Institute of Metals Division - Measurement of Relative Interface Energies in Twin Related Crystals

IN recent papers on interface energies in metals¹,² the concept of an equilibrium of forces has been used in the measurement of interfacial tensions. Mathematically the equilibrium of three forces is given by the relations: where the 2's are the interfacial tensions (interface energies per unit area of interface) of three phases or grains meeting at angles 21 22 and 23 (212 relates to the 1-2 boundary formed by grains 1 and 2;223 to the 2-3 boundary, etc.). If these relationships are valid, one can calculate relative values of the interface energies from measured boundary angles on specimens which have been brought to equilibrium. Since Eq 1 may be derived through a mathematical process of minimizing the grain boundary energy under the restriction that the ?'s be independent of boundary orientation: the concepts of surface tensions and equilibrium of forces are not particularly essential. To deal with the general case involving dependence of boundary energy on boundary orientation, it becomes desirable to drop certain concepts formerly held (namely that energy per unit area is identical with force per unit length or surface tension) and to use the energy concept alone. It is appropriate, therefore, to use E in place of ? in deriving and using the more general equations that express the anisotropic nature of the grain boundary energy. Herring3# ' recently has derived the required general equations using the principle of minimization of energy. Specifying the boundary orientations in terms of angles (4) measured with reference to crystal axes within the grains, Herring's equations take the form E23 + E21 cos?8 + E12 cos?2 — sin ?12 + sin?2-?E12/?12 =O [2] where the partial derivatives are measured in the direction of counterclockwise rotation of the boundaries. Eq 2 implies that an infinitesimal displacement of the triple point along the 2-3 boundary produces no change in the boundary energy. Similar equations apply for infinitesimal displacements along each of the other two boundaries. From such equations Herring obtains the following general relationship (1 + E1 E2) sin?8 + (E2 — E1) cos?3 ________ E38______________ = (1+ E2 E8) sin ?1+ (E3— E2) cos ?1 __________^5____________[3] (1 + E sin ?2, + (E1 — E8,) COS ?2 1 E28 where E1 = 1/E23- E25 and PC and c8 have similar expressions. It is apparent that Eq 3 reduces to Eq 1 when the c's vanish. Eq 3 also reduces to Eq 1 when the E'S are all equal. Shockley and Read' have pointed out one limitation to these equations. When E plotted versus 4 appears as an energy cusp, as their theory requires, for example, for the energy near a twin boundary position, then Eq 3 (and therefore Eq 1 also) no longer applies if one boundary is in the minimum energy position. One of the equations of form Eq 2, however, applies if the infinitesimal displacement is along the boundary associated with the energy cusp. From the foregoing treatment it will be apparent that equilibrium angles alone are not sufficient in general for calculating relative interface energies.