Coarse-Grained Model of Glycosaminoglycans in Aqueous Salt

Article pubs.acs.org/JPCB

Coarse-Grained Model of Glycosaminoglycans in Aqueous Salt Solutions. A Field-Theoretical Approach Andrei L. Kolesnikov,†,‡ Yurij A. Budkov,§,∥ and Evgenij A. Nogovitsyn*,‡ †

Institut fur Nichtklassische Chemie e.V., Universität Leipzig, 04109 Leipzig, Germany Department of Physics, Ivanovo State University, Ermaka 39, 153025 Ivanovo, Russia § Institute of Solution Chemistry, Russian Academy of Sciences 153045, Academicheskaya 1, Ivanovo, Russia ∥ National Research University Higher School of Economics, 101000 Moscow, Russia ‡

ABSTRACT: We present results of self-consistent field calculations of thermodynamic and structural properties of glycosaminoglycans (chondroitin sulfate, hyaluronic acid, and heparin) in aqueous solutions with added monovalent and divalent salts. A semiphenomenological coarse-grained model for semiflexible polyelectrolyte chains in solution is proposed. The coarsegrained model permits one to focus on the essential features of these systems and provides significant computational advantages with respect to more detailed models. Our approach relies on the method of Gaussian equivalent representation for the calculation of the partition functions in the form of functional integrals. This method provides reliable thermodynamic information for polyelectrolyte solutions over wide ranges of monomer concentrations. In the present work, we use the comparison and fitting of the experimental osmotic pressure with a theoretical equation of state within the Gaussian equivalent representation. The degrees of ionization, radii of gyration, persistence lengths, and structure factors of chondroitin sulfate, hyaluronic acid, and heparin in aqueous solutions with added monovalent and divalent salts are calculated and discussed.

1. INTRODUCTION

tissues. GAGs are involved in a variety of extracellular and sometimes intracellular functions.8 During recent years, systems composed of chondroitin sulfate (CS) have been the subject of various experimental and theoretical investigations, both in solution9−13 and in cartilage.14 In ref 15, the counterion condensation phenomenon in CS solutions at different monomer and salt concentrations was investigated, and the influence of solvation effects on the frictional−compressive properties of CS polyelectrolytes in solution and cartilage were considered. This investigation was carried out using the field-theoretical approach for flexible polyelectrolyte chains introduced by us in ref 16. According to the criterion of Manning,17 complete ionization occurs for the infinite-line-charge model of polyelectrolytes when the charge density parameter ξ is less than unity, where ξ is defined by ξ = zmλB/b, zm is the charge on a monomer, λB = e2/ϵkBT is the Bjerrum length, and b is the length of a monomer. ξ must be less than 1 to meet this criterion. If ξ > 1, Manning’s theory predicts increasing counterion condensation in the near vicinity of the polyion, which partly shields the polyion charge. Therefore, this effect decreases the effective charge of polyion. In ref 15, it was shown that, at physiological salt concentrations, chondroitin sulfate solutions exhibit optimal frictional−

Biological systems abound with polyelectrolytes, because polyelectrolytes, being water-soluble, are natural for aqueous environments.1 Even though they have been extensively investigated, polyelectrolytes are much less understood than neutral polymers.2−4 Unfortunately, these systems are too complex for fully atomistic computer simulations.5 A reasonable alternative to a fully atomistic computer simulation is a coarsegrained, particle-based approach in which atoms or groups of atoms are lumped into larger “particles”.6 Coarse-grained models permit one to focus on essential features of biomolecular systems, while averaging over less important details. They provide significant computational advantages with respect to more detailed models.7 In this article, we develop a field-theoretic methodology within a coarse-grained model of semiflexible polyelectrolyte chains in solution to describe the properties of glycosaminoglycans (chondroitin sulfate, hyaluronic acid, and heparin) in aqueous solutions with added salts. Glycosaminoglycans (GAGs) are complex polysaccharides that exist both on the cell surface and free within the extracellular matrix. GAGs are long unbranched polysaccharides that are composed of repeating disaccharide units and are also called mucopolysaccharides because of their viscous and lubricating properties, just like in mucous secretions. They are essential to life and important components of connective © XXXX American Chemical Society

Received: April 16, 2014 Revised: October 15, 2014

A

dx.doi.org/10.1021/jp503749a | J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

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sequences.26 Based on ref 27, we used the most widespread monomer of heparin in our research. The characteristics of the polymer chains that we used in calculations for CS, HA, and HEP are summarized in Table 1.

compressive properties, which indicates that the phenomenon of counterion condensation onto the chondroitin sulfate chains has a major influence on the mechanical behavior of articular cartilage. In the present work, we continue a theoretical exploration of CS chains in aqueous solution with added NaCl using a fieldtheoretic approach for semiflexible polyelectrolyte chains. On the basis of experimental data on osmotic pressure,9 we calculate the static structure factor, degree of dissociation, persistence length, and radius of gyration for CS chains at various concentrations of polyelectrolyte and added salt. The abundance of hyaluronic acid (HA) (also called hyaluronan) and its multiple roles in biological tissues are unique among biopolymers.11,18−21 HA is a component of connective tissue whose function is to cushion and lubricate. HA, or commercial preparations containing HA, are in use, or being studied for use, to prevent, treat, or aid in the surgical repair of many types of problems that people with connective tissue disorders tend to have, such as fractures, hernias, glaucoma, detached retinas, and osteoarthritis. HA is a primary constituent of the extracellular matrix and also participates in a variety of cell-to-cell interactions. Among its many functions, HA plays a critical role in cartilage, where the collagen network enmeshes large aggrecan−HA complexes that provide resistance to compressive loads.22 HA is similar in chemical structure to chondroitin, but it is typically of very high molecular weight, ranging from 105 Da to more than 106 Da, equivalent to ∼250−2500 disaccharide repeat units, and is not sulfated. HA is among a class of polyacids that, even when fully neutralized in water (pH 7), are completely ionized, or nearly so.18,19 HA is one of the few available polyions that meet the criterion for complete ionization, because conformational calculations suggest an average ionizable site separation of about ∼10 Å,11,19,23 depending on the energy parameters chosen to represent the molecule. Available experimental evidence supports the assumption, based on the Manning criterion, that hyaluronate is completely ionized in solution.18,19 HA in neutral aqueous solution, at or near a physiological concentration of NaCl, generally behaves as a typical semiflexible polymer molecule.20 In ref 24, we studied the thermodynamic properties of HA aqueous solution using a field-theoretic approach for flexible polyelectrolyte chains. In the present work, we investigate HA in aqueous solutions with added salts, namely, NaCl and CaCl2, using the fieldtheoretic approach for semiflexible polyelectrolyte chains. We base our work on experimental data for osmotic pressure.18 The static structure factor, degree of dissociation, persistence length, and radius of gyration of HA chains at various concentrations of polyelectrolyte and added salt are calculated. Heparin (HEP) is a naturally occurring anticoagulant produced by basophils and mast cells. HEP acts as an anticoagulant, preventing the formation of clots and extension of existing clots within the blood. Heparin is a polysaccharide that contains a large number of linear and polydisperse chains. The molecular weight of natural heparin ranges from 3 × 103 to 3 × 105 Da, whereas medical-grade unfractionated HEP has an average molecular weight of (12−16) × 103 Da.25 HEP is a more complicated compound then HA or CS. In HEP, the order of the monomers is implicitly defined. HEP species can contain a large number of different disaccharide units, arranged either in block structures or in less well ordered complex

