Infinite dilution scattering

The various physical methods in use at present involve measurements, respectively, of osmotic pressure, light scattering, sedimentation equilibrium, sedimentation velocity in conjunction with diffusion, or solution viscosity. All except the last mentioned are absolute methods. Each requires extrapolation to infinite dilution for rigorous fulfillment of the requirements of theory. These various physical methods depend basically on evaluation of the thermodynamic properties of the solution (i.e., the change in free energy due to the presence of polymer molecules) or of the kinetic behavior (i.e., frictional coefficient or viscosity increment), or of a combination of the two. Polymer solutions usually exhibit deviations from their limiting infinite dilution behavior at remarkably low concentrations. Hence one is obliged not only to conduct the experiments at low concentrations but also to extrapolate to infinite dilution from measurements made at the lowest experimentally feasible concentrations. 

Figure 7 Schematic Zimm plot The determined Kc/(it values ( ), the extrapolated values for fixed concentrations to q — 0 ( ), and the extrapolated values for fixed scattering vector to infinite dilution (°) c — 0.

Therefore we expect Df, identified as the fast diffusion coefficient measured in dynamic light-scattering experiments, in infinitely dilute polyelectrolyte solutions to be very high at low salt concentrations and to decrease to self-diffusion coefficient D KRg 1) as the salt concentration is increased. The above result for KRg 1 limit is analogous to the Nernst-Hartley equation reported in Ref. 33. The theory described here accounts for stmctural correlations inside poly electrolyte chains. 

The z-averag translational diffusion coefficient aj infinite dilution, D, could be determined by extrapolating r/K to zero scattering angle and zero concentration as shown typically in Figs. 4 and 5. D is related to the effective hydrodynamic radius, by the Stokes-Einstein relation ... 

Fact (1) dictates that we must deal with a kind of copolymer, and fact (2) leads to the familiar problem of preferential adsorption of one solvent component on the polymer chain over the other. Either of these introduces ambiguity in the determination of from angular dependence of scattered light at infinite dilution. 

The heats of solution of lithium perchlorate in aqueous acetonitrile were measured at concentrations between 0.01 and 0.1m. The concentration dependence was small compared with the experimental scatter of about 0.1-0.2 kcal mole-1. AHs values are given in Table II. The heats of solution in anhydrous acetonitrile were corrected to infinite dilution using measured heats of dilution (6), and the corrected values were averaged. The heats of dilution were measured for lithium perchlorate in the mixed solvent containing 90% MeCN. 

The anisotropy of the diffusion process is most pronounced in the limit of infinite dilution in the absence of intermolecular collisions that scatter... 

Flory has recently summarized the experimental evidence pertaining to local correlation and their effects on chain dimensions (49). There is experimental support for local alignment from optical properties such as stress-optical coefficients in networks (both unswelled and swelled in solvents of varying asymmetry), and from the depolarization of scattered light in the undiluted state and at infinite dilution. The results for polymers however, turn out to be not greatly different from those for asymmetric small molecule liquids. The effect of... 

Analysis of low angle X-ray scattering data gives a value for the radius of gyration at infinite dilution of 18.3 A (254)- The scattering curve can best be explained on the basis of a cylinder or prolate ellipsoid... 

The single-chain structure factors calculated in the previous sections correspond to the infinite dilution limit. This limit also corresponds to zero scattering intensity and is not useful so that concentration effects have to be included in the modeling of polymer solutions. First, Zimm s single-contact approximation [5] is reviewed for dilute polymer solutions then, a slight extension of that formula which applies to semidilute solutions, is discussed. 

With respect to the virtually very low sample loads, one could argue that the determined diffusion coefficient is at infinite dilution. However, there is no possibility in FFF techniques to perform a safe extrapolation to infinite dilution as in analytical ultracentrifugation or dynamic light scattering. In addition there is a severe problem for all particles deviating from the spherical shape (see Sect. 4.2.3 for a detailed discussion), as the evaluation of D from the FFF experiments using the conventional theory is inappropriate. 

There are no thermodynamic consistency tests that can be applied to the data. For each system in the infinite dilution VLE data base, the weight fraction activity coefficients were plotted as a function of temperature. In many cases considerable scatter was observed. Some data were found that were significantly outside the anticipated range or which showed contradictory behavior with temperature from that expected. These points were kept in the data base but are indicated by an "R", for Rejected, if they were judged to be clearly erroneous or by an "N", for Not recommended, if they appeared questionable but not obviously incorrect. Similarly, in the finite concentration VLE data base some points were judged to be significant outliers and are indicated accordingly. 

Since the square of the specific refractive index increment (dn/dc) appears in the light scattering equations, this value must be accurately known in order to measure Mv,- (An error of x% in dn/dc will result in a corresponding error of about 2x% in M. ) The value of dn/dc is needed at infinite dilution, but there is very little concentration dependence for polymer concentrations in the normal range used for light scattering. The required value can therefore be obtained from... 

Dynamic Light Scattering. The hydrodynamic radius, Rh, is defined as the Stokes radius from the mutual diffusion coefficient at infinite dilution (Do)... 

The value of c, is measured using a concentration-sensitive detector, which is usually a differential refractometer. As in the case of light scattering, polymer concentration during an SEC experiment is considered to be close to infinite dilution thus, extrapolation to zero concentration is not required. Providing that universal calibration is valid for a given SEC system, data of the polymer at each elution volume increment Mi can be determined ... 

The calculated correlation volumes and energetic parameters for the alcohol—water and hydrocarbon—water systems are hsted in Tables 4 and 5. The calculated volumes are compared with the sizes of clusters in several alcohol/water systems determined by small-angle X-ray scattering or light scattering " at low concentrations (Table 6). Table 6 shows that there is reasonable agreement between them and the calculated correlation volumes at infinite dilution. 

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