Functional integral approach to the dissolution of polyelectrolyte complexes

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Functional integral approach to the dissolution of polyelectrolyte complexes

P. Haronska

To cite this version:

P. Haronska. Functional integral approach to the dissolution of polyelectrolyte complexes. Journal de

Physique, 1989, 50 (14), pp.1827-1841. �10.1051/jphys:0198900500140182700�. �jpa-00211033�

Functional integral approach ^to the dissolution of polyelectrolyte complexes

P. Haronska

Akademie der Wissenschaften der D.D.R., Institut für Polymerenchemie « Erich Correns »,

D.D.R.-1530 Teltow-Seehof, ^D.D.R.

(Reçu ^{le 30 mars} 1988, accepté ^sous forme définitive ^{le 29 mars} 1989)

Résumé.

Nous présentons ^une étude basée sur l’intégrale fonctionnelle de la dissolution des

complexes polyélectrolytes complètement symétriques. La dissolution elle-même peut être causée par l’addition de sel ^ou par ^une variation de la température. Nous utilisons l’intégrale

fonctionnelle pour transformer le hamiltonien de Landau-Ginzburg-Wilson qui ^{décrit le} système

en une représentation ^{à deux} champs. ^La paire ^de champs ^a plusieurs propriétés qui font penser

au paramètre ^{d’ordre en} supraconductivité. ^Le hamiltonien transformé nous permet de calculer

l’énergie libre de Helmholtz et la chaleur spécifique ^{dans la} région critique. La chaleur spécifique

a une discontinuité au point critique.

Abstract.

A functional integral approach ^to the dissolution of fully symmetric polyelectrolyte complexes ^is presented. The dissolution itself may be caused by adding of low molecular salt or by varying ^the temperature. ^The zero-component Landau-Ginzburg-Wilson Hamiltonian, ^{which is} used for describing ^the system ^under consideration, is transformed to a pair ^field representation exactly via functional integrals. ^The pair field has many features reminding ^{of the} gap-parameter in superconductivity. The transformed Hamiltonian allows the Helmholtz free energy and the

specific heat in the critical region ^to be calculated. The specific heat shows a discontinuity ^{at the}

critical point.

Classification

Physics ^Abstracts 05.40 - 36.20

’

1. Introduction.

Experimental studies indicate an abrupt dissolution of some polyelectrolyte complexes ^above

a critical concentration of added low molecular salt [1]. ^{This kind of} dissolution has therefore features of a critical phenomenon [2].

De Gennes showed that the statistical properties ^of linear, ^flexible polymers ^{can be}

deduced from zero-component Landau-Ginzburg-Wilson ^field theory [3]. Polymer properties

which are proportional ^{to some} power of the molecular weight has been derived within this framework and within the conformation-space renormalization-group approach [4-8]. ^{In the}

case of dissolution we are interested in properties ^which depend ^{on the} concentration of added low molecular salt or the temperature rather than the molecular weight. ^This

circumstance requires another theoretical approach which is different from that mentioned before.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500140182700

The dissolution phenomenon ^is mainly ^caused by ^the attractive coulombic interchain interaction between oppositely charged polyions. ^{Therefore we make use of a} change ^of integration ^{variables in} functional integrals which has the advantage ^of replacing ^the original

zero-component Landau-Ginzburg-Wilson Hamiltonian by ^{another one} in which all funda- mental fields appear ^as pair ^{fields. The} pair ^field directly describes the pairing ^of oppositely charged polyions. Such situation is quite ^{familiar to a} type ^II superconductor [9], ^{where the}

gap-parameter plays ^{the same role as the} pair ^field. Moreover, ^the pair field formulation will

give ^{us the} possibility ^to calculate the Helmholtz free energy in the critical region.

This article is organized ^as follows. In section 2, ^{we recall the} expressions ^of polymer physical quantities of interest in the framework of the continuous model, and in the framework of the zero-component Landau-Ginzburg-Wilson ^field theory. In section 3, ^a pair

field Hamiltonian is introduced, and its connection to the Landau-Ginzburg-Wilson ^Hamilto-

nian is established. In section 4, ^the concept ^{of an} anomalous correlation function is

developed in the framework of polymer physics. The last section is devoted to the calculation of the free energy and the specific ^{heat at} the transition point.

