in deuterated benzenes generated evidence suggesting 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, , of 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 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, and , 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.">
Comparison of the Molecular Dynamics of in the Solid and Liquid Phases
R. M. Hughes,^{1}P. Mutzenhardt,^{2} and A. A. Rodriguez^{3}
^{1}Department of Biochemistry, Nanaline Duke Building, Duke University Medical Center, Durham, NC 27710, USA
^{2}Laboratoire de Méthodologies RMN (UMR 7565 CNRS), Faculte des Sciences et Techniques, Université H. Poincaré, Boulevard des Aigillettes, BP 239, 54506 Vandoeuvre-lés-Nancy Cedex, France
^{3}Department of Chemistry, East Carolina University, Greenville, NC 27858, USA
Academic Editor: Clare M. McCabe
Received21 Nov 2008
Accepted07 Feb 2009
Published15 Apr 2009
Abstract
A previous study of in deuterated benzenes generated evidence suggesting 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, , of 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 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, and , 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 's molecular
dynamics have been studied extensively, its close family member has received limited attention [1–10]. Early solid-state NMR measurements indicated that 's reorientational
motion was anisotropic near room temperature [9]. Later differential scanning calorimetry
measurements on solid indicated that '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 molecules rotated isotropically [11]. In a later study, Tycko and coworkers found
that solid 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 in several deuterated benzenes revealed evidence
suggesting '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 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. Spin-Lattice Relaxation
spin-lattice relaxation in fullerenes, both in the solid and liquid-phase, is
known to be dominated by the chemical shift anisotropy, (CSA). The spin-rotation mechanism, (SR), may also contribute in the liquid phase
if the temperature is above 320 K [13–16]. Therefore, the overall spin-lattice relaxation rate in
can be expressed as sum of these two
mechanisms [17]:
While it is possible for the spin-rotation
mechanism to be present in the solid state, (1) can be reduced to 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]
where is the carbon
magnetogyric ratio, represents the field strength, is the CSA,
and is the effective reorientational correlation
time. The CSA is obtained from the three
principal components (i.e., ) of
the CST as defined by (3):
Although solid-state NMR measurements can allow the
determination of ,
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 [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 . Alternatively, can be computed with a high degree of accuracy
using standard computational methods [19–26]. The values used in this work were calculated using a Gaussian software package [27], employing the
B3LYP exchange-correlation energy density functional, the 6-31 basis set [28, 29] and employing the gauge-independent atomic orbital (GIAO) method [30–33]. The calculated values are given in
Table 1. It is worth noting the
similarity of in to
the value in (178 ppm) [15], suggesting that anisotropies in
fullerenes fall within this range. Substitution of (2) into (1) gives
In the liquid state, a fit of the
overall relaxation rate against two or more field strengths () allows the separation of the CSA from the SR contribution which then leads to the
determination of .
Carbon
(ppm)
(ppm)
(ppm)
(ppm)
(degrees)
1
−33.56
−0.97
162.06
179.32
13.94
2
−24.35
0.04
164.65
176.81
51.99
3
−32.24
3.04
165.95
180.55
62.15
4
−22.30
−2.62
167.19
179.65
72.88
5
−8.62
11.56
181.77
180.30
90.00
2.2. Reorientational Dynamics
Reorientation dynamics in liquids is described
by either diffusion constants, , or reorientational correlation
times, , since these two parameters are closely correlated. is the diffusion rate about a
given molecular axis while is the time period required for the angular correlation function to decay
to of its initial value [34, 35]. For
symmetric-top molecules, such as , two diffusion constants, and , are usually required to characterize the overall
motion. and represent
rotational diffusion about and of the top axis, respectively. The overall motion is now characterized by an
effective reorientational correlation time, , that, in the limit of small-step diffusion,
is given by [36]
For CSA relaxation, is the orientation of the CST tensor
relative to the molecular symmetry axis. In principle it is possible to determine and for a symmetric-top molecule provided and values are known for different nuclei in the
molecule. We employed the solution and
solid-state experimental correlation times, along with the Gaussian generated values, in (5) to simultaneously solve an equation
for each carbon and obtained the best-fit values for and 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-,
chlorobenzene-, o-dichlorobenzene-, and were purchased from the Aldrich Chemical Company, USA, and were used as received. Solution and solid-state NMR
spectra of 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.
