5.1. Validation with existing experimental and theoretical studies

We have validated our algorithm by comparing with several existing theoretical and experimental studies. In figure 2(a), we compare our PNP model with the experimental study by López-Viota et al. (Reference López-Viota, Mandal, Delgado, Toca-Herrera, Möller, Zanuttin, Balestrino and Krol2009) and the theoretical study by Yeh et al. (Reference Yeh, Fang, Hsu and Tseng2011) for a soft particle with a step-like PEL ( $\nu \to 0$ ) with low surface charge density. Our results show good alignment with these studies.

Figure 2(b) shows the comparison of the radial distribution of the induced potential at equilibrium for a soft particle with a step-like coating ( $\nu \to 0$ ) obtained by the present PNPC (in which ion correlations is included in the PNP model) and MNPDC models with the corresponding results of Lesniewska et al. (Reference Lesniewska, Beaussart and Duval2023) when the $\zeta$ -potential at the core is $\zeta =-5$ . Lesniewska et al. (Reference Lesniewska, Beaussart and Duval2023) considered a constant correlation length ( $\delta _{c}=2$ ) and the boundary condition is same as (2.15). We find that our results match exactly with Lesniewska et al. (Reference Lesniewska, Beaussart and Duval2023). Overscreening of the surface charge, indicated by the sign reversal in potential, is observed in the PNPC model where the dielectric decrement and finite ion size effects are ignored. However, in the MNPDC model no coion-dominated region forms or change of sign in electric potential profile occurs. This is due to the dielectric decrement effect, which reduces the accumulation of mobile ions within the EDL. In figure 2(b) we have also indicated results obtained by considering different forms of the boundary condition for the electric potential equation (2.8). For the PNPC model, we observe a slight contraction in the coion-dominated region when employing the mixed boundary condition (Condition-2) as adopted by de Souza & Bazant (Reference de Souza and Bazant2020). This contraction becomes more pronounced for the boundary condition $\nabla ^{2} \phi =0$ at the surface (Condition-3), as considered by Gupta et al. (Reference Gupta, Rajan, Ananth, Emily and Stone2020b ). Notably, in the case of the MNPDC model, the difference in results obtained by imposing Condition-3 and boundary condition (2.15) is insignificant. This can be attributed to the suppression of electrostatic correlation effects by the incorporation of dielectric decrement and finite ion size. Gupta et al. (Reference Gupta, Rajan, Ananth, Emily and Stone2020b ) compared results for EDL charge density based on these three different additional boundary conditions by imposing fixed values for the surface potential or surface charge density and observed that the boundary conditions governed by (2.15) yields a larger overscreening of surface charge compared with Condition-2 and Condition-3. Our results in figure 2(b) agree with this remark of Gupta et al. (Reference Gupta, Rajan, Ananth, Emily and Stone2020b ).

Figure 2. Comparison of the (a) electrophoretic mobility with experimental results of López-Viota et al. (Reference López-Viota, Mandal, Delgado, Toca-Herrera, Möller, Zanuttin, Balestrino and Krol2009) and numerical results of Yeh et al. (Reference Yeh, Fang, Hsu and Tseng2011) at $b=15$ nm, $Q_{fix}=-30.2$ ( $\rho _{_{fix}}=-0.245 \times 10^{7} \text {C m}^{-3}$ ), $\beta =5.15$ , $\Gamma =0.7$ for a soft particle with step-like PEL( $\nu \to 0$ ) in NaCl electrolyte; (b) radial distribution of equilibrium potential with Lesniewska et al. (Reference Lesniewska, Beaussart and Duval2023) when $\zeta =-5$ at the core, $\delta _{c}=2$ , $\rho _{_{fix}}=-1.93 \times 10^{7}\,\rm C\,\rm m^-{^3}$ ( $Q_{fix}=-2640.2$ ), ${C}_{0}=500\,\rm mM$ , $\Gamma =0.91296$ for a soft particle with step-like PEL( $\nu \to 0$ ) in KCl electrolyte; and (c) electrophoretic mobility with experimental results of Semenov et al. (Reference Semenov, Raafatnia, Sega, Lobaskin, Holm and Kremer2013) at $\sigma ^{*}=-3.1\,\rm mC\,\rm m^-{^2}$ for a rigid particle of diameter $2.23\,\unicode{x03BC}$ m in LaCl $_3$ electrolyte (blue line) and a mixture of LaCl $_3$ with 0.01 M KCl electrolyte (red line). In (b) and (c), symbols $\Delta$ , Condition-2; O, Condition-3.

We have also compared our computed electrophoretic mobility based on the MNPDC model with the experimental results of Semenov et al. (Reference Semenov, Raafatnia, Sega, Lobaskin, Holm and Kremer2013) for a rigid colloid ( $\Gamma =1$ ) in multivalent LaCl $_3$ as well as mixed electrolyte composed of LaCl $_3$ and KCl in figure 2(c). Our results show a good agreement with the experimental results, and a change of sign in electrophoretic mobility at a higher range of the bulk ionic concentration in the trivalent LaCl $_3$ electrolyte is evident. However, no sign reversal in $\mu _{_E}^{*}$ is found for a mixture of KCl and LaCl $_3$ electrolytes. It may be noted that the coupling parameter $\xi$ for the LaCl $_3$ electrolyte becomes 1.6 creating the ion–ion correlations relevant, which reduces due to the addition of the monovalent salt KCl. We have elaborated the impact on $\mu _{_E}^{*}$ of a soft particle due to the mixture of monovalent salt with trivalent electrolytes later in this section. We find that $\mu _{_E}^{*}$ based on the MNPDC model becomes identical with the MNPC model as the dielectric decrement effect becomes negligible for this lower range of charge density.

