Mark-Houwink exponents

The viscosity average molecular weight is not an absolute value, but a relative molecular weight based on prior calibration with known molecular weights for the same polymer-solvent-temperature conditions. The parameter a depends on all three of these it is called the Mark-Houwink exponent, and tables of experimental values are available for different systems. 

At 25°C, the Mark-Houwink exponent for poly(methyl methacrylate) has the value 0.69 in acetone and 0.83 in chloroform. Calculate (retaining more significant figures than strictly warranted) the value of that would be obtained for a sample with the following molecular weight distribution if the sample were studied by viscometry in each of these solvents ... 

If changes in the molecular weight distribution can be neglected, substitution of the Mark-Houwink equation into Eq. 7 leads to Eq. 8, where a is the Mark-Houwink exponent. 

Fig. 4. Dependence of the Mark-Houwink exponent, a, on comonomer ratio, y=[MMA](,/ [12]o, for the copolymerization of the inimer 12 with MMA. Linear PMMA ( ) =0.68. (Reproduced with permission from [28], Copyright 2001 American Chemical Society.)...

Another investigation involved the SC VP of a macroinimer 8 via ATRP [46]. GPC/viscosity measurements indicated that the intrinsic viscosity of the branched polymer is less than 40% of that of the linear one at highest MW area (Fig. 6). A significantly lower value for the Mark-Houwink exponent (a=0.47 compared to a=0.80 for linear Pf-BuA) was also observed, indicating the compact nature of the branched macromolecules. 

The molecular weight calibration curves obtained for PVC are shown plotted in Figure 3. Table III shows an investigation of the effect of the peak broadening parameter (a) assumed when a single broad MWD PVC standard is used. The corrections for imperfect resolution for PV2 and PVC with a a = 0.5 are now reduced to about k% for both standards. It is of interest to note that with a reduced correction for imperfect resolution the Mark-Houwink exponent obtained is closer to published literature values for PVC in THF (13). The use of the associated molecular weight calibration curve for PVC would reproduce the M j and M of the PVC standards with errors of about 15. ... 

The universal calibration approach ([n]. M vs elution volume) for polystyrene standards and narrow molecular triacetate fractions show slight deviation from linearity. This departure from linearity has been attributed to differences in both hydrodynamic behavior and the Mark-Houwink exponent a for the two polymers in question. 

Somewhat improved reductions can be obtained by using cMa as the independent variable, where a is chosen empirically. In these cases a is usually rather close to the Mark-Houwink exponent for the system (129-133). 

Figure 5.6 shows the same data plotted as a function of cM0 6S to test the low concentration reduction scheme based on c rf] with a typical value of the Mark-Houwink exponent for good solvents. The data have been shifted vertically to achieve superposition at high molecular weights. It is clear that the cM variable produces a better superposition of data at all molecular weights and concentrations. The apparent variation in the values of cM at the intersections in Fig. 5.4 (Table 5.1) is largely due to a lack of data to define the limiting behavior at low molecular weights at some concentrations. The intersection on the superposed plot in the composite Fig. 5.5 is cM = 30000, giving Mc = 30600 for undiluted polystyrene (q = 0.98 at T = 217° C, in good agreement with the value 31200 reported by Berry and Fox (16).

It can be seen then that one need not use Equation 7 at all, since n can be obtained from Equation 6 and w /n from Equation 17, thus permitting W to be calculated.. If one is dealing with an unknown whole polymer, Equation 6 permits the determination of n with no knowledge of solute-solvent interaction. To find w/n and hence W requires a knowledge of (e) for the application of Equation 17. This is easily obtained from GPC provided two or more samples of a different molecular weight can be found. Equation 6 permits the determination of n for such samples from which the Mark-Houwink exponent (a) can be determined from the relationship in Equation 9, i.e.,... 

In Table 9.5, calculated values of the Mark-Houwink exponent a are compared with literature values. There is a reasonable agreement, except if the solvent has hydrogen bonding properties considerably different from that of the polymer. 

The dilute solution properties of branched polymers differ from those of linear polymers of the same composition. Generally, the Mark-Houwink exponent a is lowered by branching (Zimm and Stockmayer, 1949 Zimm and Kilb, 1959). The relationship... 

This phenomenon can be detected experimentally by a very high value of the Mark-Houwink exponent a, the value of which varies between 0.5 and 0.8 for coiled molecules. Theoretical investigations (Flory, 1953) predict a value a 1.8 for rigid stretched molecules. This value is indeed found experimentally in some cases. Another indication of rod-like behaviour of macromolecules is a high ratio of radius of gyration to molar mass. 

Fig. 9.18 shows the influence of the ionic strength I on the Mark-Houwink exponent a. Although there is a large amount of scatter, the data clearly indicate a decrease of a with... 

The effects of sulfonic acids on starch are discussed later (p. 375). Dimethyl sulfoxide and carbon disulfide are the only other sulfur-containing compounds that have been examined with respect to their complex formation with starch. For example, a complex of potato starch with carbon disulfide was prepared via the starch-acetone complex on refluxing. It was reported that this complex contains 5.8-5.9% of CS2.682 Dimethyl sulfoxide causes expanded coiling of amylose without the formation of a helix.378 Banks and Greenwood385 reviewed the Mark-Houwink exponent for Me2SO-starch solutions. Reported variations in this exponent are believed... 

Figure 2. Signal tracings from the three detectors showing excess LS intensity, specific viscosity, and concentration signals, for a sample with a Flory-Schulz MWD, polydispersity of 2, and a Mark-Houwink exponent of 0.725.

Diblock copolymers of PS-PDMS were chosen for study because PS and PDMS homopolymers in good solvents have the same molar mass calibrations in SEC (23). For PS and PDMS homopolymers in tetra-chloroethylene, it can be shown from data for intrinsic viscosity that the Mark-Houwink exponent for both of these polymers is near 0.8 (9). Equations for universal calibration (24) indicate that an M(PS-PDMS) diblock copolymer calibration should therefore follow that for the corresponding homopolymers. Consequently, there should be a narrow range of masses at each elution volume, so that the term containing Pi in equation 7 can be ignored and can be replaced by Mi, giving... 

Intrinsic viscosities of polystyrenes in three solvents. Cyclohexane is a 6 -solvent (v = 0, filled circles, from Y. Einaga et al., J. Polym. Sci., Polym. Phys. 17, 2103, 1979), with Mark-Houwink exponent a = 1/2. Methyl ethyl ketone is a better solvent ... 

Mark-Houwink exponent, [dimensionless], p. 34 tube diameter, [m], p. 265 degeneracy, [dimensionless], p. 206... 

If the measurement of [r ] is not performed under 0 conditions, the power law exponent a , which is usually called the Mark-Houwink exponent, becomes different from 0.5, and its value depends on the magnitude of the deviation from 0 conditions. Experimental data on intrinsic viscosities can be correlated very roughly by using the following empirical expressions [20] ... 

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