3.1. $\zeta -$ potential, Brinkmanlet, electrophoretic mobility and their qualitative validation

The present study considers $\sigma$ as a function of $\epsilon$ . However, the surface zeta potential of the charged biocolloids must depend on $\sigma$ . The zeta potential is an essential factor for the stability of an electrolyte medium. The $\zeta$ potential on the biocolloid surface can be presented in dimensional form as

(3.5) \begin{align} \zeta = \frac {\sigma _0 a}{\varepsilon _0\varepsilon _s}\left [ \frac {1}{\kappa a + 1} + \epsilon \left ( \frac {1}{\kappa a + 1} + \frac {(\kappa a)^2 + 3\kappa a + 3}{(\kappa a)^3 + 4(\kappa a)^2 + 9\kappa a + 9}\right ) \left (\frac {(\kappa a)^2 + 2\kappa a + 2}{\kappa a + 1} + \frac {\sigma _1}{\sigma _0} \right ) \right ]\!. \end{align}

Depending on the zeta potential, a charged biocolloid can achieve faster internalisation through the surface of a targeted cell. After adsorption, the biocolloid size regulates the uptake rate through endocytosis. In biocolloid-based drug delivery for cancer treatment, biocolloid size and zeta potential determine cytotoxicity. Therefore, controlling size and zeta potential is essential for the effectiveness of nanostructures (biocolloids) in drug delivery. We discuss how the shape of a biocolloid affects its zeta potential. By controlling the shape of a biocolloidal entity, it is possible to achieve a higher surface zeta potential, which can lead to improved outcomes in biocolloid-based drug delivery. Consequently, in this study, we demonstrate that a slight deformation of the biocolloid’s shape alters its surface zeta potential. Figure 2 shows the variation of $\zeta$ potential for varying Debye lengths corresponding to different $\epsilon$ . In this context, a charged biocolloidal entity demonstrates its prolate (oblate) nature when its major (minor) axis is oriented along the $z$ -axis, which aligns with the direction of the electric field (see figure 1). It is noticed that a biocolloid with $-1\lt \epsilon \lt 0$ exhibits a higher surface $\zeta -$ potential as compared with a biocolloid with $0\lt \epsilon \lt 1$ having the same magnitude of $\epsilon$ . For a spherical biocolloid $(\epsilon =0)$ , the magnitude of $\zeta -$ potential lies between them. In other words, for aspect ratio $b/a$ greater than unity, i.e. for a more elongated biocolloid (at higher negative values of $\epsilon$ ) along the $z$ -axis, the surface $\zeta$ -potential gets enhanced. In contrast, a more flattened biocolloid perpendicular to the $z$ -axis (with a higher positive value of $\epsilon$ ) exhibits a decrease in $\zeta$ -potential. However, during electrophoretic migration within a host, this alignment cannot be externally controlled. In this context, it is evident that a biocolloid with a prolate nature exhibits smaller surface energy, making it more stable in suspension. Therefore, under an applied electric field, an oblate-nature biocolloid tends to reorient itself to achieve a state of lower surface energy. However, this behaviour arises naturally from the physical characteristics of biocolloid shape, even though the present mathematical model does not explicitly account for biocolloid reorientation. Moreover, a decrease in surface energy reduces aggregation tendencies among the biocolloids and weakens interactions with the surrounding medium. Lastly, in figure 2, the $\zeta $ -potential obtained from the present study for $\epsilon =0$ is shown to coincide with that reported in a study by Li & Hill (Reference Li and Hill2013), where electrophoresis of a charged spherical biocolloid through a Brinkman medium is discussed.

Figure 2.

Scaled $\zeta$ -potential versus scaled Debye length $\kappa a$ : with $a/\ell =0.2$ ; $\sigma _0=-0.5\,{\unicode{x03BC}}$ C m $^{-2}$ and $\sigma _1=-0.2\,{\unicode{x03BC}}$ C m $^{-2}$ for $\epsilon =-0.2$ (prolate), $\epsilon =-0.1$ (prolate), $\epsilon =0$ (spherical), $\epsilon =0.1$ (oblate), $\epsilon =0.2$ (oblate).

We examine the influence of biocolloid shape on the hydrodynamic field in the E -problem using the quantity $C_{{{E}}} = \lim _{r\rightarrow \infty }r^{2} (h_{10}+\epsilon h_{11} )$ called Brinkmanlet which quantifies electric field induced velocity and the short-range hydrodynamic screening that arises from the resistance of the polymer matrix of hydrogel (Hill Reference Hill2006a ; Hill & Ostoja-Starzewski Reference Hill and Ostoja-Starzewski2008; Li & Hill Reference Li and Hill2013; Leiderman & Olson Reference Leiderman and Olson2016). In the present context of gel electrophoresis, $C_{{{E}}}$ quantifies how the applied electric field affects flow patterns, pressure distribution and the mobility of charged biocolloids through biologically porous environments. The analytical expression for $C_{{{E}}}$ is derived as in the dimensionless form,

(3.6a) \begin{align} C_{{{E}}} &= \sigma _0 \Bigg [\mathcal{B}(\kappa a, a/\ell ,\gamma ,\rho _f/\sigma _0) \nonumber\\ &\quad -\,\mathcal{F}\!\left (\kappa a,\frac {a}{\ell },\gamma \right ) +\,\epsilon \Bigg \{ -A^{*}_{11} \left ( 3+\frac {3a}{\ell }+\frac {a^2}{\ell ^2} \right ) + \frac {1}{5} \left ( 1+\frac {a}{\ell } \right ) H_c(a) \nonumber\\ &\qquad - \frac {2}{3}\mathcal{\tilde {Q}}_{a} \left ( 1+\frac {3\ell }{a}+\frac {3\ell ^2}{a^2} \right ) -\mathcal{G}\!\left (\kappa a,\frac {a}{\ell },\gamma \right )\Bigg \}\Bigg ], \end{align}

which is valid for all possible values of $\kappa a$ and $a/\ell$ . Note that $H_c(a) = \tilde {h}^{*\,\prime \prime }_{10}(1)$ is obtained from (2.36b ) ( $\tilde {h}^{*}_{nk}$ is the scaled form of $\tilde {h}_{nk}$ , explicitly mentioned in Appendix D), and $\mathcal{B}(\kappa a, a/\ell ,\gamma ,\rho _f/\sigma _0)$ , $A^{*}_{11}$ , $\mathcal{\tilde {Q}}_{a}$ , $\mathcal{F} (\kappa a, {a}/{\ell },\gamma )$ and $\mathcal{G} (\kappa a,{a}/{\ell },\gamma )$ are explicitly given in Appendix D. Moreover, in the case of an uncharged hydrogel, i.e. $\rho _{\!f}=0$ , bulk ionic strength $I$ is solely influenced by the gel’s mobile charge ions, due to vanishing $I^{{{f}}}$ as given in (2.25d ). Hence, $C_{{{E}}}$ becomes

(3.6b) \begin{align} C_{{{E}}} &= \sigma _0 \Bigg [ \mathcal{B}(\kappa a, a/\ell ,\gamma ) - \mathcal{F}\!\left (\kappa a,\frac {a}{\ell },\gamma \right ) \nonumber\\ &\quad +\, \epsilon \, \Bigg \{ -A^{*}_{11} \left ( 3+3\frac {a}{\ell }+\frac {a^2}{\ell ^2} \right ) + \frac {1}{5} \left ( 1+\frac {a}{\ell } \right ) H_{uc}(a) \nonumber\\ &\qquad - \frac {2}{3}\mathcal{\tilde {Q}}_{a} \left ( 1+3\frac {\ell }{a}+3\frac {\ell ^2}{a^2} \right ) -\mathcal{G}\!\left (\kappa a,\frac {a}{\ell },\gamma \right )\Bigg \} \Bigg ], \end{align}

where

(3.6c) \begin{align} H_{uc}=(1+\gamma )(\kappa a)^{2}\left (\frac {\ell ^{2}}{a^{2}}-1\right )\left (1+3\frac {\ell }{a}+3\frac {\ell ^{2}}{a^{2}}\right )\!. \end{align}