Figure 2(c) shows that our solutions based on the present boundary condition (2.15) and the boundary conditions governed by Condition-2 as well as Condition-3 produce the same $\mu _{_E}^{*}$ for the considered range of ${C}_{0}$ and $\sigma =-3.1$ mC m $^{-2}$ . It may be noted that for this present range of ${C}_{0}$ for the larger particle size $b=1.115\,\unicode{x03BC}$ m of smaller $\sigma$ , the particle curvature has less significance as $\kappa b \gt 2500$ and ion steric interactions are negligible for this lower $\sigma$ . For this, the results based on these different additional boundary conditions do not vary by a large extent.

Figure 3. Variation of $\mu _{_E}$ (a) for different boundary conditions in the MNPC model when $Q_{fix}=-1.5$ , $\sigma ^{*}=-10\,\rm mC\,\rm m^-{^2}$ and in different models when (b) ${C}_{0}=1\,\rm mM$ ( $\kappa b=12.6$ ), $Q_{fix}=-1.5$ , (c) ${C}_{0}=50\,\rm mM$ ( $\kappa b=89$ ), $Q_{fix}=-1.5$ and (d) ${C}_{0}=1\,\rm mM$ , 50 mM, $\sigma ^{*}=-5\,\rm mC\,\rm m^-{^2}$ . The circular and square symbols in (b)–(d), respectively, represent the results of present WF analysis for MNPDC model and Ohshima (Reference Ohshima2000) for a step-like PEL.

We find in figure 3(a) for the moderate range of $C_0$ and $\sigma$ through our numerical simulations for MNPC model in which $\varepsilon$ is considered to be constant that, the overscreening and sign reversal in mobility occurs at a higher range of $\sigma$ when we impose Condition-2 or Condition-3 as the additional boundary condition. Similar observation is also made by Gupta et al. (Reference Gupta, Rajan, Ananth, Emily and Stone2020b ) and remarked that the choice of $l_c$ for the quantitative agreement in results obtained by different boundary conditions is challenging. Condition-2 may become questionable for $\kappa b \sim O(1)$ and at a higher range of $\sigma$ for which ion crowding is non-negligible. Condition-3 becomes identical with Condition-2 for $l_c=0$ on the charged surface. It may be noted that the boundary condition (2.15) yields vanishing ion correlations at the boundary, which is justified as the ion crowding in the close proximity of the highly charged surface diminishes the effects of correlations (Storey & Bazant Reference Storey and Bazant2012). It is also noted before that the global electroneutrality condition can be satisfied for any choice of $\kappa b$ by using the boundary condition (2.15). In addition, we find from figure 3(a) that the pattern of variation of $\mu _{_E}$ with ${C}_{0}$ , particularly for its higher range, obtained by either of these three types of additional boundary conditions are similar. For this, in the present study we use the widely adopted boundary condition (2.15) to analyse the ion–ion correlations.

5.2. Mobility reversal and polarity inversion of charge density in LaCl $_3$

The dependence of mobility on the core surface charge density at different values of the bulk ionic concentration ( $\kappa b$ ) is presented in figure 3(b,c). We obtain the mobility based on different models. The PNP model shows an occurrence of a local maxima in mobility for low to moderate range of $\kappa b$ . However, a monotonic pattern of $\mu _{_E}$ with $\sigma$ is found for $\kappa b \gg 1$ . This local maxima in mobility arises due to the stronger surface conduction. The occurrence of a local maxima in mobility of a colloid when varied with the $\zeta$ -potential is found by O’Brien & White (Reference O’Brien and White1978), which has been justified through the double layer polarization (DLP) leading to the surface conduction. For a soft particle, the DLP becomes significant due to the stronger electrosmotic flow induced by the PEL fixed charge. However, for a thinner Debye length ( $\kappa b \gt 2$ ), the Debye layer induced by the core does not extend the PEL and the core charge density has an insignificant effect on its electrophoresis as $\kappa b$ is increased. Our computed solution for the PNP-model follows the similar pattern as that of $\mu _{_E}$ determined by the theoretical model by Ohshima (Reference Ohshima2000) for the step-like PEL under the weak applied field consideration. A deviation of our results for $\nu _{1} \neq 0$ from the results of Ohshima (Reference Ohshima2000) for $\nu \to 0$ at a higher range of $C_0$ and $\sigma$ is justified as the electrophoresis becomes sensitive to the PEL electrostatic property at this higher $\kappa b$ and $\sigma$ . It may be noted that the theoretical analysis of Ohshima (Reference Ohshima2000) is identical with our WF analysis for the PNP model when a step-like PEL is considered.

