Methyl Group Dynamics in Polycrystalline and ... - ACS Publications

Jan 5, 2009 - ReceiVed: August 26, 2008; ReVised Manuscript ReceiVed: October 28, 2008. We present a quasi-elastic neutron scattering (QENS) study on ...
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J. Phys. Chem. B 2009, 113, 916–922

Methyl Group Dynamics in Polycrystalline and Liquid Ubiquinone Q0 Studied by Neutron Scattering Christoph Smuda,†,‡ Sebastian Busch,†,‡ Rene´ Schellenberg,§ and Tobias Unruh*,†,‡ Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM II), Technische UniVersita¨t Mu¨nchen, Lichtenbergstrasse 1, D-85747 Garching b. Mu¨nchen, Germany; Lehrstuhl fu¨r Experimentalphysik IV, Technische UniVersita¨t Mu¨nchen, Physik Department E13, James-Franck-Strasse 1, D-85747 Garching b. Mu¨nchen, Germany; and Technische UniVersita¨t Bergakademie Freiberg, Institut fu¨r Physikalische Chemie, Leipziger Strasse 29, D-09599 Freiberg, Germany ReceiVed: August 26, 2008; ReVised Manuscript ReceiVed: October 28, 2008

We present a quasi-elastic neutron scattering (QENS) study on the methyl group dynamics of ubiquinone Q0 in the solid and liquid state. For solid ubiquinone Q0, the dynamics can be described with three Lorentzian functions in the framework of a jump model among three equidistant sites on a circle. According to the known molecular structure of Q0 in the solid state, this is consistent with three nonequivalent methyl groups in the molecule. From the temperature-dependent analysis of the QENS spectra, the activation energies were determined. The barrier heights could be evaluated from librational bands in the inelastic part of the spectra. The results from neutron spectroscopy are compared to Gaussian 03 calculations leading to an assignment of the activation energies to the different methyl groups in Q0. The dynamics of Q0 in the liquid state is evaluated with a scattering function taking into account three different molecular motions. It is demonstrated that the temperature dependence of the long-range diffusion and isotropic rotational diffusion exhibit an Arrheniuslike behavior, whereas the process of methyl group rotation in the liquid phase is virtually free of a barrier. 1. Introduction Ubiquinone Q0 (cf. Figure 1) is an essential structural element of ubiquinones. All these ubiquinones (coenzymes Q) contain the Q0 moiety bonded to an isoprenoid side chain of varying length where the number of isoprene monomers is indicated as subscript n in the name “coenzyme Qn”. The relevance of coenzymes covers biophysical aspects as well as applications of Q10 in pharmaceutical formulations (nanodispersions) as drug carrier.1 In this context, we measured the self-diffusion coefficients of the oligoisoprene derivative Q10 in nanodroplets and bulk by QENS2 at the neutron time-of-flight spectrometer TOFTOF (FRM II, Germany).3,4 The results showed that, for a detailed understanding of the complex dynamics of molecular liquids which are covered in the QENS spectra, it is necessary to reduce the number of unknown fit parameters for the respective motions in the scattering function used. According to the various methyl groups in the Qn molecules, a detailed study of the dynamic behavior is indicated. Therefore, we studied the methyl group rotation in glassy, polycrystalline, and liquid coenzyme Q10 using time-of-flight neutron spectroscopy since almost half of the hydrogen atoms in coenzyme Q10 are bound in methyl groups.5 A fundamental scattering contribution to the quasi-elastic intensity is therefore caused by this local motion. Assuming a logarithmic Gaussian distribution of jump rates, a mean activation energy of 4.8 kJ/ mol for the methyl group rotation of polycrystalline and glassy coenzyme Q10 was determined. A comparison of the average * Corresponding author: phone +49 (0)89 289 14735; e-mail [email protected]. † Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM II), Technische Universita¨t Mu¨nchen. ‡ Lehrstuhl fu¨r Experimentalphysik IV, Technische Universita¨t Mu¨nchen. § Institut fu¨r Physikalische Chemie.

