Molar mass scaling laws

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. 

In Ref. [107] it has been demonstrated how, based on the scaling law for the diffusion coefficient, molar mass distributions can be calculated from time correlation functions obtained from scattering experiments. 

In order to calculate the molar mass distribution, the scaling law (Equation (52))... 

The more time-consuming task is the establishment of the scaling law, which requires a series of polymer samples of narrow molar mass distribution and known molar mass. Their sedimentation coefficients have to be measured as a function of concentration and extrapolated back to c — 0 in order to obtain So(M) (Figure 18). 

The same authors then discuss the determination of the entire molar mass distribution from sedimentation velocity runs via scaling laws for the polymer polystyrene in cyclohexane, where the scaling law is also known [78] ... 

We hope that this chapter on the molecular weight determination of synthetic polymers has illustrated that in the case of a complex polymer it is preferable to use several experimental methods for the molecular weight determination to obtain a full picture. Owing to the different sensitivity of the various methods some are blind for low molar masses while others are blind at low concentrations. As exemplified, often scaling laws can be utilized to compare results of different methods and different sensitivities. 

Of special interest are dilute polymer solutions, where the thermal diffusion coefficient, Dr = Sr D, is independent of molar mass [11,21]. For the diffusion coefficient D, a scaling law D M a holds, with the exponent for linear flexible... 

Other measures of the solubility of a gaseous solute are readily derived from its mole fraction. The Henry s law constant is KH lim(pjxj30) atp2 —> 0, which becomes on the molal scale Km lim(1000 p2/Mjc2) with Mj the molar mass of the solvent in g mob1 for use in Eq. (2.13) below. The Ostwald coefficient is the limit of y2 = RT(xJp lVl atp2 -> 0 and is related to the mass fraction w2 by ... 

Based on these indications, 0.1 M NaCl and 1.5 M NaCl were selected for measurement of the molar mass dependences of the light scattering parameters under two limiting solvent conditions respectively. The following scaling laws were found ... 

Section 12.1 introduces the concept of pressure and describes a simple way of measuring gas pressures, as well as the customary units used for pressure. Section 12.2 discusses Boyle s law, which describes the effect of the pressure of a gas on its volume. Section 12.3 examines the effect of temperature on volume and introduces a new temperature scale that makes the effect easy to understand. Section 12.4 covers the combined gas law, which describes the effect of changes in both temperature and pressure on the volume of a gas. The ideal gas law, introduced in Section 12.5, describes how to calculate the number of moles in a sample of gas from its temperature, volume, and pressure. Dalton s law, presented in Section 12.6, enables the calculation of the pressure of an individual gas—for example, water vapor— in a mixture of gases. The number of moles present in any gas can be used in related calculations—for example, to obtain the molar mass of the gas (Section 12.7). Section 12.8 extends the concept of the number of moles of a gas to the stoichiometry of reactions in which at least one gas is involved. Section 12.9 enables us to calculate the volume of any gas in a chemical reaction from the volume of any other separate gas (not in a mixture of gases) in the reaction if their temperatures as well as their pressures are the same. Section 12.10 presents the kinetic molecular theory of gases, the accepted explanation of why gases behave as they do, which is based on the behavior of their individual molecules. 

It is well known that the rio parameter is correlated by a scaling law with the molar mass of the sample, and knowing 7 c, it is possible to... 

Below the gel point, the system is self-similar on length scales smaller than the correlation length with a power law distribution of molar masses with Fisher exponent r = 5/2 [Eq. (6.78)]. Each branched molecule is a self-similar fractal with fractal dimension 27 = 4 for ideal branched mole-cules in the mean-field theory. The lower limit of this critical behaviour is the average distance between branch points (= ). There are very few... 

In practice, the gel point is often difficult to determine with sufficient precision to test these scaling laws. Instead, viscosity and relaxation time can be correlated with weight-average molar mass (M e with 7 1.82) [Eq. (6.103)] ... 

The reliable characterization of molar mass, as well an availability of a series of chemical identically polymer samples with different molar masses, remain indispensable prerequisites for an efficient determination of the scaling parameters of a macromolecule. Ultimately, the scaling parameters are required to understand the physical relationships that are valid for hb polymers, and which clearly differ from those of linear polymers, taking into account their strong influence on the material s properties. The scaling laws form the basis for any characterization of the global molecular parameters of polymers, and are based on their size (mostly expressed as the radius of gyration Rg) or on their intrinsic viscosity [/j] as a function of the molar mass M ... 

Solution activity data obtained by osmometry on dilute solutions showed that the second virial coefficient is dependent on molar mass, contradicting the Flory-Huggins theory. These problems arise from the mean-field assumption used to place the segments in the lattice. In dilute solutions, the polymer molecules are well separated and the concentration of segments is highly non-uniform. Several scaling laws were therefore developed for dilute (c < c is the polymer concentration in the solution, c is the threshold concentration for molecular overlap) and semi-dilute (c > c ) solutions. In a good solvent the threshold concentration is related to molar mass as follows ... 

It is well known that linear chain owns the simplest chain topology and the characteristic of self-similar objectives, namely, there exists a classical scaling law between the chain molar mass and chain size. Moreover, both theoretical and experimental studies have revealed that cyclic polymer is also fractal objective, just like linear chain [58]. 

It has been proved that, as analogues of hyperbranched polymers, dendrimer-like polymers are not a self-similar objective, i.e., the classic scaling laws between their sizes (/ ), intrinsic viscosities ([) ]) and molar masses (Af) are not valid for dendrimer-like polymers, which is partially because the hydrodynamic radius (l h) is not proportional to the radius of gyration (i g) [59-61]. There exists a maximum in the plot of [r] versus M, i.e., [jj] first increases with the number of generation (G) and then decreases after G reaches a certain value, as shown in Fig. 1.4 [59]. 

Until recently, the nature of the ordinary phase was also controversely discussed as the strong increase of the diffusion coefficient with increasing polyion concentration at constant salt concentration was interpreted in terms of scaling laws for semi-dilute poiyelectrolyte concentrations. As it is now proven that the onset of the cross-over region does not depend on molar mass, the coupled mode picture unambiguously describes the ordinary phase best. See Sect. 3.2.3... 

Detailed osmotic studies by Mandel et al. [140] on NaPSS yield an exponent of 9/8 at low concentration and 9/4 at high concentration of the osmotic pressure-concentration power law (n c ) which again was interpreted as a dilute-semidilute concentration transition. A recent literature study [66] confirmed the experimental scaling exponents but clearly demonstrated that the cross-over concentration does not depend on the molar mass of the polyions. 

Fig. 2.17. Relation between the radius of gyration Rg and the molar mass M, observed in light scattering experiments on dilute solutions of PS in toluene. The continuous straight line corresponds to the scaling law Eq. (2.83). Data from Win-termantel et al. [5]...

This simply shows that there is a physical relationship between different quantities that one can measure in a gas system, so that gas pressure can be expressed as a function of gas volume, temperature and number of moles, n. In general, some relationships come from the specific properties of a material and some follow from physical laws that are independent of the material (such as the laws of thermodynamics). There are two different kinds of thermodynamic variables intensive variables (those that do not depend on the size and amount of the system, like temperature, pressure, density, electrostatic potential, electric field, magnetic field and molar properties) and extensive variables (those that scale linearly with the size and amount of the system, like mass, volume, number of molecules, internal energy, enthalpy and entropy). Extensive variables are additive whereas intensive variables are not. 

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