Dipole-driven interlude of mesomorphism in polyelectrolyte solutions

Significance

Control of self-assembled structures of charged macromolecules in aqueous solutions is vital in myriads of natural phenomena. In the simplest situation, uniformly charged polyelectrolyte chains in electrolyte solutions are homogeneously distributed due to their electrostatic repulsion. In stark contrast to this well-established result, we find that the same system exhibits rich phase behavior consisting of precipitation, an interlude of self-assembled mesomorphic structures that can self-poison, and homogeneous solution, in the presence of a small organic anion. Using conceptual arguments, our experiments reveal that this phenomenon originates from dipole–dipole interactions overwhelming charge–charge repulsion. The discovered principle of dipole-directed assembly is essential to understand and control macromolecular structures in the broader context of polyzwitterions, polyelectrolyte–surfactant complexation, coacervation, and membraneless organelles.

Abstract

Uniformly charged polyelectrolyte molecules disperse uniformly in aqueous electrolyte solutions, due to electrostatic repulsion between them. In stark contrast to this well-established result of homogeneous polyelectrolyte solutions, we report a phenomenon where an aqueous solution of positively charged poly(L-lysine) (PLL) exhibits precipitation of similarly charged macromolecules at low ionic strength and a homogeneous solution at very high ionic strength, with a stable mesomorphic state of spherical aggregates as an interlude between these two limits. The precipitation at lower ionic strengths that is orthogonal to the standard polyelectrolyte behavior and the emergence of the mesomorphic state are triggered by the presence of a monovalent small organic anion, acrylate, in the electrolyte solution. Using light scattering, we find that the hydrodynamic radius R_h of isolated PLL chains shrinks upon a decrease in electrolyte (NaBr) concentration, exhibiting the “anti-polyelectrolyte effect.” In addition, R_h of the aggregates in the mesomorphic state depends on PLL concentration c_p according to the scaling law, $R_h~c_p^{1/6}$. Furthermore, at higher PLL concentration, the mesomorphic aggregates disassemble by a self-poisoning mechanism. We conjecture that all these findings can be attributed to both intra- and interchain dipolar interactions arising from the transformation of polycationic PLL into a physical polyzwitterionic PLL at higher concentrations of acrylate. The reported phenomenon of PLL exhibiting dipole-directed assembly of mesomorphic states and the anti-polyelectrolyte effect are of vital importance toward understanding more complex situations such as coacervation and formation of biomolecular condensates.

Charged macromolecules in aqueous media display myriads of self-organized structures and functions in various biological systems that are vital to life as well as in technologies relevant to the healthcare industry (1, 2). Even the simplest situation of a uniformly charged polymer, such as DNA, poly(styrene sulfonate), or poly(L-lysine), etc., dispersed in an aqueous electrolyte solution exhibits a variety of phenomena that cannot be simply extrapolated from the fundamental concepts developed for uncharged macromolecules (3–5). This departure is due to the confluence of several forces emanating from long-ranged topological connectivity of the macromolecule, electrostatic and hydrophobic interactions among all species in the solution, and organization of solvent molecules and electrolyte ions around the macromolecule. The consequences of this confluence are even more amplified when multiple charged macromolecules can interact as in nondilute solutions or when different kinds of charged macromolecules are present.

One example of such complex scenarios is the coacervation phenomenon exhibited by oppositely charged polyelectrolytes leading to not yet fully understood liquid–liquid phase separation (LLPS) (6–10). Drawing an analogy with the LLPS in coacervate systems, there is growing interest in understanding the molecular basis of membraneless organelles (biomolecular condensates) formed by solutions of intrinsically disordered proteins as well as in the presence of RNA and crowding agents (11–17). Another class of complexing systems is constituted by oppositely charged polyelectrolytes and surfactants where the hydrophobic tails of the surfactants dominate over electrostatic interactions and generate well-ordered morphologies (18–22), which belongs to a different universality class from the LLPS of coacervation and biomolecular condensate formation.

The key physical molecular origin of self-assembly by macromolecules possessing oppositely charged domains is the formation of dipoles, where each dipole is made by two oppositely charged repeat units of the macromolecules. Furthermore, the dipoles from two ion pairs (created by two different chain domains) can form quadrupolar physical cross-links due to dipole–dipole interactions. An understanding of the role of dipolar interactions in these systems of intermolecular complexation is only at its infancy. Even in the simplest case of uniformly dipolar polymers such as polyzwitterions, where the focus has been on the effects arising from hydrophobic effects, the role of dipolar interactions in the formation of their intrinsic structures and their complexes with polyelectrolytes remains unexplored (23–30).

Toward the goal of understanding the role of dipolar interactions in macromolecular assemblies and to contrast with the more familiar polyelectrolyte behavior dictated by (monopolar) ionic groups, it is desirable to study the simplest system where only charge–charge and dipole–dipole interactions are dominant without additional complications from hydrophobicity effects and solvent quality. With this purpose, we have identified a simple monodisperse polycation and transformed it into a physical polyzwitterion (polydipole) using a monovalent organic counterion. Despite the simplicity of the chosen system, our experimental results reveal a different universality class of dipole-driven mesomorphism in polyelectrolyte complexation. While counterion release is the dominant driving force for complexation between polyelectrolytes, dipole–dipole attraction dominates complexation by physical polyzwitterions. In addition, the concepts originating from dipolar interactions form the basis for deeper understanding of the phase behavior of all other abovementioned systems of coacervation and biomolecular condensation of higher complexity.

