Mark-Houwink plots

Highly branched PMMA synthesized by SCVCP of MMA with the inimer 12 via GTP was characterized by GPC using universal calibration and MALS [28]. The corresponding Mark-Houwink plots, log[ ]] versus logM, and contraction factor,g =[ ]]b nched/[fl]iinear> a function of logM are presented in Fig. 3 for different comonomer ratio y=[MMA]o/[12]o. For M>10 the viscosity of the... 

Fig. 3. Mark-Houwink plot (a) and contraction factors (b),g =[ j]b,a ched/[ lliineaD as a function of the molecular weight for the copolymerization of the methacrylate-type inimer 12 with MMA under different comonomer ratios, y=[MMA]o/[12](,=1.2 (+), 5.2 (0), 9.8 (V), 26 (A), 46.8 (0),86.5 ( ),respectively.TheintrinsicviscositiesofPMMA(—) are given for comparison. (Reproduced with permission from [28]. Copyright 2001 American Chemical Society.)...

Fig. 5. Mark-Houwink plots (a) and RI signals (b) for polymers obtained by SCVCP of t-BuA with the inimer 1 CuBr/PMDETA at 60°C (O) and 100°C (A) CuBr/Bipy at 60°C ( ) and 100 °C (A). The viscosity result for a linear Pf-BuA (—) is given for comparison. (Reproduced with permission from [31], Copyright 2001 American Chemical Society.)...

Fig. 6. Mark-Houwink plots ( ) and contraction factors (A) of branched Pt-BuA (a=0.47) obtained by SCVP of the macroinimer 8. ( ) linear Pt-BuA (a=0.80). (Reproduced with permission from [46]. Copyright 2000 WILEY-VCH.)...

Fig. 7. a Mark-Houwink plot of highly branched PMMA obtained by SCVCP of MMA with the inimer 12. (-)RI signal ( ) intrinsic viscosity of feed, ( ) intrinsic viscosity of linear PMMA (O) contraction factor, g. b Separation of feed polymer into fractions by preparative SEC. (-) RI signal of fractions (-) accumulated RI signals (.) RI signal of feed polymer. (Repro-... 

Fig. 8. a Mark-Houwink plot for various fractions of highly branched PMMA obtained by SCVCP of MMA with the inimer 12. M unical (xlO ) = 12.06 ( ), 9.64 (O), 7.2 (A), 5.58 (V), 3.52 (0),3.26 (+),2.18 (x),2.00 ( ) (—) unfractionated feed polymer ( ) linear PMMA. b Mark-Houwink plot obtained from intrinsic viscosities of each fraction of branched PMMA. (Reproduced with permission from [88]. Copyright 2001 American Chemical Society.)... 

A Mark-Houwink plot results in a = 0.3 0.1 in the range of approximately 1000-40,000 g/mole, clearly indicating a high degree of branching. The deter-... 

Figure 2. Mark-Houwink plot for polymers obtained from trichlorophenoxide (O) and 4-bromo-2,6-dichlorophenoxide (" )...

Henderson-Hasselbalch equation lahn-Teller effect Lee-Yang-Parr method Lineweaver-Burk method Mark-Houwink plot Meerwein-Ponndorf theory Michaelis-Menten kinetics Stem-Volmer plot van t Hoff-Le Bel theory Wolff-Kishner theory Young-Laplace equation Ziegler-Natta-type catalyst... 

Figure 6. Mark-Houwink plots of intrinsic viscosities of starch standards (amylose and amylopectin) compared with wheat and cornflours.

The Mark-Houwink plot for the pullulans is displayed in Figure 9 and indicates a smooth relationship with little scatter. A slope of 0.64 was obtained from the best fit. Figure 10 displays a double logarithmic plot of the radius of gyration versus the molecular weight for pullulans, and this plot has a slope of 0.37. The theoretical values of Rg were calculated by using the Ptitsyn-Eisner equation (as follows) and are shown in the same figure 11) ... 

Figure 12 displays the Mark-Houwink plot for dextrans. The graph exhibits a slope of 0.39, which suggests extensive branching as reported elsewhere (12) and described later. 

Figure 12. Mark-Houwink plot for dextran. Intrinsic viscosities were obtained from the viscosity detector and (M ysfrom the light scattering one.

