Electrophoretic mobility and electric conductivity in suspensions of charge-regulating porous particles

Abstract

The electrophoresis and electric conduction in a homogeneous suspension of charge-regulating porous spheres such as polyelectrolytes with uniformly distributed ionogenic functional groups and hydrodynamic resistance segments in an arbitrary electrolyte solution are analyzed. The charge regulation relationship between the local fixed-charge density and electric potential because of association and dissociation reactions of the functional groups is linearized and a unit cell model allowing the overlap of the electric double layers of adjacent particles is employed to take account of the effect of particle interactions under an applied electric field. The equilibrium and perturbed electric potential distributions are ascertained by solving the linearized Poisson-Boltzmann equation, whereas the external and internal fluid velocity fields are determined by solving the modified Stokes and Brinkman equations, respectively. Explicit formulas for the mean electrophoretic mobility of the particles and the effective electric conductivity of the suspension are obtained. The variation in the concentration of the charge-determining ions in the bulk solution can cause a reversal in the direction of the electrophoretic velocity and in the difference of the electric conductivity from its reference value for a suspension of neutral porous particles. The magnitude of the electrophoretic mobility decreases with increases in the volume fraction and resistance parameter of the porous particles, but the electric conductivity in general is not a monotonic function of the particle volume fraction.

Appendix

For conciseness, some coefficients in Eqs. (22) and functions in Eqs (27) are listed here.

$$ \begin{array}{l}{C}_1=\frac{W_{\mathrm{H}}}{3{\left(\lambda a\right)}^2}\left\{-6\left[10{\left(\lambda a\right)}^2{\varphi}^2-3\left(3{\lambda}^2{a}^2+10\right){\varphi}^{5/3}-{\left(\lambda a\right)}^2\right]{I}_{\alpha }(a)+{\left(\lambda a\right)}^2\right.\left\{\left[{B}_2\right.\right.\hfill \\ {}\left.\kern7.5em +60\alpha \left(\lambda a\right)\right]{\varphi}^2-3\left[{\left(\lambda a\right)}^3 \cosh \left(\lambda a\right)-5{\left(\lambda a\right)}^2 \sinh \left(\lambda a\right)+20\lambda a \cosh \left(\lambda a\right)\right]{\varphi}^{5/3}\hfill \\ {}\left.\kern7.5em +3{\left(\lambda a\right)}^2\alpha \left(\lambda a\right){\varphi}^{1/3}-2{\left(\lambda a\right)}^3 \cosh \left(\lambda a\right)\right\}{I}_3(a)-3{\left(\lambda a\right)}^4\left\{2\alpha \left(\lambda a\right){\varphi}^2\right.\hfill \\ {}\left.\kern7.5em -\left[\lambda a \cosh \left(\lambda a\right)+ \sinh \left(\lambda a\right)\right]{\varphi}^{5/3}\right\}\left[{J}_5(a)-{\varphi}^{-5/3}{J}_0(a)\right]+3{\left(\lambda a\right)}^2\left[{B}_9{\varphi}^2\right.\hfill \\ {}\left.\left.\kern7.5em +{\left(\lambda a\right)}^2\alpha \left(\lambda a\right){\varphi}^{1/3}\right]\left[{J}_3(a)-{\varphi}^{-1/3}{J}_2(a)+3{\varphi}^{-1/3}U\right]\right\}+\frac{2}{3{\left(\lambda a\right)}^2}{I}_0(a),\hfill \end{array} $$

(A1a)

$$ \begin{array}{l}{C}_2=\frac{W_{\mathrm{H}}}{{\left(\lambda a\right)}^5}\left\{2{\left(\lambda a\right)}^7\left(1-{\varphi}^{5/3}\right)\left[3U-{J}_2(a)\right]-2{\left(\lambda a\right)}^3\left[10{\left(\lambda a\right)}^2{\varphi}^2-3\left(3{\lambda}^2{a}^2+10\right){\varphi}^{5/3}\right.\right.\hfill \\ {}\left.\kern7em -{\left(\lambda a\right)}^2\right]{I}_3(a)+2{\left(\lambda a\right)}^5\left[{\left(\lambda a\right)}^2{\varphi}^2-\left({\lambda}^2{a}^2+3\right){\varphi}^{5/3}\right]\left[{J}_5(a)-{\varphi}^{-5/3}{J}_0(a)\right]\hfill \\ {}\kern6.6em +2{\left(\lambda a\right)}^7\left({\varphi}^{1/3}-{\varphi}^2\right){J}_3(a)-2\left[{B}_6{\left(\lambda a\right)}^2{\varphi}^2-{B}_8{\varphi}^{5/3}+3{\left(\lambda a\right)}^4\beta (x){\varphi}^{1/3}\right.\hfill \\ {}\left.\left.\kern7em -{\left(\lambda a\right)}^2{B}_5\right]{I}_{\alpha }(a)\right\}+\frac{2}{{\left(\lambda a\right)}^5}{I}_{\beta }(a),\hfill \end{array} $$

