Development of the Theory of a Self Consistent Field

ISSN 00360244, Russian Journal of Physical Chemistry A, 2011, Vol. 85, No. 8, pp. 1363–1368. © Pleiades Publishing, Ltd., 2011. Original Russian Text © E.A. Nogovitsyn, Yu.A. Budkov, 2011, published in Zhurnal Fizicheskoi Khimii, 2011, Vol. 85, No. 8, pp. 1477–1483.

PHYSICAL CHEMISTRY OF SOLUTIONS

Development of the Theory of a SelfConsistent Field for Polyelectrolyte Solutions E. A. Nogovitsyna, b and Yu. A. Budkovc a

Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany b Ivanovo State University, Ivanovo, 153000 Russia c Dubna International University of Nature, Society, and Man, Dubna, 141980 Russia email: [email protected], [email protected] Received June 9, 2010

Abstract—A new theoretical approach to calculating the thermodynamic and structure functions of poly electrolyte solutions is proposed, based on the method of Gaussian equivalent representation for calculating the functional integrals. Formulas for the meanforce potential, osmotic pressure, and complete monomer– monomer pair distribution functions are presented. A sodium polystyrene sulfonate solution with NaCl addi tives is considered as an example. Keywords: thermodynamic functions, solutions, polyelectrolytes, osmotic pressure, meanforce potential, functional integrals. DOI: 10.1134/S0036024411080255

INTRODUCTION Studies of the physicochemical properties of mac romolecular systems are currently one of the highest priorities in physical chemistry, biophysics, medicine, and nanotechnologies. A special role is played by the oretical methods which, on the one hand, should explain the behavior of individual systems and, on the other hand, describe their physicochemical properties on the basis of fundamental physical principles. For solutions consisting of relatively small mole cules, the dominant methods of theoretical studies are numerical simulation (molecular dynamics, the Monte Carlo method) and the numerical solution of integral equations. For solutions of macromolecules in which one molecule consists of thousands or tens of thousands of atoms, conventional numerical methods are inefficient. At the same time, the thermodynamic properties of macromolecular systems are to a large extent determined by the multimolecular nature of interactions [1–3]: one polymer molecule simulta neously interacts with hundreds of others, while a molecule in ordinary liquids has only about ten neigh bors that cannot move independently of each other. Methods based on the selfconsistent field idea and the theoretical field approach [3] are therefore more appropriate for theoretical descriptions of macromol ecules. The theoretical field approach is usually under stood as methods based on collective variables, func tional integration, and renormalization group equa tions [2–4]. Within these methods, a number of mod

els have been developed that differ in the details of considering the structure of the system under study [3]. The main advantages of the theoretical field approach are the possibilities for studying and predict ing physicochemical properties of macromolecular systems without substantial computing power. The main disadvantages are associated with the problem of calculating functional integrals. In general, calcula tions are performed in the mean field approximation, which is reduced to the saddle point method [2–4] and does not always lead to accurate results even at the qualitative level [3–8]. For lowmolecular solutions, these theoretical tools certainly cannot compete in content of informa tion with numerical methods in which the system is considered at the atomistic level, though they do allow us to obtain analytical relations between various phys icochemical quantities and to determine important systematic features following from the ab initio princi ples of statistical physics [4]. For polymer solutions, however, such approaches seem to be most informative ones developed so far. Multiscale methods that simul taneously combine both atomistic and field approaches have also been widely accepted [3, 5–7]. Among polymers, polyelectrolytes in which Cou lomb interactions play an important role are of special interest. The existence of the longrange electrostatic potential leads to considerable differences in both the experimental behavior and the theoretical description of polyelectrolytes from ordinary uncharged neutral polymers. For strongly charged polymer chains, the

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longrange Coulomb interaction dominates and con trols the system behavior. In [9, 10], we proposed a new theoretical field approach to describing the thermodynamic properties of polyelectrolyte solutions, based on the method of Gaussian equivalent representation for calculating functional integrals, going beyond the mean field approximation [11, 12]. In [11], we analyzed in detail the results obtained within the methods of Gaussian equivalent representation, meanfield approximation, and Monte Carlo numerical simulation, and showed the advantages of the proposed method by examples of systems with simple pair potentials. The aim of this paper is to present the results from the further development of the method of Gaussian equivalent representation for describing the thermo dynamic and structural properties of polyelectrolyte solutions. Explicit analytical expressions are given for the meanforce potential, pair distribution functions, and osmotic pressure of polymer chain solutions. As an example, we consider an aqueous solution of sodium polystyrene sulfonate (NaPSS) with low molecular salt (NaCl) additives.

