrylate) copolymer (HMA:LMA), poly(butyl methacrylate-co-lauryl methacrylate) copolymer (BMA:LMA), poly(ethyl methacrylate-co-lauryl methacrylate copolymer and thoroughly mixed. The flask was heated to 100˚C and the monomer/initiator mixture was added dropwise over approximately 90 min. The temperature was allowed to drop to 82.5˚C and the reaction was held at temperature until the monomer conversion was complete. After polymerization, the polymers were diluted to constant oil (50 wt%) and the molecular weights were analyzed by gel permeation chromatography (GPC, Supplementary Figure S1). These results are shown in Table 1.
The values of statistical coil size were calculated using relation where n = 200 is the number of bonds per chain and 0.15 nm l = is the bond length [11]. Introducing comonomers with shorter alkyl chains (e.g., butyl, ethyl, or methyl) makes chains more flexible so that the chain volume of LMA is approximately 1.4 times greater than these copolymers. However, inclusion of hexyl methacrylate in the copolymer results in a characteristic ratio much closer to that of LMA homopolymer.

Blending
Polymer samples (at 50 wt% in Ultra-S3 base oil) were added to a jar and diluted with the appropriate amount of Ultra-S3 base oil (final sample wt% ranged from 1 wt% to 25 wt% polymer). The mixture was heated to 70˚C and stirred for 1 h until the sample was homogenous.

Gel Permeation Chromatography (GPC)
GPC was performed using a Waters 2695 instrument equipped with a refractive index detector.

Kinematic Viscosity
Kinematic viscosity was determined by glass capillary viscometer where the temperature was maintained within 0.02˚C [25] (Supplementary Figures S2-S4).
The method utilized has a precision of 1% of the tested value. The kinematic viscosities of polymer solutions and a solvent (ν and ν s , respectively) were used to calculate specific viscosities ( 1 sp s

Viscosity Index
The VI of a fluid is a calculation to measure the variation in viscosity with respect to temperature. Defined by Dean and Davis in the 1920s and utilized by the petroleum lubrication industry since, viscosity index is an important measure of thermal effects on the viscous properties of the lubricating fluid [26]. To improve the overall efficacy of these oils, viscous properties of a fluid must be maintained over a wide range of temperatures. At lower temperatures, oils must maintain a low viscosity to permit oil pumping and cranking. At higher temperatures, oils must maintain a higher viscosity at to provide a full-film lubrication regime to protect parts from wear and prevent overheating. Although the lubrication industry eventually developed methods to determine viscosities at elevated and reduced temperatures [27] [28], the viscosity index continues to be a standard measure for a fluid's ability to meet certain performance criteria.
The standard practice for calculating VI was published by ASTM [29], where a kinematic viscosity (KV) method by glass capillary viscometer was utilized for the determination. VI is calculated using Equation (1) where Covitch [30] identifies the variables in Equation (2) [34]. Also, c * has the basic relationship with intrinsic viscosity, Equation (3) [ ] with units of inverse concentration, dL/g or mL/g, the intrinsic viscosity of a fluid represents the random-coil volume per unit mass that the polymer takes.
This concept derives from the use of Einstein's relationship of non-interacting, non-deformable spheres to predict viscosity, Equation (4) where η R is the relative viscosity or ratio of solution viscosity to solvent viscosity and ϕ which is the volume fraction of spheres [35].
Converting Einstein's relationship for the use in polymer applications to a virial expansion expression, Equation (5) [ From that expansion we can derive the Huggins Equation, Equation (6), which relates the reduced viscosity (η sp /c) to the concentration (g/dL), intrinsic viscosity (dL/g), and the Huggins parameter, k H , which is related to the intramolecular interaction and solvent quality [36].
Determining the intrinsic viscosity requires dilution of a polymer system as limit of c goes to 0 g/dL, 0 lim sp c c η → , or the y-intercept η sp /c vs. c; and k H is determined by the slope of η sp /c vs. c.
The poly(alkyl methacrylate) systems prepared herein, with solutions from 1 wt% to 5% at 1% increments, then at increasing increments to 25 wt% were prepared with the concentration being converted to g/dL for analysis. The concentrations provided an adequate number of solutions below c * as well as samples above the expected overlap concentration.
Another method to calculated intrinsic viscosity with a corresponding interaction parameter is to use the Martin Equation, Equation (7) (8), where V 0 is the viscosity of the unsheared oil, V s is the viscosity of the sheared oil, and V b is the viscosity of the base oil.
Therefore, an SSI of 0 represents zero loss of oil viscosity due to polymer shear degradation, whereas an SSI of 100 indicates complete loss of the viscosity contribution from the viscosity modifier due to shearing.

Results and Discussion
The with the excluded volume exponent ν = 3/5 for good solvents [11].     with the same degree of polymerization, but with different alkyl side chains, help explain the differences in VI measured for solutions of these copolymers at various concentrations. Indeed, the dependence of VI on concentration ( Figure 4) demonstrates a significant variation in behavior based on the size of the alkyl group. The MMA:LMA and EMA:LMA copolymers exhibit a peak VI, after which increased polymer concentration results in a rapidly decreasing VI. Alternatively, the polymers containing monomers with longer side chains (i.e., HMA:LMA copolymer and LMA homopolymer) show a VI plateau at a similar concentration. The BMA:LMA copolymer exhibits only a slight decrease in VI is observed past the maximum. These trends have also been identified and discussed by Willett, et al. [21]. A closer look at the concentration-dependence of the kinematic viscosities clearly shows the origin of this behavior. As shown in  . This effect is also seen to a lesser extent in the BMA:LMA copolymer. For these polymers, the viscosity index is controlled not only by the chain volume but also by the hydrodynamic and attractive interactions between the chains. This effect is more pronounced when the value of c * is estimated from [η] measured at T = 40˚C, which is related to the strong temperature dependence of Martin constants for these polymers (Table 2).
To further understand the correlation between VI, the chemical structure of copolymers, and their associated rheological characteristics, we can shift the dependences of VI on c c * using horizontal shift factor a k and relate the values of a k and K M for each polymer at a given temperature. Advances in Chemical Engineering and Science    (Table 3), the former polymer is expected to more resistant to shear degradation than the latter. The shear stability of LMA and MMA:LMA polymers was examined using the KRL shear stability test at various solution viscosities (i.e., multiple polymer concentrations). At similar KV100, the MMA:LMA copolymers are noticeably more shear stable than the LMA homopolymer ( Figure 7). As mentioned previously, this difference is explained by the different sizes of macromolecules in solution; at 60˚C (the measurement temperature), the LMA homopolymer is significantly more soluble than the MMA:LMA copolymer, and therefore is more extended and more susceptible to cleavage events. The relationship between the solubility and the polymer coil size has been outlined by Rubenstein (11); briefly, the excluded volume , where b is the size of monomer unit, χ is the Flory-Huggins parameter characterizing the polymer-solvent interactions. The value χ is related to the solubility parameters of polymer segment and solvent molecules. For θ-solvents, 1 2 χ = , 0 v = , and the coil size is . Therefore, the side chain length (and more specifically the polymer solubility) directly impacts the shear stability of a macromolecule.