Theoretical and computational analysis of the electrophoretic polymer mobility inversion induced by charge correlations

Electrophoretic (EP) mobility reversal is commonly observed for strongly charged macromolecules in multivalent salt solutions. This curious effect takes place, e.g., when a charged polymer, such as DNA, adsorbs excess counterions so that the counterion-dressed surface charge reverses its sign, leading to the inversion of the polymer drift driven by an external electric field. In order to characterize this seemingly counterintuitive phenomenon that cannot be captured by electrostatic mean-field theories, we adapt here a previously developed strong-coupling-dressed Poisson-Boltzmann approach to the cylindrical geometry of the polyelectrolyte-salt system. Within the framework of this formalism, we derive an analytical polymer mobility formula dressed by charge correlations. In qualitative agreement with polymer transport experiments, this mobility formula predicts that the increment of the monovalent salt, the decrease of the multivalent counterion valency, and the increase of the dielectric permittivity of the background solvent suppress charge correlations and increase the multivalent bulk counterion concentration required for EP mobility reversal. These results are corroborated by coarse-grained molecular dynamics simulations showing how multivalent counterions induce mobility inversion at dilute concentrations and suppress the inversion effect at large concentrations. This re-entrant behavior, previously observed in the aggregation of like-charged polymer solutions, calls for verification by polymer transport experiments.

Figure 1

Left panel: A schematic of the CG DNA located at $(x,y)=(0,0)$ along the $z$ axis of the simulation box with total volume $L_x\times L_y\times L_z$. Right panel: Ions distributed within a cylindrical volume of radius $r$ centered at $(x,y)=(0,0)$. The red, green, and orange spheres represent trivalent cations, monovalent anions, and monovalent cations, respectively. In this snapshot, $L_x=L_y=24$ nm, and $L_z=20$ nm. The concentration of multivalent salt (${\mathrm{TCl}}_3$) is $n_{\mathrm{cb}}=200$ mM and that of monovalent counterions (e.g., ${\mathrm{Na}}^{+}$) neutralizing the CG DNA charge 17 mM.

Figure 2

Radial functions related to charge inversion from the MD simulations. (a) Normalized density profiles of the ${\mathrm{T}}^{3+}$ cations (solid curves) and ${\mathrm{Cl}}^{-}$ anions (dashed curves). (b) Cumulative charge density in Eq. (3). (c) Electrostatic potential in Eq. (4) at various trivalent salt concentrations $n_{\mathrm{cb}}$ indicated in the legend of (b). The inset in (a) displays the dimensionless ${\mathrm{Cl}}^{-}$ densities on a smaller linear scale. The dashed vertical line in (c) marks the no-slip boundary located at $r=a+\delta$. The gray regions indicate the radius of the polymer $a$. The relative liquid permittivity is ${\varepsilon }_{\mathrm{w}}/{\varepsilon }_0=78$.

Figure 3

EP mobility ${\mu }_{\mathrm{p}}$ from MD simulations as a function of the multivalent counterion concentration $n_{\mathrm{cb}}$ for various values of (a) the cation valency $q_{\mathrm{c}}$, (b) the dielectric constant, and (c) the monovalent salt concentration $n_{\mathrm{sb}}$. The dashed curves correspond to a quadratic fitting function (see main text), and the crosses represent the reversal concentrations $n_{\mathrm{cb}}^{*}$, which are reported in Table 2.

Figure 4

EP mobility from MD simulations in the presence of trivalent counterions of different intramolecular structures. The reversal concentrations are found at $n_{\mathrm{cb}}^{*}=7,70,80,140$, and 260 mM for sphere, rod A, B, C, and D, respectively. The reversal concentrations $n_{\mathrm{cb}}^{*}$ are shown in Table 2. Here the concentration of monovalent salt is $n_{\mathrm{sb}}=0$. In the inset the red circle denotes a spherical trivalent counterion, whereas the yellow circles are monovalent spherical counterions located at different sites in the seven-bead rodlike molecule.

Figure 5

(a) Polymer mobility ${\mu }_{\mathrm{p}}$ in Eq. (21) versus the multivalent counterion concentration $n_{\mathrm{cb}}$ for various values of the (a) monovalent salt concentration $n_{+\mathrm{b}}$, (b) the electrolyte permittivity ${\varepsilon }_{\mathrm{w}}$, (c) the multivalent charge valency $q_{\mathrm{c}}$, and (d) the polymer permittivity ${\varepsilon }_{\mathrm{p}}$. The model parameters are $a=1.2$ nm, $\delta =0.1$ nm, and ${\sigma }_{\mathrm{p}}=0.783e/{\mathrm{nm}}^2$. The charge valency in (a), (b), and (d) is $q_{\mathrm{c}}=3$, and the salt concentration in (b)–(d) is $n_{+\mathrm{b}}=300$ mM.

Figure 6

(a) Dimensionless multivalent ion density $k_{\mathrm{c}}\left(r\right)=n_{\mathrm{c}}\left(r\right)/n_{\mathrm{cb}}$ against the separation distance from the polymer axis at the model parameters in Figs. 5–5.

Figure 7

(a) Polymer mobility (30) and [(b) and (c)] critical concentration (32) versus the electrostatic coupling parameter $\mathrm{\Xi }$. The polymer radius is $\overline{a}=10.0$ and the no-slip length is $\overline{\delta }=1.0$. The remaining parameters are given in the legends.

Figure 8

Normalized trivalent ion density at the bulk concentrations $n_{\mathrm{cb}}=50$ mM (left) and $n_{\mathrm{cb}}=5$ mM (right), and for different dimensions of the simulation box (see the legends) displaying reduction in the finite-size effects at $L_x\gtrsim 60$ nm (left) and $L_x\gtrsim 112.8$ nm (right). The relative permittivity and the longitudinal system size are ${\varepsilon }_{\mathrm{w}}/{\varepsilon }_0=78$ and $L_z=20$ nm.

Figure 9

Cumulative charge density with trivalent counterions of bulk concentrations $n_{\mathrm{cb}}=1$ mM (left) and $n_{\mathrm{cb}}=50$ mM (right), and different longitudinal system sizes (see the legend) indicating the numerical convergence at $L_z\gtrsim 10$ nm. The relative permittivity is ${\varepsilon }_{\mathrm{w}}/{\varepsilon }_0=78$, and the transverse system size is $L_x=112.8$ nm (left) and $L_x=60$ nm (right).