Evidently, (3.6a ) and (3.6b ) are the equations that represent Brinkmanlet for a slightly deformed spherical non-uniformly charged biocolloid inclusion for charged and uncharged hydrogels, respectively. Hence, for a uniformly charged spherical biocolloid inclusion within an uncharged hydrogel ( $\rho _f=0$ ), $C_{{{E}}}$ becomes

(3.6d) \begin{align} C_{{{E}}} =\;& \frac {\sigma _0(\kappa a)^2}{\kappa a+1} \left (\frac {(a/\ell )^2}{3}+\frac {a}{\ell }+1\right ) \Bigg \{ \gamma \Bigg [ \frac {1}{3} - \frac {a}{2\ell } + \frac {\kappa a}{3} + \left (\frac {a}{\ell }\right )^2 - \frac {(\kappa a)^2}{3} - \frac {\kappa a}{2}\frac {a}{\ell } \Bigg ] \notag\\ &\qquad + \gamma E_1(\kappa a)\exp (\kappa a) \Bigg [ 1-\left (\frac {a}{\ell }\right )^3 -\frac {(\kappa a)^3}{3} +\frac {(\kappa a)^2}{2}\left (\frac {a}{\ell }\right ) \Bigg ] \notag \\ &\qquad + \exp \left (\frac {\kappa a((a/\ell )+1)}{a/\ell }\right ) E_1\!\left (\frac {\kappa a((a/\ell )+1)}{a/\ell }\right ) \Bigg [ -\frac {\kappa a}{a/\ell } +\left (\frac {a}{\ell }\right )^2\kappa a +\left (\frac {a}{\ell }\right )^3\gamma +1 \Bigg ]\Bigg \}. \end{align}

We compare the analytical expression $C_{{{E}}}$ obtained in the present study, valid for $\kappa a$ lying in the range (0.01, 10), as mentioned earlier, with the corresponding expressions reported by Hill (Reference Hill2006a ) and Doane et al. (Reference Doane, Cheng, Babar, Hill and Burda2010), which are given as follows:

(3.7a) \begin{align} C_{{{E}}} = \frac {3}{2} \zeta \frac {\left (a/\ell \right )} {\left (\kappa a + a/\ell \right )}, \end{align}

valid for $\kappa a \gg 1$ , $a/\ell \gg 1$ and the low surface $\zeta -$ potential, and

(3.7b) \begin{align} C_{{{E}}} = \frac {\zeta }{\kappa a} -\frac {3}{2}(\kappa a) \left ( \rho _{\!f} \right ) \left \{ \left (\frac {\ell }{a}\right )^{2} +\left (\frac {\ell }{a}\right ) +\frac {1}{3} \right \}\!, \end{align}

which is valid for $\kappa a \ll 1$ and $\kappa \ell \ll 1$ . The graphical results in figure 3(a) depict a comparison between the study by Hill (Reference Hill2006a ) with our computed result in a qualitative way. The obtained result given by (3.6d ) is valid for a finite range of Debye screening length lying in the range of (0.01,10). To validate this result, it is compared with the result given by (3.7a ) for $\kappa \ell \gg 1$ and $\kappa a \gg 1$ . Slight quantitative deviations between the Brinkmanlet coefficient ( $C_{{{E}}}$ ) obtained in the present model and the same obtained by Hill (Reference Hill2006a ) arise from higher-order corrections captured in this present model beyond the leading-order thin-layer treatment. These differences become negligible in the asymptotic limit $\kappa a \to \infty$ . In other words, the magnitude difference for the spherical biocolloid in figure 3(a) tends to diminish further for higher values of $\kappa a$ . Similarly, for figure 3(b), we compare with the result given by (3.7b ) in the $\kappa \ell \ll 1$ and $\kappa a \ll 1$ limit, and obtained that in both cases, the Brinkmanlet is influenced by biocolloid shape, i.e. the decaying of far-field flow is distorted by the biocolloid geometry. In this study, $C_{{{E}}}$ is calculated for a bare, charged biocolloid suspended in a Brinkman medium. The marginal difference in magnitude given by the blue dashed line and red solid line for the spherical biocolloid given in figure 3(b) reflects the increased frictional coupling experienced by the PEGylated biocolloid (Doane et al. Reference Doane, Cheng, Babar, Hill and Burda2010) compared with the bare biocolloid.

Figure 3.

(a) Comparison of $C_{{ {E}}}$ from the current study with that obtained by Hill (Reference Hill2006a ), in which $\kappa \ell \gg 1$ and $\kappa a \gg 1$ with $\rho _{\!f}=0$ , and $\sigma _0=-0.5\,{\unicode{x03BC}}$ C m $^{-2}$ , $\sigma _1=-0.2\,{\unicode{x03BC}}$ C m $^{-2}$ . (b) Comparison of $C_{{{E}}}$ from the current study with that obtained by Doane et al. (Reference Doane, Cheng, Babar, Hill and Burda2010), where $\kappa \ell \ll 1$ and $\kappa a \ll 1$ with $\rho _{\!f}=-10^{3}$ C m $^{-5}$ , and $\sigma _0=-0.5\,{\unicode{x03BC}}$ C m $^{-2}$ , $\sigma _1=-0.2\,{\unicode{x03BC}}$ C m $^{-2}$ .

Figure 3. Long description

Panel A: A line graph compares the electrophoretic coefficient from the current study with results obtained by Hill in 2006. The x-axis represents the scaled Brinkman screening length, ranging from 10 to 100. The y-axis represents the Brinkmanlet Strength, ranging from 0 to 1. The graph includes multiple lines representing different values of a parameter epsilon (0, 0.1, -0.1) and a dashed line representing Hill’s 2006 model. Trends show that as the ratio increases, the electrophoretic coefficient also increases, with variations depending on the value of epsilon. Panel B: Another line graph compares the electrophoretic coefficient from the current study with results obtained by Doane et al. in 2010. The x-axis represents the scaled Brinkman screening length, ranging from 1 to 10. The y-axis represents the Brinkmanlet Strength, ranging from 0 to 1. The graph includes multiple lines representing different values of a parameter epsilon (0, 0.1, -0.1) and a dashed line representing Doane et al.’s 2010 model. An inset zooms in on the range from 4 to 10 on the x-axis and 0.05 to 0.35 on the y-axis. Trends show that as the ratio increases, the electrophoretic coefficient decreases, with variations depending on the value of epsilon.

The present study is consistent with the study by Ohshima (Reference Ohshima2019), which describes the electrophoretic migration of a spherical charged biocolloid in a charged hydrogel with a uniform surface charge density and accounts for long-range hydrodynamic interactions between the biocolloid and the gel. Due to the limited availability of directly comparable analytical or experimental results of the present coupled configuration concerning the electrophoresis of a slightly deformed spherical biocolloid with non-uniform surface charge density in a charged hydrogel medium, a fully quantitative validation is hardly possible. Therefore, the present results are validated through comparison with the corresponding theoretical results available for spherical biocolloids with uniform surface charge density. The observed agreement in the qualitative trend of the electrophoretic mobility provides confidence in the validity and physical consistency of the present analysis. The approximated analytical expression of the electrophoretic mobility $(\mu _{{{E}}})$ presented in the study by Ohshima (Reference Ohshima2019) (given by (75)) is compared with that obtained from the present study through figure 4(a). The variation in both electrophoretic mobilities is presented concerning $\kappa a$ within the range $[0.01, 10]$ for uniform surface charge density $\sigma _0=-0.5\,{\unicode{x03BC}}$ C m $^{-2}$ and hydrogel with $(a\ll \ell )$ . Due to the significant differences in the magnitude of electrophoretic mobility between the current study and that of Ohshima (Reference Ohshima2019), the mobility data from Ohshima (Reference Ohshima2019) are scaled to allow for a qualitative comparison with the present study without altering the trends. Ohshima (Reference Ohshima2019) derived an analytical expression for the electrophoretic mobility that involves exponential integral functions, making it cumbersome to evaluate. To address this, Ohshima (Reference Ohshima2019) proposed an approximate expression with a controlled relative error. Moreover, the analytical expression obtained in the present work does not rely on any approximation of exponential integral functions and follows directly from the governing electrokinetic equations. The complex expressions in the present model are computed in MATLAB without any approximations. The mobility expression (( $75$ ) of Ohshima Reference Ohshima2019) is itself an approximation; a direct quantitative comparison between the two results is not strictly feasible. When $\kappa a\ll 1$ , the two graphs for spherical biocolloid as given by the black dashed line (present study) and the purple solid line (Ohshima Reference Ohshima2019) coincide. Increasing $\kappa a$ value increases the deviation between the approximated solution and the complete general solution for a spherical biocolloid. Nevertheless, the qualitative agreement observed in the figure 4(a) confirms that both approaches capture the same underlying electrokinetic trends. Figure 4(a) illustrates the variation in $\mu _{{{E}}}$ obtained from the present study with respect to three values of $\epsilon = -0.1,\,0,\,0.1$ . Magnitude differences in $\mu _{{{E}}}$ are attributed to geometric variation of the biocolloid. These three mobility curves qualitatively agree with Ohshima (Reference Ohshima2019). The biocolloid geometry influences mobility through its uniform surface charge density.

