J . Phys. Chem. 1985,89, 5759-5766
5759
An Analysis of Dielectric Models of Solvatochromism James E. Bradyt and Peter W. Carr*$ Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, and Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received: May 8, 1985; In Final Form: August 22, 1985)
The manner in which orientational and distortional polarization contributions to solvatochromic effects are isolated within the McRae-Bayliss model of solvatochromismis shown to be untenable. An alternate separation scheme, christened modified McRae-Bayliss, is developed and predicts that distortional contributionsto solvatochromic effects diminish in absolute magnitude as the dielectric constant of the solvent increases, an effect which is suported by experimental observation. The correlating ability of McRae-Bayliss, modified McRae-Bayliss, and alternate formulations discussed in the text is tested with both nonpolar and general solvent groups. In the nonpolar solvent limit, the various models are equivalent and suffer a common fate in which systematic class deviations (e.g., perfluorinated or alkane solvent groups) are clearly evident. It is clear that no model which relies solely on bulk dielectric properties will be able to correlate solvatochromic behavior in even nonpolar solvents without systematic lack of fit by solvent class. General solvent correlations are a more difficult issue. Development of a detailed correlation model incorporating suppression of distortional contributions to solvatochromism as the static permittivity increases would lead to the introductionof additional, highly correlated, parameters; this course was deemed untenable. However, correlations based on the two major model parameters are examined. Despite major differences in physical make-up and initial presumptions, it is virtually impossible to distinguish the correlating ability of the McRaeBayliss, modified McRae-Bayliss, or alternate models. As with the nonpolar solvents, lack of fit patterning with respect to solvent class is discernible as is global patterning which appears related to the distribution of the input data.
Introduction In a comprehensive series of studies on solvatochromic effects Kamlet, Taft, and co-workers presented the development, rationalization, and applications of the T*,a,and p scales of solvent dipolarity-polarizability, hydrogen-bond acidity, and hydrogenbond basicity.’-9 Two of the more significant aspects of their work are the use of multiple solvatochromic probe solutes to identify and avoid solute specific anomalies and the concerted effort to experimentally separate the general dielectric contributions to solvation from hydrogen-bonding components. The latter point is particularly important to solvatochromic modeling studies based on continuum or molecular dielectric theory (e.g., reaction field1*21 or discrete dipolez2 models). Preliminary physical modeling of the r* scale by Abboud and Taft14 focused on a group of solvents termed “select” and a dielectrically saturated reaction field model developed earlier by Block and Walker.23 Restriction of the correlation analysis to the select solvent set suppresses differential contributions due to solute dipole-solvent-induced dipole interactions. This is immediately apparent from the very narrow range in refractive index spanned by the select solvent set which excludes aromatic and polyhalogenated liquids. In effect, select solvent correlations most directly test models of only the dipole-dipole component of solvation, and thus do not encompass all aspects of dielectric solvation to which the r* scale is sensitive. The original model of r* forwarded by Abboud and Taft is the basis for later analyses by other workers. An early generalized model for a*,that is not restricted to select solvents, presented by us13 performs quite well in simply fitting the data available. However, we consider the ad hoc nature of the model to be a major deficiency. A correlation model loosely based on Onsager’s reaction field and Buckingham’s analysis of solvent effects on infrared spectra26 presented by BekarekI6 also appeared to perform well in fitting available solvatochromic data. However, Bekarek’s result is difficult to reconcile with solvatochromic data obtained in perfluorinated solvent^"^^' or in the gas phase.28 Similarly, our generalized model appears to be at variance with gas-phase spectral data. An earlier multiparameter model which we used” and later rejected on the basis of systematic structure in the residual plot seems to accommodate the currently available gas-phase data (see recent analysis by Abboud et aL9). Despite ‘University of Pittsburgh. :University of Minnesota.
early success in analyzing select solvent behavior with the Block and Walker reaction field model, Kamlet, Taft, and co-workers have more recently advocated the used of more intuitively based “reducing functions”29in pursuing general solvent correlations.
(1) Kamlet, M. J.; Abboud, J. L.; Taft, R. W. Prog. Phys. Org. Chem. 1980, 13,485-630.
(2) Kamlet, M. J.; Abboud, J. L.; Abraham, M. H.; Taft, R. W. J . Org. Chem. 1984,48, 2877-87. (3) Fong, C. W.; Kamlet, M. J.; Taft, R. W. J . Org. Chem. 1983, 48, 822-5. (4) Kamlet, M. J.; Doherty, R. M.; Taft, R. W.; Abraham, M. H. J . Am. Chem. SOC.1983, 105, 6741-3. ( 5 ) Abraham, M. H.; Kamlet, M. J.; Taft, R. W.; Weathersby, P. K. J . Am. Chem. SOC.1983, 105, 6797-801. (6) Kamlet, M. J.; Abraham, M. H.; Doherty, R. M.; Taft, R. W. J . Am. Chem. SOC.1984, 106, 464-6. (7) Kamlet, M. J.; Doherty, R. M., Taft, R. W.; Abraham, M. H.; Koros, W. J. J . A m . Chem. Soc. 1984, 106, 1205-12. (8) Taft, R. W.; Abboud, J. L.; Kamlet, M. J. J . Org. Chem. 1984, 49, 200 1-5. (9) Abboud, J. L.; Guiheneuf, G.; Essfar, M.; Taft, R. W.; Kamlet, M. J. J . Phys. Chem. 1984, 88, 4414-20. (10) Carr, P. W. J . Chromatgr. 1980, 194, 105-19. (11) Brady, J. E.;Carr, P. W. Anal. Chem. 1982, 54, 1751-7. (12) Brady, J. E.; Bjorkman, D.; Herter, C. D.; Carr, P. W. Anal. Chem. 1984, 56, 278-83. (13) Brady, J. E.; Carr, P. W. J . Phys. Chem. 1982, 86, 3053-7. (14) Abboud, J. L.; Taft, R. W. J . Phys. Chem. 1979,83, 412-9. (15) Bekarek, V. Collect. Czech. Chem. Commun. 1980, 45, 2063-9. (16) Bekarek, V. J . Phys. Chem. 1981, 85, 722-3. (17) Bekarek, V.; Jurina, J. Collect. Czech. Chem. Commun. 1982, 47, 1060-8. (18) Kolling, 0. W. Trans. Kansas Acad. Sei. 1981, 84, 32-8. (19) Ehrenson, S. J . Comput. Chem. 1981, 2, 41-52. (20) Ehrenson, S . J . Am. Chem. SOC.1981, 103, 6036-43. (21) Abe, T.; Tero-Kubota, S.; Ikegami, Y . J . Phys. Chem. 1982, 86, 1358-65. (22) Reddoch, A. H.; Konishi, S. J . Chem. Phys. 1979, 70, 2121-30. (23) Block, H.; Walker, S. M. Chem. Phys. Left. 1973, 19, 363-4. (24) Onsager, L. J . A m . Chem. SOC.1936, 58, 1486-93. (25) Kirkwood, J. G. J . Chem. Phys. 1934, 2, 351-61; 1939, 7, 911-9. (26) Buckingham, A. D. Proc. R . SOC.London A 1958, 248, 169-82. (27) Brady, J. E. Ph.D. Thesis, University of Minnesota, 1984. (28) Essfar, M.; Guiheneuf, G.; Abboud, J. L. J . A m . Chem. SOC.1982, 104, 6786-87.