(A1b)

$$ \begin{array}{c}\hfill {C}_3=\frac{W_{\mathrm{H}}}{15}\left\{5{\left(\lambda a\right)}^2{B}_3{\varphi}^{1/3}\left[3{\varphi}^{-1/3}U+{J}_3(a)-{\varphi}^{-1/3}{J}_2(a)\right]-{\left(\lambda a\right)}^2\left[3{\left(\lambda a\right)}^2\alpha \left(\lambda a\right){\varphi}^{1/3}\right.\right.\hfill \\ {}\hfill \left.\kern5.5em -{B}_1\right]{J}_5(a)-30\left[{\left(\lambda a\right)}^2{\varphi}^{1/3}-\left(3+{\lambda}^2{a}^2\right)\right]{I}_{\alpha }(a)+15{\left(\lambda a\right)}^2\left\{2\alpha \left(\lambda a\right){\varphi}^{1/3}\right.\hfill \\ {}\hfill \left.\left.\kern2.2em -\left[\lambda a \cosh \left(\lambda a\right)+ \sinh \left(\lambda a\right)\right]\right\}{I}_3(a)-\left[{\left(\lambda a\right)}^2{B}_2{\varphi}^{1/3}-{B}_4\right]{J}_0(a)\right\},\hfill \end{array} $$

(A1c)

$$ \begin{array}{l}{C}_4=\frac{W_{\mathrm{H}}}{3}\left\{-\left[{B}_4{\varphi}^{5/3}+{\left(\lambda a\right)}^2{B}_1\right]{J}_3(a)-{\left(\lambda a\right)}^2\left[{B}_2{\varphi}^{5/3}+3{\left(\lambda a\right)}^2\alpha \left(\lambda a\right)\right]\left[3U-{J}_2(a)\right]\right.\hfill \\ {}\kern4.5em +{\left(\lambda a\right)}^2{B}_3{\varphi}^{5/3}\left[{J}_5(a)-{\varphi}^{-5/3}{J}_0(a)\right]-6{\left(\lambda a\right)}^2\left(1-{\varphi}^{5/3}\right){I}_{\alpha }(a)\hfill \\ {}\left.\kern5em -3\left[{B}_9{\varphi}^{5/3}+{\left(\lambda a\right)}^2\alpha \left(\lambda a\right)\right]{I}_3(a)\right\},\hfill \end{array} $$

$$ {C}_5=-{\varphi}^{1/3}\left\{{C}_4+{W}_{\mathrm{H}}{\left(\lambda a\right)}^2\left[{B}_2{\varphi}^{5/3}+3{\left(\lambda a\right)}^2\alpha \left(\lambda a\right)\right]U\right\}+{W}_{\mathrm{H}}\left[{B}_4{\varphi}^{5/3}+{\left(\lambda a\right)}^2{B}_1\right]U, $$

(A1e)

$$ {C}_6=-{\varphi}^{5/3}{C}_3 $$

(A1f)

for the Happel model, and

$$ \begin{array}{l}{C}_1=\frac{W_{\mathrm{K}}}{3{\left(\lambda a\right)}^2}\left\{-3{\left(\lambda a\right)}^2\alpha \left(\lambda a\right)\left[2{\varphi}^2+\varphi \right]\left[{J}_5(a)+5{\varphi}^{-1}{J}_2(a)-15{\varphi}^{-1}U\right]-30\left(2{\varphi}^2\right.\right.\hfill \\ {}\left.\kern5em -\varphi -1\right){I}_{\alpha }(a)+\left\{\left[{B}_2+60\alpha \left(\lambda a\right)\right]{\varphi}^2-10{\left(\lambda a\right)}^2\left[\lambda a \cosh \left(\lambda a\right)-3 \sinh \left(\lambda a\right)\right]\varphi \right.\hfill \\ {}\left.\kern5em +18{\left(\lambda a\right)}^2\alpha \left(\lambda a\right){\varphi}^{1/3}-10{\left(\lambda a\right)}^3 \cosh \left(\lambda a\right)\right\}{I}_3(a)+3\left\{{B}_9{\varphi}^2+5{\left(\lambda a\right)}^2\left[\lambda a \cosh \left(\lambda a\right)\right.\right.\hfill \\ {}\left.\left.\kern5em + \sinh \left(\lambda a\right)\right]\varphi +6{\left(\lambda a\right)}^2\alpha \left(\lambda a\right){\varphi}^{1/3}\right\}{J}_3(a)-3\left\{{B}_9\varphi -12{\left(\lambda a\right)}^2\alpha \left(\lambda a\right){\varphi}^{1/3}\right.\hfill \\ {}\left.\kern5em +5{\left(\lambda a\right)}^2\left[\lambda a \cosh \left(\lambda a\right)+ \sinh \left(\lambda a\right)\right]\right\}\left.{J}_0(a)\right\}+\frac{2}{3{\left(\lambda a\right)}^2}{I}_0(a),\hfill \end{array} $$

