48 INFLUENCE OF DISLOCATIONS ON CRYSTAL GROWTH THE INFLUENCE OF DISLOCATIONS ON CRYSTAL GROWTH BY F. C. FRANK Received 16th February, 1949 The kinetic theory of the nucleation of new phases, developed especially by Volmer,l by Farkas,2 by Kaischew and Stranski3 and by Becker and DOring4 indicates that under typical conditions * the self-nucleation from vapour of new crystals, new liquid drops and fresh two-dimensional mono- layers of molecules on a “ saturated ” crystal face require respectively supersaturations of the vapour by factors of typically 10, 5 and 1.5 respec- tively in order to proceed at appreciable rates. Experimentally, the first two of these figures are apparently correct : but the third is much larger than the actual supersaturation required to cause further growth of a crystal already formed.In fact, the existence of a critical finite supersaturation for further growth has only been established for a few materials, and then for individual faces of individual crystals, being different from case to case ; at the most it is about I yo. Volmer and Sch~ltze,~ who found a critical supersaturation of 0.8 yo for the growth of an iodine crystal from the vapour, interpreted this as the critical supersaturation for two-dimensional nuclea- tion : but the quantitative discrepancy is far too great (for details of the growth rate formulzJ see the contributions of Burton and Cabrera to this Discussion). However, this discrepancy is not in the least surprising. One ought not to expect that any visible crystal will exhibit a completed perfect face needing fresh two-dimensional nucleation in order to grow.Investigation of the mechanical properties of solids shows that no macroscopic specimen ever exhibits the theoretical strength of the perfect crystal; and this enormous discrepancy (a factor of 100, say, and more for “good” 1 Volmer and Weber, 2. physik. Chem., 1926, 1x9, 277. Volmer, Kinetik dev Phasen- 2 Farkas, 2. physik. Chem., 1927, 125, 236. 3 Kaischew and Stranski, 2. physik. Chem. B, 1934, 26, 317 ; Physik. Z . , 193 j, 36, 393. 4 Becker and Doring, Ann. Physik, 1935, 24, 719. bildung (Leipzig, 1939). Volmer and Schultze, 2. physik. Chem. A, 1931, 156, I. * Typical conditions may be taken as such that the vapour pressure lies within a few powers of 10 of I mm. Hg : or the temperature between about 0.5 and 0.S times Th, the boiling point in “K.F.C. FRANK 49 crystals) is attributed to the presence of dislocations. In the early stages of the development of dislocation theory by Polanyi,6 Orowan and Taylor,8 only one aspect of the dislocation was recognized, in which the displacement direction was normal to the dislocation line. In 1939 Burgers drew attention to the ‘ I screw ’’ form assumed by the dislocation when the displacement is parallel to the dislocation line, and the developments of dislocation theory by Mott and Nabarro,lo Frank l1 and others emphasize the fact that disloca- tion lines can be curved and exhibit any orientation. Fig. I shows (a) as a continuous defonnation of a plane, (b) in a block model, the form of a simple cubic crystal when a screw dislocation emerges normally at the FIG.r.-The end of a screw dislocation. cube-face. I t is clear that when dislocations of this type are present, the crystal face always has exposed molecular terraces on which growth can continue, and the need for fresh two-dimensional nucleation never arises. If just one dislocation of this type emerges at the centre of the face, that crystal face can grow perpetually “ up a spiral staircase.” If there are two, respectively right- and left-handed, we can show that the terrace connecting them will grow indefinitely if the supersaturation is raised to such a value that the diameter, Zo, of the critical two-dimensional nucleus is less than the distance between them. More precisely, since the critical nucleus is not 6 Polanyi, 2.Physik, 1934, 89, 660. 7 Orowan, 2. Physik, 1934, 8g, 634. 8 Taylor, Proc. Roy. SOC. A , 1934, 145, 362. Burgers, Proc. Kon. Ned. Akad. Wet., 1939, 42, 293. 10 Mott and Nabarro, The Strength of SoZids (Physical Society, London), 1948, p. I , 11 Frank, ibid., p. 46.50 INFLUENCE OF DISLOCATIONS ON CRYSTAL GROWTH circular, we should say that growth occurs when the critical nucleus, correctly oriented, will pass between two points in the positions of the two dislocations. Likewise when a single dislocation is close to the boundary of the face, growth occurs if the critical nucleus will pass between the dislocation and the boundary. If it will not pass, the terrace rests in a position corresponding to a portion of the boundary of the critical nucleus (Fig.2). For rough calculations we may approximate the boundary as a circle of radius iZo, and we have a close analogy to the equilibrium (or non-equilibrium) of a bubble at an orifice. Measuring in molecular spacings, we have lo = cp/kT In a, where 0: is the saturation ratio : since the supersaturation is small, (a - I) may be written in place of In a. (p is the