Acta Chim. Slov. 2006, 53, 331–337
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Scientific Paper
Water-Soluble Highly Charged Fullerene Derivatives as
Spherical Electrolytes. Electrostatic Contributions to Thermodynamic Functions of Systems Containing Mixtures of Counterions Differing
in Charge and Size†
Jože Škerjanc
Faculty of Chemistry and Chemical Technology, University of Ljubljana, Aškerčeva 5,
1000 Ljubljana, Slovenia
Received 08-05-2006
Dedicated to the memory of Prof. Dr. Davorin Dolar
Abstract
The Poisson-Boltzmann equation has been solved to obtain the electrostatic potential for the spherical cell model of a fullerene molecule at which charged functional groups are attached. The fullerene macroion is surrounded by mixtures of small counterions differing in size and charge. The electrostatic internal and free energies, degree of binding of counterions, osmotic coefficient, and distribution of counterions around the charged fullerene shell are calculated. These properties are presented as functions of radii, charges and mole fractions of counterions.
Key words: Poisson-Boltzmann equation, spherical cell model, thermodynamic functions, polyelectrolyte solutions, fullerene electrolytes, fullerenehexamalonic acid
1. Introduction
The influence of size and charge of small ions (counterions) on thermodynamic properties of solutions containing charged macromolecules (polyions) of opposite charge is well known.1,2 Quite a few papers have been devoted also to studies of polyelectrolyte solutions containing mixtures of counterions. It has been observed theoretically as well as experimentally that polyelectrolyte solutions with mixtures of counterions display interesting dependence of equilibrium properties on the mole fraction of a counterion species. For example, the osmotic coefficient of solutions of polystyrenesulfonates with mixtures of monovalent and divalent3 or trivalent4 counterions show distinct maxima when plotted against mole fraction. On the other hand, corresponding curves for the apparent molar enthalpies4,5 and volumes4,6 for the same systems have a sigmoidal shape. This striking behavior has been also interpreted theoretically in the light of the cylindrical cell model.3-7
Recently, we have reported8-13 on the first physicochemical studies of aqueous solutions of fullerene electrolytes, new type electrolytes that fill up the gap between simple electrolytes and polyelectrolytes. Th symmetric derivative of fullerene C60, at which
six malonic acid molecules have been attached, Th-C60(C(COOH)2)6, has been isolated and characterized by some fundamental data and properties. It has been found that fullerenehexamalonic acid, C66(COOH)12, has properties of a diprotic acid. The first carboxylic group of the malonic acid functional group behaves as an almost strong acid, whereas the second group is a weak acid. Its sodium salt, C66(COONa)12, has the properties of a highly charged strong electrolyte. For experimentalists as well as theoreticians working in the field of electrolyte solutions, such an electrolyte might be attractive at least for two reasons. One of them is that this solute is a good example of a highly asymmetric electrolyte due to the high charge number of the bulky ion. Another one is the charming spherical symmetry of the fullerene macroion that justifies the use of the spherical model for theoretical interpretation of this system.
The solution of the Poisson-Boltzmann equation in spherical geometry has been recently discussed14-17 for solutions of colloidal particles containing only counterions, or counterions and coions. The electrostatic potential and ion concentration profiles around or within a charged sphere have been presented, in same cases also the electrophoretic mobility of a spherical particle in these media.
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In this article we shall extend our previous calculations on solutions of fullerene electrolytes containing one kind of counterions10 to solutions with mixtures of counterions which differ in charge and size. We shall apply the Poisson-Boltzmann equation and the spherical cell model to calculate some fundamental thermodynamic properties: the electrostatic internal and free energies, degree of binding, osmotic coefficient, and the distribution functions of counterions around the fullerene macroion. These properties will be presented as functions of the counterion radii and charge, as well as there mole fraction in the mixture.