Figure 4.

Scaled electrophoretic mobility as a function of $(\kappa a,\epsilon )$ when (a) $a/\ell =0.1$ , $\sigma _0=-0.5\,{\unicode{x03BC}}$ C m $^{-2}$ , $\sigma _1=0.0\,{\unicode{x03BC}}$ C m $^{-2}$ (uniform charge density) and $\rho _{\!f}=-10^{3}$ C m $^{-3}$ , and comparison with the scaled electrophoretic mobility of charged biocolloid in gel-electrophoresis as reported in the study by Ohshima (Reference Ohshima2019). (b) Scaled electrophoretic mobility as a function of $(\kappa a)$ , when $\sigma _0=-0.5\,{\unicode{x03BC}}$ C m $^{-2}$ , $\sigma _1=0.0\,{\unicode{x03BC}}$ C m $^{-2}$ (uniform charge density), and $\rho _{\!f}=0$ and comparison with the scaled electrophoretic mobility of charged spherical ( $\epsilon =0$ ) biocolloid in an uncharged gel experimentally reported in the study by Hanauer et al. (Reference Hanauer, Pierrat, Zins, Lotz and Sönnichsen2007).

When $\epsilon \lt 0$ , the biocolloid exhibits its prolate nature and can overcome the hydrogel resistance due to its elongated shape along the $z$ -direction. Moreover, the major axis of the spheroidal biocolloid is parallel to the applied electric field. It shows higher mobility compared with the case when $\epsilon \gt 0$ , where the biocolloid demonstrates its oblate nature with its minor axis parallel to the applied electric field. The mobility in the case of $\epsilon = 0$ (spherical charge biocolloid) lies between them. In all the cases presented in figure 4(a), $\mu _{{{E}}}$ decreases as $\kappa a$ increases. As $\kappa a$ rises, the ionic strength of the hydrogel’s electrolyte solution also increases. Consequently, the number of mobile counterions in the solution increases, enhancing the screening of the negatively charged biocolloid by the surrounding ions. The effective charges of the biocolloid reduce due to the counteraction of the nearby ions. As a result, the electrostatic force on the biocolloid reduces. At higher ionic strengths, the fixed charges within the gel lose their overall charge due to interactions with mobile counterions. These aligned dipoles create an opposing electric field, which in turn decreases the net electric field strength surrounding the fixed charges. Consequently, the combined impact of the electrostatic force and the hydrodynamic force resulting from the applied electric field on the biocolloid is substantially reduced, resulting in the surface geometry not affecting biocolloid mobility. However, in the case of $\kappa a\ll 1$ , the number of mobile counterions is lower, and the effective dipole moment between fixed ions and mobile counterions is reduced. Also, the screening effect on the biocolloid would be less. As a result, the net electrical force and the electric field-derived hydrodynamic force would be sufficient to move the biocolloid with higher mobility. In this case, a deformed spherical biocolloid and a spherical one can be distinguished by differences in their mobilities. When $\epsilon = -0.1$ , the biocolloid exhibits higher mobility as compared with that of $\epsilon = 0.1$ . The biocolloid mobility corresponding to $\epsilon = 0$ lies in between them. Therefore, the electrophoretic mobility variation depends significantly on the surface geometry of the biocolloid within hydrogels saturated with low-ionic-strength electrolytes.

Experimental validation. The electrophoretic mobility of a biocolloid is further validated by an experimental study that demonstrates the separation of gold and silver biocolloids by agarose gel electrophoresis based on their size and shape. In that experiment, the biocolloids are coated with a charged polymer and subsequently immersed in a $0.2\,\%$ agarose gel in $0.5\times$ TBE (Tris-borate EDTA) buffer (pH $\sim 9$ ). Hanauer et al. (Reference Hanauer, Pierrat, Zins, Lotz and Sönnichsen2007) calculated the mobility $\mu ^{{\textit{{H}}}}_{{{E}}}$ for the fixed $\zeta$ -potential using the Henry formula in dimensional form:

(3.8) \begin{align} \mu ^{H}_{E}=\frac {\varepsilon _0\varepsilon _s\zeta }{\eta }\,\left (\frac {2}{3}\mathcal{H}(\kappa a)-1\right )\!, \end{align}

where $\mathcal{H}(\kappa a)$ is the Henry correction factor accounting for the finite thickness of the Debye layer, and it is given by

(3.9) \begin{align} \mathcal{H}(\kappa a)=1+\frac {1}{2\left (1+\frac {\delta '}{\kappa a}\right )^3}, \end{align}

with

(3.10) \begin{align} \delta '=\frac {2.5}{1+2\exp (-\kappa a)}. \end{align}

In the experiment corresponding to $\kappa a \approx 5$ , the ionic strength is approximately $20$ mM, and agarose gel concentration $\lt 0.5\,\%$ , giving pore diameters in the gel ( $200{-}400\,\mathrm{nm}$ ) that are larger than the biocolloid radius. Therefore, the experimental configuration proposed by Hanauer et al. (Reference Hanauer, Pierrat, Zins, Lotz and Sönnichsen2007) provides a suitable basis for comparing and validating the mobility expression derived here for spherical biocolloids. In the experimental measurements, the biocolloid mobility is interpreted using the Henry mobility expression. At the same time, electroosmotic flow generated by the surface potential contributes primarily as a constant background mobility offset of order $\varepsilon _0\varepsilon _s\zeta /\eta$ . Therefore, to validate the mobility expression derived in the present study against the experimental results (Hanauer et al. Reference Hanauer, Pierrat, Zins, Lotz and Sönnichsen2007), a uniform surface charge density is assumed for the spherical biocolloid. To validate the result, the $\zeta$ -potential is calculated from surface charge density using the relation given by (3.5). Furthermore, for the comparison shown in figure 4(b), $\sigma _0=-0.5\,{\unicode{x03BC}}$ C m $^{-2}$ is chosen, ensuring consistency throughout the present study. The present study treats a spherical biocolloid as a bare charged biocolloid, without explicitly accounting for the polymer coating. In contrast, the experimental study involves biocolloids with a polyethylene glycol layer. The polymer coating effectively modifies the frictional coupling, which can lead to a slight deviation in the magnitude of the mobility, as observed in figure 4(b). Nevertheless, the qualitative agreement indicates that the experimental configuration closely corresponds to the theoretical framework for the uncharged gel in the Stokes limiting regime considered in the present study.