neighbour-neighbour binding energy of the crystal. For a rule of thumb we may use Trouton’s rule and take y / k T as 3-5 Tb/T. Thus at an absolute temperature of 0.6 of the boiling point of the material, and a supersaturation of I %, I, is about 600 molecular spacings : or 6000 at 0.1 74. In the theory of mechanical defonna- tion, it is commonly estimated that in an annealed metal crystal there are about 1oS dislocation lines intersecting each square centimetre.This appears to be the limiting perfection attainable in metals, and would be classed, in- deed, as high perfection from the point of view of X-ray diffraction. (This limit may well be connected with the handed screw dislocations ending in order-of-magnitude estimate, and does a crystal face. not necessarily apply to other mate- rials than metals: but it serves as a guide, and suggests that critical supersaturations of the order of a fraction of I yo are reasonably to be expected. We are perforce limited for quantitative discussion to the case of growth from the vapour of homopolar crystals, since the classical nucleation rate theory has not been quantitatively developed for any other case.But experimental indications, and such theoretical guesswork as we can make, suggest that the conditions governing the growth from solution of crystals, including ionic crystals, are substantially similar. It is possible, but less certain, that the same is true of growth from the melt. Perhaps the most distinctively different case is that of the growth, either from vapour or solution, of crystals of highly non-equiaxed organic molecules. There are indications that in such cases growth proceeds through the formation of adsorption films, dense, but differing in molecular orientation from the bulk crystal (e.g., liquid, or liquid-crystalline) within which subsequent rearrange- ment occurs. The general importance of dislocations for crystal growth accounts immediately for many observations, such as the individual behaviour of each crystal face, particularly on the microscopic scale, leading sometimes to such unexpected results as the formation of lath-shaped crystals for a lattice of cubic symmetry.Under steady uniform supersaturation the terrace attached to an isolated growth-promoting dislocation in a crystal otherwise perfect in its neighbour- hood will grow outwards in a spiral of which the spacing between turns, and the rate of their advance, will be uniform at a considerable distance from FIG. 2.-One right-handed and two left- impurity ‘Ontent-) This is Only anF. C. FRANK 51 the centre. Near the centre the rate of advance must be less, since the radius of curvature of the spiral terrace-line must remain less than +Zo.At given supersaturation and distance from other terrace-lines, the rate of advance, o, is an increasing function of the radius of curvature, being o when it is +?, and, say, v, when the terrace becomes straight. Supposing it increased very steeply to voo for a very small increase of the radius of curvature above +Zc, the inner portion of the spiral would be an arc of a circle of this radius, and the spiral would make vm/xZ, turns per second. The spacing of turns in the outer region of the spiral would then be nZ,. Actually v will not increase infinitely steeply to vm, so that the number of turns per second will be less, and the spacing greater, but still of the same order of magnitude : let us guess zxZo. Macroscopically this spirally terraced hill would appear to be a flat cone, with its sides inclined at an angle of $do radians to the true lattice surface-say, I min.for a supersaturation of I yo, and in other cases pro- FIG. 3.-The influence of random dislocations on crystal growth. portional to the supersaturation. We have so far neglected the dependence of the rate of advance of the terrace on its direction. On this account, instead of a flat cone, we shall have a more or less sharply defined flat pyramid having the symmetry appropriate to the crystal face. When growth spreads from an isolated pair of dislocations, respectively right- and left-handed, the observable result will be practically the same provided that the supersaturation suffices to make Z, less than their separa- tion. If it is not, no growth occurs at all.If it is, the terrace-line connecting them repeatedly spreads out on one side, wraps round, and meets itself on the other side : thus forming a short connecting terrace-line again, and an outward-growing closed loop. The rate of formation of such loops will be the same in order of magnitude as the rate of formation of turns of the spiral from a single dislocation. The result will be a flat cone or pyramid indis- tinguishable from the previous case. In the case that there is a random distribution of dislocations a variety of phenomena occur, exhibited pictorially in Fig. 3 ( I to 12). In these pictures there is supposed to be positive supersaturation, so that each terrace makes a curve convex on its cliff side, running from one dislocation t o