ARTICLE pubs.acs.org/Langmuir
Molecular Simulation of the Frictional Behavior of Polymer-on-Polymer Sliding Y. K. Yew, Myo Minn, S. K. Sinha, and V. B. C. Tan* Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore ABSTRACT: Molecular simulations of the sliding processes of polymer-on-polymer systems were performed to investigate the surface and subsurface deformations and how these affect tribological characteristics of nanometer-scale polymer films. It is shown that a very severe deformation is localized to a band of material about 2.5 nm thick at the interface of the polymer surfaces. Outside of this band, the polymer films experience a uniform shear strain that reaches a finite steady-state value of close to 100%. Only after the polymer films have achieved this steady-state shear strain do the contacting surfaces of the films show significant relative slippage over each other. Because severe deformation is limited to a localized band much thinner than the polymeric films, the thickness of the deformation band is envisaged to be independent of the film thickness and hence frictional forces are expected to be independent of the thickness of the polymer films. A strong dependency of friction on interfacial adhesion, surface roughness, and the shear modulus of the sliding system was observed. Although the simulations showed that frictional forces increase linearly with contact pressure, adhesive forces contribute significantly to the overall friction and must therefore be accounted for in nanometerscale friction. It is also shown that the coefficient of friction is lower for lower-density polymers as well as for polymers with higher molecular weights.
’ INTRODUCTION The successful deployment of microelectromechanical (MEMS) and nanoelectromechanical systems (NEMS) devices requires effective solutions to tribological issues facing contacting surfaces of moving parts.1,2 Minimizing frictional and wear deformation is necessary to enhance the performance of such devices. Magnetic recording systems and miniature motors with small loads are examples of MEMS and NEMS components that are severely affected by friction and wear because their area to volume ratio is orders of magnitude larger than their conventional counterparts. Because of the small length scales involved in MEMS/NEMS devices, conventional tribological solutions might not be applicable and may be much less sufficient. Applying a polymer coating to MEMS devices has become a popular means to lower friction as well as to increase wear durability. Friction and wear are dependent on many parameters including mechanical properties, molecular structures, and the topography of the contact region. Although significant advances have been made in polymer tribology research, there is still a lack of understanding in phenomenological events in the tribological interface and subsurface regions. The study of nanotribology, encompassing physical, chemical, and mechanical phenomena at the interface of moving surfaces on the molecular length scale, is intimately related to the development of MEMS and NENS. Atomistic- and molecularlevel modeling and simulation have been proven to be effective tools in providing information on the nanometer length scale and are being utilized to supplement experimental investigations.3,4 r 2011 American Chemical Society
With the advances in computing technology over the past few decades, computational simulations are closer to being able to predict physical phenomena accurately. Simulations to study tribological processes are also becoming popular. The flexibility of controlling parameters independently of one another, for example, geometry, sliding conditions, and interactions between atoms, is especially useful in understanding physical mechanisms during sliding contact on the molecular level. However, simulations are still lagging behind experimental findings, especially for soft materials such as polymers. Molecular mechanics simulations are carried out to investigate the tribology of polymer-on-polymer sliding. Because polymers are usually characterized by their density and molecular weight, these two parameters are varied in the simulation models.
