Mark-Houwink coefficient

This relationship with a = 1 was first proposed by Staudinger, but in this more general form it is known as the Mark-Houwink equation. The constants k and a are called the Mark-Houwink coefficients for a system. The numerical values of these constants depend on both the nature of the polymer and the nature of the solvent, as well as the temperature. Extensive tabulations of k and a are available Table 9.2 shows a few examples. Note that the units of k are the same as those of [r ], and hence literature values of k can show the same diversity of units as C2, the polymer concentration. 

Table 9.3 lists the intrinsic viscosity for a number of poly(caprolactam) samples of different molecular weight. The M values listed are number average figures based on both end group analysis and osmotic pressure experiments. Tlie values of [r ] were measured in w-cresol at 25°C. In the following example we consider the evaluation of the Mark-Houwink coefficients from these data. 

Table 9.2 Values for the Mark-Houwink Coefficients for a Selection of Polymer-Solvent Systems at the Temperatures Noted...

Evaluate the Mark-Houwink coefficients for poly (caprolactam) in w-cresol at 25°C from the data in Table 9.3. 

Since viscometer drainage times are typically on the order of a few hundred seconds, intrinsic viscosity experiments provide a rapid method for evaluating the molecular weight of a polymer. A limitation of the method is that the Mark-Houwink coefficients must be established for the particular system under consideration by calibration with samples of known molecular weight. The speed with which intrinsic viscosity determinations can be made offsets the need for prior calibration, especially when a particular polymer is going to be characterized routinely by this method. 

It is apparent from an examination of Table 9.2 that the Mark-Houwink a coefficients fall roughly in the range 0.5-1.0. We conclude this section with some qualitative ideas about the origin of these two limiting values for a. We consider a polymer molecule consisting of n repeat units, and two different representations of its interaction with solvent. 

Our primary objective in undertaking this examination of the coil expansion factor was to see whether the molecular weight dependence of a could account for the fact that the Mark-Houwink a coefficient is generally greater than 0.5 for T 0. More precisely, it is generally observed that 0.5 < a < 0.8. This objective is met by combining Eqs. (9.55) and (9.68) ... 

For conditions of intermediate solvent goodness, a shows a dependence on M which is intermediate between the limits described in items (1) and (2) with the corresponding intermediate values for the Mark-Houwink a coefficient. 

If Mark-Houwink coefficients were supplied at setup time, the chromatogram may be converted into the differential molecular weight distribution of the specimen. Various averages characterizing this molecular weight distribution are then calculated. The molecular weight distribution may be written to a file. 

For polymer-solvent systems with known Mark-Houwink coefficients, K and a, the polymer intrinsic viscosity value [n] can be estimated from the SEC-MW data using the following equation ... 

The results showed that good accuracy can be obtained from SEC calculated [x]] values when reliable Mark-Houwink coefficients are available. 

In some cases the relationship between polymer intrinsic viscosity ([n]) and molecular weight (M) has been established for the SEC solvent and temperature conditions i.e., the empirical Mark-Houwink coefficients (2)(K,a) in the equation... 

In order to estimate the branching factor e for polyvinyl acetate we have analyzed the SEC data obtained on sample PVAc-E4 using the MWBD method with various e values. This sample was synthesized under kinetically controlled conditions (isothermal, T = 60°C, [AIBN] = 10"5 g-mole/1, conversion level of 48.5 percent). The SEC measurements were made at 25°C in tetrahydro-furan. The Mark-Houwink coefficients used for linear polyvinyl acetate are those suggested by Graessley (21), namely K = 5.1 x 10"5 dl/gm and a = 0.791. The whole polymer M, Mj, and B j values obtained are listed in Table II. 

