Path dependence

Figure 12. Vibrational levels for the first-excited electronic state of H3 calculated [4] using Longuet-Higgins phase /4(R) = cp/2 Eq, (A.14) with a path-dependent phase A(R) =y(p,9,(p), The extra levels arising in one calculation but not in the other are indicated by longer line segments,...

We also want to point out the difference between simple rate-dependent phenomena and path-dependent effects. Simple rate dependence means that the internal micromechanical state (as possibly represented by some meso-scale variables) depends only on the current deformation and current rate of deformation the material has no memory of the past. In terms of dislocation dynamics and (7.1), a simple rate-dependent constitutive description would be one in which... 

Here t has no memory of the rate of deformation in getting to the final plastic strain y. This differs from a path-dependent law, in which the strength (for example) depends in detail on the path followed in getting to the current macroscale state. 

Two examples of path-dependent micromechanical effects are models of Swegle and Grady [13] for thermal trapping in shear bands and Follansbee and Kocks [14] for path-dependent evolution of the mechanical threshold stress in copper. 

The importance of emphasizing the essential difference between simple rate-dependent and path-dependent processes is that in the former case one does not have to follow the actual time-resolved deformation path in numerical computation, while in the latter case it is essential. 

The lone remaining aspect of this topic that requires additional discussion is the fact that the mechanical threshold stress evolution is path-dependent. The fact that (df/dy)o in (7.41) is a function of y means that computations of material behavior must follow the actual high-rate deformational path to obtain the material strength f. This becomes a practical problem only in dealing with shock-wave compression. 

Although the difference in final strength f, integrated through both the actual shock wave and the computational shock wave, will be mitigated by dynamic recovery (saturation) processes, this is still a substantial effect, and one that should not be left to chance. These are very important practical considerations in dealing with path-dependent, micromechanical constitutive models of all kinds. 

Combining Eqs. (12)-(14) yields Eq. (7) directly. Although such a free energy decomposition is path-dependent [8], it provides a useful and rigorous framework for understanding the nature of solvation and for constructing suitable approximations to the nonpolar and electrostatic free energy contributions. 

Notice that AE, like AH, is a state property it has the same value regardless of how or where or why the reaction is carried out. In contrast, q and w are path-dependent their values vary depending on whether the reaction is carried out in the atmosphere, an engine, or an electrical cell. 

Students often ask, What is enthalpy The answer is simple. Enthalpy is a mathematical function defined in terms of fundamental thermodynamic properties as H = U+pV. This combination occurs frequently in thermodynamic equations and it is convenient to write it as a single symbol. We will show later that it does have the useful property that in a constant pressure process in which only pressure-volume work is involved, the change in enthalpy AH is equal to the heat q that flows in or out of a system during a thermodynamic process. This equality is convenient since it provides a way to calculate q. Heat flow is not a state function and is often not easy to calculate. In the next chapter, we will make calculations that demonstrate this path dependence. On the other hand, since H is a function of extensive state variables it must also be an extensive state variable, and dH = 0. As a result, AH is the same regardless of the path or series of steps followed in getting from the initial to final state and... 

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

The rate of dissociation increases rapidly above 2000°C. It also increases with decreasing pressure.P" ] The rate of recombination (i.e., the formation of the molecule) is rapid since the mean-free-path dependent half-life of atomic hydrogen is only 0.3 sec. 

If a laser beam produces in the outer atmosphere a spectrum spanning from the ultraviolet to at least the red, then the return light will follow different optical paths depending on the wavelength (Fig. 19). The air refraction index is a function of air temperature T and pressure P ... 

The oxidation by Mn(Iir) in a perchlorate medium follows two kinetic paths depending on the bromide concentration, viz. 

The purification obtained in electrorefining is based on the selectivity provided by the process itself. Electrorefining may, in principle, be carried out along two different paths, depending on the nature of the impurities to be removed. Either the impure metal forms the anode and the pure metal is concentrated in the cathode, or the impurities are selectively dissolved from the anode so that the purity of the metal constituting it increases. Although literature records electrorefining processes based on both these approaches, the former seems to dominate in commercial practice. 

The first step corresponds to the formation of a Lennard-Jones cavity with the shape of the solute the charges are included in the second step. This free energy decomposition is, of course, path dependent different (divergent) results would be obtained if the electrostatic coupling were included first. 

As noted above, the curl of the expression on the right-hand side of Equation 7.47 vanishes. However, it does not mean that the Coulombic and non-Coulombic components—the former is the electric field produced by the Fermi-Coulomb hole and the latter arises from the kinetic energy tensor—of this field also have vanishing curl. Thus the potential Wxc of Equation 7.38 may sometimes be path dependent [21]. 

As illustrated in Figure 26, which is a varied presentation for a single pore from the scheme shown in Figure 19, there are five possible phases in the current path in which significant potential drops may occur. The distribution of the applied potential in the different phases of the current path depends on doping type and concentration, HF concentration, current density, potential, illumination intensity and direction. The phases in the current path... 

In the application of the principle of microscopic reversibility we have to be careful. We cannot apply this concept to overall reactions. Even Eqs. (4.43) - (4.45) cannot be applied unless we know that other reaction steps (e.g., diffusional transport) are not rate controlling. In a given chemical system there are many elementary reactions going on simultaneously. Rate constants are path-dependent (which is not the case for equilibrium constants)and may be changed by catalysts. For equilibrium to be reached, all elementary processes must have equal forward and reverse rates... 

---