In the present study, we have chosen poly(L-lysine) (PLL) as the cationic polyelectrolyte in NaBr solutions and challenged the polymer with sodium acrylate (NaAcr) where the counterion acrylate binds strongly with PLL. By tuning the charge ratio r of molar concentration of NaAcr to that of PLL, $r=[\text{NaAcr}]/[\text{PLLBr}]$, the electrostatic nature of PLL is transformed from polycationic at lower r to polydipolar (physical polyzwitterionic) at higher r, as sketched in Fig. 1A. Based on the pK_d of lysine–acrylate (31), the fraction f_b of the bound lysine–acrylate pair is given in Fig. 1B as a function of r (SI Appendix, section 3.1). The experimental variables are the degree of polymerization N and PLL concentration C_p, concentration of added sodium bromide C_s, and the molar charge ratio r. We have investigated two different values of degree of polymerization (N = 800 and 100, denoted as PLL800 and PLL100 with the corresponding molecular weights of 167 and 21 kDa, respectively) for C_p in the range of 1 to 15 mg/mL, C_s in the range of 0 to 5.3 M, and r in the range of 0 to 10, all at room temperature of 22 ${}^{°}$C.

Fig. 1.

(A) Schematic of acrylate counterion binding more strongly with PLL compared to bromide counterion and transformation of polycationic PLL into polydipolar physical polyzwitterion. r is the molar charge ratio of sodium acrylate to PLL bromide. (B) Fraction f_b of dipoles in the chemical equilibrium for the formation of PLL acrylate as a function of r (SI Appendix, section 3.1). (C) Emergence of a mesomorphic state at intermediate NaBr concentrations in solutions of PLL800 (C_p $=$ 1 mg/mL) containing sodium acrylate with r = 10. Three regimes are present: 1) precipitation at low salt concentrations, 2) intermediate mesomorphic state of spherical aggregates, and 3) unaggregated homogeneous state at high salt concentrations.

Results and Discussion

Phase Behavior of PLL–Acr Complexation.

First, in the absence of NaAcr, the behavior of PLL in NaBr solutions, as observed using static and dynamic light scattering, is characteristic of uniformly charged flexible isolated polyelectrolyte chains (3, 32–34). The PLL solutions with NaBr in the investigated ranges of C_p and C_s are transparent to the eye without any precipitation. As a typical example of PLL characterization (SI Appendix, Fig. S1), for 1 mg/mL PLL800 in 1.2 M NaBr solution, the diffusion coefficient D of PLL is $(1.14\pm 0.03)\times {10}^{-7}$ cm²/s, hydrodynamic radius R_h $=$ 19.3 nm, radius of gyration R_g $=$ 29.0 nm, and $R_g/R_h\simeq$ 1.5, indicating that the PLL chain conformation is a Gaussian coil. In stark contrast to the homogeneous optically clear behavior of such solutions, addition of NaAcr results in a rich behavior as a function of NaBr concentration. An example is portrayed in Fig. 1C for PLL800 at C_p $=$ 1 mg/mL and r = 10, exhibiting three regimes.

Regime 1. Unaggregated homogeneous state (high salt).

For $c_s\hspace{0.17em}>$ 1.4 M, the solution is homogeneous with a single diffusive mode of relaxation, and the PLL chains are in their isolated chain (random coil) conformation: for example, for C_s $=$ 2.4 M NaBr, R_h $=$ 19.2 nm, R_g $=$ 30.5 nm, and $R_g/R_h\simeq$ 1.5 (SI Appendix, Fig. S2). These values are the same within experimental error bars as in the case of absence of NaAcr. Thus, at higher NaBr concentrations, the dipolar nature of PLL is electrostatically screened by NaBr in this regime.

Regime 2. Intermediate mesomorphic state.

For 0.93 M $<C_s<$ 1.4 M, large-scale approximately monodisperse spherical structures are observed. For example, at C_s $=$ 1.2 M NaBr, the normalized electric-field correlation function $g_1(t)$ measured using dynamic light scattering (DLS) obeys single exponential decay with the correlation time t (Fig. 2A) with zero residuals, and the decay rate Γ is quadratic in the scattering wave vector q (Fig. 2B). From the slope of the line in Fig. 2B and the Stokes–Einstein relation, $D=(2.32\pm 0.04)\times {10}^{-8}$ cm²/s, and R_h $=$ 96.1 nm. Using static light-scattering intensity $I_s(q)$ and the Guinier plot $I_s(q)=I_s(0)]\exp (-q^2{R}_g^2/3)$, R_g $=$ 82 nm (Fig. 2C). As the NaBr concentration is increased from 0.96 M, the hydrodynamic radius of these aggregates decreases from 322.5 nm at C_s $=$ 0.96 M to 96.1 nm at C_s $=$ 1.2 M, as shown in Fig. 2D. Meanwhile, the shape factor is in the range of 0.8 to 0.85, indicating their spherical shape (Fig. 2D). Furthermore, the size distribution of these spherical aggregates is approximately monodisperse as shown in Fig. 2E. The aggregates in the mesomorphic state are extremely stable against centrifugation and supersonic shaking. Upon further increase in C_s to 1.5 M and above, these large-scale structures are unstable and single-chain behavior as in the PLL solutions without NaAcr is observed with R_h around 19 nm and $R_g/R_h$ around 1.5 to 1.6 (SI Appendix, Figs. S1 and S2). The transition from the intermediate state to the single-chain solution state at higher salt concentrations is continuous as observed from the static scattering intensity (SI Appendix, Fig. S3A). Concomitant to the formation of aggregates in the intermediate state, the static scattering intensity increases sharply upon lowering C_s (SI Appendix, Fig. S3A). The low salt concentration boundary between the mesomorphic state and precipitation is obtained by determining the C_s at which the extrapolated zero-angle scattering intensity diverges (SI Appendix, Fig. S3 B and C). The result of this boundary is C_s $=$ 0.93 M NaBr, at which the mesomorphic state becomes unstable against precipitation.

