N. F. MOTT FORMS OF EQUILIBRIUM OF CRYSTALS BY I. N. STRANSKI Received 17th February, 1949 A knowledge of the forms of equilibrium of crystals is important for an understanding of the processes on crystal surfaces, independent of whether the crystal is immediately concerned in these, or merely functions cata- lytically. Furthermore, a comparison between theoretically and experi- mentally deduced forms of equilibrium makes it possible to check the assumptions used in the former, and supplies valuable clues to alteration in structure and changes within the individual lattice surfaces, The following observation should first be made. The theoretical treatment falls into two parts. First the underlying ideas must be developed, then the mathematical work can be started. This is directed by the knowledge at the time of the force functions, and must of necessity bring new problems in its train.In the following I will confine myself to the part dealing with the underlying ideas. The treatment of the forms of equilibrium of crystals has been developed on the basis of two fundamentally different ideas. The older one, historically, made use of an analogy to liquid surfaces. The surface tension here was replaced by the idea of the specific surface energy 0. The values of cr for crystals are dependent upon direction, so that in general the form of equilibrium is a polyhedron which must satisfy Gibbs’ condition : h i . Fi = minimum, at constant volume. If one ignores the edges and corners, it is known that here, also, one arrives at the same relation as for vapour pressure, which is completely analogous t o Thomson’s equation and has the following form : 1 The following recent papers on the Thomson-Gibbs relation are mentioned : Volmer, Kinetik der Phasenbildung (Dresden and Leipzig, 1g3g), p.87 et seq. v. Laue, 2. Krist., 1943, 105, 124. Stranski, 2. Krist., 1943, 105, 91. Honigmann, Moliere and Stranski, Ann. Physik, 1947, I, 181.I4 FORMS OF EQUILIBRIUM OF CRYSTALS vo represents the volume of a crystal unit and ra the centre distance, ci the specific surface energy of the i-th face. $, and $, are the sublimation pressures of the finite- and infmite-sized crystals respectively. Wulff’s method for the construction of equilibrium forms of crystals follows directly from eqn. (I). An exact relation, capable of general application, cannot be derived in this way.For if we wish to take into account the fact that the crystal also possesses edges and corners, and that the specific surface energy and the specific energy of the edges and comers which must further be introduced, ( 5 ) 1 possibly depend on the size of the crystal as 7 well, a relation can only be derived at first FIG. div division of a crystal by for simplified models. This is to be shown in three planes* Two Of these the following for the case where the form are shown as lines where they cut the plane of the of equilibrium of the crystal is represented diagram, the third lies in the by a simple crystalline form, i.e., it is sur- diagram plane itself. The rounded by only one kind of face. For figures in brackets denote this purpose let us refer to the definition of the sections below the plane of the diagram, those with- the specific surface energy, and give the out brackets the sections definition of the specific edge and corner above it.energies in reference to Born and Stern2 The specific edge energy x is defined as the work which must be done in order to separate the crystal sections I and 3, z and 4 respectively (see Fig. I), divided by twice the length of the edge, and given a negative sign. Correspondingly, the specific corner energy E is half the work required to separate two crystal sections situated diagonally in space, with their corners touching, e g . , I from 7, or 3 from 5 (see Fig. I). Assuming that these values are independent of the dimensions of the crystal, one obtains in place of eqn.(I) : Thus, as a result of the existence of edges, an additional term appears as correction. The corners are without influence. In order to be able to discuss the dependence of the values 0, x and E upon the size of the crystal at all, the definitions of these values for finite crystals had first to be found. The definitions given by me at that time will be explained for a simple case with the aid of Fig. 2. If the form of equilibrium is represented by a cube, cU is equal t o the work of separating such a small crystal from a cube face of the infinite crystal, divided by twice the area of one cube face of the small crystal. xu is correspondingly equal to the work of separating such a cube from the infinite crystal quadrant lying diagonally opposite divided by twice the length of a single edge and FIG.2.