Solute−Solvent Frictional Coupling in Electrolyte Solutions. Role of Ion

J. Phys. Chem. B 1997, 101, 2339-2347

2339

Solute-Solvent Frictional Coupling in Electrolyte Solutions. Role of Ion Pairs N. Balabai and D. H. Waldeck* Department of Chemistry, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15260 ReceiVed: August 30, 1996; In Final Form: January 17, 1997X

Rotational diffusion of the organic anion resorufin in different electrolyte solutions is studied. The rotational relaxation time τ increases as the concentration of electrolyte increases for all electrolytes studied. Although τ follows the viscosity in NaNO3 aqueous solution, the τ in Mg(NO3)2 solution is too large to be explained only by the increase of the viscosity. This effect is more significant in DMSO electrolyte solutions. The increase of the rotational relaxation time is well correlated with the charge-to-size ratio of the cation. The analysis of the experimental data using theoretical models that incorporate the different mechanisms of friction in electrolyte solutions strongly indicates the existence of ion pairs in DMSO electrolyte solution.

Introduction Electrolyte solutions are an important medium in biological and chemical processes, and their study has been a central area in physical chemistry since its origin as a discipline. Early theories of electrolytes are based on the works of Debye, Hu¨ckel, and Onsager. They considered a dilute solution of salt completely dissociated into rigid, spherically symmetric ions. The interaction between the ions was computed by Coulomb’s law assuming the medium to have the dielectric constant of the pure solvent. The qualitative agreement between the predictions of the theory and experimental results for several electrolytes demonstrated that the electrolytes were in a completely dissociated state in the dilute solutions considered. The study of dilute electrolytes that deviate from ideal Debye-Onsager behavior because of ion associations is the main subject of many theories. Modern theories, largely based on integral equation methods, are able to treat such solutions.1 A simple but useful model is the Bjerrum treatment.2,3 In this model the curve of probability for finding an oppositely charged ion at a given distance from a central ion has a flat minimum at a distance where the work of separating the two ions is 4 times as great as the mean kinetic energy per degree of freedom. This minimum distance (or Bjerrum length, b) is given by

b)

|z1z2|e2 2kT

(1)

For ions that are so large that their centers cannot approach closely to b, the Debye-Hu¨ckel theory should be satisfactory. However, small ions are able to approach distances below this critical value such that their work of separation increases rapidly and can become very large. Bjerrum regarded a pair of ions within this range as associated to form an “ion pair”. The associated ion pairs could be treated as neutral molecules in equilibrium with the free ions; i.e., C+ + A- / C+A-, and the fraction of ion pairs may be determined by the value of the association constant

Kass )

[C+A-] [C+][A-]

(2)

In this work the role of ion pairs is examined by studying the rotational diffusion, and therefore the friction, of a probe X

Abstract published in AdVance ACS Abstracts, March 1, 1997.

S1089-5647(96)02687-9 CCC: $14.00

molecule in different electrolyte solutions. Because of the rapid collision rate, the rotational motion of the solute is well described by Debye’s model in which the molecule reorients in random steps of small angular displacements. This diffusion model allows one to obtain a direct relationship between the diffusion coefficient and the relaxation time. By use of the Einstein relation,

D ) kT/ζ

(3)

where k is the Boltzmann constant, T is a temperature, and ζ is a friction coefficient, one can connect the rotational diffusion coefficient with the friction coefficient. Hence, the measured relaxation time connects to the friction coefficient. Comparison of the experimental relaxation times with those predicted by models for the solute-solvent frictional coupling allows the adequacy of those models to be assessed. Details of this modeling and analysis are discussed below. Although the mechanisms of friction in polar and nonpolar solvents have been widely discussed in recent years,4-16 only a few studies of rotational diffusion in electrolyte solutions have been performed.5,12,14 These studies show that ion pair formation can play a significant role in these solutions. This idea was supported by recent studies of Hartman and Waldeck.5 They studied the rotational diffusion of resorufin in aqueous and DMSO solutions of LiNO3. They calculated the hydrodynamic, dielectric, and ion atmosphere friction in the absence and presence of ion pairs and were unable to explain their data without the inclusion of an ion-paired species. They concluded that the larger friction associated with ion pairs is the major contribution to the long rotational relaxation times observed in DMSO solutions. Their results are consistent with previous data of Kenny-Wallace and co-workers for resorufin in methanol and water14 in which a significant dependence of the rotational relaxation time on LiCl concentration was observed. Some related studies of solvation dynamics17-19 in electrolyte solutions have also been performed. Recently, Chapman and Maroncelli19 studied solvation dynamics in a wide range of electrolyte solutions consisting of different cations (Li+, Na+, Mg2+, Ca2+, Sr2+). Their analysis required the formation of associations between cations and the probe solute molecule to explain the experimental results. The present work represents an extension of previous studies of this group into the role of solute-solvent interactions on the frictional coupling experienced by a probe molecule (resorufin anion) in different electrolyte solutions. The primary experi© 1997 American Chemical Society

2340 J. Phys. Chem. B, Vol. 101, No. 13, 1997

Balabai and Waldeck tion the rotational diffusion coefficient for rotation about axis i, Di is given by

Di ) Figure 1. Molecular structure of resorufin. The b axis is out of the molecular plane.

mental goal of this work is to characterize the rotational relaxation time in highly concentrated (0.1-2 M) electrolyte solutions. These studies explore and show the transition from the ion pair regime to the free rotation regime and its dependence on the charge-to-size ratio of the cation. Salts with different charge and size of cation were chosen, KNO3, NaNO3, and Mg(NO3)2, and were compared with results of previous studies of this group on the rotational diffusion of resorufin anion in LiNO3 electrolyte solution. The solvents used in these studies are water and dimethyl sulfoxide (DMSO). Background The quantity measured in the experiment is the orientational anisotropy r(t). In general, for the case of a solute represented as an asymmetric rotor, r(t) is given by a sum of five exponentials. The resorufin molecule studied in this work (see Figure 1) can be approximated as an asymmetric ellipsoid with its transition dipole moment along the long axis so that the model predicts a double-exponential decay.20 Experimentally, r(t) is found to be well approximated as a single exponential. In this case the characteristic relaxation time τor (i.e., τor ≡ ∫r(t) dt) is given by

