Efficient electrochemomechanical energy conversion in nanochannels grafted with polyelectrolyte layers with pH-dependent charge density

Abstract

Nanochannels, functionalized by grafting with a layer of charged polyelectrolyte (PE), have been employed for a large number of applications such as flow control, ion sensing, ion manipulation, current rectification and nanoionic diode fabrication. Recently, we established that such PE-grafted nanochannels, often denoted as “soft” nanochannels, can be employed for highly efficient, streaming-current-induced electrochemomechanical energy conversion in the presence of a background pressure-driven transport. In this paper, we extend our calculation for the practically realizable situation when the PE layer demonstrates a pH-dependent charge density. Consideration of such pH dependence necessitates consideration of hydrogen and hydroxyl ions in the electric double layer charge distribution, cubic distribution of the monomer profile, and a PE layer-induced drag force that accounts for this given distribution of the monomer profile. Our results express a hitherto unknown dependence of the streaming electric field (or the streaming potential) and the efficiency of the resultant energy conversion on parameters such as the pH of the surrounding electrolyte and the $\hbox {pK}_{\mathrm{a}}$ of the ionizable group that ionizes to produce the PE charge—we demonstrate that increase in the pH and the PE layer thickness and decrease in the $\hbox {pK}_{\mathrm{a}}$ and the ion concentration substantially enhance the energy conversion efficiency.

Appendix: Derivation of the governing equations

The free energy can be expressed as:

$$\begin{aligned} \Delta {F}^\prime =\int \Delta f\left[ \psi ,n_{\pm },n_{\mathrm{H}^+},n_{\mathrm{OH}^-}\right] d^3{\mathbf {r}}, \end{aligned}$$

(27)

where $\Delta f$ is the free energy density, expressed as (written in expanded form and using Eq. (6) to express $n_{\mathrm{A}^-}$):

$$\begin{aligned} \Delta f &= -\frac{\epsilon _0\epsilon _r}{2}|\nabla \psi |^2+e\psi \left( n_+-n_-\right) +e\psi (n_{\mathrm{H}^+}-n_{\mathrm{OH}^-})-e\frac{K_a^\prime \gamma _a \varphi }{K_a^\prime +\quad n_{\mathrm{H}^+}}\psi \\&+\,k_BT\left[ n_+\left( \ln {\left( \frac{n_+}{n_{+,\infty }}\right) }-1\right) +n_-\left( \ln {\left( \frac{n_-}{n_{-,\infty }}\right) }-1\right) +n_{\mathrm{H}^+}\left( \ln {\left( \frac{n_{\mathrm{H}^+}}{n_{\mathrm{H}^+,\infty }}\right) }-1\right) \right. \\&\left. +\,n_{\mathrm{OH}^-}\left( \ln {\left( \frac{n_{\mathrm{OH}^-}}{n_{{\mathrm{OH}^-},\infty }}\right) }-1\right) \right] \\&\quad \quad \,[\text {for}\,-h\le y\le -h+d], \\&\Delta f=-\frac{\epsilon _0\epsilon _r}{2}|\nabla \psi |^2+e\psi \left( n_+-n_-\right) +e\psi (n_{\mathrm{H}^+}-n_{\mathrm{OH}^-}) \\&+\,k_BT\left[ n_+\left( \ln {\left( \frac{n_+}{n_{+,\infty }}\right) }-1\right) +n_-\left( \ln {\left( \frac{n_-}{n_{-,\infty }}\right) }-1\right) +n_{\mathrm{H}^+}\left( \ln {\left( \frac{n_{\mathrm{H}^+}}{n_{\mathrm{H}^+,\infty }}\right) }-1\right) \right. \\&\left. + \,n_{\mathrm{OH}^-}\left( \ln {\left( \frac{n_{\mathrm{OH}^-}}{n_{{\mathrm{OH}^-},\infty }}\right) }-1\right) \right] \\&\quad \quad \,[\text {for}\,-h+d\le y\le 0]. \end{aligned}$$

(28)

The equilibrium conditions will be obtained by minimizing Eq. (28) with respect to $\psi$, $n_+$, $n_-$, $n_{\mathrm{H}^+}$, $n_{\mathrm{OH}^-}$. Below we discuss this minimization procedure in detail.