Figure 3(b,c) shows that for the MNPC and MNPDC models, in which the ion–ion electrostatic correlation is incorporated, the mobility of a negatively charged particle changes sign from negative to positive (mobility reversal) as the surface charge density is varied. At a higher range of $\sigma$ the electrostatic coupling parameter $\xi$ becomes 51.5. In this case the counterion distribution in the vicinity of the charged surface is governed by the mutual repulsion and it may not obey the mean-field theory. This leads to the development of a condensed layer of counterions in the vicinity of the charged surface. This condensed layer normalizes the surface charge and attracts coions at the outer layer, which again attracts a layer of counterions and so on, leading to an alteration in the polarity of the EDL charge density. The formation of a counterion condensed layer in the vicinity of the core and the coion-dominated region in the EDL is discussed later in this section. The overscreening of the surface charge due to the excess accumulation of counterions in the condensed layer may lead to the mobility of the particle to reverse its direction (figure 3 b,c). The electric force on the particle may alter its direction when the electrostatic correlation effects are accounted for. For example, we find that the electric force on the soft particle for $\sigma ^{*}=-100$ mC m $^{-2}$ and $Q_{\textit{fix}}=-1.5$ at $C_0$ = 1 mM by the PNP model is $F_E=-14.486$ (corresponding to $\mu _{_E}=-0.684$ ), which becomes $F_{E}=32.19$ ( $\mu _{_E}=1.52\gt 0$ ) by MNPC model. The occurrence of such layered structure of EDL and mobility reversal cannot be explained by the mean-field-based electrokinetic models.

In both MNPC and MNPDC models the hydrodynamic steric interaction of ions is considered, which creates a saturation in counterions near the highly charged core. The dielectric decrement which is incorporated in the MNPDC model creates a Born energy difference of ions, which drags the ions towards the lower electric field region. This leads to a saturation in ions near a charged surface and an extension of the EDL. In addition, the spatial variation of the dielectric permittivity due to the dielectric decrement modifies the Maxwell stress and the Korteweg–Helmholtz force appears in the momentum equation. We find that at a higher ${C}_{0}$ for which the impact of the dielectric decrement is prominent, the MNPDC model determines the mobility reversal at a higher $-\sigma$ compared with the MNPC model. In the MNPDC model the dielectric permittivity of the medium becomes smaller near the charged surface, which creates a reduction in the charge storage capacity of the EDL, leading to an attenuation in counterion density. This amplifies the surface potential due to the attenuation of screening of surface charge as well as decrement in the correlative force of the counterions as the density of counterions reduces. For this, the impact of the ion–ion correlation manifests at a relatively higher $-\sigma$ when the dielectric decrement is considered (MNPDC-model). It is evident from the results that the impact of the dielectric decrement is prominent at a higher range of ${C}_{0}$ , i.e. $\kappa b \ge 1$ as the Born force amplifies at a higher ${C}_{0}$ .

Figure 3(d) shows that when the core and PEL are charged with the same polarity, an increment in $Q_{fix}$ at a fixed $\sigma$ does not induce a mobility reversal. The fixed charge density of the PEL attracts counterions and repels coions to increase the net space charge density and the electric force on the particle. The PEL charges have less relevance in forming the condensed layer of counterions. On the contrary, an increase in $-Q_{fix}$ enhances the electric force on the particle, leading to a higher $-\mu _{_E}$ . At a smaller Debye length ( $\kappa b\gg 2$ ) for which EDL does not extend beyond the PEL, the core charge density has a negligible influence on the electrophoresis of the soft particle and the mobility is dominated by the $Q_{fix}$ . For a rigid colloid, the mobility approaches zero as $\kappa b$ approaches infinity. However, the mobility of a charged soft particle attains a saturation limit as the Debye length tends to zero. For lower value of $\kappa b$ (larger Debye length), the mobility is proportional to the net charge density of the particle. The screening of core charge density enhances with the increase of $\kappa b$ , which creates a reduction in the mobility. Figure 3(d) shows that the variation of $Q_{fix}$ at a fixed $\sigma$ does not produce any significant difference in $\mu _{_E}$ between the MNPDC model and the MNPC-model implying that the dielectric decrement is relatively little effected by $Q_{fix}$ . At a lower $C_{0}$ the variation in $Q_{fix}$ does not affect appreciably the ion–ion correlations as well as counterion saturation. For this we find that the difference in $\mu _{_E}$ between MNPC and PNP models are unaffected as $Q_{fix}$ is varied at $C_0$ = 1 mM. However, at a higher $C_0$ = 50 mM a difference in $\mu _{_E}$ obtained by different models occurs by a small margin, i.e. 14.6 % at $Q_{fix}$ = –10.

In figure 3(b,c) we compared our WF analysis for the MNPDC model with the exact numerical solution. It may be noted that the scaled electric field $\Lambda =0.01938$ when $E_{0}=10^{4}\,\rm V\,m^-{^1}$ . Thus, the WF analysis is not expected to differ by a larger margin for the considered range of the imposed electric field. However, as is seen figure 3(b), we find that WF analysis overestimates the exact numerical solution at higher $\sigma ^{*}$ . At higher $\sigma ^{*}$ the nonlinear effects like surface conduction and ion relaxation become significant, which the WF analysis based on the first-order perturbation may not capture accurately. Therefore, our WF analysis, which is computationally cost effective, is valid for low-to-moderate range of surface charge density and weak applied fields.