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- prohibited the inclusion of this
peak across all solvents. Optimum room
temperature concentrations of in the solvents were calculated
from published data [37]. To eliminate
the potential for clustering, solutions were prepared with a
very low mole fraction of . 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
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.
Carbon 1
Carbon 2
Carbon 3
Carbon 4
T
(K)
(s)
(s)
(s)
(s)
283
15
27
27
28
293
25
28
28
29
303
29
30
31
30
313
24
24
25
23
323
18
17
17
15
Carbon 1
Carbon 2
Carbon 3
Carbon 4
T
(K)
(s)
(s)
(s)
(s)
283
18
21
27
23
293
22
27
28
25
303
26
29
31
28
313
29
39
37
33
323
25
29
35
29
Carbon 1
Carbon 2
Carbon 3
Carbon 4
T
(K)
(s)
(s)
(s)
(s)
283
14
23
23
20
293
20
25
25
24
303
22
27
28
25
313
20
26
27
24
323
16
19
24
23
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.
Carbon 1
Carbon 2
Carbon 3
Carbon 4
Carbon 5
T
(K)
(s)
(s)
(s)
(s)
(s)
283
13
8
8
8
11
293
16
12
11
12
16
303
20
20
20
20
29
313
20
22
24
24
30
323
28
35
22
30
42
4. Discussion
The temperature behavior of the relaxation times of in benzene-, chlorobenzene-, o-dichlorobenzene- 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 in the solid state, in benzene, chlorobenzene,
and 1,2-dichlorobenzene, respectively.
Carbon 1
Carbon 2
Carbon 3
Carbon 4
Carbon 5
T
(K)
(ps)
(ps)
(ps)
(ps)
(ps)
(1/s)
(1/s)
283
45
75
72
73
52
2.2
2.5
293
37
52
50
51
36
3.4
3.6
303
29
30
29
29
20
5.4
7.5
313
29
27
24
24
19
5.5
9.9
323
20
17
26
19
13
8.0
10.5
Carbon 1
Carbon 2
Carbon 3
Carbon 4
T
(K)
(ps)
(ps)
(ps)
(ps)
(1/s)
(1/s)
283
25
14
13
13
0.6
2.9
293
14
12
12
12
1.2
1.7
303
11
11
10
10
1.5
1.9
313
8
8
7
7
2.0
2.7
323
7
7
5
6
2.4
3.5
Carbon 1
Carbon 2
Carbon 3
Carbon 4
T
(K)
(ps)
(ps)
(ps)
(ps)
(1/s)
(1/s)
283
20
17
13
15
0.8
1.6
293
13
10
12
13
1.4
1.6
303
9
9
9
10
1.8
1.8
313
7
8
6
7
2.3
2.6
323
6
6
4
5
2.6
3.9
Carbon 1
Carbon 2
Carbon 3
Carbon 4
T
(K)
(ps)
(ps)
(ps)
(ps)
(1/s)
(1/s)
283
26.1
15.9
15.9
17.9
0.6
1.9
293
13.5
13.8
13.7
14.2
1.2
1.2
303
10.3
11.3
10.5
11.7
1.5
1.5
313
8.9
7.6
8.0
8.5
1.9
2.3
323
7.8
6.5
6.2
6.2
2.1
3.6
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 , where faster rapid
rotational motion is observed in the solid state than in solution, the values of in the solid state
are found to be from 2 to 3 times longer than in solution [14].
The slower
rotational dynamics in solid , as compared to solid ,
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, and , at
the two lowest temperatures, which is indicative of a quasi-isotropic
reorientation condition, the observed drastic drop in all the 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, 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 and in
solution are smaller than in the solid state suggesting that, although solvent
displacement is needed in solution, the energetic difference between and type motion in solution is lower than in the solid
phase. It is interesting to note that
diffusion values of solid-state 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, and , for in the various solvents generate some interesting observations. Since the value for is slightly
lower in the more viscous solvent of 1,2-dichlorobenzene- than in benzene- or chlorobenzene-, this observation suggests that the viscosity
parameter is not the dominant factor giving rise to the observed spinning
behavior of in these solvents. The data suggest that, in chlorobenzene- and 1,2-dichlorobenzene-,
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-,
this suggests that the available free space is more important than
intermolecular interactions in determining the spinning rate of in these solvents. The tumbling motion,
, is slowest in 1,2-dichlorobenzene-; 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, ,
of in the solid-state are 2 to 3 times longer than in
solution. A comparison of the rotational
dynamics of solids and indicates that 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 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, and , 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.
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