Electrokinetic Mixing and Displacement of Charged Droplets in Hydrogels

Abstract

Mixing in droplets is an essential task in a variety of microfluidic systems. Inspired by electrokinetic mixing, electric field-induced hydrodynamic flow inside a charged droplet embedded in an unbounded polyelectrolyte hydrogel is investigated theoretically. In this study, the polyelectrolyte hydrogel is modeled as a soft, and electrically charged porous solid saturated with a salted Newtonian fluid, and the droplet is considered an incompressible Newtonian fluid. The droplet-hydrogel interface is modeled as a surface, which is located at the plane of shear, with the electrostatic potential $\zeta$. The fluid inside the droplet attains a finite velocity owing to hydrodynamic coupling with the electroosmotic flow arising from the droplet and polymer charge. The fluid velocity inside the droplet is linearly proportional to the electroosmotic flow velocity in the charged gel and the electroosmotic flow velocity beyond the electrical double layer of a charged interface. It is found that the polymer boundary condition at the droplet surface and the viscosities of the fluids inside and outside the droplet significantly modulate the interior fluid flow. The ionic strength and the permeability of the polymer network impact the flow differently depending on whether the flow arises from the droplet or polymer charge. Finally, the displacement of a charged droplet embedded in a gel under the influence of an external electric field is undertaken. This work is motivated by experimental attempts, which can register sub-nanometer-scale inclusion displacements in hydrogels, to advance electrical microrheology as a diagnostic tool for probing inclusion-hydrogel interfaces. In the absence of polymer charge, a close connection is found between the electrical response of a charged droplet when it is immobilized in an uncharged incompressible gel and when it is dispersed in a Newtonian electrolyte.

Appendices

Appendix 1: Equilibrium Base State

As there exist electrostatic interactions between the hydrogel and the droplet, the equilibrium fixed charge density $\rho ^{f\circ }$ is not constant, and the equilibrium displacement ${\varvec{v}}^\circ$ is not zero. The equations governing equilibrium (in the absence of external stimuli) are

$$\begin{aligned}&-\epsilon _\circ \epsilon _s \nabla ^2\psi ^\circ = \rho ^{m\circ } + \rho ^{f\circ }\end{aligned}$$

(92)

$$\begin{aligned}&{\varvec{u}}^\circ = \mathbf{0}\end{aligned}$$

(93)

$$\begin{aligned}&-\varvec{\nabla }p^\circ = \rho ^{m\circ } \varvec{\nabla }\psi ^\circ \end{aligned}$$

(94)

$$\begin{aligned}&\frac{\fancyscript{E}}{2(1+\nu )}\Big [ \nabla ^2{\varvec{v}}^\circ +\frac{1}{1-2\nu } \varvec{\nabla }(\varvec{\nabla } \cdot {{\varvec{v}}^\circ }) \Big ] -\rho ^{f\circ } \varvec{\nabla }\psi ^\circ = \mathbf{0}\end{aligned}$$

(95)

$$\begin{aligned}&\rho ^{f\circ } = \rho ^{f}_r (1-\varvec{\nabla } \cdot {{\varvec{v}}^\circ })\end{aligned}$$

(96)

$$\begin{aligned}&{\varvec{u}}_d^\circ =\mathbf{0}\end{aligned}$$

(97)

$$\begin{aligned}&p_d^\circ = \text {constant}\end{aligned}$$

(98)

$$\begin{aligned}&\psi _d^\circ = \text {constant}. \end{aligned}$$

(99)

The boundary conditions satisfying the fields are $\psi ^\circ = \zeta$, and ${\varvec{v}}^\circ = \mathbf{0}$ at $r=a$, and $\psi ^\circ \rightarrow 0$, ${\varvec{v}}^\circ \rightarrow \mathbf{0}$, $\rho ^{m\circ }\rightarrow -\rho ^f_r$, $\rho ^{f\circ }\rightarrow \rho ^f_r$, and $p^\circ \rightarrow 0$ as $r\rightarrow \infty$.

To obtain a general solution for the equilibrium condition, we seek an asymptotic approximation for all of the unknowns $\psi ^\circ$, $\rho ^{m\circ }$, $\rho ^{f\circ }$, ${\varvec{v}}^\circ$, and $p^\circ$. Thus,

$$\begin{aligned} \psi ^{\circ } = \psi ^{\circ }_0 + \left( \frac{e\zeta }{k_B T}\right) \psi ^{\circ }_1 + \fancyscript{O}(\zeta ^2) \end{aligned}$$

