Scattering from dilute solutions

The size and shape of a polymer chain in dilute solution is best studied using scattering methods. Each monomer absorbs the incident radiation 

This power law extends from the monomer size b to the size of the chain R, provided that the entire chain has the same fractal dimension V. Scattering on such small scales is dominated by intra-molecular scattering from 

The inter-molecular scattering dominates the scattering function at wave-vectors q smaller than the reciprocal distance between chains (c /AO, where c is the number density of monomers in solution. The inter-molecular regime is controlled by concentration fluctuations arising from the difference in the number of chains in volumes j(. Assuming there are no interactions between chains (strictly valid only in very dilute solutions), the mean-square fluctuation in the number of chains in the volume jq is of the order of the average number nJNi c l Nq ). The fluctuation in the 

Note that this value matches the low- end of the fractal regime. It is hardly surprising that the scattering function contains information about the chain length, since in Chapter 1 we demonstrated how light scattering can 

---

McIntyre, D., Gornick, F. (Eds.) Light Scattering from Dilute Solution, New York Gordon Breach 1964... 

Scattering techniques for measuring various static and thermodynamic properties of polymers, such as molar mass, size, conformations, interaction parameters, etc. were described in experimental sections of Chapters 1-5. In addition to static properties, scattering can provide important information about dynamic properties of polymeric systems. This section focuses on dynamic scattering from dilute solutions, but similar methods are used in semidilute and concentrated solutions."... 

Scattering from dilute solutions was discussed in Section 3.5. The inter-... 

A detailed experimental study of the isotropic component of light scattered from dilute solutions of tobacco mosaic virus (a rod-like molecule with L = 3000 A and cross section diameter = 180 A) has been perfomed by Cummins et al. (1969) using spectrum analysis techniques. These authors found that the measured spectrum fit the theory described above rather well. Wada et al. (1971) repeated these experiments using an autocorrelator with similar results. 

The concentration fluctuation contribution to the scattering from dilute solutions is given by... 

Small-angle neutron and X-ray scattering from dilute solutions of star polymers 64... 

Dynamic light scattering from dilute solutions provides the value of the diffusion coefficient, which can be converted to hydrodynamic radius J h,star of the star polymer. The ratio Rh/(Rg) characterizes the compactness of the macromolecule for the uniform hard sphere impenettable for the flow, it is Rb/Rg= (5/3) 1.29, whereas for the Gaussian coil, Rh/(Rg) = 3a- / /8 0.66. For ideal stars (without excluded-volume interactions), the ratio Rh/(Rg) can be derived within the Kirkwood-Riseman approximation, which gives the value of Rh/(Rg) 0.93. Reported experimental values of the Rh/(Rg) ratio for star polymers and starlike block copolymer micelles are usually found close... 

Further work with application to large molecules or particles in solution was done by Smoluchowski (J), Einstein (4), Debye (5, 6) and Zimm (7). For many years now the measurement of the intensity of the light scattered from dilute solutions of macromolecules as a function of concentration, and scattering angle has provided much important information on the size, shape, and thermodynamic properties of these... 

P. Debye, in Light Scattering from Dilute Polymer Solutions (D. McIntyre and F. Gornick, eds.), p. 13, Gordon and Breach, New York, 1964. 

The equilibrium structure is considered of flexible polymer chains within the RIS model. This model is solved by an irreducible tensor method which is somewhat different from, and simpler than, the approach of Flory and others. The results are used to compute the light scattering intensities from dilute solutions of flexible polymer chains, and the angle dependence is found... 

T. Imae and S. Ikeda, Intermicellar correlation in light-scattering from dilute micellar solutions of dimethyloleylamine oxide, J. Colloid Interface Sci. 98 (1984) 363-372. 

M. J. Menosveta and D. A. Hoagland, Light scattering from dilute poly(styrene) solutions in uniaxial extensional flow, Macromolecules, 24,3427 (1991). 

A Some ambiguity must be attached to this statement in view of the conclusions reached by Luzzati el al. (1961) on the basis of small angle X-ray scattering studies from dilute solutions of polybenzyl-L-glutamate in dimethylformamide, pyridine, and TO-cresol. These authors conclude that the conformation of the polypeptide in these solutions is that of the 3io-helix (Donohue, 1953) and not that of the a-helix. The value of 6o = —630° in Eq. (13) was obtained from optical rotatory dispersion measurements with similar solutions, and therefore if the conclusions of Luzzati et al. are correct, this value characterizes the 3io-helix. It would then be necessary to determine whether the 6o value for the a-helix is significantly different. 

The Debye function describes the q dependence of scattering data from dilute solutions of ideal chains. Such dilute solutions can either be obtained in a (9-solvent or by having a dilute solution of ordinary chains in a melt of perdeuterated chains. Small-angle neutron scattering data for four dilute concentrations of poly(methyl methacrylate) (PMMA with 2.50 000 g/mol ) in perdeuterated PMMA are shown to be fit by the Debye... 

Molar mass dependence of the radius of gyration from light scattering in dilute solutions for polystyrenes in a 0-solvent (cyclohexane at 0 = 34.5 °C, circles) and in a good solvent (benzene at 25 C, squares). Data are compiled in L. J. 

Dautzenberg H, Rother G, Hartmann J. Light scattering studies of polyelectrolyte complex formation effect of polymer concentration. In Schmitz KS, ed. Macroion Characterization From Dilute Solution to Complex Fluids. ACS Symposium Series 548. Washington, DC American Chemical Society, 1994 210-224. 

More difficult, but also proniising to be honoured with success proves the isolation of pure, integral membrane proteins and their structure analysis by diffraction techniques. One avenue is given by defined solubilization with detergents and the evaluation of the small-angle (particle) scattering pattern from dilute solution (for reviews on this method, see Refs. and ). This has so far been attempted with bovine rhodopsin, the major protein component of retinal rod outer sement membranes with the Ca -dependent ATPase from sarcoplasmic reticulum... 

Fig. 15.23. Plot of p/l q) (in arbitrary units) against q2(in nm 2). Result obtained by neutron scattering. Semi-dilute solution of deuterated polystyrene (Mw = 1.1 x 10s) in CS2 (from Farnoux16), Mass-concentration of the solute p = 25 g/l. The quantity f 2 is determined by linear extrapolation of the points which are near to qz = 0. We also marked "2 calculated from eqn (13.2.122).

---