2. Model.

Due to the complexity ^of polyelectrolytes simplifications ^are necessary ^to develop ^a

theoretical approach. ^{We shall use} the model of uniformly charged polyelectrolyte chains with

charges ^smeared along the chains. Furthermore, ^{we shall assume} that the chain backbone is

sufficiently ^{flexible to} utilize the random flight ^model [10]. ^{This can be} only justified ^{if the} spacing ^between charge ^centres along the chain is greater ^{than or} approximately equal ^{to the} Debye screening length of the low molecular salt [11].

In the case of 1-1 low molecular salt the Debye screening length ^{is devined as} [12]

where n is the concentration of the salt, e is the electrical charge, Eo is the permittivity ^of

vacuum, D is the dielectric constant of the medium, kB is Boltzmann’s constant, and T is the absolute temperature.

Owing ^{to the} importance of interchain interaction between oppositely charged polyions

with regard ^to complex ^{formation we} shall confine ourselves to this interaction only. ^The

attractive contribution to the monomer-monomer interaction potential ^{is taken as a} Debye potential given by

where we have used K

À D 1 for convenience. Note that the Debye potential ^is only ^{valid for}

weak electrical fields which is exactly ^{the condition} when the dissolution occurs. Beside of this attractive interaction we can also find a repulsive short-range interaction. In its simplest approximation ^this potential ^{is a} 5-function [7, 8].

A suitable continuous model of polymer ^{chains is the} Edwards model [13]. ^The

conformation of a chain is given by ^{a vector} position ^function R (’T) (0 :5 ’T :5 L ), ^{where L is}

the length of the chain. The total interaction potential ^{of a} poly-anion ^{1 and} poly-cation ^{2 is}

that after averaging ^{over the} microscopic details. We have

where v (v > 0 ) represents the excluded-volume parameter [7, 8], while V is given by equation (2.2), and ai i is the spacing between the charge ^centres along the chain. Let

li be the Kuhn segment length, then the full Hamiltonian for this two-polymer system ^reads

[7, 8]

The probability distribution function G12 (rl, ri, r2, r2 ; Li, L2) is obtained from averages

over all possible ^chain conformation with fixed end-point ^vectors ri, ri and r2, r’2. ^This averaging ^can be written in terms of a functional integral ^as

Here f U)c represents ^{the summation over all} possible configurations ^{of a} polymer ^chain [10].

The Helmholtz free energy F for this two-polymer system ^reads [14]

There is another approach ^for calculating statistical properties ^of polymer chains which makes use of the correspondence ^between polymer physics ^{and field} theory [3, 4, 8]. ^The

correlation function is after the Laplace ^transform

that of a O (n ) ^field theory in the limit n -. 0. Starting from the Hamiltonian

where

and

thereby the 0 ^{i fields are to be} understood as

we find for the correlation function [4]

In a more elegant way the correlation function ^can be expressed by using ^{the so-called} generating functional which is given by [15, 16]

Then the correlation function reads

Here the symbol ⁸ denotes the functional derivative with respect ^{to a} vanishing external field

hi.

In appendix ^A is shown how the free energy may be re-expressed by the saddle points si ^and S2*. ^We find for the saddle points

Here _çi is an unknown function and si* is connected with the chain length via the relation

where f ^is given by (see appendix A)

3. The pair field Hamiltonian.

In spite of the model character of the Hamiltonian (2.8) ^{an exact} calculation of, ^for instance,

the free energy is beyond ^our computational abilities. Hence, ^it is necessary ^to carry ^out calculations within the perturbation theory. ^A conventional perturbation expansion ^is performed ^{around an ideal state with} respect ^{to a small} parameter ^which corresponds ^{to the} strength ^of interaction. Note, however, ^{that even in the case of a} small attractive potential U,

that is, ^{U 0 the} perturbation expansion is invalid. This can be immediately ^seen by inspecting the Boltzmann probability weight exp [- 3C/kB T] ^{for U 0. To overcome this} deficiency ^{we shall use an} interesting analogy between the theory ^of type ^II superconductors

and the present problem [9].

Let us first demand that the polyions ^are fully symmetric which is in good coincidence with

experimental findings [1]. We then have

and

From equation (2.15) ^{it follows} (see appendix A)

Furthermore, ^{we shall} approximate ^the potential ^U by ^a &-potential, ^{that is,}

where y is

We have good ^{reasons that} approximation (3.4) should work well, first, because in the case of dissolution the potential ^is short-ranged ^indeed and, second, scaling arguments show that for

spatial dimensionality d > 3 the large-scale ^{behaviour of a} polymer ^is mainly governed by ^{a a-} potential [7, ^8, 15].