0022-3654/85/2089-5759$01.50/00 1985 American Chemical Society
5760 The Journal of Physical Chemistry, Vol. 89, No. 26, 1985
This approach appeared statistically equivalent to our reaction field model.13 Subsequent solvatochromic examination of the n-alkanenitrile s e r i e ~ ~demonstrates ’.~~ that the reducing function approach, as currently implemented, leads to incorrect trends within the n-alkanenitrile series. This could have been anticipated on the basis of the discrete dipole-dipole interaction model proposed and discussed by Reddoch and Konishi.22 The necessary modification to the Kamlet-Taft-Abboud reducing function is clear from their discussion. We emphasize that the failure of the Kamlet-Taft-Abboud reducing function only becomes apparent when well-controlled solvent groups, e.g., homologous series, are examined. Finally, Ehrenson has presented a dielectrically saturated reaction field model which sacrifices the analytic solution of Block and Walker’s approach.lg The necessary numerical evaluation is described in the original paper. Ehrenson subsequently analyzed the correlation ability of this and other reaction field models employing select solvent dataZoand concluded that the dipole-dipole model due to Reddoch and Konishi was superior to the continuum based reaction field models. This result is almost certainly tied to incomplete removal of differential dipole-induced dipole effects from the continuum reaction field models tested. The Reddoch-Konishi model depends in no way on solvent refractive index while reaction field models which rely on bulk permittivity implicitly depend in some manner on solvent refractive index. Naturally, the relative impact of this contribution increases as the dipolarity of the liquid decreases. The modeling studies mentioned above possess major theoretical and/or conceptual difficulties aside from inherent problems associated with viewing the solvent as an isotropic continuum. Our original model is an ad hoc quilt that cannot be traced to a unified parent theory of equilibrium solvation. The model advocated by Bekarek and the reducing function approach of Kamlet et al. suffer a similar liability. Critical comment on Ehrenson’s model on this point is not in order since no attempt has been made to formally extend the model to the nonequilibrium solvation relevant to the solvatochromic experiment even though it has been applied in analyzing this type of data. However, one can easily discern a major difficulty of the type of intuitive approach, aside from obvious problems of extending the model to applications beyond pure curve fitting, which we and others have pursued. This is the lack of a strong model test. Correlation analysis does not inherently provide strong indications of model validity. The sole approach to verification is through a priori calculation of the experimental observable and comparison with experiment. As we will point out below, this test can be relaxed somewhat if combined with correlation analysis. The strategy employed in this work relies on the establishment of bounds on the ratios of coefficients of the correlation equation which eliminate unknown variablesz7 Adherence of the correlation parameters to the required bounds can then be tested. The development of a realistic model of solvatochromic effects rests on achieving two goals: (1) construction of a good equilibrium model of dipole-dipole and dipole-induced dipole interactions; (2) extraction of the two components from the parent theory and extension to nonequilibrium solvation. Models based on Block and Walker’s approach are physically more plausible than are models based on Onsager’s theory. Yet the only model of solvatochromic effects coherently developed to date is the Onsager-based model presented some time ago by McRae and B a y l i ~ s . ~ Reasonable ~,~~ success at correlating solvatochromic effects with solvent parameters emerging from the McRae-Bayliss model explains its continued use to the present day (see e.g., ref 33-35). Further, despite the physically more (29) Taft, R. W.; Abboud, J. L.; Kamlet, M. J. J . Am. Chem. S o t . 1981, 103, 1080-6.
(30) Brady, J. E.; Carr, P. W. J . Phys. Chem. 1985,89, 1813-22. (31) Bayliss, N. S.; McRae, E. G. J . Phys. Chem. 1954, 58, 1002-6. (32) McRae, E. G . J . Phys. Chem. 1957, 61, 562-72. (33) Hilliard, L. J.; Foulk, D. S.; Gold, H.. S.; Rechsteiner, C. E. Anal. Chim. Acta 1981, 133, 319-27. (34) Salem, L. “Electrons in Chemical Reactions: First Principles”; Wiley-Interscience: New York, 1982.