Figure 1. Structural formula of ubiquinone Q0.

activation energy for the methyl group rotation in chain deuterated polyisoprene-d5 (9.6 ( 0.3 kJ/mol) shows that the value for Q10 is smaller by a factor of 2.6 This discrepancy was the motivation for a closer look to methyl group rotation in Qn molecules in order to understand its origin. In a gedankenexperiment, one can decompose a Q10 molecule in the oligoisoprene chain and ubiquinone Q0. With neutron scattering measurements of Q0, a broad spectrum of questions can be answered. First, measurements in the solid state are intended to determine activation energies for the methyl group rotation. The inelastic neutron spectra contain further information on this motion and shall be compared to the spectra of Q10 already measured. Second, it is of interest to separate local motions from the long-range diffusional motion. The quasi-elastic scattering contribution of methyl groups as side groups in different organic compounds is for the solid state well understood. However, the corresponding analysis of the QENS spectra of the liquid phase are realized in very different ways.7,8 The substance Q0 offers the possibility to study methyl group rotation without spurious influence of other hydrogen atoms in the molecules probed by neutrons. The first part of this paper addresses the rotational dynamics of methyl groups of Q0 in the solid polycrystalline state. The results obtained from neutron scattering experiments are compared with a Gaussian 03 calculation9 which supports the interpretation of the experimental data concerning the assign-

10.1021/jp807601g CCC: $40.75  2009 American Chemical Society Published on Web 01/05/2009

Polycrystalline and Liquid Ubiquinone Q0

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ment of the activation energies to the respective methyl group in the molecule. A comparison with methyl group dynamics of Q10 is drawn. Finally, the correlation times of the methyl groups in liquid Q0 are extracted from an approved scattering function characterizing the dynamics of methyl groups in liquids of small molecules.10

The interaction between methyl groups and their environment is usually described by the single particle model.11 In this model, the methyl group is regarded as a rigid rotor whose rotation is hindered by environmental interaction. The potential that hinders the reorientation depends only on a single angular coordinate φ which represents the rotation angle of the methyl group with respect to its fixed environment. The total rotational potential energy depends on the symmetry of the methyl group as well as on the symmetry of the environment. For a methyl group as a 3-fold symmetric top, the periodic rotational potential reads ∞

V3n (1 - cos(3nφ)) 2 n)1

∑

(1)

Usually, this sum is truncated after the second term, and the rotational potential is described by a combination of a 3-fold and a 6-fold cosine potential. The maxima of V(φ) represent the height of the barrier for the hopping process. The potential V(φ) determines the quantized so-called torsional or librational levels. There are several kinds of transitions between the energy levels, e.g., tunneling transitions in the µeV range between split ground librational states (pωt) and transitions between different librational levels in the meV range, including the transition between the ground state and the first excited librational state (E01). The activation energy EA is the difference between the top of the barrier and the ground state.12 Assuming a rotational motion of the methyl group protons performing jumps among three equidistant sites on a circle of radius r, the scattering function Smethyl(Q,ω) (powder average) for the incoherent scattering of the protons is given by13

Smethyl(Q, ω) ) A0(Q)δ(ω) + [1 - A0(Q)]

Γ 1 π ω2 + Γ2

(2)

with the half-width at half-maximum (hwhm) Γ and the elastic incoherent structure factor A0(Q)

1 A0(Q) ) [1 + 2j0(√3Qr)] 3

(3)

In this equation, j0 denotes the spherical Bessel function of zeroth order, r the distance of the protons to the 3-fold symmetry axis, pQ the momentum transfer, and pω the energy transfer of the neutron to the sample or vice versa. The temperature dependence of the hwhm Γ for a thermally activated process is expected to obey an Arrhenius equation:

Γ(T) ) Γ∞ exp(-EA /kBT)

N

S(Q, ω) ) A0(Q)δ(ω) + [1 - A0(Q)]

∑ pnLn(Γn,ω) (5)

n)1

2. Methyl Group Rotation by Neutron Scattering

V(φ) )

If there exist N nonequivalent methyl groups in the molecule and in the crystal packing, this leads to a superposition of N Lorentzians Ln(Γn,ω):14

(4)

where EA is the activation energy for the rotation and Γ∞ is related to the attempt frequency.

