Electroosmotic flows with simultaneous spatio-temporal modulations in zeta potential: cases of thick electrical double layers beyond the Debye Hückel limit

Abstract

We devise a mathematical model for analyzing the effects of spatio-temporal perturbations in zeta potential on electroosmotic transport in narrow fluidic confinements, considering thick electrical double layer limits. The spatial perturbations in zeta potential may be attributed to surface charge patterning, either designed or manifested as a natural artifact of the surface inhomogeneities. The time-dependent variations in zeta potential, as considered in this work, may stem from the temporal perturbations in the bulk ionic concentrations in the end-channel reservoirs or ‘wells’. Overcoming the simplifications routinely employed in the literature, we develop here an improved analytical formalism, without imposing any constraints on the magnitude of the zeta potential. Using these solutions, we highlight the possibilities of obtaining designed rotationalities in the flow structure with simultaneous spatial variations in the zeta potential and temporal variations in the well concentrations. We show that such combinations of spatial and temporal variations, in effect, render the flow system to be capable of shedding vortex structures that are not otherwise obtainable with spatial variations in zeta potential alone.

Appendices

Appendix A: Derivation of different order perturbation terms (Table 1)

Using end-channel well modeling of nanochannel flow and under the assumption of electrically neutral end-channel wells we obtained the modified PB equation for nanochannels as

$$ \alpha \varepsilon^{2} \left( {\varepsilon_{1}^{2} \frac{{\partial^{2} \psi }}{{\partial x^{2} }} + \frac{{\partial^{2} \psi }}{{\partial y^{2} }}} \right) = \frac{{n_{ - w} \exp (\alpha \psi ) - n_{ + w} \exp ( - \alpha \psi )}}{{2n_{0} }} $$

$$ \Rightarrow \alpha \varepsilon^{2} \left( {\varepsilon_{1}^{2} \frac{{\partial^{2} \psi }}{{\partial x^{2} }} + \frac{{\partial^{2} \psi }}{{\partial y^{2} }}} \right) = \sinh (\alpha \psi ) + \varepsilon_{1} \left( {\frac{{n_{ - w}^{(1)} }}{{2n_{0} }}\exp (\alpha \psi ) - \frac{{n_{ + w}^{(1)} }}{{2n_{0} }}\exp ( - \alpha \psi )} \right) + O(\varepsilon_{1}^{2} ) $$

$$ n_{w} = n_{w}^{(0)} + \varepsilon_{1} n_{ \pm w}^{(1)} + \varepsilon_{1}^{2} n_{ \pm w}^{(2)} + \cdots , $$

$$ \psi = \psi^{(0)} + \varepsilon_{1} \psi^{(1)} + \varepsilon_{1}^{2} \psi^{(2)} + \cdots $$

$$ \alpha \varepsilon^{2} \left( {\frac{{\partial^{2} \psi^{(0)} }}{{\partial y^{2} }} + \varepsilon_{1} \frac{{\partial^{2} \psi^{(1)} }}{{\partial y^{2} }} + O(\varepsilon_{1}^{2} )} \right) = T_{1} + \varepsilon_{1} T_{2} + O(\varepsilon_{1}^{2} ) $$

$$ T_{1} = \sinh (\alpha \psi^{(0)} + \varepsilon_{1} \alpha \psi^{(1)} + \varepsilon_{1}^{2} \alpha \psi^{(2)} + \cdots ) = \sinh (\alpha \psi^{(0)} ) + \varepsilon_{1} \{ \alpha \psi^{(1)} \cosh (\alpha \psi^{(0)} )\} + O(\varepsilon_{1}^{2} ) $$

