A. Efforts to bridge the gap between experiments and models

In , the most important results from NEMD simulations of PB tribology have been presented, along with accompanying scaling theories that seem to describe the friction behavior of the simulated bilayers quite well. However, it becomes crucial to ask, how well do these computational and theoretical approaches capture what is observed experimentally? Answering this question is rather complex for multiple reasons, mostly arising from computational resource limitations, which make it a challenge to simulate sheared bilayers that correspond exactly to those studied experimentally. In this section, we will discuss the sources of discrepancy as well as methods and efforts to bridge the gap between simulation results and experimental data.

1. Challenges arising from coarse-graining

As explained earlier, there is a need to use some form of the CG model for the chains as well as the solvents, if taken into account explicitly, in order to be able to simulate the dynamics of large enough bilayers at the required timescales using NEMD. Coarse-graining involves reducing the degrees of freedom of the polymer chain in order to increase its fundamental time scale,205205. G. S. Grest, K. M. Salerno, B. L. Peters, T. Ge, and D. Perahia, Handbook of Materials Modeling, 2nd ed., edited by W. Andreoni and S. Yip (Springer International Publishing, Cham, 2020), p. 1397). https://doi.org/10.1007/978-3-319-44677-6_34 accomplished by grouping segments of the polymer chains into beads that interact with one another through interactions such as those presented in . Regardless of the CG model employed, there is an associated loss of information as the dynamics of real polymers depend on a wide range of time and length scales. Thus, an efficient but still effective coarse-graining procedure should aim to retain the smallest length scales needed to correctly represent the dynamics of the system but no smaller.205205. G. S. Grest, K. M. Salerno, B. L. Peters, T. Ge, and D. Perahia, Handbook of Materials Modeling, 2nd ed., edited by W. Andreoni and S. Yip (Springer International Publishing, Cham, 2020), p. 1397). https://doi.org/10.1007/978-3-319-44677-6_34A consequence of using CG models is that the simulations are performed, and the results are presented, in dimensionless units, whether they be LJ or DPD units. While this provides scalability to the results, as the CG polymer chains lose their specific atomic structure, it adds a further barrier to comparison with experiments, namely, what is the best choice for the various parameters to map a DPD or LJ system to a specific polymer and its chemisty, such that a meaningful comparison to experimental data can be made? Choosing the optimal value for the interaction parameters can itself be a rather involved task depending on the approach adopted, ranging from simply trusting values found in the existing literature to, performing a separate set of simulations and tuning the parameters to match to experimentally available data or full-atom simulations. This can be accomplished, for example, by simulating a melt of CG chains and matching to experimentally available densities or by simulating a free CG chain in solution and matching to data for the radius of gyration of the chains for a known degree of polymerization.108108. M. K. Singh, P. Ilg, R. M. Espinosa-Marzal, M. Kröger, and N. D. Spencer, Langmuir 31, 4798 (2015). https://doi.org/10.1021/acs.langmuir.5b00641 Incorporating a bond angle potential into the CG model can allow for matching to the persistence length for a semiflexible or stiff polymer. The aformentioned structural matching is used to obtain the chain lengths and bead sizes for the CG model. The mass of the CG bead will depend on the particular polymer chain being modelled and the degree of coarse-graining. We notice in this regard that while the choice of mass is completely irrelevant when one is interested in equilibrium quantities (at least in situations where classical physics applies), this is not the case when intrinsically non-equilibrium (dynamic) properties like friction coefficients are involved. The energy scale, on the other hand, can be set, for instance, by matching the normal stress under compression with experimental data.108108. M. K. Singh, P. Ilg, R. M. Espinosa-Marzal, M. Kröger, and N. D. Spencer, Langmuir 31, 4798 (2015). https://doi.org/10.1021/acs.langmuir.5b00641 The recent reviews by Dhamankar et al.,206206. S. Dhamankar and M. A. Webb, J. Polym. Sci. 59, 2613 (2021). https://doi.org/10.1002/pol.20210555 Grest et al.,205205. G. S. Grest, K. M. Salerno, B. L. Peters, T. Ge, and D. Perahia, Handbook of Materials Modeling, 2nd ed., edited by W. Andreoni and S. Yip (Springer International Publishing, Cham, 2020), p. 1397). https://doi.org/10.1007/978-3-319-44677-6_34 and the earlier review by Merchant et al.207207. B. A. Merchant and J. D. Madura, Annu. Rep. Comput. Chem. 7, 67 (2011). https://doi.org/10.1016/b978-0-444-53835-2.00003-1 delve into the process of coarse-graining polymers and its limitations in far greater detail than we do here and are highly recommended.