2. Electrostatic Potential
As stated above, 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.18,19 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 ? 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 t1 = ln (b/a) and t2 = ln (c/a). The Poisson-Boltzmann equation for this model has three domains:
(1) (2)
V2<&! =0,    a<r<b
V2c&
2    4jr/azawaexp(za<&2),    b<r<c
V2<&3=47i;/J!^z,«,0exp(z,<I>3),   c<r<R       (3)
i
where i in eqn. (3) stand for B or C, and dimensionless quantities <D, t, and y and the charge parameter 0, are given by
Q = —e0\\i/kT,    r = aexp(t),    R = aexp(y).      (4)
0
 ve;/ekTa
vlB/a
(5)
In these equations \|/ is the electrostatic potential, e0 is the proton charge, e is the dielectric constant of the solvent, ni 0 is the number density of counterions i at \|/ (R) = 0, lB (= e02/ekT) 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
{d®Jdi)l=Q=-Q                           (6)
Oj (?1) = 02 (tx),    (d011 dt\ = (d®21 dt)h         (7)
02(t2) = 0,(t2),   {d®2ldt\={d®ildi)h       (8)
<D3O) = 0,    (JO3/^=r = 0                (9)
Equation (1) can be solved analytically, with the result
Ol = Ol (0) - 0(1 - exp(—?)),   L/<!>[ / eft = -0 exp(—0   (10)
where (t1(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
d1<&2ldt1 + d<$>2ldt-kB exp(2? + zB®2) = 0    (11)
d2®, /dt2 + d®, Idt-^ki exp(2? + z,<&3) = 0   (12)
i
where i in eqn. (12) stand for B or C and the constants kB and kC are given by
/1—22          0
k = 4rce0a ZjHj
^c =
/1—22           0
4rce0a zc»c        (13)
For later use we introduce the equivalent fraction Xi of the species i, i.e. the fraction of the macroion charge compensated by the species i:
X;    =   Z;     j
AT. / v = z,«,^ / v = (z, / v) |«,JF
(14)
In eqn. (14) Ni is the total number of counterion species i, nj is its average number density, and Vi is the volume accessible to the species i, i.e. VB = 4n(R3 – b3)/3 and VC = 4n(R3 - c3)/3. It follows from eqn. (14) that the equivalent fractions XB and XC are related to the constants kB and kC by eqn. (15) and (16):
kB
f 2 exp(3t + zB$>2)dt+ \yexp(3t + zB$>3)dt
h                                                                              %2
exp(3? + zc03        = QXr
&X   (15)
(16)
In all computations we used for the radius of the fullerenehexamalonate macroion the value8 a = 0.77 nm, which gives for water solutions at 25 0C (e = 78.54) and for v = 12 the value of the charge parameter 0 = 11.12. The analytical concentration c and the concentration parameter y are related by
c exp(3y) = 3v / 4%aiNA
(17)
where NA is the Avogadro constant. From this relation we get c exp(3y) = 10.42 mol COO dm3.
V
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3. Fraction of Bound Counterions
Figure 1 shows the radial distribution function of counterions throughout the elementary cell for the equal equivalent fractions of monovalent and divalent counterions, i.e. for mixtures of two monovalent and one divalent counterions, for various values of their radii. The fraction of the total number of counterion species i, Ni, per spherical shell of thickness dr is dNi /Ni dr = ni ( r ) dV/Ni dr. Taking into account eqs. (4), (5), (13), and (14), and since dr = rdt, and the ratio ki /® = 4na3 zini0 /v, we get for the radial distance b <r<c, accessible only to the smaller counterions B, the expression
--------- = kKexp(3t + zH<&,)/ XKQ           (18)
For the radial distance c<r<R, accessible to both counterions B and C we get
1   dN
and
NB	dt
1	dNc
Nc	dt
 kB exp(3? + zB<L>?)/ XB®
 kc exp(3? + zc<J>3)/Xc0
(19)
(20)

0,01
Figure 1. The radial distribution functions of counterions around the spherical macroion of radius a = 0.77 nm vs. the reduced radial distance t, for a mixture of two monovalent and one divalent counterions. Parameters: ? = 12, zB = 1, zC = 2, ? = 3, XB = 0.5.
Figure 1 shows that the distribution functions given by eqs. (18)-(20), have minima at rm = a exp(tm). The value of tm can be indirectly determined from eqn. (19) and (20) by applying the analytical condition for the minimum, d(dNi/Ni dt)/dt = 0, which gives
^3
dt
3
(21)
The osmotic coefficient, ?, and the fraction of free counterions, fi, are measures of the total and
individual concentrations of counterions, respectively, at the surface of the cell.218 By analogy, with the practice adopted in the field of polyelectrolyte solutions containing only one counterion species,18 we can define the osmotic coefficient of solutions with a mixture of counterions by
<P = E»,7E»,                        (22)
which for the mixture of two counterionic species B and C yields
)[SxV(37)-l]VBVc
<P =
\ZBkC
3e(zBvBxc
zcVcXb)V
(23)
The fraction of free counterions has been
defined2,7,18 by
f,
ni     vXilzyi
(24)
Taking into account eqs. (5), (13) and (14) and dimensionless quantities t1 and t2 we get
and
/b
fc
kB[SxV(37)-SxV(%)]
30X
Lc[exp(3y)-exp(3?2)]
30X
Evidently, the degree of binding, bi, is
b,=\-f,
(25)
(26)
(27)
The influence of the radius and charge of counterions on the osmotic coefficient of the mixed fullerene electrolyte solutions is illustrated in Fig. 2 – 4. It can be seen from Fig. 2 that ? is not very sensitive to the size of counterions of the same charge. On the other hand, the dependence of ? on the charge of counterions is considerable (Fig. 3 and 4). All curves in Fig. 4 show distinct maxima, which are shifting to lower values of the equivalent fraction of the monovalent counterion B with the increasing charge of the multivalent counterion C. It has to be mentioned that such behavior of ? has been observed experimentally as well as theoretically also with solutions of linear polyelectrolytes containing mixtures of counterions.3,4 The osmotic coefficient increases with the increasing zC for all values of XB except for the values of XB close to zero.