-T0 define the values cro, xa and for a finite crystal cube with an edge-length a. 2 Born and Stern, Ber. Berlin Akad., 1919, 48, 91 ; Stranski, 2. Krist., 1943, 105, 287. Stranski, Ber. Wien. Akad.. math.-natzrrwi. Kl., 1936, IIb, 145, 840 ; Mh. Chem., 1936, 69, 234.I. N. STRANSKI with a negative sign. Lastly the corner energy E, is equal to half the work of separating a small crystal from the infinite crystal octant lying diagonally opposite in space. Thus the total surface energy of a small crystal with edges of length a is @, = 6a20, + 12 a xa + Sc,. - (3) In this case it is possible to obtain the form of equilibrium of a small crystal simply by taking an infinite crystal to pieces, and CD, can also be defined as the work of separating a small crystal from a crystalline half- crystal position (see Fig.3 ) . However, it should be mentioned that is generally given in the following relation : Nu @u = Na - ~ p 1 / ~ - t ~ p ” . * (4) v = 1 pin is the work of separating a crystal unit from 3.-crysta11ine half- crystal position. @, the half-crystal position (see below). The second is equal to the work term is the work obtained in building up the small of separation of the crystal from its Nu individual crystal units. crystal cube with The example dealt with in the last section is edge-length a, in the especially simple. The important thing is, that as shown positions in the this case already shows that it is not possible to diagram.specify the exact sublimation pressure of a small crystal from the forms of equilibrium, with the aid of the values o,, xu and E ~ , now assumed to be variable. For this purpose, the differentiation of the eqn. (3) is necessary : P a da do, da dx, dE, = 12 c, a - + 6n2 ~ + IZ xu-+ 12 a -+ 8 - P m dN dN dN dN dN dN (5) kTln - = The values mu, o,, Xa and E, would thus have to occur as continuous functions of the number of crystal units N . That is not the case, however, for they present themselves as a series of isolated points. The following possibilities can be discussed. (I) Curves are drawn through these points and differentiated. The result could give the sublimation pressure with sufficient exactitude. (2) The dependence of the values o,, xu and &a upon N can be found to be so small that it can be neglected.Neither possibility, however, can be proved for no standard of comparison exists at present, which gives us the correct pressure values. We will return to these questions below. The advantages of the method using the values G, x and E are not to be denied, for by means of it, all considerations which had been made on liquid systems could be applied in a comparatively simple way, and with little alteration, to crystal systems. Special attention is here drawn to the fact that, on the whole, Volmer’s theory on the frequency of nucleus formation also reproduces the conditions correctly for crystal systems. By continuing the nucleus idea, introducing, namely, the idea of a two-dimensional nucleus, the growth of a crystal could be submitted for the first time to a mathematical method.Many different questions could be answered comparatively simply. The interpretation of Ostwald’s step-rule may be mentioned as an example.5 The values o, x and E do not refer at all to elementary stages of growth and reduction, and the relations which are obtained with their aid can only be applied under certain conditions to kinetic considerations on crystals, and remain difficult to visualize. As is known, the application of Thomson-Gibbs’ But the disadvantage of this method must also be enumerated. Volmer and Weber, 2. physik. Chem., 1926, 1x9, 277 ; Volmer, loc. cit. Stranski and Totomanow, 2. physik. Cham. A , 1933, 163, 399.16 FORMS OF EQUILIBRIUM OF CRYSTALS equation has led to numerous, and often crass, misunderstandings.This method, by simulation of completion, has also prevented many equilibrium questions from being asked and answered a t the right time. The second treatment began to take form as a result of work by Kossel on the one hand, and myself on the other.' The work of separating individual crystal units from the crystal surface was estimated, NaCl being taken as the first example, and with the help of this it was possible to draw a picture of the molecular processes connected with growth and solution. The logical starting point for these considerations is the determination of the length of time a crystal unit remains in the so-called half-crystal position. The crystal unit in the half-crystal position possesses a work of separation which amounts to half of that of a crystal unit in the inside of the crystal.It is thus equal to the negative value of the lattice energy per crystal unit, and determines the vapour pressure of the infinite-sized crystal. Elementary reasons can be given for this conclusion if the position