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Triple Junction Motion for Faceted Interfaces

We give one example of the usefulness of the variational framework for cases where interfacial energy is discontinuous or non-differentiable. Consider a material composed of many grains, of perhaps differing phases, and surface motion that does NOT conserve the amount of material in a phase or grain. Let the total energy of a system to be the sum of the surface energies of the interfaces plus the bulk energies of the phases:


Gradient flow of surfaces with respect to the mobility-weighted tex2html_wrap_inline1016 inner product on the surface results in normal velocity


where M is a mobility function (probably depending on normal direction), tex2html_wrap_inline1352 the weighted mean curvature, and tex2html_wrap_inline1422 is the bulk energy per unit volume of the phase behind the interface minus the bulk energy per unit volume of the phase in front of the interface.

In two dimensions, force-balance uniquely determines equilibrium angles when all the Wulff shapes tex2html_wrap_inline1424 have no facets or corners, but otherwise there can be several different sets of angles. For example, in the isotropic equal-energy case triple junctions are at 120 tex2html_wrap_inline1426 angles, and fixing the angle of one interface (and the direction of the triple junction, in tex2html_wrap_inline1428 ) fixes the other two as well. But when all tex2html_wrap_inline1430 are the same regular octagon (Figure 4.1), there are three different equilibrium configurations when one segment is held fixed.

* Figure tex2html_wrap1897 . tex2html_wrap1891

The number of triple equilibrium triple junctions is even larger when tex2html_wrap_inline1432 are the same regular hexagon:

* Figure tex2html_wrap1898 . tex2html_wrap1892

It can be seen that several different configurations can all have the same surface energy while being the same away from that triple junction but different near it (Figure 4.3)

* Figure tex2html_wrap1899 . tex2html_wrap1893 tex2html_wrap_inline1434 .

In general, one can use the variational formulation to determine how such configurations move. The fundamental idea is that the minimum, for fixed tex2html_wrap_inline1436 , of


occurs at tex2html_wrap_inline1438 , by calculus. Also, the energy change tex2html_wrap_inline1440 in moving an initial surface S by the vector field v to a new surface tex2html_wrap_inline1446 is approximately tex2html_wrap_inline1448 .

So one can construct a discrete time step flow as follows. Starting with some initial configuration S(0) and a time step tex2html_wrap_inline1436 , set tex2html_wrap_inline1454 to be the minimizer tex2html_wrap_inline1456 of


Then find tex2html_wrap_inline1458 by minimizing starting from tex2html_wrap_inline1460 , etc. As the time step tex2html_wrap_inline1436 goes to zero, the flows should converge to an appropriate limit flow.

Consider surface energy functions for which the Wulff shape W is a polygon in the plane and grain boundaries which are polygonal curves with normal directions that are normals of W. This is a type of numerical discretization where it is the normal directions that are discretized. Two segments of the same interface can be adjacent only if their normal directions are also adjacent in W. And three segments from three different interfaces can meet at a triple junction only if the configuration is force-balanced.

Given a line segment tex2html_wrap_inline1468 in some interface tex2html_wrap_inline1470 in 2D, with normal tex2html_wrap_inline1472 and length tex2html_wrap_inline1474 , and with the segment preceding it having normal tex2html_wrap_inline1476 and that following it having normal tex2html_wrap_inline1478 , the energy change in moving segment tex2html_wrap_inline1468 in its normal direction a distance tex2html_wrap_inline1482 is computed to be tex2html_wrap_inline1484 , where the f are contributions that derive from the geometry of the corners as illustrated in Figure 4.4.


and tex2html_wrap_inline1488 [resp. tex2html_wrap_inline1490 ] is 1 if tex2html_wrap_inline1492 follows [resp., tex2html_wrap_inline1494 precedes] tex2html_wrap_inline1496 as a normal of W and is -1 if tex2html_wrap_inline1492 precedes [resp., tex2html_wrap_inline1494 follows] tex2html_wrap_inline1496 as a normal of tex2html_wrap_inline1432 .

* Figure tex2html_wrap1900 . tex2html_wrap1894 v. For this case, equation 21 can be written as tex2html_wrap_inline1755 . The interpretation of tex2html_wrap_inline1512 in equation 21 comes from the fact that tex2html_wrap_inline1759 and tex2html_wrap_inline1761 and whether the corner is convex with respect to the Wulff shape. Formulae similar to equation 21 written in terms of the trigonometric functions can be found in Gibbs [11] for the three-dimensional case.

In the absence of triple junctions, the quantity to be minimized in the time-stepping procedure outlined in equation 19 becomes


Minimizing this over the distances tex2html_wrap_inline1518 to move the segments tex2html_wrap_inline1520 results in


which is Euler's method for motion by weighted mean curvature plus a constant, since for polygonal Wulff shapes tex2html_wrap_inline1522 . Observe that the motion of each segment over this time step is decoupled from the rest: tex2html_wrap_inline1482 does not depend on any tex2html_wrap_inline1526 or tex2html_wrap_inline1528 for tex2html_wrap_inline1530 . (Of course, the motions of the segments change the lengths, so the lengths at the next time step are indeed dependent on the motions at the current time step of the neighboring segments.)