$$ \begin{aligned} T_{2} = & \left( {\frac{{n_{ - w}^{(1)} }}{{2n_{0} }}\exp (\alpha (\psi^{(0)} + \varepsilon_{1} \psi^{(1)} + \varepsilon_{1}^{2} \psi^{(2)} + \cdots )) - \frac{{n_{ + w}^{(1)} }}{{2n_{0} }}\exp ( - \alpha (\psi^{(0)} + \varepsilon_{1} \psi^{(1)} + \varepsilon_{1}^{2} \psi^{(2)} + \cdots ))} \right) \\ = & k_{2} \sinh (\alpha \psi^{(0)} + \varepsilon_{1} \alpha \psi^{(1)} + \varepsilon_{1}^{2} \alpha \psi^{(2)} + \cdots ) + k_{3} \cosh (\alpha \psi^{(0)} + \varepsilon_{1} \alpha \psi^{(1)} + \varepsilon_{1}^{2} \alpha \psi^{(2)} + \cdots ) \\ = & k_{2} \sinh (\alpha \psi^{(0)} ) + k_{3} \cosh (\alpha \psi^{(0)} ) + O(\varepsilon_{1} ) \\ \end{aligned} $$

Hence the differential equations governing the 0th and 1st order solutions are

$$ \alpha \varepsilon^{2} \left( {\frac{{\partial^{2} \psi^{(0)} }}{{\partial y^{2} }}} \right) = \sinh (\alpha \psi^{(0)} )\quad {\text{and}} $$

$$ \alpha \varepsilon^{2} \frac{{\partial^{2} \psi^{(1)} }}{{\partial y^{2} }} = \alpha \left( {\psi^{(1)} + \frac{{k_{3} }}{\alpha }} \right)\cosh (\alpha \psi^{(0)} ) + k_{2} \sinh (\alpha \psi^{(0)} )\,{\text{respectively,}} $$

where $k_{2} = \frac{{n_{ - w}^{(1)} + n_{ + w}^{(1)} }}{{2n_{w}^{(0)} }},k_{3} = \frac{{n_{ - w}^{(1)} - n_{ + w}^{(1)} }}{{2n_{w}^{(0)} }}$

Appendix B: Derivation of Eq. I (Table 1) and Eq. 16

We begin with the differential equation governing the zeroth order potential variation in thick EDL limit:

$$ \alpha \varepsilon^{2} \left( {\frac{{\partial^{2} \psi^{(0)} }}{{\partial y^{2} }}} \right) = \sinh (\alpha \psi^{(0)} ) $$

(25)

Defining $Y = \frac{y}{\varepsilon }$, we get

$$ \alpha \left( {\frac{{\partial^{2} \psi^{(0)} }}{{\partial Y^{2} }}} \right) = \sinh (\alpha \psi^{(0)} ) $$

(26)

Multiplying both sides by $\alpha \frac{{{\text{d}}\psi^{(0)} }}{{{\text{d}}Y}}$ and integrating, we get

$$ \frac{{{\text{d(}}\alpha \psi^{(0)} )}}{{{\text{d}}Y}} = \sqrt {2(\cosh (\alpha \psi^{(0)} ) - \cosh (\alpha \psi_{c}^{(0)} ))} $$

(27)

Integrating once again and using the definition of Elliptic integral of first kind given in Appendix C, we get

$$ \frac{y}{\varepsilon } = \sqrt m \left\{ {F\left( {\frac{{i\alpha \psi^{(0)} }}{2}, - m} \right) - F\left( {\frac{i\alpha }{2}, - m} \right)} \right\} $$

(28)

Let us now consider the differential equation governing the first order potential variation in thick EDL limit:

$$ \alpha \varepsilon^{2} \frac{{\partial^{2} \psi^{(1)} }}{{\partial y^{2} }} = \alpha \left( {\psi^{(1)} + \frac{{k_{3} }}{\alpha }} \right)\cosh (\alpha \psi^{(0)} ) + k_{2} \sinh (\alpha \psi^{(0)} ) $$

(29)

where $k_{2} = \frac{{n_{ - w}^{(1)} + n_{ + w}^{(1)} }}{{2n_{w}^{(0)} }},k_{3} = \frac{{n_{ - w}^{(1)} - n_{ + w}^{(1)} }}{{2n_{w}^{(0)} }}$.