of growth of a repeatable growing crystal face is chosen as model of the half-crystal position. For, in this case, the position as such is retained after any number of separa- tions or addition of crystal units. Thus the crystal would only be in equili- brium with its surroundings, if the probability of a separation of a crystal unit from the half-crystal position is found to be equal to the probability of an addition on this. With the help of the different works of separation, it has already been possible to draw a series of conclusions which, at that time, were justifiably regarded as completely new-fangled.Only as a consequence of these was it asked whether certain faces in equilibrium can be retained as such, or whether their surface structures would have to undergo alterations of a coarsening nature. It should be emphasized here that these questions could have been asked earlier, as a result of the determination of the values, or merely the signs, of the specific peripheral energies of the lattice surfaces concerned. That they have not been asked up to this time is to be explained exclusively by the fact that the older theories were difficult to visualize. Because of its importance the criterion might be given here by reason of which one can decide whether a certain face appears in the equilibrium form of the infinite-sized crystal or can remain as crystal face.If the specific peripheral energy of this lattice face shows the value zero or a negative value, in one direction only, this face cannot appear as a form of equilibrium. Should this condition be fulfilled for one direction only, the face concerned will grow over one-dimensional nuclei and show a typical chain formation. A chain formation alone, on the other hand, is not sufficient argument against the face belonging to the equilibrium form. If this condition is fulfilled for two directions, the one-dimensional nucleus formation is also eliminated. An example of the first case is (011) on the NaCl crystal and of the second, (111) on the same crystal.Another question could also be answered with the help of the work of separation, namely, with what kind of face must the infinite-sized crystal be surrounded ? For it is evident that the only possible form of equilibrium is one in which all comer crystal units are bound a t least as f i r m l j 7 as in the half-crystal position. So that by starting with a simple form, and systema- tically removing all crystal units which are less firmly bound, one could arrive a t forms which no longer exhibit such crystal units, and which then mirror the equilibrium form, in that they possess all the faces of same. In order to arrive at an expression which represents the sublimation Kossel, NacR. Ges. Wiss. Giittingen, 1927, 135 ; Leipziger Vortrlige, 1928, I.Stranski, 2. physik. Chem., 1928, 136, 259.17 I. N. STRANSKI pressure of finite crystals, those crystal units will be taken into consideration which, on evaporating, produce a deviating value for the work of separation. In the case of a single crystal face, that is a very simple matter.8 The mean value Fa appears here in place of = p/, (" work " of separation of the crystal unit in the half-crystal position), where the mean is taken so as to include all crystal units of the uppermost lattice face, and for a process also, in which the crystal units of the lattice face are removed. The logarithm of the relation between this vapour pressure and that of an infinitely extended lattice face is then simply This simple result can be explained as follows : for the faces concerned in equilibrium, the probability that the uppermost lattice face is removed by solution must be equal to the probability that, after removal, it is re- formed by means of a condensation process.But this stipulation is connected with the fact that the work of formation of a lattice face nucleus (two- dimensional nucleus) by condensation is exactly equal to that by superficial solution of an uppermost lattice face. Let us imagine a position where a crystal unit is bound in such a way that the work of separation has the exact value (F;.) required for the position to be occupied by a crystal unit for not more than exactly one-half of a very long observation period. We could then undertake the formation of a lattice face nucleus by condensation, by allowing the crystal units to attach them- selves first at this point from the vapour phase, and forming the nucleus by bringing them each time from there on to the face.If the nucleus contains m crystal units and the whole lattice face n, the following work is necessary for the production of the nucleus : m t (Fa - Cpi) . I By superficial solution of an existing lattice face, on the other hand, the work of formation of the nucleus amounts