Calculated surface stress

Fig. 8 Left schematic of the model to explain the negative cantilever inchnation, and range of action of the forces involved. Right experimental evaporation curve (circles), calculated curve assuming CCR evaporation (solid line), and calculated surface stress (dashed line)...

Surface area, 0, is determined empirically from an auxiliary study of surface area versus shrinkage. The net unit surface energy, y, is assumed to be essentially constant with a value on the order of 10 ergs/cm. The calculated surface stress, / y0, is added to the effective applied stress, to obtain a total stress which takes into account the significant driving forces due to remnant surface area. 

Table 8. Calculated surface stress on substitutional and adsorbate covered Si and Ge surfaces. Calculations were performed for T = 0 K.

Fig. 12. Calculated surface stress vs. H coverage on Pt(lll) (coverage 1.0 1.50x10 atoms/cm ). Data from [97Fei].

Calculated Surface Energies and Surface Stresses (ergs/cm )... 

Surface Stresses and Edge Energies. Some surface tension values, that is, values of the surface stress t, are included in Table VII-2. These are obtained by applying Eq. Vll-5 to the appropriate lattice sums. The calculation is very sensitive to the form of the lattice potential. Earlier calculations have given widely different results, including negative r [43, 51, 52]. 

Strained set of lattice parameters and calculating the stress from the peak shifts, taking into account the angle of the detected sets of planes relative to the surface (see discussion above). If the assumed unstrained lattice parameters are incorrect not all peaks will give the same values. It should be borne in mind that, because of stoichiometry or impurity effects, modified surface films often have unstrained lattice parameters that are different from the same materials in the bulk form. In addition, thin film mechanical properties (Young s modulus and Poisson ratio) can differ from those of bulk materials. Where pronounced texture and stress are present simultaneously analysis can be particularly difficult. 

The results presumably were distorted by the unavoidable heating of the microcrystals occurring in the electron beam and other side effects. However, if the A a detected in the (111) reflexions is real, then the surface stress or rather surface tension 7S causing the compression of the lattice can be calculated. If R i and R2 in Eq. (37) are supposed to be identical and equal to 0.5 D, see above, then Pc= 4 ys/D. Let k be the compressibility of the micro-crystal, i.e., -dV/V.dp. Then the relative compaction under the pressure, that is -dV/V, is 4 ysk/D. The ratio dV/V may be approximated as 3 A aja. Hence... 

DFT calculations were used to quantify the different interactions. All interactions were found to be repulsive. The most repulsive interactions were along the copper rows (in the [lIO] direction), due to a through-surface interaction between carboxylate groups of different tartaric acid molecules binding next to each other. In addition to these interactions we have proposed the existence of an adsorbate-induced surface stress which reduces the binding energy when more than three tartaric acid molecules bind to the same copper row. This surface stress causes the empty troughs in the (9 0,1 2) ordered structure. ... 

The second method is to calculate a stress that is appropriate to a particular situation of interest. An example of this would be the stress acting on a drop of material due to its own weight as it rests on a support medium. The force of gravity tends to make the drop spread out into a film, while its surface tension tends... 

Table 8.3 Calculated surface tensions 7 and surface stresses T of ionic crystals for different surface orientations compared to experimental results [325,327]. All values are given in mN/m. (a) from cleavage experiments, (b) extrapolated from liquid.

The as-grown cells are usually extremely warped toward the cathode side at room temperature. To reduce the warp, the cells are flattened at 1400°C with a distributed load on the surface. The flattening treatment could reduce the warp at room temperature indicating that the sample is in a plastic state at 1400°C. In contrast, it was reported that the sintering of YSZ does not proceed well below 1250°C [25], Thus, it is considered that the temperature at which both the electrolyte and anode are constrained is between 1250°C to 1400°C. When the temperature at which the electrolyte and anode are constrained is assumed to 1400°C, the calculated residual stress is close to the measured value. 

For the calculation of the thermal shock-induced stresses, we consider the plate shown in Fig. 15.1 with Young s modulus E, Poisson s ratio v, and coefficient of thermal expansion (CTE) a, initially held at temperature /j. If the top and bottom surfaces of the plate come into sudden contact with a medium of lower temperature T they will cool and try to contract. However, the inner part of the plate initially remains at a higher temperature, which hinders the contraction of the outer surfaces, giving rise to tensile surface stresses balanced by a distribution of compressive stresses at the interior. By contrast, if the surfaces come into contact with a medium of higher temperature Tm, they will try to expand. As the interior will be at a lower temperature, it will constrain the expansion of the surfaces, thus giving rise to compressive surface stresses balanced by a distribution of tensile stresses at the interior. 

Mathematically, this problem bears some resemblance to those considered above. The governing partial differential equation is still Eq. (6), and on the surfaces boundary conditions of constant potential, constant charge density or linear regulation [i.e., Eq. (45)] must be imposed. However, a further constraint arises from the need to satisfy mechanical equilibrium at the interface, and it is this new condition that provides the mathematical relation needed to calculate the interface shape. The equation is the normal component of the surface stress balance, and it is given by [12]... 

Fig. 11 Calculated surface profiles of the octahedral shear stress at yield assuming a modified Von Mises criterion (a), and of the octahedral shear stress for a glass/epoxy contact under gross sliding condition (b). The grey area delimits the region at the leading edge of the contact where the octahedral shear stress is exceeding the limit octahedral shear stress at yield (a is the radius of the contact area) (from [97])...

E is the modulus of elasticity, d is the thickness of the plate, r is the radius of curvature, and v is the Poisson ratio (typically 0.3). For a discussion of Eq. (40) see Refs. 137 and 138. Spontaneous bending was used to calculate the difference in surface stress between opposite faces of several crystals, including InSb, GaAs, InAs. GaSb. and AlSb [134,139] and aluminum nitride [140]. 

One way to calculate the stresses is to imagine that the cellular contents are removed, leaving only the cell wall, which has a uniform hydrostatic pressure acting perpendicular to its inside surface. The projection of this P over the appropriate area gives the force acting in a certain direction, and the reaction to this force is an equal force in the opposite direction in the cell wall. By dividing the force in the cell wall by the area over which it occurs, we can determine the cell wall stress. 

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