Gyration tensor

FIG. 11 Eigenvalues of the radius of gyration tensor (dots largest, squares middle triangles smallest) of micelles vs aggregation number N in an oif-lattiee model of H2T2 surfaetants. The mieelle size distribution for this partieular system has a peak at 28. (From Viduna et al. [144].)... 

Almond and Axelson have introduced an approach of some resemblance to TRAMITE, but based on the molecule s gyration tensor, which more closely reflects molecular shape and thus provides higher accuracy for molecules of highly anisotropic shape.104 They propose that the eigenvalues of the order tensor can be predicted to an accuracy which nearly matches SSIA, according to the simple relationship,... 

Example 15.4-2 For a crystal to exhibit optical activity the gyration tensor [gy] with , / = 1,2,3, which is a symmetric axial second-rank tensor, must have at least one non-zero element. Determine the form of the gyration tensor for C4V and D2d symmetry. 

We can see that a set of constitutive equations for dilute polymer solutions contains a large number of relaxation equations. It is clear that the relaxation processes with the largest relaxation times are essential to describe the slowly changing motion of solutions. In the simplest approximation, we can use the only relaxation variable, which can be the gyration tensor (S i5J), defined by (4.48), or we can assume the macromolecule to be schematised by a subchain model with two particles. The last case, which is considered in Appendix F in more detail, is a particular case of equations (9.3) and (9.4), which is followed at N = 1, Ai = 2,... 

Shape Analysis from the Radius of Gyration Tensor... 

For a rigid array of n + 1 particles, the radius of gyration tensor, S, can be expressed as a symmetric 3x3 matrix... 

The shape deduced from the analysis of the radius of gyration tensor also changes as the generation number increases, as shown in Table 1.3. The entries in this table were calculated from the results for the moments of inertia that were reported by Mansfield and Klushin J24 Even as soon as the second generation, these model dendrimers (1) are closer to being spherically symmetric than are any of the conformations listed in Table 1.2. 

Table 1.3 Ratios of the averaged principal moments of the radius of gyration tensor, asphericity, acylindricity, and anisotropic shape factor for model dendrimers (1, Fig. l.l)a).

Fig.20. Order parameter profiles m(z)=([pA(z)-pB(z)])/([pA(z)+pB(z)]), where pA(z), pB(z) are densities of A-monomers or B-monomers at distance z from the left wall, for LxLx20 films confining a symmetric polymer mixture, polymers being described by the bond fluctuation model with N=32, ab=- aa=- bb=8 and interaction range 6. Four inverse temperatures are shown as indicated. In each case two choices of the linear dimension L parallel to the film are included. While for e/kBT>0.02 differences between L=48 and L=80 are small and only due to statistical errors (which typically are estimated to be of the size of the symbols), data for e/kBT=0.018 clearly suffer from finite size effects. Broken straight lines indicate the values of the bulk order parameters mb in each case [280]. Arrows show the gyration radius and its smallest component in the eigencoordinate system of the gyration tensor [215]. Average volume fraction of occupied sites was chosen as 0.5. From Rouault et al. [56].

In addition to the optical rotation tensor fi, the gyration tensor is often used as the basis of computing optical rotations, since it is more straightforward to define working equations for it in the frequency domain. The relation to the OR tensor is... 

The notation G indicates the isotropic average of the G tensor, G = (1/3) YhuG uu, which is related to the OR parameter via G (m) = —oj/i(oj). The gyration tensor is the imaginary part of the mixed electric-magnetic dipole response, relating the perturbed dipole moments of (1) in the frequency domain directly to the field amplitudes. 

The measured crystal optical activity, in general, can be either of molecular origin or due to the chiral helical arrangement of chiral or achiral molecules in the crystal, or both. The two factors are difficult to separate. Kobayashi defined a chirality factor r = (pc — ps)/pc = 1 — pslpc, where pc is the rotatory power per molecule of a randomly oriented crystal aggregate derived from the gyration tensors determined by HAUP, and ps that in solution [51]. It is a measure of the 4 crystal lattice structural contribution to the optical activity and represents the severity of the crystal lattice structural contribution to the optical activity, and represents the severity of the restriction of the freedom of molecular orientation by forming a crystal lattice. Quartz is a typical example of r = 1, as it does not contain chiral molecules or ions and its optical activity vanishes in random orientation (ps = 0). 

One can also analyze the rotational relaxation of the adsorbed molecules.140 Figure 27a shows a time sequence of a single molecule with an overlay of the unit vector u(t) defined as the direction of the longer principal axis of the gyration tensor. An instantaneous polymer configuration may be described by an ellipse, and therefore, the simplest conformational change is the rotational motion of an ellipse. The time correlation function of u(t) decays exponentially where zr denotes the rotational relaxation time, °c exp(-f/rr). 

As an alternative to the moment of inertia, second moment of atomic distribution, also known as the gyration tensor, was proposed to approximate the molecular alignment.192 Both tensors share a common PAF, while the latter provides more realistic values of the order parameters, in particular for highly elongated molecular shapes. 

For amorphous fibers composed of C,ooH202 and with thicknesses of 5-8nm, the anisotropies of the individual chains, as measured by the principal moments of the radius of gyration tensor, are comparable with those expected for chains in the bulk. However, the radius of gyration tensors tend to be oriented within the fiber, with the nature of the orientation depending on the distance of the center of mass from the axis of the fiber. Chains with their centers of mass close to the fiber axis tend to have the largest component of their radius of gyration tensor ahgned with the fiber axis. 

The number of terms (up to five, but only three of them independent) and values of pre-exponential factor A, depend on the fluorophore symmetry and on the orientations of and in the molecule. The rotation correlation times, Tc,i, reflect the main components of the gyration tensor only. For the parallel orientation of both dipole moments, the initial anisotropy in a fluid system has the highest possible value, ro = A = 0.4. In the case of a spherical rotor, fluorescence anisotropy reduces to a single exponential function. For a symmetric rotor, r t) is either single-or double-exponential, depending on the orientation of dipole moments with respect to the long axis. 

The two examples of numeric DLCA and RLCA aggregates in Fig. 4.1 illustrate that the particles are not homogenously distributed in space and that there is no spherical symmetry. Even more, a systematic deviation fiom spherical (macro) shape can generally be observed for both aggregate types. This asphericity was first examined by Thouy and Jullien (1997), who introduced a gyration tensor... 

To characterize the shape of the globules occurring in the simulation upon decreasing the temperature we measured the three principle moments of inertia Mi, M2, M3 (eigenvalues of the gyration tensor) and constructed the following shape parameters from them... 

Figure S. The rotational velocity as function of the shear rate. The black dots mark the rotational velocity evaluated from the angular momentum, the gray dots Stand for the values inferred from the gyration tensor. The large and small s rmbols are associated with the average of ratios euid with the ratio of averages, respectively.

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