N with the normalization ∑n)1 pn ) 1. The incoherent scattering function S(Q,ω) is convolved with the instrumental resolution function R(Q,ω) and fitted to the experimental QENS spectra:

Sexp(Q, ω) ) R(Q, ω) X S(Q, ω)

(6)

The instrumental resolution function was determined by a vanadium measurement. Performing the convolution of the model function with the vanadium data, the small Q dependence of the energy resolution is respected during the fitting procedure. The energy dependence of the resolution function can be neglected in the considered energy range [-1.5 meV, 1.5 meV]. 3. Experimental Section 3.1. Samples and Measurements. Ubiquinone Q0 (2,3dimethoxy-5-methyl-p-benzoquinone, 98%) was purchased from ABCR, Germany. Neutron scattering experiments were performed at the cold neutron multidisk-chopper time-of-flight spectrometer TOFTOF at the FRM II.3,4 According to the selected incident neutron wavelength of λ ) 6 Å, the investigated Q range was 0.4 e Q e 1.8 Å-1. The chopper rotation frequency was set to 12 000 rpm, leading to an instrumental resolution of about 60 µeV (fwhm). The experiments were carried out in the temperature range of 57-400 K using aluminum hollow cylinders as sample holders.15 The gap between the inner and outer cylinder of 0.1 mm was filled with 0.5 mL of the sample, yielding a 10% scatterer. The measurement with a resolution of about 4 µeV (fwhm) was performed with a chopper rotation speed of 16 000 rpm and a wavelength of λ ) 14 Å. For this measurement, a flat aluminum sample holder filled with Q0 with a sample thickness of 0.5 mm was mounted at an angle of 45° to the transmitted neutron beam measured toward the detectors. The instrumental resolution function of the spectrometer was determined by a vanadium measurement in both cases. The empty sample container was measured at 100 and 240 K. 3.2. Data Analysis. Neutron data analysis was carried out using the program IDA.16 The time-of-flight spectra were corrected for detector efficiency, normalized to vanadium, converted to the energy scale, and corrected for self-absorption. The S(2θ,ω) spectra were converted into S(Q ) const, ω) spectra. The spectra of Q values in the region of Bragg reflections of the sample were excluded from data evaluation. The diffraction pattern of Q0 is displayed in Figure 2 as measured at the TOFTOF spectrometer. The applied model functions were convolved with the resolution function of the instrument and fitted to the experimental data. An error weighted average of the widths of the Lorentzians obtained from the fits to the experimental scattering function Sexp(Q,ω) was calculated for each temperature. This procedure is only reasonable if the half-widths are mainly independent of Q, which is fulfilled for the investigations presented here.

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{

S1(Q, ω) ) F(Q) [cfix + (1 - cfix)A0]δ(ω) + (1 - cfix)

Figure 2. Neutron diffraction patterns of polycrystalline (115 K, solid line) and molten (355 K, dashed line, shifted) Q0 obtained at TOFTOF. No phase transition for solid Q0 was observed below the melting point (333 K). The first structure factor maximum of liquid Q0 appears at Q ≈ 0.9 Å-1.

Figure 3. Molecular structure of Q022 visualized using Mercury23 in the ball-and-stick style. The three methyl groups are designated with letters A, B, and C for a simple differentiation in this paper. Gray balls represent carbon atoms, red balls oxygen atoms, and the white ones hydrogen atoms.

