ISSN 00360244, Russian Journal of Physical Chemistry A, 2013, Vol. 87, No. 4, pp. 638–644. © Pleiades Publishing, Ltd., 2013. Original Russian Text © Yu.A. Budkov, E.A. Nogovitsyn, M.G. Kiselev, 2013, published in Zhurnal Fizicheskoi Khimii, 2013, Vol. 87, No. 4, pp. 652–658.
STRUCTURE OF MATTER AND QUANTUM CHEMISTRY
The Model of a Polyelectrolyte Solution with Explicit Account of Counterions within a SelfConsistent Field Theory Yu. A. Budkova, E. A. Nogovitsynb, and M. G. Kiseleva a
Krestov Institute of Chemistry of Solutions, Russian Academy of Sciences, Ivanovo, 153045 Russia b Ivanovo State University, Ivanovo, 153025 Russia email:
[email protected] Received March 26, 2012
Abstract—A theoretical approach to calculating the thermodynamic and structural functions of solutions of polyelectrolytes based on Gaussian equivalent representation for the calculation of functional integrals is pro posed. It is noted that a new analytical result of this work is the direct assumption of counterions, along with an equation for the gyration radius of a polymer chain as a function of the concentrations of monomers and added lowmolecular salt. An equation of state is obtained within the proposed model. Our theoretical results are used to describe the thermodynamic and structural properties of an aqueous solution of sodium polysty rene sulfonate with additions of NaCl. Keywords: Gaussian chain, selfconsistent field, gyration radius, statistical mechanics, polyelectrolytes, func tional integrals. DOI: 10.1134/S0036024413040079
INTRODUCTION Polyelectrolytes, which include basic biopolymers (e.g., polysaccharides and nucleic acids) and many artificial synthesized polymers with important appli cations (e.g., in medicine and the transformation of solar energy) form a special group among polymers. Any theoretical description of polyelectrolytes is com plicated by the presence of longrange Coulomb inter action in a system and the strong effect of external fac tors (e.g., the pH of the medium and its dielectric per mittivity) [1]. Despite the great number of scientific works on the topic, our understanding of the processes that occur in polyelectrolyte solutions is far from complete. The requirement for calculations or estimates of functional integrals is an intrinsic feature of each strict continuum theory of solutions or melts of flex ible polymer chains, based on the principles of statis tical mechanics. The conventional methodology for calculating path integrals in the physics of macro molecules is approximating the mean field theory or saddle point approximation or saddle point [2–4]. Equations for a selfconsistent field through which the Gibbs energy of the system can be determined are usually obtained within this approach [4, 5]. Unfortunately, mean field approximations do not always describe the thermodynamic properties of a system correctly [1, 2]. In the low concentration range and in the regime of a semidiluted solution, the mean field theory in some cases does not represent the real
behavior of a system even qualitatively [3, 6]. It is therefore necessary to modify this procedure. We pro pose an approach based on Gaussian equivalent repre sentation (GER) [7], which was initially designed for solving problems of quantum physics formulated using functional integrals, as one such modification [8]. GER is now widely used to describe the thermody namic and structural properties of simple liquids and solutions of polymers [6, 7, 9–15]. In [10–13], we proposed a onecomponent model model for describing the thermodynamic and struc tural functions of solutions of polyelectrolytes within GER. In this model, counterions were not considered as individual components of a solution. Their concen tration was considered to be an external parameter, and was included only in the screening constant in the potential of pair interaction between monomers. In [12], we formulated an effective model of a solu tion of polyelectrolyte in which counterions and added salt were not considered directly. In this work, we propose a better statistical model for a solution of polyelectrolyte with the addition of salt, in which counterions are considered directly. The solvent is considered a continuous medium with a given dielec tric permittivity ε, and the electrical charge on the polymer chains is distributed evenly. The concentra tion of lowmolecular ions of the salt are treated as an external parameter and was taken into account only in the ionic strength of the solution in constant of screen ing. The equation of state for a solution of polyelectro lyte was obtained using this model.
