Polydisperse systems with solvent

This equation appears to have a number of names, of which the Mark-Houwink equation is the most widely used. In order to use it, the constants K and a must be known. They are independent of the value of M in most cases but they vary with solvent, polymer, and temperature of the system. They are also influenced by the detailed distribution of molecular masses, so that in principle the polydispersity of the unknown polymer should be the same as that of the specimens employed in the calibration step that was used to obtain the Mark-Houwink constants originally. In practice this point is rarely observed polydispersities are rarely evaluated for polymers assigned values of relative molar mass on the basis of viscosity measurements. Representative values of K and a are given in Table 6.4, from which it will be seen that values of K vary widely, while a usually falls in the range 0.6-0.8 in good solvents at the 0 temperature, a = 0.5. 

There is a large body of literature on phase diagrams for pseudo-binary and pseudo-ternary polymer systems (pseudo- is to indicate polydispersity of molecular weights) with solvents [Flory, 1953 ... 

The physicochemical properties of alkylated polysaccharides have received some attention, and details of the structures and conformations of 0-methylcelluloses, and their interaction in micelle junctions, have been included in a thorough discussion of polysaccharide gels and networks. Information on the polydispersity of samples of partially methylated cellulose may be obtained from column fractionation and by fractional precipitation from a solvent-nonsolvent system, but, for a more complete characterization of polydispersity, fractionation with a series of solvent-nonsolvent systems is necessary. The solubility, in water, of polysaccharides that are mainly methylated may be considerably improved by introduction of a few suitable ionizing groups, for example, by reaction with monochloroacetic acid to introduce carboxymethyl ether groups. The general sorption and diflFusion features of hydrocarbons and other... 

However, traditional chemical thermodynamics is based on mole fractions of discrete components. Thus, when it is applied to polydisperse systems it has been usual to spht the continuous distribution function into an arbitrary number of pseudo-components. In many cases dealing, for example, with a solution of a polydisperse homopolymer in a solvent (the pseudobinary mixture), only two pseudo-components were chosen (reproducing number and mass averages of molar mass of the polymer) which, indeed, are able to describe some main features of the liquid-liquid equilibrium in the polydisperse mixture [1-3]. In systems with random copolymers the mass average of the chemical distribution is usually chosen as an additional parameter for the description of the pseudo-components. However, the pseudo-component method is a crude and arbitrary procedure for polydisperse systems. 

Here hence denotes the position of monomer with label i (i= 1,..., N) in the feth chain molecule (fe = 1,..., N ). For simplicity, we have specialized here to a monodisp>erse system of linear homopolymers hut the generalization to polydisperse systems or to heteropolymers or to branched architecture is straightforward, as well as to multicomponent systems (including solvent molecule coordinates, for instance). Typically, the volume in which the S3 tem is considered is a cubic LxLxL box (in d = 3 dimensions, or a square LxL box in d = 2 dimensions), and one chooses periodic boundary conditions to avoid surface effects but if the latter are of interest, the corresponding change of boundary conditions is straightforward. All of what has been said so far applies to lattice models as well as to models in the continuum. 

The polarizability, in turn, is proportional to the particle volume, a oc and the difference between the refractive indices of the particle and the solvent, with the latter quantity being accessible from differential refractometry measurements. For homogeneous spheres of radius / , the ratio between the scattering intensity and the particle volume fraction, //0, is proportional to / , and this quantity can be used to estimate the particle size. For a polydisperse system, the signal is proportional to the following ratio of moments ... 

Size exclusion chromatography (SEC) polymer elution profiles yield information regarding the molecular size distributions of polydisperse macromolecules. Polymer molecular weight distribution (MWD) represents an intrinsic property which provides direct correlation with many end-use physical properties and a universal criterion for polymer characterization (1). In order to convert elution profiles or chromatograms into MWD information proper calibration methods are required. SEC molecular weight calibration techniques represent experimental approaches for transformation of polymer elution profiles into MWD information and are dependent upon instrumentation, columns, and the polymer/solvent system under study. 

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