Parameters of rotational potentials were calculated using the program TUNCALC provided by M. Prager (FZ Ju¨lich, Germany). 3.3. Gaussian 03 Calculations. A vibrational analysis was performed using the Gaussian 03 program package.9 The gas phase geometry was optimized with the density function method B3LYP17-19 and the 6-311++G(d,p) basis set.20,21 Frequencies were calculated with the same method and characterize the stationary point of the optimized geometry as a minimum (absence of imaginary frequencies) on the potential energy surface. 4. Results and Discussion 4.1. Quasi-Elastic Scattering in the Solid State. The molecular structure of ubiquinone Q0 was already successfully solved some decades ago.22 Three nonequivalent methyl groups were found which can be expected according to the molecular structure in the packing. Furthermore, it was assumed that they reveal different dynamics, i.e., different activation energies for the CH3 rotational process. According to our labeling of the methyl groups in Q0 (cf. Figure 3), methyl group A is directly bonded to the benzoquinone ring, whereas methyl group B is the out-of-plane (oop) methoxy methyl group and methyl group C is the (approximately) in-plane (ip) methoxy methyl group. The plane is given by the six-membered ring plus the carbon atom of methyl group A (cf. ref 22). In order to study the temperature dependence of the methyl group rotation, QENS spectra were collected at several different temperatures and analyzed using the model function

}

1 - A0 N 1 L (Γ , ω) (7) π n)1 N n n

∑

where F(Q) denotes a scaling factor which includes the wellknown Debye-Waller factor. Parameter cfix represents the ratio of the number of immobile (fixed) hydrogen atoms over the total number of protons in Q0. The fitting range was restricted to an energy transfer from -1.2 to 1.2 meV. The parameter r was kept constant at 1.03 Å, which is the distance of the hydrogen atoms of the methyl groups from the rotation axis. Because of different rotational barriers of the three methyl groups, it is expected that the number of rotationally active methyl groups N which can be resolved on the time scale will increase successively with increasing temperature. For temperatures below 115 K, QENS spectra could be fitted with a single Lorentzian function, whereas above 115 K a second Lorentzian function has to be included in the scattering function indicating the onset of the rotation of the second type of methyl group. Accordingly, the parameter cfix changes. This change in the dynamics can be tracked by leaving the parameter cfix free: it decreases step by step with increasing temperature. For temperatures g190 K, three Lorentzians are required to describe QENS data; i.e., all three methyl groups rotate. The line widths Γ1 for temperatures T g 190 K were extrapolated from line widths determined from QENS spectra at T < 140 K to increase the stability of the fits for the QENS spectra at T g 190 K. This is based on the assumption that the Arrhenius behavior continues up to the melting point as long as no phase transition takes place which is the case for Q0 in the considered temperature range. It could already be shown that this procedure is justified since for other compounds a perfect Arrhenius behavior over a large temperature range was observed for the process of methyl group rotation.10,24 Examples of QENS spectra of Q0 for different temperatures are shown in Figure 4 including the best fits according to eq 7. The extracted half-widths were in good approximation independent of Q as expected for a local motion due to the jump model. The temperature dependences of the hwhms of each of the three methyl groups exhibit an Arrhenius-like behavior. The Arrhenius plot of all line widths Γn is displayed in Figure 5. From the slopes, the activation energies EA ) 2.7, 7.0, and 13.5 kJ/mol were calculated. The obtained prefactors Γ∞ ) 2.8, 5.1, and 10.8 meV are typical values for methyl group rotation.10,12,14 4.2. Gaussian 03 Calculation. For the assignment of the obtained activation energies to the respective methyl group in the Q0 molecule, a vibrational analysis using Gaussian 03 was performed. One has to take into account that Gaussian 03 calculations are based on a single molecule. Unlike the experimentally determined structure in the solid phase (cf. Figure 3), the optimized isolated Q0 molecule showed another conformation. Methyl group C was out-of-plane, and methyl group B was nearly in-plane. Nevertheless, qualitatively meaningful results can be derived from the calculation. The calculated frequencies for the methyl torsional vibrations of the three methyl groups were 16.1 meV for methyl group A, 17.9 meV for the oop methoxy methyl group B, and 23.6 meV for the ip methoxy methyl group C. Hence, one can assign the smallest activation energy to the methyl group A directly coordinated to the benzoquinone ring, whereas the barriers of the two methoxy methyl groups are larger.

