Radius of gyration defined

The mean square radius of gyration (defined by Eq. (5.4)) for freely rotating chain is... 

Another measure of the size of a polymo- chain, besides the root-mean-square end-to-end distance, is the radius of gyration, defined as follows ... 

One of the most important fiinctions in the application of light scattering is the ability to estimate the object dimensions. As we have discussed earlier for dilute solutions containing large molecules, equation (B 1.9.38) can be used to calculate tire radius of gyration , R, which is defined as the mean square distance from the centre of gravity [12]. The combined use of equation (B 1.9.3 8) equation (B 1.9.39) and equation (B 1.9.40) (tlie Zimm plot) will yield infonnation on R, A2 and molecular weight. 

For a body that consists of n masses mj, each separated by a distance rj from the axis of rotation of the array, the radius of gyration is defined... 

We may therefore think of r as the weight average value of r, by analogy with Eq. (1.12). As a reminder of how the radius of gyration comes to be defined this way, recall that the moment of inertia I of this same body is given by... 

There exists some radial distance from the axis of rotation at which all of the mass could be concentrated to produce the same moment of inertia that the actual distribution of mass possesses. This distance is defined to be the radius of gyration. According to this definition,... 

Another important characteristic aspect of systems near the glass transition is the time-temperature superposition principle [23,34,45,46]. This simply means that suitably scaled data should all fall on one common curve independent of temperature, chain length, and time. Such generahzed functions which are, for example, known as generalized spin autocorrelation functions from spin glasses can also be defined from computer simulation of polymers. Typical quantities for instance are the autocorrelation function of the end-to-end distance or radius of gyration Rq of a polymer chain in a suitably normalized manner ... 

The principal defining properties of the molten globule are as follows (Arai and Kuwajima, 2000) (1) substantial secondary structure (2) no significant tertiary structure (3) structure only slightly expanded from the native state (10—30% increase in radius of gyration) (4) a loosely packed hydrophobic core with increased solvent accessibility. The first two criteria are readily assessed by far- and near-UV CD, respectively. Therefore, CD has been extensively applied to the detection and characterization of molten globules. 

Basic definitions Let X be some property of a polymer chain such as the degree of polymerization, molar mass, radius of gyration, or comonomer content of a copolymer, etc. In general, the polymer is heterogeneous with respect to X, which can assume discrete values X,. We now define for molecules with X = X,-. 

The first of Eq. 5 defines the radius of gyration, R being the distance from the center of mass to the k segment The second equation is a form useful for the scattering problem. Expansion of Eq. 4b and substitution of Eq. 5 leads to... 

Approximate limits to the adsorbed layer thickness can be defined. The lower limit is about twice the radius of gyration for particles of the appropriate size. This particle size can b culated from the radius of gyration and the relationship <5aR The adsorbed layer thickness increases with increasing particle size, and the measured thicknesses are always greater than twice the radius of gyration, the difference increasing with increasing particle size. The upper limit cannot be defined at present. Moreover, these limits are conjectural and require more experimental evidence for their verification. 

The constant in Equation (5.112) cannot be readily evaluated using scaling theory. Our transformation applies equally well to the radius of gyration or the root mean square end-to-end length, only the numerical constant changes. We would like to be able to apply this idea to the role of concentration in semi-dilute and concentrated polymer regimes. In order to do this we need to define a new parameter s, the number of links or segments per unit volume ... 

Table 2). All the radii have a certain molar mass dependence. The magnitudes of these radii, however, can deviate strongly from each other. These differences result from the fact that they are physically differently defined. The radius of gyration, R, is solely geometrically defined the thermodynamically equivalent sphere radius, R-p is defined by the domains of interaction between two macromolecules, or in other words, on the excluded volume. The two hydrodynamic radii R and R result from the interaction of the macromolecule with the solvent (where the latter differs from R by the fact that in viscometry the particle is exposed to a shear gradient field).

The leading parameter that characterizes the distributions of randomly branched samples is an exponent Tthat is defined in the next section. The average procedures for the z-average of the mean square radius of gyration and the weight average molar mass results in the relationship [7]... 

It Is convenient to expand the sin (qr) term in eq. 1 which, when compared to the usual expressions for isolated particles, allows us to define a scattering radius of gyration ... 

For a rigid particle consisting of mass elements of mass mt, each located at a distance r from the centre of mass, the radius of gyration, s, is defined as the square root of the mass-average of r for all the mass elements, i.e.. 

Parameter characterizing the effect of long-chain branches on the size of a branched molecule in solution and defined as the ratio of the mean-square radius of gyration of a branched molecule, si), to that of an otherwise identical linear molecule si), with the... 

The molecular weight of a polymer which begins to approach R), can be approximated by calculating the radius-of-gyration of a macromolecule which is defined as the... 

The branching coefficient, of star polymers is defined as the ratio of the radius of gyration of branched and linear polymers having the same molecular weight [67],i.e.,... 

---