When the polyions ^are fully symmetric, ^a slight modification of the CPi fields, ^that is,

will give ^{us the} possibility ^{to extend our} consideration to the case of 2 N oppositely charged polyions. ^The Hamiltonian

is fully equivalent ^{to the} original ^{one of} equation (2.8), ^{so that the case of two or more} polyions ^can be treated on equal footing. Thereby G-1 (r, r’) ^{stands for}

We are now in the position ^{to introduce an} auxiliary ^{field fi} (r, r’ ). It will be clear later that this field is strongly connected with the pairing ^of polyions ^with opposite signs and, therefore,

it will be called pair field in the following. ^{The shorthand} notation il (r, r’ ) ^{is to be}

understood as (see appendix B)

m (r) is assumed to have the property

Here k and f run from 1 to n and t and s run from 1 to N. In appendix ^B (see ^also [9]) ^{is shown}

that the following identity ^holds

Here N is an irrelevant constant (see appendix B ; [9]). ^After inserting ^the identity (3.11) ^into

the generating functional (2.13) ^and using the fact that the integration ^over 0 is Gaussian the

following expression ^{is obtained}

where

and

Furthermore the symbols ^{Tr and tr stand for}

where A is a symmetric operator ^{and tr denotes a trace} which involves a summation over all discrete indices. The logarithm log A may be expanded in the standard fashion as

Let us now introduce a pair field Hamiltonian in the following ^manner

The definition of the pair field Hamiltonian given ^{above is} suggested by comparing ^the generating functional (3.12) ^{with the} original ^{one of} equation (2.13). ^{The main feature is that}

a mean value of an arbitrary product ^of Oi ^{fields can be} expressed ^{as a} corresponding ^{one of} pair ^fields. Using the notations

and

we find, ^for instance, ^{that the mean value}

takes the form

Thereby the indices 1 and 2 indicate the corresponding ^element of the K matrix which is given by ^{the relation}

where K-1 is taken from equation (3.14).

4. The anomalous correlation function.

In the last section we have seen that the correlation function (3.20), ^{in the} following ^called

anomalous correlation function, is connected with a mean value of pair fields. It can be easily

verified that the mean value (3.21) ^{involves an} expansion in odd powers of the pair ^field only.

We have

Here is

where G is given according ^to equation (3.22). ^{Note that m is} independent ^{of r due to}

translational invariance.

When the interaction in the Hamiltonian (3.7) ^{is a} repulsive one, that is, y > 0 no

anomalous correlation function can occur. This statement is easily checked within the

perturbation theory [15, 16]. ^{On the other hand,} if the interaction is an attractive _one, that is,

y 0 no perturbation expansion ^works and, therefore, the appearance of ^an anomalous

mean value such as (4.1) may be possible. ^{In some} respects ^{this is} analogous ^to superconductivity ^{where an anomalous} propagator exists due to the attractive interaction between electrons caused by exchange ^{of a} phonon [9].

In the case of polymers the appearance of the anomalous correlation function is connected with the concept ^{of the} O(n ) field Oi. ^Before going ^{into details we confine ourselves to the}

case of two polyions only, ^{that is, one} poly-anion ^{and one} poly-cation. ^{Then the} following

mean value

yields the concentration of monomers which belong ^to the chain i at point ^r [14]. ^{In this}

respect the anomalous correlation function

may be interpreted ^{as the} concentration of monomers which belong ^to the chains 1 and 2 at

point ^{r. Due to the different} chains the correlation function (4.4) ^{will be} normally ^{zero, while}

a nonzero value of the anomalous correlation function (4.4) ^{indicates a} pairing ^{of both} polyions. Obviously, the results stated above can be easily generalized ^{to the case of an} arbitrary (even) ^{number of} polyions.

It is worth coming ^{back to} equation (4.1) once more. An inspection ^{of this} equation ^shows

that the appearance of ^an anomalous correlation function is connected with an odd _power

dependence ^on fi and vice versa. On the other hand, by expanding ^the logarithm ^{in the} pair

field Hamiltonian (3.17) according ^to equation (3.16) ^{it can be} easily verified that the series involves an expansion ^{in even} powers of the pair ^field only. The appearance of ^m is hard to understand if there is a symmetry ^{for fi --+ - n. Let us therefore introduce a} small external

field b ^{and let us further} investigate the linear response of the system ^{to this field.}

Starting ^{from the} generating ^functional

and expanding

one finds to first order in 4 ^{that the} following ^relation

holds. Note

that ({(p)}==i)& w ^{is analogous to an induced} magnetization, ^{while m} corresponds ^{to a} spontaneous ^one. Equation (4.7) establishes a connection between the

magnetization, the correlation function, and the external field. It can be immediately ^seen

from equation (4.7) ^{that the} spontaneous magnetization m, ^that is, (P ) s (1, = 1) 1 t = 1 e, 4-0

only appears when the correlations become of infinite range. This is ^an important

characteristic of a second-order transition [2, 15].