Brady and Carr appealing assumptions of the Block and Walker reaction field model, the Onsager model remains a useful benchmark of idealized dielectric behavior.36 In this regard the Onsager model remains a legitimate point of analysis. In this work we show the dipole-dipole-induced separation scheme utilized by McRae-Bayliss in developing their model of solvatochromism is in direct conflict with major implicit assumptions of the parent Onsager theory as well as general mathematical principles. An alternate separation scheme which successfully addresses these difficulties is presented. One prediction of this modified McRae-Bayliss model is the suppression of dipole-induced dipole contributions to solvation as the bulk dielectric constant increases. This prediction is in direct conflict with the original McRae-Bayliss formulation and necessarily entails the inductive component of solvation following different functional forms in the ground and excited states. The prediction is qualitatively tested with available experimental data. Complications in the development and interpretation of correlation analyses arise if dipole-induced dipole effects follow different functional dependences on dielectric constant and refractive index in the ground- and excited-state situations. This will become clear as our analysis proceeds. The dipolar-inductive separation scheme leading to the modified McRae-Bayliss model is extended to a modification of the Block and Walker model which we recently pre~ented.~’ The developments outlined yield two models of solvatochromism possessing reasonable theoretical coherence and a definable relationship to equilibrium solvation models. In an effort to examine these and other proposed models of solvatochromic effects two particularly simple systems are analyzed initially: (1) the n-alkane series (pentane through hexadecane) and subsequently with other branched and cyclic hydrocarbons and nonpolar liquids; (2) a series of very weakly dipolar (pseudo-nonpolar) methyl/phenyl siloxane polymers. The McRae-Bayliss, modified McRae-Bayliss, and modified Block and Walker reaction field models are equivalent in the nonpolar solvent limit. The results of this part of the study are therefore of general validity. We then examine the practical issue of general solvent correlations. The models, as well as our conclusions, are on much looser ground here. This work has two major goals. The first is a presentation and correlation analysis of coherently developed reaction field models of solvatochromic effects. Our second goal addresses the value of correlation analysis in establishing the superior basis of coherently developed models over those founded on a more intuitive basis.
A Reexamination of the McRae-Bayliss Model The McRae-Bayliss model of sol~atochromism~~~~~~~~~~~ evolves directly from Onsager’s reaction field t h e ~ r y . Onsager’s ~ ~ , ~ ~ theory provides a means for calculating the energy of equilibrium dielectric solvation of an ideal nonpolarizable dipole immersed in a dielectric continuum. Due to the nonequilibrium solvation of the excited state, treatment of solvatochromic data requires the function describing the equilibrium solvation energy to be split into dipolar and inductive components. Upon excitation one assumes the inductive component of solvation is in equilibrium with the excited-state dipole while the dipolar component remains at the ground-state value. We will start with the result obtained by McRae and Bayliss. According to the McRae-Bayliss model, the transition energy of an absorption band of solute i in solvent j is
(35) Capomacchia, A. C.; White, F. L.; Sobol, T.L. Spectrochim. Acta, Part A 1982, 38A, 5 13-21. (36) Bishop, W. H. J . Phys. Chem. 1979, 83, 2338-44. (37) Brady, J. E.; Carr, P. W., submitted for publication in J . Phys. Chem. (38) Nicol, M. F. Appl. Spectrosc. Rev. 1974, 8, 183-227. (39) Reichardt, C. ‘Solvent Effects in Organic Reactions”;Verlag Chemie: Weinheim, West Germany, 1979. (40) Riddick, J. A.; Bunger, W. B. “Organic Solvents”, 3rd ed.; WileyInterscience: New York, 1970.
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5761
Analysis of Dielectric Models of Solvatochromism TABLE I: Effective“ Dipolar Permittivity of Polar Liquids Based on the McRae-Bayliss Separation Scheme solvent hexane benzene diethyl ether tetrahydrofuran pyridine benzonitrile nitrobenzene dimethyl sulfoxide dimethylacetamide butyrolactone dimethylformamide formamide acetone nitromethane acetonitrile
tb
ndb
Csffc
1.8799 2.275 4.335 7.58 12.4 25.2 34.82 46.68 37.78 39. 36.7 1 111.0 20.7 35.87 37.5
1.3723 1.4979 1.3495 1.4050 1.5074 1.5257 1.5500 1.4773 1.4356 1.4348 1.4282 1.4468 1.3560 1.3796 1.3416
1.0 1.0 1.759 2.089 2.108 2.341 2.354 2.674 2.808 2.821 2.836 2.948 2.994 3.096 3.373
The mathematical inadequacy of the McRae-Bayliss model can be examined in a more general fashion as follows. Consider any function F(z). Assume F ( z ) is known but we are interested in determining the contributions to F ( z ) due to component variables x and y , z = f ( x , y ) ,cf. equilibrium solvation decomposed into dipolar and inductive components. For simplicity assume z = x y . Clearly, unless F ( z ) = kz ( k = constant), F ( x ) F b ) # F ( z ) . In general, F ( z ) will possess interaction terms between x a n d y . Let
+
+
F(z) =
+ Cay1+ &,I
= &xl
&,zi
I
I
(7)
I
I
where I represents the interaction terms from the binomial expansion of ( x + y ) ! . Comparing eq 7 to the Onsager result and separation scheme employed by McRae and Bayliss leads one to conclude the McRae-Bayliss separation scheme holds component due to x = F ( x ) + &,I
(8)
I
The “effective” dipolar permittivity computed in accord with the McRae-Bayliss model as described in text. bLiterature sources of t and nd values are given in ref 27; see also ref 40. Equation 5 .
