Center-of-mass structure factor

The difference in fhe moment of inertia vectors is equal to the difference in the absolute location vectors of the subchains 4 - Sy = 4. The second average is then equal to the scattering fxmction for a single molecule, S i(ij) (Equation 3.7). The total scattering fxmction is then expressed as the product of the center-of-mass structure factor and the intramolecular structure factor ... 

The center-of-mass structure factor can be expressed in terms of the pair correlation fxmction, g[Ru), for the centers of mass of different molecxiles. Within the limit of infinite dilution, the pair correlation is defined exactly in terms of the pair potential of mean force ... 

At higher concentrations, the effect of the interactions of three molecules must be taken into accoxmt. The center-of-mass structure factor is given by ... 

The dynamical properties of polymer molecules in solution have been investigated using MPC dynamics [75-77]. Polymer transport properties are strongly influenced by hydrodynamic interactions. These effects manifest themselves in both the center-of-mass diffusion coefficients and the dynamic structure factors of polymer molecules in solution. For example, if hydrodynamic interactions are neglected, the diffusion coefficient scales with the number of monomers as D Dq /Nb, where Do is the diffusion coefficient of a polymer bead and N), is the number of beads in the polymer. If hydrodynamic interactions are included, the diffusion coefficient adopts a Stokes-Einstein formD kltT/cnr NlJ2, where c is a factor that depends on the polymer chain model. This scaling has been confirmed in MPC simulations of the polymer dynamics [75]. 

Properties of a Simulated Supercooled Polymer Melt Structure Factors, Monomer Distributions Relative to the Center of Mass, and Triple Correlation Functions. 

The experimental observation that one has different Debye temperatures for the three distinct surface sites of the AU55 cluster makes the use of a continuum-model picture for discussing the thermal behavior questionable. Indeed, for such small particle sizes, where the surface structure is so manifest, the use of the concept of surface modes becomes dubious, and is certainly inadequate to explain the observed temperature dependence of the f-factors. None the less, it has proven possible to describe the low temperature specific heat of AU55 quite well using such a continuum-model, when the center-of-mass motion is taken into account [99],... 

A simple theory of the concentration dependence of viscosity has recently been developed by using the mode coupling theory expression of viscosity [197]. The slow variables chosen are the center of mass density and the charge density. The final expressions have essentially the same form as discussed in Section X the structure factors now involve the intermolecular correlations among the polyelectrolyte rods. Numerical calculation shows that the theory can explain the plateau in the concentration dependence of the viscosity, if one takes into account the anisotropy in the motion of the rod-like polymers. The problem, however, is far from complete. We are also not aware of any study of the frequency-dependent properties. Work on this problem is under progress [198]. 

Since we are considering a 2D system, we note that this is a monomolecular system and all of the centers of mass of the molecules are in the same plane (even if individual nuclei in the molecule are not). S Q), introduced earlier in this chapter, therefore only depends on the component of Q projected onto the scattering plane, Qparaiiei Just as Warren did, the case of lamellar systems like graphite and mica can now be considered, assuming that the scattering system has random orientations of the crystalhtes about an axis normal to the basal (or 2D) plane. Because the magnitude of the parallel component of the structure factor 5 (Q) is the relevant quantity... 

For concentrated solutions or polymer melts the correlations of segments on different molecules must be considered. Calculation of the summation on the right-hand side of Eq. (8.2.1) is very difficult in this case since the evaluation of intermolecular form factors demand a detailed knowledge of the solution structure. For instance, for a solution of rod-shaped molecules the average relative orientation and center-of-mass positions of pairs of rods must be known (Zimm, 1946, 1948a). 

We consider the dynamic structure factor of a rodlike molecule. The long-time behavior is rather trivial. The orientational distribution will be averaged, and the center-of-mass diffusion alone will survive. Then,... 

Figure 28a displays a typical three-dimensional plot of the neutron intensity scattered by a nematic lyotropic solution in the (qv,qvv)-plane. The data were obtained on the SDS/Dec calamitic phase at 50 s (concentration c = 29.5 wt. % and R = [Dec]/[SDS] = 0.33). As shown in the iso-intensity contour plot (Fig. 28b), the patterns are characterized by two crescent-like peaks aside from the velocity axis. The maximum scattering corresponds to the first order of the structure factor, from which the distance between the center-of-mass of the micelles can be estimated (here 6 nm for a radius of nm). The modulation of the azimuthal intensity is also of interest since it reflects the distribution of micellar orientations. The spectra were analyzed in terms of angular distribution of the scattered intensity. The scattering was integrated over an elementary surface dgvdgvv = where corresponds typically to the half width at half... 

The response time of the z-scanner (tz is expressed by Tz = Qzli tf ], where and /z are the quality factor and resonant frequency of the z-scanner, respectively, /z is almost solely determined by the resonant frequency of a piezoactuator used and by the way to hold the piezoactuator, as far as the structural resonance arising from the designed framework is completely suppressed. In the most recent z-scanner, a piezoactuator with a resonant frequency of 500 kHz is held at the four rims parallel to the displacement direction. In this way of holding, the center of mass of the piezoactuator is not displaced (hence, no impulsive force is exerted on the supporting mechanism, and importantly, the resonant frequency of the z-piezoactuator is nearly unchanged. The maximum displacement is... 

An important complication arises here because the intermolecular structure factor as introduced in Eq. (2.13) has now become a function of the form factor P(q), i.e., the distribution of polyions depends on their mutual orientation and their shape and vice versa. It is only in the case of spherical polyions that S(q) and P(q) are separable by the use of center-of-mass coordinates. For rod-like polyions the mutual orientation and the spatial distribution are correlated, and for flexible polyions the chain conformation and the spatial distribution of chains depend on each other. Assuming weak interactions, several approximations were introduced to separate form- and structure factor. However, for strong, long-range electrostatic interactions intra- and intermolecular correlations cannot yet be properly separated [28]. This is an important limitation to all current theories except for Monte Carlo simulations. 

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