another of opposite sense, or to the boundary of the crystal face.The supersaturation is supposed to be low, and fluctuating, so that in general52 INFLUENCE OF DISLOCATIONS ON CRYSTAL GROWTH only one terrace-line-normally the longest-moves at a time. In each picture the terrace due to move next is specially marked. Shading indicates the area covered by fresh growth after picture 1. Among points to be specially noticed are : (i) the holding up of one terrace-line behind another (pictures 2, 8, 9, 10) or at a dislocation connected to another terrace (9, 10, 11, 12) ; (ii) the way in which a close pair of dislocations connecting a terrace facing the advancing terrace break it and so impede its passage (many examples, especially pictures 4, 5, 6 ) . In these two ways, a dislocation pair holding a terrace facing either way is an obstruction to the passage of an advancing terrace ; (iii) the way in which the obstructions of type (i) are broken down (10, 11, a) ; .(iv) the relatively impregnable region in the bottom left-hand corner. In 6 growth into this region has ceased by obstruction of type (ii) : a new advancing terrace is held up again (at 9) by obstruction of type (i) : but this obstruction is more easily broken down, and by 12 all but a small portion of the crystal face has increased in thickness by at least one mono- layer. A third advancing terrace, now commencing, will finally overrun this strong-point : however, a continuous line of closely spaced dislocations could be totally impenetrable below a critical supersaturation. Obstruction of type (i) requires a little more consideration.One way in which type (i) obstruction can break down is the following. Each " pinned " terrace-line holds its equilibrium form by small statistical fluctuations back and forth. When two terraces lie together, the lower cannot fluctuate back under, nor the upper fluctuate forward over the other. Thus they exert a small effective repulsion on each other, and the lower one may be pushed beyond the critical curvature when the upper one arrives. Secondly, there is only a small region of seriously deformed crystal structure and hence of seriously reduced binding energy for new molecules, in the immediate neigh- bourhood of the dislocation. When the two terraces at such a point of obstruction face each other at an acute angle, as in pictures 9, 10, 11, only a small number of molecules need condense in these unfavourable positions to enable the terrace to link across and pass on.If the obstruction arises from a straight row of dislocations spaced I , apart, this angle becomes smaller as I, decreases, diminishing rapidly from 120' when I, becomes less than 21,. The obstruction will be ineffective when E, is significantly smaller than this value. Even when it is greater, a comparatively small number of mole- cules (say, 6 to 10) in unfavourable positions suffice to overcome the obstruc- tion, so that statistical fluctuations (negligible with regard to obstruction of type (ii)) can be effective. Thus obstruction of type (i) probably only imposes a delay rather than a total prohibition of growth.In view of this, one might suppose it a particularly important question whether the numbers of right-handed and left-handed dislocation-ends in a face are equal or not : the surplus of one kind might have very long terrace- lines linking them to the boundary. This is not the case for reasonably uniformly or randomly distributed dislocations, unless the relative numbers are very different. If, in an area A , n dislocations of one kind and m of another are formed randomly, like head and tail throws of a coin, the probable excess of one kind, n - m, is 1-35 (n + m)'/z 1.9 nl/a : but if it does not exceed m z ' l ~ (where n > m) the dislocations linked to the boundary can all be close to it, and the longest necessary terrace inappreciably longer than in the case n = m, i.e., still of the order of magnitude (A/n)*!a.In a systematically deformed crystal it is possible to have n > m, and then there is always a terrace-line at least as long as the (m + r)th furthest dislocation of the n from the boundary.F. C . FRANK 53 The most important aspect of the type (i) obstruction by a fence of dis- locations all of the same kind is that every boundary between two crystal blocks inclined at a small angle to each other constitutes such a fence (see Fig. 4, and Fig. IZ in ref. 9). The distance between the dislocations in the fence, measured in molecular spacings, is equal to the reciprocal angle of rotation between the two blocks, whether for " tilt," " twist " or more general boundaries, so long as the angle is small.A possible cause of the visible growth terraces sometimes seen advancing over a crystal face is that a number of molecular terraces have accumulated behind such a fence, and then been set free by a rise in supersaturation: but it is possible to think of alternative causes which can bring about the same " bunching " of molecular growth terraces into visible ones. I FIG. 4.