Fig. 2.

Dependence of hydrodynamic radius R_h, radius of gyration R_g, and the shape factor on NaBr concentration at $C_p=$ 1 mg/mL in the presence of sodium acrylate (r = 10). (A–C) C_s $=$ 1.2 M NaBr. (A) Normalized electric-field correlation function showing single exponential decay with correlation time (blue triangles denote the residuals for fitting the experimental data) at scattering angle 30${}^{°}$. (B) Relaxation rate is quadratic in scattering wave vector q, giving $D=(2.32\pm 0.04)\times {10}^{-8}$ cm²/s and R_h $=$ 96.1 nm. (C) Guinier analysis of static scattering intensity gives R_g $=$ 82 nm. (D) Plot of R_h and the shape factor $R_g/R_h$ versus NaBr concentration. The hydrodynamic radius decreases from 322.5 nm at 0.96 M NaBr to 96.1 nm at 1.2 M NaBr. In this mesomorphic regime, the aggregates are spherical as deduced from their value (0.8 to 0.85) of the shape factor. At NaBr concentrations at 1.5 M and higher, the large-scale structures become unstable and return to the single-chain random coil behavior (with shape factor of about 1.5 to 1.6) as observed in the absence of sodium acrylate. (E) Plot of the distribution function of the decay time (obtained from CONTIN analysis at scattering angle 30${}^{°}$) at different NaBr concentrations showing that the spherical aggregates are monodisperse. For comparison, the distribution function for single chains at 2.4 M NaBr is included.

Regime 3. Precipitation (low salt).

For $C_s\le$ 0.93 M, two coexisting phases are seen, analogous to phase separation of uncharged polymers in poor solvents (35). In the supernatant polymer-poor phase, the DLS signal is too weak to detect any polymer for C_s between 0 and 0.87 M, indicating that almost all polymer chains are in the precipitate. For C_s $=$ 0.9 and 0.93 M, single PLL chains can be detected using DLS, with $R_h=11.5$ and 10.7 nm, respectively (SI Appendix, Fig. S6). R_h at these low NaBr concentrations is significantly smaller than 19.2 nm observed at C_s $=$ 2.4 M NaBr. The shrinkage in the PLL chain size as C_s is lowered clearly indicates that it is not a polyelectrolyte effect. Such an “anti-polyelectrolyte effect” is congruent with our supposition that PLL–Acr functions as a physical polyzwitterion controlled by intrachain dipolar interactions. This feature is also in conformity with the formation of aggregates due to interchain dipolar interactions, which is discussed below.

Role of Charge Ratio and Transformation from Polycation to Physical Polyzwitterion.

For the same above example (Fig. 1B) of C_p $=$ 1 mg/mL and C_s $=$ 0.96 M NaBr, where the intermediate aggregate phase forms at r = 10, we have varied the value of r to monitor the transformation of polycationic PLL to polydipolar PLL. The number of relaxation modes and their corresponding diffusion coefficients have been determined using DLS. For r < 1 and r > 1, we found only one mode, and for r = 1, we found two relaxation modes in DLS (SI Appendix, Fig. S4). The corresponding diffusion coefficients and hydrodynamic radii are given in Fig. 3 A and B. For r < 1, D is around $1.12\times {10}^{-7}$ cm²/s and R_h $=$ 19.7 nm. These results are close to those in the absence of NaAcr (SI Appendix, Fig. S1), confirming that PLL behaves as a polycation for r < 1. For r = 1, where the molar charge concentrations of PLL and acrylate are roughly matched (Fig. 1B), there are two populations with R_h values of 15.9 and 104.2 nm. We attribute the smaller R_h (obtained from the fast mode in DLS) to unaggregated single chains and the larger value (obtained from the slow mode in DLS) to aggregated structures described above for regime 2. Consistent with results for the supernatant phase in regime 3, R_h of single chains is smaller than the value at r = 0 due to additional intrachain dipole–dipole attraction arising from the complexed lysine–acrylate ion pairs. This is also consistent with the observation that the shape factor $R_g/R_h$ for the unaggregated chain decreases from 1.5 at r = 0 to 1.3 at r = 1 (Fig. 3C). Thus, the chain conformation of the unaggregated chain changes from random coil to slightly shrunken coil. The value of $R_g/R_h$ for the aggregate at r = 1 is 0.81, indicating that the aggregate is roughly spherical. Analysis of DLS data using two exponential fits (with zero residuals) and evaluation of the relative areas for these two modes in the decay-time distribution functions show that the weight fractions of unaggregated polycationic chains and spherical aggregates from polyzwitterionic chains are roughly the same (52% single chains and 48% spherical aggregates) (Fig. 3D). The representation of the polycationic population decreases progressively as r increases from 0 to 2. When r > 1, only one relaxation mode corresponding to the mesomorphic state is observed. As shown in Fig. 3C, the value of $R_g/R_h$ is between 0.81 and 0.85, indicating again that these structures are spherical.