’ SIMULATION MODEL The sliding of a polymeric material over another polymeric material is simulated. The phenomenological behaviors of polymeric materials pose a challenge for multiscale simulations because capturing such mechanisms span several length and time scales ranging from electron interactions to entire macroscopic structures and from femtoseconds to years of reaction period. However, a full atomistic molecular simulation with detailed chemical interactions is computationally intractable for studying dense Received: August 23, 2010 Published: April 25, 2011 5891
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Table 1. Parameter Values for Interatomic Potentials of the Polymeric System parameters
approximated value for CH2
σ
5Å
ε R0
30 meV 1.5σ
k
30 ε/σ2
rcutoff
2.2σ
systems consisting of many long polymer chains over a long relaxation period. To reduce the complexity of the molecular model and yet be able to span the entire hierarchy of length and time scales, a commonly adopted approach is to describe polymer chains as interactions of groups of atoms rather than discrete atoms.59 Variations of the scheme are mainly distinguished by the method of discretizing polymer chains, distributing the mass center of interaction of the new coarse-grained unit and incorporating different interatomic force fields. Coarse graining refers to the mapping of a group of atoms to a beadspring model (i.e., the beadspring model represents a polymer chain as a collection of beads connected by elastic springs). Each bead typically represents one or a few monomer units, and the springs reproduce the Gaussian distribution of separations between monomers connected through a number of chemical bonds. Because some of the information about physical and chemical properties will be lost through coarse graining, the choice of the force field governing the interactions of the beadspring model is crucial in determining the practicability and accuracy of the simulations. Despite lacking fine chemical details, simplified general models are capable of predicting many physical properties and phenomena of polymer networks, solutions, and melts.5,8,1014 Such results help identify key parameters (e.g., chain, length, architecture, and stiffness) that are mainly responsible for particular properties. The coarse-grained model introduced by Kremer and Grest6 has been applied to various polymer studies with great success. It allows the treatment of longer length and time scales, which would save at least an order of magnitude of computational time compared to a corresponding fully atomistic model to get the same results. The same force field model is employed in our present tribological simulations. In our simulations, all of the beads in the polymer system interact through a truncated Lennar-Jones (LJ) potential, 0 ! !6 1 12 σ σ A, rij < rcutof f ð1Þ VLJ ðrij Þ ¼ 4ε@ rij rij where rij is the distance between two interacting beads i and j and ε and σ are the energy and length at which VLJ = 0. A cutoff radius of rcutoff = 21/6σ and geometric mean combination rules are employed. The parameter values used in the simulations are listed in Table 1. For chemically bonded beads, an additional potential is needed. The finite extensible nonlinear elastic (FENE) potential that reproduces a simple harmonic potential for small extensions is added for the beads connected along the sequence of the chains, 8 " 2 # > < 0:5kR 2 ln 1 rij , rij e R0 0 R0 ð2Þ VFENE ðrij Þ ¼ > : ¥, rij > R0
Here, k = 30ε/σ2 and R0 = 1.5σ are the spring constant and maximum bond extension, respectively. The chosen spring constant was strong enough to maintain the maximum extensions of bonds to be less than 1.2σ. As a result, bond crossings are energetically unfeasible. Mapping of the potential with these specific parameter values to typical hydrocarbons yields values of between 25 and 45 meV for ε and between 0.5 and 1.3 nm for σ.6 The influence of adhesion at the contacting surfaces can be controlled by varying the LJ potential cutoff distance for the intermolecular interaction. A value of rcutoff = 21/6σ gives a purely repulsive potential representing zero adhesive interaction. The larger the cutoff distance, the longer the attractive tail of the LJ potential and the stronger the adhesive effects. Because the focus of the simulations was on interfacial friction, a cutoff radius of rcutoff = 2.2σ was applied for nonbonded beads in all simulations. This cutoff was large enough for the attractive portion of the potential to be active without sacrificing computational efficiency. To provide some indication of how the computational model compares with actual polymers, the mechanical properties of bulk polyethylene15 and values computed from our model are presented in Table 2. The simulation model replicates a symmetric system of a polyethylene surface sliding on another polyethylene substrate.16 It comprises an upper slider block and a lower substrate block of beadsprings as depicted in Figure 1. Periodic boundary conditions were applied to the vertical surfaces while the contacting surfaces of the two blocks were stress free prior to contact. A layer of beads at the bottom of the substrate block was fixed, and a layer of beads at the top of the slider block was prescribed first to displace the slider block downward to attain contact pressures in the range of 0.5 to 5.5 GPa before they were displaced horizontally to simulate sliding. Considering that the substrate surfaces of the models were molecularly flat and had no defects, the compressive loads were nearly constant throughout the entire surface. This is in contrast to the typical experimental systems where the surfaces contain random asperities that characterize the surface roughness. Experiments show that the real area of contact is often much smaller than the apparent surface area and the mean contact pressure can be as large as 3 orders of magnitude larger than the normal applied pressure.17,18 Therefore, the contact pressure range of 0.5 to 5.5 GPa studied in this article corresponds approximately to physical loading pressures on the order of 100 MPa. Conjugate gradient energy minimization was performed throughout the simulations (i.e., dynamics and thermal fluctuations were not included in the simulations). This greatly reduces noises from thermostat fluctuation to facilitate the detection of delicate changes in the energy profile and an investigation into the mechanisms of the system. Although we have not taken into consideration heat dissipation at the surfaces during sliding, this is very small at low loads.19
’ RESULTS Simulations were performed on beadspring computational models of different polymer densities and molecular weights to study the effects of these two parameters. Starting with a polymer model of normalized density 0.85 and molecular weight 3500, the density of the polymer was then changed to normalized values of 0.75 and 0.95 while keeping the molecular weight at 3500. A normalized density of 1 corresponds to 0.95 g/cm3. Another series of simulations were performed using polymer models of 5892
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Table 2. Comparison of Young’s Modulus, Yield Stress, and Tensile Strength between Computational Predictions and Experimental Values for LLDPE and HDPE Young’s modulus (MPa) experiment
yield stress (MPa)
simulation
experiment
simulation
tensile strength (MPa) experiment
simulation
low density
137520
568
6.211.5
9.45
920
14.46
high density
60290
310
1832
11.35
1060
29.40
Figure 1. Schematic drawing of the computational model for polymeron-polymer sliding.
