Intrinsic viscosity Mark-Houwink-Sakurada equation

The Mark-Houwink-Sakurada equation relates tire intrinsic viscosity to tire polymer weight ... 

The specific viscosity )jsp of a dilute solution of spheres is directly related to their hydrodynamic volume VV Nl denotes Avogadro s number. Typically the intrinsic viscosity [tj] follows a scaling law, the so-called Mark-Houwink-Sakurada equation ... 

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. 

Not only good solubility but also solution behavior differs for hyperbranched polymers compared to linear polymers. For example, hyperbranched aromatic polyesters, described by Turner et al. [71,72], exhibit a very low a-value in the Mark-Houwink-Sakurada equation and low intrinsic viscosities. This is consist-... 

Intrinsic viscosity is related to the relative viscosity via a logarithmic function and to the specific viscosity by a simple algebraic relationship. Both of these functions can be plotted on the same graph, and when the data are extrapolated to zero concentration they both should predict the same intrinsic viscosity. The specific viscosity function has a positive slope and the relative viscosity function has a negative slope, as shown in Fig. 3.7. The molecular weight of the polymer can be determined from the intrinsic viscosity, the intercept of either function, using the Mark-Houwink-Sakurada equation. 

The relation between number molecular weight, Mn and intrinsic viscosity, [t ], for poly(penLachlorophenyI methacrylate) (PPCIPh) can be represented by the Mark - Houwink - Sakurada equation [44],... 

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 ... 

Osmotic pressure measurements for the determination of MW were used in 1900 to characterize starch. Twenty years later, the solution viscosity measurements were introduced by Staudinger for this purpose. However, it was Mark and his collaborators who developed the concept of the intrinsic viscosity ([r ]) and demonstrated that it provides information on the volume of individual colloidal particles, thus on MW. For the freely rotating chains the dependence (today known as Mark-Houwink-Sakurada equation) was obtained [Guth and Mark, 1934] ... 

Dilute Solution Properties. The rheology of dilute polymer solutions has been used extensively to gain insight into the structure and conformation of polymers in solution (11). The intrinsic viscosity provides a measure of the molecular weight of a polymer through a relationship such as the Mark-Houwink-Sakurada equation. Earlier studies of polyacrylamide (PAM) systems and details of the complexity of the characterization of high-molecular-weight water-soluble systems can be found in references 9, 13, and 14. 

This equation applies for polymeric solutions under theta conditions. Theta conditions are those at which excluded volume effects (expansion of the dimensions of the ideal coil) are exactly compensated by polymer solvent interactions (Chapter 25). The dependence between intrinsic viscosity and MW is given by the Mark-Houwink-Sakurada equation (see also Chapter 1) ... 

Routinely, molecular weights of polymers are conveniently estimated from intrinsic viscosity measurements using the Staudinger (also known as the Mark-Houwink-Sakurada) equation... 

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 ... 

According to the Mark-Houwink-Sakurada equation, the intrinsic viscosity [jj] might be related to the molecular weight (M) of a linear pol3mier ... 

The logarithms of intrinsic viscosities of fractionated samples are plotted against log or log Mn. The constants a and K of the Mark-Houwink-Sakurada equation are the intercept and the slope, respectively, of that plot. Except for the lower molecular weight samples, the plots are linear for linear polymers. Many values of K and a for different linear polymers can be found in the literature [66]. 

Viscosimetry is the mostly utilized method to determine the molecular weight of chitosan due to its simplicity. The method has the disadvantage of not being absolute because it relies on the correlation between the values of intrinsic viscosity with those of molecular weight, as determined hy an absolute method for fractions of a given polymer. This relationship is given by the well-known Mark—Houwink—Sakurada equation. 

The viscosity of polymer solutions depends on both the concentration and the molecular weight of the polymer. Thus, measurement of the viscosity of a solution of polymer can be a way to determine the molecular weight of the dissolved polymer. This can actually be the case providing that the coefficients K and a of the Mark-Houwink-Sakurada equation (Equation 2.5) giving the relationship between the intrinsic viscosity of the polymer, [tj], and the viscosity average molecular weight, M, are known. 

This also means that calibration is required in methods based on viscosity measurements before the method can be applied to characterise the molecular weight of a given polymer. Such calibration is tedious and requires that the molecular weights of a series of the polymer have been characterised by another technique in order to establish the relationship with the intrinsic viscosity. At present, this can be performed using the SEC method coupled with triple detection. It is noteworthy that calibration of the Mark-Houwink-Sakurada equation is valid only for one couple including a polymer and a solvent at a given temperature. 

Much of the above discussion indicates that, to study excluded volume effects, an accurate determination of unperturbed dimensions is required. For this, a common procedure is to extrapolate intrinsic viscosity of known molecular weight samples to zero molecular weight. Several extrapolations have been used, notably the Stockmayer-Fixman plot. Dondos and Benoit have now introduced a modified version of this, which appears to be linear over a wider range of molecular weights. It introduces a parameter D, which is shown to be linearly related to the exponent of the Mark-Houwink-Sakurada equation. 

In Table II, the large variations in the parameters for the Mark-Houwink-Sakurada equation [Eq. (I)] relating intrinsic viscosity [/y] to molecular weight M (AT and a are parameters which have to be determined for each polymer-solvent system). 

Due to dieir compact, branched structure and to die resulting lack of chain entanglement, dendritic polymers exhibit much lower melt and solution viscosity dian their lineal" counterparts. Low a-values in die Mark-Houwink-Sakurada intrinsic viscosity-molar mass equation have been reported for hyperbranched polyesters.198 199 Dendrimers do not obey diis equation, a maximum being observed in die corresponding log-log viscosity-molar mass curves.200 The lack of chain entanglements, which are responsible for most of the polymer mechanical properties, also explains why hyperbranched polymers cannot be used as diermoplastics for structural applications. Aldiough some crystalline or liquid... 

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. 

Viscosity molecular weight is calculated using the Mark-Houwink-Sakurada (MHS) equation of [rj] = Km Mv, where 7] is intrinsic viscosity, Mv is viscosity molecular weight, a is the MHS exponential factor (material and system-specific, between 0.9 and 1.0 for metal complex-based solvents), and Km is a constant. Moiecuiar weight distributions resuit from GPC experiments. 

Even without the use of a UC curve (one must be generated for each series of measurements), measurement of [t7o] is believed by some to yield an intrinsic viscosity-weighted molar mass [2]. Most importantly, there is a historic interest in the relation of [t ] to molar mass and/or size. Indeed, the study and explanation of UC has occupied the theorists for some time and, accordingly, there are various formulations describing such relationships [2]. For linear polymers, the most popular empirical relationship between [77] and molar mass is the Mark-Houwink-Sakurada (MHS) equation... 

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