Viscosity correlation functions

Dynamic information such as reorientational correlation functions and diffusion constants for the ions can readily be obtained. Collective properties such as viscosity can also be calculated in principle, but it is difficult to obtain accurate results in reasonable simulation times. Single-particle properties such as diffusion constants can be determined more easily from simulations. Figure 4.3-4 shows the mean square displacements of cations and anions in dimethylimidazolium chloride at 400 K. The rapid rise at short times is due to rattling of the ions in the cages of neighbors. The amplitude of this motion is about 0.5 A. After a few picoseconds the mean square displacement in all three directions is a linear function of time and the slope of this portion of the curve gives the diffusion constant. These diffusion constants are about a factor of 10 lower than those in normal molecular liquids at room temperature. 

For liquids, few simple and widely accepted theories have been developed. The shear viscosity can be related to the way in which spontaneous fluctuations relax in an equilibrium system, leading to the time correlation function expression " " ... 

Chain-growth polymerizations are diffusion controlled in bulk polymerizations. This is expected to occur rapidly, even prior to network development in step-growth mechanisms. Traditionally, rate constants are expressed in terms of viscosity. In dilute solutions, viscosity is proportional to molecular weight to a power that lies between 0.6 and 0.8 (22). Melt viscosity is more complex (23) Below a critical value for the number of atoms per chain, viscosity correlates to the 1.75 power. Above this critical value, the power is nearly 3 4 for a number of thermoplastics at low shear rates. In thermosets, as the extent of conversion reaches gellation, the viscosity asymptotically increases. However, if network formation is restricted to tightly crosslinked, localized regions, viscosity may not be appreciably affected. In the current study, an exponential function of degree of polymerization was selected as a first estimate of the rate dependency on viscosity. 

All these features were observed experimentally for solutions of 3-amino-/V-methylphthalimide, 4-amino-/V-methylphthalimide, and for nonsubstituted rhoda-mine. The results were observed for cooled, polar solutions of phthalimides, in which the orientational relaxation is delayed. Exactly the same spectral behavior was observed [50] by picosecond spectroscopy for low viscosity liquid solutions at room temperature, in which the orientational relaxation rate is much higher. All experimental data indicate that correlation functions of spectral shifts Av-l(t), which are used frequently for describing the Time Dependent Stokes Shift, are essentially the functions of excitation frequency. 

The diffusion coefficient is estimated using Stokes law, D = kTjQmrja, where rj is the viscosity and a the radius of the rotating sphere. This rough model allows a calculation of the correlation function. It turns out that a Boltzmann distribution remains a Boltzmann distribution so that the system of equations (12)... 

A particular question of interest is whether the DNA torsional motions observed on the nanosecond time scale are overdamped, as predicted by simple Langevin theory, and as observed for Brownian motions on longer time scales, or instead are underdamped, so that damped oscillations appear in the observed correlation functions. A related question is whether the solvent water around the DNA exhibits a normal constant viscosity on the nanosecond time scale, or instead begins to exhibit viscoelastic behavior with a time-, or frequency-, dependent complex viscosity. In brief, are the predictions for... 

Different equilibrium, hydrodynamic, and dynamic properties are subsequently obtained. Thus, the time-correlation function of the stress tensor (corresponding to any crossed-coordinates component of the stress tensor) is obtained as a sum over all the exponential decays of the Rouse modes. Similarly, M[rj] is shown to be proportional to the sum of all the Rouse relaxation times. In the ZK formulation [83], the connectivity matrix A is built to describe a uniform star chain. An (f-l)-fold degeneration is found in this case for the f-inde-pendent odd modes. Viscosity results from the ZK method have been described already in the present text. 

The complex viscosity, i.e., the viscosity observed in the presence of an oscillatory shear rate, is a dynamic property that can be straightforwardly obtained from the Rouse, or Rouse-Zimm theory as the Fourier transform of the stress time-correlation function. Thus, these theories give [15]... 

The transport coefficients of diffusion, heat conductivity, and viscosity can all be computed by the method of correlation functions. 

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. 

It appears that the formal theories are not sufficiently sensitive to structure to be of much help in dealing with linear viscoelastic response Williams analysis is the most complete theory available, and yet even here a dimensional analysis is required to find a form for the pair correlation function. Moreover, molecular weight dependence in the resulting viscosity expression [Eq. (6.11)] is much too weak to represent behavior even at moderate concentrations. Williams suggests that the combination of variables in Eq. (6.11) may furnish theoretical support correlations of the form tj0 = f c rjj) at moderate concentrations (cf. Section 5). However the weakness of the predicted dependence compared to experiment and the somewhat arbitrary nature of the dimensional analysis makes the suggestion rather questionable. 

