Molar mass distribution measure

Determine the moment of the molar mass distribution measured by intrinsic viscosity of a polydisperse sample. 

The effect is that the polymer molecules are separated into fractions. These are measured by an appropriate detector located at the end of the column, and the detector records the response as a peak on a chart. The chromatogram thus consists of a series of peaks corresponding to different elution volumes, the shortest elution volume being due to the largest molar mass polymer molecules within the sample. Details of the molar mass distribution can be determined from the size and number of the individual peaks in the chromatogram. An example of a gel permeation chromatogram is shown as Figure 6.4. 

The viscosity level in the range of the Newtonian viscosity r 0 of the flow curve can be determined on the basis of molecular models. For this, just a single point measurement in the zero-shear viscosity range is necessary, when applying the Mark-Houwink relationship. This zero-shear viscosity, q0, depends on the concentration and molar mass of the dissolved polymer for a given solvent, pressure, temperature, molar mass distribution Mw/Mn, i.e. 

In Ref. [107] the procedure above has been employed for the measurement of the molar mass distribution of a broad molecular weight polystyrene, obtained by radical polymerization with ethylacetate as solvent. The scaling parameters for this polystyrene in this marginal solvent have been determined to be a 2.8 x 10-4 cm2/s and b 0.52 [107]. The upper curve in Figure 17 shows the resulting molar mass distribution in comparison with the one obtained by SEC. 

The more time-consuming task is the establishment of the scaling law, which requires a series of polymer samples of narrow molar mass distribution and known molar mass. Their sedimentation coefficients have to be measured as a function of concentration and extrapolated back to c — 0 in order to obtain So(M) (Figure 18). 

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 ... 

The Schulz-Zimm distribution would be found for/end-to-end coupled linear chains which obey the most probable distribution, as well as for/of such chains which are coupled onto a star center. This behavior demonstrates once more the quasi-linear behavior of star branched macromolecules. In fact, to be sure of branching, other structural quantities have to be measured in addition to the molar mass distribution. 

The melt flow index is a useful indication of the molar mass, since it is a reciprocal measure of the melt viscosity p. p depends very strongly on 77 ( ) (doubling of results in a 10.6 times higher 77 ). This relation is valid for the zero-shear viscosity the melt index is measured at a shear stress where the non-Newtonian behaviour, and thus the width of the molar mass distribution, is already playing a part (see MT 5.3.2). The melt index is a functional measure for the molar mass, because for a producer of end products the processability is often of primary importance. 

The GPC technique was used to determine gel formation. Figure 13 shows the soluble fraction of aromatic polysulfone I measured with this technique after irradiation at 30 and 150°C. The relative radiation resistance of different polymers can be obtained by comparison of the gel doses (the highest dose for complete solubility of the polymer) provided that the initial molar masses of the of the polymers are known, or from G(S) and G (X) values these can be derived from the dose dependence of the soluble fractions beyond the gel dose, using a Charlesby-Pinner, or Saito-type plot with allowance for the molar mass distribution. 

Further important information about these products comes from measurements of their molar masses. Of the available methods, vapour pressure osmometry (VPO) and gel permeation chromatography (GPC) are the most widely used, but, because of its sensitivity to low-molar-mass components, VPO tends to yield results which are too low. In contrast, GPC provides a detailed picture of the molar mass distribution. 

Molar mass distributions of PDADMAC were also determined by fractionation using dioxane/methanol system [144] and from sedimentation velocity measurements in 1 m NaCl solution [134]. The molar mass calculations were based on the s-M relation [134] ... 

In this contribution, the experimental concept and a phenomenological description of signal generation in TDFRS will first be developed. Then, some experiments on simple liquids will be discussed. After the extension of the model to polydisperse solutes, TDFRS will be applied to polymer analysis, where the quantities of interest are diffusion coefficients, molar mass distributions and molar mass averages. In the last chapter of this article, it will be shown how pseudostochastic noise-like excitation patterns can be employed in TDFRS for the direct measurement of the linear response function and for the selective excitation of certain frequency ranges of interest by means of tailored pseudostochastic binary sequences. 

