Comparison of the Molecular Dynamics of C_𝟕𝟎 in the Solid and Liquid Phases

Abstract

A previous study of C₇₀ in deuterated benzenes generated evidence suggesting C₇₀ exhibited unique reorientational behavior depending on its environment. We present a comparison of the dynamic behavior of this fullerene, in the solid and solution phases, to explore any unique features between these two phases. The effective correlation times, 𝜏^eff_𝐶, of C₇₀ in the solid state are 2 to 3 times longer than in solution. In the solid state, a noticeable decrease in all the carbons' correlation times is seen between 293 K to 303 K; suggesting a transition from isotropic to anisotropic reorientational behavior at this temperature change. Although C₇₀ in solution experiences van der Waals type interactions, these interactions are not strong enough to slow the solution-state motion below what is observed in the solid state. All observed differences in the diffusion constants, 𝐷_𝑋 and 𝐷_𝑍, in solution are smaller than in the solid state suggesting a lower energy of activation between these two modes of reorientation in the liquid phase. A small-step diffusion “like” condition appears to be thermally generated in the solid phase at 323 K.

1. Introduction

Molecular reorientational dynamics are sensitive to several factors including temperature, pressure, free volumes, solute-solvent interactions, and, henceforth, are commonly employed to probe a solute's rotational energetics or the effects of the immediate environment on the rotational motion. While fullerene C₆₀'s molecular dynamics have been studied extensively, its close family member C₇₀ has received limited attention [1–10]. Early solid-state NMR measurements indicated that C₇₀'s reorientational motion was anisotropic near room temperature [9]. Later differential scanning calorimetry measurements on solid C₇₀ indicated that C₇₀'s dynamic behavior dependent on the thermally stable solid-phase structure (e.g., fcc, rhomobohedral, or monoclinic phase) at a given temperature while zero-field measurements suggested C₇₀ molecules rotated isotropically [11]. In a later study, Tycko and coworkers found that solid C₇₀ molecules reorientational motion was anisotropic between 223–330 K (monoclinic and rhombohedral phases) but became isotropic at temperatures beyond 330 K (face-centered phase)  [10]. In terms of molecular dynamics in solution, our recent investigation into the molecular dynamics of C₇₀ in several deuterated benzenes revealed evidence suggesting C₇₀'s reorientational motion oscillated between anisotropic and isotropic behavior depending on the solvent and the temperature [12]. In order to explore any unique features of the dynamic behavior of C₇₀ in the solid and liquid phase, we present a comparison of our solution measurements to values obtained in the solid state [10].

2. Theoretical Background

2.1. ¹³C Spin-Lattice Relaxation

¹³C spin-lattice relaxation in fullerenes, both in the solid and liquid-phase, is known to be dominated by the chemical shift anisotropy, 𝑅^CSA₁ (CSA). The spin-rotation mechanism, 𝑅^SR₁ (SR), may also contribute in the liquid phase if the temperature is above 320 K [13–16]. Therefore, the overall ¹³C spin-lattice relaxation rate in C₇₀ can be expressed as sum of these two mechanisms [17]: 𝑅₁=1𝑇₁=𝑅^CSA₁+𝑅^SR₁. (1) While it is possible for the spin-rotation mechanism to be present in the solid state, (1) can be reduced to 𝑅₁=𝑅^CSA₁ at high magnetic fields and at moderate temperatures. Under extreme narrowing arguments, and assuming axial symmetry of the chemical shift tensor (CST), the CSA relaxation process is described by [18] 𝑅^CSA₁2=(15)𝛾²_C𝐵²₀Δ𝜎²𝜏^eff_𝐶, (2) where 𝛾_Cis the carbon magnetogyric ratio, 𝐵₀ represents the field strength, Δ𝜎 is the CSA, and 𝜏^eff_𝐶 is the effective reorientational correlation time. The CSA is obtained from the three principal components (i.e., 𝜎_𝑧𝑧≥𝜎_𝑦𝑦≥𝜎_𝑥𝑥) of the CST as defined by (3): Δ𝜎=𝜎_𝑧𝑧−0.5(𝜎_𝑥𝑥+𝜎_𝑦𝑦). (3) Although solid-state NMR measurements can allow the determination of Δ𝜎, the approach can be experimentally challenging and is primarily limited to chemical systems with high symmetry. In fact, Tycko and coworkers experienced these challenges when attempting to evaluate the CST for the carbons in C₇₀ [10]. The relative high noise in their measurements only allowed for an average value of 200 ppm to be determined for each of the carbons in C₇₀. Alternatively, Δ𝜎 can be computed with a high degree of accuracy using standard computational methods [19–26]. The Δ𝜎 values used in this work were calculated using a Gaussian software package [27], employing the B3LYP exchange-correlation energy density functional, the 6-31G^∗ basis set [28, 29] and employing the gauge-independent atomic orbital (GIAO) method [30–33]. The calculated Δ𝜎 values are given in Table 1. It is worth noting the similarity of Δ𝜎 in C₇₀ to the value in C₆₀ (178 ppm) [15], suggesting that anisotropies in fullerenes fall within this range. Substitution of (2) into (1) gives 𝑅₁2=(15)𝛾²_C𝐵²₀(Δ𝜎)²𝜏^eff_𝐶+𝑅^SR₁. (4) In the liquid state, a fit of the overall relaxation rate against two or more field strengths (𝐵²₀) allows the separation of the CSA from the SR contribution which then leads to the determination of 𝜏^eff_𝐶.

