Orientation correlation time

The orientational diffusion observed by intermolecular Raman techniques is not the diffusion of single molecules, but rather arises from collective effects. For symmetric tops, the collective orientational correlation time rcon is related to the single-molecule orientational correlation time rsm via... 

To investigate the nature of ordering in liquids, we have studied the temperature dependence of the OHD-RIKES response of a number of symmetric-top liquids, including acetonitrile-d3, benzene, benzene-d6, carbon disulfide, chloroform, and methyl iodide (56). These liquids were chosen in particular because data on rsm were available from other sources, including NMR data for acetonitrile-d3 (57), Raman data for benzene and benzene-d3 (45), NMR data for carbon disulfide (58), NMR data for chloroform (59), and Raman data for methyl iodide (45). Since the OHD-RIKES data were not all obtained at the same temperatures as the rsm data, we used the fact that the single-molecule orientational correlation time generally obeys the Arrhenius equation to interpolate (and, where necessary, extrapolate) values of rsm at temperatures matching those of the icon data. 

The Debye-Stokes-Einstein (DSE) equation (60) predicts that the orientational correlation time of a spherical object in a continuum liquid is... 

Figure 7 Single-molecule orientational correlation times (circles) from NMR data (57) and collective orientational correlation times (triangles) from OHD-RIKES data (56) as a function of rj/T and estimated static orientational correlation parameter (squares) as a function of temperature for acetonitrile-d3.

Since the collective orientational correlation time depends on the structure of a liquid, it is plausible that the rate of structural evolution of the liquid is proportional to this quantity. Thus, at lower temperatures rcon is longer and therefore the structural fluctuations are slower. As a result, motional narrowing is less effective as the temperature is lowered. While less motional narrowing would normally lead to a slower decay in the time domain, in this case the spectral density goes down to zero frequency. Thus, motional narrowing can reduce the spectral density at low frequencies and thereby decrease the intermediate relaxation time. 

The mean diffusional structure (structure D) is that obtained with an exposure time of the order of the orientational correlation time (or of the translational counterpart) of the molecule. This structure has also been explored by computer simulation experiments,but the agreement between the results and experiment is not always satisfactory. 

A. De Santis, M. Nardone, M. Sampoli, P. Morales, and G. Signorelli. Raman spectra of fluid nitrogen Intermolecular torques and orientational correlation times. Molec. Phys., 59 913-921 (1980). 

The decay of the orientational correlation function is highly nonexponential and one needs at least four exponentials to fit it. The average orientational correlation time, t, is slower by about a factor of 20 than that of its bulk value. The orientational correlation function for the interfacial water molecules will, of course, decay in the very long time (of the order of tens of nanoseconds), either because of "evaporation" of the interfacial water molecules or rotation of the micelle. 

Figure 20. (a) Orientational correlation time t in the logarithmic scale as function of the inverse of the scaled temperature, with the scaling being done by the isotropic to nematic transition temperature with Ti-N. For the insets, the horizontal and the vertical axis labels read the same as that of the main frame and are thus omitted for clarity. Along each isochor, the solid line is the Arrhenius fit to the subset of the high-temperature data and the dotted line corresponds to the fit to the data near the isotropic-nematic phase boundary with the VFT form, (b) Fragility index m as a function of density for different aspect ratios of model calamitic systems. The systems considered are GB(3, 5, 2, 1), GB(3.4, 5, 2, 1), and GB(3.8, 5, 2, 1). In each case, N = 500. (Reproduced from Ref. 136.)... 

Figure 24. Breakdown of Arrhenius behavior of the single-particle orientational correlation times for the model calamitic system GB(3, 5, 2, 1) (N = 256). The inverse temperature dependence of the single-particle orientational correlation times ij(r), / = 1 (filled) and / = 2 (opaque), in the logarithmic scale. The straight lines are the Arrhenius fits to the subsets of data points, with each set corresponding to an isochor p = 0.31 (circle), p = 0.32 (square), and p = 0.33 (triangle). (Reproduced from Ref. 144.)...

