Normal Modes of Motion

So far we have analysed the shape change of polymer molecules in terms of conformational changes in small rotating sections of the chain. However, it is also possible to analyse the deformations of the whole chain. When the temperature is 

The first such treatment considers the coiled macromolecule as a soft ball. This becomes particularly useful in the discussion of the viscosity of polymer melts and solutions. To do this, we follow the work of Rouse and of Zimm, who returned to some observations by Isaac Newton. 

So long as there are enough freely rotating units in the chain to permit these modes of motion, the dynamic properties depend on the molar mass, but not the chemical nature, of the chain. We shall be treating this in more detail when we come to look at the property of flow viscosity. 

Let us take a 1.00-kg oscillator and couple it with an identical oscillator by means of a coupling spring of k = 1.00Nm . The force constants of the lateral springs are also 1.00Nm . Now the positive, real solutions for the frequencies are, from Eq. (5-15) 

Two degrees of freedom lead to two modes of motion. These two modes of motion, synchronous and antisynchronous, are the normal modes of motion for this system. If only synchronous motion is excited, the antisynchronous mode will never contribute to the motion. The same is true for the pure antisynchronous mode (Fig. 5-2b) there will never be a synchronous conPibution. Under these conditions, but only under these conditions, energy does not pass from one mass to the other. 

If the masses are displaced in an arbiPary way or arbiPary initial velocities are given to them, the motion is asynchronous, a complex mixture of synchronous and antisynchronous motion. But the point here is that even this complex motion can be broken down into two normal modes. In this example, the synchronous mode of motion has a lower frequency than the antisynchronous mode. This is generally Pue in systems with many modes of motion, the mode of motion with the highest symmePy has the lowest frequency. 

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In this respect, another insufficiency of Lodge s treatment is more serious, viz. the lack of specification of the relaxation times, which occur in his equations. In this connection, it is hoped that the present paper can contribute to a proper valuation of the ideas of Bueche (13), Ferry (14), and Peticolas (13). These authors adapted the dilute solution theory of Rouse (16) by introducing effective parameters, viz. an effective friction factor or an effective friction coefficient. The advantage of such a treatment is evident The set of relaxation times, explicitly given for the normal modes of motion of separate molecules in dilute solution, is also used for concentrated systems after the application of some modification. Experimental evidence for the validity of this procedure can, in principle, be obtained by comparing dynamic measurements, as obtained on dilute and concentrated systems. In the present report, flow birefringence measurements are used for the same purpose. 

In order to show that this procedure leads to acceptable results, reference is briefly made to the normal coordinate transformation mentioned at the end of Section 2.2. By this transformation the set of coordinates of junction points is transformed into a set of normal coordinates. These coordinates describe the normal modes of motion of the model chain. It can be proved that the lowest modes, in which large parts of the chain move simultaneously, are virtually uninfluenced by the chosen length of the subchains. This statement remains valid even when the subchains are chosen so short that their end-to-end distances no longer display a Gaussian distribution in a stationary system [cf. a proof given in the appendix of a paper by Ham (75)]. As a consequence, the first (longest or terminal) relaxation time and some of the following relaxation times will be quite insensitive for the details of the chain... 

There is no diflBculty in the imdamped plane motion which corresponds to the wagging of a rigid dipolar group elastically connected to the rest of a molecule which rotates freely as a whole, since in this case normal modes of motion exist, and are indeed obvious. For simplicity we take the mass centre of the polar group to be fixed in position within the molecule. If the angular position of the molecular body is given by a and that of the... 

The long-range order in a crystal structure imposes correlation on the dipole moments of neighbouring structural units, and the normal modes of motion of the structure are running or standing waves of definite M. Mandel, Physica, 1972, 57, 141. 

Recent numerical experiments by the method of molecular dynamics have shown that, for a chain model consisting of particles joined by ideally rigid bonds, the Van der Waals interactions of chain units cause only a little change in the dependence of relaxation times on the wave vector of normal modes of motions, i.e. in the character and shape of the relaxation spectrum. It was found that for the model chain the important relationship... 

The expansion of V given in Equation 37-4 is not valid except when the nuclei stay near their equilibrium positions. That is, we have assumed that the molecule is not undergoing translational or rotational motion as a whole. Closely related to this is the fact, which we shall not prove, that zero occurs six1 times among the roots i of the secular equation. The six normal modes of motion corresponding to these roots, which are not modes of vibration because they have zero frequency, are the three motions of translation in the x, y, and z directions and the three motions of rotation about the x, y, and z axes. 

Figure 1 The normal modes of motion for the three stretch modes of DCCH. (Adapted from T. A. Holme and R. D. Levine, Chem. Phys. Lett. 150 393 (1988).) The displacement of the atoms in each mode is shown by an arrow. Note that while all atoms contribute to all modes, the respective contributions do vary and the v, mode is almost a localized CH stretch. For recent studies of the overtone spectroscopy of HCCH and its isotopomers see J. Lievin, M. Abbouti Temsamani, P. Gaspard, and M. Herman, Chem. Phys. Lett. 190 419 (1995) M. J. Bramley, S. Carter, N. C. Handy, and 1. M. Mills, J. Mol. Spectrosc. 160 181 (1993) B. C. Smith and J. S. Winn, J. Chem. Phys. 89 4638 (1988) K. Yamanouchi, N. Ikeda, S. Tuschiya, D. M. Jonas, J. K. Lundberg, G. W. Abramson, and R. W. Field, J. Chem. Phys. 95 6330 (1991).)...

