Scaling features of the tribology of polymer brushes of increasing grafting density around the mushroom-to-brush transition

Nonequilibrium coarse-grained, dissipative particle dynamics simulations of complex fluids, made up of polymer brushes tethered to planar surfaces immersed in a solvent yield nonmonotonic behavior of the friction coefficient as a function of the polymer grating density on the substrates, $\mathrm{\Gamma }$, while the viscosity shows a monotonically increasing dependence on $\mathrm{\Gamma }$. This effect is shown to be independent of the degree of polymerization, $N$, and the size of the system. It arises from the composition and the structure of the first particle layer adjacent to each surface that results from the confinement of the fluid. Whenever such layers are made up of as close a proportion of polymer beads to solvent particles as there are in the fluid, the friction coefficient shows a minimum, while for disparate proportions the friction coefficient grows. At the mushroom-to-brush transition (MBT) the viscosity scales with an exponent that depends on the characteristic exponent of the MBT (6/5) and the solvent quality exponent ($\nu =0.5$, for θsolvent), but it is independent of the polymerization degree ($N$). On the other hand, the friction coefficient at the MBT scales as $\mu \sim N^{6/5}$, while the grafting density at the MBT scales as $\mathrm{\Gamma }\sim N^{-6/5}$ when friction is minimal, in agreement with previous scaling theories. We argue these aspects are the result of cooperative phenomena that have important implications for the understanding of biological brushes and the design of microfluidics devices, among other applications of current academic and industrial interest.

Figure 1

Snapshots for two different grafting densities of the fluid made up of polymer brushes: (a) $\mathrm{\Gamma }=0.2$ and (b) $\mathrm{\Gamma }=0.6$ with $N=10$; the solvent is removed for clarity. The heads of the polymers grafted to the membranes are shown in blue, the rest of the brush chains are in yellow. For both cases, $L_x=L_y=50$ and $D=5$, and the shear rate is ${\gamma }^{\prime }=0.028$, in reduced DPD units. The shear velocity $v_0$ imposed on the (blue) grafted beads on each wall is also indicated.

Figure 2

Viscosity ($\eta$, squares) and friction coefficient ($\mu$, solid circles) of polymer brushes as functions of the grafting density ($\mathrm{\Gamma }$ ), for different values of the degree of polymerization of the brushes: (a) $N=7$, (b) $N=8$, (c) $N=10$, (d) $N=12$, (e) $N=17$, and (f) $N=21$. In all cases, $L_x=L_y=D=7$, and the shear rate is ${\gamma }^{\prime }=0.028$. Error bars are smaller than the symbols’ size; dashed lines are only guides for the eye.

Figure 3

Viscosity ($\eta$, empty squares) and friction coefficient ($\mu$, solid circles) of polymer brushes as functions of the grafting density ($\mathrm{\Gamma }$), for different values of the degree of polymerization of the brushes: (a) $N=5$, (b) $N=7$, (c) $N=10$, (d) $N=20$. In all cases, $L_x=L_y=50$ and $L_z=D=7$, and the shear rate is ${\gamma }^{\prime }=0.028$. All quantities are reported in reduced units. Error bars are smaller than the symbols’ size; dashed lines are only guides for the eye.

Figure 4

Density profiles for the solvent (symbols) and brush monomers (lines) of the larger system at increasing values of the grafting density $\mathrm{\Gamma }$, for polymerization degree equal to $N=10$. The density profiles for all polymerization degrees modeled can be found in Fig. 9 in the Appendix. All quantities are reported in DPD units.

Figure 5

Viscosity $\eta$ as a function of $\varphi =N_m/N_s$ for four values of the polymerization degree, $N$. The solid line represents the fit $\eta \sim {\varphi }^{\alpha }$, where the exponent $\alpha =0.70$.

Figure 6

(a) The extrapolation length $b$ as a function of the grafting density $\mathrm{\Gamma }$ for $N=10$. The line is the best linear fit. The results for all values of $N$ modeled can be found in Fig. 10 in the Appendix. All quantities are reported in reduced units. (b) Slope $S$ (symbols) of the linear fits of $b$ vs $\mathrm{\Gamma }$, shown in Fig. 6, as a function of $N$. The solid line is the fit $S\sim N^{-\nu }$, with $\nu =0.5$.

Figure 7

(a) Scaling behavior for the extrapolation length $b$, multiplied by the polymerization degree to the 6/5 power, as a function of $\varphi$ for different values of N. All quantities are reported in reduced units. The solid line is the fit $N^{6/5}b\sim {\varphi }^{\alpha }$, with $\alpha =0.7$, as in Fig. 5. (b) Friction coefficient $\left(\mu \right)$ of polymer brushes as a function of the degree of polymerization (Ν). In all cases, $L_x=L_y=D=7$, the shear rate is $\dot{\gamma }=0.028$, and the grafting density is $\mathrm{\Gamma }=0.30$. The line is the fit $\mu \sim N^{6/5}$.

Figure 8

Grafting density at the minimum value of the friction coefficient $\left(\mu \right)$, extracted from Figs. 2 and 3, ${\mathrm{\Gamma }}_{min}$, as a function of the polymerization degree $N$. The line is the fit to the scaling appropriate for the mushroom-to-brush transition (MBT), obtained when the scaling exponent of $N$ is $-6/5$; see Eq. (2).

Figure 9

Density profiles for the solvent (symbols) and brush monomers (lines) of the larger system at increasing values of the grafting density $\mathrm{\Gamma }$, for four values of the polymerization degree, namely $N=5$ (a), 7 (b), 10 (c), and 20 (d), as indicated in each graph. All quantities are reported in DPD units.

Figure 10

The extrapolation length $b$ as a function of the grafting density $\mathrm{\Gamma }$ for different values of $N$: (a) $N=7$, (b) $N=8$, (c) $N=10$, (d) $N=12$, (e) $N=17$, and (f) $N=21$. The lines are the best linear fits. All quantities are reported in reduced units.