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Acta Chim. Slov. 1998, 45(2), pp. 111-123 (Received 12.5.1998)
SHORT-RANGE INTERACTION IN A POLYELECTROLYTE SOLUTION CONTAINING A MIXTURE OF MONO- AND DIVALENT
COUNTERIONS
D. Dolar and M. Bester Rogac
Faculty of Chemistry and Chemical Technology, University of Ljubljana,
1000-Ljubljana, Aškerceva 5, Slovenia
ABSTRACT
The osmotic coefficient and the excess free energy have been calculated for a polyelectrolyte solution with mixtures of mono and divalent counterions. The results have been obtained by applying the cylindrical cell model in the Poisson-Boltzmann approximation. The short-range interaction between polyion and counterions, described by a square-well potential, has also been taken into account. The results of osmotic coefficient are presented as functions of the equivalent fraction of monovalent counterions for different values of ionic radii, depth of the potential well, and concentration.
INTRODUCTION
The cylindrical cell model which considers the electrostatic interactions among ions has usually been applied for the interpretation of thermodynamic properties of polyelectrolyte solutions with a fair success. Several times it appears, however, that the
112
discrepancy between experimental results and theoretical calculations is rather large [1,2,3] and consequently, it has stimulated some attempts [4,5,6] in order to improve the calculations by introducing the non-Coulombic, short-range interaction into the model.
In the present study we are interested in a polyelectrolyte solution containing a mixture of mono and divalent counterions. The first theoretical approach to this problem has been published long ago [7] followed by an experimental verification [8,9,10]. The same system has also been treated by applying the line charge model [11]. In both cases the electrostatic interaction has only been taken into account. In this contribution the influence of short-range interactions between polyion and counterions, represented by a square-well potential, on the osmotic coefficient and excess free energy will be presented.
THE MODEL AND THE POTENTIAL
The polyelectrolyte solution is represented as an ensemble of cylindrical cells with radius R and length h (h >> R). In the axis of each cell is fixed a cylindrical polyion of radius a and length h = vb, where b is the lenght of the monomer unit. The charge of the polyion is -v e0, supposed to be
spread uniformly over its surface. In the free volume of the cell is a mixture of mono and z-valent counterions with the total charge equal in number but opposite in sign to the charge of the polyion. By denoting the radius of monovalent counterions with q and z-valent with r2, it follows for the distances of closest approach a^ = a + q and a2 = a + r2. The short-range interactions of the non-Coulombic type are represented by a square-well potential Vnc for monovalent counterion:
Vnc = °°,        0 < r < ai
Vnc=-E1,     a^r^bi                                                                                  (1)
Vnc=0,         b1<r<R
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and for z-valent counterions:
nc = °°,	0< r <a2
nc = _E2>	a2 <r<b2
nc = 0,	b2 <r<R
(2)
where r is the cylindrical coordinate and  Ej   and  E2   are positive quantities. By supposing the additivity of Coulombic and non-Coulombic potential and by denoting
E,/kT          ,        „         E2/kT
T|1 = e l          and      r\2 = e 2
the Poisson-Boltzmann equation for this system reads
(3)
1 d(rdV j =     eo r dr ^   dr J       e0e
„fexp-^k+z„Sexp-^L
kT
kT
(4)
with the boundary conditions
d\|/
ver
dr Ja    27re0eah
d\|/ "d7
= 0
(5)
R
The values of the parameters r^ and r|2 are given in equations (1) and (2). In equations above 0 is the electrostatic potential, k the Boltzmann constant, T the absolute temperature, e0 the elementary charge, e0 the vacuum and e e0 the solvent
permittivity, n° and n° the number density of mono and z-valent counterions at \|/ = 0.
