Mark-Houwink expression

The relationship between molecular weight, M, and intrinsic viscosity, [tjJ, of a polymer is given by the widely used Mark-Houwink expression... 

Table I compares results obtained in this manner with those reported by Arro Laboratories for 3 broad-distribution PVC standards. Table II does the same for several PVC commercial suspension resins from Sha-winigan, using a Mark-Houwink expression as a basis for comparison of W values.

In the absence of molecular weight data for the itaconates, it can be shown by combining the Mark-Houwink expression relating intrinsic viscosity and molecular weight, [ j] — KM , with the expression for the kinetic chain length to be expected from pure bimolecular termination, v= kpmj2 fkaki)°- that for bulk polymerization ... 

To solve Eq. (6), the denominator must be known. Substituting into the denominator the Mark-Houwink expression [r ] = KM for the investigated polymer and rearranging, we obtain... 

In many cases the analyst does not know a priori whether the particular polymer is compositionally uniform. Some information on this point can be obtained from a CV detector in-line with the SEC apparatus. As explained earlier, universal calibration with the CV and narrow molecular weight distribution polymers provides a relation between [r ]Mand elution volume. The CV measures [r ] of the unknown polymer, from which Mean be derived. A log-log plot of the measured values of [r ] versus the corresponding M data provides the parameters of the Mark-Houwink expression... 

Goedhart and Opschoor (21) first applied the principle of universal calibration to calculate the Mark-Houwink constants for PVAc in THF. By measuring the intrinsic viscosity of fractions leaving the siphon of a Waters 200 GPC using an automatic capillary tube viscometer, they determined the following Mark- Houwink expression for PVAc in THF at room temperature ... 

The weight averaged molecular weight (M ) can subsequently be obtained from the Mark-Houwink expression as shown in (6.9), where K and a are the empirical Mark-Houwink constants, depending on the molecular weight range and molecular weight distribution. 

The intrinsic viscosity values (rf), deduced from data in Fig. 26, were plotted as function of M in Fig. 26. This data was fitted to Mark-Houwink expression [97, 105]... 

Fig. 3. Solution viscosity vs concentration for ethylene oxide polymers (10). The molecular weight of the polymer is indicated on each curve. The dependence of the intrinsic viscosity [Tj] on molecular weight M for these polymers can be expressed by the Mark-Houwink relationship ...

Relationships between dilute solution viscosity and MW have been determined for many hyperbranched systems and the Mark-Houwink constant typically varies between 0.5 and 0.2, depending on the DB. In contrast, the exponent is typically in the region of 0.6-0.8 for linear homopolymers in a good solvent with a random coil conformation. The contraction factors [84], g=< g >branched/ <-Rg >iinear. =[ l]branched/[ l]iinear. are another Way of cxprcssing the compact structure of branched polymers. Experimentally, g is computed from the intrinsic viscosity ratio at constant MW. The contraction factor can be expressed as the averaged value over the MWD or as a continuous fraction of MW. 

Many polymer properties can be expressed as power laws of the molar mass. Some examples for such scaling laws that have already been discussed are the scaling law of the diffusion coefficient (Equation (57)) and the Mark-Houwink-Sakurada equation for the intrinsic viscosity (Equation (36)). Under certain circumstances scaling laws can be employed advantageously for the determination of molar mass distributions, as shown by the following two examples. 

Staudinger realized that for macromolecules [77] depends characteristically on the molar mass which can be expressed by the Kuhn-Mark-Houwink-Sakurada (KMHS) relationship... 

An appropriate formalism for Mark-Houwink-Sakurada (M-H-S) equations for copolymers and higher multispecies polymers has been developed, with specific equations for copolymers and terpolymers created by addition across single double bonds in the respective monomers. These relate intrinsic viscosity to both polymer MW and composition. Experimentally determined intrinsic viscosities were obtained for poly(styrene-acrylonitrile) in three solvents, DMF, THF, and MEK, and for poly(styrene-maleic anhydride-methyl methacrylate) in MEK as a function of MW and composition, where SEC/LALLS was used for MW characterization. Results demonstrate both the validity of the generalized equations for these systems and the limitations of the specific (numerical) expressions in particular solvents. 

The dependence of the intrinsic viscosity [r ] on molecular weight M for these polymers can be expressed by the Mark-Houwink relationship ... 

