Number-average molar masses

The knowledge of the distribution of molar mass of a polydisperse polymer is of extreme importance. Many of the physical properties of polymers depend upon this and a correct distribution is essential in any polymer which is used in a practical situation. One of the most important points in the distribution is the number average molar mass M , and there are several techniques which can be used to measure it. All of these methods involve determining the number of molecules in a given mass of polymer. The polymer is usually dissolved in a solution of known concentration and hence the mass of polymer per unit volume is defined. The most important methods involve the use of the colligative properties of dilute solutions although for a limited number of polymers it is possible to measure M by end-group analysis. 

The colligative properties of a solution are those which depend only upon the number of solute molecules in unit volume of the solution and not upon the nature of the molecules. They are insensitive to variables such as branching or copolymer composition and provide a useful method of measuring M . Colligative properties include osmotic pressure, vapour 

Osmosis is a phenomenon of great importance in many applications and particularly so in polymer science. It is a very accurate and sensitive method of determining M and provides a useful application of the theories of solution thermodynamics developed in Section 3.1. The apparatus used for measuring osmotic pressure is shown schematically in Fig. 3.9. It consists essentially of a chamber with two compartments separated by a semi-permeable membrane. In one compartment there is pure solvent and in the other there is a polymer solution and both are at the same temperature. The membrane is permeable only to solvent molecules and is not permeable to polymer molecules. If both of the compartments are at the same pressure initially it is found that solvent molecules tend to diffuse from the pure solvent through the membrane and the diffusion stops only when the pressure in the solution compartment is increased by either applying an external pressure or allowing a pressure head to develop. The pressure which is required to stop solvent diffusion across the membrane is called the osmotic pressure. 

The reason for the diffusion taking place can be found by considering the chemical potential of the solvent in the two compartments in Fig. 3.9. If compartment A contains pure solvent and is initially at standard atmospheric pressure (101 kNm ) Po, then the chemical potential of the solvent in A will be px and given by 

The term (p - po) is known as the osmotic pressure fl and using Equation (3.53) it follows that 

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The number-average molar mass of thermoplastic polyesters varies from about 20,000 for film and fiber applications to 40,000 for injection-molding or blowmolding resins. Relationships between intrinsic or molten viscosity and molar mass have been published for PET,131-136 PBT,135,137 and PEN.138... 

The rigid and soft blocks used in polyesteredier diermoplastic elastomers (polyesteretlier TPEs) are typically PBT and poly(oxyteti amethylene) (PTMO), respectively, witii block number-average molar mass varying between 1000 and 3000. They are obtained by the melt reaction between dimethyl terephthalate, butanediol, and dihydroxy-terminated PTMO in the conditions typical of a PBT syndiesis. 

Thermoplastic linear polyesters, 18 Thermoplastic polyesters, 20-29, 31 commercial (table), 21-22 number-average molar mass of, 45... 

Gelation occurs at relatively low conversions of monomer to polymer hence the number-average molar mass at the gel point is low. By contrast, however, the weight-average molar mass becomes infinite at the gel point. 

The methods by which polymers are prepared result in a mixture of molecular sizes whose properties depend on the average size of the molecules present. In principle there are a number of ways in which such an average can be calculated. The most straightforward is the simple arithmetic mean, usually called the number average molar mass, M. This is defined by the expression... 

In the case of a polydisperse polymer it is still the total number n of solute molecules that is measured and the total mass m of solute molecules that is known from sample preparation, resulting in the number average molar mass M = ... 

M) is the number average molar mass of the solute (polymer), R — 8.315 J/K mol the gas constant, and c the polymer concentration in polymer mass per volume solution. 

In this section, we review the properties of a series of PNIPAM-b-PEO copolymers with PEO blocks of varying length, with respect to the PNIPAM block. Key features of their solutions will be compared with those of PNIPAM-g-PEO solutions. PNIPAM-b-PEO copolymers were prepared by free-radical polymerisation of NIPAM initiated by macroazoinitiators having PEO chains linked symmetrically at each end of a 2,2/-azobis(isobutyronitrile) derivative [169,170]. The polydispersities of PEOs were low, enabling calculations of the number-average molar mass for each PNIPAM block from analysis of their H-NMR spectra (Table 2). 

Mass spectrometry can be used to measure the molar mass distribution (MMD) of a polymer sample by simply measuring the intensity, Nt, of each mass spectral peak with mass m . This is due to the fact that mass spectrometers are equipped with a detector that gives the same response if an ion with mass 1 kDa or 100 Da (actually any mass) strikes against it. In other words, the detector measures the number fraction and this implies that Nt also represents the number of chains with mass m,. Thus, the number-average molar mass, Mn, is given by ... 

