Nonlinear Electro-Osmosis of Uncharged Polymer Solutions with Low Ionic Strength

Abstract

Nonlinear electro-osmotic behaviour of dilute non-adsorbing polymer solutions with low salinity is investigated using Brownian dynamics simulations and a kinetic theory. In the Brownian simulations, the hydrodynamic interaction between the polymers and a no-slip wall is considered using the Rotne-Prager approximation of the Blake tensor. In a plug flow under a sufficiently strong applied electric field, the polymer migrates toward the bulk, forming a depletion layer thicker than the equilibrium one. Consequently, the electro-osmotic mobility increases nonlinearly with increasing electric field and becomes saturated. This nonlinear mobility does not depend qualitatively on the details of the rheological properties of the polymer solution. Analytical calculations using the kinetic theory for the same system quantitatively reproduce the results of the Brownian dynamics simulation well.

Appendix 3

3.1.1 3.A Derivation of the Velocity for Brownian Dynamics Simulation

In this appendix, the derivation of Eq. (3.25) is explained. The velocity field induced by the polymer is given by

$$\begin{aligned} \delta u(z)=-\frac{1}{\eta _0}\int ^z_0\sigma ^\mathrm {p}_{xz}(z)\mathrm{d}z, \end{aligned}$$

(3.55)

and the polymeric part of the stress tensor is obtained by averaging those of the microscopic expression in the lateral directions as

$$\begin{aligned} \sigma ^\mathrm {p}_{\alpha \beta }=\frac{1}{L^2}\int \mathrm{d}x\mathrm{d}y \hat{\sigma }^\mathrm {p}_{\alpha \beta }(\varvec{x}). \end{aligned}$$

(3.56)

Here the microscopic expression of the stress tensor is given by

$$\begin{aligned} \hat{\sigma }^\mathrm {p}_{\alpha \beta }(\varvec{x})=-\frac{1}{2}\sum _{n}\sum _{m\ne n}F_{nm,\alpha } x_{nm,\beta }\delta ^\mathrm {s}_{nm}(\varvec{x}), \end{aligned}$$

(3.57)

where $\varvec{F}_{nm}$ is the force exerted on the n-th bead from the m-th bead and $\delta ^\mathrm {s}_{nm}(\varvec{x})$ is the symmetrized delta function given by

$$\begin{aligned} \delta _{nm}^\mathrm {s}(\varvec{x})=\int ^1_0 \mathrm{d}s\delta (\varvec{x}-s\varvec{x}_n-(1-s)\varvec{x}_m). \end{aligned}$$

(3.58)

The symmetrized delta function is integrated in the lateral directions as

$$\begin{aligned} \bar{\delta }_{nm}^\mathrm {s}(z)= & {} \int \mathrm{d}x \mathrm{d}y \delta _{nm}^\mathrm {s}(\varvec{x})=\int ^1_0 \mathrm{d}s\delta (z-sz_n-(1-s)z_m)\nonumber \\= & {} \frac{\theta (z-z_m)-\theta (z-z_n)}{z_n-z_m}, \end{aligned}$$

(3.59)

where $\theta (z_n-z)=1-\theta (z-z_n)$. Then we obtain

$$\begin{aligned} \int ^z_0 \mathrm{d}z'\bar{\delta }^{nm}_\mathrm {S}(z')= & {} \frac{(z-z_n)\theta (z-z_n)-(z-z_m)\theta (z-z_m)}{z_m-z_n}\nonumber \\= & {} \frac{\min (z,z_n)-\min (z,z_m)}{z_n-z_m}, \end{aligned}$$

(3.60)

where $\min (z,z_n)=z\theta (z)-(z-z_n)\theta (z-z_n)$. Finally, the velocity increment is expressed by

$$\begin{aligned} \delta u(z)= & {} \frac{1}{2\eta _0 L^2}\sum _{n,m}F_{nm,1}(z_n-z_m)\int ^z_0\mathrm{d}z'\bar{\delta }_{nm}^\mathrm {s}(z')\nonumber \\= & {} \frac{1}{2\eta _0 L^2}\sum _{nm}F_{nm,1}[\min (z,z_n)-\min (z,z_m)]\nonumber \\= & {} \frac{1}{\eta _0 L^2}\sum _n\min (z,z_n)F_{n,1}. \end{aligned}$$

