© Copyright 2002 by the American Chemical Society
VOLUME 106, NUMBER 48, DECEMBER 5, 2002
ARTICLES Reorientational Dynamics of the Model Compound 1,2,3,4-Tetrahydro-5,6-dimethyl1,4-methanonaphthalene in Neat Liquid from Temperature-Dependent 13C Nuclear Magnetic Relaxation Data: Spectral Densities and Correlation Functions Andreas Do1 lle* Institut fu¨ r Physikalische Chemie, Rheinisch-Westfa¨ lische Technische Hochschule, 52056 Aachen, Germany ReceiVed: March 29, 2002; In Final Form: July 2, 2002
The reorientational motion of the hydrocarbon 1,2,3,4-tetrahydro-5,6-dimethyl-1,4-methanonaphthalene (5,6Me2THMN) was studied over a wide range of temperature by the evaluation of 13C spin-lattice relaxation rates and NOE factors. The data from measurements at 22.63, 75.47, and 100.62 MHz were fitted to spectral densities introduced by Bloembergen, Purcell and Pound (1948), Davidson and Cole (1951), and for the first time, by a new spectral density derived from a Tricomi correlation function recently introduced by Zeidler (1991). For the temperature dependence of the correlation times, a Vogel-Fulcher-Tammann equation was applied. To obtain a reasonable fit with either Bloembergen-Purcell-Pound or Cole-Davidson spectral densities, it was necessary to apply the generalized order parameter from the approach by Lipari and Szabo (1982). The same quality of fit was achieved by the use of the Tricomi spectral density, although in this case it was not necessary to employ the Lipari-Szabo approach.
Introduction The liquid state of matter is of great importance in nature and technology. Almost all reactions in biological and chemical systems proceed in solution or liquidlike environments. Therefore, it is of interest to develop further the existing models and theories for describing the molecular structure and dynamics of liquids. These models and theories should mediate a better understanding, for example, of the arrangement of the molecules relative to each other or of the dynamic behavior of the molecules and thus of the route of chemical reactions in liquid systems. Whereas it is often relatively easy to describe the molecular structure and dynamics of the gas or solid state, this is not true for the liquid state. The model that is usually applied for the description of molecular reorientational processes in liquids is the rotational diffusion model. A particularly important experi* E-mail:
[email protected].
mental method to obtain information on the reorientational dynamics of molecules is the measurement of nuclear spinlattice relaxation rates. Most of the experimental findings on the reorientation of molecules and molecular segments by means of nuclear relaxation rates were obtained in the extreme narrowing region where the product of the resonance frequency and the reorientational correlation time is much less than unity. Outside the extreme narrowing region, this condition is not valid anymore, and the relaxation rates become frequency-dependent. Bloembergen, Purcell, and Pound1 employed in their study of relaxation effects on nuclear magnetic resonance phenomena the model of rotational diffusion by Debye.2 However, the interpretation of experimental relaxation data with the BPP model by Bloembergen, Purcell, and Pound was often hampered by the fact that the data could not be reproduced with this model when it was assumed that the reorienting molecules or molecular segments behave like totally or partially rigid bodies.3,4 Thus, it is questionable whether the rotational diffusion model for rigid bodies is suited for the determination
10.1021/jp0208438 CCC: $22.00 © 2002 American Chemical Society Published on Web 11/12/2002
11684 J. Phys. Chem. A, Vol. 106, No. 48, 2002 of molecular reorientational dynamics. Hertz3,4 introduced criteria by which it should be possible to prove the rigidity of the reorienting molecular entities. To explain the failure of the rotational diffusion model, several other models were developed, many of which were described in the comprehensive review by Beckmann.5 One of the most successful attempts5 is the continuous distribution of correlation times by Cole and Davidson,6 which found broad application in the interpretation of relaxation data from crystalline and glassy solids and viscous liquids. It is, however, a longstanding discussion whether the deviation from the BPP spectral density stems from the fact that there exists a distribution of correlation times or if the correlation functions are just not exponential. At least for highly viscous, glassy systems, the experimental results indicate a distribution of correlation times (cf. to the review by Sillescu7), whereas the situation in liquids is still not clear. Another way to describe the deviations from the BPP spectral density is the so-called model-free approach by Lipari and Szabo,8 which corresponds to a discrete distribution of correlation times and is a generalization of the two-step model by Wennerstro¨m et al.9 It was originally developed for taking internal motions in macromolecules into account without giving an explicit model when calculating the relaxation data.10 Although the model-free approach was first applied mainly for the interpretation of relaxation data of macromolecules, it is now also used for fast internal