Polycrystalline and Liquid Ubiquinone Q0

Figure 4. Neutron scattering spectra of Q0 for different temperatures (Q ) 1.4 Å-1). The solid lines show the best fits (eq 7) to the experimental data. The instrumental resolution is indicated by the dashed line.

Figure 5. Arrhenius plot of the hwhm Γn for the three methyl groups in solid Q0. The solid lines represent best fits of eq 4 to the data to determine the activation energies and Γ∞.

This is similar to the situation for the rotational barriers of methyl and methoxy methyl groups bound to a benzene ring where the activation energies for the methyl group rotation of direct bound CH3 groups are considerably smaller than the ones for the methoxy methyl groups.10,25,26 Furthermore, the results of the Gaussian 03 calculation are consistent with the assumption that due to the less steric hindrance, the rotational barrier of the oop methyl group B should be smaller in comparison to the one of the ip methyl group C. 4.3. Inelastic Neutron Scattering (INS). For every nonequivalent methyl group, a CH3 torsional vibration can be observed in the inelastic part of the neutron spectra (if the resolution allows it). The corresponding energy E01 of this band is directly proportional to the activation energy EA and potential barrier. In addition to the determined activation energies by QENS spectra analysis, potential parameters for the methyl group dynamics were calculated based on the observed methyl torsional bands in the inelastic part of the neutron spectra. The time-of-flight spectra of Q0 recorded at 85, 140, 190, and 310 K (solid state) and at 355 K (liquid state) are displayed in Figure 6. From these spectra, the broadening of the elastic line as well as the increase of the inelastic scattering contribution with increasing temperature can be extracted. However, the interpretation and assignment of these different transitions is in the most cases not completely unambiguous. For the identification of methyl torsion bands in neutron spectra, it is helpful to know that they show a high intensity and a rapid broadening with temperature.27 Correspondingly,

J. Phys. Chem. B, Vol. 113, No. 4, 2009 919

Figure 6. Time-of-flight spectra of polycrystalline Q0 at four different temperatures and of liquid Q0 at 335 K (inset). In the liquid phase, the discrete bands are smeared out and broadened. The neutron energy transfer scale was obtained by subtracting the energy of the incident neutrons (E ) 2.27 meV) from the energy of the scattered neutrons. The bands assigned to methyl torsional vibrations are indicated by arrows.

the strong bands at 19.5 and 28.8 meV (cf. Figure 6) are attributed to the 1 f 0 transitions (E01) of the torsional vibration of two methyl groups. With INS measurements on different partially deuterated Q0, it would be possible to check isotopic shifts of the attributed methyl torsional bands and to prove the mentioned assignments.14,28 However, these substances were not commercially available. Using the program TUNCALC, the rotational parameters were obtained by solving the Schro¨dinger equation and calculating the energy levels for a given potential which is the input parameter (V3 and V6). The activation energy EA, 1 f 0 methyl torsional band E01, and tunneling energy pωt can be expressed as transitions between characteristic energy levels of the molecules of the sample.12 To get a suitable start parameter V3 for the refinement of the rotational potential, one can estimate V3 from the methyl torsional band which can directly be obtained from the inelastic part of the neutron spectrum.29,30 Based on the supposed torsional bands at 19.5 and 28.8 meV, V3 potentials were calculated and used in the program TUNCALC to harmonize all parameters mentioned above. The results are listed in Table 1. The rotational potentials of the methyl groups B and C can be described as virtually pure V3 potentials. The calculated rotational parameters are in good agreement with values determined experimentally. However, for methyl group A it was not possible to decide which band might be the methyl torsional bands from the spectra. Therefore, the energy of this band E01 will be a calculated result of the rotational potential. Because of the low activation energy of methyl group A, it was possible to observe a corresponding tunneling line with a high-resolution measurement at TOFTOF. It was detected with pωt ) 15.4 µeV at 4 K with a resolution of 4 µeV (fwhm) (cf. Figure 7). With this additional information, a suitable potential could be found which is consistent with experimental values. For the methyl group A, an additional 6-fold term is necessary to describe the potential. On the basis of the given V3 and V6 potentials in Table 1, the band at 7.3 meV was assigned to be the E01 transition. While the tunneling line at 15.4 µeV was detected at the TOFTOF spectrometer, it is beyond the scope of this article to