(3.61)

3.1.2 3.B Approximated Expressions for Kinetic Theory

3.1.2.1 3.B.1 Hookian Dumbbell

Equation (3.54) can be rewritten in a closed form for the second moment of the spring coordinates in a steady state with an imposed plug flow. The solution is given by [15]

$$\begin{aligned} \langle \varvec{q}\otimes \varvec{q}\rangle _q=\frac{k_\mathrm {B}T}{H}\left( \begin{array}{ccc} 1+2\theta ^2&{}0&{}\theta \\ 0&{}1&{}0\\ \theta &{}0&{}1 \end{array}\right) , \end{aligned}$$

(3.62)

where

$$\begin{aligned} \theta =\tau \frac{\mathrm{d}u_0}{\mathrm{d}z}=\tau \kappa \mu _0 E e^{-\kappa z}. \end{aligned}$$

(3.63)

Therefore, we have

$$\begin{aligned} \langle \varvec{q}\otimes \varvec{F}^\mathrm {s}\rangle _q=H\langle \varvec{q}\otimes \varvec{q}\rangle _q, \end{aligned}$$

(3.64)

and the polymeric stress tensor is

$$\begin{aligned} {\sigma }^\mathrm {p}= & {} c\langle \varvec{q}\otimes \varvec{F}^\mathrm {s}\rangle _q-ck_\mathrm {B}T\mathbf {I}\nonumber \\= & {} ck_\mathrm {B}T\left( \begin{array}{ccc} 2\theta ^2&{}0&{}\theta \\ 0&{}0&{}0\\ \theta &{}0&{}0 \end{array}\right) . \end{aligned}$$

(3.65)

The Kirkwood diffusion constant can be estimated by

$$\begin{aligned} \langle \mathrm {D}^\mathrm {K}_{zz}\rangle _q= & {} \frac{k_\mathrm {B}T}{12\pi \eta _0 a}\left[ 1+\frac{3a}{4}\left\langle \frac{1}{q}\left( 1+\frac{q_z^2}{q^2}\right) \right\rangle _q\right] \nonumber \\\approx & {} \frac{k_\mathrm {B}T}{12\pi \eta _0 a}\left[ 1+\frac{3a}{4}\frac{\langle q^2+q_z^2\rangle _q}{\langle q^2\rangle _q^{3/2}}\right] , \end{aligned}$$

(3.66)

where the second term is split into the second order moments, and thus, we obtain

$$\begin{aligned} \langle \mathrm{D}_{zz}^\mathrm {K}\rangle _q=\frac{k_\mathrm {B}T}{12\pi \eta _0 a}\left[ 1+\frac{3a}{4\delta }\frac{2(\theta ^2+2)}{(2\theta ^2+3)^{3/2}}\right] . \end{aligned}$$

(3.67)

It is differentiated with z as

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}z}\langle \mathrm{D}_{zz}^\mathrm {K}\rangle _q=\frac{k_\mathrm {B}T}{12\pi \eta _0 a}\frac{3a}{4\delta }\frac{4\kappa \theta ^2(\theta ^2+3)}{(2\theta ^2+3)^{5/2}}. \end{aligned}$$

(3.68)

The migration velocity can be estimated using the splitting approximation of the averages as

$$\begin{aligned} u_\mathrm {mig}(z)= & {} \frac{3k_\mathrm {B}T}{64\pi \eta _0 z^2}\left\langle \omega (q_x^2+q_y^2)-2\omega \right\rangle _q\nonumber \\\approx & {} \frac{3k_\mathrm {B}T}{32\pi \eta _0 z^2}\langle \omega \rangle _q\langle q_x^2+q_y^2-2\rangle _q\nonumber \\= & {} \frac{3k_\mathrm {B}T}{32\pi \eta _0 z^2}\langle \omega \rangle _q\theta ^2, \end{aligned}$$

(3.69)

where

$$\begin{aligned} \langle \omega \rangle _q= & {} \left\langle \left( 1+\frac{q_x^2+q_y^2}{4z^2}\right) ^{-5/2}\right\rangle _q\nonumber \\\approx & {} \left( 1+\frac{\theta ^2+1}{2z^2}\right) ^{-5/2}. \end{aligned}$$