Number-average molar masses were determined using a vapor pressure osmometer (VPO) (Hitachi 117 Molecular Weight Apparatus) at 54.8 0.1°C in toluene (Fisher Scientific, certified A.C.S.) which was distilled from freshly crushed CaH2. The VPO apparatus was calibrated with pentaerythritol tetrastearate (Pressure Chemical). Gel permeation chromatographic (GPC) analyses were performed in tetrahydrofuran by HPLC (Perkin-Elmer 601 HPLC) using six y-Styragel columns (106, 105, 10l, 103, 500, and 100 A) after calibration with standard polystyrene samples. 

Polymers are therefore mainly characterised in terms of two molar masses. These are the Number Average Molar Mass, M, (Eq. 5.6) and the Mass Average Molar Mass, M, (Eq.5.7)... 

Eq. 5.17 when written in terms of number average molar mass yields ... 

Interestingly, our own studies have revealed that both the shape of the macromolecule and the glass transition temperature, Tg, change with irradiation time. For example, the irradiation of a bimodal commercial sample of polyvinylcarbazole (PNVK) (Fig. 5.31a) in dichloromethane occurred with an initial increase in (the number average) molar mass (M ) and an apparent loss in the bimodal nature of the polymer (Tab. 5.16, Fig. 5.31b). A similar initial increase in has been observed by Price [39] during a sonically induced polymerisation. 

Thus, the result of an osmotic pressure experiment with a mixture of solute macromolecules yields the number-average molar mass M . 

Table 15.1 contains osmotic pressure data calculated from the work of Browning and Ferry [3] for solutions of polyvinyl acetate in methyl ethyl ketone at 10°C. Plot H/vv against w, fit the data to a quadratic polynomial, and calculate the number-average molar mass from the intercept with the n/w axis. 

Number-average molar mass of polymer chains between two adjacent crosslinks or junction points in a polymer network. 

A reactor block consisting of 16 reactors was divided into 4 zones with 4 different CTA to initiator ratios, and 4 different acrylates or methacrylates were used in each set of experiments. The polymerization of tert-buiyl methacrylate was repeated four times to demonstrate the reproducibility of the polymerization in an automated parallel synthesizer. Structural analysis of the polymers revealed that there was less than 10% deviation in the number average molar mass (Mn) and the PDI values. 

Fig. 15 Number average molar masses (Mn opc) and PDI values obtained for the first blocks and for the final copolymers of PMA, PnBA, PMMA, or PDMAEMA (25 units) with PEEA (25, 50, 75, and 100 units for 100% conversion). AH Mn pc values are calculated against PMMA standards. SEC eluent CHClsiNEtsii-PrOH. (Reprinted with permission from [87]. Copyright (2005) American Chemical Society)...

Table 12 Theoretical number average molar masses (M ) and polydispersity indices for the for ...

Table 13 Number of incorporated monomer units into the 30 triblock copoly(2-oxazoline)s resulting from combined H NMR spectroscopy analyses (top) of the model [A and AB (block co) polymers] and final polymers as well as the measured number average molar masses (Mn,SEC/PDI bottom). H NMR spectra were recorded in CDCI3 or CD2CI2 (PhOx containing polymers) and GPC analyses were performed using DMF (with 5 mM NH4PF6) as eluent. Mn,GPC was calculated utilizing poly(methyl methacrylate) (PMMA) standards...

The molar mass of the products is very dependent on the final conversion achieved and thus, on the final quality of the vacuum. Under the described conditions number average molar masses as determined by GPC in DMAc/LiCI (3 g/l)/water (2 vol%) will reach values between 2,000-5,000 g/mol. 

Measurements of osmotic pressure provide an absolute determination (without calibration) of the number-average molar mass. This is independent of the type of solvent for each solvent the extrapolation to zero concentration results in the same value H/c = RT/Mn. With viscosimetry the determination of M is not absolute dependent on the solvent and the temperature one finds a valne for the intrinsic viscosity, [rj], which is not unique but which has to be calibrated. 

Here, /u ° and ju are, respectively, the chemical potentials of pure solvent and solvent at a certain concentration of biopolymer V is the molar volume of the solvent Mn=2 y/M/ is the number-averaged molar mass of the biopolymer (sum of products of mole fractions, x, and molar masses, M, over all the polymer constituent chains (/) as determined by the polymer polydispersity) (Tanford, 1961) A2, A3 and A4 are the second, third and fourth virial coefficients, respectively (in weight-scale units of cm mol g ), characterizing the two-body, three-body and four-body interactions amongst the biopolymer molecules/particles, respectively and C is the weight concentration (g ml-1) of the biopolymer. 

It is important for us to keep in mind that biopolymers are generally not monodisperse components. Proteins are typically paucidisperse — mixtures of monomers, dimers and multimers. And polysaccharides are polydisperse their chain lengths and molar masses can be represented as a continuous distribution. For this reason the virial coefficients appearing in equations (5.16) and (5.17) should be interpreted as averages. So the inverse of the number-averaged molar mass of component / is given by... 

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