τor )

Da + Db + 4Dµ 1 12 DaDb + DbDµ + DaDµ

(4)

where Dµ is the diffusion coefficient for rotation about the molecular axis that contains the transition dipole, and Da and Db are the diffusion constants for rotation about the other two axes of the molecule.16 The analysis will use models to determine the friction coefficient ζ that is then used to construct a relaxation time from eqs 3 and 4. Equations 3 and 4 provide the connection between the rotational diffusion coefficient, which is determined by the solute-solvent interaction in the solution, and the rotational relaxation time measured in the experiment. In this way, the relaxation time reflects the intermolecular interactions that control the molecular motion in the liquid. The friction experienced by the solute in the solvent can be characterized according to the type of intermolecular interactions present in solution. The three mechanisms for the frictional coupling used here are mechanical (or hydrodynamic) friction ζhyd, dielectric friction ζdiel, and ion atmosphere friction ζia. The total friction ζ is used in eq 3, however. As in previous work, the total friction is taken to be the sum of these three components

ζ ) ζhyd + ζdiel + ζia

(5)

Although this assumption limits the rigor,21 it greatly simplifies the analysis. Mechanical Friction. The mechanical friction is identified with the hard repulsive collisions between the molecules and is largely determined by the size and shape of the solute and the viscosity of the solvent. A common model for mechanical friction is based on the well-known Debye-Stokes-Einstein (DSE) theory, which treats the solvent surrounding the probe molecule as a continuous viscous medium. In this approxima-

kT 6Vησi

(6)

where V is the volume of the solute, η is the viscosity of solvent, and σi is a parameter that reflects the hydrodynamic boundary condition and the shape of the molecule.6,20,22,23 Two different hydrodynamic boundary conditions are common, stick and slip. Previous work6-9,20 indicates that the slip boundary condition gives good agreement with experimental results for the rotational diffusion of nonpolar molecules whose size is comparable or less than that of the solvent molecules in nonpolar noninteracting solvents.20 As the size of a solute molecule increases (as compared to the solvent molecules), the effect of roughness features of the molecule becomes significant, and the boundary condition is better described using the stick condition.7,8 The resorufin molecule (see Figure 1) used in these experiments can be modeled as an asymmetric ellipsoid with a volume of 190 Å3 and axial radii of 6.5:3.5:2.0 Å. If the hydrodynamic boundary condition is considered to be slip and other friction mechanisms are ignored, then the characteristic relaxation time for the orientational correlation function is calculated to be 26 ps/cP at room temperature.5 Dielectric Friction. Another type of friction that is present in polar solvents and electrolyte solutions is called dielectric friction.15,16,24-28 In contrast to mechanical friction, which relates to short-range interactions, the dielectric friction is associated with long-range electrostatic interactions of charged or dipolar probe molecules with the surrounding solvent. The simplest and the earliest model for rotational dielectric friction was proposed by Nee and Zwanzig.24 This and other models used point sources to describe the electrostatic coupling between the solute and the solvent. It has been found that these models may underestimate the magnitude of the friction by a 100 times for a medium-sized molecule such as that studied here.15,28,29 A more realistic model,27,28 which treats the solute as an extended charge distribution with partial charges on the atomic sites, is used here. In this model the interaction of the charge distribution and its reaction potential creates a torque that is used to obtain the dielectric friction. The friction coefficient is given by

ζdiel )

(s - 1) (2s + 1)2 8qiqj ri a

N

τD

N

a

L

∑ ∑∑ ∑ j)1 i)1 L)1M)1

( )( ) L

∞

rj a

( )

2L + 1 (L - M)! 2 M × L + 1 (L + M)!

L

PLM(cos θi)PLM(cos θj)cos(MΦij) (7)

where s is the static dielectric constant, τD is the Debye relaxation time, qi is the charge on atomic site i, (ri, θi, φi) is the position of atom i in spherical coordinates, Φij ) φi - φj, N is the number of the atomic sites in the solute molecule, and PLM(x) is the (LM)th Legendre polynomial. Experimental studies16,30 indicate that this model describes the frictional coupling reasonably well in unassociated solvents. However, the value of the friction coefficient is a sensitive function of the radius parameter a, as well as of the charge distribution. In previous studies from this laboratory the dielectric friction of resorufin in DMSO and water was modeled using eq 7. When the slip boundary condition is used to model the mechanical friction, the best fit cavity radius was found to be 6.87 Å in DMSO and 5.98 Å in water. Although this value is quite realistic in DMSO, the electrostatic model fails to describe the friction in water. This behavior in water may be caused by the

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J. Phys. Chem. B, Vol. 101, No. 13, 1997 2341

strong hydrogen bonding in water. If the solvent is “attached” to the solute for times comparable to the rotational period, then the observed relaxation would reflect the rotation of the “hydrated ” molecule. This sort of mechanism has often been used to explain long rotation times in hydrogen-bonding solvents.20,31,32 Ion Atmosphere Friction. The presence of ions in the solution complicates the picture. First, adding the electrolyte to the solution changes the hydrodynamic and dielectric properties of the solution and, therefore, the solute-solvent frictional coupling. Second, since oppositely charged ions attract each other, the cations and anions are not uniformly distributed in the solution. Although the whole system is electrically neutral, a net charged region, called the ion atmosphere, exists about any given ion.33 The ion atmosphere has two effects on the solute-solvent interaction in electrolyte solutions. First, when the solute rotates, the ion atmosphere responds to it by translational diffusion of ions. Since this response is not instantaneous, an ion atmosphere torque that opposes the motion is created. This additional mechanism of friction is called the ion atmosphere friction. Second, the ion atmosphere screens the electrostatic potential of the solute, which decreases the electrostatic solute-solvent interaction. The parameter that characterizes the extent of this screening is the Debye length given by

