Polyelectrolyte brush bilayers in weak interpenetration regime: Scaling theory and molecular dynamics simulations

We employ molecular dynamics (MD) simulations and develop scaling theories to quantify the equilibrium behavior of polyelectrolyte (PE) brush bilayers (BBLs) in the weakly interpenetrated regime, which is characterized by $d_0<d_g<2d_0$, where $d_g$ is the gap between the opposing plates where the PE brushes are grafted and $d_0$ is the unperturbed height of a PE brush grafted at a single plate. Scaling predictions establish that, for the weakly interpenetrated osmotic PE BBLs $\delta \sim N^{1/2}{(2-d_g/d_0)}^{1/2}$ (where $\delta$ is the interpenetration length and $N$ is the number of Kuhn segments in PE brush). MD simulations excellently recover this dependence of $\delta$ on $N$ and the extent of interpenetration (quantified by $d_g/d_0$). These predictions, unlike the existing studies, establish a finite interpenetration for all values of $d_g/d_0$ as long as $d_g<2d_0$. Finally, we quantify the monomer and counterion concentration distributions and point out that these two distributions may quantitatively deviate from each other at locations very close to the channel centerline, where the interpenetration-induced monomer concentration can be significantly low.

Figure 1

Snapshots of the interpenetrating PE brushes for $N=40$ for (a) $d_g/d_0=1.75$, (b) $d_g/d_0=1.4$ and (c) $d_g/d_0=1.2$. The PE brushes are shown with yellow and red, while the counterions are shown with blue.

Figure 2

Snapshots of the interpenetrating PE brushes for $N=75$ for (a) $d_g/d_0=1.75$, (b) $d_g/d_0=1.4$ and (c) $d_g/d_0=1.2$. The PE brushes are shown with yellow and red, while the counterions are shown with blue.

Figure 3

Variation of interpenetration length $\delta$ (in units of ${\sigma }_{LJ}$) with $d_g/d_0$ for different $N$ for polymer (P) and polyelectrolyte (PE) brushes. The markers are the MD results, while the continuous lines are the scaling predictions of the form $\delta ={\delta }_{\mathrm{OB}}=C_1{N}^{1/2}{(2-d_g/d_0)}^{1/2}$ [this form of equation is obtained from Eq. (27)]. The fit is obtained for nearly constant values of $C_1$ for all the three values of $N$ [${\left(C_1\right)}_{N=75}=0.86, {\left(C_1\right)}_{N=40}=0.94, {\left(C_1\right)}_{N=20}=0.95$], establishing the rigour of the scaling law. In the inset of the figure, we provide this comparison between the MD and the scaling predictions on a log-log scale. Clear $1/2$-power dependence of $\delta$ on $2-d_g/d_0$ is evident. Also in the main figure, we provide the $\delta$ variation obtained from the phenomenological scaling prediction of Ref. [75], which states $\delta \sim {(d_g/d_0)}^{-3/2}[1-{(d_g/1.35d_0)}^2]$ (i.e., they assume that there is no IP for $d_g/d_0>1.35$).

Figure 4

Monomer concentration distribution ($c_{\mathrm{monomer}}$) for different $d_g/d_0$ values (see the legend) for (a) $N=20$, (b) $N=40$, and (c) $N=75$. $c_{\mathrm{monomer}}$ is normalized, i.e., ${\int }_0^{d_g}{c}_{\mathrm{monomer}}dy=1$. In the inset of each figure, we magnify the monomer concentration distribution near the channel centerline.

Figure 5

Counterion concentration distribution ($c_{counterion}$) for different $d_g/d_0$ values (see the legend) for (a) $N=20$, (b) $N=40$, and (c) $N=75$. $c_{counterion}$ is normalized, i.e., ${\int }_0^{d_g}{c}_{counterion}dy=1$. In the inset of each figure, we magnify the counterion concentration distribution near the channel centerline.