The radius of gyration

We have already seen that the radius of gyration in the dilute region varies as in a good solvent for polymers of high (infinite) molecular weight. What 

Here the superscripts D and SD refer to the dilute and semi-dilute regimes respectively and universal scaling exponent. Substitution of equation (4.36) into this expression yields 

In the semi-dilute range, is seen to be a function of the polymer concentration and the chain length. The scaling law theories assume that in the semi-dilute range This challenging assumption, which appears 

This demonstrates that decreases slowly with the polymer concentration in the semi-dilute regime. Of course, in a 0-solvent, the corresponding value of y leads to i=0, i.e. Rg N.  

Another measure of the space taken up by a polymer in solution is the radius of gyration. The radius of gyration is the effective radius of the polymer coil in solution and can be defined as 

In the previous sections, we described a polymer as a free chain, interacting only with the solvent molecules around it. However, this model is only really useful in dilute solutions. As the concentration of polymer c in 

FIGURE 4.10 Parameters defining the radius of gyration Rg for a polymer molecule. 

FIGURE 4.11 Representative cartoons ofthethree different regimes in polymer solutions with increasing 

Dilute In a true dilute polymer solution, the concentration is low, and the individual polymer molecules are considered not to interact with each other. Molecules behave as self-avoiding chains and are folded up into a globular configuration, the size of which may be characterized by the radius of gyration or the end-to-end distance R. The total solution volume is greater than the number of molecules multiplied by Rg. 

The average end-to-end distance is not experimentally observable. Directly measurable (for example via light scattering) is the radius of gyration Rq, The radius of gyration Rq is correlated to the end-to-end distance [73]  

According to Fig. 8.4 only 92% of the polymer segments are inside a sphere with the radius 8% of the segments are outside of this sphere. 

Solvents where a Q has the value one are so-called theta -solvents with the coil in its unperturbed dimensions. In general, all solvents for a certain polymer can behave as a theta-solvent at theta -temperature, since the solvent polymer interactions depend on the temperature. Theta-temperatures are listed for several polymer-solvent systems [13,45]. The values for C ,listed in the polymer handbook are determined for theta-systems. Only for these systems, a calculation of the radius of gyration according to Eq. (8.15) is acceptable. The parameter a is experimentally not accessible and therefore not tabulated. Since a also depends on the molar mass M, the radius of gyration for non-theta systems is generally determined directly from the molar mass via an i Q-M-relationship. 

For theta systems, the radius of gyration Rq can be correlated directly with the molar mass M. Combining Eq. (8.15) with Eq. (8.17) yields a correlation of the end-to-end distance of a coil in its unperturbed dimensions with the molar mass  

Since the only variable on the right side of the equation is the molar mass, a substitution of the end-to-end distance r o with the radius of gyration Rq according 

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Estimate the thickness of a polymer layer from the loop profile in Eq. XI-20. Assume x = 0,Xs = 2.,= 0.01, and N = Ifr. Calculate the second moment of this profile (this is often measured by ellipsometry) and compare this thickness to the radius of gyration of the coil Rg = VN/6. 

A graphical method, proposed by Zimm (thus tenned the Zinnn plot), can be used to perfomi this double extrapolation to detemiine the molecular weight, the radius of gyration and the second virial coefficient. An example of a Zinnn plot is shown in figure Bl.9.6 where the light scattering data from a solution of poly... 

The radius of gyration of tire whole particle, R can be obtained from the distance distribution fimction p(r) as... 

Figure C2.5.6. Thennodynamic functions computed for the sequence whose native state is shown in figure C2.5.7. (a) Specific heat (dotted curve) and derivative of the radius of gyration with respect to temperature dR /dT (broken curve) as a function of temperature. The collapse temperature Tg is detennined from the peak of and found to be 0.83. Tf, is very close to the temperature at which d (R )/d T becomes maximum (0.86). This illustrates...

For a continuous distribution, summation may be replaced by integration and by assuming a Gaussian distribution of size, Stoeckli arrives at a somewhat complicated expression (not given here) which enables the total micropore volume IFo, a structural constant Bq and the spread A of size distribution to be obtained from the isotherm. He suggests that Bq may be related to the radius of gyration of the micropores by the expression... 

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,... 

As should be expected, both (fg ) and r show the same dependence on the degree of polymerization or molecular weight. Since the radius of gyration can be determined experimentally through the measurement of viscosity or light scattering, it is through this quantity that we shall approach the evaluation of 1. 