( ) skT

κ-1 )

4π

1/2

∑i qi ci

(8)

2

where qi and ci are the charge and concentration of ion i, respectively. This screening decreases the solute-solvent frictional coupling. Both these effects of the ion atmosphere were considered by van der Zwan and Hynes34 who proposed a quantitative treatment of ion atmosphere friction ζia. They treated a solute molecule as a point dipole µ that rotates inside a spherical cavity of radius a that is surrounded by an ion atmosphere. In their model the ion atmosphere friction is given by

ζia )

sx 3µ2 2 D(2 + 1)(2 + 1 +  x)k 2a s s s

(9)

Figure 2. Experimental decay curves for DMSO solution of resorufin shown for δ ) +1° (O), δ ) -1° (0), and their difference ([). See the text for details.

the molecule. The relaxation of the anisotropy as a function of time is detected by the absorbance of a polarized probe beam. The change of polarization of the probe beam is monitored as a function of time delay between the pump and probe pulses. The sensitivity and selectivity of polarization spectroscopy can be significantly improved by using optical heterodyned polarization spectroscopy (OHPS),35-37 and this detection scheme was used here. The heterodyned signal is generated by rotating the analyzer polarizer’s transmission axis a small angle δ from the null position. This procedure causes a large increase in the signal level, which is proportional to δ. When decay curves at +δ and -δ are measured and subtracted, the contribution of the nonheterodyned signal can be eliminated. Decay curves of this type are shown in Figure 2. When only the ground state is resonant with the probe pulse and the excited-state relaxation is much longer than the rotational relaxation, the signal S(t) is given by

S(t) ∝ r(t)K(t)

where r(t) is the anisotropy decay in the ground electronic state and K(t) is the population decay of the excited state. For resorufin these assumptions hold quite well. The excited-state absorption is small at the pump wavelength,38 and the excitedstate lifetime is a few nanoseconds. If r(t) and K(t) are single exponential with time constants τor and τf, respectively, then S(t) decays as a signal exponential with time constant τm and

1 1 1 ) + τm τor τf

where

x)

(κa)2 1 + κa

(10)

and D is the ionic diffusion constant. They pointed out that their model should be applied at low electrolyte concentrations where ion-solute associations are not important. Recent studies of the solvation of dye molecules in electrolyte solutions17-19 showed that the relaxation predicted by van der Zwan and Hynes is 5-50 times faster than the observed dynamics. Such slow dynamics cannot arise from pure diffusion but must involve the formation of ion pairs in electrolyte solutions. Experimental Methods The time-resolved polarization spectroscopy technique was used to study the rotational diffusion of resorufin in solution. In this technique a linearly polarized pump beam is used to create an anisotropy in the solution by exciting the molecules of a particular orientation. The probability of excitation is proportional to cos2 θ where θ is the angle between the direction of polarization of the pump beam and the transition moment of

(11)

(12)

By measurment of τm and τf independently, eq 12 can be solved to determine τor . The fluorescence lifetime τf was measured in a separate experiment using the time-correlated single photon counting method.20,39 The time-resolved polarization spectrometer has been described previously.37 The instrument consists of a picosecond laser system (CW mode-locked Nd:YAG laser (Spectra Physics Series 3000) and a home-built dye laser), a Michelson interferometer for performing pump-probe experiments, and a data acquisition system. The original Nd:YAG laser was modified to obtain better characteristics. Stable pulses were obtained at 1064 nm with an average power of 7 W, pulse width of 100110 ps, and a repetition rate of 82 MHz. The pump laser was then frequency doubled in a 5 mm long KTP crystal so that the power at 532 nm (green) is about 0.9-1.0 W. This output is used to synchronously pump a rhodamine 6G dye laser that has an average power of 100-120 mW at 590 ( 5 nm. The time-correlated single photon counting (TCSPC) method was used to measure the fluorescence lifetime of resorufin’s excited state20,39 in the different electrolyte solutions. The

2342 J. Phys. Chem. B, Vol. 101, No. 13, 1997

Figure 3. Rotational relaxation time is plotted versus electrolyte concentration in aqueous electrolyte solutions ((b) Na+, (4) Li+ (taken from ref 5), (1) Mg2+).

Balabai and Waldeck

Figure 5. Rotational relaxation time normalized by viscosity versus electrolyte concentration in aqueous electrolyte solutions ((b) Na+, (4) Li+ (taken from ref 5), (1) Mg2+).

TABLE 1: Rotational Relaxation Times for Resorufin in 1 M Nitrate Solutions H2O

Figure 4. Rotational reorientation time versus electrolyte concentration in DMSO electrolyte solutions ((0) K+, (b) Na+, (4) Li+ (taken from ref 5), (1) Mg2+).