molecular weights 700 and 14 000 while keeping the density at 0.85. For each model, the simulations were conducted at different normal contact pressures by controlling the vertical displacement of the slider block (i.e., displacement in the applied load direction in Figure 1). Snapshots of a typical sliding simulation are presented in Figure 2. Beads within the slider and substrate are colored blue and red, respectively. Two initially straight and collinear columns of beads shown in Figure 2a—one within the slider in yellow and another within the substrate in cyan—were tracked during the simulations to aid visualization. Steady-state shear was attained for the bulk of the polymer slider and substrate after the slider had traversed a certain distance. The shear is quantified by the shear angle θ as labeled in Figure 2d. The shear stress, determined from the sum of lateral forces per unit area of contact acting on the beads of the substrate in the sliding direction of the slider block, is plotted against the slider displacement in Figure 3 for three different contact pressures for polymer models with a normalized density of 0.95 and a molecular weight of 3500. After some sliding distance, the shear stress for each contact pressure fluctuates about a mean value that is taken as the frictional stress. The frictional stress as a function of contact pressure is presented in Figures 4 and 5 for the different polymer densities and molecular weights respectively.
’ DISCUSSIONS The highlighted columns of beads at different instances of the simulation in Figure 2 show that there was significant shear deformation of the polymer prior to slipping at the slider substrate interface. In reported studies of tribological mechanisms,
most were concerned with the shearing at the contacting surfaces.20,21 Shearing within the bulk system is rarely reported. It is seen in Figure 2b, when the slider had already been displaced by 100 Å, that the two columns of beads remained stuck together even though they had undergone large shear deformation. This demonstrates that bulk material shearing dominated without noticeable sliding at the contacting surfaces. It is only in Figure 2c, when the top of the slider had been displaced horizontally by 200 Å, then the two columns started to become disjointed. The subsurface shearing continued to increase with further displacement of the slider until Figure 2d, when interface sliding was very advanced. Beyond that stage, there was no further increase in shearing as quantified by shear angle θ. In the initial sliding stage, the molecules were able to move more freely because there were fewer chain entanglements. During shearing, some molecular chains locked up with one another. When chain entanglements severely reduced chain mobility, a minimal increase in subsurface shearing was observed. The sliding distances before steady-state shearing angles were achieved were 220, 320, and 360 Å for contact pressures of 1.58, 2.55, and 3.93 GPa, respectively. For all of the simulations in this study, the steady-state shear angles ranged from 42 to 53, suggesting that the sliding of two contacting thin polymeric films is preceded by stiction for sliding distances that are approximately equal to the total thickness of the films before slipping of the surfaces takes place. Stiction over large sliding distances poses important implications in sliding MEMS components if polymer coatings are very thick compared to the thickness of individual components or if the relative motion of contacting components is of the same order of magnitude as the coating thickness. In two review articles,22,23 Briscoe and his colleagues suggested that there is a narrow regime adjacent to the interface where a large deformation is concentrated. It is also seen in Figure 2 that the deformation mechanism beyond steady-state shearing is largely a localized plastic deformation within a narrow band at the contacting interface of about 25 Å thickness as indicated by t. The discontinuity at the contacting junctions manifest that surface deformation for interfacial sliding was highly localized and much greater in magnitude than the subsurface shear. The subsurface beads were displaced in the sliding direction in a manner similar to lamellar flow during the simulations, which is a characteristic of viscoelastic or rubbery materials. To compare bulk shearing and sliding at the interface quantitatively, the average displacement of the slider beads within the frictional interface t relative to the average displacement of interfacial substrate beads is plotted against slider displacement in Figure 6 for three different contact pressures. It is seen that when the slider has been displaced by 400 Å, the actual interfacial sliding ranges from only 10% (or 40 Å for a contact pressure of 3.93 GPa) to 20% (or 80 Å for a contact pressure of 1.58 GPa) of the slider displacement. The plots display an obvious increase in the gradient at about 250 Å of slider 5893
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Figure 2. Snapshots from the simulation of polymer-on-polymer sliding for a normalized density of 0.75 and a molecular weight of 3500. Blue and red beads are from the slider and substrate polymer blocks, respectively, as described in Figure 1. Beads at the bottom were fixed, and beads at the top were displaced laterally to the right of the image by (a) 0, (b) 100, (c) 200, and (d) 400 Å. The columns of yellow and cyan beads highlight the level of shear deformation and interfacial sliding. The shear angle is indicated by θ, and severe localized deformation is contained within a thickness of t.