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TABLE 1: Calculated Rotational Potentials and Parameters in Comparison with Experimental Values for the Three Methyl Groups in Q0 methyl group

V calc (meV) 3

V calc (meV) 6

E calc 01 (meV)

E meas (kJ/mol) 01

E Acalc (meV)

E Ameas (kJ/mol)

pωtcalc (µeV)

pωtmeas (µeV)

A B C

8 74.9 154.9

36 0.5 0

7 19.5 28.7

7.3 19.5 28.8

2.6 6.2 13.5

2.7 7.0 13.5

15.8 0.11

15.4

TABLE 2: Diffusion Coefficients D and Parameters Dr, ΓM, τ, and Jump Lengths l of Q0 Obtained from Fits Using Models Eq 9 and Eq 10, Respectively temp (K)

D

Dr (µeV)

ΓM (meV)

τ (ps)

l (Å)

335 355 375 400

0.88 ( 0.002 1.26 ( 0.01 1.67 ( 0.01 2.38 ( 0.01

96 ( 7 126 ( 7 161 ( 8 209 ( 6

1.095 ( 0.016 1.126 ( 0.020 1.119 ( 0.033 1.079 ( 0.025

10.1 5.9 3.7 2.6

2.3 2.1 1.9 1.9

identify the corresponding tunneling line of methyl group B using the backscattering technique. Solid Q10 exhibits three torsion bands at 6.4, 12, and 21.4 meV.5 A potential assignment to methyl groups bonded to the benzoquinone ring could be that the methoxy methyl group B is related to the band at 21.4 meV. On the other hand, because of the unknown molecular structure of Q10, a comparison is difficult: already a structural phase transition leads, in general, to a change of the dynamics as already shown for toluene31 for the process of the methyl group rotation. 4.4. Quasi-Elastic Scattering in the Liquid State. The dynamics of liquid Q0 as well as other molecular liquids is complicated due to the existence of other reorientational motions, e.g., rotational diffusion and long-range diffusion. The observed increased quasi-elastic broadening compared to QENS spectra of Q0 in the solid state is caused by further motional

components in the melt. However, the separation of superimposed motions is not straightforward. In our recent study,10 the evaluation of QENS spectra of small organic molecules containing methyl groups in the melt, namely pentafluoroanisole and pentafluorotoluene, is discussed in detail. Summarizing, different models for the description of the superimposed internal and diffusional motions were tested. A satisfactory description of QENS data was only achieved when methyl group rotation, translational diffusion of the whole molecule [ST(Q,ω)], and isotropic rotational diffusion (IRD)13 of the molecule [SIRD(Q,ω)] were taken into account. In this case, the scattering function is a superposition of different kinds of motions independent of each other which corresponds to a convolution of the respective scattering functions in the frequency domain:

