J . Phys. Chem. 1989, 93, 4664-4669
4664
TABLE 111: Comparison of ESR Parameters with Conductivity at 25 OC for Various Polvaniline Derivatives polyaniline A/B log conductivity, derivatives ratio AH,,, G I value (ohmam-' 2.001 0 4 3.1 1 1 90.00 PAP 1.134 2.666 19.50 2.00465 PATFB 1.714 2.00475 1.919 PATS 1.753 0.75
In Figure 1 I , the ESR spectrum of the PATS sample a t room temperature shows a single peak like that of other conjugated conducting polymers. The measured peak-to-peak line width, AIB ratio, and g value are 0.75 G, 1.7533, and 2.004 75, respectively. For comparison of the ESR parameters and conductivity for other polyaniline derivatives, polyaniline tetrafluoroborate (PATFB) and polyaniline perchlorate (PAP) were synthesized by the electrochemical and chemical oxidation of aniline, respectively. For these synthesized polyaniline derivatives, the ESR and conductivity measurements were performed by the same methods as in the case of PATS. The values of the ESR parameters and conductivity a t 25 OC are summarized in Table 111. Table I11 shows that the conductivities of various polyaniline derivatives increase with increasing AHppand decreasing AIB ratio and decreasing g value. Conclusions The polarography and cyclic voltammetry results show that the oxidation reaction of aniline containing p-toluenesulfonate anions
a t the anodic electrode is an irreversible reaction relating one electron. In other words, polymerization should take place through a previous production of an aniline x-radical cation, formed by one electron transfer, which would act as the monomer for polymerization by reaction with neighboring aniline molecules. Then, the p-toluenesulfonate anion could play a role in the initiation of polymerization. From the temperature dependence of the electrical conductivity, E , is obtained to be 0.038 eV, and the hopping conduction model is suggested. As the ESR measurements show the presence of singly charged paramagnetic centers, the conduction mechanism for PATS is suggested the small polaron hopping conduction. That is, following doping of anions as an electron acceptor, a polaron with 0.038 eV gap is formed near the conduction band. The electron in the level causes a local polarization near the positively charged center, and hence the electron becomes a small polaron with x-cation radical. The observed conduction is due to polarons, charge carriers, that hop from state to state.
Acknowledgment. We are grateful to the Ministry of Education of Korea for financial support and to Professor J. W . Park for assistance with the electrochemical measurements. We also thank Professor H . S. So for ESR measurements. Registry No. TEATS, 733-44-8; polyaniline, 25233-30-1; p-toluenesulfonate, 16722-5 1-3; aniline, 62-53-3; tetrafluoroborate, 14874-70-5; perchlorate, 14797-73-0; polythiophene, 25233-34-5; poly(3-methylthiophene). 84928-92-7.
Electrostatics of Ion-I on Interactions in Solution Alexander A. Rashin Department of Physiology and Biophysics, Mount Sinai School of Medicine of the City University of New York, New York, New York 10029 (Received: September 6, 1988; In Final Form: January 9, 1989)
Potentials of mean force (PMF) between ions in solution are calculated with a method based on a continuum representation of the solvent, and employing the boundary element technique. PMFs calculated with this method for Li'CI-, Na'CI-, and K T - are in quantitative agreement with corresponding PMFs from microscopic theories. The agreement suggests that the double minimum shape of PMF is due to the fact that the energy of unscreened Coulombic interaction and the energy of hydration have different dependence on ion-ion distance. I t also suggests that contributions from dielectric saturation, a specific structure of solvent around ions, and nonelectrostatic effects-which were not included in our calculations-are not dominant for PMFs. Differences in details of PMF for like charged ions calculated with macroscopic and microscopic methods can be attributed to the errors involved in calculations of large hydration energies of double-charged systems, or to a slight shrinking of the cavity containing the double charge. It is shown that the larger association constant for an ion pair in a nonassociated solvent than in an associated solvent with the same dielectric constant can be due to larger cavity radii in nonassociated solvents.