Fig. 3.

Effect of molar charge ratio r on the onset of the mesomorphic state at fixed C_p $=$ 1 mg/mL and C_s $=$ 0.96 M. (A–C) Diffusion coefficient D(A), hydrodynamic radius R_h (B), and shape factor $R_g/R_h$ versus r (C). For $r<1$, there is only one diffusion coefficient corresponding to polycationic chains with random coil conformation. At r = 1, slightly shrunken polycationic chains coexist with large spherical aggregates in a single phase. For r > 1, only one diffusive mode corresponding to the mesomorphic state is observed. (C, Inset) $R_g/R_h$ for the aggregates. (D) Distribution function of decay time in DLS obtained from CONTIN fit at scattering angle ${40}^{°}$ for $r=0.1,1,$ and 2.

Discussion.

All of the above experimental results lead to the conclusion that a mesomorphic state made of large-scale spherical aggregates intervenes between the two-phase liquid–liquid phase separation at lower salt concentrations and a homogeneous solution containing individual chains at higher salt concentrations. In the supernatant solutions and homogeneous solutions outside the mesomorphic regime, the experimental results suggest that individual chains undergo globule-to-coil transition upon an increase in salt concentration exhibiting the anti-polyelectrolyte effect, which is a hallmark of polyzwitterions. The mesomorphic state is stable only if $r\ge 1$. To explain these findings, we propose the phase diagram sketched in Fig. 4 and address the salient features using a mean-field theory with predictions on the dependence of the mesomorphic state on the concentrations of PLL and NaBr, consistent with the above experimental findings.

Fig. 4.

Sketch of $C_s-C_p$ phase diagram for PLL–acrylate complexation and the role of intra- and interchain dipole–dipole interactions.

As described in Fig. 1A, complexation between acrylate ions and the lysine repeat units of PLL results in the transformation of polycationic PLL into a physical polyzwitterion, where each repeat unit (ion pair) is a dipole. Thus, each polylysine chain functions as a polydipolar chain instead of a polyelectrolyte. Since dipole–dipole interactions are attractive and become progressively weaker as the salt concentration C_s increases due to electrostatic screening, dipolar chains in dilute solutions can exhibit globule-to-coil transition as C_s is increased. In addition, presence of salt can weaken dipole formation to begin with. Furthermore, as the PLL concentration is increased multiple chains can aggregate where the interchain physical junctions arise from the quadrupoles formed by pairs of ion pairs. Therefore, we envisage a critical aggregation concentration (CAC) of PLL for the aggregate formation. Since the dipole–dipole interaction becomes weaker at higher salt concentrations, CAC is expected to increase with an increase in C_s. Upon further increase in PLL concentration, the physical aggregates become bigger and eventually reach a percolation threshold beyond which the mesomorphic aggregates become a physical gel. The PLL concentration at the percolation threshold is expected to increase with C_s due to electrostatic screening. At lower salt concentrations, the dipole–dipole interactions are so strong that the solution undergoes macrophase separation into polymer-rich and polymer-poor phases. The above dipole-dominated mechanism for the emergence of the mesomorphic intermediate aggregation state in the complexation of the PLL–acrylate system is sketched in Fig. 4. As indicated in Fig. 4, the polymer-rich phase at lower salt concentrations is a gel phase and the meeting point of the percolation line and the coexistence curve for the polymer-rich and polymer-poor phases is reminiscent of the tricritical point in the phase behavior of associating polymers (36). The rich unexpected phenomenon described above is so complex that theoretical formulation at the monomer-level details is prohibitively difficult at the current state of the art in polymer theory. Here, we present a qualitative argument that captures qualitative (scaling-law–like) features of the emergence of the mesomorphic state by focusing on the characteristics of the dipole-driven physical aggregates and isolated chains as functions of C_p and C_s.

Isolated single chains.

The Helmholtz free energy F₁ of a single dipolar chain of N segments, each of length $\mathcal{l}$ and dipole moment $\overrightarrow{p}$, is given by

$$\begin{array}{l}{e}^{-F_1/k_{B}T}=\hspace{0.17em}{\displaystyle \int \mathcal{D}}[\overrightarrow{R}(s)]\exp \{-\frac{3}{2{\mathcal{l}}^2}{\displaystyle {\int }_0^{N}d}s{\left(\frac{\partial \overrightarrow{R}(s)}{\partial s}\right)}^2\\ -\frac{1}{2}{\displaystyle {\int }_0^{N}d}s{\displaystyle {\int }_0^{N}d}s\prime [U_{\text{exc}}(\overrightarrow{R}(s)-\overrightarrow{R}(s\prime ))\\ +U_{dd}(\overrightarrow{R}(s)-\overrightarrow{R}(s\prime ))]\},\end{array}$$ [1]

where $\overrightarrow{R}(s)$ denotes the position vector of the sth segment and the functional integral denotes the sum over all allowed conformations (and k_BT is the Boltzmann constant times absolute temperature). The first term, $U_{\text{exc}}$, and $U_{dd}$ in the exponential denote chain connectivity, short-range excluded volume interaction, and dipole–dipole interactions, respectively. Using the standard procedure (35), the excluded volume interaction is parameterized as