The determination of the viscosity law in GPC-viscometry is even more important for branched polymers. Branches reduce the sizes of a macromolecule, including its hydrodynamic volume H. This size reduction is reflected by the changes in the shape and position of the viscosity law plot for a branched polymer. Short-chain branches usually do not change the linearity and slope of the Mark-Houwink plot and just decrease the value of parameter K, whereas the long-chain branches cause bending of the corresponding plot. 

Henderson—Hasselbalch equation Jahn—Teller effect Lineweaver—Burk method Mark-Houwink plot Meerwein—Ponndorf theory Michaelis—Menten kinetics Stern—Volmer plot van t Hoff—Le Bel theory Wolff—Kishner theory Young—Laplace equation Ziegler—Natta-type catalyst... 

Fig. 10. Mark-Houwink plots of polyraethyldivinylsilane(B), polymethyldiallylsilane ( ), polytriallylsilane ( ), and polymethyldiundecenylsilane ( ).

To determine the interdetector delay volume for a viscometer, a broad molecular weight distribution standard can be injected and a Mark-Houwink plot, that is, log [t ] versus log M, generated using universal calibration. The interdetector volume is adjusted until the expected Mark-Houwink exponent is obtained (36). 

A) Mark-Houwink plot of log [u ] versus log M for a linear and a branched polystyrene. (B) Plot of branching index g as a function of molecular weight for the randomly branched polystyrene. (Adapted from Reference 36 and used with permission from the American Chemical Society.)... 

The intrinsic viscosity and Mark-Houwink constants of standards can be determined from a static capillary viscometer or an on-line viscometer detector in an SEC system. If the intrinsic viscosity is to be used for constructing a universal calibration curve, it is important to use the identical conditions in performing the SEC analysis and the intrinsic viscosity measurement. A Mark-Houwink plot for five PAM standards and one PAA standard is shown in Figure 4. The intrinsic viscosity of PAM may decrease with time and becomes constant after about one week. It is recommended that the PAM solution be analyzed while still fresh. 

Mark-Houwink plot for high -molecular-weight, fully hydrolyzed PVA in 0.10 M... 

Figure 10 also includes an overlay of the Mark-Houwink plots for the PVAc paste and reacetylated PVAc, showing how the PVAc paste curve deviates slightly from linearity because of the presence or branched polymer. The loss of long-chain branches in the hydrolysis and the corresponding decrease in molecular weight of the PVAc are reflected in this small difference in the Mark-Houwink plot. 

Molecular weight distributions and Mark-Houwink plots of PVAc paste and reacetylated PVA from SEC-viscometry in THF. 

Long-chain branching can be characterized by SEC-viscometry (in THF) of the starting PVAc used for the hydrolysis to PVA. The Mark-Houwink plot of log [t ] versus log molecular weight for fully hydrolyzed PVA supports a linear structure for the polymer. 

Rg. 1 Overlay of the Mark-Houwink plots of linear and branched polyethylene (PE) samples. See Ref. for experimental details. 

These features of the viscosity-law plots for branched polymers are demonstrated in Fig. 1 with two NIST (National Instimte of Standards and Technology, U.S.A.) polyethylene standards as examples high-density linear polyethylene PE1475 and low-density branched polyethylene PE1476. This last one contains both short- and long-chain branches. Dashed straight line 3 represents the Mark-Houwink plot for linear polyethylene, parallel solid line 4 takes into account the short-chain branches, and polyethylene with both types of branches (PE 1476) is described by solid curve 5. 

Fig. 2 (a) Mark-Houwink plot for linear polyethylene NBS1475. (b) Mark-Houwink plot for branched polyethylene NBS1476. 

Fig. 9. Mark-Houwink plot from SEC with triple detection indicating linear polyethylene...

When an on-line viscometer is used together with the refractive index detector to generate the intrinsic viscosity [t]] in order to build the universal calibration curve. Sec. II.B, the intrinsic viscosity [t]] can also be used to determine the presence and degree of branching. This is done by plotting the log of [t]] versus log molecular-weight for each slice of the distribution. This plot is called the viscosity law plot, or the Mark-Houwink plot. It is described by the equation... 

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