The Universal Calibration Curve

in perhaps the most important discovery in this field, Benoit and coworkers50 recognized that separations occur on the basis of hydrodynamic volume rather than molecular weight and made the connection to intrinsic viscosity, which also depends 

Rearranging Equation 12-60 tells you that the product of the molecular weight and the intrinsic viscosity of a polymer is proportional to the hydrodynamic volume (Equation 12-61)  

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PMMA, on the unmodified porous glass and silica gel, and the universal calibration curves for polystyrenes and poly(methyl methacrylates) did not coincide (10,12,19). 

From this it follows that plotting a graph of log M[7]] against elution volume should give a straight line for all polymer-solvent combinations. This was shown by Benoit et al to be true for a large number of systems and is referred to as the universal calibration curve. It is illustrated in Figure 6.6. 

In reality, the universal calibration curve is not quite universal. Certain exceptions exist, such as polymers with very rigid backbones. However, the... 

Once a value for M[ is assumed, this leaves one unknown b which can be determined from the SEC chromatogram and the measured whole polymer intrinsic viscosity in the following manner. First, one estimates a value for b, and calculates M CV) and [n](V) across the chromatogram using the universal calibration curve and equation (3). Then the whole polymer intrinsic viscosity is obtained from... 

Figure 12 were superimposable on those for detector 2. Therefore, when the plot shown in Figure 14 is linear over the range of compositions involved in the sample, then (according to equations (1-4) ) the composition of the sample is the same at each retention volume. If the variation with retention volume is negligible the copolymer can then possibly be treated as is a homopolymer in GPC interpretation. In particular, intrinsic viscosity measurements could then lead to estimates of molecular weight via the universal calibration curve. 

Figure 19 shows the Universal Calibration Curve obtained for the coupled GPCs in terms of the hydrodynamic volume in THF (Ref Table II). The addition of the n-Heptane caused a dramatic shift downstream of the polystyrene standards. 

Other analysis methods dependent on multiple detectors can be implemented using this automated system. Two methods under development are the use of a continuous viscometer detector with a refractive index detector to yield absolute molecular weight and branching, utilizing the universal calibration curve concept (4), and the use of a UV or IR detector with the refractive index detector to measure compositional distribution as a function of molecular weight. 

Consequently, a relatively low molecular weight polymer (MM=1.39x 10 ) has an intrinsic viscosity (25.6 dl/g) equivalent to a PS or PDMS more than lOx higher in MW. (Earlier GPC work on PBIC (9) and PHIC (7) had shown no deviations from the universal calibration curve for GPC at high molecular weights). The primary role of chain flexibility in GPC degradation rather than simple molecular hydrodynamic volume is conclusively shown by these results. 

Using the intrinsic viscosity data and the universal calibration curve( ) a secondary molecular weight calibration curve can be constructed for the polymer of interest as shown by the following equation ... 

Figure 10 shows DRI and viscometer traces for the NBS 706 polystyrene standard. Based on the information from these two chromatograms in conjunction with the universal calibration curve, one can calculate the intrinsic viscosity EnJCv) and molecular weight M(v) at each retention volume as shown in... 

As illustrated in Figure 15, the search for the MW calibration of an unknown polymer from the universal calibration curve requires an on-line SEC viscosity detector. Since narrow MW standards are not available for most commercial and experimental polymer samples, the determination of the polymer [b] calibration would not be possible without an on-line SEC viscosity detector. Once the [q] calibration for the unknown polymer is established, the polymer MW calibration can then be deduced from the universal calibration curve as indicated by the approaches shown in Figure 15. 

Values for system peak parameters were found using a narrow distribution polystyrene standard (PS68K) before calculating MWD data for the lignin samples from universal calibration. To check software and instrument operation, several narrow MWD polystyrene and one broad MWD polymethylmethacrylate standards were treated as unknown samples and subjected to analysis with the universal calibration curve assembled from all polymer standards files. It was found that the MWD could be estimated for the recalculated polymer standards with errors between 5 and 10% of the original value indicated by the supplier of the standard (e.g., Mw for PS11K and Mw and M for PMMA17K-6). 