3.2. Electrostatic potential and nonuniform surface charge density

Without an external electric field, the electrostatic potential around the biocolloid is shaped by its surface charge, the fixed charges of the polymer skeleton of the hydrogel and its geometry. In the case of $\epsilon \neq 0$ , the biocolloid exhibits the highest curvature near its poles; consequently, they can accumulate surface charges densely in these regions, while the surface charges near the equator are more spread out. On the one hand, when $\epsilon = 0$ , the uniform curvature in the biocolloid geometry leads to a uniform accumulation of surface charges. Figures 5(a), 5(c) and 5(e) illustrate the electrostatic potential surrounding the biocolloid when $\kappa a \ll 1$ , corresponding to $\epsilon = 0.2$ , $\epsilon = 0.0$ and $\epsilon = -0.2$ , respectively. When $\kappa a \ll 1$ , in figures 5(c) and 5(e), the counterions are less dense, creating a diffuse layer of ions. Consequently, counterions tend to neutralise surface charges, leading to weak electrostatic screening and a high potential in the immediate vicinity of the surface. On the other hand, figures 5(b), 5(d) and 5(f) illustrate the electrostatic potential around the biocolloid for the case where $\kappa a \gg 1$ , representing values of $\epsilon = 0.2$ , $\epsilon = 0.0$ and $\epsilon = -0.2$ . From figures 5(d) and 5(f), it can be seen that increasing salt concentration in gel ( $\kappa a \gg 1$ ) can increase the neutralisation of surface charges more effectively. As a result, the strong localised screening effect can induce a comparatively small surface potential.

Figure 5.

Electrostatic potential for the combinations of scaled Debye length $\kappa a$ and deformation parameter $\epsilon$ when (a) $\kappa a\ll 1$ and $\epsilon =0.2$ (oblate shape), (b) $\kappa a\gg 1$ and $\epsilon =0.2$ (oblate shape), (c) $\kappa a\ll 1$ and $\epsilon =0$ (spherical shape), (d) $\kappa a\gg 1$ and $\epsilon =0$ (spherical shape), (e) $\kappa a\ll 1$ and $\epsilon =-0.2$ (prolate shape), (f) $\kappa a\gg 1$ and $\epsilon =-0.2$ (prolate shape), given $\sigma _0=-0.2\,{\unicode{x03BC}}$ C m $^{-2}$ , $\sigma _1=-0.1\,{\unicode{x03BC}}$ C m $^{-2}$ , $\rho _{\!f}=-10^{3}$ C m $^{-3}$ and $a\ll l$ .

Figure 5. Long description

Panel A: A heat map showing electrostatic potential for a combination of scaled Debye length and deformation parameter with an oblate shape. The X and Z axes range from -2.0 to 2.0. The color scale ranges from 0 to 3.0 times 10^-3, with red indicating higher values and blue indicating lower values. The potential is highest around the central oblate shape and decreases outward. Panel B: A heat map showing electrostatic potential for a different combination of scaled Debye length and deformation parameter with an oblate shape. The X and Z axes range from -2.0 to 2.0. The color scale ranges from 0 to 0.010, with red indicating higher values and blue indicating lower values. The potential is highest around the central oblate shape and decreases outward. Panel C: A heat map showing electrostatic potential for a combination of scaled Debye length and deformation parameter with a spherical shape. The X and Z axes range from -2.0 to 2.0. The color scale ranges from 0 to 9 times 10^-3, with red indicating higher values and blue indicating lower values. The potential is highest around the central spherical shape and decreases outward. Panel D: A heat map showing electrostatic potential for a different combination of scaled Debye length and deformation parameter with a spherical shape. The X and Z axes range from -2.0 to 2.0. The color scale ranges from 0 to 9 times 10^-3, with red indicating higher values and blue indicating lower values. The potential is highest around the central spherical shape and decreases outward. Panel E: A heat map showing electrostatic potential for a combination of scaled Debye length and deformation parameter with a prolate shape. The X and Z axes range from -2.0 to 2.0. The color scale ranges from 0 to 0.12, with red indicating higher values and blue indicating lower values. The potential is highest around the central prolate shape and decreases outward. Panel F: A heat map showing electrostatic potential for a different combination of scaled Debye length and deformation parameter with a prolate shape. The X and Z axes range from -2.0 to 2.0. The color scale ranges from 0 to 0.020, with red indicating higher values and blue indicating lower values. The potential is highest around the central prolate shape and decreases outward.

However, this scenario changes significantly in figures 5(a) and 5(b), due to the high curvature dominance around the equatorial region of biocolloids ( $\epsilon \gt 0$ ). In the case of $\kappa a \ll 1$ , thin charge clouds are observed at poles and equators. However, when $ \kappa a \gg 1$ , the charge clouds rotate by $45^\circ$ with respect to the $ z$ -axis and they become thicker. Each of the purple-, green-, indigo blue- and yellow-shaded regions has increased thickness due to variations in the salt concentration added to the hydrogel network. In the thin Debye limit, it shows a higher localised surface potential than in the thick Debye limit, though it rapidly decays beyond the Debye layer thickness. However, in figures 5(c) and 5(d), the spherical biocolloids accumulate uniform surface charge density ( $\sigma _0=-0.2\,{\unicode{x03BC}}$ C m $^{-2}$ , $\sigma _1=0.0 \,{\unicode{x03BC}}$ C m $^{-2}$ ), resulting in a uniform potential distribution irrespective of Debye layer thickness limit. In later sections, an elaborate discussion of how the non-uniformity of surface charges is influenced by the external field strength, the immobile charge in the gel and their combined effect on electrophoretic mobility is presented.

3.3. Non-uniform surface charge density and biocolloid mobility

In figure 6(a,b), the electrophoretic mobility in the dilute gel ( $a\ll \ell$ ) varies under an external electric field depending on the value of $\left |\sigma _{1} - \sigma _{0}\right |$ , where $\rho _{\!f} \sim 10^{3}$ C m $^{-3}$ . As previously stated, $\sigma _0$ represents the baseline uniform surface charge density, while $\sigma _1$ is introduced due to a perturbation in the geometry. In figure 6(a), the relative shift of surface charge density from baseline is given by $\left |\sigma _{1} - \sigma _{0}\right |$ . Now, when $\sigma _{1} - \sigma _{0}\gt 0$ , then the perturbed surface charge enhances the net charge density. The local electric field concentrates at the pole along the positive $z$ direction for the biocolloid with $\epsilon =-0.2$ , resulting in a boost in the electrostatic potential. In contrast, for $\epsilon =0.2$ , the local electric field becomes comparatively weaker near the surface region of the biocolloid. As a result of both non-uniform surface charge distribution and surface geometry variation, a significant variation in the net electrical forces (purely electrostatic force and electric field-induced hydrodynamic force) corresponding to $\epsilon =-0.2$ and $\epsilon =0.2$ occurs, which results in a marked difference in the electrophoretic mobility. However, when $\sigma _{1}-\sigma _{0}\leq 0$ , the baseline uniform surface charges $\sigma _{0}$ try to dominate the charged biocolloid movement. Both tend to lose non-uniform charge domination, resulting in a small mobility difference for $\epsilon =-0.2$ and $\epsilon =0.2$ .

Figure 6.

Scaled electrophoretic mobility as a function of $\kappa a$ and shape deformation parameter $\epsilon$ : for the various situations corresponding to non-uniform surface charge densities displayed in panels (a), (b) and (c). Three situations corresponding to different values of $\sigma _{1}/\sigma _{0}$ for a biocolloid with prolate and oblate shapes are displayed in all the figures where $\rho _{\!f}= -1\times 10^{3}$ C m $^{-3}$ .

A visual focusing on $\sigma _{1}/\sigma _{0}$ is presented in figure 6(b). For biocolloids with $\epsilon =-0.2$ and $\epsilon =0.2$ , if the baseline surface charge density is fixed to a specific value, then parameter $\sigma _{1}$ dictates whether the non-uniformity of charge distribution is amplified or suppressed. Therefore, an increase in $\sigma _{1}/\sigma _{0}$ values results in a higher difference in the mobility between the cases $\epsilon =-0.2$ and $\epsilon =0.2$ . The mobility behaviour corresponding to a ratio of $\sigma _{1}/\sigma _{0}$ up to $2$ is presented here. However, beyond $\sigma _{1}/\sigma _{0}=2$ , this phenomenon continues to persist up to a much higher magnitude of $\sigma _{1}/\sigma _{0}$ . Nevertheless, when $\sigma _{1}/\sigma _{0}=0.01$ , the influence of non-uniform surface charge distribution is very low. In this case, both biocolloids with $\epsilon =-0.2$ and $\epsilon =0.2$ carry the same effective net charge density, and when $\kappa a \geqslant 1$ (thin double layer regime), shape effects on the biocolloid’s surface potential are negligible due to electrostatic screening by the counterions in the hydrogel.

