Colloquium: Star-branched polyelectrolytes: The physics of their conformations and interactions

Recent progress in the field of a versatile and common system in soft matter physics, namely, star-shaped polyelectrolytes, is reviewed. These charged macromolecules combine in their properties aspects of polymer physics, colloidal science, and the rich physics of charged matter, rendering them into versatile building blocks for new materials as well as prototypes for studying the effects of softness in the behavior of colloidal suspensions. The approaches to the problem are manifold. Theoretical methods typically involve scaling theory, self-consistent field theory, and variational free-energy calculations, while computer simulations play an important role in enhancing our understanding of the physics involved. Experiments, based mostly on scattering, offer insights from a different point of view. Finally, the flexibility in the synthesis of novel types of macromolecules has added to the richness and the increased activity in the field in the last few years. This review puts emphasis on theory and simulation but takes into account the key experimental findings in critically discussing the merits and shortcomings of the former. The perspectives opened for the future, with emphasis on the possibilities to steer the behavior of novel materials, are also discussed.

Figure 1

Blob picture of a linear polymer chain (a) and of polymer chains in starlike architecture (b). The blob sizes are characterized by the correlation length of the monomers ${\xi }_0$ in the undisturbed polymer case and $\xi \left(r\right)$ for the polymer segments in the starlike architecture.

Figure 2

(Color online) Conformations of polyelectrolyte stars. (a) Sketch of a PE star consisting of $f=5$ stretched arms and radius $R$ inside a spherical Wigner-Seitz cell of radius $R_W$. There are three different states $i=1,2,3$ for the counterions, with populations $N_i$. In particular, $N_1$ are condensed within tubes of total volume $V_1$, $N_2$ are trapped in the volume $V_2$ inside the star, and $N_3$ counterions are free in the outside volume $V_3$ (42, 43). (b) Simulation snapshot of a $f=5$ PE star with $N=50$ monomers per arm and a charge fraction $\alpha =1$. Connected spheres denote charged monomers, whereas the star counterions are shown as disconnected spheres.

Figure 3

(Color online) Double logarithmic plot of density profiles of monomers (black line) and counterions [red (gray) line] $\rho \left(r\right)$ as a function of the distance $r$ from the central monomer. The results stem from MD simulations of a star which consists of $f=10$ arms with degree of polymerization, $N=50$, and a charge fraction $\alpha =1∕3$. The slope indicates strong stretching of the arms, close to the full stretching limit, in which $\rho \left(r\right)\sim r^{-2}$ (42, 43).

Figure 4

(Color online) The fraction of trapped counterions $N_{\mathrm{in}}∕Q_b$ (where $N_{\mathrm{in}}=N_1+N_2$) plotted vs the star functionality $f$. The simulation results, denoted by the circles, are compared with variational free-energy calculations of the three-state model, black solid line (42, 43), and of the two-state model, black dashed line (48). The fraction of condensed counterions $N_1∕Q_b$ calculated from simulations (squares) and that from the three-state model (dash-dotted line) are shown as well. The Manning condition for the fraction of condensed counterions is $1-1∕\left(u^2\alpha \right)$ and is denoted by the horizontal line. (a) Star with degree of polymerization per arm $N=50$, $\alpha =1∕3$, and cell radius $R_W=55.83a$, simulation results from (42, 43). (b) Star with $N=60$, $\alpha =1∕2$, and $R_W=99.26a$.

Figure 5

The star radius $R$ vs the degree of polymerization $N$ for a star with functionality $f=10$, $\alpha =1∕3$, and cell radius $R_W=136.48a$. The simulation results, denoted by the circles (42, 43), along with a power-law fit of the same (dash-dotted line), are compared with variational free-energy calculations of the three-state model (solid line) (42, 43) and the two-state model (dashed line) (48).

Figure 6

Sizes of polyelectrolyte stars. The star radius $R$ vs functionality $f$. The simulation results, denoted by the circles, along with a power-law fit of the same (dash-dotted line), compared with variational free-energy calculations of the three-state model (solid line) (42, 43), and the two-state model (dashed line) (48). (a) Star with degree of polymerization per arm $N=50$, $\alpha =1∕3$, and cell radius $R_W=55.83a$, simulation results from 42, 43 (b) Star with $N=60$, $\alpha =1∕2$, and $R_W=99.26a$, simulation results from 40.

Figure 7

Theoretical prediction, solid line, of the salt dependence of a ten arm star with $N=50$, $\alpha =1∕3$, compared with simulation results, circles (43). Above the threshold $c_s>\alpha c_m\left(R_0\right)$, the scaling relation is $R\sim c_s^{-0.16}$.