Table 1. Input Parametersa CS HA HEP

Mw (Da)

m (Da)

Zm

a (Å)

20 × 103 9 1.2 × 106 18 16 × 103 28

5139 40018 59328

−2 −1 −4

9b 95 95

a

Mw, weight of polymer chain; m, weight of monomer; Zm, charge of monomer; a, characteristic size of excluded volume of monomer. b Computer simulation by the program Avogadro.

This article is organized in the following manner: In section 2, we briefly review the basic derivation of field theory for semiflexible polymer chains and follow this with a derivation of Gaussian equivalent representation (GER) theory. In this section, the expression for osmotic pressure and self-consistent equations for the potential of mean force are presented. The details of calculations within the GER are considered in Appendix A. A calculation technique for the structure factor and the persistence length of polymer chains in solution is explained in Appendix B. To describe the intermonomer interactions, we use the Derjaguin−Laudau−Verwey−Overbeek (DLVO) potential. The features of this potential and corresponding potential of mean force are considered in section 3. In section 4, we use two parameters to approximate the theoretical osmotic pressure to the experimental data. These parameters are the persistence length of a single polymer chain in solution (l0) and the degree of dissociation of a polymer chain in solution (α). Using these parameters, we calculate the radius of gyration and persistence length of polymer chains in solution. Herein, we present and discuss the numerical results of our calculations for aqueous GAG solutions at various monomer and salt concentrations.

2. EQUATION OF STATE OF SEMIFLEXIBLE POLYMER CHAINS IN SOLUTION The field-theoretical formalism for polymer systems has been detailed in many works.2−4,29,30 The traditional methodology for the calculation of functional integrals in polymer physics is mean-field (MF) theory, which is based on the saddle-point approximation.2 MF theory approximates the equation of state of a polymer solution in the high-concentration regime only, because fluctuations near the saddle points are neglected. Thus, it is incorrect for the semidilute regime of polymer solutions, which is characterized by high fluctuations in concentration.2,31 To carry out calculations for the dilute and semidilute regimes, it is necessary to have reliable methods for describing the thermodynamic properties of polymer solutions beyond the MF level of approximation.2,16 Successful attempts to go beyond MF approximation have been undertaken in several works.29,32−41 In these works, a combination of variational methods based on Gibbs− Bogolyubov inequality and the random phase approximation (RPA) was used to describe the thermodynamic properties of polymer solutions. The Gaussian equivalent representation (GER) method is an approach of approximate calculations of functional integrals beyond the MF aprroximation and is a generalization of the variational method.42 The corrections to lowest approximation B



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where S(k)⃗ is a static structure factor2 and Φ̃(k)⃗ is the Fourier transform of the initial potential. The next two equations define the equaiton of state of the polymer system. They can be written in parametric form as

can be calculated through cumulant expansion.42−44 As a rule, the second correction accounts for no more than 10%.43 The GER method has been developed and successfully applied in quantum physics,42 and it can be applied to complex functional integrals.45 Presently, the GER is actively used for describing the thermodynamic and structural properties of polyelectrolyte solutions.15,16,24,46 Semiflexible polymers comprise a wide class of macromolecules. They include many important biopolymers, which are also semiflexible. Unfortunately, there are no rigorous methods for calculating any thermodynamic properties of semiflexible chains.47 We shall describe the stiffness following refs 47 and 48. Consider a solution of semiflexible polymer chains in the framework of the grand canonical ensemble. The solvent is taken into account implicitly. The grand canonical partition function for a solution of semiflexible polymer chains has the form ∞

Ξ=

∑ n=0

zn Q n! n

βP(ψGER ) =

1 + 12π 2

∫

∫

⎛ ⎞ n + rn⃗ exp⎜⎜ −β ∑ H0[ rj⃗] − βHint⎟⎟ ⎝ ⎠ j=1

∫0

βH0[ r ⃗] =

3 4l0

∫0

2

(2)

D( r ⃗) =

is the Hamiltonian of an ideal semiflexible polymer chain, and βHint

β = 2l0 2

n

∑∫ i,j=1

0

L

ds1

∫0

(3) 47,48

D̃ (k ⃗) =

(9)

(10)

The derivations of the structure factor S(k) and persistence length of the polymer chain l1 in solution at finite concentration are presented in Appendix B. Here, we present only the final result

L

ds2 Φ[ ri (⃗ s1) − rj(⃗ s2)] (4)

S (k ) =

1 Ll0

∫0

L

ds1

∫0

L

⎧ ⎪ 2k 2l12 ⎡ |s1 − s2| ⎢ − ds2 exp⎨ ⎪ 3 ⎢⎣ l1 ⎩

⎤⎫ ⎛ |s − s 2 | ⎞ ⎪ + exp⎜ − 1 ⎟ − 1⎥⎬ ⎪ l1 ⎠ ⎝ ⎦⎥⎭

(11)

where l1 is the renormalized persistence length, obtained from the equation β 1 1 − = l0 l1 9Ll0 2

∫0 ⃗

L

ds1

∫0

L

ds2 (s1 − s2)2 ⎧ ⎪ 2k 2l12 ⎡ |s1 − s2| ⎢ ⎪ 3 ⎢⎣ l1 ⎩

∫ (2dπk)3 k2D̃(k ⃗) exp⎨−

⎫ ⎤⎪ ⎛ |s − s 2 | ⎞ ⎬ + exp⎜ − 1 ⎟ − 1⎥⎪ l1 ⎠ ⎝ ⎦⎥⎭

(5)

(12)

The set of eqs 6−12 allows one to calculate the shift parameter ψGER, the osmotic pressure P(ρm), the structure factor S(k), the persistence length l1, the potential of mean force D(r), and the pair correlation function g(r). We use eq 7 to fit the experimental data for osmotic pressure.9,18,28

Φ̃(k ⃗) Φ̃(k )⃗ 1 + ψGER S(k ⃗) Φ̃(0)

(8)