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The gyration radius is the most important struc tural characteristic of a polymer chain in solution. The value of the gyration radius determines how much the chain is compressed or extended, relative to the size of an ideal chain. The gyration radius of a polymer chain is defined as the rootmeansquare distance from points of the polymer chain to its center of mass. A new result with respect to [12] is calculat ing the gyration radius of a chain of polyelectrolyte as a function of the concentrations of monomers and lowmolecular salt. Our theoretical results were used to describe the thermodynamic and structural properties of an aque ous solution of sodium polystyrenesulfonate with the addition of NaCl salt.
rootmeansquare length of a segment of a Gaussian chain. Equations for effective potentials of interac tions expressed in kBT units can be determined using the following equations: −κr U m−m(r ) = qm2 A 2(κ, a)λ B e , kBT r
(3)
−κr U c−m(r ) = qmqc A(κ, a)λ B e , kBT r
(4)
−κr U c−c(r ) = qc2λ B e , kBT r
(5)
where q m and q c are the effective charges of the mono mer and counterion, respectively;
EQUATION OF STATE FOR A SOLUTION OF POLYELECTROLYTE
A(κ, a) =
The large grand partition function for a twocom ponent system composed of chains of polyelectrolyte and counterions is written as ∞
∞
c
p
where zc and z p are the activities of the counterion and chain of polyelectrolyte, respectively, and β = 1/ kBT is the inverse temperature. The configuration integral Qncnp ( β , V) with which integration is performed for all possible positions of the counterions and conforma tions of the chains of polyelectrolyte, has the form
∫
∫
∫
∫
Qncnp (β,V ) = D r1.. D rnp d x 1.. d x nc e −β Wtot .
is a multiplier that determines the excluded volume of monomer [16]; κ = (8πλ BI )1/2 is a parameter of 2 screening; I = 1 q j ρ j is the ionic strength of the j 2 solution; and λ B = e 2 /(4πεε 0kBT ) is the Bjerrum length. It should be stressed that our equations for the potentials of pair interactions (3)–(5) took the form of screened Coulomb potentials in order to allow for the presence of lowmolecular salt ions in solution via the screening constant κ. Introducing the local density of the electrical charge nc
ρ ncnp ( x) = qc λ B
(1)
The full potential energy of the system can be written as np
W tot =
1
∑∫ 0
2 +N 2
1
1 1
∫∫ 0 0
nc , np
∫ ∑
+ N ds 0
( )
d r (s) 3 ds k 2 ds 2Nb β k =1
∑U
+ q m A(κ, a) λ B
− r j (s 2 ))
k
np
k =1 1
k =1
0
(6) k
and using the Habbard–Stratonovich transformation
Dϕ −12(ϕU ϕ)+i(ρϕ) , e det U we obtain the following equation for the large grand partition function: e
m − m (ri (s1)
∑δ(x − x )
∑N ∫ dsδ(x − r (s))
2
np
ds1ds 2
e κa /2 1 + κa/2
∑
np
z cnc z p Ξ(z c, z p,β,V ) = Qn n (β,V ), n ! np! c p n = 0 n =0 c
∑∑
639
(2)
i, j
−1(ρU ρ) 2
=
−1
∫
nc
∑
U c − m (rk (s) − x j ) + 1 U c −c(x k − x j ) . 2 k≠ j k, j
The first component of Eq. (2) reflects the entropy and determines the flexibility of the polymer chains [1]; the three others are the potential energy of monomer– monomer, counterion–monomer, and counterion– counterion pair interactions, respectively; N is the index of the polymerization of chains; and b is the RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A
Ξ(z c, z p,β,V ) =
Dϕ −12(ϕU e det U
∫
−1
ϕ)+W I [ϕ]
where
∫
W I [ϕ] = z c dx : e i
λ B qcϕ(x)
:U
1
∫
+ z p d σ[r] : e Vol. 87
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:U ,
,
(7)
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d σ[r] = D r e
( )
d r (s ) − 3 2 κ ds 2 Nb
U (x − y ) = e
where
2
is
the
Wiener
measure;
Nb p 72 ⎛ e − 6 − 1⎞ G(p) = 12 + ⎜ ⎟ ⎟ p 2b 2 Nb 4 p 4 ⎜⎝ ⎠ 2 2
−κ| x − y |
is the Green function of the Gauss | x − y| ian measure, which corresponds to the potential of pair interaction; the colons containing the exponents denote the normal product [8]. In contrast to [10–12], Eq. (6) for the local density of the system’s charge is contains two components: the local densities of the charges of counterions and monomers, respectively.