Table 1 

Average values for the calculated tensor components, chemical shift anisotropies, and the chemical shift tensor (CST) orientation of carbons in C₇₀.

2.2. Reorientational Dynamics

Reorientation dynamics in liquids is described by either diffusion constants, 𝐷_𝑖, or reorientational correlation times, 𝜏_𝐶, since these two parameters are closely correlated. 𝐷_𝑖 is the diffusion rate about a given molecular axis while 𝜏_𝐶 is the time period required for the angular correlation function to decay to 1/𝑒 of its initial value [34, 35]. For symmetric-top molecules, such as C₇₀, two diffusion constants, 𝐷_𝑍 and 𝐷_𝑋, are usually required to characterize the overall motion. 𝐷_𝑍 and 𝐷_𝑋 represent rotational diffusion about and of the top axis, respectively. The overall motion is now characterized by an effective reorientational correlation time, 𝜏^eff_𝐶, that, in the limit of small-step diffusion, is given by [36] 𝜏^eff_𝐶=0.25(3cos²𝜃−1)²6𝐷_𝑋+3sin²𝜃cos²𝜃5𝐷_𝑋+𝐷_𝑍+0.75sin⁴𝜃2𝐷_𝑋+4𝐷_𝑍. (5) For CSA relaxation, 𝜃 is the orientation of the CST tensor relative to the molecular symmetry axis. In principle it is possible to determine 𝐷_𝑍 and 𝐷_𝑋 for a symmetric-top molecule provided 𝜏^eff_𝐶 and 𝜃 values are known for different nuclei in the molecule. We employed the solution and solid-state experimental correlation times, along with the Gaussian generated 𝜃 values, in (5) to simultaneously solve an equation for each carbon and obtained the best-fit values for 𝐷_𝑍 and 𝐷_𝑋 at each temperature. While (5) assumes a small-step diffusion process, one can employ this equation to characterize reorientational motion outside this limit provided the calculated values are viewed as rough or base-line estimates of the diffusion process.

3. Experimental Methods

Benzene-d₆, chlorobenzene-d₅, o-dichlorobenzene-d₄, and C₇₀ were purchased from the Aldrich Chemical Company, USA, and were used as received. Solution and solid-state ¹³C NMR spectra of C₇₀ showed the 5 unique carbon resonances at approximately 151 ppm, 147 ppm, 148 ppm, 146 ppm, and 131 ppm. These carbon resonances correspond to the carbons labeled in Figure 1.

Figure 1 

Carbon assignments in C70. Carbon 1 (151 ppm), carbon 2 (146.5 ppm), carbon 3 (147.5 ppm), carbon 4 (145.5 ppm), and carbon 5 (131 ppm). The approximate carbon CSA tensor orientations (𝜃), relative to the molecular symmetry axis, are indicated by the various arrows.

