Challenging scaling laws of flexible polyelectrolyte solutions with effective renormalization concepts

Abstract

Recasting a many-particle problem in a field-theoretic formalism is nowadays a well-established theoretical tool used by scientists across a wide spectrum of research areas, ranging from polymer physics to molecular electronic structure theory. It has shown to provide useful results in many complex situations, where the physics of the system involves many degrees of freedom and a multitude of different length scales, generally rendering its numerical treatment on a detailed level computationally intractable. To reduce the computational burden, field-theoretic methodologies usually take advantage of the mean-field approximation. This approximation technique is known to give reliable information about the system in the high concentration regime, where the interactions are highly screened. However, it is well established that the ranges of physical interest in most biological and technological applications lie in the intermediate to low concentration regimes, where fluctuations beyond the mean-field level of approximation become important and dominate the overall physical behavior. In this work we introduce a new self-consistent field theory for flexible polyelectrolyte chains, in which the monomers interact via a pair potential of screened Coulomb type, and derive suitable thermodynamic expressions for all concentration regimes. Our approach combines the renormalization concepts of tadpole renormalization, which has recently been successfully employed in calculations of prototypical neutral polymer and polyelectrolyte solutions, with the Hartree renormalization procedure. By comparing our approach to experimental measurements as well as alternative theoretical approaches, we demonstrate that it provides useful osmotic pressure results for polyelectrolyte systems composed of sodium poly(styrene-sulfonate) without and with added salt over the whole range of monomer concentrations.

Introduction

Charged macromolecules are well known to play a vital role in nature and technology [1]. Of special importance among them are a special type of macromolecules called polyelectrolytes (PEs). They consist of long polymeric chains, possessing a multitude of ionizable groups along their backbone that may dissociate in a polar solvent by producing charged species [2]. Among the most prominent examples are the nucleic acids DNA and RNA, which are highly charged biopolyelectrolytes controlling the development and functioning of living cells. In addition to their central role played in biological systems, PEs find widespread use as solubilizing agents, phase separation agents, and rheological property modifiers in daily life and technological applications [3]. However, despite of their importance, PE systems still remain only poorly understood [4], [5]. This relates to the fact that their chemistry and physics is influenced by many controlling parameters, such as molecular weight, salt concentration, pH of the solution, etc. Another important characteristic of PE systems is the coexistence of long-range Coulomb and short-range excluded volume interactions. The presence of long-range interactions generally renders their simulation particularly difficult because of the need for computationally expensive techniques, like the Ewald summation [6]. Moreover, their often highly polymeric nature introduces additional complexity by severely slowing down their equilibration [7].