Introducing the local density of monomers, 1

∫ ∑ δ ( r – r ( s ) ),

ρ(r) =

βN ds

∞

Ξ ( z, β, V ) =

n n

z  n!

∑ ∏ ∫ δr e

n=0

βW 0 [ r i ] + βW I [ r i, Φ ]

i

.

(1)

i=1

The intramolecular configuration energy is deter mined by the potential 1

3k B T dr i ( s )⎞ 2 W 0 [ r i ] = –  2 ds ⎛   , 2Nb 0 ⎝ ds ⎠

∫

and using the functional relation 2

1

1

βN exp –  ds 1 ds 2 2

2

1

W I [ r i, Φ ] = – N  ds 1 ds 2 2

n

0

0

∫ ∫ ∑ ∑ Φ(r (s ) – r (s )) 0

0

i

1

j

2

i = 1j = 1

3 βNnΦ (0) 1 3 = exp –  d r d r'ρ (r)Φ (r – r')ρ (r') +  2 2

∫ ∫ V

=

V

(6)

Dφ

e ∫ ( det Φ )

–1 ( 0) – 1 ( φΦ φ ) + i ( ρφ ) + βNnΦ  2 2

1/2

n 1

∫

=

dμ Φ [ φ ] : exp i βN

∑ ∫ dsφ ( r ( s ) )

:Φ ,

i

i=10

the statistical sum can be written as the functional integral Ξ ( z, β, V ) =

∫ dμ

Φ [ φ ] exp

∫

z δre

2 3  ds ⎛ dr –  (s)⎞ ⎝ ds ⎠ 2 2Nb 0

(7)

1

1

∫

∫

i βN dsφ ( r ( s ) ) 0

:e

:Φ ,

where δφ  exp – 1 ( φΦ –1 φ ) dμ Φ [ φ ] =  1/2 2 ( det Φ )

–1

( φΦ φ ) =

Φ ( r i ( s 1 ) – r j ( s 2 ) ) (3)

(8)

∫ d r ∫ d r'φ ( r )Φ 3

V

3

–1

( r, r' )φ ( r' ),

(9)

V

∫ d rΦ ( r', r )Φ 3

–1

3

( r, r'' ) = δ ( r' – r'' ).

(10)

V

Colons denote the normal product [13]:

i = 1j = 1 1

describes the interactions between monomers. Inte gration in (1) is performed over all spatial curves of polymer chains, β = 1/k B T,

n

is the Gaussian measure,

n

∫ ∫ ∑∑

n

(2)

where N is the polymerization index and b is the statis tically average length of the polymer chain segment. The potential 1

(5)

i

i=1

0

GAUSSIAN EQUIVALENT REPRESENTATION FOR POLYMER SOLUTIONS The statistical sum for the grand canonical ensem ble of Gaussian polymer chains can be written as [3, 9]

n

2πm⎞ 3/2 e βμ z = ⎛  ⎝ βh 2 ⎠

(4)

where ri ∈ R3 is the activity, and n is the number of polymer chains in the system.

