Form factor star polymer

Figured displays simulated form factors for a multiarm star polymer of varying functionality and a hard sphere [41], The high-g asymptotic behavior, characteristic of the coil structure, is absent in the latter case. A handicap in the experimental determination of P(g) is often the narrow-g range accessible by the scattering techniques that can be overcome through the combination of low-g light scattering and high-g X-ray and/or neutron scattering (utilized on the same system). Size and shape also determine the translational diffusion Dq of the nanoparticles in dilute solution, and hence Dq can prove the consistency of the scattering results.

Fig. 19 (a) Normalized SANS curves in different D20/DMF-d7 compositions at a polymer volume fraction of 0.25%. Solid lines represent fits with a spherical core shell model. Data in pure DMF-dv were fitted with a Beaucage form factor, (b) Aggregation number P plotted versus interfacial tension, y. The solid line depicts the power-law dependence, P rj as predicted by Halperin for star-like micelles [40]. Reprinted with permission from [45]. Copyright (2004) American Chemical Society... 

For the calculation of the form factors of unperturbed central regions of the stars, one has to take into account that (1) the central-symmetrical unperturbed polymer density profile c(r)... 

We now calculate the form factor Pstar( ) for an riA-arm star polymer with a uniform arm length Ni. When calcnlating the average of exp[ik-(r - r )], it is necessary to distinguish two cases for r and r (1) being on the same arm and (2) being on different arms. The former takes place with a probability of l/n. Then,... 

Problem 2.21 Verify that the form factor of a two-arm star polymer (% = 2 in Eq. 2.98) reproduces the form factor of a Gaussian chain. 

Fig.43 Single star form factors at 71 °C for the PE-PEP and PEP-PE stars in d-decane for a polymer concentration poi = 0.24%. The solid lines represent a fit with the Gaussian star form factor...

Fig. 9.9 Form factor P q) versus qu for four-star polymers in a good solvent with/=10-50 for JV = 50 simulated by MD for a purely repulsive Lennard-Jones interaction between nonbonded monomers from Ref. 96. Also shown are the results for two linear polymers with 50 and 100 monomers. The data have been offset for clarity.

In the case of perfect networks, combination of equations (47), (107), (112) and (113) yields equation (124). Thus, k depends on the inverse square-root of the phantom modulus and is independent of swelling. The factor is Avogadro s number, which appears in equation (124) since n in equation (112) is the number of junctions and not the number of moles of junctions. In the case of randomly cross-linked networks, use of equation (61) yields equation (125). In the case of networks formed by random cross-linking of star polymers, equation (63) is used instead of equation (61) to derive the expression for k. The other parameter C is the result of the relationship between k and network inhomogeneities and its magnitude is estimated by experiment. 

Because of the fundamental nature of the size of polymer coils, considerable effort has been devoted to the experimental measurement of the size of regular star polymers both in good solvents and under 6-conditions. The size of a polymer coil is obtained from the angular dependence of scattered radiation at low angles. The angular dependence of the intensity of light (or neutrons) scattered by a polymer solution is analyzed in terms of the product of the form factor, P Q), and the structure factor, S Q) ... 

TTie form factor of a regular star polymer has been derived under the assumption of the Gaussian segment distribution [22,23] ... 

Figure 1 Zero-concentration intramolecular scattering function or form factor of a 32-arm polybutadiene star obtained by light scattering in cyclohexane with g

A detailed analysis of the form factor over the whole Q range was undertaken by Dozier et al. [68]. They analyzed results on 8- and 12-arm star polymers in good solvent and 0-solvents and concluded that the form factor of stars consists of two regions. In the low Q range, the form factor is characteristic of the whole molecule and is described by a Guinier relation in terms of the radius of gyration, P(Q) = exp[-(l/3) Rq], Eq. (20). At high values of Q, on the order of the inverse of the size of the outermost blob, -Rq [see Eq. (5)],... 

Figure 5 Zero-concentration form factors of star polymers with different functionalities. Data obtained in a good solvent, methylcyclohexane-di4 with SANS. From bottom to top /= 8 (polyisoprene),/=18 (polyisoprene),/= 32 (polybutadiene),/= 64 (polybutadiene), and/= 128 (polybutadiene). The data are offset vertically for clarity. The solid line represents Eq. (27). Arrows indicate Q=the onset of the asymptotic regime in which scattering is caused by the swollen blobs in the corona. (From Ref 31.)...

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