The average space charge density of mobile ions at a distance $r$ from the origin is determined as

(5.1) \begin{equation} \rho _{e}^{S}=\frac {1}{2} \displaystyle \int _{0}^{\pi } \rho _{e} \sin \theta \, \rm d\theta. \end{equation}

Figure 4(a–c) shows that the ion steric and dielectric decrement effects create a saturation in counterions, leading to a saturation in $\rho _{e}^{S}$ near the particle core. It is evident that the space charge density near the charged surface obtained by the PNP model becomes enormous. Results based on the MNPDC model show that a distinct compact layer (condensed layer) of large $\rho _{e}^{S}\gt 0$ develops at the outer surface of the core. At the outer region of this compact layer $\rho _{e}^{S}$ rapidly decreases and changes sign from positive to negative before it eventually approaches to zero in the electrolyte region. As surface charge density is increased, $|\rho _{e}^{S}|$ becomes larger. We find that the PNP model shows no change of sign in polarity of the EDL charge density. In this case $\rho _{e}^{S}$ remains positive, approaches zero and no condensed layer develops. By comparing the $\rho _{e}^{S}$ obtained by the MNPDC and PNP models we find that the ion steric interaction, dielectric decrement and the ion–ion correlations modify the electric field near the core-shell interface significantly whereas, the shell-electrolyte interface ( $r=1+2.3 \nu _{1}=1.575$ ) is relatively little affected. Figure 4(c) shows that the variation in $Q_{fix}$ has an insignificant effect on $\rho _{e}^{S}$ in the condensed layer, which corroborates our previous finding that an increment in $Q_{fix}$ at a fixed $\sigma$ does not induce a reversal in $\mu _{_E}$ .

Figure 4. Radial distribution of $\rho _{e}^{S}$ at different (a) $\sigma ^{*}$ $(=-10,\,-50,\,-100$ mC m $^{-2}$ ) for ${C}_{0}$ = 1 mM, (b) $\sigma ^{*}$ $(=-10,\,-50,\,-100\,\rm mC\,\rm m^-{^2}$ ) for ${C}_{0}$ = 50 mM when $Q_{fix}=-1.5$ ( $\rho _{_{fix}}=-1.1\times 10^{4}\,\text {Cm}^-{^3}$ ) and (c) $Q_{fix}$ (= –1, –10, –20) when $\sigma ^{*}$ = $-100\,\rm mC\,\rm m^-{^2}$ , $C_{0}=1\,\rm mM$ . The inset shows the zoomed view of the charge density inversion. The dashed lines are the PNP model and the solid lines are the MNPDC model.

Figure 5(a) shows the variation of $\mu _{_E}$ with the bulk ionic concentration at a fixed value of the core charge density and the PEL volume charge density. The results based on the PNP model show that for a lower range of ${C}_0$ , $-\mu _{_E}$ enhances as ${C}_0$ is increased as the polarization effect is lower in this case, which manifests as ${C}_0$ becomes higher so as to have $\kappa b$ is of order 1. This leads to a local maxima in $\mu _{_E}$ by the PNP model. Results based on the MNPC and MNPDC models show that $\mu _{_E}\gt 0$ for a lower range of ${C}_0$ for which the surface potential is higher. In this case a larger accumulation of counterion occurs near the charged surface, creating a stronger ion–ion correlations. We find in the MNPDC model the dielectric decrement diminishes the counterion density near the charged surface, which attenuates the screening of charge density and enhances the electric force on the particle. This leads to a negative $\mu _{_E}$ at a higher range of ${C}_0$ when the dielectric decrement is accounted for.

The radial distribution of $\rho _{e}^{S}$ in the vicinity of the particle is depicted in figure 5(b). The occurrence of the counterion saturation by the MNPC and MNPDC models is evident from the results. A larger number of counterions accumulate in the PEL, which may reduce the Donnan potential in the shell. A condensed layer of counterions develops near the core ( $r=\Gamma$ ), which overcompensates the surface charge and draws a layer of coions in the EDL (figure 5 b). This may lead to a reversal in the soft particle mobility, as is seen in figure 5(a). Contrary to the PNP model, in which the concentration of counterions gradually approaches the bulk value as we move away from the particle surface, the concentration of counterions in the MNPC as well as MNPDC models undershoots the bulk value outside the condensed layer and coions overshoots the bulk value. This multilayer formation of ions refers to the ion crowding effect (Storey & Bazant Reference Storey and Bazant2012).

Figure 5. (a) Variation of $\mu _{_E}$ in different models with ${C}_{0}$ when $\sigma ^{*}$ = –100 mC m $^{-2}$ and $Q_{fix}=-1.5$ . Radial profile of (b) $\rho _e^{S}$ in different models when ${C}_{0}=10\,\rm mM$ , $\sigma ^{*}$ = –100 mC m $^{-2}$ and $Q_{fix}=-1.5$ ( $\rho _{_{fix}}=-1.1\times 10^{4}\,\text {C m}^{-3}$ ). (c) Variation of $\mu _{_E}$ with $\Gamma$ for $\sigma ^{*}=-100\,\rm mC\,\rm m^-{^2}$ at different $C_{0}$ when $Q_{fix}=-1.5$ .

In figure 5(b) we have quantified the impact of dielectric decrement on the counterion saturation by comparing with the result based on the MNPC model. It is evident that the MNPDC model provides a lower counterion concentration near the charged surface. Outside the condensed layer the MNPDC model shows a higher accumulation of counterions and a lower concentration of coions. As pointed-out by Nakayama & Andelman (Reference Nakayama and Andelman2015) and Gupta & Stone (Reference Gupta and Stone2018) both the dielectric decrement and ion steric interactions create a saturation in counterions near a charged surface, which justifies the occurrence of lower counterion concentration $n_1$ near the charged surface by the MNPDC model. Reduced concentration of $n_1$ attenuates the electrostatic correlations, which creates a small difference in $n_1$ and $n_2$ distribution between the MNPC and MNPDC models outside the condensed layer. We find that outside the condensed layer a slightly lower accumulation of coions and a higher concentration of counterions occurs in the MNPDC model. Nakayama & Andelman (Reference Nakayama and Andelman2015) determined a condition under which the counterion saturation in the EDL is dominated either by the dielectric decrement effect or by the ion steric effect. Due to the nonlinearity of the ion transport equation in the present case of electrophoresis we cannot provide conditions for which the dielectric decrement dominates over the ion steric interactions. In addition, the dielectric decrement creates a Maxwell stress and modifies the electric force. However, it is evident that both these effects manifests when the ion density in the EDL becomes stronger. From the expression of the Born force we find that the impact of the dielectric decrement augments for a multivalent counterion.