5. The phase transition.

Sufficiently ^{close to the} transition point, ^{when the} correlation length ^becomes large ^{to the} microscopic structures, ^one obtains after a coarse graining procedure ^{an effective Hamilto-}

nian [2, 15], ^{which is more} convenient for analytical calculation than the original ^{one of} equation (3.17). ^In appendix ^{C an} expression ^{for this effective} Hamiltonian is derived. We have

The q ^{field is to} be understood as

and the parameters ^{« and w are} shortland notations for

and

The dimensionless coupling ^{constant g is defined} by

The non-Gaussian contribution to the Hamiltonian (5.1) ^{is now} well controlled by ^the

dimensionless coupling ^{constant u.} When g ^is sufficiently ^small the n 4 -term may be omitted.

The correlation length ^{then reads} [2]

To be sure that the coarse graining procedure ^is justified ^{and a} spontaneous magnetization

may exist, ^{we must examine} whether the correlation length (5.6) ^is large indeed for small a ^or

not [2, 15]. ^Before doing this, ^{we restrict ourselves to the} high polymer limit, ^that is,

L --+ oo. In accordance with relation (2.16) ^{we can then} replace ^{s *} by f. Using ^{this, we obtain} for e

Experimental findings ^show that after dissolution the bulk concentration is a very dilute one, that is, ^the complex ^{formation takes} place ^{for a} very dilute polymer ^solution [1]. ^{So the} polymer concentration must be handled in the limit

where Vol is the volume in which the polymers ^{are confined.} If g > 0 the condition (5.8)

states that fi ^is influenced only by ^{the fixed} point ^of the dimensionless coupling constant, ^that is, fi ^is independent ^{of IL.} Note, however, that this statement holds only ^{if we restrict}

ourselves to the high polymer limit L --> oo [15]. ^{Then it follows from} equation (5.7) that e

cannot be large ^{even in the case of a} small, ^but positive, dimensionless coupling ^constant.

Physically speaking ^{this means that a} pairing ^of polyions ^with opposite signs ^{cannot take} place

for a repulsive interaction and, furthermore, ^{the coarse} graining procedure fails, ^that is, ^the

Hamiltonian (5.1) is irrelevant for describing ^the system under consideration.

On the other hand, ^if IL 0 no fixed point ^{of the} dimensionless coupling ^{constant exists} [2, 14, 15] and, therefore, fi may be considered ^{as a} nontrivial function of IL. From

superconductivity ^{we have} good ^{reason that the} pairing effect should take place for any small,

but negative, coupling ^constant [9]. Bearing ^{this in mind, we} may therefore conclude that a

change ^of sign of the dimensionless coupling ^constant corresponds ^{to the} dissolution transition.

At non-vanishing magnetization ^the system ^is characterized by ^two length ^scales, namely,

the longitudinal correlation length ^{and the} transverse correlation length [2, 15]. ^Note,

however, that in the case of a vanishing q 4 -term Ward identities predict ^the identity ^{of both}

lengths [15]. ^{Due to} the fact that the transverse correlation length ^{tends to} infinity ^{when the}

spontaneous magnetization appears the inverse (longitudinal) correlation length -1 ^{must be}

therefore considered as a vanishing ^{one for} small g 0, ^that is,

Then it follows immediately ^from equation (5.7)

or in a more general ^{form one} gets

Here (ff ) * is the fixed point ^of fi.

Equation (5.5) ^and the definition (3.5) ^{of y} suggest ^an expression for g ^{in the form}

where n, is the concentration of low molecular salt for which the counterbalance of attractive screened coulombic interaction and repulsive ^excluded volume effect take place. Clearly,

such kind of counterbalance can also be accomplished by varying ^the temperature. ^Let T, ^be the critical temperature, ^{then the} Helmholtz free energy reads for temperatures T very

close to Te in accordance with equation (2.17)

Here F * is the fixed point of the free energy of the polyions, ^that is, the value which is

independent of the dimensionless coupling parameter and, therefore, ^the temperature. ^The specific ^heat C, ^an experimentally measurable quantity, is related to the free _{energy via}

Using this, ^we find with the help ^of equation (5.13)

where C’ is the specific heat for T Tc. ^The specific heat shows a discontinuity ^{at the critical}

temperature Tc. Unfortunately, ^no measurements were carried out for temperature depen-

dence.

