Stress autocorrelation function

As for the properties themselves, there are many chemically useful autocorrelation functions. For instance, particle position or velocity autocorrelation functions can be used to determine diffusion coefficients (Ernst, Hauge, and van Leeuwen 1971), stress autocorrelation functions can be used to determine shear viscosities (Haile 1992), and dipole autocorrelation functions are related to vibrational (infrared) spectra as their reverse Fourier transforms (Berens and Wilson 1981). There are also many useful correlation functions between two different variables (Zwanzig 1965). A more detailed discussion, however, is beyond the scope of this text. 

Figure 8 The integrated stress-stress autocorrelation function as described in Eqs. [121] for SPC/E water at 303.15 K as described in Ref. 42. Note the convergence of the integral over time deteriorates owing to insufficient data sampling. The experimental value of the shear viscosity is 7.97 x 10 Pa s, whereas the calculated value from this curve 6.6 0.8 x 10 Pa s.

Das and Bhattacharjee236 derive the frequency and shear dependent viscosity of a simple fluid at the critical point and find good agreement with recent experimental measurements of Berg et al.237 Ernst238 calculates universal power law tails for single and multi-particle time correlation functions and finds that the collisional transfer component of the stress autocorrelation function in a classical dense fluid has the same long-time behaviour as the velocity autocorrelation function for the Lorentz gas, i.e. 

This surprising result prompted Mazenko, Ramaswamy and Toner to examine the anharmonic fluctuation effects in the hydrodynamics of smectics. We have already shown that the undulation modes are purely dissipative with a relaxation rate given by (5.3.39). To calculate the effect of these slow, thermally excited modes on the viscosities, we recall that a distortion u results in a force normal to the layers given by (5.3.32). This is the divergence of a stress, which, from (5.3.53), contains the non-linear term 0,(Vj uf. Thus, there is a non-linear contribution (Vj uf to the stress. Now the viscosity at frequency co is the Fourier transform of a stress autocorrelation function, so that At (co), the contribution of the undulations to the viscosity, can be evaluated. It was shown by Mazenko et that Atj(co) 1 /co. In other words, the damping of first and second sounds in smectics, which should go as >/(oo)oo , will now vary linearly as co at low frequencies. 

Earlier we mentioned that Voth and co-workers conducted equilibrium MD simulations on [C2mim][N03] at 400 K and computed the self-diffusivity and shear viscosity using both a fixed charge and polarizable force field. They computed the viscosity not from integrating the stress-stress autocorrelation function as is normally done, but rather from integrating the so-called transverse current correlation function, details of which are foimd in a work by Hess. ° They used the standard Einstein formula (Eq. [15]) for the self-diffusivity and were careful to ensure that diffusive behavior was achieved when computing the self-diffusivity. Their calculated values of ca. 1 x 10 m /s for the polarizable model and ca. 5 x 10 m /s are reasonable. The finding that the polarizable model yielded faster dynamics than with the nonpolarizable model... 

The top plot in Figure 17 contains the stress-stress autocorrelation function for [Cimim][Cl] at 425 K calculated by Bhargava and Balasubrama-nian. ° The rapid oscillations are due to high-frequency intramolecular motions of the cation. The correlation function shows a rapid short-time decay but a very slow long-time decay, as can be seen in the bottom graph in... 

Each GK expression is an integral of an autocorrelation function. GK was historically the first technique to calculate TTCs. A theoretically important and numerically disadvantageous property of the autocorrelation functions is that their decay is not exponential.Rather the autocorrelation functions have power law long time tails which in two-dimensional systems produce diverging GK integrals. It is beyond the scope of this article to discuss this difficult and controversial subject, but as a practicality we mention that in three-dimensional calculations the slow decay of the stress-stress autocorrelation function causes severe difficulties, especially for liquids of complex molecules. 

Heisenberg formalism. The Heisenberg view leads to an expression equivalent to the Schrodinger formalism that stresses the time evolution of quantum systems it has a clear correspondence with classical mechanics it is most conveniently expressed in terms of the dipole autocorrelation function (Gordon 1968). 

