The Martin Equation

Equation (4.9) is a logarithmic equivalent of the Huggins equation that obscures the exaggerated effect of the exponent and yields a wider linear range  

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Fig, 5.3. Viscosity at various concentrations and molecular weights in the low to moderate concentration range. Polystyrene-decalin and polymethyl methacrylate-xylene are theta or near-theta systems the remainder are good solvent systems (121,177). Note that the c[i/] reduction is somewhat better in theta solvents, and that the Martin equation [Eq. (5.9)], which would give a straight line in the figure, is a somewhat better representation for... 

Let us suppose that the experimental contribution of polymer to solution viscosity at low concentration, rj0 — t]s, can be represented to a sufficient approximation by the Martin equation [Eq.(5.9)]. An alternative which is presumably always available to the solution is free draining behavior [Eq.(4.16)]. As concentration is increased, then, according to the minimum dissipation principle, the behavior would be expected to change from the Martin form to the free draining form in the range where the two expressions for rj0 — rjs become equal ... 

The retention factors of the polymers in 60 % THF were calculated from gradient elution and discussed together with those directly measured for oligomers with 2-12 repeat units or 2000, 4000, and 9000 g/mol molar mass. On a lOOnm-pore packing, the Martin equation was fulfilled for n = 19 repeat units. At higher values, the curve... 

Different empirical equations relating the viscosity ratio excess to the intrinsic viscosity were tested, and the Martin equation (50) correlated the data with an error of 1% with respect to the results obtained using Equation 43 ... 

By comparing RM values of compounds that differ by only a methylene group or a functional group, a table of RM values can be compiled. Since they are proportional to the free energies for these groups, they should be additive, and the total RM value for any analyte can be predicted by summing the values for the groups of which the molecule is composed. This idea is known as the Martin equation ... 

Example 12.2. Potential mass transfer-induced instability in olefin hydroformylation [14]. The rate of olefin hydroformylation with cobalt hydrocarbonyl catalysts in a liquid phase obeys in good approximation the Martin equation... 

The solution viscosity is calculated using the Martin equation... 

The Martin equation (also known as the Bungenberg-de Jong equation) utilizes the common logarithm of the reduced viscosity ... 

For concentrated polymer solutions, the Martin equation is applicable ... 

This equation—sometimes referred to as the Martin equation—has been applied by Chauveteau (1982) in correlating the viscosity-concentration behaviour of xanthan in the semi-dilute regime. A number of other empirical equations which correlate polymer viscosity over a wide range of concentrations are reviewed by Ott (1955). 

We have more to say about entanglement later. Here we note that in practice, the coil overlap considerations leading to eq. 11.3.9 seem to be the dominant type of intermolecular interaction in the concentration range 1 < c[//] < 10. Figure 11.3.1 shows some data correlated with the use of the Martin equation (eq. 11.3.10). Above c[t]] 10, this second mode of interaction known as entanglement begins to dominate, and other viscosity correlations are more effective. 

Two-thirds of a century ago, Martin proposed an empirical extension to elevated concentration, now generally known as the Martin equation( )... 

The Martin equation often provides a good first approximation to r (c) over a wide range of concentrations. 

Koenderinck, et al. report r]/r)Q of aqueous 4 MDa xanthan as measured with a controlled-stress cone-and-plate viscometer(48). The original authors noted that their measurements are described well by the Martin equation(7) at small c and by a power law at larger c. Figure 12.2 Ic shows a transition at c < 0.1 wt% between these forms. The root-mean-square fractional experimental scatter in each fit is 16%, so the transition concentration is not determined with great accuracy. Koenderinck, et al. estimated c from Rg to be 0.008 wt%, so the crossover concentration is c+/c 13. 

In studying the viscous properties of dilute solutions of rigid-chain polymers [10-16], it was shown that the Martin equation [9] applies to them, as to solutions of flexible-chain polymers, for low concentrations c ... 

A sample of polyacrylamide forms solutions in water that can be fitted by a Huggins equation plot over the range of 1.0 < iir< 1.5 with a slope of 0.700 (g/dl) and an intercept of 1.35 dl/g. If the data from the Huggins equation at relative viscosities of 1.0 and 1.5 are used to fit the Martin equation, specific viscosity can be predicted at a concentration of 5.0 g/dl ... 

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