We consider in figure 5(c) the variation of mobility as the thickness of the PEL is varied at a fixed particle size $b$ . Increase of $\Gamma$ implies an increase of the core and consequently a reduction in the PEL. We find that the impact of the ion–ion correlation magnifies as the core size is increased, creating a positive mobility for a higher value of core surface potential, which occurs at a higher $\sigma$ for a lower range of ${C}_{0}$ .

Figure 6. Contour plot for the mobile space charge distribution $\rho _{e}=\sum _{i} z_{i} n_{i}$ and streamlines in (a) PNP-model at $C_{0}=1\,\rm mM$ and $Q_{fix}=-1.5$ ; (b) MNPDC-model at $C_{0}=1\,\rm mM$ and $Q_{fix}=-1.5$ ; and (c) MNPDC-model at $C_{0}=50\,\rm mM$ and $Q_{fix}=-50$ when $\sigma ^{*}=-100\,\rm mC\,\rm m^-{^2}$ . The dashed circle indicates the interface of the PEL–electrolyte.

Figure 7. Variation of $\mu _{_E}$ with $Q_{fix}$ for various $\sigma ^{*}$ = –10, –50, –100 mC m $^{-2}$ at ${C}_{0}=50\,\rm mM$ (solid lines) and ${C}_{0}=1$ mM (dashed lines) in MNPDC model. (b) Critical value of $\sigma ^{*}_{c}$ at different $Q_{fix}$ for which $\mu _{_E}=0$ .

The streamline pattern and the mobile charge distribution around the soft particle obtained by different models is shown in figure 6(a–c). For the considered values of the parameters the mobility is negative based on the PNP model and it is positive when evaluated by the MNPDC model. The development of the EDL around the core is evident from the results. We find that in the PNP model the charge density of mobile ions is positive, i.e. counterion-dominated region develops around the negatively charged core and the negatively charged PEL. In this case the charge density gradually decreases and approaches to electroneutrality as we move away from the core. The negatively charged particle moves along the $z\lt 0$ direction and the mobile counterions around the particle creates an electro-osmotic flow along $z\gt 0$ direction. Results based on the MNPDC model (figure 6 b) show the development of a condensed layer of counterions around the outer surface of the core. This condensed layer of counterions neutralizes the core charge density and draws coions in the EDL, which leads to reversal in polarity of the EDL. The mobility based on the MNPDC model is positive and the coions in the EDL creates an electroosmosis along the $z\lt 0$ direction. At a higher C $_{0}$ (figure 6 c) the screening of the core charge density magnifies, creating a weaker condensed layer of counterions and the density of coions in EDL becomes lower than the case considered in figure 6(b). We find that the streamlines penetrate the PEL without deformation. This shows that the resistance created by the porous layer is low enough to induce a boundary layer at the fluid-PEL interface. For the present highly permeable PEL the velocity within the PEL region (Darcy velocity) evolves the interface velocity within a region of thickness of the order of $1/\lambda _{0}$ . Thus, no boundary-layer-type region develops at the interface between the PEL and electrolyte at this high screening length $\lambda _{0}^{-1}$ . For this, no EDL develops at the PEL–electrolyte interface.

Figure 7(a) shows that for a higher range of $\sigma$ in which the electrostatic correlations become significant, a positive mobility for the considered range of ${-}Q_{fix}$ is created. This positive $\mu _{_E}$ increases linearly as $Q_{fix}$ increases from negative to positive value. This positive $\mu _{_E}$ declines with the increase of $C_0$ and reverses its sign from positive to negative as the screening of the core charge is amplified at a higher ${C}_0$ . Results show that $\mu _{_E}\gt 0$ decreases as ${-}Q_{fix}$ enhances as the particle experiences a larger electric force, which can overcome the overscreening of core charge occuring due to the formation of condensed layer of counterions on the charged core. We find that $-\mu _{_E}$ is higher than $\mu _{_E}$ as the electric force on the soft particle becomes higher when both $\sigma$ and $Q_{fix}$ are of same sign. Figure 7(a) also shows that at a thinner Debye length, i.e. at a higher $C_{0}$ the dependence of mobility on the surface charge density $\sigma$ is less prominent than the case of thicker Debye length. At higher $C_{0}$ for which $\kappa b\gt 2$ , the EDL does not extend beyond the PEL. Consequently, the impact of the core charge density gradually diminishes with the increase of $\kappa b$ , leading to the impact of PEL charge density to dominate the $\mu _E$ . At a smaller Debye length the variation of $\mu _{_E}$ with $Q_{fix}$ is linear as the DLP effect is diminished and the ion transport within the PEL is governed by a linear diffusion mechanism.