where Y is the solute transition energy in the gas phase or solvent i, a the solute cavity radius, p* the excited-state dipole moment, p the gound-state dipole moment, and 4 the angle between the ground- and excited-state dipoles. L(e) and L(n2) are reaction field functions defined by eq 2 with t and n the bulk static relative permittivity and refractive index, respectively. L ( x ) = 2(x - 1)/(2x + 1) (2) In applying correlation analysis [ L ( t )- L(n2)]and L(n2)are the relevant solvent variables. The remaining terms (p,p*, 4, a) are assumed to be solute-dependent constants independent of the solvent. This is not strictly true since any real solute dipole is polarizable and therefore affected by the reaction field. While a priori calculation of the coefficients of the correlation model from eq 1 requires the availability of generally unknown variables (p*,4, a; p assumed known), the ratio of the inductive to dipolar coefficient must be greater than 2 for bathochromic dipolar and inductive shifts. ~2
-p
2 ) / m [ ( p p *
COS
= (k2- l ) / ( k - 1)
k
+ 1;
k =p*/p
component due to y = ( y / z ) F ( z ) as appropriate. The problem remains to implement this in terms of physical observables relevant to reaction field theory. Recall the definition of relative permittivity t = €’/to = 1 x(u) (10)
+
to is the permittivity of free space and x ( w ) the frequency-dependent electric susceptibility. The electric susceptibility of the solvent medium is the causative physical factor in dielectric polarization and therefore generator of the reaction field. Assume x(w) is a linear sum of components due to dipole orientation (~~(0 and ) )electronic distortion (Xd(W)).
x(w) = xo(w) + x d ( w )
>1
- F2) (3)
This situation covers the experimental results employed below. Realistic values of the coefficient ratio should fall in the range of 4-10. We will focus on the dipolar reaction field component, [ L ( t ) - L ( n 2 ) ] .This component is formally equivalent to an effective dipolar permittivity, teff,implicitly defined below. = L(c) - L(n2)
(4)
[4tn2
+ 5 t - n2 + 1]/[4tn2 - 4t + 8n2 + 11
(5)
The remarkably weak dependence of cCff on t is immediately evident. In the high dielectric limit teff depends only on n2 (eq 6). teff(€ a) = (n2 + 1.25)/(n2 - 1) (6)
-
This is counterintuitive. Calculation of teff values for actual solvents, Table I, reveals this anomaly should be noticeable for physically realizable values of e and n. Further, the magnitude of typical teff values are unreasonably small. The McRae-Bayliss model is based on the Onsager reaction field theory which relies on an essentially macroscopic view of dielectric solvation. One would anticipate macroscopic expectations to hold. Thus, intuitively, dipolar effects should contol dielectric solvation in high dielectric liquids. The McRae-Bayliss separation scheme ultimately fails on the basis of this physically intuitive point.
(11)
The appropriate weighting factors are therefore x,(w)/x(w) and Xd(W)/X(u).
At high frequency dipole reorientation is unable to follow the applied electric field and xo(w
- m)
(12)
0
From eq 10, at low frequency
x(w) = t while at high frequency
(t
1
(13)
= n2)
xd(w) = n2 - 1
from which teff =
Fb)
In other words all interaction terms are subsumed to one component. In a numerical sense this is the root of the problems noted above. A simple analysis based on the series expansion example given above2’ suggests a separation scheme of component due to x = ( x / z ) F ( z ) (9)
4 - p2)/~31 > (Ir*2 - K 2 ) / ( W *
L(teff)
component due to y =
(14)
Combining eq 11-14,
x,(w)/x(w)
= (e - n 2 ) / ( t - 1)
(15)
Xd(u)/X(u)
= (n2 - I ) / ( € - 1)
(16)
Application of this weighting scheme to the Onsager reaction field model leads to the dipolar reaction field component as L,(t,n2) = 2(t
- n 2 ) / ( 2 t + 1)
(17)
while the inductive component is Ld(6,n2)
= 2(n2 - 1)/(2e
+ 1)
(18)
As is readily apparent, eq 17 and 18 are numerically consistent with the parent Onsager model (i.e., the sum of eq 17 and 18 yields Onsager’s equilibrium result). Further, since both equations depend on t and n, the two components of solvation are coupled. Equation 18 predicts a decrease in the absolute magnitude of the inductive component of solvation decreases as the bulk static
5762
Brady and Carr
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 A
TABLE 11: Inductive Contribution to Solvatochromic Effects with Increasing Solvent Dipolarity“ solvent € nd P* An*/Andb
Group A perfluorooctane perfluoroheptane pentane hexane benzene to I u en e
1.85 1.765 1.844 1.879 2.275 2.379
tetrahydrofuran dimethoxyethane ethyl formate diethyl malonate tributyl phosphate l,l,l-trichloroethane
7.58 7.2 7.16 7.87 7.96 7.53 7.39
1.271 1.258 1.355 1.372 1.498 1.494
-0.41 -0.39 -0.09 -0.08 0.59 0.54
4.40 (3.27)
Group B
butyl chloride
hexanenitrile phenylacetonitrile 2-butanone cyclohexanone acetophenone ethyl chloroacetate acetic anhydride
Group 19.71 18.7 18.51 18.3 17.39 20 20.7
1.405 1.378 1.360 1.414 1.442 1.436 1.400
0.58 0.53 0.61 0.64 0.65 0.49 0.39
C 1.395 1.521 1.376 1.451 1.534 1.423 1.390
0.67 0.99 0.67 0.76 0.90 0.70 0.76
2.02 (1.46)
Group D acetonitrile but yrolactone dimethylformamide dimethylacetamide nitromethane nitrobenzene
37.5 39 36.71 37.78 35.87 34.82
1.342 1.435 1.428 1.436 1.380 1.550
0.75 0.87 0.88 0.88 0.85 1.01
1.35
a Literature sources of t and nd values given in ref 27; see also ref 40. bValues given without parentheses correspond to the ratio of the net range of P* to nd for each solvent group. Values in parentheses use the difference in nd for the two solvents with the maximum and minimum
P*
B
3.42 (1.58)
values as And.