A comprehensive discussion of the sign variation associated with the non-uniform surface charge contribution, $\sigma _{1}$ , is presented based on figure 6(c). For biocolloids with $\epsilon =-0.2,\,0.2$ and $\sigma _{0}=-0.5\,{\unicode{x03BC}}$ C m $^{-2}$ , the non-uniform charge distribution parameter $\sigma _{1}$ plays a decisive role in determining the role reversal aspect of the electrophoretic mobility based on the biocolloid orientation in the applied electric field. In the case of $\epsilon =-0.2$ (prolate), when $\sigma _{1}\lt 0$ , the perturbation counteracts the baseline distribution by reducing charge density at the poles, thereby weakening the curvature-induced field amplification. However, for $\sigma _{1}\gt 0$ , the scenario changes as $\sigma _{1}$ reinforces $\sigma _{0}$ , amplifying the curvature-induced field effects. However, in the case of $\epsilon =0.2$ (oblate), the phenomenon gets reversed. Note that we consider $\sigma _{1}=0.3\,{\unicode{x03BC}}$ C m $^{-2}$ and $\sigma _{1}=-0.2\,{\unicode{x03BC}}$ C m $^{-2}$ for the first and second cases, respectively. When $\sigma _{1}=0$ , the distribution of surface charges is uniform, both biocolloids with $\epsilon =-0.2$ and $\epsilon =0.2$ carry the same effective net charge density, and when $\kappa a \geqslant 1$ (thin double layer regime), the influence of biocolloid shape on the surface potential becomes negligible, since the electrostatic screening suppresses curvature effects due to the increased counterions in the charged hydrogel. In cases of $\sigma _{1} \lessgtr 0$ , the difference in mobility between biocolloids with $\epsilon = -0.2$ and $\epsilon = 0.2$ is found to be greater than that of a biocolloid with a uniform surface charge density $(\sigma _{1} = 0)$ . However, in the $\kappa a \geqslant 1$ regime, the geometric influence on mobility becomes negligible for prolate and oblate biocolloids in the case of uniform surface charge density.

Figure 7.

Scaled electrophoretic mobility as a function of $a/\ell$ and deformation parameter $\epsilon$ , for the combinations (a) $\kappa a\ll 1$ and $\rho _{\!f}=-1\times 10^{5}$ C m $^{-3}$ high, (b) $\kappa a\ll 1$ and $\rho _{\!f}=-1\times 10^{3}$ C m $^{-3}$ low, and (c) $\kappa a\gg 1$ and $\rho _{\!f}=-1\times 10^{5}$ C m $^{-3}$ high, (d) $\kappa a\gg 1$ and $\rho _{\!f}=-1\times 10^{3}$ C m $^{-3}$ low, when $\sigma _0=-0.2\,{\unicode{x03BC}}$ C m $^{-2}$ and $\sigma _1=-0.1\,{\unicode{x03BC}}$ C m $^{-2}$ .

Figure 7. Long description

The image contains four line graphs labeled (a), (b), (c), and (d), each depicting the scaled electrophoretic mobility (μE) as a function of the scaled Brinkman screening length (a/ℓ). Panel A: The line graph shows the relationship for high ρf and low ka. The x-axis represents the scaled Brinkman screening length (a/ℓ) on a logarithmic scale ranging from 0.1 to 10, and the y-axis represents the scaled electrophoretic mobility (μE) ranging from 0.4 to 1.8. Three lines are plotted for different values of ε (-0.2, 0, and 0.2), showing a decreasing trend with increasing a/ℓ. Panel B: The line graph shows the relationship for low ρf and low ka. The x-axis represents the scaled Brinkman screening length (a/ℓ) on a logarithmic scale ranging from 0.1 to 10, and the y-axis represents the scaled electrophoretic mobility (μE) ranging from 0 to 30. Three lines are plotted for different values of ε (-0.2, 0, and 0.2), showing a decreasing trend with increasing a/ℓ. Panel C: The line graph shows the relationship for high ρf and high ka. The x-axis represents the scaled Brinkman screening length (a/ℓ) on a logarithmic scale ranging from 0.1 to 10, and the y-axis represents the scaled electrophoretic mobility (μE) ranging from 1.7 to 1.9. Three lines are plotted for different values of ε (-0.2, 0, and 0.2), showing a non-monotonic trend with a minimum around a/ℓ approximately equal to 2. Panel D: The line graph shows the relationship for low ρf and high ka. The x-axis represents the scaled Brinkman screening length (a/ℓ) on a logarithmic scale ranging from 0.1 to 10, and the y-axis represents the scaled electrophoretic mobility (μE) ranging from 15 to 35. Three lines are plotted for different values of ε (-0.2, 0, and 0.2), showing a non-monotonic trend with minima around a/ℓ approximately equal to 0.38 and 1.88.

3.4. Brinkman screening length and biocolloid mobility

The electrophoretic mobility typically decreases as the biocolloid radius increases relative to the Brinkman screening length in Brinkman-type biological media, particularly for spherical charged biocolloids (Li & Hill Reference Li and Hill2013). The Brinkman screening length characterises the sharp decay of the velocity field in a Brinkman medium. In addition to electromigration of the biocolloid under an applied electric field, it experiences drag due to the surrounding motion of the hydrogel. For instance, when $ a/\ell \ll 1$ , the flow around the biocolloid is less hindered, allowing for more unrestricted movement. In contrast, when $ a/\ell \gg 1$ , the flow decays rapidly around the biocolloid, resulting in greater resistance to its motion. Therefore, this classical result indicates that the electrophoretic mobility of a biocolloid decreases as the $a/\ell$ ratio increases. As reported by Li & Hill (Reference Li and Hill2013), dimensionless scaled electrophoretic mobility for a charged spherical colloidal biocolloid is found to decrease with $a/\ell$ within a Brinkman medium through the relation

(3.11) \begin{align} \mu _{{{E}}}\sim \frac {1}{\left [1+\left (\frac {a}{\ell }\right )+\frac {1}{9}\left (\frac {a}{\ell }\right )^{2}\right ]}. \end{align}

