158 



NATURE 



[June i8, 1891 



attraction of the molecules. The energy will therefore 

 be a minimum for a surface in which the molecules are 

 as closely set as possible. 



Now, if you draw a surface through a heap of balls 

 packed so that each is touched by twelve others, you 

 will find that the surfaces which have the greatest 

 number of centres of balls per unit area are all plane sur- 

 faces. That in which the concentration is greatest is 

 the surface of a regular octahedron, next comes that of a 

 cube, then that of a rhombic dodecahedron, and so on 

 according to the law of indices of crystallographers. 



The relative numerical values of these concentrations 

 are as follows, taking that of the faces of the cube as 

 unity : — 



Tetrakishexahedron 0*4472 

 Eikositessarahedron 0-4083 

 Triakisoctahedron ... 0*3333 



Octahedron ... 



Cube 



Dodecahedron 



1-1547 

 i-oooo 

 0-7071 



We do not know that the surface-tension is exactly in 

 the inverse proportion to the concentration, all that we 

 can at present say is that it increases as the concentration 

 diminishes. 



If, then, the molecules occupy spherical spaces, the 

 bounding surface will tend to be a regular octahedron. 



But we have another point to consider. If a solid is 

 bounded by plane surfaces, there must be edges where 

 these planes meet. At such an edge the surface-tensions 

 will have a resultant (see Fig. 5) tending to compress the 

 mass, which must be met by a corresponding opposite 

 pressure, and unless there is some internal strain there 

 must be a corresponding resultant of the tensions on the 

 opposite side of the crystal. Hence, if one face of a form 



is developed the opposite face will also be developed ; and 

 generally, if one face of a form be developed all the faces 

 will be developed ; and if one edge, or angle, be truncated, 

 all the corresponding edges, or angles, will be truncated. 

 Were it otherwise, there would not be a balance between 

 the surface-tensions in the several faces. But there is 

 another point to be taken into account. The surface 

 energy may become less in two ways — either by reducing 

 the tension per unit surface, or by reducing the total 

 surface. When a liquid separates from another fluid, as 

 chloroform from a solution of chloral hydrate on adding 

 an alkali, or a cloud from moist air, the liquid assumes 

 the form which, for a given mass, has the least surface — 

 that is, the drops are spherical. If you cut off the pro- 

 jecting corners and plane away the projecting edges of a 

 cube or an octahedron, you bring it nearer to a sphere, 

 and if you suppose the volume to remain constant, you 

 still diminish the surface. And if the diminution of the 

 total surface is not compensated by the increased energy 

 on the truncations, there will be a tendency for the 

 crystals to grow with such truncations, The like will be 

 true in more complicated combinations. There will be a 

 tendency for such combinations to form, provided the 

 surface energy of the new faces is not too great as com- 

 pared with that of the first simple form. 

 ■ But it does not always happen that an octahedron of 



NO. II 29, VOL. 44] 



alum develops truncated angles. This leads to another 

 point. To produce a surface in a continuous mass re- 

 quires a supply of energy, and to generate a surface in 

 the interior of any fluid is not easy. Air may be super- 

 saturated with aqueous vapour, or a solution with a salt, 

 and no cloud or crystals be formed, unless there is some 

 discontinuity in the mass, specks of dust, or something 

 of the kind. In like manner, if we have a surface already, 

 as when a supersaturated solution meets the air or the 

 sides of the vessel containing it, and if the energy of 

 either of these surfaces is less than that of a crystal of 

 the salt, some energy will have to be supplied in order to 

 produce the new surface, but not so much as if there 

 were no surface there to begin with. Hence, crystals 

 usually form on the sides of the vessel or at the top of the 

 liquid. When a solid separates from a solution there is 

 generally some energy available from the change of state, 

 which supplies the energy for the new surface. But at 

 first when the mass deposited is very small the energy 

 available will be correspondingly small, and since the 

 mass varies as the cube of the diameter of the solid, 

 whereas the surface varies as the square of the diameter, 

 the first separated mass is liable to be squeezed into liquid 

 again by its own surface-tension. This explain s the usual 

 phenomena of supersaturated solutions. A deposit occurs 

 most easily on a surface of the same energy as that of the 

 deposit, because the additional energy required is only 

 for the increased extent of surface. It explains, too, the 

 tendency of large crystals to grow more rapidly than 

 small ones, because the ratio of the increase of surface 

 to that of volume diminishes as the crystal grows. 



While speaking of the difficulty of creating a new sur- 

 face in the interior of a mass, the question of cleavage 

 suggests itself. In dividing a crystal we create two new 

 surfaces — one on each piece, and each with its own 

 energy. The division must therefore take place most 

 readily when that surface energy is a minimum. Hence 

 the principal cleavage of a crystal made up of mole- 

 cules having their motions comprised within spherical 

 spaces will be octahedral. As a fact, we find that the 

 greater part of substances which crystallize in the octa- 

 hedral, or regular system, have octahedral cleavage. But 

 not all ; there are some, like rock salt and galena, which 

 cleave into cubes, and a very few, like blende, have 

 their easiest cleavage dodecahedral. These I have to 

 explain. I may, however, first observe that some sub- 

 stances — as, for instance, fluor-spar — which have a very 

 distinct octahedral cleavage are rarely met with in the 

 form of octahedra, but usually in cubes. In regard to 

 this, we must remember that the surface energy depends 

 upon the nature of both the substances in contact at the 

 surface, as well as on their electrical condition, their tem- 

 perature, and other circumstances. The closeness of the 

 molecules in the surface of the solid determines the 

 energy, so far as the solid alone is concerned ; but that 

 is not the only, though it may be the most important 

 factor conducing to the result. It is therefore quite pos- 

 sible that, under the circumstances in which the natural 

 crystals of fluor were formed, the surface energy of the 

 cubical faces was less than that of the octahedral, 

 although when we experiment on them in the air it is 

 the other way. This supposition is confirmed by the 

 well-known fact that the form assumed by many salts in 

 crystallizing is affected by the character of the solution. 

 Thus alum, which from a solution in pure water always 

 assumes the octahedral form, takes the cubic form when 

 the solution has been neutrahzed with potash. 



To return to the cubic and dodecahedral cleavages. If 

 we suppose the excursions of the parts of the molecule 

 10 be greater in one direction than in the others, the figure 

 within which the molecule is comprised will be a prolate 

 spheroid ; if less, an oblate spheroid. Now, as already 

 explained, the spheroids will be packed as closely as 

 j possible if the axes are all parallel and each is touched 



