Curvature elasticity of a grafted polyelectrolyte brush

The curvature elasticity of a polyelectrolyte brush monolayer attached to curved surface is investigated theoretically. An analytical method based on the strong-stretching theory for a Gaussian chain is developed to calculate the elastic modulus induced by a polyelectrolyte brush. In particular, the scaling relations for the bending or Gaussian modulus with respect to system parameters related to the electrostatic interaction (degree of ionization and salt concentration) are derived. Using the numerical self-consistent-field theory, the inner structural, free-energy, and elastic moduli are computed for the polyelectrolyte brush with excluded-volume interactions. Compared to the analytical result, the curvature elasticity has a weaker dependence on the system parameters, which is attributed to the linearization for the Poisson-Boltzmann equation in the analytical treatment. Furthermore, our results are compared to the curvature elasticity of a bare charged surface, wherefrom the unique polyelectrolyte brush effect on the surface elasticity is clarified clearly. The scaling relations derived in our paper can serve as a guide to experimental studies on the related systems.

Figure 1

Structure of a polyelectrolyte brush grafted on a flat surface at different conditions. The parameters are fixed at $N=500$, $\sigma =0.1$, and $u_0=0.1$. Shown are spatial distributions of polymer segments and reduced electrostatic potential at (a) and (b) different fractions of charged segments $(\alpha =0.02,0.04,0.06,0.08,\mathrm{and}0.10)$ for a given salt concentration of $c_0=100\mathrm{mM}$ and (c) and (d) different salt concentrations $(c_0=20,50,100,200,\mathrm{and}400\mathrm{mM})$ with a given fraction $\alpha =0.10$. All lengths are expressed in units of $b$.

Figure 2

(a) Bending modulus $K$ of a polyelectrolyte brush layer as a function of the fraction $\alpha$ and grafting density $\sigma$ with $c_0=100\mathrm{mM}$ and $u_0=0.1$. After subtracting the contribution from the neutral polymer brush $K_0$, the linear relationship is found between $K-K_0$ and ${\left(\alpha \sigma \right)}^2$. The charge fraction changes as $\alpha =0.02,0.04,0.06,0.08,\mathrm{and}0.10$, while the grafting density changes as $\sigma =0.025,0.050,0.075,\mathrm{and}0.100$. The different chain lengths are displayed for $N=100,200,300,400,\mathrm{and}500$. (b) All data points in (a) are combined into a unified function of ${\alpha }^2{\sigma }^2{N}^{2.8}$. The linear fitting for $ln(K-K_0)$ and $ln\left({\alpha }^2{\sigma }^2{N}^{2.8}\right)$ produces a slope of $\sim 1.0$. (c) The Gaussian modulus of the polyelectrolyte brush $\overline{K}-{\overline{K}}_0$ also has a linear relation with ${\left(\alpha \sigma \right)}^2$. All parameters are the same as in (a). (d) The absolute value of the Gaussian modulus $|\overline{K}-{\overline{K}}_0|$ is proportional to ${\alpha }^2{\sigma }^2{N}^{2.9}$, as shown on a log-log scale. All moduli are expressed in units of $k_{B}T$.

Figure 3

Bending modulus $K-K_0$ as a function of salt concentration on a log-log scale. The salt concentration $c_0$ changes from $10\mathrm{to}10000\mathrm{mM}$. The other parameters are $\sigma =0.1$ and $u_0=0.1$. (a) The chain length changes as $N=100,200,300,400,\mathrm{and}500$ with a fixed charge fraction of $\alpha =0.10$. (b) The charge fraction changes as $\alpha =0.02,0.06,\mathrm{and}0.10$ with a chain length of $N=500$.

Figure 4

Bending modulus $K$ as a function of surface charge density $|{\sigma }_e|$. The solid line corresponds to the expression of the PB equation (72), while the dashed line corresponds to the linearized PB equation using the Debye-Hückel approximation (73).