⃗

∫ (2dπk)3 D̃(k ⃗)eik ⃗r ⃗

g (r ) = exp[−βD(r )]

is the total effective potential energy of pair interactions between monomers renormalized by presence of the solvent. The parameter l0 is the persistence length of an ideal wormlike polymer chain, β = 1/(kBT) is the inverse temperature, L = Nl0 = Npa is the length of a polymer chain, N is the effective number of segments with length l0, and Np = Mw/m is the 2 degree of polymerization. We note that r ⃗ ̈ (s) is related to the 47−49 local curvature of a space curve. In other words, this term defines the local stiffness of a polymer chain. The derivation of the equation of state is introduced in Appendix A; here, we present only basic formulas. To obtain the equation of state, we use a field-theoretical representation of the grand partition function. Then, we rewrite it in the form of a functional integral using a Hubbard−Stratonovich transformation. We calculate this functional integral by using the GER. After some mathematical manipulation (see Appendix A), we can write two equations that define the so-called shift parameter ψGER and the Fourier transform of the potential of mean force D̃ (k)⃗ ψGER = z1βN Φ̃(0)e−NψGER

[1 + ψGER u(k)]2

where D̃ (k)⃗ is defined by eq 6. The pair correlation function of monomers takes the form

2

ds [ r ⃗ ̇ (s) + l0 2 r ⃗ ̈ (s)]

u 2 (k )k 2 d q

∞

where P(ψGER) is osmotic pressure, ρm(ψGER) = ⟨ρ⟩Np is the average monomer number density and u(k) = S(k) Φ̃(k)/Φ̃(0), k = |k|.⃗ Equations 7 and 8 determine the osmotic pressure for a polymer solution as a function of monomer concentration and temperature. The potential of mean force of monomer−monomer interactions within our approximation can be written as43

is the partition function with fixed number n of polymer chains, L

2 3 du(k) ψGER k u(k) dk dk [1 + ψGER u(k)]2

NψGER 2 ψGER ρm (ψGER ) = × − 2π (1 + ψGER N ) β Φ̃(0)

(1)

+ r1⃗ ···

∫0

∞

(7)

where z is the activity Qn =

2ψGER + NψGER 2 2Nβ Φ̃(0)

(6) C



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3. MONOMER INTERACTION MODEL To describe intermonomer interactions, we use the Derjaguin− Laudau−Verwey−Overbeek (DLVO) potential,16,50,51 whose electrostatic part arises as an approximate solution to the Poisson−Boltzmann equation. Neglecting the long-range attractive contribution, we can write it as Φ(r ) = (zmα)2 A(kD , a)

λB exp−kDr β r

(13)

where ⎡ exp(kDa /2) ⎤2 A (k D , a) = ⎢ ⎥ ⎣ 1 + kDa /2 ⎦

(14)

is a geometry factor that depends on the size and form of the monomer and α is the degree of dissosiation. The parameters a and kD are the characteristic size of the excluded volume of the monomer and the Debye screening parameter, respectively. The solvent is considered implicitly through the Bjerrum length λB = e2β/(4πϵϵ0) and the Debye screening parameter kD. Taking into account electroneutrality, it is possible to write kD =

4πλB(2zm 2αρm + 2z s12ρs + z s2(z s2 + 1)ρs ) 1

2

Figure 1. Interaction potential Φ(r) and potential of mean force D(r) as a function of the dimensionless parameter r/a at different polymerization indexes. The potential D(r) has a minimum at Np > 10, which corresponds to the appearance of an effective attraction between monomers.

In Figures 2−4, solid lines denote polynomial fits of the experimental data, and symbols represent theoretical calcu-

(15)

where zm is the charge on a monomer; zs1 is the charge of the cation of NaCl; zs2 is the charge of the cation of CaCl2; ρm = NACm10−27 Å−3 is the monomer density; and ρs1 = NACs10−27 Å−3 and ρs2 are the densities in solution of NaCl and CaCl2, respectively. Cm is the concentration of monomer per unit volume and has units of moles of monomer per liter. NA is Avogadro’s number. Fourier transformation of the potential of mean force takes the form D̃ (k ⃗) =

4πzm 2A(kD , a)λBkBT 2

k ⃗ + kD2[1 + ψGER S(k ⃗)]

(16)

The potential of mean force can be calculated as the inverse Fourier transformation D(r ) =

1 2π 2

∫0

∞

sin(kr ) D̃ (k) k dk r

(17)

Figure 2. Osmotic pressure of CS as a function of polyelectrolyte concentration at different NaCl concentrations (Cs). Solid lines denote fits of the experimental osmotic pressure taken from ref 9, and symbols represent theoretical data.

The dependence of the potential in eq 17 on the dimensionless parameter r/a is given in Figure 1 at different values of Np for hyaluronic acid in aqueous solution (Cm = 0.2 mol/L, Cs1 = 0.1 M, Cs2 = 0) . The potential D(r) has a minimum at Np > 10, which corresponds to the appearance of an effective attraction between monomers, resulting from collective effects.52

lations according to eq 7. Experimental data for CS, HA, and HEP were taken from the articles by Chahine el al.,9 Horkay el al.,18 and Peitzsch el al.28 As one can see, eq 7 correctly describes the experimental data. The osmotic pressure increases as a function of monomer concentration. The osmotic pressure is the highest in HEP solution, and it is the lowest in HA solution, as the monomer of HEP has the highest charge (see Table 1). From our numerical analysis, one can conclude that the obtained set the α and l1 is unique, and these parameters are insensitive to the little alterations of the experimental data of the osmotic pressure. 4.2. Dissociation Degree. At first, we represent the dependence of the degree of dissociation on salt concentration for HA and CS solutions. For HA and CS solutions, the degree of dissociation very poorly depends on monomer concen-

4. NUMERICAL RESULTS AND DISCUSSION 4.1. Osmotic Pressure. In our work, we use two fitting parameters to approximate the osmotic pressure calculated by eq 7 to the corresponding experimental data. These parameters are the persistence length at infinite dilution (l0) and the degree of dissociation (α). Using these values, we calculate the persistence length of polymer chains (l1) and the radius of gyration (Rg) at various concentrations of polyelectrolytes and added salts. Our calculations are performed only within the experimental ranges of the monomers and salt concentrations9,18,28 at 298 K. D



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The Manning parameter is significantly larger (ξ > 1) for CS and HEP polyions, and these polyions are not fully ionized in solution. We consider that, in CS solution, counterion condensation takes place.17,53 Counterions (H+) and ions (Na+) of salt condense on the polymer backbone and reduce the charge density of the polyelectrolyte chain. Thus, this effect explains the significantly low values of α in CS solutions. The degree of dissociation of HEP in solution depends strongly on the polymer concentration. We consider this to be a consequence of the high charge on the monomer (zm = −4). The dependence of the effective degree of dissociation on the monomer concentration for HEP solution is presented in Figure 6. As can be seen, α decreases with increasing concentration of added salt. The effects of varying polyelectrolyte degree of ionization in solutions of different concentrations and a description of overcharging were discussed in ref 54. 4.3. Persistence Length of Chains in Solution. Now, we consider the persistence length l1 of CS, HA, and HEP in solution as a function of the electrostatic interactions, specifically, its dependence on the Debye screening parameter kD. This is an important issue, because the persistence length characterizes the stiffness of the polymer chain. Many properties of polymers are related to stiffness.55 Odijk56 and Skolnik and Fixman57 used the equation