1

∫

∫

i βN dsφ ( r ( s ) )

:e

0

:Φ = e

i βN dsφ ( r ( s ) ) β NΦ ( 0 ) 2 0

e

,

: φ ( r i ( s ) )φ ( r j ( s' ) ) : Φ = φ ( r i ( s ) )φ ( r j ( s' ) ) – βΦ ( r i ( s ) – r j ( s' ) ),

(11) (12)

1

∫

i βN dsφ ( r ( s ) )

∫ dμ

Φ

:e

0

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DEVELOPMENT OF THE THEORY OF A SELFCONSISTENT FIELD

To define measure (8), it is sufficient to satisfy the condition of the positive Fourier transform of the potential ˜ ( p ) = d 3 rΦ ( r )e i ( pr ) > 0. Φ

The function D(r) and the shift parameter c are unknown. In the last term, W2[φ], and there are no lin ear and quadratic terms in field φ. We choose D(r) and c so that the linear and quadratic terms in the integra tion field φ would be absent in the exponent WD of expression (14), and their contribution was contained in the new Gaussian measure

∫

To calculate the statistical sum (7) in the form of the functional integral, we use the method of Gaussian equivalent representation. We turn to a new Gaussian measure dμD, introducing the effective potential D(r)

1

1/2

∫ dμ

D [ φ ]e

WD

,

(14)

2

1

+ z 1 i βNe

.

∫

– Nc

(17)

∫ ds ∫ dσ [ r ]φ ( r ( s ) ) = 0,

–1 –1 1 –  ( φ [ Φ – D ]φ ) 2

(15)

1 + i βN dsφ ( r ( s ) )

(16)

0

1 (18) 2 – Nc 1 z 1 N βe –   ds 1 ds 2 dσ [ r ]φ ( r ( s 1 ) )φ ( r ( s 2 ) ) = 0, 2

0

∫ ∫ ∫

1

– βN  ds 1 ds 2 φ ( r ( s 1 ) )φ ( r ( s 2 ) ) : D + W 2 [ φ ]. 2

∫ ∫ 0

φ)

1

1

∫ dσ [ r ] :

–1

i  ( cΦ –1 φ ) –  β

–1 –1 1 i  ( cΦ –1 φ ) +  1 ( cΦ –1 c ) –  ( D [ Φ – D ] ) –  2 2β β

+ zAe

–  ( φD 2 δφ  e 1/2 ( det D )

The following conditions

–1 –1 W D = – 1 : ( φ [ Φ – D ]φ ) : D 2

– Nc

∫

dμ D [ φ ] =

and shift the integration variable φ → φ + ic/ β in the complex plane, det D Ξ ( z, β, V ) = ⎛ ⎞ ⎝ det Φ⎠

1365

0

0

must then be satisfied, where

0

1

1

3 dr ( s ) 2 3 dr ( s ) 2 dσ [ r ] = δr exp – 2 ds ⎛ ⎞ / δr exp – 2 ds ⎛ ⎞ , 2Nb 0 ⎝ ds ⎠ 2Nb 0 ⎝ ds ⎠

∫

∫

∫

(19)

1

3 dr ( s ) 2 exp [ βμ + ( β/2 )N [ Φ ( 0 ) – D ( 0 ) ] ] z 1 =    δr exp – 2 ds ⎛ ⎞ , 3 3/2 2Nb 0 ⎝ ds ⎠ h ( β/2πm )

∫

∫

:D ,

e 2 = e – 1 – x – x /2.

(20)

1

∫

i βN dsφ ( r ( s ) )

W 2 = zAe

– Nc

∫

dσ [ r ] : e 2

0

The set of Eqs. (17) and (18) represents the self consistency condition that defines the dependence of the effective potential D(r) (a selfconsistent field) on the external parameters β and z. Equation (17) can be rewritten as ˜ ( 0 )e –Nc , c = zβNAΦ (21) ˜ (0) = Φ

∫ d rΦ ( r ). 3

(22)

=

1

0

2

1

2

0

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A

∫

2

1

3

1

2 2

Nq b

 s1 – s2 6 d q  φ˜ ( q ) 2 ds ds e –   1 2 3 ( 2π ) 0 0

∫ ∫

∫

(23)