-'' Twist " boundary- crossed grid of screw dislocations. I t remains to elaborate the concept of dislocations somewhat, to free it from the simplifications introduced by considering only simple cubic crystals, with cube faces. In the general case the important property of a dislocation is its displacement vector (or Burgers vector). If this vector has a component nonnal to the crystal face on which the dislocation line ends, there will be an associated molecular terrace in the face, promoting crystal growth.But there is also an important distinction between perfect dislocations, whose displacement vectors are lattice vectors, and imperfect dislocations whose displacement vectors lead in general from a lattice position to a twin-lattice I I 1: position. Such is the dislocation with displacement vector (-3, -3, -j), produced in the close-packed cubic crystal by omission of part of a (111) close-packed plane of atoms. This dislocation cannot glide,l2 but must lie in its (111) plane : ending in a (II~) or similar surface, it attaches a mole- cular growth terrace in the usual way-but every time this growth terrace reaches the trace in the surface of the missing plane of atoms, the terrace must pause unless there is a definite supersaturation.At this boundary the lattice is not continuous but has a translation-twin relationship. Instead of the usual 6 contacts per added atom, characteristic of cubic close-packing, one row of atoms added at this boundary make 7 contacts each, and the next row only 5. The latter loosely bound row will only be formed at a definite supersaturation, or by statistical fluctuation after delay. la Frank, Proc. Physic. Soc. A , 1949, 62, 202.54 INFLUENCE OF DISLOCATIONS ON CRYSTAL GROWTH We must give a brief account of the origin of the dislocations which, it is suggested, dominate crystal growth. The chief origins which have been thought of so far are : (i) Surface nucleation of layers in improper (e.g., twin) positions and proper positions simultaneously on the same face.Where these meet there is a dislocation. It must be remembered that the initial nucleation of the crystal always takes place at high supersaturation, much more than adequate for the Becker-Doring condition for surface nucleation on a perfect face. (ii) Formation of one-dimensional dislocations in the edge row of the growing terrace (cf., van der Menve’s contribution to this Discussion). Such one-dimensional dislocations have an energy similar to the latent heat of evaporation of a molecule, and consequently exist in thermal equilibrium. During rapid growth at high supersaturation they can be trapped in an edge row, developing into two-dimensional and thence into three-dimen- sional dislocations.(iii) The development of curvature in the growing crystal owing to the presence of impurities (a subject to be treated at length elsewhere). This ultimately leads to stress in the surface which demands a certain super- saturation for further growth, which can then continue if, and only if, dislocations are formed. (iv) When systems of dislocations are present (particularly su b-grain boundaries) it is probable that the stress they would cause in perfect crystal compels the formation of further members of the system in growth : i.e., sub-grain boundaries are propagated in lineage structure. (v) Aggregation of molecular vacancies into flat collapsed cavities (the edges of which are dislocation loops) whenever the temperature of a crystal is lowered.This is very likely the process responsible for the intensification of X-ray reflection (the so-called establishment of mosaic structure) when an organic crystal is plunged in liquid air. (vi) Plastic yield under mechanical stress : this is believed not to create dislocations ab initio but to multiply those already present.ll These various considerations indicate that the initial dislocations necessary for growth are formed inevitably in the conditions needed for nucleation : and that to secure the best attainable perfection thereafter we must ensure small supersaturation (this involves good stirring, or there will be large supersaturation at the corners when it is small at the centre of a face), high purity of materials, steady temperature and absence of mechanical stresses. The effect which dislocations have upon crystal growth produces a rather odd natural selection both of imperfection and perfection in crystals. The nucleation stage with high supersaturation makes a population of initial nuclei of varied, mostly rather great, imperfection. If growth is now carried out at low supersaturation, only a few of these seeds, in which dislocations are relatively far apart, will grow. The lower the supersaturation at this stage, the fewer and the more perfect the seeds which will actually grow. But the completely perfect crystal will not grow in any circumstances : the conditions which could cause it to grow would also soon make it imperfect. H . H. Wills Physical Laboratory, Royal Fort, Bristol 8.