Minimizing with respect to $\psi$ yields:

$$\begin{aligned} \frac{\delta (\Delta F^\prime )}{\delta \psi }&= 0 \Rightarrow \frac{\partial (\Delta f)}{\partial \psi }-\frac{\hbox {d}}{\hbox {d}y}\left( \frac{\partial (\Delta f)}{\partial \psi ^\prime }\right) \Rightarrow \frac{\hbox {d}^2\psi }{\hbox {d}y^2} \\&= \frac{-e(n_+-n_-)+e\frac{K_a^\prime \gamma _a \varphi }{K_a^\prime +n_{\mathrm{H}^+}}-e\left( n_{\mathrm{H}^+}-n_{\mathrm{OH}^-}\right) }{\epsilon _0\epsilon _r} ~ \\&\quad \quad [\text {for}-h\le y\le -h+d], \\ \frac{\delta (\Delta F^\prime )}{\delta \psi }&= 0\Rightarrow \frac{\partial (\Delta f)}{\partial \psi }-\frac{\hbox {d}}{\hbox {d}y}\left( \frac{\partial (\Delta f)}{\partial \psi ^\prime }\right) \Rightarrow \frac{\hbox {d}^2\psi }{\hbox {d}y^2} \\ &= \frac{-e(n_+-n_-)-e\left( n_{\mathrm{H}^+}-n_{\mathrm{OH}^-}\right) }{\epsilon _0\epsilon _r}\,~[\text {for}-h+d\le y\le 0]. \end{aligned}$$

(29)

Minimizing with respect to $n_{\pm }$ yields:

$$\begin{aligned} \frac{\delta (\Delta F^\prime )}{\delta n_\pm }=0\Rightarrow n_\pm =\left( n_{\pm ,\infty }\right) \exp {\left( \mp \frac{e\psi }{k_BT}\right) }\,[\text {for}\;y\ge -h]. \end{aligned}$$

(30)

Minimizing with respect to $n_{\mathrm{OH}^-}$ yields:

$$\begin{aligned} \frac{\delta (\Delta F^\prime )}{\delta n_{\mathrm{OH}^-}}=0\Rightarrow n_{\mathrm{OH}^-}=\left( n_{\mathrm{OH}^-,\infty }\right) \exp {\left( \frac{e\psi }{k_BT}\right) }~[\text {for}\;y\ge -h]. \end{aligned}$$

(31)

Minimizing with respect to $n_{\mathrm{H}^+}$ yields:

$$\begin{aligned}&\frac{\delta (\Delta F^\prime )}{\delta n_{\mathrm{H}^+}}=0\Rightarrow n_{\mathrm{H}^+}=\left( n_{\mathrm{H}^+,\infty }\right) \exp {\left[ -\frac{e\psi }{k_BT} \left( 1+\frac{K^\prime _a \gamma _a \varphi }{{\left( K^\prime _a+n_{\mathrm{H}^+}\right) ^2}} \right) \right] }~ \\&\quad \quad \quad \,~[\text {for}-h\le y\le -h+d], \\&\frac{\delta (\Delta F^\prime )}{\delta n_{\mathrm{H}^+}}=0\Rightarrow n_{\mathrm{H}^+}=\left( n_{\mathrm{H}^+,\infty }\right) \exp {\left( -\frac{e\psi }{k_BT}\right) }~ \\&\quad \quad \quad \,~[\text {for}-h+d\le y\le 0]. \end{aligned}$$

(32)

Equations (7–9) are the dimensionless forms of Eqs. (30, 31, 32). Equation (32) establishes that $n_{\mathrm{H}^+}$ distribution within the PE layer deviates from that predicted by the Boltzmann partitioning. This stems from the fact that the pH-dependent charge density of the PE induces a particular kind of $n_{\mathrm{H}^+}$-dependent term in the free energy functional. In virtually all the previous studies on mean-field modelling of the electrostatics of grafted PE layer with pH-dependent PE charge density, while this term was included in the free energy density, the hydrogen ion equilibrium was not obtained by minimizing this free energy density with respect to $n_{\mathrm{H}^+}$; rather in a most ad hoc and erroneous fashion, it was assumed to obey the Boltzmann distribution. Please note that we do recover the Boltzmann distribution for ${\mathrm{H}}^+$ ion concentration outside the PE layer and for $\hbox {OH}^-$ ion in the entire system. Of course, we shall have $\hbox {OH}^-$ ions deviating from Boltzmann distribution for cases where the PE is positively charged and demonstrate a $p\hbox {OH}$-dependent charge density; for that case, the $\hbox {H}^+$ ion will obey the Boltzmann distribution in the entire system. To summarize, therefore, this deviation of $\hbox {H}^+$ ion concentration from the Boltzmann distribution occurs by virtue of the fact that the PE layer demonstrate pH-dependent charging, and the equilibrium $\hbox {H}^+$ ion concentration must be obtained (something that, most erroneously, has not been done by other researchers) from minimization of the free energy change with respect to $n_{\mathrm{H}^+}$.