2. Challenges comparing to experiments

Even after coarse-graining, the time and length scales accessible through simulations remain limited, which makes it difficult to compare to experimental conditions. To illustrate this, let us consider a Kremer–Grest bead-spring CG model for a polymer brush composed of hydrocarbon chains. Typically used LJ bead parameters, where a bead would represent about three chemical (CH2) monomers, are σ=0.5nm and m=38.6Da≈6.41×10−26kg,164164. K. Kremer and G. S. Grest, J. Chem. Phys. 92, 5057 (1990). https://doi.org/10.1063/1.458541 assuming room temperature T=300K allows one to estimate a rough energy scale through ε=kBT=4.14×10−21J, which then would together determine the time τ=σ2mε≈2ps and velocity σ/τ≈254m/s units for the simulation. A common MD timestep for such a model is Δt=0.005τ≈9.84fs and the bilayers are usually sheared for time periods of the order (107Δt≈100ns). It is already clear then that simulations could at most capture time scales in the range of microseconds or milliseconds at most with typical computing facilities. This fact severely limits the chain lengths accessible, since longer chains have larger relaxation times. Indeed, chain lengths in most PB experiments are about an order of magnitude larger than those in simulations. Furthermore, this makes simulating bilayers at experimentally observed shear rates incredibly computationally demanding. In simulations, the bilayers are typically sheared at relative wall velocities in the range v=[0.0002−1]σ/t≈0.05−254m/s, keeping in mind that lower velocities will need to be sheared for a longer period. This is because for statistical accuracy and to reduce finite size effects during the simulation time the system needs to sample a large number of conformations, which requires a total displacement in the shear directions by the length of the simulation box multiple times. Velocities in experiments vary depending on the technique employed. Traditional SFA/SFB experiments typically only manage velocities up to 0.450nm/s while AFM velocities can go up to about 5mm/s and a PoD tribometer a bit higher still at about 19mm/s, all of which are various orders of magnitude below the range accessible in simulations. On the other hand, MTM experiments cover velocities in the range of 0.025–2.5 m/s which is in fact comparable with the lower range velocities accessible only through long simulations. Note that the values presented above are only rough estimates based on the assumed parameters above as well as experimental data available in the literature. While the MTM is capable of performing mixed sliding/rolling at high speeds reaching into the hydrodynamic lubrication regime, the slower pure-sliding PoD offers more control and may be preferred for investigating the boundary and mixed lubrication regimes.3737. M. Müller, S. Lee, H. A. Spikes, and N. D. Spencer, Tribol. Lett. 15, 395 (2003). https://doi.org/10.1023/B:TRIL.0000003063.98583.bbIt is important to highlight that by removing some degrees of freedom, the effect on dynamic quantities due to coarse graining is difficult to gauge. For example, it has been shown by Zwanzig that the smoothening of the underlying energy scale might lead to faster dynamics than what would be expected in the full-atom systems.208,209208. R. Zwanzig, Proc. Natl. Acad. Sci. U.S.A. 85, 2029 (1988). https://doi.org/10.1073/pnas.85.7.2029209. T. H. Gray and E. H. Yong, Phys. Rev. E 102, 022138 (2020). https://doi.org/10.1103/PhysRevE.102.022138 In practice, this means that the correct timescale to simulate in a CG system might be shorter than the one calculated by the simple arguments presented above, at least partially closing the gap between experiments and simulations. However, it is difficult to estimate exactly how much faster the dynamics become due to coarse-graining. In other words, even if one had the necessary computational resources to simulate much slower rates, the idea to match experimental and simulation timescales might be flawed in the first place by lack of correspondence between real and simulated time in a CG system. In this regard, a somewhat better approach, and one more in line with the general spirit of coarse graining, would be to try to look at trends as a function of shear rate rather than comparing at an arbitrary value. However, most experimental studies of PBBs under shear are limited to a single or a few sliding velocities,194194. J. P. Ewen, D. M. Heyes, and D. Dini, Friction 6, 349 (2018). https://doi.org/10.1007/s40544-018-0207-9 whereas only a few look at the effect of shear velocity explicitly. This is even more important considering that the measured friction coefficient depends strongly on the shear rate. Spencer et al. have performed shear MTM experiments covering almost four orders of magnitude for poly(L-lysine)-graft-poly(ethylene glycol) (PLL-g-PEG) and PLL-g-dex bilayers in various HEPES/glycerol mixtures, the friction coefficient was found to decrease with increasing shear velocity up to a minimum at a velocity of the order v≈1m/s , after which μ begins to increase with increasing shear velocity.74,7674. C. Perrino, S. Lee, and N. D. Spencer, Tribol. Lett. 33, 83 (2009). https://doi.org/10.1007/s11249-008-9402-676. P. C. Nalam, J. N. Clasohm, A. Mashaghi, and N. D. Spencer, Tribol. Lett. 37, 541 (2010). https://doi.org/10.1007/s11249-009-9549-9 This finding is in good agreement with the simulation results, where an increase in μ is observed with increasing shear velocity [Fig. 9(a)], assuming the velocity ranges accessed in simulations would generally lie above v>1m/s, regardless of CG effects.A further complication arises from the fact that the various above-mentioned techniques exhibit different curved geometries117117. P. Mocny and H.-A. Klok, Mol. Syst. Des. Eng. 1, 141 (2016). https://doi.org/10.1039/C5ME00010F that may deviate quite strongly from the idealized flat substrates used in most PBB simulations. Although the SFA/SFB interacting surfaces that are in a crossed-cylinder configuration with radii of curvature on the order of centimeters may be somewhat comparable to flat substrates at the simulation scale,59,21059. J. Klein, D. Perahia, and S. Warburg, Nature 352, 143 (1991). https://doi.org/10.1038/352143a0210. J. Klein, Nature 288, 248 (1980). https://doi.org/10.1038/288248a0 AFM incorporates a spherical tip that is about 5 μm in diameter mounted on a cantilever, which scans the sample at a resolution of nanometers.73,7973. K. Kitano, Y. Inoue, R. Matsuno, M. Takai, and K. Ishihara, Colloids Surf. B 74, 350 (2009). https://doi.org/10.1016/j.colsurfb.2009.08.00479. M. Kobayashi, M. Terada, and A. Takahara, Faraday Discuss. 156, 403 (2012). https://doi.org/10.1039/c2fd00123c Both the PoD tribometer and MTM consist of a ball, that is, millimeters in diameter, which comes into contact with a rotating disk.74,7674. C. Perrino, S. Lee, and N. D. Spencer, Tribol. Lett. 33, 83 (2009). https://doi.org/10.1007/s11249-008-9402-676. P. C. Nalam, J. N. Clasohm, A. Mashaghi, and N. D. Spencer, Tribol. Lett. 37, 541 (2010). https://doi.org/10.1007/s11249-009-9549-9 Indeed, these experimental techniques are intended to probe fundamentally different length scales, and it may be difficult to mimic their geometries in simulations.Moreover, the longer chains explored in experiments could lead to increased entanglements in the bilayer within a single brush as well as between chains on opposing brushes, entanglements that are neglected completely for simple DPD models and otherwise not commonly observed for the relatively short grafted chains (N<200) used in simulations of sheared bilayers.4747. T. Kreer, Soft Matter 12, 3479 (2016). https://doi.org/10.1039/C5SM02919H These entanglements as well as possible microcontacts due to the roughness of physical substrates could lead to damage and chain detachment that is also not typically accounted for in simulations, where the chains are irreversibly bound to the substrate. In experiments, the synthesis technique will determine the chain grafting mechanism, while brushes synthesised through grafting-from approaches may indeed be covalently bound to the substrates, those synthesized through grafting-to approaches may be loosely adsorbed to the surface and more prone to detachment. For brushes where chain adsorption is electrostatic in origin, shearing leading to chain detachment can be followed by subsequent reattachment from a polymer-rich solution arising due to the detached chains, this is often referred to as the “self-healing” capacity of the polyelectrolyte-anchoring approach.118118. W. Lin and J. Klein, Adv. Mater. 