Molecular relaxations averages

Goldflam R., Kouri D. J. On accurate quantum mechanical approximations for molecular relaxation phenomena. Averaged... 

We compare the orientational relaxation of central deuterated part of the PS HDH 188 copoljraer with that of the end part of the PS DH 184 copolymer. Both types of chains have almost the same molecular weight as well as deuterated blocks of comparable length. The master curves of the orientational relaxation average (calculated as the weight average of the measured orientation of each block) can be superimposed as shown in Figure 1. 

The degree of molecular mobility (assessed as the average molecular relaxation time r) of amorphous systems in the region near Tg follows a non-Arrhenius temperature dependence. This so-called fragility (dr/dr at Tg) of amorphous materials is a defining characteristic. ... 

To fully understand the performance of amorphous materials, it is necessary to be able to measure the molecular mobility of the samples on interest. This is because at temperatures as far as 50 K below the glass transition temperature, pharmaceutical glasses exhibit significant molecular mobility that can contribute to both chemical and physical instability.The main techniques that have been developed for monitoring molecular motions in amorphous materials are nuclear magnetic resonance (NMR) and calorimetric techniques (e.g., DSC and isothermal microcalorimetry). Average molecular relaxation times and relaxation time distribution functions obtained from these... 

Fig. 8 The temperature dependence of average molecular relaxation times for amorphous pharmaceutical materials. (From Ref.. )...

Fig. 7. Scheme of molecular relaxation processes after Flygare8 ). The characteristic times shown are average values for room temperature and atmospheric pressure... 

A closer look at this concentration dependence of the first positive maximum N, as well as the shear rate at which this occurs, was conducted by Baek et al [46]. Both were seen to be monotonically increasing functions up to concentrations of 40%. They demonstrated qualitative agreement with the predictions of Doi theory with the Hinch-Leal closure and the Maier-Saupe potential. They also note that the ratio of shear rate at which becomes negative to the shear rate of the first positive maximum remains constant at about 3.5. (Our data from 1978 and 1980 yielded an average ratio of 3.2 with a standard deviation of 0.9 for nine PBG solutions and an average ratio of 2.2 with a standard deviation of 0.3 for three PCBZL solutions.) They determined that the rapid increase in (dy/dt) with concentration cannot be attributed to a decrease in molecular relaxation time with... 

After choosing an adequate model for each different component of the system and integrating them into a final atomistic model that will be simulated, an important issue is the selection of a discretization scheme to implement the computer representation of the ion channel and its environment. Within the framework of a computer experiment, the adjective realistic is strictly related to the phenomena one wants to study, and to the resolution required to reproduce those phenomena. The basic idea for modeling many-body systems is to build a set of rules that apply to each component and let the system evolve dynamically. Ensemble and time averages are then computed to obtain observables that are compared with experiment to validate the model. A characteristic of ion channel systems is that the measurable quantities of direct biological interest evolve in times up to 12 orders of magnitude larger than the smallest atomic or molecular relaxation times (milliseconds versus femtoseconds). In comparison, solid state many-body systems collectively relax in a faster fashion, and the difference between the microscopic... 

Cox [11] has discussed the relaxation mechanisms in this type of system in detail with experimental results and simulations to clearly demonstrate the various aspects. The most likely candidate that would account for the relaxation due to molecular dynamics is shown to be the fluctuation of the hyperfine interaction or the so called Fermi contact term. The hyperfine constant is in fact a thermal average over the different vibrational modes of the molecule. Therefore the molecular vibrations or librations will modulate the hyperfine constant. Hyperfine interaction is in general anisotropic, except in the gas or liquid state when the fast molecular tumbling averages out the anisotropy leaving the isotropic part. One can think of two separate mechanisms of relaxation depending on the modulation of either the isotropic part or the anisotropic part. 

It is well known that the linear viscoelastic properties of polymer melts and concentrated solutions are strong function of molecular structure, average molecular mass and molecular mass distribution (MWD). The relaxation time spectrum is a characteristic quantity describing the viscoelastic properties of polymer melts. Given this spectrum, it is easy to determine a series of rheological parameters. The relaxation time spectrum is not directly accessible by experiments. It is only possible to obtain the spectrum from noisy data. 

The friction coefficient is the inverse particle s relaxation time, jS = 9py/(2pp ), where py is the fluid s dynamic viscosity. Since the Langevin equations are linear, particle velocity and position may be formally solved as functionals of the random force, and in the diffusive limit f >> i. e., for times much larger than the particle relaxation time, they allow for the analytical evaluation of ensemble averaged products of particle position and velocity and two-point correlation functions, in terms of the random-force strength q. The authors carefully justify why they use the classical (equilibrium) form of the fluctuation-dissipation theorem (FDT) in a Langevin description the time scale of the white noise is considered to be much shorter than the time scale of the imjxjsed flow. Thus, the non-equilibrium corrections would be of the order of the ratio of the fluid molecular relaxation time to the time scale of the imposed shear and may be neglected. In this case both the time scales are clearly separated and q may be determined solely from the classical form of the FDT,... 

It follows from this treatment that the magnitude of the temperature derivative of fo is smaller for very low molecular weights than for high (when compared at the same temperature), and that the molecular weight average which controls this effect is Jf . Thus, for example, the apparent activation energy for viscoelastic relaxation at a temperature To can be expressed by combining equations 35,56, and 44 ... 

This is physically the ratio of energy lost to energy stored per deformation cycle. A peak in tan 8 occurs when the impressed frequency matches the frequency of molecular relaxation through thermally activated processes. If X is the average molecular relaxation time at temperature T, then a loss peak will be observed at this temperature if the impressed vibration frequency (/max) satisfies the relation ... 

It is important to note that the effects described by Equation [6] are only observed in rigid solids. Both in solution and in highly mobile solid phases, random molecular motions average out all anisotropic contributions, leaving only Equation [5] (a further motional effect may be a fast quadrupole relaxation on nucleus S, which would erase the multiplet structure of the I signal). 

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