Figure 3. Osmotic pressure of HA as a function of polyelectrolyte concentration at different CaCl2 concentrations (Cs) and at a fixed NaCl concentration (0.1 M). Solid lines denote fits of the experimental osmotic pressure taken from ref 18, and symbols represent theoretical data.

l1 = ln +

α 2λB 4(kDb)2

(18)

to describe solution properties of flexible strongly charged polyelectrolytes with the electrostatic interaction parameter λBα2/b ≈ 1. The first term in eq 18 (ln) is the persistence length of the neutral polymer chain. In this case, the electrostatic contribution (the second term) to the chain persistence length l1 is the main factor controlling a chain’s bending rigidity. The Odijk−Skolnik−Fixman (OSF) result was extended to flexible weakly charged polyelectrolytes with λBα2/b ≪ 1 by Khokhlov and Khachaturian.58 On the other hand, Barrat and Joanny55 found that the OSF approach, although correct for intrinsically rigid polyelectrolytes, breaks down for flexible chains. Dobrynin59 showed that, for semiflexible and strongly charged flexible polyelectrolytes, the electrostatic part of the chain persistence length is proportional to the Debye screening length. In our investigation, the salt concentrations (Cs1 and Cs2; equivalently, the densities ρs1 and ρs2) are fixed in eq 15, and the Debye length (1/kD) is considered as a function of monomer concentration. The persistence length of the polymer chain in solution is calculated according to eq 12. The electrostatic persistence length of CS as a function of inverse Debye parameter at different NaCl salt concentrations is presented in Figure 7. Symbols are theoretical data for persistence length, and solid lines are linear fits. The Debye screening length kD−1 decreases with increasing monomer and salt concentrations. We found that the electrostatic persistence length l1 has a linear dependence on the Debye screening length kD−1. This behavior corresponds to the results of Dobrynin59 for semiflexible and strongly charged flexible polyelectrolytes. The same tendency holds for HEP, as shown in Figure 8. However, in this case, the persistence length depends weakly on the Debye screening parameter. It should be emphasized that the linear dependence breaks down for HA chains in solution (Figure 9). One can see that

Figure 4. Osmotic pressure of HEP as a function of polyelectrolyte concentration at different NaCl concentrations (Cs). Solid lines denote fits of the experimental osmotic pressure taken from ref 28, and symbols represent theoretical data.

tration. However, it strongly depends on salt concentration. The dissociation degree is visualized in panels a and b of Figure 5 for CS and HA, respectively. The α decreases at the increase of salt concentration. The degree of dissociation is greater in the HA solution than in the CS solution. As noted elsewhere,18,19,23 HA is one of the few available polyions that can be completly ionized. The monomer unit of HA is ∼10 Å, zm = 1, and the Manning parameter is ξ ≈ 0.7 < 1. Thus, the effect of the condensed cations is absent in HA solution. However, as was shown by Muthukumar in ref 29, the Manning criterion should be invalid in the case when a polyion at large scale has a coiled conformation. Nevertheless, Manning’s theory should give qualitatively correct trends even for coiled polyions.29 Our results for hyaluronic acid confirm the general result of Muthukumar and predict that, at sufficiently high concentration of CaCl2, the degree of ionization can be less than unity. E



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Figure 5. Effective degree of dissociation as a function of the salt concentration for (a) CS at various NaCl concentrations and (b) HA at 0.1 M NaCl and various CaCl2 concentrations. The degree of dissociation of CS is smaller in comparison with that of HA. Symbols are theoretical predictions of the degree of dissociation at the salt concentrations used in the experiments.9,18

Figure 6. Effective degree of dissociation as a function of the polyelectrolyte concentration for HEP at different NaCl concentrations (Cs). α decreases with increasing concentration of added salt.

Figure 7. Electrostatic persistence length as a function of Debye parameter at different NaCl concentrations for CS. Symbols are theoretical data for persistence length, and solid lines are linear fits. The Debye screening length kD−1 decreases with increasing monomer and salt concentrations. We found that the electrostatic persistence length l1 has a linear dependence on the Debye screening length kD−1.

this dependence has an inflection point. In the case Cs2 = CCaCl2 = 0, the inflection point corresponds to a monomer concentration of Cm ≈ 0.15 mol/L. One can assume that le ≈ kD−1 at Cm < 0.15 mol/L and le ≈ kD−1 at Cm > 0.15. At low monomer concentrations, HA has some stiffness and a large radius of gyration (see Figure 10b). At Cm > 0.15 mol/L, the chains of HA are flexible, and the radius of gyration decreases considerably. The chain of HA is enough long to appear as electrostatic blobs58 in this concentration range. Also, the influence of changes in the internal polyelectrolyte dielectric constant and in the dipole−dipole interactions on the persistence length and radius of gyration might be significant.60,61 4.4. Radius of Gyration. The radius of gyration can be calculated according to eq 67 (Appendix B) or the formula for a wormlike chain

Rg 2 =

⎛ L ⎞⎤ Ll1 2l 3 2l 4 ⎡ − l12 + 1 − 12 ⎢1 − exp⎜ − ⎟⎥ 3 L L ⎢⎣ ⎝ l1 ⎠⎥⎦

(19)

where L = Mwa/m (Table 1) is the contour length of the polymer chain in solution. The results in both of these cases are very similar. The dependencies of the radii of gyration of CS and HA chains in solution on monomer concentration are presented in Figure 10. In both cases, the radius of gyration decreases with increasing polyelectrolyte concentrations. However, one can observe that, at some polyelectrolyte concentration, there is approximately no difference between the values of the radii of gyration in different salts. Thus, starting from some concentration of salt, the radius of gyration of the polyelectrolyte chain depends poorly on the polyelectrolyte concentration. F



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M NaCl were calculated by means of the Flory−Fox equation.65 In Table 3, we compare these experimental data with the results of our calculations. As one can see, basically, the theoretical values for the radius of gyration are smaller than the experimental values. We propose that this is a consequence of insufficient accounting for excluded volume in the DLVO potential.12

