Mark-Houwink relationships for

Table 3. Mark-Houwink relationships for polystyrene in different solvents at 25 °C as com-... 

The Mark-Houwink relationship for poly(CPP-SA) was calculated from the viscosity data and the Mw values, as determined by universal calibration of the GPC data using polystyrene standards ... 

The molecular weight and intrinsic viscosity for our PAM sample was also consistent with the other Mark-Houwink relationships for PAM in water containing various levels of salt (13). Thus, the presence of salt in the water did not appear to affect the measured intrinsic viscosity or molecular weight (determined by light scattering) of PAM polymers. This observation is reasonable because, in solution, PAM behaves as an uncharged random-coil polymer. 

The molecular weight of polyanhydrides were determined by iscosity measurements and gel permeation chromatography (GPC) (Ron et al., 1991). The weight average molecular weight (Mw) of polyanhydrides ranges from 5,000 to 300,000 with a polydispersity of 2 to 15 which increases with the increase in Mw. The intrinsic viscosity [f ] increases with the increase in Mw. The Mark-Houwink relationship for poly (CPP-SA) was calculated from the viscosity data and the Mw values as determined by universal calibration of the GPC data using polystyrene standards. 

Figure 5. Mark-Houwink relationships for high molecular weight PLAs .

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. 

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 ...

The molecular weights were estimated by applying the Mark-Houwink relationship [r/ = KMa. The intrinsic viscosity [r/ was determined in m-cresol at 20°C. The values for K in the literature vary from 1.35 to 18.10 x 10-4 and the a values vary from 0.654 to 0.96 for polyamide-cresol systems. 

The rheological behaviour of polymeric solutions is strongly influenced by the conformation of the polymer. In principle one has to deal with three different conformations, namely (1) random coil polymers (2) semi-flexible rod-like macromolecules and (2) rigid rods. It is easily understood that the hydrody-namically effective volume increases in the sequence mentioned, i.e. molecules with an equal degree of polymerisation exhibit drastically larger viscosities in a rod-like conformation than as statistical coil molecules. An experimental parameter, easily determined, for the conformation of a polymer is the exponent a of the Mark-Houwink relationship [25,26]. In the case of coiled polymers a is between 0.5 and 0.9,semi-flexible rods exhibit values between 1 and 1.3, whereas for an ideal rod the intrinsic viscosity is found to be proportional to M2. 

The viscosity level in the range of the Newtonian viscosity r 0 of the flow curve can be determined on the basis of molecular models. For this, just a single point measurement in the zero-shear viscosity range is necessary, when applying the Mark-Houwink relationship. This zero-shear viscosity, q0, depends on the concentration and molar mass of the dissolved polymer for a given solvent, pressure, temperature, molar mass distribution Mw/Mn, i.e. 

Also, a good correlation between Mw (light scattering) and intrinsic viscosity is observed for [Ph(Me)PN]n over the molecular weight range of the samples studied. Thus, the Mark-Houwink relationship, [rj] = K(AOa, yields values of K = 1.44x1 o 4 (with [rj] in dL/g) and a = 0.66. These data again indicate a well solvated, extended-chain structure of the polymer in THF. 

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 ... 

Flexible polymer chains expand with increasing solvent power of the medium, leading to an increase in [77] with increasing polymer solvation. For chains of a similar kind, varying in length (homologous series), the relationship between [77] and molecular weight, M, may be represented by the Mark-Houwink relationship [19-22],... 

In order to estimate g, it is necessary to measure [tj]b and MwB (or MnB). Also, a Mark-Houwink relationship of the linear polymer must be known for the conditions employed in the measurement of [tj]b. In general, if the log [77] versus log M relation is known for the linear species, the deviation from this relationship may be used to estimate the number of branches, provided certain assumptions can be made concerning the distribution of branches. 

Therefore, when dealing with polyelectrolytes, countless Mark-Houwink relationships must be established. This shows the increasing problems of simple molecular weight determinations on charged systems (for more details, see Kulicke. Horl (1985)). Thus, the molecular weight determination of polymer molecules or additives is an important fact in the drag reduction area. Molecules or particles which are effective have molecular weights above 105 g/mol (the polymer backbone chains should be linear, flexible, and unbranched). 

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

The intrinsic viscosity is a function of molar mass via the Mark-Houwink relationship, wherein K and a are coefficients for a given polymer in a given solvent at a given temperature. 

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) ... 

Mark-Houwink Relations. The Mark-Houwink relationships obtained for pullulans and dextrans in 0.2 M NaNOa are as follows ... 

The GPC-viscometry with universal calibration provides the unique opportunity to measure the intrinsic viscosity as a function of molecular weight (viscosity law, log [17] (it versus log M) across the polymer distribution (curves 3 and 4 in Fig. 1). This dependence is an important source of information about the macromolecule architecture and conformations in a dilute solution. Thus, the Mark-Houwink equation usually describes this law for linear polymers log[i7] = ogK+ a log M (see the entry Mark-Houwink Relationship). The value of the exponent a is affected by the macromolecule conformations Flexible coils have the values between 0.5 and 0.8, the higher values are typical for stiff anisotropic ( rod -like) molecules, and much lower (even negative) values are associated with dense spherical conformations. 

Mark-Houwink relationships are also important for the application of the universal calibration procedure in GPC-SEC of polymer molecules, where the... 

Let us note, that the dependences m(MM) with fractional exponent gother polymers as well. So, in work [4] the constants of Kuhn-Mark-Houwink equation for -solvent in case of the branched polyarylate D j and its linear analog have been adduced, that allows to calculate intrinsic viscosities [tj] 0 and [Tjjj g, accordingly, for arbitrary MM. Then, the value g can be estimated according to the relationship [4] ... 

Martin equation is usually valid in the range of redueed eoneentrations, c < 10. For evaluation of [r ], the Mark-Houwink relationship is reeommended... 

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