Molar mass dispersity

Conversely, under the assumption of very low molar-mass dispersity (0, ) of primary chains, the weight-average chain length of the primary chains can be approximated by its number-average value, that is. 

Due to the high importance of chain growth polymerization for polymer production, huge efforts have been undertaken to understand and elucidate the kinetics involved in the chain growth process and the parameters influencing molar mass, dispersity D, and the nature of Ihe resulting macromolecules. In Figure 3.4, the kinetic steps, exemplified for a free radical process, are outlined. 

The molar mass dispersity is the best known of all dispersities in moleeular char-acteristies of synthetie polymers and it is most frequently determined. Assessment of dispersities in ehemieal straeture and in physieal arehiteeture is much more demanding. Let us stress again that molar mass dispersity fairly affects numerous utihty properties of industrial polymers. The width of molar mass dispersity is usually expressed with the ratio of particular molar mass averages. This is habitually sufficient for a sound estimate of suitabihty of particular polymer for most particular applications. Molar mass dispersity function quantitatively reflects amount of macromolecules with certain molar mass present in the sample. It can be represented in the integral or in the differential form. The latter possesses a more informative nature and it is more frequently used. 

Considering molar mass of a polymer, amount of macromolecules with given molar mass M. can be expressed by their number N. or by their mass MN.. Hence the molar mass dispersity function is a relation between the number fractionp. or the mass (weight) fraction q. and logarithm of molar mass M. of particular macromolecules. It holds for the number fraction p. 

The ratio A//A/ is called polydispersity (the previously used term was poly-molecularity) of a polymer and it gives important information on the width of its molar mass dispersity. Less frequently, the viscosity average molar mass, A/, and the artificial z and z+1 averages of molar mass. A/ and Af are considered. M is defined... 

Unfortunately, average values of molecular characteristics of polymers say little about the type and shape of the molar mass dispersity fimctioa For example the MJM values may be identical for both the broad unimodal and the mnlti-modal distributed polymers. 

FIGURE 1 Typical courses of molar mass dispersity functions, a, b, c—differential, and d- integral representation a- broad molar mass dispersity polymer b-narrow molar mass dispersity polymer c- polymer with bimodal dispersity. 

Distinct shapes of molar mass dispersity functions exist for synthetic polymers of different origin log normal, Schultz-Zimm, and so on. Graphical representations of quantitative dispersity functions are less commonly employed than simple ratios Af/Af otM/M. ... 

Often, size exclusion chromatograms (SEC) (compare section 11.7, Size Exclusion Chromatography) of polymers under study are expressed as differential representations of molar mass dispersity. The chromatographic retention volumes are directly transformed into the molar masses. This approach renders useful immediate information about tendencies of molar mass evolution in the course of building or decomposition polyreactions but the absolute values of molar mass can be only rarely extracted from it. As a rale, polystyrene calibrations are applied for molar mass calculation so that one deals with the polystyrene equivalent molar masses, not with the absolute values. The resulting dispersity (distribution) functions may be heavily skewed because the linear part of the calibration dependence for the polymer under study may exhibit well different slope compared with the polystyrene calibration, which was employed for the transformation of retention volumes into molar masses. 

The important feature of many synthetic polymers is the simultaneous presence of dispersities in two and even in several molecular characteristics. Polymers exhibiting multiple dispersities - while it must be accentuated again that molar mass dispersity is always present( ) - are called complex polymers. The representative complex polymers are different kinds of copolymers and functional polymers. A mixture of a complex polymer with another macromolecular substance is denoted complex polymer system. Typical complex polymers systems are many polymer blends, as well as the block copolymers, which contain their parent homopolymers. Molecular characterization of constituents of complex polymers and complex polymer system constitutes a particular challenge. Different approaches to address this task will be presented in this chapter. 

In order to determine dispersity of their molecular characteristics, polymers must be separated, fractionated. There are only few exceptions from this rule. For example, concerning molar mass dispersity, valuable information can be acquired from the rheological measurements. 

In the first approximation, the column packed with small particles can be considered an array of capillaries. Hydrodynamic processes augment the exclusion based separation (see section 11.7.2, Retention Mechanisms and Accompanying Processes in SEC) and form the ground of a liquid chromatography-like method called hydrodynamic chromatography, HDC. HDC found application in separation of veiy large macromolecules, particles and dispersions. Its separation selectivity is rather low and it was partially substituted by the group of methods termed fieldflow fractionation (see seetion 11.3.3, Molar Mass Dispersity). 