apparatus for TCSPC has been described previously. The emission decay profiles obtained by this technique are a convolution of the molecular excited-state decay law and the instrument function. The instrumental response function was obtained by replacing the sample with a scattering solution. A typical instrument function had a 100 ps full width at halfmaximum. The fluorescence lifetimes measured for resorufin ranged from 2820 ( 90 ps in water to 4880 ( 110 ps in DMSO electrolyte solutions. Conductivity measurements were performed to determine the association of the anion of resorufin with the cation of Na+ to form an ion pair in DMSO solution. The equivalent conductance Λ of a dilute solution of the sodium salt of resorufin in DMSO was measured on a conductivity meter using a glass electrode (YSI Model 35). The viscosity and density values for each solution were measured. The values are reported in the Supporting Information. Density measurements were made for the solutions of resorufin in DMSO at different concentrations of electrolyte using a pycnometer of capacity 10 mL (Thomas Scientific). Viscosity measurements were carried out in a water bath at 298 K using a calibrated Ubbelohde bulb viscometer (Thomas Scientific). All values were measured five times and had an accuracy of 2-3%. Resorufin was used as a probe molecule and was studied in both aqueous and DMSO solutions of KNO3, NaNO3, and Mg(NO3)2. Resorufin (sodium salt) was used as received (from Aldrich). KNO3 and NaNO3 were dried for 24 h before use. Mg(NO3)2 was obtained as Mg(NO3)2‚6H2O. This salt was dried in the oven for 24 h and then over P2O5 in a desiccator for several days. DMSO (Aldrich, 99.9%) was treated with molecular sieves before use. Deionized water (18 MΩ) was used to prepare aqueous solutions. Results and Analysis Rotational Diffusion Data. The rotational diffusion times for resorufin in both water and DMSO in the presence of different electrolytes are provided in Figures 3 and 4. All data were measured five times, and the average value was calculated. The salts were chosen so that the charge-to-size ratio could be

no salt K+ Na+ Li+ Mg2+

a

q/r au/Å

r Å

0.72 0.98 1.35 2.77

1.38 1.02 0.74 0.72

ηb

DMSO

cP

τ ps

η cP

τ ps

0.89

66 ( 5

0.95 1.06 1.003

71 ( 4 84 ( 4 102 ( 7

1.99 3.30 3.34 4.35 10.05

91 ( 5 250 ( 12 264 ( 13 406 ( 64 664 ( 66

a Values for the effective ionic radii are obtained from the following reference. Shannon, R. D.; Prewitt, C. T. Acta Crystallogr. 1970, B26, 1076. b Viscosity of aqueous ionic solutions is taken from the following reference. Landolt-Bo¨rnstein. Transport Phenomena; Springer-Verlag: New York, 1967; Band II, Teil 5.

varied from 0.72 for K+ to 2.77 for Mg2+. For all the salts studied, an increase in the electrolyte concentration causes an increase in the rotational relaxation time. This trend holds true for concentrations up to 2 M. This increase in relaxation time is much smaller for water than it is for DMSO. First, consider the aqueous electrolyte solutions. For cations with smaller charge density (Na+ and Li+) the increase in rotation time with concentration follows the viscosity, but for Mg2+ the increase is bigger than can be explained by the increase in viscosity of the solution. The rotational relaxation time in water increases by 5% from 0 to 1 M NaNO3, by 25% for 1 M LiNO3, and by 79% for 1 M Mg(NO3)2 (Table 1). Figure 5 plots the relaxation times normalized by viscosity versus the concentration of electrolyte for the different aqueous electrolyte solutions. For the case where the frictional coupling is controlled by the viscosity, this plot would be horizontal, i.e., independent of electrolyte concentration. The fact that the lines are horizontal for the monocations suggests that the changing friction is well captured by the change in viscosity. In contrast, the data for Mg2+ show that the frictional coupling here does not correlate well with the shear viscosity. In DMSO the rotational relaxation time increases more dramatically for all the cations. Figure 6 plots the rotational relaxation time normalized to the shear viscosity versus the cation concentration. For cations with smaller charge densities, K+ and Na+, the rotational relaxation times increase by 2.7 and 2.9 times, respectively. For Li+ the relaxation time is 4.5 times higher, and for Mg2+, which has the largest charge density, it is 7.3 times larger than the relaxation time in pure DMSO. In addition, the data show that the viscosity of the electrolyte solution does not fully account for the dramatic increase in the rotational relaxation time. Mechanisms other than the changing of the viscosity must be considered to explain this complex behavior. Conductivity Measurements. The measurement of the conductivity and its dependence on salt concentration is a classical method for determining the association constant of the

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J. Phys. Chem. B, Vol. 101, No. 13, 1997 2343

Figure 6. Rotational reorientation time normalized by viscosity versus electrolyte concentration in DMSO electrolyte solutions ((0) K+, (b) Na+, (4) Li+ (taken from ref 5), (1) Mg2+). The curves are drawn as a device to guide the eye.

TABLE 2: Association Constants KAss (in Salts in Water and DMSO a

RfNa RfLia NaNO3 LiNO3 a

M-1)

for Different

H2O

DMSO

4.1 4.5 0.2 0.2

12 23 21 10

Figure 7. Molecular structure for a possible ion-paired species.

a relaxation time τip. In this case the two species are treated as independent and a relaxation time is computed for each, in the manner outlined above. This procedure assumes a structure for the ion pair that is rigid on the time scale of the overall relaxation. The limitations of this approximation are discussed later. The total rotational relaxation time was computed as the properly weighted sum of relaxation times for each species,

Rf is resorufin anion.