(A2a)

$$ \begin{array}{l}{C}_2=\frac{W_{\mathrm{K}}}{{\left(\lambda a\right)}^5}\left\{-2{\left(\lambda a\right)}^5\left(\varphi -{\varphi}^2\right)\left[{J}_5(a)+5{\varphi}^{-1}{J}_2(a)-15{\varphi}^{-1}U\right]-10{\left(\lambda a\right)}^3\left(2{\varphi}^2-\varphi \right.\right.\hfill \\ {}\left.\kern4.5em -1\right){I}_3(a)+2{\left(\lambda a\right)}^3\left[{\left(\lambda a\right)}^2\varphi -6{\left(\lambda a\right)}^2{\varphi}^{1/3}+\left(5{\lambda}^2{a}^2+15\right)\right]{J}_0(a)-2{\left(\lambda a\right)}^3\left[{\left(\lambda a\right)}^2{\varphi}^2\right.\hfill \\ {}\left.\kern4em +5\left({\lambda}^2{a}^2+3\right)\varphi -6{\left(\lambda a\right)}^2{\varphi}^{1/3}\right]{J}_3(a)-2\left[{B}_6{\varphi}^2-10{B}_7\varphi +18{\left(\lambda a\right)}^2\beta \left(\lambda a\right){\varphi}^{1/3}\right.\hfill \\ {}\left.\left.\kern4.5em -5{B}_5\right]{I}_{\alpha }(a)\right\}+\frac{2}{{\left(\lambda a\right)}^5}{I}_{\beta }(a),\hfill \end{array} $$

(A2b)

$$ \begin{array}{l}{C}_3=\frac{W_{\mathrm{K}}}{15{\left(\lambda a\right)}^2}\left\{5{\left(\lambda a\right)}^2\left(5{B}_3-{B}_2\varphi \right)\left[3U-{J}_2(a)\right]-5\left[{B}_4\varphi -6{B}_3{\left(\lambda a\right)}^2{\varphi}^{1/3}\right]{J}_3(a)\right.\hfill \\ {}\kern5em +{\left(\lambda a\right)}^2\left[5{B}_3\varphi -18{\left(\lambda a\right)}^2\alpha \left(\lambda a\right){\varphi}^{1/3}+5{B}_1\right]{J}_5(a)+30\left[{\left(\lambda a\right)}^2\varphi -6{\left(\lambda a\right)}^2{\varphi}^{1/3}\right.\hfill \\ {}\left.\kern5em +5\left({\lambda}^2{a}^2+3\right)\right]{I}_{\alpha }(a)-15\left\{{B}_9\varphi -12{\left(\lambda a\right)}^2\alpha \left(\lambda a\right){\varphi}^{1/3}+5{\left(\lambda a\right)}^2\left[\lambda a \cosh \left(\lambda a\right)\right.\right.\hfill \\ {}\left.\left.\left.\kern5em + \sinh \left(\lambda a\right)\right]\right\}{I}_3(a)-\left[6{\left(\lambda a\right)}^2{B}_2{\varphi}^{1/3}-5{B}_4\right]{J}_0(a)\right\},\hfill \end{array} $$

(A2c)

$$ \begin{array}{l}{C}_4=\frac{W_{\mathrm{K}}}{3}\left\{-15\alpha \left(\lambda a\right)\left(2\varphi +1\right){I}_3(a)+3{\left(\lambda a\right)}^2\alpha \left(\lambda a\right)\varphi \left[{J}_5(a)+5{\varphi}^{-1}{J}_2(a)-15{\varphi}^{-1}U\right]\right.\hfill \\ {}\left.\kern4em -5\left({B}_3\varphi +{B}_1\right){J}_3(a)-30\left(1-\varphi \right){I}_{\alpha }(a)+\left({B}_2\varphi -5{B}_3\right){J}_0(a)\right\},\hfill \end{array} $$

