Rayleigh instability of charged aggregates: Role of the dimensionality, ionic strength, and dielectric contrast

We extended a previous analysis of the classical Rayleigh instability of spherical charged droplets in the presence of neutralizing monovalent counterions [M. Deserno, Eur. Phys. J. E 6, 163 (2001)], by generalizing the problem for suspensions of aggregates with $D$-dimensional symmetry, corresponding for $D=2$ to infinite (rodlike) cylindrical charged bundles and for $D=3$ to spherical charged droplets. In addition, we include the effects of added monovalent salt and of dielectric contrast between the charged aggregate and the surrounding solvent. The electrostatic energy taking the microion screening into account is estimated using uniform profiles within the framework of the cell model. We verify the robustness of these results by also considering Debye-Hückel-type microion profiles that are obtained by the minimization of a linearized Poisson-Boltzmann free-energy functional. In the case when the microions can enter inside the charged aggregates, we confirm the occurrence of a discontinuous phase change between aggregates of finite size and an infinite aggregate, which takes place at a collapse temperature that depends on their volume fraction $\varphi$ and on the salt content. Decrease of $\varphi$ shifts the phase-change temperature toward higher values, while salt addition has an opposite effect. We obtain analytical expressions for the phase-separation line in the asymptotic limit of infinite dilution $(\varphi \to 0)$, showing that the collapse temperature depends logarithmically on $\varphi$. As an application for $D=3$ we discuss the stability of the pearl-necklace structures of flexible polyelectrolytes in poor solvents. The case $D=2$ is applied to the problem of finite bundle sizes of stiff polyelectrolytes that attract each other—via, e.g., multivalent counterions—leading to an effective surface tension.

Figure 1

Two-dimensional representation of a Wigner-Seitz cell of radius $R=a{\varphi }^{-1∕D}$ containing a charged aggregate in its center (represented by the dark gray region) of cross-section radius $a$ and dielectric constant ${ϵ}_{<}={ϵ}_{>}ϵ$, where ${ϵ}_{>}$ is the dielectric constant of the solvent in the region $r>a$. For $D=2$ the cell is cylindrical with an infinite length—the geometry shown corresponds to a cross section perpendicular to the symmetry axis. For $D=3$ the cell has a spherical symmetry and the figure represents a cross section at a plane containing the center of the cell. Therefore, the generalized radial coordinate $r$ corresponds to the radial distance from the symmetry axis for $D=2$ and to the distance to the center of the cell for $D=3$. We will assume the existence of a condensed layer of microions within the intermediate shell $a<r<a\lambda$ (represented by the light gray region).

Figure 2

Phase-separation cross-section radius ${\tilde{a}}^{*}$ as a function of ${\tilde{l}}_B$ for aggregates of spherical symmetry $(D=3)$ in the absence of dielectric contrast $(ϵ=1)$ and several values of $s$ and $\varphi$. Dashed lines are obtained for constant amount of salt (varying from $s=0$ to 100, from right to left), while dot-dashed lines are obtained for constant volume fraction (varying from $\varphi ={10}^{-1}$ to ${10}^{-6}$, from top to bottom). Note the existence of a lower bound for the phase-separation line as the salt concentration (measured by the parameter $s$) increases. The solid lines correspond to the equilibrium-size salt-free curves $(s=0)$ for fixed values of $\varphi$, and terminate at the salt-free phase-separation dashed line. For values of $\varphi$ below the critical value ${\varphi }^{\mathrm{crit}}=4.413\dots \times {10}^{-5}$ the equilibrium-size salt-free curves are in fact discontinuous functions of ${\tilde{l}}_B$ (see magnification in Fig. 3). The details shown in Fig. 3, especially the gray coexistence region, could not be seen if they would be plotted here in unmagnified scale.

Figure 3

Magnification of the salt-free $(s=0)$ phase-separation droplet radius ${\tilde{a}}^{*}$ as a function of ${\tilde{l}}_B$ (dashed line) for aggregates of spherical symmetry $(D=3)$ in the absence of dielectric contrast $(ϵ=1)$. The solid lines correspond to the equilibrium-size salt-free curves for two small values of $\varphi$, showing that they are in fact discontinuous functions of ${\tilde{l}}_B$. In the gray region distinct thermodynamic phases, characterized by finite droplets with two different radii, may coexist. In the inset we show this region in the vicinity of the critical point (black dot), located at $({\tilde{l}}_B^{\mathrm{crit}},{\log }_{10}{\varphi }^{\mathrm{crit}})=(6.503\dots ,-4.355\dots )$. For volume fractions below this critical value, $\varphi <{\varphi }^{\mathrm{crit}}$, besides the macroscopic phase separation that occurs at lower temperatures (dashed line), there is a discontinuous phase change associated with this two-phase coexistence. We emphasize that this magnification corresponds to a subtle detail of Fig. 2—note the scales of the axes. The banana-shaped phase coexistence region is very narrow and it is not presented in Fig. 2, since it would not be possible to identify it on the unmagnified scale of Fig. 2.