To simplify the notation the dimensionless quantities are introduced
e0Ą kT '
-i„a
ti =ln
X
t = ln-
,   bl tn=ln^
y = In ,    t2=ln
R
a
a
122 = ln
b2
ve;
47TŁ0ekTh
(6)
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The parameter Ŕ is proportional to the linear charge density of the polyion which is the basic property of each polyelectrolyte. Furthermore, we introduce the equivalent
fractions N° and N° in the place where \|/ = 0 and their average values Ni and Nz
N
o _       nx
n^ +znz
N?
zn,
n^ +znz
-,    Ni
n, + znz and the ratio       s =
o , ^„o z
nf + zn
n, + zn;
N,
znz
n, + zn2
(7)
The Poisson-Boltzmann equation and the boundary conditions now read
d2y
4X
.2t
dt2     (e2^-l)^
TěiNfey+TěěN^
dt
-2Ŕ
dt
(8)
The average values Ni and Nz may be obtained from
Nl=^^Le^+2tdt ^(e2Y-l)Ě
N7 =
2N
o    y
Š(e2*-1)
Jn2e^+2tdt
(9)
Within the integration interval, 0 < t < y , each of the parameters r\\ and r|2 assumes three different values as indicated in equations (1) and (2). Let us repeat this condition in the new notation
Til
0,     0 < t < tx
Tib   tl < t< tu        T|2
1,      tn<t<y
0,    0 < t < t2
T|b   t2<t<t22
1,     t22 < t < y
(10)
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Because equation (8) cannot be solved analytically we applied the Runge-Kutta method of the third order for numerical computation. The values of N° and N° were chosen arbitrarily and the parameter i; was found by an appropriate iteration procedure.
THE FREE ENERGY AND THE OSMOTIC COEFFICIENT
The excess free energy Aex of a polyelectrolyte has three contributions
Aex -Unc+Uc -TScf
(11)
where Unc is the non-Coulombic contribution, Uc the Coulombic contribution and
Scf is the configurational entropy [12]. Unc was calculated from
bi
Unc =-El JnjV^dV^ J n^e^dV
(12)
The final expression is
Unc=-2vkTVm^
41                                          l22
TlilnTlin? } ey+2tdt + ri2lnri2n° J ezy+2tdt
(13)
2
where   Vm = % a b is the volume of the monomer unit. The Columbic contribution
'm
Uc was calculated in two different ways, according to equations (14) and (15), giving different but equivalent expressions
Uc=^-Jp\|/dV + ^Jo\|/(a)dS
(14)
V
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Uc=^j"(gradV)2dV
(15)
V
where p is the local volume charge density and G is the charge density on the surface of the polyion. The configurational entropy Scf due to nonuniform distribution of counterions within the cell was calculated from
Scf- = -kXj" nilnni-ni lnni   dv i=l  v                                '
(16)
where nj  is the local concentration of mono and z-valent counterions and  n^   its
average value.
From equation (14) it follows
Uc=-vkTy(0)-vkTVm
and from equation (15)
jninfe^ydt + j^n^Vdt
(17)
TT      vkTJfdy^
iei
(18)
In both integrals of equation (17) the condition (10) has to be taken into account. The expression for Scf is too long and for that reason it is not reproduced here.
The calculation of Aex is a very long procedure and it will not be presented here. Therefore, we give only the final expression of the excess free energy per monomer unit by introducing equation (14) into equation (11)
( A     ^ vkT
1
N
y(0) + Niln 1                   ŠNi
+ -
N5
In
N
ŠN,
¦ +
VmTilnf ] Qy+2ty dt + Vmil2n° ] ^+2tz y dt 0                               o
(19)
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and equation (15) into equation (11)
(  A        \         1     \( A„\2        __        "MP      XT.       M°
A
ex
vkT
Nl   , Nz,„ Nz
= ±Ě(ČL)
4^J0UtJ
y
2VmTělnf j ey+2ty dt + 2VmTi2n° j" ez^+2tz y dt
dt + Niln^- + —ln^^ + U                                                                                                       (20)
y                                    y
o                                                 o
In equations (19) and (20) the fraction (e ^ -e   1 )/(e ^ -e   2) was approximated
by 1 which simplifies the writing and does not make an appreciable error. By multiplying equation (19) by 2 and substracting equation (20) from it, we obtain the simple expression
^ = y(0)-^If^i2dt + Niln^L + ^ln^l       ,                      (21)
very suitable for numerical evaluation.