3

where G(q2) is the Debye structure factor [2], 2 2

Nb q

 ⎞ 6 12 72 ⎛ –  G ( q ) =     e – 1⎟ . + ⎜ 2 2 4 4 ⎠ q b Nb q ⎝ 2

1

∫ ds ∫ ds ∫ dσ [ r ]φ ( r ( s ) )φ ( r ( s ) )

x

d q  φ˜ ( q ) 2 G ( q 2 ), = 1  N ( 2π ) 3

Omitting the calculation details, we write the sec ond term of the lefthand side of Eq. (18) in the form 1

x

(24)

At the same time, the first term of the lefthand side of Eq. (18) can be written in the form Vol. 85

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POLYELECTROLYTE SOLUTION MODEL Let us consider a solution of Gaussian polymer chains with the distribution function

–1

( φ [ D – Φ ]φ ) =

∫

3

d q  φ˜ ( q ) 2 [ D ˜ –1 ( q 2 ) – Φ ˜ –1 ( q 2 ) ],  3 ( 2π )

1

3  ds ⎛ dr i ( s )⎞ , W 0 [ r i ] = –   2 2Nb 0 ⎝ ds ⎠

˜ ( q2 ) Φ ˜ ( q 2 ) =  . D 2 ˜ ( q 2 )/Φ ˜ (0)) 1 + cG ( q ) ( Φ

(25)

Taking into account selfconsistent Eqs. (17) and (18), the expression for the statistical sum of the grand canonical ensemble takes the form Ξ ( z , β, V ) = e

βPV

(0)

= e

– βΩ GER

∫ dμ

D [ φ ]e

W2 [ φ ]

.

(0) ⎛ ∂Ω GER⎞

z 1 〈 ρ〉 = 1    . 0 ) ⎝ ∂z ⎠ β, V V Ω (GER 1 Having performed transformations that include (21), this expression can be rewritten as 2

∞

2

∞

2 3

0

For the effective potential D(r), which is also the meanforce potential within the described approxima tion [13, 15], we obtain ∞ ˜ ( q ) sin ( qr ) 1 Φ D ( r ) = 2  q dq. 2π 0 1 + cu ( q ) r

∫

exp ( κa/2 ) 2 A ( κ, a ) = ⎛ ⎞ ⎝ 1 + κa/2 ⎠

(31)

is the geometrical factor defined by monomer sizes and shapes [9], a is the characteristic size of the excluded monomer volume, and κ = (8πλBI)1/2 is the Debye screening parameter corresponding to a solu tion with ionic strength I. The Fourier transform for the potential is given by 2

m A ( κ, a )λ B ˜ ( q ) = 4πz  , Φ 2 2 q +κ 2

P du ( q ) c q u ( q ) 1  dq  2c + Nc +   . (28)  =  2 ˜ kB T dq [ 1 + cu ( q ) ] 2 2NΦ ( 0 ) 12π

∫

(30)

where r = |r| is the distance between monomer centers, λB = e2/(4πε0kBT) is the Bjerrum length (λB = 7.14 Å, ε = 78.4 for water at 298 K), and e and ε0 are the ele mentary charge and permittivity, respectively;

(27)

˜ (q)/Φ ˜ (0) and ρ = 〈ρ〉N is the where u(q) = G(q2)Φ m average monomer density. The calculation scheme is as follows: Setting the average monomer density ρm, we determine the shift parameter c from Eq. (27). We can then calculate the thermodynamic and structure functions. For the pressure, we have the following expression: 2

– κr

2 e Φ ( r ) = z m A ( κ, a )λ B  , r

2

c Nc u ( q )q dq ρ m =  –  , 2 ˜ 2π ( 1 + cN ) βΦ ( 0 ) [ 1 + cu ( q ) ] 0

∫

whose monomer units have the charge zm screened due to the presence of opposite charges (counterions) in the system. We consider the solvent as a continuous medium with permittivity ε. The interactions between monomers in both one chain and several chains is described by the potential

(26)

Since W2 contains the field contributions only above the second degree, we set W2 ≈ 0. The accuracy of such an approximation was estimated previously [13, 14]. In a wide range of external parameters, the approxi mation error does not exceed 10%. In this case, the (0) pressure is P ⯝ – Ω GER /V and the average density of polymer chains is

2

∫

and, for the Fourier transform of the effective poten tial D, we obtain the equation

(29)

The monomer–monomer pair distribution function is given by g(r) = exp[–βD(r)].