The corresponding values for the degree of binding of counterions calculated from eqn. (25) – (27) are presented in Figs. 5 and 6 for the same mixtures as before. It can be seen in Fig. 5 that the degree of binding of the less bound monovalent ions B decreases and consequently that of the more bound divalent ions C increases with increasing the ionic radius of monovalent counterions, keeping the radius of divalent counterions constant. The influence of the charge of counterion on bi
0
0
n
n
10

0,1

z
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0,58
0,52
0,0
0,2
0,4
0,6
0,8
1,0
X
Figure 2. Dependence of the osmotic coefficient, ?, on the mole fraction XB, for a mixture of two monovalent counterions B and C, calculated for the exclusion radii from the center of the macroion to the center of the smaller and larger counterions, b and c, respectively, as indicated. Parameters: c = 1.1 nm, zB = zC = 1, ? = 2.
0,60
0,50
0,40
0,30
b = 1.1 nm 1.0 nm 0.9 nm
XB
Figure 3. Osmotic coefficient of a fullerene electrolyte solution containing a mixture of monovalent and divalent counterions, B and C, respectively, vs. the mole fraction XB, for the values of b as indicated. Parameters: c = 1.1 nm, zB = 1, zC = 2, ? = 2.
0,8
0,6
f   0,4
0,0
X
Figure 4. Osmotic coefficient of a fullerene electrolyte solution containing a mixture of monovalent and zC-valent counterions vs. XB. Parameters: b = c = 0.77 nm, zB = 1, ? = 2.
			D^-------
		b.     ^<	<^^^-^'
0,8-			—-—"a
0,6-		b	
	A		—¦---------------
0,4-	S———~		
0,2-	D		
0,0
0,2
0,4
0,6
0,8
1,0
X
Figure 5. Degrees of binding, bB and bC, of the monovalent and divalent counterions B and C, respectively, against XB for series of distances b and c. Parameters: A (b = 0.77 nm), B (b = 0.9 nm), C (b = 1.0 nm), D (b = 1.1 nm); c = 1.1 nm, zB = 1, zC = 2, ? = 2.
is for mixtures of monovalent and zC - valent counterions C presented in Fig. 6. As expected the degree of binding of monovalent ions, bB, decreases and of the multivalent ions, bC, increases with the valency of the multivalent counterions. Also, it can be seen that the degree of binding of the less bound monovalent ionic species increases and of the more bound multivalent ionic species decreases with increasing the corresponding equivalent fractions, a situation observed also with polyelectrolytes solutions.
It has been shown for polyelectrolyte solutions20 that the fraction of free counterions fi obtained from transport measurements can be approximated by the ratio Di/Di0, where Di and Di0 are the macroscopic self-diffusion coefficients of counterions in the presence and absence of the polyion, respectively. It can be estimated21,22 from eqn. (28)
Di D0
l\Vih
vtfzpdVj^ |Vexp(-zicI>)dVj       (28)
0,60
0,56
Y
0,54
0,2
0,50
0,0
0,2
0,4
0,6
0,8
1,0
0,65
0,55
b
9
0,45
0,35
0,0
0,2
0,4
0,6
0,8
1,0
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0,6
0,4
0,2
0,0
0,2
0,4
0,6
0,8
1,0
X
Figure 6. Degrees of binding, bB and bC, of the monovalent and multivalent counterions B and C, respectively, against XB. Values of parameters as in Figure 4.
applying to the i-th ion in the counterion mixture. Supposing that the same relation holds also for counterions in the high electrostatic field of the fullerene macroion, we get
and
DB D0
DC D0
/i
exp(3t + zB<S?2)dt exp(3t - zB<S?2)dt
exp(3? + zc<53)<i? exp(3? - zc<53)<i?
