Acta Chini. Slov. 2001, 48, 279-288.
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CONTRIBUTIONS OF THE MACROION AND COUNTERIONS TO THE INTERNAL ENERGY OF FULLERENE ELECTROLYTE SOLUTIONS*
Jože Škerjanc
Faculty of Chemistry and Chemical Technology, University of Ljubljana, 1000 Ljubljana, Slovenia
Dedicated to Professor Davorin Dolar on his 80  birthday
Received 29-05-2001
Abstract
From the solution of the Poisson-Boltzmann equation for the spherical cell model of a fullerene electrolyte solution, the contributions of the fullerene macroion and counterions to the electrostatic energy of the solution are calculated. Computations are made for solutions containing mixtures of counterions differing in size and charge. The results are presented as functions of the macroion concentration and mole fractions of the counterions.
Introduction
Theoretical predictions for behavior of charged particles in solutions have been tested by various experimental properties that are related to various derivatives of the electrostatic free energy of the solution. Relevant thermodynamic properties are activity and osmotic coefficients, enthalpies of dilution, apparent molar volumes, etc. Although theoretical expressions for simple electrolytes demonstrate very distinctly the contribution of individual ionic species to the absolute value of these properties, in the relevant theoretical formulas for polyelectrolyte solutions the contributions of counterions and the polyion are not so explicitly evident. It has been demonstrated that for high molecular weight polyelectrolytes only counterions contribute to the osmotic and activity coefficients. On the other hand, we have shown that both the polyion and the counterions contribute to the enthalpy of dilution. So for anionic polyelectrolytes the positive polyion contribution to the electrostatic energy of the solution is partly compensated by the negative counterion contribution.
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In this paper we present the relevant theoretical results for solutions of fullerene electrolytes, a new type electrolytes that fill up the gap between simple electrolytes and polyelectrolytes. Thus far we have reported " on the first thermodynamic and transport studies of aqueous solutions of 7/, symmetric derivative of fullerene Coo, at which six malonic acid molecules have been attached, 7VC6o(C(COOH)2)6, and its sodium salt, 7V Có6(COONa)i2, which has the properties of a highly charged ideally-spherical strong electrolyte. For this macroelectrolyte we shall show that contributions of the macroion and counterions to the internal energy of the solution is similar as those found for linear polyelectrolytes, although not so explicitly expressed.
Electrostatic Potential
The general equations derived here refer to a spherical cell model of a fullerene electrolyte solution containing mixtures of counterions. The approximations and assumptions concerning the use of this model are well known. ' The volume of the solution is divided by the number of fullerene particle to obtain the average volume of the solution per fullerene particle. This volume is assumed to be spherical, and the spherical fullerene ion, which we shall called macroion, is located in the center of the cell of radius R. The fullerene macroion of radius a carries v negative ionized groups which are supposed to be uniformly smeared over its surface. The cell contains the neutralizing number of counterions B and C with radii rB and rc and with the charge numbers zB and zc, respectively. The exclusion radii from the center of the macroion to the center of the smaller and larger counterions are denoted by b (= a + rB) and c (= a + rc), or according to eqn. (4) by dimensionless quantities t\ = In {bla) and tj = In (c/a). The Poisson-Boltzmann equation for this model has three domains:
V2O,-0,    a<r<b                                             (1)
V202 =4TilBzBn°BQxV{zfi>2),   b<r<c                                (2)
V203 =4nlBYJzln°Qxv(zi<3>3),   c<r<R                              (3)
where dimensionless quantities O, t, and y and the charge parameter 0, are given by
O = -e0\|/ / kT,   r -a exp(/),   R-a exp(y ).                            (4)
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0 = vel I zkTa - vlBa                                             (5)
In these equations ^is the electrostatic potential, eo is the proton charge, sis the relative permittivity of the solvent, n   is the number density of counterions at y/(R) = 0, /# (= eo /skT) is the Bjerrum length, r (a < r < R) is the radial distance from the center of the cell and k and T have their usual significance. The boundary conditions are
(dOl/dt)t=0=-®                                               (6)
0,(0 = O2(0,   (dO, ldt\ = (J02 l(dt)h = -0exp(-O                   (7)
02(t2) = 03(t2),   (d02/dt)ti=(d03/(dt)ti                             (8)
O3(y) = 0,   (®3/dt\__y=0                                        (9)
Equation (1) can be solved analytically, with the result
O, - 0,(0) - 0(1 - exp(-O),   dOy/dt = -0exp(-f)                     (10)
where Oi(0) is the dimensionless potential at the surface of the macroion.
