Normal mode theory

If a sinusoidal vibration is applied to a longitudinal rod, and both ends of the rod are then fixed, the wave will be reflected at these fixed points or nodes. A series of discrete standing waves, known as the normal modes, can be established in the rod, with wavelengths given by 

Consideration of the restoring force when a bead is displaced from its equilibrium position leads to the expression 

The 3 m equations have to be uncoupled using a normal coordinate transformation, to obtain eigenfunctions that are linear combinations of the positions of the submolecules. Each eigenfunction then describes a single viscoelastic element with characteristic time-dependent properties. 

It can be shown, for example, that the stress relaxation modulus is given by 

For the Rouse model the submolecule is the shortest length of chain that can undergo relaxation, and the motion of segments within a submolecule is ignored. This limitation implies that the theory is applicable only when m 1, and means that the equation for xp reduces to 

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Goodyear G and Stratt R M 1996 The short-time intramoleoular dynamios of solutes in liquids. I. An instantaneous-normal-mode theory for friotion J. Chem. Phys. 105 10050-71... 

Cao, J., Voth, G.A. The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties. J. Chem. Phys. 100 (1994) 5093-5105 II Dynamical properties. J. Chem. Phys. 100 (1994) 5106-5117 III. Phase space formalism and nalysis of centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6157-6167 IV. Algorithms for centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6168-6183 V. Quantum instantaneous normal mode theory of liquids. J. Chem. Phys. 101 (1994) 6184 6192. 

Refs. 25 and 130. The lines were calculated from a recently developed exact normal-mode theory for finite <... 

Our instantaneous-normal-mode theory of vibrational energy relaxation suggests a rather interesting physical picture. The solvent surrounding our... 

Gilbert, F. Dziewonski, A. M. 1975. An application of normal mode theory to the retrieval of structural parameters and source mechanism from seismic spectra. Philosophical Transactions of the Royal Society of London, Series A, 278, 187-269. 

Buchner, M., Ladanyi, B. M., and Stratt, R. M., The short-time dynamics of molecular liquids. Instantaneous-normal-mode theory. J. Chem. Phys. 91,8522 (1992). 

At low ionic strength (kR 1), other effects connected with the finite diffusivity of the small ions in the EDL surrounding the particle are present. The noninstantaneous diffusion of the small ions (with respect to the Brownian motion of the colloid particle) could lead to detectable reduction of the single particle diffusion coefficient, Dq, from the value predicted by the Stokes-Ein-stein relation. Equation 5.447. For spherical particles, the relative decrease in the value of Dq is largest at k/ 1 and could be around 10 to 15%. As shown in the normal-mode theory, the finite diffusivity of the small ions also affects the concentration dependence of the collective diffusion coefficient of the particles. Belloni et al. obtained an explicit expression for the contribution of the small ions in Ac)... 

Although theories of the Rouse-Bueche-Zimm type have been very successful in rationalizing the behavior of polymeric systems from a molecular point of view, another class of theories is presently commanding the most attention. These theories treat the motion of polymer molecules in terms of reptation, a reptile-like diffusive motion of each polymer molecule through a matrix formed by its neighbors. To a considerable extent, this new approach has overcome some of the most important shortcomings of the normal-mode theories, which... 

Molecular theories describing the dielectric relaxation behavior of polymers have been developed and are summarized in references 5, 12 and 13. Again, if the dipoles are rigidly attached along the chain contour, normal mode theories such as those of Rouse and Bueche described in Chapter 3 for the mechanical case might be expected to be applicable. In addition, the time-temperature superposition principle also generally applies. 

The dynamic structural solution for the thrust loading is obtained using a modified-predictor-corrector-integration technique and normal mode theory. 

NORMAL MODE THEORY (THE IDEAL OR SIMPLE HARMONIC OSCILLATOR)... 

Normal Mode Theories Based on Motion of Isolated Flexible Chains... 

In the introductory section on amorphous polymers (Section 7.1.1), we considered the relaxation spectrum of amorphous polymers and noted that it was quite complex. The normal mode theories, now to be discussed, attempt to predict the relaxation spectrum for amorphous polymers, as well as the time-temperature equivalence. 

The next major contributions to the progress of theory used normal mode [34-36] calculations for various simple liquids. Normal mode theory was... 

The failure of INM theory is a strong indication of the sensitivity of the fifth-order Raman signal to anharmonicities. Ma and Stratt modified INM theory to allow for dephasing of the modes which they termed adiabatic instantaneous normal mode theory [42]. This allowed them to incorporate the dynamical anharmonicity which was missing from their previous calculation (see (1.9)). In an approach similar to Okumura and Tanimura, the response was broken into its component parts to investigate their dependence on the various signal contributions (see Fig. 1.10). When treated individually the dynamical anharmonicity... 

Keyes, T. (1996). Normal mode theory of two step relaxation in liquids Polarizability dynamics in CS2. J. Chem. Phys. 104 9349-9356 Keyes, T. (1997). Instantaneous normal mode approach to liquid state dynamics. J. Phys. Chem. 101 2921-2930. 

Ma, A., Stratt, R.M. (2002). The molecular origins of the two-dimensional Raman spectrum of an atomic liquid. II. Instantaneous-normal-mode theory. 7. Chem. Phys. 116 4972 984. 

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