The osmotic coefficient can be obtained from the partial derivative of Aex with respect to volume [13]. The most convenient expression [12], adapted for this case is
O = ^—^-                                                                                                           (22)
ll
n, +nz
2^0 +n°
which reads in our notation           O = —1—=-----=?-r-   .                                          (23)
^(zNi + Nz)
RESULTS AND DISCUSSION
All equations in the previous section were developed for the general case of mono and z-valent counterions. The numerical calculations have been made, however, for mixtures of mono and divalent counterions which are more frequently encountered in biological and industrial systems.   For all calculations the following values of the
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parameters characterizing the polyion were applied: a = 0.8 nm, b = 0.252 nm, and Ŕ = 2.84. For counterions the following values of ionic radii were taken:   ri = 0.33 nm (e.
g. Cs+) which gives a^ = 1.13 nm, and r2 = 0.46 nm (e. g. Be +) [14] which gives a2 = 1.26 nm. In Figure 3 we have for comparison a^ = 0.8 nm; 1.0 nm; 1.13 nm and a2 = 0.8 nm; 1.1 nm; 1.26 nm. The relation between the concentration parameter y and concentration cm in moles of monomer units per liter is
Y = ln^ = -^ln(l03NAVm)-^lnc
m>
(24)
where NA is the Avogadro number. Thus, we have y = 1.5 (cm= 0.163 mol/l), y = 2 (cm= 0.06 mol/l),   y = 3 (cm= 0.00812 mol/l),   and   y = 4 (cm= 0.0011 mol/l). For almost all calculations the value  y = 2 was applied. The range of the short-range interaction,  b^ = 1.4 nm and  b2 = 1.6 nm, was the same in all cases.
0.50 0.45 0.40 0.35 ® 0.30 0.25 0.20 0.15 0.10
7 = 2
z = 2
a = 0.8 nm
a1 = a2 = 0.8 nm
b1 = 1.4 nm, b2 = 1.6 nm
FIGURE 1. The influence of the short-range interaction on the osmotic coefficient. Counterions are point charges.
119
0.50-r
0.45-
0.40-
0.35-
F  0.30-
0.25
0.20
0.15
0.10 -i---------¦---------1---------¦---------1---------¦---------1---------¦---------1---------¦---------1
0.0               0.2               0.4               0.6               0.8               1.0
Ni
FIGURE 2. The influence of the short-range interaction on the osmotic coefficient. Radii of counterions are r1 = 0.33 nm and r2 = 0.46 nm.
The greater part of the figures presented in this article is devoted to the osmotic coefficient because, it can be obtained directly from the experiment, and gives an approximate information about the distribution of counterions inside the cell. The concentration of counterions at the border of the cell is decisive for the value of the osmotic coefficient, as shown in equation (22). A low value of F signifies that counterions are gathering around the polyion and, as a result, a decrease of concentration is produced at the border of the cell. A comparison of Figures 1 and 2 reveals that a lower depth of the potential well ( higher values of h and h2) gives rise to a stronger attraction of counterions to the polyion, irrespective of the ionic size and the value of Ni. Furthermore, larger counterions are less attracted by the polyion causing a higher value of F, as seen in Figure 3. The dependence of the osmotic coefficient on concentration is presented in Figure 4 and 5 for two different cases. For lower concentrations (higher values of the parameter g ) the well expressed maxima appear what has been confirmed by the experiments [8].
1.13 nm, a2 = 1.26 nm 1.4 nm, b2 = 1.6 nm
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0.50 0.45 0.40 0.35 F 0.30 0.25 0.20 0.15 0.10
O    a1 = a2 = 0.8 nm •    a1 = 1.0, a2 = 1.1 nm D    a1 = 1.13, a2 = 1.26 nm
0.0
-o—"
-D-
-o-
-CD-
Hfl
o~^
g = 2
z = 2
a = 0.8 nm
b1 = 1.4 nm, b2 = 1.6 nm
h1  = 2, h2 =  3
0.2
0.4
0.6
0.8
1.0
N1
FIGURE 3. The influence of radii of counterions on the osmotic coefficient.