to n mtr E ((pi - Cpu) * By ba1ancing:the two work equations, one obtains directly The following should also be taken into account. The conditions of equilibrium deduced quite generally apply to both lattice faces and single lattice rows. In the case of the lattice face nuclei, the peripheral rows must be in equilibrium with the surroundings, i.e., the mean work of separation per peripheral row of the lattice face nucleus must show the same value pa on all sides.If we now consider a three-dimensional crystal which is in equilibrium with its surroundings, this implies that the same conditions must be fulfilled for each of its faces. From which it further follows that the surroundings are supersaturated as regards all rows on the edges of the crystal (for the uppermost lattice face of an equilibrium form is greater than the corre- sponding lattice nucleus) ; in the same way, the surroundings are also super- saturated as regards each single point on the surface of the crystal, andr8 FORMS OF EQUILIBRIUM OF CRYSTALS therefore a150 as regards the corner crystal units.This conclusion is instruc- tive. But it is also fundamental for the consideration of the equilibriom of a crystal. It leads directly to an easy method of construction of the form of equilibrium of crystals. In order to obtain the form which corre- sponds to a certain pressure, in the vapour phase, one proceeds as follows : the value of (pa corresponding to the pressure PI is calculated. Then starting from any simple form of the crystal, all crystal units which show a work of separation smaller than (9. are eliminated, one after the other, from its surface. Lastly, the areas of all faces are varied until each single mean work of separation reaches the value Cp,. Another conclusion from the thermodynamic deduction of the sublimation pressure of a small crystal is made especially clear.That is theconclusion which can be drawn directly from eqn. (2) : the vapour pressure is simply a function of the relation between c and the centre distance of any face. Provided the latter remains the same this quotient must remain unchanged independent of whether the face concerned occurs in a simple form or in a combination. When drawn from the thermodynamic deduction, this conclusion is not clear, as the deduction includes only the form of equilibrium itself, and is tied to the assumption that for small evaporation and growth processes the form remains similar. The following explanation, based on the mean work of separation, can be given for this conclusion. To this end, let us begin with a simple form and study a definite face. This form is now allowed to develop into a combination, the centre distance of the face under consideration remaining unchanged.The area of the face decreases but the deviations of the individual works of separation also decrease to the same degree, for the rows on the edges of the Combination border upon more lattice neighbours than the rows on the edges of the simple form. Lastly, eqn. (6) provides the possibility of deciding the question which cropped up on a previous page. It supplies the vapour pressure in a manner which is quite independent of that in eqn. (I) or (5). It is also possible, in this manner, to carry out the calculation for a definite example, namely, for a sim#,?ified NaCl crysta1.l I t showed, though only for this case, that the second possibility is realized, namely, that it is not necessary to include the dependence of the specific energy values 0, x and E on the number of crystal units, in the calculation, for all crystal sizes which actually come into question.It is comparatively easy to obtain the form of equilibrium theoretically for typical ionic crystals, if simplifying assumptions are made. In all cases dealt with up to now, it has been found to be a simple form. It is a cube for NaCl ', a rhombic dodecahedron for CsCl 9, an octahedron for CaF, 10, a rhombohedron for CaCO, or NaNO,. It is also independent of the size of the small crystal. Thus, form of equili- brium and form of growth are here identical (for low supersaturations). In this case the greater the range of the forces between the crystal units, and the nearer Cp.approaches CpW, the greater the number of faces appearing in the form of equilibrium. Table I gives a list of (infinitely great) forms of equilibrium for a few simple lattices as functions of the said range and calculated under the assump- tion that the work of separating one crystal unit from another is always The conditions in the case of non-polar crystals are different. 9 Kleber, Zbl. Miner., Geol., Paliiont. A , 1938, 363. lo Bradistilov and Stranski, 2. Krist., 1940, 103. I.FIG. 4.