TABLE 3: Ion Pair Fraction YIp for Sodium Resorufin and Lithium Resorufin DMSO Solutions c (mol/L) Na Li+

+

0.10

0.25

0.50

0.75

1.00

1.25

1.50

2.00

0.35 0.58

0.45 0.72

0.59 0.79

0.63 0.82

0.67 0.86

0.68 0.88

0.71 0.89

0.75 0.90

salt. The sodium salt of resorufin was used in conductivity measurements, and the data were compared with previous results for the lithium salt of resorufin.5 The association constant K1 for the reaction

R- + M+ a MR

(13)

was calculated using the Ostwald dilution law and Kolrausch’s law,33 which is in reasonable agreement with the data over the concentration range 10-3-10-4 M (R- is the resorufin anion). The analysis used here considers a second reaction to occur in the solution, namely,

NO3- + M+ a MNO3

(14)

with the association constants K2. The fraction of free resorufin, Yfr, and ion-paired resorufin, Yip, were estimated using both these equilibria. The specifics of this modeling is provided in the Supporting Information. Values of K1, K2 ,Yip, and Yfr for both the lithium salt and the sodium salt of resorufin are given in Tables 2 and 3. Because of difficulties in preparing the Mg salt of resorufin with a well-defined stoichiometry, conductivity studies could not be performed on it. Calculation of the Friction Using Theoretical Models. Two models for the rotational relaxation were investigated. The first model considers the case of one species, free solute, in the solution with a relaxation time τfr. Resorufin was modeled as an asymmetric ellipsoid, and the total friction coefficient was computed for each of the principal axes. The hydrodynamic, dielectric, and ion atmosphere friction were calculated separately, and the total friction was computed as their sum. The total friction coefficients were used to compute the diffusion coefficients via eq 3 and subsequently the relaxation time via eq 4. The second model considers the presence of two species in the solution, the free solute, and an ion-paired species with

τrot ) Yfrτfr + Yipτip

(15)

The DSE model with a slip boundary condition was used to calculate the hydrodynamic part of the friction. The resorufin molecule was modeled as an asymmetric ellipsoid with axial ratios 6.5:3.5:2.0 Å and a volume of 190 Å.5 The ion-paired species is hypothesized to have the cation coordinated to one of the carbonyl oxygens of resorufin and two DMSO molecules. This hypothesis was chosen because it is consistent with literature reports of the coordination number of these cations in DMSO.40 A stable ion-paired species was identified using the Macromodel software package. Geometry optimization of the ion pair structure was performed with the Amber united atom force field and a distance dependent dielectric constant.41 This procedure attempts to account for the spatially dependent dielectric electrostatics and functions as a crude approximation for polarization effects. Figure 7 shows a view of the structure that is obtained by this procedure. This structure was approximated as an asymmetric ellipsoid for the friction calculation. The sodium-resorufin ion pair has axial radii 8.5:5.0: 3.5 Å and a volume of 312 Å3. The lithium-resorufin ion pair has axial radii 8.5:4.9:3.4 Å and a volume of 311 Å3. This structure for the ion pair should be considered as a qualitative picture that reflects one of the possible geometries. The value of σ for the slip boundary condition was calculated for both the free resorufin and the ion pairs using these parameters and tabulated values.42 This calculation results in a lower limit for the friction coefficient, since the roughness features of real molecules will increase the solute-solvent coupling. The dielectric friction was computed using eq 7. Previous studies by this group showed that the extended charge distribution model gives the most realistic values for the dielectric contribution to the friction. The charge distribution was calculated for the free and ion-paired resorufin in a dielectric whose dielectric constant matched that of DMSO (see the Supporting Information). For this calculation (identification of the electrostatic field), the ion pair was modeled as the cation coordinated to the carbonyl oxygen of resorufin. Ab initio calculations were performed using the Gaussian-94 program. Geometry optimization of both the free resorufin and the ionpaired resorufin was performed by the Hartree-Fock method with a 3-21G basis set. The cation-to-oxygen atom distance

2344 J. Phys. Chem. B, Vol. 101, No. 13, 1997

Balabai and Waldeck

changed by less than 10% between this geometry optimization and that used in Macromodel (1.9 versus 2.1 Å for Na+). The partial charges used for the charge distribution were obtained from electronic structure calculations using the Merz-SinghKollman method.43 The solvent was treated as a dielectric continuum with the dielectric constant of the electrolyte solution using the Tomasi method.44 Since this model of dielectric friction (eq 7) is very sensitive to the value of the cavity radius about the solute, the cavity radius was varied until the measured rotational relaxation time in pure solvent matched the relaxation time computed with the model. This “best fit” radius was then used to calculate the dielectric friction of free resorufin in the electrolyte solutions (the dielectric parameters used in the calculations are given in the Supporting Information). For the case where the slip boundary condition is used to model the mechanical contribution to the friction, the value of the cavity radius is estimated to be 6.92 Å for resorufin in DMSO. This result is consistent with previous studies of this group.4,5 The ion-paired species was treated in a similar manner. The cavity radius for the ion pair in DMSO was chosen to be 9.0 Å because it is consistent with the optimized geometries using Macromodel and because it provides a good fit to the data. Because the dielectric friction constitutes less than 20% of the total friction for the ion pair, it is difficult to accurately estimate the radius. However, the choice of 8 or 10 Å for the radius results in noticeably worse fits to the data. Because it neglects the coordination of DMSO molecules to the cation, this modeling provides an upper bound on the magnitude of the dielectric friction for the ion-paired species. Nevertheless, it is much smaller than the hydrodynamic part of the friction. The ion atmosphere friction was calculated using eq 9. In this model the diffusion constant of ions in electrolyte solution was determined from the Nernst-Einstein relation33 assuming D ) D+ ) D- so that

Λ0M )