These results confirm the observation that polyelectrolyte aqueous solutions show two separate decay modes in the autocorrelation function and support our contention that ionic polymer systems generally behave similarly in polar solvents [23], To support this, it may be added that similar dynamic scattering behavior was recently reported for another type of ionomer, polyurethane ionomer, dissolved in a polar solvent, dimethylacetamide (e = 38) [92], Finally, it should be stressed that the explanation given above for light scattering (both static and dynamic) behavior of salt-free polyelectrolytes is based on the major role of intermolecular electrostatic interactions in causing characteristic behavior. No intramolecular interactions are explicitly included to explain the behavior. This is in accord with our contention that much of the polyelectrolyte behavior, especially structure-related aspects, is determined by intermolecular interactions [23]. 

Fig. 6.10. Time dependence of curvature fluctuations of a giant lipid vesicle with stress-sensitive alamethicin channels in its membrane. Inside the vesicle there is a ferricyanide solution undergoing a photochemical reaction under illumination, which produces a pH gradient and a photopotential across the membrane. The graph shows the second Legendre polynomial amplitude of the angular autocorrelation function of the vesicle radius as a function of time. Brief episodes (peaks) of extensive thermal fluctuations in a tension-free membrane are separated by long periods of a tensed, non-fluctuating vesicle membrane. (V. Vitkova, A.G. Petrov, unpublished.)...

When the property calculated is a single particle property, such as the velocity autocorrelation function, or the particle s mean squared displacement, averaging over the particles does help a lot. Averaging over even a moderate number of particles of about 1000 decreases the error by more than an order of magnitude. Unfortunately, this is not possible for all properties. For instance, the calculation of viscosity calls for the stress correlation function [Eq. (67)], which is not a single particle property. This is why self-diffusion coefficients are usually estimated with much better accuracy than viscosity. 

The fact that GeTe is getting more stressed-rigid upon doping is not the only parameter that influences the stability of the compound. If we compute the vibrational density of states by performing a Fourier transform of the velocity autocorrelation function computed for the AIMD trajectory, we obtain curves plotted in Fig. 18.16. As already found experimentally [62], amorphous GeTe is at the same time elastically... 

Examples of linear response functions (susceptibilities) include the frequency dependent electrical conductivity (the Fourier transform of an equilibrium current autocorrelation function), dielectric susceptibility, which is the transform of a dipole moment autocorrelation function, along with stress, heat flux, and an assortment of velocity correlation functions. 

S. T. Cui, P. T. Cummings, andH. D. Cochran, Mol. Phys., 88,1657 (1996). The Calculation of the Viscosity from the Autocorrelation Function Using Molecular and Atomic Stress Tensors. 

Magnetic resonance methods have been used extensively to probe the structure and dynamics of thermotropic nematic liquid crystals both in the bulk and in confined geometry. Soon after de Gennes [27] stressed the importance of long range collective director fluctuations in the nematic phase, a variable frequency proton spin-lattice relaxation Tx) study [32] showed that the usual BPP theory [33] developed for classical liquids does not work in the case of nematic liquid crystals. In contrast to liquids, the spectral density of the autocorrelation function is non-Lorentzian in nematics. As first predicted independently by Pincus [34] and Blinc et al. [35], collective, nematic type director fluctuations should lead to a characteristic square root type dependence of the spin-lattice relaxation rate rf(DF) on the Larmor frequency % ... 

MD simulation can gain insight into the viscoelastic behavior of nanopartide-polymer composites. The shear stress relaxation modulus can be calculated using the time autocorrelation function of the stress tensor, while the viscosity is calculated based on the Einstein relations. Compared to conventional composites, the viscoelastic properties are strongly perturbed by the nanopartides and depend upon the nature of nanopartide-polymer interactions. The viscosity and dynamic shear modulus can be dramatically increased for composites with attractive... 

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