Since the pioneering works of Edwards [8] and de Gennes [9], it has been well acknowledged that concepts originally introduced in quantum field theory (QFT) [10], like e.g. functional integrals or renormalization group theory, have substantially contributed to major breakthroughs in the field of polymer science [7], [11], [12]. For instance, the groundbreaking idea of Edwards to use functional integral methods to investigate the physics of polymers and complex fluids has led in the last few years to a rapid development of analytical calculation and computer simulation tools, suitable for describing structure and properties of a wide variety of important polymer systems, including polymer melts, blends, and block copolymers [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. A standard approximation strategy for functional integral approaches is the mean-field (MF) approximation, which consists in replacing the many-body interaction term in the action by a term where all bodies of the system interact with an average effective field. This approach reduces any multi-body problem into an effective one-body problem by assuming that the partition function integral of the model is dominated by a single field configuration. A major benefit of solving problems with the MF approximation, or its numerical implementation commonly referred to as the self-consistent field theory (SCFT), is that it often provides some useful insights into the properties and behavior of complex many-body systems at relatively low computational cost. Successful applications of this approximation strategy can be found for various systems of polymers and complex fluids, like e.g. strongly segregated block copolymers of high molecular weight, highly concentrated neutral polymer solutions or highly concentrated block PE solutions [7], [11], [12], [13]. There are, however, a multitude of cases for which SCFT provides inaccurate or even qualitatively incorrect results [7]. These comprise neutral polymer or polyelectrolyte solutions in dilute and semidilute concentration regimes, block copolymers near their order–disorder transition, polymer blends near their phase transitions, etc. In such situations the partition function integral defining the field-theoretic model is not entirely dominated by a single MF configuration and field configurations far from the saddle point can make important contributions, which require the use of more sophisticated calculation techniques beyond the MF level of approximation. One possibility to face the problem is to calculate higher-order corrections to the 0th-order MF approximation. Tsonchev et al. developed a MF strategy including leading-order (one-loop) fluctuation corrections, to gain new insights into the physics of confined PE solutions [22]. However, in situations where the MF approximation is bad many computationally demanding higher-order corrections to the integral are necessary to get the desired accuracy. Another possibility is to use Monte Carlo (MC) algorithms and to sample the full partition function integral in field-theoretic formulation. However, in a recent work Baeurle demonstrated that MC sampling in conjunction with the original field-theoretic representation is impracticable due to the so-called numerical sign problem [23]. The difficulty is related to the complex and oscillatory nature of the resulting distribution function, which causes a bad statistical convergence of the functional integral averages of the desired thermodynamic and structural quantities. In such cases special analytical and numerical techniques are necessary to accelerate their statistical convergence [23], [24], [25], [26], [27]. To make the methodology amenable for computation, Baeurle proposed to shift the contour of integration of the partition function integral through the homogeneous MF solution using Cauchy's integral theorem, which was previously successfully employed by Baer et al. in field-theoretic electronic structure calculations [28]. Baeurle demonstrated that this technique provides a significant acceleration of the statistical convergence of the functional integral averages in the MC sampling procedure [23], [29]. An alternative theoretical tool to cope with strong fluctuation problems occurring in field theories has been provided in the late 1940s by the concept of renormalization, which has originally been devised to calculate functional integrals arising in QFTs [10], [30]. In QFTs a standard approximation strategy is to expand the functional integrals in a power series in the coupling constant using perturbation theory. Unfortunately, generally most of the expansion terms turn out to be infinite, thereby rendering such calculations impracticable [30]. A way to remove the infinities from QFTs is to make use of the concept of renormalization [31]. It mainly consists in replacing the bare values of the coupling parameters, like e.g. electric charges or masses, by renormalized coupling parameters and requiring that the physical quantities do not change under this transformation, thereby leading to finite terms in the perturbation expansion. A simple physical picture of the procedure of renormalization can be drawn from the example of a classical electrical charge, Q, inserted into a polarizable medium, such as electrolytes. At a distance r from the charge due to polarization of the medium, its Coulomb field will effectively depend on a function Q(r), i.e. the effective (renormalized) charge, instead of the bare electrical charge, Q [30]. At the beginning of the 1970s, Wilson further pioneered the power of renormalization concepts by developing the formalism of renormalization group (RG) theory, to investigate critical phenomena of statistical systems [32]. The RG theory makes use of a series of RG transformations, each of which consists of a coarse-graining step followed by a change of scale [7], [33], [34]. In case of statistical–mechanical problems the steps are implemented by successively eliminating and rescaling the degrees of freedom in the partition sum or integral that defines the model under consideration. De Gennes used this strategy to establish an analogy between the behavior of the zero-component classical vector model of ferromagnetism near the phase transition and a self-avoiding random walk of a polymer chain of infinite length on a lattice, to calculate the polymer excluded volume exponents [9]. Adapting this concept to field-theoretic functional integrals implies to study in a systematic way how a field theory model changes while eliminating and rescaling a certain number of degrees of freedom from the partition function integral [7], [33]. An alternative approach is known as the Hartree approximation or self-consistent one-loop approximation [35], [36]. It takes advantage of Gaussian fluctuation corrections to the 0th-order MF contribution, to renormalize the model parameters and extract in a self-consistent way the dominant length scale of the concentration fluctuations in critical concentration regimes [7]. In a more recent work Efimov and Nogovitsin showed that an alternative renormalization technique originating from QFT, based on the concept of tadpole renormalization, can be a very effective approach for computing functional integrals arising in statistical mechanics of classical many-particle systems [37], [38]. They demonstrated that the main contributions to classical partition function integrals are provided by low-order tadpole-type Feynman diagrams, which account for divergent contributions due to particle self-interaction. The renormalization procedure performed in this approach effects on the self-interaction contribution of a charge (like e.g. an electron or an ion), resulting from the static polarization induced in the vacuum due to the presence of that charge [39]. As evidenced by Efimov and Ganbold in an earlier work [40], [41], the procedure of tadpole renormalization can effectively be employed to remove the related divergences from the action of the original field-theoretic representation of the partition function, which leads to an alternative functional integral representation called the Gaussian equivalent representation (GER). They showed that the procedure provides functional integrals with significantly ameliorated convergence properties for analytical perturbation calculations. In subsequent works Baeurle applied [23], [24], [25], [26], [29] the concept of tadpole renormalization in conjunction with advanced Monte Carlo (MC) techniques in the grand canonical ensemble, and demonstrated that this approach efficiently accelerates the statistical convergence of the desired ensemble averages. Very recently, Baeurle et al. developed effective low-cost approximation methods based on the tadpole renormalization procedure, which have shown to deliver useful results for prototypical polymer and PE solutions [14], [15], [42].

In this work we develop a new field-theoretic methodology, which combines the concept of tadpole renormalization with the Hartree renormalization procedure, for solving statistical–mechanical problems of PE solutions over the entire range of monomer concentrations. We demonstrate the effectiveness of our approach on the example of a system of flexible PE chains, where the monomers interact via a Derjaguin–Landau–Verwey–Overbeek (DLVO) type of pair potential. We test the reliability of our method with regard to alternative theoretical approaches as well as experimental data, obtained from osmotic pressure measurements of sodium poly(styrene-sulfonate) (NaPSS) PE solutions without and with added salt in various concentration regimes.

Our paper is organized in the following way. In Section 2 we review the basic derivation of the field theory for flexible polymer chains, followed by the derivation of the GER formalism in conjunction with the Hartree renormalization procedure. Then, in Section 3 we show applications of the method on the example of NaPSS PE solutions, and demonstrate that the Hartree renormalized 0th-order GER methodology is an effective low-cost approximation strategy for evaluating thermodynamic information of systems composed of flexible PE chains over the whole range of monomer concentrations. Finally, we end our paper with a brief summary and conclusions.