-Cd mono-crystal, formed in the fused liquid and allowed to grow further in the vapour. The small circular face a t the bottom left-hand corner is { I IZO), the irregular coarsened face above it {I 12 I}. (Eisenloeffel.)l5 To jute page 191I.N. STRANSKI positive and only dependent upon the distance between them.ll The majority of crystals with simple lattices would seem to represent transition stages between the polar and non-polar type. The metals constitute a special class. I t is worthy of note that the experimental data for metals l2 l3 also agree well, on the whole, with the results in the Table, inasmuch as they give the correct order of the faces. It was possible to make a more accurate experimental investigation especially in the case of Zn l4 and recently also for Cd1415, and these results were confirmed. In both cases the experi- ments on growth, carried out accurately on spherical rudimentary forms consisting of one crystal, which grow from supersaturated vapour without TABLE I FORMS OF EQUILIBRIUM The range of the forces between the lattice crystal units embraces Simple cube .. B o d y - c e n t r e d , cubic Face-centred cubic Diamond lattice. . Hexagonal closely packed spheres Adjacent crystals units only 00 I 0 1 I 111, 0 0 1 111, 001 OOOI, roir I oio Next crystal unit but one, also 001, 011, I 1 1 011. 001 111, 001, 011 111, 001, 011 0005, 1oi1 Ioio, II%O roiz Next crystal unit but two, also 001, 011, I11 I I2 011, 001, I12 I11 111, 001, 0 1 1 113, 0122 I35 111, 001, 011 113 0001, IoiI Ioio, IIVO r o i z Investi- gated on W, urotro- Al, Ag, Pt ~~ pine __ diamond Be, Mg, Zn, Cd __ The most far-reaching effect was found in every case to embrace the next crystal unit but two. The underlined examples have been investigated more thoroughly.any signs of coarsening, gave the faces : (OOOI), (IO~I), (IOTO) ; (IIZO), (10E2). It is also very significant that the faces (1120) and ( I O ~ Z ) ~ which are to be traced back to the influence of nearest neighbours but one, exhibit a con- siderably smaller area in the case of Cd, than in the case of Zn (see Fig. 4). This is probably connected with the greater screening capacity of the Cd atoms in the crystal lattice. It is also very noteworthy that W l6 l3 and urotropine l7 l8 which both have the same lattice (cubic body-centred) but belong otherwise to quite different valency types, exhibit exactly the same equilibrium form faces : { 0111 ; (001) ; { 112). In both cases, of the two faces which are to be traced back to the effect of nearest neighbours but two ((IIZ) and (III}), only { IIZ } appears.Concerning further properties of the urotropine crystal, 11 Stranski, 2. fihysik. Chem. B, 1931, 11, 342 ; Ber., 1939, 72, 141 ; Stranski and l2 Straumanis, 2. Physih. Cltem. B, 1931, 13, 317 ; 1932, 19, 64 ; 1934, 26, 246. 1s Stranski and Suhrmann, Ann. Physik, 1947, I, 153. 1 4 Kaischew, Keremidtschiew and Stranski, 2. Metullkunde, 1942, 34, 201. l6 Eisenloeff el, Dissertatim (Techn. Universitat Berlin-Charlottenburg, 1948). l6 Muller, 2. Physik, 1937. 106, 541 ; 1938, 108,668 ; 1943, 120, 270. 17 Kaischew, Jahyb. Univ. Sojia, fihys. math. Fak., XLIII, 1946147, 2, gg. 1 8 Stranski and Honigmann, Nuturwiss., 1948, 35, 156. Kaischew, 2. Krist., 1931,78, 373 ; Ann. Physik, 1935, 23, 330.20 FORMS OF EQUILIBRIUM OF CRYSTALS whose lattice can be considered approximately as homopolar with snperim- posed dipolar forces, see later.A brief study of the relation between form of equilibrium and form of growth (more exactly, final growth form) will be inserted here. The crystals which are investigated are, almost without exception, the product of a growth process. If the form of equilibrium is not a simple crystalline form, the resulting growth form contains only the slow-growing faces large enough to be visible ; the quick-growing faces remain the same size as the same faces in the form of equilibrium, which in general is sub- microscopic. The form of equilibrium here is to be ascribed to the pressure prevailing during the process of growth. Intermediate stages of growth of rounded single crystal forms provide the possibility of making all equilibrium faces visible l1 l4 l5 I t is, unfor- tunately, always possible that in the course of this faces also appear which do not belong to the equilibrium form.The appearance of { 012 ) and { 111 ) in Spangenberg’s l9 and Neuhaus’s 2o experiments on growth with spherical, polished NaCl crystals from aqueous solution may be recalled, for example, although the only form of equilibrium here is { OOI ). It is therefore of great importance to develop a thoroughly reliable method for the experimental production of equilibrium forms. This was achieved for the first time for urotropine, following on observations by Kaischew,l’ Honigmann l* and myself. At low temperatures, at which the transfer of matter takes place