Characteristic lengths surface extrapolation

If one expected to sinq)ly extrapolate the properties of nanostructures from the size scales above or below, then there would be little reason for the current interest in nanoscience/nanotechnology. There are three reasons for nanostructured materials to behave very differently large surface/interface to volume ratios, size effects (where cooperative phenomena like ferronmgnetism is con romised by the limited number of atoms/molecules) and quantum effects. Many of the models for nmterials properties at the micron and larger sizes have characteristic length scales of nanometers (see Table II). When the size of the structure is nanometer, diose paran ters will no longer be adequate to model/predict the property. One can expect surprises - new materials behavior that may be technologically exploitable. 

Boundary conditions in Eq. (3.11) include characteristic length si and surface polarization P, induced by misfit tension between film and substrate, as well as by piezoelectric effect, that exists even in cubic lattices near the surface, where there is no inversion symmetry in the direction normal to surface. Note, that at P, = 0 the characteristic length X.5, coincides with extrapolation length (see e.g. Ref. [13]). It follows from Eq. (3.12), that s can be both positive and negative depending on the signs and values of the parameters 0i2, 12, r = p.(5n -F 512) (see Ref. [11] for details). 

The observed dependence of the range of repulsion on the number of spread particles can be attributed to the previously mentioned surface pressure gradient along the layer that manifests itself in a gradient of the surface concentration of the particles. Hence, the characteristic lengths of repulsion could be determined by extrapolating the 77l values in Fig. 9 to zero particles. 

Indeed, the shear stress at the solid surface is txz=T (S 8z)z=q (where T (, is the melt viscosity and (8USz)z=0 the shear rate at the interface). If there is a finite slip velocity Vs at the interface, the shear stress at the solid surface can also be evaluated as txz=P Fs, where 3 is the friction coefficient between the fluid molecules in contact with the surface and the solid surface [139]. Introducing the extrapolation length b of the velocity profile to zero (b=Vs/(8vy8z)z=0, see Fig. 18), one obtains (3=r bA). Thus, any determination of b will yield (3, the friction coefficient between the surface and the fluid. This friction coefficient is a crucial characteristics of the interface it is obviously directly related to the molecular interactions between the fluid and the solid surface, and it connects these interactions at the molecular level to the rheological properties of the system. 

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