Radii of gyration

Let be the number of monomers in a concentration blob. Since the number density inside the blob must be equal to the concentration (p/a of the solution, we find 

This is much larger than the number of monomers gr = 1/t in the temperature blob studied in Section 1.6. 

When the chain is seen as a whole with blobs as the structural repeat unit, the excluded-volume effect by monomer interaction is screened by the presence of the blobs of other chains, and hence considered to behave as an ideal chain. If we assume such a screening effect, the radius of gyration is given by 

This result is called the Daoud radius of gyration, since it was found by Daoud [24]. 

This equation suggests that, if the concentration is increased still further at a fixed temperature t, the radius of gyration becomes the same order as the Gaussian chain R = na at the concentration satisfying the condition 

If the particle is of a constant scattering length density in its entirety, Equation (5.2) is simplified to 

As examples of the radius of gyration of particles of well-defined geometric shape, we obtain for a solid sphere of radius R 

For a solid rod having length L and circular cross section of radius R it becomes 

The radius of gyration of such a Gaussian chain, calculated according to (5.6) (see Flory9), is 

Note that the instantaneous value of the radius of gyration can change as the polymer conformation undergoes continual change with time and the above expression (5.13) holds only for the radius of gyration ( R j averaged over time (or, equivalently, over an ensemble of identical chains). 

At first sight, it would appear that this direct method would be the preferred one in this group, since s2 or at least Sq can be related fairly directly to the degree of branching, if the type of branching is known, as described in Section 3, whereas the remaining methods described in this subsection involve hydro-dynamic problems that are hard to solve. However, this method has not been used to any great extent, for several reasons  

Moore and Millns (40) and also Hama and co-workers (42) have attempted to overcome this difficulty by estimating the weight average value So w from the z-average value using an assumed MWD of breadth indicated by measurements of Mw and Mn on the (polyethylene) fractions they studied, but their procedure involves the assumption that s2 is proportional to M, which may not be accurate for branched molecules. However, their procedure is probably the best currently available for the estimation of the weight-average value of s2. 

It is worth remarking that though Moore (/35), who used light scattering to study polyethylene, stated the theory of this method, the estimates of the degree of branching that he presented were actually derived from values of via Beasley s theory (S). 

Much the most usual method of estimating branching is from the intrinsic viscosity ratio g1, defined in Eq. (4.2). In addition to the intrinsic viscosity of the sample, its (viscosity-average) MW must also be known, so that the denominator of the fraction (4.2) can be found. It is usual to use the light scattering technique for this, assuming that Afw = or if any information on the MWD is available, a correction may be applied to M . Recently it has become common to combine GPC measurements with those of viscosity see Subsection 9.2.4. 

Thurmond-Zimm or the Zimm-Kilb rules, and it seems to be a reasonable compromise between them. 

Where r is the position of site a, r is the position of the molecular center of mass, m is the mass of site a and the angle brackets denote an ensemble average. The value of the squared radius of gyration is defined as the trace of the tensor 

The size of linear chains can be characterized by their mean-square end-to-end distance. However, for branched or ring polymers this quantity is not well defined, because they either have too many ends or no ends at all. Since all objects possess a radius of gyration, it can characterize the size of polymers of any architecture. Consider, for example, the branched polymer sketched in Fig. 2.6. The square radius of gyration is defined as the average square distance between monomers in a given conformation (position 

The position vector of the centre of mass of the polymer is the number-average of all monomer position vectors  

Therefore, the expression for the square radius of gyration takes the form 

We assume that all the monomers have the same mass Mj = Mo for all j. 

In solution, polymeric chains can form different conformations, depending upon the solvent. When the solvent is such that the chains are fully solvated, they are relatively extended and the molecules are randomly coiled. The polymer-solvent interaction forces determine the amount of space that the molecular coil of the polymer occupies in solution. While quite extended in a good solvent, if the solvent is a poor one, the chains are curled up. A measure of the size of the polymer molecule in solution, or the amount of space that a polymer molecule occupies in solution is determined as radius of gyration., ox root mean square radius of gyration, S. Qualitatively, it is the average distance of the mass of the molecule from the center of its mass (from its center of gravity). The following equation 

The distance between the chain ends is often expressed in terms of unperturbed dimensions (So orRo) and an expansion factor (a) that is the result of interaction between the solvent and the polymer 

The unperturbed dimensions refers to molecular size exclusive of solvent effects. It arises from intramolecular polar and steric interactions and free rotation. The expansion factor is the result of solvent and polymer molecule interaction. For linear polymers, the square of the radius of gyration is related to the mean-square end-to-end distance by the following relationship  

This follows from the expansion factor, a is greater than unity in a good solvent where the actual perturbed dimensions exceed the unperturbed ones. The greater the value of the unperturbed dimensions the better is the solvent. The above relationship is an average derived at experimentally from numerous computations. Because branched chains have multiple ends it is simpler to describe them in terms of the radius of gyration. The volume that a branched polymer molecule occupies in solution is smaller than a linear one, which equals it in molecular weight and in number of segments. 

The volume that these molecules occupy in solution is important in determinations of molecular weights It is referred to as the hydrodynamie volume. This volume depends upon a variety of factors. 

Let us now turn to the concentration crossover in the excluded volume limit. Equations (13.35) yield 

If compared to Eq. (9 17) the second equation shows that N/Nr is to be identified with the number of segments n per concentration blob, whereas Nr — Nfn is the number of blobs per chain. The first equation shows that we find a smooth crossover from the dilute limit w — 1 to the semidilute limit w — 0. In the latter limit Eqs. (14 13), (14.14) yield the expected power law 

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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 ... 

MEAN RADIUS OF GYRATION, ANGSTROMS DIPCLF MOMENT, DEBYES... 

MEAN RADIUS OF GYRATION OF COMPONENT I I A I. CRITICAL TEMPERATURE OF COMPONENT I (DEGREES K). TEMPERATURE OF MIXTURE (DEGREES Kl. 

Umesi, N.O. (1980), Diffusion coefficients of dissoived gases in iiquids -Radius of gyration of solvent and solute . M.S. Thesis, The Pennsylvania State University, PA. 

Fig. XI-6. Polymer segment volume fraction profiles for N = 10, = 0-5, and Xi = 1, on a semilogarithinic plot against distance from the surface scaled on the polymer radius of gyration showing contributions from loops and tails. The inset shows the overall profile on a linear scale, from Ref. 65.

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. 

For free particles, the mean square radius of gyration is essentially the thennal wavelength to within a numerical factor, and for a ID hamionic oscillator in the P ca limit. 

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... 

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. 

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

The above radius of gyration is for an isotropic system. If the system is anisotropic, the mean square radius of gyration is equal to... 

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...

Altliough tire tlieories of colloid stability and aggregation kinetics were developed several decades ago, tire actual stmcture of aggregates has only been studied more recently. To describe tire stmcture, we start witli tire relationship between tire size of an aggregate (linear dimension), expressed as its radius of gyration and its mass m ... 

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. 

The summation in Eq. (2.52) corresponds to n times the square of a two-dimensional radius of gyration r 2d there is no rotation in the z direction. 

The three-dimensional radius of gyration of a random coil was discussed in Sec. 1.10 and found to equal one-sixth the mean-square end-to-end distance of the polymer [Eq. (1.59)]. What we need now is a connection between two-and three-dimensional radii of gyration. Since the molecule has spherical symmetry r, r> = V + r + r, = 3r . If only two of these contributions are present, we obtain (2/3)rg 3 = rg2o- this result and Eq. (1.59) to... 

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. 

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