A Perturbative Approach to Polyelectrolyte Configuration - ACS

Chapter 3 A Perturbative Approach to Polyelectrolyte Configuration 1

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Downloaded by CALIFORNIA INST OF TECHNOLOGY on December 24, 2017 | http://pubs.acs.org Publication Date: December 13, 1993 | doi: 10.1021/bk-1994-0548.ch003

D. Bratko and K. A. Dawson

Department of Chemistry, University of California, Berkeley, CA 94720

A simple self-consistent perturbative approach is applied to study the configuration of an isolated ringlike polyion. The calculation is based on the application of Edwards Hamiltonian for a self avoiding macromolecule, supplemented by screened electrostatic interactions among ionized units of the chain. The structure is described in terms of mean square distances between the beads, calculated using the probability distribution of a reference Hamiltonian with purely harmonic interactions. The method provides analytic scaling relations for mean square distances among distant units and lends itself in a form, suitable for numerical solution. The swelling of a strongly ionized polyion is studied as a function of its length L and the ionic strength of the supporting solution. Extension of the method to systems with attractive intramolecular forces between nonadjacent units is considered and numerical results for a helix-forming polymer are presented.

The configuration of an ionized macromolecule in solution is determined by an interplay of chain elasticity, steric effects and long-ranged electrostatic repulsion. These interactions have been the subject of a variety of theoretical (7-27) and simulation (8,22-30) studies o f polyelectrolyte solutions. In view of the complexity o f real systems, different simplifications were introduced to facilitate the theoretical analysis. A conventional model of polymer theory describes the bonds among monomer units as harmonic springs and treats the excluded volume forces in terms of the Dirac δ function

1

Also affiliated with J. Stefan Institute, University of Ljubljana, Ljubljana, Slovenia Current address: Department of Chemistry, University College Dublin, Belfield, Dublin 4, Ireland

2

0097-6156/94/0548-0034$06.00/0 © 1994 American Chemical Society

Schmitz; Macro-ion Characterization ACS Symposium Series; American Chemical Society: Washington, DC, 1993.


3.

BRATKO & DAWSON

Perturbative Approach to Polyelectrolyte Configuration 35

(31). Electrostatic effects may be introduced as an additional term in the polymer Hamiltonian, in the simplest case represented by a sum of pairwise additive Y u k a w a interactions among the charged groups on the chain. In the case of long neutral macromolecules, valuable insights are known to be obtained by studying polymer rings which behave in a similar way as linear chains and, in the case of the above Hamiltonian, lend themselves to a useful analytic solution (32-34). In the preceding works (21,35), a generalization of the ringlike polymer model to charged systems has been suggested and a numerical scheme based on the variational method for the polyelectrolyte structure has been proposed (21,35,36). In the present paper, the variational treatment of these works is shown to be equivalent to a special case of a more general perturbative approach (37). For the particular type of the Hamiltonian, used in our system, the first order perturbation approximation is equivalent to the application of the Gibbs-Bogoliubov bound for the free energy of the polymer. In either approach, the polymer configuration is determined by averaging with respect to an approximate probability distribution of a reference system with a Gaussian Hamiltonian of a collection of Harmonic oscillators. The true Hamiltonian of our system and the perturbative analysis will be described in the following Section. The results will be used to establish scaling relations for mean square distance between distant beads in a number of characteristic situations. In the last Section, we describe the numerical solution of the system of nonlinear equations for reference Hamiltonian couplings and present numerical results for a set of strongly charged dilute polyelectrolytes in a good solvent and at varying degree of polymerization and concentration of the simple electrolyte. Finally, we consider extensions to anisotropic systems and polymers with specific attractive intramolecular interactions.

