Dynamics of polyelectrolyte solutions under a constant gradient of base concentration

Phase-separation dynamics of weakly charged polyacid solutions under a constant gradient of base concentration is studied both theoretically and numerically. The time-evolution equation of polymer volume fraction is derived by assuming that the chemical equilibrium of the dissociation reaction is locally established. Numerical simulations of the system in contact with two reservoirs in which the base concentrations differ are performed. The numerical results show that the polymer volume fraction can be transported by the concentration gradient of the base, which leads to the dynamic behavior of mesophase domain structures.

Figure 1

The growth rate ${\mathrm{\Gamma }}_q$ of the Fourier mode ${\varphi }_q$ as a function of the wave number $q$ for $\chi =0.64$ (bottom), 0.65 (middle), and 0.66 (top). Other parameter values (in dimensionless units, see next section) $N=1000, c_0=n_0=0.001, K=100, m_0=0.022$, and $\overline{\varphi }=0.17$ are fixed.

Figure 2

The stability diagram in the parameter space $(A,B)$, where $A\equiv -f_{\mathrm{FH}}^{\prime \prime }\left(\overline{\varphi }\right)/\left(C{\kappa }^2\right)$ and $B\equiv 4\pi l_{\mathrm{B}}{\left(m_0\overline{\zeta }\right)}^2/\left(C{\kappa }^4\right)$. The homogeneous state is stable for the parameters in the upper region [$B>A$ for $A\le 1$; $B>{(1+A)}^2/4$ for $A>1$]. In the lower region ($B<A$) macrophase separation occurs. In the middle (striped region) microphase separation occurs.

Figure 3

Steady-state profiles of $\varphi \left(x\right)$ (left) and $\psi \left(x\right)$ (right) for (a), (b) $\chi =0.0, c_L=c_R=0.001$ and (c), (d) for $\chi =0.0, c_L=0.003, c_R=0.001$.

Figure 4

Profiles of $\varphi \left(x\right)$ (thick lines on the left), $\zeta \left(x\right)$ (thin lines on the left) and $\psi \left(x\right)$ (thick lines on the right) at $t=4\times {10}^5$ for (a), (b) $\chi =0.65, c_L=c_R=0.001$ and (c), (d) for $\chi =0.65, c_L=0.003, c_R=0.001$.

Figure 5

Snapshots of $\varphi \left(\mathit{r}\right)$ at $t=2\times {10}^5$ for $\chi =0.65, c_L=c_R=0.001$.

Figure 6

Time evolution of the system. Snapshots of $\varphi \left(\mathit{r}\right)$ for $\chi =0.65, c_L=0.003, c_R=0.001$ at $t=1\times {10}^3, 1.2\times {10}^4, 1.3\times {10}^4, 1\times {10}^5$, and $2\times {10}^5$ (from top to bottom) are shown.

Figure 7

Phase stability diagram in $c_0-\overline{\varphi }$ plane. Left (wavy and striped) and right regions correspond to unstable and stable regions of the uniform state, respectively. In the striped region microphase separation occurs. Two arrows indicate the directions of local parameter changes in time at two different points.