Memory function dynamics

The stress depends on the extent of reaction, p(tf), which progresses with time. However, it is not enough to enter the instantaneous value of p(t ). Needed is some integral over the crosslinking history. The solution of the mutation problem would require a constitutive model for the fading memory functional Gf Zflt, t p(t") which is not yet available. This restricts the applicability of dynamic mechanical experiments to slowly crosslinking systems. 

Rostov, K.S. Freed, K.F., Mode coupling theory for calculating the memory functions of flexible chain molecules influence on the long time dynamics of oligoglycines, J. Chem. Phys. 1997,106, 771-783... 

In generalized Rouse models, the effect of topological hindrance is described by a memory function. In the border line case of long chains the dynamic structure factor can be explicitly calculated in the time domain of the NSE experiment. A simple analytic expression for the case of local confinement evolves from a treatment of Ronca [63]. In the transition regime from unrestricted Rouse motion to confinement effects he finds ... 

The description of the chain dynamics in terms of the Rouse model is not only limited by local stiffness effects but also by local dissipative relaxation processes like jumps over the barrier in the rotational potential. Thus, in order to extend the range of description, a combination of the modified Rouse model with a simple description of the rotational jump processes is asked for. Allegra et al. [213,214] introduced an internal viscosity as a force which arises due to a transient departure from configurational equilibrium, that relaxes by reorientational jumps. Thereby, the rotational relaxation processes are described by one single relaxation rate Tj. From an expression for the difference in free energy due to small excursions from equilibrium an explicit expression for the internal viscosity force in terms of a memory function is derived. The internal viscosity force acting on the k-th backbone atom becomes ... 

There have been a number of attempts to calculate time-correlation functions on the basis of simple models. Notable among these is the non-Markovian kinetic equation, the memory function equation for time-correlation functions first derived by Zwanzig33 and studied in great detail by Berne et al.34 This approach is reviewed in this article. Its relation to other methods is pointed out and its applicability is extended to other areas. The results of this theory are compared with the results of molecular dynamics. 

The memory function equation for the time-correlation function of a dynamical operator Ut can be cast into the form of a continued fraction as was first pointed out by Mori.43 We prove this in a different way than Mori. In order to proceed it is necessary to define the set of memory functions K0 t),. .., Kn t). .., such that... 

The single relaxation time approximation corresponds to a stochastic model in which the fluctuating force on a molecule has a Lorentzian spectrum. Thus if the fluctuating force is a Gaussian-Markov process, it follows that the memory function must have this simple form.64 Of course it would be naive to assume that this exponential memory will accurately account for the dynamical behavior on liquids. It should be regarded as a simple model which has certain qualitative features that we expect real memory functions to have. It decays to zero and, moreover, is of a sufficiently simple mathematical form that the velocity autocorrelation function,... 

In this article the memory function formalism has been used to compute time-correlation functions. It has been shown that a number of seemingly disparate attempts to account for the dynamical behavior of time correlation functions, such as those of Zwanzig,33,34 Mori,42,43 and Martin,16 are... 

In the critical phenomena, the contribution from the different hydro-dynamic modes to the transport coefficients are calculated. On the other hand, in the extended hydrodynamic theory, only the Enskog values of the transport coefficients are used. Thus, while the critical phenomena considers only the long-time part of the memory function, the extended hydrodynamic theory uses only the short-time part of the memory function. None of the theories involve any self-consistent calculation. 

To write down the expression for the dynamic structure factor, we need explicit expressions for the components of the frequency matrix, memory function matrix, and the normalization matrix C(q). 

It has been discussed in the previous section that the long-time part in the memory function gives rise to the slow long-time tail in the dynamic structure factor. In the case of a hard-sphere system the short-time part is considered to be delta-correlated in time. In a Lennard-Jones system a Gaussian approximation is assumed for the short-time part. Near the glass transition the short-time part in a Lennard-Jones system can also be approximated by a delta correlation, since the time scale of decay of Tn(q, t) is very large compared to the Gaussian time scale. Thus the binary term can be written as... 

As discussed by Kirkpatrick [30], T( ) can be replaced by T( m) since there is a marked softening near this wavenumber. The maximum contribution from Eq. (225) in the long time comes when both the dynamic structure factors are evaluated near qm. The Gaussian part of the dynamic structure factor can also be neglected in the asymptotic limit. Thus the long-time part of the memory function now contains the vertex function and a bilinear product of the dynamic structure factors, all evaluated at or near qm. To make the analysis simpler, the wavenumber dependence of all the quantities are not written explicitly. 

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... 