However, in this article, we examine changes in these classical scenarios under various circumstances. We discuss the scaled mobility behaviour of the biocolloid in relation to the ratio $a/\ell$ across four scenarios: (i) $\kappa a \ll 1 \ \text{ and high}\,\rho _{\!f}$ ; (ii) $\kappa a \ll 1 \text{ and low}\,\rho _{\!f}$ ; (iii) $\kappa a \gg 1\,\text{and high}\,\rho _{\!f}$ ; and (iv) $\kappa a \gg 1\,\text{and low}\,\rho _{\!f}$ , through figure 7(a–d), respectively. Three values of $\epsilon$ are considered here: $\epsilon = -0.2 \,\text{(prolate with major axis parallel to } z\text{-axis)}$ , $\epsilon = 0 (\text{spherical})$ and $\epsilon = 0.2 \,\text{oblate with minor axis parallel to }z\text{-axis}$ . For all these values, the mobility decreases within $a/\ell \in [0.1,\,10]$ . When $\kappa a \ll 1\, \text{and}\,\rho _{\!f}$ is high, figure 7(a) illustrates the occurrence of higher and lower mobility for $\epsilon = -0.2$ and $\epsilon = 0.2$ , respectively, for $0.1\leq a/\ell \leq 2.0$ . Beyond $a/\ell = 2.0$ , this trend becomes opposite, whereas for a spherical biocolloid ( $\epsilon = 0$ ), mobility lies between the mobilities of biocolloids with $\epsilon = -0.2$ and $\epsilon = 0.2$ . Moreover, the porous nature of the gel matrix acts as a sieving medium, slowing the passage of biocolloids based on their size. Specifically, $a\lt \ell$ corresponds to larger pore sizes in gels, which provide less resistive drag force for fluid transport and, consequently, allow for greater mobility. The alignment of biocolloids $(\epsilon \lt 0)$ with their major axis aligned with the electric field minimises hindrance, resulting in higher mobility. Moreover, biocolloid mobility is hindered by the strong repulsion between the fixed charges on the gel network and the biocolloid surface charge. However, when $\kappa a \ll 1$ , the diffused ionic cloud surrounding the biocolloid becomes thinner in the case of $\epsilon = -0.2$ compared with when $\epsilon = 0.2$ . The higher fixed-charge density of the gel strengthens the electrostatic field surrounding the gel, thereby enhancing the attraction between the fixed charges and counterions in the surrounding solution. This stronger attraction leads to more significant counterion condensation near the gel’s charged surfaces, as outlined previously (Manning Reference Manning2011; Hsu & Lee Reference Hsu and Lee2012; Dey et al. Reference Dey, Raja Sekhar and Burada2018). As a result, the interaction between the ionic cloud (Debye layer) containing the counterions and the fixed charges on the gel skeleton becomes less in the former case than in the latter. Additionally, the biocolloid with $\epsilon \lt 0$ can accumulate more surface charges, interacting more with the surrounding medium’s fixed charges, which leads to a more substantial electric repulsion on the biocolloid compared with more compact shapes like $\epsilon \geq 0$ . However, as $a/\ell \gt 2.0$ , the gel-pore size is reduced. Now, due to the increase in the interaction between the counterions and the fixed charges on the gel skeleton, the mobility hindrance due to the repulsion effect between the charged biocolloid and the gel fixed charges becomes reduced. Therefore, the biocolloid with $\epsilon = 0.2$ shows higher mobility as compared with that with $\epsilon = -0.2$ , when $a/\ell \gt 2.0$ . Therefore, when $\kappa a \ll 1\, \text{and}\,\rho _{\!f}$ is high, there exists a threshold value $a/\ell =2$ such that the impact of biocolloid geometry on the mobility can be illustrated. It is also worth mentioning that the biocolloid with $\epsilon = 0.2$ should experience a larger drag as compared with $\epsilon = -0.2$ due to a larger surface area exposed towards the hydrodynamic force (induced by the applied electric field) along the $z\text{-}$ direction (i.e. the direction of the applied electric field). Overall, the discussion can be concluded through a remark that when $\kappa a \ll 1\, \text{and}\,\rho _{\!f}$ is high, the biocolloid mobility is shape dependent. Figure 7(b) shows the variation in biocolloid mobility with $a/\ell$ when $\kappa a \ll 1$ and $\rho _{\!f}$ is low, indicating that biocolloid mobility cannot be distinguished based on shape. Within the range $a/\ell \in [0.1,\,10 ]$ , the biocolloid in this context shows identical mobility corresponding to $\epsilon = -0.2,\,0,\,0.2$ . In this situation, a weak repulsion arises between the charged biocolloid and the fixed charges on the gel network, as the Debye layer surrounding the biocolloid thickens. Overall, we observe a high mobility magnitude for all these cases.

Next, we consider the situation of biocolloid mobility variation with respect to $a/\ell$ when $\kappa a \gg 1\,\text{and}\,\rho _{\!f}$ is high in figure 7(c). Due to the high concentration of added salt, the screening effect between the gel counterions and the fixed charges becomes significantly stronger, as does the screening effect between the counterions and the charged biocolloid. In this situation, the electrical force on the biocolloid drops significantly. In this case, the weak repulsion effect between the gel-fixed charges and the charged biocolloid also prevents the biocolloid mobility from being hindered. However, we observe a reduced magnitude of mobility in all cases of $\epsilon = -0.2,\,0,\,0.2$ , with the highest and lowest mobilities of the biocolloid corresponding to $\epsilon = -0.2$ and $\epsilon = 0.2$ , respectively, throughout the aforementioned range of $a/\ell$ . However, the biocolloid cannot experience a sufficient force to produce significant mobility. In such a case, the biocolloid mobility can be differentiated by the deformation parameter $\epsilon$ . However, the variation in the mobility magnitudes is relatively less with respect to $a/\ell$ . They drop slightly with increasing $a/\ell$ up to a specific value, $a/\ell \sim 5$ , after which they tend to saturate to some value.

Figure 7(d) combines the phenomena as seen through figure 7(a–c), where $\kappa a \gg 1 \text{and}\,\rho _{\!f}$ is low. The large number of counterions from the added salt can easily neutralise the low-charged gel network. Addditionally, these counterions screened the biocolloid; however, it experienced a significant drag force while moving through the hydrogel network, without experiencing any retardation due to repulsion. Hence, a high mobility magnitude is observed. In this case, we may locate three regimes of $a/\ell$ such as $[0.1,\,0.37)$ , $[0.38,\,1.87)$ and $[1.88,\,10]$ , where we can experience three variations in mobility. For instance, when $a/\ell$ is in the range $[0.1,\,0.37)$ , the biocolloid with $\epsilon = -0.2$ exhibits the highest mobility, whereas that with $\epsilon = 0.2$ displays the lowest. The case where $\epsilon = 0$ lies in between these two extremes. This situation becomes opposite in the second case when $a/\ell \in [0.38,\,1.87)$ . Again, the first situation repeats when $a/\ell \in [1.88,\,10]$ , but a larger difference in the mobility due to the variation in $\epsilon$ can be seen. The alteration in biocolloid mobility due to shape variation is primarily attributed to changes in the Brinkman screening length.

Figure 8.

Scaled electrophoretic mobility as a function of $a/\ell$ , when (a) $\kappa a\ll 1$ and (b) $\kappa a\gg 1$ , for the various combinations of deformation parameter $\epsilon$ and fixed charge density $\rho _{\!f}$ of the polymer skeleton within the gel-like media, corresponding to the scale of $-10^{3}$ C m $^{-3}$ , given $\sigma _0=-0.2\,{\unicode{x03BC}}$ C m $^{-2}$ and $\sigma _1=-0.1\,{\unicode{x03BC}}$ C m $^{-2}$ .

In the four graphical results, we examine four limiting cases of the parameters $\kappa a$ and $\rho _{\!f}$ . Next two graphical results figure 8(a–b) illustrate the behaviour of $\rho _{\!f}$ for two limiting cases: $\kappa a\ll 1$ and $\kappa a\gg 1$ . A specific fixed value of $\rho _{\!f}$ yields different mobility curves for these cases. For a wide region of $a/\ell$ in $[0.1,\,10]$ , we find similar phenomena influencing the geometry-dependent biocolloid mobility. Here, we illustrate the simultaneous influence of deformation parameter $\epsilon$ and fixed charge density $\rho _{\!f}$ for $\sigma _0=-0.2\,{\unicode{x03BC}}$ C m $^{-2}$ and $\sigma _1=-0.1\,{\unicode{x03BC}}$ C m $^{-2}$ in a thick and thin Debye layer limit. Irrespective of the ionic concentration of the hydrogel, the increase in fixed charges in the gel leads to an increase in electrostatic repulsion near the poles for both biocolloids, with $\epsilon \lt 0$ and $\epsilon \gt 0$ . In the former case, the poles accumulate a surface charge density that is screened by gel counterions. So, the net electric influence on the surface gets decayed. Similar phenomena are seen for the latter. However, when $\kappa a \ll 1$ , the screening effect becomes insignificant to distinguish the biocolloid mobility with $\epsilon = -0.2$ from that with $\epsilon = 0.2$ due to a smaller number of counterions. We observe a marginal mobility difference for biocolloids with $\epsilon = -0.2$ and $\epsilon = 0.2$ throughout the range $a/\ell$ in $[0.1,\,10]$ , attributed to the hydrodynamic influence. When the ionic concentration is high, the applied electric field can distinguish the mobilities of biocolloids, and the continuous competition between electrostatic repulsion and electrokinetic effects results in a cross-over in the mobility curves. However, the existence of crossovers becomes prominent in the case of $\rho _{\!f}=-0.5\times 10^{3}$ C m $^{-3}$ , but not in the other cases except when $a/\ell \gt 1$ . The electrohydrodynamic drag influences the biocolloid mobility in this case. The biocolloid with $\epsilon = -0.2$ exhibits better pore mobility through the charged polymer skeleton of the hydrogel in comparison with $\epsilon = 0.2$ . Even one can see the increase in pore mobility of the biocolloid irrespective of the shape with a higher scaled Brinkman screening length for $\rho _{\!f}=-0.5\times 10^{3}$ C m $^{-3}$ . Therefore, for $\kappa a\gg 1$ , the shape influence continues to affect biocolloid mobility throughout the region of $a/\ell$ in $[0.1,\,10]$ . However, for $\kappa a\ll 1$ , mobilities coincide for different values of $\rho _{\!f}$ irrespective of shape dependency.