The differential eqn. (2) and (3) cannot be solved analytically and are transformed into eqn. (11) and (12) which are appropriate for numerical computation
d202/dt2 + d02I dt - kB exp(2t + zB02) = 0                          (11)
d203/dt2 + dO, /dt - Y,k, exp(2ć + z,.03) - 0               (12)
i
where i in eqn. (12) stands for B or C and the constants kß and kc are given by
/I       2    2           0                                          a       2    2           0
_ 4ne0a zBnB                 _ 4ne0a zcnc
kß-       skT       '           kc~       skT       •                           (13)
For later use we introduce the equivalent fraction Nf of the species i
where V, is the volume accessible to counterion species i,  i.e.,   Vb = 4% (R3 - b3)/3 and Vç = 4%(R - c )/3, and ni is its average number density. The equivalent fraction Afg and
Nc are related to the constants kß and kc by eqn. (15) and (16):
kB
\2exp(3L + zBQ>2)dt + [ exp(3t + zB03)dt
Jr.                                                         J r,
= ®NB                   (15)
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= ®NC                                   (16)
that follow from the electroneutrality condition
vN,=jvz,n,dV.                                        (17)
In all computations we used for the radius of fullerenehexamalonates macroion the valued a = 0.77 nm, which gives for water solutions at 25 C (e= 78.54) the value of the charge parameter 0 = 11.12. The analytical concentration c and the concentration parameter fare related by
cQxp(3y) = 3v/4na3NA                                         (18)
where Na is the Avogadro constant. From this relation we get c exp(3y) = 10.42 mol C007dm3.
Results and Discussion
The electrostatic energy of the system U calculated per macroion, which is the sum of the macroion and counterion contributions, Um and Uc, respectively, is related to the potential by the expression
U = Um+Ue=-^-\' (gradyfdV                             (19)
871Ji"
where Um and Uc are given by
Um=ilsoy(a)dS                                       (20)
and
Uc=\\ywdV                                         (21)
Here, a (= -veo/S) and p are the surface and volume charge densities, and S and V are
the surface and volume per macromolecule, respectively. Evidently, p = eopn, where pn,
the number density of charges, is given by the sum expression in eqn. (3). Upon insertion
of dimensionless quantities O, t, and y from eqn. (4) we get
\ikT Um=^-Om                                   (22)
The contribution Uc is the sum of the contributions of the smaller and larger counterions, Ucb and Ucc- According to eqn. (21) we get
*c
f Qxp(3t + zc03)dt
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50  -

10 -
10 -        U
-20
0.0
0.2
0.4
0.6
N
0.8
1.0
Figure 1. Contributions Uc, Ucb , and Ucc to the electrostatic energy U vs. NB , the mole fraction of counterions B, for a mixture of two kinds of monovalent counterions B and C differing in size. The exclusion radius from the center of the macroion to the center of the larger counterion C is c = 11 Â, the concentration parameter y = 1 corresponds to the concentration about 0.026 mol COOVdm .
U.
vkT
cB-       2Q*B
f 2 02 exp(3L + zB02 ) dt + f 03 exp(3L + zB03 ) dt
hl                                                    Jt2
and
UcC - -^A J7 °3exp(3^ +zc®y)dt
(23)
(24)
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Dependence of Um, Ucb, Ucc, Uc, and U on the equivalent fraction NB is presented in Figure 1 for a mixture of two kinds of monovalent counterions B and C differing in size. In all computations we used for the charge parameter 0 the value: 0 = 11.12, typical for fullerenehexamalonates. Figure shows that for any value of NB, Um and U
become more positive, and Ucb and Uc less negative, by increasing the radius of the smaller counterion B, and keeping the radius of the larger counterion C constant. It is
40-1-------------------------------------------------------------------------------------------1
-20 H---------1---------1---------1---------1---------1---------1---------1---------1---------1--------
0.0             0.2             0.4             0.6             0.8             1.0
N
B
Figure 2. Dependence of Um , Uc, U, and the electrostatic energies of monovalent and multivalent point-charged counterions, B and C, respectively, on the equivalent fraction of the monovalent counterion species   Nb, for a fullerenehexamalonate solution with a
mixture of monovalent and zc valent counterions.
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well known that the degree of binding of counterions decreases by increasing their radius. Hence, the bulkier counterions are spread to larger distances from the macroion domain, i.e., to the regions of weaker electrostatic field which are energetically less favored {cf. discussion below). As a consequence, the absolute value of the electrostatic energy of these counterions Ucb and thus also of all counterions becomes smaller and of the less screened macroion Um larger.