1. Introduction Many chemical and biochemical phenomena take place in solution, making the evaluation of solvent effects a critical element for understanding these phenomena.'s2 Ion-ion interactions constitute a particular class of such phenomena crucial for chemical reactions in solution and for the structure and function of biological molecules.'-'0 Interionic potentials of mean force ( I ) Warshel, A,; Russell, S . T.; Quarr. Rev. Eiophys. 1984, 17, 283. (2) Computer Simulation of Chemical and Biochemical Systems. Beveridge, D. L., Jorgensen, W. L., Eds.; Ann. N Y Acad. Sci. 1986, 482. (3) Hirata, F.; Rossky, P. J.; Petitt, B. M. J . Chem. Phys. 1983, 78, 4133. (4) Berkowitz, M.; Karim, 0. A,; McCammon, J. A,; Rossky, P. J. J . Chem. Phys. 1984, 105, 577. (5) Petitt, B. M.; Rossky, P. J. J . Chem. Phys. 1986, 105, 577. ( 6 ) Jorgensen, W. L.; Buckner, J. K.; Huston, S . E.; Rossky, P. J. J . Am. Chem. SOC.1987, 109, 1891. (7) Klapper, 1.; Hangstrom, R.; Fine, R.; Sharp, K.; Honig, B. Proreins 1986, I , 47. (8) Dang. L. X . ; Petitt, B. M. J . Am. Chem. SOC.1987, 109. 5531
0022-3654/89/2093-4664$01.50/0
(PMF) can provide important insights into thermodynamics and kinetics of ion-ion interactions in solution. Calculations of P M F have been performed with integral equation t e c h n i q ~ e s , ~ .with ~,~*~~'~ the Langevin dipole model of solvent,' and with full microscopic simulations of the ion-ion interactions in ~ a t e r . ~ All . ~ .these ~ calculations yield two major minima in P M F for unlike ions: the solvent separated minimum and the contact minimum, with a barrier between them. It has been suggested that this shape of P M F is determined by a specific structure of water around the ion pairs and by saturation effect^,^.^,^ and thus can be predicted only by approaches that explicitly account for the molecularity of the solvent. R E M a p p r o a ~ h , ~in- ~particular, ,~.~ has been devised as a correction to the continuum theory a t short ion-ion separations, at which continuum theory allegedly does not work. Results (9) Hirata, F.; Levy, R. M. J . Phys. Chem. 1987, 91, 4788. ( I O ) Levesque, D.; Weiss, J. J.; Patey, G.N. J . Chem. Phys. 1980, 7 2 , 1887; Patey, G. N.: Carnie, S. L. Ibid. 1983, 78, 5183.
0 1989 American Chemical Society
Ion-Ion Interactions in Solution
2
I 0 Figure 1. Thermodynamic cycle for evaluation of PMF between ions 1 and 2 in solution.
Figure 2. Approximation of the shape of a cavity formed in solvent by
for like ion pairs also show oscillatory behavior, but beyond this similarity the results of different studies are different. P M F for the CI-CI- pair from ref 5 predicts a net stability of such a pair in water near contact distance with respect to infinite separation. Free energy perturbation calculations8 with the same model parameters have led to the same conclusion. However, no such stabilization has been found for pairs of positive ions in water.5 Calculations in methanol reverse this p i ~ t u r e .Stabilization ~ for either pair was observed in L H N C typelo calculations with model polar solvents. The quantitative accuracy of these results has been questioned by the author^,^ and it is conceivable that the counterintuitive prediction of stabilization of like charged ion pairs is erroneous. Results of R E M calculations in methanol and nonassociated dipolar liquid9 agree with experimentally observed large differences in association constants for particular salts in various solvents with similar dielectric constant^.^ Previous applications of continuum theory failed to explain these experimental findings (ref 9 and references therein). Significant success has been recently achieved in quantitative treatment of thermodynamics of hydration within the framework of the continuum We undertake here a study of the ion-ion interactions based on this success of the continuum electrostatic theory to elucidate whether it can lead to the same results as those from the microscopic approaches. As our approach invokes an intuitively simple physical model that involves only one empirical constant and does not consider saturation effects and a detailed molecular structure of the solvent, it may also help to elucidate the physical basis of the effects observed in microscopic calculations based on more complicated models with large numbers of adjustable parameters.
Each of these terms contains electrostatic and nonelectrostatic contributions; e.g.
11. Outlines of the Computational Approach Here we give only outlines of our computational approach. TO avoid a disruption of the main flow of ideas involved in this study, a brief description of the methodological technicalities is transferred to the Appendix as suggested by the reviewer. For a full description of some methodological details, one can consult ref 1 1 and 12. A . Contributions to PMF. The free energy of two ions at separation R in a solvent relative to their infinite separation can be evaluated via the following thermodynamic cycle (Figure 1). First we transfer both ions at infinite separation from the solvent to vacuum, which yields the negative sum of solvation free energies AG,). In the second step we bring for two isolated ions: -(AG,, two ions to separation R in vacuum. This yields the regular Coulombic energy of two charges at distance R , AGI2,, which should be corrected for the energy of repulsion, AG,,, a t short interionic separations. In the third step we transfer two ions at separation R from vacuum to the solvent. This yields the energy of solvation of the complex: AG,,,. These three terms determine the P M F of two ions in solution, AG12(R):
+
A G I ~ ( R= ) AGizAR)
+ AGizc(R) + AG,,(R) - AGis - AG2s (1)
( I I ) Rashin, A. A,; Honig, B. J . Phys. Chem. 1985, 89, 5588. (12) Rashin, A. A,; Namboodiri, K. J . Phys. Chem. 1987, 91, 6003.
two ions when solvent molecules do not separate them.