In Section XI we discussed the calculational method of the dynamic structure factor in the supercooled regime. We also discussed that the memory function F// needs to be calculated self-consistently with the dynamic structure factor itself. Near the glass transition, the dynamic structure factor is expected to diverge. This leads to an infinite loop numerically formidable calculation. 

Note that the above study is performed for a simple system. There exists a large body of literature on the study of diffusion in complex quasi-two-dimensional systems—for example, a collodial suspension. In these systems the diffusion can have a finite value even at long time. Schofield, Marcus, and Rice [17] have recently carried out a mode coupling theory analysis of a quasi-two-dimensional colloids. In this work, equations for the dynamics of the memory functions were derived and solved self-consistently. An important aspect of this work is a detailed calculation of wavenumber- and frequency-dependent viscosity. It was found that the functional form of the dynamics of the suspension is determined principally by the binary collisions, although the mode coupling part has significant effect on the longtime diffusion. 

We remark that the simulation scheme for master equation dynamics has a number of attractive features when compared to quantum-classical Liouville dynamics. The solution of the master equation consists of two numerically simple parts. The first is the computation of the memory function which involves adiabatic evolution along mean surfaces. Once the transition rates are known as a function of the subsystem coordinates, the sequential short-time propagation algorithm may be used to evolve the observable or density. Since the dynamics is restricted to single adiabatic surfaces, no phase factors... 

The equation (3.11) is the equation for the dynamics of a single macromolecule in the case of linear dependence on the co-ordinates and velocities.4 Let us note that, if memory functions (3(s) and viscous liquid, which was used in Chapter 2 to describe the dynamics of a macromolecule in this case. 

The set of stochastic equations given by (3.37) is equivalent (in the linear case) to equations (3.11) with the memory functions defined in Section 3.3, but, in contrast to equations (3.11), set (3.37) is written as a set of Markov stochastic equations. This enables us to determine the variables that describe the collective motion of the set of macromolecules. In this particular approximation, the interaction between neighbouring macromolecules ensures that the phase variables of the elementary motion are co-ordinates, velocities, and some other vector variables - the extra forces. This set of phase variables describes the dynamics of the entire set of entangled macromolecules. Note that the Markovian representation of the equation of macromolecular dynamics cannot be made for any arbitrary case, but only for some simple approximations of the memory functions. We are considering the case with a single relaxation time, but generalisation for a case with a few relaxation times is possible. 

In accordance with equations (3.15), the memory functions / (s) and y>(s) in the dynamic equations are given by their one-sided transforms... 

To calculate the dynamic modulus, we turn to the expression for the stress tensor (6.46) and refer to the definition of equilibrium moments in Section 4.1.2, while memory functions are specified by their transforms as... 

One can notice that the dissipative terms in the dynamic equation (3.11) (taken for the case of zero velocity gradients, z/jj = 0) have the form of the resistance force (D.3) for a particle moving in a viscoelastic liquid, while the memory functions are (with approximation to the numerical factor) fading memory functions of the viscoelastic liquid. The macromolecule can be considered as moving in a viscoelastic continuum. In the case of choice of memory functions (3.15), the medium has a single relaxation time and is characterised by the dynamic modulus... 

We note once more that the Markovian representation of the equation of macromolecular dynamics cannot be made for any arbitrary case, but only for some simple approximations of the memory functions. The above system describes the situation when the medium is characterised by the only relaxation time, but generalisation for few relaxation times is possible. 

Equation (139) was already discussed elsewhere [22,23,31] as a phenomenological representation of the dynamic equation for the CC law. Thus, Eq. (139) shows that since the fractional differentiation and integration operators have a convolution form, it can be regarded as a consequence of the memory effect. A comprehensive discussion of the memory function (137) properties is presented in Refs. 22 and 23. Accordingly, Eq. (139) holds for some cooperative domain and describes the relaxation of an ensemble of microscopic units. Each unit has its own microscopic memory function m (t), which describes the interaction between this unit and the surroundings (interaction with the statistical reservoir). The main idea of such an interaction was introduced in Refs. 22 and 23 and suggests that mg(f) JT 8(f,- — t) (see Fig. 50). 

One drawback of this method is that it requires the use of an extra parameter, t, the time constant for the decay of the memory function. In practice, this should not be a problem as long as t is longer than the period of the longest motions which are important for NOE averaging. A limitation of the method is that, in contrast to the two-state model of Scarsdale et al.,47 the procedure can only average over conformations that can interchange during the short time of a molecular dynamics simulation. 

From the concept of separability, the memory function of the linear viscoelasticity is required. This memory function can be related to a discrete relaxation time spectrum, available firom dynamic experiments, given by ... 

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