2F2νz2D RT

(16)

where Λ0M is the limiting molar conductivity, F is the Faraday constant, ν is the stoichiometric coefficient, and z is the magnitude of the charge. The model used to obtain the ion atmosphere friction (eq 13) treats the charge distribution of the solute as a point dipole. Since a rigorous treatment of the ion atmosphere friction with an extended charge distribution is not available, it was accounted for in an approximate manner. For high dielectric solvents the point dipole model and the extended charge distribution model for the dielectric friction have the same solvent dependence 5 but differ by a numerical factor related to the different description of the charge distributions. The approximation used here was to assume a similar form for the ion atmosphere friction; i.e., the dependence on solvent parameters is the same. The extended charge distribution was accounted for by assuming that the ion atmosphere friction from the extended charge distribution increased over that found for a point source model by the same numerical factors found for the dielectric friction coefficients (see ref 5 for more details). Discussion The experiments show that the rotational relaxation time of the solute increases as the concentration of the electrolyte increases, for all the salts studied. In addition, the strength of the concentration dependence is affected by the charge-to-size ratio of the cation and by the solvent. Qualitative Findings. A correlation exists between the charge density of the cation and the increase of the relaxation

time of resorufin in DMSO. For K+, which has the smallest charge-to-size ratio, the increase in relaxation time is smallest, and for Mg2+ with the largest charge-to-size ratio the effect is most dramatic. At a particular salt concentration, the trend is a slower rotational relaxation time for a larger charge-to-size ratio of the cation. At lower concentrations of salt the increase in relaxation time is too large to arise purely from the viscosity of the solution (see Figure 6). By contrast, for high concentrations of Mg2+ τ/η drops dramatically. This observation is consistent with conductivity measurements of Mg(NO3)2 in DMSO solution. The conductivity shows a maximum at the same range of concentration of about 0.6 M. With further increase in concentration the conductivity decreases again, and this decrease has been attributed to the considerable increase of viscosity in these concentrated salt solutions.40 This effect is left for a later study. In water the relaxation time for resorufin follows the viscosity quite closely for both the sodium salt and lithium salt solutions (see Figure 5). This observation does not hold true for the magnesium solution. The normalized relaxation time increases significantly for the range of concentrations studied. The sensitivity of the rotational relaxation time of the solute to the charge-to-size ratio of ions indicates that structural features of the solution should be included in any realistic modeling. Quantitative Findings. A more quantitative analysis was performed for the case of resorufin in Na+ and Li+ salt solution. A quantitative treatment of the Mg2+ has not yet been performed because of difficulty in preparing the Mg(resorufin)2 salt for conductivity studies. To analyze the experimental data more clearly, two models were considered. The first model assumes that only free resorufin is present in the solution, and the second model assumes that two species are present in the solution, free and ion-paired resorufin. In the first case the hydrodynamic, dielectric, and ion atmosphere components of the friction were calculated for free resorufin in DMSO solution with Na+ and Li+ ions. Although this separation of friction is arbitrary, it approximates the experimental values quite well in many cases. Incorporating all the friction types improves the correspondence with the experimental results, but it still underestimates the value of rotational relaxation times by more than 1.5 times at high concentrations (see Figure 8A). Figure 8A shows the concentration dependence of the experimental rotational relaxation times. The broken curves 1-3 show the magnitude of the relaxation time expected for each of the three friction types. The solid curve 4 is the relaxation time for the total friction coefficient. The cavity radius in the dielectric friction model has been adjusted to match the observed relaxation time in the pure solvent. Nevertheless, the modeling fails to describe the concentration dependence. In the second case, both free and ion-paired resorufin are included. The evidence for ion pair formation in DMSO is obtained from the conductivity data, which gives an association constant for the sodium resorufin of 13 M-1 and an association constant for the lithium resorufin of 23 M-1.5 The ion-paired species is considered to rotate as a rigid unit. This model results in a significant increase of the mechanical contribution to the rotation time, especially at higher concentrations of electrolyte. The mechanical friction dominates over the dielectric friction and the ion atmosphere friction for the ion pairs in the solution. By contrast, the mechanical friction is comparable with dielectric friction in the case of free resorufin. The second method gives very good agreement with the experimental data (see Figures 8B and 9). These plots are similar to Figure 8A except that each curve of relaxation times corresponds to the properly weighted sum of relaxation times for the free solute and the

Frictional Coupling

Figure 8. Calculated and experimental relaxation times in NaNO3/ DMSO solution: (A) assumes only free resorufin is present in the solution; (B) assumes both free and ion-paired resorufin are present in the solution. In each case 1 (ion atmosphere), 2 (dielectric), and 3 (hydrodynamic) are the different friction contributions, and 4 is the total friction. The 9 are the experimental values.

Figure 9. Calculated and experimental relaxation times in LiNO3/ DMSO solution: (1) calculated relaxation times, assuming only free resorufin is present in the solution; (2) calculated relaxation times assuming both free and ion-paired resorufin are present in the solution; (9) experimental values. Data are taken from ref 5.

ion-paired solute. Inclusion of the ion-paired species significantly improves the agreement with the experimental values. In both LiNO3 and NaNO3 electrolyte solutions of resorufin, the inclusion of ion pairs, which rotate as a unit in the analysis, allows the average relaxation time to be modeled very well. The model used here is certainly an oversimplification. The actual solution will have a distribution of ion pair geometries and coordination numbers. Nevertheless, the success of the simple model in reproducing the rotational diffusion data indicates that the ion-paired species may play a significant role in the molecular dynamics. In the context of this model, these data indicate that the ion pair exists for time scales that are long compared to the rotational relaxation time. General Comments. The comparison of the data for LiNO3 and NaNO3 salts in DMSO indicates that the relaxation times depend strongly on the charge-to-size ratio of the cation. The increase of the relaxation time with the cation of larger chargeto-size ratio can be explained by the increase in the ion-solute interaction. This trend correlates with the trend in the conduc-