Knowledge of the particle charge density for which $\mu _{_E}$ becomes zero is important in particle trapping and several other situations (Morales et al. Reference Morales, Lin and Zahn2012; Raafatnia et al. Reference Raafatnia, Hickey and Holm2014,Reference Raafatnia, Hickey and Holm2015; Moussa et al. Reference Moussa, Caillet, Town and Duval2015; Shin et al. Reference Shin, Ault, Toda-Peters and Shen2020; Cardenas-Benitez et al. Reference Cardenas-Benitez, Jind, Gallo-Villanueva, Martinez-Chapa, Lapizco-Encinas and Perez-Gonzalez2020). We have shown that the particle can have positive mobility even when both $\sigma$ and $Q_{fix}$ are negative when ion–ion correlations are incorporated. Figure 7(b) shows the critical $-\sigma$ at a fixed $Q_{fix}( =-1.5, 0 )$ at which mobility changes from negative to positive at different $C_{0}$ . We find that this reversal from negative $\mu _{_E}$ to positive $\mu _{_E}$ occurs at an increased value of $-\sigma$ when ${C}_0$ is increased. The critical $-\sigma$ increases as $-Q_{fix}$ is increased as the electric force on the particle is increased.

Figure 8. Contour plot for space charge distribution and streamlines in MNPDC model at (a) $C_{0}=1\,\rm mM$ ( $\kappa b=12.6$ ) and $\rho _{_{fix}}=-0.1\times 10^{7}\,\rm C\,\rm m^-{^3}$ ( $Q_{fix}=-136.8$ ); (b) $C_{0}=100\,\rm mM$ and $\rho _{fix}=-0.1\times 10^{7}\,\rm C\,\rm m^-{^3}$ ; and (c) $C_{0}=100\,\rm mM$ ( $\kappa b=126$ ) and $\rho _{fix}=-3.5\times 10^{7}\,\rm C\,\rm m^-{^3}$ ( $Q_{fix}=-4788$ ), when $\sigma ^{*}=-100\,\rm mC\,\rm m^-{^2}$ .

The charge density contours and streamlines in and around the soft particle for a higher $Q_{fix}$ is presented in figure 8(a–c). The soft particle mobility remains negative at this higher $-Q_{fix}$ . We find from figure 8(a) that for $\kappa b=12.6$ , a compact layer of counterions forms adjacent to the core and a coion-dominated region develops at the outer layer of this compact layer. However, the electric force on the soft particle at this larger $Q_{fix}$ overwhelmed the overscreening of the core charge so that the mobility remains negative. The negative fixed charge of the PEL draws counterions beyond the coion-dominated region to make $\rho _{e}\gt 0$ . At this higher $C_0$ the EDL contracts, which leads to a contraction of the condensed layer and also, the coion dominated region (figure 8 b). At this higher $C_0$ the shielding of PEL fixed charges amplifies and a Donnan potential develops at the PEL, leading to a constant space charge density within the PEL. We find that no coion-dominated region forms at a higher $-Q_{fix}$ when EDL is considered to confine within the PEL, i.e $\kappa b\gg 2$ . At this higher $-Q_{fix}$ the coions are repelled from the PEL region. For this $\rho _{e}$ remains positive and gradually decays to zero.

5.3. Effects of counterion size and mixture of monovalent electrolyte

The correlation length does not depend implicitly on the ion size however, the number density of ions varies with its size due to the consideration of steric interactions. Thus, the ion size is expected to have a strong impact on mobility in the modified model. The ion size effect on the electrostatic correlations for a PE is studied by Hsiao (Reference Hsiao2008) in which the anionic PE dissociates monovalent cations and tetravalent cations whereas, monovalent anions are dissociated from the salt. In figure 9(a–c) we consider three salts of trivalent counterions with different ion size. We find that the impact of the ion correlations on mobility is stronger for a lower sized counterions and the positive $\mu _{_E}$ becomes higher. However, the negative mobility, in which correlation effect is insignificant, becomes higher for larger-sized counterions as the shielding effect attenuates due to the reduction in counterion density in EDL. In addition to the lower steric interactions, the smaller-sized ions reduce the dielectric decrement. These together create a higher counterion concentration in the EDL as its size becomes smaller. The average charge density as presented in figure 9(c) shows a larger accumulation of counterions in the EDL as the counterion size is reduced. A larger accumulation of counterions in the condensed layer draws a larger coion density at the outer side of the condensed layer leading to a stronger reversal in the sign of $\rho _{e}^{S}$ , as is evident from figure 9(c). It is evident that in the MNPDC model, overscreening of surface charge attenuates and hence, the occurrence of mobility reversal is suppressed. This is justified from the distribution of the mobile charge density (figure 9 c), which shows a reduction in counterion density in the compact layer when the MNPDC model is considered.