$$U_{\text{exc}}(\overrightarrow{r})=v{\mathcal{l}}^3\delta (\overrightarrow{r})=(1-2\chi ){\mathcal{l}}^3\delta (\overrightarrow{r}),$$ [2]

where χ is the Flory–Huggins parameter. Using an equivalent procedure (high-temperature expansion) of parameterizing the interactions between two randomly oriented dipoles of dipole moment pe (e is the electronic charge), $U_{dd}(\overrightarrow{r})$ is given as $v_{dd}\delta (\overrightarrow{r})$, where (4, 37)

$$v_{dd}=-\frac{\pi }{9}\frac{{\mathcal{l}}_B^2{p}^4}{{\mathcal{l}}^6}{e}^{-2\kappa \mathcal{l}}\left[4+8\kappa \mathcal{l}+4{(\kappa \mathcal{l})}^2+{(\kappa \mathcal{l})}^3\right],$$ [3]

where ${\mathcal{l}}_B$ is the Bjerrum length ($=e^2/4\pi {ϵ}_0ϵk_{B}T$), ϵ₀ is the permittivity of vacuum, ϵ is the dielectric constant of the solvent, and κ is the inverse Debye length ($\kappa ~{\sqrt{C}}_s$). Analogous to the definition of the excluded volume parameter v, v_dd is a pseudopotential parameterizing segmental orientations and dipole orientations by denoting the intermonomer second virial coefficient at high temperatures. v_dd is attractive and its strength becomes weaker as C_s is increased. In the following discussion we illustrate results by choosing only one set of values for the parameters (p = 1 nm, $\mathcal{l}=1$ nm, ${\mathcal{l}}_B=0.7$ nm, and the shortest distance between two antiparallel distances r = 0.25 nm), because we are interested in only the qualitative trends associated with the phenomenon. The dependence of v_dd on C_s is given in SI Appendix, Fig. S8. Using a variational procedure (38), Eq. 1 yields a modified Flory result for F₁ for a chain with end-to-end distance R as

$$\begin{array}{l}\frac{F_1}{k_{B}T}=\frac{3}{2}\left(\frac{R^2}{N{\mathcal{l}}^2}-1-\ln \frac{R^2}{N{\mathcal{l}}^2}\right)\\ \hspace{1em}+\frac{4}{3}{\left(\frac{3}{2\pi }\right)}^{3/2}\frac{(v+v_{dd})N^2{\mathcal{l}}^3}{R^3}+w\frac{N^3{\mathcal{l}}^6}{R^6},\end{array}$$ [4]

where w denotes the three-body interaction parameter introduced to stabilize the globular state against collapsing into unphysical size. Minimization of the above equation with respect to R gives the equilibrium value of R as

$$\begin{array}{l}{\left(\frac{R^2}{N{\mathcal{l}}^2}\right)}^{5/2}-{\left(\frac{R^2}{N{\mathcal{l}}^2}\right)}^{3/2}=\frac{4}{3}{\left(\frac{3}{2\pi }\right)}^{3/2}(v+v_{dd})\sqrt{N}\\ \hspace{1em}+2w{\left(\frac{N{\mathcal{l}}^2}{R^2}\right)}^{3/2}.\end{array}$$ [5]

The free energy F₁ of the equilibrated dipolar chain in the limits of $v+v_{dd}>0$ and $v+v_{dd}<0$ is given by Eqs. 4 and 5 as (SI Appendix)

$$\frac{F_1}{k_{B}T}=\{\begin{array}{ll}1.8{(v+v_{dd})}^{3/5}{N}^{1/5},\hfill & v+v_{dd}>0\hfill \\ \frac{3}{4w}|v+v_{dd}{|}^{2}N,\hfill & v+v_{dd}<0\hfill \end{array}.$$ [6]

For $v>0$ and v_dd < 0, Eq. 5 shows that the chain undergoes coil-to-globule transition when C_s is reduced, as the anti-polyelectrolyte effect, which is validated by experiments as shown below (Fig. 5B).

Fig. 5.

(A) Visual display of regimes 1 and 3 and absence of the mesomorphic regime 2 for PLL100 at C_p $=$ 1 g/L and r = 10. (B) Dependence of R_h of unaggregated chains on NaBr concentration exhibiting the anti-polyelectrolyte effect.

Aggregated mesomorphic state.

When chains intermingle, a quadrupole formed by two interchain dipolar segments at a separation distance r between their centers can act as a physical cross-link. The energy ($-u_0$) of this quadrupole is typically in the range of $10k_{B}T$ in aqueous solutions. As an example, using the abovementioned single choice of the parameter values, the C_s dependence of u₀ for antiparallel orientation of dipoles is given in SI Appendix, Fig. S9. If u₀ is sufficiently strong, multichain aggregates can form with many quadrupole junctions. Following the works of Tanaka (39) and Semenov and Rubinstein (40), the fraction of dipoles p_q associated as quadrupoles is given by $p_q/{(1-p_q)}^2=f{\varphi }_0\exp (u_0)$ (SI Appendix, sections 3 and 4), where f is the fraction of the dipoles that have correct antiparallel orientation to form quadrupoles, and ${\varphi }_0(=1/\sqrt{N})$ (41) is the volume fraction of a chain in its ideal state of Gaussian chain statistics. The number of cross-links N_c in the system is $(p_q/2)f{N}_0$ (where N₀ is the total number of monomers) and the minimum number of monomers per chain to form a network is $N_{\text{min}}=1/(f{p}_q)$ (35) (SI Appendix, sections 3 and 4). Using this Gaussian network as the reference state, and assuming that the time required for the dissociation of the whole collection of quadrupoles (sticky Rouse time) is sufficiently long compared to the time required for reaching osmotic equilibrium, we follow the classical Flory–Dusek–Patterson–Tanaka theory (35, 41, 42) to obtain the free energy F_n of an osmotically equilibrated aggregate with $n=2N_c$ elastically effective strands by considering energy gain due to quadrupoles, free energy of mixing, elasticity of strands, and dipole interactions among all un–cross-linked monomers. These contributions are, respectively, given by (35, 36, 39–49)