S2(Q, ω) ) Smethyl(Q, ω) X ST(Q, ω) X SIRD(Q, ω) (8) Recently, we demonstrated that a single Lorentzian function is sufficient for the description of the methyl group rotation of molecules in the liquid state.5,10 Therefore, the sum of Lorentzians in eq 5svalid for the description of the methyl group rotation in the solid statesreduces to a single Lorentzian function. With Al(Q) ) jl2(Qa)(2l + 1), the (quasi)elastic structure factors in the model of the isotropic rotational diffusion, the scattering function reads

{

S2(Q, ω) ) F(Q) [cfix + (1 - cfix)A0] × Al(Q) l(l + 1)Dr + ΓT + 2 π ω + [l(l + 1)D + Γ ]2 l)0 r T (1 - cfix)(1 - A0) × N

∑

Al(Q) l(l + 1)Dr + ΓT + ΓM 2 π ω + [l(l + 1)Dr + ΓT + ΓM]2 l)0 N

∑ Figure 7. Tunneling spectrum of Q0 at 4 K at Q ) 0.5 ( 0.05 Å-1 obtained at TOFTOF.

Figure 8. Neutron scattering spectra of liquid Q0 at four different temperatures at Q ) 1.2 Å-1. The data were fitted with model eq 9 (solid line). The spectra were shifted along the linear y-axis.

}

(9)

with the hwhm ΓT for the translational diffusion which is Q-dependent, jl are spherical Bessel functions, and Dr is the broadening of the elastic line due to rotational diffusion. Fixing the radius r ) 1.03 Å10 and cfix ) 0.1, fits were carried out with four free parameters: F(Q), ΓT, Dr, and ΓM. The parameter cfix takes into account that only 9/10 of the hydrogen atoms in Q0 perform methyl group rotation, whereas all hydrogen atoms are involved in molecular rotation and longrange diffusion. The number of Lorentzians in the sums was set to N ) 20. In fact, only the first two terms are dominant in the considered Q range. The fitting range of the energy transfer was restricted to the quasi-elastic region [-1.5 meV, 1.5 meV] with respect to fit spectra symmetrically. At higher energy transfers, the inelastic contributions become non-negligible. The obtained fits with model eq 9 are displayed in Figure 8 for the studied temperatures. The values of the parameters obtained from the fitting procedure are summarized in Table 2.

Polycrystalline and Liquid Ubiquinone Q0

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Figure 9. Translational diffusion of Q0. Left: hwhm ΓT obtained from model eq 9 versus Q2. Data were fitted with a jump model (eq 10) that assumes a Gaussian distribution of jump lengths. Right: temperature dependence of the diffusion coefficients. The solid line represents an Arrhenius fit. Statistical errors are smaller than the symbol size.

Figure 10. Arrhenius plots of the parameters Dr and ΓM obtained from model eq 9. The solid line represents an Arrhenius fit to the data.

The ΓT vs Q2 curves are bent and exhibit an asymptotic behavior for larger Q values (cf. Figure 9, left) as also observed for other molecular liquids.10 Accordingly, the data can be well described by a translational jump diffusion mechanism:32

ΓT )

p [1 - exp(-DQ2τ)] τ

(10)

observed in contrast to the translational diffusion and rotational diffusion. Therefore, we can conclude that in liquid phase the process of the methyl group rotation is virtually free of a barrier as already found for other molecules.5,10 A correlation time defined as the mean residence time between two consecutive infinite fast jumps τ ) 3p/(2ΓM) ) 0.9 ps can be calculated from the mean value of ΓM ) 1.1 meV.13 5. Conclusion