The electrophoresis in a mixture of monovalent (LiCl) and trivalent ( ${\rm LaCl}_{3}$ ) electrolytes is illustrated in figure 10(a,b). Existing experimental studies (Quesada-Pérez et al. Reference Quesada-Pérez, González-Tovar, Martín-Molina, Lozada-Cassou and Hidalgo-A Ivarez2005; Lenz & Holm Reference Lenz and Holm2008; Semenov et al. Reference Semenov, Raafatnia, Sega, Lobaskin, Holm and Kremer2013) as well as the theoretical analyses (Pianegonda et al. Reference Pianegonda, Barbosa and Levin2005; Chen et al. Reference Chen, Ni, Wang, Xu and Li2008; Agrawal & Wang Reference Agrawal and Wang2022; Yang et al. Reference Yang, Buyukdagli, Scacchi, Sammalkorpi and Ala-Nissila2023) demonstrate that the inversion in polarity of the EDL charge density near a charged surface significantly modifies when a monovalent salt is added to a multivalent salt. Figure 10(a) shows that at a lower range of $\sigma$ for which the correlation effect is low, an addition of the monovalent LiCl electrolyte enhances $-\mu _{_E}$ . At a larger concentration of LiCl, i.e. $C_{\text {LiCl}}$ the reversal in mobility is absent and the mobility remains negative for the considered range of $-\sigma$ . At an enhanced $C_{\text {LiCl}}$ the EDL is dominated by the monovalent counterions, which leads to the lowering of the electric coupling parameter $\xi$ , resulting in a reduction in the impact of the ion–ion electrostatic correlations (Mukhina et al. Reference Mukhina, Hemmerle, Rondelli, Gerelli, Fragneto, Daillant and Charitat2019). For this, an increment in $C_{\text {LiCl}}$ gradually reduces the overscreening and the mobility reversal. We find that at a lower $C_{\text {LiCl}}$ where $\mu _{_E}\gt 0$ an increase in $-\sigma$ increases $\mu _E\gt 0$ . Increment in $-\sigma$ also increases the trivalent counterions in the EDL, leading to a stronger condensed layer in the mixed electrolyte, which results in positive $\mu _E$ , which increases gradually with $-\sigma$ . This also justifies the occurrence of decrement in $-\mu _{_E}$ with $\sigma$ at a higher $C_{\text {LiCl}}$ due to the interplay between the monovalent and trivalent counterions in the EDL.

Figure 9. Variation of $\mu _{_E}$ with (a) $\sigma ^{*}$ at ${C}_{0}=10\,\rm mM$ , and (b) ${C}_{0}$ at $\sigma ^{*}=-100\,\rm mC\,m^{-2}$ for different trivalent salts when $Q_{fix}=-1.5$ . (c) Radial distribution of $\rho _e^{S}$ at ${C}_{0}=10\,\rm mM$ and $\sigma ^{*}=-200\,\rm mC\,m^{-2}$ . Here MNPC are shown with dashed lines and MNPDC with solid lines.

Figure 10. Variation of $\mu _{_E}$ obtained by the MNPDC model in a mixture of electrolytes with (a) $\sigma ^{*}$ for 10 mM of LaCl $_3$ at various $C_{\text {LiCl}}$ (= 0, 10, 50, 150 mM), and (b) $C_{\text {LiCl}}$ at $\sigma ^{*}=-100\,\rm mC\,\rm m^-{^2}$ for various trivalent salts of bulk concentration 10 mM when $Q_{fix}=-1.5$ . (c) Radial profile of $\rho _e^{S}$ for 10 mM of LaCl $_3$ at $\sigma =-100\,\rm mC\,\rm m^-{^2}$ for various $C_{\text {LiCl}}$ (= 0, 10, 50, 150 mM).

The variation of mobility with the concentration of $C_{\text {LiCl}}$ at a fixed $\sigma$ in a mixed electrolyte consisting LiCl and different salts of trivalent counterion of different size is shown in figure 10(b). Increase in $C_{\text {LiCl}}$ reduces the positive mobility whereas, it increases $-\mu _{_E}$ . As noted before, with the increase of $C_{\text {LiCl}}$ the accumulation of monovalent counterions in the EDL enhances, which declines the overscreening of the surface charge and coion accumulation in the EDL. This creates an increment in $-\mu _{_E}$ . The average space charge density $\rho _{e}^{S}$ distribution shows that the ion strength in the condensed layer reduces as $C_{\text {LiCl}}$ is increased. The reversal in the space charge density attenuates as $C_{\text {LiCl}}$ is increased (figure 10 c). This justifies the reduction of positive $\mu _{_E}$ with the increase of $C_{\text {LiCl}}$ .

5.4. Thin EDL consideration

In the limit of a very thin Debye layer ( $\kappa b \gg 1$ ) screening cloud is confined to very close to the particle surface, and beyond this region, the fluid is electroneutral, the local radius of curvature of the surface can be considered much larger than the Debye length. Under this approximation, the surface conduction, DLP and relaxation effects can be neglected and the variation of ion distribution along the EDL is negligible compared with the variation along the normal direction. Thus, the ion distribution in EDL is governed by the EDL potential (López-García et al. Reference López-García, Horno and Grosse2019) and hence, follow its equilibrium distribution (4.1). Consequently, the modified Poisson equation for the electric potential can be derived from (2.8) as

(5.2) \begin{equation} \bigg (\frac {l_{c}}{b}\bigg )^{2}\frac {d^{2} }{\rm d\it x^{2}}\left (\varepsilon \psi \right )- \frac {\rm d}{\rm d\it x}\left (\varepsilon \frac {\rm d \phi }{\rm d\it x}\right ) = \left [ \frac {(\kappa b)^{2}}{2} \sum _{i} z_{i} n_{i}^{0} + g (x) Q_{fix} \right ] \end{equation}

where

(5.3) \begin{equation} \frac {d^{2} \phi }{\rm d\it x^{2}}=\psi. \end{equation}

Here, the $x$ -coordinate is along the normal to the charged surface. The boundary conditions for the electric potential at the charged core surface ( $x=0$ ) can be derived from (2.15) as

(5.4) \begin{equation} \frac {\rm d(\varepsilon \psi )}{\rm d\it x} - \varepsilon \frac {\rm d \phi }{ \rm d\it x}=\sigma,\,\,\,\,\,\,\frac {\rm d(\varepsilon \psi )}{\rm d\it x}=0 \end{equation}

and far from the surface ( $x \to \infty$ )

(5.5) \begin{equation} \frac {\rm d \phi }{ \rm d\it x}=0,\,\,\,\,\,\,\psi =0. \end{equation}

