Singular surfaces

If one is interested in the configuration of a singular surface or a vicinal surface close to the singular orientation, the surface is better characterized by the local height hj. The description assumes that there should be no vacancy in the crystal, no floating-solid atoms in the ambient phase, nor... 

Whitaker, S, The Species Mass Jump Condition at a Singular Surface, Chemical Engineering Science 47, 1677, 1992. 

SMOOTHING OF A GROOVED SINGULAR SURFACE WHOSE NEIGHBORING ORIENTATIONS ARE UNSTABLE... 

Proof. In order to avoid cumbersome evaluation of the potential ip near the singular surface r = R, we employ the following standard construction. Recall the identity (see, e.g., [21]) ... 

A crystal growing from the vapor phase possesses a singular surface with two screw dislocations intersecting it. The dislocations are very close to one another (relative to the dimensions of the surface) and have opposite Burgers vectors. Describe the form of the step structure that is produced because of the presence of the two dislocations. 

As surface corrugations grow, they will eventually touch each other and then folded parts will form on the surface. If the gel is sufficiently soft, the free energy can be lowered below the value in the homogeneous state by formation of a periodic pattern [20,21,89], The folded parts are singular surfaces like cracks and we encounter great mathematical difficulty. Nevertheless, we may analytically calculate the folded patterns in the highly compressible case 1... 

Based on the TLK (terrace, ledge and kink) model, growth on a perfect vicinal and singular surface proceed by a sequence of steps involving adsorption from vapor to form a surface adatom. This adatom diffuses to a kink site of the surface and incorporate into the crystal at the kink site. In order to determine the growth rate, the rate of formation of stable cluster must be determined. This is the rate at which cluster of radius r grow by the addition of one incremental atom from the adlayer. The rate is given by. 

Fig. 23. The relative surface tension difference -Ay/y(T) between components of six poly-olefinic blends grouped in three microstructurally identical pairs Xj/x2 (x1>x2) 66/52 (Q, ), 86/75 (A, ) and 75/66 (V, Y). Open and solid symbols correspond to blends with deu-terated more (x,) and less (x2) branched components, respectively. Large and small symbols correspond to previously determined whole segregation isotherms and singular surface excess data, respectively [16,120,145]. y(T) is given by Eq. (45) for polyethylene. Solid lines denote average values for each blend at Tref=100 °C (thick bar)...

Whitaker S (1985) A Simple Geometrical Derivation of the Spatial Averaging Theorem. Chemical Engineering Education, pp. 18-21 and pp. 50-52 Whitaker S (1992) The species mass jump condition at a singular surface. Chem Eng Sci 47(7) 1677-1685... 

In this section the classical heat and mass transfer theories are examined. The singular surface jump conditions for the primitive quantities, as derived in the framework of the standard averaging procedures, are approximated by the classical chemical engineering stagnant film theory normally used in chemical reactor models. The relevant transport phenomena solutions and the classical theories on heat and mass transfer considering both low- and high mass transfer rates are summarized in the subsequent subsections. 

Whitaker S (1992) The species mass jump condition at a singular surface. Chem Eng Sci 47(7) 1677-1685... 

As the temperature of a surface is raised above the roughening temperature Tr, its stiffness is reduced and it no longer appears as a facet in the equilibrium shape. On the microscopic level, this corresponds to a shift from the low entropy, low energy singular surface toward the high entropy surface populated by islands and adatoms.18-20 At ambient temperature, low-index copper surfaces are below the roughening temperature in vacuum. However, in... 

Vol. 73 K Hayami A Projection Transformation Method for Nearly Singular Surface Boundary Element Integrals X, 456 pages. 1992... 

Let f2(0 be a region in with the piecewise smooth boundary 3i2(0 depending on time t, and let S(t) C Q(0 be a singular surface across which the volume density of a physical quantity, y, experiences the jump discontinuity [y] = y —y. The superscripts + and — indicate the different sides of S f). Further, assume that S f) is moving with the displacement velocity w, and that is the unit normal vector on S. 

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