$$\begin{array}{l}\frac{F_n}{k_{B}T}=-\frac{n}{2}{u}_0+\frac{V}{v_1}\left[(1-\varphi )\ln (1-\varphi )+\chi \varphi (1-\varphi )\right]\\ \hspace{1em}+\frac{3n}{2}\left[{\varphi }_0^{2/3}{\varphi }^{-2/3}-1+\frac{1}{3}\ln (\varphi /{\varphi }_0)\right]+\frac{1}{2}\frac{V}{v_1}{v}_{dd}{\varphi }^2,\end{array}$$ [7]

where $\varphi$ is the volume fraction of the polymer in the aggregate of volume V, and v₁ is the volume of a solvent molecule. Minimization of F_n with respect to $\varphi$ gives the internal volume fraction of the polymer in the equilibrated aggregate as (35)

$${\varphi }_{\text{agg}}={\gamma }^{1/5}{N}_e^{-4/5}{\left(\frac{1}{2}-\chi +\frac{1}{2}{v}_{dd}\right)}^{-3/5}.$$ [8]

Here, N_e is the number of monomers per cross-linked monomers and $\gamma =N_e/N$. Substitution of the above equation in Eq. 7 gives the free energy of equilibrated aggregates in the physically relevant limit of small $\varphi$ as

$$\begin{array}{l}\frac{F_n}{k_{B}T}=-\frac{n}{2}{u}_0+\frac{5}{2}n{\gamma }^{1/5}{\left(\frac{1}{2}-\chi +\frac{1}{2}{v}_{dd}\right)}^{2/5}{N}_e^{1/5}\\ \hspace{1em}+\text{constant}.\end{array}$$ [9]

Using the standard theory of micellization (and aggregation) (50), the mole fraction X_n of aggregates with n elastically effective chains is given by

$$X_n=n{\left[X_1{e}^{\left(F_1-\frac{F_n}{n}\right)/k_{B}T}\right]}^n,$$ [10]

with the constraint of conservation of the total mole fraction X as $X={\displaystyle {\sum }_{n=1}^{\infty }{X}_n}$. Choosing the constant term such that $F_n\to F_1$ for n = 1, we get (SI Appendix, sections 3 and 4)

$$\frac{1}{k_{B}T}\left(F_1-\frac{F_n}{n}\right)=\left(1-\frac{1}{n}\right)\mathrm{\Theta },$$ [11]

where Θ follows from Eqs. 6 and 9, for $v+v_{dd}>0$, as

$$\mathrm{\Theta }=\frac{u_0}{2}+\left(2.375-2.5{\gamma }^{1/5}\right){\left(\frac{1}{2}-\chi +\frac{1}{2}{v}_{dd}\right)}^{2/5}{N}_e^{1/5}.$$ [12]

Substituting Eq. 11 in Eq. 10, we obtain

$$X_n=n{\left(X_1{e}^{\mathrm{\Theta }}\right)}^n{e}^{-\mathrm{\Theta }};\hspace{0.17em}{X}_1=\frac{(1+2X{e}^{\mathrm{\Theta }})-\sqrt{1+4X{e}^{\mathrm{\Theta }}}}{2X{e}^{2\mathrm{\Theta }}}.$$ [13]

The populations of aggregates and unaggregated chains for a given mole fraction X of the polymer are obtained from this equation in terms of Θ given in Eq. 12. This shows that the critical aggregation concentration CAC occurs at $X{e}^{\mathrm{\Theta }}=1$, and above CAC stable aggregates form. As an example, see SI Appendix, Fig. S11A. As C_s increases, v_dd becomes less negative so that Θ decreases (as an example, see SI Appendix, Fig. S10). Therefore, CAC increases in conformity with the proposed phase diagram in Fig. 4. According to the classical theory of micellization (50), for $X>X_{\text{CAC}}$, the average number of chains in the aggregate for the particular shape of Eq. 11 is

$$\langle n\rangle =2\sqrt{X}{e}^{\mathrm{\Theta }/2},$$ [14]

where X is proportional to polymer concentration C_p in the parent solution. Since Θ decreases with C_s, the average number of chains in the aggregate decreases with C_s. Furthermore, as mentioned above, aggregates do not form for N values smaller than $N_{\text{min}}=1/(f{p}_q)$. Taking the polymer volume fraction in the aggregate as

$${\varphi }_{\text{agg}}=\frac{\langle n\rangle N_e{\mathcal{l}}^3}{\frac{4}{3}\pi R_{g,\text{agg}}^3},$$ [15]

where $R_{g,\text{agg}}$ is the radius of gyration of the aggregate and using Eq. 8, $R_{g,\text{agg}}$ follows as

$$R_{g,\text{agg}}~C_p^{1/6},$$ [16]

where the proportionality factor depends on $\chi ,N$, and C_s. The theoretical predictions of the anti-polyelectrolyte effect for PLL single chains, $R_{g,\text{agg}}~C_p^{1/6}$, and absence of mesomorphic state for low molecular weight PLL are consistent with experiments described below.