J. Phys. Chem. B, Vol. 101, No. 13, 1997 2345 tivity data. Kass for Li-resorufin is almost twice as big as Kass for Na-resorufin (see Table 2). Hence, the concentration of ion-paired species is higher for Li+ than for Na+ at the same electrolyte concentration (see Table 3). Thus, the transition from ion pair rotation to free molecule rotation occurs at lower salt concentrations in the case of cations with larger charge-to-size ratio. The correlation between the charge density of the cation and the increase in the reorientation time supports the hypothesis that solute-ion association takes place in electrolyte solutions of DMSO. By contrast, the data in water show that the relaxation time of resorufin in sodium nitrate and lithium nitrate solutions correlates well with viscosity. Two different explanations can be offered for the data.5 First, the relative lifetimes of ion pair species in water are comparable to the rotational relaxation time of the free solute in water. By use of the measured Kass and the Smoluchowski model for the formation rate of the ion pair, the lifetime of the ion-paired species was estimated to be 0.7 ns in a NaNO3 solution of DMSO and 1.87 ns in a LiNO3 solution of DMSO.5 By contrast the lifetime of the ion pair in water was estimated to be 172 ps for the LiNO3 solution. These values indicate that the ion pairs are bound on the rotational time scale in DMSO solution but not in water where the lifetime of the ion pair is much smaller. These results support the hypothesis that resorufin in electrolyte solutions of DMSO forms ion pairs that undergo rotational motion. In water the ion pair is more likely to dissociate, and the resorufin may rotate as a free molecule in the solution. The second explanation considers resorufin in pure water as a hydrated molecule whose size is close to the size of the ion-paired resorufin. This idea is supported by calculations of the hydrodynamic parameters for the ion-paired (ip) and hydrated (H2O) resorufin in aqueous solution of NaNO3. One finds that (Vσ)ip ) (Vσ)H2O ) 0.93, which is in good agreement with calculations for LiNO3 aqueous solution, as reported by Hartman and Waldeck.5 Hence, the mechanical friction values for the ion pair and hydrated resorufin are comparable and are not easily distinguished experimentally. Thus, modeling of resorufin’s rotation time in water is troublesome and the existence of the ion pairs is not as clear in this case as it is with DMSO. On a qualitative level, however, it appears that ion pair formation is important in aqueous solutions of Mg(NO3)2 because τ/η depends on the electrolyte concentration. Friedman45 has introduced a “corresponding states diagram” to classify the states of an electrolyte solution, where the states of the solution relate to the degree of ion pairing. This diagram is qualitatively in accord with the observations reported here. His F-T diagram uses a set of reduced variables. The reduced temperature Tr is

Tr )

R 2b

(17)

where R is the distance of closest approach of the centers of ions. This definition compares the thermal kinetic energy in the medium to the potential energy of attraction between the two ions when they are at contact. Hence, when Tr is on the order 1, the available kinetic energy is high enough that ions in proximity would be able to easily dissociate. As Tr decreases in size, the ions would remain bound for longer times. He also defines a reduced density Fr

Fr )

(π/6)(N+ + N-)R3 V

(18)

where N+ is the number of cations, N- is the number of anions, and V is the solution volume. This density provides a measure

2346 J. Phys. Chem. B, Vol. 101, No. 13, 1997

Balabai and Waldeck

TABLE 4: Parameters for the Electrolyte Solutions water R+ , Å b, Å Fra R- ) 2.6 Å R- ) 3.5 Å R- ) 6.5 Å Tr R- ) 2.6 Å R- ) 3.5 Å R- ) 6.5 Å a

DMSO

Na+

Li+

Mg2+

Na+

Li+

Mg2+

1.02 3.58

0.74 3.58

0.72 7.20

1.02 5.91

0.74 5.91

0.72 11.85

0.030 0.058 0.267

0.024 0.048 0.238

0.022 0.047 0.236

0.030 0.058 0.267

0.024 0.048 0.238

0.022 0.047 0.236

0.50 0.63 1.04

0.46 0.59 1.00

0.23 0.29 0.50

0.31 0.38 0.64

0.28 0.36 0.61

0.14 0.18 0.30

The Fr are evaluated at 1 M concentration.

of the average spacing between ions. When Fr is on the order 1, the ions are in direct contact. This diagram allows one to predict the state of a particular electrolyte solution by knowing its characteristic parameters. It has been shown that the DebyeHu¨ckel treatment is an accurate approximation for electrolyte solutions with Fr e 10-4 (which corresponds to concentrations below 0.001 M) and Tr g 0.5. In addition, ion pairing should be important for electrolyte solutions with Tr e 0.15 and solvents with small dielectric constants. The reduced temperature and density of the electrolyte solutions used in this study were computed. The Bjerrum length, reduced temperature, and reduced density for the NaNO3, LiNO3, and Mg(NO3)2 in water and DMSO are given in Table 4. The quantity R is the distance of closest approach of the ions and does not depend on the concentration. In the table three different radii are used for the anion R- in order to illustrate its effect on the reduced temperature. The largest value of R- is consistent with the long axis radius of resorufin. The middle value is consistent with the short, in-plane axis of resorufin, and the smallest value of R- is consistent with the radius of NO3-. By the Bjerrum definition ion pairs do not exist in solution for Tr g 0.5 because b is less than R, where R ) R+ + R-. The values of Tr in aqueous solution show a systematic decrease with increasing charge density of the ion. The same trend is evident in DMSO solutions. However, the overall values of Tr are lower in DMSO than in water. The parameter Tr predicts that ion pair formation in DMSO will be more important than in water. From the values in the table one might also conclude that the formation of ion pairs by Mg2+ in water would be more likely than for Na+ and Li+ in DMSO. However, the state diagram given is appropriate to symmetrical electrolytes (1:1, 2:2, ...), not asymmetric (2:1, ...) electrolytes. On a qualitative level, the trend in Tr corresponds to the observations on the rotational diffusion of resorufin in the electrolyte solutions studied. Despite the success of the modeling represented by Figures 8B and 9, it must be too simple to provide a realistic description. The existence of two well-defined species with such different relaxation times should result in an observed orientational relaxation that is not a single exponential. Rather, the observed response should reflect the inhomogeneous distribution of relaxation times. The reason for the failure to observe this nonexponentiality is likely to have experimental and physical origins. Experimentally, the limited signal-to-noise ratio of the data37 restricts the ability to resolve different components of the distribution. Physically, the distribution of ion pair species is likely to be very broad and their relaxation times should encompass the range of time scales defined by the free molecule and the ion-paired species reported here. The richness of the distribution and times could very well lead to an averaging of the response.