3.5. Electrophoretic mobility and relative permittivity

To study a variety of biocolloids and hydrogel systems, we select $\lambda = \varepsilon _{p}/\varepsilon _{s}$ within the range $0 \lt \lambda \lt 10$ . This approach allows us to explore the dependence of electrophoretic mobility on dielectric constants. A detailed discussion will focus on $\epsilon$ , $a/\ell$ , $\rho _{\!f}$ , $\sigma _{0}$ and $\sigma _{1}$ in the two limiting cases of $\kappa a \ll 1$ and $\kappa a \gg 1$ , as well as for a wide range of $\kappa a$ . Figures 9(a) and 9(b) compellingly showcase the relationship between $\mu _{{{E}}}$ and the relative permittivity $\varepsilon _{p}/\varepsilon _{s}$ for different $\rho _{\!f}$ values, corresponding to $\kappa a \ll 1$ and $\kappa a \gg 1$ , respectively. At low values of $\rho _{\!f}$ , the larger ionic atmosphere is highly sensitive to the dielectric constant of biocolloids. This results in enhanced mobility trends due to curvature effects caused by deformation. In contrast, at high $\rho _{\!f}$ , the ionic environment is primarily influenced by ion-induced dipoles. In this case, mobility is marginally dependent on the biocolloid shape. However, an extensive range of relative permittivity is considered in the two above-mentioned graphical results. The shape of a biocolloid influences its mobility and the nonlinear mobility arises from variations in relative permittivity. This variation becomes significant when the ratio $\varepsilon _{p}/\varepsilon _s$ exceeds $10$ . Irrespective of ionic concentration, saturation in electrophoretic mobility is attained at a lower relative permittivity when the hydrogel’s fixed charge density is low. Here, dielectric constants modify electric fields near-surface weakly, so mobility shows dependence on $\varepsilon _{p}/\varepsilon _s$ . However, strongly charged gels provide additional electro-osmotic assistance, raising the baseline mobility relative to weakly charged gels. Geometric curvature dominates the overall mobility. Therefore, the mobility difference between the two biocolloids tends to increase with increasing fixed charge, approaching the thick- and thin-Debye-layer limits. The mobility magnitude is higher in the latter case than in the former.

Figure 9.

Scaled electrophoretic mobility as a function of the permittivity ratio $(\varepsilon _{p}/\varepsilon _{s})$ and the fixed charge density $\rho _{\!f}$ of the polymer skeleton within the gel-like media, when (a) $\kappa a \ll 1$ , (b) $\kappa a \gg 1$ , for the various combinations of deformation parameter $\epsilon$ and fixed charge density $\rho _{\!f}$ of the polymer skeleton within the gel-like media, corresponding to the scale of $-10^{3}$ C m $^{-3}$ , given $\sigma _0=-0.2\,{\unicode{x03BC}}$ C m $^{-2}$ and $\sigma _1=-0.1\,{\unicode{x03BC}}$ C m $^{-2}$ .

Figure 10.

Scaled electrophoretic mobility as a function of permittivity ratio $(\varepsilon _{p}/\varepsilon _{s})$ with non-uniform surface charge densities as described in terms of $\sigma _{1}/\sigma _{0}$ for the biocolloid with $\epsilon =-0.2$ and $\epsilon =0.2$ shapes is described for the situations (a) $\kappa a \ll 1$ , (b) $\kappa a \sim 1$ and (c) $\kappa a \gg 1$ . (d) Scaled electrophoretic mobility for various $\kappa a$ in the case of the prolate and oblate shaped biocolloid with uniform surface charge density.

Figure 10. Long description

The image contains four panels, each depicting a line graph that shows the scaled electrophoretic mobility as a function of the permittivity ratio. Panel A: This line graph shows the scaled electrophoretic mobility for biocolloids with non-uniform surface charge densities. The x-axis represents the permittivity ratio, and the y-axis represents the scaled electrophoretic mobility. The graph includes multiple lines representing different surface charge density ratios and electric field strengths. Panel B: This line graph also shows the scaled electrophoretic mobility for biocolloids with non-uniform surface charge densities but under different conditions. The x-axis represents the permittivity ratio, and the y-axis represents the scaled electrophoretic mobility. The graph includes multiple lines representing different surface charge density ratios and electric field strengths. Panel C: This line graph shows the scaled electrophoretic mobility for biocolloids with non-uniform surface charge densities under different conditions. The x-axis represents the permittivity ratio, and the y-axis represents the scaled electrophoretic mobility. The graph includes multiple lines representing different surface charge density ratios and electric field strengths. Panel D: This line graph shows the scaled electrophoretic mobility for biocolloids with uniform surface charge densities. The x-axis represents the permittivity ratio, and the y-axis represents the scaled electrophoretic mobility. The graph includes multiple lines representing different shapes of biocolloids and varying electric field strengths.

The influence of non-uniform surface charge density on biocolloid mobility, resulting from changes in dielectric constants for $\kappa a \ll 1$ , $\kappa a \sim 1$ and $\kappa a \gg 1$ , is illustrated graphically in figure 10(a–c). The non-uniformity of surface charge density is given by (2.7), while the ratio $\sigma _1/\sigma _0$ is assumed to lie within the range $(0.01,\,10)$ . In the absence of an applied electric field, dielectric constants and surface charge density satisfy a relationship at the boundary given by (2.24). Since the induced electrostatic potential is directly proportional to $\gamma ={(1-\lambda )}/{(2+\lambda )}$ , the relative permittivity directly controls both the sign and magnitude of induced multipolar corrections to the surface potential. These induced corrections amplify or suppress the exterior potential generated by any free surface non-uniformity of $\sigma$ . When $\kappa a \ll 1$ , strong nonlinear mobility trends emerge because the diffuse ionic cloud extends significantly, allowing non-uniform charge distributions to couple with the electrostatic field strongly. This pronounced nonlinearity is characteristic of the $\kappa a \ll 1$ regime. Now, increasing $\sigma _1/\sigma _0$ from 0.1 to 10, the mobility for $\epsilon =-0.2$ decreases, but for $\epsilon =0.2$ , the trend reverses, i.e. mobility increases because of the curvature influence. The difference in mobility between biocolloids with $\epsilon =-0.2$ and $\epsilon =0.2$ is significantly greater for $\sigma _1/\sigma _0=0.1$ compared with $\sigma _1/\sigma _0=10$ .

In the scenario where $\kappa a \sim 1$ (see figure 10 b), the mobility trends with varying dielectric constants for $\epsilon =-0.2$ and different ratios of $\sigma _1/\sigma _0$ show a consistent pattern. This pattern is similar to what is observed for $\epsilon =-0.2$ in figure 10(a), where $\kappa a \ll 1$ . However, for $\epsilon =0.2$ , we do not see a comparable relationship between the cases $\kappa a \ll 1$ and $\kappa a \sim 1$ . The change in $\sigma _1/\sigma _0$ becomes highly sensitive to biocolloid mobility when studying variations in medium dielectric constants, as $\varepsilon _{p}/\varepsilon _{s}$ ranges within $(0.01,\,10)$ . Here, biocolloid shape, dielectric mismatch and ionic screening all play equal roles in determining electrophoretic mobility. Specifically, in the cross-over regimes (as shown in figure 10 b), mobility trends are sensitive to parameter changes. Corresponding to $\kappa a\gg 1$ , figure 10(c) shows that the electric double layer is strongly compressed. The ionic cloud forms a thin layer that tries to neutralise the overall surface charge. We observe that mobility trends become linear with respect to the relative permittivity. The difference in biocolloid mobility for $\epsilon =-0.2$ and $\epsilon =0.2$ at $\sigma _1/\sigma _0=0.1$ becomes small because the influence of non-uniform surface charge density $\sigma _1$ on biocolloid geometry becomes negligible. Conversely, when $\sigma _1$ becomes more dominant over $\sigma _0$ , the difference in mobility for $\epsilon =-0.2$ and $\epsilon =0.2$ increases. Across these three graphs, we can see an increase in mobility magnitude as the added salt concentration increases.