We shall use GER to calculate the large statistical sum (7).
is the Debye structural factor [1]; c1 = λ Be −qic ziU (0), c2 = N λ Be −Nqmc zcU (0); and D (p) and U (p) are Fourier images of U (r) and D(r) functions. The potential of the average strength can be calculated as ∞
D(r ) = 12 dpp sin( pr )D (p). 2π r
∫ 0
Let us write the large grand partition function as
Following the standard procedure in [8], we obtain the following selfconsistent equation: = e kBT
1
+ iz p λ Be
∫ ∫
N ds d σ[r]ϕ(r(s)) = 0 ,
−1
q m A (κ, a)λ Be 2 2
−
2
−qcc
q c λ Be 2 2
−1
− D ]ϕ) −
− Nq mc
zc
zc
∫ dx ϕ (x) 2
qc2c12 + qm2 A 2(κ, a)c22 + 2qcqm A(κ, a)c1c2 2λ BU (0) ⎡ (c q 2 + c q 2 A 2(κ, a)G(p))U (p) 1 i 2 m U (0) dp ⎢ +1 ⎢ 2 (2π)3 ⎢1 + (c q 2 + c q 2 A 2(κ, a)G(p))U (p) 1 i 2 m ⎢⎣ U (0) +
1 1
N
2
∫ ∫ ds ds 1
2
∫
0 0
∫
× d σ[ r]ϕ(r(s1))ϕ(r(s 2 )) = 0 .
Solving these equations in thermodynamic limit V → ∞ , we obtain
c = qcλ Be
−qcc
D (p) =
−Nq c zcU (0) + qm A(κ, a)N λ Be m zpU (0) = qcc1 + qm A(κ, a)c2,
U (p)
, U (p) 2 2 2 1 + (c1qc + c2q m A (κ, a)G(p)) U (0)
ρc =
∫
−1
Π0 c1 c2 = + kBT λ BU (0) λ BNU (0)
0
− 1 (ϕ[U 2
Π 0V
Dϕ −12(ϕD ϕ)+W I e kBT , e det D where Π is the osmotic pressure. In the zeroth order of Gaussian equivalent representation, the osmotic pres sure takes the form
− i (cU −1ϕ) + izc λ B e −qcc dxϕ(x) λB −q m N c
ΠV
Ξ(z c, z p,β,V ) = e kBT Π 0V
∫
(10)
(8)
(9)
(11)
⎤ ⎛ U (p)⎞⎥ 2 2 2 − ln ⎜1 + (c1qi + c2qm A (κ, a)G(p)) ⎟. U (0) ⎠⎥ ⎝ ⎥⎦ Equations (8), (9), and (11) contain two parame ters: c1 and c2 . When c1 = 0 , we obtain the same equa tion as in [12] for our model of a solution of polyelec trolyte without explicit account of counterions. Let us calculate the average concentrations of counterions and chains using standard thermody namic relations:
zc ∂ c (1 + q m2 NA 2(κ, a)c2 ) f1(c1, c2 ) − qcq m NA(κ, a)c1c2 f 2 (c1, c2 ) , ln Ξ = 1 2 2 2 V ∂ zc 1 + qi c1 + q m NA (κ, a)c2
(12)
zp ∂ c f (c , c ) − q q A(κ, a)c1c2 f1(c1, c2) ln Ξ = 2 2 1 2 2 c m2 , 2 V ∂zp 1 + qi c1 + qmNA (κ, a)c2
(13)
ρp =
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THE MODEL OF A POLYELECTROLYTE SOLUTION WITH EXPLICIT ACCOUNT
where f1(c1, c2 ) =
1 + qc2c1 + q cq m A(κ, a)c2 λ BU (0)
2 U (p) (c1qc2 + c2q m2 A 2(κ, a)G(p))⎛⎜ ⎞⎟ dp ⎝U (0) ⎠ , − 2 2 (2π)3 ⎡ U (p)⎤ 2 2 2 c q c q A a G p 1 + ( + ( κ, ) ( )) 1 c 2 m ⎢⎣ U (0)⎥⎦
qc2
∫
1 + Nq m2 A 2(κ, a)c2 + Nqcq m A(κ, a)c1 f 2(c1, c2 ) = N λ BU (0)
−
q m A (κ, a) dp 2 (2π)3 2
2
∫
2 U (p) (c1qc2 + c2qm2 A 2(κ, a)G(p))G(p)⎛⎜ ⎞⎟ ⎝U (0) ⎠ . × 2 ⎡1 + (c q 2 + c q 2 A 2(κ, a)G(p))U (p)⎤ 1 i 2 m ⎢⎣ U (0)⎥⎦
with the use of Eq. (10). Taking the potential of mean force strength (10) as the potential of interaction, we effectively take into account for the interaction between a chain and its surroundings, i.e., counterions and other polymer chains. The partition function sum determining the normalization in Eq. (14) can be writ ten in the form of a path integral: N