dynamics of small and mediumsized molecules. It was the aim of the present study to find out by measurements of relaxation data on a model compound also outside the extreme narrowing region whether the rotational diffusion of partially or entirely rigid bodies is in principle applicable to the description of the reorientational motions of molecules or molecular segments in liquids. For such an investigation, a substance had to be found, the molecules of which consist mainly of rigid structural units. The reorientational processes have to be slow enough to leave the extreme narrowing region at experimentally readily accessible temperatures. These requirements are met by the model substance 5,6-dimethyl-1,2,3,4tetrahydro-1,4-methanonaphthalene11,12 (5,6-Me2-THMN, 1). The neat hydrocarbon 1 is liquid at room temperature and does not crystallize over a wide temperature range. The basic structure of 1,2,3,4-tetrahydro-1,4-methanonaphthalene consists of rigid structural units. The notion “rigid” is used here in accordance with Sykora et al.13 for molecules or molecular segments consisting of units that cannot perform rotations about single bonds. Their internal motions should be restricted to vibrations about the equilibrium structure. In the study by Do¨lle and Bluhm,12 the rotational diffusion was a good model process to describe the anisotropic rotational dynamics of 5,6-Me2-THMN in neat liquid. In the present study, we investigated whether this also holds outside the extreme narrowing region. The resonance frequencies of the nuclei are given by the accessible magnetic field strengths via the resonance condition. Since the magnets usually applied for NMR spectroscopy have only fixed field strengths, the correlation times (i.e., the rotational dynamics) have to be varied to leave the extreme narrowing regime. To vary the correlation times and thus also the spectral densities and relaxation data, either the pressure (cf. to the review by Lang and Lu¨demann14) or the temperature has to be changed. In the present study, 13C spin-lattice relaxation rates and nuclear Overhauser factors in temperature ranges in and outside the extreme narrowing region were measured to observe the temperature dependence of the spectral
Do¨lle densities. Subsequently, several models were fitted to the experimental relaxation data.
Theoretical Background Longitudinal Relaxation of 13C Nuclei. The relaxation of nuclei in medium-sized molecules and for moderate magnetic fields is generally determined by dipolar interactions with directly bonded protons. When the relaxation times are measured under 1H decoupling conditions, the cross relaxation term vanishes,15 and the intramolecular dipolar longitudinal 13C nucleus i relaxation rate (1/T DD 1 )i for the relaxation of the by proton j is related to the molecular reorientations by15
13C
( ) 1 T1DD
ij
)
1 (2πDij)2[J(ωC - ωH) + 20 3J(ωC) + 6J(ωC + ωH)] (1)
with the dipolar coupling constant
Dij )
µ0 γ γ (p/2π)r-3 ij 4π C H
(2)
where µ0 is the magnetic permeability of the vacuum, γC and γH are the magnetogyric ratios of the 13C and 1H nuclei, respectively, p ) h/2π with the Planck constant h, and rij is the length of the internuclear vector between i and j. The J(ω) are the spectral densities with the resonance frequencies of the 13C and 1H nuclei, ωC and ωH, respectively. Aromatic 13C nuclei relax even in moderate magnetic fields partially via the chemical-shift anisotropy (CSA) mechanism. The corresponding longitudinal relaxation rate of 13C nucleus i is given by
( )
1 1 ) γC2H20(∆σi)2J(ωC) T1CSA i 15
(3)
with the magnetic field strength H0 and the chemical-shift anisotropy ∆σ ) σ|| - σ⊥ for an axially symmetric chemicalshift tensor with σ|| and σ⊥ as the components parallel and perpendicular, respectively, to the main principal axis of the chemical-shift tensor. When discussing the various mechanisms15 contributing to the 13C relaxation of 5,6-Me2-THMN, it is reasonable to assume that the aliphatic non-methyl 13C nuclei relax exclusively via the dipolar relaxation mechanism. The experimental relaxation times can then be equated to the dipolar ones, whereas the CSA mechanism also contributes to the relaxation of the aromatic methine 13C nuclei. For the fast rotational motions of small, symmetric molecules or molecular segments such as methyl groups about their symmetry axis, relaxation via the spinrotation (SR) mechanism also becomes important. Thus, in general, the total longitudinal relaxation rate of 13C nuclei is determined by the relation
() ( ) ( ) ( )
1 1 1 1 ) + + +... DD CSA T1 T1 i T1 T1SR i i i
(4)
Dynamics of a Model Naphthalene Compound
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The nuclear Overhauser (NOE) factor ηi of carbon atom i for relaxation by nH protons j is given by15,16 nH
ηi )
∑ j)1
γH γC
JBPP(ωλ) )
σij (5)
nH
Fij + ∑ j)1
F/i
where σij is the cross relaxation rate and Fij is the dipolar relaxation rate for the interaction between 13C nucleus i and proton j in eq 1. Except the seldom-observed scalar interaction, only the dipolar mechanism contributes to cross relaxation. Therefore, the contributions from all other mechanisms to the relaxation of 13C nucleus i are summarized by the so-called leakage term F/i , which reduces the NOE factor. Usually, intermolecular dipolar contributions can be neglected for 13C nuclei with directly bonded protons. The contribution of nondirectly bonded protons in the same molecule is also neglected because of the rij dependence of the relaxation rate. For the relaxation of 13C exclusively via the intramolecular dipolar interaction, F/i ) 0, the NOE factor reaches its maximum value and depends only on the reorientational molecular dynamics:15,16
