Wedge disclinations in cellular structures

Scripta METALLURGICA

Vol. ii, pp. 415-416, 1977 Printed in the United States

Pergamon Press,

Inc.

WEDGE DISCLINATIONS IN CELLULAR STRUCTURES J.E. MORRAL Department of Metallurgy and I n s t i t u t e of Materials Science U n i v e r s i t y of Connecticut, Storrs, Connecticut 06268

(Received March 14, 1977) The topology of c e l l u l a r s t r u c t u r e s , such as those found in soap forms, b i o l o g i c a l t i s s u e , and p o l y c r y s t a l l i n e materials, was recently described by Morral and Ashby ( I ) using a l a t t i c e graph construction. The l a t t i c e graph provided a convenient way to determine the number of sides, edges and corners of c e l l s near dislocations in an otherwise perfect array of c e l l s . It was also possible to f o l l o w how d i s l o c a t i o n movement changed these topological features. In the present communication the l a t t i c e graph is applied again to imperfect c e l l u l a r s t r u c t u r e s , t h i s time to characterize the topology of c e l l s near d i s c l i n a t i o n s . The geometry of wedge d i s c l i n a t i o n s have been described in a recent review by Harris (2). They can be constructed by making a cut in a m a t e r i a l , adding or removing wedges of perfect material, and then r e p a i r i n g the cut. In order to form a pure wedge d i s c l i n a t i o n in a l a t t i c e , i t is necessary that the d i s c l i n a t i o n core l i e s along an s - f o l d symmetry axis and that the wedge angle, ~, be a m u l t i p l e of 2~/s. Since the l a t t i c e graph f o r c e l l u l a r structures is cubic, one would expect d i s c l i n a t i o n s along the 4 - f o l d axes, ; 3 - f o l d axes, < I I I > ; and 2fold axes, . Before applying the l a t t i c e graph to d i s c l i n a t i o n s a few of i t s features w i l l be recalled. A perfect array of 14-sided c e l l s is represented on the l a t t i c e graph by an array of points which forms a body-centered cubic l a t t i c e . The f i r s t and second-neighboring l a t t i c e points are connected by l i n e s ; the fourteen such l i n e s correspond to the fourteen sides of each perfect c e l l . By convention the f i r s t - n e i g h b o r l i n e s are drawn as s o l i d l i n e s and the second-neighbor ones as dashed l i n e s . The l i n e s divide the l a t t i c e graph i n t o tetrahedra which represent cell corners (four c e l l j u n c t i o n s ) and i t s t r i a n g u l a r sides represent c e l l edges (three c e l l junctions). Figure 1 i l l u s t r a t e s how a wedge d i s c l i n a t i o n along a [ I 0 0 ] axis appears on the l a t t i c e graph. Each open c i r c l e on the graph represent a c e l l l y i n g on a (200) and the closed c i r c l e s represent c e l l s on the next (200) below. Each of the open c i r c l e s away from the d i s c l i n a t i o n core is seen to be connected to four second-neighbor c e l l s in i t s own plane and four f i r s t neighbor c e l l s in the plane below. Not shown on the graph is that i t is also connected to four f i r s t - n e i g h b o r c e l l s in the plane above and to one second-neighbor c e l l d i r e c t l y above and one d i r e c t l y below i t , the t o t a l being fourteen l i n e s which represents the fourteen sides of a perfect c e l l . However, at the d i s c l i n a t i o n core shown in Figure l ( a ) where a x/2 wedge has been added (note: This is a wedge d i s c l i n a t i o n of tyDe, m = - ~/2), the c e l l has an addit i o n a l second-neighbor in i t s own plane and an additional f i r s t - n e i g h b o r in both the plane above and the one below. These three extra near-neighbors correspond to three additional sides. Therefore, the d i s c l i n a t i o n core is seen to consist of a row of seventeen-sided c e l l s . I f a 7/2 wedge is removed (~ = x / 2 ) , three near-neighbors are l o s t and the d i s c l i n a t i o n core is a row of eleven sided c e l l s , Figure l ( b ) . This r e s u l t can be generalized f o r perfect d i s c l i n a t i o n s by describing them as a row of p-sided c e l l s along a f o r which p = 14 - 6m/~

(I)

Equation ( I ) applies to < I I I > wedge d i s c l i n a t i o n s as well and t h i s is i l l u s t r a t e d in Figure 2. Here the open c i r c l e s represent c e l l s on a (222), closed c i r c l e s represent c e l l s on the f i r s t (222) below, and the open c i r c l e s containing an X represent c e l l s on the second (222) below. There are no near-neighbors in the plane of the c e l l but there are three f i r s t - n e i g h b o r s in the plane below and three second-neighbors in the second plane below. There is also another f i r s t - n e i g h b o r d i r e c t l y below each c e l l , which is not shown on the graph. These seven neighbors below the plane of each c e l l are repeated above the plane f o r a t o t a l of fourteen near-neighbors or fourteen sides f o r each perfect c e l l . By adding or removing a wedge of angle 2x/3, Figures 415

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WEDGE DISCLINATIONS

Vol.

Ii, No.

S

2(a) and {b), the number of near-neighbors is seen to change by four at the disclination core, which is the number predicted by equation (1). Equation (1) can be deduced somewhat i n t u i t i v e l y in the following way. I t applies to disclinations which form along near-neighbor directions. For this reason, the two near-neighbor lines intersecting cells along the disclination core are not affected when wedges are added or removed. However, the remaining twelve near-neighbor lines are increased or decreased in proportion to the fractional wedge angle, m/2~. Hence, the fourteen sides of a perfect cell are increased or decreased by 12 x m/2x or 6m/s. For perfect disclinations which form along directions which do not contain near-neighbors, such as the , then a l l fourteen sides are affected when wedges are added or removed and equation (2) applies: p' = 14 (l - ~/2~)

(2)

In summary, perfect wedge disclinations in cellular structures appear as rows of cells with other than fourteen sides that l i e along symmetry axes of the structure, the number of sides gained or lost being proportional to the wedge angle. Adjacent cells retain the fourteen sides found in perfect arrays of cells. As in the case of dislocations, the topology of these defects can be characterized on a l a t t i c e graph from which the number of cell sides, edges, and corners can be deduced. References I. 2.

J . E . Morral and M. F. Ashby, Acta Met. 22, 567 (1974). W.F. Harris, Surf. Defect Prop. Solids, 3, 57 (1974).

'Y I/

!

(a)

(b)

FIG. l wedge disclination in a BCC lattice-graph (a) m = - 7/2, (b) m = 7/2. The number of nearneighbor (corresponding to cell faces) for sites at the disclination core are indicated on each figure.

_

-

/

%

-1

-

,

(a)

,

(b)

FIG. 2 wedge disclination in a BCC l a t t i c e graph. (a) w = -27/3, (b) m = 27/3.