THE CALCULATION OF THE GYRATION RADIUS OF A POLYMER CHAIN The gyration radius for a Gaussian chain is calcu lated exactly and is determined by an equation from [17]:
R2
where R 2 = Nb 2 is the average value of the square of the distance between the ends of polymer chain. The rootmeansquare distance between the ends of a nonideal flexible polymer chain (i.e., when there are interactions between monomers) takes the form N
2
∫
∫
∫
C Z = 1 Dr e C 0 C1
1
2
2
2
∫
− 3 2 dsr 2 (s ) 2b1 0
N ⎡ ⎛ ⎤ ⎞ 3 1 1 exp ⎢− ⎜ 2 − 2 ⎟ dsr 2(s)⎥ ⎢⎣ 2 ⎝ b b1 ⎠ 0 ⎥⎦
∫
⎡ q2 λ N N d p 2 ip(r(s1)− r(s2 )) ⎤⎥ × exp ⎢− m B , ds1ds 2 D( p )e 3 2 (2π) ⎢⎣ ⎥ ⎦ 0 0
∫∫
∫
−3/2
−3/2
⎛ ⎞ ⎛ ⎞ where C1 = ⎜ 3 2 ⎟ and C0 = ⎜ 3 2 ⎟ are ⎝ 2πNb1 ⎠ ⎝ 2πNb1 ⎠ constants of normalization. The potential of average strength D(r) is expressed via the Fourier integral D(r) =
dp
∫ (2π) D (p)e 3
ipr
.
Let us now introduce the variable ξ(s) via the relation (16) r(s) = R s + ξ(s), N and consider that random function ξ(s) is low in abso lute value,
| R| s , N and satisfies the boundary conditions
(17)
ξ(0) = ξ(N ) = 0.
The ξ(s) value determines the arbitrary deviation from the line described by the first term in Eq. (16). Using approximation (17), we can rewrite the statisti cal sum of the polymer chain as
N
∫ ∫ ds ds D(r(s )−r(s )) 1
2
2
Z ≈
2
0 R 2 = 1 Dr R e , (14) Z C0 where R = r(N ) − r(0), D(r) is the potential energy of interaction between monomers, which we determine
∫
N
∫ ∫ ds ds D(r(s )−r(s ))
| ξ(s)|
2 = Nb , 6 6
q λ − 3 2 ds r 2(s)− m B 2 2 2b 0
∫
0 . (15) Z = Dr e 0 C0 Let us calculate the rootmeansquare distance between the ends of a chain according to GER. From a physical point of view, calculating the partition func tion of a single polymer chain according to GER is reduced to finding a new measure that effectively takes into account for the interaction between monomers. Let us therefore move to a new Gaussian measure with a new rootmeansquare distance for segment b1:
qcρc + qmρ m = 0.
=
q2 λ − 3 2 dsr 2(s)− m B 2 2b
N
To determine the c1 and c2 parameters at densities ρ c and ρp, we must solve the system of transcendent Eqs. (11)–(13). The dependence on the concentra tions is implicitly contained in the c1 and c2 parame ters. Equations (10), (12), and (13) represent the equation of state for a solution of polyelectrolyte in parameter form. By varying the values of c1 and c2 parameters, we obtain the dependence of osmotic pressure on the concentrations of chains and counte rions. For a complete description, the condition of electroneutrality must be added to the equation of state of the solution
Rg2
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C1 d σ1[r] C0
⎛ ⎞ 2 q2 λ − 3 ⎜ 12 − 12 ⎟R − m B 2N ⎝ b b1 ⎠ 2
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∫
NN
∫∫ 0 0
ds1ds2
∫
d 3 p 2 ip(r(s1) − r(s2 )) D( p )e 3 (2π)
(18) .
