Conformations and Charge Fluctuations in Polyelectrolyte Solutions

Introduction: Charged Systems

Charged Molecules in Nature

Our lives abound in polymers and colloids - insoluble large molecules and micron-sized particles "suspended" in solution. In addition to the formation of nanoparticles by polyelectrolyte complexation, another interesting class of materials formed from polyelectrolytes are complex coacervates.

Figure 1.1: Reproduced from Bungenberg de Jong and Kruyt, a picture of a coacer- coacer-vate phase formed by gum arabic and serum albumen [26].

Simple Electrolytes: Fluctuations and Debye-Hückel

It is then immediately found that the Coulomb energy of the mean field is identical everywhere, independent of the beam length and the charge density. A physically motivated approach to describing such correlations makes clever use of mean field theory.

Theory of Polyelectrolytes

Attempts to connect experiment with theoretical predictions have often started with VO theory [62, 65] or ideas of Debye-Hückel screening applied in various ways. The most recent example of the failure of the VO theory is its predictions for polyelectrolyte coacervation bond lines in the sol-polymer plane.

Thesis Outline

The self-consistent equations thus also require a self-consistent solution of the single-chain structure. A crucial effect of the added pair interactions in the reference ensemble is to change the chain structure.

Figure 1.4: Schematic of two main factors dictating charged systems: chain connec- connec-tivity and strength of electrostatic coupling

The Polyelectrolyte Self Energy

Introduction

The internal energy contribution to the self-energy is similar to Eq. 2.3) and involves the single-chain structure factor, which reflects the spatial extent of the polyelectrolyte chain. In Section II, we present a complete derivation of the RGF theory for polyelectrolyte solutions.

General Theory

The first cumulant in the exponent can be easily evaluated due to the Gaussian nature of the fluctuating field, and is given by: The final piece needed to implement our theory is an evaluation of the single-chain partition function and the corresponding intramolecular charge structure.

Self-Consistent Calculation of Flexible Chain Structure

An important feature of the theory is the necessity of self-consistently determining the chain charge structure ˜Schgp (k) Eq. We also note that the expression for the interaction energy Eq. 2.67) is an improvement over typical scaling estimates of the Coulomb energy.

Numerical Results and Discussion

To understand the extent of the overestimated self-energy of fg-RPA, we consider the limit of infinite dilution. 2.6 we plot the chain length dependence of the critical Bjerrum length (∼inverse temperature) lcband of the critical monomer density ρcp for chain lengths up to N = 104. In contrast, the magnitude of the contribution of counterion condensation to the correlation energy would be much larger if the correlation energy followed the behavior of fg - RPA.

Figure 2.1: End-to-end distance in the single-chain limit, with parameters l b = 0.7, f = 1, z p = 1, at different salt concentrations ρ ±

Conclusions

We discuss our proposed models for the charge structure and the physical content of the electrostatic binding constant arising in the theory. This delay of the counterion condensation peak applies to the discrete backbone model (dotted red). Ψ·ρˆ, and can be averaged within the configuration integral using the identities in the appendix.

Polyelectrolyte Phase Behavior

Introduction

50] In light of the RGF theory, many qualitative deficiencies of the VO, TPT-1, and fg-RPA theories have been demonstrated to be related to their erroneous treatment of chain structure. In this chapter, we apply RGF to study the influence of added salt on the phase behavior of polyelectrolyte systems, with a focus on the role of chain structure. While we would have liked to compare the RGF with 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 limit and nor to generalize single cut.

Theory

In Section 3.3 we study the phase behavior of solutions of one type of polyelectrolytes, while in Section 3.4 we consider solutions with symmetrical, oppositely charged polyelectrolytes. Using defined interactions, it is easy to write the corresponding field theory, a grand canonical partition function using standard field theory identity transformations [60] that introduces a fluctuating electrostatic potential field Ψ [50]. This partition function is approximated by a non-perturbative RGF procedure. 2vhs, where vhs is the volume of the sphere, the volume fraction of the species γ is then φγ = ργ/ρ0.