5. CONCLUSIONS The approach presented in this work goes beyond simple simulations of a single chain and the mean-field approximation. In contrast to the simulations of an isolated coarse-grained polymer chain, our formalism allows for the treatment of the many-chain problem, so that monomer concentration can be varied in a wide range. Moreover, in contrast to the simple mean-field approach, our formalism takes into account collective effects arising from the chain structure of macromolecules. We describe the polyelectrolyte chain in solution with a simple Gaussian chain model with a DLVO-type potential for monomer−monomer interactions. We also assume that remaining effects, such as couterion-condensation, excluded-volume, and solvent effects, can be taken into account through fitting parameters. We treat the grand canonical ensemble of polymer chains and use the Gaussian equivalent representation method for the calculation of the osmotic pressure, electrostatic persistence length, and structure factor of polymer chains in solution. We note, however, that there is a weak point in our approach. In the present model, the counterions are taken into account only through ionic strength. In other words, within our model, it is impossible to describe the behavior of counterions in the neigborhood of the polymer chains explicitly. Models that explicitly account for counterions were developed in our recent works.40,66 Finally, as already mentioned above, our approach does not take into account the detailed structure of macromolecules. In our opinion, one of the more valuable results of this work is the demonstration of counterion condensation in CS and HEP solutions and the concomitant contraction of the charge density on these macromolecules. On the other hand, we conclude that the effect of counterion condensation is absent in HA solutions, so that HA chains have a high degree of dissociation. We found that the electrostatic persistence length l1 depends linearly on the Debye screening length 1/kD for CS and HEP solutions. The persistence length very poorly depends on the concentration of HEP monomers and on the Debye screening parameter. However, the linear dependence of the electrostatic persistence length on the Debye screening length breaks down for HA. We assume that, in this case, there are two different concentration regimes of behavior of persistence length. The valence of the salt’s ion exerts a specific effect on the HA molecule: With increasing CaCl2 concentration, the radius of gyration decreased in two regimes. It became approximately constant at the highest CaCl2 concentrations. The theoretical value of the radius of gyration of HA was close to the experimental one. The root-mean-square end-to-end distance of CS chains was also found to give satisfactory agreement with the experimental results. We conclude that our simple theoretical model can be used for the description of the thermodynamics and structure of complex polymer systems by means of fitting a sufficiently small numbers of adjustable parameters (in our case, the persistence length of an ideal wormlike polymer chain l0 and the degree of

Figure 8. Electrostatic persistence length as a function of Debye parameter at different NaCl concentrations for HEP. Symbols are theoretical data for persistence length, and solid lines are linear fits. The Debye screening length kD−1 decreases with increasing monomer and salt concentrations. We found that the electrostatic persistence length l1 has a linear dependence on the Debye screening length kD−1.

Figure 9. Electrostatic persistence length as a function of Debye parameter at 0.1 M NaCl and different CaCl2 concentrations (Cs) for HA. A deviation from linear dependence occurs in HA solutions. We assume that there are two different regimes of behavior of the persistence length.

In the case of HA, a different behavior of the radius of gyration as a function of monomer concentration takes place. In this solution, the additional salt CaCl2 is present at different concentrations. The valence of the salt’s ion exerts a specific effect on the HA molecule: With increasing CaCl2 concentration, the radius of gyration decreased in two regimes. Here, we do not present a radius of gyration for HEP, because the persistence length very poorly depends on the concentrations of HEP and salt (Rg ≈ 36.5 Å). In refs 62 and 63, light-scattering experiments in aqueous hyalunorate solutions with added NaCl are described. In Table 2, we compare these experimental data with the results of our calculations. One can see a good agreement between the theoretical and experimental data. In ref 64, experiments were carried out by viscometry. The end-to-end distances of CS in aqueous solutions containing 0.2 G



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Figure 10. Radius of gyration as a function of the polyelectrolyte concentration for (a) CS at different NaCl concentrations and (b) HA at different CaCl2 concentrations. The dependence of the radius gyration on the monomer concentration in HA solution has two different concentration regimes.

■

Table 2. Experimental and Theoretical Root-Mean-Square Radius of Gyration, Rg, for HA

GER of the Grand Partition Function

To derive the basic field-theoretical representation of the grand partition function,1 we introduce the segment density operator of the polymer system

Rg (Å) Mw (kDa)

NaCl (M)

experimental

theoretical

difference (%)

104 69 215 350 85 610 900 2000

0.5 0.5 0.5 0.5 0.1 0.1 0.1 0.1

228a 173a 341a 453a 270 ± 20b 600 ± 50b 760 ± 50b 1200 ± 100b

220 176 322 414 198 550 670 1000

−3 +1.7 −5 −9 −26 −9 −12 −16

a

ρ( r ⃗) =

a

experimental

theoretical

difference (%)

0.2 0.2

273a 408a

225 308

−18 −24

l0

n

L

∑ δ[ r ⃗ − ri (⃗ s)]

ds

(20)

i=1

∫

R (Å) NaCl (M)

∫0

n ⎧ ⎫ L L ⎪ ⎪ β exp⎨− 2 ds1 ds2 ∑ Φ[ ri (⃗ s1) − rj⃗(s2)]⎬ ⎪ ⎪ 0 i,j=1 ⎩ 2l0 0 ⎭ ⎡ 1 βNnΦ(0) ⎤ = exp⎢ − (ρΦρ) + ⎥ ⎣ 2 ⎦ 2 ⎡ 1 +ψ = exp⎢ − (ψ Φ−1ψ ) + i(ρψ ) ⎣ 2 det Φ ⎡ βNnΦ(0) ⎤ βNnΦ(0) ⎤ + ⎥ = exp⎢ ⎥ dμΦ [ψ ] ⎦ ⎣ ⎦ 2 2

Table 3. Experimental and Theoretical Root-Mean-Square End-to-End Distance, R ≈ 61/2Rg, for CS

27 44

β

Using the well-known Hubbard−Stratonovich transformation,2 we can recast the integrand exponent in the grand partition function in the form of a functional integral,2,16 which results in the expression

Hayashi et al.63 bEsquent and Buhler.62

Mw (kDa)

APPENDIX A

∫

∫

Tanaka.64

∫

⎧i β ⎨ exp⎪ ⎩ l0

dissociation α). This possibility is related to the fact that the initial theoretical model and the Gaussian equivalent representation method, which do not use adjustable parameters, allow for the description of the general trends that can be observed in real polyelectrolyte systems. The variation of adjustable parameters simply allows one to obtain agreement between theoretical and experimental values of osmotic pressure. As an additional result, we obtained the dependencies of the degrees of dissociation of HA, HEP, and CS (related to the average charge of the macromolecules in solution) on the monomer and salt concentrations. The latter itself is extremely valuable, as direct measurements of the average charge of macromolecules in solution is a quite sophisticated task.

n

⎪

∑∫

L

0

i=1

⎫ ⎬ ds ψ [ ri (⃗ s)]⎪ ⎭ ⎪

(21)