6. Conclusion.

We have presented ^{here a theoretic treatment of the} dissolution of fully symmetric polyelectrolyte complexes which allows us to calculate the Helmholtz free energy in the critical region.

Using arguments ^{which are similar to those in} superconductivity, ^we may identify ^the

dissolution effect as counterbalancing ^of attractive screened coulombic interaction and

repulsive ^excluded volume of the chains. The counterbalance _{may be} caused by varying ^the

concentration of low molecular salt or the temperature.

Appendix ^A

The inverse Laplace transform is defined as

where Ci ^{is a contour in the} complex si -plane running from - i oo to ₊ i oo and lying ^{to the} right ^{of the} singularities ^of G12 (r1, ri, r2, r2 ; si, s2 ). Now the free energy ^can be expressed ⁱⁿ

the form

When Li ^is sufficiently large it should be possible ^to approximate ^F by steepest ^descent.

Therefore it is necessary ^to calculate the saddle points si ^and s2*. ^{We have}

and

The inversion of the equations (A3) ^and (A4) produces ^a solution for st and S2* in the form

The complexity ^of G12 ^{makes an} explicit calculation of SI* and S2* impossible. ^{Due to the fact}

that the fraction F/(L1 ⁺ L2 ) is an intensive physical quantity, ^that is, F/(L1 ⁺ L2) may be considered as independent ^of L1 ⁺ L2 ^{in the} high polymer limit, ^the following relation holds for large L1 ⁺ L2

Here f ^{is a constant.} Starting ^{from the} identity

we can by comparing ^with equation (A2) ^{conclude that}

holds. Thereby s, stands for

By inserting ^the right-hand ^{side of} equation (A8) ^into equation (A3) ^or (A4) ^{we find}

Obviously, the results stated above can easily ^be generalized ^{to the case of 2 N} oppositely, ^but fully symmetric, charged polyions. ^Then f is related to F via

Equation (A10) takes the form

Appendix ^B

Here we derive the identity (3.11). Starting ^{from the} symmetric ^field

where k and f run from 1 to n and t and s run from 1 to N, ^we may introduce another field in the following ^manner

which is connected with a matrix A via

Obviously, ^{À stands for}

and

respectively. ^{The matrix A} is therefore a symmetric ^one. Note that the differential + is not

changed by ^relation (B2), ^{that is,}

Here fi is

Let us now consider the product ^{A2 which reads in} accordance with equation (B3)

The trace over all discrete indices yields

Defining ^{the constant JV in the} following ^manner

one finds by using ^the identity (B6) ^and equation (B9)

Here f2 (r, r’) ^{stands for}

Equation (B11) ^is nothing ^{but the} identity (3.11).

Appendix ^C.

In this appendix ^{we shall derive the effective} Hamiltonian (5.1). ^{Close to} the critical region

the expansion ^of

may be truncated after the fourth !1-term without loss of information as far as long-range properties ^{are concerned} [2, 15]. ^Before going on, let us decompose the matrix K-1 in the

following ^manner

The first term of the right-hand ^{side is} independent of 12 and may be therefore absorbed in the irrelevant normalization JY’ . The last term of the right-hand side may be expanded ^{in the}

standard fashion via equation (3.16) up ^to the fourth order in ,l2. We find the following expression by noticing ^that Ko ^is nothing ^{but K for} vanishing f2

The quadratic ^{term becomes}

Using ^the fact that

the Fourier transform

yields

Equation (C4) ^{then reads}

The computation of

is standard and can be found elsewhere [16]. ^{As far as} long-range properties ^{are considered}

terms up ^to the second order _{in p} are of interest only [2, 15]. The result reads

r

and Tr (Ko il)2 ^becomes

For Tr (Ko n)4 ^{one finds}

An analogous ^{treatment as} before yields exactly ^for

Using ^ths shorthand notation

then Tr ( log ( I 1/2 Ko il ) ) may be written in the form

Setting IL

y li ^{we get} by adding ^of

to equation (C15) ^{the effective} Hamiltonian (5.1)

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