permittivity of the liquid increases for constant n. The McRaeBayliss model maintains this component is independent oft. This prediction can be tested by examining the relative spread of a normalized measure of dielectric solvation, e.g., T * , with refractive index (or L(n2))for liquids of similar static permittivity, see Table 11. In qualitative agreement with our prediction, A.**/An decreases as € increases. For ease of referral below we will term this effect “suppression of inductive solvation”. Although suppression of inductive solvation is evident in many studies,29distinctly different rationalizations have been forwarded. Such an effect is a posteriori now evident in our previous modeling study.I3 In that work we observed a three-variable model employing variables related to dipolarity, polarizability, and the product of these two factors improved the statistical fit of the correlation beyond that expected on the basis of increased model freedom. Further, the sign of the “cross term” coefficient indicated a decrease in the polarizability contribution to dielectric solvation for constant refractive index as the static permittivity increased (conversely, one could argue the dipolarity contribution for constant static permittivity decreased with increasing refractive index-this is implicit in eq 17; however, the effect predicted is smaller in magnitude). At that time we forwarded, as did Taft et the suggestion that the solute chromophore (dipole) was, on the average, sought out by the dipolar and presumably less polarizable terminus of the solvent molecules. As shown above this effect can emerge from a continuum model-a model which is not predicated on the existence of discrete solvent molecules. Our previous suggestion clearly does not survive a strict application of Ockham’s razor. An elementary physical explanation for suppression of inductive solvation is clear. The magnitude of inductive solvation depends on the dipole electric field integrated over a distance weighted volume element. Since the magnitude of the dipole field at any point in space will be inversely proportional to e, equilibrium
KUC i
Figure 1. Definition of terms for dipole and reaction field vectors used in eq 19-21.
inductive solvation must drop with increasing t . Suppression of inductive solvation complicates extension of the reaction field model to situations of nonequilibrium solvation. Formally is is possible to break the excited-state dipole into ground-state and incremental vector components, see Figure 1A. We will assume the incremental vector component senses a solvent medium of permittivity n2. The excited-state reaction field is then the vectoral sum of component reaction fields due to the ground and incremental dipoles, Figure 1B. Simple geometric relations allow calculation of the magnitude and direction of the excitedstate reaction field (see Figure 1 for definition of symbols appearing in eq 19-21).
+
E = (Y2a3){p2[2(t- n2)/(2t + 1)12 pa2[2(n2- I ) / (2n2 1)2 - 2(cos 6)pp*[2(€ - n2)/(2t + 1)][2(n2 - 1)/ (2n2 1)]]’/2 (19)
+
+
0 = arctan
([pa sin
1)/(2
6 [2(n2 - 1)/(2n2 + l)]]/[p[2(n2 COS 6 [ 2 ( n 2- 1)/(2n2 + l)]]] (20)
+ I)] -
Note the direction of the excited-state dipole is
4 = arctan
(pa sin O / [ p - pn
cos 01 1
(21)
and therefore the inductive component of the excited-state reaction field is no longer collinear with the excited-state dipole. For the purposes of correlation analysis eq 19-21 make the following important points: 1. Although a more comprehensive model could be developed, the additional variables would generally be highly correlated and therefore render the result susceptible to statistical instability and variance inflation. Further, the increase in model freedom would diminish our confidence in the physical meaning of the result. 2. The terms dependent on p2[2(n2- 1)/(2n2 1)12 in eq 19 will be significant for compounds generally employed in solvatochromic studies. Thus, the inductive component of the excited-state reaction field will still appear to follow (or approximately follow) a 2(n2 - 1)/(2n2 + 1) proportionality. 3. A rigorous lower bound on the inductive to dipolar coefficient ratio, eq 3, is no longer calculable. In general, a value lower than the range of 4-10 anticipated on the basis of eq 3 is expected. The last point is the most significant for users of correlation analysis since a potentially powerful tool useful in discriminating between competing models is no longer available.
+
Extension to Dielectrically Saturated Models
The details of the preceding analysis apply only to the Onsager reaction field model. This is due to the clear identities relating the local and macroscopic dielectric variables. Extension to a
Analysis of Dielectric Models of Solvatochromism
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5763
TABLE 111: Relative Contributions of Orientational and Distortional Polarization to Dielectric Solvation with the Modified Block and Walker Model" %
solvent c hexane 1.88 diethyl 4.335 ether pyridine 12.4 acetone 20.7 phenylacet- 18.7 onitrile benzonitrile 25.2 acetonitrile 37.5 nitro35.87 methane nitro34.82 benzene propylene 69 carbonate formamide 111
z
nd
% dipolar inductive
a*
1.3723 1.3495
1.88 2.485
0.0 44.7
100 55.3
-0.08
1.5074 1.3560 1.5209
4.356 4.917 5.290
62.1 78.6 69.4
37.9 21.4 30.6
0.87 0.72 0.99
1.5257 1.3416 1.3796
6.108 6.558 6.583
74.0 85.6 83.8
26.0 14.4 16.2
0.90 0.75 0.85
1.5500
7.272
77.6
22.4
1.01
1.4212
9.600
88.1
11.9
0.81
90.7
9.3
0.98
1.4468 12.76
0.27
'Literature sources of c and nd are listed in ref 27; see also ref 40. dielectrically saturated reaction field model clearly requires determinationa of a bcal permittivity as well as a local balance of dipolar and inductive contributions. One approach to this problem is to define an average cybotactic weighted integral of the permittivity-distance permittivity as a rrn profile.