Using eq 21, one can represent the grand partition function in the form ⎛

−

β 2l0 2

∫0

⎝

L

ds1

: exp{i β N H

⎧

∫ dμΦ[ψ ] exp⎜⎜z∫ + r ⃗ exp⎨⎩−βH0[r ⃗]

Ξ(z , β , V ) =

∫0

∫0 L

L

⎫ ds2 Φ[ r (⃗ s1) − r (⃗ s2)]⎬ ⎭

⎞ dsψ [ r (⃗ s)]}:Φ ⎟⎟ ⎠

(22)



Article

where we introduced the concept of the normal product42,43 according to the given Gaussian measure dμ Φ [ϕ] = +ψ / det Φ exp[−1/2(ψΦ−1ψ)] ⎧i β :exp⎨ ⎩ l0 ⎪

⎪

∫0

L

⎫ ds ψ [ r (⃗ s)]⎬:Φ ⎭ ⎪

⎪

⎧i β = exp⎨ ⎩ l0 ⎪

∫0

⎪

⎧ β exp⎨ 2 ⎩ 2l0

1 1 WD = − : [ψ (Φ−1 − D−1)ψ ]:D − [D(Φ−1 − D−1)] 2 2 i 1 −1 (ψ Φ ψ ) + (ψ Φ−1ψGER ) − 2β GER β GER

∫0

L

+ zA e−NψGER

⎫ ds ψ [ r (⃗ s)]⎬ ⎭

⎪

−

⎪

L

ds1

∫0

L

⎫ ds2 Φ[ r (⃗ s1) − r (⃗ s2)]⎬ ⎭

⎧i β ⎪

∫ dμΦ : exp⎨⎩ ⎪

l0

∫0

L

⎪

⎪

(24)

(ρψ ) =

∫V dx ⃗ ρ(x ⃗) ψ (x ⃗)

×

⃗

∫ (2dπk)3

1

(28)

⎛ ×⎜ ⎜ ⎝

ds1

∫0

ds1

(29)

L

2 ⎡⎛ ⎛ d2 r (⃗ s) ⎞2 ⎤ d r (⃗ s) ⎞ ⎟ + l 0 2⎜ ds ⎢⎜ ⎟⎥ ⎢⎣⎝ ds ⎠ ⎝ ds 2 ⎠ ⎥⎦

⎞ ⎫ ⎪ ⎬ ds2 D[ r (⃗ s1) − r (⃗ s1)] ⎟ ⎪⎟ ⎭⎠

⎪

⎩

L

0

2 ⎡⎛ ⎛ d2 r (⃗ s) ⎞2 ⎤ d r (⃗ s) ⎞ ⎟ + l 0 2⎜ ds ⎢⎜ ⎟⎥ ⎢⎣⎝ ds ⎠ ⎝ ds 2 ⎠ ⎥⎦

∫0

L

ds1

∫0

L

⎞ ⎫ ⎪ ds2 D[ r (⃗ s1) − r (⃗ s1)]⎬⎟ ⎪⎟ ⎭⎠

(35)

z1 = A e βμ

−

⎧ ⎪

∫ + r ⃗ exp⎨− 43l ∫0

β 2l0 2

⎪

⎩

∫0

L

ds1

W2 = z1e−NcGER

and switch to the new Gaussian measure dμD[ψ], where ψGER and D(x,y) are functions to be determined. The grand partition function now takes the form

∫ dμD[ψ ]eW

∫ dσ[r ⃗] ψ [r (⃗ s)] = 0

−1

(30)

det D det Φ

∫0

L

∫0

⎧ ⎪

iψGER

Ξ(z , β , V ) =

ds

∫ dσ[r ⃗] ψ [r (⃗ s1)] ψ [r (⃗ s2)] = 0

ds 2

L

β − 2l0 2

e ik r ⃗

β

L

∫0

∫ + r ⃗ exp⎨− 43l ∫0

⃗

Φ(k ⃗)

L

β − 2l0 2

The ordinary strategy for approximate calculation of the functional integral in eq 22 is a mean-field approximation. However, there are many cases where the mean-field approach provides either inaccurate or even qualitatively wrong results.2 Our aim is to obtain the equivalent representation of the functional integral that is suitable for any external parameters. First, we perform the displacement of the functional variable

ψ→ψ+

∫0

⎛ ⎧ ⎪ 3 dσ[ r ⃗] = ⎜+ r ⃗ exp⎨ − ⎜ ⎪ 4l 0 ⎩ ⎝

is the sufficient condition for the existence of the functional integral in eq 22 and the thermodynamic limit of the grand partition function: n → ∞, V → ∞, and n/V → ρ. The inverse potential Φ−1(|r|⃗ ) is the generalized function Φ−1( r ⃗) =

L

where

(27)

∫V d r ⃗ Φ( r ⃗) ei(k ⃗r ⃗) > 0

∫0

(34) (26)

A positivity of the Fourier transformation for the potential Φ Φ̃(k ⃗) =

ds2 ψ [ r (⃗ s1)] ψ [ r (⃗ s2)]}:D +W2[ψ ]

z β e−NψGER 1 − [ψ (Φ−1 − D−1)ψ ] − 1 2 2l0 2

The following short-hand notations are introduced in eq 22

∫V d r ⃗ ∫V d r ′⃗ ψ ( r ⃗) Φ−1( r ⃗ , r ′⃗ ) ψ ( r ′⃗ )

ds ψ [ r (⃗ s)]

(33)

(25)

(ψ Φ−1ψ ) =

L

L

∫0

ds1

z i β e−NψGER i (ψGER Φ−1ψ ) + 1 l0 β

−

This provides the correct account of the potential in zero and leads to the summation of so-called tadpole diagrams.42 The inverse operator satisfies the relation → ⎯ → ⎯ → ⎯ → ⎯ d r ⃗ Φ(r′, r ⃗) Φ−1( r ⃗ , r″) = δ(r′ − r″)

∫V

L

l0

∫0

Functional W2[ψ] is written in a form that does not have linear and quadratic terms in the integration variable ψ(x)⃗ . Our basic idea is that the main contribution to the functional integral is concentrated in the new Gaussian measure. This means that linear and quadratic terms in the integration variable ψ(x⃗) should be absent from the integrand exponent. Thus, we obtain two self-consistent equations

(23)

⎫ ds ψ [ r (⃗ s)]⎬:Φ = 1 ⎭

∫0

β

(32)

: ψ [ ri (⃗ s)] ψ [ rj(⃗ s′)]:Φ = ψ [ ri (⃗ s)] ψ [ rj⃗(s′)] − β Φ[ ri (⃗ s) − rj⃗(s′)]