(29)
(30)
Equations (33) and (34) have been used in the interpretation of the experimental data presented recently in our paper12 on the molar conductivity of fullerenehexamalonate aqueous solutions containing mixtures of sodium and calcium counterions. All details have been given there.
4. Electrostatic Energy and Free Energy
The electrostatic energy of the solution, U, calculated per macroion is related to the electrostatic potential, ?, by the expression
U
%K   v
\  (gradwfdV
(31)
which upon insertion of dimensionless quantities ? and t from eqn. (4) transforms into
U
_vkT r
d&
d7,
exp(t)dt
(32)
The integral in eqn. (32) is the sum of three integrals in accordance with the three domains of the
model [cf. eqn. (1) – (3)]:
[    -----    exp(t)dt = \
tAd<$>
<H   dt
i;

d7
exp(t)dt
exp(t)dt
\t
y(d®
t 2     dt
exp(t)dt      (33)
The first integral can be solved analytically by applying for the derivative d<&1/dt eqn. (10):
1------    exp(t)dt = 0 [l-exp(-?])]         (34)
whereas the second and third integral must be solved numerically. The final expression for U reads
U
where
vkT
------
0
(35)
« = —0   l-exp(-?,)+—       -------
2                     u    2 J*, ^  cfr
1    r(d®3
2   h \   dt
exp(t)dt
exp(t)dt   (36)
The electrostatic energy of the system [/is simply related23 to the total electrostatic internal energy Ue
Ue=U(\ + d\m/d\nT)                     (37)
which can be compared with the experiment.
The electrostatic contribution^ to the Helmholtz free energy can be obtained by the method24,25 that combines the electrostatic energy U with the configurational entropy Sconf
Ae = U-TSconf                            (38)
where Sconf represents the entropy change due to the ordering of counterions in a charged system. The final result reads
Ae =
-U + vkTi
where
0
f
J
y = <&1(0) + V—iln^ f = -u + Qv
(39)
(40) (41)
and the dimensionless potential at the surface of the macroion, ?1(0), can be calculated from
<!>[ (0) = Q>1 (tj + 0[l - exp(-^)J
(44)
Fig. 7 and 10 represent the electrostatic energy and free energy functions, u and f, respectively, for mixtures of monovalent B and divalent C counterions as a function of equivalent fraction XB, for various values
1,0
0,8
+
b
+
0,0


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Acta Chim. Slov. 2006, 53, 331–337
0,0               0,2               0,4               0,6               0,8               1,0
X
B
Figure 7. Dependence of the electrostatic energy function u of a fullerene electrolyte solution containing a mixture of monovalent and divalent counterions, on the mole fraction XB for series of distances b and c. Parameters: , zB = 1, zC = 2, ? = 2.

Xs
Figure 9. Electrostatic contribution to the enthalpy of dilution Au vs. XB, for mixtures of monovalent and zC-valent counterions. Concentration parameters y = 1 and y = 3 correspond to the initial to final concentrations 0.519 and 0.00129 mol COO dm-3, respectively. Parameters: b = c = 0.77 nm, zB = 1.
X
Figure 8. Dependence of the electrostatic energy function u on XB, for mixtures of monovalent and zC-valent counterions. Parameters as in Fig. 4.
f   40
Figure 10. Electrostatic free energy function f vs. XB, of a fullerene electrolyte solution containing a mixture of monovalent and divalent counterions, for a series of the distances of closest approach b. Parameters as in Figure 3.
of distances b and c, the exclusion radii of counterions from the center of the macroion. The influence of the charge number of counterions on the electrostatic energy function u is for mixtures of monovalent and zC-valent point-charged counterions demonstrated in Fig. 8. It is seen that with the increasing zC the electrostatic energy u decreases for all values of XB. Fig. 9 presents Au (y :1^3), the electrostatic contribution to the heat of dilution from y = 1 to y = 3, i.e., from the initial to final concentrations 0.519 to 0.00129 mol COO dm-3, respectively. The most remarkable feature
of the results in Fig. 9 is the fact that the enthalpy of dilution is practically the same for various counterion mixtures when XB exceeds a given value. This critical value decreases with the increasing charge number zC, so that for zC > 4 the differences in the values of ?u become insignificant and are lying within the limit of the expected experimental errors. We mentioned in the introduction of this paper, that experimentally observed as well as theoretically predicted curves for the heat of dilution of solutions of linear polyelectrolytes containing mixtures of counterions, display well-defined sigmoidal
40
25
35
20
30

15
25
u
10
20
5
15
0
10
0,0
0,2
0,4
0,6
0,8
1,0
44
25
20
42
15
u
10
38
5
36
0
0,0
0,2
0,4
0,6
0,8
1,0
X
B
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shapes.4,5 Contrary to those observations, we can see in Fig. 9 that in the case of spherical fullerene electrolytes the inflection point is only slightly indicated.
