2025 Volume 11 Article ID: 2024-0036
To elucidate the micro-scale wear mechanisms of concentrated polymer brushes (CPB), we developed a coarse-grained molecular dynamics (CGMD) model of CPB-against-slider sliding simulation with an explicit solvent. Under good solvent conditions, the CPB swelled. The thickness of the swollen CPB reached approximately 77% of the fully stretched length, which is in close agreement with the experimentally determined range for swollen film thicknesses of CPB. The validity of the developed model was evaluated through a sliding simulation. The simulation analysis showed that the wear of the CPB can involve multiple factors, such as contact with the slider, solvent flow, and interactions between polymer chains. This indicates that the simulation model allows us to analyze not only the wear caused by direct contact with the slider but also the wear caused by solvent flow.
Globally, approximately 20% of energy is expended to overcome friction, resulting in substantial CO2 emissions and economic inefficiencies [1]. In light of mounting environmental concerns, developing low-friction materials to reduce energy losses is essential.
Polymer brushes (PB) are ultrathin polymer coatings where polymer chains are anchored to the substrate surface at one end, while the other end extends away from the surface to avoid overlapping with neighboring polymer chains, forming a brush-like structure [2]. In recent years, concentrated polymer brushes (CPB), distinguished by a graft density that is at least an order of magnitude higher (> 0.1 chains/nm2) than that of conventional PB with moderate densities (e.g., 0.001–0.05 chains/nm2), have garnered considerable attention as promising lubricant materials [3,4,5]. Notably, CPB, when swollen in a good solvent, exhibits superior tribological properties such as high compression resistance and ultra-low friction [5, 6].
However, to apply CPB to practical components with reliable performance, their durability must be enhanced. To this end, a comprehensive understanding of their wear mechanisms is imperative. Several experimental studies have examined the wear process of CPB [7, 8]. Nevertheless, detailed insights into the micro-scale wear mechanism are still lacking.
To reveal the molecular-level dynamics of PB, coarse-grained molecular dynamics (CGMD) simulations have been employed. CGMD simulations effectively minimize computational costs compared to all-atom molecular dynamics simulations while maintaining the structural and dynamic characteristics of large-scale complex systems such as polymers. Singh et al. investigated the tribological behavior of cross-linked PB against walls using CGMD simulations with an implicit solvent model, finding that cross-linking enhances shear resistance [9]. Liu et al. conducted CGMD sliding simulations to examine the effect of cross-linked structure and cyclic chain structure on the wear behavior of CPB in a CPB/CPB interface model. They demonstrated that these structures prevent interpenetration of polymer chains on the counter surface, leading to a reduction in frictional forces and consequently mitigating wear [10, 11].
However, to the best of our knowledge, few studies have investigated the wear behavior and the friction mechanisms of CPB in the presence of an explicit solvent. Given that the superior properties of CPB are attributed to its swollen structure, it is imperative to perform simulations with an explicit solvent to accurately assess the wear mechanisms of CPB.
In this study, we have developed a CGMD simulation model to investigate the wear mechanisms of CPB in a slider-CPB friction system with an explicit solvent. Our objective is to uncover molecular-level insights into the wear process and to contribute to the design of durable low-friction CPB.
A CGMD simulation model was developed based on a bead-spring model. In this model, the units of time are designated as τ, length as σ, and energy as ϵ. The model consists of four components: polymer, substrate, slider, and solvent. In the simulation model, polymer chains were fixed at one end to the substrate surface and swollen in a solvent, with a slider gliding over them. All the simulations were conducted using the MD program Laich, which was developed in our group.
The detailed procedure for constructing the slider-CPB-solvent system developed in this study is schematically illustrated in Figure 1. Initially, a face-centered cubic (FCC) substrate with dimensions of 320 × 40 × 10 σ3 and a number density of 0.95 σ−3 was constructed. As depicted in Figure 1 (a), one substrate bead was selected at random from the surface and tethered to one end of polymer chains having a fully stretched structure. The polymer chains had an average length of 120 σ, corresponding to an average of 120 polymer beads per chain, with a dispersion length of ± 5 σ. This process was repeated until a grafting density of 0.3 chains/σ2 was achieved (Figure 1 (b)). The constructed CPB model was then positioned at the base of a simulation cell measuring 320 × 40 × 168 σ3. The solvent molecules were randomly arranged using Packmol [12] to achieve a number density of 0.33 σ−3 in the bulk phase. To hinder close packing and incorporate rotational and vibrational degrees of freedom, LJ dimers were employed as solvent particles instead of monomers [13]. To facilitate an intuitive understanding of the model, the length scale can be converted using the relation 1 σ = 1.17 nm, which is derived by comparing the experimentally reported grafting density of CPB, 0.22 chains/nm2 [7], with the grafting density used in our simulations, 0.3 chains/σ2. Using this conversion, the average chain length of 120 σ corresponds to 140 nm, and the slider radius of 50 σ corresponds to 59 nm.