β 2l0 2

∫ dσ[r ⃗]: {1 + i

∫0

0

L

L

2 ⎡⎛ ⎛ d2 r (⃗ s) ⎞2 ⎤ d r (⃗ s) ⎞ ⎟ + l 0 2⎜ ds ⎢⎜ ⎟⎥ ⎢⎣⎝ ds ⎠ ⎝ ds 2 ⎠ ⎥⎦

⎫ ⎪ ds2 D[ r (⃗ s1) − r (⃗ s2)]⎬ ⎪ ⎭

⎧i β ⎪

∫ dσ[r ⃗]: exp2⎨⎩ ⎪

l0

∫0

L

(36)

⎫ ds ψ [ r (⃗ s)]⎬:D ⎭ ⎪

⎪

(37)

where exp2 (x) = e2 x = e x − 1 − x −

D

(31) I

x2 2

(38)



{

A = exp

Article

β [Φ(0) − D(0)] 2

}

In our case, one can write the static structure factor in the form

(39)

1 S(k ⃗ ) = l0L

The system consisting of eqs 33 and 34 can be written in the form (40)

(41)

=

where S(k)⃗ is the static structure factor of the polymer chain in solution.2 The grand partition function can be written as Ξ(z , β , V ) = e

=e

−β Ω(0) GER

0 ⎞ z1 ⎛ ∂ΩGER ⎟ ⎜ V ⎝ ∂z1 ⎠

∫ dμD[ψ ] e

(42)

Z=

∞

u 2 (k )k 2 d k [1 + ψGER u(k)]2

Z=

∫0

L

ds1

∫0

L

⎫ ds2 D[ r (⃗ s1) − r (⃗ s2)]⎬ ⎭

dr ⃗

⎡

0

⎣

∫0

L

ds1

∫0

∑q≠ 0 rq⃗ (e iqs

1

q≠0

L

ds 2

rq⃗ r −⃗ q ⎤ ⎥ g0̃ (q) ⎥⎦

⃗

∫ (2dπk)3 D̃(k ⃗)

⎫ ⎤⎪ ⎬ − e iqs2)⎥⎪ ⎦⎭

(50)

2l0 3q2(1 + l0 2q2)

(51)

is the propagator of the Gaussian measure in q representation. We now calculate the partition function Z using the GER method. In this case, there is no need to perform the displacement of the integration variable r(⃗ s) because the functional in the integrand exponent is symmetric.42 We now switch to the Gaussian measure with the new propagator g̃(q). Thus

(46)

(47)

The derivation of the structure factor of a polymer chain in solution is presented in Appendix B.

■

(49)

∫ ∏ [2πg ̃ (qq)]3/2 exp⎢⎢− 21L ∑

g0̃ (q) =

and the pair correlation function of monomers takes the form

g (r ) = exp[−βD(r )]

⎫ ds′ D[ r (⃗ s) − r (⃗ s′)]⎬ ⎭

where

(45)

dk ⃗

∫ (2π )3 D̃(k ⃗)eik ⃗r ⃗

∫0

⎡ k⃗ × exp⎢i ⎣ L

The potential of mean force for monomer−monomer interactions within our approximation (W2 = 0) can be written as43 D( r ⃗) =

β 2l0 2

⎧ β ⎪ ⎨− 2 exp⎪ ⎩ 2l0

ψGER 2k3u(k) [1 + ψGER u(k)]2

L

∫0

⎧

q≠0

2ψGER + NψGER 2 βP = 2Nβ Φ̃(0) dk du(k) dk

ds

is the partition function of the polymer chain in solution. As mentioned above, the propagator D(r )⃗ of the renormalized Gaussian measure has the sense of a potential of mean force for the intermonomer interactions. The partition function Z can be written as a functional integral over the Fourier components of the random function r(⃗ s) = (1/ L)∑q≠0rq⃗ eiqs [q = (2πm)/L), m = ±1, ± 2, ...] corresponding to a conformation of the polymer chain. Using the Fourier transformation of propagator D(r)⃗ , one can write

(44)

∞

L

∫ + r ⃗ exp⎨⎩−βH0[r ⃗]

−

(43)

∫0

⎧

is an average on the statistic of an effective polymer chain and

where u(k) = S(k) Φ̃(k)/Φ̃(0), k = |k|,⃗ and ρm = ⟨ρ⟩Np is the average monomer number density. The analytic expression for osmotic pressure has the form

1 12π 2

∫ dσ[ r ⃗] exp{ik[⃗ r (⃗ s1) − r (⃗ s2)]}

∫ +Zr ⃗ exp⎨⎩−βH0[ r ⃗] − 2lβ 2 ∫0

W2[ψ ]

β ,V

ψGER NψGER 2 − 2π (1 + ψGER N ) β Φ̃(0)

+

⃗ r (⃗ s1) − r (⃗ s2)]}⟩ ds2 ⟨exp{ik [(

0

Using this equation, we obtain ρm =

L

× exp{ik [⃗ r (⃗ s1) − r (⃗ s2)]}

Linear and quadratic terms in the integration variable ψ(x⃗) are absent from W2[ψ]. Therefore, we can choose W2 ≈ 0. This approximation was estimated in ref 42, and it was shown that the contribution of the subsequent terms is smaller than 10% over a wide range of thermodynamic parameters. The average polymer chain concentration within the zeroth approximation of GER can be expressed as ⟨ρ⟩ =

∫0

⟨exp{ik [⃗ r (⃗ s1) − r (⃗ s2)]}⟩ =

Φ̃(k )⃗ ψGER S(k ⃗) Φ̃(0)

βPV

ds1

where

Φ̃(k ⃗) 1+

L

(48)

ψGER = z1βN Φ̃(0)e−NψGER D̃ (k ⃗) =

∫0

Z = eW0

dr ⃗

q≠0

APPENDIX B

⎧ ⎪

∫ ∏ [2πg (̃ qq)]3/2 exp⎨− 21L ∑ ⎪

⎩

q≠0

⎫ ⎪ + WI[ r ⃗]⎬ ⎪ g (̃ q) ⎭

rq⃗ r −⃗ q

(52)

Calculation of the Static Structure Factor

where

In this appendix, we present a technique for the calculation of the static structure factor of a polymer chain in soution by the GER. A similar technique was applied to the polaron problem in quantum statistical mechanics.42

W0 = J

3 2

⎡

∑ ⎢log q≠0

⎢⎣

⎤ g (̃ p) g (̃ q) − + 1⎥ g0̃ (q) g0̃ (q) ⎦⎥

(53)



Article

and ⎡ ⎤ ∑ : rq⃗ r −⃗ q:g ⎢ 1 − 1 ⎥ − β 2 g ( q ) g ( q ) ⎢ ⎥⎦ 2l0 ̃ ̃ ⎣ 0 q≠0

1 WI[ r ⃗] = − 2L

∫

⎧ ⎪ dk ⃗ k2 D̃ (k ⃗) exp⎨ − 3 ⎪ (2π ) ⎩ L

∫0

L

ds1

∫0

∑ : rq⃗ r −⃗ q:g

L

q≠0

ds 2

(54)

g (̃ q) =

⎭

: rq⃗ rq⃗ ′:g = rq⃗ rq⃗ ′ − 3g (q)Lδq + q ′ ,0

=

∑

: rq⃗ rq⃗ ′:g

q , q ′≠ 0

∫0

L

ds1

∫0

(55)