In figure 10(d), we focus on the results as ionic strength increases from low to high, keeping the surface charge density uniform ( $\sigma _1=0$ ). This shows graphically that increasing salt concentration widens the difference between the mobility trends for $\epsilon =-0.2$ and $\epsilon =0.2$ . When $\kappa a\ll 1$ , the ions are diffused in the solution. Here, $\lambda = (\varepsilon _{p}/\varepsilon _{s} )$ has only a weak effect, because most of the potential drop happens in the diffuse region, not right at the near-surface region. In this region, the flow is mainly dominated by surface potential. So, due to the lack of influence of the non-uniform surface charge, the difference between the mobility graphs for both the biocolloids $\epsilon =-0.2$ and $\epsilon =0.2$ is minimal. The permittivity ratio’s influence on mobility becomes negligible in the $\kappa a\ll 1$ regime. However, increasing ionic strength makes the Debye layer more densely confined to the biocolloid surface. An increased number of counter-ions tends to neutralise the surface charges, but here the permittivity ratio $\lambda$ amplifies the mobility difference in $\kappa a\gg 1$ .

3.6. Stream function U-problem

The deformed geometry of the sphere alters the flow field, as captured through the following stream function:

(3.12) \begin{eqnarray} \psi ^{{{U}}}=-r\left (h_{10}\sin ^2{\theta }+\epsilon \left (h_{11}\sin ^2{\theta }+\frac {3}{2}h_{31}\sin ^2{\theta }(5\cos ^2{\theta }-1)\right )\right )\!. \end{eqnarray}

The streamlines around a deformed sphere (in figure 11) will deviate from those around a perfect sphere, showing complex flow patterns. Here, the scenario involves only viscous hydrodynamic drag. When the hydrogel pore size is much larger than the biocolloid radius ( $\ell \gg a$ ), the biocolloid moves more readily and is primarily influenced by viscous forces. In the absence of an electric field, the hydrodynamic force (represented through (2.53d )) contains an extra $\epsilon$ dependency term due to deformation in geometry. Since the drag depends on the projected area and curvature distribution relative to flow direction (here in the $z$ -direction), biocolloids with $-1\lt \epsilon \lt 0$ experience less drag due to their elongated shape near the poles. However, this scenario changes when $0\lt \epsilon \lt 1$ for a biocolloid whose major axis is perpendicular to the flow direction.

Figure 11.

Stream function $\psi ^{{{U}}}$ corresponding to the $\boldsymbol{U}$ -problem with $a\ll \ell$ for (a) $\epsilon =-0.3$ (prolate), (b) $\epsilon =0$ (spherical) and (c) $\epsilon =0.3$ (oblate).

The oblate shape experiences increased drag due to its larger surface area facing the flow. The negative streamline value near the pole region in figure 11(c) denotes opposite backflow flux. As a result, biocolloids with $0\lt \epsilon \lt 1$ in figure 11(c) experience higher hydrodynamic resistance compared with the biocolloids with $-1\lt \epsilon \lt 0$ in figure 11(a). Because of the symmetrical geometry of spherical biocolloids (where $\epsilon = 0$ ), the volumetric flux is distributed uniformly. Consequently, in figure 11(b), streamlines merely deform around the body without producing closed near-surface recirculation loops.

Figure 12.

Stream function $\psi ^{{{E}}}$ corresponding to the $\boldsymbol{E}$ -problem with $a\gg \ell$ for (a) $\epsilon =-0.3$ (prolate), (b) $\epsilon =0$ (spherical) and (c) $\epsilon =0.3$ (oblate) $\rho _{\!f}=-1\times 10^{3}$ C m $^{-3}$ , when $\sigma _0=-0.2\,{\unicode{x03BC}}$ C m $^{-2}$ and $\sigma _1=-0.1\,{\unicode{x03BC}}$ C m $^{-2}$ .

Figure 12. Long description

Panel A: A contour plot shows the stream function for a prolate-shaped biocolloid. The X and Z axes range from -3 to 3. The contours are labeled with numerical values indicating the stream function levels. Panel B: A contour plot displays the stream function for a spherical biocolloid. The X and Z axes range from -3 to 3. The contours are labeled with numerical values indicating the stream function levels. Panel C: A contour plot illustrates the stream function for an oblate-shaped biocolloid. The X and Z axes range from -3 to 3. The contours are labeled with numerical values indicating the stream function levels.

3.7. Stream function E-problem

When an external electric field is applied along the $z$ -direction, the motion of biocolloids depends on the accumulation of surface charges, interactions with the hydrogel and the thickness of the Debye layer. In the $\boldsymbol{E}$ -problem, the stream function around a slightly deformed sphere immersed in a charged Brinkman medium is given by

(3.13) \begin{eqnarray} \psi ^{{{E}}}=-r\left (\tilde {h}_{10}\sin ^2{\theta }+\epsilon \left (\tilde {h}_{11}\sin ^2{\theta }+\frac {3}{2} \tilde {h}_{31}\sin ^2{\theta }(5\cos ^2{\theta }-1)\right )\right )\!. \end{eqnarray}

Therefore, electro-hydrodynamic effects influence the velocity field surrounding the biocolloid. The stream function given by (3.13) accounts for both the electrostatic and hydrodynamic contributions under an external electric field. In the low-permeability charged hydrogel, $a\gg \ell$ indicates that the electric field significantly influences the electrolyte medium that saturates the hydrogel. The biocolloid with $-1\lt \epsilon \lt 0$ has significant curvature along the poles, which intensely concentrates the electric field lines, but a small tangential component because the electric field lines are perpendicular to the surface, resulting in a negligible tangential component. It drives a strong electro-hydrodynamic flow outwards from the poles, as shown in figure 12(a). Conversely, a biocolloid with $0\lt \epsilon \lt 1$ has a large lateral surface area oriented perpendicular to the applied electric field. Due to this orientation, the motion is dominated by the hydrodynamic force induced by the electric field in the region $a\gg \ell$ . The characteristics of the local volumetric flux generated by electro-hydrodynamic flow are described by the sign and magnitude of the stream function presented in (3.13). In figure 12(c), the streamlines show positive values in the equatorial region, indicating that fluid flows outward from the equator towards the poles. Therefore, increased local volumetric flux results in larger streamline magnitudes that enclose the biocolloid surface. Consequently, the intense electro-hydrodynamic flow generates strong recirculation zones. As this fluid flow converges towards the poles, the streamline pattern shows flow separation and a narrow wake behind the biocolloid. Eddy structures and back-circulation loops are present in these wakes. At the poles, the tangential electric field is weak, leading to stagnant zones where streamlines detach from the surface. In contrast, a biocolloid with $-1 \lt \epsilon \lt 0$ shows negative streamlines near the poles, indicating flux in the opposite direction. Here, the electric field is aligned with the major axis, reducing the tangential electric field strength and, therefore, weakening the net electro-hydrodynamic force. This phenomenon leads to narrower wakes with weaker eddy structures. Both shapes have stagnation zones where the flow velocity is zero and the streamlines do not touch the surface. For spherical biocolloid $\epsilon =0$ , the stream function is given by

(3.14) \begin{align} \psi ^{{{E}}}=-r\tilde {h}_{10}\sin ^2{\theta }. \end{align}

As a sphere possesses uniform curvature throughout, the higher-order terms of $\epsilon$ disappear, as shown in (3.14). Consequently, the flow patterns do not include any closed recirculation loops, as depicted in figure 12(b). Therefore, due to the symmetry of geometry, the hydrodynamic flow induced by the electric field influences the spherical biocolloid in all directions uniformly. Therefore, no eddies formed, i.e. the streamline pattern is distributed in such a way that the local fluid velocity does not reverse direction relative to the main flow.