Williams has derived the molecular weight and concentration dependence of a viscoelastic time constant t0 (actually the characteristic time governing the onset of shear rate dependence in the viscosity) from his theory (217-219). Employing a dimensional argument, he equates the parameters which control the shear rate dependence of chain configuration and the intermolecular correlation function. The result agrees with the observed form of characteristic relaxation time in concentrated systems [Eq.(6.62)] ... 

Time-dependent correlation functions are now widely used to provide concise statements of the miscroscopic meaning of a variety of experimental results. These connections between microscopically defined time-dependent correlation functions and macroscopic experiments are usually expressed through spectral densities, which are the Fourier transforms of correlation functions. For example, transport coefficients1 of electrical conductivity, diffusion, viscosity, and heat conductivity can be written as spectral densities of appropriate correlation functions. Likewise, spectral line shapes in absorption, Raman light scattering, neutron scattering, and nuclear jmagnetic resonance are related to appropriate microscopic spectral densities.2... 

One source of information on intermolecular potentials is gas phase virial coefficient and viscosity data. The usual procedure is to postulate some two-body potential involving 2 or 3 parameters and then to determine these parameters by fitting the experimental data. Unfortunately, this data for carbon monoxide and nitrogen can be adequately represented by spherically symmetric potentials such as the Lennard-Jones (6-12) potential.48 That is, this data is not very sensitive to the orientational-dependent forces between two carbon monoxide or nitrogen molecules. These forces actually exist, however, and are responsible for the behavior of the correlation functions and - In the gas phase, where orientational forces are relatively unimportant, these functions resemble those in Figure 6. On the other hand, in the liquid these functions behave quite differently and resemble those in Figures 7 and 8. 

As discussed by Kirkpatrick [10], this slow mode is important in the theories that include mode coupling effects. Such theories have been used to quantitatively understand the anomalous long-time tails of the stress-stress correlation function and the shear-dependent viscosity [3, 30, 34], observed in computer simulations. As mentioned earlier, a theory of glass transition has also been developed based on the softening of the heat mode. 

Figure 2. A pictorial representation of the mode coupling theory scheme for the calculation of the time-dependent friction (f) on a tagged molecule at time t. The rest of the notation is as follows Fs(q,t), self-scattering function F(q,t), intermediate scattering function D, self-diffusion coefficient t]s(t), time-dependnet shear viscosity Cu(q,t), longitudinal current correlation function C q,t), longitudinal current correlation functioa...

Equation (228) is the normalized density correlation function in the Fourier frequency plane and has the same structure as Eq. (210), which is the density correlation function in the Laplace frequency plane. i/ (z) in Eq. (228) is the memory function in the Fourier frequency plane and can be identified as the dynamical longitudinal frequency. Equation (228) provides the expression of the density correlation function in terms of the longitudinal viscosity. On the other hand, t]l itself is dependent on the density correlation function [Eq. (229)]. Thus the density correlation function should be calculated self-consistently. To make the analysis simpler, the frequency and the time are scaled by (cu2)1 2 and (cu2)-1 2, respectively. As the initial guess for rjh the coupling constant X is considered to be weak. Thus rjt in zeroth order is... 

Mode coupling theory provides the following rationale for the known validity of the Stokes relation between the zero frequency friction and the viscosity. According to MCT, both these quantities are primarily determined by the static and dynamic structure factors of the solvent. Hence both vary similarly with density and temperature. This calls into question the justification of the use of the generalized hydrodynamics for molecular processes. The question gathers further relevance from the fact that the time (t) correlation function determining friction (the force-force) and that determining viscosity (the stress-stress) are microscopically different. 

It is worthwhile to discuss the relative contributions of the binary and the three-particle correlations to the initial decay. If the triplet correlation is neglected, then the values of the Gaussian time constants are equal to 89 fs and 93 fs for the friction and the viscosity, respectively. Thus, the triplet correlation slows down the decay of viscosity more than that of the friction. The greater effect of the triplet correlation is in accord with the more collective nature of the viscosity. This point also highlights the difference between the viscosity and the friction. As already discussed, the Kirkwood superposition approximation has been used for the triplet correlation function to keep the problem tractable. This introduces an error which, however, may not be very significant for an argon-like system at triple point. 

The observation that the rate constant may be expressed in terms of an auto-time-correlation function of the flux, averaged over an equilibrium ensemble, has a parallel in statistical mechanics. There it is shown, within the frame of linear response theory, that any transport coefficients, like diffusion constants, viscosities, conductivities, etc., may also be expressed in terms of auto-time-correlation functions of proper chosen quantities, averaged over an equilibrium ensemble. 

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