Let $\psi_{1}^{(1)} = \psi^{(1)} + \frac{{k_{3} }}{\alpha }$

Since $\alpha \left( {\frac{{\partial^{2} \psi^{(0)} }}{{\partial Y^{2} }}} \right) = \sinh (\alpha \psi^{(0)} )$ and $\frac{{\partial^{3} \psi^{(0)} }}{{\partial Y^{3} }} = \cosh (\alpha \psi^{(0)} )\frac{{\partial \psi^{(0)} }}{\partial Y}$, it follows:

$$ \left( {\frac{{\partial^{2} \psi_{1}^{(1)} }}{{\partial Y^{2} }}} \right)\frac{{\partial \psi^{(0)} }}{\partial Y} = \psi_{1}^{(1)} \frac{{\partial^{3} \psi^{(0)} }}{{\partial Y^{3} }} + k_{2} \frac{{\partial \psi^{(0)} }}{\partial Y}\frac{{\partial^{2} \psi^{(0)} }}{{\partial Y^{2} }} $$

(30)

Adding $\frac{{\partial^{2} \psi^{(0)} }}{{\partial Y^{2} }}\frac{{\partial \psi_{1}^{(1)} }}{\partial Y}$ to both sides we have

$$ \left( {\frac{{\partial \psi^{(0)} }}{\partial Y}\frac{{\partial \psi_{1}^{(1)} }}{\partial Y}} \right) = \left( {{\psi_{1}^{(1)} \frac{{\partial^{2} \psi^{(0)} }}{{\partial Y^{2} }}}} \right) + \frac{{k_{2} }}{2}\left( {\frac{{\partial \psi^{(0)} }}{\partial Y}} \right)^{2} + C $$

(31)

Integrating both sides, we have

$$ \left( {\frac{{\partial \psi^{(0)} }}{\partial Y}\frac{{\partial \psi_{1}^{(1)} }}{\partial Y}} \right) = \left( {\psi_{1}^{(1)} \frac{{\partial^{2} \psi^{(0)} }}{{\partial Y^{2} }}} \right) + \frac{{k_{2} }}{2}\left( {\frac{{\partial \psi^{(0)} }}{\partial Y}} \right)^{2} + C $$

(32)

where $C = \frac{{ - \psi_{c1}^{(1)} \sinh (\alpha \psi_{c}^{(0)} )}}{\alpha }$ (utilizing the centerline boundary condition $\frac{{\partial \psi^{(0)} }}{\partial Y} = 0,\alpha \frac{{\partial^{2} \psi^{(0)} }}{{\partial Y^{2} }} = \sinh (\alpha \psi_{c}^{(0)} )$. Further, since $\alpha \frac{{\partial \psi^{(0)} }}{\partial Y} = - \sqrt {2(\cosh (\alpha \psi^{(0)} ) - \cosh (\alpha \psi_{c}^{(0)} ))}$, we have

$$ \frac{{\partial \psi_{1}^{(1)} }}{\partial Y} = \psi_{1}^{(1)} \frac{{\frac{{\partial^{2} \psi^{(0)} }}{{\partial Y^{2} }}}}{{\frac{{\partial \psi^{(0)} }}{\partial Y}}} + \frac{{k_{2} }}{2}\left( {\frac{{\partial \psi^{(0)} }}{\partial Y}} \right) - \frac{{\psi_{c1}^{(1)} \sinh (\alpha \psi_{c}^{(0)} )}}{{\alpha \frac{{\partial \psi^{(0)} }}{\partial Y}}} $$

(33)

It is important to note that the above is linear in $\psi_{1}^{(1)}$, for solving which one may employ the following integrating factor:

$$ \int { - \frac{{\frac{{\partial^{2} \psi^{(0)} }}{{\partial Y^{2} }}}}{{\frac{{\partial \psi^{(0)} }}{\partial Y}}}{\text{d}}Y = } \int { - \frac{{\sinh \left( {\alpha \psi^{(0)} } \right)}}{{\alpha \frac{{{\text{d}}\psi^{(0)} }}{{{\text{d}}Y}}}}{\text{d}}Y = \int { - \frac{{\sinh \left( {\alpha \psi^{(0)} } \right)}}{{\alpha \left( {\frac{{{\text{d}}\psi^{(0)} }}{{{\text{d}}Y}}} \right)^{2} }}{\text{d}}\psi^{(0)} } } $$

