Gyration, mean radius, equation

VSTR = O Connell characteristic volume parameter, cm /g-mol ZRA = Rackett equation parameter RD = mean radius of gyration, A DM = dipole moment, D R = UNIQUAC r Q = UNIQUAC q QP = UNIQUAC q ... 

To reveal fine phenomena in the properties of polymer solutions, des Cloizeaux calculated the mean square gyration radius of niacromolecules by means of (Equation 5.4 103, 104). In particular, it was found that... 

For the application system involving acetic acid in Chapter 9, the above representation in Eq. (2.3) is not accurate enough. The reason is that acetic acid can exhibit association leading to dimerization in the vapor phase. Hayden and O Connell used the virial equation of state tmncated after the second term. They developed a correlation for the second virial coefficient of polar, nonpolar, and associating species based on the critical temperamre and pressure, the dipole moment, and the mean radius of gyration. Association of like and unlike molecules is described with the adjustable association parameter. Pure component and binary values for the association parameter are available in the Aspen Physical Property System. For detail calculations in the vapor phase, please refer to the above Hayden and O Connell paper and also to Aspen Physical Property System. ... 

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. 

Bueche (16,172) proposed that the viscosity is proportional to the fourth power of the polymer concentration and a complex function of the free volume of the mixture. Kraus and Gruver (170) find that the 3.4 power fits experimental data better than does the fourth power. They used equation (58) with (r2) replaced by the mean-square radius of gyration (s2). The term r2)/(rf) indicates that poor solvents should lower the viscosity more than a good solvent. As the temperature increases, the factor increases as a function of the ratio (T - 7 (tJJ)/(7 - 7 ). The glass transition temperatures of the polymer and diluent are andT o, respectively. 

Figure 28a shows the result of SAXS on sample BrlOOO. We used Guinier s formula (see eq. 6) for the small angle scattering intensity, I(k), from randomly located voids with radius of gyration, Rg. Although Guinier s equation assumes a random distribution of pores with a homogeneous pore size, it fits our experimental data well. The slope of the solid line in Fig. 28b gives R - 5.5 A and this value has been used for the calculated curve in Fig. 28a. This suggests a relatively narrow pore-size distribution with an equivalent spherical pore diameter of about 14A. Similar results were found for the other heated resin samples, except that the mean pore diameter changed from about 12 A for samples made at 700°C to about 15 A for samples made at 1100°C.

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 ... 

The equation for the unperturbed mean-square radius of gyration, 0, of copolymers is obtained for two cases by using the RIS method. For one case it is assumed that the total mass of each structural unit of the chain is situated on the skeletal atom. For the other case the deviation is considered of the center of mass of each structural unit from each skeletal atom. 

In the last chapter, equations were derived for the particle-scattering factor, the mean-square radius of gyration, the diffusion coefficient and the first cumulant of the dynamic structure factor. All these have the common feature that, for homopolymers at least, they can be written in the following form ... 

The modeling of a polymerization process is usually understood as formulation of a set of mathematical equations or computer code which are able to produce information on the composition of a reacting mixture. The input parameters are reaction paths and reactivities of functional groups (or sites) at monomeric substrates. The information to be modeled may be the averages of molecular weight, mean square radius of gyration, particle scattering factor, moduli of elasticity, etc. Certain features of polymerizations can also be predicted by the models. 

The root-mean-square z-averaged radius of gyration Rg,z was measured with both instruments. The BI200SM photogoniometer was used with software from Brookhaven Instruments to determine Kc/Re for 11 angles (15° < 0 < 150 °), and these values were fitted to the equation... 

Equation (2.2.12) may be directly obtained from minimizing the elastic free energy under the constraint that the mean-square radius of gyration has a fixed value [see Eqs. (2.1.63) and (2.1.39), C q) -+ ot q)C q) [10]. The physical meaning of this result is that under chain compression the free energy due to the interatomic contacts is basically a function of only, no matter what are the individual values of the a (q). As a consequence, all the mean-square distances (r (k)) may be expressed under a general form [53]. Defining... 

Several size parameters can be used to describe the dimensions of polymer molecules radius of gyration, end-to-end distance, mean external length, and so forth. In the case of SEC analysis, it must be considered that the polymer molecular size is influenced by the interactions of chain segments with the solvent. As a consequence, polymer molecules in solution can be represented as equivalent hydrodynamic spheres [1], to which the Einstein equation for viscosity may be applied ... 

In Equation 14.31 is the position of the aggregate center of mass. The radius of gyration is a root-mean-square radius, which is often a useful point of view. Given Rg, a fractal of N monomers or primary particles obeys... 

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