Wulff

Since the crystal shape, or habit, can be determined by kinetic and other nonequilibrium effects, an actud crystal may have faces that differ from those of the Wulff construction. For example, if a (100) plane is a stable or singular plane but by processing one produces a plane at a small angle to this, describable as an (xOO) plane, where x is a large number, the surface may decompose into a set of (100) steps and (010) risers [39]. 

Fig. Vn-2. Conformation for a hypothetical two-dimensional crystal, (a) (lO)-type planes only. For a crystal of 1 cm area, the total surface firee energy is 4 x lx 250 = 1000 eigs. (b) (ll)-type planes only. For a crystal of 1-cm area, the total surface free eneigy is 4 x 1 x 225 = 900 ergs, (c) For the shape given by the Wulff construction, the total surface free energy of a 1-cm crystal is (4 x 0.32 x 250) + (4 x 0.59 x 225) = 851 ergs, (d) Wulff construction considering only (10)- and (ll)-type planes.

Since an actual crystal will be polyhedral in shape and may well expose faces of different surface tension, the question is what value of y and of r should be used. As noted in connection with Fig. VII-2, the Wulff theorem states that 7,/r,- is invariant for all faces of an equilibrium crystal. In Fig. VII-2, rio is the... 

We noted in Section VII-2B that, given the set of surface tension values for various crystal planes, the Wulff theorem allowed the construction of fhe equilibrium or minimum firee energy shape. This concept may be applied in reverse small crystals will gradually take on their equilibrium shape upon annealing near their melting point and likewise, small air pockets in a crystal will form equilibrium-shaped voids. The latter phenomenon offers the possible advantage that adventitious contamination of the solid-air interface is less likely. 

The surface tensions for a certain cubic crystalline substance are 7100 = 160 ergs/cm, 7110 = 140 eigs/cm, and 7210 = 7120 = 140 ergs/cm. Make a Wulff construction and determine the equilibrium shape of the crystal in the xy plane. (If the plane of the paper is the xy plane, then all the ones given are perpendicular to the paper, and the Wulff plot reduces to a two-dimensional one. Also, 7100 = 7010. etc.)... 

WulffProcess. The regenerative technique is best exemplified by the Wulff process, Hcensed by Union Carbide Corp. The furnace consists basically of two masses of high purity alumina refractory tile having cylindrical channels for gas flow and separated by a central combustion space as shown in Figure 10. Its cychc operation has four distinct steps, each of approximately 1 min in duration, the sequence being pyrolysis and heat in one direction followed by pyrolysis and heat in the other direction. Continuity of output is achieved by paired installations. 

Fig. 10. (a) Wulff furnace design, (b) Checker detail of Wulff furnace refractory. 

The regenerative nature of the Wulff operation permits the recovery of most of the sensible heat in the cracked gas. The gases leave the furnace at temperatures below 425°C, thus obviating the need for special high temperature alloys in the switch valve and piping system. 

Acetylene Winning With Wulff Chem. Week, 89 (Apr. 16, 1966). 

Process Costs, Wulff Acetylene," Chem. Process. Eng. 47(2), 71 (1966). 

W. G. Moffatt, G. W. Pearsall, andj. Wulff, The Structure and Properties of Materials, Vol. 1, John Wiley Sons, Inc., New York, 1964. 

In 1964, two competing series of slender volumes appeared one, the Macmillan Series in Materials Science , came from Northwestern Morris Fine wrote a fine account of Phase Transformations in Comlen.ted Systems, accompanied by Marvin Wayman s Introduction to the Crystallography of Martensite Transformations and by Elementary Dislocation Theory, written by Johannes and Julia Weertman. The second series, edited at MIT by John Wulff, was entitled The Structure and Properties of Materials , and included slim volumes on Structure, Thermodynamics of Structure, Mechanical Behaviour and Electronic Properties. 

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