Under a stationary frame of reference fixed at the charged surface the electrolyte velocity field is $\boldsymbol {u}=u_{z}(x)\boldsymbol {e}_{z}$ , which is tangential to the particle surface along which the electric field is applied. Thus, $u_{z}=0$ on the surface $x=0$ and at a distance far away from the surface ( $x \to \infty$ ) the velocity approaches $-U_E$ , i.e. $\boldsymbol {u}=-U_{E}\boldsymbol {e}_{z}$ when $x\gg 1$ . The fluid flow equation is governed by the balance of the viscous force with the electric force, which is

(5.6) \begin{equation} \frac {d}{\rm d\it x}\left (\eta \frac {\rm d\it u_{z}}{\rm d\it x}\right )-\beta ^{2} \eta g(x) u_{z}=\Lambda \bigg [\frac {d}{\rm d\it x}\bigg ( \varepsilon \frac {\rm d \phi }{{\rm d}\it x}\bigg )-\bigg (\frac {l_c}{b}\bigg )^{2} \frac {d^{2}}{\rm d\it x^{2}}(\varepsilon \psi )+g(x) Q_{fix}\bigg ]. \end{equation}

A closed-form solution for this nonlinear problem may not be possible. Therefore, we use the finite difference method to solve for $\phi$ and subsequently, (5.6) for $u_z$ . Based on the velocity field $u_z$ , the electrophoretic mobility is obtained as $\mu _{_E}=-\displaystyle \lim _{x\to \infty } u_{z}/\Lambda$ .

However, for a rigid particle $\Gamma =1$ , which corresponds to $g(x)=0$ , the fluid flow equation (5.6) can be integrated to evaluate the electroosmotic velocity of the electrolyte as

(5.7) \begin{align} u_{z} & =\Lambda \bigg [\bigg (\frac {\varepsilon \phi }{\eta }-\frac {\varepsilon _{_S} \zeta }{\eta _{_S}}\bigg )-\bigg (\frac {l_c}{ b}\bigg )^{2}\bigg ( \frac {\varepsilon \psi }{\eta } -\frac {\varepsilon _{_S}\psi _{_S}}{\eta _{_S}} \bigg )\bigg ]\nonumber\\& \quad - \Lambda \bigg [\int _{0}^{x} \phi \frac {d}{\rm d\tau }\left ( \frac {\varepsilon }{\eta } \right ) \,{\rm d}\tau -\bigg (\frac {l_c}{ b}\bigg )^{2} \int _{0}^{x} \varepsilon \psi \frac{d}{\rm d\tau }\left (\frac {1}{\eta }\right ) \,\rm d\tau \bigg ]. \end{align}

Note that $\varepsilon _{_S}=\varepsilon |_{x=0}$ , $\eta _{_S}=\eta |_{x=0}$ and $\psi _{_S}=\psi |_{x=0}$ . Thus, the electrophoretic mobility of the rigid particle can be expressed in a closed form as

(5.8) \begin{equation} \mu _{_E}=-\lim _{x\to \infty }\frac {u_{z}}{\Lambda }=\frac {\varepsilon _{_S}\zeta }{\eta _{_S}}-\int _{0}^{\infty } \phi \frac {d}{{\rm d}\tau }\left ( \frac {\varepsilon }{\eta } \right ) \,{\rm d}\tau -\bigg (\frac {l_c}{b}\bigg )^{2}\bigg [\frac {\varepsilon _{_S}\psi _{_S}}{\eta _{_S}}-\int _{0}^{\infty } \varepsilon \psi \frac {d}{{\rm d}\tau }\left (\frac {1}{\eta }\right ) \,{\rm d}\tau \bigg ]. \end{equation}

If the electrostatic correlation effect is relaxed ( $l_{c}=0$ ), this solution matches exactly with the solution as derived by López-García et al. (Reference López-García, Horno and Grosse2019). For the PNP-model, (5.8) reduces to

(5.9) \begin{equation} u_{z}=\Lambda [\phi -\zeta ] \end{equation}

which is the well-known Helmholtz–Smoluchowski (Hunter Reference Hunter2013) velocity. The corresponding electrophoretic mobility is $\mu _{_E}=\zeta$ .

This thin-layer model can easily be implemented to obtain the mobility for a soft particle based on the MNPDC model. We have shown in figure 11(a) by comparing with our exact numerical solutions that this thin-layer model is in excellent agreement when $\kappa b=218$ . It is evident from figure 11(a) that the impact of the moderately charged core ( $\zeta \lt 1$ ) is diminished at a smaller Debye length and the mobility varies linearly with the PEL charge density. A linear increment of $\mu _{_E}$ at a higher range of $\sigma$ (i.e. $\zeta \gt 1$ ) is evident from figure 11(b). We have already shown based on the exact numerical solutions (figure 3 c) that at a higher $C_{0}$ the dielectric decrement suppress the mobility reversal. Similar trend in mobility based on the thin layer analysis is evident from figure 11(b).

Figure 11. Thin-layer model for $\mu _{_E}$ (a) comparison with the present numerical solution of MNPDC model for different $\sigma ^{*}=-10,\,-20,\,-40\,\rm mC\,\rm m^-{^2}$ at ${C}_0=0.3\,\rm M$ ; (b) as a function of $Q_{fix}$ for $\sigma ^{*}(=-100,\,-200,\,-300\,\rm mC\,\rm m^-{^2}$ ) at ${C}_0=0.5\,\rm M$ .