Spring bond vector

The elastic dumbbell model studied iu Chapter 6 is both structurally and djmamicaUy too simple for a poljmier. However, the derivation of its constitutive equation illustrates the main theoretical steps involved. In this chapter we shall apply these theoretical results to a Gaussian chain (or Rouse chain) containing many bead-spring segments (Rouse segments). First we obtain the Smoluchowski equation for the bond vectors. After transforming to the normal coordinates, the Smoluchowski equation for each normal mode is equivalent in form to the equation for the elastic dumbbell. 

Figure 12 (a) A coarse-grained chain of Mg=6 superunits (groups of g=5 consecutive repeat units) with grouped bond vectors (b) The beads-and-springs model of a Gaussian chain h.rz. are the position vectors of the beads. 

Thus, the bond vectors li = ri+i ri are characterized by independent and uncorrelated Gaussian distributions with (1 ) = In view of eqn [ 17], the bonds can be considered as elastic springs (the beads-and-springs model, see Figure 12(b)). 

Dimensions of Ideal Chains Now we obtain Rp and R for ideal chains whose conformations are given as trajectories of random walkers. They include a random walk on a lattice, a freely jointed chain, a bead-spring model, and any other model that satisfies the requirement of Markoffian property (Eq. 1.19). The bond vector r, - r, i of the ith bond is then the displacement vector Ar, of the ith step. We assume Eq. 1.19 only. Then the end-to-end distance is Nb. To calculate / g, we note that a part of the ideal chain is also ideal. The formula of the mean square end-to-end distance we obtained for a random walk applies to the mean square distance between the ith and/th monomers on the chain just by replacing N with i - j. ... 

In anisotropic materials, the electronic bonds may have different polarizabilities for different directions (you may think of different, orientation-dependent spring constants for the electronic harmonic oscillator). Remembering that only the E-vector of the light interacts with the electrons, we may use polarized light to test the polarizability of the material in different directions, lno is one of the most important electro-optic materials and we use it as an example. The common notations are shown in Figure 4.7. If the E-vector is in plane with the surface of the crystal, the wave is called a te wave. In this example, the te wave would experience the ordinary index na of LiNbOs (nG 2.20). If we rotate the polarization by 90°, the E-ve ctor will be vertical to the surface and the wave is called tm. In lno, it will experience the extraordinary index ne 2.29. Therefore these two differently polarized waves will propagate with different phase velocities v c/n. In the example of Figure 4.7, the te mode is faster than the tm mode. 

Consider the Rouse chain containing N identical beads as shown in Fig. 3.2. Each bead in the Rouse chain plays the same role as the bead in the elastic dumbbell. Denote the Hooke constant of the spring by ZkT/b where represents the mean square length of the spring or bond (b ). Relative to a certain point O in space, the position vectors of the beads are denoted by R = (Ri, R2j j Riv)- Then the center-of-mass position of the chain is given by... 

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