(100)

with similar expansions for $\rho ^{m\circ }$, $\rho ^{f\circ }$, ${\varvec{v}}^\circ$, and $p^\circ$. Substituting Eq. (100) into Eqs. (92)–(99), and solving the resulting equations results in

$$\begin{aligned} \psi _0^\circ = 0, {\varvec{v}}_0^\circ = \mathbf{0}\text {, }\rho ^{m\circ }_0 = -\rho ^f_r, p_0^\circ = \text {constant}\text {, and }\rho ^{f\circ }_0 = \rho ^f_r \end{aligned}$$

(101)

for zeroth-order terms, and

$$\begin{aligned}&\displaystyle \psi _1^\circ = \frac{k_B T}{e} \frac{a}{r} e^{-D(r-a)}\end{aligned}$$

(102)

$$\begin{aligned}&\displaystyle {\varvec{v}}_1^\circ = -\frac{k_B T}{e} \frac{\rho ^f_r}{\fancyscript{E}}\frac{(1+\nu )(1-2\nu )}{(1-\nu )}\left[ e^{-Dr} \frac{Dr+1}{D^2 r^2} - e^{-Da} \frac{Da+1}{D^2 r^2} \right] {\varvec{e}}_r\end{aligned}$$

(103)

$$\begin{aligned}&\displaystyle \rho ^{m\circ }_1 = - \epsilon _\circ \epsilon _s \kappa ^2 \frac{k_B T}{e} \psi _1^\circ \end{aligned}$$

(104)

$$\begin{aligned}&\displaystyle p_1^\circ = \rho ^f_r \psi _1^\circ \end{aligned}$$

(105)

$$\begin{aligned}&\displaystyle \rho ^{f\circ }_1 = - \frac{(1+\nu )(1-2\nu )}{(1-\nu )} \epsilon _\circ \epsilon _s \beta ^{2} \psi _1^\circ \end{aligned}$$

(106)

for first-order terms. Note that $D^2 = \kappa ^2+\beta ^2 (1+\nu )(1-2\nu )/(1-\nu )$, with $\beta ^2={\rho ^{f}_r}^2/(\epsilon _\circ \epsilon _s \fancyscript{E})$; $\beta ^{-1}$ is a length scale that characterizes the ratio of electrostatic repulsion force among fixed charge to the tensile elastic force. $D^{-1}$ is the effective electrical double layer thickness, which has contributions from mobile ions and fixed charges. In the limit of incompressible polymer skeleton pursued in this work, $D\rightarrow \kappa$.

Surprisingly, if the fixed charge and surface potential are like-signed, a layer with an effective thickness $D^{-1}$ forms, which has a higher polymer segment density than the bulk. Otherwise, a depletion layer with thickness $D^{-1}$ forms. This unexpected behavior can be explained by the screening effect of counter ions. If the gel and the droplet are like-signed, the net mobile charge density $\rho ^{m\circ }$ is higher than in the bulk ($=-\rho ^{f}_r$), so the electrostatic repulsion among fixed charges is more effectively screened than in the bulk. Otherwise, with oppositely signed charge, the mobile charge density is lower than in the bulk, and the screening effect is reduced.

Appendix 2: Equations Governing Zeroth and First-Order Terms

The equations prevailing zeroth-order terms in $e\zeta /(k_B T)$ are

$$\begin{aligned}&\epsilon _\circ \epsilon _s \fancyscript{L} \hat{\psi }^X_{0} + \sum _{j=1}^{M+N} z_j e \hat{n}^X_{j0} = 0 \end{aligned}$$

(107)

$$\begin{aligned}&\fancyscript{L} \hat{n}^X_{j0} + \frac{n_j^\infty e z_j}{k_B T} \fancyscript{L} \hat{\psi }^X_{0} = 0~(j=1\ldots M+N)\end{aligned}$$

(108)

$$\begin{aligned}&\left( \fancyscript{L}-\frac{1}{\ell ^{2}}\right) \fancyscript{L} \frac{dk^X_{0}}{dr} = 0 \end{aligned}$$

(109)

$$\begin{aligned}&\fancyscript{L}\fancyscript{L}\hat{\phi }^X_{0} - \frac{\rho ^f_r}{\mu } \fancyscript{L} \hat{\psi }^X_{0} = 0 \end{aligned}$$

(110)

$$\begin{aligned}&\fancyscript{L} \fancyscript{L} \frac{df^X_{0}}{dr} + \frac{1}{\ell ^2}\frac{\eta }{\mu } \fancyscript{L} \frac{dk^X_{0}}{dr} = 0\end{aligned}$$

(111)

$$\begin{aligned}&\fancyscript{L}\hat{\psi }^X_{d0} = 0 \end{aligned}$$

(112)

$$\begin{aligned}&\fancyscript{L} \fancyscript{L} \frac{dM^X_{0}}{dr} = 0\end{aligned}$$