Single Polyelectrolyte Species

The dashed lines (dotted blue) are for N=100 and indicate a lower salt concentration in the polymer-rich phase. We also note that under the presented conditions (lbf/b = 1), the TPT-1 theory does not predict a loop in the phase diagram. The electrostatic correlation energies in the TPT-1 theory are much weaker than in the RGF theory.

Figure 3.1: ρ s vs. ρ p phase diagram for single-species polyelectrolyte solution at l b = 1 < l c b , b = f = 1

Symmetric Polyelectrolyte Species

Away from the critical point, the dense polyelectrolyte branch approaches the N-independent curve (Figure 3.6) with increasing chain length. Not only is the dilute branch of the binodal too dilute, the dense branch can also be too dense. This overestimation of the electrostatic energy is worst for the dilute phase, but the overestimation is still quantitatively significant in the dense phase.

Figure 3.4: Scaling of the critical charge fraction f c (blue) and critical monomer density ρ c p (green) for salt-free solutions of oppositely charged, symmetric polyelec-trolytes with l b = b = 1

Effects of Chain Structure

The change in the chain structure with the polyelectrolyte and salt concentrations has large effects on the electrostatic correlation energies. At a crosslinking polymer concentration that increases with increasing salt, the electrostatic environment is eventually dominated by the polyelectrolytes, causing a negative change in the correlation energy. In the salt-free limit, the fg-RPA was shown to overestimate the correlation energy by up to a factor of √.

Figure 3.9: Polyelectrolyte end-to-end distance relative to the Gaussian chain for chain length N = 100 at l b = b = f = 1, along the binodal of supernatant (dashed) and dense (solid) phases in solutions of single polyelectrolyte species (blue) and opposit

Conclusions

The self-consistent calculation of the chain structure in RGF essentially renormalizes the entire chain structure and in doing so. We conclude by discussing the relationship of RGF to other self-consistent fluctuation theories. The resulting action ("Hamiltonian") in the statistical weight is usually complex-valued, and the evaluation of .

Ionic Atmosphere vs. Counterion Condensation: From Weak to

Introduction

This careful development of the electrostatic coupling coefficient allows us to estimate the effects of screening and chain connectivity on electrostatic coupling and reduces the risks of double counting physical effects when assembling different electrostatic theories into a polyelectrolyte theory of counterion condensation. To emphasize the physical content of the electrostatic coupling constant, we discuss how other theories of counterion condensation in the literature can be understood in terms of our expression for the electrostatic coupling constant, using Voorn Overbeek-type binding constant estimates as a concrete example. We then follow the exchange of counterions between tightly bound and loosely bound populations to assess the transition from weak to intermediate fluctuation effects and present the contribution of the counterion condensation osmotic coefficient compared to (renormalized) linearized fluctuations.

Theory

For transparency of the physical assumptions in the cB-cC model and its relations to the new charge models that we will propose, it is useful to decompose the charge structure of the polyelectrolyte-plus-counterion type into explicitly polyelectrolyte-polyelectrolyte (pp), polyelectrolyte-counterion (pc) and counterion -counterion (cc). Physically, when charges condense (become “neutralized”), the GDH term only accounts for the loss of favorable correlation energy, leading to an electrostatic binding constant that opposes the binding of counterions. Theoretical estimates of the binding constant have been estimated for simple ion pairing [34] and estimated in the polyelectrolyte literature to be ~ lb/b [12].

Figure 4.1: Charge model S chg p of the combined polyelectrolyte-condensed counteri- counteri-ons object

Numerical Results

The concentration dependence of the rate of counterion condensation shown in Figure 4.3 can be understood by examining Eq. 4.40), which shows that there is a competition between the translational entropy and the electrostatic driving force. The peak in the condensed fraction hxi coincides with the salt concentration at which the rate of change of the translational entropy penalty matches the rate of change of the electrostatic driving force. Consequently, counterion condensation decreases as the average condensation fraction increases, increasing the translational entropy penalty.

