Chemical potential stiffness

For a polydisperse polymer, analysis of sedimentation equilibrium data becomes complex, because the molecular weight distribution significantly affects the solute distribution. In 1970, Scholte [62] made a thermodynamic analysis of sedimentation equilibrium for polydisperse flexible polymer solutions on the basis of Flory and Huggins chemical potential equations. From a similar thermodynamic analysis for stiff polymer solutions with Eqs. (27) for IT and (28) for the polymer chemical potential, we can show that the right-hand side of Eq. (29) for the isotropic solution of a polydisperse polymer is given, in a good approximation, by Eq. (30) if M is replaced by Mw [41],... 

Here the sum of y and its second derivative with respect to orientation is called the surface stiffness. For the special case of a 2-dimensional monatomic island on a single crystal surface, the surface energy is replaced by the step energy, and the eqttation for the chemical potential of the island edge, eqtrivalent to Eq. (17), is... 

Following the analysis proposed by Giesen et al. [30], at any point on its perimeter an island in equilibrium has a constant chemical potential ji, which is related to the stiffness coefficient y and the local curvature. In addition, the stiffness coefficient depends on the step orientation 0 (see Fig. 3.5) and the line tension y. The authors derive a simple relation between the shape coordinates of islands and the energy parameters. Using the coordinate system and the island orientation shown in the figure, a point of minimum curvature exists at y, V = 0, and for reasons of symmetry, also at -y, x = 0. Line tension and step stiffness are then related by... 

Consider the chemical potential expression (8.137) for the special case when the surface energy density Us depends on orientation 9 but not on surface strain e. Verify that the chemical potential expression reduces to x = f/ — n Us 4- Dg), where the prime denotes differentiation with respect to 6. The quantity in parentheses is sometimes called surface stiffness its existence presumes the function Us 9) is indeed twice differentiable. 

When considering relea.se mechanisms, the physical and chemical heterogeneity of the adhesive/release interface cannot be ignored. At its most basic level, roughness of the release and PSA surface, the stiffness of the PSA and the method in which the PSA and release surface are brought together define the contact area of the interface. The area of contact between the PSA and release material defines not only the area over which chemical interactions are possible, but al.so potential mechanical obstacles to release. In practice, a differential liner for a transfer adhesive can be made to depend in part on the substrate roughness for the differences in release properties [21],... 

For some reactions the rate constant kj can be very large, leading potentially to very rapid transients in the species concentrations (e.g., [A]). Of course, other species may be governed by reactions that have relatively slow rates. Chemical kinetics, especially for systems like combustion, is characterized by enormous disparities in the characteristic time scales for the response of different species. In a flame, for example, the characteristic time scales for free-radical species (e.g., H atoms) are extremely short, while the characteristic time scales for other species (e.g., NO) are quite long. It is this huge time-scale disparity that leads to a numerical (computational) property called stiffness. 

The force constant is a measure of the stiffness of a chemical bond. Larger values of k imply sharply curved potential energy functions and are often associated with deeper potential wells and stronger bonds. Molecular force constants are typically in the range of 200-2000 N/m, remarkably, not very different from those for bedsprings. From the solution for the harmonic oscillator, we identify the ground-state vibrational energy, with quantum number u = 0, as... 

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