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BUDKOV et al.
p/RT × 103, mol/L 10
b, A 4.8 4.6
8
4.4 1 2 3 4
6 4
4.2 4.0
2 0
1 2
3.8 30
20 0
0.04
0.08
Fig. 1. Osmotic pressure depending on the concentration of NaPSS (mol/L): (1, 2) 0.005 and (3, 4) 0.05; (1, 3) experiment and (2, 4) GER.
Let us write all equations in normal form, relative to the new Gaussian measure [7]:
R 2 = : R 2: + R 2 = : R 2: + Nb12 and −
∫
e2x = e x − 1 − x − x 2 /2. The condition of the simplification of square compo nents in the exponent of Eq. (18) leads to the selfcon sistency equation for calculating the b1 parameter:
N N
×
− 3 2 ds r 2(s) 2b1
∫∫ 0 0
: e ip(r(s1)−r(s2)): = 1.
0
−1/3
tance between monomers raverage = nm ; cs = (1) 0.005 and (2) 0.05 mol/L.
2 1 − 1 = q mλ B b 2 b12 18N
N
∫
∫
Z = e −β F0 d σ1[r]eW1[r],
0
2 2
2
Nb p ⎛ − 1 6 e + 48 + 1 2 ⎜ p 4b16 ⎜⎝
⎞ 4N − Nb1 p ⎟+ 2 4e 6 . ⎟ p b1 ⎠ Solving Eq. (19), we obtain the values of the renor malized rootmeansquare length of a segment at various Cm and C s . The rootmeansquare distance between the ends of a chain in the zero approxima tion of GER is 2 2
⎛b2 b2 ⎞ −β F0 = − 3 ⎜ 12 − 1 − ln 12 ⎟ 2 ⎝b b ⎠ N N
∫∫
ds1ds 2
0 0
q2 λ W1[r] = − m B 2
∫
(20)
⎛ − Nb1 p ⎞ 6 B(b1, p ) = 864 e − 1 ⎜ ⎟ ⎟ Np 6b18 ⎜⎝ ⎠
,
1
q m2 λ B 2
.
where
∫ d σ [r] = 1,
−
(19)
∫
N
d σ1[r] = D r e C1
∫
p 2b12 |s 2 − s1| 6
dp 1 − 1 = q2 λ D(p)B(b1, p 2 ), m B 2 2 3 b b1 (2π)
where − 3 2 ds r 2(s) 2b1
− dp ds1ds 2(s 2 − s1) D(p) p 2e 3 (2π) 2
This equation, which is analogous to Eq. (19), was obtained in [18] with the use of the Bogolyubov ine quality. Calculating the integrals in the righthand side results in the equation
We thus have
∫
50
Fig. 2. Correlation between parameter b and average dis
p 2b12 | s2 − s1| 6
e ip(r(s1)−r(s2)) = : e ip(r(s1)−r(s2)): e , so that the following relationship is valid: Dr e C1
40
0.12
dp 2 − D( p )e 3 (2π)
2 2
p b1 | s 2 − s1| 6
,
2 2
N N
dp ∫ ∫ ds ds ∫ (2π) 1
0 0
2
D ( p ): e2 2
3
i p(r(s1)− r(s 2 ))
:,
R
2
∫
= d RR e 2
2 − 3R 2 2 Nb1
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A
∫ dRe
2 − 3R 2 2 Nb1
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THE MODEL OF A POLYELECTROLYTE SOLUTION WITH EXPLICIT ACCOUNT gcc(r)
643
Rg/b 1 2 3 4
1.2 5 0.8 4 1 2 3
0.4
3 0.1
0 0
0
4
8
12
The gyration radius is determined as
≈
Nb12 /6.
0.3
16
Fig. 3. Counterion–counterion pair correlation function at cs = 0.05 mol/L and different concentrations of mono mers: (1) 0.012, (2) 0.059, and (3) 0.126 mol/L.