TABLE I V Nonpolar Solvent Systems: Correlation Summary (a* against (n2 - 1)/(2n2+ + 1))
solvent set alkanes' alkanesb alkanesC n-alkanes nonpolard nonpolarc nonpolar' OV seriesg OV seriesh
ni
r'
slope
std dev slope
intercept'
std dev interceDt
16 15 14 12 28 24 21 7 5
0.942 0.982 0.995 0.998 0.949 0.935 0.971 0.987 0.999
5.46 5.71 5.53 5.75 10.65 13.52 8.17 10.51 12.32
0.52 0.30 0.16 0.12 0.69 1.09 0.46 0.77 0.33
-1.06 -1.10 -1.06 -1.11 -2.03 -2.63 -1.58 -1.96 -2.34
0.10 0.06 0.03 0.02 0.14 0.23 0.09 0.17 0.07
On-Alkanes (C5-C16), cycloalkanes (C5, C6, C8), and isooctane. *n-Alkanes(C5-C16), cycloalkanes (C5, C8), isooctane. cn-Alkanes (C5-C16), cycloalkanes (C5, C8). dn-Alkanes(C5-C16), cycloalkanes (C5, C6, C8), isooctane, nonpolar aromatics (benzene, toluene, M - and p-xylene, cumene, mesitylene), perfluorinated solvents (FC-43, - 104, -84, -48), carbon tetrachloride. Solvent group (d) with perfluorinated solvents excluded. 'Solvent group (d) with nonpolar aromatics excluded. gOV series methyl/phenylsiloxane polymers (OV-101, -3, -7, -11, -17, -22, -25). hOV series with OV-22 and OV-25 excluded. ' n and r correspond to the number of solvents in the correlation and correlation coefficient, respectively. !Experimental value is -1.06 (ref 27).
where a is the solute cavity radius. The choice of appropriate values for m will generally depend on the distance dependence of the interaction probed. The general solution of eq 22 for the modified Block and Walker permittivity-distance profile previously developed, t(r) = t exp(-(a In ( c / n 2 ) ) / r )is, 2
z = ( m - l)!t/[ln (t/n2)]rn-111- n 2 / t ( C [ l n (t/n2)]'/r!) r.=n
The solute dipole field falls as r-3, and therefore m = 3 is an appropriate distance weighting factor. Values of z replace t in eq 15 and 16 and may be used to calculate the local weighting factors to the modified Block and Walker equilibrium solvation reaction field. Representative values of z and relative percentages of dipolar and inductive contributions to the total reaction field for selected liquids appear in Table 111. A complicated interplay between dipolar and inductive solvation for liquids of moderate dielectric constant is seen with complexities most pronounced in liquids with t < 15. Model Testing-Nonpolar
Systems
The first group of liquids used to examine the various models will be the nonpolar liquids. In this case all reasonable models (e.g., McRae-Bayliss, modified McRae-Bayliss, modified Block and Walker) are equivalent and indicate the solute transition energy, or any linearly related free energy variable, should be linear with 2(n2 - 1)/(2n2 1). We recently obtained extensive solvatochromic data in the n-alkane series.30 This series will serve as a basis nonpolar set and is particularly suited for model testing since the refractive index is changed in a controlled fashion without relying on gross changes in the chemical nature of the constituents. We will employ the Kamlet-Taft K* parameter as a convenient linear free energy variable. Solvent ?r* values are the averaged results obtained from a number of solute probes and therefore much less influenced by experimental imprecision or solute specific anomalies. The slope and intercept of the correlation of K* against (n2 1)/(2n2 + 1) for the n-alkane series, pentane through hexadecane, appears in Table IV. Table IV also contains correlation results obtained when additional nonpolar liquids (perfluorinated,Io po-
+
I/, a14
, 0.16
,
, ai8
,
, 0.20
,
, a22
,
,
,
J
0.24
(Illly(2nLII)
Figure 2. Plot of a* against (n2 - 1)/(2n2 + 1) for nonpolar solvent systems. Symbols used are (0)perfluorinated solvents, ( 0 )alkanes, (A)
carbon tetrachloride, (A) nonpolar aromatics, ( X ) OV serries methyl/ phenylsiloxane polymers. Lines shown are A = case c of Table IV; B = case g of Table IV; C = case d of Table IV. lychlorinated, aromatic) are added to the n-alkane basis set as well as the correlation results observed for the pseudo-nonpolar methyl/phenylsiloxane polymer series," the OV type gas-liquid chromatographic stationary phases, which have been analyzed in detail elsewhere. A plot of K* against (n2 - 1)/(2n2 1) for the various nonpolar liquids is shown in Figure 2. The wide range in correlation slope and intercept displayed by the solvent groups in Table IV is immediately apparent. This signals fundamental problems. A factor of 2 change in the slope of the correlation is much too large for a group of simple nonpolar liquids if the fitted model is valid. Further, it is difficult to devise any trivial adjustment of the reaction field function which will significantly improve the situation. Yet it is readily apparent from Figure 2 that despite the disagreement in correlation slope among the various nonpolar solvent families, a strong linear global trend is discernible. Note also that the trend established within the perfluorinated alkane solvents, while based on very limited data, closely parallels the alkane solvent line though with a significant offset. A number of difficult issues related to the manner in which linear free energy correlations are best pursued and interpreted are implicitly raised in the consideration of these results. It is
+
Brady and Carr
5764 The Journal of Physical Chemistry, Vol. 89, No. 26, I985 inappropriate to parade our own prejudices on this issue here, save to underscore that which should be in the vocabulary of all correlation users: caveat emptor. In a recent paper on the solvatochromic characterization of the OV series of methyl/phenylsiloxane polymers*’we noted good agreement between the experimental dependence of a* on (n2 1)/(2n2 1) and that calculated on the basis of a previously published generalized correlation. The results just presented imply this agreement is happenstance. We conclude the coefficient of (n2- 1)/(2n2 1 ) in a set of solvents containing both hydrocarbon and aromatic members is determined by the cluster type behavior of these two groups of solvents. We should also point out inspection of a* values for the nonpolar aromatic solvents alone points to difficulties in correlating solvatochromic results with any function of bulk refractive index. The a* values of benzene, toluene, m-xylene, p-xylene, cumene, and mesitylene span the range 0.41-0.59 while the bulk refractive index covers a range of only 1.4889-1.4979 (cf. T * range of -0.08-0.08 and refractive index range of 1.3547-1.4325 for the n-alkanes). A reasonable explanation for this behavior lies in the high anisotropy of the polarizability of aromatic molecules. However, regardless of cause, consideration of the bulk physical properties of aromatic solvents would not foreshadow this result. While a number