In solution, resonances for carbons 1–4 were only used for the analysis since slight solvent peak interference and weak peak intensity of the carbon 5 in chlorobenzene-d₅ prohibited the inclusion of this peak across all solvents. Optimum room temperature concentrations of C₇₀ in the solvents were calculated from published data [37]. To eliminate the potential for C₇₀ clustering, solutions were prepared with a very low mole fraction of 1.0×10⁻⁴. Samples were contained in 5 mm tubes, connected to a vacuum line and thoroughly degassed by several freeze-pump-thaw cycles to remove molecular oxygen. The tubes were then sealed under vacuum.

Solution ¹³C spin-lattice relaxation measurements were performed on instruments operating at 11.75 and 7.05 Tesla. Experiments were conducted at five different temperatures (283 K, 293 K, 303 K, 313 K, and 323 K). Temperature accuracy for these measurements is ±0.1 K. Relaxation times were obtained using the standard inversion-recovery pulse sequence as described in our earlier work [1–5]. Relaxation times in the various solvents are shown in Tables 2, 3, and 4.

Table 2 

Experimental ¹³C relaxation times for the different carbons at various temperatures for C₇₀ in benzene-d₆. (Relaxation times measured at 11.7 Tesla.)

Table 3 

Experimental ¹³C relaxation times for the different carbons at various temperatures for C₇₀ in chlorobenzene-d₅. (Relaxation times measured at 11.7 Tesla.)

Table 4 

Experimental ¹³C relaxation times for the different carbons at various temperatures for C₇₀ in o-dichlorobenzene-d₄. (Relaxation times measured at 11.7 Tesla.)

Solid-state relaxation times were obtained from the work of Tycko and coworkers and their temperature-dependent data, ranging from 223 to 342, was interpolated to the corresponding solution phase temperatures to permit the comparisons [10]. These relaxation times are given in Table 5.

Table 5 

Experimental ¹³C relaxation times for the different carbons at various temperatures for C₇₀ in the solid state. (Values interpolated from [10].)

4. Discussion

The temperature behavior of the relaxation times of C₇₀ in benzene-d₆, chlorobenzene-d₅, o-dichlorobenzene-d₄ and in the solid-state are illustrated in Figures 2, 3, 4, and 5; respectively. Tables 6, 7, 8, and 9 contain the reorientational correlation times of the various carbons, as well as the calculated diffusion coefficients, of C₇₀ in the solid state, in benzene, chlorobenzene, and 1,2-dichlorobenzene, respectively.

Table 6 

Experimental correlation times and predicted rotational diffusion constants for the different carbons at various temperatures for C₇₀ in the solid state.

Table 7 

Experimental correlation times and predicted rotational diffusion constants for the different carbons at various temperatures for C₇₀ in benzene-d₆.

Table 8 

Experimental correlation times and predicted rotational diffusion constants for the different carbons at various temperatures for C₇₀ in chlorobenzene-d₅.

Table 9 

Experimental correlation times and predicted rotational diffusion constants for the different carbons at various temperatures for C₇₀ in o-dichlorobenzene-d₄.

Figure 2 

13C relaxation times for C70 carbons at various temperatures in benzene-d6.

Figure 3 

13C relaxation times of C70 carbons at various temperatures in chlorobenzene-d5.

Figure 4 

13C relaxation times of C70 carbons at various temperatures in dichlorobenzene-d4.

Figure 5 

Relaxation times of carbons in C70 in the solid state.

The reorientational times for all carbons, both in the solid and in solution, are seen to systemically decrease with rising temperature indicating more rapid reorientational motion with escalating temperature. Unlike in C₆₀, where faster rapid rotational motion is observed in the solid state than in solution, the 𝜏^eff_𝐶 values of C₇₀ in the solid state are found to be from 2 to 3 times longer than in solution [14].