Figure 3. Shear stress of the substrate as a function of sliding distance at varying pressures of 1.58, 2.55, and 3.93 GPa with a normalized density of 0.95 and a molecular weight of 3500.
Figure 4. Shear stress plotted against contact pressure for different normalized densities.
displacement, indicating a transition from mainly bulk shearing to mainly interface sliding. It is noted that stiction (indicated by plateaus in the plots) did not cease completely after the transition and slippage (indicated by steps in the plots) did not start only after the transition. Stickslip occurred throughout the sliding simulation with the magnitude of the slippages increasing after about 250 Å of slider displacement. During the stickslip events, contacting junctions are constantly being formed and deformed. In the brief periods that a junction exists, the atoms are pushed
and pulled toward new equilibrium positions. Reorientation and restructuring occur continuously in the interface zone. Therefore, the intensity of deformation is large and localized. As illustrated in Figure 2, except for the band of severe deformation t, the subsurface shear is uniform. Given that the band thickness t is much smaller than the slider and substrate thicknesses, it is expected that t is purely determined by the local interfacial environment and would be independent of the thickness of the polymer model. Similarly, the steady-state subsurface shear 5894
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Figure 5. Shear stress plotted against contact pressure for different molecular weights.
Figure 7. (a) Plot of shear angle as a function of contact pressure and (b) comparison of coefficient R and the intrinsic shear flow stress for films of different thicknesses.
Figure 6. Displacement of interface beads of a slider relative to the displacement of interface beads of the substrate for a normalized density of 0.95 and a molecular weight of 3500.
angles should also remain unchanged as long as the polymer substrate and slider are considerably thicker than t. To verify this, a model with a substrate that is twice as thick (i.e., 31.12 nm) was constructed, and sliding simulations were performed for contact pressures of 1.59, 2.61, and 3.92 GPa. In all cases, the steady-state shear angles were in the range of 46 to 53, whereas the coefficient of R and intrinsic shear flow stress were calculated to be 0.107 and 1.07 GPa, repectively. These values are very close to those obtained for the model with a substrate thickness of 15.56 nm, which gave values of between 42 and 53 for the steady-state shear angle, with a corresponding coefficient of R and intrinsic shear flow stresses of 0.109 and 1.04 GPa. The comparisons are shown in Figure 7. Hence, it is proposed that the frictional forces are independent of polymer thickness. With respect to the sliding forces presented in Figure 3, it is inferred that the outcome of thicker polymer
models will simply be manifest as a larger sliding distance before the sliding forces converge more gradually to the same frictional forces as for thinner models to attain the same steady-state shear angle. Interatomic van der Waals forces pulled the slider and substrate together when the two surfaces were brought into contact. This is the origin of the adhesion experienced by most materials in contact, resulting in sticking of the two contacting surfaces in the initial stage. For atomically smooth surfaces, this stiction force is non-negligible.2327 To commence sliding, an external lateral force is needed to overcome frictional forces as well as the adhesive force. The trend lines of the interface shear stress in Figure 3 start from zero and increases gradually to converge to different values for the three different contact pressures. The steady-state sliding forces correspond to the frictional forces that exist after interface slippages were established and the shearing of the polymer subsurfaces had reached steady states. The interface shear stresses from all of the simulations are plotted in Figures 4 and 5 against contact pressure. The plots in Figure 4 show the changes in shear stress with polymer density whereas Figure 5 shows the effects of polymer molecular weight. The frictional stresses increased linearly with contact pressures as generally observed physically. There is a finite shear stress at zero load, which indicates that friction is not entirely load-controlled but also has significant adhesion-controlled contributions.24 From the plots of the graphs, we can express the linear dependency of the 5895
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Figure 8. Effects of molecular weight on the frictional stress of polymeric material. Coefficient R and intrinsic shear flow stress as a function of molecular weight.
