Mark-Houwink coefficients Equation

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. 

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 ... 

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... 

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 ... 

For strictly linear chains, universal calibration is extremely useful because the Mark-Houwink coefficients have been tabulated for all common linear polymers. The calibration curve allows [q M to be deter-mined from the elution volume. The Mark-Houwink equation [Eq. (1.100)] then allows the SEC measure of [r]]M to determine the molar mass of the polymer ... 

In the above equation the R group designates the particular acrylate. Ethyl, iso-propyl, and / -butyl acrylates were produced yielding the polymers poly(ethyl acrylate)-di (PEA-di), poly(wo-propyl acrylate)-di (PIPA-di) and poly(n-butyl acrylate)-di. For PIPA the viscosity average molecular mass was measured to be 98,000 daltons.(5) The molecular masses of the other polymers were not estimated directly because their Mark-Houwink coefficients were not known, but they had similar intrinsic viscosities. At these molecular weights, we expect the relaxation times to be independent of molecular mass.(i)... 

The Mark-Houwink equation relates the molecular weight of the polymer to [ ] if the Mark-Houwink constants K and "a" for the solvent, polymer, temperature, and polymer molecular weight range are known. These constants are routinely determined fiom viscosity data of solutions of narrow molecular weight polymers samples and are tabulated in the Polymer Handbook (18). A listing of seleeted Mark-Houwink coefficients for the polymers discussed in this book is given in Table 7. 

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. 

Mark-Houwink equation n. Also referred to as Kuhn-Mark-Houwink-Sakurada equation allows prediction of the viscosity average molecular weight M for a specific polymer in a dilute solution of solvent by [77] = KM, where K is a constant for the respective material and a is a branching coefficient K and a (sometimes a ) can be determined by a plot of log [77] versus logM" and the slope is a and intercept on the Y-axis is K. Kamide K, Dobashi T (2000) Physical chemistry of polymer solutions. Elsevier, New York. Mark JE (ed) (1996) Physical properties of polymers handbook. Springer-Verlag, New York. Ehas HG (1977) Macromolecules, vols 1-2. Plenum Press, New York. 

When equation 4.9 is applied on a chain-length basis, using the data from table 4.1, a new estimate for the Mark-Houwink coefficients of EA is obtained K= 17.3 lO dLg and a = 0.68. By using these latter data, the tendency in the hydrodynamic volume of the homologous series of acrylates seems to be the most reasonable of all three options discussed here. 

Coincidental or not, also in a PLP study of several methacrylates, the ethyl monomer EMA was found to have the highest activation energy of propagation [15, 28]. (2) Because of the high uncertainty in the exact values of the Mark-Houwink coefficients of EA, the corresponding confidence intervals of equations 4.12 to 4.14 have not been calculated. 

Before kinetic information can be extracted from this number MWD, two criteria must be met (i) the MWD must be of the correct shape and (ii) it must also be scaled correctly, according to equation 4.3. The former criterion, however, could be systematically violated in several ways. This can be the result of inaccurate SEC calibration, inaccurate Mark-Houwink coefficients, non-linear baseline drift, the occurrence of column broadening during a SEC analysis and the non-linear optical properties of a series of oligomers. It goes beyond the scope of this thesis to discuss all these effects. Here, the discussion will be limited to only two of these effects column broadening and non-linear optical properties of oligomers. 

When a value of r = 0.78 is used, the data points for MSM copolymers (shown as crosses) coincide with the curve for polystyrene. According to Equation 4, r may be calculated from the coefficient K for homopolymers. When applied to the data on polystyrene and poly (methyl methacrylate) (3), a value of 0.72 is obtained, in good agreement with the above value of 0.78. The slight difference is probably the result of the polydispersity in the copolymers since the Mark-Houwink equation requires monodispersed polymers. 

The relationship can also be cast in terms of distributions of molecular weights for polymer systems. This transformation is achieved by relating the decay constant to its corresponding diffusion coefficient D which in turn is related to molecular weight through a Mark-Houwink like relationship. Equation 3 shows the fundamental relationship between the autocorelation function and polymer molecular weight distributions. 

In analogy with the Mark-Houwink equation the dependence of the sedimentation coefficient on molar mass can be expressed as ... 

It is clear that the interpolation between the calibration lines cannot be applied to mixtures of polymers (polymer blends). If the calibration lines are different, different molar masses of the homopolymers will elute at the same volume. The universal calibration is also not capable of eliminating the errors which originate from the simultaneous elution of two polymer fractions with the same hydrodynamic volume, but different composition and molar mass. Ogawa [33] has shown by a simulation technique that the molar masses of polymers eluting at the elution volume Ve are given by the corresponding coefficients K and a in the Mark-Houwink equation. 

Determination of D is the first step in studying macromolecular coils by fractal analysis. D is usually estimated by finding the exponents in the Mark-Kuhn-Houwink type equation, which relate the characteristic viscosity [r ], the translational diffusion coefficient Dq, or the rate sedimentation coefficient Sq) with the molecular weight (M) of polymers [3] ... 

This can be done as follows. Using experimentally determined rjred values, [rj] can be calculated according to Shultze-Braschke empirical equation (Eq. (6) of Chapter 1) and then to determine the coefficient in Kuhn-Mark-Houwink equation according to the Eq. (58) of Chapter 1. Farther the value MM is determined according to the Eq. (57) of Chapter 1 and polymerization degree N is calculated as follows ... 

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