These are called the orientational correlation times. From Eq. (7.6.12b) we thus find... 

Thus if we know the free rotor functions C f t) which is always possible we can evaluate through their Laplace transforms the orientational correlation times r. ... 

In order to interpret these NOEs in terms of the bound conformation, the contributions from the free state have to be negligible. The NOE effect vanishes for molecular orientation correlation times in the order of the inverse of the resonance frequency (i.e., a few nanoseconds for 300-600 MHz H frequency) [15]. This is the case for molecules with molecular masses of ca. 100-1000 Da (depending on temperature, solvent viscosity and spectrometer frequency). 

The increase in the value of to over the rotational correlation time can be understood quantitatively in terms of the micro-macro relation, which provides a relationship between the orientational correlation time (a microscopic, single-particle property) and the DR (a collective phenomenon). Simple continuum model arguments give the following relation between the two relaxation times [11],... 

The comparison of simulation data with experimental data is shown in Fig. 21. It has to be noted that a strongly simplified polymer model was used and therefore only semiquantitative agreement was expected and achieved. The problem of mapping the simulation model onto a real polymer has been extensively discussed in the original paper [87]. The simulated and experimental orientational correlation times exhibit the same type of behavior, i.e., they progressively decrease with increasing degree of dissociation. The results of MD simulations thus support our intuitive interpretation of fluorescence experiments on PMA performed about two decades earlier. 

It is noted that a similar approach carried out for polyethytene leads to an activation energy of 13.6 0.4 kJ/mol, for the relaxation of short (6-8 bonds) chain segments, regardless of the constraints imposed by chain connectivity [105]. This is in good agreement with the value 13.8kJ ol obtained by Brownian dynamics simulations of PE chains, and with NMR measurements of perfluoroalkane chains by Matsuo and Stodonayer, in which the mean orientational correlation times for C-F and C-H bends are found to have equal activation energies of about 12 kJ/mol. The barrier for internal rotation about typical CH2-CH2 bonds is slightly over 12 kJ/mol, from spectroscopic data and conformational analysis [3,106,107]. This value is exactly reproduced in the present theoretical treatment... 

The conventional, and very convenient, index to describe the random motion associated with thermal processes is the correlation time, r. This index measures the time scale over which noticeable motion occurs. In the limit of fast motion, i.e., short correlation times, such as occur in normal motionally averaged liquids, the well known theory of Bloembergen, Purcell and Pound (BPP) allows calculation of the correlation time when a minimum is observed in a plot of relaxation time (inverse) temperature. However, the motions relevant to the region of a glass-to-rubber transition are definitely not of the fast or motionally averaged variety, so that BPP-type theories are not applicable. Recently, Lee and Tang developed an analytical theory for the slow orientational dynamic behavior of anisotropic ESR hyperfine and fine-structure centers. The theory holds for slow correlation times and is therefore applicable to the onset of polymer chain motions. Lee s theory was generalized to enable calculation of slow motion orientational correlation times from resolved NMR quadrupole spectra, as reported by Lee and Shet and it has now been expressed in terms of resolved NMR chemical shift anisotropy. It is this latter formulation of Lee s theory that shall be used to analyze our experimental results in what follows. The results of the theory are summarized below for the case of axially symmetric chemical shift anisotropy. 

FIGURE 2.6 (Upper panel) Plot for the reorientational correlation function against time for a representative composition (Xj = 0.1). (Lower panel) Product of the translational diffusion coefficient Dj. and the average orientational correlation time x, of the first-rank correlation function as a function of composition. Note that the solid line and dashed line indicate the hydrodynamic predictions with the stick and slip boundary conditions, respectively. 

Composition Dependence of the Orientational Correlation Time of Rank = 1 and 2, and the Ratio between the Two... 

TABLE 5.10 Rotational orientation Correlation Times x/ps of Several Polyatomic Ions in Various Solvents... 

---