M A ( κ, a )λ B ˜ ( 0 ) = 4πz Φ  , 2 κ

(32)

2

2 κ . u ( q ) = G ( q )  2 2 q +κ

The set model parameters are λB, N, ρm, ρs, zm, zs, a, and b. The Bjerrum length (λB) is uniquely defined by the temperature and permittivity of a pure liquid solvent. The numerical densities of monomers and lowmolecular salt added to the solution are uniquely defined by their concentrations. If Ang stroms (Å) are chosen as the dimension of length, then ρm = NACm/1027 (Å–3) and ρs = NACs/1027 (Å–3), where the polyelectrolyte monomers (Cm) and added salt (Cs) concentrations are given in (mol/l), NA = 6.02 × 1023 (1/mole is the Avogadro number). The 2 Debye screening parameter κ = (4πλB(2z m ρ m + 2

2z s ρ s ))1/2 (Å–1). The parameters a and b for the solu tion are, generally speaking, unknown or known approximately. They may therefore be considered as fitting parameters for describing an individual system.

RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A

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DEVELOPMENT OF THE THEORY OF A SELFCONSISTENT FIELD Φ(r), D(r) 2

P/RT, mol/l

0.008 1

1367

1

2

0.004 2 0

0 0

10

20

30

40

50

Fig. 1. (1) Mean force D(r) and (2) initial Φ(r) potentials; CNaCl = 0.005 mol/l and Cm = 0.059 mol/l.

0.04

0.08

0.12 Cm, mol/l

Fig. 2. Dependences of the osmotic pressure on the NaPSS concentration at NaCl concentrations of (1) 0.005 and (2) 0.1 mol/l.

g(r)

b, Å 10 8

0

60 r, Å

1.2 1 0.8

2

6 4

1 2 3

0.4

2

0

0.04

0.08

0.12 Cm, mol/l

Fig. 3. Dependence of the statistically average length of the chain segment on the NaPSS concentration at two various NaCl concentrations. See the notations in Fig. 2.

AQUEOUS SOLUTION OF NAPSS WITH NACL ADDITIVES Sodium polystyrene sulfonate (NaPSS) is a poly electrolyte, i.e., a linear homopolimer with monomer charge zm = –1. Sodium polystyrene sulfonate is widely used in chemical technologies and medicine to form polyelectrolyte shells on colloidal particles of dif ferent nature to obtain microcells, nanotubes, to treat hyperkalemia, and others. Using the experimental results of [16] for osmotic pressure at 298 K in various polyelectrolyte and added RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A

0

10

20

30

40

50

60 r, Å

Fig. 4. Monomer–monomer pair distribution functions at various polyelectrolyte concentrations: (1) 0.012, (2) 0.059, (3) 0.126 mol/l and identical NaCl salt concentrations (0.005 mol /l).

NaCl salt concentrations, we fixed the parameters a and b and calculated monomer–monomer pair func tions. Calculations were performed for N = 3155, which corresponds to the molar mass of the polymer chain M = 6.5 × 105 g/mol. Figure 1 shows the effective potential D(r) whose behavior differs considerably from the behavior of the initially set potential Φ(r). The potential D(r) has a minimum; hence, there exists a range of distances within which polymer chain units exhibit weak mutual attractions. Vol. 85

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NOGOVITSYN, BUDKOV

CONCLUSIONS The proposed theoretical approach to calculating thermodynamic and pair distribution functions can be generalized to models that explicitly consider mono mer–counterion, monomer–solvent molecule, coun terion–counterion, counterion–solvent molecule, and solvent–solvent interactions. The set of selfcon sistency equations (17), (18) is then complicated, and its solution requires numerical methods.

g(r) 1.2 1 0.8

2

ACKNOWLEDGMENTS We are grateful to M.V. Fedorov for his attention and helpful advice.