(A2d)

$$ \begin{array}{l}{C}_5=\frac{W_{\mathrm{K}}}{3{\left(\lambda a\right)}^2}\left\{{\left(\lambda a\right)}^2\left({B}_3{\varphi}^2+{B}_1\varphi \right)\left[15{\varphi}^{-1}U-{J}_5(a)\right]-{\left(\lambda a\right)}^2\left[{B}_2{\varphi}^2-5{B}_3\varphi \right.\right.\hfill \\ {}\left.\kern6.3em +18{\left(\lambda a\right)}^2\alpha \left(\lambda a\right){\varphi}^{1/3}\right]{J}_2(a)+\left[{B}_4{\varphi}^2+6{\left(\lambda a\right)}^2{B}_1{\varphi}^{1/3}\right]{J}_3(a)\hfill \\ {}\kern6.5em -6\left[{\left(\lambda a\right)}^2{\varphi}^2+5\left({\lambda}^2{a}^2+3\right)\varphi -6{\left(\lambda a\right)}^2{\varphi}^{1/3}\right]{I}_{\alpha }(a)+3\left\{{B}_9{\varphi}^2\right.\hfill \\ {}\left.\kern6.2em +5{\left(\lambda a\right)}^2\left[\lambda a \cosh \left(\lambda a\right)+ \sinh \left(\lambda a\right)\right]\varphi +6{\left(\lambda a\right)}^2\alpha \left(\lambda a\right){\varphi}^{1/3}\right\}{I}_3(a)\hfill \\ {}\left.\kern6.5em -\left[{B}_4\varphi -6{\left(\lambda a\right)}^2{B}_3{\varphi}^{1/3}\right]{J}_0(a)\right\},\hfill \end{array} $$

(A2e)

$$ {C}_6=\frac{\varphi }{5}{C}_4 $$

(A2f)

for the Kuwabara model, where

$$ {W}_H={\left[{\left(\lambda a\right)}^2{B}_2{\varphi}^2-{B}_4{\varphi}^{5/3}+3{\left(\lambda a\right)}^4\gamma \left(\lambda a\right){\varphi}^{1/3}-{\left(\lambda a\right)}^2{B}_1\right]}^{-1}, $$

(A3a)

$$ {W}_K={\left[{B}_2{\varphi}^2-10{B}_3\varphi +18{\left(\lambda a\right)}^2\gamma \left(\lambda a\right){\varphi}^{1/3}-5{B}_1\right]}^{-1}; $$

(A3b)

$$ {B}_1=3\alpha \left(\lambda a\right)+2{\left(\lambda a\right)}^3 \cosh \left(\lambda a\right), $$

(A4a)

$$ {B}_2=2\left\{15\alpha \left(\lambda a\right)+{\left(\lambda a\right)}^2\left[\lambda a \cosh \left(\lambda a\right)-6 \sinh \left(\lambda a\right)\right]\right\}, $$

(A4b)

$$ {B}_3=6\alpha \left(\lambda a\right)+{\left(\lambda a\right)}^2\left[\lambda a \cosh \left(\lambda a\right)-3 \sinh \left(\lambda a\right)\right], $$

(A4c)

$$ \begin{array}{l}{B}_4=3\left\{30\alpha \left(\lambda a\right)+2{\left(\lambda a\right)}^2\left[7\lambda a \cosh \left(\lambda a\right)-12 \sinh \left(\lambda a\right)\right]\right.\hfill \\ {}\kern2.1em \left.+{\left(\lambda a\right)}^4\left[\lambda a \cosh \left(\lambda a\right)-5 \sinh \left(\lambda a\right)\right]\right\},\hfill \end{array} $$

(A4d)

$$ {B}_5=3\beta \left(\lambda a\right)+2{\left(\lambda a\right)}^3 \sinh \left(\lambda a\right), $$

(A4e)

$$ {B}_6=2\left\{15\beta \left(\lambda a\right)+{\left(\lambda a\right)}^2\left[\lambda a \sinh \left(\lambda a\right)-6 \cosh \left(\lambda a\right)\right]\right\}, $$

(A4f)

$$ {B}_7=6\beta \left(\lambda a\right)+{\left(\lambda a\right)}^2\left[\lambda a \sinh \left(\lambda a\right)-3 \cosh \left(\lambda a\right)\right], $$

(A4g)

$$ \begin{array}{c}\hfill {B}_8=3\left\{30\beta \left(\lambda a\right)+2{\left(\lambda a\right)}^2\left[7\lambda a \sinh \left(\lambda a\right)-12 \cosh \left(\lambda a\right)\right]\right.\hfill \\ {}\hfill \left.\kern10.8em +{\left(\lambda a\right)}^4\left[\lambda a \sinh \left(\lambda a\right)-5 \cosh \left(\lambda a\right)\right]\right\},\hfill \end{array} $$

(A4h)

$$ {B}_9=2\left\{15\alpha \left(\lambda a\right)+{\left(\lambda a\right)}^2\left[2\lambda a \cosh \left(\lambda a\right)-7 \sinh \left(\lambda a\right)\right]\right\}. $$

(A4i)