Figure 4

Phase-separation cross-section radius ${\tilde{a}}^{*}$ as a function of ${\tilde{l}}_B$ for aggregates of cylindrical symmetry $(D=2)$ in the absence of dielectric contrast $(ϵ=1)$ and several values of $s$ and $\varphi$. Dashed lines are obtained for constant amount of salt (varying from $s=0$ to 10, from right to left), while dot-dashed lines are obtained for constant volume fraction (varying from $\varphi ={10}^{-1}$ to ${10}^{-6}$, from top to bottom). As in the case of spherical aggregates, the position of the phase-separation line reaches a saturation value as the salt concentration increases. Note, however, that the phase-separation line in the infinite-dilution limit has no lower bound, ${\lim }_{\varphi \to 0}4\pi {\tilde{l}}_B^{*}{\tilde{a}}^{*3}\to 0$. On this scale, the position of the phase-separation line for $s=100$ is indistinguishable from the one for $s=10$. The solid lines correspond to the equilibrium-size salt-free curves $(s=0)$.

Figure 5

Same as in Fig. 4, but using the Manning parameter $\xi \equiv \pi {\tilde{l}}_B{\tilde{a}}^2$ as variable for the $y$ axis. The solid lines correspond to the salt-free $(s=0)$ equilibrium Manning parameter curves, while the dashed and dot-dashed lines correspond to the phase-separation Manning parameter with varying salt content $s$ and volume fraction $\varphi$, respectively. Note that the infinite-dilution limit of the phase-separation Manning parameter does not vanish, ${\lim }_{\varphi \to 0}{\xi }^{*}=\pi {\tilde{l}}_B^{*}{\tilde{a}}^{*2}=2∕3$.

Figure 6

Comparison of the salt-free $(s=0)$ droplet equilibrium radius as a function of ${\tilde{l}}_B$ for aggregates of spherical symmetry $(D=3)$ in the absence of dielectric contrast $(ϵ=1)$ for decreasing volume fractions, varying from $\varphi ={10}^{-1}$ to ${10}^{-6}$ (from top to bottom). Solid lines are obtained with the uniform-profile model [27], and dashed lines with the linearized Poisson-Boltzmann profiles for the outer region derived in Appendix E [29]. Except for the spurious discontinuous phase change between two phases associated with finite droplets with different radii (see also Fig. 3) the uniform-profile model describes well the main features of the system. The inset shows the unstable branch of the droplet equilibrium radius for the lowest volume fraction, $\varphi ={10}^{-6}$. Curves extending to the right (at lower temperatures) are obtained when the counterions cannot enter inside the droplets, while the curves terminating at a phase-separation line (represented by the thick diagonal curves) are obtained when the counterions are allowed to enter inside the droplets. At this phase-separation value, droplets of finite radius would coalesce into a single droplet of infinite size, leading to macroscopic phase separation.

Figure 7

Comparison of the salt-free $(s=0)$ equilibrium cross-section radius as a function of ${\tilde{l}}_B$ for aggregates of cylindrical symmetry $(D=2)$ in the absence of dielectric contrast $(ϵ=1)$ for decreasing volume fractions, varying from $\varphi ={10}^{-1}$ to ${10}^{-6}$ (from top to bottom). The meaning of the curves is the same as in Fig. 6. The uniform-profile model still yields a spurious discontinuous phase change for very dilute systems, but the effect is less pronounced than in the spherical case. Note the narrow unstable branch of the equilibrium cross-section radius for the lowest volume fraction, $\varphi ={10}^{-6}$, denoted by the large arrow.

Figure 8

Salt-free $(s=0)$ equilibrium (thin lines) and phase-separation cross-section radii (thick diagonal curves) as a function of ${\tilde{l}}_B$ for aggregates of cylindrical symmetry $(D=2)$ and decreasing volume fractions, varying from $\varphi ={10}^{-1}$ to ${10}^{-5}$ (from top to bottom) for two different dielectric ratios $ϵ=1∕10$ (solid lines) and $1∕20$ (dot-dashed lines). The solvation-energy difference was estimated using the Born equation (7) with a fixed ratio $l_B∕r_0=2$. To allow a comparison with the results in the absence of dielectric contrast, we also presented the $ϵ=1$ curves with penetration of the counterions inside the aggregates (dashed lines).