In this work we concentrate on electrostatics of ion-ion interactions and, thus, on electrostatic terms in AG12(R). B. Calculation of Solvation Energies. It has been shown12 that, if polarization of the ions (e.g., by the solvent) is neglected, the electrostatic contribution to their solvation energy, AG,,,equals the energy of interaction of ionic charges with the reaction field due to polarization of the solvent by ions. The reaction field can be obtained from the numeric solution of the Poisson equation with the boundary element methodI2 for a system of charges Qk in one or a few cavities with dielectric permittivity Di= 1, imbedded in a continuum solvent of dielectric permittivity Do.The reaction field can be represented as the field of polarization charges at the dielectric boundary. The polarization charges and the reaction field depend on the shape and size of the dielectric boundary. Calculation of polarization charges with the boundary element method is described in the Appendix. The dielectric boundary is defined as a closed surface (or surfaces) around the ions' nuclei a t which the polarizable electron density of the solvent rises sharply and which contains inside a negligible electron density of the solvent".l2 (ionic cavity). For individual spherical ions these surfaces are spherical. For polyatomic ions or for two individual ions close to each other, the dielectric boundary should contain parts of the sharp corners between the overlapping spheres of individual ions (or atoms), as such corners cannot be completely filled by the solvent molecules having a shape and finite size. Therefore the boundary between the cavity and the solvent, represented as a dielectric, can be definedI2 as a "molecular surface".I3 This surface is formed by rolling a spherical probe on a surface formed by the overlapping spheres of ionic The values of the cavity radii are consistently determined from crystallographic or N M R data'lJ5 and are transferrableI2 (see Appendix). According to our definition the probe radius describes a sphere delimited by the electron density of the solvent molecule high enough to determine the dielectric boundary of the ionic cavity,I2 and therefore can be smaller than the van der Waals radius of the solvent.11s12Most of the results presented have been obtained with a probe radius of 0.8 A.12 A real molecule of water is not a sphere, and the radius of the approximating spherical probe should somewhat vary, depending on its energetically preferred orientations. Dependence of results on the robe radius has been checked with probe radii of 0.6, 1, and 1.27 (for the CI-CI- pair). Probe radii smaller than 0.6 A would correspond to energetically unfavorable orientations of water molecules relative to ions.I2 Two ionic cavities were considered "solvent separated" if a probe representing the solvent molecule12 could freely pass between them. For two solvent separated ions the cavity for each ion is represented by a sphere with the radius defined in ref 11. When probes can no longer touch the line connecting the centers of two ionic cavities, a single cavity is formed (Figure 2). The molecular surface of such a cavity includes a part of toroidal surface.13 Details of the
w
-
(13) Connolly,
M.L. Science 1983, 221, 709.
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Rashin
The Journal of Physical Chemistry, Vol. 93, No. 11, 1989 4.0 .
1I i 1
1
I
2.0
c1-c1-
- ... . Ne+Na+
-2.0 -4.0
1 I
2.
a
v I
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r in
I
I
1
6.
8.
0 0 1 2.
1
I
-
L
6.
4.
r in
-
L
.
0.
A
Figure 4. PMFs for like charged ion pairs in water calculated with the use of continuum representation of the solvent.
A
7 -
-- K'C1-
E-
x
\
F
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'
2.
b
I
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Figure 3. PMFs for alkali-metal halide ion pairs in water: (a) continuum model of this work; (b) RISM results. Reprinted with permission from ref 5. Copyright 1986 American Institute of Physics.
generation of molecular surfaces used in this work are given in the Appendix. C. Repulsion at Short Separations. T o account for repulsive forces a t short distances and to make our results comparable to the results of RISM calculation^,^ the values of vacuum repulsive potentials, G,,(R), from ref 5 were added to the values of AG12el(R).These potentials obtained from quantum calculations also account for the mutual polarization of ions. This polarization is probably modified by the solvent, but we neglected this effect to make our results comparable to those of ref 5. D. Ion Pairs in Nonassociated Solcents. It has been suggested that the cavity radius for the same ion in hydrogen-bonding and non-hydrogen-bonding solvents could be different due to differences in ion-solvent distances in these solvents." In C,H,NO, or (CH&202, for example, the cavity radius for CI- will be larger than in water. Radii of these cavities are not known a t present." A model system was devised here to evaluate effects of the change in the solvent on the P M F of an ion pair. It was assumed that in water there is a cation with the cavity radius of CI- and anion with the cavity radius of Na', and that in some non-hydrogenbonding solvent with dielectric constant D = 78, the cavity radius of this anion increases to that of K+ in water. This model allows one to use AG12el(R)calculated for Na+CI- and K'CI-. in water and to add to them the vacuum potential of Na'CI- to represent PMFs of our hypothetical ion pair in water and a non-hydrogen-bonding solvent. 111. Results
A . Pairs of Oppositely Charged Ions. PMFs for Li'Cl-, Na'CI-, and K'CI- ion pairs calculated in this work are presented in Figure 3a. The figure shows a water-separated and a contact minimum and a barrier between them for each ion pair. The height of this barrier decreases and shifts to larger interionic distances with increasing size of the cation. The depth of the
contact minimum increases in the same direction. Results for the Cu++CI- ion pair (not shown) follow the pattern observed in the series K'CI-, Na'CI-, Li'CI-. with decreasing value of AG,, + G2. The height of the barrier is over 10 kcal/mol, and the minimxm a t contact separation is very shallow. The results of RISM calculations for the same ion pairs are shown in Figure 3b. The two parts of Figure 3 exhibit striking similarity. Small ( < k n differences in the heights of barriers and small (