33, 2005513 (2021). https://doi.org/10.1002/adma.202005513 DPD simulations of sheared bilayers in the μVT ensemble where chain detachments were introduced in a controlled manner show that the detached chains are expelled from the brush into the overlap zone under shear inhibiting mutual brush interpenetration and hence reducing the measured friction.104,152104. F. Goujon, P. Malfreyt, and D. J. Tildesley, Soft Matter 6, 3472 (2010). https://doi.org/10.1039/c002204g152. C. Pastorino, K. Binder, T. Kreer, and M. Müller, J. Chem. Phys. 124, 064902 (2006). https://doi.org/10.1063/1.2162883Another discrepancy arises because simulations are usually carried out at a constant number density and substrate separation which would correspond to the thermodynamic NVT ensemble at equilibrium, whereas in experiments, bilayers are sheared at a fixed normal load and the overall density may in fact change with increased load.9898. G. S. Grest, MRS Proc. 464, 71 (1996). https://doi.org/10.1557/PROC-464-71 Goujon et al. attempt to take this discrepancy into consideration by performing the bilayer shearing simulations at a constant chemical potential (μVT ensemble) as described in .149,211149. F. Goujon, A. Ghoufi, P. Malfreyt, and D. J. Tildesley, Soft Matter 9, 2966 (2013). https://doi.org/10.1039/c3sm27641d211. F. Goujon, P. Malfreyt, and D. J. Tildesley, ChemPhysChem 5, 457 (2004). https://doi.org/10.1002/cphc.200300901 Furthermore, as pointed in , in experiments, isothermal conditions are not necessarily ensured across the sheared bilayer due to viscous heating effects in the overlap region, but rather, expected only for the substrates, this may promote the use of thermostating schemes that apply a thermostat only on the explicit wall beads but not the brush or solvent beads for comparison to experimental data.172,180172. X. Yong and L. T. Zhang, J. Chem. Phys. 138, 084503 (2013). https://doi.org/10.1063/1.4792202180. R. Khare, J. D. Pablo, and A. Yethiraj, J. Chem. Phys. 107, 2589 (1997). https://doi.org/10.1063/1.474570

Due to all the aforementioned challenges, many computational studies aim for a qualitative rather than exact agreement with experimental results.

It is important to note, however, that even qualitatively, there are stark differences between some simulation predictions as presented in and experimental outcomes. For instance, it is well established from many experiments of sheared PBBs immersed in solvents of varying quality that the higher the solvent quality the lower the observed friction coefficient.8,55,72,73,76,77,828. Z. Zhang, A. J. Morse, S. P. Armes, A. L. Lewis, M. Geoghegan, and G. J. Leggett, Langmuir 27, 2514 (2011). https://doi.org/10.1021/la104384855. M. T. Müller, X. Yan, S. Lee, S. S. Perry, and N. D. Spencer, Macromolecules 38, 5706 (2005). https://doi.org/10.1021/ma050154572. M. A. Brady, F. T. Limpoco, and S. S. Perry, Langmuir 25, 7443 (2009). https://doi.org/10.1021/la900371k73. K. Kitano, Y. Inoue, R. Matsuno, M. Takai, and K. Ishihara, Colloids Surf. B 74, 350 (2009). https://doi.org/10.1016/j.colsurfb.2009.08.00476. P. C. Nalam, J. N. Clasohm, A. Mashaghi, and N. D. Spencer, Tribol. Lett. 37, 541 (2010). https://doi.org/10.1007/s11249-009-9549-977. A. Nomura, K. Okayasu, K. Ohno, T. Fukuda, and Y. Tsujii, Macromolecules 44, 5013 (2011). https://doi.org/10.1021/ma200340d82. P. C. Nalam, S. N. Ramakrishna, R. M. Espinosa-Marzal, and N. D. Spencer, Langmuir 29, 10149 (2013). https://doi.org/10.1021/la402148b For example, Müller et al. performed shear experiments of PLL-g-PEG bilayers immersed in solvents of decreasing quality (water, methanol, ethanol, and 2-propanol) where the exact degree of areal solvation was determined from the difference between the wet and dry adsorbed masses obtained from QCM-D and OWLS measurements, respectively. The obtained friction coefficient was found to decrease with increasing degree of solvation of the brushes.5555. M. T. Müller, X. Yan, S. Lee, S. S. Perry, and N. D. Spencer, Macromolecules 38, 5706 (2005). https://doi.org/10.1021/ma0501545 All these experiments are in contrast to simulation results (e.g., Fig. 4), which show the opposite trend, attributed to reduced interpenetration between the brushes as described in .Furthermore, experimentally, a much lower friction is usually observed for charged brushes when compared to neutral ones at similar grafting densities,118118. W. Lin and J. Klein, Adv. Mater. 33, 2005513 (2021). https://doi.org/10.1002/adma.202005513 in contrast to simulation results which suggest that the friction is of the same order of magnitude for both cases, with the charged brushes generally exhibiting a higher friction under most conditions. It is difficult to isolate the effects of electrostatic interactions experimentally and any comparison will likely involve a change of polymer species.

