Moment of inertia of a circle given radius of gyration
However, recent bubble inflation 128,129,396 and surface forces apparatus 69,126 measurements of polymer melts evidence a ‘stiffening’ in very thin films that would seemingly not support a reduced entanglement density. 395 These measurements were supported by Rowland and co-workers who saw a decreased resistance to viscous flow of the polymer melt under a nanoscale punch in high molecular mass polymers when the thickness of the film became less than the R g of the polymer. Si and co-workers recently used the degree of necking in the thickness direction that occurs in a polymer film under tensile deformation to predict a reduced thin film entanglement density. There are now a few measurements starting to address this issue. However, it is difficult to quantify the degree of entanglement in a thin polymer film as quantitative rheology measurements are difficult. 393,394 This is an attractive concept, depicted schematically in Figure 9, to explain dynamic observations where increased molecular mobility is evidenced in thin films. If in extremely thin films the chains are forced to fold back in on themselves, the result would be that intermolecular penetration and entanglements are lost. This has to be true to generate the well-understood intermolecular entanglement junctions that are evidenced in rheological experiments. We know that chains of an amorphous polymer are highly entangled and interpenetrating. When a random walk approaches an interface, it simply changes directions and reflects back on itself. The observation above of an average chain diameter in the plane of the film that does not flatten out and expand with decreasing film thickness also suggests that interfaces of the film act as reflecting interfaces.
Entanglements are constraints that resist viscous flow.
It is well known that dynamic properties like flow, viscosity, and other relaxation processes are influenced by the entanglement density of a polymer. However, there is a second possible explanation that would have an impact on the dynamical response of the polymer. The explanation of the R g measurements above does not strongly relate to dynamics. An average radius of gyration can be determined from the angular dependence of the intensities of scattered light. The radius of gyration, r g, of a polymer in solution will depend on the molecular weight of the macromolecule, on its constitution (whether or not and how it is branched), and on the extent to which it is swollen by the solvent. (The radius of gyration and other measures of macromolecular size and shape are considered in more detail in Section 1.13.) The radius of gyration is one measure of the size of the random coil shape which many synthetic polymers adopt in solution or in the amorphous bulk state. For a polymer chain, this is also the root-mean-square distance of the segments of the molecule from its center of mass. Alfred Rudin, Phillip Choi, in The Elements of Polymer Science & Engineering (Third Edition), 2013 3.2.6 Radius of Gyration from Light-Scattering DataĪ radius of gyration in general is the distance from the center of mass of a body at which the whole mass could be concentrated without changing its moment of rotational inertia about an axis through the center of mass.