Figure 2 represents various contributions to U as functions of NB for mixtures of
j*
3
Figure 3. Variation of the contributions Um, Uc, Ucb , and Ucc to the to the electrostatic energy U, of a mixture of two monovalent and one divalent point-charged counterions, B and C, respectively, with the concentration parameter y.
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monovalent counterions B and multivalent counterions C with the charge number zc. It is seen that the values of counterion contributions Ucb, Ucc, and Uc depend only slightly on the charge number zc, whereas dependence of the electrostatic energy of the macroion Um, and consequently, of the entire system [/onzcis more explicit.
Concentration dependence of various contributions to U is presented in Figure 3 for mixtures of two monovalent and one divalent counterions. It can be seen that electrostatic contributions of monovalent as well as divalent counterions become negligible at concentrations at which the binding of counterions become insignificant (for monovalent counterions B at y » 4, and for divalent counterions C at y » 8). As a consequence, Um and [/join each other at extremely dilutions, and tend to the limiting value,6 U/kT= 66 J.
Interesting is the distribution of counterion energy among the counterion species throughout the elementary cell. Evidently, the radial counterion-energy distribution functions around the spherical macroion are given by
1   dUcB =         Q2 exp(3f +zgP2)+ P, exp(3f +zB®3)
Ucb    dt       \202Qxp(3t+zB02)dt+\Y O^Qxp(3t+zB0^)dt
and
1   dUcC =     P3exp(3f+zcP3)
UcC    dt       fY03exp(3ć+zc03)afr
Jt2
a results that follow from eqn. (23) and (24).
The results of computations are presented in Fig. 4 for a mixture of two monovalent and one divalent counterions, B and C, respectively, and for various ionic radii. It can be seen that 99% of the energy of divalent counterions is possessed by the ions which are laying within the radial distances 0.13 R (t = 1), from the surface of the macroion. This volume element represents only 0.2% of the total cell volume AnR /3. Due to the exclusion of the monovalent counterions B to regions farther from the macroion their energy distribution function is not so sharp. We can thus conclude that the contribution of so-called "osmotically active" counterions to the total energy U is negligible. Obviously, the overall energy change accompanying for example the dilution process, that is experimentally demonstrated as the heat of dilution, is governed by the
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energy of ions   in the   immediate vicinity of the macroion,   and not by the energy of "free
T3_
3
3
¦D
b = c = 7.7Â
b = 9Â,   c= 11 Â
b = c= 11 Â
dU   /U   dt            dU   /U   dt
cC       cC                                    cB.      cB
3
(Y)
Figure 4. The radial electrostatic energy distribution functions of counterions around the spherical macroion, for a mixture of two monovalent and one divalent counterions, B and C, respectively, vs. the dimensionless radial distance t. The concentration parameter
r=3.
counterions" that determine the values of the osmotic and activity coefficients. Similar
finding has been found also for linear polyelectrolytes.
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References
1.    A. Katchalsky, Z. Alexandrowicz, O. Kedem, in Chemical Physics of Ionic Solutions; ed. B. E. Conway and R. G. Barradas, Wiley, New York, 1966, pp 295-346.
2.     J. Škerjanc, J. Chem. Phys. 1990, 93, 6731-6737.
3.     J. Cerar, J. Cerkovnik, J. Škerjanc, J. Phys. Chem. B 1998,102, 7377-7381.
4.     J. Cerar, J. Škerjanc, J. Phys. Chem. B 2000,104, 727-730.
5.     J. Škerjanc, J. Chem. Phys. 1999,110, 6890-6895.
6.     J. Škerjanc, D. Dolar, Acta Chim. Slov. 1999, 46, 523-530.
7.     A. Katchalsky, Pure Appi. Chem. 1971, 26, 327-373.
8.     H. A. Scheraga, A. Katchalsky, Z. Alterman, J. Am. Chem. Soc. 1969, 91, 7242.
9.     D. Dolar, J. Škerjanc, J. Polym. Sci. Polym. Phys. Ed. 1976,144, 1005-1013.
Povzetek
Z rešitvijo Poisson-Boltzmannove enačbe za sferični celični model raztopine fulerenovega elektrolita smo izračunali prispevek fulerenskega makroiona in protiionov k elektrostatski energiji raztopine. Račune smo naredili za raztopine z mešanico protiionov, ki se ločijo tako po velikosti kot tudi po naboju. Rezultate podajamo v odvisnosti od koncentracije makroelektrolita in molskega ulomka protiionov.
J. Škerjanc: Contributions of the macroion and counterions to the internal energy of fuller ene...