This model yields diffusion coefficients D, the local residence times τ, and jump lengths l ) (6Dτ)1/2. During the fitting procedure, data points at Q ) 1.0 Å-1 were not includedsalthough it is only a small effectsdue to deviations arising from the structure factor maximum (cf. Figure 2, de Gennes narrowing).13 The temperature-dependent increase in the diffusion coefficients of Q0 follows an Arrhenius equation D ) D0 exp(-EA/ RT), where EA is an apparent activation energy, R the gas constant, T the absolute temperature, and D0 a pre-exponential factor (cf. Figure 9, right). The calculated apparent activation energy of 17.0 ( 0.1 kJ/mol is a typical value for molecular liquids.33 The expected Q independence is reasonably fulfilled for Dr for all Q values; however, the errors decrease with increasing Q. The error-weighted average values of Dr exhibit an Arrhenius temperature dependence (cf. Figure 10, left). An activation energy of 13.2 ( 1.1 kJ/mol can be calculated for Dr. The obtained values for the methyl group rotation ΓM were independent of Q at all temperatures as expected. The temperature dependence of ΓM is displayed in Figure 10 (right). Within the experimental errors, no temperature dependence can be

The methyl group rotation was studied in solid Q0 on a picosecond time scale which is the single dominant process in the solid state giving rise to a quasi-elastic broadening of the elastic line. The QENS spectra of solid Q0 were fitted using an uniaxial jump diffusion model which includes up to three Lorentzian functions depending on the respective temperature. With increasing temperature, the number of rotating methyl groups entering the time window of the spectrometer increases. By evaluation of the temperature dependencies of the quasielastic line widths for each of the three methyl groups in Q0, activation energies for the CH3 group rotation were determined. They showed an Arrhenius behavior over the whole temperature range studied here. The respective activation energies 2.7, 7, and 13.5 kJ/mol are quite different, which is caused by intramolecular interactions as well as interactions between molecules in the packing. On the basis of experimental values for the activation energies and methyl torsional bands observed in the inelastic part of the time-of-flight spectra, for all three methyl groups a rotational potential was determined which well describes the observed parameters.

922 J. Phys. Chem. B, Vol. 113, No. 4, 2009 Energies of methyl torsional bands obtained from Gaussian 03 calculations were calculated for a qualitative estimation of the energies of the methyl torsions bands. So it was possible to assign the activation energies to the respective methyl groups in Q0. QENS spectra of liquid Q0 were interpreted with a model function that took into account a long-range diffusion, isotropic rotational diffusion, and methyl group rotation. The temperature dependencies of the diffusion coefficients and the rotational diffusion coefficients exhibited an Arrhenius-like behavior which was not observed for the line widths representing the CH3 group rotation. Therefore, it can be concluded that the methyl group rotation in liquid Q0 is free of a barrier. This result is analogous to our study with pentafluoroanisole and pentafluorotoluene.10 The small improvement of the description of the data when introducing three Lorentzians representing methyl group rotation of each methyl group in the model function for liquid Q0 does not justify the increased number of free fit parameters. Thus, within the experimental errors, there is no difference of the rotational frequency of the three different methyl groups. From this study, we conclude that the small value of the mean activation energy of 4.8 kJ/mol for methyl group rotation in amorphous Q10 does not arise from the methyl groups of the ubiquinone ring. Therefore, we suggest the reduced activation energy in Q10 is due to the short length of the isoprenoid side chain or a free volume effect of Q10. Acknowledgment. We wish to express our deep gratitude to Michael Prager (FZ Ju¨lich, Germany), who tragically passed away during the publication of this article. We gratefully acknowledge him for the many helpful discussions. He also provided the program TUNCALC for the calculation of potential parameters. References and Notes (1) Siekmann, B.; Westesen, K. Pharm. Res. 1995, 12, 201. (2) Unruh, T.; Smuda, C.; Gemmecker, G.; Bunjes, H. Quasi-Elastic Neutron Scattering Conference 2006 (QENS2006). Mater. Res. Soc. 2007, 137. (3) Unruh, T.; Meyer, A.; Neuhaus, J.; Petry, W. Neutron News 2007, 18, 22. (4) Unruh, T.; Neuhaus, J.; Petry, W. Nucl. Instrum. Methods A 2007, 580, 1414; erratum: 2008, 585, 201.

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