$$ = \int {\frac{{ - \alpha \sinh (\alpha \psi^{(0)} )}}{{2(\cosh (\alpha \psi^{(0)} ) - \cosh (\alpha \psi_{c}^{(0)} ))}}{\text{d}}\psi^{(0)} = } - \frac{1}{2}\ln \left(\left| {\cosh (\alpha \psi^{(0)} ) - \cosh (\alpha \psi_{c}^{(0)} )} \right|\right) $$

$$ {\text{Integrating factor}} = {\text{e}}^{{ - \frac{1}{2}\ln (\left| {\cosh (\alpha \psi^{(0)} ) - \cosh (\alpha \psi_{c}^{(0)} )} \right|)}} = \frac{1}{{\sqrt {\left| {\cosh (\alpha \psi^{(0)} ) - \cosh (\alpha \psi_{c}^{(0)} )} \right|} }} $$

(34)

Accordingly, we obtain the following solution for Eq. (33), after applying the pertinent boundary condition at y = 0:

$$ \begin{aligned} \psi^{(1)} + \frac{{k_{3} }}{\alpha } = & \left( {\zeta^{(1)} + \frac{{k_{3} }}{\alpha }} \right)\frac{{\sqrt {(\cosh (\alpha \psi^{(0)} ) - \cosh (\alpha \psi_{c}^{(0)} ))} }}{{\sqrt {(\cosh (\alpha ) - \cosh (\alpha \psi_{c}^{(0)} ))} }} - \frac{{k_{2} }}{\alpha }\frac{y}{\varepsilon \sqrt 2 }\sqrt {(\cosh (\alpha \psi^{(0)} ) - \cosh (\alpha \psi_{c}^{(0)} ))} \\ & \quad - 2\left( {\psi_{c}^{(1)} + \frac{{k_{3} }}{\alpha }} \right)\frac{{\sinh^{2} \left( {\frac{{\alpha \psi_{c}^{(0)} }}{2}} \right)}}{{\sinh (\alpha \psi_{c}^{(0)} )}}\sqrt \frac{m}{2} \left\{ {E\left( {\frac{i\alpha }{2}, - m} \right) - E\left( {\frac{{i\alpha \psi^{(0)} }}{2}, - m} \right)} \right\}\sqrt {(\cosh (\alpha \psi^{(0)} ) - \cosh (\alpha \psi_{c}^{(0)} ))} \\ & \quad + \left( {\psi_{c}^{(1)} + \frac{{k_{3} }}{\alpha }} \right)\frac{{\sinh (\alpha \psi^{(0)} )}}{{\sinh (\alpha \psi_{c}^{(0)} )}} - \left( {\psi_{c}^{(1)} + \frac{{k_{3} }}{\alpha }} \right)\frac{1}{{\sinh (\alpha \psi_{c}^{(0)} )}}\sinh (\alpha )\frac{{\sqrt {(\cosh (\alpha \psi^{(0)} ) - \cosh (\alpha \psi_{c}^{(0)} ))} }}{{\sqrt {(\cosh (\alpha ) - \cosh (\alpha \psi_{c}^{(0)} ))} }} \\ \end{aligned} $$

(35)

Appendix C: Definition of elliptic integrals for imaginary arguments

Elliptic integrals of 1st and 2nd kind are defined as

$$ \begin{gathered} F(\phi ,m) = \int\limits_{0}^{\phi } {\frac{1}{{\sqrt {1 - m\sin^{2} (t)} }}} \;{\text{d}}t \hfill \\ E(\phi ,m) = \int\limits_{0}^{\phi } {\sqrt {1 - m\sin^{2} (t)} \;} {\text{d}}t \hfill \\ \end{gathered} $$

For purely imaginary argument, the integral may be simplified as

$$ \begin{gathered} F(i\phi ,m) = i\int\limits_{0}^{\phi } {\frac{1}{{\sqrt {1 + m\sinh^{2} (t)} }}\;} {\text{d}}t \hfill \\ E(i\phi ,m) = i\int\limits_{0}^{\phi } {\sqrt {1 + m\sinh^{2} (t)} \;} {\text{d}}t \hfill \\ \end{gathered} $$

We numerically evaluate these integrals using ‘quad’ function in MATLAB.