Schematic diagram illustrating: (a) polymer grafting, (b) CPB before immersion in the solvent, (c) equilibration process using MD, (d) placement of the slider, (e) compression process using MD, and (f) MD sliding simulation, induced by the movement of the slider across the CPB.
To achieve system equilibrium and facilitate CPB swelling, a molecular dynamics (MD) simulation was conducted under the NVT (constant number, volume, and temperature) condition, followed by a MD simulation under NPT (constant number, pressure, and temperature) condition (Figure 1 (c)). In the NVT simulation, the position of all polymer chains was constrained, and the temperature was set to T = 2.0 ϵ/kB which is twice that of the sliding simulation to enhance mixing between the CPB and the solvent. In the following NPT simulation, the constraint on the polymer chains was removed, allowing the CPB to swell in the solvent. After relaxation, a semicylindrical slider with a radius of 50 σ was placed above the swollen CPB. During this process, a portion of the solvent molecules was removed to position the slider (Figure 1 (d)). Subsequently, an additional equilibration simulation was conducted under the NPT condition at a temperature of T = 1.0 ϵ/kB. After equilibration, a compression simulation was performed under the NVT condition, in which a normal load was applied on the particles in the upper part of the slider at T = 1.0 ϵ/kB to apply pressure to the CPB (Figure 1 (e)). Finally, the sliding simulation was performed under the NVT condition by imposing a constant normal load and a constant sliding velocity in the +x-direction on the top particles of the slider, while maintaining the system temperature at T = 1.0 ϵ/kB (Figure 1 (f)). In this study, the model consisted of a total of 1,385,333 particles, including 156,040 slider particles, 128,000 substrate particles, 643,006 solvent particles, and 458,287 polymer particles. The conditions of temperature, simulation duration, load, sliding velocity, simulation cell size, polymer length, and distribution, and the density of polymer and solvent presented here are illustrative examples and can be modified as needed to suit specific research objectives.
To demonstrate the validity of our developed model, a wear simulation of CPB swollen in a good solvent, where it exhibits excellent lubrication properties, was conducted using the following force fields. In this model, the substrate beads and slider beads were treated as a single type of bead. The potential energy of the system, Utotal is expressed by the following equation:
| (1) |
where Ubond, which includes two types, Ubond-breaking and Ubond-harmonic, represents the bond potential, Ubend denotes the bending potential, and ULJ corresponds to the Lennard-Jones potential.
To address polymer chain breakage, Ubond-breaking was applied to bonds between polymer-polymer particle pairs:
| (2) |
where rij represents the distance between the ith and jth particles, Kr, ra, and rb are parameters. In equation (2), we used Kr = 3600 ϵ/σ4, ra = 1.376 σ, and rb = 0.8747 σ which gives the energy barrier for bond breaking of 24 ϵ. Here, we employed a rather lower energy barrier compared to the previous work [14] to observe the bond-breaking event in the timescale utilized in the MD simulation. The same force field was applied to the substrate-polymer particle pairs as well.
Ubond-harmonic was employed for all other particle pairs not defining the breakage:
| (3) |
where Kb and r0 are parameters. In equation (3), Kb was set to 360 ϵ/σ2. r0 was set to 1.0 σ for solvent-solvent particle pairs and 0.8660 σ for substrate (or slider)-substrate (or slider) particle pairs.
The bending potential Ubend is expressed as follows:
| (4) |
where θ is the angle between three connected beads, whereas Kθ and θ0 are parameters. To ensure structural stability, equation (4) was applied to the interactions between substrate or slider particles, where Kθ and θ0 were set to 360 ϵ/rad2 and 1.91 rad, respectively.
The 12–6 Lennard-Jones potential ULJ, as expressed by the following equation, was employed to describe the non-bonding interactions:
| (5) |
where έij represents the potential well depth, aij denotes the equilibrium interatomic distance, and rcutoff indicates the cutoff distance. When rij > rcutoff, ULJ is set to zero.
