Chapter IV: Ionic Atmosphere vs. Counterion Condensation: From Weak to
4.1 Introduction
Charge correlations underpin the phenomenology and behavior of charged macro- molecules. A particularly provocative effect is when the correlations of the small ions lead to effective attraction [1, 2] and even phase-separation between like- charged macromolecules.[3–7] Physically, charge correlations reflect the fact that
charges are preferentially surrounded by opposite charges (conversely, like-charges are repelled), reducing the effective charge seen away from the charge.
The earliest acknowledgment of such a charge correlation effect is by Debye and Hückel (DH), who calculated the linear density response of simple electrolytes about tagged charges;[8] this formalized the notion of an “ionic atmosphere” and the screened Coulomb (Yukawa) potential. The DH linear response calculation can be applied to integrate out small ion degrees of freedom in colloidal and polymer systems,[5, 8–10] and results in the same pairwise-additive screened Coulomb po- tential between macromolecular charges. We note that the linear response approach, also known as the Random Phase Approximation can also be used to integrate out macromolecular degrees of freedom.[11–13] In this chapter, we will formally re- fer to the electrostatic response calculated via linear response approaches as the
“linearized fluctuations” or more informally the (generalized) “ionic atmosphere.”
It is not hard to see that such linear response theories will be inadequate when stronger electrostatic interactions and correlations are involved. At high enough in- teraction strengths there will necessarily be counterion condensation, an effect first elucidated by Manning and Oosawa for charged polymers, or polyelectrolytes.[14–
18] They idealized the polyelectrolyte as an infinitely long cylinder and found that above a certain linear charge density, counterions are condensed onto the polyelec- trolyte and renormalize the effective charge of the cylinder.
Leading-order effects of counterion condensation on thermodynamic quantities like the osmotic coefficient can be readily understood – the condensed counterions lose their translational entropy, resulting in an osmotic coefficient plateau proportional to the fraction of uncondensed counterions; conversely the release of counterions leads to a free energy change of kT per released ion. Overall, counterion condensation has informed many studies of polyelectrolytes – it serves as a way to reason about or heuristically treat the strong-correlations between counterions and highly charged polyelectrolytes, reducing the system to a simpler one with only modestly charged polyelectrolytes.[19–22]
Due to the historic importance and wide application of DH theory, the ionic atmo- sphere is often used synonymously with electrostatic correlations, but we emphasize again that both counterion condensation and the linearized electrostatic response that underpins the DH ionic atmosphere are differentformsof electrostatic correlations.
Even once the translational entropy reduction effect of counterion condensation is considered, the linearized electrostatic fluctuations (of both small ion and macro-
molecular degrees of freedom), which also reduce the osmotic pressure, must still be accounted for in order to produce effective attractions and/or phase separation.[4, 5, 23, 24]
A complicating factor for flexible polyelectrolytes is that, well before counterion condensation sets in, electrostatic fluctuations and correlations first act to change the conformation of polyelectrolyte chains,[23, 25–27] which can also significantly affect thermodynamic properties. Intriguingly, after the renormalization of the chain conformations, the remaining linear-response fluctuations of unbound counterions and electrostatically stiffened chains can produce thermodynamic signatures charac- teristic of counterion condensation, such as the aforementioned osmotic coefficient plateau.[23, 27] This raises a conceptual question: if both counterion condensation and the ionic atmosphere characterize the accumulation of opposite charges next to each other, and both effects result in similar thermodynamic effects, how should one distinguish between the two? It is an outstanding challenge to construct a theory that self-consistently accounts for the many different ways that charge correlations can manifest: from weak-coupling ionic atmospheres, [5, 8–10] to chain conformational changes that accompany intermediate couplings,[23, 25, 26] to the onset of strong coupling effects like counterion condensation.[14–18]
The most common theoretical treatment of counterion condensation introduces a tightly-bound population of counterions in the vincinity of monomers, alongside an “unbound” population.[9, 12, 28–32] Theories that invoke a tightly-bound pop- ulation of counterions often introduce the notion of a local binding constant. The binding constant can range from a phenomenological parameter,[33] to an intrinsic piece describing remaining effects after others (i.e. VO-type screening, excluded volume capture) have been estimated,[32] to identification with a bare Coulomb in- teraction energy or backbone-solvent dielectric mismatch.[12, 30, 31] The simplest estimate of the binding constant lnKestimate it to be driven by an interaction energy of orderlb/σ, theunscreenedelectrostatic interaction of two charges,[12] where lbis the Bjerrum length characterizing the strength of electrostatic interactions and σ is the average ion size. However, this estimate neglects the long-ranged nature of electrostatics and chain connectivity – there will necessarily be interactions with other monomers in the chain and condensed counterions, meaning that counterion binding should be a non-local and cooperative binding effect that reflects chain connectivity and conformations.