Appendix 9.B — Rouse Motions in an Entanglement Strand The Rouse-Mooney Normal Modes of Motion ... 

If we consider that the three major absorption bands of the amide group given above actually represent normal modes of motion of the whole amide linkage, along... 

Thus, the restoring force is proportional to the extension and the onedimensional chain behaves as a Hookean spring. This important result simplifies the analysis of the normal modes of motion of a polymer. Polymer chain models can be treated mathematically by the much simpler linear differential equations because second order effects are absent. (It should be noted diat, while the elastic equation for a polymer chain is identical in form with Hooke s law, the molecular origin of the restoring force is very different). 

Fig. 2-2. Normal modes of vibration for equilateral triangular type molecules, (a) One choice of the normal modes of vibration, (h). Alternative choice of the normal modes of motion for the degenerate frequencies. 2 = 8 2, 3 = 3 — 2.

The diagrams which represent normal modes of motion can also be used to represent normal coordinates if the arrows are drawn so as to represent displacements, not in ordinary units, but in the mass-adjusted scale of coordinates qi. Then the component of an arrow along the direction of the coordinate qi is proportional to kj, and since = kk, the diagram... 

In Sec. 5-8, it has already been indicated how the number of normal modes of motion of each possible symmetry cati be obtained. This section will treat the problem more thoroughly. The fact that the transformations of the displacement coordinates of the atoms of the molecule form a reducible representation of the symmetrj pohit group of the molecule has been discussed in Sec. 5-6. It follows directly from this that the 3. normal coordinates (including translation and rotation) also form a basis of a representation of the group, since the normal coordinates Qk are linear combinations of the displacement coordinates (see Sec. 2-4). 

Consequently, the coefficients kk which specify the normal modes of motion are identical with the coefficients of the transformation from the normal coordinates Qk to the original coordinates g,-, while the roots k of the secular equation are the coefficients of Ql in the expression for 2F. 

Consequently, there are six roots with value zero as originally stated, and the six corresponding normal modes of motion are the translations and rotations. Furthermore, (Ri,. . . , Gle are a set of corresponding normal coordinates. 

The moduli are thus determined by a discrete spectrum of relaxation times, each of which characterizes a given normal mode of motion. These normal modes are shown schematically in Figure 6.17. In the first mode, corresponding top = I, the ends of the molecule move, with the centre of the molecule remaining stationary. In the second mode there are two nodes in the molecule, and in the general case the pitv mode has p nodes, with the motion of the molecule occurring in (p + 1) segments. 

Rouse solved the m — 1 simultaneous equations [Eqs. (45)] by transforming them into a set of uncoupled equations for the normal modes of motion of the chain. 

The moduli of a dilute solution containing n chains per unit volume may then be expressed in terms of a discrete relaxation time spectrum, where each relaxation time corresponds to one of the normal modes of motion [Figure 14.9(b)]. For simple shear, this leads to Eqs. (46) and (47), with Xp given by Eq. (4S) [13-16]. 

When we looked at the time required for each normal mode of motion, we saw in Chapter 3 that the first mode required the most time, with the higher modes requiring progressively shorter times. In general terms, the time required for each mode is given, for the ith mode, by ... 

For low molar mass polymers having no entanglements, the normal modes of motion of the polymer chain are the main contribution to the observed viscosity. 

Fig. 5. Schematic illustration of some normal modes of motion of a polymer chain first mode corresponds to rotational diffusion...

Independent of the significance of normal mode behavior is the use of normal or collective coordinates to describe polymer motion. For example, with a simple type-A polymer in which all monomer dipoles are inserted parallel to each other, the net dipole moment can be represented as proportional to a collective variable, the end-to-end vector ryv - ro. Collective coordinates supply sets of 3A -I- 3 collective variables, all vectors that are properly normalized and orthogonal, with no intent of claiming that they correspond to normal modes of motion or are eigenvectors of some linearized problem. 

Equations (21a) and (21b) are empirical, as introduced, therefore the derived relations (23) and (24) are also empirical. For molecules moving in a barrier system by thermal activation or a polymer chain undergoing normal modes of motion, 0(t) takes the form of equation (21a) where the coefficients gi are functions of the parameters of the model. Clearly, experimental dielectric... 

An additional type of scattering is obtained due to the fact that molecules themselves are vibrating at frequencies corresponding to various normal modes of motion. These characteristic vibrational frequencies can mix with the exciting light to form sum and difference frequencies in the scattered radiation. Although weak, these frequencies may be detected as shifts from the Rayleigh frequency and are called normal Raman spectra. The measure of these shifts reflects the characteristic vibrations of the molecule and may be utilized as a complement to infrared spectroscopy. 

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