(113)

$$\begin{aligned}&\hat{p}^X_0 = \rho ^f_r \hat{\psi }^X_0 + \frac{\eta }{\ell ^2} \frac{d}{dr}\left( r\frac{dk_0^X}{dr} \right) , \end{aligned}$$

(114)

and the equations prevailing first-order terms in $e\zeta /(k_B T)$ are

$$\begin{aligned}&\displaystyle \epsilon _\circ \epsilon _s \fancyscript{L} \hat{\psi }^X_1 + \sum _{j=1}^{M+N} z_j e \hat{n}^X_{j1} = 0 \end{aligned}$$

(115)

$$\begin{aligned}&\displaystyle \fancyscript{L} \hat{n}^X_{j1} + \frac{n_j^\infty e z_j}{k_B T} \fancyscript{L}\hat{\psi }^X_1 = \frac{n_j^\infty e z_j^2}{k_B T} \left[ \frac{dG(r)}{dr} \frac{d\hat{\psi }^X_{0}}{dr} + G(r) \fancyscript{L} \hat{\psi }^X_{0} \right] \nonumber \\&\displaystyle -z_j \Bigg [ \frac{dG(r)}{dr} \frac{d\hat{n}^X_{j0}}{dr} + \frac{1}{r^2}\frac{d}{dr}\left( r^2 \frac{dG(r)}{dr} \right) \hat{n}^X_{j0} \nonumber \\&\displaystyle -\frac{n_j^\infty }{D_j} \frac{2}{r} \frac{dG(r)}{dr} \frac{dk^X_{0}}{dr}\Bigg ]~(j=1\ldots M+N)\end{aligned}$$

(116)

$$\begin{aligned}&\displaystyle \left( \fancyscript{L}-\frac{1}{\ell ^{2}}\right) \fancyscript{L} \frac{dk^X_1}{dr} = \frac{k_B T}{e\eta } \frac{1}{r}\frac{dG(r)}{dr} \left[ \epsilon _\circ \epsilon _s \kappa ^2 \hat{\psi }^X_{0} + \sum _{j=1}^{M+N} z_j e \hat{n}^X_{j0}\right] \end{aligned}$$

(117)

$$\begin{aligned}&\displaystyle \fancyscript{L} \fancyscript{L} \hat{\phi }^X_1 - \frac{\rho ^f_r}{\mu } \fancyscript{L} \hat{\psi }^X_1 = 0\end{aligned}$$

(118)

$$\begin{aligned}&\displaystyle \fancyscript{L} \fancyscript{L} \frac{df^X_1}{dr} + \frac{1}{\ell ^2}\frac{\eta }{\mu }\fancyscript{L} \frac{dk^X_1}{dr} = 0\end{aligned}$$

(119)

$$\begin{aligned}&\displaystyle \fancyscript{L}\hat{\psi }^X_{d1} = 0\end{aligned}$$

(120)

$$\begin{aligned}&\displaystyle \fancyscript{L} \fancyscript{L} \frac{dM^X_1}{dr} = 0 \end{aligned}$$

(121)

$$\begin{aligned}&\hat{p}^X_1 = \rho ^f_r \hat{\psi }^X_1 + \frac{\eta }{\ell ^2} \frac{d}{dr}\left( r\frac{dk_1^X}{dr} \right) + \epsilon _\circ \epsilon _s \kappa ^2 \frac{k_B T}{e} G(r) \hat{\psi }^X_0. \end{aligned}$$

(122)

Appendix 3: Functional Form of the Velocity Field Outside the Droplet

For determining the velocity field outside the droplet, function $k_1^E$ is required, which is given by

$$\begin{aligned} \frac{dk_1^E}{dr}&= - u_{d1} \left[ i a e^{a/\ell } h_1^{(1)}\left( ir/\ell \right) + \frac{a\ell ^2(a/\ell +1)}{r^2} \right] \nonumber \\&+\, \epsilon _\circ \epsilon _s \frac{k_B T}{e\eta } e^{\kappa a} \kappa ^2 a^2 \frac{\ell }{96} \left( \frac{a}{r}\right) ^2 \left( \frac{\ell }{r}\right) ^4 \left[ \delta _1 + \delta _2 + \delta _3 - \delta _8 (\delta _4-\delta _5 ) + \delta _6 - \delta _7 \right] ,\nonumber \\ \end{aligned}$$

(123)

where

$$\begin{aligned} \delta _1&= 2 \left( r/\ell \right) ^3 e^{-\kappa r} \left[ \kappa ^3 r^3-\kappa ^2 r^2 + 2 \kappa r + 2 +16(r/a)^3 - \kappa ^4 r^4 e^{\kappa r} \text {E}_1(\kappa r)\right] , \end{aligned}$$