In the presence of triple junctions, one might just move a triple junction from its position tex2html_wrap_inline1532 at time tex2html_wrap_inline1534 to a new position tex2html_wrap_inline1536 . One can then write explicitly the energy as a function of x and y, and use the relationship tex2html_wrap_inline1542 ( tex2html_wrap_inline1544 , tex2html_wrap_inline1546 ) for each segment tex2html_wrap_inline1468 with an endpoint at the triple junction. If none of the tex2html_wrap_inline1468 coming into the triple junction has its other endpoint at another triple junction, then the motion of the three segments is decoupled from that of all the other segments but coupled to each other. If we assume that the tex2html_wrap_inline1468 at the triple junction have i= 1,2,3 and that the normals tex2html_wrap_inline1556 each point in a clockwise direction around the triple point (see Figure 4.5, then to determine x and y one minimizes


* Figure tex2html_wrap1901 . tex2html_wrap1895 tex2html_wrap_inline1562 , of a grain of type tex2html_wrap_inline1564 embedded within a grain of type tex2html_wrap_inline1566 , is the central inversion of tex2html_wrap_inline1568 .

Setting the partial derivatives with respect to x and y to zero gives the pair of simultaneous equations






The solution is therefore


where tex2html_wrap_inline1574 is clearly positive.

* Figure tex2html_wrap1902 . tex2html_wrap1896

Gradient flow, however, carries the obligation of considering all possible paths. An obvious alternative motion is to add a small segment between the endpoint of one of the original segments, say tex2html_wrap_inline1576 , and the new position of the triple junction. If the normal direction direction of such a segment is a normal of tex2html_wrap_inline1578 and is adjacent to tex2html_wrap_inline1580 in tex2html_wrap_inline1578 , then we can label that normal direction as tex2html_wrap_inline1584 and define tex2html_wrap_inline1586 as before. Now the energy is minimized with


as if tex2html_wrap_inline1576 had not been at a triple point, except that there is the constraint tex2html_wrap_inline1590 if tex2html_wrap_inline1592 or tex2html_wrap_inline1594 if tex2html_wrap_inline1596 . To determine x and y, one minimizes


Again, there is a unique solution for the values of x and y; if tex2html_wrap_inline1606 does not satisfy its constraint, then that says that there is no minimum with such a small segment being added.

Adding more than one segment to an interface cannot further decrease energy. Adding it with normal adjacent to tex2html_wrap_inline1584 but not equal to tex2html_wrap_inline1610 would result in the segment with normal tex2html_wrap_inline1584 having weighted curvature of order tex2html_wrap_inline1614 , thereby forcing it up against its constraint and eliminating the second small segment. Adding it with normal equal to tex2html_wrap_inline1610 would result in a configuration with the same energy and not affect the minimizing position of the triple junction.

One might also try to add new little segments to two of the interfaces. In this case, one finds that now the equations for x and y are dependent; either there are no minimizers within the constraints, or, provided the surface energy functions satisfy a linear relationship with each other, there might be a whole continuum of minimizers. The latter situation has probability zero of holding for surface energy functions corresponding to random grain orientations.

If one tries to add little segments to three of the interfaces, then there is no quadratic term in x and y and hence any minimum occurs at a constraint, proving that one cannot decrease energy by adding three such segments.

The net result is that if the surface energy functions for all the interfaces do not satisfy that particular relationship, then at most one of the six possible ways of adding a single small segment satisfies its constraint, and one cannot add more than one segment. If one can add one segment while satisfying its constraint, then that is the overall minimizer; if it doesn't, then the minimum is obtained by not adding any small segments. In either case, there results an explicit formula for x and y as well as for each of the tex2html_wrap_inline1482 .

In case the surface energy functions for all the interfaces do satisfy that particular relationship, then this variational formulation is inadequate to determine the triple junction position. It is necessary to replace the inner product, an integral over the surface, by integrals over the region between the surface and the comparison surface. This formulation is an extension of that of [12] and related to, but different from, that of [13], in which the mobility critically determines which of the equal-surface-energy minimizers should be used. (In [14] it was shown that the formulation of [12] with arbitrary comparison curves gives the same motion as that obtained by minimizing within the restricted class of comparisons having the same normal directions as those of the Wulff shapes, a result that one hopes will carry over to triple junction motion as formulated here.) In fact, this integrating-over-regions is also required in order to handle the case that all the surface energy functions are zero and only the bulk energy differences tex2html_wrap_inline1422 provide the driving forces (a special case of the particular relationship!), and it shows how and why the motions can be different for the assumptions of zero surface energy and vanishingly small surface energy. But that is a subject for another paper.

next up previous
Next: Discussion Up: Variational Methods for Microstructural Previous: Level Sets

W. Craig Carter
Tue Sep 30 16:07:27 EDT 1997