Figure 4.2: Probability distribution of counterion condensation fraction, at l b / b = 2 for chains with N = 100, 300, 1000, 3000, 10000, f=1, with the dB-dC charge model.

Conclusion

In this paper, we explore the relation of the variational Gaussian procedure to the first-order self-consistent perturbation (which we will call sc1P for short). We have thus shown, within the grand canonical ensemble, the equivalence of the self-consistency conditions derived from the RGF and from the sc1P procedures. Another interesting work considers the derivatives of the grand canonical separation function with respect to fugacity and an external source term [2].

Field Theoretic Techniques to Study Fluctuations

Introduction

The fields represent a common description of the system, usually representing fields of density or auxiliary (chemical) potential. One of the most powerful tools for describing fluctuations is the variational Gaussian or renormalized Gaussian fluctuation (RGF) approaches based on the Gibbs-Feynman-Bogoliub (GFB) limit [29]. Despite the success of various variational Gaussian theories, it is interesting to note that the GFB free energy bound on which these variational theories are built only holds if the action in the partition sum is realistically valued.

Field Theoretic Framework

In Section II, we outline the field-theoretic formalism for a generic system of polymers interacting with a pair potential, and summarize the RGF procedure in the grand canonical ensemble. This introduces an auxiliary potential Ψ(r) and the interaction energy is written as. 5.5) Note that the argument of the exponential is now complex-valued, but is only linear in the density ρ. One sees that the canonical and grand canonical field-theoretic partition functions differ only in the term with the chain partition functionQ.

Renormalized Gaussian Fluctuations

The saddle point approximation of the partition function is found by taking the value of the action in the mean field. Importantly, Green's function G depends on the single chain structure factor, which in turn implicitly depends on GthroughhQi. However, the specific way one evaluates the single-chain partition function is not necessary to the discussions in this article, and we refer interested readers to the above references or Ch.

Self-Consistent First Order Perturbation (sc1P)

Finally, the term in parentheses is set to zero, giving 0=− hρ1i+∫. which is the same expression as Eq. 5.19) obtained previously for RGF. 12ρˆ(11) · G12 · ρˆ(21)] as RGF, and thus also contains the same self-consistent renormalization of chain structure and G as described in RGF. It is important that there is a complicated self-consistency hidden in that the chain structure hδρδρir e f depends on G and ψ, and vice versa.

Connection to Exact Field Theory and RPA

U−1−G−1+hρˆρˆir e f − hρˆir e f hρˆir e f (5.67) To a large extent, the hρˆi hρˆiterm is unimportant for the Green's function equation because it leads to a delta function in Fourier space and thus only changes the mode k = 0 , and arises as a consequence of keeping the number of particles constant when working in the canonical ensemble. The Green's function is then essentially determined by the density-density pair correlation function hρˆρiˆ r e f (related to the structure factor. RPA attempts to approximate the chain structure expression with a scaled chain structure with one mean field.

Discussion and Conclusion

A key feature of this perturbation approach is that the reference system is multi-chain as opposed to the effective single-chain reference system of the grand canonical ensemble. We compare our results with a previous first-order perturbation theory applied to the canonical ensemble [11, 24]. We also highlight that our canonical ensemble sc1P theory includes renormalizations of the inter-chain structure factor.

Appendix

Morse, "Renormalization of the one-loop theory of fluctuations in polymer blends and diblock copolymer melts", Phys. Palyulin, “Complexation of oppositely charged polyelectrolytes: Effect of discrete charge distribution along the chain”, Phys. Potemkin, “Explicit description of complexation between oppositely charged polyelectrolytes as an advantage of the random phase approximation over the scaling approach”, Phys.