(124)

$$\begin{aligned} \delta _2&= (3/\beta _4)\left( 1-r/\ell \right) e^{-\kappa r} \Big [{\beta _{3}} {\beta _4}+16(r/a)^3(\beta _4+\kappa r)\nonumber \\&-{\beta _4}^3 {\beta _5}^2 (r/\ell )^4 e^{\beta _4 (r/\ell )} \text {E}_1\left[ \beta _4 (r/\ell )\right] \Big ],\end{aligned}$$

(125)

$$\begin{aligned} \delta _3&= 16 (r/\ell )^3 (r/a)\Big [3 e^{-\kappa a} \left( \kappa ^2 a^2+2 \kappa a + 2\right) \nonumber \\&-(a/r) e^{-\kappa r} \left[ \kappa ^2 a^2 + 2 (r/a) \left( \kappa ^2 r^2+3 \kappa r + 3\right) \right] \Big ]/(\kappa ^2 a^2),\end{aligned}$$

(126)

$$\begin{aligned} \delta _4&= (\ell /r)^4 e^{-{\beta _5} (r/ \ell )} \Big [{\beta _{2}} {\beta _5} + 16 (r/a)^3 ({\beta _5}-\kappa r)\nonumber \\&-{\beta _4}^2 {\beta _5}^3 (r/\ell )^4 e^{\beta _5 (r/ \ell )} \text {E}_1\left[ \beta _5 (r/\ell )\right] \Big ],\end{aligned}$$

(127)

$$\begin{aligned} \delta _5&= (\ell /a)^4 e^{-\beta _5 (a/ \ell )} \Big [18 {\beta _5} + 2 ({\beta _6}-10 \kappa \ell ) (a/\ell ) - {\beta _4}^2 {\beta _5} (a/\ell )^2 + {\beta _4}^2 {\beta _5}^2 (a/\ell )^3 \nonumber \\&-{\beta _4}^2 {\beta _5}^3 (a/\ell )^4 e^{\beta _5 (a/\ell )} \text {E}_1\left[ \beta _5 (a/\ell )\right] \Big ],\end{aligned}$$

(128)

$$\begin{aligned} \delta _6&= ({3}/{\beta _4}) (r/a)^4 e^{-\kappa a} \left[ -{\beta _{1}} e^{(a-r)/\ell }+2 (a/\ell )\right] \Bigg [18 {\beta _4}+2 ({\beta _6}+10 \kappa \ell )(a/\ell ) \nonumber \\&-{\beta _4} {\beta _5}^2 (a/\ell )^2 + {\beta _4}^2 {\beta _5}^2 (a/\ell )^3 -{{\beta _4}^3 {\beta _5}^2 (a/\ell )^4 e^{\beta _4 (a/\ell )} \text {E}_1\left[ \beta _4(a/\ell )\right] }\Bigg ], \end{aligned}$$

(129)

$$\begin{aligned} \delta _7&= 2 e^{- \kappa a} (r/a)^3 (r/\ell )\left[ (a/\ell )^2 + 3 (a/\ell ) + 3 -3 \beta _{1} e^{(a-r)/\ell }\right] \nonumber \\&\times \left[ 18+2 \kappa a - \kappa ^2 a^2 + \kappa ^3 a^3 - \kappa ^4 a^4 e^{\kappa a} \text {E}_1(\kappa a)\right] ,\end{aligned}$$

(130)

$$\begin{aligned} \delta _8&= 3 (\beta _{1}/\beta _5) (r/\ell )^4 e^{-r/\ell }, \end{aligned}$$

(131)

with

$$\begin{aligned} \beta _{1}&= r/\ell + 1, \end{aligned}$$

(132)

$$\begin{aligned} \beta _{2}&= \kappa ^3 r^3\!-\!\kappa ^2 r^2 \!+\! 2 \kappa r \!+\! 2 \!-\! (r/\ell )^2 (\kappa r \!+\! 1) \!+\! (r/\ell ) \left[ \kappa ^2 r^2 - 2 \kappa r -2 - (r/\ell )^2 \right] , \end{aligned}$$

(133)

$$\begin{aligned} \beta _{3}&= \kappa ^3 r^3-\kappa ^2 r^2 \!+\! 2 \kappa r \!+\! 2 \!-\! (r/\ell )^2 (\kappa r \!+\! 1) \!-\! (r/\ell ) \left[ \kappa ^2 r^2 \!-\! 2 \kappa r \!-\!2 \!-\! (r/\ell )^2 \right] ,\qquad \end{aligned}$$

(134)

$$\begin{aligned} \beta _4&= \kappa \ell + 1, \end{aligned}$$

(135)

$$\begin{aligned} \beta _5&= \kappa \ell - 1, \end{aligned}$$

(136)

$$\begin{aligned} \beta _6&= \kappa ^2 \ell ^2 + 1. \end{aligned}$$

(137)