of plausible explanations suggest themselves as candidates for rationalizing the results shown in Table IV, there is not sufficient data to choose betwen the various options at this time. The most commonly voiced deficiency revolves around the isotropic continuum assumption of the reaction field approach. Yet, the large number of successful linear free energy correlations suggests the continuum assumption per se might not be the cause of failures. Introduction of molecular entities must be augmented with the realization that thermal agitation will smear out the discrete nature of the solvent molecules; it is not a big step from a smeared average of molecular configurations to effective continuum. Of course, if one pursues the effective continuum description the question of how the effective continuum relates to molecular or bulk properties arises. Eventually one is led to the conclusion that at some point molecular discreteness must be incorporated into the model if only to provide a means of determining the relationship between molecular and bulk properties and the physical observables of the experiment. Certainly, the results presented here are not consistent with any relation incorporating only bulk physical properties as the models input parameters. We should hasten to emphasize that although only nonpolar solvent systems were considered in the present section, a sizeable fraction of the experimentally accessible range of the a* scale was spanned by the solvents examined; we are not dealing with minor local deficiencies superimposed on a generally wellbehaved model. One interesting point involving the correlations shown in Table IV is the very good agreement between the experimental gas-phase a* value (-1.06, ref 28) and that predicted from the alkane solvent correlation. Unfortunately, given the predictions of the remaining solvent sets it is impossible to determine whether this agreement is real or happenstance.
+
+
Model Testing-General Systems The examination of nonpolar solvents by the various reaction field models has been a clear disappointment. We do not expect extension to more general groups of solvents will be of necessarily better quality or more meaningful physical insight; however, we believe this exercise will have value on two counts. First, general solvent correlations are the type of analyses which will eventually have practical utility. Even if the fundamental merit of the correlation equations is not clear, the ability to perform well in a predictive sense is the paramount issue in many applications. Second, considerable effort has been expended in the development of correlation models concerned with solvent Generally there is very little to guide one in determining the validity of a given model. One typically bases conclusions on the gross statistical characteristics of the correlation such as patterning of the residuals and general quality of the correlation. An important
**
..
“c
0’45
?? .*.
t
-0.
c I
..
.
e*.
* *
I
0.E
I
1
0.25
I
I
I
045
e(s,) Figure 3. Plot of the L(c) - L(n2)against e(€,) illustrating the approximate numerical incorporation of dielectric saturation in the McRaeBayliss model of solvatochromism.
subsidiary issue is the sensitivity of correlation analysis in providing insight into model performance. We must ask ourselves whether better statistical performance of a model is necessarily the consequence of a better physical model. Although our analysis of the McRae-Bayliss model above leads us to believe a two-variable correlation model of solvent effects is, in principle, inadequate, we will consider a number of models of this form since more exact treatments appear to reduce to correlation models which reflect the parameterization of the McRae-Bayliss approach. The use of two solvent variables strikes a good balance between theoretical expectations and excessive model freedom. Furthermore, aside from general worries of the effects of excessive model freedom in the apparently more complete models, high correlation among the additional solvent parameters is unavoidable-recall the basic form of eq 19. We will examine a large number of two-variable models. A detailed justification of the origin of these models is not necessary for our present purposes and has been touched upon elsewhere.27 The modified McRae-Bayliss and Block and Walker models introduced above will be included, as will the original versions. The two-variable version of our generalized model and two-variable extensions of a dipole-dipole interaction model presented by Reddoch and Konishi will also be considered. We wish to emphasize the models are not of equal rigor. Furthermore, rather than consider solvent a* values as above, we will examine correlations involving transition energies pertaining to specific solutes. The parameters used in the various model correlations are shown in Table V with correlation results appearing in Table VI; a detailed listing of all model-solute correlation permutations is a~ailable.~’Of the numerically consistent models presented in Table V (B, C, D, E, F, G, H), only the McRae-Bayliss (B) and hybrid Block/Walker (D) models do not violate the predicted lower limit of 2 for the inductive to dipolar coefficient ratio. Above we pointed out physical inconsistencies of the McRae-Bayliss model that severely compromise the integrity of the model. Also somewhat surprising is the marginally superior correlating ability of the McRae-Bayliss model. The physical models leading to the Onsager and Block and Walker reaction field models are distinctly different, yet there appears to be little to discriminate them in the present analysis. Some insight into this result can be obtained by comparison of the respective equilibrium solvation parameters with the parameters used to model the orientational components of solvation, Figure 3. We can see that although the plot involving the parent equilibrium parameters shows the curvature expected for the nonsaturated vs. saturable reaction field comparison, the orientational components are reasonable linear. In effect it appears the improper separation of orientational and distortional components of solvation within the McRae-Bayliss scheme leads to a result which is numerically equivalent to incorporation of dielectric saturation. A more rational approach to examining the correlation results is via an inspection of residual plots. Figure 4 contains residual plots for all models examined by using the results obtained for
Analysis of Dielectric Models of Solvatochromism
The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 5765
v
I
b
0
C
I
I
/
L
I
b
d
L
L
2/
L
2/
I:
1
.
I I
I
I
.
kl I
. I
I
1
i II
II
I
I .
I %J
r
I
.
1
.