Rg2
0.2
(21)
It should be noted that within the limits of our model, the approximation for gyration radius (21) can be improved by calculating the following corrections according to perturbation theory, in contrast to the approach proposed in [18, 19]. MODEL OF AN NAPSS AQUEOUS SOLUTION: NUMERICAL RESULTS As an example, we considered an aqueous solution of sodium polystyrenesulfonate (NaPSS) with an addition of NaCl salt. We took the values of the follow ing parameters from [20]: a = 8.14 Å, q m = −1, and qc = 1. Bjerrum length λ B = 7 . 14 Å, corresponding to temperature T = 298 K. The index of the polymeriza tion of chains corresponded to N = 3155, i.e., to a chain molar mass of M = 6. 5 × 105 g/mol. We moved from the numerical densities of particles to molar con centrations in the same manner. Calculations were then performed as follows: For each value of the experimental concentration of monomers, the intrinsic value of free parameter b was selected in order to achieve consistency between the theoretical and experimental values of osmotic pres sure [21]. Plots of the dependence of osmotic pressure on the concentration of monomers at various concen trations of salt are given in Fig. 1. As a result, we obtained the correlations between the values of parameter b and the average distance between monomers raverage = 1/ 3 nm at various concen trations of salt. As follows from Fig. 2, the correlation between these values is close to linear. It should be also RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A
Fig. 4. Comparison of (1, 2) theoretical results with (3, 4) MD simulations for a gyration radius as a function of the concentration of monomers at different values of index of polymerization N = (1, 3) 20 and (2, 4) 40.
noted that the value of parameter b, ~4 Å, changes lit tle over the broad range of values of raverage . The coun terion–counterion pair correlation function was cal culated on the basis of Eq. (12) for the potential of medium strength at various values of concentration of monomers: 2
−q λ D(r ) (22) g cc(r ) = e c B . As follows from Fig. (3), the most likely distance between counterions in solution that correspond to the peaks on our pair functions is ~5 Å, or approximately the distance between adjacent monomers. This could indicate that counterions group predominantly along a polymer chain, screening its charge. Plots of the dependence of the gyration radius on the concentration of salt C s at two values of the num bers of monomers N in a chain are given in Fig. 4. As follows from Fig. 4, our theory is in good agreement with the results from computer simulations given in [22]. The value of the gyration radius decreased upon an increase in the concentration of salt, indicating the folding of polymer coils. This effect can be explained by the increase in screening of the charged monomers of the polymer chain upon an increase in the concen tration of lowmolecular salt ions.
CONCLUSIONS The prospects for the development of GER to describe the properties of solutions of polyelectrolytes depend on how it is combined with other methods for the theoretical description of macromolecular sys tems. These are primarily computer methods; more specifically, quantum chemistry and molecular dynamics. Combining GER with computer methods would enable us to consider in detail the properties of Vol. 87
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a solvent and the features of interaction between the components of a solution. In computer methods, effective potentials of the interaction between mono mers can be calculated by considering the effect of counterions and the surrounding molecules of a sol vent and used to calculate the thermodynamic and structural quantities of a solution within statistical mechanics according to GER. REFERENCES 1. A. Yu. Grosberg and A. R. Khokhlov, Statistical Physics of Macromolecules (Nauka, Moscow, 1989). 2. G. H. Fredrickson, V. Ganesan, and F. Drolet, Macro molecules 35, 16 (2002). 3. G. H. Fredrickson, The Equilibrium Theory of Inhomo geneous Polymers (Clarendon Press, Oxford, 2006). 4. Q. Wang, T. Taniguchi, and G. Fredrickson, J. Phys. Chem. B 108, 6733 (2004). 5. I. Borukhov, D. Andelman, and H. Orland, Eur. Phys. J. B 5, 869 (1998). 6. S. A. Baeurle, M. Charlot, and E. A. Nogovitsin, Phys. Rev. E 75, 011804 (2007). 7. G. Efimov and E. Nogovitsin, Phys. A 234, 506 (1996). 8. M. Dineykhan, G. V. Efimov, G. Ganbold, et al., Oscil lator Representation in Quantum Physics, Lecture Notes in Physics (Springer, Berlin, 1995), Vol. 26.
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