Energy per unit length

The line can only behave in this way by being flexible. It has an energy per unit length (line tension). This energy per unit length has two parts ... 

Thus, the quantum of circulation of an individual quark vortex is Mfi/fi, and the vortex velocity vs = 3(1/2fir, where r is the radius of a circular contour. Then we obtain for the energy per unit length of an isolated vortex and for the critical velocity of the vortex appearance the following expressions ... 

For the calculation of the step free energy we consider first the partition functions G, and G, of straight segments of steps along the x and y direction. The segments have energies per unit length equal to 6 and 8y. In Fourier space one has ... 

Incoherent Nucleation. Consider first incoherent nucleation on dislocations [19]. For linearly elastic isotropic materials, the energy per unit length Ei inside a cylinder of radius r having a dislocation at its center is given by... 

The vibrational energy per unit length along the string, or wave energy density [Morse, 1981] is given by the sum of potential and kinetic energy densities ... 

Dimensionally intermediate between spheres and planes, the interactions of cylinders reveal properties possessed by neither. In particular, there is always torque as well as force the energy of interaction depends not only on separation zbut also on mutual angle 0. For parallel cylinders of length indefinite compared with thickness and separation, the interaction is expressed as force or energy per unit length. As with spheres, there are few exact expressions for cylinder-cylinder van der Waals forces. The many approximate expressions must be used circumspectly (see Fig. LI.50). 

In the opposite limit, cylinders near to touching, the interaction energy per unit length goes to an inverse 3/2-power dependence on surface-to-surface separation / (z -> Ri + R2) (see Table C.3.b.3) ... 

Here g(l, 9 = 0) is an energy per unit length each rod in A, say, interacts with all rods in B. Think of the second derivative as the interaction between two infinitesimally thin slabs at separation Z. If we say that these slabs are parallel to the x, y plane and that the parallel rods all point in the direction x, then we must integrate over direction y parallel in order to collect all interactions between a rod in slab A with rods in the apposing slab B. 

Problem Pr.4 Try something harder than spheres. Consider parallel cylinders of radius R, fixed length L, and surface separation Z. Use the tabulated energy per unit length... 

TABLE 1 Perfect and partial dislocations in the hexagonal crystal structure and their energy per unit length which is proportional to I b 12. This assumes a perfect hexagonal lattice where c/a = (8/3)1/2. 

Figure 1.20. Reduced free energy per unit length of triple line of a meniscus on the solid surface shown in Figures 1.18 and 1.19 plotted as a function of the macroscopic contact angle. The physical parameters used in the calculation are p = 7 x 103kg.m 3,<7LV = 1 J.m 2,0Y = 40°, g = 9.81 m.s 2, fi = 10° and Lf = 100 /im. Results from (Eustathopoulos and Chatain 1990) [1],...

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