As stated in section 11.5, methods of polymer HPLC are not absolute. This also holds for the most important one of them, SEC. Therefore, the instraments have to be calibrated or molecular characteristics of analyzed macromolecules that leave the separation column are to be monitored with an independent method (compare sections 11.6.1.4 and 11.7.3.2). Polymers with the known molecular characteristics serve for the colunm calibration. They are called - not always entirely correctly - macromolecular standards. The sets of polymers with different known molar masses and as a rale with low molar mass dispersity are available from several producers. They have been previously characterized with other physicochemical method(s) or just by the independent SEC measurements. In some cases, molecular characteristics of above standards are derived or even only estimated taking into account method of their preparation. As a result, quality of macromo-... 

There were also attempts to calibrate the SEC columns with help of broad molar mass dispersity poplymers but this is less lehable. The most common and well credible SEC cahbration standards are linear polystyrenes, PS, which are prepared by the anionic polymerizatioa As indicated in section 11.7, according to lUPAC, the molar mass values determined by means of SEC based on PS calibration standards are to be designated polystyrene equivalent molar masses . Other common SEC calibrants are poly(methyl methaciylate)s, which are important for eluents that do not dissolve polystyrenes, such as hexafluoroisopropanol, further poly(ethylene oxide)s, poly(vinyl acetate)s, polyolefins, dextrans, pullulans, some proteins and few others. The situation is much more complicated with complex polymers such as copolymers. For example, block copolymers often contain their parent homopolymers (see sections 11.8.3, 11.8.6 and 11.9). The latter are hardly detectable by SEC, which is often apphed for copolymer characterization by the suppliers (compare Figure 16). Therefore, it is hardly appropriate to consider them standards. Molecules of statistical copolymers of the same both molar mass and overall chemical composition may well differ in their blockiness and therefore their coils may assume distinct size in solution. In the case of complex polymers and complex polymer systems, the researchers often seek support in other characterization methods such as nuclear magnetic resonance, matrix assisted desorption ionization mass spectrometry and like. 

SEC is presently the most important method for separation and moleeular characterization of synthetic polymers. The method enjoys enormous popularity and most institutions involved in research, production, testing and apphea-tion of synthetie polymers are equipped at least with a simple SEC instrument. Size exclusion chromatograms are often directly transformed into the molar mass dispersity functions (compare section 11.3.3, Molar Mass Dispersity). Often, the molar mass data presented are not absolute, beeause polystyrene or other polymer standards distinct from polymer under study have been employed for the column calibration (see sections 11.6.3 and 11.7.3.1). Still, the data equivalent to the polymer applied to the column cahbration, more or less precisely represent the tendencies of molar mass evolution in the course of building-up or decomposition polyreactions. 

V shonld be similar for the calibration standards and for the sample. Apparently, this advice is convenient to follow experimentally, however it may comphcate treatment of wide and flat chromatograms of broad molar mass dispersity polymers. The effect of v on the peak apex position of narrow cahbration standard differs from the influence of v. on the retention volumes of fractions comprised in broad molar mass dispersity sample. Qnantitative evaluation of the above difference is a complex matter... 

FIGURE 16 Real chromatograms of model 4 1 blends of poly styrene and PMMA. Molar masses of narrow molar mass dispersity polymers were 270 and 293 kg mob, respectively. For detailed explanation, see the text. 

The ratio of [tj]/[tj]0 is called the viscosity expansion factor a. The viscosity-averaged molar mass (A/ ) obtained from the MHSS equation may also be used to get information about the degree of molar mass dispersion from the comparison of the M values of a polymer sample in two different solvents [5,6],... 

Dispersity is an appropriate word to describe a numerical attribute of the dispersion of a distributiondispersion of a distribution. The use of the term polydispersity index or other words involving the word polydispersity is strongly discouraged [09IUP1]. The general symbol Dj, pronounced D-stroke , is introduced for molar mass dispersity. 

After defining the average molar masses in the next section (Section 3.4.2), the various methods of quantifying the molar-mass dispersity will be presented. 

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