The slower rotational dynamics in solid C₇₀, as compared to solid C₆₀, can be attributed to differences in the orientational ordering of their lattice structures. Also of interest in the solid state is the noticeable decrease in all the carbons' correlation times in going from 293 K to 303 K. Noting the closeness of the two diffusion constants, 𝐷_𝑋 and 𝐷_𝑍, at the two lowest temperatures, which is indicative of a quasi-isotropic reorientation condition, the observed drastic drop in all the 𝜏^eff_𝐶 values suggest a definite transition from isotropic to anisotropic reorientational behavior at this temperature change. The enhanced difference between the two diffusion coefficients as temperature rises is particularly interesting since this correlates to the phase transition occurring from rhombohedral to face-centered cubic suggesting in increase in molecular disorder. We must however be cautious in overly interpreting this observation since diffusion values were calculated via (5) which assumes small-step diffusion and, as indicated by Tyco and coworkers, molecular reorientation in this phase is attributed to “thermally activated” orientational jumps rather than to step-wise diffusion. Nonetheless, the calculated diffusion values, and their differences, indicate preferred axial reorientation, becoming more pronounced with rising temperature.

As pointed out above, solution correlation times are significantly shorter than in the solid phase indicating faster reorientational motion in the liquid environment. Unlike the solid-state, C₇₀ in solution is bathed in solvent molecules which, through van der Waals type interaction, experiences solute-solvent interactions which can affect the overall motion. However, these interactions do not appear to be strong enough to slow the solution-state motion below what is observed in the solid state. Additionally, at any common temperature, one observes that the differences between 𝐷_𝑋 and 𝐷_𝑍 in solution are smaller than in the solid state suggesting that, although solvent displacement is needed in solution, the energetic difference between 𝐷_𝑋 and 𝐷_𝑍 type motion in solution is lower than in the solid phase. It is interesting to note that diffusion values of solid-state C₇₀ at 323 K are comparable to values observed at 283 K in solution suggesting that, at 323 K, enough thermal energy is present to promote small-step diffusion “like” behavior in the solid phase.

A comparison of the diffusion coefficients, 𝐷_𝑍 and 𝐷_𝑋, for C₇₀ in the various solvents generate some interesting observations. Since the value for 𝐷_𝑍 is slightly lower in the more viscous solvent of 1,2-dichlorobenzene-d₄ than in benzene-d₆ or chlorobenzene-d₅, this observation suggests that the viscosity parameter is not the dominant factor giving rise to the observed spinning behavior of C₇₀ in these solvents. The data suggest that, in chlorobenzene-d₅ and 1,2-dichlorobenzene-d₄, there is a balance between the strength of intermolecular forces and solvent structure which determines the reorientational behavior. Since the free volume is greater in 1,2-dichlorobenzene-d₄, this suggests that the available free space is more important than intermolecular interactions in determining the spinning rate of C₇₀ in these solvents. The tumbling motion, 𝐷_𝑋, is slowest in 1,2-dichlorobenzene-d₄; consistent with the higher viscosity of this solvent. One must however be cautious of oversimplifying this observation since, as we saw for the other type of motion, other solvent-related factors are also present.

5. Conclusions

Reorientational times for all carbons, in solution and in the solid phase, decrease with rising temperature indicating faster rotational motion with escalating temperature. The effective correlation times, 𝜏^eff_𝐶, of C₇₀ in the solid-state are 2 to 3 times longer than in solution. A comparison of the rotational dynamics of solids C₆₀ and C₇₀ indicates that C₇₀ reorients slower which can be attributed to differences in the orientational ordering of their lattice structures. A noticeable decrease in all the carbons' correlation times is observed between 293 K to 303 K; suggesting a definite transition from isotropic to anisotropic reorientational behavior at this temperature change. Solution correlation times are seen to be significantly shorter than in the solid phase indicating faster reorientational motion in the liquid environment. Although C₇₀ in solution experiences van der Waals type interactions, these interactions do not appear strong enough to slow the solution-state motion below what is observed in the solid state. All observed differences in the diffusion constants, 𝐷_𝑋 and 𝐷_𝑍, in solution are smaller than in the solid state suggesting a lower energy of activation between these two modes of reorientation in the liquid phase. A small-step diffusion “like” condition appears to be thermally generated in the solid phase at 323 K.

Acknowledgment

The authors are grateful to the National Science Foundation for support of this project under Grant CH-9707163.