Figure 8

(Color online) Relative brush thickness for a spherical polyelectrolyte brush with core size $r_c=6a$, $f=40$, $N=30$, and $\alpha =1$. The concentration ratio of multivalent to monovalent ions $c^{q+}∕c^{+}$ is varied. Compared are theoretical results (lines) with data obtained from MD simulations (symbols) for systems with valencies $q=2$ (spheres and solid line), $q=3$ (squares and dashed line), and $q=4$ (diamonds and dash-dotted line). From 61.

Figure 9

Two stars of radius $R$ at center-to-center distance $D$. At the bisecting plane the chains retract from each other. The surface section $\mathcal{S}$ where the two stars meet is indicated by a solid black line in the bisecting plane (dashed line). The two stars are within a cell of a fused double-sphere shape of radius $R_W$. The volume of the two fused stars is $V_{\mathrm{in}}\left(D\right)=2\Omega \left(D\right)$ and that of the shell outside the stars is $V_{\mathrm{out}}\left(D\right)$.

Figure 10

(Color online) Simulation snapshot of two PE stars with $f=18$ arms and $N=50$ monomers per arm each at a center-to-center separation $D\approx 0.2R$. The light-colored, connected spheres denote neutral monomers, whereas the dark, connected ones denote charged monomers on each star. Disconnected spheres are the star counterions. Note the retraction of the arms of each star toward the half-space on which the star lies.

Figure 11

The Pincus force between two stars vs distance $D$. Compared are the Pincus force resulting for a $D$-independent normalization coefficient $A$ [Eq. (24)], solid lines, with the corrected Pincus force [Eq. (26)], dashed lines.

Figure 12

Effective force $F\left(D\right)=-d{V}_{\mathrm{eff}}∕dD$ between two PE stars versus distance $D$. Shown are results for 10 arm stars (solid line) and 18 arm stars (dashed line) with $\alpha =1∕3$ and $N=50$. For comparison, the effective core diameter $2r_c$, which is implemented in the simulation model (results represented by symbols), needs to be subtracted (42, 43).

Figure 13

(Color online) Approximated effective force $F\left(D\right)=-d{V}_{\mathrm{eff}}∕dD$ between two stars with $V_{\mathrm{eff}}\left(D\right)$ from Eq. (30) compared with MD results (41). Parameters are the same as in Fig. 12. Also shown is the corrected Pincus force according to Eq. (26).

Figure 14

Effective force $F=-d{V}_{\mathrm{eff}}∕dD$ between two stars with and without added salt, lines, as compared with MD results, symbols (41). Star parameters: $\alpha =1∕3$, $f=10$, $N=50$. From 43.

Figure 15

(Color online) Sketch of the Wang-Denton model for two interacting PE stars. Each arm is modeled as a rigid freely rotating rod, on which monomers are located in regular separations $a$ from each other. The monomer-monomer interaction is modeled by a Yukawa potential to take into account the screening by counterions and salt [see Eq. (32)]. From 81.

Figure 16

Comparison of forces $F\left(D\right)$ between two PE stars of radius $R$ as a function of their center-to-center separation $D$ from the Yukawa-segment MC simulations of 81 and the fully microscopic Coulomb MD simulations of 43. Courtesy of A. R. Denton.

Figure 17

The LRT result for the effective interaction potential between two PE stars of diameter $2R=100\mathrm{nm}$ and bare charge $Q_b=100$ (15). The results pertain to a system with packing fraction $\pi \rho {\sigma }^3∕6=0.01$ at room temperature, ${\lambda }_B=0.71\mathrm{nm}$. Courtesy of A. R. Denton.

Figure 18

The effective interaction potential between PE stars with $f=15$ arms and degree of charging $\alpha =1∕3$ as a function of center-to-center separation $D$ scaled over the star diameter $\sigma$. Results are shown for four different values of the star density $\rho {\sigma }^3$ as indicated in the legend. From 38.

Figure 19

The structure factor of concentrated PE-star solutions for different star densities $\rho {\sigma }^3$ as indicated in the legend. As in Fig. 18, the results pertain to stars with $f=15$ arms and charge fraction $\alpha =1∕3$. From 38.

Figure 20

The phase diagram of PE stars with charge fraction $\alpha =1∕3$ plotted on the inverse functionality vs density representation. Phase boundaries between coexisting phases are calculated at the discrete $f$ values indicated by the points. The width of the two-phase coexistence regions is negligible on the scale shown, whereas all lines are guides for the eye connecting the calculated phase boundaries at the points shown. From 38.

Figure 21

The phase diagram of ionic microgels, plotted in the bare charge vs density representation (25, 26). Note that the quantity $Q_b$ on the vertical axis plays for microgels a role analogous to $f$ for PE stars. The phase diagram here thus has a topology similar to that of Fig. 20 if looked “upside down.”