1 L

βL 3l0 2

∫

⎤ ⎥ − g ( q )(1 cos qs ) ̃ ∑ ⎥ ⎦ q≠0 (58)

1 L2

L

ds1

∫0

L

ds2 e iqs1C(|s1 − s2|)e−iq ′ s2

1 Ll0

∫0

g (s ) =

1 L

g (̃ q) ≈

ds1

∫0

L

ds1

∫0

L

⎡⎛ ⎢ ds2 ⎢⎜ ⎜ ⎢⎣⎝

∫

⎧ ⎪

∫ ∏ d rq⃗ exp⎨− 21L ∑ q≠0

⎪

⎩

⎡ 1 ∏ d rq⃗ exp⎢⎢− ⎣ 2L q≠0

q≠0

∑ q≠0

rq⃗ r −⃗ q ⎫ ⎪ ⎬ ⎪ g (̃ q) ⎭

−1⎤ rq⃗ r −⃗ q ⎤⎫ ⎪ ⎥⎬ ⎥ ⎥ g (̃ q) ⎥⎦⎪ ⎭ ⎥⎦

ds2 exp{− k 2[g (0) − g (|s1 − s2|)]}

∑ g (̃ q)eiqs ≈ ∫

∞

−∞

q≠0

(63)

dq g (̃ q)e iqs 2π

(64)

2l1 2

3q (1 + l12q2)

(65)

L L β 1 1 dk ⃗ 2 ̃ ⃗ − = ds1 ds2(s1 − s2)2 k D(k ) 2 0 l0 l1 9Ll0 0 (2π )3 ⎧ ⎤⎫ ⎛ |s − s2| ⎞ ⎪ ⎪ 2k 2l12 ⎡ |s1 − s2| ⎢ − + exp⎜− 1 exp⎨ ⎟ − 1⎥⎬ ⎪ ⎪ ⎢ ⎥ 3 l l ⎝ ⎠ ⎣ ⎦⎭ 1 1 ⎩ (66)

∫

ds C(s)e

L

where l1 is the renormalized persistence length of the polymer chain. The equation for the calculation of the renormalized persistence length l1 has the form

where

∫0

∫0

L

where Rg is the mean-square radius of gyration of a polymer chain.31 As a result, we have the following approximate expression for the propagator in the q representation

Cq̃ , q ′ = Cqδq + q ′ ,0

L

(62)

dq (·) 2π

⎛ k 2R g 2 ⎞ ⎟ S(k ⃗) ≈ N ⎜⎜1 − ⎟ 3 ⎝ ⎠

Because the function C(|s1 − s2|) is translation-invariant, we have

1 Cq = L

⎤

dq

As k → 0, we have

(59)

∫0

∞

where

where Cq̃ , q ′ =

−∞

q≠0

=

: rq⃗ rq⃗ ′:g (Cq̃ + q ′ ,0 − Cq̃ , −q ′)

q , q ′≠ 0

∞

⎞⎧ ⎪ exp{ik [⃗ r (⃗ s1) − r (⃗ s2)]}⎟⎨ ⎟⎪ ⎠⎩

(57)

⎡ ⎤ ∑ : rq⃗ r −⃗ q:g ⎢ 1 − 1 ⎥ g (̃ q) ⎥⎦ ⎢⎣ g0̃ (q) q≠0

∑

∑ (·) → ∫

1 S(k )⃗ ≈ Ll0

ds2 C(|s1 − s2|)

Therefore, we have

=

ds (1 − cos qs)

⎡

L

⎡ 2 dk ⃗ ̃ (k ⃗)k 2 exp⎢− k D ⎢ L (2π )3 ⎣

L

The nonlinear integral in eq 61 can be solved with respect to the function g̃(q) by an iterative method. Hence, we can obtain an expression for the static structure factor of a polymer chain in solution in the zeroth approximation of the GER as

where the function C(s) has the form C(s) =

∫0

We stress that eq 62 becomes exact in the limit of L → ∞. Because we consider the case when L → ∞, we can replace the sum by an integral according to the standard rule

(56)

{exp[i(q + q′)s1] − exp[iqs1 + iq′s2]}

β 3l0 2

⃗

⎡ 1 1 ⎤⎥ ⎢ − g (̃ q) ⎥⎦ ⎢⎣ g0̃ (q)

q≠0

(61)

∫ (2dπk)3 k2D̃(k ⃗) exp⎢⎣−k2 ∫−∞ 2π g (̃ q)(1 − cos qs)⎥⎦

The basic contribution to the value of the functional integral is concentrated in a new Gaussian measure with the new propagator g(s). This means that quadratic terms in variables rq⃗ should be absent from WI[r]⃗ . This corresponds to the following condition

∑ : rq⃗ r −⃗ q:g

1 − Σ(q) g0̃ (q)

Σ(q) = C0 − Cq =

⎫ ⎪

q≠0

g0̃ (q)

where

:exp{ik [⃗ r (⃗ s1) − r (⃗ s2)]}:g = exp{ik [⃗ r (⃗ s1) − r (⃗ s2)]}

∑ g (̃ q)[1 − cos q(s1 − s2)]⎬⎪

(C0 − Cq)

q≠0

After some algebra, we obtain the following integral equation for the calculation of propagator g(s)

In eq 54, we introduce the symbol :(·): for the normal product,42 which, for this case, yields

⎧ ⎪ k2 × exp⎨− ⎪ ⎩ L

∑ : rq⃗ r −⃗ q:g

(60)

⎫ ⎪ ∑ g (̃ q)[1 − cos q(s1 − s2)]⎬⎪ q≠0 ⎭

: exp{ik [⃗ r (⃗ s1) − r (⃗ s2)]}:g

⎡ 1 1 ⎤⎥ ⎢ − = g (̃ q) ⎥⎦ ⎢⎣ g0̃ (q)

−iqs

Hence, we obtain K

∫

∫



Article

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Using eqs 63 and 64, we obtain an expression for the radius of gyration of an effective polymer chain in solution through the function g(s) Rg 2 = =

3 L2

6 L2

∫0

∫0

L

ds1

∫0

L

ds2 [g (0) − g (|s1 − s2|)]

L

ds (L − s)[g (0) − g (s)]

(67)

Now, one can substitute eq 65 into this equation and obtain eq 19.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank reviewers for useful comments which helped us to improve our manuscript. REFERENCES

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Coarse-Grained Model of Glycosaminoglycans in Aqueous Salt

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