shear stress on the contact pressure as τ ¼ τ0 þ RP
ð3Þ
where τ and P are the shear stress and contact pressure, respectively. Here, the gradient of the frictionload curve R is similar to the coefficient of friction as defined by Amontons’ law.25 τ0 is the intrinsic shear flow stress arising from the adhesion force between the two contacting surfaces that has to be overcome to initiate sliding.28 This is also the force responsible for the sticking of two smooth, clean surfaces even without any externally applied pressure. This linear relationship agrees well with the commonly observed linear dependence of the interface friction force on the normal load.21,25,26 For very low external loads, the adhesive force will be the dominant term in eq 3.27 The notion that friction should be related to adhesion seems fairly natural. The same bonding mechanism that makes it difficult to pull surfaces apart should also make it difficult to slide them over each other. The interface shear stress becomes directly proportional to the contact pressure when the contact pressure becomes significantly large. Hence, eq 3 is an extension of the laws of friction of Amonton to accommodate atomically smooth surfaces where adhesion is evident. As reflected in Figures 4 and 5, changes in the density (i.e., the number of beads per unit volume) or molecular weight of the polymer model influence the frictional behavior of polyethylene. Plots of the coefficient R and intrinsic shear flow stress τ0 against molecular weight and density are presented in Figures 8 and 9, respectively. As shown in Figure 8, increasing the molecular weight while keeping the polymer density constant led to an increase in τ0 but a decrease in coefficient R. The difference in the surface structure was believed to be the main reason for the increase in the coefficient of friction for lower-molecular-weight polymers. The atomic configurations near the interface of the slider and substrate in Figure 10 illustrate the degree of surface interpenetration of three models with different molecular weights. The interpenetration is quantified by d, the vertical distance between the topmost substrate bead and the bottommost slider bead. The interpenetration depths for the models of 14 000, 3500, and 700 molecular weight were found to be 2.8, 5.95, and 6.64 Å, respectively.
Figure 9. Plots of coefficient R and intrinsic shear flow stress as a function of the polymer density.
The inverse relationship between molecular weight and diffusion depth has been reported previously.30 In addition to the higher interpenetration depth, the number of dangling ends of the molecular chain for polymers of lower molecular weight, as highlighted in green and cyan in Figure 10, was also higher. The number of chain ends that protruded across the interface was also higher. Even after sliding had progressed for some distance, more chain penetration and atom diffusion at the interface were observed for the model with the lowest molecular weight. The depths of interpenetration at the end of the simulations were measured to be 8.27, 8.8, and 12.07 Å for molecular weights of 14 000, 3500, and 700, respectively. Such observations suggest that slider and substrate models with lower molecular weights should experience higher levels of locking. However, the models with the highest molecular weights had the smoothest surfaces, which meant that the contacting surfaces were in close proximity to each other over a larger area and consequently there was greater adhesion because of the van der Waals interaction. Hence, for atomically smooth surfaces, such as the simulations presented, adhesion forces can dominate to give the highest τ0 (Figure 8) for the model with the highest molecular weight. However, adhesion contributes less to friction when surfaces are not as smooth or when contact pressures are increased. Friction forces then become dominated by coefficient R. Therefore, coefficient R is highest for the model with the lowest molecular weight (Figure 8). Chen et al.,30 who carried out tribological experiments on polyethylene, also found a decrease in the gradient of the frictionload curve (coefficient R) with increasing molecular weight. Therefore, it is suggested that applying a high-molecularweight polymer as a tribological coating can reduce friction and prolong wear life. The effects of polymer density on friction were found to be different from the effects of molecular weight. For a constant molecular weight of 3500, a decrease in the normalized density from 0.95 to 0.85 to 0.75 significantly changes the surface roughness characteristics of the polymer models. The contact junctions at the surface asperities are also more compliant for lower-density polymers, thereby encouraging more molecularscale interdiffusion and bulk flow to the contact boundary.25,31 These were projected by the increases in interpenetration depth from 3.05 to 5.95 to 14.21 Å with decreasing polymer density. 5896
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Figure 10. Molecular configurations of two contacting polymer blocks before sliding for three different models with molecular weights of (a) 14 000, (b) 3500, and (c) 700. The atoms in the lower block are red, and the atoms in the upper block are blue. Cyan atoms are the ends of chains belonging to the lower block, whereas green atoms are from the upper block. The black lines are the peak of the surface for both lower and upper blocks.
