Viscometry Mark-Houwink equation

Vj is the volume of sample at concentration c1 and v is the volume of solvent to be added to accomplish the next dilution to e2. Nine or ten dilutions should be made, but no less than three, so that T p/c, vs ct contains a minimum of four points, excluding ci = 0 where extrapolation yields [r)]. In either single-point or dilution viscometry, the basic Tirel is measured first, precedent to the secondary equations in Table I in Chapter 4. Having plotted and extrapolated "f]sp/ct vs c, to [nq] and knowing K and v beforehand, substitutions may be made in the Mark-Houwink equation [Eq. (4.47)] Mv can thus be calculated. 

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. 

The most popular method to measure the molecular weight of chitosan is viscometry, which is based on the Mark-Houwink equation... 

One of the easiest means of determining the molar mass of a polymer is via viscometry. This yields the viscosity average molar mass, Mt . The viscosity of a polymer solution can be measured using capillary viscometers. The time for flow of the polymer solution through a given distance is measured and this is proportional to viscosity. The viscosity obtained by this method is expressed relative to that of the pure solvent. To determine molar mass, the intrinsic viscosity is required. It has this name because it relates to the intrinsic ability of a polymer to increase the viscosity of a solvent. The intrinsic viscosity [ j] is related to the specific viscosity jsp = 1 — rj/rjo, where rj is the viscosity of the polymer solution and rjo is that of the solvent via a virial equation in concentration resembling Eq. (2.9). It has been found empirically that for many polymer solutions the Mark-Houwink equation for intrinsic viscosity is obeyed ... 

Viscosity measurements alone cannot be directly used in the Mark-Houwink-Sakurada equation to relate absolute viscosity and polymer molecular weight, since additional unknowns, K and a must be determined. Therefore, viscometry does not yield absolute molecular weight values it rather gives only a relative measure of polymer s molecular weight. Viscosity measurements based on the principle of mechanical shearing are also employed, most commonly with concentrated polymer solutions or undiluted polymer these methods, however, are more applicable to flow properties of polymers, not molecular weight determinations. 

Equation (5) is an important practical molecular weight average derived from viscometry. In order to calculate this average, the exponent a of the Mark-Houwink relationship relating intrinsic viscosity [p] to molecular weight must be known ... 

In the viscometry method " one measures the viscosity, rf, of a polymer solution at various degrees of dilution. The intrinsic viscosity, [rj], of the polymer-solvent system is obtained by plotting the quantity l (jj - J7o)/ o vs. C (here rjo is the viscosity of the solvent and C is e concentration of the polymer solution) and extrapolating the resulting curve to zero concentration. Then one applies the Mark-Houwink-Sakurada (MHS) equation ... 

An interesting extension of the GPC viscometry technique is that it is possible to measure the exponent a in the Mark-Houwink-Sakurada equation. Since both ln[i/] and InM are known, a plot of ln[ /] versus InM has a slope equal to a. This plot can either be constructed from a mixture of monodisperse polymer samples or from a broad molecular weight distribution polymer sample. 

In addition, there is a great utility of having reliable values for both Mark-Houwink and Schultz-Blaschke parameters in that the combination enables the measurement of an average molecular weight in a very simple viscometry experiment. Equation 10.17 gives an explicit formula for the intrinsic viscosity based only on the relative flow times for a single concentration experiment. 

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