J
Figure 4. Residual plot obtained from correlations listed in Table VI, transition energy of N,N-diethyl-4-nitroaniline.The plots are of the studentized residual against predicted transition energy. Model designations are given in Table V. Lines drawn indicate patterned residuals by solvent group. For plots a, b, and d-i the lines from left to right refer to dipolar, alkane, and perfluorinated solvent groups, respectively. Plot c indicates trends for alkane (left) and perfluorinated (right) solvents, while plots j-l indicate trends for dipolar (left) and alkane (right) solvents.
N,N-diethyl-4-nitroaniline; these plots are quite representative of those obtained for all solutes. Casual inspection of these figures would suggest little or no structure in the residuals. One can arguably discern a weak inverted wedge in most of the plots; however, the structure is not pronounced. However, very distinct (and troubling) trends are apparent if we focus our attention on specific solvent families. Very significant linear correlation in the residuals is apparent for virtually all models within the alkane, perfluorinated, and (apparently) dipolar solvent families. The
trend is most distinctive in the alkane series. There are two possible explanations for this type of behavior. First, and most obvious, this is a signal of severe problems with the models examined at a very fundamental level. Second, this behavior could be a manifestation of a reasonably complete model being unduly influenced by the distribution of available data; for example, see the data distribution plot in Figure 5 . While the latter explanation can be assessed only via additional (and well planned) experiments, the results we have provided for the nonpolar systems considered
Brady and Carr
5766 The Journal of Physical Chemistry, Vol. 89, No. 26, 1985 TABLE V Model Summary" solvent effect component orientational distortional
model designtn A
B
c D E F G H I J K L
2(n2 - 1)/(2n2 + 1) 2(e - 1)/(2t + 1) - (2(n2 2(n2 - 1)/(2n2 + 1) - 1)/(2n2 + 1) 2(n2 - 1)/(2t + 1) 2(€ - n2)/(2t + 1) 8(e)(n2 - l ) / ( t - 1) B(t)(e - n2)/(t - 1) e(z,n2)(n2 - I)/(z- 1) O ( Z , ~ ~ ) ( Z- n2)/(r - 1) B(e,n2)- 2 ( d - 1)/(2n2+ 2(n2 - 1)/(2n2 + 1) 1) (n2 - l)/(~- 1) B(t,n2) - (n2 - l)/(t - 1 ) (n2 - )/(2n2 + 1) 2(n2 - 1)/(2n2 + 1) Nt,) B(e,n2) - 2(n2 - 1)/(2n2 + 1) 4v 2(n2 - 1)/(2n2 + 1) w/V 2(n2 - 1)/(2 + I ) (!J/V)'' 2(n2 - 1)/(2n2 + I) (P/Vos 2(n2 - 1)/(2 + 1) o(c,)
t
E
0.76
f
0
;0.55 I-
W
2
5
0.35
3
50
0.15
I
I
2S.3
283
I
I
2x3
28.3
"See ref 28.
8(c) = 3c In
(t)/(t
In
(c)
- t + 1 ) - 6/ln
(e) - 2
1/(6),kK
8(c,n2) = {In ( e / n 2 ) [ t n Z In (t/n2) - 2t(n2 1) - 2nZ] + 2(2 + n 2 ) ( e - n 2 ) } / (In (e/n2)[tn2 ln(e/n2) + t(1 - 2n2) + n2] - 2(1 - n2)(e - n 2 ) )
Figure 5. An approximate representation of the cumulative distribution of solvatochromic data for N,N-diethyl-4-nitroaniline.This plot is representative of the ?r* data base as a whole.
In models I-L, !J and V are the solvent dipole moment and molar volume, respectively.
Summary In the discussion above we have considered the modeling of solvatochromic data from perspectives of widely varying rigor. Inadequacies in the McRae-Bayliss model have been pointed out and an alternate development presented. We have also extended this modified orientational-distortional separation scheme to a modified Block and Walker reaction field model recently presented by us. One outcome of this development is the recognition of the coupled nature of the orientational and distortional components of solvation. Although hindsight provides a clear physical rationalization for this effect, no models of solvatochromism have explicitly recognized it. The implications for correlation analyses lie in the magnitude of the coefficients of the correlation equation and not on the dominant veriables. In the nonpolar solvent limit, in which virtually all reaction field models presented to date are equivalent, servere problems in obtaining satisfactory correlation results were encountered. Variability in correlation slope with the solvent groups considered well beyond what can be accepted as tolerable was observed. We do not have a firm explanation for this behavior although a number of reasonable suggestions were forwarded. Results for the general solvent correlations were equally disappointing. No model is distinguishably superior in a correlation sense even though a number of the models tested were clearly more plausible on physical grounds.
+
TABLE VI: General Solvent Systems: Correlation Summary. Correlation of the Transition Energy of N,N-Diethyl-4-nitroaniline against Model Parameters model" nc rc bla vm..(pred)b 2.15 0.964 32.22 A 82 3.27 32.94 0.970 82 B 1.85 34.65 0.974 82 C 2.73 32.12 0.941 82 D 1.09 32.46 0.960 82 E 1.43 32.95 0.966 82 F 1.14 32.89 0.963 82 G 1.22 33.11 0.965 H 82 32.47 2.62 0.917 79 I -0.96 25.89 0.823 17 J 3 1.92 9.30 0.961 K 79 28.10 1.26 71 0.897 L Models defined in Table V. bPredicted gas-phase transition energy, experimental value = 30.39 kK, ref 27. and r refer to the number of solvents in the correlation and correlation coefficient, respectively.
alone clearly point to the former as the most likely situation. Note also comparison of predicted gas-phase transition energies with experimentally observed valuesz8runs counter to what would have been anticipated from the apparently weak structure present in the residual plot, supporting the view that global trends in the residual plot are more apparent than real.
Acknowledgment. J.E.B. was a recipient of an ACS Analytical Division Full Year Fellowship sponsored by the Upjohn Co., Kalamazoo, MI. this work was supported by National Science Foundation Grant CHE-8205187.