ηi,max )
γH 6J(ωH + ωC) - J(ωH - ωC) (6) γC J(ωH - ωC) + 3J(ωC) + 6J(ωH + ωC)
Time Autocorrelation Functions and Spectral Densities.5 For the case of isotropic liquids, the observed NMR relaxation rates are related after normalization to the molecular dynamics by reduced spectral densities, which are obtained from the reduced correlation functions by Fourier transformation:
J(ωλ, {xi}) )
∫-∞∞C(t, {xi}) exp(iωλt) dt
(7)
where ωλ is the transition frequency and {xi} are the characteristic parameters in the reduced correlation functions C(t). Their time integrals are defined as correlation times τ. Often a so-called effective correlation time is introduced that contains the different correlation times characterizing the molecular rotational motions such as the correlation times of overall anisotropic rotational diffusion and internal motions. Then, these effective correlation times are in the extreme narrowing case the sum of the single correlation times weighted by structural factors.15,17,18 In the present study, the effective correlation time is taken to be the area under the reduced correlation function:
τeff )
C(t)
∫0∞C(0) dt
(8)
Exponential Correlation Functions and BloembergenPurcell-Pound Spectral Density (BPP). When the reorientational motions are of a stochastic (i.e., diffusive) nature and when only one ensemble of reorienting units is present, the correlation functions are given by an exponential function:5
CBPP(t) ) exp(-t/τBPP)
The result of the Fourier transformation for the case of the exponential correlation function is
with t g 0
(9)
2τBPP 1 + (ωλτBPP)2
(10)
which corresponds to a Lorentzian function and was used for the first time by Bloembergen, Purcell, and Pound1 for the interpretation of NMR relaxation data. The parameter τBPP is the correlation time for rotational diffusion (i.e., for an exponential correlation function). As limiting values are obtained,
JBPP ) 2τBPP for ωλτBPP , 1
(11)
as the extreme narrowing condition, and -1 ωλ-2 JBPP ) 2τBPP
for ωλτBPP . 1
(12)
which is the slow-motion limit. The maximum value is
JBPP ) ωλ-1
for ωλτBPP ) 1
(13)
Nonexponential Correlation Functions.5 For many liquids, especially for viscous liquids and for solids, it was found that the experimental data did not behave like Lorentzian spectral densities. These findings have also been known in the case of nuclear magnetic relaxation data for a long time.19 Two reasons for such behavior are discussed: • The rotational motion is not a stochastic process as it was assumed in the preceding section, but it issat least for some timescorrelated. As an example, the cooperative motion in liquid crystals is mentioned here.20 The result of the cooperative motion is a correlation function that does not decay exponentially. This case is called homogeneous distribution.21 • Within an ensemble of reorienting units, subensembles can exist, consisting of molecules for which the rotational dynamics are described by an exponential correlation function and a corresponding Lorentzian spectral density. The characteristic correlation time for each of these subensembles is different, and for the whole ensemble, a heterogeneous distribution of correlation times21 is found. Such a distribution is assumed for amorphous solids and supercooled liquids, for which a distribution of different surroundings for the rotating units is postulated. From a formal point of view, one cannot differentiate between both cases,21 and experimentally, no discrimination is possible by conventional measurements of transversal or longitudinal relaxation data.22 Quite recently, a number of sophisticated experimental methods have been developed to detect dynamic heterogeneities (cf. to the review by Sillescu7 and the literature cited therein). Furthermore, several theoretical concepts have been developed to explain the occurrence of the distributions of correlation times.7 The correlation functions are related to distributions Λ(ξ, {xi}) of the correlation times ξ by
C(t, {xi}) )
∫0∞Λ(ξ, {xi}) exp(-t/ξ) dξ
(14)
in which {xi} is a set of parameters specific for the distribution. The following normalization condition is valid:
∫0∞ Λ(ξ, {xi}) dξ ) 1
(15)
The distribution does not depend on the observation frequency; it is a property characteristic for the investigated system. The
11686 J. Phys. Chem. A, Vol. 106, No. 48, 2002
Do¨lle is derived from the Tricomi function25 and has after normalization the form
corresponding spectral densities are
J(ω, {xi}) )
∫0∞Λ(ξ, {xi})1 +2ξ(ωτ)2 dξ
(16) CT(t) )
For the BPP spectral density, a distribution consisting of the Dirac δ function is valid:
ΛBPP(ξ, τBPP) ) δ(ξ - τBPP)
(17)
From this the Lorentzian spectral density is obtained. Cole-Davidson Spectral Density (CD). The spectral density introduced by Davidson and Cole6 is given here in the form by Beckmann.5 The distribution function is
{
(
)
β sin(βπ) 1 , z