A universal calibration curve was established by plotting the product of the limiting viscosity numbers and molecular weight, Mw[iy], vs. the elution volume, EV, for a variety of characterized polymers. The major usefulness of the universal calibration curve was to validate individual molecular-weight values and to provide extended molecular-weight calibration at the ends of the calibration curve where fractions of narrow dispersion of the polymer being analyzed are not available. The calibration curve was monitored daily with polystyrene fractions certified by Pressure Chemicals. The relationship between the polyethylene fractions and polystyrene fractions was determined using the universal calibration curve. 

Thus, from a log-log plot of any of the above equations, and W and n obtained from the universal calibration curve and the GPC trace, the dispersity index D can be easily determined. 

The linear segment of the universal calibration curve is customarily expressed as... 

Since size exclusion chromatography separates polymer molecules by their size (especially hydrodynamic size), plotting the molecular size vs. the retention volume should be universal, regardless of the polymer molecular weight. The universal calibration curve is given as ... 

Figure 7.14 shows the universal calibration curve in tetrahydrofuran. 

The procedure to determine the molecular weight of a branched polymer is as follows. If for a certain GPC column the universal calibration curve is measured with the aid of a series of monodisperse linear polymers of known molecular weight, the next step is to determine the elution volume and the intrinsic viscosity of the unknown branched fraction. Then the product [rj]M corresponding to the mean elution volume of a branched fraction is read from the universal calibration curve this value divided by the determined intrinsic viscosity gives the molar mass of the fraction. At the same molar mass one can also calculate the intrinsic viscosity of the linear polymer by using the Mark-Houwink equation. 

In a plot of log (M[r ]) vs. Ve identical calibration lines should be found for the two polymers 1 and 2, irrespective of their chemical composition. This universal calibration approach was predicted and experimentally proved by Benoit et al. [62]. As a consequence, using the universal calibration curve established with... 

If the Mark-Houwink coefficients are not available, a universal calibration curve is established using polystyrene calibration standards and the SEC-vis-cometer combination. The basic steps involved in the MMD analysis are summarized in Fig. 11. First, the universal calibration curve of the SEC separation system has to be established by using narrow molar mass standards as indicated by the top arrow pointing to the right. Once the universal calibration curve is established, the procedure can then be reversed, by going from right to left following the bottom arrow, to obtain the molar mass calibration curve of any unknown... 

A comparison of these separation equations with the universal calibration curve (Eq. 22) leads to the prefactors of Eq. (22) ... 

Benoit et al. then reasoned that all polymers, regardless of chemical structure and chain architecture, should fit on the same plot of rj)M versus elution volume. And most of them do, as shown on the plot in Figure 12-42, which is called the universal calibration curve. (It is actually not quite universal, as data from things like liquid crystalline polymers that have extended chain rather than coil conformations in solution do not fall on this curve.)... 

The utility of the universal calibration curve is that it allows us to calculate the molecular weight of a polydisperse polymer sample for which we do not have monodis-perse standards. First, a universal calibration curve is prepared for a given set of columns using, say, monodisperse polystyrene samples. This is plotted as log J versus elution volume (Figure 12-43) where for the rth sample (Equation 12-62) ... 

J. = Mfy], from the universal calibration curve. We have assumed that the intrinsic viscosity of a branched polymer is equal to a branching function g expressed in terms of a parameter A equal to the (number) average number of branch points per unit molecular weight (see Equation 12-68). 

Briefly describe how SEC (size exclusion chromatography) separates polymers according to size. Explain how the universal calibration curve can be used to determine molecular weight. 

The combination of the differential refractive index (RI) detector and on-line viscometer allows the direct use of the universal calibration and thus true molecular weight determination. The RI detector is concentration-sensitive, and the viscometer records specific viscosity. The ratio of the specihc viscosity to the concentration is equivalent to intrinsic viscosity (as discussed in Section 6.1), and the continuous dependence of this ratio versus the retention volume could be related to the universal calibration curve, thus allowing the correlation of each point on the chromatogram with the true molecular weight. 

Here the Wj are available from the ordinates of the gel permeation chromatogram and the 7, from the universal calibration curve of elution volume against... 

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