An example of this method of determining e is shown in Figure 3 where the Am has been calculated as a function of e for three HP-LDPE resins, designated LDPE A, B, and C. LDPE A was produced at the highest conversion and LDPE C at the lowest conversion. The SEC data used were obtained at 140°C in 1,2,4-trichlorobenzene. The Mark-Houwink coefficients used for linear polyethylene were K = 5.1 x 10-4 dl/gm and a = 0.706. The 13c NMR Xm values are indic ted by open triangles. 

Strictly speaking, monodisperse samples would be required for the determination of the Mark-Houwink coefficients. Since, however, the poly-dispersities of the nine individual fractions are only moderate (Mw/Mn 2) and since both Mw and [tj] are measured as weight averages with the same statistical weights, the error introduced by the incorrect treatment of the polydispersity could be neglected. 

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. 

Markham and Benton model, 1 628 Mark-Houwink coefficients for cellulose, 20 558t for PBT, 20 64t for PET, 20 58 for PTT, 20 69t Mark-Houwink constants, for poly(ethylene oxide), 10 677t Mark-Houwink equation, 19 717, 839 Mark-Houwink relationship, 10 675 ... 

L.Dq/u.Rq dimensionless number Mark-Houwink-Sakurada constant Axial diffusion coefficient... 

Mark-Houwink-Sakurada constant Mass transfer coefficient around gel Fractional reduction in diffusivity within gel pores resulting from frictional effects Solute distribution coefficient Solvent viscosity nth central moment Peak skewness nth leading moment Viscosity average molecular weight Number of theoretical plates Dimensionless number... 

From this information the absolute molecular weight distribution and the intrinsic viscosity-molecular weight plot can be constructed. From this plot the solvent and temperature dependent Mark-Houwink coefficients for linear polymers and information for polymer chain-branching of non-linear polymers can be obtained. 

Figure 12 shows the classical method of obtaining the Mark-Houwink coefficients, K and a, by plotting the log [n](v) vs. log M(v) for this polymer in THF at 50°C. The data points used for the plot in Figure 12 are indicated by the area between the arrows in Figure 10. Linear regression analysis of the data resulted in K o =1.86x10" and a o =0.662 with a correlation coefficient or t =u.9996 for NBS 70o polystyrene.

EXAMPLE 4.5 Empirical Determination of the Mark-Houwink Coefficients for a Polymer Solution. The molecular weights of various polycaprolactam preparations were determined by end-group analysis (see Example 3.2) intrinsic viscosities of the various fractions in m-cresol were measured at 25°C. The following values are representative of the results obtained (Reim-schussel and Dege 1971) ... 

The practical significance of the result of this example lies in the great ease with which viscosity measurements can be made. Once the k and a values for an experimental system have been established by an appropriate calibration, molecular weights may readily be determined for unknowns measured under the same conditions. Extensive tabulations of Mark-Houwink coefficients are available, so the calibration is often unnecessary for well-characterized polymers (see Table 4.5). 

TABLE 4.5 Mark-Houwink Coefficients for Some Typical Polymer-Solvent Systems at the Indicated Temperatures... 

Even though the two polymers have the same molecular weight, the cellulose triacetate has an intrinsic viscosity more than eight times greater than the polyisobutene. Note that the Mark-Houwink coefficient a is primarily responsible for this the intrinsic viscosities would be ranked oppositely if k were responsible. 

Stress-strain relationship from a concentric-cylinder viscometer Capillary viscometers versus concentric-cylinder viscometers Inherent viscosity at low volume fractions Extent of hydration from intrinsic viscosity measurements Empirical determination of the Mark-Houwink coefficients Variation of viscosity with polymer configuration... 

Simha and Zakin (126), Onogi et al (127), and Comet (128) develop overlap criteria of the same form but with different numerical coefficients. Accordingly, flow properties which depend on concentration and molecular weight principally through their effects on coil overlap should correlate through the Simha parameter c[ /], or cM , in which a is the Mark-Houwink viscosity exponent (0.5 < a < 0.8). If coil shrinkage, caused by the loss of excluded volume in good... 

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