Polyelectrolytes: From Seminal Works to the Influence of the Charge Sequence

Abstract

We propose a selected tour of the physics of polyelectrolytes (PE) following the line initiated by de Gennes and coworkers in their seminal 1976 paper. The early works which used uniform charge distributions along the PE backbone achieved tremendous progress and set most milestones in the field. Recently, the focus has shifted to the role of the charge sequence. Revisited topics include PE complexation and polyampholytes (PA). We develop the example of a random PE in poor solvent forming pearl-necklace structures. It is shown that the pearls typically adopt very asymmetric mass and charge distributions. Individual sequences do not necessarily reflect the ensemble statistics and a rich variety of behaviors emerges (specially for PA). Pearl necklaces are dynamic structures and switch between various types of pearl-necklace structures, as described for both PE and PA.

Keywords: polyampholytes; polyelectrolytes.

Figure 1

(a–c) PE solutions. (a) Left—Single PE chain in good solvent which is stretched by electrostatics with a few counterions. Right—PE solution in good solvent where the correlation length corresponding to typical distances between chains is represented as the mesh size, without physical crosslinks. (b) Left—Overall stretched pearl-necklace structure of a single PE chain in poor solvent. Right—PE solution in poor solvent where meshes interconnected by pearls appear. (c) Semi-dilute solution: The depicted polyelectrolyte configuration (2D illustration) gains entropy by selecting orientations at each ‘crossing’. In the case of an intrinsically flexible polyelectrolyte, the correlation length, illustrated as the distance between chains, can be qualitatively equated to the electrostatic persistence length. The thick shadow line represents a conformation of a test chain. (d) PE gel. Crosses (×) indicate permanent (chemical) crosslinks. The osmotic pressure of trapped counterions tends to swell the gel. (e) PE Complexes. Left—Ladder structure with an open loop. Right—Scrambled-egg structure. (f) Multilayers. Sketch of a periodic multilayer structure. (g) PE precipitation mediated by multivalent ions. (h,i) PE adsorption on a substrate. (h) Intrinsically flexible PE. (i) rigid PE. Left—DNA plasmid on a planar substrate. Right—ds-DNA on a colloid/histone. (j) PE brushes. PEs are grafted to a solid substrate by one end and form a so-called brush. The brush of flexible PEs is swollen by the osmotic pressure of the trapped counterions.

Figure 2

Pearl-necklace conformations of PE in poor solvent conditions (left) and pearl necklaces of random PA sequences in weakly poor solvent conditions (right). The necklace structure consists of pearls (indicated by dashed circle) and connecting strings. The number of pearls increases with increasing net charges. The conformations are obtained from MD simulations for PA and PE chains consisting of N = 202 monomers in which every 3rd monomers can carry charges ($p=3$). All charge sequences are created by unbiased Markovian processes. Blue and red colors represent different types of charges. The net charges of the chains increase from left to right: $Q_{PE}$ = 16, 22, 28 and 34 for PE chains and $Q_{PA}$ = 12, 16, 20 and 24 for PA. (See, Supplementary Materials for simulation model).

Figure 3

Density of states $p(x,y)$ of a 2-pearl structure of PE chain consisting of N = 202 monomers and net charge Q = 22. Charge sequences have positive charge correlation with average block length 4. The color bar in [0,1] scale represents the density of visited states realized in MD simulations. Relative excess mass $x=m/M_p-1/2$ and excess charge $y=q/Q_p-1/2$ measure the asymmetry between the two pearls. The region where $x>0$ corresponds to larger pearls, i.e., $x_1$, while the region where $x<0$ corresponds to smaller pearls, i.e., $x_2$. The line $x=y$ represents the uniform charge distribution where each pearl has a charge proportional to its mass. States below (above) the line indicate undercharged (overlarged) large (small) pearl states. Note that the data points exhibit point symmetry with respect to the origin, a property given by the constraints, as shown in Equation (2).

Figure 4

Density of states $p(m_s,q_s)$ of PE conformations in pearl necklaces where the number of pearls is n = 3. The states are evaluated according to the relative mass and charge allocated at strings, $m_s=N_s/N$ and $q_s=Q_s/Q$. Ensemble of PE chains consisting of N = 202 monomers and carrying net charges of (a) Q = 22 and (b) Q = 34 are considered. Each charge sequence has charge–charge correlations with the average block length of four. The color bar in [0,1] scale represents the density of visited states realized in MD simulations. With increasing net charges, more mass is allocated to the string. A greater population above the line $m_s=q_s$ indicates that strings carry more charges than what corresponds to their shares.

Figure 5

Simplex representations for PE with blocky charge sequences, Q = 22, N = 202. Mass asymmetry among n-pearl states are shown on $(n-1)$ simplex representation: 1-simplex representation for two pearl states, 2-simplex representation for three pearl states, and 3-simplex representations for four pearl states. The color bar in [0,1] scale represents the density of visited states realized in MD simulations. Representative conformations are shown for each $(n-1)$ simplex. Weakly charged PE occupies the conformational space near the vertex $V_{1,n-1}$ where one large pearl is dominant. In the presence of positive charge correlations, the asymmetry of the pearl structure is further accentuated compared to uncorrelated sequences, as demonstrated in Ref. [65]. Mass asymmetry of 4-pearl states can be shown as internal points in tetrahedron. A shaded triangular face of the tetrahedron corresponds to a 2-simplex. The thick orange line in a 2-simplex corresponds to 1-simplex. Each vertex $V_{n_l,n_s}$ of tetrahedron is identified with number $n_l$ of large pearls (shown as blue disks).

Figure 6

Structural changes accompanying changes in pearl number are shown for (a) PE and (b) PA. We retain the sequences from the sub-ensembles with the given net charges. (a) Fluctuation of PE with Q = 22 and N = 202. The conformation of pearl-necklaces is monitored with time interval $10\tau$ and the change of the number of pearls $n\left(t\right)$ over $230\tau$ is shown in the upper left panel. Small pearls appear and disappear on the tail/string. Snapshots are taken every 40$\tau$ ($\tau$ is the MD time unit.). (b) PA pearl-necklaces with Q = 20 (44 majority charges and 24 minority charges) and N = 202. The change in the number of pearls $n\left(t\right)$ is shown in the upper left panel over $600\tau$. Snapshots are taken every 100$\tau$ from MD simulations. In multiple simplexes (right panels), 1-pearl states ($n=1$) are represented as a point, the 2-pearl states ($n=2$) are depicted as a 1-simplex $x_1\in [0,0.5)$, 3-pearl states ($n=3$) are shown as triangular 2-simplex $\{x_1,x_2\}$. The states with a larger number of pearls ($n\ge$ 4) are shown as a point above the triangular 2-simplex for convenience. The chronologically numbered PE and PA forms are displayed in the corresponding simplex according to the number of pearls in the multiple simplexes. If the pearl number in the previous state is 1, 2, 3, or 4, the points on the simplex are displayed in blue, orange, green, and red, respectively. PA switches from the vicinity of V${}_{2,1}$ to the vicinity of V${}_{1,2}$ along a complex path through different number of pearls. A direct path inside the triangle (3-pearl states) is largely suppressed due to the very sparsely populated intermediate states within the triangle.