5. References
1.  R. W. Armstrong, U. P. Strauss, in: H. F. Mark, N. G. Gaylord and N. M. Bikales (Eds.): Encyclopedia of Polymer Science and Technology, vol. 10, Interscience, New York, 1969, pp. 781–861.
2.  A. Katchalsky, Z. Alexandrowicz, O. Kedem, in: B. E. Conway, R. G. Barradas (Eds.): Chemical Physics of Ionic Solutions, Wiley, New York, 1966, p. 295–346.
3.  D. Dolar, D. Kozak, Osmotic Coefficients of Polyelectrolyte Solutions with Mono- and Divalent Counterions, in: Abstract of the IUPAC Symposium on Macromolecules (Leyden), Vol. 1, Inter Scientias, Hague, 1970, pp. 363–366.
4.  J. Škerjanc, D. Kozak, S. Hočevar, G. Vesnaver, B. Jus, D. Dolar, Biophys. Chem. 1977, 6, pp. 9–13.
5.  D. Dolar, J. Škerjanc, J. Chem. Phys. 1974, 61, 4106–4109.
6.  J. Škerjanc, D. Dolar, J. Chem. Phys. 1975, 63, 515–517.
7.  D. Dolar, A. Peterlin, J. Chem. Phys. 1969, 50, 3011–3015.
8.  J. Cerar, J. Cerkovnik, J. Škerjanc, J. Phys. Chem. B 1998, 102, 7377–7381.
9. J. Cerar, J. Škerjanc, J. Phys. Chem. B 2000, 104, 727–730.
10.  J. Škerjanc, J. Chem. Phys. 1999, 110, 6890–6895.
11.  J. Škerjanc, D. Dolar, Acta Chim. Slov. 1999, 46, 523–530.
12.  A. Vrhovšek, J. Cerar, M. Bešter-Rogač, J. Škerjanc, Phys. Chem. Chem. Phys. 2001, 3, 2650–2654.
13.  J. Cerar, J. Škerjanc, J. Phys. Chem. B 2003, 107, 8255–8259.
14.  R. Tuinier, J. Colloid Interface Sci. 2003, 258, 45–49.
15.  J. J. Lopez-Garcia, J. Horno, C. Grosse, J. Colloid Interface Sci. 2002, 251, 85–93.
16.  H. K. Tsao, Langmuir 1999, 15, 4981–4988.
17.  H. Ohshima, J. Colloid Interface Sci. 2003, 262, 294–297.
18.  A. Katchalsky, Pure Appl. Chem. 1971, 26, 327–373.
19.  H. A. Scheraga, A. Katchalsky, Z. Alterman, J. Am. Chem. Soc. 1969, 91, 7242–7249.
20.  A. Schmitt, R. Varoqui, J. Chem. Soc., Faraday Trans. 2, 1973, 62, 1087–1103.
21.  S. Lifson, J. Jackson, J. Chem. Phys. 1962, 36, 2410–2414.
22.  J. Jackson and S. R. Coriell, J. Chem. Phys. 1963, 38, 959–968; 1963, 39, 2418–2422.
23.  N. Bjerrum, Z. Phys. Chem. 1926, 119, 145–157.
24.  R. A. Marcus, J. Chem. Phys. 1955, 23, 1057–1068.
25.  J. Škerjanc and D. Dolar, J. Chem. Phys. 1989, 91, 6290–6294.
Povzetek
Z rešitvijo Poisson-Boltzmannove enačbe za sferični celični model smo izračunali elektrostatski potencial okrog fulerenove molekule, na kateri so pripete nabite funkcionalne skupine. Fulerenov makroion obdajajo majhni protiioni, ki se razlikujejo tako po velikosti kot po naboju. Izračunali smo elektrostatsko notranjo in prosto energijo, stopnjo vezanja protiionov, osmozni koeficient in porazdelitev protiionov okrog nabite fulerenove molekule. Prikazali smo odvisnost teh količin od velikosti, naboja in molskega ulomka protiionov.

Škerjanc  Water-Soluble Highly Charged Fullerene Derivatives as Spherical Electrolytes