(3.70)

3.1.2.2 3.B.2 FENE Dumbbell

The second moment of the spring coordinate for a FENE dumbbell can be obtained by pre-averaged closures of p-FENE model [14, 15]. It is given by

$$\begin{aligned} \langle \varvec{q}\otimes \varvec{q}\rangle _q=\frac{k_\mathrm {B}T}{H}\frac{\varTheta }{\theta }\left( \begin{array}{ccc} 1+2\varTheta ^2&{}0&{}\varTheta \\ 0&{}1&{}0\\ \varTheta &{}0&{}1 \end{array}\right) , \end{aligned}$$

(3.71)

and

$$\begin{aligned} \langle \varvec{q}\otimes \varvec{F}\rangle _q=k_\mathrm {B}T\left( \begin{array}{ccc} 1+2\varTheta ^2&{}0&{}\varTheta \\ 0&{}1&{}0\\ \varTheta &{}0&{}1 \end{array}\right) , \end{aligned}$$

(3.72)

where

$$\begin{aligned} \varTheta =6\sqrt{\frac{3+b}{54}}\sinh \left\{ \frac{1}{3}\mathrm {arcsinh}\left[ \frac{b\theta }{108}\left( \frac{3+b}{54}\right) ^{-3/2}\right] \right\} . \end{aligned}$$

(3.73)

The polymer stress tensor is

$$\begin{aligned} {\sigma }^\mathrm {p}= & {} ck_\mathrm {B}T\left( \begin{array}{ccc} 2\varTheta ^2&{}0&{}\varTheta \\ 0&{}0&{}0\\ \varTheta &{}0&{}0 \end{array}\right) . \end{aligned}$$

(3.74)

The Kirkwood diffusion constant is

$$\begin{aligned} \langle \mathrm{D}^\mathrm {K}_{zz}\rangle _q=\frac{k_\mathrm {B}T}{12\pi \eta _0 a}\left[ 1+\frac{3a}{4\delta }\sqrt{\frac{\theta }{\varTheta }}\frac{2(\varTheta ^2+2)}{(2\varTheta ^2+3)^{3/2}}\right] , \end{aligned}$$

(3.75)

and its derivative is

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}z}\langle \mathrm{D}_{zz}^\mathrm {K}\rangle _q=\frac{k_\mathrm {B}T}{12\pi \eta _0 a}\frac{3a}{4\delta }\times \kappa \sqrt{\frac{\theta }{\varTheta }}\nonumber \\&\times \left[ \theta \frac{\mathrm{d}\varTheta }{\mathrm{d}\theta }\frac{4\varTheta (\varTheta ^2+3)}{(2\varTheta ^2+3)^{5/2}}+\left( \frac{\theta }{\varTheta }\frac{\mathrm{d}\varTheta }{\mathrm{d}\theta }-1\right) \frac{2(\varTheta ^2+2)}{(2\varTheta ^2+3)^{3/2}}\right] ,\nonumber \\ \end{aligned}$$

(3.76)

where

$$\begin{aligned} \frac{\mathrm{d}\varTheta }{\mathrm{d}\theta }= & {} 2\sqrt{\frac{b+3}{54}}\cosh \left\{ \frac{1}{3}\mathrm {arcsinh}\left[ \frac{b\theta }{108}\left( \frac{3+b}{54}\right) ^{-{3/2}}\right] \right\} \nonumber \\&\times \frac{b}{108}\left[ \left( \frac{b\theta }{108}\right) ^2+\left( \frac{b+3}{54}\right) ^3\right] ^{-1/2}. \end{aligned}$$

(3.77)

Finally the migration velocity is obtained as

$$\begin{aligned} u_\mathrm {mig}\approx \frac{3k_\mathrm {B}T}{32\pi \eta _0 z^2}\langle \omega \rangle _q\varTheta ^2, \end{aligned}$$

(3.78)

where

$$\begin{aligned} \langle \omega \rangle _q\approx \left( 1+\frac{\varTheta }{\theta }\frac{\varTheta ^2+1}{2z^2}\right) ^{-5/2}. \end{aligned}$$

(3.79)