In practice, these clear qualitative discrepancies point to the fact that the current simulation approaches are missing some key ingredient controlling the physics of PB friction and finding the solution to this conundrum a particularly interesting problem.

B. Effect of brush thickness and chain architecture on friction

We now turn to another important area in which further developments are needed to enable simulations to be bridged with experiments, therefore providing a route to adopt molecular simulations as a design tool for the next generation of solutions for friction control based on polymer brushes. In simulations, we have precise, and individual, control of chain lengths N, grafting densities σg, chain architectures, charge fraction, or stiffness of the chains composing the brushes; however, experimentally, achieving such strict control is not as straightforward and may involve a change of synthesis technique, polymer species, or even characterization methods. For instance, “grafting-to” approaches allow significant control over the adsorbed chain architectures and thus allow easier characterization of the molecular parameters of the brush, but due to increased steric repulsion, it becomes difficult to prepare dense brushes composed of long chains when compared to “grafting-from” approaches, which themselves would then suffer from weaker control over the polydispersity and chain architecture. Furthermore, it is not possible to obtain the grafting density directly through measurements, but rather it is usually estimated through characterization of the dry brush thickness (via VASE) or adsorbed mass (via OWLS) of the chains along with prior knowledge of the grafted polymer species.73,78,9073. K. Kitano, Y. Inoue, R. Matsuno, M. Takai, and K. Ishihara, Colloids Surf. B 74, 350 (2009). https://doi.org/10.1016/j.colsurfb.2009.08.00478. L. J. T. Landherr, C. Cohen, P. Agarwal, and L. A. Archer, Langmuir 27, 9387 (2011). https://doi.org/10.1021/la201396m90. B. Pidhatika and P. C. Nalam, J. Appl. Polym. Sci. 136, 47659 (2019). https://doi.org/10.1002/app.47659Shear experiments of PBBs where the molecular parameters of the brushes are varied over a wide range for the same grafted polymer species are quite rare. That being said, significant efforts have been made to isolate the effect molecular parameters on friction using both “grafting-to”71,7471. S. S. Perry, X. Yan, F. T. Limpoco, S. Lee, M. Müller, and N. D. Spencer, ACS Appl. Mater. Interfaces 1, 1224 (2009). https://doi.org/10.1021/am900101m74. C. Perrino, S. Lee, and N. D. Spencer, Tribol. Lett. 33, 83 (2009). https://doi.org/10.1007/s11249-008-9402-6 and “grafting-from”8,788. Z. Zhang, A. J. Morse, S. P. Armes, A. L. Lewis, M. Geoghegan, and G. J. Leggett, Langmuir 27, 2514 (2011). https://doi.org/10.1021/la104384878. L. J. T. Landherr, C. Cohen, P. Agarwal, and L. A. Archer, Langmuir 27, 9387 (2011). https://doi.org/10.1021/la201396m approaches. The experimental data suggest that there is a decrease in the friction coefficient μ with increasing brush thickness at moderate loads.88. Z. Zhang, A. J. Morse, S. P. Armes, A. L. Lewis, M. Geoghegan, and G. J. Leggett, Langmuir 27, 2514 (2011). https://doi.org/10.1021/la1043848 What is less clear is the particular dependence on degree of polymerization N or grafting density σg as an increase in either should lead to an increase in brush thickness. To this end, Perry et al. have performed shear experiments on PLL-g-PEG brushes with different grafting ratios as well as lengths of PEG side chains, which suggest a decrease in friction with an increase in either N or σg, within the regime (0.5<L/2Rg<1) approaching the mushroom-to-brush transition (L≈Rg).7171. S. S. Perry, X. Yan, F. T. Limpoco, S. Lee, M. Müller, and N. D. Spencer, ACS Appl. Mater. Interfaces 1, 1224 (2009). https://doi.org/10.1021/am900101m On the other hand, Klein et al. found that the brush thickness had little effect on μ when comparing highly dense poly((2-(methacryloyloxy)ethyl)phosporylcoline) (pMPC) brushes synthesized through surface-initiated atom transfer radical polymerization (SI-ATRP).1010. O. Tairy, N. Kampf, M. J. Driver, S. P. Armes, and J. Klein, Macromolecules 48, 140 (2014). https://doi.org/10.1021/ma5019439 More recently, Benetti et al. have conducted experiments comparing PEOXA brushes composed of linear and cyclic chain architectures at different grafting densities in the range of 0.05–0.37 chains/nm2. They confirm the finding of decreased friction for thicker brushes that are due to a higher grafting density but also find greatly reduced friction for cyclic compared to linear bilayers over a wide range of applied loads as can be seen from the AFM measurements in Fig. 10(a), where we see that with increasing contact pressure, linear bilayers showed a monotonic increase in μ, whereas cyclic bilayers in fact showed a slight decrease. The authors attribute this enhanced lubricity of cyclic brushes to their greater steric repulsion leading to reduced bilayer interpenetration under compression [Figs. 10(b) and 10(c)] as well as an increased measured hydration than that of linear brushes.121121. M. Divandari, L. Trachsel, W. Yan, J.-G. Rosenboom, N. D. Spencer, M. Zenobi-Wong, G. Morgese, S. N. Ramakrishna, and E. M. Benetti, ACS Macro Lett. 7, 1455 (2018). https://doi.org/10.1021/acsmacrolett.8b00847The most complete theoretical description of sheared bilayers is laid out in the instructive review by Kreer,4747. T. Kreer, Soft Matter 12, 3479 (2016). https://doi.org/10.1039/C5SM02919H which has served as a major inspiration for the present review. Kreer combines the scaling approaches92,19792. A. Galuschko, L. Spirin, T. Kreer, A. Johner, C. Pastorino, J. Wittmer, and J. Baschnagel, Langmuir 26, 6418 (2010). https://doi.org/10.1021/la904119c197. L. Spirin, A. Galuschko, and T. Kreer, Macromolecules 44, 9399 (2011). https://doi.org/10.1021/ma2014029 of for the shear force with their own scaling theory for compressed bilayers212212. T. Kreer and S. M. Balko, ACS Macro Lett. 2, 944 (2013). https://doi.org/10.1021/mz4004387 to obtain scaling relations as a function of molecular parameters (N,σg) of the brushes for lateral chain extension, viscosity, and friction coefficient μ for both melt and semidilute bilayers under strong compression. For everyone’s convenience, Kreer has calculated all of the exponents and compiled the predictions for both the Newtonian and shear-thinning regimes in Table 1 of Ref. 4747. T. Kreer, Soft Matter 12, 3479 (2016). https://doi.org/10.1039/C5SM02919H. The result for the friction coefficient of a SD bilayer μ(sd) is as follows:4747. T. Kreer, Soft Matter 12, 3479 (2016). https://doi.org/10.1039/C5SM02919H μ(sd)∼{N−0.77σg−0.77D2.77γ˙(W≪1),N−1.82σg−0.91D2.40γ˙0.54(W≫1),(29)As can be observed from exponents, the theory agrees with experiments on the fact that friction decreases with increasing brush thickness through increases in either the chain length N or grafting density σg or with decreasing wall separation (increasing pressure) for a fixed system, which is in line with the simulation results of .In the MD study by Galuschko et al. presented earlier in , it was observed that bilayers with an implicit DPD solvent exhibited high μ that indeed decreased with increasing σg, whereas the opposite trend (low μ that increases with σg) was observed for bilayers with explicit solvent dimers. This result is not as surprising as it may seem. Because for the explicit solvent treatment, the total number density is kept constant across different brushes by removing mobile solvent particles in order to increase the number of grafted (immobile) chains. Both cases (impicit and explicit) are identical only at the highest σg when there are no more solvent particles and all of the beads belong to grafted chains, as illustrated in Fig. 9(a). However, the difference in number density for all other points means the two solvent treatments are not strictly comparable. Again, we see a nontrivial effect of choosing certain coarse-graining.In the previous study, the simulated bilayers were all above the mushroom-to-brush transition (MBT) (σg>1.1σg∗).9292. A. Galuschko, L. Spirin, T. Kreer, A. Johner, C. Pastorino, J. Wittmer, and