Figure 16 shows the reconstruction of the molar mass distribution from two measurements in Fig. 13 according to Eq. (43) for an exposure time Tp = 0.025 s, which is shorter than the inverse of the fastest rate in the system, and for Tp = 1.5 s, which is longer than the inverse of the slowest rate. The inserts show the respective rate distributions. It is evident that the slow mode is much more pronounced in P(T) for the long exposure time. The peak positions in the differential molar mass distributions c(M) agree reasonably well with the nominal molar masses, which are marked by the dashed vertical lines. The two peaks are somewhat broader and less resolved in the case of the shorter exposure. The peak areas in the molar mass distributions are, contrary to the rate distributions, identical for both measurements, as can be seen from the plateaus at I(M) 0.5 in the cumulative molar mass distribution... 

Figure 18 shows the molar mass distributions as computed from the rate distributions according to Eq. (42) for the two measurements in Fig. 17 and from an additional long exposure TDFRS measurement with Tp = 2s. Superimposed is the molar mass distribution obtained from size exclusion chromatography. 

Obviously, none of the three scattering experiments is able to reproduce the true molar mass distribution exactly. The two TDFRS measurements show a bi-modal structure, which must be regarded as an artifact. Otherwise, the general shape of the distribution, with the peak position around 200 kg/mol and a tail towards low M, is rather well reproduced. In principle unimodal distributions could be enforced by the choice of a larger regularization parameter. Purposely,... 

While polydisperse model systems can nicely be resolved, the reconstruction of a broad and skewed molar mass distribution is only possible within certain limits. At this point, experimental techniques in which only a nonexponential time signal or some other integral quantity is measured and the underlying distribution is obtained from e.g. an inverse Laplace transform are inferior to fractionating techniques, like size exclusion chromatography or the field-flow fractionation techniques. The latter suffer, however, from other problems, like calibration or column-solute interaction. 

This procedure results in a concentration - volume curve, from which, after previous calibration, the molar mass distribution can be derived. Calibration can be carried out with known monodisperse polymers, and is needed only once for a certain type of polymer on a certain column. The measurement takes only a few hours. From the measured MMD the various averages can be computed easily. It is also possible to characterise the eluted polymer solution not only on concentration, but also on molar mass, e.g. by laser light scattering. In this way the calibration can be avoided. 

The difference between A and B is due to a difference in molar mass distribution MMD for A this is broader than for B (and for C). A broader MMD causes the polymer to deviate more strongly from Newtonian behaviour. A first approximating measure of the width of the distribution is Mw/Mn as far as the effect on the non-Newtonian behaviour is concerned, Mz/Mw would be a better measure. 

The ratio Mw/ Mn must by definition be greater than unity for a polydisperse polymer and is known as the polydispersity or heterogeneity index. Its value often is used as a measure of the breadth of the molar mass distribution, though it is a poor substitute for knowledge of the complete distribution curve. Typically Mw/ Mn is in the range 1.5-2, though there are many polymers which have smaller or very much larger values of polydispersity index. A perfectly monodisperse polymer would have Mw/ Mn = 1.00. 

Measurements of sedimentation behaviour of polymer molecule in solution can provide a consideratble amount of information, e.g., hydrodynamic volume, average molar masses and even some indication of molar mass distribution. Such measurements have been extensively used to characterise biologically-active polymers which often exist in solution as compact spheroids or rigid rods. However, sedimentation methods are rarely used to study synthetic polymers and so will be given only brief non-theoretical consideration here. 

Gel permeation chromatography (GPC) is a widely used technique for determining molar mass and molar mass distribution of polymers. In its usual form it is not an absolute method, though by making the appropriate measurements it may be made to be so. 

The separation of chains according to their hydrodynamic volume inside SEC columns is accompanied by their dilution. Consequently, the molar mass distribution is shifted toward lower values during the measurement. Usually, the dissociation of hydrogen-bonded chains is fast compared to the elution time, so that only quahtative information can be derived [115,121,141]. However, if the dissociation is sufficiently slow, it is possible to measure rehable molar mass values [186,187]. 

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