At first glance, the contents of Chap. 9 read like a catchall for unrelated topics. In it we examine the intrinsic viscosity of polymer solutions, the diffusion coefficient, the sedimentation coefficient, sedimentation equilibrium, and gel permeation chromatography. While all of these techniques can be related in one way or another to the molecular weight of the polymer, the more fundamental unifying principle which connects these topics is their common dependence on the spatial extension of the molecules. The radius of gyration is the parameter of interest in this context, and the intrinsic viscosity in particular can be interpreted to give a value for this important quantity. The experimental techniques discussed in Chap. 9 have been used extensively in the study of biopolymers. 

We saw in Chap. 1 that the random coil is characterized by a spherical domain for which the radius of gyration is a convenient size measure. As a tentative approach to extending the excluded volume concept to random coils, therefore, we write for the volume of the coil domain (subscript d) = (4/3) n r, and combining this result with Eq. (8.90), we obtain... 

Both the intrinsic viscosity and GPC behavior of random coils are related to the radius of gyration as the appropriate size parameter. We shall see how the radius of gyration can be determined from solution viscosity data for these... 

According to one point of view, the entire domain of the coil is unperturbed by the flow. The coil in this case behaves effectively like a rigid body whose volume is proportional to r/, where r is the radius of gyration of the coil. 

Random coils. Equation (9.53) gives the Kirkwood-Riseman expression for the friction factor of a random coil. In the free-draining limit, the segmental friction factor can, in turn, be evaluated from f. In the nondraining limit the radius of gyration can be determined. We have already discussed f in Chap. 2 and (rg ) in this chapter and again in Chapter 10, so we shall not examine the information provided by D for the random coil any further. 

We shall see in subsequent sections that measuring Rg as a function of 0 can be used to evaluate the radius of gyration of the scattering molecules, thereby providing more information about the polymer in addition to M and B. 

At this point we return to Chap. 1 to connect Eq. (10.85) with the radius of gyration. Although we have not encountered the form rj, explicitly... 

In Example 10.5 we extracted both the molecular weight and the radius of gyration from Ught-scattering data. There may be circumstances, however, when nothing more than the dimensions of the molecule are sought. In this case a simple alternative to the analysis discussed above can be followed. This technique is called the dissymmetry method and involves measuring the ratio of intensities scattered at 45° and 135°. The ratio of these intensities is called the dissymmetry ratio z ... 

Table 10.1 Relationships Between the Radius of Gyration and Geometrical Dimensions for Some Bodies Having Shapes Pertinent to Polymers...

Mi ellaneous Characterizing Constants The radius of gyration (R) is a simultaneous size-shape fac tor varying with the manner in which mass is distributed about the center of gravity of the molecule. For planar molecules, the radius of gyration is... 

AB and ABC are the products of the principal moments of inertia. Moments of inertia are calculated from bond angles and bond lengths. Many values are given by Landolt-Bornsteiu. is Avogadro s number, and M is the molecular weight of the molecule. Stuper et al. give a computerized method for prediction of the radius of gyration. 

For predicting liquid diffiisivities of binary nonpolar liquid systems at high solute dilution, Umesi " developed a method that only depends on the viscosity of the solvent (2) and the radius of gyration of the solvent (2) and the solute (1). The Technical Data Book— Petroleum Refining gives the method and values of the radii of gyration for common hydrocarbons. Errors average 16 percent but reach 30 percent at times. 

Umesi-Danner They developed an equation for nonaqueous solvents with nonpolar and polar solutes. In all, 258 points were involved in the regression. Rj is the radius of gyration in A of the component molecule, which has been tabulated by Passut and Danner for 250 compounds. The average absolute deviation was 16 percent, compared with 26 percent for the Wilke-Chang equation. 

Component reliability will vary as a function of the power of a dimensional variable in a stress function. Powers of dimensional variables greater than unity magnify the effect. For example, the equation for the polar moment of area for a circular shaft varies as the fourth power of the diameter. Other similar cases liable to dimensional variation effects include the radius of gyration, cross-sectional area and moment of inertia properties. Such variations affect stability, deflection, strains and angular twists as well as stresses levels (Haugen, 1980). It can be seen that variations in tolerance may be of importance for critical components which need to be designed to a high reliability (Bury, 1974). 

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