J. Baschnagel, Langmuir 26, 6418 (2010). https://doi.org/10.1021/la904119c On the other hand, Goicochea et al. used a DPD model with explicit solvent particles at a constant number density to study the effect of both chain length and grafting density on the friction coefficient of sheared bilayers that are around the MBT. The bilayers are composed of rather short chains (N=7−21) and the number of grafted chains is varied over a wide range, covering the MBT for different chain lengths considered. The simulations are performed at a single shear rate (γ˙=0.028) and wall separation (D=7) (DPD units). The results, reproduced in Figs. 9(c)–9(f), show a nonmonotonic behavior of μ with increasing σg, in particular, for bilayers of shorter chains, first a drastic decrease leading up to a clear minimum σgmin followed by a shallow increase with greater σg. By varying N at constant σg [Fig. 9(f)], the authors observed an increase in μ for increasing chain length N to which they fit a scaling law σg∼N6/5, an outcome in disagreement with the predicted exponents of Eq. (29). Furthermore, the authors report a decrease in σgmin with increasing chain length, which follows σgmin∼N−6/5 and matches theoretically predicted scaling for the critical grafting density σg∗,44. P. G. de Gennes, Macromolecules 13, 1069 (1980). https://doi.org/10.1021/ma60077a009 suggesting to them that the minimum in friction observed occurs at the MBT (σgmin=σg∗).151151. E. Mayoral, J. Klapp, and A. G. Goicochea, Phys. Rev. E 95, 012505 (2017). https://doi.org/10.1103/PhysRevE.95.012505Similarly, Singh et al. have performed MD simulations employing the modified LJ pair interaction [Eq. (6)] with explicit solvent beads using interaction parameters that model PBBs in good solvent conditions.110110. M. K. Singh, P. Ilg, R. M. Espinosa-Marzal, N. D. Spencer, and M. Kröger, Polymers 8, 254 (2016). https://doi.org/10.3390/polym8070254 The temperature is maintained constant through a profile unbiased thermostatting (PUT) velocity-rescaling scheme. In these simulations, the bilayers are composed of chains of length N=30 and a range of grafting densities (σg≈0.6−6σg∗) with σg∗≈0.025 (LJ units). When comparing two brushes (σg=0.075,0.015) over a range of shear velocities but a constant wall separation (D=30), the authors found that the higher density brush exhibited a higher friction coefficient μ. However, when μ was extracted for bilayers of different σg from the slopes of the shear vs normal stress curves obtained for a single velocity (vs=0.001) at different wall separations (0.5<d=D/2h<1), μ is found to decrease with increasing grafting density, within the range of σg explored [see Fig. 9(b)]. Additionally, the model used in this study incorporates a cosine angle potential (17) to study the effect of chain-stiffness, which was used to show that, for the range bending stiffnesses explored (K=0,1,2,3), the friction coefficient μ decreases with increasing persistence length Lp.110110. M. K. Singh, P. Ilg, R. M. Espinosa-Marzal, N. D. Spencer, and M. Kröger, Polymers 8, 254 (2016). https://doi.org/10.3390/polym8070254More recently, Goicochea et al. have performed an extensive DPD study of the effect of grafting density and solvent quality on the tribology of bilayers composed of mixtures of two CG polymers immersed in solvent particles and free chains at a constant number density. The results are reported in physical units assuming a CG degree of three for a wide range of grafting densities (σg≈0.1−1.2chains/nm2) covering the upper range of experimentally observed values. In agreement with the previous results, the friction coefficient is found to decrease with increasing grafting density as well as with increasing number of free chains in the solution for a given grafting density, in line with the predictions due to chain detachment of . Interestingly, the authors find that the friction coefficient follows a scaling law that depends only on the solvent quality but not the brush composition or the number of free chains and that mixed brushes may result in reduced friction compared to monochain brushes.213