Anti-Polyelectrolyte Coil-to-Globule Transition.

The above-described experiments resulting in Figs. 1C and 2 for PLL800 (N is 800, M_w $=$ 167 kDa) were repeated with PLL100 ($N=100,M_w$ $=$ 21 kDa) as a function of C_s at fixed C_p $=$ 1 g/L and r = 10. As a reference, R_h of PLL100 alone with no NaAcr at C_s $=$ 2.4 M NaBr is 3.7 nm. Analogous to the situation with PLL800 (Fig. 1C), regime 3 of precipitation occurs for PLL100 at low C_s exhibiting coexistence of a polymer-rich phase and the supernatant polymer-poor phase (Fig. 5A). For C_s below 0.72 M NaBr, no light-scattering signal can be detected in the supernatant solution, indicating that most of the PLL100 chains are in the polymer-rich phase. At C_s $=$ 0.72 M, R_h of PLL100 in the supernatant solution is 2.8 nm. This value clearly indicates that the chain is shrunken in comparison with its size of 3.7 nm observed in the reference state of C_s $=$ 2.4 M and r = 0. As C_s is increased from 0.72 to 1.08 M, the amount of the polymer-rich phase decreases and R_h of the chain in the supernatant solution also increases, as shown in Fig. 5B.

Remarkably, and in stark contrast with Fig. 1C, at C_s $=$ 1.08 M and above, the whole system becomes transparent to the eye without any symptom of the mesomorphic state. For this lower molecular weight, there are only regimes 1 and 3, and regime 2 is absent, indicating that a critical chain length of PLL is required for the emergence of the mesomorphic regime consistent with the above argument. Upon further increase beyond C_s $=$ 1.08 M, R_h increases toward the plateau (Fig. 5B). A summary of the C_s dependence of R_h in the supernatant solution for $C_s\le 1.08$ M and in the homogeneous solution for $C_s>1.08$ M is given in Fig. 5B. As evident from Fig. 5B, the PLL100 chains shrink upon lowering C_s, exhibiting the anti-polyelectrolyte effect, which is analogous to the behavior in the supernatant solutions in regime 3 of PLL800.

Polymer Concentration Dependence of Mesomorphism and Self-Poisoning Effect.

Using DLS, we have examined the sensitivity of the mesomorphic state to PLL800 concentration (in the range of 1 to 15 g/L) at fixed NaBr concentration C_s $=$ 1.2 M and molar charge ratio r = 10. The dependence of the hydrodynamic radius as determined from DLS on C_p is summarized in Fig. 6A. For C_p up to 3 g/L, there is only one relaxation mode and the corresponding R_h values are in the range of 64 to 115 nm. Remarkably, in this range, R_h of the aggregate depends on C_p following the power law, $R_h~C_p^{0.17}$, in agreement with Eq. 16. We label this regime the mesomorphic regime. When C_p is increased to 5 to 8 g/L, R_h of the mesomorphic structures becomes increasingly larger in the range of 164 to 296 nm, indicating merger of several aggregates, and we label this regime the coalescence regime. Upon further increase in C_p to 10 g/L and above, DLS exhibits two diffusive modes (fast and slow) of relaxation (SI Appendix, Fig. S5). For C_p $=$ 10 g/L, the corresponding hydrodynamic radii are $R_{h1}$ $=$ 14 nm and $R_{h2}$ $=$ 287.6 nm, as shown in Fig. 6A. For C_p $=$ 15 g/L, the hydrodynamic radii are $R_{h1}$ $=$ 10 nm and $R_{h2}$ $=$ 116.5 nm. We attribute the fast mode to unaggregated shrunken chains and the slow mode to the mesomorphic structure. By evaluating the relative areas for the two modes in the decay time distribution function (Fig. 6B), the weight fraction of the slow mode is found to decrease from 89% at C_p $=$ 10 g/L to only 26% at C_p $=$ 15 g/L. Therefore, the mesomorphic structures in the coalescence regime are becoming progressively self-poisoned upon an increase in polymer concentration and disassemble into individual unaggregated chains and smaller aggregates. We label this regime at $C_p\ge 10$ g/L the self-poisoning regime.

Fig. 6.

(A) Double-logarithmic plot of the dependence of R_h on PLL800 concentration C_p at molar charge ratio r = 10 and C_s $=$ 1.2 M NaBr. Three regimes of mesomorphic state, coalescence, and self-poisoning emerge progressively with an increase in C_p. In the mesomorphic regime, $R_h~C_p^{0.17}$ in agreement with the theoretical prediction of Eq. 16. In the coalescence regime, several aggregates merge together. In the self-poisoning regime, there are two relaxation modes. The red circles denote the fast mode corresponding to unaggregated chains and the black squares denote the slow mode corresponding to aggregated structures. (B) Decay time distribution functions at scattering angle ${30}^{°}$ for C_p $=$ 10 and 15 g/L showing the diminishing proportion and the characteristic decay time of the self-poisoned aggregates at higher C_p.