Model and Method The cyclic polyelectrolyte molecule is pictured as a necklace comprising L monomer units. The unit length in the absence of steric and electrostatic forces is 1. The total charge of the polyion is uniformly spread over all beads, each carrying an average charge q. The bonds between neighbouring units are treated as harmonic springs and we describe the excluded volume interactions in terms of a virial expansion for the Dirac δ function form of the potential between the particles. The electrostatic effects are approximated by the Debye-Huckel screened potential among the beads of the polyion. The solution is sufficiently diluted that there are no significant intermolecular interactions. The configuration dependent part of the Hamiltonian is

—r.l

-K\r s

3!

s,s ,s

s,s

s

Ir -r ,1 s

s


(1)

36

MACRO-ION CHARACTERIZATION

where the energy is given in units k T , k is the Boltzmann constant and Τ the temperature, κ is the Debye screening parameter, V = q ^ / 8 n e k T is one half of the Bjerrum length, ε the permittivity, s the relative position on the chain and the summations are carried out over all L units of the polyion. A t nonzero values of the two particle excluded volume parameter U2, the three body term can usually be omitted but w i l l become important (35) at theta conditions where it represents the dominant short


ranged interaction. The structure of the polyion is described in terms o f mean square distances is obtained by expanding the δ functions in plane waves and Q

then averaging with respect to the reference Hamiltonian _3

Jdkjdk'e ^ 6 2

Σ s,s',s"

2

-[ f l - c o s q(s-s')] + s,s' z


2

z

- — 6 ΤΊ-Σ

(16)

2

2

D

2

*"

and 2 1I2 are Όpure -2 ν constants obtained by integration over Is-s'l. A p p l y i n g the

where

2


method of dominant balance (39) at small q we find the solution

V

f

3.

V

5 α - 1 = 2 , a = - , D : — - — a,1 —~- + — u~ 5 3 κ 2 2

I

Ï

(Π)

2

Over long distances l s - s ' l » l a well screened polymer resembles a self-avoiding ring with a modified second virial coefficient. A more careful analysis (35) avoiding the approximations of the preceding paragraph yields similar results but with Reiss exponent 2a=4/3 replacing the Flory result. A t shorter distances or in the absence of simple electrolyte, when we may ignore the screening of Coulombic interactions, an alternative expansion of equation 12 gives

q

3u„ +— attractive forces between the units would lead to the formation of collapsed blobs with α = 1 / 3 . A detailed analysis along these lines will be reported (35). Numerical

Results

The system of L / 2 nonlinear equations 12 for coefficients g(q) is solved by direct iteration. The procedure is initiated by assuming an ideal Gaussian behavior of the ring and using the corresponding mean square distances are determined and the procedure is repeated until a self consistent solution of a sufficient accuracy is obtained. The use of mixing parameter of rather low value between 10"^ and 10"! is needed to ascertain the convergence of the iteration. The number of iterations needed depends on 2

the length of the chain L , the charge q and the screening parameter κ. About 10 -10^ cycles were usually sufficient to obtain the accuracy better than 0.01% in calculated 2 mean square distances /dln L for various degrees of polymerization L and the ionic strengths are collected. A t very strong screening, numerical data approach the results for self avoiding cyclic polymers (32,33). A s pointed out by des Cloizeaux (32-34),

the self-avoiding rings under

present approximations are characterized by the Reiss exponent 2ct=4/3 and this is confirmed by the present results. The nonscreened polyions, on the other hand, behave as fully extended rings. In a few extreme cases, a slightly exceeds unity reflecting not


3. BRATKO & DAWSON

Perturbative Approach to Polyelectrolyte Configuration 41

only the full expansion but also the extension of individual links due to the increase of the electrostatic repulsion with growing L . The problem is not seen in studies of intramolecular exponent measured at constant L . Figure 1 illustrates the dependence of 9

the renormalized intramolecular scaling exponent 2 a ' =

(dln/dlnL, renormalized exponents

2 a ' ( L / 2 ) , and mean square distances zlR =5| _ -| 5. Figure 2 a

s

s



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MACRO-ION CHARACTERIZATION

IS-S'I Figure 1

The renormalized intramolecular scaling exponent 2 α ' as a

function of the separation Is-s'l at L=2000 and at ionic strengths (top to 3

2

1

3

bottom): I = 1 0 ' , Ι Ο " , 10" or 1.0 mol d m " .

IS-S'I F i g u r e 2 The mean square distance

A Perturbative Approach to Polyelectrolyte Configuration - ACS

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