Pressure tensor anisotropy

Sinee our system in the slit is anisotropie and inhomogeneous, it makes sense to eonsider the loeal pressure tensor, Eq. (42), whieh depends on the distanee z from the adsorbing wall. If one defines Px = Pxx Py = Pyy ... 

The change in stress state between 0.7 ps and 0.8 ps is caused by a change in the structure of the material. The transformation consists of a homogeneous rotation of half the SiOe octahedra that persists for the remainder of the simulation (Fig. 8). The new structure has a (—H-i-) pattern of octahedral rotation which corresponds to Pmmn symmetry. The change with respect to the Pbnm phase is more subtle than phase transformations that had been contemplated in previous theoretical and experimental work none of the three octahedral rotations vanish in the Pmmn structure and the magnitude of the octahedral rotations is similar to that in Pbnm. The increase in mean stress at the transition means that the Pmmn phase has a slightly larger volume than Pbnm at the same pressure. The anisotropy of the stress tensor reflects the differences in equilibrium axial ratios between the two phases. 

Figure 13 Anisotropy, AP, of the pressure tensor versus r — a, where r is the radial distance and a = 100 run is the droplet radius Curve 1—benzene drop in air curve 2—benzene drop in water.

The equilibrium molecular dynamic simulations done have permitted to obtain usefull informations on physico-chemical properties at a liquid liquid interface. In particular, the anisotropy of the diffusion near the interface and its relation with the anisotropy of the pressure tensor is an original and interesting result. 

A characteristic of small sample simulations of dense polymers under constant volume conditions is that the average pressure tensor is often anisotropic with substantial differences between the on-diagonal components and, in addition, nonzero off-diagonal terms. This has led to the use of molecular dynamics algorithms which control the pressure tensor, P, and which allow these dynamic anisotropies to be relaxed out. Once implemented such methods have the added advantage of being easily adapted to measure the response of such systems to externally applied pressure fields as we shall show later. 

At this point, we emphasize that the use of Eq. (42) is not the ttaditional standard method to estimate interfacial tensions from computer simulations Usually, one simulates a system prepared in a slab configuration such as shown in Fig. 4, using the canonical (nVT) ensemble or the microcanonical ensemble (when one carries out Molecular Dynamics (MD) rather than MC simulations). The interface free energy is then found from the anisotropy of the pressure tensor [138,139]... 

For an ideal elastic solid, pdV must be replaced in Eq. (4.4) by the product of the pressure tensor and the differential of the deformation tensor. In the limiting case of a fluid phase, pressure anisotropies disappear. In the treatment that follows, we will neglect pressure anisotropies and assume sufficient mobilities of the components [91,92], which then allows the introduction of a scalar chemical potential for the components without any problem. We will return to this point again in Section 4.3.7 and in Section 5.4.4. 

In addition, intensity changes under increasing pressure have been observed. For example, the most intense Raman line at STP conditions is the flg component of v ( 220 cm ), but at about 2 GPa the intensity decreases in favor of the ag component of Vi ( 475 cm ) which on further compression gains more intensity (about a factor of 2 at 5 GPa) [120]. This behavior was explained by the anisotropy of the crystal s compressibihty [139] and differences in the components of the Raman tensor of the two modes [87] with respect to the crystal axes [109]. 

The flow velocity, pressure and dynamic viscosity are denoted u, p and fj and the symbol (...) represents an average over the fluid phase. Kim et al. used an extended Darcy equation to model the flow distribution in a micro channel cooling device [118]. In general, the permeability K has to be regarded as a tensor quantity accounting for the anisotropy of the medium. Furthermore, the description can be generalized to include heat transfer effects in porous media. More details on transport processes in porous media will be presented in Section 2.9. 

Here, ay is the thermal expansion coefficient which is assumed to be equal in both lattices and 6 is the dilatation coefficient (Khs — ls)/ ls> where Fls and Fhs are the unit cell volumes of the pure LS and HS species at 0 K, respectively. In order to account for the anisotropy of the lattice, the thermal expansion v and dilatation e coefficients must be introduced as tensors instead of scalars. Similarly, an equivalent expression could be defined for pressure-induced spin conversions. 

So far we considered phases with sufficient atomic mobihties and vanishing pressure anisotropies such that we could use the term -pdV to describe the mechanical energy increment (see also footnote 6). Generally, for elastic deformations (i.e. usually small deformations), this increment has to be expressed in terms of the stress tensor components Sy and the differential strain tensor components dey ... 

The end-to-end vectors of the subchains have distributions in their length and orientation. Equation [10] clearly indicates that the deviatoric part (measurable part) of the stress tensor due to the entropy elasticity of the polymer chains, hereafter referred to as the polymeric stress, reflects the orientational anisotropy of the subchains specified by the configuration tensor S(n,t). Consequently, the polymeric stress relaxes, even though the material keeps its distorted (e.g., sheared) shape, when the orientational anisotropy induced by tbe applied strain relaxes tbrougb tbe tbermal motion of tbe cbains. (In tbis relaxed state, S(n,t) is equal to 1/3 and tbe subcbain tension is transmitted isoUopically in aU tUrecrions to balance tbe isotropic pressure.) Thus, tbe relaxation time of the polymeric stress is identical to the orientational relaxation time of the polymer chains. 

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