After introducing a description of counterion condensation, it is necessary for a the-
ory to account for the correlations of uncondensed charges. (We again note previous work’s imprecise terminology separating counterion condensation and electrostatic correlations) These remaining electrostatic correlations have been modeled using a variety of approaches, including Voorn-Overbeek-type approximations [32], Ran- dom Phase Approximation [12], and double screening theory [30, 31]).
However, there are still two important contributions to electrostatic correlations. As discussed, one of these important effects is chain conformational changes – this is not a widely appreciated fact and has seen only limited study.[30, 31] Secondly, the electrostatic fluctuations of thecondensedcharges are also important, and will necessitate a more detailed consideration of the charge structure of the bound state.
These residual charge fluctuations are important when treating the ion pairing and phase separation of simple electrolytes,[34] and they have also been implicated in the coil-globule phase transitions of polyelectrolytes.[35–39] Yet for simplicity, the tightly-bound counterion population is often accounted for by identically neutraliz- ing both the condensed charges and sites.[12, 29, 31, 32] Such a strict neutralization of the counterion and backbone charges eliminates much of the charge fluctuations of the counterion and its condensation site. To recover these residual charge fluc- tuation effects, theories have had to supplement their free energy expressions with density-functional expressions,[37, 40] or with added dipole interaction terms.[30, 39]
In this chapter, we strive for a unified exposition of electrostatic interactions that accounts for the aforementioned effects: chain connectivity and fluctuations ofboth condensed and uncondensed charges. In section 4.2 we describe how to combine fluctuations, described using the recently-proposed renormalized Gaussian fluctu- ation (RGF) framework, which can account for changes in chain structure, with a tight-binding picture. In the vein of colloid charge-renormalization theories,[4, 5, 41–44] we give a thermodynamic interpretation for the loosely bound counterions as those providing the linear density response to the (conformation-adapted)[23, 27]
macromolecule and its strongly “bound” population of condensed counterions. Our theory allows us to separate the thermodynamic effects associated with the transla- tional entropy loss of bound counterions from the remaining linearized fluctuations, and compare their relative contributions.
To understand the fluctuations of condensed charges we relax the assumption of identically neutralizing the condensed charges and sites. We retain the discreteness of both the counterion and the charged site, and each condensed pair is characterized
by a length parameter d and in the far field contributes a dipolar electrostatic response; no new dipole interaction terms need to be introduced. Small values ofd suppress charge fluctuations and is expected to stabilize the solution against phase separation, while larger (but still reasonable) values ofdallow for phase separation.
The extent of residual charge fluctuations is thus seen to be an important determinant of phase separation. We also compare to models with continuous backbones and either smeared or discrete condensed counterions.
Crucially, our derivation allows the transparent identification of an effective elec- trostatic binding constant that captures the direct interaction between condensing charges as well as the non-local effects due to chain connectivity and correla- tion/screening. This careful development of the electrostatic binding coefficient allows us to appreciate the effects of screening and chain-connectivity on the electro- static binding, and mitigates risks of double-counting physical effects when piecing together different electrostatic theories into a polyelectrolyte theory of counterion condensation. To emphasize the physical content of the electrostatic binding con- stant, we discuss how other counterion condensation theories in literature can be understood in terms of our expression for the electrostatic binding constant, using Voorn Overbeek-type estimates of the binding constant as a concrete example.
Following, in section 4.3 we present numerical results, first studying the dependence of counterion binding on salt and polymer concentration. Using the self energy we explain differences in counterion binding between different chain structures and charge models, and we decompose the binding constant to distinguish between a bare Coulombic and finite-concentration correlation contributions to the binding constant. We then track the exchange of counterions between the tightly-bound and loosely-bound populations to evaluate the transition from weak to intermediate fluctuation effects, and present the osmotic coefficient contribution of counterion condensation versus the (renormalized) linearized fluctuations. Finally, we discuss the extent to which counterion condensation stabilizes polyelectrolyte solutions, and the degree to which explicit charge structure is needed to produce expected phase separations in polyelectrolyte solutions.