almost entirely via the adsorption layer, the growth form, which in the case of urotropine is a rhombododecahedron, re-forms the faces (001) and (112) (see Fig.5). Specially accurate investigations were carried out a t 0°C. (If one subjects the crystal to small fluctuations in temperature the same form appears much more quickly.) In answer to the question why, up to now, one had neither observed a spontaneous appearance of equilibrium forms of visible size (microscopic) ~ nor considered this possible theoretically, one can say that the relative differences in the vapour pressures of faces of an already visible crystal, which are not in proper ratio to one another, are in fact very small. In spite of this, it is not so much the smallness of the differences of the relative vapour pressure which is responsible for retarding the course of the reaction as Volmer’s work of formation of the two-dimensional nucleus connected with the supersaturation.This must appear in the formation of new lattice faces, and as the supersaturation disappears, converges towards infinity. If therefore one succeeds in removing the energy threshold of the work of formation of the two-dimensional nucleus, the process of alteration leading to the equilibrium form on a crystal of the growth form should be possible. It is possible to remove this energy threshold, or to lower it considerably, by the construction of hollow edges starting from which single lattice faces can develop. Only the few crystals whose crystal units show a comparatively high mobility within the adsorption layer at low temperatures will qualify for this.The discrepancy between theory and experiment, already mentioned, evinces itself with urotropine, by the appearance of (112) of the faces referred to the nearest neighbour but two, but not of {XXI). As is to be set forth elsewhere by Honigmann and myself, the experimental result can be explained by the fact that a profound alteration in lattice takes place in the uppermost lattice face of { 112). This is probably a Iattice alteration which is also stable a t a low temperature. With urotropine another phenomenon can be studied. Certain alterations 1s Spangenberg, N . Jahrb. Miner., Miigge-Festbd. A , 1928, 57, I 197. 20Neuhaus, 2. Krist., 1928, 68, 15.( a ) Growth form ( 0 1 1 ) .( 3 ) The form which is formed on tempering (011, 001, 112). (Honigmann.) FIG. 5.-Urotropine crystals. [To jure Page 20I. N. STRANSKI 21 i n the surface lattice do not appear until the temperature is high, i.e., altera- tions also exist which show the character of two- and three-dimensional changes. Above 170' C the mechanism of growth and evaporation, as well as the form of equilibrium of the urotropine crystal, changes fundamenta1ly.l8 The growth and reduction take place now in multimolecular (visible) layers, whose border is completely rounded ; the form of equilibrium is a rhombic dodecahedron whose corners and edges are also rounded. This phenomenon is obviously connected with the fact that new degrees of freedom (rotations) of crystal units of certain lattice surfaces, edges or peripheral rows are aroused by temperatures considerably lower than those in the inside of the crystal. In closing, the question may further be asked, how the equilibrium form of a crystal changes when it is surrounded by a liquid instead of its own diluted vapour. The simplest case would be to suspend the small crystals in their own fused liquid. The specific interface energy of a certain face ~ h k l would be given here by the following relation : where 1c and 2chkJ are the corresponding values for the liquid and the crystal relative to vacuum, and 1 p h k i the work which would be obtained by the contact of unit areas of crystal and liquid. It is seen that dhkl is not only very small if p h k l is very small, but also when 12Ghkl is especially large. The latter is all the more likely to be true, the more continuous the transition from crystal via the interface to liquid. The growth form of Cd which is produced from the fused liquid1415 may be quoted here as an example. It is seen that the face (IIZI} appears here, which as a rule is coarsened on continuing to grow in vapour, as it does not belong to the equilibrium form of the crystal surrounded by vapour phase (see Fig. 4). The general case of an equilibrium form surrounded by a phase of any desired composition has not yet been accurately treated, either experi- mentally or theoretically. Up to the present a certain amount of attention has only been paid to the occasional growth forms showing great deviations, which precipitate from various solutions. Gh/d = lcT + zghkl - 1 2 ~ h k l 1 - (8) Institut fur Physikalische Chemie wad Elektrochemie, Berlin-Charlottenbuurg 2, Hardenbergstrasse 34, Germany.