Figure 10. Calculated and experimental relaxation times of NO3- in aqueous NaNO3 solution: (1) calculated relaxation times, assuming only free anion is present in the solution; (2) calculated relaxation times assuming both free and ion-paired species are present in the solution; (9) experimental values. Data are taken from ref 46.

Connection with Other Studies. R. E.Wasylishen and coworkers measured the rotational correlation time of the nitrate anion in aqueous NaNO3 electrolyte solutions of 0-4 M of NaNO3.46 Their results indicate that the motion of NO3- is significantly slower than that expected by the changing viscosity of the solution. They suggested that Na+ and NO3- associate strongly in aqueous solution in this range of concentrations; i.e., ion pairs form. The theoretical models described above were used to analyze their data. Two cases were considered. First is that only free anions are present in the solution, and second is that both free and ion-paired species exist. The ion pair is supposed to consist of the hydrated cation coordinated to one of the oxygens. The mechanical friction was calculated assuming that the free anion and the ion pairs can be approximated as ellipsoids with axial radii 2.6:2.6:2.0 Å and a volume of 54 Å3, and axial radii 4.7:3.0:2.0 Å and a volume of 84 Å3, respectively. The charge distribution for the free and ion-paired anion in water were obtained in a manner similar to that described for resorufin. This charge distribution was used to calculate the dielectric friction. The best fit cavity radius was 3.6 and 5.0 Å for the free anion and ion pair, respectively. Comparison of the calculated values of relaxation times with experimental values (see Figure 10) shows that accounting for ion pairing improves the correlation between the experimental and calculated value significantly. At higher concentrations (2-4 M) however, the values of the rotation time are larger than can be predicted by this model. These differences may be related to the formation of associations of higher order (i.e., multimers) at such high concentrations. Conclusions The rotational diffusion times of resorufin in DMSO and aqueous electrolyte solutions have been measured. By use of different salts, the charge-to-radius ratio of the cation was varied systematically. These measurements and their analysis leads to the following conclusions. Although quantitative differences exist, the cations change the rotational relaxation times in a similar way in all electrolyte solutions of DMSO. Increasing the electrolyte concentration causes an increase in the rotation times, which cannot be explained by the viscosity of the solution. Analysis of the experimental data strongly indicates the presence of ion pairs in DMSO solutions of NaNO3. These conclusions are consistent with previous data for resorufin in DMSO solutions of LiNO3. At low electrolyte concentrations the rotation times increase in a systematic way with respect to the charge-to-size ratio of the cation. For a given concentration of a particular salt, the bigger charge-to-radius ratio of the cation results in stronger

Frictional Coupling solute-ion interactions and, hence, larger relaxation times. These observations show the importance of structural aspects of the solution and support the suggestion that the dramatic increase of the rotational time in electrolyte solutions of DMSO arises from ion pair formation. The success of the model that incorporates the different contributions to the friction in the case of different electrolyte solution of DMSO demonstrates that this model is able to provide a qualitative and semiquantitative picture of solute-solvent interactions in a particular kind of solvent. Acknowledgment. We thank Zhe Lin for help with drying the Mg(NO3)2‚6H2O salt, Maria Kurnikova for help with calculating the charge distribution, and Ian Read for assistance in work with TCSPC. This work was supported by a grant from the National Science Foundation (CHE-9416913). Supporting Information Available: Tables listing solution parameters used in the data analysis, a description of charge distributions for free resorufin and the iron pairs, and a description of the procedure used to compute the mole fraction of ion pairs from the conductivity data (4 pages). Ordering information is given on any current masthead page. References and Notes (1) (a) Friedman, H. L. Annu. ReV. Phys. Chem. 1981, 32, 179. (b) Wang, J.; Haymet, A. D. J. J. Chem. Phys. 1994, 100, 3767, and references therein. (c) Kremp, D.; Beskrownij, W. J. Chem. Phys. 1996, 104, 2010. (d) Justice, J.-C. J. Solution Chem. 1991, 20, 1017. (e) Justice, J.-C. J. Phys. Chem. 1996, 100, 1246. (f) Zhai, Y.; Stell, G. J. Chem. Phys. 1995, 102, 8089. (2) (a) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworth: London, 1959. (b) Harned, H. S.; Owen, B. B. The Physical Chemistry of Electrolytic Solutions; Reynhold Publishing Corporation: New York, 1950. (3) Davies, C. W. Ion Association; Butterworth: Washington, 1962. (4) (a) Alavi, D. S.; Hartman, R. S.; Waldeck, D. H. Ultrafast Phenomena; Springer: Berlin, 1990. (b) Alavi, D. S.; Hartman, R. S.; Waldeck, D. H. J. Phys. Chem. 1991, 95, 7872. (5) Hartman, R. S.; Konitsky, W. M.; Waldeck, D. H. J. Am. Chem. Soc. 1993, 115, 9692. (6) Dote, J. L.; Kivelson, D.; Schwartz, R. N. J. Phys. Chem. 1981, 85, 2169. (7) Ravi, R.; Ben-Amotz, D. Chem. Phys. 1994, 183, 385. (8) Ben-Amotz, D.; Drake, J. M. J. Chem. Phys. 1988, 89, 1019. (9) Ben-Amotz, D.; Scott, T. W. J. Chem. Phys. 1987, 87, 3739. (10) Jiang, Y.; Blanchard, G. J. J. Phys. Chem. 1994, 98, 6436. (11) Simon, J. D.; Thomson, P. A. J. Chem. Phys. 1990, 92, 2891. (12) Philips, L. A.; Webb, S. P.; Clark, J. H. J. Chem. Phys. 1985, 83, 5810.

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