If the determination of the intrinsic viscosity of each polymer fraction obtained from the GPC syphon counter is infeasible, use can be made of [77], Mn, Mw, and the GPC trace of the whole polymer sample for the determination of the Mark-Houwink parameters. Provder and co-workers (I, 2) have derived expressions for Mn, Mw, and [77] in terms of c, K, and Z which are summarized below ... 

Using the Mark-Houwink parameters for PMMA in tetrahydrofuran and in TFE leads to the following expressions relating Z5 A (i>tfe) to Z F... 

By use of both the appropriate value for in Equation 2 and the Mark-Houwink viscosity expression, one may write... 

If the Mark-Houwink constants of the polymer are unknown, then the two broad MWD standards method, with known Mn and Mw, is used. The molecular weight averages of the two polymer standards are expressed as ... 

This is the Mark-Houwink equation, where the formal dimension of K is equal to cm3/g x (g/mol)a. In general, however, K is tacitly expressed and tabulated in the same dimension as [r ] and the resulting polymer molecular weight M is then expressed in g/mol. For polydis-perse polymers, M is replaced by Mv the so-called viscosity-average molar mass, so that... 

So a prediction method for the limiting viscosity number can at best give the order of magnitude of this quantity. Such a method will be described on the following pages. The method is based on the empirical relationship between [rj] and M the Mark-Houwink Eq. (9.32). In principle, the Stockmayer-Fixman Eq. (9.38) could be used as well, but the majority of the literature data has been expressed in the constant K and the exponent a of the Mark-Houwink equation. 

In analogy with the Mark-Houwink equation the dependence of the sedimentation coefficient on molar mass can be expressed as ... 

The molecular weight dependence of fr ] can be expressed by the semi-empirical Mark-Houwink-Sakurada (MHS) equation,... 

This expression is known as the Mark-Houwink equation, and K and a are constants for any given pair of polymer and solvent. These constants are tabulated and could be found for most known polymers in reference 2. 

The Mark-Houwink relation for polypropylene in o-dichlorobenzene at 130°C was calibrated as follows. A series of sharp fractions of the polymer was obtained by fractionation, and the molecular weight of each fraction was determined by membrane osmometry in toluene at 90"C. The samples were then dissolved in o-dichlorobenzene at I30°C and their intrinsic viscosities ([ ]) were measured. The resulting data fitted an expression of the form... 

Intrinsic viscosity is the most useful of the various viscosity expressions because it can be related to molecular weight by the Mark-Houwink-Sakurada equation ... 

The dependence of the intrinsic viscosity on the molecular weight for a homologous series has customarily been expressed by the modified Stau-dinger equation, often called the Mark-Houwink equation (Mark, 1938 Houwink, 1940) ... 

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] ... 

The intrinsic viscosities are obtained by making viscosity measurements at different polymer concentrations and by plotting the above expression against the concentration. The limit that this quantity assumes as the infinite dilution is approached is the value desired. Polymer solutions exhibit a Newtonian behavior at low shear rates, changing to a non-Newtonian flow at higher shear rates (6). The intrinsic viscosities should be determined at the low shear rate range. The most commonly used equation that relates the intrinsic viscosity and the molecular weight of a macromolecule is the Mark-Houwink equation ... 

Figure 2 shows the plot of log [g] vs. log M (values of [g] from reference 2 of the levan hydrolysates for [g] measured in water at 25 C. It is noted from this figure, that there are two linear segments. The 1 inear.segments can be expressed by the Mark-Houwink equation, [g] = K where the exponent a is obtained from the slope. The exponents calculated from Figure 2 are 0.50 for levan hydrolysates of Rp, < 4 x 10 and 0.11 for values... 

Introducing the v values evaluated by assuming v =(a+l)/3 from the exponent a in the Mark-Houwink-Sakurada equations, [j7]=KM [9], into eq 7, we have the calculated results denoted by the solid lines in Figure 4. As shown in the figure 7 °u appears to be expressed as a universal function of C[ 7 ] and its experimental dependence appears to agree with the calculated line in 0. 5M NaCl solutions, whereas 7 is not expressed as a universal function and its dependences are lower than the calculated lines in O.OIM NaCl solutions. 

To detemiine the intrinsic viscosity, both inherent and reduced viscosities are plotted against concentration (Q on the same graph paper and extrapolated to zero. If the intercepts coincide, then this is taken as the intrinsic viscosity. If they do not, then the two intercepts are averaged. The relationship of intrinsic viscosity to molecular weight is expressed by the Mark-Houwink-Sakurada equation ... 

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