The parameters employed in equation (5) are shown in Table 1. In this study, we employed the ULJ parameters for substrate (or slider)-polymer, polymer-polymer, polymer-solvent, and solvent-solvent particle pairs utilized by de Beer and Müser [15], which are designed to achieve good solvent conditions and miscibility in PB system. The only repulsive terms were applied to interactions between the substrate (or slider)-substrate (or slider) particle pairs, substrate (or slider)-polymer particle pairs, as well as substrate (or slider)-solvent particle pairs, utilizing a cutoff length of 21/6 σ.
| Combination | έij [ϵ] | aij [σ] | rcutoff [σ] | |
| Substrate (Slider) | Substrate (Slider) | 1.0 | 1.0 | 21/6 |
| Substrate (Slider) | Polymer | 1.0 | 1.0 | 21/6 |
| Substrate (Slider) | Solvent | 1.0 | 1.0 | 21/6 |
| Polymer | Polymer | 1.0 | 1.0 | 1.60 |
| Polymer | Solvent | 1.2 | 1.0 | 2.50 |
| Solvent | Solvent | 0.5 | 1.0 | 2.50 |
To validate the constructed simulation model, a sliding simulation was conducted. In this study, a semicylindrical slider moved over the CPB in the +x-direction at a constant speed of 1.2 σ/τ, under a constant load of 2346 ϵ/σ in the –z-direction. The duration of the simulation was set to 400 τ, with a timestep of 0.005 τ. A Langevin thermostat was applied to the beads of the slider and substrate [16].
Figure 2 (a) provides a snapshot of the CPB before the sliding simulation (t = 0 τ). The swollen film thickness attained approximately 92 σ, corresponding to ~77% of the fully stretched length. This outcome closely aligns with experimental findings, where the swollen film thickness of CPB ranges from 80% to 90% of their fully stretched length [4]. The slider penetrated the CPB, leading to compression of the polymer chains in the contact region along the slider surface. Figure 2 (b) presents a snapshot of the CPB during the sliding simulation (t = 270 τ). The CPB significantly deformed due to the sliding.
Snapshots of CPB (a) before the sliding simulation (t = 0 τ) and (b) during the sliding simulation (t = 270 τ).
Polymer chain breakage was observed during the sliding process. Figure 3 (a) presents a snapshot of CPB at the end of the sliding simulation (t = 400 τ), showing severed polymer chains and their corresponding breakpoints. This outcome demonstrates the efficacy of the developed simulation model in replicating the micro-scale wear process of CPB. Notably, chain breakage occurred randomly at locations distant from the slider and deep within the CPB as well as at the contact area with the slider. This finding suggests that polymer chain breakage is driven not only by direct forces exerted by the slider but also by other factors, such as solvent flow and inter-dependent motion of polymer chains. Furthermore, our results are consistent with the experimental observations reported by Okubo et al. [7], who demonstrated that random chain breakage events occur at the sliding interface. Figure 3 (b) depicts a snapshot of CPB at the end of the sliding simulation (t = 400 τ), showing polymer chain fragments that were broken during sliding and remain dispersed within the simulation cell. While the majority of the polymer chain fragments remained entrapped within the CPB structure, a subset of these fragments was expelled into the solvent area during sliding, either floating freely or accumulating on the CPB surface. These findings demonstrate that CPB wear is governed by intricate dynamics involving mutual interactions between the solvent, slider, and CPB. By explicitly incorporating all these components, our model enables precise analysis of these interactions.
Snapshots at the end of the sliding simulation (t = 400 τ): (a) The broken polymer chains that are broken just at t = 400 τ are highlighted. (b) All polymer chain fragments that have been broken during t = 0–400 τ are highlighted. The solvent beads are not shown.
In this study, we developed a CGMD-based simulation model for the frictional behavior of CPB. This model explicitly incorporates the solvent, slider, and CPB, enabling detailed analysis of their complex wear dynamics. Under good solvent conditions, equilibration resulted in a swollen CPB film thickness of approximately 77% of the fully stretched length, which aligns well with earlier experimental studies on the swollen film thickness of CPB. To validate the model, a sliding simulation was performed. The analysis of the breakpoints indicated that the wear of the CPB can involve multiple factors, such as contact with the slider, solvent flow, and interactions between polymer chains. This finding underscores the efficacy of the developed simulation model, as it enables the analysis of both the wear caused by direct contact with the slider and that caused by solvent flow.
This work was supported by JST, CREST (Grant Number JPMJCR2193). The simulation was performed with the MAterial science Supercomputing system for Advanced MUltiscale simulations toward NExt-generation Institute for Materials Research (MASAMUNE-IMR) of the Center for Computational Materials Science, Institute for Materials Research, Tohoku University (Proposal Number 2312SC0511).
