Chapter III: Polyelectrolyte Phase Behavior
3.1 Introduction
Polyelectrolytes are commonly found in both natural [1–5] and synthetic [6–12]
polymer systems. Many applications (i.e. food, [4, 13, 14] pharmaceutical, [15–21]
adhesives [8, 22–25]) of polyelectrolytes rely on their propensity to phase separate
into complexes, which can be fluid or solid, microscopic or macroscopic, and can often be structured. [20, 21, 26–29] In this chapter we focus on the electrostatic- correlation-induced liquid-liquid phase separation of polyelectrolytes in water or other polar solvent.
Electrostatic correlations in polyelectrolyte solutions continue to attract consid- erable theoretical efforts, as summarized in recent reviews.[28–30] One class of theories follows the spirit of Voorn and Overbeek (VO), and begins with a physical picture where the electrostatic correlations are based on an equivalent solution of disconnected charges. [31] The original VO work used the Debye-Hückel free en- ergy density[32] for the electrostatic correlation, while more recent work use more detailed thermodynamic models of simple electrolytes,[33] or include the leading connectivity effects via a first-order thermodynamic perturbation (TPT-1) about the simple electrolyte reference.[34–39]
Another class of theories attempts to account for chain connectivity from the out- set. [40–47] These theories use chain density correlation functions to characterize connectivity effects on electrostatic fluctuations. In the most basic form,[40, 44, 45] most commonly applied to flexible chains, a fixed, Gaussian chain structure (hereafter referred to as fg-RPA for fixed-Gaussian random phase approximation) is assumed for all concentrations. This approximation is only valid when the chain density is above ρ∗b3 ∼ (lbf2/b)1/3,[40] where lb/b characterizes the strength of electrostatic interactions and is O(1) for typical aqueous solutions. At a mod- est charge fraction f . 1 where no Manning condensation occurs, this condition requires very high densities ρ∗b3≈ 1.
For typical systems, the fg-RPA approximation greatly overestimates the correlation contribution to the free energy, especially at low concentrations.[46] One insightful attempt to correct the failings of the fg-RPA recognized that fluctuations in fg-RPA are treated improperly on short length scales. However, rather than tackling the root problem associated with the chain structure, this “modified RPA theory” advocated for completely suppressing the short length scale fg-RPA fluctuations through the use of ad hoc cutoffs. [46] Another theory that is shown to yield significantly improved predictions of the critical parameters and osmotic coefficients was based upon decomposing the system into two one-component plasmas [48, 49] and using a neutral semi-dilute solution as the reference structure factor. [43]
We recently proposed a renormalized Gaussian fluctuation (RGF) theory using a variational, non-perturbative framework, that naturally prescribes a self-consistent
calculation of the chain structure. [50] In light of the RGF theory, many qualitative shortcomings of the VO, TPT-1, and fg-RPA theories have been demonstrated to be connected to their incorrect treatment of the chain structure. [50] The RGF was shown to correctly capture the crossovers in overall chain scaling as function of salt and polymer concentration, similar to some previous work that self-consistently accounted for the concentration-dependence of chain structure. [50–57]
Using the RGF we further elucidated the effects of conformation-concentration cou- pling on thermodynamics. A key observation was that the electrostatic correlation per monomer for linear flexible chains can, at most, increase logarithmically in chain length (stemming from the electrostatic energy of a linear object, which scales as lnN per unit length). A consequence of capturing this expected N-dependence is that, when applied to salt-free solutions of a single polyelectrolyte species, the RGF correctly predicts that there exists a critical Bjerrum length below which the solution is stable for all chain lengths, in agreement with simulations [58] and in qualitative contrast to the prediction from the fg-RPA theory. Below this critical Bjerrum length, the entropic penalty of partitioning counterions to a dense phase is too great for any chain length to overcome.
In this chapter, we apply the RGF to study the influence of added salt on the phase behavior of polyelectrolyte systems, with a focus on the role of the chain structure.
An important conclusion is that small wavelength fluctuations are highly coupled to the chain structure at short length scales; stiff chains (semiflexible rods) have lower correlation energies and consequently narrower two-phase regions than flexible chains. Although in semidilute or high-salt solutions flexible polyelectrolytes are overall Gaussian, we find that their phase behavior is nevertheless much closer to that of semiflexible rods than that of ideal Gaussian chain. By renormalizing the chain structure, the RGF properly treats fluctuations on intrachain length scales and naturally suppresses the higher-wavenumber fluctuation modes without the introduction of an artificial cut-off as in “modified RPA". Thus, our theory can treat systems across a significantly wider density window and, importantly, can treat systems at much higher charge densities (up tolbf/b'1) than previously possible with the commonly used fg-RPA theory or any other field theories that involve a bare Gaussian chain structure approximation. While we would have liked to compare the RGF to predictions from the “modified RPA” theory, the latter was developed and illustrated only for salt-free polyelectrolyte solutions, and no attempt was made to give a quantitative estimate of the cutoff nor to generalize the single cut-off
to solutions with multiple polyelectrolytes or small ions, making fair quantitative comparisons difficult. We thus do not attempt to compare the RGF to the “modified RPA.”
The rest of the chapter is organized as follows. In Section 3.2 we present the essential results of the RGF relevant to bulk solutions. In Section 3.3 we examine the phase behavior of solutions of a single polyelectrolyte species, while in Section 3.4 we consider solutions with symmetric, oppositely charged polyelectrolytes. In both sections we compare phase diagrams predicted by the RGF for flexible chains with self-consistent chain structure to diagrams predicted for chains with fixed- Gaussian and semiflexible-rod structures. We follow in Section 3.5 by discussing the renormalization of chain structure and its effect on correlation energies. Finally, in Section 3.6 we conclude with a summary of the key results.