F
0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10
0.0
-D-Ű
^W
~~W~
z = 2
a =0. 8 nm
a1 = 1.13, a2 = 1.26 nm    D    ^
> .  .       > ^                •    g = 2
b. = 1.4, b, = 1.6 nm
1               ^                          Bg = 3
hl= h2 = 1                           O     g = 4
o<S
0.2
0.4               0.6
N1
0.8
1.0
FIGURE 4. Dependence of the osmotic coefficient on concentration in the absence of the short-range interaction.
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F
z = 2
a = 0.8 nm
a1 = 1.13 nm, a2 = 1.26 nm
b, = 1.4 nm, b, = 1.6
0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10
FIGURE 5. Dependence of the osmotic coefficient on concentration in the presence of the short-range interaction.
In Figures 6 and 7 the excess free energy is plotted against Ni for different values of g and of the parameters h1 and h. In this case, a simple interpretation which would be analogous to that applied with osmotic coefficient is not possible.
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8.0 7.5 7.0 6.5 6.0 5.5
z = 2
a = 0.8 nm
a1 = 1.13 nm, a2 = 1.26 nm
b1 = 1.4 nm, b2 = 1.6 nm
h1 = 3,  h1 = 4
o
g = 1.5 g = 2 g = 3 g = 4
^O?
JCr~~~
5.0 tLço-O0"
4.5
4.0 +—¦¦¦"¦'
3.5
3.0
2.5
-CD-
jCO&>
^co-
-©o-
«----------1>
—D-------DO
0.0
0.2
0.4
0.6
N1
FIGURE 6. Dependence of Aex /íkT on concentration.
0.8
1.0

4.4 4.2 4.0 3.8 3.6 3.4 3.2
3.0 -
0.0
0.2
0.4
0.6
0.8
1.0
N1
FIGURE 7. The influence of short-range interaction on Aex /íkT.
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REFERENCES
[I]   P. W. Hales, G. Pass, Eur. Polym. J. 1981, 17, 657-659.
[2]   G. Vesnaver, M. Rudež, C. Pohar, J. Škerjanc, J. Phys. Chem. 1984, 88, 2411-2414. [3]   G. Vesnaver, Z. Kranjc, C. Pohar, J. Škerjanc, J. Phys. Chem. 1987, 91, 3845-3848. [4]   D. Dolar, IX. Yugoslav Symposium on Electrochemistry, Dubrovnik, Book of Abstracts,
Serbian Chem. Soc, Belgrade, 1985, pp 47- 50. [5]    V. Vlachy, Eur. Polym. J.  1988, 24, 737-740. [6]    D. Dolar, M. Bešter, J. Phys. Chem. 1995, 99, 4763-4767. [7]    D. Dolar, A. Peterlin, J. Chem. Phys. 1969, 10, 3011-3015. [8]    D. Dolar, D. Kozak, IUPAC International Symposium on Macromolecules, Leiden, Book of
Abstracts, Inter Scientias Inc., The Hague, 1970, Vol. 1, pp 363-366. The results of this article
are reproduced in Ref. 9 and 10. [9]    A. Katchalsky, Pure Appl. Chem. 1971, 26, 327-373. [10] D. Dolar, in Polyelectrolytes, ed. E. Selegny, M. Mandel, U. P. Strauss, D. Reidel, Dordrecht,
The Netherlands, 1972, pp 97-113.
[II]   G. S. Manning, ibid. pp 9-37.
[12] R. A. Marcus, J. Chem. Phys. 1955, 23, 1057-1068. [13] S. Lifson, A. Katchalsky, J. Polym. Sci. 1954, 13, 43-55.
[14]   B. E. Conway, Ionic Hydration in Chemistry and Biophysics, Elsevier Scientific Publishing Company, 1981, pp 73.
POVZETEK
Za raztopino polielektrolita z mešanico eno- in dvovalentnih protioiinov smo na osnovi celicnega modela s cilindricno simetrijo in z uporabo Poisson-Boltzmannove enacbe izracunali osmozni koeficient in presežno prosto energijo. Pri racunu smo upoštevali tudi kratkosežne interakcije med poliionom in protiioni, opisane s pravokotnim potencialom. Vrednosti osmoznega koeficienta smo podali kot funkcijo ekvivalentnega ulomka enovalentnih protiionov za razlicne vrednosti ionskih radijev, globine potencialne jame in koncentracije.