0.4

0

10

20

30

40

50

60 r, Å

Fig. 5. Monomer–monomer pair distribution functions at the various NaCl concentrations: (1) 0.005, (2) 0.1 mol/l and identical NaPSS concentrations (0.059 mol/l). CNaCl = (1) 0.005 and (2) 0.1 mol/l.

Figure 2 shows the experimental and theoretical data for the dependence of the osmotic pressure on the NaPSS concentration at NaCl concentrations Cs = 0.005 and 0.1 mol/l. We can see that the proposed method allows efficient approximation of the experi mental data. The value of the parameter characteriz ing the excluded monomer volume was chosen as a = 8 Å. The concentration dependence of parameter b is shown in Fig. 3. Salt additives and changes in the poly electrolyte concentration slightly (within the mea surement error of the osmotic pressure) affect the sta tistically average segment length. Knowing the param eters a, b, c, and the function D(r), other thermodynamic functions of the system can be calcu lated. Figures 4 and 5 show the pair distribution functions for one salt additive and various polyelectrolyte con centrations, and for one polyelectrolyte concentration and various salt additives, respectively. We can see that increases in the salt additive and polyelectrolyte con centrations shift the distribution function maxima to the left. This can be explained by partial polymer coil ing due to additional screening of the polyion charges, or by the counterion condensation effect [2, 17]. The calculated pair distribution functions are in qualitative agreement with the numerical simulation results [18] and are in fact independent of the polymerization index at N > 100.

REFERENCES 1. M. Doi and S. Edwards, The Theory of Polymer Dynam ics (Mir, Moscow, 1998; Clarendon, Oxford, 1986). 2. A. Yu. Grosberg and A. R. Khokhlov, Statistical Physics of Macromolecules (Nauka, Moscow, 1989; Amer. Inst. Phys., Ithaca, 1994). 2. G. H. Fredrickson, The Equilibrium Theory of Inhomo geneous Polymers (Clarendon, Oxford, 2006). 4. S. Ma, Modern Theory of Critical Phenomena (Mir, Moscow, 1980; Benjamin, Reading, Mass., 1976). 5. M. V. Fedorov, Doctoral Dissertation in Chemistry (IKhR, Ivanovo, 2007). 6. M. V. Fedorov, G. N. Chuev, Yu. A. Kuznetsov, and E. G. Timoshenko, Phys. Rev. E 70, 051803 (2004). 7. G. N. Chuev and M. V. Fedorov, J. Chem. Phys. 131, 074503 (2009). 8. S. Baeurle, G. Efimov, and E. Nogovitsin, Europhys. Lett. 75, 378 (2006). 9. S. A. Baeurle and E. A. Nogovitsin, Polymer 48, 4883 (2007). 10. S. A. Baeurle, M. G. Kiselev, E. S. Makarova, and E. A. Nogovitsin, Polymer 50, 1805 (2009). 11. S. A. Baeurle, M. Charlot, and E. A. Nogovitsin, Phys. Rev. E 75, 0118041 (2007). 12. E. A. Nogovitsyn, E. S. Gorchakova, and M. G. Kiselev, Zh. Fiz. Khim. 81 (11), 1 (2007) [Russ. J. Phys. Chem. A 81, 1799 (2007)]. 13. G. Efimov and E. Nogovitsin, Phys. A (Amsterdam), 234, 506 (1996). 14. G. V. Efimov and E. A. Nogovitsin, Commun. JINR E1795217 (Dubna, 1995). 15. G. V. Efimov and E. A. Nogovitsyn, Zh. Fiz. Khim. 76, 2062 (2002) [Russ. J. Phys. Chem. A 76, 1877 (2002)]. 16. R. S. Koene, T. Nicolai, and M. Mandel, Macromole cules, 231 (1983). 17. G. S. Manning, J. Chem. Phys. 51, 924 (1969). 18. C. Holm, J. F. Joanny, and K. Kremer, Adv. Polym. Sci. 166, 67 (2004).

SPELL: OK

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