The origin of self-poisoning of the mesomorphic state lies in additional electrostatic screening of dipole–dipole interactions. Since the molar charge ratio is fixed at r = 10, as C_p is increased, the sodium acrylate concentration is also increased proportionately with a huge excess of sodium acrylate beyond the required amount to transform the polycationic PLL into physical polyzwitterions. Thus, the excess sodium acrylate functions as an additional salt (just like NaBr) and considerably decreases the electrostatic screening length (Debye length). Due to this electrostatic screening, the energy of quadrupoles from ion pairs is weakened, prohibiting the interchain associations of PLL–Acr. We attribute this electrostatic screening from the excess sodium acrylate to the self-poisoning mechanism.

Conclusions

It is well known that solutions of cationic PLL in water containing dissociated NaBr salt are homogeneous without any tendency to phase separate. This feature and the accompanying chain swelling upon a decrease in salt concentration C_s arise from electrostatic repulsion between the various repeat units. We show that such a typical behavior of similarly charged polyelectrolytes in aqueous solutions is completely transformed into a different set of phenomena when a small organic anion, acrylate, is introduced as a strong counterion to adsorb on PLL. Presence of adequate acrylate ions transforms the monovalent charges of lysine repeat units into ion pairs (dipoles). As a result, polycationic PLL is physically transformed into a poly(dipole) that can function as a physical polyzwitterion. Instead of the electrostatic repulsion between the monomers in the polycationic form, the polydipoles are subjected to both intra- and interchain dipole–dipole attractions, which lead to rich phase behavior as a function of C_s. At very low C_s, the PLL solutions exhibit phase separation, while at very high C_s, the PLL solutions are homogeneous. At intermediate C_s, stable mesomorphic states made of essentially uniform spherical aggregates emerge.

We have interpreted the emergence of the mesomorphic state and the anti-polyelectrolyte coil-to-globule transition using an approximate theory to account for dipole–dipole interactions. A scaling law for the size of the mesomorphic structures, $R_{g,\text{agg}}~C_p^{1/6}$, is predicted consistent with experimental results. The mean-field theory used in deriving this power law might not be airtight, and it is hoped that the present results would inspire more rigorous theories in the future. Upon further increase in C_p, the mesomorphic structures can coalesce into larger structures, but then self-poison themselves due to electrostatic screening by the large excess acrylate ions before reaching the gelation condition. The additional scope of the dependence of the phase behavior on temperature and variations in the large experimental parameter space is relegated to future work.

Our results emphasize the importance of dipolar interactions even in nominal uniformly charged polyelectrolyte systems. The dipole–dipole interactions are omnipresent in the broader contexts involving intrinsically disordered proteins, chemical polyzwitterions, complexation between oppositely charged polymers, polyelectrolyte–surfactant complexes, and coacervation in general. The present discovery of dipole-driven rich phase behavior and mesomorphic self-assembly of physical polyzwitterions provides additional opportunities to formulate more rigorous theories and to better understand liquid–liquid phase separation in biological systems such as membraneless organelles as well as myriads of complexation phenomena involving surfactants, chemical polyzwitterions, proteins, DNA, RNA, and synthetic polyelectrolytes.

Materials and Methods

More detailed descriptions of data analysis methods, additional figures for light-scattering data, details for derivation of equations, and typical qualitative results predicted by the conjectured model are provided in SI Appendix.

Materials.

Poly(L-lysine hydrobromide) with $M_{\text{w}}=167$ kDa (polydispersity index $=$ 1.06), 21 kDa (polydispersity index $=$ 1.07), and 2,100 Da was bought from Alamanda Polymers and was used as received. Sodium acrylate (NaAc) was purchased from Sigma-Aldrich and used as received. Sodium bromide was obtained from Acros Organics. Deionized water was obtained from a Milli-Q water purification system (Millipore). The resistivity of deionized water used was 18.2 M$\mathrm{\Omega }\hspace{0.17em}·\hspace{0.17em}$cm. Hydrophilic polyvinylidene fluoride (PVDF) filters with pore size 220 nm were purchased from Millex Company.

Methods.

Sample preparation.

Light-scattering experiments are extremely sensitive to dust. For the DLS tubes to be dust-free, they were first washed several times in deionized water and acetone separately, and then they were dried in the oven overnight. Then the dried tubes were wrapped in aluminum foil and were mounted on an acetone fountain setup and further cleaned by distilled acetone, where each tube was thoroughly washed out by the distilled acetone for 15 min and then dried in the oven overnight to obtain dust-free tubes. For light-scattering measurements, each species with a specific amount was filtered by a hydrophilic PVDF filter with pore size of 220 nm into a dust-free tube and then the samples were sonicated for 5 min. The sample preparation work was conducted in a superclean bench to avoid the dust. After preparation, all the samples were equilibrated for 2 wk at room temperature before the measurements.

Light-scattering measurement.

Dynamic and static light-scattering measurements were performed on a commercial ALV/CGS-3 Compact Goniometer System equipped with a multidigital time correlator (ALV/LSE-4) purchased from ALV-GmbH. The laser source is the solid-state laser with wavelength 660 nm and an output power of 120 mW (Cobolt Flamenco; 660 nm). The light scattering cell is held in an index-matching thermostat vat filled with purified and dust-free toluene. For each sample, the intensity at the different scattering angles was correlated, and each data point was obtained by averaging over three samples. All the experiments were conducted at 22 ${}^{°}$C. For data analysis, both the multiple exponential fitting method and the CONTIN method were used. The details of the fitting methods are in SI Appendix, section 1.

Data, Materials, and Software Availability

All study data are included in the article and SI Appendix.