Continuum theories

The next developments of the FC approach were in papers by (R. A.) Marcus,41 49 and a later series from the Soviet Union. About the same time Hush50 introduced other concepts, to be discussed below. The early work of Marcus41 considered the Inner Sphere to be invariant with frozen bonds and vibrational coordinates up to the time of electron transfer. The classical subsystem for ion activation has its ground state floating on a continuum of classical levels, i.e., vibrational-librational-hindered translational motions of solvent molecules in thermal equilibrium with the ground state of the frozen solvated ion. 

The most probable X and X were obtained by setting the differentials of the Gibbs energy of X (5F ) and the difference between those of X and X (8F - 5F) equal to zero. The sum 8F + 

By conservation of energy, F - F must be equal to AG° - TASe + w, where AG° is the standard Gibbs energy of reaction, ASe is any electronic entropy change on electron transfer, and w is the work required to bring the activated reactants and activated products from infinity. The second minimizing condition, i.e., 8F - 8F = 0 gives  

The electrode case may also be thought of as involving two ions, one having infinite radius. However, Marcus considered that there was also an image charge situated at R, twice the distance of the reacting ion from the electrode.44 46 Thus, A,elec was given by  

We should note that for water, l/e0 is negligible, and the infrared value of 1/n2 is about 0.55. The overall rate is exp-m2A./kT multiplied by a frequency equal to the collision number and by the concentrations of the reactants. 

---

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

Leslie F M 1998 Continuum theory for liquid crystals Handbook of Liquid Crystals Vol 1. Fundamentals ed D Demus, J Goodby, G W Gray, Fl-W Speiss and V Vill (New York Wiley-VCFI)... 

Consider an alchemical transformation of a particle in water, where the particle s charge is changed from 0 to i) (e.g., neon sodium q = ). Let the transformation be performed first with the particle in a spherical water droplet of radius R (formed of explicit water molecules), and let the droplet then be transferred into bulk continuum water. From dielectric continuum theory, the transfer free energy is just the Born free energy to transfer a spherical ion of charge q and radius R into a continuum with the dielectric constant e of water ... 

The rigid nature of the mesophase pitch molecules creates a strong relationship between flow and orientation. In this regard, mesophase pitch may be considered to be a discotic nematic liquid crystal. The flow behavior of liquid crystals of the nematic type has been described by a continuum theory proposed by Leslie [36] and Ericksen [37]. 

Khan, A.S. and Huang, S. (1995) Continuum Theory of Plasticity (Wiley, New York). 

The continuum theory of deformation of elastic solids is old and well developed [65T01, 74T01], and, in its linear version, is widely applied. Nonlinear theory is of much more recent origin. Most application of nonlinear theory has been to the behavior of highly deformable materials such as rubber or to the explanation of subtle effects observed by precise ultrasonic... 

Recall that in the usual continuum theory, the amplitude K a,ff) for going from point a to /3 is an integral over all paths from a to with each path weighted by a factor exp i5c, where Sc == 5c[x(t)] is the classical action integral (equation 12.32) ... 

Kn = 0.01-0.1 Slip flow rarefaction effects that can be modeled with a modified continuum theory with wall slip taken into consideration... 

As revealed through experimental works, however, the flow of lubricants in TFL provides a hint that the macroscopic properties, such as the viscosity and the elastic modulus remain to be a measurement of the fluid characteristics. In addition, the transition from EHL to TFL is inherently progressive, wherein no abrupt transform in lubrication states are found. Thus, the continuum theory is validated to some extent. Furthermore, one can arrives at a continuum viewpoint, but in a different way from conventional fluid mechanics, by considering the material to be a continuum one in an ensemble averaged, rather than a spatial averaged, sense. 

This section provides an alternative measurement for a material parameter the one in the ensemble averaged sense to pave the way for usage of continuum theory from a hope that useful engineering predictions can be made. More details can be found in Ref. [15]. In fact, macroscopic flow equations developed from molecular dynamics simulations agree well with the continuum mechanics prediction (for instance. Ref. [16]). 

From experimental results, the variation of film thickness with rolling velocity is continuous, which validates a continuum mechanism, to some extent in TFL. Because TFL is described as a state in which the film thickness is at the molecular scale of the lubricants, i.e., of nanometre size, common lubricants may exhibit microstructure in thin films. A possible way to use continuum theory is to consider the effect of a spinning molecular confined by the solid-liquid interface. The micropolar theory will account for this behavior. 

As expected from continuum theory, the friction and diffusion coefficients are replaced In Inhomogeneous fluid by tensors whose symmetry reflects that of the Inhomogeneous media. 

Apart from obvious features such as laminarity, there are speculations that flows in micro channels exhibit a behavior deviating from predictions of macroscopic continuum theory. In the case of gas flows, these deviations, manifesting themselves as, e.g., velocity slip at solid surfaces, are comparatively well understood (for an overview, see [130]). However, for liquid flows on a length scale above 1 pm, there is no clear theoretical foundation for deviations from continuum behavior. Nevertheless, various unexpected phenomena such as friction factors deviating from the continuum prediction [131-133] have been reported. A more detailed discussion of this still unsettled matter is given in Section 2.2. At any rate, one has to be careful here since it may be that measurements in small systems lack precision, essentially because of the incompatibility of analysis in a confined space and with large measuring equipment... 

It is not the purpose of chemistry, but rather of statistical thermodynamics, to formulate a theory of the structure of water. Such a theory should be able to calculate the properties of water, especially with regard to their dependence on temperature. So far, no theory has been formulated whose equations do not contain adjustable parameters (up to eight in some theories). These include continuum and mixture theories. The continuum theory is based on the concept of a continuous change of the parameters of the water molecule with temperature. Recently, however, theories based on a model of a mixture have become more popular. It is assumed that liquid water is a mixture of structurally different species with various densities. With increasing temperature, there is a decrease in the number of low-density species, compensated by the usual thermal expansion of liquids, leading to the formation of the well-known maximum on the temperature dependence of the density of water (0.999973 g cm-3 at 3.98°C). 

Chipman M (2002) Computation of pKa from Dielectric Continuum Theory. J Phys Chem A 106 7413-7422. 

Simonson, T. Briinger, A.T., Solvation free energies estimated from macroscopic continuum theory an accuracy assessment, J. Phys. Chem. 1994, 98, 4683-4694... 

Baumgartner, L. P. Rumble III, D. (1988). Transport of stable isotopes I Development of a kinetic continuum theory for stable isotope transport. Contrib. Mineral. Petrol., 98, 417-30. 

---