The intrinsic shear flow stress τ0 also increases accordingly (Figure 9) because the actual contact area becomes larger with increasing interpenetration. Figure 9 shows that the lowest-density polymeric material has the lowest coefficient R. For surfaces to slide over each other, contacting asperities must undergo elastic and plastic deformation, resulting in localized shear stresses within the contacting junctions.32 From Table 2, we can expect highdensity polymers to exhibit higher coefficients of friction than lowdensity polymers because of their higher strength.
’ CONCLUSIONS For the dry interfacial sliding of two molecularly flat films of polymeric materials, two distinct layers of deformation were observed: localized surface deformation and subsurface shearing. These two regimes of deformation are largely dependent on the surface geometry and mechanical properties of the two surfaces in contact. Localized plastic deformation was confined within a narrow band of molecular-scale thickness at the contacting junction. Simulation results also showed that the localized deformation region is independent of film thickness. The beads in this localized region were actively reoriented and restructured with the pulling and pushing of beads, as manifested in stickslip sliding. However, the deformation in the subsurface was predominantly shearing, which attains a uniform, steady-state value during which the macromolecules rearrange themselves in a manner similar to lamellar flow in viscoelastic materials. As the shear angle θ converged to steadystate values in the range of 42 to 53, interfacial slipping was minimal, suggesting that large-scale slipping will commence only for sliding distances equivalent to the polymeric coating thickness. It was also shown that whereas friction scales linearly with contact pressure in accordance with Amonton’s law, the adhesive force contributes significantly to frictional forces on the nanometer scale, especially for the interfacial sliding of atomically flat surfaces. The effects of the density and molecular weight of the polymer on the frictional behavior were also studied. The studies predicted that a very low frictional sliding of the polymer-on-
polymer system can be obtained with two blocks of low-density and high-molecular-weight polyethylene.
’ AUTHOR INFORMATION Corresponding Author
*To whom correspondence should be addressed. Email: mpetanbc@ nus.edu.sg.
’ ACKNOWLEDGMENT The authors would like to acknowledge the financial support given to this study by the National Research Foundation (NRF), Singapore (Award No. NRF-CRP 2-2007-04). ’ REFERENCES (1) Hirano, M. Atomistics of friction. Surf. Sci. Rep. 2006, 60, 159–201. (2) Szlufarska, I.; Chandross, M.; Carpick, R. W., Recent advances in single-asperity nanotribology. J. Phys. D: Appl. Phys. 2008, 41,. (3) Landman, U.; Luedtke, W. D.; Gao, J. P. Atomic-scale issues in tribology: Interfacial junctions and nano-elastohydrodynamics. Langmuir 1996, 12, 4514–4528. (4) Robbins, M. O.; M€user, M. H. Computer simulations of friction, lubrication and wear. In Modern Tribology Handbook; Bhushan, B., Ed.; CRC Press: Boca Raton, FL, 2001; Vol. 1, pp 717765. (5) Muller-Plathe, F. Coarse-graining in polymer simulation: From the atomistic to the mesoscopic scale and back. ChemPhysChem 2002, 3, 754–769. (6) Kremer, K.; Grest, G. S. Dynamics of entangled linear polymer melts - a molecular-dynamics simulation. J. Chem. Phys. 1990, 92, 5057–5086. (7) Faller, R.; Muller-Plathe, F.; Heuer, A. Local reorientation dynamics of semiflexible polymers in the melt. Macromolecules 2000, 33, 6602–6610. (8) Tschop, W.; Kremer, K.; Batoulis, J.; Burger, T.; Hahn, O. Simulation of polymer melts. I. Coarse-graining procedure for polycarbonates. Acta Polym. 1998, 49, 61–74. (9) Baschnagel, J.; Binder, K.; Doruker, P.; Gusev, A. A.; Hahn, O.; Kremer, K.; Mattice, W. L.; Muller-Plathe, F.; Murat, M.; Paul, W.; Santos, S.; Suter, U. W.; Tries, V. Bridging the gap between atomistic and coarse-grained models of polymers: Status and perspectives. In Advances 5897
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